BUNDESANSTALT FUR¨ WASSERBAU Karlsruhe • Hamburg • Ilmenau
Technical Report
Mathematical Model UnTRIM
Validation Document
– Version June 2004 (1.0) – BUNDESANSTALT FUR¨ WASSERBAU Karlsruhe • Hamburg • Ilmenau
Technical Report
Mathematical Model UnTRIM
Validation Document
– Version June 2004 (1.0) –
Contributors: Vincenzo Casulli (Trento University, Italy) G¨unther Lang (BAW, Germany)
Version-Date: June 2004
Version-no.: 1.0
Trento ¡ Hamburg — August 2004
¡ ¡ The Federal Waterways Engineering and Research Institute ¡ Wedeler Landstr. 157 D-22559 Hamburg ([+49] (0)40 81908 - 0 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Summary
This document is the validation document for the mathematical model UnTRIM. The docu- ment is organized conforming to Guidelines for documenting the validity of computational mod- elling software [IAHR, 1994]. The subject of this document is the validation of a computational model. The term compu- tational model refers to software whose primary function is to model a certain class of phys- ical systems, and may include pre- and post-processing components and other necessary ancillary programmes. Validation applies primarily to the theoretical foundation and to the computational techniques that form the basis for the numerical and graphical results pro- duced by the software. In the context of this document, validation of the model is viewed as the formulation and substantiation of explicit claims about applicability and accuracy of the computational results. This preface explains the approach that has been adopted in organizing and presenting the information contained in this document.
Standard validation document
This document conforms to a standard system for validation documentation. This system, the Standard Validation Document, has been developed by the hydraulic research indus- try in order to address the need for useful and explicit information about the validity of computational models. Such information is summarized in a validation document, which accompanies the technical reference documentation associated with a computational model. In conforming to the Standard, this validation document meets the following require- ments:
1. It has a prescribed table of contents, based on a framework that allows separate quality issues to be clearly distinguished and described.
2. It includes a comprehensive list of the assumptions and approximations that were made during the design and implementation of the model.
3. It contains claims about the performance of the model, together with statements that point to the available substantiating evidence for these claims.
4. Claims about the model made in this document are substantiable and bounded: they can be tested, justified, or supported by means of physical or computational experi- ments, theoretical analysis, or case studies.
5. Claims are substantiated by evidence contained within this document, or by specific reference to accessible publications.
6. Results of validation studies included or referred to in this document are reproducible. Consequently the contents of this document are consistent with the current version of the software.
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7. This document will be updated as the process of validating the model progresses.
Organization of this document
Chapter 1 contains a short overview of the computational model and introduces the main issues to be addressed by the validation process. The model overview includes information about the purpose of the model, about pre- and post-processing options and other software features, and about reference versions of the software. Validation priorities and approaches are briefly described, and a list of related documents is included. Chapter 2 summarizes the available information about the validity of the computational core of the model. In this chapter, claims are made about the range of applicability of the model and about the accuracy of computational results. Each claim is followed by a brief statement regarding its substantiation. This statement indicates the extent to which the claim has in fact been substantiated and points to the available evidence. Chapter 3 contains such evidence, in the form of brief descriptions of relevant validation studies. Each description includes information about the purpose and approach of the study, and a summary of main results and implications. A glossary and complete list of references are contained in this document too.
A word of caution
This document contains information about the quality of a complex modelling tool. Its pur- pose is to assist the user in assessing the reliability and accuracy of computational results, and to provide guidelines with respect to the applicability and judicious employment of this tool. This document does not, however, provide mathematical proof of the correctness of re- sults for a specific application. The reader is referred to the License Agreement for pertinent legal terms and conditions associated with the use of the software. The contents of this validation document attest to the fact that computational modelling of complex physical systems requires great care and inherently involves a number of uncer- tain factors. In order to obtain useful and accurate results for a particular application, the use of high-quality modelling tools is necessary but not sufficient. Ultimately, the quality of the computational results that can be achieved will depend upon the adequacy of available data as well as a suitable choice of model and modelling parameters.
Electronic standard validation document
This document is also available in electronic form in Portable Document Format (PDF). The electronic version may be read using the ACROBAT READER software which is available for many computer platforms.
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Acknowledgements
Ralph T. Cheng is responsible for the development of a version of the UnTRIM at the U. S. Geological Survey (USGS). Continuing discussions and exchanging of modeling ideas with regards to UnTRIM validations and applications with Ralph T. Cheng are acknowledged. During the first International UnTRIM Users Workshop in Verona, June 7 – 9, 2004, it was decided to update this document, to conform to the actual version of the software (June 2004). Positive and stimulating feedback from the users to continue working on this type of document is herewidth gratefully acknowledged too.
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Contents
1 Introduction 1 1.1 Model overview ...... 1 1.1.1 Purpose ...... 1 1.1.2 Properties of the computational model ...... 2 1.1.3 Unstructured orthogonal grid ...... 2 1.1.4 Pre- and post-processing and other software features ...... 2 1.1.5 Version information ...... 4 1.2 Validation priorities and approaches ...... 4 1.3 Related documents ...... 4
2 Model validity 5 2.1 Physical system ...... 5 2.2 Model functionality ...... 6 2.2.1 Applications ...... 6 2.2.2 Processes ...... 11 2.3 Conceptual model ...... 14 2.3.1 Governing equations ...... 14 2.3.2 Assumptions and approximations ...... 17 2.3.3 Claims and substantiations ...... 18 2.4 Algorithmic implementation ...... 21 2.4.1 Unstructured orthogonal grid ...... 21 2.4.2 Assumptions and approximations ...... 24 2.4.3 Claims and substantiations ...... 33 2.5 Software implementation ...... 35 2.5.1 Implementation techniques ...... 35 2.5.2 Claims and substantiations ...... 37
3 Validation studies 40 3.1 Analytical test cases ...... 41 3.1.1 Wave propagation: rectangular basin ...... 41 3.1.2 Free barotropic oscillations: rectangular basin ...... 42 3.1.3 Free barotropic oscillations: circular basin ...... 43 3.1.4 Free oscillations: non-hydrostatic pressure ...... 44 3.1.5 Tidal forcing: flat bottom ...... 46 3.1.6 Tidal forcing: varying bathymetry ...... 47 3.1.7 Wind driven flow: flat bottom ...... 48 3.1.8 Internal seiches ...... 49 3.1.9 Steady density induced flow ...... 50 3.1.10 Wetting and Drying ...... 51 3.1.11 Solitary wave: flat bottom ...... 52 3.1.12 Solitary wave: varying bottom ...... 53
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3.2 Laboratory test cases ...... 54 3.2.1 Wave flume ...... 54 3.3 Schematic test cases ...... 56 3.3.1 Lock exchange flow: hydrostatic pressure ...... 56 3.3.2 Lock exchange flow: non-hydrostatic pressure ...... 58 3.3.3 Wave pattern in a square basin ...... 59 3.3.4 Short waves in a harbour basin ...... 60 3.3.5 Advection in a curved channel ...... 63 3.4 Examples from real-world applications ...... 66 3.4.1 Hydrostatic and non-hydrostatic flow in Venice Lagoon ...... 66 3.4.2 Tidal flow and salt transport ...... 70 3.4.3 River flow ...... 71 3.4.4 Storm surge ...... 72 3.4.5 Suspended sediment transport ...... 73 3.4.6 Transport of cooling water from a power plant ...... 74
References 75
A Glossary 78
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List of Figures
1 Unstructured orthogonal grid ...... 3 2 Definition of total water depth ...... 15 3 Orthogonal unstructured grid ...... 22 4 Special unstructured orthogonal grids ...... 22 5 Mixed grid with grid refinement ...... 23 6 Software concept ...... 38 7 Free oscillations: analytical, hydrostatic and non-hydrostatic solution . . . . . 45 8 Wave flume: flume geometry ...... 54 9 Wave flume: results at 13.5 m from open boundary ...... 55 10 Wave flume: results at 15.7 m from open boundary ...... 55 11 Wave flume: results at 19 m from open boundary ...... 55 12 Lock exchange flow: hydrostatic computation ...... 57 13 Lock exchange flow: non-hydrostatic computation ...... 58 14 Harbour basin: grid ...... 61 15 Harbour basin: incoming waves ...... 61 16 Harbour basin: diffraction of waves ...... 62 17 Harbour basin: diffraction and reflection of waves ...... 62 18 Advection in U-channel: square grid ...... 64 19 Advection in U-channel: flow aligned grid ...... 64 20 Advection in U-channel: upwind on a square grid ...... 65 21 Advection in U-channel: flux limiter on a square grid ...... 65 22 Advection in U-channel: flux limiter on a flow aligned grid ...... 65 23 Venice Lagoon: unstructured orthogonal grid ...... 68 24 Venice Lagoon: vertical velocity for hydrostatic/non-hydrostatic pressure . . 69
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List of Tables
1 Venice Lagoon: model performance ...... 66
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List of Abbreviations
Abbreviation Full Name BAW The Federal Waterways Engineering and Research Institute USGS U. S. Geological Survey r. h. s. right hand side terms of a linear system
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List of Variables
Name Unit Description C – tracer concentration
Ci k – C for the i-th polygon and k-th layer
CB – prescribed bottom concentration
CB i – CB for i-th polygon
CT – prescribed surface concentration
CT i – CT for i-th polygon 1/2 Cz m /s Chezy coefficient 3
Di k m /s diffusive flux coefficient for i-th polygon and k-th layer 3
D j k m /s diffusive flux coefficient for j-th side and k-th layer
I3 – number of computational prisms £ I3 – I3 at open boundaries (with prescribed water level) Ii – number of computational prisms above polygon i
J3 – number of computational faces £ J3 – J3 at open boundaries (with prescribed water level) Jj – number of computational faces above side j H m total water depth
Hi m H for the i-th polygon
Hj m H for the j-th side
Hmin m minimum allowed water depth H Kh m2/s horizontal eddy diffusivity Kh m2/s Kh for j-th side and k-th layer j k Kv m2/s vertical eddy diffusivity v 2 v
K 1 m /s K at i-th polygon and top/bottom of the k-th layer ¦ i k 2 L m basin length M – index for the surface z-layer
Nc – number of species
Nd – number of source/sink locations
Np – number of polygons/linear tridiagonal systems
Npr – number of red polygons (red-black-sorting) £ Np – Np along open boundaries (with prescribed water level) Ns – number of sides
Nsi – number of internal sides (share two polygons)
Ns f – number of last side with Dirichlet boundary condition Nv – number of vertices
Nz – number of level surfaces Nτ – number of substeps
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List of Variables (continued)
Name Unit Description 2 Pi m area of the i-th polygon 3
Qi k m /s advective flux coefficient for i-th polygon and k-th layer 3
Q j k m /s advective flux coefficient for j-th side and k-th layer
Si – number of sides for the i-th polygon
S · – sides j for outflow-faces of polygon i, layer k i k S – sides j for inflow-faces of polygon i, layer k i k
νv ∆ ∆
¦ ¦
a j k 1 m abbreviation for t/ z j k 1 ¦ j k 1 c m/s wave celerity 3
di k m /s flux coefficient for polygon i and k-th layer 3
d j k m /s flux coefficient for side j and k-th layer f 1/s Coriolis parameter s fdep – probability of deposition for suspended sediments s fres – probability of resuspension for suspended sediments g m/s2 gravitational acceleration h m bathymetric depth
h j m h for the j-th side
hL m depth for permanently dry land
i – index for the i-th polygon of a grid
µ
i´ j 1 – index for left polygon of the j-th side
µ i´ j 2 – index for right polygon of the j-th side
j – index for the j-th side of a grid
µ j ´i l – side index for l-th side of i-th polygon
k – index for the k-th layer of a grid µ
kb ´i – bottom layer index k for polygon i µ
kt ´i – surface (top) layer index k for polygon i µ
kb ´ j – bottom layer index k for side j µ kt ´ j – surface (top) layer index k for side j 1/3 kStr m /s Strickler coefficient l – loop index m – index for the bottom z-layer s 2 mres kg/(m s) resuspension rate for suspended sediments n – index for the n-th time step nε – maximum number of iterations for iterative solvers
ns f – number of sides with Dirichlet boundary condition
pi l – neighbour polygon to polygon i sharing side l p m2/s2 normalised pressure
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List of Variables (continued)
Name Unit Description 2 2
pi k m /s p for i-th polygon and k-th layer 2 2 pa m /s normalised atmospheric pressure 2 2
pa i m /s pa for i-th polygon q m2/s2 non-hydrostatic normalised pressure component 2 2
qi k m /s q for i-th polygon and k-th layer
2 2 q˜i k m /s qi k non-hydrostatic pressure correction
rT – wind friction coefficient
rT j – rT for the j-th side
rB – bottom friction coefficient
rB j – rB for the j-th side
r j k – ratio of consecutive gradients ¦
si l – sign function 1 for i-th polygon and l-th side t s time
tn s time level
tr s radiation time ∆t s numerical time step u m/s velocity in x-direction
u j k m/s normal velocity component for j-th side and k-th layer
u˜ j k m/s u j k after hydrostatic pressure step £
u m/s u interpolated at time level t j k j k n ua m/s wind velocity in x-direction
ua j m/s ua for j-th side
v m/s velocity in y-direction
v j k m/s velocity component orthogonal to u j k
v˜j k m/s v j k after hydrostatic pressure step £
v m/s v interpolated at time level t j k j k n va m/s wind velocity in y-direction w m/s velocity in z-direction
w 1 m/s w for i-th polygon at top/bottom of k-th layer ¦ i k 2
w˜ 1 m/s w 1 after hydrostatic pressure step
¦ ¦
i k 2 i k 2 £
w 1 m/s w 1 interpolated at time level t
¦
n ¦
i k i k 2 2 ws m/s settling velocity of sediments s s
w 1 m/s w for i-th polygon at top/bottom of k-th layer ¦ i k 2 x,y,z m cartesian coordinates
z 1 m z-coordinate of the k-th horizontal level surface k · 2
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List of Variables (continued)
Name Unit Description
z0 m roughness length ∆z m thickness of z-layer
∆zk m thickness of the k-th z-layer ∆ ∆
zi k m z for i-th polygon and k-th layer ∆ ∆
z j k m z for j-th side and k-th layer Ω 1/s angular speed of rotation of earth about its axis Φ – geographic latitude Φ, φ – flux limiter functions Ψ – contribution to flux limiter function due to ratio of con- secutive gradients α B m/s bottom flux parameter α α B i m/s B for i-th polgon α T m/s surface flux parameter α α T i m/s T for i-th polgon β B m/s bottom flux parameter β β B i m/s B for i-th polgon β T m/s surface flux parameter β β T i m/s T for i-th polgon δ m distance between adjacent polygon centers
δ j m δ for the j-th side
δmin m minimum allowed distance between polygon centers η m water surface elevation
η £ m specified water level at open boundaries
ηi m η for the i-th polygon
η˜ i m ηi after hydrostatic pressure step εη – tolerance for free-surface iterative solver
εq – tolerance for non-hydrostatic pressure iterative solver
Ô 2 2 γ · B m/s bottom friction factor rB u v γ γ B j m/s B for j-th side
Ô 2 2
µ ·´ µ γ ´ T m/s wind friction factor rT ua u va v γ γ T j m/s T for j-th side κ – von Karman constant λ m length of the side of a polygon, wave length
λ j m λ for the j-th side νh m2/s horizontal eddy viscosity νv m2/s vertical eddy viscosity
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List of Variables (continued)
Name Unit Description νv m2/s νv for i-th polygon and k-th layer i k νv 2 νv
1 m /s for j-th side at top/bottom of k-th layer ¦ j k 2 ρ kg/m3 water density ρ 3 ρ
i k kg/m for i-th polygon and k-th layer 3 ρa kg/m density of air 3 ρ0 kg/m reference density of water ρs kg/m3 density of suspended sediments σ – substep (subcycle) index θ – impliciteness factor τ 2 B N/m bottom shear stress τx 2 τ B N/m x-component of B τy 2 τ B N/m y-component of B τ 2 T N/m wind shear stress τx 2 τ T N/m x-component of T τy 2 τ T N/m y-component of T τr s relaxation time (radiation boundary condition) ∆τ s time substep size
ωl – weighting factor
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List of Operators
Name Unit Description 2 ∆h 1/m discretisation of the horizontal Laplacian F – explicit finite difference operator
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List of Vectors and Matrices
Name Unit Description
A j m positive definite matrix for j-th side 2 G j m /s vector with r. h. s. terms for j-th side
U j m/s vector of velocities u j k for j-th side
U˜ j m/s U j after hydrostatic pressure step ∆ ∆
Z j m vector of thicknesses z j k for j-th side
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1 Introduction
This chapter alludes to the model as a complete software product, and clarifies the relation of that which is being validated to the rest of the software. It includes brief descriptions of pre- and post-processing options, as well as an explanation of the modular structure of the computational core of the model.
