Velocity Structure of Density Currents Propagating Over Rough Beds

water

Article Velocity Structure of Density Currents Propagating over Rough Beds

Reza Nasrollahpour 1, Mohamad Hidayat Jamal 2,3,*, Zulhilmi Ismail 2,3, Zulkiﬂee Ibrahim 2, Mazlin Jumain 2, Mohd Ridza Mohd Haniffah 2 and Daeng Siti Maimunah Ishak 3

1 Civil Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada; [email protected] 2 Water and Environmental Engineering, School of Civil Engineering, Faculty of Engineering, Universiti Teknologi Malaysia, Johor Bahru 81310, Malaysia; [email protected] (Z.I.); [email protected] (Z.I.); [email protected] (M.J.); [email protected] (M.R.M.H.) 3 Center for Coastal and Ocean Engineering (COEI), Research Institute of Sustainable Environment (RISE), Universiti Teknologi Malaysia, Johor Bahru 81310, Malaysia; [email protected] * Correspondence: [email protected]

Abstract: In most practical cases, density-driven currents flow over surfaces that are not smooth; however, the effects of bottom roughness on these currents have not been fully understood yet. Hence, this study aims to examine the velocity structure of density currents while propagating over rough beds. To this end, alterations in the vertical velocity profiles within the body of these currents were investigated in the presence of different bottom roughness configurations. Initially, laboratory experiments were carried out for density currents flowing over a smooth surface to provide a baseline for comparison. Thereafter, seven bottom roughness configurations were tested, encompassing both Citation: Nasrollahpour, R.; Jamal, dense and sparse bottom roughness. The bottom roughness consisted of repeated arrays of square M.H.; Ismail, Z.; Ibrahim, Z.; Jumain, cross-section beams covering the full channel width and perpendicular to the flow direction. The M.; Mohd Haniffah, M.R.; Ishak, primary results indicate that the bottom roughness decelerated the currents and modified the shape D.S.M. Velocity Structure of Density of velocity profiles, particularly in the region close to the bed. Additionally, a critical spacing of the Currents Propagating over Rough roughness elements was detected for which the currents demonstrated the lowest velocities. For Beds. Water 2021, 13, 1460. https://doi.org/ the spacings above the critical value, increasing the distance between the roughness elements had 10.3390/w13111460 little impact on controlling the velocity of these currents. Moreover, using dimensional analysis, equations were developed for estimating the mean velocities of the currents flowing over various Academic Editors: Jochen Aberle and configurations of the bottom roughness. The findings of this research could contribute towards Robert Bose better parameterization and improved knowledge of density currents flowing over rough beds. This can lead to a better prediction of the evolution of these currents in many practical cases as well as Received: 23 April 2021 improved planning and design measures for the control of such currents. Accepted: 20 May 2021 Published: 23 May 2021 Keywords: density current; velocity profile; acoustic doppler velocimeter; bottom roughness

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- 1. Introduction iations. Density currents are generated when the fluid of one density is released into another fluid with a different density. The density difference can result from temperature gradients, dissolved contents, suspended particles, or a combination of them. A wide range of flows are classified as density currents, and it emphasizes the importance of studying them. The Copyright: © 2021 by the authors. most usual type of these currents is an underflow produced when a flow is introduced Licensee MDPI, Basel, Switzerland. into an ambient fluid of a lower density. In oceanic and river systems, these currents occur This article is an open access article because some of the water in an estuary, ocean, lake is colder, saltier, or contains more distributed under the terms and suspended sediment than the surrounding water [1,2]. The density-driven currents are conditions of the Creative Commons Attribution (CC BY) license (https:// known as the main agent for sediment transport in reservoirs; these can carry incoming creativecommons.org/licenses/by/ suspended sediments and existing sediment deposits over the reservoir bed to the area 4.0/). near the dam. These currents decelerate as they approach the dam, and this can give rise to

