MARITIME SOLUTIONS FOR A CHANGING WORLD #156 - AUTUMN 2019 TERRA ET AQUA #156 - AUTUMN 2019
TRENCHING MODEL MODELLING THE CABLE TRENCHING PROCESS ON BENEFICIAL USE Promoting viable uses for SAND DUNES sediments to stakeholders
EXPERTISE EXPORTER Engineer takes expertise in nature-first development abroad IADC HIGHLIGHT TRENCHING MODEL MODELLING THE CABLE TRENCHING PROCESS ON SAND DUNES
Numerous offshore wind farms have been recently installed in the southern part of the North Sea. Their infield and export cables are buried for protection against dropped or dragged objects. In sandy soils, burial is carried out by remotely operated tracked vehicles. Two swords with waterjets are used to fluidise the sand and generate a backward flow of the water-sediment mixture. The area’s highly variable seabed topography – characterised by sand waves and mega-ripples – can significantly influence the trenching process. At the moment, it is not possible to make an accurate estimate of the influence of sand dunes on the trenching process.
The trench formation process is split into two parts: a front section where the seabed is eroded by waterjets (erosion model) and a rear section where the sand grains are settling in a backward flow (sedimentation model). A team of authors from Delft University of Technology and DEME Offshore present the combined fluidisation, sedimentation and cable model which is validated against full-scale field data. Read more on page 34.
2 TERRA ET AQUA #156 - AUTUMN 2019 3 CONTENTS
ENVIRONMENT
Sustainable management of the beneficial use of sediments: a case studies review CEDA’s Working Group on Beneficial Use aims to inspire sediment stakeholders and practitioners by describing the importance of sediments in the context of sustainable development and sharing a curated selection of case studies.
06 18
SAFETY
14 innovations face off for Safety Award 2019 Each year, IADC gives a safety award to one exceptional innovation which proves it is the best in its class. This year, 14 innovations addressing challenges faced within the dredging industry have been nominated.
27
INTERVIEW
‘If we understand how to build with nature, then creating new nature through infrastructural development is the next step.’ 34 After 30 years at Witteveen+Bos, Henk Nieboer has moved on from the engineering and consultancy firm to TECHNICAL focus his expertise into his current roles as director of EcoShape and Modelling the waterjet founder of Adaelta. cable trenching process on sand dunes Cables for offshore wind farms in the North Sea EVENTS are buried for protection. A highly variable seabed topography influences the trenching process Develop professional in sandy soils. A model has been developed to networks abroad estimate the influence of sand dunes on the Head to Mumbai to participate in trenching process. IADC’s next Seminar on Dredging and Reclamation. Save the date for the next and 23rd WODCON to be hosted by CEDA 49 in one of Copenhagen’s harbours.
4 TERRA ET AQUA EDITORIAL CAN THE DREDGING INDUSTRY BE MADE SAFER?
Safety awareness remains a top priority in the dredging Speaking of sustainability, in this issue of Terra, industry. By encouraging the creation of safety CEDA’s Working Group on Beneficial Use presents solutions as well as bringing them into public discourse a categorised collection of case studies which via publication in Terra et Aqua, the IADC continues its demonstrate sustainable solutions for working mission to improve safety in the dredging industry and with sediment, a resource which is increasingly its projects on water and land by sharing strategies. addressed in the industry’s dialogue. Director of Ecoshape Henk Nieboer discusses the importance In that context IADC has unveiled the safety of Building with Nature and the future of the innovations nominated for its annual Safety Award programme. And lastly, a model to estimate the 2019. At fourteen innovations, this year’s group is influence of sand dunes on the trenching process the largest to be nominated in a single year. These is presented, a situation encountered while burying submissions address all facets of the industry, from cables for offshore wind farms. Frank Verhoeven land to sea and to the office: Equipment to enable President, IADC safer inspections underwater or from the air, a protective barrier during regular processes on deck, a programme to instill a safety mindset at any moment of the day, and an application with step-by-step instructions to save lives during medical emergencies at sea or in remote areas.
Looking over the span of four years, more than forty safety innovations have been nominated for IADC’s Safety Award.
Of course, when it comes to safety, quality always wins over quantity, but the numerous Safety Award nominations prove you can have both. And with more innovations, from engineers, safety experts, and crewmembers, the greater the safety awareness in the industry and the fewer accidents and incidents. This year’s Safety Award winner will be announced at IADC’s Safety awareness Annual General Meeting in New Delhi, India in October. remains a top priority In addition, in conjunction with the AGM, IADC is hosting a conference on Dredging for Sustainable in the dredging industry. Infrastructure in New Delhi, bringing its publication Dredging for Sustainable Infrastructure – made in collaboration with the Central Dredging Association – to an international audience in Asia.
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SUSTAINABLE MANAGEMENT OF THE BENEFICIAL USE OF SEDIMENTS: A CASE STUDIES REVIEW
6 TERRA ET AQUA Over the last few decades there has been an increasing recognition that dredged sediment is a resource which Through its latest publication, the Central Dredging should be utilised Association (CEDA) Working Group on the Beneficial beneficially for human Use of Sediments informs stakeholders and practitioners about the recent advances, ongoing development activities international initiatives and programmes, and best and/or enhancement management practices for the beneficial use and of ecological habitats. value of sediments using relevant case studies.
Sediment as a resource of experience in this environmental area. remains below its overall potential. Technical By describing the importance of sediments This review intentionally focused on the aspects are often outweighed by country- in the context of sustainable development technical aspects of these case studies to specific legislation, policy, economics and impact of climate change, this article demonstrate feasibility. This article does and public and industry perception (Brils aims to inspire international government not address legislation, economical, or et al. 2014). This complexity hampers the agencies and policy makers, contractors, governance aspects in detail. While very beneficial use of (dredged) sediments. project proponents and international important, these are often country-specific, Therefore, we recommend addressing these donors (i.e., World Bank) to encourage the which would distract from the central scope important aspects in a future publication in implementation of sustainable sediment of this article. an effort to promote beneficial use practice management strategies. to a level in which full potential can be Based on the case studies collected, realised and further in line with sustainable This review elaborates on previous literature beneficial use examples range from dredged human development. and experience on this topic (e.g., PIANC materials affected by anthropogenic 2009; IADC 2009; CEDA 2010). The sources and natural sediments, to be used This article intends to demonstrate that Working Group on the Beneficial Use for construction applications, or to help beneficial use applications exist for clean, of Sediments (WGBU) researched and restore freshwater and marine habitats, as well as, sediments contaminated collated details from 38 case studies. The with nature-based solutions becoming a with low-level pollutants. Dealing with studies collected involved contaminated, prominent driver for sustainable sediment contamination is perceived as challenging as well as, clean sediments. These included use in the last decade. In this article, we (both operationally and publicly); therefore, studies that have been undertaken in 11 define beneficial use as the use of dredged a separate complementary Position Paper countries over the last 30 years, specifically or natural sediment in applications that are by the Central Dredging Association (CEDA) focusing on the last decade. These case beneficial and in harmony to human and was produced in 2019, by this same WGBU, studies highlight many effective methods natural development. that focuses on the risk management and for beneficial use, supported by specific beneficial use opportunities of sediments pilot and commercial project applications, While this article illustrates technical with various degree of contamination. For assembled by an active community of feasibility and success, to date, beneficial this article, case studies were made available practitioners with more than two decades use of natural and dredged sediment by the WGBU members and their industrial
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contacts. Because the overview given in the This likely, exaggerated, overstatement suspected to increase turbidity in major rivers article is not exhaustive, the authors openly indicates that human interaction with natural (Winterwerp and Wang 2013; Winterwerp invite the professional community to share processes is significant. Humans move et al. 2013) and erosion of banks. Where their experiences with the CEDA community. sediment to enable and optimise: disturbance to natural processes of sediment The archival platform and email contact are • transport and logistics (e.g., dredging of accumulation and erosion occurs, it can available on the CEDA website to facilitate ports and waterways for navigation); contribute to increase the vulnerability of submission of additional and future case • space for living and commercial activities natural systems and human developments, studies, and mutual knowledge exchange (e.g., fill for land reclamation and such as: coastal erosion and loss of land, regarding the beneficial use of sediments remediation/brownfields); flooding from sea or rivers, decrease of worldwide. • flood safety and water management (e.g., productivity and environmental quality of construction of dykes, breakwaters, dams); ecosystems (Winterwerp and Wang 2013; Why sediment matters and Winterwerp et al. 2013). Climate change, The surface of the earth is being fractionised • natural ecosystem protection and resulting in more frequent and more intense into sand, silt, and clay by the natural enhancement (e.g., contaminated sites or events (i.e., storms and hurricanes) and sea process of rock weathering. The fractions, or wetland restoration, improving water clarity level rise, aggravates these risks and impacts sediments, are (re)distributed over the earth’s and quality). further. surface through erosion and sedimentation processes induced by ice, water, and air. In Human interventions interact with the natural Dredging of sediments this way, nature shapes the landscapes of dynamics of sediment accumulation and Humans move most sediment by dredging. the Earth through continuous and episodic erosion processes, which often disturbs the Unlike natural processes, like those that build events. However, the impact of Man on these natural dynamic. Examples include: sediment and reduce shorelines seasonally, man-made dynamic natural processes has increased trapped behind dams is not available to feed infrastructure is static and less tolerant of tremendously in the last century, especially downriver floodplains or nourish a beach near dynamic sediment processes. The largest due to the development of (land or waterborne) the river mouth (Vörösmarty et al. 2003); driver for dredging comes from the need to infrastructural works. In 2017, The Economist sediment from river mouths, reallocated remove accumulated sediment from ports, stated ‘humans now move more sediments offshore, is not available to nourish a wetland harbours, and shipping channels in order to than the natural processes of erosion’. anymore; excess deepening of estuaries is maintain their function as the backbone of our economy. Historically, the most common sediment management approach employed in many countries has been aquatic disposal of the dredged sediments at sea, or simply relocated in mid-river. This is particularly true for finer silts that are maintenance dredged from ports and harbors. In the UK alone, for example, around 22 to 44 million cubic meters (m3) of sediment is dredged from ports and harbours every year (ABPmer 2017).
Over the last few decades there has been an increasing recognition that dredged sediment is a resource which should be utilised beneficially for human development activities and/or enhancement of ecological habitats. The need to seek beneficial use opportunities was identified as a priority within the International Maritime Organisation (IMO) (London Convention and London Protocol (IMO, 2014) and other dredged material management reviews and guidance (IADC 2009; CEDA 2010; OSPAR 2014; and HELCOM 2015). In 1992 and 2009, the World Association for Waterborne Transport Infrastructure (PIANC) established workgroups focused on preparing guidance regarding the beneficial use of dredged FIGURE 1 material (PIANC 1992; PIANC 2009). The The restoration of an eroded coastline in Northern Java, Indonesia. Photo EcoShape PIANC (2009) report by the PIANC EnviCom
8 TERRA ET AQUA FIGURE 2 Dyke reinforcement underway in Hamburg, Germany. Photo Julia Gebert, TUD
Beneficial use may involve clean or contaminated sediments, when appropriately managed or treated, and when they provide added value.
Working Group 14 (chaired by CEDA) provided increasingly considered an integral part of and in partial collaboration with the a forum for the development of guidance, dredging projects from an early stage. These EwN, the LLM is a living platform that for future consideration, of uses for dredged advances highlight the central role of sediment brings together various EcoShape pilots material on a routine basis. Since the management and have facilitated the related to sustainability, with nature (fine) publication of the PIANC paper, many new development and implementation of innovative sediments management to facilitate cross- examples and initiatives have focused on sediment uses. Several international pilot and international knowledge and the beneficial use of dredged sediments, as programmes and initiatives seek to support experience exchange. reported in this review. An appendix to this the sustainable development of infrastructure • Working with Nature (WwN) is similar report provides wide-ranging case studies that through improved alignment and integration of to Building with Nature (BwN), EwN and demonstrate how dredged material has been engineering and natural systems. PIANC, promoting the development of used successfully worldwide. navigation-related projects based on International initiatives and the ‘with nature’ concept (PIANC 2008). Beneficial use of sediment programmes Integrated and circular dredged sediment Beneficial use of sediment is herein defined There are several world-wide initiatives and use is a central theme of this initiative. as the use of dredged or natural sediment in programmes that are centered on sustainable, • SEABUDS (Precipitating a SEA Change applications that are beneficial and in harmony and nature-based, development of hydraulic in the Beneficial Use of Dredged to human and natural development. Beneficial and civil infrastructures. Beneficial use of Sediment) which was led by the UK’s Royal use may involve clean or contaminated sediment is a key, constant, theme across Society for the Protection of Birds (RSPB), sediments, when appropriately managed or these programmes. Some of the most recent involves reviews and meetings by key treated, and when they provide added value. initiatives include: regulators and advisors to evaluate policy Considered in the context of the three pillars • Engineering with Nature (EwN) and practice in the field of beneficial use of sustainability (economic value, social gain was initiated by the US Army Corps with a view to implementing more projects and environmental benefit), many beneficial of Engineers’ Engineer Research in the future (Ausden M et al., 2018). use projects typically achieve at least two of Development Center (ERDC). The • Solent Forum (BUDS) Regional these objectives. Those projects which focus EwN programme has a specific focus Strategic Review is a project which on habitat restoration have the potential to on developing knowledge and practical is underway to strategically identify directly deliver all three. Since the mid- to experience regarding the use, and re-use, beneficial use project sites in the Solent late-1900s, knowledge about the natural of dredged sediment in light of resilience (south coast of the UK) which has been environment – and its processes and dynamics and nature restoration. Their work is underpinned by an innovative new study – has advanced significantly. Environmental documented in many completed and (by ABPmer http://www.abpmer.co.uk/ considerations, nature-based approaches, ongoing case studies. buzz/cost-benefitanalysis-of-using- value engineering and win-win solutions • The Living Lab for Mud (LLM) is hosted dredged-sediment-to-restore-andcreate- (i.e., benefits/value for all stakeholders) are by EcoShape (EcoShape 2017). Similarly, intertidal-habitat/) which reviews the costs
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and benefits of using dredged sediment changing the perception of sediments from a 3. Reclamation : creating new, or expanding for marine habitat restoration, based on ‘waste’ to a sustainable resource. existing, land mainly for human/commercial examples in Europe. development activities. • Using Sediment As a Resource (USAR) Several case studies and information included 4. Restoration : creation of habitat to support and Promoting Integrated Sediment in this article are derived from these initiatives aquatic organisms and wetlands to improve Management (PRISMA), are two European and therefore are concrete examples of natural value. Union, North Sea Region initiatives covering achieving socially acceptable, economically 5. Resiliency : shoreline nourishment and England, France, The Netherlands and viable and environmentally sustainable (dyke) reinforcement for defence against Belgium (Flanders). These programmes projects. floods and extreme climatic events. centre on developing alternative options, at no added cost, for the processing, treatment Classification of beneficial use of It is certainly recognised that some, in fact and beneficial use of sediments in estuaries sediments most, beneficial use applications fulfil more and coastal waterways, from dredging to There are many different types of beneficial than one function. For example, dredged recycling, in lieu of the circular economy. use applications, as well as different material can be a substitute for raw material, • European Sediment Network (EU nomenclature and terminology associated which can be used as a top layer of a landfill SedNet) Working Group on Sediment with it. Therefore, adopting a unified classifying closure project or for dyke reinforcement; a Quantity Management – Sediments on approach is not simple. For example, it is contaminated site can be remediated as part the Move From the Mountains to the Sea quite common to frame beneficial use of land reclamation for further redevelopment; (https://sednet.org/), with main objectives potential in terms of geotechnical/structural a coastal nourishment can create habitat to increase the general awareness for material types (e.g., clay, rock, sand and and improve flood safety and sea level rise sediment quantity management with the silts). Alternatively, beneficial uses may be resiliency; remediation of a mine can be part entire watershed system and to promote separated into categories based on final of a reclamation and restoration function to the sharing of experiences and best objective and end-use (i.e., engineering and/ repair and mitigate a century of environmental management practice in this field, in line or environmental) or based on the dredging impacts. In all these cases, the various with the CEDA WGBU. equipment/technique used (e.g., backhoe applications are categorised following the bucket mechanical dredge, trailing suction major function, yet mentioning, and perhaps Over the years, other beneficial use sediment hopper dredge). In this article, beneficial integrating, the other functions explicitly. programmes have contributed to the overall uses are categorised according to five knowledge base, focusing on materials science end-use functions the project fulfils (i.e., the Furthermore, the various beneficial use (e.g., structural or geotechnical aspects) and application) and to the general operational applications can be divided into four broad sediment treatment (i.e., in the context of technique used in the application. techniques categorising the method used to destroying or immobilising contaminants). Five major functions are here defined as ‘the implement the activity. These techniques are: Five Rs’: A. On Land, Natural or Enhanced Treatment: These include: SEDI.PORT.SIL, CEAMas, 1. Raw Material: substitution for virgin sediment is pumped and treated on land, SETARMS, SEDILAB, GeDSET, the manufactured soil or building materials, such as drying/dewatering and ripening Sedimateriaux Approach and the USEPA/ such as tiles or aggregates. fields and dewatering plants (see Figure 3). NJDOT New York and New Jersey Harbour 2. Remediation : clean-up of contaminated B. In Water, Reallocated at a Final Location: Sediment Decontamination Programme. These sites, brownfields or closure of landfills and sediment is transported and pumped, programmes have been at the forefront of mines. or deposited, at final locations, such as
FIGURE 3 In Technique A, sediment is deposited on land, and in this illustrative case, into drying cells. Sediment is possibly treated and reclaimed for other subsequent beneficial uses such as dyke reinforcement.
10 TERRA ET AQUA FIGURE 4 In Technique B, sediment is reallocated in water at the final location. Demonstrated in this case, the final location is an island with a major function of nature restoration.
FIGURE 5 In Technique C, sediment is reallocated in water at a strategic location. Tidal flow and waves transport the sediment to the final location. In this illustrative case, the function is wetland restoration in front of a sea dyke with a consequential reduction of flood risk.
FIGURE 6 In Technique D, the trapping of sediment is enhanced. This illustrative case demonstrates the use of permeable dams to favour wetland restoration through mangroves.
It is certainly recognised that some, in fact most, beneficial use applications fulfil more than one function.
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nourishments, land reclamation, waterfront systems and engineering tools as knowledge about the sustainable use of fine redevelopment (see Figure 4). sediment management measures. sediments. C. In Water, Reallocated at a Strategic Location: sediments are disposed at a Human intervention decreases from Case studies strategic location, letting the local natural techniques A through D, with techniques Case studies were collected during the processes (e.g., hydrodynamic forces) C and D mostly relying on nature-based preparation of this article by WGBU members transfer and trap the sediment at the final approaches. Technique A often involves and associates. In total 38 case examples, location, such as sand or mud engine (see the use of chemical or physical treatments undertaken in 11 countries over the last 30 Figure 5). to sequester contaminants or improve years, with the focus on the last decade were D. In Water, Enhanced Trapping: improving sediment properties. Techniques A through collected. All case studies are described in the trapping capacity of the natural system, D (Figures 3 through 6) are consistent with standard two-page summaries, all of which for example strategic mangrove or wetland those proposed by the EcoShape – Building are available on the CEDA website at: https:// restoration projects (see Figure 6). In with Nature Initiative, Living Lab for Mud dredging.org/resources/ceda-publications- this case sediments are not dredged or (EcoShape 2017). EcoShape is working with online/beneficial-use-of-sediments-case- transported by humans but use natural their partners on five pilot projects to develop studies
TABLE 1 Case studies classified after Function (Rows) and Technique (Columns). Rows 1 through 5 refer to Function and columns A through D refer to Technique. Case study nomenclature includes a reference to the Function, Technique, the year at project start, and the country location of the project. Underlining indicates contamination present. Case studies in italics indicate treatment (see Position Paper for details on treatment techniques).
Technique Function A. On Land, Natural or B. In Water, Reallocated C. In Water, Reallocated D. In Water, Enhanced Enhanced Treatment at Final Location at Strategic Location Trapping
1. Raw Material R1A_1985_DE R1A_1993_DE R1A_1996_DE R1A_2006_DE R1A_2006_NL R1A_2012_FR R1A_2015_US R1A_2017_IT R1A_2018_US
2. Remediation R2A_1988_DE R2A_1995_NL R2A_2015_DE
3. Reclamation R3B_2006_NZ R3A_2016_US R3B_2010_NO R3A_2018_NL R3B_2018_SE
4. Restoration R4B_2002_US R4B_2005_US R4C_1999_NL R4B_2008_US R4C_2002_US R4A_2010_NL R4B_2016_NL R4C_2007_US R4B_2016_UK(A) R4C_2016_NL R4B_2016_UK(B)
5. Resiliency R5A_2004_DE R5B_1990_UK R5A_2005_BE R5B_2006_NL R5A_2013_FR R5C_2008_US R5D_2015_ID R5B_2010_US R5A_2018_NL R5A_2019_BE
12 TERRA ET AQUA TABLE 2 List of case studies by title and classification code.
Classification Code Case Study Title R1A_1985_DE Production of raw material through dewatering fields, Hamburg – DE R1A_1993_DE Production of raw material through a dewatering plant, Hamburg – DE R1A_1996_DE Use in ceramic industry through industrial treatment, Hamburg – DE R1A_2006_DE Use as agricultural soil after dewatering, Ihrhove – DE R1A_2006_NL Reclamation of clean sand through sand separation, Rotterdam – NL R1A_2012_FR Use in road construction after immobilisation and stabilisation, Dunkirk – FR R1A_2015_US Use in civil and environmental applications after stabilisation via Pneumatic Flow Tube Mixing, New Jersey – US R1A_2017_IT Use in civil and environmental applications after multiple phase cleaning and sorting process, Palermo – IT R1A_2018_US Production of grade cement after thermo-chemical high temperature treatment and immobilisation,New Jersey – US R2A_1988_DE Use as sealing material after dewatering, Hamburg – DE R2A_1995_NL Use as landfarming through bioremediation, Oostwaardhoeve – NL R2A_2015_DE Use as substitute for sand to backfill former harbour-basins, Hamburg – DE R3A_2016_US Raise elevation of near-shore agricultural fields after natural dewatering, Ohio – US R3A_2018_NL Raise elevation of low-lying peatlands and production of high value soil through blending with local organic waste, Krimpenerwaard – NL R3B_2006_NZ Use in expansion of port terminal after blending with cement, Auckland – NZ R3B_2010_NO Use in expansion of port terminal after blending with cement and stabilisation contaminated sediments, Oslo – NO R3B_2018_SE Use in civil applications after testing with various binders, Gothenburg – SE R4A_2010_NL Raise elevation of low-lying peatlands after natural dewatering in confined facilities, Jisperveld – NL R4B_2002_US Creation of natural habitat and morphological stabilisation through strategic deposition, New Jersey – US R4B_2005_US Counter subsidence and creation of natural habitat through strategic deposition, California – US R4B_2008_US Habitat restoration through creation of islands, Wisconsin – US R4B_2016_NL Habitat restoration through creation of islands, Lelystad – NL R4B_2016_UK(A) Habitat and wetland restoration through strategic deposition, Brightlingsea – UK R4B_2016_UK(B) Habitat and wetland restoration in three locations through strategic deposition, Hampshire – UK R4C_1999_NL Feeding the natural system through natural dispersive processes, Wadden Sea – NL R4C_2002_US Creating islands through natural dispersive processes, Louisiana – US R4C_2007_US Beach replenishment and lagoon restoration through natural dispersive processes, California – US R4C_2016_NL Wetland enhancement through of natural dispersive processes, Harlingen – NL R5A_2004_DE Use in dyke construction reinforcement to enhance flood resilience after industrial dewatering, Hamburg – DE R5A_2005_BE Use in dyke construction reinforcement to enhance flood resilience after dewatering and treatment, Dendermonde – BE R5A_2013_FR Use in breakwater components to enhance flood resilience after dewatering and treatment, Dunkirk – FR R5A_2018_NL Use in dyke construction reinforcement to enhance flood resilience after natural ripening, Delfzijl – NL R5A_2019_BE Use in dyke construction reinforcement to enhance flood resilience after dewatering and treatment, Waasmunster – BE R5B_1990_UK Coastal defence and habitat restoration through strategic disposal, Essex - UK R5B_2006_NL Making room from rivers through various beneficial uses, various location in NL R5B_2010_US Use for coast defence and nature restoration through strategic placement, Mississippi – US R5C_2008_US Use for coast defence and nature restoration through strategic placement and use of natural processes, California – US R5D_2015_ID Use for coast defence and local economy enhancement through natural trapping, Demak – ID
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These case studies include general the total – depending on annual sedimentation Nature-based case studies, the information about a specific project, technical behaviour). The beneficial use output, of focus of the 21st century information of the beneficial use application, the METHA plant, was used for reclamation Since the early 2000s more case studies and illustrations. Should the reader be and restoration projects as well as for the implement nature-based techniques interested in more information, a contact manufacturing of bricks and ceramics. The and focus on restoration and resilience reference is also provided. All case studies remaining clean sediment is reallocated functions (see Table 1). Nature-based were classified after function and technique, downstream of the Elbe river. Two decades solutions (NBS) rely on natural processes as described in the previous section of later, the Port of Antwerp followed with a similar (i.e., currents, waves, the deposition and this article, uniquely named, and included plant, the AMORAS. In France similar sediment erosion of sediment, and plant growth) that in the summary table (see Table 1). The output is utilised as a sub-base material for are directly incorporated in the design and nomenclature of the case studies includes road construction. Sediment treatment, such construction methods (Borsje et al. 2011; De the year of project initiation and the country. as mixing with Portland cement and/or other Vriend and van Koningsveld 2012; De Vriend binders, has been successfully implemented et al. 2015). This requires an understanding This table also identifies those case studies for the stabilisation of contaminants and of the specific natural system, its main that involved contaminated sediments and modification of the geotechnical properties forces, their variation, the ecosystem, and (chemical/physical) treatment. For further of the dredged material, mostly fines, in order the societal and governance structure. clarity, Table 2 provides a list of the case to meet geotechnical specifications for For this reason, there is not a ‘one solution studies by title and cross-referenced against specific project applications in remediation, fits all’ but instead an appropriate solution their classification. and redevelopment projects (including port needs to be strategically considered for each development) in the United States, Norway and site, river basin, estuary, coastal system, Historical and enhanced beneficial Sweden. Stabilisation focuses on minimising community and country. Nature-based use case studies segregation of different grain sizes, increasing projects must therefore be integrated in Table 1 shows that beneficial sediment strength and reducing water content and the large-scale, long-term development of use is not a new concept but began in the permeability. Stabilisation is not only used to the social and physical (eco)system. NBS 1960s with the flushing fields at the Port stabilise contaminated sediments, but also has does not mean green or nature-based only of Hamburg, Germany, and updated in the a role in coastal resiliency in the construction but are often a combination of green and 1980s with dewatering fields being an iconic of seawalls, levees and dykes. For dredged grey (i.e., conventional approaches) with the example. In the 1990s, the Port of Hamburg materials not suited for aquatic placement, proportion of each depending on the project built a large-scale facility for the Mechanical upland stabilisation for geotechnical objective, specific environment, the (natural Treatment of Harbour Sediments (METHA construction purposes, mine reclamation, and social) ecosystem and the potential for plant) for enhanced dewatering and treatment road subbase, landfill and brownfield caps, are sustainable outcomes. The beneficial results of the (mildly) contaminated portion of the examples of routine value-added beneficial use of nature-based sediment use are often to dredged sediment in the harbour (5%-20% of applications. be achieved and appreciated in the longer term and larger scale. Design, planning, construction, testing, long-term monitoring, and adaptive management should account for appropriate time and spatial scales.
Given the scarcity and cost of sand, many case studies begin to explore the effective implementation of soft fine sediments (or mud). These case studies are often brought forward by the international initiatives mentioned before (i.e., Building/Engineering/ Working with Nature, USAR, PRISMA). These initiatives rely heavily on NBS and fine sediments management. Sediment and beneficial use are critical considerations for all types of NBS, and the link between NBS and beneficial sediment (re-)use is intrinsically strong. Examples of nature-based projects, based on beneficial use, collected during this study are varied in scope. They include: • using natural products and processes such as manure, vegetation and ripening, FIGURE 7 to stabilise sediments (e.g., Kleirijperij or Depositing dredged sediment to enhance wetlands. Photo Exo Environmental Krimpenerwaard in The Netherlands);
14 TERRA ET AQUA • using stabilised sediment directly or The practice of beneficial use is well- This article focused on technical feasibility, indirectly for land reclamation, raising established, particularly in relation to only indirectly touching on legislation or subsiding land or strengthening dykes production of alternative raw material to economic components (which are often (e.g., Vlassenbroek in Belgium, Auckland support civil infrastructural projects. More country-specific). However, case studies in New Zealand, Sandvika in Norway, recently, innovative applications and pilot did generally discuss these project aspects, Hamburg in Germany, and Lowlands in projects have been explored on how to best use and non-technical challenges critical for the The Netherlands, see Figure 2); natural forces and processes, implementing success of a beneficial sediment use project, • depositing of dredged sediments in thin NBS, that incorporate beneficial use of especially when implementing NBS. These or thick layers on marine wetlands and sediment. Successful projects include wetland are, for example: definition of beneficiaries retreating or vulnerable coastlines (e.g., at restoration and coastal nourishment studies and funding mechanism; clear policy and legal Horsey Island, Lymington, or Brightlingsea to improve resilience against coastal flooding framework to regulate, permitting design, in the UK, see Figure 7); and extreme climatic events. A community implementation and maintenance; and • creation of artificial nature islands to of practitioners lies behind these numerous managing institutional and public perception. improve flood safety and/or improve the successful applications, with over two decades In early 2019 PIANC, initiated WG 214 on the habitat biodiversity and the natural value of experience to draw upon. The collected same topic of beneficial sediment use. WG 214 of the specific area (e.g., Marker Wadden case studies unequivocally demonstrate that includes various CEDA members who worked Restoration Project in The Netherlands, applications of beneficial use of sediments, on this article, which serves as a solid technical Cat Island and Deer Island in the United contaminated by low-level pollution, are baseline. It is the ambition of the PIANC WG to States); implementable. A parallel Position Paper is include a wider analysis of the non-technical • attempting to extend coastal wetlands produced that describes how to evaluate and success or failure factors, to provide a broader by depositing dredged material at a mitigate risk, to successful beneficial use, when perspective on how to consistently implement strategic location and relying on coastal contamination is present. beneficial sediment use in large scale processes for transport (e.g., Koehoal in applications. The Netherlands); and This number of applications demonstrate that • implementation of old Dutch techniques many possibilities for beneficial use exist, Finally, as a call for ongoing collaboration, the to trap sediments (i.e., permeable dams) offering the opportunity for its prioritisation in authors invite the reader and the professional in front of eroding coastlines, to trap dredging and sediment management activities. community to share their experience, sediment and restore mangrove forest, so In some instances, the benefits of beneficial knowledge and further case studies by improving the resilience against flooding use applications may only be realised long after sending them to [email protected]. As of rural communities (e.g., Demak in project implementation, or may be less directly identified in Murray (2008), ongoing active Indonesia, see Figure 1). quantifiable, such as indirect ecosystem communication on this subject is vital in order service benefits. Successful applications may to see more and larger projects achieved. Given their integration with natural processes, also require long-term maintenance or adaptive Therefore, CEDA will provide a platform for the selection of the location of nature-based management approaches. This is the logical ongoing knowledge and experience exchange solutions is critical. Strategic reviews are being consequence of implementing NBS, where on the subject of beneficial sediment use. carried out to actively explore where projects natural processes intrinsically need time to can be best located. One recent example respond and adapt to changes. includes the UK Solent Forum Study, which identified economic criteria for site selection. An online map for potential project locations, in the Southampton area, was developed (ABPmer 2018). These sites should be taken forward to affirm economic and ecological merits. Luca Sittoni Luca is a Senior Adviser and Program Manager Conclusions Nature-Based Solutions at Deltares and This article demonstrates that dredged Management Team at EcoShape – Building with sediment is a valuable resource, reinforcing Nature. Before Deltares, Luca worked as a the findings from past reviews on this subject. hydraulic engineer at Barr Engineering in Sediments can be used to support the Minneapolis, USA. Luca has a MSc. in Civil/ sustainable development of many important Hydraulic Engineering from the University of human activities in harmony and in integration Minnesota and the University of Trento. Luca is with nature. Vice versa, failure to do so will likely an expert in beneficial sediment use and soft reduce resiliency and increase the vulnerability sediments processes, with applications to the to natural forces. The numerous case studies dredging and mining industry, nature-based provided in this article demonstrate that solutions, and contaminated sediments technical knowledge and experience with remediation. beneficial use of sediment is significant.
#156 - AUTUMN 2019 15 ENVIRONMENT
Summary
This article is based on a paper which has been prepared by the Central Dredging Association (CEDA) Working Group on the Beneficial Use of Sediments (WGBU). The WGBU was initiated by CEDA’s Environmental Commission in 2017. This article intends to inform sediment stakeholders and practitioners about the recent advances, on-going international initiatives and programmes, and best management practices regarding the beneficial use of sediments and the value of sediments as a natural resource in the context of sustainable development using relevant case studies.
This article was first published as an Information Paper presented by the Central Dredging Association (CEDA), an independent, international organisation with an extensive professional network, a centre of expertise on dredging and reclamation, and an easy-to-access forum for knowledge exchange. The article has been prepared by a working group of international experts of broadly diverse backgrounds and range of expertise, under the remit of the CEDA Environment Commission.
Citation CEDA (2019) Sustainable Management of the Beneficial Use of Sediments. Information Paper. [Online] Available at: http://www.dredging.org/ media/ceda/org/documents/resources/cedaonline/2019-05-BUS-ip.pdf
Members of the CEDA Working Group on the Beneficial Use of Sediment
Luca Sittoni (Chair) Will Manning Peter Seymour EcoShape Centre for Environment Fisheries & Ecocem Materials Ltd Aquaculture Science Nick Buhbe Eric Stern Mission Environment, LLC Helmut Meyer Tipping Point Resources Group, LLC Federal Waterways and Shipping Agency William Coulet David Tenwolde Exo Environmental Ivo Pallemans Dredging Marine Offshore Services Jan De Nul Heinz-Dieter Detzner Envisan Thomas Vijverberg Hamburg Port Authority Boskalis Hans Quaeyhaegens Rebecca Gardner De Vlaamse Waterweg nv Marco Wensveen Anchor QEA Port of Rotterdam Chris van Schalm Dafydd Lloyd Jones Rijkswaterstaat Arjan Wijdeveld Marine Space Deltares Colin Scott Delft University of Technology Joost Koevoets Associated British Ports Marine & Royal IHC Environmental Research (ABPmer)
Some members from the CEDA BU WG are concurrently participating in the ongoing PIANC EnviCom WG 214 on Beneficial Sediment Use.
16 TERRA ET AQUA REFERENCES
ABPmer (2017) CEDA (2019) Murray, L. (2008) White Paper: Using dredge sediment Assessing the Benefits of Using of Dredged material as a resource. Terra et for habitat creation and restoration: A Contaminated Sediments. Position Paper. Aqua, September, Number 112, pp. 3-10. cost benefit review, A summary of the Available at: http://www.dredging.org/ https://www.iadcdredging.com/ul/cms/ techniques, costs and benefits associated media/ceda/org/documents/resources/ terraetaqua/document/2/4/0/240/240/1/ with using fine dredge sediment to cedaonline/2019-05-BUC S-pp.pdf article-dredged-material-as-a-resource- ‘recharge’ intertidal habitat, ABPmer [Accessed: May 2019]. terra-et-aqua-112-1.pdf [Accessed: May Internal White Paper, Report No. R.2865. 2019]. De Vriend, H. J. and Van Koningsveld, M. ABPmer (2018) (2012) OSPAR (2014) Beneficial use of dredge sediment in the Building with Nature: thinking, acting and Guidelines for the management of dredged Solent (BUDS), Phase 1 project scoping interacting differently. EcoShape, Building material. Available at: www.ospar.org/ and partnership building, ABPmer Report with Nature, Dordrecht, The Netherlands. documents?d=34060 No. R.2845. [Accessed: May 2019]. De Vriend, H. J., Van Koningsveld, M., Ausden M, Dixon M, Lock L, Miles R, Aarninkhof, S. G. J., De Vries, M. B. and PIANC (1992) Richardson N & Scott C. (2018) Baptist, M. J. (2015) Beneficial uses of dredged material: A SEA Change in the Beneficial Use of Sustainable hydraulic engineering through practical guide. Report of PIANC Working Dredged Sediment. Royal Society for the building with nature. Journal of Hydro- Group 19, PTC II. Protection of Birds. environment Research, 9, pp.159-171. PIANC (2008) Borsje, B.W., van Wesenbeeck, B.K., Dekker, EcoShape (2017) Working with Nature. PIANC Position F., Paalvast, P., Bouma, T.J. and de Vries, The living lab for mud. Available at: https:// Paper. M.B. (2011) www.ecoshape.org/en/projects/living-lab- How ecological engineering can serve in mud/ [Accessed: May 2019]. PIANC (2009) coastal protection – a review. Ecological Dredged material as a resource: Options Engineering, 37, pp. 113-122. HELCOM (2015) and constraints. Report of PIANC EnviCom HELCOM Guidelines for management Working Group 14, No 104. Bridges, T. S., Bourne, E.M., King, J.K., H. of dredged material at sea. Adopted Kuzmitski, H.K., Moynihan, E.B. and Suedel, by HELCOM 36-2015 on 4 March The Economist (2017) B.C. (2018) 2015. Available at: http://www.helcom. Improving the Oceans: Getting serious Engineering with Nature: an atlas. fi/Lists/Publications/HELCOM%20 about overfishing, Printable Edition, May ERDC/EL SR-18-8. Vicksburg, MS: Guidelines%20for%20Management%20 27th, 2017. U.S. Army Engineer Research and of%20Dredged%20Material%20at%20 Development Center. http://dx.doi. Sea.pdf [Accessed: May 2019]. Vörösmarty, C.J., Meybeck, M., Fekete, org/10.21079/11681/27929. [Accessed: B., Sharma, K., Green, P. and Syvitski, J.P. May 2019]. IADC (2009) (2003) Facts about dredged material as a Anthropogenic sediment retention: Brils J., de Boer P., Mulder J., de Boer E. resource. An information update from the major global impact from registered river (2014) IADC – Number1–2009. impoundments. Global and planetary Reuse of dredged material as a way to Available at: https://www.iadc-dredging. change, 39(1), pp.169-190. tackle societal challenges. Journal of Soils com/ul/cms/fck-uploaded/documents/ and Sediments, DOI 10.1007/s11368-014- PDF%20Facts%20About/facts-about- Winterwerp, J.C. and Wang, Z.B. (2013) 0918-0. dredgedmaterial-as-a-resource.pdf Man-induced regime shifts in small [Accessed: May 2019]. estuaries – I: theory. Ocean Dyn., 63 (11–12), CEDA (2010) pp. 1279-1292. Dredged Material as a Resource: Options IMO (2014) and Constraints. Information Paper. Revised Specific Guidelines for the Winterwerp, J.C, Wang, Z.B., van Braeckel, A., Available at: http://www.dredging. assessment of dredged material. Tech. van Holland, G. and Kösters, F. (2013) org/documents/ceda/downloads/ Rep., International Maritime Man-induced regime shifts in small publications-2010-6-ceda_information- Organization, London, UK. 146, 152. estuaries – I: a comparison of rivers. Ocean paperdredgedmaterialasaresource.pdf Dyn., 63 (11–12), pp. 1293-1306. [Accessed: May 2019].
#156 - AUTUMN 2019 17 SAFETY 14 INNOVATIONS
FACE OFF FOR SAFETY AWARD 2019
18 TERRA ET AQUA Each year IADC gives the Safety Award to one The Safety Award is innovation which proves it is the best in its class. This an annual initiative year, fourteen innovations addressing challenges faced within the dredging industry have been conceived by IADC. nominated to compete for the Safety Award 2019.
Combatting risk by increasing safety As an association, IADC is committed to promoting safety in the The Safety Award is an annual initiative conceived by IADC. By dredging industry. Dredging activities can be risky operations with giving this award, IADC intends to encourage the development hidden dangers amongst heavy machinery. In response, the dredging of safety skills on the job by rewarding individuals and companies industry proactively maintains a high level of safety standards. demonstrating diligence in safety awareness in the performance As a representative of contractors in the dredging industry, IADC of their profession. A particular project, product, ship, team encourages its own members as well as non-members participating or employees can receive the award, serving to recognise in the global dredging industry to establish common standards the exceptional safety performance which their innovation and a high level of conduct in their worldwide operations. The demonstrates. IADC’s members are committed to safeguarding their employees, continuously improving to guarantee a safe and healthy work Nominated for IADC’s Safety Award 2019, fourteen solutions environment and reducing the number of industry accidents and address the safety of equipment as well as routine processes and incidents to zero. situations encountered in the workforce of the dredging industry.
Boskalis has integrated real-time sonar imaging on diving helmets to increase sight in zero-visibility conditions
Diving for marine projects is not a risk-free technology. Some helmet suppliers are activity. Divers work in hazardous situations starting to provide options and Boskalis has subsea and one of the basics for safe diving is taken the opportunity to be an innovator and understanding where you are and where you early adopter by embracing this technology need to go, especially in an emergency. Divers now. Technically speaking, the idea is feasible sometimes encounter zero visibility under because it integrates existing technology and water. By integrating real-time sonar imaging componentry into dive helmets. technology in existing diving helmets, Boskalis is creating a helmet that provides vision in Within Boskalis (and within the industry) zero-visibility conditions, access to images and diving incidents are in the top three risks task plans subsea. This will keep divers safer and this innovation will improve safety. This and be more efficient and cost effective. innovative helmet is in the early engineering stages, as suppliers investigate technical To understand, just imagine Google glasses for restrictions on components available such diving helmets. By integrating a HUD (Heads as power requirements, data transfer and Up Display) into a diving mask and linking this capture. Suppliers need to identify the best to a sonar imaging system (mounted on the components on the market to integrate into diving helmet), divers will be able to work safely the system (considering equipment cost, and efficiently in zero-visibility conditions. reliability, image quality and ease of integration. By providing a display to the divers, other Once a design is complete, Boskalis plans information becomes available as well. For to procure the components and integrate example, real-time images of the online NAV them into its existing helmet in order to test screen, and drawings, dive plans and sketches the equipment subsea. Extensive training will FIGURE 1 at the worksite which otherwise would not be be needed. Development of a new product By integrating a HUD (Heads Up Display) available on the subsea worksite. is an intensive process but well worth it as into a diving mask and linking this to a sonar an integrated diving helmet will increase the imaging system (mounted on the diving The idea originated with the United States divers’ efficiency and help complete jobs more helmet), divers will be able to work safely and Navy, which has been investigating the quickly and safely. efficiently in zero-visibility conditions.
#156 - AUTUMN 2019 19 SAFETY
Bender Benelux's monitoring box for mobile generators prevents electrical shocks
During construction of offshore wind farms, earth-leakage circuit breakers windmills have no electrical source. The tower (ELCB) sensitive to 230V. The ECLB teams, who are installing cables leading to is a safety device used in electrical these windmills at sea, must carry their own installations to prevent shock. It mobile generators. But there is a problem – detects small stray voltages on FIGURE 2 these generators are not sufficiently grounded the metal enclosures of electrical The new line insulation monitoring box by Benelux Bender because the yellow transition pieces are equipment, interrupting the circuit is a safety device for work in the vicinity of water. sealed in a coating meant to last for 25 years if dangerous voltage is detected. in a saline underwater environment. As a result, Bender Benelux determined that the standard circuit breaker is not adequate these mobile generators did not and this can lead to unsafe situations for meet the safety requirements for the workers as well as for equipment. installation of cables in the sea. less likely to injure a worker if something malfunctions. Recently, when Boskalis wanted to update Bender concluded that instead of 230V their mobile generators, Bender Benelux, for the generator, the maximum should be In principle, these line insulations monitoring supplier of these generators, decided to 110V and that the standard ELCBs should boxes can be used in every industry as a investigate UK law and requirements. The be replaced with a line insulation monitoring standard precaution. Specifically, this new line supplier found that the present systems did box. This unique monitoring box continuously insulation monitoring box is a safe application not meet safety requirements so they set to measures the insulation resistance in the for work in the vicinity of water. work to build a new ‘line insulation monitoring electrical circuit. As soon as the resistance box’ with specifications that meet present-day becomes too low (and voltage too high), the The Bender line insulation monitoring box has safety standards. circuit is automatically interrupted. It serves been installed in all Boskalis mobile generators as an extra fuse and, whereas the old system and once installed, it did not entail any In building the wind towers at sea, small reacted after there was a short circuit, this additional conditions of use. This innovative electrical winches to lift equipment box offers a proactive solution. It reacts device could and should be easily applied to all to the windmill were running on before a shock occurs and using lower voltage standard generators or electrical installations mobile generators equipped with is also safer because a shock from 110V is industry-wide.
Cspect’s state-of-the-art drones help in safely inspecting inaccessible spaces
CSpect’s flying robot is making inspections bigger than 500 mm; spuds from the inside; by rope access, scaffolding and cherry pickers inside transition pieces and towers of wind obsolete. The CSpect drone is an intuitive, turbines; cranes; storage, ballast tanks; jetties; reliable and precise indoor inspection tool areas at heights; bridges; large boilers; and that reduces the number of personnel needed towers. and alleviates the administrative burden associated with inspections. By eliminating the CSpect’s technology includes cutting-edge human interface, operations with practically drone data capture capabilities, which ensures zero risk can be achieved. By using CSpect’s flawless inspections from the very first flight. drones, the workforce stays out of harm’s way A typical drone-based inspection starts with while reducing downtime and inspection costs. a reconnaissance flight, which identifies all areas of interest deserving a closer look. CSpect drones enable remote visual CSpect’s experience gathered through a inspection in any indoor environment and keep wide variety of missions has shown that for workers away from hazardous areas such as most infrastructures 10 minutes is sufficient confined spaces, extreme heights, and places to perform the reconnaissance flight. Based with energised equipment. CSpect robots can on the information gathered during the be prepared for visuals within a minute and an reconnaissance flight, further flights are entire inspection is performed in a matter of planned to more deeply inspect defined points FIGURE 3 hours instead of days. CSpect drones have of interest through the capture of close-up CSpect’s drones enable the workforce to stay conducted inspections in complex areas, such images. After each segment of the inspection out of harm’s way. as: hopper walls; pipe lines with a diameter the drone is brought back to the operators to
20 TERRA ET AQUA review the images in detail and refine/update inaccessible places up to multiple hundreds as ballast tank inspections. The combination the inspection plan on-the-go based on actual of metres beyond the line of sight. The video of above and below water inspections makes data. output is directly available to third parties, who CSpect drones unique in the market and can analyse the live footage. suitable for use in the dredging and maritime In addition, the CSpect drone is the first industries. Most importantly, CSpect drones collision-tolerant drone and is equipped with The fact that the same scope of inspections are approved by the major Classification an innovative wireless communication system is also available below water as above has Societies: Bureau Veritas, Rina and ABS, to that provides a live video feedback. This been a bonus, providing sea-water cross- perform inspections by means of Remotely allows the pilot to bring the drone to the most over inspections on board of vessels as well Inspection Techniques (RIT).
MedAssist is an app for medical support at sea
No doctor on board? Ship owners agree: MedAssist gives needed medical support to ships at sea. The MedAssist Skills Application FIGURE 5 provides offline step-by-step instructions for basic medical skills The Skills app contains the 18 and procedures on board a ship when there is no doctor present or most important (STCW) medical the ship is at a remote location. The app is a low-cost way for the procedures that a captain captain to improve his crew’s medical care when they are far away or officer must be able from professional medical staff and facilities. It also helps maritime to perform. employers to comply with international safety regulations and legislation for medical care.
The initiative for this long-distance medical support grew from the The app presents information in an intuitive and simple way, using experiences of doctors at the Emergency Control - Maritime Training instructional audio, video and photos that give a step-by-step guide (ECMT) - Training Center in Rotterdam, The Netherlands. At ECMT to the safe and professional preparation and execution of medical each year around 500 captains and officers from various companies procedures and after-care. The instructions are based on the use of are trained to perform medical procedures. The requests from clients medical resources available on board. for digital training and support materials for their ships, led to the development of this ‘Skills app’. The Skills app contains the 18 most Another application is the Heart App, which consists of an easy-to-use important (STCW) medical procedures that a captain or officer must heart rate monitor and the accompanying software on a tablet. With this be able to perform, such as stitching a wound, setting-up a drip, or app, a captain or officer can make a hospital-quality electrocardiogram stabilising a neck. in a straightforward way, resulting in a PDF file. With one click this PDF can be sent to a doctor onshore to help making a faster and better diagnosis.
These apps also provide support for on board training for the crew. Personnel should take note of the topics on the apps and, for instance, with the electrocardiogram, time should be taken to practice doing this. The app also provides an overview of important phone numbers for contact with various Radio Medical Services and other practical information that may be urgently needed on board. The app also works offline and can be made available in 45 languages. At present, a patented 2-Way-Augmented Reality Application – called MedAssist Live- is being developed, so an onshore doctor can really work together with the captain to solve a medical problem in real-time.
The cost of the Skills app was only 200 euros per tablet per vessel per year, a reasonable price to pay to safeguard crew members. Ship owners have used these apps and see them as a useful addition to the mandatory medical training that their officers complete on a regular FIGURE 4 basis. Medassist.online’s apps combine medical know-how, practical The MedAssist app presents information in an intuitive and simple way, nautical experience and IT knowledge in a simple and effective way, using instructional audio, video and photos that give a step-by-step taking into account the often limited bandwidth on board ships. The guide to the safe and professional preparation and execution of medical apps can be made available on a ship’s server or on dedicated tablets in procedures and after-care. a rubber encasing.