1.1 Model overview 1.1.1 Purpose
The primary purpose of the computational model UnTRIM is to solve various one-, two- and three-dimensional, time-dependent, non-linear differential equations related to hydrostatic and non-hydrostatic free-surface flow problems on an unstructured orthogonal grid to cover problems with complicated geometry. The equations solved are mathematical descriptions of physical conservation laws for
¯ water volume (continuity equation),
¯ linear momentum (Reynolds-averaged Navier-Stokes (RANS) equations), and
¯ tracer mass (transport equation), e. g. for
– salt, – heat (temperature), and – suspended sediments or passive pollutants applied to different types of waters (rivers, lakes, estuaries, coastal seas, etc.). The above
mentioned equations are solved numerically. The following physical quantities can be ob-
µ
tained in dependence on three-dimensional space ´x y z and time t:
η ´ µ
¯ water surface elevation x y t with regard to a reference surface (e. g. mean sea level),
´ µ ´ µ ´ µ
¯ current velocity u x y z t , v x y z t , w x y z t ,
´ µ
¯ non-hydrostatic pressure component q x y z t , and
´ µ ¯ tracer concentration C x y z t , e. g. temperature, salinity, concentration of suspended sediments or passive pollutants.
When the computational model UnTRIM is used in one- or two-dimensional mode (with one z-layer in vertical direction) the results for u, v and C will be the respective depth averaged values for current velocity and tracer concentration.
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1.1.2 Properties of the computational model
The computational model UnTRIM can be characterized by means of the following distin- guished properties:
¯ complicated boundaries can be fitted and local grid refinements can be made to meet the needs of resolving fine spatial resolution in various numerical modelling tasks, which results in an accurate description of geometry (unstructured orthogonal grid, see Figure 1 on the following page);
¯ can be applied for one- and two-dimensional vertically averaged as well as hydrostatic or non-hydrostatic three-dimensional problems;
¯ terms that control the numerical stability are treated implicitely, and the remaining terms explicitely (semi-implicit finite differences);
¯ the equations are solved in the original plane without invoking neither horizontal nor vertical co-ordinate transformations;
¯ fluid and tracer mass is conserved locally and globally;
¯ proves to be computationally efficient and robust;
¯ the software consists of a computational core and a separate user interface, to ease adaption to local needs as well as integration with proprietary software.
1.1.3 Unstructured orthogonal grid
µ The horizontal computational domain ´x y must be covered with a set of non-overlapping convex polygons. Each side of a polygon is either a boundary line or a side of an adjacent polygon. Moreover, it is assumed that within each polygon there exists a point (hereafter called a center) such that the segment joining the centers of two adjacent polygons and the side shared by the two polygons have a non-empty intersection and are orthogonal to each other (for further details see Figure 1 on the next page).
1.1.4 Pre- and post-processing and other software features
A brief, integral description of the whole model as a stand-alone piece of software is given in the following. This includes optional add-on modules, as well as components which do not belong to the computational core of the model but which are essential to the computational model as a whole.
Software concept The core of the computational model UnTRIM can be regarded as a separate software package. Encapsulated methods and data of the core can be only accessed by means of a user interface module (see Section B.4 on page 66 for further details). The latter interconnects user data and/or user software to the computational core of the model. The
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CENTER POINTS of polygons circumcircle polygon 2 INTERSECTION POINT
circumcircle polygon 1 shared side of two polygons
segment joining centers of two adjacent polygons
Polygon 1
orthogonal intersection
circumcircle polygon 1 Polygon 2
VERTICES of polygons
Figure 1: Unstructured orthogonal grid. Some key terms of the unstructured grid concept are ex- plained inside the figure. Water surface elevation, non-hydrostatic pressure component and tracer concentration are computed at the center points and are assumed to be constant inside each polygon. Normal velocity components are computed at the intersection points and are assumed to be constant along each side. different routines of the user interface module (e. g. user set initial conditions)are called from the computational core during progression of the running program. For transfer of data (e. g. boundary data or computed results) as well as other services a comprehensive set of easy-to-use get- and set- functions is available to the user (see Section B.2 on page 24 as well as Section B.1 on page 1 for further details). In this way the user is free to shape the predefined routines of the user interface according to his or her needs. This gives plenty of freedom to connect the computational core with different user data structures or user specific software packages.
Related Software The computational model UnTRIM does not depend on other stan- dard mathematical software packages. But there are some software components (indepen- dent programs) which are essential to the computational model as a whole. Examples of the latter are:
¯ grid generator: a specific grid generator does not belong to the UnTRIM model; any grid generator who is able to generate unstructured orthogonal grids can be used — but use of the following grid generator is highly recommended anyway:
– JANET1 .
More informations about this grid generator can be found at SMILECONSULT2 .
1JANET: http://www.baw.de/vip/abteilungen/wbk/Publikationen/pkb/janet/janet-de.html 2SMILECONSULT: http://www.smileconsult.de/en/index en.html
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¯ plot generator: due the existence of the user interface the computed results can be converted to any desired file format and therefore different plot generators can be ap- plied to display UnTRIM results; up to now the following plot generators were used successfully:
– UnTRIM Studio, a 3D visualization and graphical user interface package was developed by OSWALDO LANZ3 , supported by USGS contract under Ralph T. Cheng, which is freely available for UnTRIM users and may run on different com- puter platforms;
– and see postprocessing4 of BAW-AK.
For a detailed description of any of these items please refer to separate documents or reports available on the web5.
1.1.5 Version information
The contents of this document is consistent with the current version of the UnTRIM software, released June 2004.
1.2 Validation priorities and approaches
The main issue is the description of the theoretical and numerical foundations of the com- putational model UnTRIM. At first the physical processes are described by means of the governing differential equations used together with some fundamental assumptions and/or simplifications. Possible consequences with respect to the applicability of the model will be discussed (conceptual model). After that the algorithmic implementation is presented which deserves as a prerequisite for the subsequent numerical treatment of the physical problem. Finally the software implementation is explained to demonstrate the conversion of the al- gorithmic implementation into a computer program. In the last chapter the computational model is validated against different analytical solutions. But also results from other applica- tions are shown to demonstrate the applicability of the model to real-world problems.
1.3 Related documents
Further documents related to the current version of the computational model UnTRIM can be found on the web6.
3OSWALDO LANZ: mailto:[email protected] 4http://www.baw.de/vip/abteilungen/wbk/Methoden/hnm/untrim/hnm untrim-de.html#pos 5http://www.baw.de/vip/abteilungen/wbk/Methoden/hnm/untrim/hnm untrim-de.html 6http://www.baw.de/vip/abteilungen/wbk/Methoden/hnm/untrim/hnm untrim-de.html#doku
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2 Model validity
This chapter summarizes all available information pertaining to the validation of the com- putational core of the model. This includes the assumptions and approximations that were introduced during the design and implementation of the model. It further includes claims about the applicability and/or accuracy of (aspects of) the model, together with statements about the substantiations of those claims. The nature of a claim and its substantiation varies depending on the subject, as explained below, under the headings of the various subsections in which they appear. Claims should be as explicit as possible and provide useful information about model validity. Substantiation should be thorough but brief, which can be achieved by using references. Note that a substantiation may be incomplete, due to the nature of the claim, or because the evidence is not (yet) available. In such cases it is important to admit this rather than to invent a substantiation that appears convincing. These claims and substantiations together comprise the essential information in this doc- ument. The remainder of the document serves either to provide context, necessary back- ground material or substantiating evidence.
2.1 Physical system
This section describes the physical system or systems being modelled. It is described what is being modeled, rather than how it is being modeled. The computational model UnTRIM can be considered as an abstract model for the follow- ing time-dependent, non-linear physical system:
¯ a water body with a free surface and fully developed turbulent flow, taking into ac- count
– conservation of volume, – propagation of short and long waves at the free surface, – advective transport of linear momentum, – horizontal turbulent diffusion of linear momentum, – vertical turbulent diffusion of linear momentum, – hydrostatic or non-hydrostatic pressure, – Coriolis acceleration, – horizontal and vertical gradients of water density, – energy losses due to bottom friction (bottom shear stress), – wind friction at the free surface (wind shear stress), – sources and sinks of water;
¯ transport of dissolved substances, e. g., salt, heat (temperature) or suspended sedi- ments, in the water body, where
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– advective transport, – horizontal turbulent diffusion, – vertical turbulent diffusion, – settling of suspended sediments (settling velocity), – flux of suspended sediments at the bottom (deposition and resuspension), – heat flux at the free surface, – sources and sinks of salt, heat (temperature) and suspended sediments in the wa- ter column
are taken into account;
¯ the system can have complex bathymetry and the modelling domain may be of any shape;
¯ drying and wetting of parts of the modelling domain (e. g. tidal flats) can be part of the problem.
2.2 Model functionality
This section describes the functionality of the model by referring to specific instances or configurations of the physical system described in Section 2.1 on the page before. It consists of claims about what the model is actually able to represent, and (to the extent that this is possible) how well it does so. For the purposes of this section the model can be regarded as a black box, taking input information and producing computational results.
2.2.1 Applications
This section presents an overview of the domain of applicability of the model. This is done by making claims about the types of practical and realistic situations in which the model can be employed, and showing the nature and quality of the information that the model is capable of generating in those situations. The purpose of providing the reader with an inventory of application types is that it allows him to quickly recognize whether the model is indeed suitable for the application he has in mind.
Tidal dynamics of estuaries
¯ input data:
– bathymetry of the modelling domain, given on a unstructured orthogonal grid; – bottom roughness for all grid points; – data at open boundaries:
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£ water surface elevation and salinity in dependence on space and time along the seaward open boundary,
£ fresh water flow (river inflow), and
£ other sources and sinks inside the modelling domain, together with the salin- ity of the sources; – time- and spatially varying wind speed and/or atmospheric pressure; – model steering parameters and physical parameters.
¯ computational results (for all grid points and each time step):
1. water surface elevation, 2. three-dimensional depth dependent current velocity, and 3. three-dimensional depth dependent salinity.
¯ example validation studies:
1. Hydrostatic and non-hydrostatic flow in Venice Lagoon, Section 3.4.1 on page 66; 2. Tidal flow and salt transport, Section 3.4.2 on page 70.
Notice: the computational model can be applied in the same way for two-dimensional depth-averaged studies of tidal dynamics of estuaries as well.
Short waves in a harbour basin
¯ input data:
– bathymetry of the modelling domain, given on a unstructured orthogonal grid; – data at open boundaries:
£ water surface elevation (incoming waves) along the open boundary of the harbour basin; – model steering parameters and physical parameters.
¯ computational results (for all grid points and each time step):
1. water surface elevation, 2. three-dimensional depth dependent current velocity, and 3. three-dimensional depth dependent non-hydrostatic pressure component.
¯ example validation studies:
1. Short waves in a harbour basin, Section 3.3.4 on page 60.
2. Model validity Page 7 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Stationary flow in rivers
¯ input data:
– bathymetry of the modelling domain, given on a unstructured orthogonal grid. – bottom roughness for all grid points. – data at open boundaries:
£ water surface elevation at the downstream open boundary,
£ fresh water flow (river flow) at the upstream boundary, and
£ other sources and sinks inside the modelling domain. – model steering parameters and physical parameters.
¯ computational results (for all grid points and each time step):
1. water surface elevation, and 2. three-dimensional depth dependent current velocity.
¯ example validation studies:
1. River flow, Section 3.4.3 on page 71.
Notice: the computational model can be also applied to study two-dimensional depth- averaged stationary river flow problems.
Transport of suspended sediments in rivers and estuaries
¯ input data:
– bathymetry of the modelling domain, given on a unstructured orthogonal grid; – bottom roughness for all grid points; – critical shear stresses for erosion and deposition for all grid points in dependence on the suspended sediment fraction; – erodability of deposited sediments for all sediment fractions at all grid points; – settling velocity of all sediment fractions for all computational points; – data at open boundaries:
£ water surface elevation, salinity (if required) and suspended sediment con- centration for all sediment fractions in dependence on space and time along the seaward open boundary,
£ fresh water flow (river inflow) and concentration of suspended sediment (for all sediment fractions) as well as salinity (if required), and
£ other sources and sinks inside the modelling domain, together with the salin- ity (if required) as well as the suspended sediment concentration for the sources.
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– time- and spatially varying wind speed and/or atmospheric pressure. – fluxes of sediment from/towards the bottom. – model steering parameters and physical parameters.
¯ computational results (for all grid points and each time step):
1. water surface elevation, 2. three-dimensional depth dependent current velocity, 3. three-dimensional depth dependent salinity (if required), and 4. three-dimensional depth dependent suspended sediment concentration for all sediment fractions.
¯ example validation studies:
1. Suspended sediment transport, Section 3.4.5 on page 73.
Notice: the computational model can be also applied to study two-dimensional depth- averaged suspended sediment transport problems as well.
Storm surges in estuaries and coastal seas
¯ input data:
– bathymetry of the modelling domain, given on a unstructured orthogonal grid; – bottom roughness for all grid points; – data at open boundaries:
£ water surface elevation and salinity in dependence on space and time along the seaward open boundary,
£ fresh water flow (river inflow), and
£ other sources and sinks inside the modelling domain, together with the salin- ity of the sources. – time- and spatially varying wind speed and atmospheric pressure. – model steering parameters and physical parameters.
¯ computational results (for all grid points and each time step):
1. water surface elevation, 2. three-dimensional depth dependent current velocity, and 3. three-dimensional depth dependent salinity.
¯ example validation studies:
1. Storm surge, Section 3.4.4 on page 72.
Notice: the computational model can be applied in the same way for two-dimensional depth-averaged studies of storm surges.
2. Model validity Page 9 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Free oscillations of lakes due to atmospheric forcing
¯ input data:
– bathymetry of the modelling domain, given on a unstructured orthogonal grid; – bottom roughness for all grid points; – model steering parameters and physical parameters; – time- and spatially varying wind speed and atmospheric pressure.
¯ computational results (for all grid points and each time step):
1. water surface elevation, 2. three-dimensional depth dependent current velocity.
¯ example validation studies:
1. Free barotropic oscillations in a rectangular basin, Section 3.1.2 on page 42; 2. Free barotropic oscillations in a circular basin, Section 3.1.3 on page 43.