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the area near the dam. These currents decelerate as they approach the dam, and this can givesedimentation rise to sedimentation [3]. Reservoir [3]. Reservoir sedimentation sedimentation is a worldwide is a worldwide issue hindering issue hindering the sustainable the sustainableuse of reservoirs use of reservoirs and sediment and balance sediment of balance impacted of rivers impacted [4]. Therivers loss [4]. of The storage loss of capacity stor‐ agein damcapacity reservoirs in dam due reservoirs to sedimentation due to sedimentation has been an issuehas been of great an issue concern of great and aconcern topic of andresearch a topic [5 of–8 research]. [5–8]. DetailsDetails regarding regarding velocity velocity profile profile and and friction friction coefficients coefficients in in open open channel channel flows flows can can bebe found found in in [9]. [9 ].In In the the case case of ofdensity density currents, currents, however, however, the the flow flow is confined is confined to the to thebed bed at theat bottom the bottom and normally and normally bounded bounded by the by stationary the stationary ambient ambient fluid at fluidthe top. at The the top.velocity The vanishesvelocity at vanishes the bed and at the interface bed and between interface the between current and the ambient current andfluid; ambient thus, the fluid; velocity thus, distributionthe velocity attains distribution a peak attains (maximum) a peak (maximum)velocity um at velocity a distance um at h am distanceabove the hm bed.above This the formsbed. Thisa boundary forms a between boundary the between inner and the outer inner layers and outer of the layers velocity of the profiles velocity [10]. profiles As seen [10 ]. inAs Figure seen in1, the Figure vertical1, the downstream vertical downstream velocity of velocity density of currents density currentsis divided is into divided two intore‐ gions:two regions: an inner an region inner below region the below location the of location maximum of maximum velocity, which velocity, has which a steep has increase a steep inincrease velocity in from velocity the bottom from the to the bottom maximum to the maximumvelocity, and velocity, an outer and region an outer above region the max above‐ imumthe maximum velocity with velocity a negative with a velocity negative gradient. velocity gradient.

FigureFigure 1. 1. SketchSketch of of a atypical typical vertical vertical velocity velocity profile. proﬁle.

TheThe general general approach approach for for density density current current studies studies has has been been simplifying simplifying the the situation situation byby regarding regarding the the bed bed as as smooth smooth [2,11–16]. [2,11–16 ].However, However, in in most most practical practical cases, cases, these these currents currents usuallyusually flow flow over over surfaces surfaces that that are are not not smooth, smooth, such such as, as, mobile mobile beds, beds, obstacles, obstacles, grain grain roughnessroughness (e.g., (e.g., sand sand or or gravel), gravel), and and form form roughness roughness (e.g., (e.g., ripples ripples or or dunes). dunes). Particularly, Particularly, there is still a gap in knowledge on the interaction between bottom roughness and the there is still a gap in knowledge on the interaction between bottom roughness and the continuous flow of density-driven currents. In studying obstacles for mitigation of density- continuous flow of density‐driven currents. In studying obstacles for mitigation of den‐ driven currents, the bulk of literature concerns the case of an isolated (single) roughness sity‐driven currents, the bulk of literature concerns the case of an isolated (single) rough‐ element or obstacle, for example [17–20]. There have also been limited investigations in ness element or obstacle, for example [17–20]. There have also been limited investigations respect to the effect of form roughness on these currents, including Negretti et al. [21], in respect to the effect of form roughness on these currents, including Negretti et al. [21], Peters [22], Tanino et al. [23], Tokyay [24], Chowdhury [25], Bhaganagar [26], Howlett Peters [22], Tanino et al. [23], Tokyay [24], Chowdhury [25], Bhaganagar [26], Howlett et et al. [27], Nasrollahpour et al. [28], and Köllner et al. [29]. These works have been mostly al. [27], Nasrollahpour et al. [28], and Köllner et al. [29]. These works have been mostly focused on the frontal region of the density currents. Overall, the behavior of the continuous focused on the frontal region of the density currents. Overall, the behavior of the contin‐ flow of density-driven currents over nonplane beds is complex and not fully understood uous flow of density‐driven currents over nonplane beds is complex and not fully under‐ yet, despite its important implications. The interaction of density currents with submarine stood yet, despite its important implications. The interaction of density currents with sub‐ installations (e.g., porous screens, dykes, oil and gas pipelines, cables) can lead to disastrous marinedamages installations [30,31]. As (e.g., an example,porous screens, a natural dykes, turbidity oil and current gas pipelines, was captured cables) in can the lead Fraser to disastrousRiver delta damages slope (Canada) [30,31]. As that an wasexample, powerful a natural enough turbidity to carry current a one-ton was captured observatory in theplatform Fraser River and severe delta aslope heavily (Canada) armored that cable was [powerful32]. enough to carry a one‐ton obser‐ vatoryThe platform bottom and roughness severe a canheavily be representative armored cable of [32]. various scenarios where density cur- rentsThe flow bottom over roughness nonplane surfaces.can be representative Additionally, of the various roughness scenarios elements where can density be a cur good‐ rentsapproximation flow over nonplane to submerged surfaces. dams Additionally, employed for the controlling roughness sediment elements deposition can be a in good reser- approximationvoirs and to obstacles to submerged installed dams in riversemployed for retardation for controlling of the sediment incoming deposition density currents. in res‐ ervoirsMoreover, and to they obstacles can represent installed obstacles in rivers used for retardation for partially of or the entirely incoming stopping density density currents. cur- Moreover,rents transporting they can represent pollutants obstacles or hazardous used for materials. partially Therefore, or entirely studying stopping these density currents cur‐ rentsover transporting roughness arrays pollutants can have or hazardous considerable materials. benefits forTherefore, human andstudying environmental these currents safety overpurposes roughness and lead arrays to more can accurate have considerable management benefits of various for industrial human and naturalenvironmental scenarios.