#156 - AUTUMN 2019 21 SAFETY
Jan De Nul’s Full Mission Simulator safely prepares crews for risky conditions at sea
The Full Mission Simulator (FMS) is a limited power on the propellers. In total five 360° simulator of a dredger’s bridge where sessions were organised to include masters real situations can be practised in a safe and Officers of Watch (OOWs) of two environment. The simulator trains officers TSHDs. The worst possible conditions were on project-specific ship power management simulated in regards to power management, of a designated vessel by setting up the bridge resource management and third party parameters as they are known for a specific pleasure vessels. project area and scope of work. In this way, the crew gains an understanding of the Using the Full Mission Simulator helped ship and the project and can assess best the crews be better prepared for the actual approaches before operating in the real project risks, resulting in better operational FIGURE 6 world. control and thus improved safety. Based on Using the Full Mission Simulator helped the the positive experiences of Jan De Nul and crews be better prepared for the actual project In a cooperative operation amongst VDAB its partners, more of such exercises should risks, resulting in better operational control (the Flemish government), Jan De Nul and be conducted when dredging close to the and thus improved safety. others at Zeebrugge, a dredging simulator operational limits. This is no easy task, just as was used to simulate a specific project risk. the daily work of dredging crews is not easy, In this case, trailing suction hopper dredgers and competent instructors are crucial to the (TSHDs) needed to discharge full power successful use of the simulator. delivery in 2020. The cost of the FMS is through a spray pontoon on a Dynamic not prohibitive for the dredging industry Positioning (DP) track. The TSHDs needed The FMS, which was used to recreate and the results as reported by Jan De Nul, to sail with the same speed and heading, conditions at Zeebrugge, dates from 2005 ‘no incidents, no damages and no delays’ taking into account the floating pipeline and the success of the operation led Jan makes the FMS as a safety tool worth the forces, wind and current, in combination with De Nul to order a new model with expected investment.
Boskalis’ remote control Floating Line Connecting System eliminates dangerous manual operations
get close to the pipelines, eliminating manual operations entirely. This results in fewer crew transfers and fewer safety risks.
The high risk operation of connecting pipelines was identified by the crews doing the work and the consensus was, there must be a better way. The first step was developing a self-floating pipeline that could handle sharp materials. The flexibility of this pipeline meant that 100-metre-long pieces could be placed instead of 20 metres long as is normal with steel pipes. This meant an immediate reduction in the number of connection points so fewer people were put in risky situations less often. But still people were a necessary element. Brainstorm sessions led to various designs and demands, but it took ten years before all the pieces fell in place and a final design was made. A patent has now been applied for and the Boskalis inventors continue to look at ways to improve the design.
The FLCS is based on another Boskalis innovation, the ‘mooring FIGURE 7 actuator’ and the coupling pontoons are specially designed for remote The FLCS is based on another Boskalis innovation, the ‘mooring actuator’ coupling. The pontoons are brought into position without people and the coupling pontoons are specially designed for remote coupling. entering the ‘line of fire’. In so many ways, the new system represents a tremendous safety innovation. People are no longer at risk of injuring hands or fingers or come close the waterline. The system has already been applied to the project in Duqm, Oman and it will soon be rolled out Boskalis’ in-house technical department has developed an innovative and be applied to future Boskalis projects. But it is indeed a generic Floating Line Connecting System (FLCS), where floating pipes are application for floating pipes that could be of value to others in the connected safely by remote control, without the need for people to dredging industry.
22 TERRA ET AQUA A
At DEME, every day – rain or shine – begins with a Safety Moment
The search for ways to improve a safety culture is continuous. presentation, employees become more aware DEME's management team came up with a direct, simple proposal of the risks they face every day at work and the to make sure everyone is always aware of safety. The idea is to safety measures they should be taking. By sharing start every meeting with a ‘Safety Moment’ in which in the first few safety experiences and knowledge with others, minutes of every meeting, a safety topic of choice will be discussed employees start seeing their jobs and those of B with all participants. To facilitate this idea, the QHSE-S department their colleagues in a different way. developed a straightforward tool to guide colleagues reminding them of safety when they enter the meeting room. DEME's ‘Safety Moment’ at the start of every meeting is a simple direct way to remind The instrument is simple: In the meeting room, a board with employees about safety and to emphasise the instructions, together with a branded cotton bag is placed on the need for leadership, communication collaboration wall. When participants enter the room they see the bag immediately. and engagement in staying safe. As this sight is not traditional, it immediately stands out, sparking conversation and interest. Next an individual takes the bag and passes it around the room for every employee to deposit his/her ID badge in the bag. An innocent hand then draws the lucky one who FIGURE 8 gets to present his/her prepared safety moment. To remind colleagues of safety when they C enter the meeting room, colleagues see a Like a pop quiz at school, you must be prepared in advance. Since branded cotton bag placed on the wall [A]. An the presenter is drawn randomly, all employees should have a safety individual takes the bag and passes it around moment of their choice ready at all times, so they can present it the room for every employee to deposit his/her when chosen. While safety moments are by no means new in the ID badge in the bag [B]. By choosing a topic dredging industry, DEME's approach adds an element of surprise. and researching it to give a short presentation, It forces all employees to be prepared at all times to hold a safety employees become more aware of the risks moment, which means that safety must be on everyone’s mind they face every day at work and the safety all the time. By choosing a topic and researching it to give a short measures they should be taking [C].
Van Oord’s ‘Safety News Alert’ is a hard-hitting film followed by open, honest team discussions
An increase in accidents during Q1 2019 led Van highlighted and discussed by colleagues. These way. By involving both projects/vessels/yards Oord to seek a new approach to improving safety were accidents to which many colleagues and office, the tools were available for an open awareness worldwide. To start, an analysis was could relate and the film was followed by team and honest discussion that resulted in a variety done of the root causes of the accidents, which meetings during which colleagues from all of personal and relatable team agreements. occurred under all sorts of circumstances, during offices, projects, vessels and yards discussed Response and participation exceeded full scale work, during routine and non-routine safety statements in an open and honest expectations. jobs and in all business units. Extensive research way and made clear team agreements. These resulted in a ‘Safety News Alert’ that was agreements and group photos were actively The strength of the entire Safety News Alert released throughout the entire organization. This shared on the Van Oord intranet. The involvement (both film and follow-up team meeting) is the was a combination of a hard-hitting film meant and commitment on all levels – from senior usability for all employees. Supplying local to inform, make an impact and inspire, followed management to colleagues at all work locations managers, who act as moderators, with a clear by open and honest team discussions that were throughout the entire organisation made this and easy to use manual enabled them to playful, inspirational and motivating. Safety News Alert an across the board success. organise the Safety News Alert at any location.
Via a worldwide kick-off on 14 May 2019 in a The statements and toolkits provided by Van The Safety News Alert concept can easily be ‘news cast’ film, the Executive Board emphasised Oord were tailored to a team’s work location and used by others in the dredging industry. It is the importance of safety and 5 accidents were were set up in a creative, playful and interactive important to decide on a proper script about
FIGURE 9 Released throughout the entire organisation, the 'Safety News Alert' was a combination of a hard-hitting film meant to inform, make an impact and inspire, followed by open and honest team discussions that were playful, inspirational and motivating. Photo Van Oord
#156 - AUTUMN 2019 23 SAFETY
situations that have occurred within the information within the organisation about the awareness has increased and people are giving company and to film these situations without any answers and actions taken by different teams. more feedback. Registrations for the safety judgment, to determine dilemmas to stimulate leadership training have increased and loads of safety discussions, to createa platform that At Van Oord the results have been clear: Since data from the team meetings has been delivered, encourages participation, and to exchange the kick-off, safety numbers are improving, safety giving Van Oord tangible data to act upon.
Falling overboard is always a danger but Manual lifting is a thing of the past with IHC’s hydraulic Jan De Nul's hopper crew found a solution release shackles
No one knows better than the crew of a dredger where the IHC Handling Systems has developed an easy-to-use hydraulic release shackle dangers lie on board a vessel. When the crew of the Capitan device for lifting modules, templates, jackets and other objects, which is safer and Nunez, a Jan De Nul trailing suction hopper dredger, pointed more efficient than traditional manual handling of shackles. out that the area where the trunnion is located is a risky spot, management was listening. The trunnion is a part of a rotating A traditional shackle is a hinged metal loop secured with a quick-release locking pin joint that is inserted into cylinder that has moving parts. There that is used for lifting. The lifting operation is usually executed manually, often with are also open gaps in the railings at that spot because it is the multiple people working together to lift. It is a job that can be a safety hazard and exact location where the dredge pipe goes over the vessel’s which, because of the physical input from workers, increases occupational health side. It is also the spot where human intervention, such as risks significantly by increasing the chance of injuring hands and feet and putting checks of the pipe heads, is needed. This gap in the railing extra burdens on backs and spines. at a point where people are actively working presents an obvious risk to someone falling overboard. IHC’s hydraulic release shackle device reduces these health risks to workers. By adding this hydraulic tool to a shackle it makes lifting objects in the range of 25- Acting on this observation, the crew of the hopper Capitan 1,500t shackles easier and safer to handle and operate a shackle. The device is Nunez took it upon themselves to solve the problem and designed to be lifted above or beneath water and it is available in two versions – as protect their own. Working with Jan De Nul’s Technical a standard type, with Department, they invented a system to close the openings maximum depth to A in the railing. They designed, constructed and welded a 500 metres and as a railing that opens and closes simultaneously with every deep-water type, with movement of the dredge pipe. It creates a safe barrier a maximum to 3,000 between the working zone and the sea, without limiting the metres. Several different movements of the pipe. This system is automatic and needs backup systems are no manual handling. possible (hot stab receptacle, double Recognising this is the most dangerous area on their ship, cylinders, accumulators the crew solely invented this system and constructed this and separate hydraulic risk control. Crew of Capitan Nunez is rightly proud of this circuits) as well as lifting system and happy it is currently being implemented on pad eyes. The success B board of other trailing hoppers as well. of the device has led to the suggestion to make it available for a broader range of shackle weights. It is already available in the offshore industries for even bigger shackles. The device is able to work in salt laden and corrosive wind driven dust and heavy rain. And FIGURE 11 IHC is able to modify the By adding this hydraulic tool to a shackle, it makes lifting hydraulic shackle device objects in the range of 25-1,500t easier and safer to for a wide number of handle and operate a shackle [A]. The device is designed FIGURE 10 shackle brands. to be lifted above or beneath water and it is available in On Capitan Nunez, an automated system creates a safe two versions – as a standard type, with maximum depth to barrier between the working zone and the sea, without 500 metres and as a deep-water type, with a maximum to limiting the movements of the pipe. 3,000 metres. [B].
24 TERRA ET AQUA Safer and flexible, the CSpect mini ROV replaces risky diving teams
CSpect’s mini ROVs are faster, cheaper and light system with a maximum intensity of more flexible than the mobilisation of a dive 5000 lumens, which provides superb video team. Diving is and will always be a risky footage of the inspected area. A stabilised operation. Diving fatalities have a major impact camera system is controlled by an Inertial on the family left behind, loss of income, lost Measurement Unit (IMU), in contrast to a business, insurance premium increases and camera attached to the face mask of a diver high litigation costs. which is subject to the movements inherent to a diver. In addition, the video and images FIGURE 12 CSpect uses and co-engineers world-leading taken are not disturbed by the air bubbles of The mini ROV can be hand carried onto a plane mini Remotely Operated Vehicle technology the diver, eliminating cloudy images and video and no Port Permits are required so the mini to deliver a safe and cost-effective solution recordings. ROV can be deployed almost immediately upon compared to inspections performed with arrival on the worksite. divers. Their mini ROVs, which are merely The mini ROV can dive up to 150 metre the size of a basketball, are used to visually depth, can sustain 2.5 knots (whereas a inspect underwater structures such as diver is limited to 0.5 knots) and can stay marine warranty; in- and out-hire surveys; hull submerged for longer periods than a diver. ROV which results in more stability. With an inspections on damages, fouling, bottom doors, The mini ROV can be hand carried onto a on board power supply, the mini ROV can work spud cans; quay inspections; tank inspections; plane and no Port Permits are required, so with an extremely thin tether cable, making touchdown monitoring; oceanography, seabed practically speaking, the mini ROV can be the drag of the cable very limited compared to inspections and oil spill monitoring; and fish deployed almost immediately upon arrival on standard ROVs in the market. farming cages surveys. In addition, underwater the worksite, without the heavy administrative inspections in lieu of dry dock for pontoons burden attached to diving missions. In addition, Compared to dive teams, where a three- and vessels provided with azimuth thrusters the mini-ROV is approved by the major Class person operation is mandatory, the mini (vessels where no clearances of rudder and Societies (Bureau Veritas, Rina and ABS) to ROV can be deployed by one person, at any propulsion shaft bearings can be measured) perform underwater inspections. time, day or night, and can stay submerged can be carried out. This is a direct saving for for longer periods. Increased currents and the customer because dry-docking a vessel With the size and shape of a basketball, the depths are no threat and the risk of human for inspection can be replaced by an in-water mini ROV has a symmetric drag profile, which injuries is decreased because there is no survey, which is much cheaper to perform. ensures that when the vehicle turns, it will hold human interface underwater. The practical its ground and not get swept downstream. It result is a zero risk of diving injuries and The mini ROVs are equipped with a GPS can also be equipped with a gripper, thickness fatalities because fewer people are involved positioning system and Low-Light High gauge, spot cleaning system and micro sonar. in the day-to-day operations and a safer work Definition cameras supported by a dimmable Power is supplied by a battery system on the environment.
Jan De Nul’s hopper crew designed, built and installed a safety platform to protect their crewmates
After each dredge cycle on a trailing suction railings are installed on the dredge pipe at hopper dredger, the pipe operator must grease the exact location of the underwater block, and/or inspect the underwater block of the which reduces the chance of trips, slips and hoisting dredge pipe. This is a dangerous job falls. on a slippery and uneven surface. To carry out the work, the pipe operator normally needs This intervention tremendously increases to wear a ‘fall arrest’ harness and prepare the safety level for the pipe operators and extensive risk management paperwork, a time allows them to work more efficiently as well. consuming task. This process on Jan De Nul’s According to the experienced crew of Charles hopper Charles Darwin is now a thing of the Darwin, this was the best safety improvement past. onboard in the last year. Furthermore, it can easily be implemented onboard of all hopper The crew of the Charles Darwin has dredgers without major costs. The installed FIGURE 13 designed, built and installed a platform, platforms are designed to withstand all The crew of the Charles Darwin has designed, with railings, access ladders and grip polls dredging conditions during project execution. built and installed a platform, with railings, that allow safer access to the dredge pipe’s The original design, sketched by the crew of access ladders and grip polls that allow safer underwater block. The platform, ladders and Charles Darwin, was calculated and approved access to the dredge pipe’s underwater block.
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by the Fleet Management to insure that the are perfectly capable of implementing this The work of the pipe operator can now be installation would be built to last and get the system as well. done in 5 minutes instead of 20. Efficiency proper material, and to set up a maintenance is higher and the risk ‘cost’ is lower. Simple regime. After approval, the crew set to welding, Implementation of this improvement requires safety solutions are often the most constructing and installing the platform that the ship has a particular dredge pipe effective ones. Listening to the people in themselves. This working process can easily configuration and room to build the access the field and giving them the freedom to be communicated to all crews on hopper platform. The costs are limited, needing some come up with ideas is often the best way to dredgers. They are all crews with similar steel and grid mesh plus a day’s work for the improve safety in their working environment experience level as on the Charles Darwin and ship’s welder. and conditions.
Van Oord’s pre-assembled onshore ‘CPS Storage & Handling Boskalis reduces risk by securing containers system’ avoids the risks of crane lifting with simple container twist locks
When a subsea power cable is laid, there is an area where the cable may be subjected to When mobilising for (short-term) projects, there are always increased dynamic forces, which the cable is not necessarily designed to survive over a few things that cost time, and always recur, and are the lifetime of the installation. A Cable Protection System (CPS) is used to protect the almost always last minute. One of these things is sea- subsea power cables against these negative impacts over the long term. The installation fastening and securing containers twist locks. Normally a of this protective CPS is traditionally done using an on board crane. The crane picks up bolt or a piece of solid round welding is used to safeguard all parts of the CPS independently from an open top container and then places the parts a container’s twist lock. This process takes approximately on the cable highway for installation. This is an operation that has many risks including 45 minutes per container. Boskalis has adopted new and dropping objects and suspended loads falling on workers. To improve safety and eliminate simple system for securing container twist locks that these risks during the lifting operations, including those caused by the assembly of CPS requires no welding and no grinding, saves time and can be on deck, a special ‘CPS Storage & Handling system’ has been designed and mobilised on implemented safely during mobilisation. Van Oord’s cable laying vessel ‘Nexus’. This new system eliminates the need for offshore craning to install the CPS protection during cable installation. A small thin plate of 1.5 or 2mm slides into the dovetail where the twist lock comes on top. As soon as the With this new CPS Storage & Handling system the CPS are completely assembled container is placed on it, you simply bend the lip up, and the onshore and placed as a whole in racks. Then, when the vessel is in port, the complete container twist lock is sea fastened and secure. The idea to rack is placed on the vessel. This saves time and handling of different components implement this came after doing several quick mobilisations and mitigates many risk factors. By eliminating offshore lifting there are no more in recent years. Although all containers were placed on the suspended loads, no one walks below the suspended load and there can be no right spot, they still had to be secured. Realising that this is dropped objects. The system also features a vertical sliding ramp. This ramp, placed a recurring task that frequently has to be done at the last in front of the CPS rack is movable, and is pulled by a winch, avoiding the need for minute, Boskalis sought a more efficient method and found manual handling, and allowing workers to slide out any CPS at any time. In addition, the answer by looking at the Logistics industry. the system is weatherproof because no lifting by crane is necessary, operations can continue regardless of whether wind speeds exceed the crane limits. All these factors The system can be used on every project where dovetails have created a safer work environment during day-to-day operations. and twist locks are used and container twist locks need to be sea fastened. It is especially useful on projects where The CPS rack can store up to 48 CPS systems, in different layers. Any CPS size can the deck lay-out changes often. By using this simple secure be installed at any time. This means there plate, this task can be done more quickly, saving time and is more flexibility during cable installation, money, and the process is safer. Applying it to the dredging which results in higher production. industry is straightforward and it is already being used by Boskalis on several projects. The system was designed digitally (in 3D) and after completion of the design, and all calculations were done, the system was directly built and installed on the Nexus. The system is operational and works as expected. Although the CPS is more FIGURE 15 efficient and installation is done more To save time and work more safely quickly, which leads to higher production during mobilisation, Boskalis has rates and a shorter installation cycle, the adopted new and simple system for FIGURE 14 real plus is the increased safety for crew. securing container twist locks that The CPS rack can store up to 48 CPS requires no welding and no grinding. systems, in different layers.
The winner of the IADC's Safety Award 2019 will be announced at the IADC's Annual General 26 TERRA ET AQUA Meeting in New Dehli, India on 17 October 2019. INTERVIEW
DIRECTOR OF ECOSHAPE AND ADAELTA HENK NIEBOER
‘WE HAVE TO PUT MORE EFFORT INTO Engineering and THE DEMAND SIDE, entrepreneurship, preferably in an international setting, have determined the course of Henk FIND WAYS TO BRIDGE Nieboer’s path. As a director of Witteveen+Bos, he showed how to conquer new markets in the THE VALLEY OF DEATH.’ field of hydraulic engineering across borders. With the innovation programme Building with Nature, he proved that sustainability and engineering can go hand-in-hand providing added value for society. Self-employed in Adaelta, he now focuses on the next challenge: to convince decision makers that nature-based solutions are the answer.
Interview by Astrid Kramer
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You started working for Witteveen+Bos right after your graduation in 1987. In 2017, you stepped down as director and in July of this year, you left the company completely. How does it feel to say goodbye to the company where you worked over 30 years? Well, it feels quite good. It has been great working for Witteveen+Bos for such a long time and it has been great until the end. But although it was a really great time, a good experience to work there and I look back with pride and much pleasure, it is also good that I left.
When you became a director of Witteveen+Bos, you defined three goals: internationalisation, increasing entrepeneurship and putting Witteveen+Bos on the map in the field of delta technology. Did you leave Witteveen+Bos because you accomplished your goals? Well, with these goals, there is of course pass the aqueduct Hardersluis between based solutions was apparent. Especially the always more work to be done. So, I wouldn’t Harderwijk and Flevoland and the naviduct, dredging industry was having trouble with say ‘mission accomplished’. My main reason a special class of navigable aquaduct, at environmental regulation and legislation. for leaving was that it was time for a new Enkhuizen. In the early nineties, I was the So, already at that time we were thinking challenge. I had worked in many different project leader of the team that came up with about what to do about that. But in the end, positions, first as a specialist engineer, then the concepts of these objects and they were we didn’t proceed with starting EcoShape as a project leader, group leader followed actually built. because we didn’t have the means and by business unit leader. In 2006, I became concluded that the level we were talking at a member of the board of directors and in I am also very proud of the Kapuk project on was too low. 2017, I was advisor to the board. When you Java, Indonesia. This involved a reclamation have been in charge of a whole company, it is area of 1100 hectares which was designed And then the dredging industry a very special experience to step down and into a residential area. A construction of stepped in? remain in the company. In the end, I felt that five polders, areas with managed ground Building with Nature escaped from my view my career within the company was complete. water tables which we made using the old- for a few years until we, from the hydraulic fashioned Dutch art of constructing ring engineering sector, were called to the Which project during your dykes, building pumping stations, pumping office of Van Oord in Rotterdam in 2006. Witteveen+Bos career are you most the water out, letting the soil ripen and bring Frank Verhoeven of Boskalis – and IADC’s proud of? in a drainage layer. If you go there now, it current president – and John van Herwijnen Actually, there are quite a few. When I travel is mainly middle-class housing areas. I am of Van Oord told us they wanted to initiate from the east of The Netherlands to the proud to walk around there and see all the an innovation programme focusing on the northwest, I am always very proud when I families living happily in their houses. environmental aspects of our work. We had brainstorms and tendered for subsidy from In August 2015, you became Economic Affairs. We lost, resubmitted director of EcoShape. When did and lost again. But then in the coalition You have to think you become aware of Building with agreement of the cabinet Balkenende-Bos, Nature? innovation money was reserved from which about which In the early 2000s there was a platform in we received a contribution starting in 2008. The Netherlands called Waterfront where The programme at that time was financed factors are driving a group of people informally discussed 50-50% by the public and private sector. the organisation and improvement of the the system. knowledge infrastructure in our sector. We I always thought it was an important already spoke about Building with Nature initiative of the dredging sector, not only because at that time, the need for nature- because the theme is important, but also
28 TERRA ET AQUA learn. A lot of progress is possible. What I also like is the challenge that it is still quite difficult to get nature-based solutions Meet Henk Nieboer accepted by clients. We need to overcome that. In the run-up to becoming a director of Witteveen+Bos in 2005, Henk Nieboer defined his ambition to bring internationalisation, entrepreneurship and a top The content, the challenge is interesting, but position in the field of delta engineering to the company. Over the span of a decade, the way of working is also really interesting. he successfully taught employees worldwide how to start something new from To work with a small group of focused people, scratch. driving a much larger group of people, creating the context so that they can do their research. In 2019, he decided it was time to set a new goal and start something new himself. It is a wonderful job. Guided by his entrepreneurial spirit and passion for hydraulic engineering, he left Witteveen+Bos and founded his own company Adaelta. He is now dedicated to Despite the positive results of the enabling the showcasing of nature-based solutions worldwide in the field of climate Building with Nature pilots and adaptation projects. the effort made by the EcoShape partners to promote the concept, Additionally, he currently holds the appointed position of Honorary Consul of the full-scale applications seem rare. Republic of Kazakhstan in The Netherlands and director of EcoShape, the foundation If you look in the Engineering with Nature based in Dordrecht, The Netherlands, which runs the innovation programme Building Atlas by the US Army Corps of Engineers, you with Nature. see dozens of project examples that have been realised. Some of them do not meet our criteria of Building with Nature solutions but many of them do. Application of the concept does exist.
However, one of the problems is: everybody wants it, but nobody buys it. There is still the perception among infrastructure managers that Building with Nature solutions are because it was a good opportunity for have his PhD. After completion of the relatively unpredictable. So, if I buy it, what am knowledge institutes and public and private first programme, I was a member of the I buying? What will it be in five- or ten-years’ parties to work together. I had a lot of international usability review board who time? Because it is a natural system, it is experience with the innovation programmes assessed whether the results were useful difficult to predict how it will behave and what of the 1990s which were financed by our or not. When it was decided to start the the management efforts are to maintain it. In government from natural gas revenues. I was second phase in 2012, I thought my time our global society, there is a lot of willingness always very disappointed in them because was up. I asked another representative of to invest a lot of money up front in a project these programmes were always managed Witteveen+Bos to follow up and decided but there is no willingness to compete for by either knowledge institutes or public to concentrate on being a director of cash flow to do long-term maintenance. parties, and the private sector could only Witteveen+Bos. But three years later, Capital expenditure is okay but operational learn from the results in reports or courses. EcoShape came back and asked if I wanted expenditure should be as low as possible. In my opinion, this was not very efficient. It to fulfil the position of director. My primary was my ambition to create opportunities reaction was: ‘this is not possible because At the EcoShape conference last year, there where private and public parties could work I am a director of Witteveen+Bos’ but while was a presentation by Cees Brandsen, one of together with knowledge institutes on saying it, I realised I was going to do it the directors of Rijkswaterstaat. He said he innovation questions or challenges. I saw anyway because I wanted it. wants nature-based solutions, but he needs the innovation programme Building with to know the predictability. That is something Nature as a good opportunity to prove that What do you like about working at we cannot give him yet; not with the same this would work. EcoShape? reliability as with grey infrastructure. We do First of all, it is a great topic to be working on. not know what the mangrove will look like So in 2008, the Building with A completely new way of looking at solutions after five years. Therefore, we should work Nature movement really started up? and products in our sector and trying to get with asset managers and convince them, Yes, I put a lot of effort in the initial a grip on that. How does it work? What do show them or experiment together with brainstorming and formation of ideas. we need to know? We now have developed adaptive management so that they learn to Later on, in the programming of the topics, quite a lot of knowledge. Of course, we are cope with and appreciate the unexpected I became a member of the scientific nowhere as far as we are with the so-called developments that nature-based solutions advisory board – the only one who didn’t ‘grey infrastructure’ so there is still a lot to will demonstrate.
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Are there other issues with Building with Nature? Well, one thing is: what am I buying? The other thing is, defining the term. There is no common perception on what a Building with Brundtland Report’s Nature solution is. People have different ideas about it. Sometimes you are talking to Definition of people and when you get down to the details, you understand that you have been talking Sustainability about different things. We need to have some kind of common language and visualise it. In 1983, the then Secretary General of the United We need to create showcases – exactly what Nations approached Gro Harlem Brundtland to assume EcoShape did – to prove to people that it an enormous undertaking: forming and chairing the World works. You can show it, take people there so Commission on Environment and Development (WCED). you can make them experience it. The independent commission was tasked with conceiving ‘a global agenda for change’, addressing topics of long-term There is no transactional language environmental strategies for the upcoming millennium, for nature-based solutions. With a ways to encourage collaboration between countries at monofunctional design you can determine, diverse stages of economic and social development, and calculate or show the dimensions and ways to deal with environmental concerns among others. determine if the desired function will be This would require defining perceptions of the issues at fulfilled for an acceptable amount of money. hand. But nature-based solutions are always multi- purpose, they bring different benefits, but it The ensuing 1987 publication of Our Common Future, is very hard to quantify these benefits and widely known as the Brundtland Report, defined sustainable very often these benefits are not a benefit for development as: ‘development that meets the needs of the client that you are working for. Because the present without compromising the ability of future they produce multiple benefits; they also generations to meet their own needs’. This definition laid influence more stakeholders compared with the fundamental groundwork for future initiatives. After a traditional solution. This also means that in the Brundtland Report’s publication, the commission was a planning process you need to involve more dissolved and replaced with an organisation named after stakeholders which is complicated in the the report, Center for Our Common Future, to address planning process. the findings of the report. Although this organisation ceased activity in 2002, other initiatives culminated with One of the arguments used for the United Nations Sustainable Development Goals (UN lack of upscaling is that money SDGs) in 2015. is available, but that it cannot be reached. Do you agree with this? Yes, and this is exactly what I want to dedicate my further career to. We need to go to these people, the financers. We need to go out and talk to them, find out what is keeping them from investing in it and try to remove these barriers for them. To connect them to other parties and show that nature-based solutions will work.
Is there a good tool or method to calculate how much is lost and how much is earned? A societal cost benefit analysis is such a tool. You have to make a cost benefit analysis across the whole spectrum, not only monetary aspects but also other aspects. However, this is far from settled science. People are still investigating a lot but there is not yet consensus on how to measure all
30 TERRA ET AQUA these positive aspects. As long as we do not have consensus on the method, we cannot compare projects. You cannot say this project has more value than that. Like in tendering, you cannot say this contractor is better than that one. If we understand how to build with nature, Sustainability has become an important concept in engineering then creating new nature through over the last two decades. What is the key to sustainability for you? infrastructural development is the next step. I am always very much impressed by the original Brundtland definition stating that we should provide the needs of the present generation without compromising the opportunities for future generations to meet their needs. At the moment, we are not there yet. Therefore, we have to take a look at all our processes, all our activities to see where years, then it could be quite dangerous to Is there a role for EcoShape to train we can optimise these in such a way that we rely on a nature-based solution. The system people in Building with Nature? can be convinced that in the future, people analysis determines the scale you have to We contribute to several courses, for instance also meet their needs. That is, for me, the key look at. to the Building with Nature curriculum at the aspect of sustainability. TU Delft and TU Twente. I also spoke at the IHC What do you think is the relevance summer school this year. We do contribute but Both the book Dredging for of Building with Nature for the it is a relatively modest role. Sustainable Infrastructure and the industry and society in general? Building with Nature concept talk I see Building with Nature as an opportunity Good news is that we have now have a about placing a project in a bigger for our sector not only to innovate or to get professor in ecological engineering, Peter picture. How big should this picture a better image but also to reach out to new Herman. He knows exactly what Building with be? potential clients and stakeholders. It brings Nature is and he is training a new generation This fully depends on the assignment us new products with which we can solve of engineers. Stefan Aarninkhof and Mark van or question you are working on. societies’ challenges. If you work in a limited Koningsveld have also become professors. Nature-based solutions can exist at relatively segment of society, for instance the oil and They both worked at EcoShape for several small scales but also on enormous scales. gas industry or infrastructure, and you tailor years and are among the founding fathers of With nature-based solutions, your ambition or optimise your product for that segment, this concept in The Netherlands. is to make use of the natural system as much it will be very difficult to reach out to new as possible, so you have to understand the clients. What about the training of system very well. decision makers and clients For society as a whole, I see that willingness outside The Netherlands? Therefore, you have to start each project with to work with nature means that our projects One of the results of this phase of the a good system analysis. For instance, where are going to contribute to the conservation EcoShape programme will be a book of is the sediment coming from? What is driving of nature. As soon as human functions concepts. This is not a technical book or a the sediment? Where does it silt up? What become dependent on the conservation book with guidelines but a book showing is keeping the sediment in place? What does of nature, the natural processes, humans people what can be achieved with Building with the accumulation lead to? You have to think will do their best to conserve this nature Nature solutions. We want to present it in such about which factors are driving the system. and to keep the processes ongoing. It is a way that decision makers get passionate and The area that you want to study may be small, an opportunity for conservation but also inspired. They do not have to live it through but the sediment source may be a river 20 for restoration. If we understand how to first-hand, but they have to facilitate it. We kilometres away. build with nature, then creating new nature want to touch them with inspiration. through infrastructural development is the If the functioning of your nature-based next step. This is needed because so much Then again, this book will not be enough. solution depends on the supply of sediment, has already been lost. If we learn how to You have to go out there yourself and talk you also have to take a good look at where work with nature in the future, then we may to people, and I think that is mainly a task the river gets its sediment from. How will be able to see every infrastructure project for the consortium members of EcoShape. that be in the future? If you see that there as an opportunity to revive something that After all, EcoShape is not a goal in itself, it is is quite a lot of sediment runoff from an has been lost instead of creating even more a supporting vehicle for the ambitions of our area that will be urbanised in the coming ten loss. partners.
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What is EcoShape?
Founded in 2008, EcoShape is the foundation that manages the public-private innovation programme known as Building with Nature (BwN). After four years of consultations, BwN was launched in 2012 to shift the direction of engineering solutions toward concepts that encourage building with natural materials as well as utilising the forces and interactions present within the natural system.
EcoShape brings together diverse parties including contractors, engineering One BwN solution is the project companies, research institutions, government and NGOs to develop and spread along the Norfolk coast in the knowledge about BwN. These parties cooperate under a common goal: to provide UK, known as Sandscaping durable solutions in the water, by letting nature to do its work. Through collaboration which is based on the Sand with ecologists and economists, innovative hydraulic infrastructure solutions can be Engine concept developed conceived which serve the environment, society, and economy. along the Delfland coast in The Netherlands. Photo Chris Taylor The foundation is dedicated to creating awareness of BwN solutions, developing tools to support the implementation and assessment of BwN solutions through field experiments, and expanding its knowledge base. Through pilot projects, EcoShape acquires knowledge which can be applied in other locations around the world, supporting its belief that knowing the system is key to designing a sustainable solution.
How do you keep expanding your But the thing is, everybody is using their a city and the monkeys in the trees would move vision on sustainability in your own terms and slightly different definitions. away. Behind the mangrove was open water, personal life? This adds to the confusion at the clients and perhaps 20 or 25 metres, followed by the levy, By reading an enormous amount of information. recipient’s side. That needs to be solved in the protecting the houses behind it. We decided to Ten years ago, we were frontrunners on this future. There are several attempts to make a keep a strip of mangrove intact as the first wall topic but currently there are so many related central platform, some more alive than others. of sea protection. What is very interesting now initiatives worldwide. I am following these is that over the years, the strip of mangrove through social media and try to keep track of the Has there been a project in your expanded to over 100 metres deep. And the publications, absorbing as much as possible. career which you would like to redo monkeys came back. according to the Building with Nature It is incredible how much information is out philosophy? If I could go back to the beginning of my career there. In Europe, you have Think Nature, OPLA, In 1989 I worked on the Kapuk project in with the knowledge I now have, I would make Enable and Horizon 2020 programmes. The Indonesia and one day I stood in a strip of a masterplan for the Bay of Jakarta using the World Bank is publishing guidelines. The ADB mangroves, perhaps some 25 metres wide. I concept of mangrove formation. Create an is doing it. IUCN is making a standard. realised this relatively quiet area would soon be enormous city inside a mangrove, make real
32 TERRA ET AQUA parks, green areas for people to walk in and create programmes executed with the revenues soil ripening. If you know this, you can use the conditions for this coast to accrete so there will from natural gas were done without any worms and the silts as construction material be a very large band of mangrove in front of it. involvement of the private sector. How was by making use of the worms. But what you the private sector able to learn what was do with the construction material, what you One of the big issues of the bay of Jakarta is done? By attending courses and reading the design and how you use it to create added that there are still 10.000 people earning their publications. That does not work. You have to value, that is the creative part. That you cannot money as traditional fishermen. Maybe we experience the whole process together and share. Here you can prove your added value. could then create living areas for these people live it in order to be able to apply it. on this spontaneously accreting land. The current Building with Nature Does large-scale application of programme ends in 2020. Is there a How do you see the future for Building with Nature fit in with future for EcoShape? nature-based solutions outside of competitive commercial hydraulic EcoShape manages the innovation programme The Netherlands? engineering? Building with Nature. If the programme stops, I see it as very bright. Everybody wants them. I see no reason why not. I think the good thing then EcoShape in its present form has no use Everybody thinks they need them. One way or about the commercial world is that if a client anymore. Therefore, the real question is: Will another they will be used and created. There is a wants to work with a commercial party, he is there be a third programme? lot of research being done. The only thing is that forced to clearly express what he needs in – but you see this in many sectors – there is a order to make a contract. If you want to realise I see potential for that. Along with many period between the supply and demand called a solution, be it nature based or not, you have people, I am convinced that it would be great the ‘valley of death’. We saw demand and we to know what you need, what you expect from if we could come up with a new programme. reacted with a new proposition. Now we have a it. Because, otherwise, the other party cannot However, experience with previous four-year proposition which has become much more than design what you need or cannot make what innovation programmes showed that it is we originally wanted but we do not see large you asked for. important that follow up programmes are scale demand for this proposition yet. We have better shaped in a different formula, because to put more effort into the demand side. That is What about sharing knowledge in a otherwise the programme may lose its charm what we still have to cope with. We have to find competitive industry? and energy. Therefore, we need to look for a ways to bridge the valley of death. Well you have pre-competitive and competitive different formula and maybe also for different knowledge. Pre-competitive knowledge, for people. With new ambitions, new insights. This It is very often the case that the science and instance, is the impact of a group of worms on is currently being discussed. policy world want it. What we need to do – besides upscaling – is to connect this world. We have to connect knowledge institutes, governmental bodies and public parties to the private sector. Resumé Why is this connection important? What I see is that they are mainly talking 2015–Present 2009–Present among themselves. It was the same in The Director of EcoShape Board Member of Witteveen+Bos Netherlands. The large knowledge development Pension Fund www.ecoshape.org 2010–2016 2015–Present Vice President of KIVI If I could go back to the Member of Supervisory Board of Deltares www.kivi.nl beginning of my career www.deltares.nl 2017–2019 with the knowledge Engineer of Witteveen+Bos 2012–Present I now have, I would Chairman of the Kazakhstan 2006–2017 Chamber of Netherlands Council for Director of Witteveen+Bos make a masterplan Trade Promotion 1987–2006 for the Bay of Jakarta www.internationaalondernemen.nl Various positions at Witteveen+Bos
using the concept of www.witteveenbos.com mangrove formation.
#156 - AUTUMN 2019 33 TECHNICAL MODELLING THE WATERJET CABLE TRENCHING PROCESS
ON SAND DUNES
34 TERRA ET AQUA Numerous offshore wind farms have been recently installed in the southern part of the North Sea. Their infield and export cables are buried for protection against dropped or dragged objects. In sandy soils, burial is carried out by remotely operated tracked Offshore cables vehicles. Two swords with waterjets are used to are commonly fluidise the sand and generate a backward flow of the water-sediment mixture. The area’s highly buried for protection variable seabed topography, characterised by sand against dropped or waves and mega-ripples, can influence the trenching dragged objects. process. At the moment, it is not possible to make an accurate estimate of the influence of sand dunes on the trenching process.
The trench formation process is split into two sediment. A numerical one-dimensional finite fluidisation, sedimentation and cable model is parts: a front section where the seabed is volume model is proposed which is solved on a validated against full-scale field data. eroded by waterjets (erosion model) and a rear staggered grid. section where the sand grains are settling in a Introduction backward flow (sedimentation model). An elastic cantilever beam model is used to Offshore cables are commonly buried for determine the cable shape as it sinks in the protection against dropped or dragged The erosion model is made based on the trench. Subsequently, the depth of lowering objects. In sandy soils, burial is carried out by assumption that the specific energy required of the cable is determined by the intersection remotely operated tracked vehicles, see Figure to fluidise sand is equal to the specific energy of the cable and trench shape. The combined 1A. Two swords with waterjets are located in required to cut sand with a blade. The blade is considered to have a small blade angle and to operate at zero meter water depth, following Miedema (2015). For a given jetting A B configuration and trench dimensions, this vt results in a limiting trencher velocity. A volume balance between situ soil, waterjet flow and entrained flow gives the backwash flow rate and concentration. The last two are used as input for the sedimentation model.
The sedimentation model relates water flow, sediment transport, bed evolution and trench width evolution, based on the shallow water equations. The governing equations represent horizontal momentum and mass FIGURE 1 conservation of the water-sediment mixture Typical waterjet trencher with the jet swords indicated in red (A) as well as an illustration of the and horizontal mass conservation of the erosion, sedimentation and cable component of the jet trenching model (B).