Notice: problems related to free oscillations can be also studied using the two-dimensional depth-averaged mode of the computational model.
Transport of cooling water from a power plant in a tidal estuary
¯ input data:
– bathymetry of the modelling domain, given on a unstructured orthogonal grid; – bottom roughness for all grid points; – data at open boundaries:
£ water surface elevation, salinity (if required) and temperature in dependence on space and time along the seaward open boundary,
£ fresh water flow (river inflow) as well as salinity (if required) and tempera- ture, and
£ other sources and sinks inside the modelling domain, e. g. for intake and outtake related to the operation of the power plant, together with the increase in temperature caused by the cooling process. – time- and spatially varying wind speed, atmospheric pressure, solar radiation, water vapor content, etc. – flow of heat from/towards the bottom. – flow of heat from/towards the atmosphere. – model steering parameters and physical parameters.
¯ computational results (for all grid points and each time step):
2. Model validity Page 10 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
1. water surface elevation, 2. three-dimensional depth dependent current velocity, 3. three-dimensional depth dependent salinity (if required), and 4. three-dimensional depth dependent temperature.
¯ example validation studies:
1. Transport of cooling water from a power plant, Section 3.4.6 on page 74.
Notice: the computational model can be also applied to study two-dimensional depth- averaged suspended sediment transport problems as well.
2.2.2 Processes
This section further characterizes the domain of applicability of the model. This is done by making claims about the individual physical processes or phenomena that the model was designed to represent. The idea is to break down the physics into elements that are as simple as possible, yet still meaningful. The information contained in this section supplements that in the previous section. It is intended to allow the reader to judge whether or not the model is suitable for his purpose, by considering separately the individual processes that play a role in the application he has in mind.
Propagation of long waves
¯ description: for long waves (in shallow water) the vertical acceleration can be assumed to be negligible and the pressure to be hydrostatic. Under these assumptions the celer- ity of the wave does only depend on gravity and water depth. It is also independent of the wave length. In the linearized case the wave travels without changing shape.
¯ validation studies:
1. Wave propagation in a rectangular basin, Section 3.1.1 on page 41.
Propagation of short waves
¯ description: for short waves the vertical acceleration of the fluid can no longer be ne- glected and the pressure is non-hydrostatic. The celerity of the wave then depends on gravity, water depth as well as wave length.
¯ validation studies:
1. Free oscillations in a non-hydrostatic situation, Section 3.1.4 on page 44; 2. Solitary wave in a wave channel with flat bottom, Section 3.1.11 on page 52; 3. Solitary wave in a wave channel with varying bottom, Section 3.1.12 on page 53;
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4. Wave pattern in a square basin, Section 3.3.3 on page 59; 5. Short waves in a harbour basin, Section 3.3.4 on page 60.
Diffraction of short waves
¯ description: corners or obstacles much smaller than the wave length cause diffrac- tion of wave energy. Thus wave energy is also transferred to regions which are in the shadow of the incoming waves. Diffraction at corners or at the end of walls are common examples for this process.
¯ validation studies:
1. Short waves in a harbour basin, Section 3.1.7 on page 48.
Reflection of short waves
¯ description: at solid walls incoming wave energy is fully or partly reflected. Due to this, an increase of the maximum wave amplitude occurs in front of a reflector. Typical reflectors are solid vertical walls.
¯ validation studies:
1. Short waves in a harbour basin, Section 3.1.7 on page 48.
Wind induced currents
¯ description: wind causes a wind stress at the free surface which results in a wind induced free surface current and a corresponding near bottom return flow. In closed embayments a wind induced setup of the water level can occur.
¯ validation studies:
1. Wind driven currents in a rectangular basin with flat bottom, Section 3.1.7 on page 48; 2. Storm surge, Section 3.4.4 on page 72.
Free barotropic oscillation
¯ description: an oscillating free surface (seiche) in a natural basin is the result of a bal- ance between inertia and gravity. This phenomenon can be induced by means of at- mospheric pressure variations at the free surface of a water body.
¯ validation studies:
1. Free barotropic oscillations in a rectangular basin, Section 3.1.2 on page 42; 2. Free barotropic oscillations in a circular basin, Section 3.1.3 on page 43.
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Baroclinic flow
¯ description: baroclinic flow is a result of varying density in horizontal directions. Near the bottom the baroclinic current is oriented towards decreasing density whereas near the surface an opposing return flow is induced.
¯ validation studies:
1. Steady density induced flow, Section 3.1.9 on page 50; 2. Lock exchange flow with hydrostatic pressure, Section 3.3.1 on page 56; 3. Lock exchange flow with non-hydrostatic pressure, Section 3.3.2 on page 58.
Internal seiches
¯ description: if stratification is present in natural waters oscillatory motions with some well-defined frequency occur that cause isopycnals to tilt and oscillate around an equi- librium position. These oscillatory phenomena are referred to as internal waves.
¯ validation studies:
1. Internal seiches in a rectangular basin, Section 3.1.8 on page 49.
Wetting and drying
¯ description: in estuaries and coastal seas with significant tidal range quite often vast areas of land (tidal flats) are subsequently covered and unrecovered with water during each tidal cycle.
¯ validation studies:
1. Wetting and Drying, Section 3.1.10 on page 51.
Salt transport
¯ description: in estuaries and coastal seas the salinity is not a constant. Advection as well as turbulent mixing may cause significant variations of salinity throughout each tidal cycle in larger parts of the modelling domain.
¯ validation studies:
1. Advection in a curved channel, Section 3.3.5 on page 63; 2. Tidal flow and salt transport, Section 3.4.2 on page 70.
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Heat transport
¯ description: in estuaries and coastal seas the temperature may vary strongly. Advec- tion as well as turbulent mixing, heat exchange with the atmosphere and the bottom, in- and outtake of water due to power plants (going to be used for cooling purposes) may cause significant variations of temperature throughout each tidal cycle in larger parts of the modelling domain.
¯ validation studies:
1. Advection in a curved channel, Section 3.3.5 on page 63; 2. Transport of cooling water from a power plant Section 3.4.6 on page 74.
Suspended sediment transport
¯ description: transport of suspended sediment contributes to a large amount to the total sediment transport in estuaries and coastal seas. In the stratified and/or gradient zone of an estuary a turbidity maximum, formed by accumulation of suspended sediments, is a common phenomenon.
¯ validation studies:
1. Advection in a curved channel, Section 3.3.5 on page 63; 2. Suspended sediment transport, Section 3.4.5 on page 73.
2.3 Conceptual model
This section describes technical aspects of the conceptual model that are relevant to the vali- dation process. In particular, it addresses the differences between the conceptual model and the actual physics.
2.3.1 Governing equations
The governing three-dimensional equations describing free-surface flows can be derived from the Navier–Stokes equations after averaging over turbulent time-scales (Reynolds- averaged Navier-Stokes equations (RANS)). Such equations express the physical principle of conservation of volume, mass and momentum. The momentum equations for an incom-
pressible fluid have the following form
∂ ∂ ∂ ∂ ∂ ∂2 ∂2 ∂ ∂
u u u u p h u u v u
· · · · ν · ν u v w fv · (1)
∂t ∂x ∂y ∂z ∂x ∂x2 ∂y2 ∂z ∂z
∂ ∂ ∂ ∂ ∂ ∂2 ∂2 ∂ ∂
v v v v p h v v v v
· · · · · ν · ν u v w fu · (2)
∂t ∂x ∂y ∂z ∂y ∂x2 ∂y2 ∂z ∂z
∂ ∂ ∂ ∂ ∂ ∂2 ∂2 ∂ ∂ ρ
w w w w p h w w v w
· · · · ν · ν · u v w 2 2 g (3) ∂t ∂x ∂y ∂z ∂z ∂x ∂y ∂z ∂z ρ0
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Validation Document – Version June 2004 (1.0)
µ ´ µ ´ µ
where u´x y z t , v x y z t and w x y z t are the velocity components in the horizontal x,
µ y and vertical z-directions, respectively; t is the time; p´x y z t is the normalized pressure defined as the pressure divided by a constant reference density ρ 0; f is the Coriolis param- eter; g is the gravitational acceleration and νh and νv are the coefficients of horizontal and vertical eddy viscosity, respectively. These coefficients can be derived from an appropriate turbulence closure model. Here, it will only be assumed that νh and νv are prescribed non- negative functions of space and time and no specific closure model is required. Finally, ρ denotes the water density and ρ0 is a constant reference density. The volume conservation is expressed by the following incompressibility condition
∂u ∂v ∂w
· · 0 (4) ∂x ∂y ∂z
Integrating the continuity equation (4) over depth and using a kinematic condition at the
free-surface leads to the following free-surface equation [Casulli, V. and Cheng, R.T., 1992]
∂η ∂ η ∂ η
·
∂ ∂ udz · ∂ vdz 0 (5)
t x h y h
µ
where h´x y is the prescribed bathymetry measured from the undisturbed water surface
µ ´ µ ´ µ·η ´ µ and η ´x y t is the free-surface elevation. Thus, H x y t h x y x y t is the total water depth (see Figure 2). At open boundaries, where the water level is going to be prescribed, a radiation bound-
ary condition is taken to be
η η £
∂η ∂ ∂ η η
·
∂ ∂ udz · ∂ vdz τ (6)
t x h y h r where η £ is the specified water level an τr is a problem-specific relaxation time.
η = water level elevation (positive upwards) H = η + h h = bottom (positive downwards) η H = total water depth MSL
h
Figure 2: Relation between water surface elevation η, bathymetric depth h and total water depth H.
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The mass conservation of a scalar variable is expressed by the following differential equa-
tion
s
µ ∂´ µ ∂´ µ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂´
C uC vC w w C h C h C v C
· · · · · K K K (7) ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂y ∂z ∂z where C denotes the concentration of any scalar transported specie such as salinity, temper- ature, suspended sediment or a passive substance which may be relevant in water quality studies; ws is a specified settling velocity assumed to be nonzero when the sediment trans- port is being modeled; Kh and Kv are prescribed non-negative horizontal and vertical diffu- sivity coefficients, respectively. The system is closed by an equation of state which relates the water density to the con-
centration of each scalar variable. This equation takes the form
ρ´ µ
ρ C (8)
µ The pressure p´x y z t in equations (1)–(3) can be decomposed into the sum of its hydro- static and a non-hydrostatic component. The hydrostatic pressure component is determined
from the vertical momentum equation (3) by neglecting the convective and the viscous ac-
µ celeration terms. Thus, p´x y z t can be expressed as
η ρ
ρ 0
µ ´ µ· η ´ µ · ζ · ´ µ p´x y z t pa x y t g x y t z g d q x y z t (9)
z ρ0
µ where the first term pa ´x y t on the right hand side of equation (9) corresponds to the at-
mospheric pressure at the free surface, while the second and the third terms represent the
µ barotropic and the baroclinic contributions to the hydrostatic pressure and q´x y z t denotes the non-hydrostatic pressure component. Consequently, the momentum equations (1)–(3)
can also be written as
∂η ∂ η 2 2 ∂ ρ ρ ∂ ∂ ∂ ∂ ∂
du pa 0 q h u u v u
ζ · ν · ν fv g g d 2 2 · (10)
dt ∂x ∂x ∂x z ρ0 ∂x ∂x ∂y ∂z ∂z
∂η ∂ η 2 2 ∂ ρ ρ ∂ ∂ ∂ ∂ ∂
dv pa 0 q h v v v v
· ζ · ν · ν · fu g g d 2 2 (11)
dt ∂y ∂y ∂y z ρ0 ∂y ∂x ∂y ∂z ∂z
∂ ∂2 ∂2 ∂ ∂
dw q h w w v w
· · ν · ν (12) dt ∂z ∂x2 ∂y2 ∂z ∂z
When the hydrostatic approximation is made, equation (12) is neglected and q 0 is as- sumed throughout. In this case the non-hydrostatic component of the pressure is assumed not to have a significant effect on the resulting flow. The boundary conditions for the momentum equations at the free-surface are specified by the prescribed wind stresses as ∂ ∂
v u v v
γ ´ µ ν γ ´ µ η ν u u v v at z (13) ∂z T a ∂z T a
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where ua and va are the wind velocity components in the x and y direction, respectively, and γ
T is defined according to Õ
2 2
´ µ ·´ µ γ T rT ua u va v (14) wherein rT is a non-negative wind stress coefficient. At the sediment–water interface the bottom friction is specified by ∂ ∂
v u v v
γ ν γ ν u v at z h (15) ∂z B ∂z B
where γ is defined according to B Ô
2 2
· γ B rB u v (16) wherein rB is a non-negative bottom friction coefficient. Typically, rB can be given by the Nikuradse formula, or by fitting it to a turbulent boundary layer. The boundary conditions for the scalar transport equation at the free-surface and at the sediment–water interface, respectively, are specified by prescribing the mass fluxes as fol- lows ∂
v C s
´ µ α · β ´ µ η K w w C C C at z (17) ∂z T T T and ∂
v C s
·´ µ α · β ´ µ K w w C C C at z h (18) ∂z B B B α β α β where T , T and B , B are prescribed non-negative parameters, independent from the C and defined at the top and bottom, respectively; CT and CB are the prescribed top and bottom concentrations.
2.3.2 Assumptions and approximations
A list of assumptions and approximations that have been introduced into the formulation of the conceptual model:
1. continuous medium assumption,
2. incompressibility approximation,
3. Reynolds-averaged Navier-Stokes equations (RANS),
4. Boussinesq approximation,
5. eddy-viscosity concept,
6. Coriolis parameter,
7. parameterization of bottom friction,
8. parameterization of wind friction, and
9. no fluid-sediment interaction.
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2.3.3 Claims and substantiations
Claims about the validity of the conceptual model, and statements about the substantiation of these claims. This section serves to defend the choices that were made in formulating the conceptual model (i. e. the assumptions and approximations listed in Section 2.3.2 on the page before, and to explain the implications of those choices for applicability and/or accuracy. The claims and substantiations in this section should collectively imply that the conceptual model is indeed suitable for modelling the physics described in Section 2.1 on page 5, and that it leads to the functionality claimed and demonstrated in Section 2.2 on page 6.
Continuous medium assumption According to [Malvern, L.E., 1969] the ad- jective continuous refers to the simplifying concept underlying the analysis: we disregard the molecular structure of matter and picture it without gaps or empty spaces. We suppose that all functions are continuous functions, except at a finite number of surfaces separating regions of continuity (e. g. free surface between water and air). This implies that the deriva- tives of the functions are continuous too. This assumption is valid as long as we are dealing with phenomena much larger in size than the free length of particles between successive collisions.
Incompressibility If a fluid is treated as incompressible the density of the fluid will depend on temperature and the concentration of dissolved substances but not on pressure. Under the assumption that a fluid element (volume) does neither exchange heat nor dis- solved substances with its surroundings (isentropic deformation) the fluid can be regarded as incompressible [Batchelor, G., 1967] if the
1. fluid particle velocity is much smaller than the speed of sound, and the
2. phase speed of the disturbances (e. g. speed of the free surface waves) is also small compared to the speed of sound, and the
∂ρ ρ 3. vertical scale of motion (water depth) must be small compared to a mean value of ∂z (scale height).
These assumptions normally hold true for typical oceans and estuaries and therefore the incompressibility condition (4) can be applied together with a simplified equation of state (8). Precisely spoken this means that volume rather than mass is conserved. Therefore effects like thermal expansion or cooling cannot be reproduced exactely by the computational model.