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Water 2021, 13, 1460 3 of 16 safety purposes and lead to more accurate management of various industrial and natural scenarios. The main aim of this study is to extend previous studies by considering the effect of bottomThe roughness main aim ofon this the study velocity is to structure extend previous of the density studies currents. by considering This is the achieved effect ofthrough bottom the roughness following on specific the velocity objectives: structure (1) to of acquire the density the vertical currents. structure This is of achieved stream‐ through the following specific objectives: (1) to acquire the vertical structure of streamwise wise velocities within the body of density currents; (2) to investigate alterations in the velocities within the body of density currents; (2) to investigate alterations in the velocity velocity profiles at the presence of various bed roughness configurations; (3) to develop profiles at the presence of various bed roughness configurations; (3) to develop equations equations for estimating mean velocities of the currents flowing over various configura‐ for estimating mean velocities of the currents flowing over various configurations of tions of bottom roughness. bottom roughness.

2.2. Materials Materials and and Methods Methods 2.1.2.1. Experimental Experimental Setup Setup ThisThis study study creates creates density density currents currents in in a a lock-exchange lock‐exchange configuration configuration in in which which a a gate gate separatesseparates two two fluids fluids with with different different densities. densities. ToTo model model these these currents currents in in the the laboratory, laboratory, a a specificspecific setting setting (illustrated (illustrated in in Figure Figure2) was2) was employed employed to prepare to prepare dense dense fluids fluids and and maintain main‐ thetain steady the steady state state of density of density currents currents during during each each experiment. experiment. Prior Prior to to each each experiment, experiment, densedense fluidsfluids werewere preparedprepared in in the the two two mixing mixing tanks tanks with with a a combined combined volume volume of of 6 6 m m3 3 throughthrough continuous continuous circulation circulation in in a closed-loopa closed‐loop system. system. Furthermore, Furthermore, salt salt was was dissolved dissolved in tapin tap water water inside inside the the tanks tanks until until the the required required salinity salinity was was obtained obtained and and the the solution solution was was homogeneouslyhomogeneously mixed; mixed; dye dye was was added added to to the the dense dense fluids fluids during during the the circulation circulation process process forfor visualizationvisualization purposes.purposes. ToTo preventprevent fluctuations,fluctuations, aa headhead tanktank locatedlocated 33 m m above above the the groundground on on a a steel steel structure structure was was used used to to transfer transfer the the prepared prepared dense dense fluids fluids to to the the flume flume withwith a a constant constant head. head.

FigureFigure 2. 2.Schematic Schematic sketch sketch of of the the experimental experimental set-up. set‐up.

TheThe flume flume was was 10 10 m m long, long, 0.3 0.3 m m wide, wide, and and 0.7 0.7 m m deep, deep, capable capable of of having having different different bottombottom slopes. slopes. A A sliding sliding vertical vertical gate gate located located at at a 1.25a 1.25 m m distance distance from from the the upstream upstream end end of theof the flume flume divided divided this this into into two two sections sections of unequal of unequal length. length. Additionally, Additionally, an overflow an overflow weir wasweir at was the at downstream the downstream end of end the of flume the flume kept thekept depth the depth of lighter of lighter ambient ambient fluid fluid constant con‐ duringstant during each experiment. each experiment. The upstream The upstream of the gateof the was gate occupied was occupied by the denseby the fluiddense while fluid thewhile downstream the downstream part was part occupied was occupied by freshwater by freshwater simulating simulating a reservoir a reservoir where a densitywhere a currentdensity can current propagate. can propagate. The density The currentsdensity currents were initiated were initiated by the sudden by the sudden removal removal of the verticalof the vertical lock gate. lock Once gate. the Once gate the was gate opened, was opened, a density a density current current (heavier (heavier fluid) advanced fluid) ad‐ throughvanced through the fresh the ambient fresh waterambient (lighter water fluid) (lighter by afluid) raised by front a raised (also front known (also as theknown head) as inthe the head) form in of the an form underflow. of an underflow. After the front After reached the front the reached downstream the downstream end of the end channel, of the itchannel, was directed it was fordirected withdrawal. for withdrawal. Thereafter, Thereafter, a shallower a shallower steady andsteady quasi-uniform and quasi‐uniform layer of density current was formed known as the body (see Figure3) which is the focus of this study.