#156 - AUTUMN 2019 35 TECHNICAL
between the tracks of the vehicle, on either deflection model, the point of intersection the trencher velocity. The in-situ production side of the cable. Water is pumped through with the trench shape is determined. It is can also be defined by dividing jet power Pj by the swords to fluidise the sand and generate a assumed that cable remains fixed in this point specific energy Esp. backward flow of the water-sediment mixture, of intersection, hereby giving the achieved 1 see Figure 1B. The cable located on the seabed depth of lowering. See Figures 2 and 7A for an = = = is lowered into the water-sediment mixture illustration of the jet trenching model working 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃 due to its own weight. Due to waves, tides and principle. 𝑄𝑄𝑄𝑄(1) 𝐴𝐴𝐴𝐴 ⋅𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 ⋅𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 2 currents, sandy seabeds are often not flat but = can contain seabed features such as sand Erosion model The specific𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 energy1 𝑤𝑤𝑤𝑤 is determined𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ε by assuming 𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐 ⋅ρ ⋅𝑔𝑔𝑔𝑔�푔 ⋅𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 waves and mega-ripples. When burying a cable At the front of the trench, sand is eroded by the it to be equal to that of non-cavitating𝑘𝑘𝑘𝑘 cutting, 3 in these seabed features, the achieved depth water flow from the jet swords. It is assumed given by equation (2).10 Where ε is dilatency, 𝑚𝑚𝑚𝑚 of lowering shows an oscillating profile with the that the erosion process results in well-mixed km is mean permeability𝑘𝑘𝑘𝑘 and0 ρw is seawater ≈ ⋅𝑘𝑘𝑘𝑘 / maximum depth of lowering achieved at the backward flow of water and sediment. Via a density. Horizontalε force coefficient c1 must 4 2 peaks of these features and minimum at their volume balance the backward flow rate and be calibrated using experiments, Miedema 1 2 4 10 troughs. Models to predict the achieved depth sediment concentration is determined. Via (2015) suggests a 𝑗𝑗𝑗𝑗value of 0.12. 2This value is = ⎡ 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑝𝑝𝑝𝑝 π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 0⎤ of lowering are currently not able to account a specific energy approach the maximum based on𝑝𝑝𝑝𝑝 calibration∙𝑛𝑛𝑛𝑛� 𝑤𝑤𝑤𝑤 ∙ with�α experiments⋅𝐷𝐷𝐷𝐷� ⋅ ⋅𝑘𝑘𝑘𝑘done ⎢ ρ ⎥ 1 for the influence of seabed features, therefore trencher velocity is determined at which the jets =𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚by Combinatie⎢ = Speurwerk2 Baggertechniek= ⎥ 𝑑𝑑𝑑𝑑 1 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 0 an attempt has been made in modelling the are still able to sufficiently erode the seabed. published⎢ by Jong𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣 (1988).⋅ρ ⋅𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�𝑃𝑃𝑃𝑃𝑗𝑗𝑗𝑗 ⎥ 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐴𝐴𝐴𝐴𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡⎢ℎ ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑏𝑏𝑏𝑏0 ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⎥ effect of seabed features on the depth of It is assumed that below this trencher velocity 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 lowering. there are no problems to erode the seabed. = ⎣ ⎦ 2 = + + = , + , + , + + 5 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ε 𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐 ⋅ρ ⋅𝑔𝑔𝑔𝑔�푔 ⋅𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 To make an accurate prediction of the depth Limiting trencher velocity using a 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄(2)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 �𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑄𝑄𝑄𝑄𝑘𝑘𝑘𝑘𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤� 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 3 of lowering of a cable, modelling is divided into specific energy approach 10 2 6 three parts: a model for the erosion section, Miedema (2015) derived a theory for the For cutting,𝑘𝑘𝑘𝑘𝑚𝑚𝑚𝑚 sand= with a blade the parameter 0 𝑗𝑗𝑗𝑗 4 2 ≈ ⋅𝑘𝑘𝑘𝑘⋅ Δ𝑝𝑝𝑝𝑝 π / a model for the sedimentation section and a in-situ production of jets in a draghead. It is hi is the𝑗𝑗𝑗𝑗 layer𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ε𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 thickness. Since the𝑡𝑡𝑡𝑡 jet𝑗𝑗𝑗𝑗 swords 4 𝑄𝑄𝑄𝑄 � 𝑤𝑤𝑤𝑤 �α ⋅𝐷𝐷𝐷𝐷� model for the cable deflection. The erosion based on the assumption that the specific have numerous2 jets spacedρ vertically,1 2 4 = 10 7 model determines the maximum trencher energy required to fluidise sand is equal to the this does not𝑗𝑗𝑗𝑗 directly relate.2 A reasonable = ⎡ 𝑝𝑝𝑝𝑝 π ⎤ velocity at which the seabed can still be specific energy required to cut sand with a assumption𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗� is𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 to� consider𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗� the𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 trench0 as a ⎢𝑝𝑝𝑝𝑝 =∙𝑛𝑛𝑛𝑛 ρ𝑤𝑤𝑤𝑤ℎ∙ α=α⋅𝐷𝐷𝐷𝐷 ⋅(𝑑𝑑𝑑𝑑⋅ ⋅𝑘𝑘𝑘𝑘+⎥ ) 8 1 eroded. Furthermore, it provides flow input blade, having a small blade angle and at zero-𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 whole⎢ and therefore2 assume hi to be⎥ equal 𝑑𝑑𝑑𝑑= = 1 𝑤𝑤𝑤𝑤 = 0 values for the sedimentation model. The metre water depth. The in-situ production 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠to the trench𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 depth𝑠𝑠𝑠𝑠 ⋅ρℎ ⋅𝑔𝑔𝑔𝑔𝑠𝑠𝑠𝑠. 𝑔𝑔𝑔𝑔�The𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤ratio𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 of mean𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 0 Qsitu 𝑄𝑄𝑄𝑄 ⎢ 𝑣𝑣𝑣𝑣 ⋅ 𝐴𝐴𝐴𝐴 d𝑣𝑣𝑣𝑣max ⋅ ℎ𝑗𝑗𝑗𝑗 ⎥ℎ ⋅ 𝑏𝑏𝑏𝑏 9 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃 sedimentation model determines the shape (sand plus pore water) of the jet trencher𝑄𝑄𝑄𝑄 is 𝐴𝐴𝐴𝐴 permeability⎢ ⋅𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 to⋅𝑏𝑏𝑏𝑏 dilatancy=2⋅𝑣𝑣𝑣𝑣 𝑁𝑁𝑁𝑁 can be approximated⎥ 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 , of the trench behind the vehicle. By iteratively given by equation (1), where d and b are initial = using⎣ the Kozeny Carman equation,⎦ resulting 2 max 0 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 = + + = , 𝑄𝑄𝑄𝑄 + , ⋅ �+𝑄𝑄𝑄𝑄 , + + 5 calculating the cable shape in the cable trench depth and width respectively and vt is in the following relation.ε 𝑠𝑠𝑠𝑠=1 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐1 ⋅ρ𝑤𝑤𝑤𝑤 ⋅𝑔𝑔𝑔𝑔�푔𝑠𝑠𝑠𝑠 ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 10 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑘𝑘𝑘𝑘𝑚𝑚𝑚𝑚𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 �𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄 � 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 3 10, = 2 ( ) ( ) 6 = 𝑘𝑘𝑘𝑘, 𝑚𝑚𝑚𝑚 4 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 0𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗� 𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠 (3) ≈𝑄𝑄𝑄𝑄 ⋅𝑘𝑘𝑘𝑘α ⋅⋅π Δ𝑝𝑝𝑝𝑝 π 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢/ 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 ε𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 � 0 �α𝑡𝑡𝑡𝑡 ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� 4 2 ρ𝑤𝑤𝑤𝑤 /2 11, < = : 10( ) = 1 2 7 Total jet4 power P is given by the product of 12 𝑗𝑗𝑗𝑗 j 2 +𝑗𝑗𝑗𝑗 /2 = ⎡ 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 jet𝑝𝑝𝑝𝑝 pressureπ𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑓𝑓𝑓𝑓𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓 and𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 flow𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 rate, 0where⎤ 0 𝐷𝐷𝐷𝐷 flow�𝑘𝑘𝑘𝑘 rate 𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛�𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑤𝑤𝑤𝑤𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟∙ℎ �𝑠𝑠𝑠𝑠α ⋅𝐷𝐷𝐷𝐷α �⋅(𝑢𝑢𝑢𝑢𝑑𝑑𝑑𝑑⋅ 𝑠𝑠𝑠𝑠⋅𝑘𝑘𝑘𝑘 𝑢𝑢𝑢𝑢1 ) 1 ⎢ = ρ = +⎥ 𝑗𝑗𝑗𝑗 8 is determined by jet pressure( ) = p𝑠𝑠𝑠𝑠 and+𝐷𝐷𝐷𝐷 nozzle�𝑘𝑘𝑘𝑘 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎢ 2 ⎥ 2 j 2 𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 ℎ𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 0 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 0 𝑄𝑄𝑄𝑄⎢ 𝑣𝑣𝑣𝑣diameter𝑐𝑐𝑐𝑐⋅ 𝐴𝐴𝐴𝐴⋅𝑣𝑣𝑣𝑣 ⋅ρ Dj. By⋅𝑔𝑔𝑔𝑔𝑣𝑣𝑣𝑣 combining𝑔𝑔𝑔𝑔�⋅ ℎ𝑠𝑠𝑠𝑠 the⎥ ℎ equations⋅ 𝑏𝑏𝑏𝑏 𝑗𝑗𝑗𝑗 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 9 1 ⎢ mentioned= =2 before,𝑁𝑁𝑁𝑁 the=, maximum ⎥√ 𝑘𝑘𝑘𝑘 trench= depth 13, 1 𝑗𝑗𝑗𝑗 ⎣ dmax is found as a: function( ) =of⎦ soil parameters,𝑃𝑃𝑃𝑃 14 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄 𝐴𝐴𝐴𝐴 �⋅𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 2⋅𝑏𝑏𝑏𝑏2 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = + + = jetting, 𝑄𝑄𝑄𝑄 parameters+ , ⋅ + 𝑄𝑄𝑄𝑄and, trencher+ speed+ 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 5 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑠𝑠𝑠𝑠=1 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 0 𝐷𝐷𝐷𝐷 2 v . Since𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟the≥𝑠𝑠𝑠𝑠 sword= depth𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 is an �input,𝑢𝑢𝑢𝑢 the 10 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 t 𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝐸𝐸𝐸𝐸 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 �maximum𝑄𝑄𝑄𝑄= 𝑄𝑄𝑄𝑄trencherΔℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 ( 𝑄𝑄𝑄𝑄velocity)( ) �=( can)𝑄𝑄𝑄𝑄 be ε determined𝑄𝑄𝑄𝑄 6 , 2 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐1 ⋅ρ𝑤𝑤𝑤𝑤 ⋅𝑔𝑔𝑔𝑔�푔𝑠𝑠𝑠𝑠 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 =by the intersection of sword depth𝑘𝑘𝑘𝑘 𝑚𝑚𝑚𝑚and , 𝑠𝑠𝑠𝑠 3 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 α𝐸𝐸𝐸𝐸 ⋅π𝑗𝑗𝑗𝑗 4� 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢2𝑠𝑠𝑠𝑠10𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠� 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 maximum⋅ Δ𝑝𝑝𝑝𝑝 trenchπ depth dmax. 𝑘𝑘𝑘𝑘 𝑄𝑄𝑄𝑄. 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 � 0 �α𝑡𝑡𝑡𝑡𝑘𝑘𝑘𝑘⋅𝐷𝐷𝐷𝐷𝑚𝑚𝑚𝑚 𝑗𝑗𝑗𝑗� 15 1 1ρ𝑤𝑤𝑤𝑤 ≈ /2 ⋅𝑘𝑘𝑘𝑘0 /2 11, 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 ( ) / , = < =+ : ε= + 6.2 7 12 4 2 2 2 +2 /2+𝑗𝑗𝑗𝑗 /22 α𝐸𝐸𝐸𝐸 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 π10 1 2 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟ℎ𝑟𝑟𝑟𝑟𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓α𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 ⋅ 𝑑𝑑𝑑𝑑𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠0 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢014 1 𝐸𝐸𝐸𝐸 0 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄 π =� � 𝑠𝑠𝑠𝑠 = 𝐷𝐷𝐷𝐷 �(�𝑢𝑢𝑢𝑢 ( ) =+𝑗𝑗𝑗𝑗 ) +�𝑗𝑗𝑗𝑗𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠 2 α ⋅𝑢𝑢𝑢𝑢 ⋅𝐷𝐷𝐷𝐷�Δℎ − ⋅𝐷𝐷𝐷𝐷8 � 0 = ⎡ 𝑝𝑝𝑝𝑝 𝑗𝑗𝑗𝑗 π𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 ⎤ √ 𝑘𝑘𝑘𝑘 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑠𝑠𝑠𝑠� 𝐷𝐷𝐷𝐷 �2𝑘𝑘𝑘𝑘� 𝑡𝑡𝑡𝑡 2𝑗𝑗𝑗𝑗� 0 ⎢𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛 ρ𝑤𝑤𝑤𝑤 ∙ α ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅𝑘𝑘𝑘𝑘 ⎥ 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⋅ 𝐴𝐴𝐴𝐴𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⋅ ℎ𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ℎ𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 ⋅ 𝑏𝑏𝑏𝑏0 𝑗𝑗𝑗𝑗 FIGURE 2 𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎢ 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 ⎥ 9 𝑐𝑐𝑐𝑐1 ⋅𝑣𝑣𝑣𝑣√𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘⋅ρ𝑤𝑤𝑤𝑤 ⋅𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�0 13, =2 𝑁𝑁𝑁𝑁⎢ , 1 ⎥ Schematisation of the interaction between erosion, sedimentation and :⎢ ( ) = ⎥ 14 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 cable model with corresponding outputs. 𝑄𝑄𝑄𝑄(4) ⋅ �⎣ 𝑄𝑄𝑄𝑄 2 2 𝑗𝑗𝑗𝑗 ⎦ 𝑓𝑓𝑓𝑓𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓=1 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 0 𝐷𝐷𝐷𝐷 = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹+𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠+ 𝑢𝑢𝑢𝑢= 𝑠𝑠𝑠𝑠 , �+ 𝑢𝑢𝑢𝑢, + , + + 10 5 2 𝑠𝑠𝑠𝑠 Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 ( ) ( ) (=) 𝑄𝑄𝑄𝑄, 𝑠𝑠𝑠𝑠 = 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 �𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤 𝑓𝑓𝑓𝑓 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤� 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 2 6 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 36 TERRA ET AQUA 𝑄𝑄𝑄𝑄 α ⋅π � 𝑟𝑟𝑟𝑟 , 𝑠𝑠𝑠𝑠𝑟𝑟𝑟𝑟 ∙𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠= 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 . 𝑘𝑘𝑘𝑘 4 0 ⋅ Δ𝑝𝑝𝑝𝑝𝑗𝑗𝑗𝑗 π 2 15 1 1 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 /2 /2 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 11, 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄 � 𝑤𝑤𝑤𝑤 �α ⋅𝐷𝐷𝐷𝐷� , = < +: ( ) = ρ + 6.2 12 2 2 2 + +/2= /22 7 α𝐸𝐸𝐸𝐸 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 π 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠 0 01 1 𝐸𝐸𝐸𝐸 0 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄 π �𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹�𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑢𝑢𝑢𝑢�𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 � 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚α ⋅𝑢𝑢𝑢𝑢 ⋅𝐷𝐷𝐷𝐷�Δℎ − ⋅𝐷𝐷𝐷𝐷� 0 =( ) = 𝑗𝑗𝑗𝑗ℎ +=α𝑗𝑗𝑗𝑗 ⋅ (𝑑𝑑𝑑𝑑 + ) √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷2�𝑠𝑠𝑠𝑠𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷2�𝑘𝑘𝑘𝑘 8 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 0 𝑄𝑄𝑄𝑄 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣 ⋅ 𝐴𝐴𝐴𝐴 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝐷𝐷𝐷𝐷⋅ ℎ ℎ ⋅ 𝑏𝑏𝑏𝑏 9 √ 𝑘𝑘𝑘𝑘 13, 1 =2 𝑁𝑁𝑁𝑁 , : ( ) = 14 2 𝐸𝐸𝐸𝐸2 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷⋅𝑗𝑗𝑗𝑗� 𝑄𝑄𝑄𝑄 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑢𝑢𝑢𝑢0 𝑠𝑠𝑠𝑠=1 2 𝑠𝑠𝑠𝑠 10 ( ) = Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 , = ( ) ( ) 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 �𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄 α𝑘𝑘𝑘𝑘 ⋅π � 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 . 0 15 1 1 /2 /2 11, 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 ( ) , = + < : + = 6.2 12 2𝐸𝐸𝐸𝐸 2 2 +𝑗𝑗𝑗𝑗 /2 2 +𝑗𝑗𝑗𝑗 /2 α 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 π𝑠𝑠𝑠𝑠 0 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� �𝑢𝑢𝑢𝑢0 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 α𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠 ⋅𝑢𝑢𝑢𝑢0𝑢𝑢𝑢𝑢⋅𝐷𝐷𝐷𝐷1 𝑗𝑗𝑗𝑗�Δℎ𝑗𝑗𝑗𝑗1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� 𝑗𝑗𝑗𝑗 ( ) = +𝑗𝑗𝑗𝑗 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 2𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷2�𝑘𝑘𝑘𝑘 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗 √ 𝑘𝑘𝑘𝑘 13, 1 : ( ) = 14 2 2 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑢𝑢𝑢𝑢0 2 𝑠𝑠𝑠𝑠 ( ) =
𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 . 15 1 1 /2 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 , = + + 6.2 2 2 2 + /2 2 α𝐸𝐸𝐸𝐸 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 π 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� �𝑢𝑢𝑢𝑢0 � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 α𝐸𝐸𝐸𝐸 ⋅𝑢𝑢𝑢𝑢0 ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�Δℎ𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 1 = = = 𝑃𝑃𝑃𝑃𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐴𝐴𝐴𝐴𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑏𝑏𝑏𝑏0 ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 2
𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ε 𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐 ⋅ρ ⋅𝑔𝑔𝑔𝑔�푔 ⋅𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 𝑘𝑘𝑘𝑘 3 10 𝑚𝑚𝑚𝑚 𝑘𝑘𝑘𝑘 0 ≈ ⋅𝑘𝑘𝑘𝑘 1 ε = = / = 4 2 10 1 2 𝑃𝑃𝑃𝑃𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 4𝐴𝐴𝐴𝐴𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑏𝑏𝑏𝑏0 ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 2 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = ⎡ 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑝𝑝𝑝𝑝 π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 0⎤ 2 � 𝑤𝑤𝑤𝑤 �α= � ⎢𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛 ρ ∙ ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅𝑘𝑘𝑘𝑘 ⎥ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎢ 2 ⎥ ε 𝑑𝑑𝑑𝑑 1 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤 1 0𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ⎢ 𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝐸𝐸𝐸𝐸⋅ρ ⋅𝑔𝑔𝑔𝑔𝑐𝑐𝑐𝑐 𝑔𝑔𝑔𝑔�⋅ρ ⋅𝑔𝑔𝑔𝑔�푔⎥⋅𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 𝑘𝑘𝑘𝑘 3 ⎢ 10 ⎥ ⎣ 𝑚𝑚𝑚𝑚 ⎦ 𝑘𝑘𝑘𝑘 0 = + + = + +≈ ⋅𝑘𝑘𝑘𝑘+ + 5 , , ε , / 4 over-depth is dependent2 on trencher velocity, 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 10 𝐸𝐸𝐸𝐸 1 2 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 jetting�𝑄𝑄𝑄𝑄 power𝑄𝑄𝑄𝑄 and seabed𝑄𝑄𝑄𝑄4 permeability.� 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄After 6 calibration2 a reasonable𝑗𝑗𝑗𝑗 assumption2 was = = ⎡ 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑝𝑝𝑝𝑝 π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 0⎤ 10 , 𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛� 𝑤𝑤𝑤𝑤 ∙ �α ⋅𝐷𝐷𝐷𝐷� ⋅ ⋅𝑘𝑘𝑘𝑘 found to ⎢be α𝑗𝑗𝑗𝑗 4 = 0.03.ρ 2 ⎥ ⋅ Δ𝑝𝑝𝑝𝑝 ODπ Maximum trench depth 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎢ 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗2 ⎥ 𝑄𝑄𝑄𝑄 𝑑𝑑𝑑𝑑 � 𝑤𝑤𝑤𝑤 �α 1⋅𝐷𝐷𝐷𝐷𝑠𝑠𝑠𝑠� 𝑤𝑤𝑤𝑤 0 8 Depth of sword below seabed ρ⎢ 𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣 ⋅ρ ⋅𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� ⎥ = 7 ⎢ ⎥ 6 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂⎣ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎦ = ℎ(7) =α ⋅(𝑑𝑑𝑑𝑑 + ) 8 4 = + + = , + , + , + + 5 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 0 𝑄𝑄𝑄𝑄 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 ⋅ 𝑗𝑗𝑗𝑗𝐴𝐴𝐴𝐴 The𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 initial𝑣𝑣𝑣𝑣 trench𝐸𝐸𝐸𝐸⋅ ℎ width𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 bℎ is assumed𝑗𝑗𝑗𝑗⋅𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏 𝑗𝑗𝑗𝑗to𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 9 2 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 �𝑄𝑄𝑄𝑄 0 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 � 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 6 be approximately=2 𝑁𝑁𝑁𝑁 , the2 same as the sword , = Distance below seabed [m] 0 separation distance (outside4 to outside), plus 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 ⋅ � 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 ⋅ Δ𝑝𝑝𝑝𝑝𝑗𝑗𝑗𝑗 π 2 0 200 400 600 800 1000 a small 𝑄𝑄𝑄𝑄margin.𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠=1𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 The� in-situ flow� αrate𝑡𝑡𝑡𝑡 ⋅𝐷𝐷𝐷𝐷 is𝑗𝑗𝑗𝑗 �now ρ𝑤𝑤𝑤𝑤 10 given by equation (8). Trencher velocity [m/hr] Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 = 7 , = ( ) ( ) ℎ𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 α𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 ⋅(𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ) 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 α𝐸𝐸𝐸𝐸 ⋅π=� 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 =𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 + 8 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 /2 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 0 11, 𝑄𝑄𝑄𝑄(8) 𝑣𝑣𝑣𝑣 ⋅ 𝐴𝐴𝐴𝐴 𝑣𝑣𝑣𝑣 ⋅ ℎ ℎ ⋅ 𝑏𝑏𝑏𝑏 9 < : ( ) = 12 FIGURE 3 =2+ 𝑁𝑁𝑁𝑁 /2 , 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 01 1 Maximum trench depth based on the specific energy approach (blue line) and depth of sword𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 Entrainment𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 of𝐸𝐸𝐸𝐸 ambient water𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 ( ) = 𝑄𝑄𝑄𝑄 𝑠𝑠𝑠𝑠 +𝐷𝐷𝐷𝐷⋅𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 𝑄𝑄𝑄𝑄 below the seabed (dashed red line). Plotted example is for typical trencher parameters. A certain amount2 of ambient2𝑠𝑠𝑠𝑠=1 water will entrain 10 the flow𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 before𝑠𝑠𝑠𝑠 reaching𝑠𝑠𝑠𝑠 the𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗 transition Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 , = √ 𝑘𝑘𝑘𝑘 ( ) ( ) 13, between erosion1 and sedimentation. Due to the: complex𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠( ) =𝐸𝐸𝐸𝐸 flow pattern𝑠𝑠𝑠𝑠 and lack𝑠𝑠𝑠𝑠 of 14 𝑄𝑄𝑄𝑄 α 2⋅π2� 𝑟𝑟𝑟𝑟𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 An example of this relation is given in Figure In-situ flow rate experimental data it𝑘𝑘𝑘𝑘 is0 difficult𝐷𝐷𝐷𝐷 to say which 1 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑢𝑢𝑢𝑢0 /2 11, 3 from which can= be concluded =that the = In-situ flow rate Qsitu is defined as the maximum mechanisms are 2taking( place.𝑠𝑠𝑠𝑠) Therefore, to <( ) =: = 12 trencher should not have any problem to trench𝑗𝑗𝑗𝑗 cross sectional area times the trencher estimate the entrainment flow rate+ some /2 𝑃𝑃𝑃𝑃 1 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 = = 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐴𝐴𝐴𝐴𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= ℎ ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑏𝑏𝑏𝑏0 ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 01 1 fluidise the seabed up to a trencher velocity of velocity.𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 Maximum trench depth is given by simplifications𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �and assumptions𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 have to be 𝑃𝑃𝑃𝑃𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 2 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ( ) = 𝑠𝑠𝑠𝑠 +𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡approximatelyℎ 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 6000 m/hr.𝑠𝑠𝑠𝑠 = the depth of the swords below. the seabed made. 𝑘𝑘𝑘𝑘 2 2 𝑄𝑄𝑄𝑄 𝐴𝐴𝐴𝐴 ⋅𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 ⋅𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 15 𝐸𝐸𝐸𝐸 1 1 /2 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 ε plus a certain overdepth hOD. The𝑗𝑗𝑗𝑗 over-depth • Flow entrainment is based on entrainment = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 2 = 6 2𝑂𝑂𝑂𝑂 + +𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 6.2 𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐 ⋅ρ ⋅𝑔𝑔𝑔𝑔�푔 ⋅𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 , √ 𝑘𝑘𝑘𝑘 Volume conservation is calculated as a fraction (2αOD) of maximum2 2 calculations+ /2 for free2 non-cavitating jets. 13, ε 𝑘𝑘𝑘𝑘 α𝐸𝐸𝐸𝐸 3 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 π 1 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 To1 determine𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠the flow𝑠𝑠𝑠𝑠 rate and concentration10 trench depth d , calculated𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 by the specific 𝑗𝑗𝑗𝑗 0 Water is only entrained: ( 𝐸𝐸𝐸𝐸) at= the0 backside𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 of 𝑗𝑗𝑗𝑗 14 𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐 ⋅ρ ⋅𝑔𝑔𝑔𝑔�푔 ⋅𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 max𝑄𝑄𝑄𝑄 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 � �𝑢𝑢𝑢𝑢• � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 α ⋅𝑢𝑢𝑢𝑢 ⋅𝐷𝐷𝐷𝐷�Δℎ − ⋅𝐷𝐷𝐷𝐷� 𝑘𝑘𝑘𝑘 𝑚𝑚𝑚𝑚 𝑗𝑗𝑗𝑗 2 2 𝑗𝑗𝑗𝑗 at the interface between the𝑘𝑘𝑘𝑘 erosion and energy approach.3 By using this0 method,√ 𝑘𝑘𝑘𝑘 the 𝑠𝑠𝑠𝑠the 𝐷𝐷𝐷𝐷jet� (see𝑘𝑘𝑘𝑘 Figure 5A). On the 𝑘𝑘𝑘𝑘front𝐷𝐷𝐷𝐷 side of 10 ≈ ⋅𝑘𝑘𝑘𝑘0 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑢𝑢𝑢𝑢0 sedimentation𝑚𝑚𝑚𝑚 section a volumeε conservation / 4 2 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 ( ) = is applied.≈ The⋅𝑘𝑘𝑘𝑘0 flow rate and 2concentration are / 10 1 2 1 =ε = =4 4 used2 as input for the sedimentation𝑗𝑗𝑗𝑗 model. The2 𝑠𝑠𝑠𝑠 ⎡ 𝑝𝑝𝑝𝑝1 2 ⎤ 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 = 𝑗𝑗𝑗𝑗10 𝑗𝑗𝑗𝑗 π 𝑃𝑃𝑃𝑃𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 0 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 interface𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡4 ℎflow 𝑠𝑠𝑠𝑠rate Q𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑝𝑝𝑝𝑝i is∙𝑛𝑛𝑛𝑛 simply�0 𝑤𝑤𝑤𝑤 determined𝑠𝑠𝑠𝑠 ∙ �α ⋅𝐷𝐷𝐷𝐷 by� the⋅ ⋅𝑘𝑘𝑘𝑘 . 𝑄𝑄𝑄𝑄 𝐴𝐴𝐴𝐴 ⋅𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑2⎢ ⋅𝑏𝑏𝑏𝑏 ρ⋅𝑣𝑣𝑣𝑣 ⎥ 15 ⎡ sum𝑝𝑝𝑝𝑝 𝑗𝑗𝑗𝑗of the in-situ soil flow rate⎤ Q , waterjet𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 flow 1 1 /2 = 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 π𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 ⎢ 0 2situ 𝐸𝐸𝐸𝐸 ⎥ 2 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 ⎢𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛� 𝑤𝑤𝑤𝑤 ∙= �α ⋅𝐷𝐷𝐷𝐷� ⋅ ⋅𝑘𝑘𝑘𝑘 ⎥1 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 0 , = + + 6.2 rateρ Qj and flow rate⎢ of clear𝑐𝑐𝑐𝑐 seawater⋅𝑣𝑣𝑣𝑣 ⋅ρ entrained⋅𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� ⎥ 2 2 2 + /2 2 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗 𝑑𝑑𝑑𝑑 ⎢ 2 ⎢ ⎥ε ⎥ 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 α 𝑗𝑗𝑗𝑗 0 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 π 𝐸𝐸𝐸𝐸 0 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 in the1𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 flow𝑠𝑠𝑠𝑠 1 Q𝑤𝑤𝑤𝑤E, see𝑤𝑤𝑤𝑤 equation0 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 (5) and Figure 4. 𝑄𝑄𝑄𝑄 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 � �𝑢𝑢𝑢𝑢 � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 α ⋅𝑢𝑢𝑢𝑢 ⋅𝐷𝐷𝐷𝐷�Δℎ − ⋅𝐷𝐷𝐷𝐷� ⎢ 𝑐𝑐𝑐𝑐𝐸𝐸𝐸𝐸 ⋅𝑣𝑣𝑣𝑣 ⋅ρ𝑐𝑐𝑐𝑐 ⋅ρ⋅𝑔𝑔𝑔𝑔⋅𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔��푔 ⋅𝑣𝑣𝑣𝑣 ⎥𝑚𝑚𝑚𝑚 𝑗𝑗𝑗𝑗 ⎣ 𝑘𝑘𝑘𝑘 ⎦ 3 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 ⎢ 10 ⎥ = + + = , + , + , + + 5 ⎣ 𝑚𝑚𝑚𝑚 ⎦ 𝑘𝑘𝑘𝑘 0 = + + = 𝑠𝑠𝑠𝑠 , 𝑗𝑗𝑗𝑗+ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠,≈ +⋅𝑘𝑘𝑘𝑘𝐸𝐸𝐸𝐸 , � +𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 + 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 5 𝐸𝐸𝐸𝐸 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ε𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 / 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 4 6 2 2 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 �(5)𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 , 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤�= 𝑄𝑄𝑄𝑄10𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄1 𝐸𝐸𝐸𝐸2 4 4 2 6 2 2 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ⋅ Δ𝑝𝑝𝑝𝑝 π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 , ==⎡ 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑝𝑝𝑝𝑝 π𝑄𝑄𝑄𝑄 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 � 0⎤ �α ⋅𝐷𝐷𝐷𝐷� 𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛� 4𝑤𝑤𝑤𝑤 ∙ �α ⋅𝐷𝐷𝐷𝐷2 � ⋅ ⋅𝑘𝑘𝑘𝑘𝑤𝑤𝑤𝑤 In⎢ which Qj𝑗𝑗𝑗𝑗 ρis the total jet flow rateρ and⎥ thus the 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ⋅ Δ𝑝𝑝𝑝𝑝 π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 = 7 𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 sum⎢� of forward,�α 2inward⋅𝐷𝐷𝐷𝐷� and backwash⎥ jets, see 𝑑𝑑𝑑𝑑 ρ𝑤𝑤𝑤𝑤 1 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 0 equation⎢= (5).𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣Jet ⋅ρflow rates⋅𝑔𝑔𝑔𝑔𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑔𝑔𝑔𝑔� are calculated𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 ⎥ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 by 7 = ℎ =α ⋅(𝑑𝑑𝑑𝑑 + ) 8 using⎢ the Bernoulli theory, see equation⎥ (6) for ℎ𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂⎣ α𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 ⋅ 𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎦ = the flow= rate𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠( of a single𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠+⋅ 𝐴𝐴𝐴𝐴𝑠𝑠𝑠𝑠 nozzle,𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) ℎ ignoring 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⋅ ℎ𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤internal𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ℎ 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 ⋅ 𝑏𝑏𝑏𝑏0 8 = + + = + + + + 5 9 pressure losses., , , 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 =20 𝑁𝑁𝑁𝑁 , 𝑄𝑄𝑄𝑄 𝑣𝑣𝑣𝑣 ⋅ 𝐴𝐴𝐴𝐴 𝑣𝑣𝑣𝑣 ⋅ ℎ ℎ ⋅ 𝑏𝑏𝑏𝑏 9 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 �𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤� 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 =2 𝑁𝑁𝑁𝑁 2, 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 ⋅ � 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 6 , = 𝑠𝑠𝑠𝑠=1 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 4 FIGURE 4 10 𝑄𝑄𝑄𝑄 ⋅ � 𝑄𝑄𝑄𝑄 ⋅ Δ𝑝𝑝𝑝𝑝𝑗𝑗𝑗𝑗 π 2 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠=1 = 𝑡𝑡𝑡𝑡 Δ𝑗𝑗𝑗𝑗ℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 ( ) ( ) 𝑄𝑄𝑄𝑄 � 𝑤𝑤𝑤𝑤, �α ⋅𝐷𝐷𝐷𝐷� Flow idealisation10 of the erosion section; volume balance of (6) ρ Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 = in-situ, jet and entrained7 flow rate. , = ( ) 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸(𝑠𝑠𝑠𝑠 ) α𝐸𝐸𝐸𝐸 ⋅π � 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 α=𝐸𝐸𝐸𝐸 ⋅π � ℎ 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠=α∙𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 ⋅𝑠𝑠𝑠𝑠(𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 + ) /2 8 11, 0 < : ( ) = 12 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤/2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 0 +𝑗𝑗𝑗𝑗 /2 11, 𝑄𝑄𝑄𝑄 𝑣𝑣𝑣𝑣 ⋅ 𝐴𝐴𝐴𝐴 𝑣𝑣𝑣𝑣 ⋅ ℎ ℎ ⋅ 𝑏𝑏𝑏𝑏 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 #156 - AUTUMN 2019 37 < : ( ) =𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢01 1 12 9 =2 +𝑗𝑗𝑗𝑗 /2 ( ) = +𝑗𝑗𝑗𝑗 𝑁𝑁𝑁𝑁𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘, 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷2�𝑘𝑘𝑘𝑘 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢01 1 2 ( )𝐸𝐸𝐸𝐸= +𝑗𝑗𝑗𝑗𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄 2⋅𝑠𝑠𝑠𝑠�𝐷𝐷𝐷𝐷𝑄𝑄𝑄𝑄2�𝑘𝑘𝑘𝑘 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 𝑠𝑠𝑠𝑠=1 √ 𝑘𝑘𝑘𝑘 13, 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗 1 10 Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 : ( ) = 14 , = √ 𝑘𝑘𝑘𝑘 ( ) ( ) 2 2 13, 1 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗 : ( ) = 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 0 14 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 � 𝑢𝑢𝑢𝑢 𝑄𝑄𝑄𝑄 α ⋅π �2 𝑟𝑟𝑟𝑟2 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷 ( ) = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 0 � 𝑢𝑢𝑢𝑢0 2 𝑠𝑠𝑠𝑠 /2 11, < (: ) = ( ) = 𝑠𝑠𝑠𝑠 12 + 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 /2 � 𝑠𝑠𝑠𝑠 . 𝑗𝑗𝑗𝑗 𝑘𝑘𝑘𝑘 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 0 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 15 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠1 𝑢𝑢𝑢𝑢� 1𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢1 1 /2 𝑗𝑗𝑗𝑗 ( ) = +𝑗𝑗𝑗𝑗 . = 6 2𝑂𝑂𝑂𝑂 +𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 + 6.2 , 2 2 2 2 2 15 1 1 𝐸𝐸𝐸𝐸 /22 +𝑗𝑗𝑗𝑗 /2 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 α 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 𝑗𝑗𝑗𝑗 π , = +𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 π � � 𝑠𝑠𝑠𝑠 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝐷𝐷𝐷𝐷+𝑗𝑗𝑗𝑗� �𝑢𝑢𝑢𝑢0 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠6.2 α𝐸𝐸𝐸𝐸 ⋅𝑢𝑢𝑢𝑢 0 ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�Δℎ𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� 2 2 2 + /2 2√ 𝑘𝑘𝑘𝑘 𝑗𝑗𝑗𝑗 13, α𝐸𝐸𝐸𝐸 0𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘√ 𝑘𝑘𝑘𝑘 π 1 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� �𝑢𝑢𝑢𝑢0 : � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠( ) =α𝐸𝐸𝐸𝐸 ⋅𝑢𝑢𝑢𝑢0 ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�Δℎ𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� 14 𝑗𝑗𝑗𝑗 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 𝑘𝑘𝑘𝑘 2 2 𝑗𝑗𝑗𝑗 � 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑢𝑢𝑢𝑢0 2 𝑠𝑠𝑠𝑠 ( ) =
𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 . 15 1 1 /2 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 , = + + 6.2 2 2 2 + /2 2 α𝐸𝐸𝐸𝐸 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 π 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� �𝑢𝑢𝑢𝑢0 � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 α𝐸𝐸𝐸𝐸 ⋅𝑢𝑢𝑢𝑢0 ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�Δℎ𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 1 TECHNICAL = = = 𝑃𝑃𝑃𝑃𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐴𝐴𝐴𝐴𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑏𝑏𝑏𝑏0 ⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 1 2 A B= = = Output to sedimentation model ε 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ℎ 1𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃 𝑄𝑄𝑄𝑄 𝐴𝐴𝐴𝐴 𝐸𝐸𝐸𝐸 ⋅𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 ⋅ρ𝑑𝑑𝑑𝑑 ⋅𝑔𝑔𝑔𝑔�⋅𝑏𝑏𝑏𝑏푔 ⋅𝑣𝑣𝑣𝑣⋅𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 The flow rate and concentration are 𝑘𝑘𝑘𝑘 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 3 = 10 determined by 2equations (16) and (17). 𝑚𝑚𝑚𝑚 Where the entrained flow rate is corrected 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1 𝑘𝑘𝑘𝑘𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 0𝑠𝑠𝑠𝑠 ε 𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐 ⋅ρ ⋅𝑔𝑔𝑔𝑔≈ �푔⋅𝑘𝑘𝑘𝑘⋅𝑣𝑣𝑣𝑣 𝑚𝑚𝑚𝑚 / by a factor (1-c ). This correction results ε 𝑘𝑘𝑘𝑘 30 4 2 10 10 1 2 from the assumption that the entrained 𝑚𝑚𝑚𝑚 4 𝑘𝑘𝑘𝑘 𝑗𝑗𝑗𝑗 0 2 water is not completely sediment-free but = ⎡ 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 ≈𝑝𝑝𝑝𝑝 ⋅𝑘𝑘𝑘𝑘π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 /0⎤ 𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛ε � 𝑤𝑤𝑤𝑤 ∙ �α ⋅𝐷𝐷𝐷𝐷� ⋅ ⋅𝑘𝑘𝑘𝑘 has a concentration4 equal to the output ⎢ 2 ρ ⎥ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎢ 2 10 1 2⎥ concentration. Equation (16) is therefore 𝑑𝑑𝑑𝑑 1 4 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 0 ⎢ 𝑗𝑗𝑗𝑗𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣 ⋅ρ ⋅𝑔𝑔𝑔𝑔2 𝑔𝑔𝑔𝑔� ⎥ = ⎡ 1𝑝𝑝𝑝𝑝 π ⎤ implicit and must be solved iterative. = = = 𝑗𝑗𝑗𝑗 ⎢ 𝑗𝑗𝑗𝑗� � 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗� 0 ⎥ ⎢𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛 ρ𝑤𝑤𝑤𝑤 ∙ α ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅𝑘𝑘𝑘𝑘 ⎥ 𝑗𝑗𝑗𝑗 ⎣ ⎦ (1 ) 16 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠 𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎢ 2 ⎥ 𝑑𝑑𝑑𝑑 1 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 0 = 1 𝑄𝑄𝑄𝑄 𝐴𝐴𝐴𝐴 ⋅𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 ⋅𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = + + =𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣, ⋅ρ+⋅𝑔𝑔𝑔𝑔,𝑔𝑔𝑔𝑔�= + , + = + = + +5 (1 ) FIGURE 5 𝐸𝐸𝐸𝐸 ⎢ ⎥ 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 2 0 𝑗𝑗𝑗𝑗 (1 − 𝑛𝑛𝑛𝑛 ) ⋅ 𝑄𝑄𝑄𝑄 = 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠⎢ 𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤⎥ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑐𝑐𝑐𝑐 𝑃𝑃𝑃𝑃 16 Cross-section of jet flow where water is only𝑄𝑄𝑄𝑄 entrained𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 at the𝑄𝑄𝑄𝑄 backside�𝑄𝑄𝑄𝑄 (A),𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑄𝑄𝑄𝑄 an illustration𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑄𝑄𝑄𝑄 ℎ � 𝑠𝑠𝑠𝑠of𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑄𝑄𝑄𝑄 0 (16)𝑠𝑠𝑠𝑠 = 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 ⋅ −𝑐𝑐𝑐𝑐0 ε ⎣ 2 𝑄𝑄𝑄𝑄 𝐴𝐴𝐴𝐴 ⎦⋅𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 ⋅𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + +6 (1 ) 17 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 1 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 − 𝑛𝑛𝑛𝑛0 ⋅ 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 2 𝐸𝐸𝐸𝐸 vertical𝑐𝑐𝑐𝑐 ⋅ρ jet⋅𝑔𝑔𝑔𝑔 spacing�푔 ⋅𝑣𝑣𝑣𝑣 (B).𝑚𝑚𝑚𝑚 = + + ,= ,= + , + , = + + 0 5 𝑘𝑘𝑘𝑘 1 4 2 𝑐𝑐𝑐𝑐0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 = = = 3 ⋅ Δ𝑝𝑝𝑝𝑝𝑗𝑗𝑗𝑗 π ( ) 𝑄𝑄𝑄𝑄( 𝑄𝑄𝑄𝑄) 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ ( −𝑐𝑐𝑐𝑐− 𝑐𝑐𝑐𝑐 ) 10 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤𝑗𝑗𝑗𝑗 1 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ε = + + 1 1817 𝑗𝑗𝑗𝑗𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 �𝑄𝑄𝑄𝑄 � 𝑄𝑄𝑄𝑄𝑤𝑤𝑤𝑤 �α𝐸𝐸𝐸𝐸𝑄𝑄𝑄𝑄⋅𝐷𝐷𝐷𝐷��𝑐𝑐𝑐𝑐 ⋅ρ𝑄𝑄𝑄𝑄 ⋅𝑔𝑔𝑔𝑔�𝑄𝑄𝑄𝑄푔 ⋅𝑣𝑣𝑣𝑣+ = ( ) +2 𝑚𝑚𝑚𝑚 𝑃𝑃𝑃𝑃 2 ρ 𝑘𝑘𝑘𝑘𝑚𝑚𝑚𝑚 6 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠 = 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 7 𝐸𝐸𝐸𝐸 0 3 𝑄𝑄𝑄𝑄 𝐴𝐴𝐴𝐴 ⋅𝑣𝑣𝑣𝑣0 𝑑𝑑𝑑𝑑 ⋅𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 , = 10∂(ℎ𝑏𝑏𝑏𝑏 ) ∂𝑄𝑄𝑄𝑄(ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄) 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ − 𝑐𝑐𝑐𝑐 ≈ ⋅𝑘𝑘𝑘𝑘 / 𝐸𝐸𝐸𝐸 4 (17) 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 18 ε 𝑗𝑗𝑗𝑗 2 2 𝑚𝑚𝑚𝑚 + = (𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 ) ∙𝑏𝑏𝑏𝑏+2⋅𝑣𝑣𝑣𝑣 ⋅𝑑𝑑𝑑𝑑 = 4𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 ⋅ Δ𝑝𝑝𝑝𝑝𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂π 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑘𝑘𝑘𝑘 ( )∂𝑡𝑡𝑡𝑡 ( ∂𝑥𝑥𝑥𝑥) 2the jets, in-situ soil is loosened which is not region𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 and𝑡𝑡𝑡𝑡 ℎa valueα of k⋅ =𝑑𝑑𝑑𝑑 77𝑡𝑡𝑡𝑡 for𝑗𝑗𝑗𝑗 the empirical 0 19 1 2 𝑄𝑄𝑄𝑄= � =𝑤𝑤𝑤𝑤 �(α ⋅𝐷𝐷𝐷𝐷�+ ) ≈ ⋅𝑘𝑘𝑘𝑘+ = (1 8 ) +2 (1 ) 10 ρ ε ∂ ℎ𝑏𝑏𝑏𝑏 ∂ ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 / 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 considered𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 4 1 𝑤𝑤𝑤𝑤as entrainment𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ε but is included constant k, as= given in Nobel (2013). Also the Sedimentation𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 model∙𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 ⋅𝑑𝑑𝑑𝑑 4 𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗 𝑐𝑐𝑐𝑐 ⋅ρ ⋅𝑔𝑔𝑔𝑔2 �푔 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂2 ∂(ℎ0 𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐) ∂(ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢) 7 ⎡ 𝑝𝑝𝑝𝑝 π 𝑘𝑘𝑘𝑘⎤𝑚𝑚𝑚𝑚 𝑄𝑄𝑄𝑄 𝑣𝑣𝑣𝑣 ⋅ 𝐴𝐴𝐴𝐴 𝑣𝑣𝑣𝑣 ⋅ ℎ ℎ ⋅ 𝑏𝑏𝑏𝑏 ∂𝑡𝑡𝑡𝑡 10∂𝑥𝑥𝑥𝑥 −𝑣𝑣𝑣𝑣1 2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑛𝑛𝑛𝑛0 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 19 = 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 in the volume𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗conservation.0 development of the uniform3 flow velocity4 uu and+ The sedimentation= (1 9 model) +2 describes(1 ) 𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛� 𝑤𝑤𝑤𝑤 ∙ �α ⋅𝐷𝐷𝐷𝐷� ⋅ ⋅𝑘𝑘𝑘𝑘 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ( ∂𝑡𝑡𝑡𝑡 ) ( 2∂𝑥𝑥𝑥𝑥 ) 1 ( ) 20 ⎢ ρ 10 ⎥ = ℎ =α=2⋅(𝑑𝑑𝑑𝑑𝑁𝑁𝑁𝑁 , + ) 𝑗𝑗𝑗𝑗 + + = + + + + • Water is entrained in an individual jet over jet radius ru is different in= the⎡ 𝑗𝑗𝑗𝑗 flow𝑗𝑗𝑗𝑗 development𝑝𝑝𝑝𝑝∂ πℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡 ∂ 𝑗𝑗𝑗𝑗 ℎ𝑏𝑏𝑏𝑏the𝑢𝑢𝑢𝑢2𝑢𝑢𝑢𝑢 backward0⎤ flow28 containing water and 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⎢ 2 𝑚𝑚𝑚𝑚 ⎥ 𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛� 𝑤𝑤𝑤𝑤 ∙ �α ⋅𝐷𝐷𝐷𝐷� ⋅ ⋅𝑘𝑘𝑘𝑘2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑 1 𝑘𝑘𝑘𝑘𝑤𝑤𝑤𝑤 0 ⎢ ρ −𝑣𝑣𝑣𝑣⎥ −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 a distance𝑠𝑠𝑠𝑠 equal to the0 vertical spacing 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 region𝑠𝑠𝑠𝑠 and𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 regionℎ 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 of fully𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝐸𝐸𝐸𝐸 developed𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 flow.∂0(ℎ∂𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢) ∂(ℎ𝑏𝑏𝑏𝑏∂𝑥𝑥𝑥𝑥𝑢𝑢𝑢𝑢suspended) 1 ∂( 𝑏𝑏𝑏𝑏sediment.ℎ ) 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 Input𝑓𝑓𝑓𝑓 variables𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 20 ⎢ 𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣 ⋅ρ ⋅𝑔𝑔𝑔𝑔≈ 𝑔𝑔𝑔𝑔�⋅𝑘𝑘𝑘𝑘 ⎥ / 𝑄𝑄𝑄𝑄 𝑣𝑣𝑣𝑣 ⋅ 𝐴𝐴𝐴𝐴 𝑄𝑄𝑄𝑄 𝑣𝑣𝑣𝑣 ⋅⋅𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚�ℎ 𝑄𝑄𝑄𝑄 ⎢ ℎ ⋅ 𝑏𝑏𝑏𝑏 2 ⎥𝑔𝑔𝑔𝑔 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 ε 𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠4=1 1 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤+ 0 + 9 = + + + + between individual jets, ∆hjet (see Figure 5). Analytical expressions are derived 𝑐𝑐𝑐𝑐in ⋅𝑣𝑣𝑣𝑣Lee∂𝑡𝑡𝑡𝑡⋅ρ and⋅𝑔𝑔𝑔𝑔 𝑔𝑔𝑔𝑔�∂𝑥𝑥𝑥𝑥used2 from2 the∂𝑥𝑥𝑥𝑥 2erosion model are initial 21 ⎢ 2 ⎥ =2 𝑁𝑁𝑁𝑁 ⎢ = ⎥ 10 tan( ) 1 2 , ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ 𝑏𝑏𝑏𝑏ℎ ⎣ • Water is only entrained in ⎦the10 forward jets Chu (2003) andΔ ℎare𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 given by⎢ equations (11), trench dimensions,⎥ flow𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 rate𝑓𝑓𝑓𝑓 and𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡sediment𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 4 , = ( ) ( ) 𝑔𝑔𝑔𝑔 ∂𝑧𝑧𝑧𝑧𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑗𝑗𝑗𝑗 2 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 ⎣ ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ⎦ ∂𝑥𝑥𝑥𝑥 21 = + + = ⎡located, + on𝑝𝑝𝑝𝑝 ,the top+ half, of+ the jet swords,+⎤ the (12), (13) 𝑄𝑄𝑄𝑄and5 (14).⋅ � 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆concentration.= −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� Included푔 − tan in( theθ)� sedimentation = 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 0 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠=1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = (2 ∂𝑥𝑥𝑥𝑥 + ) 22 ⎢inward𝑝𝑝𝑝𝑝 ∙𝑛𝑛𝑛𝑛 �and𝑤𝑤𝑤𝑤 backwash∙ �α ⋅𝐷𝐷𝐷𝐷 jets� are⋅ neglected.