Reynolds-averaged Navier-Stokes equations Because of the nature of turbulent flow it does not appear practical to solve for the detailed (turbulent) velocities in rivers, estuaries, coastal seas and other similar water bodies. Around 1880 Osborne Reynolds suggested the first time, to split the variables u, v, w and p into a mean and a fluc- tuating component. After insertion into the Navier-Stokes equations and subsequent time- averaging we obtain the Reynolds-averaged Navier-Stokes equations. They contain new
2. Model validity Page 18 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0) terms, the so-called Reynolds-stresses, which describe the mean momentum transfer due to turbulence [Daily, J.W. and Harleman, D.R.F., 1966]. Solutions to these equations can only be obtained by making special assumptions about the nature of the turbulence (turbulence closure).
ρ Boussinesq approximation In natural waters variations of density ρ 0 are normally small when compared with the reference density ρ0. Boussinesq said, that, if the density variations are fairly small, to a first approximation we can neglect their effect on the mass (i. e. inertia) of the fluid but we must retain their effect on the weight. That is, we must include the buoyancy effects but can neglect the variations in horizontal acceleration due to mass variations with density, which are at most 3 – 4 % if we are dealing with natural water bodies [Pond, S. and Pickard, G.L., 1983].
Eddy-viscosity concept In the Boussinesq eddy-viscosity concept the Reynolds- stresses are parameterized by the product of a so called eddy-viscosity with the spatial gra- dient of the mean quantities. For simplicity the resulting eddy-viscosity tensor is assumed to be diagonal. The eddy-viscosity tensor depends on the state of the turbulent motion and has therefore to be determined by a separate turbulence model/parametrization.
Coriolis parameter Equations (1) – (3) are formulated in a rotating reference frame fixed to the earth’s surface. This gives rise to an apparent acceleration called Coriolis accel-
eration. In three-dimensional space the Coriolis acceleration is given by ½ ¼ Ω Φ
2Ωvsin Φ 2 wcos
Ω Φ
Coriolis acc 2 usin 2Ωucos Φ
This expression is approximated by
¼ ½
fv
Coriolis acc fu 0
Ω Φ with f 2 sin , which is a good approximation as long as the vertical velocity w is much smaller than the horizontal components u and v. It is also assumed that the vertical Coriolis acceleration can be neglected with respect to gravity g. If the horizontal area being consid- ered is not too large (100 km) then we can work in a plane tangent to the sphere. This plane is called the f -plane, because for small north-south distances the Coriolis parameter f may be taken to be constant at its value at the center of the area.
Bottom friction At the sediment–water interface the natural boundary condition for
µ´ µ the velocity equals ´u v w 0 0 0 . Because of the fact, that the details of the flow in the bottom boundary layer are not modelled by the computational model UnTRIM, an estimate
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x y
τ ´τ τ µ for the bottom shear stress B must be made. Any law of bottom friction for B B which
can be written in the form Ô
x 2 2
ρ · τ
B 0rB u v u (19) Ô
τy 2 2 ρ · B 0rB u v v (20) can be used to determine rB . Normally rB is not a constant but depends on the distance between the lowermost grid point and the bottom as well as on the local bed roughness. In the following, a few examples should illustrate the computation of rB for different bottom friction parametrizations:
g
2 : Chezy C
z
2 κ2 037H
ln : Nikuradse
rB z0 (21)
g
1 : Strickler k2 H 3 Str
Wind friction The same arguments apply at the water-air interface. Any law of wind
x y
τ τ µ
friction for ´ T T which can be written in the form Õ
x 2 2
ρ ´ µ ·´ µ ´ µ τ
T 0rT ua u va v ua u (22) Õ
y 2 2
ρ ´ µ ·´ µ ´ µ τ T 0rT ua u va v va v (23)
can be used to determine rT . As an example rT can be computed from e. g.
ρ Õ
a 2 2 3
· · rT 0 63 0 066 ua va 10 (24) ρ0 which is the parametrization for rT according to [Smith, S.D. and Banke, E.G., 1975].
No fluid sediment interaction It is assumed that any (dissolved) specie (i. e. salt, heat, suspended sediments) is transported with the velocity of the flow and no momentum transfer between water and dissolved specie occurs. In the vertical direction the velocity of the flow and that of a suspended sediment particle may differ by the settling velocity ws (see Eq. 7), but again, no momentum transfer between water and sediment is taken into account.
Flux of scalar species through the free-surface boundary At the free-surface the flux for scalar species must be prescribed setting the r. h. s. terms including
β α
T T and CT of Equation 17 on page 17 to appropriate values. E. g.
α 0
T β
T 0 (25)
CT 0 can be used to prescribe a no-flux condition. In dependence on the character of the scalar specie, as well as the physical processes taken into account at the free-surface boundary, the settings for the above mentioned variables may differ significantly from the example shown.
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Flux of scalar species through the bottom boundary Also at the sediment–water interface (bottom boundary) the flux for scalar species must be prescribed. In dependence on the type of the scalar specie considered this can be a heat-flux, a sediment-
β α flux etc. From Equation 18 on page 17 it can be seen that the r. h. s. terms including B B and
CB can be used to specify the total flux. In dependence on the scalar specie considered any appropriate parametrization can be used for that purpose. According to [Mehta, A.J., 1988], e. g. s mres s
α f B ρs res (26)
describes the resuspension of suspended sediments at the bed. In addition to that µ s s
β w f B dep (27)
CB 0 can be used to take into account deposition of suspended sediments. The total flux can be either inward, outward or zero in dependence on the relative strength of the individual processes as well as the actual concentration C near the bottom.
Gravity The gravitational acceleration g is not a constant but varies from place to place.
Actually g is determined from the (mean) geographic latitude Φ according to
¡ ¡
6 3
· ¡ Φ ¡ Φ g 9 80616 1 0 5 9 10 cos 2 637 10 cos (28) and is assumed to be constant throughout the modelling domain.
2.4 Algorithmic implementation
This section describes the technical aspects of the algorithmic implementation that are rele- vant to the validation process. In particular, it addresses the differences between the algo- rithmic implementation and the conceptual model.
2.4.1 Unstructured orthogonal grid
µ Before discretizing equations (4)–(18), the horizontal ´x y domain is covered by a set of non- overlapping convex polygons. Each side of a polygon is either a boundary line or a side of an adjacent polygon. Moreover, it is assumed that within each polygon there exist a point (hereafter called center) such that the segment joining the centers of two adjacent polygons and the side shared by the two polygons have a non empty intersection and are orthogonal to each other (see Figure 3 on the following page). A grid like this is called an unstructured orthogonal grid [Casulli, V. and Zanolli, P., 1998], [Casulli, V. and Walters, R.A., 2000]. The center of a polygon does not necessarily coincide with its geometrical center. Examples of structured orthogonal grids are, of course, the rectangular finite difference grids, as well as a grid of uniform equilateral triangles. In these particular cases the center of each polygon can be identified with its geometrical center (see Figure 4 on the next page).
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λj(i,2)
λ j(i,3) Pi
λj(i,1) δ λj(i,4) j Pi(j,1)
Pi(j,2)
Figure 3: Orthogonal unstructured grid.
Figure 4: Two special unstructured orthogonal grids. Rectangular finite difference grid (left) and grid of uniform equilateral triangles (right). Points indicate polygon centers and stippled lines show segments joining the centers of two adjacent polygons.
2. Model validity Page 22 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Figure 5: Unstructured orthogonal grid with regions of different homogeneous grid resolution and interfacing transition layer.
An example of an unstructured orthogonal grid is the Delaunay triangulation of a set
of points Qi i 1 2 N, provided that the triangulation includes only acute triangles. In this case the centers of the triangles can be identified by the vertices of the correspond- ing Dirichlet tassellation [Rebay, S., 1993]. Similarly, the Voronoi regions of a Dirichlet tas- sellation determined by Qi also form an orthogonal grid with Qi being the centers of the polygons. Using co-volumes defined by Voronoi-Delaunay triangulation, a similar orthog- onality condition has been proposed for the solution of the incompressible Navier-Stokes equations [Nicolaides, R.A., 1993]. Another example grid (see Figure 5) demonstrates that an unstructured orthogonal grid can be constructed in such a way to allow for regions of different grid resolution (using squares) within one grid. The regions are separated by a transition layer made out of trian-
gles and trapezoids.
µ
Once the ´x y domain has been covered with an unstructured orthogonal grid, one has
Np polygons, each having an arbitrary number of sides Si 3 i 1 2 Np. Let Ns be the
total number of sides in the grid and let λ j j 1 2 Ns be the length of each side. The
µ ´ µ sides of the i-th polygon are identified by an index j ´i l l 1 2 Si, so that 1 j i l Ns.
Similarly, the two polygons which share the j-th side of the grid are identified by the indices
µ ´ µ ´ µ ´ µ i´ j 1 and i j 2 so that 1 i j 1 Np and 1 i j 2 Np. The nonzero distance between the centers of two adjacent polygons which share the j-th side is denoted with δ j (see Figure 3). Along the vertical direction a simple finite difference discretization, not necessarily uni- form, is adopted. By denoting with z 1 a given level surface, the vertical discretization step
k · 2
is defined by ∆z z 1 z 1 k 1 2 N .
k · z k 2 k 2 The three-dimensional space discretization consists of prisms whose horizontal faces are the polygons of a given orthogonal grid and whose height is ∆zk. The discrete velocities
2. Model validity Page 23 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0) and water surface elevation are defined at staggered locations as follows. The water surface ηn elevation i , assumed to be constant within each polygon, is located at the center of the i-th polygon; the velocity component normal to each face of a prism, assumed to be constant over the face, is defined at the point of intersection between the face and the segment joining the centers of the two prisms which share the face; the non-hydrostatic pressure component q n i k and the concentrations Cn are located at the center of the i-th polygon and half way between i k
z 1 and z 1 . Finally, the water depth h is specified and assumed constant on each side of
· j k 2 k 2 a polygon.
2.4.2 Assumptions and approximations
A list of assumptions and approximations that have been introduced in order to convert the conceptual model into an algorithmic implementation. A semi-implicit, fractional step scheme is used in order to obtain an efficient numeri- cal algorithm whose stability is independent from the free-surface wave speed, wind stress, vertical viscosity and bottom friction. In the first step the provisional water velocity and sur- face elevation are computed by neglecting the implicit contribution of the non-hydrostatic pressure. The gradient of surface elevation in the horizontal momentum equations (10)– (11) as well as the velocity in the free-surface equation (5) are discretized by the so called θ-method [Casulli, V. and Walters, R.A., 2000]. Moreover, for stability, the wind stress, the vertical viscosity and the bottom friction will be discretized implicitly. In the second frac- tional step the provisional velocity and surface elevation are corrected by including the non- hydrostatic pressure terms, which is calculated in such a fashion that the resulting velocity field is locally and globally mass conservative [Casulli, V., 1999].
First step: hydrostatic pressure The first step of calculations is performed by neglecting the implicit contribution of the non-hydrostatic pressure. The resulting velocity field and water surface elevation at the new time level are not yet final and will be denoted by u˜, w˜ and η˜, respectively. Since equations (10)–(12) are invariant under solid rotation of the x and y axis on the horizontal plane, a consistent semi-implicit finite difference discretization for the horizontal
velocity component on each vertical face of a prism takes the following form ∆t
n·1 n n n n n
µ· ´ θµ ´η η
u˜ Fu 1 g q q
´ µ ´ µ ´ µ ´ µ j k j k i j 2 i j 1 i j 2 k i j 1 k
δ j
· · ·
n·1 n 1 n 1 n 1
∆ ∆ u˜ u˜ u˜ u˜
·
t t ·
n·1 n 1 v j k 1 j k v j k j k 1
µ·
ν ν
θg ´η˜ η˜ 1 1
´ µ ´ µ
· δ i j 2 i j 1 ∆ n j k ∆ n j k ∆ n
z 2 z 1 2 z 1
j
· j k j k 2 j k 2
n
· k m j m j 1 M j (29) where un denotes the horizontal velocity component normal to the j-th side of the grid, at j k vertical level k and time step n. The positive direction for un has been chosen to be from
j k
µ ´ µ i´ j 1 to i j 2 . F is an explicit finite difference operator which accounts for the contribu- tions from the discretization of the air pressure, Coriolis, baroclinic pressure, advection and
2. Model validity Page 24 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0) horizontal friction terms. A particular form for F can be given in several ways, such as by using an Eulerian–Lagrangian scheme (see, e. g., [Staniforth, A. and Temperton, C., 1986], [Casulli, V., 1990], [Casulli, V. and Cheng, R.T., 1992], [Casulli, V. and Walters, R.A., 2000]). When the Coriolis, the baroclinic pressure, the advection and horizontal friction terms are n n
neglected in the above scheme, F reduces to the identity operator, i. e., Fu u . Including j k j k all the terms above and by using an Eulerian–Lagrangian discretization, a form for F can be
chosen as
θ ·θ n· n M j
p p ∆
´ µ ´ µ
a i j 2 a i j 1 t
£ £
n £ h n n n
· ∆ ∆ · ∆ ν ∆ ω ρ ρ ∆
Fu u tfv t t u g ∑ z
(30)
µ ´ µ j k j k j k h j k i´ j 2 i j 1 j
δ j δ jρ0 k where u £ denotes the horizontal velocity component normal to the j-th side of the grid inter- j k polated at time t at the end of the lagrangian trajectory; v £ denotes the horizontal velocity
n j k
£
ω ω · · component orthogonal to u ; 1 2 and 1 l k 1 k 2 M . The lagrangian j k k j
n·1
µ trajectory is calculated by integrating the velocity backwards in time from node ´ j k at t n
to its location at time t . ∆h is the discretization of the horizontal Laplacian.
θ For stability the implicitness factor θ has to be chosen in the range 12 1 (see [Casulli, V. and Cattani, E., 1994]). Moreover, the vertical space increment ∆zn is defined j k as the distance between two consecutive level surfaces, except near the bottom and near the free surface where ∆zn is the distance between a level surface and the bottom or free-surface, j k respectively. Thus, in general, ∆zn depends on the spatial location and near the free-surface j k it also depends on the time step. The vertical space increment ∆zn is also allowed to van- j k ish in order to account for wetting and drying. Of course, the discrete momentum equa- n n tions (29) are not defined at the grid points characterized by ∆z 0. Finally, m and M , j k j j
n
1 m j M j Nz, denote the lower and upper limit for the k-index representing the bottom and the top finite difference stencil, respectively. As indicated, m and M depend on their spatial location and M may also change with the time level to account for the free-surface n dynamics. For notational simplicity, the subscript and the superscript to m j and M j will be omitted in the following development.