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Water 2021, 13, 1460 4 of 16 layer of density current was formed known as the body (see Figure 3) which is the focus of this study.

Figure 3.Figure(a) Head 3. (a and) Head (b) bodyand (b of) abody density of a currentdensity flowing current underflowing a layerunder of a stationarylayer of stationary ambient ambient freshwater. freshwater. 2.2. Experimental Measurements 2.2. ExperimentalA Measurements Vectrino Plus Velocimeter with a lab probe made by Nortek (Akershus, Norway) A Vectrinowas usedPlus Velocimeter to record the with velocity a lab profiles probe made in the by body Nortek of density (Akershus, currents. Norway) This is a high- was used toresolution record the 10 velocity MHZ Acoustic profiles Dopplerin the body Velocimeter of density (ADV), currents. having This a is down-looking a high‐ fixed- resolution 10stem MHZ probe. Acoustic The ADV Doppler has an Velocimeter acoustic sensor (ADV), consisting having of a onedown transmit‐looking transducer fixed‐ and four stem probe.receive The ADV transducers, has an acoustic all calibrated sensor by consisting the manufacturer. of one transmit The receivers transducer are positioned and in 90◦ four receiveincrements transducers, on all a circle calibrated around by the the transmitter, manufacturer. slanting The receivers 30◦ from theare axispositioned of the transmitter. in 90° incrementsThe sampling on a circle volume around is the the transmitter, focus point slanting of the receiver 30° from beams, the axis situated of the trans 5 cm‐ away from mitter. The thesampling probe volume to ensure is the a nearly focus nonintrusivepoint of the receiver velocity beams, measurement situated 5 [33 cm]. away The transmitter from the probesends to outensure short a nearly acoustic nonintrusive pulse pairs velocity with a known measurement frequency [33]. transmitted The transmitter along the vertical sends out shortaxis, acoustic and their pulse echoes pairs are with recorded a known by frequency the receiver transmitted arms as the along pulses the vertical travel through the axis, and theirsampling echoes volume. are recorded The echo by the is reflectedreceiver arms from theas the small pulses particles travel floating through passively the in the sampling volume.flow, assuming The echo to is have reflected a velocity from the the small same particles as the flow. floating The passively ADV works in the based on the flow, assumingDoppler to have principle, a velocity meaning the thesame shift as betweenthe flow. the The frequencies ADV works of the based transmitted on the pulse and Doppler principle,the received meaning echo the is proportional shift between to the water frequencies velocity. Theof the phase transmitted shift between pulse two and subsequent the receivedpulses echo isis thenproportional processed to to water find thevelocity. actual The velocity phase of shift the flow. between two subse‐ quent pulses is thenThe bodyprocessed of density to find currents the actual was velocity maintained of the byflow. a continuous feed of dense saline The bodyfluid of moving density under currents a layer was of maintained stagnant freshwater. by a continuous In addition, feed theof dense discharge saline rate of dense fluid movingfluid under to the a layer channel of stagnant was fixed freshwater. at the desired In addition, rate using the adischarge flowmeter. rate The of dense steady condition fluid to the waschannel considered was fixed achieved at the desired when recorded rate using velocities a flowmeter. at one The point steady in two condition minutes were the was consideredsame. achieved The depth when of the recorded current atvelocities a given stationat one waspoint also in checkedtwo minutes for remaining were the unchanged over time to ensure achieving the steady state. Local velocities were then recorded at three same. The depth of the current at a given station was also checked for remaining un‐ locations (i.e., X = 3, 4, and 5 m from the gate) along the centerline of the channel from changed over time to ensure achieving the steady state. Local velocities were then rec‐ vertical points above the channel bed. In any required section, the velocity profile was orded at three locations (i.e., X = 3, 4, and 5 m from the gate) along the centerline of the determined by changing the vertical location of the sensor. The measurements began from channel from vertical points above the channel bed. In any required section, the velocity the top part of the dense layer and continued into the lower parts by lowering the probe profile was determined by changing the vertical location of the sensor. The measurements into the current till all the necessary points were covered. About 10 points were collected began from the top part of the dense layer and continued into the lower parts by lowering for plotting a vertical profile of velocity at each station; the data recording took about 1 min the probe into the current till all the necessary points were covered. About 10 points were for each of the measurement points. A sampling frequency of 100 Hz was used, and the collected for plotting a vertical profile of velocity at each station; the data recording took samples with signal-to-noise (SNR) values less than 15 dB and correlation less than 70% about 1 min for each of the measurement points. A sampling frequency of 100 Hz was were filtered according to the manufacturer’s recommendation. used, and the samples with signal‐to‐noise (SNR) values less than 15 dB and correlation Samples were taken from the dense fluids at mixing tanks, head tank, and the region less than 70%behind were the filtered gate according to ensure theto the formation manufacturer’s of a homogeneous recommendation. dense fluid with the desired Samplesdensity. were taken Moreover, from the at the dense data fluids collection at mixing stations, tanks, sampling head tank, siphons and the were region used to collect behind the samplesgate to ensure from within the formation the currents of a homogeneous in the proximity dense of the fluid bed. with To the measure desired conductivity density. Moreover,(also known at the as data electrical collection conductivity stations, sampling EC) of the siphons taken samples,were used a WP-84to collect conductivity samples frommeter within (TPS the Pty currents Ltd., Brendale, in the proximity QL, Australia) of the with bed. a To K =measure 10 sensor conductivity was employed. Then, (also knownmass as electrical concentration conductivity C (g/L) ofEC) the of taken the taken samples samples, was derived a WP‐ from84 conductivity its conductivity value meter (TPS ECPty (millisiemens) Ltd., Brendale, utilizing QL, Australia) a calibration with a curve.K = 10 Thesensor calibration was employed. curves Then, were yielded by mass concentrationpreparing C several(g/L) of the reference taken samples samples was of known derived salt from concentration its conductivity and value recording their corresponding conductivity values. Furthermore, a DA-130N Portable Density Meter (Kyoto Electronics Manufacturing, Kyoto, Japan) was used to measure the density ϕ