⋅𝑘𝑘𝑘𝑘 ⎥ 𝑄𝑄𝑄𝑄 α =⋅π �+ 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 +∙𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠=𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 , + , + model, +is breaching+10 of trench sidewalls,5 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 ρ𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 0 2 ∂𝑧𝑧𝑧𝑧 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑄𝑄𝑄𝑄 �𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 � 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑 ⎢ 2 ⎥ , = 6𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 ( )𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ( )𝐸𝐸𝐸𝐸 /2𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑆𝑆𝑆𝑆entrainment𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤 𝑓𝑓𝑓𝑓 −𝑔𝑔𝑔𝑔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑔𝑔𝑔𝑔𝑔� ∗of ambient푔𝐸𝐸𝐸𝐸11𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡, − waterθ � and erosion/ 2 1 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 0 < 𝑄𝑄𝑄𝑄: 𝑄𝑄𝑄𝑄 ( 𝑄𝑄𝑄𝑄) = 𝑄𝑄𝑄𝑄 �𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑆𝑆𝑆𝑆� =𝑄𝑄𝑄𝑄−𝑢𝑢𝑢𝑢 (2𝑄𝑄𝑄𝑄⋅(𝑑𝑑𝑑𝑑∂𝑥𝑥𝑥𝑥 + 𝑏𝑏𝑏𝑏) ) 2322 The=⎢ total entrained𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣 ⋅ρ flow⋅𝑔𝑔𝑔𝑔 rate𝑔𝑔𝑔𝑔� is determined⎥ by 2 sedimentation= of 12the trench bottom. Breaching6 , 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 +𝑗𝑗𝑗𝑗 = /2 2 ⎢ 𝑗𝑗𝑗𝑗 4 2 ⎥ 𝑄𝑄𝑄𝑄 α 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓⋅π � 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢 0 𝑠𝑠𝑠𝑠, 𝐷𝐷𝐷𝐷𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠�𝑘𝑘𝑘𝑘 𝑓𝑓𝑓𝑓 ∗ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 the summation of entrainment per individual 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 0 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢1 1 4 is included2 𝑆𝑆𝑆𝑆 − via𝑢𝑢𝑢𝑢 the⋅ 𝑑𝑑𝑑𝑑 ρactive𝑏𝑏𝑏𝑏 wall− ρ velocity, v in 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ⋅ Δ𝑝𝑝𝑝𝑝 π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 ( ) 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ( ) wall 23 ⎣ ⎦ = 𝑗𝑗𝑗𝑗𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+𝐷𝐷𝐷𝐷/2𝑡𝑡𝑡𝑡 𝑘𝑘𝑘𝑘 ⋅ Δ𝑝𝑝𝑝𝑝 π 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑗𝑗𝑗𝑗 𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢11𝑢𝑢𝑢𝑢�, 𝑄𝑄𝑄𝑄 jet. For� a 𝑤𝑤𝑤𝑤single� jetα sword⋅𝐷𝐷𝐷𝐷� the number of forward 𝑄𝑄𝑄𝑄2 2� � �α ⋅𝐷𝐷𝐷𝐷Figure� = 7B. The sidewalls( ρ are) assumed to be = + + =ρ + + + + < : ( ) =5 𝑤𝑤𝑤𝑤 1 12 24 = , , , 7 𝑠𝑠𝑠𝑠 + /2𝑗𝑗𝑗𝑗 ρ = 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 jets at the top half of the sword is given by N. 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷= and𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 remain2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 vertical.ρ Furthermore,− ρ it is assumed7 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 0 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 𝑆𝑆𝑆𝑆 𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑤𝑤𝑤𝑤𝑓𝑓𝑓𝑓 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐸𝐸𝐸𝐸 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢1 √ 𝑘𝑘𝑘𝑘 1 2 1ρ3, − ρ ∂𝑐𝑐𝑐𝑐 Consequently,𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 the total entrained flow rate is (11, 12) ( ) 1 𝑗𝑗𝑗𝑗 that𝑡𝑡𝑡𝑡 the1 material coming( ρ from) the side walls 24 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ℎ 𝑄𝑄𝑄𝑄 α �𝑄𝑄𝑄𝑄⋅ 𝑑𝑑𝑑𝑑 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 � 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 =6 𝑠𝑠𝑠𝑠 +𝐷𝐷𝐷𝐷 �ℎ𝑘𝑘𝑘𝑘𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 α𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 ⋅ 𝑑𝑑𝑑𝑑𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑆𝑆𝑆𝑆 = − 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� = given= by equation( 2 (9).+ ) 8 : ( ) 2== 2 = ( +is mixed) instantaneous 1 14 ρ in the∂𝑥𝑥𝑥𝑥 backward8 flow, 2 2 2 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 25 , = 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 𝑗𝑗𝑗𝑗𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗 = 2 ρ − ρ ∂𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 4 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 0 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝐷𝐷𝐷𝐷0𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 hereby𝑡𝑡𝑡𝑡 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 conserving0 2 the rectangular trench 𝑄𝑄𝑄𝑄 𝑣𝑣𝑣𝑣 ⋅ 𝐴𝐴𝐴𝐴 𝑣𝑣𝑣𝑣 ⋅ ℎ 𝑗𝑗𝑗𝑗 ℎ ⋅ 𝑏𝑏𝑏𝑏 2 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢𝑄𝑄𝑄𝑄 𝑠𝑠𝑠𝑠√ 𝑘𝑘𝑘𝑘 𝑣𝑣𝑣𝑣�⋅ 𝐴𝐴𝐴𝐴𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 ⋅ ℎ 𝑆𝑆𝑆𝑆 ℎ −⋅ 𝑏𝑏𝑏𝑏𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�13, 2 ∂𝑏𝑏𝑏𝑏 9 𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ⋅ Δ𝑝𝑝𝑝𝑝 π 𝑡𝑡𝑡𝑡 𝑗𝑗𝑗𝑗 9 1 2 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 1 ρ ∂𝑥𝑥𝑥𝑥 25 𝑄𝑄𝑄𝑄 � �α ⋅𝐷𝐷𝐷𝐷� ( ) = =2 shape. All𝑆𝑆𝑆𝑆 main𝑔𝑔𝑔𝑔 parameters𝑔 are illustrated in =2 𝑁𝑁𝑁𝑁 𝑤𝑤𝑤𝑤 : ( ) = 𝑁𝑁𝑁𝑁 , = 14 ∂𝑥𝑥𝑥𝑥 , ρ 2 2 Figures 7A and2 B. Notable are the constant 26 = 7 𝑗𝑗𝑗𝑗 =2 2 ∂ 𝑏𝑏𝑏𝑏 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑘𝑘𝑘𝑘 0 𝐷𝐷𝐷𝐷 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄 ⋅ � 𝑄𝑄𝑄𝑄 concentration𝑆𝑆𝑆𝑆∂𝑏𝑏𝑏𝑏 𝑔𝑔𝑔𝑔c and𝑔 velocity u over the vertical 𝑄𝑄𝑄𝑄 ⋅ � 𝑄𝑄𝑄𝑄 . 2 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠=1 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚∂𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂=1 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 ⋅𝑣𝑣𝑣𝑣15 10 26 =(9) ℎ =α ⋅(𝑑𝑑𝑑𝑑 + ) 1 1 10 ( ) =/28 axis, and∂𝑡𝑡𝑡𝑡 the=2 distinction between initial seabed 27 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 , = + =+ Δℎ ( ) ( 6.2) = Δℎ𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠 , porosity ∂n𝑏𝑏𝑏𝑏o and re-settled𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 seabed porosity ns. = ( ) ( ) 2𝐸𝐸𝐸𝐸 2 2 +𝑗𝑗𝑗𝑗 �/2 2 𝑄𝑄𝑄𝑄, 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⋅ 𝐴𝐴𝐴𝐴𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⋅ ℎ𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ℎ𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 ⋅ 𝑏𝑏𝑏𝑏α0 𝑟𝑟𝑟𝑟𝐷𝐷𝐷𝐷𝑠𝑠𝑠𝑠�𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 π ∂𝑧𝑧𝑧𝑧 ⋅𝑣𝑣𝑣𝑣 Of which the entrainment for a 𝑄𝑄𝑄𝑄single𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 jet,π . for� a � (13,𝑠𝑠𝑠𝑠 14)𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� �𝑢𝑢𝑢𝑢0 9 𝑘𝑘𝑘𝑘�𝐸𝐸𝐸𝐸𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸α𝐸𝐸𝐸𝐸 ⋅𝑢𝑢𝑢𝑢0 ⋅𝐷𝐷𝐷𝐷𝑠𝑠𝑠𝑠𝑗𝑗𝑗𝑗�Δℎ𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠− ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� ∂𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 27 𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄 α ⋅π � 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 =15( ) 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸distance from𝑠𝑠𝑠𝑠 =2 zero𝑠𝑠𝑠𝑠 to𝑁𝑁𝑁𝑁 ∆h , is given by equation0 1 √ 𝑘𝑘𝑘𝑘 1 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷/2�𝑘𝑘𝑘𝑘 0 Shallow= water∂𝑡𝑡𝑡𝑡 + 1 equations for flow in 28 𝑄𝑄𝑄𝑄 α ⋅π � 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑jet𝑠𝑠𝑠𝑠 = 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 + + 6.2 0 , /2 ∂𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 11, 10. 2 2 The2 position+ s where/2 the transition<2 : from( flow) = a rectangular𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 channel𝑤𝑤𝑤𝑤 with variable 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 /2𝑠𝑠𝑠𝑠 α𝐸𝐸𝐸𝐸 11𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗dr,� 𝑘𝑘𝑘𝑘 π ρ 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ 12 𝑄𝑄𝑄𝑄 ⋅ � 𝑄𝑄𝑄𝑄 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 0 𝐸𝐸𝐸𝐸 0 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 +𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗 =/2 ∂𝑡𝑡𝑡𝑡 + (1 ) 28 < : ( ) = 𝑠𝑠𝑠𝑠=1 𝑄𝑄𝑄𝑄 π � � 𝑠𝑠𝑠𝑠 development𝐷𝐷𝐷𝐷 � �𝑢𝑢𝑢𝑢 region to� 𝑑𝑑𝑑𝑑developed𝑠𝑠𝑠𝑠 α ⋅𝑢𝑢𝑢𝑢 flow⋅𝐷𝐷𝐷𝐷 region�Δℎ − cross-section⋅𝐷𝐷𝐷𝐷� 12 𝑗𝑗𝑗𝑗 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 0 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 = + (1 ) +𝑗𝑗𝑗𝑗 /2 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 10𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢1 1 29 is, can be determined by s = k/2 ≈ 6.2D .( Now,) To𝑗𝑗𝑗𝑗 model the𝑠𝑠𝑠𝑠 flow of water𝑤𝑤𝑤𝑤 and sediment 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 0 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 dr j = 𝑠𝑠𝑠𝑠 +𝐷𝐷𝐷𝐷 �ρ𝑘𝑘𝑘𝑘 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 Δ𝑢𝑢𝑢𝑢ℎ1𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗 ( )1 ( ) 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 , = ( ) 𝑗𝑗𝑗𝑗 equations (10), (11), (12), (13) and (14) can be ρbehind the𝑛𝑛𝑛𝑛 trencher,⋅ ρ the− 𝑛𝑛𝑛𝑛 so-called⋅ ρ Shallow = 𝑠𝑠𝑠𝑠 +𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 = / /+ (1 / ) / / / 3029 2 2 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 +𝐷𝐷𝐷𝐷 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 combined and simplified resulting in equation𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡Water𝑡𝑡𝑡𝑡 Equations𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 (SWE)𝑡𝑡𝑡𝑡 are 𝑡𝑡𝑡𝑡used. 𝑡𝑡𝑡𝑡Since𝑡𝑡𝑡𝑡 they 𝑠𝑠𝑠𝑠 � 𝑗𝑗𝑗𝑗 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1𝑤𝑤𝑤𝑤 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠−1 213,𝑠𝑠𝑠𝑠−1 2 𝑄𝑄𝑄𝑄 𝑟𝑟𝑟𝑟α𝑠𝑠𝑠𝑠 ⋅π 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∙𝑢𝑢𝑢𝑢𝐷𝐷𝐷𝐷 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 1−ℎ𝑠𝑠𝑠𝑠ρ𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 ℎ� 𝑛𝑛𝑛𝑛 𝑢𝑢𝑢𝑢⋅ ρ 𝑏𝑏𝑏𝑏 − 𝑛𝑛𝑛𝑛− ℎ�⋅ ρ 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 (10) 0 (15). The first term gives the entrainment: ( in) the= were first proposed/ / by/ Saint Venant/ (1871),/ 14 / 30 √ 𝑘𝑘𝑘𝑘 13, 11, + 1 /2 flow development zone and the second𝑡𝑡𝑡𝑡 term+1 𝑡𝑡𝑡𝑡+ Δ𝑡𝑡𝑡𝑡is12 =2[𝑡𝑡𝑡𝑡(these𝑡𝑡𝑡𝑡)𝑗𝑗𝑗𝑗 equations𝑡𝑡𝑡𝑡( )𝑡𝑡𝑡𝑡 ] are+2𝑡𝑡𝑡𝑡 alsoΔ𝑥𝑥𝑥𝑥 referred(𝑡𝑡𝑡𝑡 ) 𝑡𝑡𝑡𝑡to as the𝑡𝑡𝑡𝑡 < : ( ) = 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝐷𝐷𝐷𝐷 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 : ( ) = 14 12 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ℎ 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 −ℎ� 𝑏𝑏𝑏𝑏0 𝑡𝑡𝑡𝑡 ℎ� 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 − ℎ� 𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 In a free turbulent2 jet,2 two regions+𝑗𝑗𝑗𝑗 can/2 be the entrainment in the developed𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠 flow𝑢𝑢𝑢𝑢 region𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢Saint-Venant equations. It is assumed that the 𝑗𝑗𝑗𝑗 𝐷𝐷𝐷𝐷 𝑘𝑘𝑘𝑘 2 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷0 � ( ) =Δ𝑡𝑡𝑡𝑡 = [(𝑣𝑣𝑣𝑣 ) − (𝑣𝑣𝑣𝑣 ) ]𝑏𝑏𝑏𝑏 +2𝑣𝑣𝑣𝑣Δ𝑥𝑥𝑥𝑥 (𝑑𝑑𝑑𝑑 ) 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟defined:𝑓𝑓𝑓𝑓𝑟𝑟𝑟𝑟𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 a region𝑢𝑢𝑢𝑢 of𝑠𝑠𝑠𝑠 flow0𝑢𝑢𝑢𝑢 development1 1 and up to a distance ∆hjet. flow is one-dimensional, hereby reducing the 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 ≥𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 (�) =𝑢𝑢𝑢𝑢 +𝑗𝑗𝑗𝑗 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 𝑡𝑡𝑡𝑡 / /𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / / 𝑡𝑡𝑡𝑡 / / / / 31 a region of fully developed2 flow. For2 the 𝑠𝑠𝑠𝑠 +𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 ( ) = 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1𝑟𝑟𝑟𝑟𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠1 �𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡− 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 flow development𝑠𝑠𝑠𝑠 region, the entrainment𝑗𝑗𝑗𝑗 . 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 −ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 ℎ� 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ − ℎ� 𝑏𝑏𝑏𝑏 15 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ 𝑠𝑠𝑠𝑠 1 1 /2 / / / / / / / / 31 coefficient𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠 is half� 𝑠𝑠𝑠𝑠of the√ fully𝑘𝑘𝑘𝑘 developed flow 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 13, + , = + 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡Δ𝑡𝑡𝑡𝑡+1 = 𝑡𝑡𝑡𝑡 (𝑡𝑡𝑡𝑡+𝑡𝑡𝑡𝑡 ) (1𝑡𝑡𝑡𝑡 )𝑡𝑡𝑡𝑡 +2𝑡𝑡𝑡𝑡 6.2𝑡𝑡𝑡𝑡(1 Δ𝑥𝑥𝑥𝑥 )(𝑡𝑡𝑡𝑡 ) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 . 𝑘𝑘𝑘𝑘 1 region entrainment( )coefficient. For a free 2 2 14 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 +𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠 /2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 2 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 : = α𝐸𝐸𝐸𝐸 15 ℎ 𝑏𝑏𝑏𝑏 𝐷𝐷𝐷𝐷𝑐𝑐𝑐𝑐�𝑘𝑘𝑘𝑘 −ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 π 𝑡𝑡𝑡𝑡ℎ 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ − ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ 1 1 /2 2 2 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐸𝐸𝐸𝐸 0 𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑗𝑗𝑗𝑗 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 non-cavitating jet, a reasonable assumption𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 � �𝑢𝑢𝑢𝑢 � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 α𝑠𝑠𝑠𝑠 ⋅𝑢𝑢𝑢𝑢 ⋅𝐷𝐷𝐷𝐷�𝑠𝑠𝑠𝑠Δℎ − ⋅𝐷𝐷𝐷𝐷� 𝑠𝑠𝑠𝑠 = + + 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷 6.2 𝑗𝑗𝑗𝑗Δ𝑡𝑡𝑡𝑡 = −(𝑣𝑣𝑣𝑣 ) (1 −𝑛𝑛𝑛𝑛 )𝑏𝑏𝑏𝑏 +2𝑣𝑣𝑣𝑣 (1 −𝑛𝑛𝑛𝑛Δ𝑥𝑥𝑥𝑥 )(𝑑𝑑𝑑𝑑 ) , 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠 � 0 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 2𝐸𝐸𝐸𝐸 2 2 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹is α𝑟𝑟𝑟𝑟+𝑟𝑟𝑟𝑟 =𝑗𝑗𝑗𝑗≥𝑠𝑠𝑠𝑠 0.085/2 in𝑢𝑢𝑢𝑢 the𝑠𝑠𝑠𝑠 fully2 developed𝑢𝑢𝑢𝑢 flow (15) α E𝐷𝐷𝐷𝐷 �𝑘𝑘𝑘𝑘 π 2 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� �𝑢𝑢𝑢𝑢0 � 𝑑𝑑𝑑𝑑(𝑠𝑠𝑠𝑠) = α𝐸𝐸𝐸𝐸 ⋅𝑢𝑢𝑢𝑢 0 ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�Δℎ𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� − 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 −𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑛𝑛𝑛𝑛0 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 𝑟𝑟𝑟𝑟𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘 . 15 1 381 TERRA/2 ET AQUA 6 2𝑂𝑂𝑂𝑂𝑗𝑗𝑗𝑗 , = + + 6.2 2 2 2 + /2 2 α𝐸𝐸𝐸𝐸 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 π 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 π � � 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� �𝑢𝑢𝑢𝑢0 � 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 α𝐸𝐸𝐸𝐸 ⋅𝑢𝑢𝑢𝑢0 ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�Δℎ𝑗𝑗𝑗𝑗𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − ⋅𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗� 0 √ 𝑘𝑘𝑘𝑘 𝑠𝑠𝑠𝑠 𝐷𝐷𝐷𝐷𝑗𝑗𝑗𝑗�𝑘𝑘𝑘𝑘 (1 ) 16 = + + (1 ) (1 − 𝑛𝑛𝑛𝑛)0 ⋅ 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 16 =0 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+ +𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸(1 )0 system of equations to only three. the active wall velocity due to breaching, = 𝑄𝑄𝑄𝑄 +𝑄𝑄𝑄𝑄 0 +𝑄𝑄𝑄𝑄 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠⋅ (1−𝑐𝑐𝑐𝑐 ) 0 − 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 17 • continuity of the total fluid volume (water n and n are the porosity of the initial and 𝑐𝑐𝑐𝑐 0 s 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 ⋅ −𝑐𝑐𝑐𝑐0 𝑄𝑄𝑄𝑄=0 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+ 𝑄𝑄𝑄𝑄+𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸(⋅1 − 𝑐𝑐𝑐𝑐)0 17 plus sediment), see equation (18) re-settled seabed respectively and (dtr the) ( ) 18 + = ( ) +2 • continuity of sediment volume, see trench depth. The source terms on the 𝑄𝑄𝑄𝑄0 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 ⋅ − 𝑐𝑐𝑐𝑐0 (∂ ℎ)𝑏𝑏𝑏𝑏 (∂ ℎ𝑢𝑢𝑢𝑢)𝑏𝑏𝑏𝑏 18 equation (19) right hand side in the momentum equation+ = ( 𝑣𝑣𝑣𝑣𝐸𝐸𝐸𝐸 −𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) ∙𝑏𝑏𝑏𝑏+2 ⋅𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ( )∂𝑡𝑡𝑡𝑡 ( ∂𝑥𝑥𝑥𝑥) c0 19 • conservation of momentum, see equation in equation (20) are given separately∂ ℎ𝑏𝑏𝑏𝑏+ in ∂ ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 = (1 h ) +2 (1 ) 𝑣𝑣𝑣𝑣𝐸𝐸𝐸𝐸 −𝑣𝑣𝑣𝑣(𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡1 bw∙𝑏𝑏𝑏𝑏) ⋅𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 16 (20) equations (21), (22), (23), (24)(∂ andℎ𝑏𝑏𝑏𝑏)𝑐𝑐𝑐𝑐∂𝑡𝑡𝑡𝑡 (25)(∂ forℎ𝑏𝑏𝑏𝑏 ∂𝑥𝑥𝑥𝑥𝑢𝑢𝑢𝑢)𝑢𝑢𝑢𝑢 = 19 + = 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(1 + )𝑠𝑠𝑠𝑠 ++2 (1𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚(1 ) )0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 improved readability.(1 ) These source terms −𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛− 𝑛𝑛𝑛𝑛𝑏𝑏𝑏𝑏0 16⋅ 𝑄𝑄𝑄𝑄 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠⋅𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 = ( ∂𝑡𝑡𝑡𝑡 ) ( ∂𝑥𝑥𝑥𝑥 ) 10 ( ) 20 ( ∂ ℎ)𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 +∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 +𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 =𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 +𝐸𝐸𝐸𝐸 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+ 0 +0 +𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 Furthermore, it is assumed that all quantities Sbed, Sf , Ssed, S+c and+ S0w account𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠1 for the bed 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ −𝑐𝑐𝑐𝑐 ( )− 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 2 −𝑣𝑣𝑣𝑣2 = −𝑛𝑛𝑛𝑛2 + 𝑏𝑏𝑏𝑏 + ⋅𝑣𝑣𝑣𝑣(1 −𝑛𝑛𝑛𝑛) 𝑑𝑑𝑑𝑑 17 are uniform over the cross-section and vertical gradient,0 1 bed friction, sediment(∂∂𝑡𝑡𝑡𝑡ℎ 𝑏𝑏𝑏𝑏exchange,)𝑢𝑢𝑢𝑢 (∂ ∂𝑥𝑥𝑥𝑥ℎ𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢) 1 (∂ 𝑏𝑏𝑏𝑏ℎ) 16 20 =𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 + + = 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+ +𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+ +𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑄𝑄𝑄𝑄+ +𝑄𝑄𝑄𝑄 (𝑄𝑄𝑄𝑄1 ⋅ ( )−𝑐𝑐𝑐𝑐 ) 2 0 𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2 𝑆𝑆𝑆𝑆𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 0 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 velocities are neglected. To simplify the solving concentration= (+10 gradient+𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) 1and divergence∂𝑡𝑡𝑡𝑡 of∂𝑥𝑥𝑥𝑥 the 2𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄∂𝑥𝑥𝑥𝑥 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄17 ⋅ − 𝑐𝑐𝑐𝑐 − 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ ℎ𝑏𝑏𝑏𝑏(𝑢𝑢𝑢𝑢 ) ( ∂ 𝑏𝑏𝑏𝑏)ℎ 16 ( ) 21 18 ( ) 0 = + = FIGURE= ( 6𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 tan)𝑓𝑓𝑓𝑓 +2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 of the momentum equation,1 it is rewritten so 𝑐𝑐𝑐𝑐 trench0𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠width𝑗𝑗𝑗𝑗 + respectively.𝑗𝑗𝑗𝑗𝐸𝐸𝐸𝐸+ 𝐸𝐸𝐸𝐸 (1016 The0) source term 𝑔𝑔𝑔𝑔 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 = ( ) 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄( 𝑄𝑄𝑄𝑄) 𝑄𝑄𝑄𝑄 (1 𝑄𝑄𝑄𝑄 ⋅0)(𝑄𝑄𝑄𝑄 −𝑐𝑐𝑐𝑐⋅𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)− 𝑐𝑐𝑐𝑐∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 ∂𝑥𝑥𝑥𝑥 16 21 ( ) = = + +− 𝑛𝑛𝑛𝑛 1⋅ 𝑄𝑄𝑄𝑄 ∂ ℎ𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡=∂ ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 ∂17𝑧𝑧𝑧𝑧 tan 18( ) that the concentration+ is+ not0 present𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠1 in the Sw0 is not present( in the) classical form of the Output values𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 to the sedimentation𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 model + 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= + 𝑗𝑗𝑗𝑗 + 𝐸𝐸𝐸𝐸 (1+20) 𝑆𝑆𝑆𝑆 −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�𝑣𝑣𝑣𝑣푔−𝑣𝑣𝑣𝑣− ∙𝑏𝑏𝑏𝑏θ � ⋅𝑣𝑣𝑣𝑣 ⋅𝑑𝑑𝑑𝑑 (1 ) 0 − 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 (116 ) 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = (2 ∂𝑥𝑥𝑥𝑥 + ) 22 = time derivative𝑐𝑐𝑐𝑐 anymore, similar to He et al. 0 shallow𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= 𝑄𝑄𝑄𝑄 water𝑗𝑗𝑗𝑗+−𝑄𝑄𝑄𝑄 equations𝐸𝐸𝐸𝐸𝑛𝑛𝑛𝑛+𝑄𝑄𝑄𝑄⋅ 𝑄𝑄𝑄𝑄⋅ (10 where−𝑐𝑐𝑐𝑐 ) the (width)∂𝑡𝑡𝑡𝑡 is ( ∂𝑥𝑥𝑥𝑥)in case ∂16of𝑧𝑧𝑧𝑧 a supercritical17 flow (u and h ) or 19 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 ( ∂) ℎ𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄=( ∂𝑄𝑄𝑄𝑄0)ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ − 𝑐𝑐𝑐𝑐 +𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = (1 ) +2 (10 ) 0 + + (1𝑄𝑄𝑄𝑄 ) 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ ( −𝑐𝑐𝑐𝑐 ) 𝑐𝑐𝑐𝑐 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+ 𝐸𝐸𝐸𝐸 𝑗𝑗𝑗𝑗(1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝐸𝐸𝐸𝐸 ) 0 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�2 푔 18− θ � (2014) 0and Cao𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= et al.+ (2004).+ 1 + considered= 𝑄𝑄𝑄𝑄( 0 𝑣𝑣𝑣𝑣 constant.𝑄𝑄𝑄𝑄−𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) 𝑄𝑄𝑄𝑄 +2⋅ ∙𝑏𝑏𝑏𝑏However,17−𝑐𝑐𝑐𝑐 ⋅𝑣𝑣𝑣𝑣 it arises ⋅𝑑𝑑𝑑𝑑 in the =𝑓𝑓𝑓𝑓 subcritical(∗2 ∂𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠 +flow𝑡𝑡𝑡𝑡 ) (Q ). 22 0 − 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 (1 ) ( )∂𝑡𝑡𝑡𝑡 ( 0∂𝑥𝑥𝑥𝑥=) −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛 +⋅ 𝑄𝑄𝑄𝑄𝑗𝑗𝑗𝑗 + 𝐸𝐸𝐸𝐸 (1 016) 𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑 1719𝑏𝑏𝑏𝑏 0 ) 𝑐𝑐𝑐𝑐 (0 ) 𝑄𝑄𝑄𝑄( 𝑄𝑄𝑄𝑄) (𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄) ⋅ − 𝑐𝑐𝑐𝑐 (∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 ) ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 23 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠=0 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0∂ ℎ𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 +∂ ℎ𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏derivation= 𝑗𝑗𝑗𝑗 of𝐸𝐸𝐸𝐸 the1 shallow0 +2 water equations1 in = 2−𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛18𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ −𝑐𝑐𝑐𝑐 + + (1 ) 𝐸𝐸𝐸𝐸 ( 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 ∗ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 = +( )+ 𝑄𝑄𝑄𝑄( (1𝑄𝑄𝑄𝑄) ) 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅0 −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐 =+𝑄𝑄𝑄𝑄 0 +17𝑄𝑄𝑄𝑄𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 =+−𝑣𝑣𝑣𝑣𝑄𝑄𝑄𝑄 ⋅𝑗𝑗𝑗𝑗(1∙𝑏𝑏𝑏𝑏−𝑐𝑐𝑐𝑐𝐸𝐸𝐸𝐸 ) ⋅𝑣𝑣𝑣𝑣+20 ⋅𝑑𝑑𝑑𝑑 ( ∂𝑡𝑡𝑡𝑡 ) ( ∂𝑥𝑥𝑥𝑥 ) 1 (17 ) 20 (1 ) − 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄( ∂)ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐( ∂) ℎ)𝑏𝑏𝑏𝑏𝑄𝑄𝑄𝑄a𝑢𝑢𝑢𝑢( channel𝑢𝑢𝑢𝑢 𝑄𝑄𝑄𝑄) of varying𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 width,⋅ 18− see𝑐𝑐𝑐𝑐 for example 𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) 23 + 0 = ( ) +2 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 16𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0+ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ρ19 =18 − ρ+ + + + = 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 +0∂ ℎ𝑏𝑏𝑏𝑏 + ∂= ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 −𝑣𝑣𝑣𝑣=(1( )−𝑛𝑛𝑛𝑛+2)𝑏𝑏𝑏𝑏 +2⋅𝑣𝑣𝑣𝑣(1 )−𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 = 2 2 2 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 ( ∂𝑡𝑡𝑡𝑡 ) 0 ( ∂𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) 1𝑗𝑗𝑗𝑗 ( 𝐸𝐸𝐸𝐸 ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 0 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� ( ) 𝑄𝑄𝑄𝑄( 𝑄𝑄𝑄𝑄) ∂ ℎ𝑄𝑄𝑄𝑄+𝑏𝑏𝑏𝑏 𝑄𝑄𝑄𝑄∂+ ℎ⋅𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 (−1=𝑐𝑐𝑐𝑐 𝑄𝑄𝑄𝑄 ) +𝑄𝑄𝑄𝑄 +𝑄𝑄𝑄𝑄 ⋅ (1−𝑐𝑐𝑐𝑐 )𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄Robert𝑄𝑄𝑄𝑄 and𝑄𝑄𝑄𝑄𝑣𝑣𝑣𝑣 Wilson−𝑣𝑣𝑣𝑣⋅ − (2011)𝑐𝑐𝑐𝑐 ∙𝑏𝑏𝑏𝑏 or Siviglia⋅𝑣𝑣𝑣𝑣 ∂ ℎ⋅𝑑𝑑𝑑𝑑 et𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 al. ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ 𝑏𝑏𝑏𝑏ℎ𝑠𝑠𝑠𝑠(𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ρ20 ) 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝐸𝐸𝐸𝐸 (𝑠𝑠𝑠𝑠1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) ( (𝑤𝑤𝑤𝑤)𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 )+ ∂𝑡𝑡𝑡𝑡( 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 () ∂𝑥𝑥𝑥𝑥)+18 = + 17+ + + 1 18ρ 19−𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ρ 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 24 + 0 = ( − 𝑛𝑛𝑛𝑛 ) ⋅ 𝑄𝑄𝑄𝑄+2 𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 ∙𝑏𝑏𝑏𝑏∂ ℎ𝑏𝑏𝑏𝑏⋅𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 +∂⋅𝑑𝑑𝑑𝑑ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 ∂=2ℎ(𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐸𝐸𝐸𝐸) 2𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+2𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 16𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑐𝑐𝑐𝑐 ( )∂𝑡𝑡𝑡𝑡 ( ∂𝑥𝑥𝑥𝑥) = + (2008).−𝑣𝑣𝑣𝑣=2 𝑣𝑣𝑣𝑣−𝑛𝑛𝑛𝑛(−𝑣𝑣𝑣𝑣1 𝑏𝑏𝑏𝑏 )∙𝑏𝑏𝑏𝑏⋅𝑣𝑣𝑣𝑣+2 ⋅𝑣𝑣𝑣𝑣−𝑛𝑛𝑛𝑛 (1⋅𝑑𝑑𝑑𝑑∂𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑 )𝑆𝑆𝑆𝑆 ∂𝑥𝑥𝑥𝑥𝑣𝑣𝑣𝑣 2The⋅𝑢𝑢𝑢𝑢 system𝑢𝑢𝑢𝑢� ∂𝑥𝑥𝑥𝑥 of equations is completed with the 21 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠0 + 𝑗𝑗𝑗𝑗 + (𝐸𝐸𝐸𝐸 ∂(1)(ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢0()) ∂ (ℎ)𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ) ( ∂)𝑏𝑏𝑏𝑏ℎ 19 1 = ( 𝑠𝑠𝑠𝑠 ρ )𝑤𝑤𝑤𝑤 tan( ) ∂ ℎ𝑏𝑏𝑏𝑏 ∂ ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏= 𝑄𝑄𝑄𝑄 ++𝑄𝑄𝑄𝑄( +)𝑄𝑄𝑄𝑄 =𝑄𝑄𝑄𝑄⋅((1−𝑐𝑐𝑐𝑐𝑄𝑄𝑄𝑄) ()1 𝑄𝑄𝑄𝑄 ) 0+2𝑄𝑄𝑄𝑄 ∂⋅∂𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝑏𝑏𝑏𝑏− 𝑐𝑐𝑐𝑐(1∂∂𝑡𝑡𝑡𝑡∂𝑥𝑥𝑥𝑥ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏) 1∂𝑥𝑥𝑥𝑥 17 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 description2 ρ20 of − the19ρ evolution∂𝑐𝑐𝑐𝑐 of trench width 24 (1 ) 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠−𝑡𝑡𝑡𝑡 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 ∂ +ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 + ∂ ℎ+𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 =𝐸𝐸𝐸𝐸 𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡=(1 𝑠𝑠𝑠𝑠𝑆𝑆𝑆𝑆+) +2𝑤𝑤𝑤𝑤+𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑆𝑆𝑆𝑆 18𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡+ 𝑆𝑆𝑆𝑆 (1 + 𝑆𝑆𝑆𝑆0) 𝑠𝑠𝑠𝑠𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡=𝑡𝑡𝑡𝑡 b 𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 +∙𝑏𝑏𝑏𝑏 0 ⋅𝑣𝑣𝑣𝑣= ( ⋅𝑑𝑑𝑑𝑑 ) +2∂𝑡𝑡𝑡𝑡 162 ∂𝑥𝑥𝑥𝑥 𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣−𝑣𝑣𝑣𝑣2 ∂𝑥𝑥𝑥𝑥∙𝑏𝑏𝑏𝑏−𝑛𝑛𝑛𝑛 ⋅𝑣𝑣𝑣𝑣𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣⋅𝑑𝑑𝑑𝑑 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� 21 ( )∂𝑡𝑡𝑡𝑡= ( ∂𝑥𝑥𝑥𝑥) ∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 ( 𝐸𝐸𝐸𝐸 ) ( 0 ) 2 2𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 ρ∂𝑤𝑤𝑤𝑤𝑧𝑧𝑧𝑧 ∂𝑥𝑥𝑥𝑥 0 (18)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠( 𝑗𝑗𝑗𝑗 ) 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑄𝑄𝑄𝑄0 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠 𝑄𝑄𝑄𝑄 ⋅∂𝑡𝑡𝑡𝑡(𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚−𝑐𝑐𝑐𝑐∂𝑡𝑡𝑡𝑡 ) ∂𝑥𝑥𝑥𝑥(0 ∂𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ) =191 ( ) tan( ) in time,1 2 see19ρ − equationρ 20 ∂𝑐𝑐𝑐𝑐 (26), and evolution of bed 25 + +𝑄𝑄𝑄𝑄=+0𝑄𝑄𝑄𝑄 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(11 𝑄𝑄𝑄𝑄 )𝑄𝑄𝑄𝑄+2⋅ −𝑣𝑣𝑣𝑣=− 𝑐𝑐𝑐𝑐 (1−𝑛𝑛𝑛𝑛+ 𝑏𝑏𝑏𝑏)+∂ ℎ 𝑏𝑏𝑏𝑏⋅𝑣𝑣𝑣𝑣𝑢𝑢𝑢𝑢(∂+1ℎ𝑏𝑏𝑏𝑏∂𝑐𝑐𝑐𝑐 ℎ)−𝑛𝑛𝑛𝑛𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢∂=ℎ𝑑𝑑𝑑𝑑𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 ∂(1𝑏𝑏𝑏𝑏ℎ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) +2𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑓𝑓𝑓𝑓 (1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤17𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ) 𝑡𝑡𝑡𝑡 0𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 =−𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�푔 − θ � ( ) ( ( ) ) ∂( ℎ𝑏𝑏𝑏𝑏 ) ∂ ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏( ) 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 +𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 + = + + + +𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�= (2 ∂𝑥𝑥𝑥𝑥 + ) 0 − 𝑛𝑛𝑛𝑛∂𝑡𝑡𝑡𝑡 ⋅ 𝑄𝑄𝑄𝑄 ∂𝑥𝑥𝑥𝑥 1 2 𝑔𝑔𝑔𝑔 −𝑣𝑣𝑣𝑣18 2𝑆𝑆𝑆𝑆−𝑛𝑛𝑛𝑛∂𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏𝑆𝑆𝑆𝑆 20 𝑆𝑆𝑆𝑆⋅𝑣𝑣𝑣𝑣 𝑆𝑆𝑆𝑆 −𝑛𝑛𝑛𝑛𝑆𝑆𝑆𝑆 𝑑𝑑𝑑𝑑 evolution2 z in time, see equation (27). 22 + = ( ) +2 𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 ∂𝑡𝑡𝑡𝑡∙𝑏𝑏𝑏𝑏( ∂𝑡𝑡𝑡𝑡 ⋅𝑣𝑣𝑣𝑣) ∂𝑥𝑥𝑥𝑥 ( ⋅𝑑𝑑𝑑𝑑∂𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏)𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 12∂𝑥𝑥𝑥𝑥 ( ) 1 2ρ 20 ∂𝑥𝑥𝑥𝑥 ∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 ∂ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸(+ )∂𝑡𝑡𝑡𝑡0 ( + ∂𝑥𝑥𝑥𝑥) = ∂ ℎ+𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 +∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢+𝑢𝑢𝑢𝑢 + 2 21∂ 𝑏𝑏𝑏𝑏 25 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡⋅ −𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠2 2 0 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2 0𝑗𝑗𝑗𝑗 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝐸𝐸𝐸𝐸 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 0+ ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑆𝑆𝑆𝑆= 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+ −𝑔𝑔𝑔𝑔∂𝑔𝑔𝑔𝑔𝑔�𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏ℎ =푔tan𝑏𝑏𝑏𝑏−𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+) 𝑓𝑓𝑓𝑓θ+19� 0𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 + 𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤= = + +−𝑣𝑣𝑣𝑣 (1−𝑛𝑛𝑛𝑛(+ ))𝑏𝑏𝑏𝑏 𝑄𝑄𝑄𝑄(⋅𝑣𝑣𝑣𝑣=𝑄𝑄𝑄𝑄) −𝑛𝑛𝑛𝑛(1𝑄𝑄𝑄𝑄 𝑑𝑑𝑑𝑑 𝑄𝑄𝑄𝑄) ⋅+2− 𝑐𝑐𝑐𝑐 (171 −𝑣𝑣𝑣𝑣(21)2 ) =𝑔𝑔𝑔𝑔−𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏(22 ∂𝑥𝑥𝑥𝑥⋅𝑣𝑣𝑣𝑣𝑆𝑆𝑆𝑆 + ) 𝑆𝑆𝑆𝑆 −𝑛𝑛𝑛𝑛𝑆𝑆𝑆𝑆 𝑑𝑑𝑑𝑑 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑓𝑓𝑓𝑓 𝑔𝑔𝑔𝑔𝑔∗ 22𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ( ∂𝑡𝑡𝑡𝑡 ) ∂ (ℎ𝑏𝑏𝑏𝑏∂𝑥𝑥𝑥𝑥 )∂∂ℎ1𝑢𝑢𝑢𝑢ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ( ∂𝐸𝐸𝐸𝐸)ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∂ 𝑏𝑏𝑏𝑏ℎ 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 202 18 𝑆𝑆𝑆𝑆 2 −𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 ) 𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 + ∙𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣= ( ⋅𝑑𝑑𝑑𝑑𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡( ∂𝑡𝑡𝑡𝑡))𝑓𝑓𝑓𝑓∂ ℎ∂𝑡𝑡𝑡𝑡+2𝑏𝑏𝑏𝑏(𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∂𝑥𝑥𝑥𝑥 )∂ 𝑡𝑡𝑡𝑡ℎ∂𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏1𝑢𝑢𝑢𝑢 𝑤𝑤𝑤𝑤 ( ) ∂ 𝑏𝑏𝑏𝑏∂𝑥𝑥𝑥𝑥ℎ∂𝑧𝑧𝑧𝑧 202 ∂𝑥𝑥𝑥𝑥 21 26 23 ( + ) ( + ) ∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 = ∂ ℎ𝑏𝑏𝑏𝑏+𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑔𝑔𝑔𝑔 + + 𝑆𝑆𝑆𝑆 + 𝑆𝑆𝑆𝑆 + 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆+𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 = = 2 + 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+tan(𝑓𝑓𝑓𝑓 +) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+ 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤 = =2 ∂𝑏𝑏𝑏𝑏 0 ∂𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2 ∂𝑥𝑥𝑥𝑥𝑗𝑗𝑗𝑗2∂𝑡𝑡𝑡𝑡 𝐸𝐸𝐸𝐸 2 ∂𝑥𝑥𝑥𝑥 0 ∂𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 𝑆𝑆𝑆𝑆 0−𝑔𝑔𝑔𝑔𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑡𝑡𝑡𝑡𝑔𝑔𝑔𝑔𝑔�2𝑔𝑔𝑔𝑔19 푔∗ − 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 θ21�𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔 ( ) 𝑄𝑄𝑄𝑄( 𝑄𝑄𝑄𝑄+) 𝑄𝑄𝑄𝑄 =𝑄𝑄𝑄𝑄 ⋅ (∂−1 ℎ𝑐𝑐𝑐𝑐𝑏𝑏𝑏𝑏 ) ∂+2ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏−𝑣𝑣𝑣𝑣 (1 −𝑛𝑛𝑛𝑛) 𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛2 𝑑𝑑𝑑𝑑𝑆𝑆𝑆𝑆 (−𝑢𝑢𝑢𝑢 ⋅ 𝑑𝑑𝑑𝑑 ) 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂( 𝑏𝑏𝑏𝑏∂𝑡𝑡𝑡𝑡ℎ ) ( ∂𝑥𝑥𝑥𝑥= ) 1 ( 𝐸𝐸𝐸𝐸 ) 𝑠𝑠𝑠𝑠tan𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ( ) ∂𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚18∂𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡= 2∂𝑥𝑥𝑥𝑥∂𝑥𝑥𝑥𝑥 +( ) ∂𝑏𝑏𝑏𝑏 ∂𝑥𝑥𝑥𝑥22 ρ2123 − ρ 26 + = ( ) +2 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡∂ ℎ𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤𝑢𝑢𝑢𝑢 ∙𝑏𝑏𝑏𝑏∂ ℎ𝑏𝑏𝑏𝑏⋅𝑣𝑣𝑣𝑣𝑢𝑢𝑢𝑢 ⋅𝑑𝑑𝑑𝑑𝑏𝑏𝑏𝑏∂𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏=ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∂𝑧𝑧𝑧𝑧𝑓𝑓𝑓𝑓 tan𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡( )20 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑔𝑔𝑔𝑔 ( +)∂𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 ( 𝑆𝑆𝑆𝑆∂𝑥𝑥𝑥𝑥+) 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 = 𝑆𝑆𝑆𝑆 + + +𝑔𝑔𝑔𝑔𝑆𝑆𝑆𝑆 + −𝑔𝑔𝑔𝑔 2 𝑔𝑔𝑔𝑔𝑔�𝑆𝑆𝑆𝑆 푔 𝑆𝑆𝑆𝑆 − 𝑆𝑆𝑆𝑆 θ �𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 (26)𝑣𝑣𝑣𝑣 ⋅𝑣𝑣𝑣𝑣⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 ∂𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 2 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∂𝑧𝑧𝑧𝑧2 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 21 ∗ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 19 ∂𝑡𝑡𝑡𝑡 ρ 27 ∂ ℎ𝑏𝑏𝑏𝑏 ∂ ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 = −𝑣𝑣𝑣𝑣 +−𝑛𝑛𝑛𝑛𝑆𝑆𝑆𝑆 tan𝑏𝑏𝑏𝑏 −𝑔𝑔𝑔𝑔( )=⋅𝑣𝑣𝑣𝑣𝑔𝑔𝑔𝑔𝑔�2 푔 (1−𝑛𝑛𝑛𝑛−∂𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑) θ+2� ∂𝑥𝑥𝑥𝑥 (1 𝑆𝑆𝑆𝑆 )∂𝑥𝑥𝑥𝑥−𝑢𝑢𝑢𝑢= ⋅ 𝑑𝑑𝑑𝑑(2 ∂∂𝑥𝑥𝑥𝑥ρ𝑧𝑧𝑧𝑧𝑏𝑏𝑏𝑏 + ) − ρ ∂𝑏𝑏𝑏𝑏 1= 21 (22 ) 24 ( ∂𝑡𝑡𝑡𝑡 ) ( ∂𝑥𝑥𝑥𝑥𝐸𝐸𝐸𝐸 ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡1∂(19)ℎ𝑏𝑏𝑏𝑏(𝑢𝑢𝑢𝑢 )∂ ℎ𝑤𝑤𝑤𝑤𝑏𝑏𝑏𝑏𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ∂ 𝑏𝑏𝑏𝑏ℎ = (22)𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 20𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ( tan( ) ) = 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚23 𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 ∙𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 ⋅𝑑𝑑𝑑𝑑= 𝑔𝑔𝑔𝑔(2 ∂𝑥𝑥𝑥𝑥 + 𝑆𝑆𝑆𝑆)𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑓𝑓𝑓𝑓 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆=𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑤𝑤𝑤𝑤−𝑔𝑔𝑔𝑔𝑣𝑣𝑣𝑣 𝑔𝑔𝑔𝑔𝑔�⋅𝑢𝑢𝑢𝑢2 𝑢𝑢𝑢𝑢�푔 − 22 θ � 2 ⋅𝑣𝑣𝑣𝑣 ( )∂𝑡𝑡𝑡𝑡 ( ∂𝑥𝑥𝑥𝑥) + + ∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 ∂𝑧𝑧𝑧𝑧=∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢+ + 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + +𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚19 0 𝑓𝑓𝑓𝑓𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡=1 ∗(2 ∂𝑥𝑥𝑥𝑥( 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 + ρ) ) ∂𝑡𝑡𝑡𝑡 ∂𝑧𝑧𝑧𝑧 2 22𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 27 + = 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(21 2 )∂𝑡𝑡𝑡𝑡+22 ∂𝑥𝑥𝑥𝑥(1 )2−𝑣𝑣𝑣𝑣 ∂𝑥𝑥𝑥𝑥 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑆𝑆𝑆𝑆 −∂𝑢𝑢𝑢𝑢𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡⋅𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 21) = 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ρ24 − ρ ∂𝑐𝑐𝑐𝑐 𝑆𝑆𝑆𝑆 −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�푔 − 𝑓𝑓𝑓𝑓 θ=� ∗ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 tan( ) 𝑆𝑆𝑆𝑆 −𝑔𝑔𝑔𝑔=𝑔𝑔𝑔𝑔𝑔�푔 −ρ θ −� ρ 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 23 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 (∂∂𝑡𝑡𝑡𝑡(𝑏𝑏𝑏𝑏ℎ) ( ∂𝑥𝑥𝑥𝑥𝑆𝑆𝑆𝑆 ) ) −𝑢𝑢𝑢𝑢1 (⋅ 𝑑𝑑𝑑𝑑 ) 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 2 20 𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔(𝑔𝑔𝑔𝑔𝑔�) = 2 ∂𝑥𝑥𝑥𝑥 +𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡( 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 ) 𝑆𝑆𝑆𝑆 = 𝑣𝑣𝑣𝑣 22𝑓𝑓𝑓𝑓(⋅𝑢𝑢𝑢𝑢 2 𝑢𝑢𝑢𝑢�∂𝑥𝑥𝑥𝑥∗ + ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 = ∂𝑡𝑡𝑡𝑡 + 221 ρ ∂𝑥𝑥𝑥𝑥 28 ∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑔𝑔𝑔𝑔 + 𝑆𝑆𝑆𝑆 +𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆= 𝑆𝑆𝑆𝑆 + + + +𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 ⋅2𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏23 ∂𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡1 25 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 2 02 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 2 ∂𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 1 ( ( ρ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡−𝑠𝑠𝑠𝑠)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ρ ∂𝑐𝑐𝑐𝑐) 23 ∂𝑡𝑡𝑡𝑡 −𝑣𝑣𝑣𝑣∂𝑥𝑥𝑥𝑥 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏2 ∂𝑥𝑥𝑥𝑥 ⋅𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑆𝑆𝑆𝑆 =−2 21𝑔𝑔𝑔𝑔 𝑔𝑔𝑔𝑔𝑔� ρ − ρ 𝑣𝑣𝑣𝑣= 24 ( ∂𝑡𝑡𝑡𝑡 ) ( ∂𝑥𝑥𝑥𝑥 ) 1 ( 𝑓𝑓𝑓𝑓) = ∂ ℎ∗𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠∂𝑡𝑡𝑡𝑡 ℎ𝑏𝑏𝑏𝑏𝑆𝑆𝑆𝑆𝑢𝑢𝑢𝑢tan( −𝑔𝑔𝑔𝑔) ∂𝑔𝑔𝑔𝑔𝑔�𝑏𝑏𝑏𝑏ℎ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠푔𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − θ � 𝑓𝑓𝑓𝑓= 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∗ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 = (27)∂𝑡𝑡𝑡𝑡 +𝑠𝑠𝑠𝑠 (1 2 ) 𝑤𝑤𝑤𝑤 28 𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 ) = (2 ∂𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏+𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠20𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 −𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤 ⋅ 𝑑𝑑𝑑𝑑⋅𝑢𝑢𝑢𝑢1 𝑢𝑢𝑢𝑢�𝑏𝑏𝑏𝑏 ρ ∂𝑥𝑥𝑥𝑥 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ 2ρ + + = + +𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 +𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 +𝑔𝑔𝑔𝑔 ρ 𝑆𝑆𝑆𝑆− ρ 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 2 23𝑆𝑆𝑆𝑆 ( 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ρ 22 23 25∂ 𝑏𝑏𝑏𝑏 2 =2 𝑆𝑆𝑆𝑆 ∂𝑧𝑧𝑧𝑧 𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� (23) 1 =2 ρ −(ρρ ∂𝑐𝑐𝑐𝑐 −) ρ 𝑤𝑤𝑤𝑤 24 2 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 ∂𝑥𝑥𝑥𝑥2 𝑡𝑡𝑡𝑡= 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2 21 = 𝑠𝑠𝑠𝑠 𝑆𝑆𝑆𝑆 + (𝑔𝑔𝑔𝑔1𝑔𝑤𝑤𝑤𝑤 ) 29 𝑆𝑆𝑆𝑆 −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�푔 − 1 𝑓𝑓𝑓𝑓=θ � ∗( ρ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ) tan( ) 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆− 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� 24 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ ∂𝑥𝑥𝑥𝑥 26 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ 𝑏𝑏𝑏𝑏ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 −𝑤𝑤𝑤𝑤𝑢𝑢𝑢𝑢 ⋅ 𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 12 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡( 2 ∂𝑏𝑏𝑏𝑏ρ ) Lastly, the definition of mixture density ρ is 𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆= (𝑆𝑆𝑆𝑆2ρ ∂𝑥𝑥𝑥𝑥=𝑆𝑆𝑆𝑆 +−)ρ𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 ( ) 221𝑤𝑤𝑤𝑤 ρ 2ρ−𝑠𝑠𝑠𝑠ρ ∂𝑥𝑥𝑥𝑥𝑤𝑤𝑤𝑤 23 =225 24 2 ∂𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔ρ − ρ ∂𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 𝑆𝑆𝑆𝑆 ∂𝑥𝑥𝑥𝑥 𝑣𝑣𝑣𝑣 (20)⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑆𝑆𝑆𝑆 21 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 ⋅𝑢𝑢𝑢𝑢=𝑢𝑢𝑢𝑢� ρ = given𝑛𝑛𝑛𝑛 ⋅ ρ by+ (equation1 − 𝑛𝑛𝑛𝑛) (28)⋅ ρ and the density of 29 = 1 tan2 (( ) ρ𝑆𝑆𝑆𝑆 ) −𝑔𝑔𝑔𝑔2𝑔𝑔𝑔𝑔𝑔�ρ −푔ρ −∂𝑐𝑐𝑐𝑐 θ � 𝑆𝑆𝑆𝑆 −22𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� ∂𝑥𝑥𝑥𝑥 / ∂𝑏𝑏𝑏𝑏 / / 26 / / / 30 𝑓𝑓𝑓𝑓 ∗ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 1 24 ( =212 ρ2 )𝑠𝑠𝑠𝑠 ρ 𝑤𝑤𝑤𝑤 ∂𝑥𝑥𝑥𝑥 24 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 𝑆𝑆𝑆𝑆⋅(𝑑𝑑𝑑𝑑 − 𝑏𝑏𝑏𝑏𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�=) (2 ρ∂𝑥𝑥𝑥𝑥 + )− ρ = 𝑡𝑡𝑡𝑡 ∂𝑏𝑏𝑏𝑏ρ − ρ ∂𝑐𝑐𝑐𝑐22 + the seabed after⋅𝑣𝑣𝑣𝑣25 it has re-settled again by 2∂𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ρ ∂𝑥𝑥𝑥𝑥 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆𝑤𝑤𝑤𝑤− 23𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑔𝑔𝑔𝑔𝑔�=𝑔 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 ρ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 ⋅ ρ𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡∂𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡− 𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 ⋅ ρ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 27 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 2 𝑠𝑠𝑠𝑠 𝑆𝑆𝑆𝑆𝑤𝑤𝑤𝑤 𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢2𝑢𝑢𝑢𝑢� 2 ∂𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠2 𝑤𝑤𝑤𝑤 25 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠+1/ 2 =𝑠𝑠𝑠𝑠+1/ 2 �𝑠𝑠𝑠𝑠−1/ 2 𝑠𝑠𝑠𝑠−1/ 2 𝑠𝑠𝑠𝑠−1/ 2 30 𝑆𝑆𝑆𝑆 −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�푔 Where− h θisρ �flow− ρ height,∂𝑐𝑐𝑐𝑐=𝑓𝑓𝑓𝑓 b is trench∗ ( width,𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ρ ) (24) 2 1 ∂𝑥𝑥𝑥𝑥𝑤𝑤𝑤𝑤2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ρ ∂𝑥𝑥𝑥𝑥ℎ 𝑏𝑏𝑏𝑏 −ℎ 𝑏𝑏𝑏𝑏 ℎ equation𝑢𝑢𝑢𝑢 (29).𝑏𝑏𝑏𝑏26 25− ℎ 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 1 𝑡𝑡𝑡𝑡 =2ρ −⋅𝑣𝑣𝑣𝑣ρ ∂∂𝑐𝑐𝑐𝑐𝑏𝑏𝑏𝑏 24 + =𝑆𝑆𝑆𝑆 (−2 ∂𝑥𝑥𝑥𝑥𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�+ ) ρ = −𝑆𝑆𝑆𝑆 ρ2 −𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 𝑆𝑆𝑆𝑆) 22− 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�𝑤𝑤𝑤𝑤∂𝑡𝑡𝑡𝑡= 𝑡𝑡𝑡𝑡+231 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 27 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 u 𝑠𝑠𝑠𝑠is𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡1 mean flowρ velocity∂𝑥𝑥𝑥𝑥 and c 2is∂ sediment𝑏𝑏𝑏𝑏 𝑆𝑆𝑆𝑆 2 𝑔𝑔𝑔𝑔𝑔 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 Δ𝑡𝑡𝑡𝑡 =𝑠𝑠𝑠𝑠 [(𝑠𝑠𝑠𝑠 )�𝑠𝑠𝑠𝑠+1(2 𝑠𝑠𝑠𝑠+1) 2∂] 𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠+1+22𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥�𝑠𝑠𝑠𝑠−1( 2 )𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑆𝑆𝑆𝑆2 𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� 𝑤𝑤𝑤𝑤 =2 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 251 =ρ 2 ∂𝑥𝑥𝑥𝑥∂𝑥𝑥𝑥𝑥 ℎ 𝑏𝑏𝑏𝑏 −ℎ 𝑏𝑏𝑏𝑏 ℎ 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏25 − ℎ 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 = 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔 2 ∂𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∂𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 26 𝑓𝑓𝑓𝑓 ∗ concentration.𝑠𝑠𝑠𝑠1𝑡𝑡𝑡𝑡 ( 𝑡𝑡𝑡𝑡Furthermore,ρ ) v is ρthe𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡−𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠ρ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∂𝑐𝑐𝑐𝑐 = 24𝑤𝑤𝑤𝑤 ⋅𝑣𝑣𝑣𝑣=2 𝑡𝑡𝑡𝑡 = 𝑡𝑡𝑡𝑡∂𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+ (1 ) 𝑡𝑡𝑡𝑡 28 𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑 2𝑏𝑏𝑏𝑏 ) 𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�∂𝑥𝑥𝑥𝑥E ρ − ρ 2 𝑆𝑆𝑆𝑆 ∂𝑧𝑧𝑧𝑧 𝑔𝑔𝑔𝑔𝑔 26 = [( )𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 ( 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) ]𝑠𝑠𝑠𝑠 +2𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚( )𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 = 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 23 ∂𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 Δ𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 − 𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏 27 Δ𝑥𝑥𝑥𝑥𝑣𝑣𝑣𝑣26 𝑑𝑑𝑑𝑑 = 𝑤𝑤𝑤𝑤entrainment2 ∂𝑏𝑏𝑏𝑏 velocity,𝑆𝑆𝑆𝑆 v𝑣𝑣𝑣𝑣 the⋅𝑢𝑢𝑢𝑢1 sedimentation𝑢𝑢𝑢𝑢� ρ ∂𝑥𝑥𝑥𝑥 ∂=𝑏𝑏𝑏𝑏 =2∂𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 25 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔2 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 sed ( ρ ) 𝑤𝑤𝑤𝑤 = ∂𝑡𝑡𝑡𝑡 + (1 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 28 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ρ − ρ∂𝑏𝑏𝑏𝑏 ∂𝑐𝑐𝑐𝑐1 = 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔 ⋅𝑣𝑣𝑣𝑣 24 𝐸𝐸𝐸𝐸 ρ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐 ⋅ ρ 𝑤𝑤𝑤𝑤−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑐𝑐𝑐𝑐 ⋅𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡ρ 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 −ρvelocity𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�− (i.e.