The values of u˜n·1 above the free-surface and below the bottom in equations (29) are elim- j k inated by means of the boundary conditions (13)–(16) which yield the following difference
formulas · n·1 n 1
u˜ u˜
·
· ·
v j M 1 j M n·1 n 1 n 1
γ ´ µ
ν 1 u u˜
(31) · j M ∆ n T j a j j M
2 z 1 · j M 2
and · n·1 n 1
u˜ u˜
·
v j m j m 1 n·1 n 1 γ
ν 1 u˜ (32)
j m ∆ n B j j m
2 z 1
j m 2 In analogy with equation (29), by neglecting the implicit contribution of the non-hydrostatic pressure, a consistent semi-implicit finite difference discretization for the vertical component of the velocity at the top face of each computational prism is derived from equation (12) and
2. Model validity Page 25 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
takes the following form ∆t
n·1 n n n
´ θµ
w˜ Fw 1 1 q q
·
1
·
i k 1 i k · i k i k ∆ n
2 2 z 1 ·
i k 2
¾ ¿
· · ·
n·1 n 1 n 1 n 1
∆ w˜ 3 w˜ 1 w˜ 1 w˜ 1
· · ·
t v i k 2 i k 2 v i k 2 i k 2
· ν ν
· ∆ n i k 1 ∆ n i k ∆ n
z 1 z z
·
· i k 1 i k i k 2
n
· k mi mi 1 Mi 1 (33) where the explicit finite difference term Fw accounts only for the contributions from the discretization of the advection and horizontal friction terms. When an Eulerian–Lagrangian
discretization is chosen, F is taken to be £
n £ h
· ∆ ν ∆
Fw 1 w 1 t hw 1 (34)
· · · i k 2 i k 2 i k 2
n·1
For each i, the set of equations (33) form a linear, tridiagonal system with unknowns w˜ 1 · i k 2 on the same water column. The coefficient matrix of these systems is symmetric and positive definite. Thus, the provisional vertical component of the velocity can be readily determined by a direct method. Equations (29) also constitute a set of linear tridiagonal systems which, however, are coupled to the unknown water surface elevation η˜ n·1. In order to determine
ηn·1 ˜ i and for numerical stability, the provisional horizontal velocity field is required to satisfy, for each i, the discrete analogue of the free-surface equation (5). Thus, a semi-implicit finite volume discretization for the free-surface equation (5) at the center of each polygon is taken
to be
Si M ·
n·1 n n n 1
η θ∆ λ ∆ η
˜
µ
P P t ∑ s ´ ∑ z u˜
µ
i i i l ´
µ
i i j i l j i l k j ´i l k
l 1 k m Si M
n n
´ θµ∆ λ ∆
µ
1 t ∑ s ´ ∑ z u (35)
µ ´ µ
i l j i l j ´i l k j i l k l 1 k m
where Pi denotes the area of the i-th polygon and si l is a sign function associated with the
orientation of the normal velocity defined on the l-th side of the polygon i. Specifically, si l 1
if a positive velocity on the l-th side corresponds to outflow, si l 1 if a positive velocity on
the l-th side corresponds to inflow to the i-th water column. Thus, si l can be written as
´ µ · ´ µ
i j i l 2 2i i j i l 1
si l (36)
´ µ ´ µ i j i l 2 i j i l 1
Equations (29) and (35) constitute a linear system of at most NzNs · Np equations. This system has to be solved at each time step in order to calculate the provisional field variables u˜n·1 j k
ηn·1 and ˜ i throughout the flow domain.
Since a linear system of NzNs · Np equations can be quite large even for modest values of Nz, Ns and Np, the system of equations (29) and (35) is first decomposed into a set of Ns independent tridiagonal systems of Nz equations and one linear system of Np equations.
2. Model validity Page 26 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Specifically, upon multiplication by ∆zn and after including the boundary conditions (31) j k
and (32), equations (29) and (35) are first written in matrix form as ¢
∆t £
· ·
n n·1 n n 1 n 1 n
θ ∆
A U˜ G g η˜ η˜ Z (37)
µ ´ µ j j j i´ j 2 i j 1 j δ j
Si
· ·
n 1 n n n 1
η θ∆ λ ∆
η ˜
˜
µ
P P t ∑ s ´ Z U
µ
i i i l ´
µ i i j i l j i l j ´i l
l 1
Si
n n
´ θµ∆ λ ∆
µ
1 t ∑ s ´ Z U (38)
µ ´ µ i l j i l j ´i l j i l
l 1
where U˜ , ∆Z, G and A are defined as follows:
¾ ¿ ¾ ¿
u˜n·1 ∆zn
j M j M
" # " #
" # " #
n·1 ∆ n
" # " #
· u˜ z
˜ n 1 j M 1 n j M 1 ∆
# " #
U j " Z j
# " #
" . .
. .
u˜n·1 ∆zn
j m j m
¾ ¿ ·
n n ∆t n n n n n·1 n 1
µ· ´ θµ ´η η · ∆ γ
∆z Fu 1 g q q t u
µ ´ µ ´ µ ´ µ
j M j M δ j i´ j 2 i j 1 i j 2 M i j 1 M T j a j
" # "
n n ∆t n n n n #
" #
´ θµ ´η η µ·
∆z Fu 1 g q q
µ ´ µ ´ µ ´ µ
n δ ´ "
j M 1 j M 1 j i j 2 i j 1 i j 2 M 1 i j 1 M 1 #
G "
j . # "
. #
n n ∆t n n n n
´ θµ ´η η µ·
∆z Fu 1 g q q
µ ´ µ ´ µ ´ µ
j m j m δ j i´ j 2 i j 1 i j 2 m i j 1 m
¾ ¿
n n n·1 n
· γ ∆
∆z · a t a 0
j M 1 j 1
T
j M 2 j M 2
" #
" #
" # #
" n n n n n
" #
∆ · ·
a 1 z a 1 a 3 a 3
j M 1
n #
" j M 2 j M 2 j M 2 j M 2
A #
j "
" #
" #
" #
n n n n·1
∆ · · γ ∆
0 a z a t
1 j m 1 j
· · B j m 2 j m 2
n v n
ν ∆ ∆
with a 1 1 t z 1 .
¦ ¦ ¦ j k 2 j k 2 j k 2
˜ n·1 Formal substitution of the expressions for U j from equation (37) into (38) yields a dis-
crete wave equation for η˜ n·1 which is given by
Si
¡
λ ¡ n
´ µ
si l j i l
· ·
n·1 θ2∆ 2 ∆ 1∆ n 1 n 1 η η η
Pi ˜ g t ∑ Z A Z ˜ ˜
´ µ ´ µ
i i j i l 2 i j i l 1
µ
δ j ´i l
´ µ
l 1 j i l
Si
¡ n
n
η ´ θµ∆ λ ∆
µ
Pi i 1 t ∑ si l j ´i l Z U
µ j ´i l
l 1
Si
¡ n
1
θ∆ λ ∆
µ
t ∑ si l j ´i l Z A G (39)
µ j ´i l
l 1 n
Since matrix A j is positive definite, its inverse is also positive definite and therefore
1 n
∆ µ ∆ ´ Z A Z j is a non-negative number. Hence equations (39) constitute a linear, sparse
2. Model validity Page 27 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
ηn·1 system of Np equations for ˜ i . This system is strongly diagonally dominant, symmetric and positive definite. Thus, its unique solution can be efficiently determined by precondi- tioned conjugate gradient iterations until the residual norm becomes smaller than a given tolerance εη (see, e. g., [Golub, G.H. and van Loan, C.F., 1996]). Once the provisional free-surface location has been calculated, equations (37) constitute
˜ n·1 a set of linear, tridiagonal systems for U j . Each of these tridiagonal systems is independent of the others and is symmetric and positive definite. Thus, they can be conveniently solved
˜ n·1 by a direct method to determine U j throughout the computational domain.
Second step: non-hydrostatic correction In the second step of calcu-
· · n·1 n 1 ηn 1 lations the new velocity field u , w 1 and the new water surface elevation are
i ·
j k i k 2 computed by correcting the provisional values after including the non-hydrostatic pressure
terms. Specifically,
∆t
· · ·
n·1 n 1 n 1 n 1
θ
u u˜ q˜ q˜ (40)
µ ´ µ j k j k i´ j 2 k i j 1 k
δ j
∆t
· · ·
n·1 n 1 n 1 n 1
θ
w 1 w˜ 1 q˜ q˜ (41)
·
· · i k i k ∆ n i k 1 i k
2 2 z 1 · i k 2 where q˜ denotes the non-hydrostatic pressure correction which, in combination with the
provisional free-surface elevation, gives the pressure p. Specifically,
· ·
n·1 n 1 n 1
´ µ·
p g η˜ z q˜ (42) i k i k i k
In each computational cell below the free-surface, the discretized incompressibility condition (4), in finite volume form, is taken to be
Si
· ·
n n·1 n 1 n 1
λ ∆ · ´ µ ·
µ
∑ s ´ z u P w 1 w 1 0 k m m 1 M 1 (43) µ
i l ´ i
µ
j i l j i l k ´
· j i l k i k 2 i k 2
l 1
At the free-surface, the finite difference approximation of equation (5) is
Si M ·
n·1 n n n 1
η η θ∆ λ ∆
µ
P P t ∑ s ´ ∑ z u
µ
i i i l j i l ´
µ
i i j i l k j ´i l k
l 1 k m Si M
n n
´ θµ∆ λ ∆
µ
1 t ∑ s ´ ∑ z u (44)
µ ´ µ
i l j i l j ´i l k j i l k l 1 k m
n·1
which, by setting w 1 0 and by using the incompressibility condition (43), can also be
i m 2
written as
Si
· ·
n·1 n n n 1 n 1
η η θ∆ λ ∆ · θ∆
µ
P P t ∑ s ´ z u tPw 1
µ
i i i l ´ i
µ
i i j i l j i l M ´
j i l M i M 2
l 1 Si M
n n
´ θµ∆ λ ∆
µ
1 t ∑ s ´ ∑ z u (45)
µ ´ µ
i l j i l j ´i l k j i l k l 1 k m
2. Model validity Page 28 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
q˜ n·1 By assuming that the pressure in the surface cells is hydrostatic, the pressure correction i M
is obtained from the following hydrostatic relation
· · ·
n·1 n 1 n 1 n 1
´η µ ´ µ·
p g z g η˜ z q˜ (46) i M i M i M i M
Hence, equation (45) becomes
Si
· · ·
n·1 n n 1 n n 1 n 1
· ´η µ θ∆ λ ∆ θ∆ η
˜
µ
P q˜ gP g t ∑ s ´ z u g tPw
1 µ
i i i l ´ i
µ
i M i i j i l j i l M ´
j i l M i M 2
l 1 Si M
n n
´ θµ∆ λ ∆
µ
g 1 t ∑ s ´ ∑ z u (47)
µ ´ µ
i l j i l j ´i l k j i l k l 1 k m
A system of equations for the non-hydrostatic pressure correction q˜ n·1 is derived by substi- i k tuting the expressions for the new velocities from (40)–(41) into (43) and (47), respectively.
The following finite difference equations are obtained
$ %
· ·
· · · S n 1 n 1 n·1 n 1 n 1 n 1
i
q˜ q˜ q˜ q˜ q˜ q˜
´ µ ´ µ
·
θ2 2 n i j i l 1 k i j i l 2 k i k i k 1 i k 1 i k ∆ λ ∆ ·
µ
g t ∑ s ´ z P
µ
i l j i l j ´i l k δ i ∆ n ∆ n
µ
´ z 1 z 1
j i l k
· l 1 i k 2 i k 2
Si
· ·
n·1 n 1 n n 1
θ∆ ´ µ θ∆ λ ∆ ·
µ
g tP w˜ 1 w˜ 1 g t ∑ s ´ z u˜ k m m 1 M 1 (48)
µ
i i l j i l ´
´ µ
· j i l k j i l k i k 2 i k 2
l 1
and
· · · S n 1 n 1 n·1 n 1 i
q˜ q˜ q˜ q˜
´ µ ´ µ
i j i l 1 M i j i l 2 M
2 2 n i M 1 i M n·1
∆ λ ∆ θ ·
µ
g t ∑ s ´ z P P q˜
µ
i l j i l j ´i l M δ i ∆ n i i M
µ
´ z 1
j i l M
l 1 i M 2
Si
· ·
n·1 n n 1 n n 1
θ∆ θ∆ λ ∆ · ´η µ η
˜
µ
g tPw˜ 1 g t ∑ s ´ z u˜ gP
µ
i i l ´ i
µ
j i l ´ i i
j i l M j i l M i M 2
l 1 Si M
n n
´ θµ∆ λ ∆
µ
g 1 t ∑ s ´ ∑ z u (49)
µ ´ µ
i l j i l j ´i l k j i l k l 1 k m
The set of equations (48)–(49) forms a sparse linear system of at most NpNz equations. This system is diagonally dominant, with strict inequality corresponding to equations (49), more- over it is symmetric and positive definite. Thus, it can be solved by preconditioned conjugate gradient iterations until the residual norm is smaller than a given tolerance εq. At the solid impenetrable boundaries, the condition of zero normal flux is imposed through equations (40)–(41) which translates to a Neumann type of boundary conditions on equations (48)–(49). At the open boundaries either the normal velocity or the non- hydrostatic pressure should be specified. Accordingly, this translates into the Neumann or Dirichlet type of boundary condition, respectively. Once the non-hydrostatic pressure is computed, the corresponding horizontal velocity field is readily determined from equation (40), while the vertical component of the velocity can be obtained, equivalently, either from (41) or from the incompressibility condition (43)
n·1
which, by setting w 1 0, gives
i m 2
1 Si
· ·
n·1 n 1 n n 1
λ ∆ ·
µ
w 1 w 1 ∑ s ´ z u k m m 1 M 1 (50) µ
i l ´
µ
j i l j i l k ´
· i k i k j i l k 2 2 Pi l 1
2. Model validity Page 29 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
This latter choice guarantees that the resulting velocity field is exactly discrete divergence free for any value of the tolerance εq. This choice enables the above algorithm to perform hydrostatic calculations simply by choosing for εq a sufficiently large value. Finally, the new free-surface elevation is obtained by the hydrostatic relation (46) as fol- lows
q˜n·1
·
n·1 n 1 i M
· η η˜ (51) i i g
On the other hand, the discrete form of the pressure equation (9) is given by
· ·
n·1 n 1 n 1
´η µ·
p g z q (52) i k i k i k thus, the non-hydrostatic pressure component qn·1 is obtained by equating the right hand i k
sides of equations (42) and (52) which yields
· ·
n·1 n 1 n 1
·
q q˜ q˜ k m m 1 M (53)
i k i k i M
The numerical algorithm presented above includes the simulation of flooding and drying
n·1 of low lying areas. At each time step the new water depths H j at the polygon’s sides are
defined as
¢ £
· ·
n·1 n 1 n 1
· η · η
H max 0 h j h j (54)
µ ´ µ j i´ j 1 i j 2
The vertical grid spacings ∆zn·1 are updated accordingly. Thus, an occurrence of zero value j k
n·1 for the total depth H j implies that all the vertical faces separating prisms between the wa-
n·1
µ ´ µ ter column i´ j 1 and i j 2 are dry and may become wet at a later time when H j becomes positive. The height of a dry face and the corresponding normal velocity are taken to be zero.
Scalar transport By using the incompressibility condition (4), the mass conservation Equation (7) can also be written in a non-conservative form which implies the maximum principle. In order to derive a conservative scheme for Equation (7) that also yield the max-min property, it must be consistent with both, the discretized continuity Equation (43) and the discretized free surface Equation (44). Assuming that the vertical component of the velocity is positive, a general semi-implicit form of such a scheme is derived. Moreover, in order not to impose a further stability restriction on the hydrodynamic model, a sub-cycling approach
is used.