Water 2021, 13, x FOR PEER REVIEW 5 of 16

EC (millisiemens) utilizing a calibration curve. The calibration curves were yielded by preparing several reference samples of known salt concentration and recording their cor‐ Water 2021, 13, 1460 5 of 16 responding conductivity values. Furthermore, a DA‐130N Portable Density Meter (Kyoto Electronics Manufacturing, Kyoto, Japan) was used to measure the density φ (g/cm3) of the collected samples. Care was taken to obtain dense fluids with approximately the same temperature(g/cm3) of the as collectedthose of the samples. clear water Care wasin the taken flume; to obtainthe maximum dense fluids temperature with approximately difference wasthe same0.5 °C. temperature This ensured as thosethat the of thedensity clear differences water in the between flume; thedense maximum and ambient temperature fluids weredifference solely was due 0.5 to ◦theC. Thissalinity ensured differences. that the density differences between dense and ambient fluids were solely due to the salinity differences. 2.3. Experimental Parameters 2.3. Experimental Parameters By definition, roughness elements should not block the flow of density currents [34]. So, thisBy study definition, used bed roughness roughness elements for which should the not elements’ block the height flow fell of density below the currents peak [ve34‐]. locitySo, this position. study For used the bed smooth roughness bed experiments, for which the the elements’ lowest position height of fell peak below velocity the peak oc‐ curredvelocity at position.1.5 cm above For the bed. smooth Therefore, bed experiments, the height of the the lowest beams position was selected of peak as velocity 1.2 cm sooccurred that it always at 1.5 cm fell above below the the bed. maximum Therefore, velocity the height height. of the The beams rough was beds selected consisted as 1.2 cmof arraysso that of it square always cross fell below‐section the beams maximum fabricated velocity from height. wood Thewith rough side dimensions beds consisted of 1.2 of cm.arrays The of elements square cross-section spanned full beams flume fabricated width perpendicular from wood with to the side flow dimensions direction, of in 1.2 a re cm.‐ peatedThe elements array extended spanned over full flumethe area width downstream perpendicular of the to gate. the flowThe desired direction, roughness in a repeated was array extended over the area downstream of the gate. The desired roughness was obtained obtained by gluing roughness elements on the bed with the required spacing. The dia‐ by gluing roughness elements on the bed with the required spacing. The diagram of the gram of the configurations of the roughness arrays is schematically shown in Figure 4 configurations of the roughness arrays is schematically shown in Figure4 where λ is the where λ is the downstream spacing between every two subsequent beams, and Kr is downstream spacing between every two subsequent beams, and Kr is roughness height. roughness height. One smooth bed and seven roughness configurations (λ/Kr = 1, 4, 8, 16, One smooth bed and seven roughness configurations (λ/Kr = 1, 4, 8, 16, 32, 64, and 128) 32, 64, and 128) were tested herein adopting both dense and sparse bottom roughness were tested herein adopting both dense and sparse bottom roughness elements. According elements. According to the classification of [35], λ/Kr = 1 corresponds to the d‐type while to the classification of [35], λ/Kr = 1 corresponds to the d-type while λ/Kr = 4, 8, 16, 32, 64, λ/Kr = 4, 8, 16, 32, 64, and 128 correspond to the k‐type bottom roughness. and 128 correspond to the k-type bottom roughness.