ρ∂𝑥𝑥𝑥𝑥 vertical= velocity2𝑤𝑤𝑤𝑤 of𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 the bed), v (25) 26∂𝑧𝑧𝑧𝑧 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 − 𝑣𝑣𝑣𝑣/(28)𝑠𝑠𝑠𝑠 /𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠26 / 𝑣𝑣𝑣𝑣 27/ 𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠 / / / / 31 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =2 ⋅𝑣𝑣𝑣𝑣 2 wall =2∂𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡= 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 + 𝑆𝑆𝑆𝑆 𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� 1 ρ ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥2𝑤𝑤𝑤𝑤 ∂𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤 25 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⋅𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡+127𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡 +1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡= 𝑡𝑡𝑡𝑡 +𝑡𝑡𝑡𝑡(1 )𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 29 ρ 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔2 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 1 ∂(𝑏𝑏𝑏𝑏 = ) 𝑡𝑡𝑡𝑡 = ρ − ρ ∂𝑐𝑐𝑐𝑐 24 = ∂𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡+ (1 ) 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ℎ� / 𝑏𝑏𝑏𝑏 / 𝑢𝑢𝑢𝑢 / 28 𝑐𝑐𝑐𝑐̂ 27/ − ℎ� / 𝑏𝑏𝑏𝑏 / 𝑢𝑢𝑢𝑢 / 𝑐𝑐𝑐𝑐̂ / 31 2 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� ∂𝑥𝑥𝑥𝑥 ∂𝑏𝑏𝑏𝑏 ∂𝑤𝑤𝑤𝑤𝑧𝑧𝑧𝑧𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚= 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝑏𝑏𝑏𝑏 26𝑐𝑐𝑐𝑐 −ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 = ⋅𝑣𝑣𝑣𝑣 ( ) + 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 2 ∂𝑏𝑏𝑏𝑏 ∂𝑧𝑧𝑧𝑧 =21 ρ ∂𝑥𝑥𝑥𝑥 =⋅𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣+ 𝑡𝑡𝑡𝑡1+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+251 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ρ 𝑛𝑛𝑛𝑛 ⋅ ρ 29 − 𝑛𝑛𝑛𝑛 ⋅ ρ 2 ∂𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤 =𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡 27=𝑠𝑠𝑠𝑠 ∂∂𝑡𝑡𝑡𝑡𝑧𝑧𝑧𝑧 + (1 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤) 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 Δ𝑡𝑡𝑡𝑡 =𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 (𝑠𝑠𝑠𝑠 )�𝑠𝑠𝑠𝑠+1(1 2 𝑠𝑠𝑠𝑠+1) 2 /𝑠𝑠𝑠𝑠+2+127 2 𝑠𝑠𝑠𝑠/+1(2821 /Δ𝑥𝑥𝑥𝑥�𝑠𝑠𝑠𝑠−1)( 2 𝑠𝑠𝑠𝑠/)−1 2 𝑠𝑠𝑠𝑠/−1 2 𝑠𝑠𝑠𝑠/−1 2 30 𝑡𝑡𝑡𝑡 ρ𝑆𝑆𝑆𝑆 −=ρ 𝑔𝑔𝑔𝑔∂𝑔𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ= − 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠ℎ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡⋅ ρ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 −ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 ℎ 𝑏𝑏𝑏𝑏+ 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ − ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ 𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� ∂𝑥𝑥𝑥𝑥= ∂𝑡𝑡𝑡𝑡 +∂𝑏𝑏𝑏𝑏(1 2 ) 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 26 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑣𝑣𝑣𝑣 28𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 =2 ⋅𝑣𝑣𝑣𝑣 2 ρ = 𝑛𝑛𝑛𝑛∂𝑡𝑡𝑡𝑡⋅+ρ((1 ))− 𝑛𝑛𝑛𝑛 ⋅ ρ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠+1𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚2 𝑠𝑠𝑠𝑠+1 2 0 𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 2𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 1 ∂A𝑧𝑧𝑧𝑧ρ ∂𝑥𝑥𝑥𝑥 𝑤𝑤𝑤𝑤 ∂𝑏𝑏𝑏𝑏 25 =∂𝑧𝑧𝑧𝑧 / 𝑠𝑠𝑠𝑠+ 1/ / 𝑤𝑤𝑤𝑤 B Δ𝑡𝑡𝑡𝑡/ =𝑠𝑠𝑠𝑠 /( 𝑠𝑠𝑠𝑠 )/ (1𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ) ℎ� +2𝑢𝑢𝑢𝑢 29 (1𝑏𝑏𝑏𝑏2830Δ𝑥𝑥𝑥𝑥 )(− ℎ� ) 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 = 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆∂𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔𝑔 ρ+ 𝑐𝑐𝑐𝑐 ⋅𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ 27 ℎ −𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 −ℎ 𝑏𝑏𝑏𝑏−𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 = 𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 = ∂𝑡𝑡𝑡𝑡∂𝑏𝑏𝑏𝑏+ (1 𝑤𝑤𝑤𝑤)ρ𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ ∂𝑥𝑥𝑥𝑥 28 ( 𝑠𝑠𝑠𝑠 ) 𝑤𝑤𝑤𝑤 26 2 ⋅𝑣𝑣𝑣𝑣 =2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠=𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 ∂𝑡𝑡𝑡𝑡ρ +�𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠=𝑐𝑐𝑐𝑐+11⋅ 𝑤𝑤𝑤𝑤ρ2 𝑠𝑠𝑠𝑠+1 +2−(𝑠𝑠𝑠𝑠 +11𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 2⋅ ρ𝑠𝑠𝑠𝑠)�𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡−1Δ𝑡𝑡𝑡𝑡 2𝑠𝑠𝑠𝑠= [( )𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ( 28𝑤𝑤𝑤𝑤) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚] 29+2 0Δ𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠(𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 ) 𝑤𝑤𝑤𝑤 ∂𝑏𝑏𝑏𝑏 ∂𝑧𝑧𝑧𝑧 ℎ 𝑏𝑏𝑏𝑏 ρ−ℎ 𝑏𝑏𝑏𝑏 𝑛𝑛𝑛𝑛ℎ ⋅ ρ 𝑢𝑢𝑢𝑢 −𝑏𝑏𝑏𝑏𝑛𝑛𝑛𝑛 ⋅ ρ− ℎ 𝑢𝑢𝑢𝑢 − 𝑣𝑣𝑣𝑣𝑏𝑏𝑏𝑏 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 30 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔 ∂𝑡𝑡𝑡𝑡 = + (1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) / 27 / / 29/ / / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 = 𝑤𝑤𝑤𝑤 ∂𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 + 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤( ) ρ 𝑐𝑐𝑐𝑐 ⋅ ρ∂𝑥𝑥𝑥𝑥 − 𝑐𝑐𝑐𝑐 ⋅ ρ = ∂𝑡𝑡𝑡𝑡 + (1 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+)1 𝑡𝑡𝑡𝑡+1 Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡26ρ=𝑡𝑡𝑡𝑡 [(𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡⋅𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡ρ)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =(𝑡𝑡𝑡𝑡 −𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐) 𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤]⋅ +ρ +21 𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠 ( 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 ) 𝑡𝑡𝑡𝑡 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 29 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 =2 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ρ �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1𝑛𝑛𝑛𝑛 2⋅ ρ𝑠𝑠𝑠𝑠+1 2 −�𝑠𝑠𝑠𝑠−1𝑛𝑛𝑛𝑛 2⋅ 28ρ𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑣𝑣𝑣𝑣 − 𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏 30 𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 ∂𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 ∂𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤 ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 −ℎ 𝑏𝑏𝑏𝑏 ℎ 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 / 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡 − ℎ/ 𝑢𝑢𝑢𝑢 27/ 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 / / = +ρ(𝑠𝑠𝑠𝑠1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) 𝑛𝑛𝑛𝑛 ⋅ ρ − 𝑛𝑛𝑛𝑛 ⋅ ρ =𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡29 +𝑠𝑠𝑠𝑠(1 𝑤𝑤𝑤𝑤 ) 30𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 / /𝑠𝑠𝑠𝑠 = / 𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤/+1 𝑡𝑡𝑡𝑡+/1 ρ𝑡𝑡𝑡𝑡/ 𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡DOL𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡⋅ ρ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 − 𝑛𝑛𝑛𝑛𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚⋅𝑡𝑡𝑡𝑡ρ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / 29 / / / / / / / 31 ∂𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ Δ𝑡𝑡𝑡𝑡 = [( ) 𝑣𝑣𝑣𝑣 ( −𝑠𝑠𝑠𝑠+1)𝑣𝑣𝑣𝑣 ]2 𝑠𝑠𝑠𝑠+1+2𝑏𝑏𝑏𝑏2 Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠+1 𝑣𝑣𝑣𝑣2( )𝑠𝑠𝑠𝑠−1𝑑𝑑𝑑𝑑 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 30 ⋅𝑣𝑣𝑣𝑣= ∂𝑡𝑡𝑡𝑡 + (1 +) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ℎ� 28/ 𝑢𝑢𝑢𝑢 / 𝑏𝑏𝑏𝑏 / − ℎ� / 𝑢𝑢𝑢𝑢 / 𝑏𝑏𝑏𝑏 / + 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∂𝑧𝑧𝑧𝑧 ℎ 𝑏𝑏𝑏𝑏 −ℎ 𝑏𝑏𝑏𝑏 + 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡ρ 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 ⋅ ρ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 − 𝑛𝑛𝑛𝑛�𝑠𝑠𝑠𝑠+1⋅ ρ2 =𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1+2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(1 �𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡+)21 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡−1+𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡1𝑠𝑠𝑠𝑠2𝑠𝑠𝑠𝑠27𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1 𝑡𝑡𝑡𝑡 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1312 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 =ℎ 𝑏𝑏𝑏𝑏/ −ℎ/ 𝑏𝑏𝑏𝑏 / ℎ /𝑢𝑢𝑢𝑢 /𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 /− ℎ 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠ρ 𝑏𝑏𝑏𝑏𝐸𝐸𝐸𝐸 [𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠( 𝑠𝑠𝑠𝑠𝑛𝑛𝑛𝑛) ⋅30𝑠𝑠𝑠𝑠ρ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 +1/( 𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)+1/−]𝑛𝑛𝑛𝑛2 𝑠𝑠𝑠𝑠𝑤𝑤𝑤𝑤+1/⋅𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ρ2 ℎ𝑠𝑠𝑠𝑠/𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠(𝑠𝑠𝑠𝑠−129𝑏𝑏𝑏𝑏 2) 𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐/−1 2 −ℎ𝑠𝑠𝑠𝑠/−1 2𝑏𝑏𝑏𝑏 /𝑐𝑐𝑐𝑐 ℎ/ 𝑏𝑏𝑏𝑏 30 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ − ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ + 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 = ∂𝑡𝑡𝑡𝑡 + (1 )ℎ 𝑏𝑏𝑏𝑏 Δ𝑡𝑡𝑡𝑡−ℎ𝑣𝑣𝑣𝑣= 𝑏𝑏𝑏𝑏−+/ 𝑣𝑣𝑣𝑣ℎ� / 𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 / 𝑏𝑏𝑏𝑏𝑣𝑣𝑣𝑣+2/Δ𝑥𝑥𝑥𝑥𝑑𝑑𝑑𝑑− ℎ� /28 𝑢𝑢𝑢𝑢 / 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ρ𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐 ⋅ 𝑡𝑡𝑡𝑡ρ 𝑡𝑡𝑡𝑡 − 𝑐𝑐𝑐𝑐 ⋅ ρ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 +𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ∂𝑧𝑧𝑧𝑧 Δ𝑡𝑡𝑡𝑡 = [( ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) ]𝑠𝑠𝑠𝑠 +2𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+Δ𝑥𝑥𝑥𝑥1𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡(+1𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ) 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2Δ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1=2 𝑠𝑠𝑠𝑠−1( 2 𝑠𝑠𝑠𝑠)−1(12 ) +2 (1 Δ𝑥𝑥𝑥𝑥 )( ) 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2ℎ 𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏2 𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐 −ℎ𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏= 𝑐𝑐𝑐𝑐[(𝑠𝑠𝑠𝑠+1 )2ℎ� 𝑠𝑠𝑠𝑠+1( 2𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1) ]2𝑢𝑢𝑢𝑢 +2𝑠𝑠𝑠𝑠−1𝑐𝑐𝑐𝑐̂ 2 𝑠𝑠𝑠𝑠−1(− 2ℎ�)𝑠𝑠𝑠𝑠−1 2𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ ℎ 𝑏𝑏𝑏𝑏 −ℎ 𝑏𝑏𝑏𝑏 ℎ𝑣𝑣𝑣𝑣 =𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 + (1ρ− ℎ ) 𝑢𝑢𝑢𝑢𝑛𝑛𝑛𝑛 ⋅ ρ𝑏𝑏𝑏𝑏ℎ 𝑏𝑏𝑏𝑏− 𝑛𝑛𝑛𝑛 −ℎ⋅ ρ Δ𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 ℎ� 𝐸𝐸𝐸𝐸/𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢 29/ 𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 / −𝑠𝑠𝑠𝑠 ℎ� / Δ𝑥𝑥𝑥𝑥𝑤𝑤𝑤𝑤𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 30/𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 / / / 31 ( ) 𝑡𝑡𝑡𝑡 ρ 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡/⋅ ρ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 / − 𝑐𝑐𝑐𝑐/ ⋅ ρ𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡 / + / 𝑣𝑣𝑣𝑣 /− 𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = ∂𝑡𝑡𝑡𝑡 + 1 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 +𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠 1 𝑡𝑡𝑡𝑡+1𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚+1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡28( 𝑡𝑡𝑡𝑡) (𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ( 𝑡𝑡𝑡𝑡 )(𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 Δ𝑡𝑡𝑡𝑡 = [( ) ( 𝑡𝑡𝑡𝑡+)1 ]𝑡𝑡𝑡𝑡+1+2𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥−𝑡𝑡𝑡𝑡 (𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡) 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 (= )𝑡𝑡𝑡𝑡 (𝐸𝐸𝐸𝐸𝑡𝑡𝑡𝑡 1) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 +2 ( 𝑤𝑤𝑤𝑤)1𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 − 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2Δ𝑡𝑡𝑡𝑡 =𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−1[ 𝑠𝑠𝑠𝑠 2𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠−1�2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠2 2𝑠𝑠𝑠𝑠+1] 2 +2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 � 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 31 ρ 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛 ⋅𝑠𝑠𝑠𝑠ρ 𝑠𝑠𝑠𝑠 −𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛 ℎ�⋅ ρ=ℎ 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 +𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐(1 −ℎ− ℎ�) 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 𝑢𝑢𝑢𝑢 ℎ𝑣𝑣𝑣𝑣𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏− /𝑣𝑣𝑣𝑣 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡/ 𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐̂ / 𝑣𝑣𝑣𝑣− ℎ/ 𝑑𝑑𝑑𝑑29𝑏𝑏𝑏𝑏 / 𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 / 𝑐𝑐𝑐𝑐̂ / / 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 ℎ / 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡 /−ℎ 𝑏𝑏𝑏𝑏/ 𝑡𝑡𝑡𝑡/ / / 𝑡𝑡𝑡𝑡 + 𝑡𝑡𝑡𝑡30𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 −𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐 ⋅𝑠𝑠𝑠𝑠 ρ 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 /𝑠𝑠𝑠𝑠 / 𝑡𝑡𝑡𝑡+1 /𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 / 𝑡𝑡𝑡𝑡 /𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 / 𝑡𝑡𝑡𝑡 /𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡𝑡𝑡31 0 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 −+ 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡 = ( 𝐸𝐸𝐸𝐸−𝑠𝑠𝑠𝑠)𝑣𝑣𝑣𝑣 (1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1)−𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 2𝑠𝑠𝑠𝑠+2𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 2 𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠+1𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑣𝑣𝑣𝑣(12 𝑠𝑠𝑠𝑠Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1𝑠𝑠𝑠𝑠 )(−𝑛𝑛𝑛𝑛2 )𝑠𝑠𝑠𝑠−1𝑑𝑑𝑑𝑑 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 31 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡 += [(𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ( 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡) ]𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 +2𝑠𝑠𝑠𝑠 Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ( 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠)𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 −𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 ℎ� 𝑡𝑡𝑡𝑡/ 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 / 𝑣𝑣𝑣𝑣𝑢𝑢𝑢𝑢 / 𝑑𝑑𝑑𝑑𝑐𝑐𝑐𝑐̂ / − ℎ� / 𝑏𝑏𝑏𝑏 / 𝑢𝑢𝑢𝑢 / 𝑐𝑐𝑐𝑐̂ / 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡=+1 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+1�(𝑠𝑠𝑠𝑠+11 𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1) 𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠+1ρ2 �𝑠𝑠𝑠𝑠−1 𝑛𝑛𝑛𝑛2 𝑠𝑠𝑠𝑠⋅−1ℎρ 2 𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏−1 2−𝑐𝑐𝑐𝑐𝑛𝑛𝑛𝑛 ⋅−ℎρ 29𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 + ℎ 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 −ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 ℎ 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 �𝑠𝑠𝑠𝑠+1 −2𝑡𝑡𝑡𝑡 ℎ𝑠𝑠𝑠𝑠+1 2 /𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡2+/𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏1+1𝑡𝑡𝑡𝑡2+1/ 𝑡𝑡𝑡𝑡�+𝑠𝑠𝑠𝑠−11 2 /𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡−1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡/𝑠𝑠𝑠𝑠−1 2𝑡𝑡𝑡𝑡/𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 30𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 / −ℎ/ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 / ℎ / 𝑏𝑏𝑏𝑏 /𝑢𝑢𝑢𝑢 /𝑐𝑐𝑐𝑐̂ /− ℎ / 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠31𝑐𝑐𝑐𝑐+1𝑠𝑠𝑠𝑠̂ 2 𝑠𝑠𝑠𝑠+1 2𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+102 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 31 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠+ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠=𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 (𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / ) 𝑡𝑡𝑡𝑡ℎ�(1/ 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 )/ +2𝑢𝑢𝑢𝑢 / 𝑐𝑐𝑐𝑐̂ (1 /Δ𝑥𝑥𝑥𝑥− ℎ�)(𝑠𝑠𝑠𝑠/ 𝑏𝑏𝑏𝑏) / 𝑢𝑢𝑢𝑢 / 𝑐𝑐𝑐𝑐̂ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = +𝑠𝑠𝑠𝑠[( 𝑤𝑤𝑤𝑤) (𝑡𝑡𝑡𝑡+1 )𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+]1 𝑠𝑠𝑠𝑠 +2𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 −( 𝑣𝑣𝑣𝑣 ) ℎ 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 −−ℎ𝑑𝑑𝑑𝑑 𝑣𝑣𝑣𝑣+𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 ( 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ) (𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠Δ𝑥𝑥𝑥𝑥 ) 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1+1 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠+11 𝑡𝑡𝑡𝑡2(+𝑡𝑡𝑡𝑡1𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)(𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡) 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡−1 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡−1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ρ 𝑛𝑛𝑛𝑛 ⋅ ρ𝑠𝑠𝑠𝑠+1Δ𝑡𝑡𝑡𝑡2 ℎ𝑠𝑠𝑠𝑠=+1− 2𝑛𝑛𝑛𝑛𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1⋅ ρ2−ℎ𝑠𝑠𝑠𝑠+11 𝑏𝑏𝑏𝑏2 ℎ�𝑠𝑠𝑠𝑠−1+22 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠−1 2 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠1−1Δ𝑥𝑥𝑥𝑥2 𝑠𝑠𝑠𝑠−−1ℎ�2 30𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 −ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 / ℎ� /𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 / 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡/𝑐𝑐𝑐𝑐̂ /− ℎ� 𝑠𝑠𝑠𝑠 / 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠Δ𝑡𝑡𝑡𝑡̂ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠= ℎ(�𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2)𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(+11 2𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠) 2𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐+2̂𝑠𝑠𝑠𝑠+1 2𝑤𝑤𝑤𝑤−𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ�(𝑠𝑠𝑠𝑠1−1 Δ𝑥𝑥𝑥𝑥2𝑏𝑏𝑏𝑏31𝑠𝑠𝑠𝑠0−1)( 2𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡)𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠 2𝑐𝑐𝑐𝑐̂𝑠𝑠𝑠𝑠−1 2 + 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 FIGURE𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 7𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ℎ/𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 / 𝑐𝑐𝑐𝑐 / −ℎ/𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡−/𝑣𝑣𝑣𝑣 / −𝑛𝑛𝑛𝑛/ 𝑏𝑏𝑏𝑏 / 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡 −𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 =𝑠𝑠𝑠𝑠 [(𝑡𝑡𝑡𝑡+𝑣𝑣𝑣𝑣 )𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠 ( )𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚] +2 0Δ𝑥𝑥𝑥𝑥 (𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 ) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 Δ𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 =𝑠𝑠𝑠𝑠 (�𝑠𝑠𝑠𝑠+1 2)𝑡𝑡𝑡𝑡+(𝑠𝑠𝑠𝑠1+11 𝑡𝑡𝑡𝑡2+1𝑠𝑠𝑠𝑠+1)𝑡𝑡𝑡𝑡+12−+2𝑣𝑣𝑣𝑣�𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡(1𝑠𝑠𝑠𝑠−1−𝑛𝑛𝑛𝑛Δ𝑥𝑥𝑥𝑥2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏)(−1 2𝑡𝑡𝑡𝑡) 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ℎ 𝑏𝑏𝑏𝑏 −ℎ 𝑏𝑏𝑏𝑏 ℎ 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 − ℎ 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1Δ𝑡𝑡𝑡𝑡 2 =𝑠𝑠𝑠𝑠+1 2( 𝑠𝑠𝑠𝑠)−1(12 𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠 2) 𝑠𝑠𝑠𝑠−1+22 𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠 2(1 Δ𝑥𝑥𝑥𝑥 )( ) 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 Trench𝑠𝑠𝑠𝑠 side𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 view𝑠𝑠𝑠𝑠 (A)ℎ�𝑡𝑡𝑡𝑡 and𝑏𝑏𝑏𝑏 cross-section𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐̂ (B),− showingℎ� −𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝑏𝑏𝑏𝑏 all main𝑢𝑢𝑢𝑢 −𝑛𝑛𝑛𝑛 parameters𝑐𝑐𝑐𝑐̂ 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 used−𝑛𝑛𝑛𝑛 in the 𝑑𝑑𝑑𝑑sedimentation model. (t0 < t1 < t2). ℎ 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐/ 𝑡𝑡𝑡𝑡 −ℎ/ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐/ 𝐸𝐸𝐸𝐸 /𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡/ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / 𝑤𝑤𝑤𝑤/ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 /𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 31 𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡 = [( ) 𝑠𝑠𝑠𝑠(𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 )+] +2𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 Δ𝑥𝑥𝑥𝑥 ( 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚) 𝑣𝑣𝑣𝑣 0− 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 − 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 = 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡( )𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛(1 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡) +2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡(1− Δ𝑥𝑥𝑥𝑥𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)(𝑠𝑠𝑠𝑠 )−𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑛𝑛𝑛𝑛0 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 −ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 ℎ� 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ −/ℎ� /𝑏𝑏𝑏𝑏 /𝑢𝑢𝑢𝑢 /𝑐𝑐𝑐𝑐̂ / / / / 31 𝑣𝑣𝑣𝑣𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 − 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 +𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = ( 𝑡𝑡𝑡𝑡+)1 (𝑡𝑡𝑡𝑡1+1 𝑡𝑡𝑡𝑡+)1 +2−𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠(1 𝑡𝑡𝑡𝑡−𝑛𝑛𝑛𝑛)(𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 ) 𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛0 𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠Δ𝑥𝑥𝑥𝑥+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 / ℎ / 𝑏𝑏𝑏𝑏 /𝑐𝑐𝑐𝑐 −ℎ/ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 / ℎ / 𝑏𝑏𝑏𝑏 / 𝑢𝑢𝑢𝑢 / 𝑐𝑐𝑐𝑐̂ − ℎ 31 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ + 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 #156 - AUTUMN 2019 39 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 − 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛Δ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 = (𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚) 𝑡𝑡𝑡𝑡(−𝑛𝑛𝑛𝑛1 0 𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+2𝑡𝑡𝑡𝑡 (1 Δ𝑥𝑥𝑥𝑥 )( ) 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 −ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ − ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐̂ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡 = ( ) (1 ) +2 (1 Δ𝑥𝑥𝑥𝑥− )(𝑣𝑣𝑣𝑣 )𝑠𝑠𝑠𝑠 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 − 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 −𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑛𝑛𝑛𝑛0 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 (1 ) 16 = ( ) + +0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠1 0 − 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 = 𝑄𝑄𝑄𝑄 +𝑄𝑄𝑄𝑄 +𝑄𝑄𝑄𝑄 ⋅ (1−𝑐𝑐𝑐𝑐 ) 17
0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 ( ) 𝑄𝑄𝑄𝑄( 𝑄𝑄𝑄𝑄) 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ − 𝑐𝑐𝑐𝑐 18 + = ( ) +2 ∂ ℎ𝑏𝑏𝑏𝑏 ∂ ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝐸𝐸𝐸𝐸 −𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∙𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ( )∂𝑡𝑡𝑡𝑡 ( ∂𝑥𝑥𝑥𝑥) 19 + = (1 ) +2 (1 ) ∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 −𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 −𝑛𝑛𝑛𝑛0 𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ( ∂𝑡𝑡𝑡𝑡 ) ( ∂𝑥𝑥𝑥𝑥 ) 1 ( ) 20 + + = + + + + 2 2 2 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ 𝑏𝑏𝑏𝑏ℎ TECHNICAL𝑔𝑔𝑔𝑔 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑓𝑓𝑓𝑓 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑤𝑤𝑤𝑤 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 ∂𝑥𝑥𝑥𝑥 21 = tan( ) ∂𝑧𝑧𝑧𝑧 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�푔 − θ � = (2 ∂𝑥𝑥𝑥𝑥 + ) 22 2 𝑓𝑓𝑓𝑓 ∗ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 ) 23 = ρ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − ρ 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� (1 ) 16 1 ( ρ ) = (1 ) 24 +16 + (1 ) = (1 ) 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 2 x= 0 − 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 16 2+ 𝑠𝑠𝑠𝑠 + 𝑤𝑤𝑤𝑤= (1 ) 𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡 ρ − 0ρ( ∂𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐 ) ( ) 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 − 𝑛𝑛𝑛𝑛 1⋅ 𝑄𝑄𝑄𝑄 + +0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠1 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 16𝑄𝑄𝑄𝑄 ⋅ ( −𝑐𝑐𝑐𝑐 ) 𝑆𝑆𝑆𝑆 −𝑐𝑐𝑐𝑐0𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�= − 𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄 = + + 1 17 1𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗ρ 0 𝐸𝐸𝐸𝐸 ∂𝑥𝑥𝑥𝑥 0( ) 25 = 𝑄𝑄𝑄𝑄 +𝑄𝑄𝑄𝑄 +𝑐𝑐𝑐𝑐+𝑄𝑄𝑄𝑄 ⋅ +(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠01−𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)1𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 = −𝑄𝑄𝑄𝑄𝑛𝑛𝑛𝑛 ⋅ 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ −𝑐𝑐𝑐𝑐 0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 17 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 2 0 = + + (1 ) ( ) 𝑄𝑄𝑄𝑄( 𝑄𝑄𝑄𝑄) 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ −17𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠∂𝑏𝑏𝑏𝑏 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 18 0𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠= 𝑄𝑄𝑄𝑄 𝑗𝑗𝑗𝑗 +𝑄𝑄𝑄𝑄𝐸𝐸𝐸𝐸 +𝑄𝑄𝑄𝑄 ⋅0(1−𝑐𝑐𝑐𝑐 ) + = ( ) +2 ( ) 𝑄𝑄𝑄𝑄𝑆𝑆𝑆𝑆( 𝑄𝑄𝑄𝑄) 𝑔𝑔𝑔𝑔𝑔 𝑄𝑄𝑄𝑄 0𝑄𝑄𝑄𝑄 ⋅ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠− 𝑐𝑐𝑐𝑐 𝑗𝑗𝑗𝑗 𝐸𝐸𝐸𝐸 0 18 17 ( ( ) ∂𝑥𝑥𝑥𝑥𝑄𝑄𝑄𝑄( ) 𝑄𝑄𝑄𝑄) 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ − 𝑐𝑐𝑐𝑐 26 + =2= +2 ∂ ℎ𝑏𝑏𝑏𝑏 ∂ ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 18 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 0 +𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑗𝑗𝑗𝑗 = (𝐸𝐸𝐸𝐸 0) +2 FIGURE 9𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 ∙𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 ⋅𝑑𝑑𝑑𝑑 ( ) 𝑄𝑄𝑄𝑄( 𝑄𝑄𝑄𝑄) 𝑄𝑄𝑄𝑄 𝑄𝑄𝑄𝑄 ⋅ − 𝑐𝑐𝑐𝑐 ( ) ( ) ∂ ℎFIGURE𝑏𝑏𝑏𝑏 ∂ ℎ8𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 18 19 ∂𝑏𝑏𝑏𝑏+ ∂ ℎ𝑏𝑏𝑏𝑏𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚−𝑣𝑣𝑣𝑣=∂( ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏∙𝑏𝑏𝑏𝑏 ) ⋅𝑣𝑣𝑣𝑣+2⋅𝑑𝑑𝑑𝑑 + Illustration= of(1 spatial) grid,+2 indicating/(1 the/ ) moving / and /fixed boundaries./ / Growth( velocity) ( ) 32 ( ) ( ) ⋅𝑣𝑣𝑣𝑣 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 + ∂𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡∂𝑥𝑥𝑥𝑥 𝑣𝑣𝑣𝑣 −𝑣𝑣𝑣𝑣 ∙𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 ⋅𝑑𝑑𝑑𝑑 27 19 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 Definition+ ∂ ℎ𝑏𝑏𝑏𝑏 of(= the∂=)ℎ∂𝑡𝑡𝑡𝑡 discrete𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 (1 ∂𝑥𝑥𝑥𝑥 variables)) +2 on the(1 staggered) ∂ ℎ 𝑏𝑏𝑏𝑏 grid,𝑐𝑐𝑐𝑐 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢of𝑢𝑢𝑢𝑢 control volumes for mass19� and𝑠𝑠𝑠𝑠+1 2momentum𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2conservation�𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 are2 𝑠𝑠𝑠𝑠 +1indicated2 𝑠𝑠𝑠𝑠+1 by𝑠𝑠𝑠𝑠 +1vt and𝑠𝑠𝑠𝑠+1 vt/2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 + 𝑣𝑣𝑣𝑣𝐸𝐸𝐸𝐸 −𝑣𝑣𝑣𝑣= 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ∙𝑏𝑏𝑏𝑏(1 ⋅𝑣𝑣𝑣𝑣) 𝑤𝑤𝑤𝑤+2𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 (1 ) −𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 −𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣ℎ𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏−𝑛𝑛𝑛𝑛0 𝑢𝑢𝑢𝑢𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 1− ℎ( )𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 (( ) )ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢( ) −ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 32 with( corresponding) ( ) control volumes. respectively. / / / / / / ∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 ∂ ℎ𝑏𝑏𝑏𝑏∂𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 ∂𝑧𝑧𝑧𝑧 ∂𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 (𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡∂𝑡𝑡𝑡𝑡 ) ( ∂𝑥𝑥𝑥𝑥 ) 1 ( ) 19 + + 20 +∂ ℎ𝑏𝑏𝑏𝑏−𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡=∂ ℎ𝑏𝑏𝑏𝑏−𝑛𝑛𝑛𝑛𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 (1𝑏𝑏𝑏𝑏 )⋅𝑣𝑣𝑣𝑣+2 −𝑛𝑛𝑛𝑛(1𝑑𝑑𝑑𝑑 ) + + 𝑡𝑡𝑡𝑡=+1 𝑡𝑡𝑡𝑡++1 𝑡𝑡𝑡𝑡++1 +𝑡𝑡𝑡𝑡 + 𝑡𝑡𝑡𝑡 2Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥2 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 2�𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ( ∂𝑡𝑡𝑡𝑡 ) ( ∂𝑥𝑥𝑥𝑥= ) ∂𝑡𝑡𝑡𝑡1+ (1( )) −𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 ⋅𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 2 20 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢/ 1−/ ℎ( /)𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢/(ℎ ) / ℎ𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏/− ℎ𝑢𝑢𝑢𝑢푢 𝑏𝑏𝑏𝑏 −ℎ( 𝑏𝑏𝑏𝑏 ) 𝑢𝑢𝑢𝑢푢 ( ) 32 +∂ ℎ𝑏𝑏𝑏𝑏𝑐𝑐𝑐𝑐 (∂+∂𝑡𝑡𝑡𝑡ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢) 𝑢𝑢𝑢𝑢 ( ∂𝑥𝑥𝑥𝑥=) 1 + ( + ) + + 28 20 = ( ) / + + ( ) / + ( ) / + ( ) / 2 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ∂ ℎ𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ 𝑏𝑏𝑏𝑏ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+𝑓𝑓𝑓𝑓1 +𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡1 𝑡𝑡𝑡𝑡+𝑡𝑡𝑡𝑡1 𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 + / 2 + −𝑣𝑣𝑣𝑣 + −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 =⋅𝑣𝑣𝑣𝑣 + −𝑛𝑛𝑛𝑛+ 𝑑𝑑𝑑𝑑 + + 𝑔𝑔𝑔𝑔 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆2Δ𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 2𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 (2 )𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2( ) 𝑡𝑡𝑡𝑡 32 ( ∂𝑡𝑡𝑡𝑡 ) ( 𝑠𝑠𝑠𝑠∂𝑥𝑥𝑥𝑥 ) 1 2( 𝑤𝑤𝑤𝑤 2) 2 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠+1�/𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1/ 2𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1/ 2 /𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ ρℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑐𝑐𝑐𝑐 ⋅ ρ ∂ −𝑏𝑏𝑏𝑏ℎ𝑐𝑐𝑐𝑐 ⋅ ρ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤 ∂𝑥𝑥𝑥𝑥 ∂𝑥𝑥𝑥𝑥 20 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢ℎ 1−𝑏𝑏𝑏𝑏 ℎ( −𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)𝑏𝑏𝑏𝑏ℎ 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏2𝑢𝑢𝑢𝑢 ( 𝑓𝑓𝑓𝑓) 𝑠𝑠𝑠𝑠+1ℎ212+ 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠+1 2−ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢푢 2 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 +∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑔𝑔𝑔𝑔 +∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 𝑆𝑆𝑆𝑆 =∂𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏ℎ +𝑆𝑆𝑆𝑆 + 𝑆𝑆𝑆𝑆 +𝑆𝑆𝑆𝑆 + = tan( ) =𝑡𝑡𝑡𝑡+(1 𝑡𝑡𝑡𝑡+) 1 𝑡𝑡𝑡𝑡++1 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 +𝑡𝑡𝑡𝑡( �𝑆𝑆𝑆𝑆 )�𝑡𝑡𝑡𝑡 + (𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 ) 𝑡𝑡𝑡𝑡 +𝑡𝑡𝑡𝑡 ( 2𝑆𝑆𝑆𝑆) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆2 2 2 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑔𝑔𝑔𝑔 + / / / / / 33 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥= +∂𝑥𝑥𝑥𝑥(21 ) 𝑔𝑔𝑔𝑔 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 which𝑆𝑆𝑆𝑆 is a 𝑆𝑆𝑆𝑆known𝑆𝑆𝑆𝑆 constant,21 see /equation/ �𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠+1 2/ 2Δ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2/𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠+1/ 𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠2+1(𝑡𝑡𝑡𝑡2 ) 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1Δ𝑥𝑥𝑥𝑥(𝑠𝑠𝑠𝑠+1) 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 32 ∂𝑡𝑡𝑡𝑡= ∂𝑥𝑥𝑥𝑥 tan (∂𝑥𝑥𝑥𝑥) 29 ∂𝑧𝑧𝑧𝑧 ℎ 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡( 𝑢𝑢𝑢𝑢 )Δ𝑥𝑥𝑥𝑥1𝑠𝑠𝑠𝑠−+1/ ℎ(= 𝑠𝑠𝑠𝑠+1)𝑏𝑏𝑏𝑏 /𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡( 𝑠𝑠𝑠𝑠 /) ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢푢 tan −ℎ𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ ℎ𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 ∂ 𝑏𝑏𝑏𝑏ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 21𝑡𝑡𝑡𝑡/ +1 𝑏𝑏𝑏𝑏/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1/ 2 𝑡𝑡𝑡𝑡 ℎ 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡/ 𝑏𝑏𝑏𝑏 +/ − 𝑠𝑠𝑠𝑠ℎ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡/ 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 (𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡2) 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1( 2/ ) 32 𝑔𝑔𝑔𝑔 = 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆tan( )𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�푔 − θ � = ( +�) �𝑠𝑠𝑠𝑠+1+2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 + (𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠2)+1 𝑡𝑡𝑡𝑡 +𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡( 𝑡𝑡𝑡𝑡) 2 𝑡𝑡𝑡𝑡+ ( ) (26). Therefore, the width is known𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 +1at 𝑡𝑡𝑡𝑡every2+1𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡2+𝑆𝑆𝑆𝑆1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1+12 𝑔𝑔𝑔𝑔𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2𝑆𝑆𝑆𝑆Δ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1/ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 2 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆+1 𝑠𝑠𝑠𝑠+1+𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+12 𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 −/𝑠𝑠𝑠𝑠𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 Δ𝑥𝑥𝑥𝑥𝑆𝑆𝑆𝑆 / / 𝑠𝑠𝑠𝑠∂𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠∂𝑥𝑥𝑥𝑥 𝑤𝑤𝑤𝑤 ∂∂𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠 = (2ℎ� 21∂𝑥𝑥𝑥𝑥 +𝑏𝑏𝑏𝑏 ) 𝑢𝑢𝑢𝑢 − ℎ(� )𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+12 𝑢𝑢𝑢𝑢2( ) ℎ𝑠𝑠𝑠𝑠+1 𝑏𝑏𝑏𝑏2/ 𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢푢22𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 −ℎ𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 2𝑢𝑢𝑢𝑢푢 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 33 ρ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑛𝑛 ⋅ ρ − 𝑛𝑛𝑛𝑛 ⋅ ρ ( ) / 𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠+11 / 2 𝑠𝑠𝑠𝑠+1𝑆𝑆𝑆𝑆2 / 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 / 2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1+1−𝑔𝑔𝑔𝑔Δ𝑥𝑥𝑥𝑥/ 2�� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1+1𝑡𝑡𝑡𝑡 2𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠(+1� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡) 𝑠𝑠𝑠𝑠+1 − ( 𝑡𝑡𝑡𝑡�𝑠𝑠𝑠𝑠𝜃𝜃𝜃𝜃)��𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠2� 𝑡𝑡𝑡𝑡 32 (29) 𝑆𝑆𝑆𝑆 / −𝑔𝑔𝑔𝑔/ 𝑔𝑔𝑔𝑔𝑔�= / 푔𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 − / θ �tan/ ∂𝑧𝑧𝑧𝑧/ grid point and every30 time step and thusℎ�( no 𝑏𝑏𝑏𝑏) / 𝑢𝑢𝑢𝑢= − ℎ�(/ ℎ𝑏𝑏𝑏𝑏)/ 𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 − ℎℎ tan𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ( 𝑢𝑢𝑢𝑢푢 ) −ℎ( 𝑏𝑏𝑏𝑏) 𝑢𝑢𝑢𝑢푢 ( ) ( ) 2 + 1 𝑏𝑏𝑏𝑏=𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡( 𝑠𝑠𝑠𝑠+1)2 / 𝑓𝑓𝑓𝑓+𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠(+1 2) 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1Δ𝑥𝑥𝑥𝑥 2 / / 𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠+1 2 / 𝑤𝑤𝑤𝑤+𝑠𝑠𝑠𝑠+1 2 / + 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡= 𝑡𝑡𝑡𝑡 2𝑆𝑆𝑆𝑆 ∂𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡+−𝑔𝑔𝑔𝑔 𝑔𝑔𝑔𝑔𝑔�𝑡𝑡𝑡𝑡 푔𝑡𝑡𝑡𝑡 − θ � 𝑓𝑓𝑓𝑓 22∗ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+2Δ𝑡𝑡𝑡𝑡1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡=2�𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡(� 𝑡𝑡𝑡𝑡) 𝑡𝑡𝑡𝑡 /𝑆𝑆𝑆𝑆2 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡( 2)𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 +𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑆𝑆𝑆𝑆 34 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 =∂𝑧𝑧𝑧𝑧 (2 ∂𝑥𝑥𝑥𝑥 + ) approximations are required. The continuity𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1+ 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡/ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1Δ𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡/ / / 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑�𝑠𝑠𝑠𝑠+1 2𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+122𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2) 𝑠𝑠𝑠𝑠+1 2 ℎ �𝑠𝑠𝑠𝑠+12Δ𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡−2 ℎ2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1𝑡𝑡𝑡𝑡 2𝑏𝑏𝑏𝑏𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠−+1𝑡𝑡𝑡𝑡 𝑧𝑧𝑧𝑧2𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 33 ℎ 𝑏𝑏𝑏𝑏 −ℎ 𝑏𝑏𝑏𝑏 ℎ� 𝑢𝑢𝑢𝑢𝑆𝑆𝑆𝑆 𝑏𝑏𝑏𝑏 2−𝑔𝑔𝑔𝑔−𝑔𝑔𝑔𝑔𝑔�ℎ� 푔 𝑢𝑢𝑢𝑢 − 𝑏𝑏𝑏𝑏 θ � ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢=𝑠𝑠𝑠𝑠+1( (12−)ℎ() �𝑠𝑠𝑠𝑠)++1𝑏𝑏𝑏𝑏=𝑠𝑠𝑠𝑠2+1𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠(+12𝑠𝑠𝑠𝑠+1+2) ( ℎ2 𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓𝑡𝑡𝑡𝑡)𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠23 𝑢𝑢𝑢𝑢푢 + �( �𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)𝑠𝑠𝑠𝑠+1−ℎ𝑡𝑡𝑡𝑡 tan𝑠𝑠𝑠𝑠+12 𝑏𝑏𝑏𝑏+2 (𝑢𝑢𝑢𝑢푢 𝑡𝑡𝑡𝑡 )𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 𝑓𝑓𝑓𝑓 ∗ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 2 = 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔 −𝑔𝑔𝑔𝑔/�/ ℎ𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏/ �/ − �ℎ𝑆𝑆𝑆𝑆/ �𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏−𝑡𝑡𝑡𝑡/2 𝜃𝜃𝜃𝜃𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 / � 𝑆𝑆𝑆𝑆/ / 𝑆𝑆𝑆𝑆 One-dimensional𝑆𝑆𝑆𝑆 −𝑢𝑢𝑢𝑢 finite= ⋅ 𝑑𝑑𝑑𝑑 (volume2 𝑏𝑏𝑏𝑏∂𝑥𝑥𝑥𝑥 + ) equation for the total volume22 and the + =(33)( 𝑓𝑓𝑓𝑓) 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 + Δ𝑥𝑥𝑥𝑥∗ 𝑠𝑠𝑠𝑠+1+2 ( )𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1+2 ( )𝑠𝑠𝑠𝑠+1 2 + ( ) ( 𝑓𝑓𝑓𝑓 ∗) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 23 Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡=𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔(�2𝑆𝑆𝑆𝑆Δ𝑥𝑥𝑥𝑥�𝑡𝑡𝑡𝑡) / 𝑡𝑡𝑡𝑡2𝑡𝑡𝑡𝑡−(2 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) / 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠�+1+ ⋅ 𝑑𝑑𝑑𝑑Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡/ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡/ � / 34 33 Δ𝑡𝑡𝑡𝑡 = [( ) ( ) ] +2 Δ𝑥𝑥𝑥𝑥 𝑆𝑆𝑆𝑆( )− 𝑢𝑢𝑢𝑢 ⋅(𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2 / 𝑡𝑡𝑡𝑡 /𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧/ scheme on= a staggered2 grid continuity equation for the sedimentρ volume−23ρ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1+12 2) 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓/ 𝑠𝑠𝑠𝑠+1=�𝑠𝑠𝑠𝑠+12Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠+1/2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 tan𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠2+1 2/ 𝑡𝑡𝑡𝑡 35 𝑓𝑓𝑓𝑓 ∗ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 ℎ 𝑏𝑏𝑏𝑏�𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡−𝑔𝑔𝑔𝑔�−� ℎ 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 � 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 − 𝑡𝑡𝑡𝑡 �𝜃𝜃𝜃𝜃��𝑆𝑆𝑆𝑆 � / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡=𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑣𝑣𝑣𝑣 ⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� = ( ( ) 𝑏𝑏𝑏𝑏)/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2+/𝑠𝑠𝑠𝑠+1=2( 𝑓𝑓𝑓𝑓)+ (/ ) /𝑠𝑠𝑠𝑠/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠++1𝑠𝑠𝑠𝑠+1(/ 2)𝑠𝑠𝑠𝑠 / 𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠+1( 2𝑡𝑡𝑡𝑡) / 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 −𝑤𝑤𝑤𝑤𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠⋅𝑡𝑡𝑡𝑡(𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 ) 𝑓𝑓𝑓𝑓 𝑔𝑔𝑔𝑔 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡∗ 𝑠𝑠𝑠𝑠+1 2 �/𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡�𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1𝑠𝑠𝑠𝑠+12𝑡𝑡𝑡𝑡 2 𝑆𝑆𝑆𝑆Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 33 The equations𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠are𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 solved onρ a one-𝑠𝑠𝑠𝑠− ρ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 are discretised explicit1 in time23( and upwindρ () �)𝑆𝑆𝑆𝑆 �𝑠𝑠𝑠𝑠+1 =2 𝑏𝑏𝑏𝑏−𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1 =2 �(𝑡𝑡𝑡𝑡 ⋅)𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠+1 2 2𝑠𝑠𝑠𝑠+1(242 tan𝑏𝑏𝑏𝑏)𝑧𝑧𝑧𝑧 −�+𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 𝑠𝑠𝑠𝑠+1 2 34 𝑣𝑣𝑣𝑣 −𝑆𝑆𝑆𝑆 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏 =⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢�𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 ρ − ρ / 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 // Δ𝑥𝑥𝑥𝑥 /−𝑔𝑔𝑔𝑔��𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 � 𝑡𝑡𝑡𝑡 / �ρ/𝑡𝑡𝑡𝑡 − / �−𝜃𝜃𝜃𝜃�ρ�𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2� 33 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = ( 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠2+1=2 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 2 𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡2𝑡𝑡𝑡𝑡 tan/𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 +1 2 35 dimensional staggered1 𝑆𝑆𝑆𝑆grid,( using𝑣𝑣𝑣𝑣ρ 𝑠𝑠𝑠𝑠) a𝑡𝑡𝑡𝑡⋅𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠finite𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢� for the fluxes, see equations2 24 (30) and (31). 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 /𝑡𝑡𝑡𝑡 �𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 �𝑠𝑠𝑠𝑠𝑣𝑣𝑣𝑣+12/2𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 𝑆𝑆𝑆𝑆/ 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆/ 31 2 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤( 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) / = ( 𝑓𝑓𝑓𝑓 ) / =/∗ 𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠+1( /2−) 𝑧𝑧𝑧𝑧 2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠(+1 2)𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1+ 2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1 2 34 / = /𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 1 ρ / −/( ρ /ρ ) / ρ − ρ 24𝑆𝑆𝑆𝑆∂ 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠+1 2 �−𝑔𝑔𝑔𝑔𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡���𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+122𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1−/ 𝑡𝑡𝑡𝑡2𝑢𝑢𝑢𝑢� 1 𝑡𝑡𝑡𝑡 � ⋅/ −𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠+1 �𝜃𝜃𝜃𝜃�𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠+1 𝑏𝑏𝑏𝑏/2�( ρ� � / ) 33 volume+ scheme.