·
µ By denoting with S the set of vertical faces belonging to the prism ´i k (prism above i k the i-th polygon in the k-th layer) through which the water is leaving the respective prism and with S the set of vertical faces through which the water is entering the same prism and,
i k
µ ´ µ
moreover, with p´i j the polygon number for the neighbour of prism i k that shares the
µ vertical face ´ j k (face located in the k-th layer above the side j), thus, for each computa-
2. Model validity Page 30 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
tional prism Equation (7) is discretized as follows:
σ· σ σ σ σ
σ·1 1
· · · · ·
n· n n n n n
θ ·θ
Nτ Nτ Nτ Nτ n· Nτ n Nτ
∆ ∆τ
P ∆z C P z C ∑ Q C ∑ Q C
i i
µ
i k i k i k i k j k i k j k p´i j k
· ¾
j S j ¾S
i k i k
σ σ σ σ
· · ·
n· n n n
θ ·θ ·θ
n· Nτ n Nτ n Nτ Nτ
∆τ · ∆τ
Q 1 C Q 1 C ∑ d C C
´ µ
· i k i k 1 j k p i j k i k
i k 2 i k 2
·
j ¾S S
i k i k
σ· σ· σ·
σ·1 1 1 1
· · ·
n· n n n
θ ·θ
n· Nτ Nτ n Nτ Nτ
∆τ
· d 1 C C d 1 C C
·
· i k 1 i k i k i k 1
i k 2 i k 2
σ σ σ σ σ σ
· · · · ·
∆τ n· n n n n n
θ ·θ
Nτ n· Nτ Nτ Nτ n Nτ Nτ
ψ ψ
1 Q 1 C C 1 Q 1 C C
·
·
· i k 1 i k i k i k 1 2 i k 2 i k 2 i k 2 i k 2
σ σ σ
· · ∆τ n· θ n n
Nτ n· Nτ Nτ
ψ ·
∑ Q C C k m m 1 M (55)
µ
j k j k p´i j k i k ·
2
j ¾S S i k i k
where
θ ·θ ·θ ·θ
n· n n n n
λ ∆ Q z u Q P w
1 1
j j k i
¦ ¦ j k j k i k 2 i k 2 v
h K 1
¦ K
n n j k n i k 2
λ ∆
D z D 1 P
j i ¦ j k j k δ i k ∆ n
j 2 z 1 ¦ i k 2
are the advective and the diffusive flux coefficients, from which
1
θ ·θ
n· n n
d 1 max 0 D 1 Q 1 and (56)
¦
¦ ¦ i k
i k 2 2 2 i k 2
1
θ ·θ
n· n n
d max 0 D Q
(57) j k j k 2 j k
σ
n· Nτ µ can be computed in a straightforward manner. ψ´r is the additional contribution to the σ
n·
· Nτ
flux limiter function that depends on the ratio of consecutive gradients. For j ¾ S , r i k j k
1 ¦ n· and r 2 can be defined as i k
σ σ σ σ
· · ·
n· n n n
θ ·θ
n· Nτ Nτ n Nτ Nτ
·
∑ Q C C Q 1 C C
´ µ
k i k p i k i k i k 1 i k 2
σ
¾
· S
n Nτ 1 i k
r σ σ and (58)
·θ ·θ · j k ·
n τ n τ n n
· N N
∑ Q Q 1
C C k i k
µ
p´i j k i k 2
¾ S i k
σ σ σ σ
· · ·
n· n n n
θ ·θ
n· Nτ Nτ n Nτ Nτ
·
∑ Q C C Q 1 C C
´ µ
k i k p i k i k i k 1 i k 2
σ
¾
· S n Nτ 1 i k
r 1 σ σ (59)
·θ ·θ
· · ·
i k n τ n τ n n
· N N
2 ∑ Q Q 1
C C k i k
·
i k 1 i k 2
¾ S i k from which
σ σ ·
n· Nτ n Nτ n
Φ φ ψ
and (60) j k j k j k
σ σ ·
n· Nτ n Nτ n
Φ φ ψ
1 1 1 (61)
¦
¦ ¦ i k 2 i k 2 i k 2
2. Model validity Page 31 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0) can be computed in dependence on the flux limiter function Φ chosen. There are many ways to determine a form for Φ. A few possible choices that have been widely studied and applied
(see, e.g., [van Leer, B., 1979]; [Sweby, P.K., 1984]; [Roe, P.L., 1985]) are
¢ £
µ φ ´ µ
Minmod : Φ´r max min 1 r ; (62)
r · r
µ φ
van Leer : Φ´r max ; (63)
1 · r
¢ £
µ φ ´ µ ´ µ
Superbee : Φ´r max min 1 2r min 2 r (64)
µ ψ´ µΦ´ µ φ ψ´ µ φ so that Φ´r 2, r r 0 and r 0 for r 0 are always fulfilled, when is
given according to
¼ ½ ½ ¼ n
n 2D 1
¦
2D
n j k n i k 2
¬ ¬ ¬ ¬
φ φ
min 1 1 min 1
and (65)
¦
j k
¬ ¬ ¬
¬ i k
θ ·θ
n· 2 n
¬ ¬ ¬
¬Q Q 1
¦ j k i k 2 For more details please refer to [Casulli, V. and Zanolli, P., 2004].
σ·1
n· The values of C Nτ above the free-surface and below the bottom in equations (55) are i k eliminated by means of the boundary conditions (17)–(18) which yield the following differ-
ence formulas σ·
σ·1 1 ·
n· Nτ n Nτ σ·
σ·1 1
· C C ·
n n ·
v i M 1 i M n s Nτ n n n Nτ
´ µ α · β ´ µ
K 1 w 1 w 1 C C C (66)
1 i i i
· · ·
i M · i M ∆ n i M i M i M T T T
2 z 1 2 2 2 · i M 2
and σ·
σ·1 1 ·
n· Nτ n Nτ σ·
σ·1 1
· C C ·
n n
θ
v i m i m 1 n· s Nτ n n n Nτ
·´ µ α · β ´ µ
K 1 w w 1 C C C (67)
1 1 i i i
i m
i m ∆ n i m i m i m B B B
2 z 1 2 2 2
i m 2 Finally, according to inequality (37) in [Casulli, V. and Zanolli, P., 2004], a time sub-step limitation on ∆τ is as follows: σ n· Nτ Pi∆z
i k
¬ ¬ ¬ ¬
∆τ (68)
¬ ¬ ¬ ¬
θ ·θ ·θ
n· n n
· ·
¬ ¬ ¬
2 ∑ ¬ Q 2 Q 1 ∑ d
·
j k j k
i k 2
· ·
¾
j S j ¾S S
i k i k i k
For each σ 0 1 2 Nτ 1, equations (34) form a set of Np linear, tridiagonal systems
σ·1
n· with unknowns C Nτ on the same water column. The coefficient matrix of these systems is i k strongly diagonally dominant. Thus, the scalar concentration Cn·1 can be readily determined i k by solving these systems with a direct method in Nτ substeps. Clearly, since a finite volume approach is being used, this scheme is mass conservative. Moreover, one can show that the numerical solution so obtained also possesses the discrete minimum-maximum property provided that the substep size ∆τ is chosen to be sufficiently small to comply with a mild stability restriction due to the explicit discretization of the hori- zontal transport and diffusivity terms.
2. Model validity Page 32 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Once the new concentrations Cn·1 have been computed for each scalar variable, the equa- i k tion of state (8) is used to update the water density ρn·1 in every computational prism. i k In summary, at each time step, the new flow variables and scalar concentrations are
ηn·1
obtained by computing the provisional free-surface elevation ˜ i from linear system (39). · n·1 n 1
Next, the provisional velocity field u˜ , w˜ 1 is obtained from equations (29) and (33). The
·
j k i k 2 non-hydrostatic pressure correction q˜n·1 is obtained by solving the linear system (48)-(49).
i k
· · n·1 n 1 ηn 1 The new velocity field u , w 1 , the new free-surface elevation and the new non-
i
· j k i k 2 hydrostatic pressure component qn·1 are obtained from (40), (50), (51) and (53), respectively. i k Finally, the new concentrations Cn·1 for the scalar variables are computed from sub-cycling
i k
the discretized conservation equations (55) for σ 0 1 2 Nτ 1. The resulting new den- sity is given by the equation of state (8).
2.4.3 Claims and substantiations
Claims about the validity of the algorithmic implementation, and statements about the sub- stantiation of those claims. This sections serves to defend the choices that were made in the algorithmic implemen- tation (i. e. the assumptions and approximations listed in Section 2.4.2 on page 24), and to explain the implications of those choices for the applicability and/or accuracy. The claims and substabtiations in this section should collectively imply that the algorithmic implemen- tation is indeed suitable for modelling the physics described in Section 2.1 on page 5, and that it leads to the functionality claimed and demonstrated in Section 2.2 on page 6. The algorithm developed above is relatively simple, yet general and robust. A number of interesting properties concerning mass conservation, numerical accuracy, stability and generality are discussed below.
1. In the present scheme the local volume conservation is assured by the finite vol- ume form (43) used to discretize the incompressibility condition (4). Also, local two- dimensional and global volume conservation is guaranteed by equation (44) which is a finite volume discretization of the free-surface equation (5). Fluid mass is also con- served locally and globally because a finite volume form (55) used to discretize the transport equation (8).
2. When the polygons of the horizontal mesh are uniform rectangles, this algorithm re- duces to the semi-implicit finite difference model presented in [Casulli, V., 1999]. The present algorithm represents an extension of the uniform grid finite difference formu- lation to unstructured grids which adds considerable flexibility to the model.
3. The highest numerical accuracy is obtained when a uniform grid, such as equilateral triangles or uniform rectangles, is used. In these cases, the normal velocity on each face of the prism is located at the center point of the face and the centers of two adjacent prisms are equally spaced from the common face. Consequently, the discretization error for the gravity wave terms in equations (37)–(38) and for the non-hydrostatic
2. Model validity Page 33 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
pressure terms in (40)–(41) is second order in space. Moreover, these equations are also second order accurate in time for θ = 1/2. Since a uniform grid, in general, does not fit the boundaries and does not allow for local mesh refinements, an unstructured, non-uniform grid can be used with a somewhat larger discretization error. This error can be minimized when the polygon size and shape vary gradually through the flow domain [Gravel, S. and Staniforth, A., 1992].
4. The stability analysis of the semi-implicit finite difference method (37)–(38) has been carried out in [Casulli, V. and Cattani, E., 1994] in the case of barotropic and hydrostatic flow on a uniform rectangular grid and under the assumptions that the governing dif- ferential equations (4)–(12) are linear, with constant coefficients and defined on an infi-
nite horizontal domain, or with periodic boundary conditions on a finite domain. The
θ analysis shows that the method is stable in the von Neumann sense if 12 1 and if the operator F used to discretize the advection and horizontal viscosity terms is itself stable. Computational results on several test cases have indicated that no additional stability restrictions are required when a non-uniform unstructured mesh is used and when the hydrostatic assumption is removed. Thus, the stability of the present algo- rithm is independent of the celerity, wind stress, vertical viscosity and bottom friction. It does depend on the discretization of the advection and horizontal viscosity terms. When an Eulerian–Lagrangian method is used for the explicit terms, a mild limitation on the time step depends on the horizontal viscosity coefficient and on the smallest polygon size. A further mild limitation on the time step is imposed in baroclinic flows because the baroclinic pressure term in the momentum equation has been discretized explicitly. This limitation is related to the speed of internal waves which is typically smaller than surface wave speed. This method becomes unconditionally stable for barotropic flows when the horizontal viscosity terms are neglected.
5. From a purely algebraic point of view, the present formulation does not require the calculation of the provisional free-surface as determined in the first fractional step by the linear system of equations (39). Indeed, by choosing a tolerance εη sufficiently large that no iterations will be required by the conjugate gradient method to solve sys- tem (39), the complete pressure field, including the water surface elevation, will be entirely determined in the second computational step by system (48)–(49) even in the case of hydrostatic flow. For computational convenience, however, accurate calculation of the provisional free-surface from the system of equations (39) provides a better start- ing point for the iterations required by the preconditioned conjugate gradient method which determines the non-hydrostatic pressure correction. This is particularly the case for hydrostatic and quasi-hydrostatic calculations.
6. It is interesting to point out that the hydrostatic solution (also obtained with the semi- implicit algorithm described in [Casulli, V. and Walters, R.A., 2000]) can be naturally 0
computed from the above non-hydrostatic algorithm by setting q 0 and by choos- i j k ing a tolerance εq sufficiently large that no iterations will be required by the conjugate gradient method to solve system (48)–(49). In this way the resulting non-hydrostatic
2. Model validity Page 34 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
pressure correction q˜n·1 remains identically zero at every time step and the hydrostatic i j k velocity field results from (40), (50) and (51) as a particular case.
7. Another interesting consideration arises from the fact that if only one vertical layer
n
∆ is specified, one has 1 m M N and the vertical spacing z represents the to- z j 1 n tal water depth H j . In this particular case the discrete incompressibility condition (43) does not apply and equations (49) form a linear and homogeneous system. This implies that the resulting pressure correction is identically zero and, accordingly, the calculations of the second fractional step are not required. Moreover, one can easily verify that in this case equations (29) and (35) are a consistent semi-implicit discretiza- tion of the two-dimensional, vertically integrated shallow water equations (see, e. g. [Casulli, V. and Zanolli, P., 1998]).
8. The above property of the present algorithm leads to a general purpose computer code that can be used for both two-dimensional vertically averaged problems as well as hy- drostatic and non-hydrostatic three-dimensional problems. More importantly, when the three-dimensional model is applied to a typical coastal plain tidal embayment char- acterized by deep channels connected to large and flat shallow areas, a great saving in computing time and memory requirement is achieved, because the deep channels are correctly represented in three dimensions while the flat shallow areas are represented only in two dimensions.
9. The resulting algorithm for advection-diffusion equation (7) is locally and globally mass conservative. Moreover it satisfies a discrete maximum principle. Accordingly, the computed results are stable and are guaranteed to have oscillation free profiles. Higher resolution can be obtained by using appropriate flux limiters and, whenever possible, by using a grid that is oriented with the expected net flow.
10. From the discretized boundary conditions Equations (66) and (67) it can be seen, that
n n
β
in case β 0 0 and/or 0 0 the respective boundary fluxes near the surface and
T i B i σ·
σ·1 1 · n· n
C Nτ C Nτ the bottom are dependent on the sub-stepping concentrations i M and i m .In situations like these, residual fluxes during time step ∆t cannot be precisely controlled by the user with respect to the amount of the outgoing flux.
2.5 Software implementation
This section describes technical aspects of the software implementation that are relevant to the validation process. In particular, it addresses the implications of software implementa- tion choices and techniques for the technical quality of the computational model as a whole.
2.5.1 Implementation techniques
The following choices were made to convert the algorithmic implementation into software:
2. Model validity Page 35 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
1. standard FORTRAN90 [Adams, J.C. et al., 1992] was chosen as the programming lan- guage for the computational model UnTRIM;
2. no further standard mathematical software or libraries are needed to run the compu- tational model;
3. the computational core of the model is regarded as a separate software package, which consists out of a
(a) data storage module which contains all global data needed during execution of the routines belonging to the computational core, a (b) master subroutine which must be called once during initialization and once for each time step from the user’s main program. This procedure calls initialization routines, computational procedures as well as user interface routines. The following actions are initiated during the initialization step: i. set paths and names for standard input files (see Section B.4.1 on page 66); ii. read user data from standard input files: A. steering data and parameters (see Section D.2 on page 93), B. grid (see Section D.1 on page 90), and C. sources and sinks (see Section D.3 on page 95); iii. allocate dynamic memory; iv. check grid consistency (see Section B.3.2 on page 65); v. set various computational constants; vi. set initial state (see Section B.4.2 on page 66); vii. update some computational data (e. g. k-layer indices for the current position of the free surface); viii. print/write computational results (see Section B.4.4 on page 74). For each time step the following actions are initiated: i. set forcing terms (see Section B.4.3 on page 67); ii. compute explicitely discretized terms; iii. compute implicitely discretized terms and solve for the first hydrostatic pres- sure step (see Page 24); iv. solve for the (second) non-hydrostatic pressure correction step (see Page 28); v. solve tracer transport, lagged one timestep (see Page 30); vi. update some computational data (e. g. k-layer indices for the current position of the free surface); vii. check continuity (see Section B.3.1 on page 65); viii. print/write computational results (see Section B.4.4 on page 74). (c) large number of get- and set-interfaces (please refer to Section B.2 on page 24 as well Section B.1 on page 1 for further details) for transfer of data from/to the data
2. Model validity Page 36 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
storage module of the computational core and external software, e. g. between the user interface (see Section B.4 on page 66) and the computational core of the software;
4. a set of user interface procedures which are called from inside the computational core. The subsequent tasks are performed:
(a) set paths and names for standard input files (see Section B.4.1 on page 66); (b) set initial conditions at start time (see Section B.4.2 on page 66); (c) set forcing terms (see Section B.4.3 on page 67); i. set data along open boundaries (see Section B.4.3 on page 67); ii. set flow and concentrations for sources and sinks (see Section B.4.3 on page 68); iii. set bottom friction data (see Section B.4.3 on page 69); iv. set atmospheric data (see Section B.4.3 on page 69); v. set density (see Section B.4.3 on page 70); vi. set turbulent viscosities and diffusivities (see Section B.4.3 on page 71); vii. set fluxes of tracers through the surface and the bottom (see Section B.4.3 on page 72); viii. set settling velocity of active tracers (set Section B.4.3 on page 72); ix. perform checks and balances (see Section B.4.3 on page 73). (d) print/write computed results for the model domain or at different specific lo- cations, and output of computed results for later restart of the model (see Sec- tion B.4.4 on page 74).