Figure 4. Sketch of the configurations of the roughness arrays. Figure 4. Sketch of the configurations of the roughness arrays. A total of 64 experiments were performed with different initial parameters listed in TableA1 total. The of creation 64 experiments of density were currents performed with differentwith different initial initial conditions parameters was to listed ensure in Tablerepeatability. 1. The creation An inlet of gatedensity opening currents height with h indifferent= 7 cm initial was tested conditions to create was subcriticalto ensure repeatability.conditions at An the inlet inlet. gate Two opening bed slopes height S = h 0.25%in = 7 cm and was 1.75% tested were to create selected subcritical based on con the‐ ditionsexperimental at the inlet. circumstances. Two bed slopes To create S = 0.25% currents and with1.75% different were selected velocities based at on the the inlet, exper two‐ imentaldischarges circumstances. of dense fluid To Qcreatein = 0.5 currents and 1 L/swith were different considered. velocities Additionally, at the inlet, two two initial dis‐ chargesmass concentrations of dense fluid of Q densein = 0.5 fluids and 1 CL/sin =were 5 and considered. 15 g/L were Additionally, chosen to examine two initial currents mass concentrationshaving different of excessdense densities.fluids Cin = These 5 and concentrations 15 g/L were chosen corresponded to examine to densities currents ofhaving 1.003 3 differentand 1.009 excess gr/cm densities., respectively. These concentrations corresponded to densities of 1.003 and 1.009 gr/cm3, respectively. Table 1. Experimental conditions. Number of Roughness h (cm) S (%) C (g/L) Q (L/s) Experiments Conditions in in in

8 Smooth 7 0.25, 1.75 5, 15 0.5, 1 8 λ/Kr = 1 7 0.25, 1.75 5, 15 0.5, 1 8 λ/Kr = 4 7 0.25, 1.75 5, 15 0.5, 1 8 λ/Kr = 8 7 0.25, 1.75 5, 15 0.5, 1 8 λ/Kr = 16 7 0.25, 1.75 5, 15 0.5, 1 8 λ/Kr = 32 7 0.25, 1.75 5, 15 0.5, 1 8 λ/Kr = 64 7 0.25, 1.75 5, 15 0.5, 1 8 λ/Kr = 128 7 0.25, 1.75 5, 15 0.5, 1 Water 2021, 13, 1460 6 of 16

To determine the flow regime of density currents, nondimensional numbers were calculated. Like the free surface flow, the driving mechanism of density current is the gravitational force, and the Froude number is considered important. However, gravitational acceleration is reduced due to the density difference, and densimetric Froude number (Frin) at the inlet is calculated as Frin = uin/ (g’ hin cosθ), where uin is the velocity of current at the inlet, g’ is reduced gravitational acceleration, hin is the inlet opening height, and θ is bottom slope angle. The reduced gravitational acceleration is g’ = g (ϕin−ϕa)/ϕa = g ∆ϕ/ϕa where g is the gravitational acceleration, ϕin is the initial density of the current, ϕa is the density of the ambient fluid, and ∆ϕ is the density difference between the current and ambient fluid. Furthermore, the inlet Reynolds number for these currents is defined as Rein = (uin hin)/νin, where uin is inlet velocity, hin is inlet opening height, and νin is the kinematic viscosity of the dense fluid at the inlet. Due to low concentrations, the inlet dynamic viscosity was set approximately equal to that of the water [36]; therefore, inlet Reynolds number for the flume with width b can be rewritten as Rein = Qin/b ν. The inlet Froude number ranged between 0.21 to 0.97; the inlet Reynolds numbers for Qin = 0.5 and 1 L/s were equal to 1956 and 3912, respectively. This indicates that the currents tested in this work were subcritical and turbulent. The boundary between the density current and ambient fluid is unclear because of the turbulence along the interface. Thus, Ellison and Turner [37] equations are generally recommended to calculate the average velocity (u) and height (h) of the density currents:

R hd u2 dz u = o (1) R hd 0 u dz

2 R hd 0 u dz h = (2) R hd 2 0 u dz

where u is local velocity at depth of z and hd is the depth where u = 0.