𝑆𝑆𝑆𝑆 In the𝑣𝑣𝑣𝑣 staggered⋅𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢� grid The variables𝑡𝑡𝑡𝑡 that are not defined on the ( 𝑡𝑡𝑡𝑡 ) 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(34)=𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1( 2 𝑠𝑠𝑠𝑠𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡−tan𝑧𝑧𝑧𝑧/ 𝑠𝑠𝑠𝑠+1( 2) 32 36 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 =𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�/ / / / / / ( ) / / �= / Δ𝑥𝑥𝑥𝑥2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡�ρ 𝑡𝑡𝑡𝑡 ( − ρ� 𝑡𝑡𝑡𝑡 )�/ � 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1/2 35 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+12 2 2𝑠𝑠𝑠𝑠(−1 2 ρ𝑠𝑠𝑠𝑠−1)2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 (𝑠𝑠𝑠𝑠+1𝑓𝑓𝑓𝑓 )2−𝑔𝑔𝑔𝑔/ 𝑠𝑠𝑠𝑠+1� 2 (𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏+1∗ )2 �/ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 − �𝜃𝜃𝜃𝜃� � 34 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 � 𝑡𝑡𝑡𝑡 1 ρ −�ρ ∂𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 1𝑡𝑡𝑡𝑡+1 24𝑡𝑡𝑡𝑡 +1ρ 𝑡𝑡𝑡𝑡+𝑆𝑆𝑆𝑆1∂𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡(𝑡𝑡𝑡𝑡 )𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣= 𝑡𝑡𝑡𝑡=𝑡𝑡𝑡𝑡(𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢+/2)𝑡𝑡𝑡𝑡2 𝑏𝑏𝑏𝑏2 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1/ 225𝑠𝑠𝑠𝑠+ Δ𝑥𝑥𝑥𝑥 / 𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1 2 ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 −ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐the flowℎ height𝑏𝑏𝑏𝑏 h𝑢𝑢𝑢𝑢 and= concentration𝑐𝑐𝑐𝑐̂ − ℎ c𝑏𝑏𝑏𝑏 are2 ρ𝑢𝑢𝑢𝑢 − ρ𝑐𝑐𝑐𝑐̂ ∂grid𝑐𝑐𝑐𝑐 are denoted with a hat and determined / / �𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡� (𝑡𝑡𝑡𝑡 −/) 𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡/𝑡𝑡𝑡𝑡− 𝑧𝑧𝑧𝑧(�22 ⋅/𝑠𝑠𝑠𝑠)+1𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 / 𝑤𝑤𝑤𝑤� 34 𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� 𝑡𝑡𝑡𝑡 = 𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠+1/ 2 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1/ 2 2𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1/𝑡𝑡𝑡𝑡2=2 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 𝑠𝑠𝑠𝑠+1/( 𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠+1) ρ� 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 /𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡(𝑠𝑠𝑠𝑠+𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠) ρ// − ρ 32 35 12 𝑆𝑆𝑆𝑆 −ρ 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔�∂𝑥𝑥𝑥𝑥 ℎ�2 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑆𝑆𝑆𝑆− ℎ� 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 −𝑔𝑔𝑔𝑔1𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡��2/𝑠𝑠𝑠𝑠+1 2𝑏𝑏𝑏𝑏ℎ𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏� 𝑡𝑡𝑡𝑡�𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠+1 2(𝑡𝑡𝑡𝑡 −ℎ− 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏�𝜃𝜃𝜃𝜃)��𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠+1 2�𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 / 36 Δ𝑡𝑡𝑡𝑡 = ( discretised) (1 )at the+2 center( 1of Δ𝑥𝑥𝑥𝑥each2)( cell.𝑠𝑠𝑠𝑠 ) The 𝑤𝑤𝑤𝑤 via an upwind approximation.25 Time steps are1 (𝑓𝑓𝑓𝑓 ) ( 𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆) (∗ 𝑠𝑠𝑠𝑠+1)+=2( − 𝑡𝑡𝑡𝑡)𝑔𝑔𝑔𝑔𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡ℎ𝑠𝑠𝑠𝑠+1 2 � 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1�ρ2−𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡− ρ� � = ρ −1ρ ∂𝑐𝑐𝑐𝑐 ρ ∂𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡2+∂1𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡+25+1 ( �𝑆𝑆𝑆𝑆)𝑡𝑡𝑡𝑡�𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2=𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1−2𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 / 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡/ �𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1(2⋅ 2𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠+1/Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡2 ) 2𝑠𝑠𝑠𝑠+1/ 𝑏𝑏𝑏𝑏2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡/� 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑓𝑓𝑓𝑓2=2 𝑠𝑠𝑠𝑠+1( 2𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 ) 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠/+1∗ 2𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢( 21𝑠𝑠𝑠𝑠+1) 𝑏𝑏𝑏𝑏 /𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠2 / 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1ρ� 22 34 flow𝑡𝑡𝑡𝑡 velocity u𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆is discretised−2 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔� at the= interfaces𝑡𝑡𝑡𝑡 indexed by superscript𝑆𝑆𝑆𝑆 ℎ�n and𝑔𝑔𝑔𝑔𝑔 𝑏𝑏𝑏𝑏space𝑢𝑢𝑢𝑢 steps2− Δ𝑡𝑡𝑡𝑡byℎ� 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏2 �/𝑆𝑆𝑆𝑆 �𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2𝑡𝑡𝑡𝑡 2−ℎ 𝑢𝑢𝑢𝑢2 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢푢 � 𝑡𝑡𝑡𝑡⋅𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥𝑑𝑑𝑑𝑑−ℎ𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡/𝑏𝑏𝑏𝑏𝑤𝑤𝑤𝑤 𝑢𝑢𝑢𝑢푢 ρ� /𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡� / 𝑠𝑠𝑠𝑠+1 2 35 37 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 21 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡ρ2𝑠𝑠𝑠𝑠 ∂𝑥𝑥𝑥𝑥 ( 1 )( 𝑠𝑠𝑠𝑠+1)𝑡𝑡𝑡𝑡 ( 𝑠𝑠𝑠𝑠+1 ) ( 𝑠𝑠𝑠𝑠)( 1𝑠𝑠𝑠𝑠) = ρ − ρ (�ρ )− ρ� � 36 − 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛∂𝑏𝑏𝑏𝑏 𝑑𝑑𝑑𝑑 25 ∂𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 ℎ𝑠𝑠𝑠𝑠+1/ =2 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡−𝑠𝑠𝑠𝑠+1ℎ�/𝑠𝑠𝑠𝑠2+12 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏2 /𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 /𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1/ 2 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠26+1 2𝑡𝑡𝑡𝑡/ 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / 35 between the cells.𝑆𝑆𝑆𝑆 Corresponding𝑔𝑔𝑔𝑔𝑔= to the 2 ∂𝑏𝑏𝑏𝑏 subscript i. =2 += ( 𝑆𝑆𝑆𝑆 )(𝑓𝑓𝑓𝑓 /( )𝑆𝑆𝑆𝑆+−) 𝑔𝑔𝑔𝑔/�ℎ=(∗ 𝑠𝑠𝑠𝑠++1�)𝑣𝑣𝑣𝑣(2 𝑐𝑐𝑐𝑐 ) /−𝑐𝑐𝑐𝑐2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡/ 𝑢𝑢𝑢𝑢(𝑠𝑠𝑠𝑠++1(2 )𝑏𝑏𝑏𝑏 2𝑠𝑠𝑠𝑠+1/) +2 ( ) 𝑠𝑠𝑠𝑠+1/ 2 𝑤𝑤𝑤𝑤 2Δ𝑡𝑡𝑡𝑡𝑔𝑔𝑔𝑔𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 �𝑠𝑠𝑠𝑠2+1𝑡𝑡𝑡𝑡2 / −𝑡𝑡𝑡𝑡=𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 /2 𝑡𝑡𝑡𝑡2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 � / 𝑡𝑡𝑡𝑡⋅ 𝑑𝑑𝑑𝑑 / Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ρ� 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠/+1𝑏𝑏𝑏𝑏/2 𝑠𝑠𝑠𝑠+1�2 𝑠𝑠𝑠𝑠ρ� 𝑡𝑡𝑡𝑡 2∂𝑥𝑥𝑥𝑥 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔 26 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡1𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 12 �2ρ 𝑠𝑠𝑠𝑠+1 2 − ρ𝑏𝑏𝑏𝑏� 𝑠𝑠𝑠𝑠−�(𝑏𝑏𝑏𝑏/ 𝑤𝑤𝑤𝑤 ) 36 =2 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1ℎ 𝑡𝑡𝑡𝑡2 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡−𝑠𝑠𝑠𝑠+1ℎ 2 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 ��𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠ρ𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡−/ρ/𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 37 35 discretised variables, the𝑤𝑤𝑤𝑤 control volume∂𝑏𝑏𝑏𝑏 ∂𝑥𝑥𝑥𝑥 ∂𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 =26𝑆𝑆𝑆𝑆 ( 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡) 𝑠𝑠𝑠𝑠+1( 2(35))+𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1(𝑓𝑓𝑓𝑓=2) /𝑆𝑆𝑆𝑆𝑢𝑢𝑢𝑢+=( 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1) 𝑠𝑠𝑠𝑠+12 𝑔𝑔𝑔𝑔2+/ℎ𝑠𝑠𝑠𝑠(+1 (𝑡𝑡𝑡𝑡 ) 𝑠𝑠𝑠𝑠+1�𝑠𝑠𝑠𝑠ρ 2+ (𝑡𝑡𝑡𝑡) 𝑤𝑤𝑤𝑤−) ρ𝑠𝑠𝑠𝑠�+1 2 � =2 ( 𝑆𝑆𝑆𝑆𝑔𝑔𝑔𝑔 ) 𝑠𝑠𝑠𝑠//𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆=𝑠𝑠𝑠𝑠+1(�/𝑆𝑆𝑆𝑆2 �𝑠𝑠𝑠𝑠)+1/ −2𝑠𝑠𝑠𝑠/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔𝑠𝑠𝑠𝑠𝑆𝑆𝑆𝑆+1�/ℎ�/2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡�/ /2𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠+1𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡−𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠2+1 /2 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 / 38 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔𝑔 ⋅𝑣𝑣𝑣𝑣 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 2𝑣𝑣𝑣𝑣 2 2𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏( 2ρ� ) 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 / for continuity equations∂𝑏𝑏𝑏𝑏 is centered∂𝑥𝑥𝑥𝑥 The momentum equation∂𝑡𝑡𝑡𝑡 is discretised on 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 =2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / ρ� 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1𝑠𝑠𝑠𝑠+12𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡2 𝑤𝑤𝑤𝑤 33 36 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 26 ( ) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 (1𝑡𝑡𝑡𝑡+1𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡−27)�𝑏𝑏𝑏𝑏 ρ� ρ� − ρ 37 ⋅𝑣𝑣𝑣𝑣=2 ∂𝑏𝑏𝑏𝑏 = ( ) 𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =𝑠𝑠𝑠𝑠+1/𝑤𝑤𝑤𝑤2=𝑠𝑠𝑠𝑠+1 /2𝑓𝑓𝑓𝑓𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1/2 2/�1𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠+1�𝑠𝑠𝑠𝑠2+1tan2ρ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1+1 2−(ρ� 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡) 2 36 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠+1𝑆𝑆𝑆𝑆2( �)𝑆𝑆𝑆𝑆2𝑆𝑆𝑆𝑆(� 𝑠𝑠𝑠𝑠)+1𝑔𝑔𝑔𝑔�2ℎ/ 𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠=−+1𝑠𝑠𝑠𝑠�2𝑔𝑔𝑔𝑔�𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠ℎ+1 2𝑠𝑠𝑠𝑠(/ �𝑆𝑆𝑆𝑆 𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡/−𝑐𝑐𝑐𝑐) 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠+1 2 around the h, c and∂𝑡𝑡𝑡𝑡 u variables, whereas ⋅𝑣𝑣𝑣𝑣the a staggered grid, see equation27 (32) for the𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 / =𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+12 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠/− 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 /𝑠𝑠𝑠𝑠+1𝑤𝑤𝑤𝑤 2𝑡𝑡𝑡𝑡 ρ� 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 2 1 Δ𝑥𝑥𝑥𝑥 2ρ 𝑠𝑠𝑠𝑠−+1ρ� ρ𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 33 38 =∂𝑏𝑏𝑏𝑏 ∂𝑡𝑡𝑡𝑡 ∂𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ( 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡27) 𝑠𝑠𝑠𝑠 +1 𝑡𝑡𝑡𝑡2 =𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠1+1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+12 2𝑧𝑧𝑧𝑧 −𝑠𝑠𝑠𝑠+1𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 2tan𝑠𝑠𝑠𝑠 (𝑡𝑡𝑡𝑡⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠) / 𝑤𝑤𝑤𝑤 37 control volume of the momentum𝑤𝑤𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚equation= discretised equation. /𝑆𝑆𝑆𝑆 � −/𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔𝑤𝑤𝑤𝑤�ℎ�/𝑠𝑠𝑠𝑠(+1 =22)𝑡𝑡𝑡𝑡� 𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡 =𝑡𝑡𝑡𝑡−𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡� �𝑏𝑏𝑏𝑏�𝑠𝑠𝑠𝑠+1/𝑡𝑡𝑡𝑡−2 𝑏𝑏𝑏𝑏 ρ − ρ 36 ⋅𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑆𝑆𝑆𝑆 ( )−𝑔𝑔𝑔𝑔� 𝑡𝑡𝑡𝑡=𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1+𝑏𝑏𝑏𝑏12 𝑡𝑡𝑡𝑡� / 𝑠𝑠𝑠𝑠(+1Δ𝑡𝑡𝑡𝑡��2− 𝑠𝑠𝑠𝑠�+1)𝜃𝜃𝜃𝜃/𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠2 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 39 ∂𝑧𝑧𝑧𝑧 = ∂𝑡𝑡𝑡𝑡 + (1 ) 𝑡𝑡𝑡𝑡 /𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 −𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1𝑔𝑔𝑔𝑔/ �Δ𝑥𝑥𝑥𝑥ℎ� 𝑔𝑔𝑔𝑔𝑠𝑠𝑠𝑠 ℎ2� 𝑐𝑐𝑐𝑐 ρ� −𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡 2 ∂𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 27 ( 𝑏𝑏𝑏𝑏 ) 2−1𝑏𝑏𝑏𝑏 𝑧𝑧𝑧𝑧(𝑡𝑡𝑡𝑡 −)2 𝑧𝑧𝑧𝑧 28=𝑡𝑡𝑡𝑡 ( Δ𝑥𝑥𝑥𝑥) 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠2 34 37 38 is centered around velocity𝑣𝑣𝑣𝑣 =u and is∂ 𝑧𝑧𝑧𝑧 thus 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 =𝑡𝑡𝑡𝑡 �(36)𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1/ 22 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+𝑤𝑤𝑤𝑤1𝑡𝑡𝑡𝑡𝑚𝑚𝑚𝑚/𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2𝑏𝑏𝑏𝑏/ 𝑤𝑤𝑤𝑤ρ� − 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 / −𝑔𝑔𝑔𝑔( �) / 𝑏𝑏𝑏𝑏= �𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1/⋅𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠12 =2− 𝑠𝑠𝑠𝑠 ��𝑠𝑠𝑠𝑠+1𝜃𝜃𝜃𝜃�� ρ2 −�ρ 37 = ∂𝑡𝑡𝑡𝑡 + (1 ) 𝑣𝑣𝑣𝑣 28 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆�𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡+12 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔�ℎ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 staggered with respect to the continuity With the discretised source𝑠𝑠𝑠𝑠 terms given𝑤𝑤𝑤𝑤 by 𝑆𝑆𝑆𝑆 2 −(Δ𝑡𝑡𝑡𝑡𝑔𝑔𝑔𝑔2)�ℎ / =Δ𝑥𝑥𝑥𝑥𝑧𝑧𝑧𝑧� 2𝑐𝑐𝑐𝑐 −𝑠𝑠𝑠𝑠+1𝑧𝑧𝑧𝑧−𝑐𝑐𝑐𝑐/ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 39 ∂𝑧𝑧𝑧𝑧 = ∂𝑡𝑡𝑡𝑡 + (1 ) ρ 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ 28 𝑓𝑓𝑓𝑓 = (𝑡𝑡𝑡𝑡 ∗) 𝑠𝑠𝑠𝑠+1/ 2 2 𝑠𝑠𝑠𝑠(𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡) 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠/ 2+ 𝑠𝑠𝑠𝑠+1/𝑣𝑣𝑣𝑣2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 34 38 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 �𝑆𝑆𝑆𝑆 �𝑠𝑠𝑠𝑠+1/ 2 𝑤𝑤𝑤𝑤−𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢 2 � 1𝑏𝑏𝑏𝑏⋅=𝑠𝑠𝑠𝑠𝑑𝑑𝑑𝑑+1(−2 𝑏𝑏𝑏𝑏 )𝑏𝑏𝑏𝑏 −𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏2 �ρ� 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡𝑔𝑔𝑔𝑔𝑡𝑡𝑡𝑡�ℎ� = � Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 +=2𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠+11 𝑠𝑠𝑠𝑠 37 40 control volume (see= Figure∂𝑡𝑡𝑡𝑡 +8).( The1 change) equations (33), (34), (35), (36)( and (37).) ( 2) 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤=𝑠𝑠𝑠𝑠+1 /2 𝑡𝑡𝑡𝑡+1 𝑠𝑠𝑠𝑠+1/𝑡𝑡𝑡𝑡⋅𝑣𝑣𝑣𝑣2𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 /− 𝑏𝑏𝑏𝑏 / 35 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 = +28 1 𝑠𝑠𝑠𝑠 / 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / 𝑡𝑡𝑡𝑡� Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡29 𝑡𝑡𝑡𝑡 38 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ − 𝑐𝑐𝑐𝑐 ⋅ ρ ( ) 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠+1=2 ( ) ∗𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠+1𝑆𝑆𝑆𝑆 −2 𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡2 Δ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1𝑔𝑔𝑔𝑔�2−𝑡𝑡𝑡𝑡ℎ 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1� 2 𝑡𝑡𝑡𝑡 39 in width is only a function of the wall velocity �𝑆𝑆𝑆𝑆 �/ − 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡/ � /⋅ =2𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡/𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 2 (𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏+1 ) �𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 38 = + (1 ) 29 𝑡𝑡𝑡𝑡+1 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1=𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 −/𝑤𝑤𝑤𝑤𝑏𝑏𝑏𝑏⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡=𝑠𝑠𝑠𝑠+1Δ𝑡𝑡𝑡𝑡2𝑡𝑡𝑡𝑡ρ� 𝑡𝑡𝑡𝑡++𝑡𝑡𝑡𝑡1�𝑠𝑠𝑠𝑠+1ρ1�2𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐ρ̂ =2 −ρ ρ��𝑠𝑠𝑠𝑠+1 /−2�𝑐𝑐𝑐𝑐̂ � 35 40 ρ 𝑐𝑐𝑐𝑐 ⋅ ρ =− 𝑐𝑐𝑐𝑐 ⋅ ρ + (1 ) ρ 𝑛𝑛𝑛𝑛 ⋅ ρ −( 𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡29)⋅𝑠𝑠𝑠𝑠ρ +1 2 = ( 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(37)/𝑆𝑆𝑆𝑆) 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠+1/𝑠𝑠𝑠𝑠2 𝑠𝑠𝑠𝑠𝑔𝑔𝑔𝑔𝑡𝑡𝑡𝑡�+𝑠𝑠𝑠𝑠+1ℎ1 2 𝑠𝑠𝑠𝑠Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡� / 𝑡𝑡𝑡𝑡 / 0 4139 / / / // / /𝑏𝑏𝑏𝑏/ − 𝑏𝑏𝑏𝑏/ 𝑧𝑧𝑧𝑧 −/ 𝑧𝑧𝑧𝑧𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 30𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑆𝑆𝑆𝑆 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏/ = 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠Δ𝑥𝑥𝑥𝑥=+1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡(2𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡) 38 = + (1 ) + 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 =2−⋅𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣ρ� 𝑤𝑤𝑤𝑤/ 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 +1 2 <0 ρ 𝑛𝑛𝑛𝑛 ⋅ ρ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡− 𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠⋅ ρ 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+1𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 30 29 𝑠𝑠𝑠𝑠+1 12 𝑠𝑠𝑠𝑠Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠++11 2 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠 �ρ 𝑠𝑠𝑠𝑠 (𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2− ρ� ) 𝑠𝑠𝑠𝑠+1�/2 36 39 / ρ/ / 𝑛𝑛𝑛𝑛 ⋅ ρ/ /− 𝑛𝑛𝑛𝑛 /𝑠𝑠𝑠𝑠 ⋅ ρ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡�𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−1+1 22 𝑠𝑠𝑠𝑠−1ρ�2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠+12 ρ2 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠̂ +1/ =2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1ρΔ𝑡𝑡𝑡𝑡𝑧𝑧𝑧𝑧2� /ℎ−−+𝑐𝑐𝑐𝑐̂𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡⋅𝑣𝑣𝑣𝑣1�𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡/ ≥ 40 + / / / ℎ 𝑏𝑏𝑏𝑏/ −ℎ/ 𝑏𝑏𝑏𝑏 / ℎ 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 𝑆𝑆𝑆𝑆− ℎ30( ) 𝑢𝑢𝑢𝑢/ =𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 / �𝑠𝑠𝑠𝑠+1(𝑏𝑏𝑏𝑏=𝑠𝑠𝑠𝑠 2Δ𝑡𝑡𝑡𝑡( )𝑡𝑡𝑡𝑡)/ 𝑠𝑠𝑠𝑠+10 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 41 39 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 The2 evolution𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡1 ℎ−𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏of trench𝑡𝑡𝑡𝑡 � widthρ� in𝑣𝑣𝑣𝑣 time𝑡𝑡𝑡𝑡 and the 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠++1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 1/ = 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑤𝑤𝑤𝑤=(𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚(𝑠𝑠𝑠𝑠 /)𝑤𝑤𝑤𝑤) 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+1ℎρ� 𝑡𝑡𝑡𝑡+1 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡⋅ ρ 𝑡𝑡𝑡𝑡− ℎ� −𝑡𝑡𝑡𝑡 𝑛𝑛𝑛𝑛 𝑢𝑢𝑢𝑢 ⋅𝑡𝑡𝑡𝑡ρ 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = [(𝑡𝑡𝑡𝑡 ) ( ) ] 30+2 ( 𝑡𝑡𝑡𝑡) 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡/𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠 1=𝑠𝑠𝑠𝑠+1 ⋅𝑣𝑣𝑣𝑣ℎ2Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ρ/<0𝑤𝑤𝑤𝑤 +− ρ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢1𝑠𝑠𝑠𝑠+1 2 /0 36 40 ℎ 𝑏𝑏𝑏𝑏 −ℎ 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / 𝑠𝑠𝑠𝑠 /𝑠𝑠𝑠𝑠+1 2 /𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1/ 2 /𝑠𝑠𝑠𝑠−1 2 Δ𝑡𝑡𝑡𝑡/𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 Δ𝑥𝑥𝑥𝑥( 𝑡𝑡𝑡𝑡) 𝑠𝑠𝑠𝑠+1/ 2 = 𝑡𝑡𝑡𝑡bed 𝑧𝑧𝑧𝑧evolutionρ�𝑠𝑠𝑠𝑠+1/𝑠𝑠𝑠𝑠−2Δ𝑡𝑡𝑡𝑡𝑧𝑧𝑧𝑧( inρ𝑠𝑠𝑠𝑠+1 time𝑐𝑐𝑐𝑐̂ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 +1is𝑠𝑠𝑠𝑠 )𝑠𝑠𝑠𝑠 given/2ρ 𝑡𝑡𝑡𝑡� in− discretised𝑐𝑐𝑐𝑐̂ / � 42 + ℎ� 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 − ℎ� 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 𝑆𝑆𝑆𝑆 − 𝑔𝑔𝑔𝑔�ℎ� ℎ � 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−𝑐𝑐𝑐𝑐 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 ≥ 𝑡𝑡𝑡𝑡/ 0 39 41 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1ℎ 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 −ℎ𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+12 2 𝑡𝑡𝑡𝑡 2𝑡𝑡𝑡𝑡 𝑧𝑧𝑧𝑧 / =𝑣𝑣𝑣𝑣−(𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) 𝑠𝑠𝑠𝑠+1/ 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = [( ) ( ) ] +2 ( ) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ℎ� 𝑡𝑡𝑡𝑡� =𝑡𝑡𝑡𝑡 = 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠ρ� 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠 <0 Δ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 version/ = 1 Δ𝑡𝑡𝑡𝑡by𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠++1 1/equations𝑠𝑠𝑠𝑠/+1+𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 1𝑠𝑠𝑠𝑠 (38)+1𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+12ρ and2𝑣𝑣𝑣𝑣−/ 𝑡𝑡𝑡𝑡ρ (39)𝑠𝑠𝑠𝑠+1 /2 𝑠𝑠𝑠𝑠+1 2 37 40 ℎ 𝑏𝑏𝑏𝑏 −ℎ 𝑏𝑏𝑏𝑏 Δ𝑡𝑡𝑡𝑡 =ℎ� [( 𝑢𝑢𝑢𝑢) (𝑏𝑏𝑏𝑏 ) −] ℎ� +2𝑢𝑢𝑢𝑢Δ𝑥𝑥𝑥𝑥 ( 𝑏𝑏𝑏𝑏 ) 𝑣𝑣𝑣𝑣 − 𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠+1ℎ𝑠𝑠𝑠𝑠 2ρ� 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢ρ 𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐̂ 𝑡𝑡𝑡𝑡 ρ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 /𝑢𝑢𝑢𝑢� <0− 𝑐𝑐𝑐𝑐̂ ≥ � 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 ( ) − 𝑔𝑔𝑔𝑔=�ℎ 𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡 /� −𝑠𝑠𝑠𝑠=+1𝑐𝑐𝑐𝑐𝑧𝑧𝑧𝑧 2Δ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−𝑐𝑐𝑐𝑐 / 𝑡𝑡𝑡𝑡/+ 𝑡𝑡𝑡𝑡 01𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 // 0 42 40 41 (30) 𝑡𝑡𝑡𝑡 respectively./ 𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐� / ℎ 𝑡𝑡𝑡𝑡� 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠ρ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖� 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢 2 ≥ 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 /2 =12�𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 2 / 2𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡=𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1 2 Δ𝑡𝑡𝑡𝑡𝑣𝑣𝑣𝑣 = [−( 𝑣𝑣𝑣𝑣) ( 𝑏𝑏𝑏𝑏 ) ]𝑣𝑣𝑣𝑣 +2𝑑𝑑𝑑𝑑 Δ𝑥𝑥𝑥𝑥 ( ) ρ�𝑡𝑡𝑡𝑡 ρℎ 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡̂ 𝑡𝑡𝑡𝑡 �ρ𝑠𝑠𝑠𝑠+1�𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐− 𝑐𝑐𝑐𝑐̂ 𝑠𝑠𝑠𝑠31 𝑠𝑠𝑠𝑠�𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖+1 𝑢𝑢𝑢𝑢 2 <0 37 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 / / / / ( / 𝑤𝑤𝑤𝑤) / = /=𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 Δ𝑡𝑡𝑡𝑡 2 / 𝑠𝑠𝑠𝑠+𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 +1 𝑠𝑠𝑠𝑠+12−𝑠𝑠𝑠𝑠1/2𝑏𝑏𝑏𝑏<0𝑤𝑤𝑤𝑤10 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1 /2 40 43 𝑣𝑣𝑣𝑣 − 𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 + 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 / 2 / ρ� �𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡/ /2ρℎ𝑐𝑐𝑐𝑐̂ 𝑡𝑡𝑡𝑡 ℎ / ρ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡 �𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖/− 𝑢𝑢𝑢𝑢𝑐𝑐𝑐𝑐̂ �≥ 41 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+𝑡𝑡𝑡𝑡1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡=2 𝑔𝑔𝑔𝑔𝑐𝑐𝑐𝑐�𝑡𝑡𝑡𝑡ℎ 𝑠𝑠𝑠𝑠+1�2𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖/ 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 ≥ / 𝑡𝑡𝑡𝑡/ 0 0 / 0 42 /𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 / 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡/ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚/ 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠/ / / 31 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡/2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ�𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡2 / 41 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐−1� 2 𝑠𝑠𝑠𝑠−1�2 𝑠𝑠𝑠𝑠=𝑠𝑠𝑠𝑠+1−1/ 2== 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1−𝑠𝑠𝑠𝑠+12/𝑏𝑏𝑏𝑏 <0𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 38 + 𝑣𝑣𝑣𝑣 − 𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏/ 𝑣𝑣𝑣𝑣/ ℎ 𝑑𝑑𝑑𝑑/ 𝑏𝑏𝑏𝑏 /𝑐𝑐𝑐𝑐 −ℎ/ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 / ℎ / 𝑏𝑏𝑏𝑏 / 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ 31− ℎ 𝑤𝑤𝑤𝑤𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠2+1𝑢𝑢𝑢𝑢 2 /𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐̂ 𝑠𝑠𝑠𝑠�𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠=2/+1𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡2 2𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠ℎ+1 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖2𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢<0 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 ρ� 𝑡𝑡𝑡𝑡+1 𝑔𝑔𝑔𝑔𝑡𝑡𝑡𝑡ℎ�ρ𝑡𝑡𝑡𝑡ℎ𝑐𝑐𝑐𝑐̂ 𝑠𝑠𝑠𝑠1−1� 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑠𝑠𝑠𝑠2 ρ𝑢𝑢𝑢𝑢 � − ≥𝑐𝑐𝑐𝑐1̂ 𝑠𝑠𝑠𝑠+1/𝑠𝑠𝑠𝑠 �−12/ <02 / 𝑠𝑠𝑠𝑠0+1 2 43 42 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠++1 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠−1 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠−1 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 �𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖/ 𝑢𝑢𝑢𝑢 �𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢0 2≥ −𝑢𝑢𝑢𝑢 � ≥ 41 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡ℎ�+1 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐̂ − ℎ� 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 =𝑐𝑐𝑐𝑐̂ ( ) (1 ) +231 (1 )( ) ℎ 𝑠𝑠𝑠𝑠 / �𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1 2 / ℎ𝑡𝑡𝑡𝑡/= Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠/+1 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 / / ≥ 0 / <0 38 ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 −ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 /𝑠𝑠𝑠𝑠 /𝑠𝑠𝑠𝑠+1 2 /𝑠𝑠𝑠𝑠+1 2 /𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1/ 2Δ𝑡𝑡𝑡𝑡 /𝑠𝑠𝑠𝑠−1 2 /𝑠𝑠𝑠𝑠−1 2 /𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 Δ𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1/−𝑐𝑐𝑐𝑐2=�𝑏𝑏𝑏𝑏� ℎ𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 2� 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 −ℎ+ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 ℎ� 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ − ℎ� 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑐𝑐𝑐𝑐̂ 𝑢𝑢𝑢𝑢푢 ℎ =2 � 𝑡𝑡𝑡𝑡 �𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚+1 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 0<0𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 22𝑠𝑠𝑠𝑠+1/2 <0𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / =𝑡𝑡𝑡𝑡 (38)𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 ⋅𝑣𝑣𝑣𝑣𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐 1/𝑠𝑠𝑠𝑠+1𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖/2𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 ≥ 42 Δ𝑡𝑡𝑡𝑡 = ( ) (1 ) +2 (1 Δ𝑥𝑥𝑥𝑥 )( ) 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡 2= 𝑠𝑠𝑠𝑠 ℎ 𝑠𝑠𝑠𝑠+11𝑠𝑠𝑠𝑠+12ℎ2𝑠𝑠𝑠𝑠−1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢+1≥ 𝑠𝑠𝑠𝑠2−1𝑡𝑡𝑡𝑡2 0 𝑠𝑠𝑠𝑠+1 2 43 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠−1 2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠/+1Δ𝑡𝑡𝑡𝑡−2 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡� 𝑢𝑢𝑢𝑢 �𝑢𝑢𝑢𝑢 �𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡−𝑢𝑢𝑢𝑢𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 � 𝑢𝑢𝑢𝑢 /� ≥ −𝑢𝑢𝑢𝑢 �0 39 42 ℎ 𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐 −ℎ 𝑏𝑏𝑏𝑏Δ𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐 =ℎ ( 𝑏𝑏𝑏𝑏) (1 𝑢𝑢𝑢𝑢 ) 𝑐𝑐𝑐𝑐̂ +2− ℎ (1 𝑏𝑏𝑏𝑏Δ𝑥𝑥𝑥𝑥 )(𝑢𝑢𝑢𝑢 −)𝑣𝑣𝑣𝑣 𝑐𝑐𝑐𝑐̂ −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 ℎ� / � = / (= 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 / 𝑠𝑠𝑠𝑠 +1/ <0/ <0𝑠𝑠𝑠𝑠+1 2 / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 /𝑠𝑠𝑠𝑠+12⋅𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐+1=2𝑠𝑠𝑠𝑠2+1 2 +𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 44 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢푢 � / = 𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡 ℎ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 ≥𝑡𝑡𝑡𝑡 1 <0𝑡𝑡𝑡𝑡 43 = 𝑠𝑠𝑠𝑠( ) (1 𝑠𝑠𝑠𝑠 )𝑡𝑡𝑡𝑡 +2 𝑡𝑡𝑡𝑡(1 𝑠𝑠𝑠𝑠)( ) 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 2Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 1 / 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+10 /2 0 39 42 (31)Δ𝑡𝑡𝑡𝑡− 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠−𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑Δ𝑥𝑥𝑥𝑥𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐� 𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠+1 2−�𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1−1 2 𝑐𝑐𝑐𝑐/𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠12+1 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1−1𝑢𝑢𝑢𝑢 2 ≥/ 𝑠𝑠𝑠𝑠+1 2 / − 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 / =𝑐𝑐𝑐𝑐𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1=𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡2( 𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖)𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 −𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 �𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏2𝑡𝑡𝑡𝑡 � 𝑠𝑠𝑠𝑠−𝑢𝑢𝑢𝑢 �<0≥ 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡+21 𝑐𝑐𝑐𝑐� 𝑡𝑡𝑡𝑡= / 𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡 1�Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠+1 Δ𝑥𝑥𝑥𝑥 <0/𝑠𝑠𝑠𝑠+1𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡2⋅ Δ𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠 �/ 𝑠𝑠𝑠𝑠 = 𝑠𝑠𝑠𝑠 + 2𝑠𝑠𝑠𝑠+1 /2 4443 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 / 𝑧𝑧𝑧𝑧= 𝑡𝑡𝑡𝑡 /Δ𝑡𝑡𝑡𝑡− 𝑧𝑧𝑧𝑧/ 𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐+ 𝑠𝑠𝑠𝑠−11𝑐𝑐𝑐𝑐2𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖/𝑢𝑢𝑢𝑢 1/ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡1𝑢𝑢𝑢𝑢≥ /𝑠𝑠𝑠𝑠−1 2𝑡𝑡𝑡𝑡0 𝑠𝑠𝑠𝑠+1 2 40 − 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 −𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 ( ) ( ) 32 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢2𝑠𝑠𝑠𝑠2𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑠𝑠𝑠𝑠 −1 �2𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 +1−𝑢𝑢𝑢𝑢2 �<0≥ 43 / / / / / / 𝑡𝑡𝑡𝑡 (39)𝑐𝑐𝑐𝑐� 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+12 � /𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡 � / / /� / 0 45 + / = 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 +𝑠𝑠𝑠𝑠+12𝑢𝑢𝑢𝑢 22𝑡𝑡𝑡𝑡 = −𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1/ 2 =𝑠𝑠𝑠𝑠−1Δ𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠Δ𝑡𝑡𝑡𝑡𝑥𝑥𝑥𝑥2𝑠𝑠𝑠𝑠+1/ 2+𝑐𝑐𝑐𝑐Δ�𝑥𝑥𝑥𝑥𝑤𝑤𝑤𝑤1𝑡𝑡𝑡𝑡1 1𝑣𝑣𝑣𝑣=𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−1⋅ 𝑠𝑠𝑠𝑠Δ𝑢𝑢𝑢𝑢+12𝑡𝑡𝑡𝑡/ +2 𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 40 44 2 2 𝑡𝑡𝑡𝑡 ρ� 𝑢𝑢𝑢𝑢 ρ/ 𝑐𝑐𝑐𝑐=̂ 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ρ ��𝑢𝑢𝑢𝑢− 𝑐𝑐𝑐𝑐̂ −𝑢𝑢𝑢𝑢� 𝑡𝑡𝑡𝑡 � ≥ 𝑡𝑡𝑡𝑡 43 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠 +1 2 / 1 0 𝑠𝑠𝑠𝑠−1/𝑠𝑠𝑠𝑠2−1<02 𝑠𝑠𝑠𝑠+10𝑠𝑠𝑠𝑠2+1 2 41 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 1− ℎ( )𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 ( ) ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 −ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡/ 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡2+ 1 /∂𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡ℎ 𝑡𝑡𝑡𝑡 /𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖∂�ℎ𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢�𝑢𝑢𝑢𝑢 −𝑢𝑢𝑢𝑢/ −𝑢𝑢𝑢𝑢�<0≥� 𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠+1 2�/ =𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 /𝑤𝑤𝑤𝑤𝑏𝑏𝑏𝑏 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏=2 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 +/ 𝑆𝑆𝑆𝑆 / 45 44 + ρ� / 𝑢𝑢𝑢𝑢푢=𝑠𝑠𝑠𝑠To+1ρ prevent2𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐̂ 𝑡𝑡𝑡𝑡 � + theΔρ𝑥𝑥𝑥𝑥 trench� =𝑡𝑡𝑡𝑡− 𝑐𝑐𝑐𝑐Δ̂ width𝑡𝑡𝑡𝑡𝑥𝑥𝑥𝑥<02 � 𝑣𝑣𝑣𝑣 from⋅ Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 diverging 2Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠−12𝑠𝑠𝑠𝑠 2 𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �𝑢𝑢𝑢𝑢1𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠−1//∂𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠2−1−𝑢𝑢𝑢𝑢02∂𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠2+1� 2 𝑡𝑡𝑡𝑡 41 46 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 =𝑢𝑢𝑢𝑢 ℎ 𝑠𝑠𝑠𝑠+1𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖=𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡�+𝑢𝑢𝑢𝑢1 ≥ 𝑠𝑠𝑠𝑠−𝑢𝑢𝑢𝑢−1𝑡𝑡𝑡𝑡( 2 ) �𝑠𝑠𝑠𝑠<0+1≥ 2 ( + ) ℎ 𝑏𝑏𝑏𝑏 − ℎ 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1�𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡2+/12unbounded,𝑡𝑡𝑡𝑡+1𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡∂/ℎ𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡= ∂theℎ𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 width+𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 evolution/ � 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 is−𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 /stopped/ � / 44 = ( ) / + + ( ) / + ( ) / + ( ) / ℎ � 𝑠𝑠𝑠𝑠+1 Δ2𝑠𝑠𝑠𝑠𝑥𝑥𝑥𝑥+1 2 <0Δ𝑥𝑥𝑥𝑥 𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 45 𝑔𝑔𝑔𝑔 / 𝑢𝑢𝑢𝑢푢 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 �𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1+𝑆𝑆𝑆𝑆=𝑡𝑡𝑡𝑡/2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡=+ 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 44 (32) 𝑡𝑡𝑡𝑡 ℎ Δwhen𝑥𝑥𝑥𝑥 ℎ the𝑡𝑡𝑡𝑡− +1ℎ trenchΔ𝑥𝑥𝑥𝑥𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 depth≥ has0 decreased up to a 42 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 ∂𝑡𝑡𝑡𝑡2 ∂𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠/ 𝑠𝑠𝑠𝑠−1 𝑏𝑏𝑏𝑏2 𝑠𝑠𝑠𝑠+1 𝑏𝑏𝑏𝑏2𝑏 2 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑏 2 46 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 ℎ =�𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡(+1𝑆𝑆𝑆𝑆�)𝑢𝑢𝑢𝑢⋅ Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 −−𝑢𝑢𝑢𝑢� ℎ𝑢𝑢𝑢𝑢( � ) − �ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 � � 45 𝑠𝑠𝑠𝑠+1 2 / =Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1 Δ𝑥𝑥𝑥𝑥 ∂ℎ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1𝑣𝑣𝑣𝑣 2∂⋅ℎΔ𝑢𝑢𝑢𝑢/+𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 = + 𝑆𝑆𝑆𝑆 �𝑆𝑆𝑆𝑆 � 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡ℎ = 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 +/ <0𝑏𝑏𝑏𝑏 / 44 𝑏𝑏𝑏𝑏 33𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 Δ𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠+1/ 2Δ𝑥𝑥𝑥𝑥 0 𝑣𝑣𝑣𝑣𝑆𝑆𝑆𝑆 ⋅ Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 42 ℎ Δ𝑥𝑥𝑥𝑥 − ℎ Δ𝑥𝑥𝑥𝑥 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∂𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥≥ 45 46 ( ) / = / / tan 𝑠𝑠𝑠𝑠+1/ 2 = 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏∂𝑏ℎ 2 ∂ℎ𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / 𝑐𝑐𝑐𝑐� �𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠+1⋅ Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 += − � ℎ𝑢𝑢𝑢𝑢=𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏+1 2 <0𝑠𝑠𝑠𝑠 (− �)ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �(𝑏𝑏𝑏𝑏𝑏+2� ) 45 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖Δ𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠+1/2𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 /𝑆𝑆𝑆𝑆 / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 Δ𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐 1𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 +∂𝑡𝑡𝑡𝑡≥ =∂𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 2 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 ∂ℎ𝑏𝑏𝑏𝑏 ∂ℎ𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 43 46 𝑆𝑆𝑆𝑆 −𝑔𝑔𝑔𝑔�� 𝑏𝑏𝑏𝑏 � − �𝜃𝜃𝜃𝜃��𝑠𝑠𝑠𝑠+1 2� ℎ 𝑐𝑐𝑐𝑐Δ� 𝑥𝑥𝑥𝑥 / − �ℎ Δ𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏 = / 𝑏𝑏𝑏𝑏 / 𝑏𝑏𝑏𝑏(𝑏 2)0 ( 𝑠𝑠𝑠𝑠 + ) 40 TERRA ET AQUA 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠1+1 𝑡𝑡𝑡𝑡 2𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢⋅∂𝑠𝑠𝑠𝑠 Δ𝑥𝑥𝑥𝑥+1ℎ 𝑆𝑆𝑆𝑆2 ∂−ℎ�𝑢𝑢𝑢𝑢 ℎ𝑢𝑢𝑢𝑢 − �/ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑏𝑏𝑏𝑏𝑏 2� / 45 Δ𝑥𝑥𝑥𝑥 = 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 ∂𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏1 +∂𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 = 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 43 = ( ) / 2 ( ) / + / 34 / Δ𝑡𝑡𝑡𝑡 1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 46 / ℎ 𝑠𝑠𝑠𝑠−1/Δ2𝑥𝑥𝑥𝑥 = − ℎ Δ𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠−1(∂𝑡𝑡𝑡𝑡/ 2𝑏𝑏𝑏𝑏) ∂𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1/ 2 ( 0+𝑏𝑏𝑏𝑏𝑏 )2 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ∂2ℎ �𝑢𝑢𝑢𝑢∂ℎ𝑢𝑢𝑢𝑢𝑆𝑆𝑆𝑆 ⋅−𝑢𝑢𝑢𝑢 Δ𝑥𝑥𝑥𝑥/ − ��ℎ𝑢𝑢𝑢𝑢<0≥ − �ℎ/ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑏𝑏𝑏𝑏𝑏 2� 46 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 = 𝑡𝑡𝑡𝑡 / = 𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡( / ) (𝑡𝑡𝑡𝑡 + ) 𝑓𝑓𝑓𝑓 ∗ 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 𝑏𝑏𝑏𝑏/ 𝑡𝑡𝑡𝑡+�1 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏1 Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡12𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 / / �𝑆𝑆𝑆𝑆 �𝑠𝑠𝑠𝑠+1 2 − 𝑢𝑢𝑢𝑢 � ⋅ 𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 � ℎ Δ𝑥𝑥𝑥𝑥 − ℎ 𝑠𝑠𝑠𝑠−1Δ𝑡𝑡𝑡𝑡𝑥𝑥𝑥𝑥2 𝑏𝑏𝑏𝑏 ∂𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1∂𝑥𝑥𝑥𝑥 2 𝑏𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠2+1 2 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏+1 2 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏 ⋅𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 Δ𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏 �−𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡−1� ℎ𝑢𝑢𝑢𝑢2 𝑡𝑡𝑡𝑡−𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1−2��ℎ𝑡𝑡𝑡𝑡<0≥𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑏𝑏𝑏𝑏𝑏 2� 46 / 𝑠𝑠𝑠𝑠35+1 2ℎ Δ𝑢𝑢𝑢𝑢𝑥𝑥𝑥𝑥 / − ℎ=Δ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑥𝑥𝑥𝑥 �𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 / ( −𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 ) / �𝑏𝑏𝑏𝑏𝑏 (2 + ) 𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢푢 𝑡𝑡𝑡𝑡+1 Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+�1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑆𝑆𝑆𝑆 ⋅ Δ𝑥𝑥𝑥𝑥 − � ℎ𝑢𝑢𝑢𝑢/ − �ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣/ � 𝑏𝑏𝑏𝑏𝑏 2� ( ) / = ( ) / / / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = +𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 44 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏Δ𝑡𝑡𝑡𝑡2 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1 2 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 𝑠𝑠𝑠𝑠+1 2 ℎ Δ𝑥𝑥𝑥𝑥 − ℎ Δ𝑥𝑥𝑥𝑥 � � 𝑠𝑠𝑠𝑠 �ρ − ρ� � 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡+1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏 ⋅𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏 − � −𝑢𝑢𝑢𝑢ℎ𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏𝑏 2 − �ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑏𝑏𝑏𝑏𝑏 2� 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1 2 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 = Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 + 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⋅ Δ𝑡𝑡𝑡𝑡 44 ρ� 𝑠𝑠𝑠𝑠+1 2 1 ( ) 36 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 ( ) = ( ) 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 45 / / Δ𝑥𝑥𝑥𝑥 +Δ𝑥𝑥𝑥𝑥 =𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 2 2 / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ρ𝑠𝑠𝑠𝑠 − ρ𝑤𝑤𝑤𝑤 45 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 − 𝑔𝑔𝑔𝑔�ℎ�𝑠𝑠𝑠𝑠+1 2� 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠+1 −𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 ∂ℎ+ ∂ℎ𝑢𝑢𝑢𝑢= ρ� 𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆 1 37 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 46 ( ) = = ∂ℎ ∂ℎ𝑢𝑢𝑢𝑢 ( ) ( + ) / / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 / / 2 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡∂𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡∂𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 46 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑏𝑏𝑏𝑏 − 𝑏𝑏𝑏𝑏 ℎ Δ𝑥𝑥𝑥𝑥 − ℎ Δ𝑥𝑥𝑥𝑥 = ( ) ( + 𝑠𝑠𝑠𝑠) 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔�ℎ� � 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏 ⋅ Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 − � ℎ𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏𝑏/ 2 − �ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑏𝑏𝑏𝑏𝑏/ 2� Δ𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 38Δ𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 =2 ℎ Δ𝑥𝑥𝑥𝑥 − ℎ Δ𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏 2 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 ⋅ Δ𝑥𝑥𝑥𝑥 − � ℎ𝑢𝑢𝑢𝑢 − �ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑏𝑏𝑏𝑏𝑏 2� 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 − 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 Δ𝑡𝑡𝑡𝑡 ⋅𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Δ𝑡𝑡𝑡𝑡 39 = ( ) 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠 − 𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 / = Δ𝑡𝑡𝑡𝑡 / + 1 / 40 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ρ�𝑠𝑠𝑠𝑠+1 2 ρ𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐̂𝑠𝑠𝑠𝑠+1 2 ρ𝑤𝑤𝑤𝑤� − 𝑐𝑐𝑐𝑐̂𝑠𝑠𝑠𝑠+1 2� / 0 41 = / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 <0 𝑡𝑡𝑡𝑡 ℎ𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1/2 ≥ ℎ�𝑠𝑠𝑠𝑠+1 2 � 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ℎ𝑠𝑠𝑠𝑠+1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1 2 / 0 42 = / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 <0 𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1/2 ≥ 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐� � 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 2 𝑐𝑐𝑐𝑐 1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 43 0 / 2 / / / = 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠−1 2 1 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �𝑢𝑢𝑢𝑢 −𝑢𝑢𝑢𝑢 �<0≥ 𝑠𝑠𝑠𝑠+1 2 / 2 / / 𝑢𝑢𝑢𝑢푢 � 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1 2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠−1 2 −𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1 2� = + 44 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 ⋅ Δ𝑡𝑡𝑡𝑡 45 + = ∂ℎ ∂ℎ𝑢𝑢𝑢𝑢 𝑆𝑆𝑆𝑆 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 46 = ( ) ( + ) 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / / ℎ𝑏𝑏𝑏𝑏 Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 − ℎ𝑏𝑏𝑏𝑏 Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏 ⋅ Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 − � ℎ𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏𝑏 2 − �ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠 �𝑏𝑏𝑏𝑏𝑏 2� Δ𝑡𝑡𝑡𝑡 / / / / / / ( ) ( ) 32 + 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 ℎ� 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 1− ℎ(� )𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 ( ) ℎ𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1 𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠+1 −ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠 + Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 2 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = ( ℎ) 𝑏𝑏𝑏𝑏+ − ℎ 𝑏𝑏𝑏𝑏+ ( ) + ( ) + ( ) 𝑔𝑔𝑔𝑔 / / / / / 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 �𝑆𝑆𝑆𝑆𝑓𝑓𝑓𝑓�𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 33 ( ) = tan / / / / / /𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡/ / // ( ) ( ) 32 𝑡𝑡𝑡𝑡 + 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+𝑡𝑡𝑡𝑡1 𝑡𝑡𝑡𝑡+𝑡𝑡𝑡𝑡1 𝑡𝑡𝑡𝑡+𝑧𝑧𝑧𝑧1𝑠𝑠𝑠𝑠+1 −𝑡𝑡𝑡𝑡𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 −𝑔𝑔𝑔𝑔ℎ�𝑠𝑠𝑠𝑠+1��𝑠𝑠𝑠𝑠+12𝑏𝑏𝑏𝑏2𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+12𝑢𝑢𝑢𝑢2𝑠𝑠𝑠𝑠+1� 2 − ℎ�𝑠𝑠𝑠𝑠+1 −2𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠(+1�2𝜃𝜃𝜃𝜃�𝑢𝑢𝑢𝑢�)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1 22� ℎ𝑠𝑠𝑠𝑠+1( 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠)+1 𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠+1 −ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠 32 / / / / / / 1Δ𝑥𝑥𝑥𝑥( ) ( ) 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 =𝑡𝑡𝑡𝑡 ( )𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡2 +( + ) + 34 / / Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡/ 𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / 2 𝑡𝑡𝑡𝑡 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 Δ𝑥𝑥𝑥𝑥 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 22 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 1− ℎ( )𝑏𝑏𝑏𝑏2 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 ( ) ℎ𝑡𝑡𝑡𝑡ℎ𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢푢 +1𝑡𝑡𝑡𝑡 − ℎ−ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢푢 𝑓𝑓𝑓𝑓 ∗ 𝑠𝑠𝑠𝑠+1 2 = (𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1) 2 +𝑠𝑠𝑠𝑠+1 2 + ( ) + ( ) + ( ) �𝑆𝑆𝑆𝑆 �+𝑠𝑠𝑠𝑠+1 2 − 𝑢𝑢𝑢𝑢 � ⋅ 𝑑𝑑𝑑𝑑 𝑔𝑔𝑔𝑔 / 𝑏𝑏𝑏𝑏 � / / / / Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 35 𝑡𝑡𝑡𝑡 ( ) = ( ℎ ) 𝑏𝑏𝑏𝑏 − ℎ 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 /= ( ) / /+ / 𝑆𝑆𝑆𝑆+/ ( ) /�𝑆𝑆𝑆𝑆+�𝑡𝑡𝑡𝑡( ) / 𝑆𝑆𝑆𝑆+ ( ) / 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔 / 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 𝑠𝑠𝑠𝑠+1 2 33 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 �ρ 𝑡𝑡𝑡𝑡 − ρ� 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 (𝑠𝑠𝑠𝑠+1 2 ) 𝑠𝑠𝑠𝑠+1/ 2 = 𝑡𝑡𝑡𝑡 / / tan 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑢𝑢𝑢𝑢�𝑆𝑆𝑆𝑆𝑓𝑓𝑓𝑓�𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 2 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 / 𝑡𝑡𝑡𝑡 ρ� 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 1 