5. grid data and computational variables are mainly stored in allocatable one-dimensional arrays. Indirect addressing is used to access variables.
6. the software implementation allows to use unstructured orthogonal grids which con- sist out of triangles, quadrilaterals or a mixture of both;
7. modularisation was used to structure the computational core.
Figure 6 on the following page gives a schematic representation of the UnTRIM software concept. The user is free to call his own methods from within the user interface. If the respective user interface methods remain void some standard assumptions are made (e. g. zero water surface elevation at start time of the model).
2.5.2 Claims and substantiations
The software implementation which has been realized for the computational model UnTRIM offers some interesting possibilities which are shortly discussed below:
2. Model validity Page 37 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Computational User User Core Interface Software
Data user_set_??? Data
Get- and Set- call get_??? Get- and Set- Routines Routines call set_??? User Input/Output out??? call user_set_??? and Others call get_???
call out??? call set_??? call out???
Figure 6: Schematic representation of the UnTRIM software concept. The computational core (left) can be accessed through various get- and set- interfaces from within different routines of the user interface module (middle) which interconnects the computational core with user data and/or user software (right).
1. the portability of the computational model UnTRIM is guaranteed by the fact that it was coded using standard Fortran90; therefore one should be able to run the model on different hardware platforms.
2. the portability of the computational model to different hardware platforms is even more facilitated by the fact that the UnTRIM software implementation is completely independent from other software packages which might eventually be unavailable for some platforms;
3. there is no need for the user to know details about software implementation and data structures used inside the computational core of the model because due to the avail- ability of a larger number of get- and set- interfaces;
4. Thanks to the present concept of the algorithmic model and its software implemen- tation the user can easily apply various parametrizations or sub-models for different physical processes. Examples of the latter are
(a) wind friction, (b) bottom friction, (c) eddy-viscosity,
2. Model validity Page 38 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
(d) eddy-diffusivity, (e) settling velocity for suspended sediments, (f) flux for (dissolved) species at the free-surface (e. g. heat flux), (g) flux for (dissolved) species at the sediment-water interface (e. g. deposition and resuspension of suspended sediments), and (h) water density;
5. one compiled version of the code can be used to study many different problems; there- fore the computational core of UnTRIM can be applied for two- and three-dimensional problems of different size because the arrays in the data storage module are dynami- cally allocated in dependence on the number of polygons, sides and z-layers;
6. the possibility to work with unstructured orthogonal grids (triangles and quadrilater- als) offers the possibility to fit complicated geometries. Beyond that, also (converted) classical finite difference grids (square cells) can be used together with the model be- cause they represent special cases of unstructured orthogonal grids;
7. the modular structure of the computational core of the model favors code-reuse and improves maintainability as well as readability.
2. Model validity Page 39 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3 Validation studies
This chapter summarizes validation studies and contributes to the substantiating evidence for the claims made in the previous chapter. Each section of this chapter corresponds to a validation study whose purpose can be clearly identified in the context of the material presented in the previous chapter. Such a study may involve case studies, theoretical analysis, comparison with measurements, com- parisons with other models, etc., as long as it is relevant to the purpose of the study. Please notice that text written in blue color is some standard text which will be replaced by specific text in the near future.
3. Validation studies Page 40 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1 Analytical test cases 3.1.1 Wave propagation in a rectangular basin
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 41 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.2 Free barotropic oscillations in a rectangular basin
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: The physical phenomenon of an oscillating free surface (seiche) in a natural basin is the result of a balance between inertia and gravity. The analytical solution for this problem in the simplest geometry a rectangular basin with horizontal bottom is well- known and can be easily implemented even in spreadsheets. This test case can be used to test the basic numerical scheme in a model without the complications of open boundaries, varying bathymetric depth, or friction. In particular, this test problem checks the discretization and implementation of the temporal acceleration and water surface slope terms in the momentum equations. It also tests the implementation of the continuity equation, both for the solution of the free-surface position and for the calculation of the vertical velocities throughout the 3-D domain. Furthermore, phase, amplitude and mass conservation errors in the model can be examined as a function of discretization parameters, allowing one to test convergence of the numerical scheme.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 42 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.3 Free barotropic oscillations in a circular basin
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: Although the underlying physics in this problem consists, as in validation study Free barotropic oscillations in a rectangular basin, of a simple balance between inertia and gravity the curved boundaries introduce an additional element of complexity. A curved boundary cannot be easily represented with Cartesian grids, and hence, this test provides a very simple way to check the ability of 3-D models that are based on unstructured orthogonal grids to represent complex plan-form geometries and its ef- fects in hydrodynamic processes. It is also possible that the imperfect representation of the boundaries might influence the phase, amplitude and mass conservation errors of the model. The effect of discretization parameters on these aspects can be checked with this test case.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 43 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.4 Free oscillations in a non-hydrostatic situation
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: This example deals with non-breaking waves resulting for a relatively large ratio
· η λ of total depth H h to the wave length . In such a case the hydrostatic pressure as- sumption does not apply and, for sufficiently small wave amplitude, the wave celerity
c can be approximated by the following dispersion relation
Ú
$ % Ù
Ù gλ 2πH
Ø c tanh (69)
2π λ
Approach: A square basin of length L 10 m and depth h 10 m is discretized using an hor-
izontal grid of Np 40 equilateral triangles of side 0.5 m and a vertical increment
∆z 0.5 m. Starting with zero initial velocity the flow is driven by an initial free-surface
of constant slope η 0 02x 0 1. By neglecting bottom friction, horizontal and vertical
viscosity, the calculation is carried out with a time step ∆t 0.01 s. The expected solu-
c tion consists of a standing wave of length λ 2L and frequency f λ , where c is given by the above dispersion relation.
Results: Figure 7 on the following page shows the water surface elevation at x 10 m obtained with and without the hydrostatic approximation. As expected, the non- hydrostatic solution is in much better agreement with the analytical result approxi- mated by the dispersion relation (69). These results compare reasonably well with those obtained with a finite difference model on a uniform rectangular grid; for more details, see e. g. [Casulli, V., 1999].
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 44 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Figure 7: Free surface wave of small amplitude. Analytical (solid), hydrostatic (chaindot) and non- hydrostatic (dash) solution
3. Validation studies Page 45 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.5 Tidal forcing with flat bottom
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: Most problems in hydrodynamic concern water bodies connected to other water masses through open boundaries. Although the interaction between them may be and usually is significant, practical considerations dictate that simulations be carried out only for body of interest, incorporating the influence of connected water masses by means of specifying the hydrodynamic variables at the open boundaries. This test case attempts to check the formulation and implementation of open boundary conditions that specify the free surface elevation, without any additional complications coming from friction or geometric complexities. The geometry, governing equations and as- sumptions are those same of validation study Free barotropic oscillations in a rectangular basin, of a simple balance between inertia, the only difference resulting from the pres- ence of an open boundary subject to period oscillations in the free surface elevation. This test is especially relevant to problems and codes aimed at simulating systems such as estuaries, affected by tidal oscillations in the boundary that connects them to the ocean.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 46 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.6 Tidal forcing with varying bathymetry
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: In validation study Tidal forcing with flat bottom the correct implementation of tidal boundary conditions was checked without any additional difficulties coming from el- ements such as basin geometry or bottom friction and vertical momentum transport. Some of these complicating effects are incorporated in this test case. First, the correct representation of more complex bathymetries and its effects in the flow field can be easily tested. The bathymetry is assumed to vary quadratically with maximum depths at the mouth of the subembayment, and the plan form of the basin is rectangular. The use of a rectangular plan form was preferred to the more complex circular geometry (as presented by Lynch and Officer 1985), to avoid the problems associated with the correct representation of lateral boundaries, which was already addressed in valida- tion study Free barotropic oscillations in a circular basin. A considerable part of the model error using curved open boundaries will result from the deficient representation of the open boundary condition. An additional effect considered, here, is the representation of the transfer of momentum from the bottom (here a linearized form of bottom friction is assumed) throughout the water column.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 47 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.7 Wind driven currents in a rectangular basin with flat bot- tom
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: Wind shear is an important process to the free surface flow when the movement of the atmosphere is significant to that of the water flow. The actual process of wind shear driven flow is very complicated: waves could be generated by the wind; the main flow near the free surface moves forwards the wind direction, and the flow near the bed would have to move backwards to offset the surface flow. The analytical so- lutions are obtained for a simplified problem: the waves and the associated processes are excluded. However, the solution could be used for examinating the correctness of the numerical model’s boundary condition at the free surface and accuracy of some of the terms of the model’s governing equations. The analytical solutions given by [Koutitas, C. and O’Connor, B., 1980] and [Huang, 1993] for wind shear driven flow in a straight shallow cavity are used. The test cases are used to determine a numerical model’s capability in predicting lin- earized vertical 2D flow field, consistency with the mathematical model, convergence to the exact solution in a grid refinement process, and quantitative accuracy. To ver- ify that the numerical model is free of error in mathematical derivations, numerical solution scheme, computational algorithms, and computer coding.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 48 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.8 Internal seiches in a rectangular basin
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: The presence of stratification in natural surface bodies to a large degree modifies and controls the hydrodynamic processes of mixing and transport. In such systems, oscillatory motions with some well-defined frequency occur that cause isopycnals to tilt and oscillate around an equilibrium position. These oscillatory phenomena are re- ferred to as internal waves. They are the result of the balance between gravity and inertial forces, and its numerical representation at the basin scale, is very dependent on the accurate discretization of the baroclinic and the scalar advection terms. The problem aims at revealing the ability of a 3-D model to simulate such basin-scale in- ternal waves. The test problem consists of the simulation of a baroclinic or internal seiche in an enclosed rectangular and narrow flat-bottom basin with a background lin- ear density profile in a non-rotating framework. Analytical solutions to this problem exist if
1. the motion is effectively 2-D in a vertical plane, 2. the fluid is assumed inviscid and 3. non-linear terms are neglected.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 49 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.9 Steady density induced flow
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: Baroclincity is one of the important forcing mechanics in estuarine flow. In this test case, we consider the vertical variations in current velocity induced by a constant horizontal gradient in a flat rectangular basin with no friction. This similar to the circulation condition in a well mixed estuary. In the steady state, the horizontal density gradient is balanced with surface slope and the vertical diffusion of momentum. The analytical solution from this test case can be used to test the ability and accuracy of a three-dimensional model in predicting the vertical variations of velocity. In addition, the numerical scheme used to solve the vertical velocity structure in a 3-D model can be examined though this simplified test case.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 50 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.10 Wetting and Drying
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 51 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.11 Solitary wave in a wave channel with flat bottom
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 52 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.1.12 Solitary wave in a wave channel with varying bottom
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 53 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.2 Laboratory test cases 3.2.1 Water surface elevation in a wave flume
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: This example is concerned with spatial evolution of steep waves propagating over a longshore bar with reference to the Scheldt Flume experiment, which was carried out at Delft Hydraulics [Beji, S. and Battjes, J.A., 1994].
Approach: The flume has an overall length of 30 m. The bottom profile is shown in Figure 8. The still water depth was 0.4 m and reduced to 0.1 m over a submerged trapezoidal bar. At the end of the flume a plane beach with a 1 : 25 slope serves as a wave absorber. The
computational domain is discretized using an horizontal grid of Np 2,400 equilateral
∆ triangles of side 2.5 cm and a vertical increment of zk 1.25 cm. A sinusoidal wave
of amplitude 1 cm and period T 2.02 s is specified at the left open boundary. The time
step chosen for this simulation is ∆t 0.025 s.
Results: The resulting water surface elevation at three stations located at 13.5 m, 15.7 m and 19 m from the open boundary is compared with the measurements in Figures 9 on the next page, 10 on the following page and 11 on the next page, respectively.
Conclusions: This example illustrates the potential of the present model in dealing with complex wave problems at laboratory scale. For this example the hydrostatic solution is totally different and, of course, unrealistic. Again, these results are in agreement with those obtained by the corresponding finite difference model on a uniform rectangular grid [Casulli, V., 1999].
Figure 8: The wave flume geometry.
3. Validation studies Page 54 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Figure 9: Measured (solid) and computed (dash) water surface elevation at 13.5 m from the open boundary.
Figure 10: Measured (solid) and computed (dash) water surface elevation at 15.7 m from the open boundary.
Figure 11: Measured (solid) and computed (dash) water surface elevation at 19 m from the open boundary.
3. Validation studies Page 55 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.3 Schematic test cases 3.3.1 Lock exchange flow with hydrostatic pressure
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different
from the original purpose of the study. Approach: A rectangular basin of length L 2 m, and depth h 0.3 m is initially filled with
3 3 ρ two fluids with different densities ρ1 1030 kg/m and 2 1000 kg/m , separated by a vertical dam located centrally in the basin. The bottom friction, horizontal and vertical viscosity are neglected. The initial velocity is zero and the initial free surface
is flat, horizontal. The computational domain is discretized using an horizontal grid
∆ of Np 200 equilateral triangles of side 2 cm and a vertical increment of zk 1 cm.
The calculation is carried out with a time step ∆t 0.01 s. The pressure was assumed to be hydrostatic.
Results: Once the dam is removed the resulting hydrostatic solution shows the develop- ment of two discontinuities moving in opposite directions (Figure 12 on the next page). In the computed results a Kelvin-Helmholtz interfacial instability can be recognised. The shape of the front is rectangular. The present results are in agreement with early calculations obtained with traditional finite difference methods ( [Casulli, V. and Stelling, G.S., 1998]).
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 56 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Figure 12: Lock exchange flow. Hydrostatic solution.
3. Validation studies Page 57 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.3.2 Lock exchange flow with non-hydrostatic pressure
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Mostly identical with study lock exchange flow – hydrostatic pressure (see Sec- tion 3.3.1 on page 56) with the exception that the pressure is now assumed to be non- hydrostatic.
Results: Once the dam is removed the resulting non-hydrostatic solution shows the devel- opment of two discontinuities moving in opposite directions (Figure 13). In the com- puted results a Kelvin-Helmholtz interfacial instability can be recognised. The shape of the front is rounded and no longer rectangular, as computed in the hydrostatic case (compare once more with result shown in Figure 12 on the page before) The present results are in agreement with early calculations obtained with traditional finite difference methods ( [Casulli, V. and Stelling, G.S., 1998]).