3. Results and Discussion 3.1. Distribution of Velocity For all the experiments, the velocity profiles were similar but scattered in a specific range. Two different algebraic expressions were used to formulate the velocity distribution in the velocity profiles. One is valid in the rigid boundary (inner or wall region) along the bed, whilst another is valid in the diffusion boundary (outer or jet region). In the wall region, below the maximum velocity (z < hm), the velocity distribution can be represented by an empirical power relation proposed by Altinakar et al. [38]:

1 u (z) z n = ( ) (3) um hm

where u(z) is the mean streamwise velocity at the distance z above the bed, um is the maximum velocity, hm is the height where the maximum velocity occurs, and n is an empirical exponent. The um and hm were obtained for all the collected velocity profiles and used to plot the normalized profiles in the wall region; examples of these profiles are shown in Figure5. Water 2021, 13, x FOR PEER REVIEW 7 of 16

u z z (3) u h

where u(z) is the mean streamwise velocity at the distance z above the bed, um is the max‐ imum velocity, hm is the height where the maximum velocity occurs, and n is an empirical exponent. The um and hm were obtained for all the collected velocity profiles and used to Water 2021 13 , , 1460 plot the normalized profiles in the wall region; examples of these profiles are shown7 of in 16 Figure 5.

FigureFigure 5. 5. DimensionlessDimensionless velocity velocity profiles profiles in the wall regionregion atat XX == 44 mm (Note:(Note: pointspoints are are experimental experi‐ mentaldata, and data, solid and lines solid are lines fitted are profiles). fitted profiles).

ToTo determine determine the the constant constant n, n, Equation Equation (3) (3) was was fitted fitted to to all all the the normalized normalized velocity velocity profilesprofiles in in the the inner inner region region for for different different bed bed configurations. configurations. The The outcomes outcomes of of the the curve curve fits fits areare listed listed in in Table Table 2 andand comparedcompared with thosethose of thethe literature.literature. Herein, the coefficientcoefficient of 2 2 determinationdetermination (R (R2) )was was used used to to verify verify the the accuracy accuracy of of the the curve curve fits; fits; the the R R2 valuesvalues were were closeclose to to unity unity showing showing that that the the power power law law well well estimated estimated the thevelocity velocity distribution distribution in the in dataset.the dataset. Given Given the constant the constant n= 5.61 n= yielded 5.61 yielded for the for smooth the smooth bed bedtests, tests, the agreement the agreement be‐ tweenbetween the theresults results of this of this study study to those to those of the of theliterature literature was was satisfactory. satisfactory. Note Note that thatin Al in‐ tinakarAltinakar et al. et [38], al. [38 Nourmohammadi], Nourmohammadi et al. et [13], al. [ 13He], et He al. et [39], al. [ 39and], andYaghoubi Yaghoubi et al. et [19] al. [the19] experimentsthe experiments were were performed performed over over a plane a plane bed. bed. Moreover, Moreover, the inlet the inlet Froude Froude numbers numbers for thesefor these studies studies ranged ranged between between 1 and 1 and2.33, 2.33, 0.6 and 0.6 and3.6, 0.48 3.6, 0.48and and0.71, 0.71, and and0.8 and 0.8 and1.55, 1.55, re‐ spectively,respectively, while while here here this this range range was was between between 0.28 0.28 and and 0.97. 0.97.