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 ( 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠)+1 2 𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 36 33 (( ) ) = 𝑆𝑆𝑆𝑆 ( −𝑔𝑔𝑔𝑔) �� tan𝑏𝑏𝑏𝑏 � − �𝜃𝜃𝜃𝜃��𝑠𝑠𝑠𝑠+1 2� / =/ // / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠+1 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠( /)𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡2 ( ) + 34 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑧𝑧𝑧𝑧 / 𝑡𝑡𝑡𝑡− 𝑧𝑧𝑧𝑧 / / / / / / / / / ( ) 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12( 2 ) �𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1+1 22 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠2+1 32𝑠𝑠𝑠𝑠 ρ − ρ � 𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 − −𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔��ℎ� 𝑏𝑏𝑏𝑏� 𝑐𝑐𝑐𝑐 �𝑡𝑡𝑡𝑡−𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡−2 𝑡𝑡𝑡𝑡 �𝜃𝜃𝜃𝜃� � 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 + 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠+1 2Δ𝑥𝑥𝑥𝑥 ρ� ∗ 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 34 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = 1( ) /�𝑆𝑆𝑆𝑆2� ( ) −/ 𝑢𝑢𝑢𝑢+ /� ⋅ 𝑑𝑑𝑑𝑑 𝑏𝑏𝑏𝑏 � 37 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 1− ℎ( )𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 ( ) ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 −ℎ 𝑏𝑏𝑏𝑏( 𝑢𝑢𝑢𝑢푢 ) / = 35 𝑡𝑡𝑡𝑡 / 2 𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / + 2 ( ) 2 = ( ) 2Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 �𝑆𝑆𝑆𝑆𝑓𝑓𝑓𝑓�𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 − 𝑢𝑢𝑢𝑢∗ 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2� /⋅𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑑𝑑𝑑𝑑+1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡−𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 2 /𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1 2�/ / 𝑡𝑡𝑡𝑡 ℎ𝑠𝑠𝑠𝑠+1 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1 − ℎ𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 �ρ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 −/ρ�𝑠𝑠𝑠𝑠+1 2� 35 = ( ) + + ( ) + ( ) 𝑆𝑆𝑆𝑆 + ( ) 𝑔𝑔𝑔𝑔�ℎ𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1� 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 / 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔 / / / ( ) / = ( /) 𝑆𝑆𝑆𝑆 Δ𝑥𝑥𝑥𝑥𝑣𝑣𝑣𝑣 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 38 𝑡𝑡𝑡𝑡 / / / / 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 =2 1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 𝑠𝑠𝑠𝑠+1 2 ( ρ� ) 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑓𝑓𝑓𝑓 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠+11 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 �ρ − ρ� � 36 𝑆𝑆𝑆𝑆 �𝑆𝑆𝑆𝑆 �𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2 (𝑠𝑠𝑠𝑠+1) 2 𝑠𝑠𝑠𝑠+1=2 𝑡𝑡𝑡𝑡 ( ) 𝑆𝑆𝑆𝑆 𝑏𝑏𝑏𝑏 𝑣𝑣𝑣𝑣 − 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏/ 2 / 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚33 ρ� 𝑠𝑠𝑠𝑠+1 2 2 𝑠𝑠𝑠𝑠 / 𝑤𝑤𝑤𝑤 / / ( /) =/ / / ( tan) ( ) 1 ⋅𝑣𝑣𝑣𝑣𝑡𝑡𝑡𝑡 32 ( 𝑡𝑡𝑡𝑡 ) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ρ − ρ 36 / / /+ 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ( / ) = Δ𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1( 2 − )𝑔𝑔𝑔𝑔�ℎ�𝑠𝑠𝑠𝑠+1 2� 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠+1 −𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 39 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡 (𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 )2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡( 𝑡𝑡𝑡𝑡)/2 = (/ )32 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1/2 𝑠𝑠𝑠𝑠+1/2 𝑠𝑠𝑠𝑠+1/2 𝑠𝑠𝑠𝑠+1/2 𝑠𝑠𝑠𝑠+1/2 𝑠𝑠𝑠𝑠+1/2 𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠−+1𝑧𝑧𝑧𝑧 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+1 2 𝑡𝑡𝑡𝑡 2 ρ� ℎ� 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠−+1ℎ(�2 )𝑏𝑏𝑏𝑏 �𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢2( 𝑠𝑠𝑠𝑠+1)+2 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 ( −ℎ(�)𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 )2𝑢𝑢𝑢𝑢푢𝑡𝑡𝑡𝑡 ( )( ) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 32𝑡𝑡𝑡𝑡 32 𝑠𝑠𝑠𝑠1 / 𝑤𝑤𝑤𝑤 37 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+/1 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡/ +/1 1 / /𝑡𝑡𝑡𝑡 /−𝑔𝑔𝑔𝑔𝑡𝑡𝑡𝑡�/ 𝑡𝑡𝑡𝑡/ 𝑏𝑏𝑏𝑏/ / �/𝑡𝑡𝑡𝑡 /𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡− 2 �𝜃𝜃𝜃𝜃�𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 (𝑡𝑡𝑡𝑡 ) ρ − ρ 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 +Δ𝑥𝑥𝑥𝑥+ 𝑠𝑠𝑠𝑠+1 2𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠/ = 𝑡𝑡𝑡𝑡 / � 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+𝑡𝑡𝑡𝑡1+1+𝑡𝑡𝑡𝑡+𝑡𝑡𝑡𝑡1+�1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 − 𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔�ℎ 𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 34 −𝑐𝑐𝑐𝑐 2𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 12Δ𝑡𝑡𝑡𝑡− ℎ( 𝑡𝑡𝑡𝑡 )=𝑏𝑏𝑏𝑏2 𝑡𝑡𝑡𝑡( 𝑢𝑢𝑢𝑢)( 𝑡𝑡𝑡𝑡/)22ℎ𝑡𝑡𝑡𝑡 ( 𝑏𝑏𝑏𝑏 ) 𝑢𝑢𝑢𝑢푢 / +Δ𝑥𝑥𝑥𝑥−ℎ/ 2𝑏𝑏𝑏𝑏 2𝑢𝑢𝑢𝑢푢 2 2 𝑣𝑣𝑣𝑣 ρ� 2 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 �𝑠𝑠𝑠𝑠+1�2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠/+12�𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1�2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 2𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12𝑠𝑠𝑠𝑠 2𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1+1 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠= 𝑠𝑠𝑠𝑠 Δ𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 1+ 1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 − 𝑏𝑏𝑏𝑏 40 37 ℎ ℎ 𝑏𝑏𝑏𝑏 +𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢ℎ1− ℎ(1−𝑏𝑏𝑏𝑏 ℎ( )𝑏𝑏𝑏𝑏2− 𝑡𝑡𝑡𝑡)𝑏𝑏𝑏𝑏ℎ 𝑢𝑢𝑢𝑢 (𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢 )( ℎ)𝑡𝑡𝑡𝑡 ℎ𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢푢 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢푢 −ℎ−ℎ𝑏𝑏𝑏𝑏/ 𝑢𝑢𝑢𝑢푢𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 / 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1/ 2 �𝑠𝑠𝑠𝑠+1 2 = 2(Δ𝑡𝑡𝑡𝑡 )𝑡𝑡𝑡𝑡 /2 +𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 +𝑡𝑡𝑡𝑡( ) / +Δ𝑥𝑥𝑥𝑥( ) / + (( )) // = / 𝑆𝑆𝑆𝑆 𝑔𝑔𝑔𝑔�ℎ � 𝑓𝑓𝑓𝑓𝑔𝑔𝑔𝑔+𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1+2 𝑠𝑠𝑠𝑠+1 ∗ 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 /2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 38 �𝑆𝑆𝑆𝑆 � ℎ2Δ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2Δ𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡−Δ𝑥𝑥𝑥𝑥𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡2− 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡ℎ2 𝑡𝑡𝑡𝑡�𝑏𝑏𝑏𝑏 ⋅𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡� Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡 2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 𝑠𝑠𝑠𝑠 = ( ) / 𝑠𝑠𝑠𝑠++1 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1+ 𝑠𝑠𝑠𝑠( 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) 𝑠𝑠𝑠𝑠/ + ( ) /ρ� + ( ) ρ 𝑐𝑐𝑐𝑐/̂ ρ � −35𝑐𝑐𝑐𝑐̂ 𝑏𝑏𝑏𝑏 �− 𝑏𝑏𝑏𝑏 =2 𝑔𝑔𝑔𝑔𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 ℎ2 ℎ 𝑓𝑓𝑓𝑓𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏−/2 ℎ− 𝑠𝑠𝑠𝑠ℎ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2/ 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+122 � 𝑠𝑠𝑠𝑠+1 2 0 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 41 ( ) 𝑆𝑆𝑆𝑆 = (==(( ) Δ𝑥𝑥𝑥𝑥)�)𝑆𝑆𝑆𝑆 +�𝑡𝑡𝑡𝑡 + 𝑆𝑆𝑆𝑆 + ( + ( ) )𝑆𝑆𝑆𝑆 + ( +)( )𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆+ ( + )( ) 𝑔𝑔𝑔𝑔�ℎ /� 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / 𝑡𝑡𝑡𝑡𝑔𝑔𝑔𝑔 𝑔𝑔𝑔𝑔 / / / / / / 𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡/𝑡𝑡𝑡𝑡 / =𝑡𝑡𝑡𝑡/ / / 33 𝑏𝑏𝑏𝑏Δ𝑥𝑥𝑥𝑥 − 𝑏𝑏𝑏𝑏 38 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑓𝑓𝑓𝑓 Δ𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠2𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 /𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 2 /𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ( ) 𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆 / = 𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆/𝑡𝑡𝑡𝑡 �𝑠𝑠𝑠𝑠+1/𝑡𝑡𝑡𝑡2 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 �ρ tan𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡− ρ� 𝑡𝑡𝑡𝑡 � 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 =2 𝑠𝑠𝑠𝑠+1/2 <0 ⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1𝑠𝑠𝑠𝑠+12𝑠𝑠𝑠𝑠+12 2 𝑠𝑠𝑠𝑠𝑓𝑓𝑓𝑓+1 2𝑓𝑓𝑓𝑓𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠+1 2 /𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1𝑡𝑡𝑡𝑡 2𝑠𝑠𝑠𝑠+1 2 ℎ𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠+11𝑤𝑤𝑤𝑤 2𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖2 𝑢𝑢𝑢𝑢 33 ≥ Δ𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 𝑣𝑣𝑣𝑣𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡�𝑢𝑢𝑢𝑢𝑆𝑆𝑆𝑆 ��𝑠𝑠𝑠𝑠𝑆𝑆𝑆𝑆+1𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏+1�2𝑠𝑠𝑠𝑠+1 2𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 �𝑠𝑠𝑠𝑠𝑆𝑆𝑆𝑆+1 2 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 39 ( ) / = / / 𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 tanρ� 𝑠𝑠𝑠𝑠+1 2 ℎ � 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1 − 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠+1 2 33 33 = ( ) 𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 −𝑔𝑔𝑔𝑔��𝑠𝑠𝑠𝑠1+1 2𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1 2 � 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 − ( �𝜃𝜃𝜃𝜃��𝑠𝑠𝑠𝑠+1)/2� ℎ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 36𝑤𝑤𝑤𝑤 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 (𝑡𝑡𝑡𝑡 ( ) )/ =/𝑡𝑡𝑡𝑡 = 𝑡𝑡𝑡𝑡 / / 𝑠𝑠𝑠𝑠+1/ / 𝑠𝑠𝑠𝑠 tan𝑡𝑡𝑡𝑡 tan ⋅𝑣𝑣𝑣𝑣/ 0 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 42 ( ) / = / 𝑧𝑧𝑧𝑧( Δ𝑥𝑥𝑥𝑥− 𝑡𝑡𝑡𝑡𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡) 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / / Δ𝑡𝑡𝑡𝑡 𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 sin( /2) 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 = 𝑠𝑠𝑠𝑠(+1 2) 𝑠𝑠𝑠𝑠+1 22 ( ) 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1+ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = 34 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 39 47 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 −𝑔𝑔𝑔𝑔𝑡𝑡𝑡𝑡 �� 2 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 / − 𝑠𝑠𝑠𝑠�𝜃𝜃𝜃𝜃�/�/ 𝑤𝑤𝑤𝑤 � / 𝑡𝑡𝑡𝑡 =𝑡𝑡𝑡𝑡( ) = 10 = 10 cot( ) 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠/𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1 2𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+12 2𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+1𝑧𝑧𝑧𝑧2 𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡− 𝑧𝑧𝑧𝑧 − 𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡+ 1 𝑡𝑡𝑡𝑡 / <0 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡 � Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 ρ 𝑡𝑡𝑡𝑡− ρ � ��𝑠𝑠𝑠𝑠�𝑡𝑡𝑡𝑡+1��2𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 = Δ𝑡𝑡𝑡𝑡 + 1 sin 40 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠+1𝑆𝑆𝑆𝑆 2 = (−𝑔𝑔𝑔𝑔2)�−𝑔𝑔𝑔𝑔�𝑠𝑠𝑠𝑠+1� 2𝑏𝑏𝑏𝑏 (𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1�) �𝑠𝑠𝑠𝑠 + 𝑡𝑡𝑡𝑡− − 𝜃𝜃𝜃𝜃 𝜃𝜃𝜃𝜃 � �𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 34 ≥/𝑡𝑡𝑡𝑡 / / sin( /2) 𝑆𝑆𝑆𝑆𝑓𝑓𝑓𝑓 / − 𝑔𝑔𝑔𝑔∗ �𝑠𝑠𝑠𝑠ℎ+1/2 � 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠−𝑐𝑐𝑐𝑐+1/Δ𝑥𝑥𝑥𝑥2 Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1/22 𝑠𝑠𝑠𝑠+1 2 𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡 − 𝑧𝑧𝑧𝑧 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 ϕ − π 0 47 �𝑆𝑆𝑆𝑆 �𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 − 𝑢𝑢𝑢𝑢= =( )(� ⋅) 𝑑𝑑𝑑𝑑2 (2 () 𝑏𝑏𝑏𝑏)ρ� + �+ 𝑐𝑐𝑐𝑐� � 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 𝑠𝑠𝑠𝑠34 34 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 − =⋅𝑘𝑘𝑘𝑘𝑡𝑡𝑡𝑡10⋅Δ⋅ − =⋅𝑘𝑘𝑘𝑘 10⋅Δ⋅ ϕ cot( ) / 2/ 𝑡𝑡𝑡𝑡1 / / 𝑡𝑡𝑡𝑡 /𝑡𝑡𝑡𝑡 / / / 𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣37 sin 𝑓𝑓𝑓𝑓 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ∗ 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 / 1 35𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 ϕ43 ( �)𝑆𝑆𝑆𝑆 �𝑠𝑠𝑠𝑠+1(2( ) −)/𝑢𝑢𝑢𝑢 = 2 �𝑡𝑡𝑡𝑡 2⋅ 𝑡𝑡𝑡𝑡/𝑑𝑑𝑑𝑑 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / = Δ𝑡𝑡𝑡𝑡 / + ρ� 1 ρ/ 𝑐𝑐𝑐𝑐̂ 0 ρ � − 𝑐𝑐𝑐𝑐̂ 0� ϕ −40π / =𝑓𝑓𝑓𝑓 𝑓𝑓𝑓𝑓 / 2 ∗ /∗ / 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / / / 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 / 0 0 41 𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠+1 22 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡2 35 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅ − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅ ϕ 48 �𝑆𝑆𝑆𝑆 ��𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡� − 𝑢𝑢𝑢𝑢− 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 � ⋅ �𝑑𝑑𝑑𝑑⋅𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1=/𝑏𝑏𝑏𝑏2 � � / = 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ϕ ( )𝑡𝑡𝑡𝑡 = (𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1)𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡2 𝑏𝑏𝑏𝑏�ρ − 𝑏𝑏𝑏𝑏 − ρ�/ � 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 <0= 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1/2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1/2 𝑠𝑠𝑠𝑠+1��/2 𝑠𝑠𝑠𝑠+1� /2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠−1ρ�/ 2 / ρ 𝑐𝑐𝑐𝑐̂ 1 𝑠𝑠𝑠𝑠−1ρ 2� −35𝑠𝑠𝑠𝑠𝑐𝑐𝑐𝑐+1̂ 352 �𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1/2 160 (1 3 ) 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 𝑣𝑣𝑣𝑣 𝑢𝑢𝑢𝑢𝑔𝑔𝑔𝑔 ℎ 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �𝑢𝑢𝑢𝑢 −𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 0�ℎ≥ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 ≥ 0 (𝑡𝑡𝑡𝑡 ( ) )/ =/ (𝑡𝑡𝑡𝑡= ( ) 𝑡𝑡𝑡𝑡 )/ / 𝑡𝑡𝑡𝑡 / / Δ𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡/𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡/𝑠𝑠𝑠𝑠+1/2𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / / 38𝑠𝑠𝑠𝑠+1 2/ / <0𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔 2 41𝑛𝑛𝑛𝑛 48 �ρ ρ� −𝑠𝑠𝑠𝑠+1ρ� 2 � = 2 ℎ� � 0 = 15 2 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡1𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡=22 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡 (𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢푢 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡)𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡/�𝑠𝑠𝑠𝑠+1/𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠+1 2/ 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 36 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 2𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷 0 3 𝑆𝑆𝑆𝑆 ( ) 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡+1 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 (𝑏𝑏𝑏𝑏 ) �ρ𝑠𝑠𝑠𝑠+1�ρ2 − ρ� − ρ� � � 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1/2 <0ℎ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 0 ⋅ν160 −𝑛𝑛𝑛𝑛(1 ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠/𝑠𝑠𝑠𝑠certain𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1=2𝑠𝑠𝑠𝑠+1 2 threshold𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1/𝑠𝑠𝑠𝑠 2𝑠𝑠𝑠𝑠 +1value.𝑠𝑠𝑠𝑠+12 2𝑠𝑠𝑠𝑠 This+1𝑠𝑠𝑠𝑠+12 threshold2𝑠𝑠𝑠𝑠+1ρ� 2 𝑡𝑡𝑡𝑡 value𝑡𝑡𝑡𝑡 is𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡 2 is used in equation𝑠𝑠𝑠𝑠−1 2 (45).𝑠𝑠𝑠𝑠+1 This2 is a simplified/ g(ρm − ρw)/ρw where2 0ρm is mixture density42 and ρw 49 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆 12 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏( ) 𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ �𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖−𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 ≥� =0 𝑔𝑔𝑔𝑔 𝑛𝑛𝑛𝑛 𝑏𝑏𝑏𝑏 − 𝑏𝑏𝑏𝑏 2 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠 / 𝑤𝑤𝑤𝑤𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠+1 2 �𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 36 𝑡𝑡𝑡𝑡 / = 15 2 ( )𝑡𝑡𝑡𝑡 set= at 0.05 metres.𝑡𝑡𝑡𝑡 ( Furthermore,𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡) ρ the− ρ mixtureρ� ρ� ℎ version=� 𝑠𝑠𝑠𝑠 +1of+ equation 𝑠𝑠𝑠𝑠 +1(18),2 with𝑡𝑡𝑡𝑡 all the source 𝑡𝑡𝑡𝑡 <0 𝑘𝑘𝑘𝑘water(1 density.44 𝐷𝐷𝐷𝐷 ) 0 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1/2 1𝑠𝑠𝑠𝑠+1/12 𝑠𝑠𝑠𝑠+1⋅𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 ( ( ) ) ℎ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 36 36 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1/2 ⋅ν −𝑛𝑛𝑛𝑛 𝑆𝑆𝑆𝑆 ( )( −) 2 𝑔𝑔𝑔𝑔�Δ𝑡𝑡𝑡𝑡ℎ� �2 𝑐𝑐𝑐𝑐 (−𝑐𝑐𝑐𝑐( ) ) 𝑡𝑡𝑡𝑡 0𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 ≥𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 49 𝑡𝑡𝑡𝑡 / =/ =𝑡𝑡𝑡𝑡 /𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1/2𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 39𝑠𝑠𝑠𝑠+1 /2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 = 42 density at the2 cell2 = faces( is) given ρρ� by− equationρ 𝑏𝑏𝑏𝑏 terms= included𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 in𝑐𝑐𝑐𝑐 variable� � S. 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠(1 ) 𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 −𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔𝑡𝑡𝑡𝑡+�ℎ1�1𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡+12−𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠/ 𝑤𝑤𝑤𝑤 / 𝑤𝑤𝑤𝑤Δ𝑥𝑥𝑥𝑥 / Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 ⋅ Δ37 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 <0𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠+1 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1 2 ρ =⋅ −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 50 (𝑡𝑡𝑡𝑡 )𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠= 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 +1 ρ2 −ρρ− ρ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1/2 1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 𝑆𝑆𝑆𝑆(40).𝑠𝑠𝑠𝑠𝑆𝑆𝑆𝑆+1 2𝑠𝑠𝑠𝑠+1/𝑧𝑧𝑧𝑧 2− −−𝑔𝑔𝑔𝑔𝑧𝑧𝑧𝑧�ℎ�𝑔𝑔𝑔𝑔𝑠𝑠𝑠𝑠+1�/ℎ�2𝑠𝑠𝑠𝑠+1� 2𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡�𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑐𝑐𝑐𝑐−𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠+1ρ� 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑐𝑐𝑐𝑐 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 ≥ 𝑣𝑣𝑣𝑣 43 12 𝑠𝑠𝑠𝑠2𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 37 𝑡𝑡𝑡𝑡 / / / 045𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ( )𝑡𝑡𝑡𝑡 = 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝑏𝑏𝑏𝑏 − 𝑏𝑏𝑏𝑏 ρ� ρ� 𝑐𝑐𝑐𝑐� � 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 2 2 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸ρ =⋅ −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 50 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1/2 1𝑠𝑠𝑠𝑠+1/12 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 + 𝑐𝑐𝑐𝑐 = =𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 37 37 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣(50) 0α.075⋅ 𝑢𝑢𝑢𝑢 51 𝑆𝑆𝑆𝑆 / = Δ𝑡𝑡𝑡𝑡2 𝑔𝑔𝑔𝑔�/ℎ� + �21 / /1 40 𝑡𝑡𝑡𝑡( )( )/ =/ 𝑡𝑡𝑡𝑡= /𝑠𝑠𝑠𝑠+1 / 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠−1 2 1 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠=+1sin2( /2) 43. / 2 2 𝑏𝑏𝑏𝑏 Δ𝑥𝑥𝑥𝑥− 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 38 𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 0�𝑢𝑢𝑢𝑢 −𝑢𝑢𝑢𝑢 (1+� ≥718 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸) 47 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 2𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+12 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ∂/ℎ ∂ℎ𝑢𝑢𝑢𝑢 / / / = 10/ / <0𝑣𝑣𝑣𝑣 =α ⋅10𝑢𝑢𝑢𝑢 cot( ) 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑔𝑔𝑔𝑔+1�ℎ�2=2 𝑤𝑤𝑤𝑤�𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 𝑆𝑆𝑆𝑆2 2 0.075 51 𝑤𝑤𝑤𝑤𝑡𝑡𝑡𝑡+𝑠𝑠𝑠𝑠1+1𝑤𝑤𝑤𝑤 2𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠�+1 2𝑠𝑠𝑠𝑠+1 𝑏𝑏𝑏𝑏2 𝑏𝑏𝑏𝑏�− 𝑏𝑏𝑏𝑏− 𝑏𝑏𝑏𝑏 / = 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢푢 �𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝐸𝐸𝐸𝐸 sin= 2 4 1 2 (40)ρ� 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆ρ 𝑐𝑐𝑐𝑐̂ 𝑔𝑔𝑔𝑔ρ�ℎ�𝑔𝑔𝑔𝑔 �ℎ�−�Δ𝑥𝑥𝑥𝑥𝑐𝑐𝑐𝑐̂ � 0 (45)∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 1 38 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 α 𝑡𝑡𝑡𝑡 46 . / 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠41−1 𝑠𝑠𝑠𝑠2+1 2 𝑠𝑠𝑠𝑠+1 2 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠𝑠𝑠−1 2 0 𝑠𝑠𝑠𝑠+1 2 ϕ −(1+π⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅718 )0 𝑏𝑏𝑏𝑏 = − 𝑏𝑏𝑏𝑏 =2 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Δ𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡 =𝑢𝑢𝑢𝑢 (𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ) �𝑢𝑢𝑢𝑢 / 𝑢𝑢𝑢𝑢 38−𝑢𝑢𝑢𝑢 (38+ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑣𝑣𝑣𝑣�)<0≥�𝑢𝑢𝑢𝑢 − ⋅𝑘𝑘𝑘𝑘−𝑢𝑢𝑢𝑢⋅Δ⋅ � − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅ ϕ 52 / 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 / / / / 𝐸𝐸𝐸𝐸 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ⋅𝑣𝑣𝑣𝑣=2 =2/ <0 𝑢𝑢𝑢𝑢푢 � 2 = 𝑡𝑡𝑡𝑡 + α ϕ 2 4 1 2 44 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 Δ𝑡𝑡𝑡𝑡−𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠𝑏𝑏𝑏𝑏+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 𝑏𝑏𝑏𝑏2 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 39𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = As stated before,ℎ𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 the𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 variables𝑤𝑤𝑤𝑤𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ ≥ Δnot𝑥𝑥𝑥𝑥 defined− ℎ Δ 𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏 For an𝑏𝑏𝑏𝑏 explicit𝑏𝑏𝑏𝑏 treatment𝑏 2 of the𝑠𝑠𝑠𝑠 source terms (51) 8 ⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ℎ�𝑠𝑠𝑠𝑠+1 2 �𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 −=𝑏𝑏𝑏𝑏(−⋅𝑣𝑣𝑣𝑣𝑏𝑏𝑏𝑏 )𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1⋅ Δ𝑥𝑥𝑥𝑥2 − �𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ𝑢𝑢𝑢𝑢 �𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠−1 −2 �−𝑢𝑢𝑢𝑢ℎ 𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢+𝑠𝑠𝑠𝑠+11 𝑣𝑣𝑣𝑣2� �𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑏 2� 48 52 𝑡𝑡𝑡𝑡+Δ𝑡𝑡𝑡𝑡1 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 = ∗ 𝑖𝑖𝑖𝑖 on the grid ℎare denoted𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 with⋅𝑣𝑣𝑣𝑣 a⋅𝑣𝑣𝑣𝑣 hat andΔ𝑡𝑡𝑡𝑡 are and the= fluxes,+ the39 discretisedΔ 𝑥𝑥𝑥𝑥 expressionΔ𝑥𝑥𝑥𝑥 𝑣𝑣𝑣𝑣 ⋅ Δ of𝑡𝑡𝑡𝑡 � = 443 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = ( )𝑡𝑡𝑡𝑡 / 0 42 160𝑢𝑢𝑢𝑢 ⋅𝑢𝑢𝑢𝑢(1 ) 𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡+1 − 𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡Δ𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 39 39 2 80 / = 𝑣𝑣𝑣𝑣 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 0 𝑔𝑔𝑔𝑔| | 𝑛𝑛𝑛𝑛 determined𝑠𝑠𝑠𝑠 via𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 an upwind= 𝑡𝑡𝑡𝑡(𝑡𝑡𝑡𝑡= approximation( ) ) (see equation𝑏𝑏𝑏𝑏 (45)𝑏𝑏𝑏𝑏 is now𝑠𝑠𝑠𝑠 given by equation (46), The friction15∗ velocity𝑖𝑖𝑖𝑖 2 is included to45 account for 53 =𝑧𝑧𝑧𝑧 Δ𝑡𝑡𝑡𝑡−𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑧𝑧𝑧𝑧+1 𝑡𝑡𝑡𝑡++1 𝑡𝑡𝑡𝑡 1𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1/2 <0 Δ𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥 40𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 + = 𝑘𝑘𝑘𝑘 = 𝐷𝐷𝐷𝐷𝑢𝑢𝑢𝑢 � 10⋅𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡/ 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠 / 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ≥/𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 n+1 ⋅ν −𝑛𝑛𝑛𝑛 equations 41,𝑧𝑧𝑧𝑧 42𝑧𝑧𝑧𝑧 −and𝑧𝑧𝑧𝑧𝑣𝑣𝑣𝑣− 43).𝑧𝑧𝑧𝑧 which can be solved for hb . the𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 friction of the flow at the bed and sidewalls 49 𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠� +1 =2 Δ𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 + 1 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 40 ∂ℎ ∂ℎ𝑢𝑢𝑢𝑢 = 𝑢𝑢𝑢𝑢 |⋅ Δ𝑡𝑡𝑡𝑡 | 53 𝑠𝑠𝑠𝑠+1/2 𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1/2 𝑤𝑤𝑤𝑤𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠2+1𝑣𝑣𝑣𝑣/2 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 (=1 45≤ ) 1 ρ� ρ 𝑐𝑐𝑐𝑐=̂ =Δ𝑡𝑡𝑡𝑡 1ρΔ𝑡𝑡𝑡𝑡�+ −+𝑐𝑐𝑐𝑐̂ 1 1� + = 43 40 40 𝑆𝑆𝑆𝑆 of the 𝑆𝑆𝑆𝑆𝑚𝑚𝑚𝑚trench.𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠− 𝐸𝐸𝐸𝐸 The friction velocity is given by 𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡/ / / / 𝑡𝑡𝑡𝑡 0 / / 0 41 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 | Δ𝑥𝑥𝑥𝑥| 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚| | 46 54 𝑠𝑠𝑠𝑠+1 2 =/ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑤𝑤𝑤𝑤 / 𝑠𝑠𝑠𝑠+1 2 / 𝑣𝑣𝑣𝑣 = equation𝑠𝑠𝑠𝑠 𝑢𝑢𝑢𝑢 =52,𝑠𝑠𝑠𝑠 where⋅ Δ𝑡𝑡𝑡𝑡 is an empirical factor and ρ� / 𝑡𝑡𝑡𝑡 ρ 𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐̂ 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 2ρ 𝑡𝑡𝑡𝑡�(𝑡𝑡𝑡𝑡− 𝑐𝑐𝑐𝑐̂) � 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡( ) ∂ℎ ∂32ℎ𝑢𝑢𝑢𝑢 = ( ) / ( 𝐶𝐶𝐶𝐶ρ+𝐹𝐹𝐹𝐹⋅𝐶𝐶𝐶𝐶 ) −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 ≤f / / / / / = /𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 /2𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2𝑠𝑠𝑠𝑠+1 𝑡𝑡𝑡𝑡 2 𝑤𝑤𝑤𝑤// <0𝑤𝑤𝑤𝑤 0𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡41 𝑡𝑡𝑡𝑡 = / 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 50 𝑡𝑡𝑡𝑡 ρ� ρ� 𝑠𝑠𝑠𝑠+ρ 𝑐𝑐𝑐𝑐̂ ρ1 𝑐𝑐𝑐𝑐̂ 𝑠𝑠𝑠𝑠+1ρ �2ρ −� 𝑐𝑐𝑐𝑐̂ − 𝑐𝑐𝑐𝑐̂ � � 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑆𝑆𝑆𝑆 𝑏𝑏𝑏𝑏 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 | 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥| 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 | | 54 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 /𝑠𝑠𝑠𝑠−1=𝑡𝑡𝑡𝑡2 ℎ 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠−1𝑡𝑡𝑡𝑡 2 2≥ 𝑠𝑠𝑠𝑠 +1𝑡𝑡𝑡𝑡 2𝑡𝑡𝑡𝑡0 𝑡𝑡𝑡𝑡0 2 ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 41 𝑡𝑡𝑡𝑡41 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢u the layer⋅ Δ𝑡𝑡𝑡𝑡 averaged𝑢𝑢𝑢𝑢 flow velocity. A bed friction 𝑡𝑡𝑡𝑡 �𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢 2 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 −𝑢𝑢𝑢𝑢<0/ / �<0≥ ℎ Δ𝑥𝑥𝑥𝑥 − ℎ Δ𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏 2𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 = 𝑠𝑠𝑠𝑠 46 = �𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠�+1𝑠𝑠𝑠𝑠+122 ℎ𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1/ �2𝑠𝑠𝑠𝑠=+1 𝑠𝑠𝑠𝑠=+1 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1+1/𝑠𝑠𝑠𝑠+1/22 𝑠𝑠𝑠𝑠/ 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 = ( ) 𝑆𝑆𝑆𝑆 ⋅ Δ𝑥𝑥𝑥𝑥( +− � )ℎ𝑢𝑢𝑢𝑢 − �ℎ 𝑢𝑢𝑢𝑢 𝐸𝐸𝐸𝐸𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 𝐸𝐸𝐸𝐸�𝑏𝑏𝑏𝑏𝑏 2� 𝑠𝑠𝑠𝑠 ℎ 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢 1−𝑢𝑢𝑢𝑢푢 ℎ( )𝑏𝑏𝑏𝑏(41)�𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 ( /ℎ) /ℎ 𝑡𝑡𝑡𝑡2𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢푢 ≥𝑡𝑡𝑡𝑡 −ℎ𝑡𝑡𝑡𝑡+1𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢푢 +1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / / factor𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡0α .of075𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚⋅ 𝑢𝑢𝑢𝑢f = 0.024𝑣𝑣𝑣𝑣 is𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 used, within the range 5551 𝑠𝑠𝑠𝑠+1 2𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 0𝑡𝑡𝑡𝑡/2𝑠𝑠𝑠𝑠+1<0/2 <0 Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 + ℎ� 𝑡𝑡𝑡𝑡 � 𝑡𝑡𝑡𝑡 / 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 42 𝑡𝑡𝑡𝑡 cosh = . / 𝑡𝑡𝑡𝑡 2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1=2 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+12 𝑡𝑡𝑡𝑡ℎ ℎ 𝑠𝑠𝑠𝑠−1𝑠𝑠𝑠𝑠+1𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖2 2𝑢𝑢𝑢𝑢𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 ℎ𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1≥2Δ𝑥𝑥𝑥𝑥≥ − ℎ Δ𝑥𝑥𝑥𝑥 𝐶𝐶𝐶𝐶(1+𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 718 𝑠𝑠𝑠𝑠 ) 𝑠𝑠𝑠𝑠 2Δ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢/ �𝑠𝑠𝑠𝑠+1�ℎ2𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡 �𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥−𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 � 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏𝑏 2 𝑠𝑠𝑠𝑠 𝑏𝑏𝑏𝑏𝑏 2 mentioned in Garcia (1990). 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 ℎ ℎ𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠� 𝑠𝑠𝑠𝑠+1� 𝑠𝑠𝑠𝑠+1 / <0𝑠𝑠𝑠𝑠+10 2𝑠𝑠𝑠𝑠+1 2 𝑆𝑆𝑆𝑆 ⋅ Δ𝑥𝑥𝑥𝑥 − �(ℎ𝑢𝑢𝑢𝑢42) = − �ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑇𝑇𝑇𝑇 �cosh tanh𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 sinh𝑣𝑣𝑣𝑣 1 + 55 ℎ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡 − ℎ 𝑠𝑠𝑠𝑠 =𝑏𝑏𝑏𝑏 ℎ ℎ+ 𝑠𝑠𝑠𝑠+1𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖/2𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 44 ⎡ �cosh 𝐸𝐸𝐸𝐸 2 4 1 2 2 ⎤ = ( ) + = 𝑐𝑐𝑐𝑐 + ( 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖) 𝑢𝑢𝑢𝑢 + ≥( ) 0 +Δ𝑡𝑡𝑡𝑡0( ) (46) � 𝐶𝐶𝐶𝐶� α 2 𝑔𝑔𝑔𝑔 / 𝑠𝑠𝑠𝑠+1/2 /𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡/ / // / 42𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 42𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sinh 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝑐𝑐𝑐𝑐� 𝑡𝑡𝑡𝑡�+1 𝑡𝑡𝑡𝑡 / <0 ⎢ ⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ⎥ 52 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏/ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1=/ =𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 −(⎢ ) =⋅ � ⋅ ⋅ �𝑇𝑇𝑇𝑇 ��cosh𝑥𝑥𝑥𝑥� − ��tanh𝑥𝑥𝑥𝑥� ⋅ ��sinh𝐶𝐶𝐶𝐶� − � − 1𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥⎥+ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓Δ𝑥𝑥𝑥𝑥 𝑐𝑐𝑐𝑐 Δ𝑥𝑥𝑥𝑥1𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢𝑣𝑣𝑣𝑣 ⋅ Δ 𝑡𝑡𝑡𝑡 ≥ 𝑡𝑡𝑡𝑡 <0<0 𝑤𝑤𝑤𝑤 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇⎡ 𝑇𝑇𝑇𝑇 �� 𝐶𝐶𝐶𝐶� 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 2𝑇𝑇𝑇𝑇 ⎤ 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠+1 2𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠� +1�𝑆𝑆𝑆𝑆2 𝑡𝑡𝑡𝑡�𝑠𝑠𝑠𝑠+1� 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡2 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 𝑠𝑠𝑠𝑠+1𝑆𝑆𝑆𝑆 /2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠+1/22 𝑆𝑆𝑆𝑆 𝑠𝑠𝑠𝑠+1 2 43 ⎢ 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸��sinh𝐶𝐶𝐶𝐶� 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑞𝑞𝑞𝑞 2⎥ 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 / 𝑠𝑠𝑠𝑠+1 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐 𝑠𝑠𝑠𝑠+1/𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 2𝑢𝑢𝑢𝑢𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 ≥/ ≥ 0 ⎢ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 8 ⎥ (42) 𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠� +1𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐2𝑠𝑠𝑠𝑠� +1 2� 𝑡𝑡𝑡𝑡12�𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 Empirical33 equations 45 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥⎣ for− ⎢ model⋅ � ⋅ closure ⋅ � �� 𝑥𝑥𝑥𝑥� − �� 𝑥𝑥𝑥𝑥� ⋅ �� 𝐶𝐶𝐶𝐶� − � − 𝑥𝑥𝑥𝑥 ⎦ 𝑥𝑥𝑥𝑥⎥ ( ) = = 𝑡𝑡𝑡𝑡 + 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠=+1𝑡𝑡𝑡𝑡 tan𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠+1 2 43 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 =𝑇𝑇𝑇𝑇 +∗ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑖𝑖𝑖𝑖 ( 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ) 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 56 / / / / / 𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐 1 𝑐𝑐𝑐𝑐 𝑡𝑡𝑡𝑡1 /𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 1𝑢𝑢𝑢𝑢𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 / / 0 The last unknowns in the set⎢ of equations�� 𝐶𝐶𝐶𝐶� 𝑢𝑢𝑢𝑢 � ⋅𝑢𝑢𝑢𝑢 ⎥ 𝑠𝑠𝑠𝑠−1 2 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 43 43 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 4 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = 𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖/𝑠𝑠𝑠𝑠 +1 � 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠 −𝑢𝑢𝑢𝑢/ / �<0≥/ / 0 0 ⎣ 2 ⎦ 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1/2 𝑡𝑡𝑡𝑡 / ∂ℎ 𝑧𝑧𝑧𝑧 ∂ℎ𝑢𝑢𝑢𝑢− 𝑧𝑧𝑧𝑧𝑡𝑡𝑡𝑡 / 𝑡𝑡𝑡𝑡 / 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 | 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 | π 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 53 56 𝑠𝑠𝑠𝑠+1𝑢𝑢𝑢𝑢푢 2 ��𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠−122 𝑠𝑠𝑠𝑠+1 2 12 𝑠𝑠𝑠𝑠−12 2 2 � �𝑠𝑠𝑠𝑠�+1𝑠𝑠𝑠𝑠+12 2 are the active wall velocity vwall, sedimentation𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞 = =(52)𝑐𝑐𝑐𝑐 +⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ 1ρ −ρ( ⋅𝑔𝑔𝑔𝑔 ) 𝑆𝑆𝑆𝑆 𝑡𝑡𝑡𝑡 −𝑔𝑔𝑔𝑔�/𝑢𝑢𝑢𝑢 =𝑡𝑡𝑡𝑡/ 𝑏𝑏𝑏𝑏= 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡� 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆− 𝑡𝑡𝑡𝑡−𝑢𝑢𝑢𝑢𝜃𝜃𝜃𝜃𝑡𝑡𝑡𝑡 �𝑡𝑡𝑡𝑡<0�≥ 𝑡𝑡𝑡𝑡 4 𝑠𝑠𝑠𝑠+1/2𝑠𝑠𝑠𝑠−1∂𝑡𝑡𝑡𝑡2𝑠𝑠𝑠𝑠−1 2∂𝑥𝑥𝑥𝑥Δ𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠−11/2 1𝑠𝑠𝑠𝑠−1 𝑠𝑠𝑠𝑠2−1+1/2 𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠+1 2 46 2 𝑠𝑠𝑠𝑠+1 2 =𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡(𝑢𝑢𝑢𝑢 ) 𝑢𝑢𝑢𝑢 𝑢𝑢𝑢𝑢2 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖( 2 �)𝑢𝑢𝑢𝑢𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖�+𝑢𝑢𝑢𝑢−𝑢𝑢𝑢𝑢�𝑢𝑢𝑢𝑢 −𝑢𝑢𝑢𝑢�−𝑢𝑢𝑢𝑢� ≥� ≥ velocity vsed34, entrainment velocity vE and friction 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 π 𝑢𝑢𝑢𝑢푢 / � 𝑡𝑡𝑡𝑡 = / / / (𝑡𝑡𝑡𝑡 /) / 𝑡𝑡𝑡𝑡 / / ( +/ )<0/ <0 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡Δ𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡 2 = + 2 2 / / 44 𝑞𝑞𝑞𝑞 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑞𝑞𝑞𝑞 𝑐𝑐𝑐𝑐 ≤⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρ −ρ ⋅𝑔𝑔𝑔𝑔 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑢𝑢𝑢𝑢푢 𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢푢 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠2+1�𝑡𝑡𝑡𝑡2 𝑡𝑡𝑡𝑡� 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠−1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 velocity u . The wall velocity for a vertical wall Boundary𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 and initial conditions 𝑓𝑓𝑓𝑓 𝑢𝑢𝑢𝑢 ∗ 𝑡𝑡𝑡𝑡 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡�𝑢𝑢𝑢𝑢 𝑡𝑡𝑡𝑡 −𝑢𝑢𝑢𝑢 � ∗ | Δ𝑥𝑥𝑥𝑥| | | 54 ℎ Δ𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠+1 2− ℎ Δ𝑥𝑥𝑥𝑥 𝑠𝑠𝑠𝑠+1𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏+𝑠𝑠𝑠𝑠+121 =2𝑠𝑠𝑠𝑠+1𝑏𝑏𝑏𝑏 2𝑡𝑡𝑡𝑡 +𝑠𝑠𝑠𝑠+1 2𝑏𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠−12𝑠𝑠𝑠𝑠+12𝑠𝑠𝑠𝑠−12 2 𝑠𝑠𝑠𝑠+1 2𝑠𝑠𝑠𝑠+1𝑠𝑠𝑠𝑠 2 �𝑆𝑆𝑆𝑆 � (43)− 𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏𝑢𝑢𝑢𝑢𝑆𝑆𝑆𝑆 �⋅𝑢𝑢𝑢𝑢 Δ𝑥𝑥𝑥𝑥⋅ 𝑑𝑑𝑑𝑑−𝑏𝑏𝑏𝑏 �𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖ℎ𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖�𝑢𝑢𝑢𝑢 𝑏𝑏𝑏𝑏�𝑢𝑢𝑢𝑢−−𝑢𝑢𝑢𝑢��ℎ −𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣� �𝑏𝑏𝑏𝑏�𝑏 2� is given by equation44 (47) (Rhee, 2015), where = On the left= (moving) boundary, the flow height, Δ𝑡𝑡𝑡𝑡 Δ𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥 = =𝑣𝑣𝑣𝑣 ⋅+Δ𝑡𝑡𝑡𝑡 + / 35 44 44 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ( ) = ( ) 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 / / 𝑏𝑏𝑏𝑏 /𝑡𝑡𝑡𝑡+1 𝑏𝑏𝑏𝑏𝑡𝑡𝑡𝑡+/1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 k0 is initial permeability, given by equation (48) width, concentration and velocity are imposed. Δ𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠⋅𝑡𝑡𝑡𝑡Δ𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 / 45 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 The𝑡𝑡𝑡𝑡 sedimentation𝑡𝑡𝑡𝑡Δ𝑥𝑥𝑥𝑥+𝑏𝑏𝑏𝑏Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏 =velocityΔ�𝑥𝑥𝑥𝑥ρ𝑏𝑏𝑏𝑏 Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠v⋅−𝑣𝑣𝑣𝑣Δ, 𝑠𝑠𝑠𝑠entrainment𝑡𝑡𝑡𝑡ρ�⋅𝑠𝑠𝑠𝑠Δ+1𝑡𝑡𝑡𝑡 2� by Adel (1987), φ is the internal friction angle, In𝑣𝑣𝑣𝑣 the⋅ Δ𝑡𝑡𝑡𝑡 case of 𝑣𝑣𝑣𝑣a subcritical flow it is sufficient 55 𝑆𝑆𝑆𝑆𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 𝑢𝑢𝑢𝑢𝑠𝑠𝑠𝑠+1 2 𝑏𝑏𝑏𝑏𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡sed 45 cosh velocity v∂ andℎ + ∂frictionℎ𝑢𝑢𝑢𝑢 = velocity ρ� 𝑠𝑠𝑠𝑠+1 u2 are defined ∆= (ρ − ρ )/ρ , ν is kinematic viscosity, n initial to specify only the flow rate instead of both 1 E ( ∗ ) s w 36w ( ) = 45 45 𝑇𝑇𝑇𝑇 0 cosh tanh sinh 1 + ( ) = ( + +𝑆𝑆𝑆𝑆 )= = ⎡ �� 𝐶𝐶𝐶𝐶� 2 ⎤ / as a function∂∂𝑡𝑡𝑡𝑡/ℎ of∂∂𝑥𝑥𝑥𝑥 ℎh𝑢𝑢𝑢𝑢, u and c on either the porosity and D1546 15th percentile⎢𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sinh grain 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸size. 𝑇𝑇𝑇𝑇the height, width𝑇𝑇𝑇𝑇 and velocity.𝑇𝑇𝑇𝑇 However𝑞𝑞𝑞𝑞 the2 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 ⎥ 2 = 2 ( 𝑆𝑆𝑆𝑆) 𝑠𝑠𝑠𝑠 / 𝑤𝑤𝑤𝑤( + ) 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 − ⋅ � ⋅ ⋅ � �� 𝑥𝑥𝑥𝑥� − �� 𝑥𝑥𝑥𝑥� ⋅ �� 𝐶𝐶𝐶𝐶� − � − 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡ℎ ∂ℎ∂ℎ𝑡𝑡𝑡𝑡𝑢𝑢𝑢𝑢∂ℎρ𝑢𝑢𝑢𝑢/ − ρ / ⎢ ⎥ 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 cell centers𝑠𝑠𝑠𝑠+1∂𝑡𝑡𝑡𝑡2 or faces.∂𝑥𝑥𝑥𝑥𝑠𝑠𝑠𝑠+1 The𝑠𝑠𝑠𝑠 explicit𝑆𝑆𝑆𝑆𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 upwind 𝑡𝑡𝑡𝑡 46 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸volume flux trough𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 the moving𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 boundary𝑇𝑇𝑇𝑇 is h 𝑇𝑇𝑇𝑇 ℎ𝑏𝑏𝑏𝑏 𝑆𝑆𝑆𝑆Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 − ℎ𝑏𝑏𝑏𝑏−Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏𝑔𝑔𝑔𝑔=�ℎ� 𝑡𝑡𝑡𝑡 � 𝑡𝑡𝑡𝑡𝑐𝑐𝑐𝑐 (−𝑐𝑐𝑐𝑐)𝑡𝑡𝑡𝑡 ( + ) sin⎢ ( /2) �� 𝐶𝐶𝐶𝐶� 47 ⎥ 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 ∂𝑡𝑡𝑡𝑡 ∂𝑡𝑡𝑡𝑡∂𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏𝑏∂𝑥𝑥𝑥𝑥/ρ� 2𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 / 46⎣ 46 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ⎦ approximations1𝑆𝑆𝑆𝑆 =⋅ Δ𝑥𝑥𝑥𝑥=− are� ℎ𝑢𝑢𝑢𝑢used( when)(− �a)ℎ variable𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣( that+�(𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏𝑏+ )2� ) = 10 = 10 cot· b(u( )+ vt), which cannot be replaced by a flow 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡𝑏𝑏𝑏𝑏+1 𝑡𝑡𝑡𝑡+1𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏1 𝑡𝑡𝑡𝑡+𝑏𝑏𝑏𝑏1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 / / / / 37 sin = + ( ) 56 ℎ Δ𝑥𝑥𝑥𝑥 Δ𝑡𝑡𝑡𝑡(− ℎ) Δ𝑥𝑥𝑥𝑥/ = 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏/ 𝑏𝑏𝑏𝑏𝑏 2 sin( 𝑠𝑠𝑠𝑠 /2)𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 is𝑏𝑏𝑏𝑏 not𝑏𝑏𝑏𝑏 defined𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏 ⋅ Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 on− 𝑡𝑡𝑡𝑡the� 𝑡𝑡𝑡𝑡ℎ𝑢𝑢𝑢𝑢𝑡𝑡𝑡𝑡 grid𝑡𝑡𝑡𝑡 is𝑡𝑡𝑡𝑡 required.𝑡𝑡𝑡𝑡− �ℎ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑏𝑏𝑏𝑏𝑏 2� sin( /2) ϕ − π 47 rate q due47 to4 the presence of vt. Therefore, ℎ ℎΔ𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥 − ℎ−Δ2ℎ𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥 2 =𝑠𝑠𝑠𝑠+1 10𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠= 𝑤𝑤𝑤𝑤10𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 cot0( ) ( ) 0 2 π Δ𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡𝑆𝑆𝑆𝑆𝑏𝑏𝑏𝑏 ⋅𝑆𝑆𝑆𝑆 Δ𝑥𝑥𝑥𝑥𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏⋅ Δ𝑥𝑥𝑥𝑥−𝑏𝑏𝑏𝑏�−−ℎ𝑢𝑢𝑢𝑢𝑏𝑏𝑏𝑏� 𝑏𝑏𝑏𝑏ℎ𝑢𝑢𝑢𝑢𝑏 𝑏𝑏𝑏𝑏2𝑏− 2�ℎ−𝑢𝑢𝑢𝑢�ℎ=𝑢𝑢𝑢𝑢𝑣𝑣𝑣𝑣 10�𝑣𝑣𝑣𝑣𝑏𝑏𝑏𝑏𝑏�𝑣𝑣𝑣𝑣𝑏𝑏𝑏𝑏2𝑏� 2� − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅= 10 cot −𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ϕ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠+1 2 sin sin ϕ 𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞 flow𝑐𝑐𝑐𝑐 velocity⋅𝐷𝐷𝐷𝐷 ⋅ u⋅ 0 ρis determined−ρ ⋅𝑔𝑔𝑔𝑔 manually as an 𝑆𝑆𝑆𝑆 Δ𝑡𝑡𝑡𝑡 Δ𝑡𝑡𝑡𝑡 𝑔𝑔𝑔𝑔�ℎ� � ϕ − π ϕ − π Treatment of𝑣𝑣𝑣𝑣 𝑤𝑤𝑤𝑤moving𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 Δ𝑥𝑥𝑥𝑥− boundary⋅𝑘𝑘𝑘𝑘0 ⋅Δ⋅ cell𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0−(47) ⋅𝑘𝑘𝑘𝑘0 ⋅Δ⋅38 ϕ 0 input, and can be tweaked to make sure the =2 𝑣𝑣𝑣𝑣 − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅ − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅ ϕ 48 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 ϕ ϕ = The trencher is moving to the left in a fixed ( 3 ) simulation is stable. 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 160 1 0 𝑏𝑏𝑏𝑏 − 𝑏𝑏𝑏𝑏 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑔𝑔𝑔𝑔 2 𝑛𝑛𝑛𝑛 48 48 grid, thus being⋅𝑣𝑣𝑣𝑣 located in different= grid cells in = 0 15 2 3 (𝑘𝑘𝑘𝑘 3 ) 𝐷𝐷𝐷𝐷 Δ𝑡𝑡𝑡𝑡 160 (1 ) 160 39 1 0 0 time. Due to= stability( ) issues, it is not possible2 0 𝑔𝑔𝑔𝑔 2 𝑛𝑛𝑛𝑛 ⋅ν −𝑛𝑛𝑛𝑛 The right boundary is an outflow49 boundary, 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 0 𝑔𝑔𝑔𝑔 15 𝑛𝑛𝑛𝑛 0 15 =2 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷 𝑘𝑘𝑘𝑘 2 𝐷𝐷𝐷𝐷 0 ( ) for𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠 the− trencher𝑧𝑧𝑧𝑧𝑠𝑠𝑠𝑠 boundary𝑡𝑡𝑡𝑡 to move exactly one (48)0 ⋅ν −𝑛𝑛𝑛𝑛 1 where a 49zero-gradient boundary condition 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 ⋅ν −𝑛𝑛𝑛𝑛 = 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 49 grid cell every time𝑣𝑣𝑣𝑣 step. It will take several= (1 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ) is imposed. For this assumption to be valid, Δ𝑡𝑡𝑡𝑡 ( ) 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 / = / + 1 / 1 40𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 ρ =⋅ −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 time steps for the trencher to move one grid𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 The𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 sedimentation velocity is given fluctuations should be minimal50 towards the 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ( 𝑠𝑠𝑠𝑠 ) 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1 2 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠+1 2 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 sin ρ =⋅ /−𝑛𝑛𝑛𝑛2 −𝑐𝑐𝑐𝑐 4750 ρ� cell.ρ As𝑐𝑐𝑐𝑐̂ a result,ρ the� − boundary𝑐𝑐𝑐𝑐̂ � cell – denotedρ =⋅= by−𝑛𝑛𝑛𝑛10 −𝑐𝑐𝑐𝑐by equation (49),= where10 𝐸𝐸𝐸𝐸S and𝐸𝐸𝐸𝐸 Ecot are( ) 50 right boundary. The initial conditions are / 0 sin 41 𝑣𝑣𝑣𝑣 0α.075⋅ 𝑢𝑢𝑢𝑢 51 /subscript= 𝑡𝑡𝑡𝑡 b – will grow𝑡𝑡𝑡𝑡 in size with velocity v for sedimentationϕ𝐸𝐸𝐸𝐸 − π𝐸𝐸𝐸𝐸 and =erosion flux respectively therefore chosen such that the bed elevation <0 𝑤𝑤𝑤𝑤𝐸𝐸𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 t 𝐸𝐸𝐸𝐸 0 𝑣𝑣𝑣𝑣 0α.075⋅ 𝑢𝑢𝑢𝑢 ( 0 . ) / 51 𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1/2 𝑣𝑣𝑣𝑣 − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅= − 1+⋅𝑘𝑘𝑘𝑘 718⋅Δ⋅ ϕ the continuityℎ 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖control𝑢𝑢𝑢𝑢 volumes≥ – see 𝑣𝑣𝑣𝑣equation0α.075⋅ 𝑢𝑢𝑢𝑢 and ρ is sedimentϕ .density./ For a detailed 51 and trench width are constant close to the �𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 = s(1+718 𝐸𝐸𝐸𝐸 ) 2 4 1 2 ℎ � 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 2 . / α (44) –ℎ and with𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 velocity𝑢𝑢𝑢𝑢 v /2 for the momentum(1+718 )explanation𝐸𝐸𝐸𝐸 of the2 sedimentation4 1 2 ⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 and erosion right boundary.48 To achieve this, the initial / t 0 α= 42 52 = 𝐸𝐸𝐸𝐸 n 2 4 1 2 160 (⋅1 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 3 ) 52 /control𝑡𝑡𝑡𝑡 volume. See𝑡𝑡𝑡𝑡 Figure 9, whereα ∆x is the fluxes, see Rhee2 (2010).0 = location of the moving boundary is set at a 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠+1/2 <0 ⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑔𝑔𝑔𝑔 𝑛𝑛𝑛𝑛 𝑡𝑡𝑡𝑡 old boundary𝑐𝑐𝑐𝑐 cell𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 size and≥ ∆xn+1 is the size of 𝑘𝑘𝑘𝑘0 =𝐷𝐷𝐷𝐷15 2 8 52 certain distance (default 5 metres) from the 𝑠𝑠𝑠𝑠+1 2 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ⋅ν 8 −𝑛𝑛𝑛𝑛0 𝑖𝑖𝑖𝑖 𝑐𝑐𝑐𝑐� � 𝑠𝑠𝑠𝑠+1 𝑠𝑠𝑠𝑠+1 2 = 𝑖𝑖𝑖𝑖 ∗ � 49 the boundary𝑐𝑐𝑐𝑐 1 cell𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑢𝑢𝑢𝑢 in the new time step. The 8 = ∗ � 𝑢𝑢𝑢𝑢 ⋅𝑢𝑢𝑢𝑢 right boundary. The bed elevation is set at 𝑢𝑢𝑢𝑢 (431 ⋅𝑢𝑢𝑢𝑢 ) | | / / / 0 ∗ 𝑖𝑖𝑖𝑖 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 53 flux going through2 the left boundary should𝑢𝑢𝑢𝑢 � ⋅𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 | | = 1 zero and53 the flow velocity, height and width are / = 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 =𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 1 𝑠𝑠𝑠𝑠−1therefore2 be1 corrected𝑠𝑠𝑠𝑠−1 2 with𝑠𝑠𝑠𝑠+1 the2 growth| velocity| (49) ρ =⋅ −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 set equal to the output values of the erosion 𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �𝑢𝑢𝑢𝑢 −𝑢𝑢𝑢𝑢 � ≥ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 53 50 / / / <0 = 1 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 𝑠𝑠𝑠𝑠+1 2 of the boundary2 cell. 