Conclusions: Comparison with experimental results indicate that the non-hydrostatic solu- tion is more realistic than the hydrostatic one.
Figure 13: Lock exchange flow. Non-hydrostatic computation.
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3.3.3 Wave pattern in a square basin
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 59 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.3.4 Short waves in a harbour basin
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version August 2001.
Purpose: This study is aimed to demonstrate that the model can be applied for short wave problems where diffraction and reflection of waves are important processes. This is usually the case inside harbour basins. Waves entering from outside are frequently reflected at solid walls or diffracted around corners or obstacles. The resulting wave pattern in the harbour basin is a complex superposition of different wave systems.
Approach: The harbour basin (see Figure 14 on the next page) was discretised in the hori- zontal plane by means of 13069 triangles with 19834 edges (typical length 5 m). In the vertical 32 z-layers were used with thicknesses 0.25 m close to the surface and 0.5 m near the bottom. Therfore the three-dimensional grid consists out of 634688 faces and 418208 prisms. The numerical time step was set to 0.25 s. At the open boundary a si- nusoidal plane wave with an amplitude of 1 m and a period of 12 s was prescribed. No bottom friction and no horizontal turbulent diffusion was used in this simulation.
Results: During the first two wave periods (24 s) the plane waves reach approximately the end of the harbour entrance channel (see Figure 15 on the following page). The wave fronts remain plane during this period of time. Within the next wave period the first wave front reaches the area of the harbour basin. Diffraction occurs at the entrance corner of the harbour basin and the left travelling wave front becomes a circular arc as expected (see Figure 16 on page 62). Two more wave periods later (60 s after the start of the simulation) diffracted, reflected and incoming waves superimpose to form a complex resulting wave pattern in the whole harbour basin (see Figure 17 on page 62). Especially in front of the diagonal wall chessboard like patterns develop due to crossing of orthogonal wave trains.
Conclusions: The results demonstrate, that the model is able to reproduce important phys- ical effects connected with short wave applications, such as diffraction and reflection in situations with complex geometry.
3. Validation studies Page 60 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
bathymetry mMSL
-19. -13. -7. 1.
0125.0 250.0 m
open boundary : incoming waves
Figure 14: Harbour basin of constant depth. Unstructured orthogonal grid superimposed. Open boundary indicated in blue. Arrow shows direction of incoming waves.
time: 08/03/2001-12:00:24
bathymetry mMSL
-2.5 0 2.5 water level mMSL
-1.25 0 1.25
0125.0 250.0 m
Figure 15: Harbour basin of constant depth. Free surface elevation shows local wave amplitudes after 24 s of simulation time. The first group of the incoming waves has reached the end of the entrance channel to the harbour basin.
3. Validation studies Page 61 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
time: 08/03/2001-12:00:36
bathymetry mMSL
-2.5 0 2.5 water level mMSL
-1.25 0 1.25
0125.0 250.0 m
Figure 16: Harbour basin of constant depth. Free surface elevation shows local wave amplitudes after 36 s of simulation time. Wave diffracten at the left corner of the end of the entrance channel can be observed.
time: 08/03/2001-12:01
bathymetry mMSL
-2.5 0 2.5 water level mMSL
-1.25 0 1.25
0125.0 250.0 m
Figure 17: Harbour basin of constant depth. Free surface elevation shows local wave amplitudes after 60 s of simulation time. Wave diffraction is clearly visible at the left corner of the end of the entrance channel. Also crossing of waves due to wave reflection can be observed in front of the diagonal solid wall opposite to the end of the entrance channel.
3. Validation studies Page 62 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.3.5 Advection in a curved channel
Date of study: 2004-06.
Date of summary: 2004-06.
Version: Version June 2004.
Purpose: This study is aimed to demonstrate that both, application of a high resolution method for advective transport of substances together with a properly devised grid plays an important role for the accuracy of the numerical result.
Approach: Two discretizations, a
1. uniform grid with square polygons with side λ 20 m (see Figure 18), and a
2. flow-aligned triangular grid, average height 20 m (see Figure 19)
for a U-shaped channel of length L 8 57 m, width W 120 m and constant depth h 10 m are used. In doing so, the influence of a non-flow-aligned grid on computational
results will be highlighted. The channel is open at both ends where a different water
η surface elevation η1 9 cm and 2 9 cm is specified in order to drive a constant flow.
A steady flow is assumed after one hour of hydrodynamic simulation using a time step ∆t 30 s. Then a tracer of constant concentration C 1 is specified at the inflow open boundary for 12 minutes and the horizontal diffusion coefficient is set to zero so that
pure transport is being considered. Results: Figure 20 shows the computed concentrations at times t 2 h (15 m), t 3 h (30 m)
and t 4 h (45 m) by using the upwind method obtained from Equation (55) without flux
limiter (Φ 0). Figure 21 shows the computed concentrations at same times obtained with the same method where the Superbee flux limiter (64) was activated. In both cases precise mass conservation and maximum principle was obtained and, as expected, use of the flux limiter leads to a less diffusive solution as shown in Figure 21. As a second example, in fact, the above curved channel is covered with a boundary fitted triangular grid and with grid lines aligned to the expected flow (see Figure 19). Figure 22 shows the computed concentrations by using again the Superbee flux limiter (64). Again, precise mass conservation and maximum principle was obtained. More- over, a comparison between Figure 21 and Figure 22 shows the importance of grid alignment with the flow: in this latter example one can clearly observe a much more realistic profile and sharper fronts.
Conclusions: This study shows that, in addition to a high resolution method, a properly devised grid plays an important role for the accuracy of the numerical result. After all, a flow that is perfectly aligned to the grid is equivalent to a set of one-dimensional transport problems for which these methods are proven to be extremely accurate (see,
3. Validation studies Page 63 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
e. g. [LeVeque, R.J., 1990]). Consequently, for practical two-dimensional problems the grid generation issue should also address the possibility of producing grids that are flow aligned, or, at least with grid lines oriented with the net flow field. More details related to this study can be found in [Casulli, V. and Zanolli, P., 2004].
Figure 18: U-channel discretization using square grid.
Figure 19: U-channel discretization using flow aligned, triangular grid.
3. Validation studies Page 64 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
Figure 20: Advection in a U-channel on a square grid. Tracer concentration obtained without flux limiter (upwind method).
Figure 21: Advection in a U-channel on a square grid. Tracer concentration obtained with flux limiter Superbee, see Equation (64).
Figure 22: Advection in a U-channel on a flow aligned triangular grid. Tracer concentration obtained with flux limiter Superbee, see Equation (64).
3. Validation studies Page 65 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.4 Examples from real-world applications 3.4.1 Hydrostatic and non-hydrostatic flow in Venice Lagoon
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: The Lagoon of Venice covers an area of about 400 km2 and consists of several inter- connected narrow channels with a maximum width of 1 km and up to 34 m deep. The Lagoon is connected to the Adriatic Sea through three narrow inlets, namely Lido, Malamocco and Chioggia. Considerable portion of the Lagoon of Venice consists of tidal flats and proper treatment of flooding and drying is essential. This study also demonstrates the differences between computed vertical velocities for hydrostatic and non-hydrostatic pressure in tidal flow problems.
Approach: The grid for the Venice Lagoon has N p 12,209 irregular triangles whose size 2 2 varies from 171 m to 324,178 m . The total number of sides is Ns 18,996 (see Fig- ure 23). The use of an unstructured orthogonal grid enables accurate resolution of the flow region including detailed representation of the tree-like structure of the main channels. The flow is driven at the three inlets, where an M2 tide of 0.45 m amplitude
and 12 h period has been specified. The time step is ∆t 90 s.
Four different runs have been made corresponding to a two-dimensional simulation (Nz 1) and a three-dimensional simulation with Nz 70. Each run was repeated with and without the hydrostatic approximation. The corresponding vertical resolu-
n n n ∆ tion is ∆z H for the two-dimensional simulations and z 0.5 m for the three- j j j k dimensional simulations.
Results: Table 1 summarizes the CPU time for each run expressed in minutes per tidal cycle. The simulations were performed on a 500 MHz DEC Alpha 21164 which has a SpecFP
Layers Nz Hydrostatic Non-Hydrostatic 1 4.14.5 70 11.318.3
Table 1: Model’s performance on the Venice Lagoon.
95 rating of 17.8. As expected, the two-dimensional calculations, with and without the hydrostatic approximation, yield the same results in about the same CPU time (some overhead is due to the non-hydrostatic setup). A fixed amount of CPU time, independent from Nz, is spent for determining the provisional free-surface elevation from system (39). Then, additional CPU time is required for computing the Lagrangian
3. Validation studies Page 66 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
trajectories needed to approximate the advection terms with an Eulerian–Lagrangian method. Finally, in non-hydrostatic calculations, a reasonable amount of additional computer time is spent for solving system (48)–(49). The vertical component of the velocity reaches 13 cm/s near Malamocco inlet where the deepest part of the Lagoon is located. The calculation repeated with the hydro- static approximation shows that no substantial difference can be observed on the wa- ter surface elevation and on the horizontal components of the velocity, but the vertical component of the velocity, though relatively small, can be higher by a factor of 10% (see Figure 24).
Conclusions: The results indicate that in the Venice Lagoon, at this time and space scale, the three-dimensional hydrostatic solution is generally acceptable, but it may not be sufficiently accurate if the vertical component of the velocity is to be used for sediment transport or to run a long term simulation for baroclinic circulation, where the effect of the vertical velocity plays an important role.
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Figure 23: Triangular unstructured orthogonal grid for the Venice Lagoon.
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Figure 24: Vertical velocity near Malamocco inlet. Hydrostatic (dash) and non-hydrostatic (solid) solution.
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3.4.2 Tidal flow and salt transport
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 70 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.4.3 River flow
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 71 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.4.4 Storm surge
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 72 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.4.5 Suspended sediment transport
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 73 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
3.4.6 Transport of cooling water from a power plant
Date of study: YYYY-MM-DD.
Date of summary: YYYY-MM-DD.
Version: Version information; e. g. the version number of the model that was used in the validation study, possibly other version numbers to which this study applies.
Purpose: One- or two-paragraph description of the main purpose(s) of this contribution to the validation document, clearly placed in the framework provided by this document. Preferably the purpose should be the verification of one of the claims or statements made in Chapter 2 on page 5 of this document. Note that this may be quite different from the original purpose of the study.
Approach: Summary of validation methods and techniques, experiments performed, etc. in order to achieve the stated objectives. Not too many details; better to point to reports or publications. However it must be possible to understand the essence of what was done by only reading this text. This means, for instance, that the most important model parameters that were used in an experiment should be listed here.
Results: Summary of results, presented graphically whenever this makes sense.
Conclusions: Conclusions that may be drawn from the results. The most important aspects are the implications to the user: how can he use the results of this study to judge model quality in his own application.
3. Validation studies Page 74 The Federal Waterways Engineering and Research Institute (BAW) Mathematical Model UnTRIM Validation Document – Version June 2004 (1.0)
References
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Batchelor, G. (1967). An Introduction to Fluid Dynamics. Cambridge University Press, London. 2.3.3
Beji, S. and Battjes, J.A. (1994). Numerical Simulation of Nonlinear Waves Propagation Over a Bar. Coastal Engineering, 23:1–16. 3.2.1
Casulli, V. (1990). A Semi-Implicit Finite Difference Methods for Two-Dimensional Shallow Water Equations. Journal of Computational Physics, 86:56–74. 2.4.2
Casulli, V. (1999). A Semi-Implicit Finite Difference Method for Non-Hydrostatic, Free- Surface Flows. International Journal for Numerical Methods in Fluids, 30:425–440. 2.4.2, 2, 3.1.4, 3.2.1
Casulli, V. and Cattani, E. (1994). Stability, Accuracy and Efficiency of a Semi-Implicit Method for Three-Dimensional Shallow Water Flow. Computers Math. Applic., 27(4):99– 112. 2.4.2, 4
Casulli, V. and Cheng, R.T. (1992). Semi-Implicit Finite Difference Methods for Three- Dimensional Shallow Water Flow. International Journal for Numerical Methods in Fluids, 15:629–648. 2.3.1, 2.4.2
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Casulli, V. and Walters, R.A. (2000). An Unstructured Grid, Three-Dimensional Model based on the Shallow Water Equations. International Journal for Numerical Methods in Fluids, 32:331–348. 2.4.1, 2.4.2, 2.4.2, 6
Casulli, V. and Zanolli, P. (1998). A Three-Dimensional Semi-Implicit Algorithm for En- vironmental Flows on Unstructured grids. In Proc. of Conf. on Num. Methods for Fluid Dynamics. University of Oxford. 2.4.1, 7
Casulli, V. and Zanolli, P. (2002). Semi-Implicit Numerical Modelling of Non-Hydrostatic Free-Surface Flows for Environmental Problems. Mathematical and Computer Modelling, 36:1131–1149.
Casulli, V. and Zanolli, P. (2004). High Resolution Methods for Multidimensional Advection- Diffusion Problems in Free-Surface Hydrodynamics. Ocean Modelling. to appear. 2.4.2, 2.4.2, 3.3.5
Daily, J.W. and Harleman, D.R.F. (1966). Fluid Dynamics. Addison-Wesley, New York. 2.3.3
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Golub, G.H. and van Loan, C.F. (1996). Matrix Computations. John Hopkins, London, 3rd edition edition. 2.4.2
Gravel, S. and Staniforth, A. (1992). Variable Resolution and Robustness. Monthly Weather Review, 20:2633–2640. 3
Huang, W. (1993). Three-dimensional numerical modeling of circulation and water quality induced by combined sewage overflow discharge. PhD thesis, Department of Ocean Engineering, Univ. of Rhode Island, Kingston, R.I. 3.1.7
IAHR (1994). Guidelines for Documenting the Validity of Computational Modelling Software. Inter- national Association for Hydraulic Research, P.O. Box 177, 2600 MH Delft, The Nether- lands. (document)
Koutitas, C. and O’Connor, B. (1980). Modeling Three-Dimensional Wind-Induced Flows. ASCE, Journal of Hydraulic Division, 106:1843–1865. 3.1.7
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A Glossary algorithmic implementation: the conversion of the conceptual model into a finite set of rules suitable for computation. This may involve spatial discretisation schemes, time integration methods, solution procedures for algebraic equations, decision algorithms, etc. conceptual model: a mathematical/logical/verbal representation of a physical system or process. This representation may involve differential equations, discrete algebraic equations, decision graphs, or other types of conceptual descriptions.
computational model: software whose primary function is to model a certain class of phys- ical systems. The computational model may include pre- and post-processing features, a user interface, and other ancillary programmes necessary in order to use the model in applications. However, this validation document primarily concerns the core of the computational model, consisting of the underlying conceptual model, its algorithmic implementation and software implementation.
software implementation: the conversion of the algorithmic implementation into a com- puter code. This includes coding of algorithms, use of standard mathematical software, design and implementation of data structures, etc. The term software implementation, for the purposes of this document, is limited to the computational core of the model. It does not include pre- and post-processing software, user interfaces, or other ancillary
programmes associated with the computational model.
µ unstructured orthogonal grid: the horizontal ´x y domain is covered by a set of non- overlapping convex polygons. Each side of a polygon is either a boundary line or a side of an adjacent polygon. Moreover, it is assumed that within each polygon there exist a center such that the segment joining the centers of two adjacent polygons and the side shared by the two polygons have a non empty intersection and are orthogonal to each other.
A. Glossary Page 78