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶≤𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≤ model. The bed elevation and trench width 𝑢𝑢𝑢𝑢푢 � 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 | Δ𝑥𝑥𝑥𝑥| | | 𝑠𝑠𝑠𝑠+1 2 𝑠𝑠𝑠𝑠−1 2 𝑠𝑠𝑠𝑠+1 2 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣| Δ𝑥𝑥𝑥𝑥0α|.075⋅ 𝑢𝑢𝑢𝑢 | | 5154 54 𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 �𝑢𝑢𝑢𝑢 −𝑢𝑢𝑢𝑢 � 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 ≤The== entrainment= velocity= is included= via are then constraint such that they will remain = + 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 (1+71844 . ) / | Δ𝑥𝑥𝑥𝑥| | equation| 𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 (50)⋅ Δ where𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 entrainment𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅ Δ𝑡𝑡𝑡𝑡 coefficient𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 54 constant in this section. 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 = = 𝐶𝐶𝐶𝐶α𝐹𝐹𝐹𝐹𝐸𝐸𝐸𝐸 𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹2𝐶𝐶𝐶𝐶4 1 2 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 (44)Δ𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥 𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 α𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚E is determined𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 via an𝑣𝑣𝑣𝑣 empirical𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 function𝑣𝑣𝑣𝑣 in 5255 55 𝑢𝑢𝑢𝑢cosh ⋅ Δ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢cosh 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑠𝑠𝑠𝑠 equation𝑠𝑠𝑠𝑠 = (51) proposed by Parker et al. (1987). Stability of the scheme ( ) = ( ) =𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡𝑇𝑇𝑇𝑇 cosh𝑣𝑣𝑣𝑣 458tanh cosh sinhtanh 1 sinh + 1 + + = ⎡ �� � 𝑇𝑇𝑇𝑇 552 ⎤ To give the treatmentcosh of the boundary cell⎡ in a 𝐶𝐶𝐶𝐶 The function�� 𝐶𝐶𝐶𝐶�𝑖𝑖𝑖𝑖 convergences to 0.075 for 2 For the scheme2 to be stable⎤ it has to satisfy ⎢𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sinh 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇∗𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 ⎥ 2 ∂ℎ ∂ℎ𝑢𝑢𝑢𝑢 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 − ⋅ � ⋅ ⎢𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸⋅ � sinh��𝑢𝑢𝑢𝑢 𝑥𝑥𝑥𝑥��− ⋅𝑢𝑢𝑢𝑢 �� 𝑥𝑥𝑥𝑥𝑇𝑇𝑇𝑇� ⋅ �� 𝐶𝐶𝐶𝐶�𝑇𝑇𝑇𝑇− � − 𝑥𝑥𝑥𝑥 𝑇𝑇𝑇𝑇 𝑥𝑥𝑥𝑥 𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 ⎥ ( )readable= expression, the𝑇𝑇𝑇𝑇 continuity⎢cosh𝑇𝑇𝑇𝑇 𝑧𝑧𝑧𝑧𝑇𝑇𝑇𝑇𝑥𝑥𝑥𝑥 equation− tanh ⋅ � ⋅ non-stratified𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸sinh ⋅ � flow.� The𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸�1 Richardson𝑥𝑥𝑥𝑥� − +𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸�� number𝑥𝑥𝑥𝑥� ⋅ is𝑇𝑇𝑇𝑇 �� 𝑇𝑇𝑇𝑇the𝐶𝐶𝐶𝐶⎥� Courant-Friedrichs-Lewy− � − 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 (CFL) condition, ⎡ 𝑆𝑆𝑆𝑆 �� 𝐶𝐶𝐶𝐶� ⎢ ⎢ 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 | | 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 2 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸⎤ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ⎥ 53 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ⎥ ∂𝑡𝑡𝑡𝑡 ∂𝑥𝑥𝑥𝑥 �� 𝐶𝐶𝐶𝐶� = 𝑇𝑇𝑇𝑇 46 ’ 12 2 ’ for shallow⎢𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 watersinh flow 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸with constant⎣ width𝑇𝑇𝑇𝑇 ⎢ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 defined𝑇𝑇𝑇𝑇 �� as𝐶𝐶𝐶𝐶 �Ri =𝑇𝑇𝑇𝑇 g h/u , with 𝑞𝑞𝑞𝑞reduced𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 gravity⎥ g = given⎦ by equation 53. Where⎥ CFL is the 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 = − ⋅ � ⋅( ) / ⋅ �( + ��) 𝑥𝑥𝑥𝑥�⎣− �� 𝑥𝑥𝑥𝑥� ⋅ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚�� 𝐶𝐶𝐶𝐶� − � − 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 56 ⎦ 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡+1 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 ⎢ 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 / = 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 +𝑢𝑢𝑢𝑢 ⋅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸Δ𝑡𝑡𝑡𝑡 ( 𝑇𝑇𝑇𝑇 ) 𝑇𝑇𝑇𝑇 ⎥ 56 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑡𝑡𝑡𝑡 ⎢ 𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 � 𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 = ≤4 + ⎥( ) ℎ Δ𝑥𝑥𝑥𝑥 − ℎ Δ𝑥𝑥𝑥𝑥 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏 𝑏𝑏𝑏𝑏�𝑏 2 𝐶𝐶𝐶𝐶� 𝑠𝑠𝑠𝑠 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 2 4 𝑆𝑆𝑆𝑆 ⎣⋅ Δ𝑥𝑥𝑥𝑥 − � ℎ𝑢𝑢𝑢𝑢 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸− �ℎ 𝑢𝑢𝑢𝑢 𝑣𝑣𝑣𝑣 �𝑏𝑏𝑏𝑏𝑏 2� 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚| Δ𝑥𝑥𝑥𝑥𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡| |π |𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 ⎦ 54 𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞 = 𝑐𝑐𝑐𝑐 ⋅𝐷𝐷𝐷𝐷=⋅ ⋅ ρ −ρ ⋅𝑔𝑔𝑔𝑔2 π 56 Δ𝑡𝑡𝑡𝑡 = + ( 𝑞𝑞𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠)𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑞𝑞𝑞𝑞𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρ𝑤𝑤𝑤𝑤 −ρ𝑠𝑠𝑠𝑠 ⋅𝑔𝑔𝑔𝑔 4 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 2 π 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 #156 - AUTUMN 2019 41 𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞 𝑐𝑐𝑐𝑐 ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρ −ρ𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡⋅𝑔𝑔𝑔𝑔 𝑣𝑣𝑣𝑣 55 cosh ( ) = 𝑇𝑇𝑇𝑇 cosh tanh sinh 1 + ⎡ �� 𝐶𝐶𝐶𝐶� 2 ⎤ ⎢𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sinh 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑞𝑞𝑞𝑞 2 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 ⎥ 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 − ⎢ ⋅ � ⋅ ⋅ � �� 𝑥𝑥𝑥𝑥� − �� 𝑥𝑥𝑥𝑥� ⋅ �� 𝐶𝐶𝐶𝐶� − � − 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥⎥ 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ⎢ �� 𝐶𝐶𝐶𝐶� ⎥ ⎣ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ⎦ = + ( ) 56 4 2 π 𝑞𝑞𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑞𝑞𝑞𝑞𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρ𝑤𝑤𝑤𝑤 −ρ𝑠𝑠𝑠𝑠 ⋅𝑔𝑔𝑔𝑔 TECHNICAL
FIGURE 10 A B
Comparison of the 0.008 0.20 dam-break analytical Numerical staggered upwind scheme Numerical staggered upwind scheme 0.007 Analytical solution Analytical solution and numerical solution for the 0.006 0.15
staggered upwind 0.005 scheme. Shown is 0.004 0.10 flow elevation (A) and velocity (B), 0.003 Flow elevation (m) Flow velocity (m/s) 0.05 snapshot at T = 9 0.002 seconds. 0.001
0 0 0 2 4 6 8 10 0 2 4 6 8 10 X coordinate (m) X coordinate (m)
FIGURE 11 A B
Comparison of 3.0 4.0 Numerical staggered upwind scheme Numerical staggered upwind scheme analytical and Analytical solution 3.5 Analytical solution numerical solution 2.5 Seabed 3.0 of the staggered 2.0 upwind scheme for 2.5
a subcritical flow 1.5 2.0 sin( /2) 47 = 10 over a bump. Shown= 10 cot( ) sin( /2) 47 sin = 10 = 10 cot( ) 1.5
is flow elevationϕ − π (A) 1.0 sin Flow velocity (m/s) 𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − ⋅𝑘𝑘𝑘𝑘0 ⋅Δ⋅ − ⋅𝑘𝑘𝑘𝑘0 ⋅Δ⋅ ϕ ϕ − π and velocityϕ (B). Bed and flow elevation (m) 𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − ⋅𝑘𝑘𝑘𝑘0 ⋅Δ⋅ − ⋅𝑘𝑘𝑘𝑘0 ⋅Δ⋅ ϕ 1.0 ϕ 0.5 48 = 0.5 48 160 (1 3 ) = 2 0 3 𝑔𝑔𝑔𝑔 𝑛𝑛𝑛𝑛 0 160 (1 ) 0 𝑘𝑘𝑘𝑘0 𝐷𝐷𝐷𝐷15 2 𝑔𝑔𝑔𝑔 2 𝑛𝑛𝑛𝑛0 0 0 5 100 15 15 20 2 25 0 5 10 15 20 25 ⋅ν −𝑛𝑛𝑛𝑛 𝑘𝑘𝑘𝑘 49 𝐷𝐷𝐷𝐷 0 = X coordinate ⋅ν(m) −𝑛𝑛𝑛𝑛 49 X coordinate (m) (1 ) = (1 ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 ρ =⋅ −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣 50 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 ρ =⋅ −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 50 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑣𝑣𝑣𝑣 0α.075⋅ 𝑢𝑢𝑢𝑢 51 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 = 𝑣𝑣𝑣𝑣 0α.075⋅ 𝑢𝑢𝑢𝑢 ( . ) / 51 1+718 = . / dimensionless CFL number, umax is the condition(1+ for718 the boundary) cell, is independent on several scenarios for which exact 𝐸𝐸𝐸𝐸 2 4 1 2 α 𝐸𝐸𝐸𝐸 maximum velocity⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 in the domain, ∆t the time ofα the 52time step. However,2 4 1 2 since a quite robust analytical solutions are known. Analytical ⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 52 step and= ∆xmin the smallest possible grid size scheme (explicit, upwind) is used, the CFL solutions are taken from the SWASHES 8 = in the domain.𝑖𝑖𝑖𝑖 number can be8 higher than one. Also, since library (Delestre et al., 2016), in which 𝑢𝑢𝑢𝑢∗ � ⋅𝑢𝑢𝑢𝑢 𝑖𝑖𝑖𝑖 there is𝑢𝑢𝑢𝑢 only∗ �one⋅𝑢𝑢𝑢𝑢 grid cell that does not fulfill numerous analytical solutions for the | | 53 = 1 the CFL condition,| | the error created dampens shallow water53 equations are summarised. 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 1 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 out in the rest𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 of the domain. Four different cases are used for this 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≤ 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 (53)| Δ𝑥𝑥𝑥𝑥| | | 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶54 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≤ verification; a dam-break, subcritical flow = = | Δ𝑥𝑥𝑥𝑥| | | 54 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = = over a bump and transcritical flow over a 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹The𝐶𝐶𝐶𝐶 trencher𝑠𝑠𝑠𝑠 boundary𝑠𝑠𝑠𝑠 moves to the left in the 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢 bump with and without a hydraulic jump. 𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 55 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 cosh grid in time, dependent on trencher velocity (54) 𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 Results are55 given in Figures 10, 11, 12 and 13. cosh ( ) = 𝑇𝑇𝑇𝑇 coshand time step.tanh In one-timesinh step, the boundary1 + ⎡ �� 𝐶𝐶𝐶𝐶� ( ) = 2 𝑇𝑇𝑇𝑇 cosh⎤ tanh sinh 1 + 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sinh 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ⎡ 𝑇𝑇𝑇𝑇 𝑞𝑞𝑞𝑞� 2 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 2 ⎤ ⎢ cell will grow by a distance equal to ∆xmin, which� 𝐶𝐶𝐶𝐶� Verification⎥ 2 Results of the verification show that the 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 − ⎢ ⋅ � ⋅ ⋅ � �� 𝑥𝑥𝑥𝑥� − �� 𝑥𝑥𝑥𝑥� ⋅ �⎢𝑞𝑞𝑞𝑞�𝐶𝐶𝐶𝐶 𝐶𝐶𝐶𝐶𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸� −sinh� − 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥⎥ 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 ⎥ 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 is also𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 the smallest𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 grid𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 size− occurring⎢ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸⋅ � ⋅ in time.𝑇𝑇𝑇𝑇 ⋅𝑇𝑇𝑇𝑇�This� section� 𝑥𝑥𝑥𝑥� − aims �to� verify𝑥𝑥𝑥𝑥� ⋅ whether�� the𝐶𝐶𝐶𝐶� − � − 𝑥𝑥𝑥𝑥 scheme𝑥𝑥𝑥𝑥⎥ is shock-capturing and performs ⎢ �� 𝐶𝐶𝐶𝐶� 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ⎥ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ⎣ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ⎢ �� 𝐶𝐶𝐶𝐶� ⎦ ⎥ = The CFL+ condition for( this cell) ⎣ is given by 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 proposed56 numerical scheme is able to capture reasonably⎦ well. Some deviation in the 4 = + ( ) 56 equation 54. From2 π this relation it can be the phenomena that4 can occur in shallow discharge is observed at the hydraulic jump 𝑞𝑞𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑞𝑞𝑞𝑞𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρ𝑤𝑤𝑤𝑤 −ρ𝑠𝑠𝑠𝑠 ⋅𝑔𝑔𝑔𝑔 2 π concluded that the CFL condition for the 𝑞𝑞𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 water𝑞𝑞𝑞𝑞𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 flows.𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 Examples⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρof𝑤𝑤𝑤𝑤 these−ρ𝑠𝑠𝑠𝑠 ⋅𝑔𝑔𝑔𝑔 phenomena location in Figure 13. However, this is not an boundary cell can only be fulfilled if the are propagating shocks and transitions issue within the current application of the trencher velocity is greater or equal to the flow between subcritical and supercritical flows scheme since an overall trench profile is the
velocity (vt ≥ |umax|). In other words, the stability and vice versa (hydraulic jumps). To do this objective and not accurate local values of of the scheme, with regard to the CFL verification, the numerical scheme is applied the discharge.
42 TERRA ET AQUA A B FIGURE 12
2 2.0 Comparison of
Numerical staggered upwind scheme Numerical staggered upwind scheme analytical and Analytical solution Analytical solution Seabed numerical solution 1.5 1.5 of the staggered
/s) upwind scheme 2 for a transcritical 1 1.0 flow over a bump,
Discharge (m Discharge without hydraulic
Bed and flow elevation (m) 0.5 0.5 jump. Shown is flow elevation (A) and discharge (B). 0 0 0 5 10 15 20 25 0 5 10 15 20 25 X coordinate (m) X coordinate (m)
A B FIGURE 13
0.8 0.40 Comparison of Numerical staggered upwind scheme Numerical staggered upwind scheme 0.35 analytical and 0.7 Analytical solution Analytical solution Seabed numerical solution 0.30 0.6 of the staggered
/s) 0.25 0.5 2 upwind scheme for a transcritical 0.4 0.20 flow over a bump, 0.15 0.3 (m Discharge with hydraulic
Bed and flow elevation (m) 0.2 0.10 jump. Shown is flow elevation (A) and 0.1 0.05 discharge (B). 0 0 0 5 10 15 20 25 0 5 10 15 20 25 X coordinate (m) X coordinate (m)
Cable deflection model 0.5 0.0 The burial depth of the cable is determined -0.5 -1.0 Trench profile by the intersection of the cable shape and -1.5 Flow elevation (h+z) Original seabed level elevation (m) -2.0 the re-settled seabed. The cable shape is Seabed and flow Seabed and flow -2.5 based on an elastic, hyperstatic cantilever 0 20 40 60 80 100 beam model which is uniformly loaded. The 3 2 1 left hand side is completely fixed and the 0 -1 right hand side is restrained in rotation. The -2
Trench width (m) Trench -3 residual lay tension in the cable is applied 0 20 40 60 80 100 at the right hand side of the cantilever 0.30 0.25 beam. It is assumed that the position of the 0.20 cable remains fixed when the touchdown 0.15 0.10 point intersects with the trench shape. The 0.05 Concentration (-) 0.00 analytical solution for the cantilever beam 0 20 40 60 80 100 is given in Vanden Berghe et al. (2011). Exact X coordinate (m) assumptions in the derivation of the equation are unknown, therefore a verification has been FIGURE 14 done by comparing the analytical solution to
Model results for D15 = 0.15mm, D50 = 0.3mm and vt = 0.1m/s. Plots show a side view of the trench numerical solutions generated by OrcaFlex, (A), a top view of the trench with the development of trench width (B) and the depth averaged which is a dynamic analysis package used sediment concentration (C). within the offshore industry.
#156 - AUTUMN 2019 43 sin( /2) 47 = 10 = 10 cot( ) sin ϕ − π 𝑣𝑣𝑣𝑣𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 − ⋅𝑘𝑘𝑘𝑘0 ⋅Δ⋅ − ⋅𝑘𝑘𝑘𝑘0 ⋅Δ⋅ ϕ ϕ sin( /2) 47 48 = 10 = 10 cot( ) = sin 160 (1 3 ) ϕ − π 𝑔𝑔𝑔𝑔 2 𝑛𝑛𝑛𝑛0 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 0 0 0 15 2 𝑣𝑣𝑣𝑣 − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅ − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅ ϕ 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷 0 ϕ ⋅ν −𝑛𝑛𝑛𝑛 49 = (1 48 ) = 160 (1 3 ) 𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 𝑔𝑔𝑔𝑔 2 𝑛𝑛𝑛𝑛0 𝑣𝑣𝑣𝑣 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 0 15 2 ρ =⋅ −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 50 𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷 0 ⋅ν −𝑛𝑛𝑛𝑛 49 = 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 (1 ) 𝑣𝑣𝑣𝑣 0α.075⋅ 𝑢𝑢𝑢𝑢 51 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 = . / 𝑣𝑣𝑣𝑣𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 (1+718 ) ρ𝑠𝑠𝑠𝑠 ⋅ −𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 −𝑐𝑐𝑐𝑐 = α𝐸𝐸𝐸𝐸 502 4 1 2 ⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 52 𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 𝑣𝑣𝑣𝑣 0α.075⋅ 𝑢𝑢𝑢𝑢 = 51 = . / 8 (1+718 ) 𝑖𝑖𝑖𝑖 ∗ � α𝐸𝐸𝐸𝐸 2 4 1 2 𝑢𝑢𝑢𝑢 ⋅𝑢𝑢𝑢𝑢 ⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 | | 52 53 = 1 = 8 𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅ Δ𝑡𝑡𝑡𝑡 TECHNICAL𝑖𝑖𝑖𝑖 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≤ 𝑢𝑢𝑢𝑢∗ � ⋅𝑢𝑢𝑢𝑢 | Δ𝑥𝑥𝑥𝑥| | | 54 = = | | 53 = 1 𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅ Δ𝑡𝑡𝑡𝑡 𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅ Δ𝑡𝑡𝑡𝑡 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡 𝑣𝑣𝑣𝑣 55 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ≤ cosh | Δ𝑥𝑥𝑥𝑥| | | sin( /2) 54 = ( ) == cosh tanh sinh 47 1 + = 10 ⎡ 𝑇𝑇𝑇𝑇 = 10 cot( ) ⎤ 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 sin �� 𝐶𝐶𝐶𝐶� 2 𝑢𝑢𝑢𝑢 ⋅ Δ𝑡𝑡𝑡𝑡 ⎢𝑢𝑢𝑢𝑢𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sinh 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑞𝑞𝑞𝑞 2 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 ⎥ � ϕ − π � � � 1.5 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 − ⎢0 𝑠𝑠𝑠𝑠⋅ ⋅ ⋅ � �0 𝑥𝑥𝑥𝑥� − � 𝑥𝑥𝑥𝑥� ⋅ � 𝐶𝐶𝐶𝐶� − � − 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥⎥ 𝑣𝑣𝑣𝑣 𝑣𝑣𝑣𝑣 ⋅ Δ𝑡𝑡𝑡𝑡− ⋅𝑘𝑘𝑘𝑘𝑇𝑇𝑇𝑇𝑣𝑣𝑣𝑣⋅Δ⋅ 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 − ⋅𝑘𝑘𝑘𝑘 ⋅Δ⋅𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ϕ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ⎢ ϕ�� 𝐶𝐶𝐶𝐶� 55 1.0 ⎥ cosh 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 ⎣ ⎦ 56 ( ) = 𝑇𝑇𝑇𝑇 cosh tanh sinh 1= + ( ) 48 0.5 ⎡ �� � = 2 ⎤ 4 𝐶𝐶𝐶𝐶 ( 3 ) 2 ⎢𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sinh 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 160 𝑇𝑇𝑇𝑇 1 0 𝑞𝑞𝑞𝑞 𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 ⎥ 2 π � � � 𝑔𝑔𝑔𝑔 �2 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑛𝑛𝑛𝑛𝑠𝑠𝑠𝑠 𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑤𝑤𝑤𝑤 𝑠𝑠𝑠𝑠 0.0 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 − ⎢ ⋅ ⋅ ⋅ � � 𝑥𝑥𝑥𝑥� − � 0 𝑥𝑥𝑥𝑥� ⋅ � 15𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶� − � −2𝑞𝑞𝑞𝑞 𝑥𝑥𝑥𝑥 𝑐𝑐𝑐𝑐 𝑥𝑥𝑥𝑥⎥⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρ −ρ ⋅𝑔𝑔𝑔𝑔 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑘𝑘𝑘𝑘𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐷𝐷𝐷𝐷𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 0 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 ⎢ �� 𝐶𝐶𝐶𝐶� (55) ⋅ν −𝑛𝑛𝑛𝑛 ⎥ 49 -0.5 ⎣ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 = ⎦ = + ( ) (1 ) 56 4 𝑆𝑆𝑆𝑆 − 𝐸𝐸𝐸𝐸 -1.0 Where2 πz is𝑣𝑣𝑣𝑣 the𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 cable deflection measured 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 depth of lowering (m) 𝑞𝑞𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑞𝑞𝑞𝑞𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρ𝑤𝑤𝑤𝑤 −ρρ𝑠𝑠𝑠𝑠 =⋅⋅𝑔𝑔𝑔𝑔 −𝑛𝑛𝑛𝑛 −𝑐𝑐𝑐𝑐 -1.5 from the seabed, q is cable weight, T is 50 lowering relative to measured
𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸 Deviation of predicted depth Cable sections residual lay tension,𝑣𝑣𝑣𝑣 L0α .is075⋅ distance𝑢𝑢𝑢𝑢 until 51 = touchdown point,(1+ EI718 is bending. ) / stiffness Measure and x is theα𝐸𝐸𝐸𝐸 distance from start2 4 1 2 of the Maximum predicted ⋅ 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 trench. The cable deflection equation is 52 Minimum predicted = incorporated in the sedimentation8 model 𝑖𝑖𝑖𝑖 and is solved each𝑢𝑢𝑢𝑢∗ time� ⋅𝑢𝑢𝑢𝑢a new grid cell is created at the moving| | boundary. The 53 = 1 cable is not lowered in clear water but in a FIGURE 15 𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅ Δ𝑡𝑡𝑡𝑡 mixture of 𝐶𝐶𝐶𝐶sediment𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 and𝑚𝑚𝑚𝑚𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 water,≤ the cable Validation of the jet trenching model by comparing predicted range to measured results during a | Δ𝑥𝑥𝑥𝑥| | | 54 weight should= thus be corrected.= A single reference project. Error bars indicate standard deviation in measured depth of lowering. reference concentration𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 ⋅ Δ𝑡𝑡𝑡𝑡 is chosen𝑢𝑢𝑢𝑢𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 to 𝐶𝐶𝐶𝐶𝐹𝐹𝐹𝐹𝐶𝐶𝐶𝐶 𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 correct the cable𝑣𝑣𝑣𝑣 weight.⋅ Δ𝑡𝑡𝑡𝑡 This𝑣𝑣𝑣𝑣 reference 55 cosh concentration is chosen to be equal to the ( ) = cosh tanh sinh 1 + ⎡ � 𝑇𝑇𝑇𝑇 2 ⎤ � concentration𝐶𝐶𝐶𝐶� at the interface of erosion 2 ⎢𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sinh 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑞𝑞𝑞𝑞 0.5𝑞𝑞𝑞𝑞𝐶𝐶𝐶𝐶 ⎥ 𝑧𝑧𝑧𝑧 𝑥𝑥𝑥𝑥 − ⎢ ⋅ � ⋅ and sedimentation⋅ � �� 𝑥𝑥𝑥𝑥� model.− �� 𝑥𝑥𝑥𝑥� ⋅ �� 𝐶𝐶𝐶𝐶� − � − 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥⎥ 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑇𝑇𝑇𝑇 0 𝑇𝑇𝑇𝑇 ⎢ �� 𝐶𝐶𝐶𝐶� ⎥ ⎣ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 -0.5 ⎦ = + ( ) 56 4 -1 2 π 𝑞𝑞𝑞𝑞𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑠𝑠𝑠𝑠 𝑞𝑞𝑞𝑞𝑤𝑤𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ⋅𝐷𝐷𝐷𝐷 ⋅ ⋅ ρ𝑤𝑤𝑤𝑤 −ρ𝑠𝑠𝑠𝑠 ⋅𝑔𝑔𝑔𝑔 -1.5 (56) -2 Cable depth of lowering (m) -2.5 0 5 10 15 20 25 30 35 Where qslurry is the cable weight in the water Distance along cable route (m) Trenching direction sediment mixture, qwater is the cable weight Measured cable position submerged in water, cref the reference Predicted cable position concentration of sediment, D is cable diameter, Seabed profile
ρw and ρs are water and sediment density respectively. FIGURE 16 Results Seabed profile used as model input and measured/predicted cable position. A typical output of the model is given in Figure 14, showing a side view, top view and the concentration development behind the trencher. 2.4 2.2 Model validation 2 To validate the jet trenching model, averages 1.8 1.6 of the depth of lowering are taken per cable 1.4 section (monopile to monopile). To account 1.2 for uncertainties in grain size and residual Cable depth of lowering (m) 1 0 5 10 15 20 25 30 35 cable tension, a minimum and maximum depth Distance along cable route (m)
of lowering case is considered. The minimum Trenching direction
case is based on (d = 0.2mm, d50 = 0.4mm, T = 15 Measured DOL 5kN ) and the maximum case on (d15 = 0.1mm, Predicted DOL
d50 = 0.25mm, T = 2kN ). Per individual cable section the average jet pressure and trencher velocity is extracted from logs recorded during FIGURE 17 trenching. Furthermore, the sword depth and Measured and predicted depth of lowering.
44 TERRA ET AQUA seabed (depth of lowering) is important. The 2.0 measured and predicted depth of lowering 1.8 X X X XXXX XX X X X are plotted in Figure 17. Difference between 1.6 X X X X X X X X X 1.4 X X X minimum and maximum depth of lowering X X X X X X X 1.2 X X X X X measured is approximately 0.8m, whereas the X X X X 1.0 X X X predicted difference is approximately 0.2m, 0.8 X X d50=0.5mm;d15=0.25mm 0.6 X d50=0.4mm;d15=0.2mm this is thus a significant underestimation. 0.4 X d50=0.3mm;d15=0.15mm X d50=0.2mm;d15=0.1mm Cable depth of lowering (m) 0.2 0.0 Model Results 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 The effect of having a higher trencher velocity Trencher velocity (m/hr) depends on grain sizes. In coarse sand, the increase in depth of lowering is larger than in fine sand for the same increase in trencher FIGURE 18 velocity (see Figure 18). Therefore, it is more
Cable depth of lowering for a range of trencher velocities and four different d50. important to have a high trencher velocity in coarse sand than in fine sand.
Sand dunes, characterised by their height and length, are modelled as a simple sinusoidal 2.00 profile. Due to the seabed profile, also the depth of lowering shows an oscillating profile. 1.90 X Maximum DOL To indicate this behaviour, the bandwidth 1.80 X Mean DOL (minimum and maximum values) and mean X Minimum DOL 1.70 depth of lowering is plotted. To investigate 1.60 X X X X X X X sensitivity of sand dune length, the height X X X X X X X 1.50 X is kept constant and only dune length is X X X XX X XXX X X XX X X X X X 1.40 X XXX X XXX X X varied (see Figure 19). The bandwidth shows X X 1.30 a clear local minimum at a wavelength of 1.20 approximately 7.5 metres. This is an interesting wavelength since it is approximately equal Cable depth of lowering (m) 1.10 1.00 to the layback of the cable (distance from 0 5 10 15 20 25 30 35 start of trench to touchdown point of cable). For a wavelength of half the layback and 1.5 Sand dune length (m) times the layback, there is a maximum in the bandwidth. Thus when the start of the trench is in phase with the touchdown point of the cable FIGURE 19 the variation of depth of lowering is minimum. Cable depth of lowering for a range of sand dune lengths and constant sand dune height of However, the mean depth of lowering is hardly 0.4 metres. influenced.
A similar plot but now for sand dune height is given in Figure 20. The bandwidth shows to nozzle configuration are known per cable measured during a reference project. The increase almost linearly with sand dune height, section. As a result a minimum and maximum selected section of seabed profile has a sand and again the mean depth of lowering is hardly depth of lowering is calculated per cable dune height and length of approximately 0.5m influenced by sand dune height. section, indicated by the red and blue lines in and 12.5m respectively (see Figure 16). The Figure 15. The results show that approximately seabed profile is somewhat smoothened by Discussion 83% of the cable sections are within the applying a moving average before importing it The amplitude at which the depth of lowering predicted range. into the model. In Figure 16, the smoothened oscillates when buried in sand dunes, is seabed profile is plotted together with the shown to be underestimated by the model Although the average depth of lowering is measured and predicted cable position. compared to field data. An explanation could hardly influenced by sand dunes in the model, The cable position predicted by the model be the relatively simple cable equation used the local depth of lowering can be higher or follows a pattern in phase with the sand in the model. This equation takes the tension lower due to the presence of sand dunes. waves, whereas the measured cable position as a constant and uses this to calculate To validate whether this effect is correctly shows an out of phase pattern with the sand the cable shape. However, it is expected included in the model, the depth of lowering dunes. However, not the absolute cable that while the cable is being lowered in the for a cable section is compared to that position but the cable position relative to the trench on sand dunes, the tension in the
#156 - AUTUMN 2019 45 cable varies. This might have a significant influence on the depth of lowering, but 2.00 requires further investigation. 1.90 Although the current model is able X Maximum DOL 1.80 to cope with hydraulic jumps, it is not X Mean DOL 1.70 X Minimum DOL X possible to include a supercritical inflow X 1.60 X at the moving boundary. The high flow X 1.50 X velocity required for this supercritical X X X X X X inflow results in instabilities initiated at 1.40 X X the boundary cell. The inflow is therefore X 1.30 X taken to be subcritical with a large flow 1.20 height and low flow velocity. By applying a Cable depth of lowering (m) 1.10 cell merging technique on the boundary 1.00 cell, the stability can be increased 0 0.2 0.4 0.6 0.8 1 1.2 allowing for a supercritical inflow. Sand dune height (m) Conclusions A jet trenching model is proposed which includes breaching, erosion, FIGURE 20 sedimentation and entrainment to make Cable depth of lowering for a range of sand dune heights, and constant sand dune length of an accurate prediction of the depth of 12.5 metres. lowering of a cable buried in sand. By verification against analytical solutions of the shallow water equations, the numerical scheme of the sedimentation model shows to be shock-capturing and performing accurately. When considering cable burial in sand dunes, an oscillating depth of lowering trend is observed, with the maximum depth Summary of lowering at sand dune crests and minimum at the troughs. This pattern is Numerous offshore wind farms have been recently installed in the resulting both from field measurements southern part of the North Sea. Their infield and export cables are buried and model simulations. However, the for protection against dropped or dragged objects. In sandy soils, burial amplitude in depth of lowering variation is carried out by remotely operated tracked vehicles. Two swords with is significantly underestimated by the waterjets are used to fluidise the sand and generate a backward flow model. Validation shows that the depth of the water-sediment mixture. The southern part of the North Sea has of lowering of approximately 83% of the a highly variable seabed topography characterised by sand waves and cable sections of a reference project is mega-ripples. These seabed features can significantly influence the within the range predicted by the model. trenching process. At the moment, it is not possible to make an accurate By investigating the effect of grain sizes estimate of the influence of sand dunes on the trenching process. and trencher velocity, it has been shown by the model that it is more important The trench formation process is split into two parts; a front section have a high trencher velocity in coarse where the seabed is eroded by waterjets (erosion model) and a sand than in fine sand. rear section where the sand grains are settling in a backward flow (sedimentation model). Both models as well as an elastic cantilever beam model – to determine the cable shape as it sinks in the trench – are delineated in this article. The combined fluidisation, sedimentation and cable model is validated against full-scale field data.
First presented as a paper at the 22nd World Dredging Congress, this article has been published in a slightly adapted version with permission of the copyright holder, WODA.
46 TERRA ET AQUA REFERENCES
Adel, H. den (1987) Parker, G., Garcia, M., Fukushima, Y., and Yu, “Heranalyse doorlatendheidsmetingen W. (1987) door middel van de forch-heimer relatie (in “Experiments on turbidity currents over Dutch)”. an erodible bed”. Journal of Hydraulic In coarse sand, Research 25(1), 123–147. Cao, C., Pender, G., Wallis, S., and Carling, P. the increase in (2004) Rhee, C. van (2010) “Computational Dam-Break Hydraulics “Sediment Entrainment at High Flow depth of lowering over Erodible Sediment Bed”. Journal of Velocity”. Journal of Hydraulic Engineering Hydraulic Engineering 130(7), 689–703. 136(9), 572–582. is larger than in
Delestre, O., Lucas, C., Ksinant, P. A., Rhee, C. van (2015) fine sand for the Darboux, F., Laguerre, C., Vo, T., James, F., “Slope failure by unstable breaching”. and Cordier, S. (2016) Proceedings of the ICE - Maritime same increase in “SWASHES: a compilation of shallow Engineering water analytic solutions for hydraulic and 168, 84–92. trencher velocity. environmental studies”. International Journal for Numerical Methods in Fluids Robert, S. and Wilson, P. (2011) 72(3), 269–300. “A well balanced scheme for the shallow water wave equations in open channels Garcia, M. (1990) with (discontinuous) varying width and “Depositing and eroding sediment-driven bed”. Austral. Mathematical Soc. 52, flows: Turbidity currents”. University of 967–987. Minnesota. Saint Venant, A. (1871) He, S., Liu, W., Li, X., and Ouyang, C. (2014) “Theorie du mouvement non-permanent “An improved coupling model for water des eaux, avec application aux crues des flow, sediment transport and bed rivieres et a l’introduction des marees dans evolution”. Journal of Geoscientific Model leur lit”. Comptes Rendus de l’Academie Development 7, 2429–2454. des Sciences 73, 147–154.
Jong, P. de (1988) Siviglia, A., Nobile, G., and Colombini, M. “Een productie optimalisatie van (2008) het gebruik van spuiters op de “Quasi-Conservative Formulation of the sleephopperzuiger Volvox Delta”. MA One-Dimensional Saint-Venant–Exner thesis. Technische Universiteit Delft. Model”. J. Hydraul. Eng. 134(10), 1521–1526.
Lee, J. and Chu, V. (2003) Vanden Berghe, J., Pyrah, J., Gooding, S., Turbulent jets and plumes: A lagrangian and Capart, H. (2011) approach. Kluwer Academic Publishers. “Development of a jet trenching model in Miedema, S. (2015). OE4607 Introduction sand”. Frontiers in Offshore Geotechnics 2, to Dredging Engineering. 2nd ed. Delft 889–894 University of Technology.
Nobel, A. (2013) “On the excavation process of a moving vertical jet in cohesive soil”. PhD thesis. Delft University of Technology.
#156 - AUTUMN 2019 47 TECHNICAL
By investigating the effect of grain sizes and trencher velocity, it has been shown by the model that it is more important have a high trencher velocity in coarse sand than in fine sand.
Sjoerd Warringa Cees van Rhee Sape Miedema Cristina Lupea Connie Visser Sjoerd obtained his Since 1985, Cees has been Sape obtained his MSc in Prior to joining DEME Connie is managing the Bachelor's degree in engaged with research for Mechanical Engineering Offshore, Cristina received engineering department for Mechanical Engineering in the dredging industry. The with honours at the Delft her MSc in Civil Engineering DEME Offshore in Breda. In 2014. As a next step, he first five years were at University of Technology in in 2013 from the Delft that role, she is involved in pursued an MSc in WL|Delft Hydraulics 1983 and his PhD in 1987. University of Technology. the Geotechnical and Offshore and Dredging (presently Deltares) and From 1987 to the present, The interest in cable Trenching engineering Engineering at the Delft then at Van Oord, a dredging he has been an assistant, trenching was sparked carried out for cable University of Technology, contractor where he was then associate, professor at during an internship at installation operations as developing a particular employed at the various the Chair of Dredging VSMC and has determined well as Research and interest in fluid mechanics departments and projects, Technology, then as a a shift of interest from Development around and dredging processes from 1990 to 2011. At the member of the management bearing to breaking trenching equipment. specifically. During his end of 2002, the author board of Mechanical capacity of soils. As a Connie obtained a Master’s graduation thesis, published obtained his PhD degree. Engineering and Marine Discipline Lead specialised degree in Hydraulic at the WODCON XXII in Since October 2007, he is Technology. From 1996 to in Geotechnical and Engineering at the faculty Shanghai, he focused on professor Dredging 2001, he was appointed Trenching Engineering, of Civil Engineering at Delft numerical modelling of Engineering at Delft educational director of Cristina has been involved University of Technology in water-sediment flow in University of Technology. Mechanical Engineering in numerous burial and 1995. She started her order to enhance cable His main scientific and Marine Technology protection projects for both professional career with an burial processes. Currently, achievements are modelling whilst remaining associate cables and pipelines as well offshore installation he is working as Research of highly concentrated professor of Dredging as trenching equipment contractor and since then and Development Engineer sediment water flows and Engineering. In 2005, he research and development. has continuously worked in on cable installation and high velocity erosion of was additionally appointed the offshore installation burial activities for DEME granular sediments. educational director of the industry. Offshore. MSc programme of Offshore Engineering.
48 TERRA ET AQUA EVENTS DEVELOP PROFESSIONAL NETWORKS ABROAD
Conferences and Special 3-Day Seminar on Dredging lectures are given by dredging experts from and Reclamation IADC member companies, whose practical seminars intended 26-28 November 2019 knowledge and experience add extra value to Radisson Mumbai Andheri MIDC Hotel the classroom lessons. Amongst the subjects for all stakeholders in Mumbai, India covered are: • Overview of the dredging market the field of dredging: For (future) decision makers and their advisors • Development of new ports and in governments, port and harbour authorities, maintenance of existing ports government officials, offshore companies and other organisations • Descriptions of types of dredging that have to execute dredging projects, IADC equipment port authorities, organises a special three-day International • Site and soil investigations Seminar on Dredging and Reclamation. This • Costing of projects and types of dredging offshore companies, time, the seminar heads to South Asia and contracts will be held from 26-28 November 2019 at researchers, scientists the Radisson Mumbai Andheri MIDC Hotel in Each participant will receive a Certificate of Mumbai, India. Achievement in recognition of the completion and dredging of the coursework. The seminar is intended for all stakeholders contractors. in the field of dredging: Government Officials Meet participants and lecturers and Port Authorities, Offshore Companies Face-to-face contact is invaluable. An and Dredging Contractors. The in-depth additional dimension to this stimulating course is a mid-week dinner where participants, lecturers and other dredging employees can interact, network and discuss the real, hands-on world of dredging.
The fee for this seminar is €1,200. This includes all tuition, proceedings, workshops and a special dinner for participants but excludes travel costs, and accommodation.
Register for the seminar at http://bit.ly/2EdpepD
For further questions contact: Ria van Leeuwen, Senior PR & Communications Officer, IADC Email: [email protected]
Photo Marco Hofsté Photography
#156 - AUTUMN 2019 49 EVENTS
WODCON XXIII: World Dredging serving Asia, Australia and the Pacific three words, “dredging is changing”, actually Congress region). have a more philosophical depth too.’ He 16-20 May 2022 added, ‘Dredging always means changing the Tivoli Congress Center The theme of the 23rd WODCON will be environment whether it is capital dredging Copenhagen, Denmark ‘Dredging is Changing’ – which has a or even maintenance. Dredging always aims www.wodcon2022.org double meaning. Not only are dredging to create a new or improved situation and activities an agent of change, modifying this aspect is quite often not recognised or On behalf of World Organization of Dredging the environment, dredging itself is also appreciated.’ Associations (WODA), the Central Dredging changing in terms of the positive impacts Association (CEDA) has announced the it has on the natural and socio-economic The WODCON organisers have chosen next World Dredging Congress, WODCON environment. Copenhagen, the capital of Denmark to host XXIII, will take place from 16-20 May 2022, the conference. With a safe and comfortable at the Tivoli Congress Center, Copenhagen, According to Johan Pennekamp, Chair of environment, cosmopolitan and yet cosy, Denmark. the WODCON XXIII Organising Committee, it is an ideal place for new trends, business ‘Apart from stating that our way of dredging and relaxation. An exciting video about WODCON is held once every three years is changing – just think: advances in the conference and the city can now be rotating among the world regions of WODA dredging technology, novel solutions for downloaded from the official WODCON XXIII members, CEDA (representing the EMEA adaptation to climate change, nature-based website. countries), WEDA (Western Dredging approaches, circular design, benefits/ Association, serving the Americas) and value for all stakeholders, corporate social CEDA is also delighted to announce the first EADA (Eastern Dredging Association, responsibility, sustainable approach – the sponsor to the congress, Rohde Nielsen, headquartered in Copenhagen. It invites the international dredging community to join in and support this major triennial event where the worldwide dredging community will gather in 2022.
In November, IADC is holding a three-day long seminar in Mumbai, India. Photo Marco Hofsté Photography
50 TERRA ET AQUA MAIN MEMBERS
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#156 #150 - -AUTUMN MARCH 20182019 516 IADC TERRA ET AQUA #156 - AUTUMN 2019 ALWAYS READY TO MEET NEW CHALLENGES
IADC stands for ‘International Association of Dredging Companies’ and is the global umbrella organisation for contractors in the private dredging industry. IADC is dedicated to promoting the skills, integrity and reliability of its members as well as the dredging industry in general. IADC has over one hundred main and associated members. Together they represent the forefront of the dredging industry.
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