TableTable 2. 2. ConstantsConstants obtained obtained for for Equations Equations (3) (3) and and (4). (4). 2 2 ReferenceReference Bed Type Bed Typen n R2 α R α mm RR2 [19] [19]Smooth Smooth 6.2 6.20.95 0.95 0.85 0.85 2 20.92 0.92 [39] [39]Smooth Smooth 5.9 5.90.97 0.97 1.42 1.42 2.34 2.34 0.97 0.97 [13] Smooth 5.8 — 0.6 2.7 — [13] Smooth 5.8 ‐‐‐‐ 0.6 2.7 ‐‐‐‐ [38] Smooth 6 — 1.4 2 0.96 [38] Smooth 6 ‐‐‐‐ 1.4 2 0.96 Smooth 5.61 0.92 1.86 2.38 0.91 Smoothλ/Kr = 15.61 4.280.92 0.98 1.86 1.91 2.38 2.63 0.91 0.92 λ/Kr =λ /Kr1 = 44.28 2.890.98 0.98 1.91 1.33 2.63 2.44 0.92 0.89 λ/Kr = 8 1.78 0.97 1.18 2.72 0.90 Present Study λ/Kr = 4 2.89 0.98 1.33 2.44 0.89 λ/Kr λ=/Kr 8 = 161.78 1.870.97 0.98 1.18 1.28 2.72 2.54 0.90 0.93 Present Study λ/Kr = 32 3.32 0.98 1.68 2.44 0.91 λ/Kr = 16 1.87 0.98 1.28 2.54 0.93 λ/Kr = 64 3.96 0.96 1.86 2.35 0.89 λ/Kr λ=/Kr 32 = 1283.32 5.450.98 0.96 1.68 1.96 2.44 2.39 0.91 0.94 λ/Kr = 64 3.96 0.96 1.86 2.35 0.89 λ/Kr = 128 5.45 0.96 1.96 2.39 0.94 As seen in Table2, the constant n varied from 5.61 for the plane surface to 1.78 for the rough bed with λ/Kr = 8. These variations stemmed from changes in the position of maximum velocity at the collected velocity profiles. Smaller constant n values represent velocity profiles where peak velocities were attained at higher positions. Given the fluctuations in the constant n for different bottom roughness configurations, this can be inferred that the

primary controlling parameter in the wall region was the bottom friction. In the jet region, above the maximum velocity (z > hm), Altinakar et al. [38] suggested a near-Gaussian relation describing the velocity distribution above the maximum velocity point as: u(z) z − h m = exp −α m (4) um h − hm Water 2021, 13, x FOR PEER REVIEW 8 of 16

As seen in Table 2, the constant n varied from 5.61 for the plane surface to 1.78 for the rough bed with λ/Kr = 8. These variations stemmed from changes in the position of maximum velocity at the collected velocity profiles. Smaller constant n values represent velocity profiles where peak velocities were attained at higher positions. Given the fluc‐ tuations in the constant n for different bottom roughness configurations, this can be in‐ ferred that the primary controlling parameter in the wall region was the bottom friction. In the jet region, above the maximum velocity (z > hm), Altinakar et al. [38] suggested a near‐Gaussian relation describing the velocity distribution above the maximum velocity point as: Water 2021, 13, 1460 8 of 16 uz zh expα (4) u h h

where u(z) is the mean streamwise velocity at the distance z above the bed, um is the max‐ where u(z) is the mean streamwise velocity at the distance z above the bed, um is the imum velocity, hm is the height where the maximum velocity occurs, h is average current maximum velocity, hm is the height where the maximum velocity occurs, h is average height calculated from Equation (2), α is an empirical constant, and m is an empirical ex‐ current height calculated from Equation (2), α is an empirical constant, and m is an ponent. The um, hm, and h were obtained for all of the collected velocity profiles and used empirical exponent. The um, hm, and h were obtained for all of the collected velocity profiles toand plot used the to normalized plot the normalized profiles in the profiles jet region; in the examples jet region; of examples the normalized of the normalizedprofiles can beprofiles seen in can Figure be seen 6. As in Figureseen, the6. Asscatter seen, of thedimensionless scatter of dimensionless velocities in the velocities jet region in thewas morejet region compared was more to those compared of the wall to those region, of the which wall is region, due to which the transitory is due to behavior the transitory of the currentsbehavior at of their the currents top edge. at their top edge.

Figure 6. Dimensionless velocity profiles profiles in the jet region at X = 4 m (Note: points points are experimental data, and solid lines are fittedfitted profiles).profiles).

In the jet region, Equation (4) was fittedfitted to thethe normalized velocity profilesprofiles for thethe currents over different bed configurations.configurations. The computed values of α α and m are listed in Table2 2 and and compared compared with with those those of theof the literature. literature. The The results results show show that thethat agreement the agreement with withthe literature the literature was rather was satisfactoryrather satisfactory given the given experimental the experimental difficulties difficulties involved. Moreover,involved. Moreover,the Gaussian the law Gaussian can be law used can to be well-describe used to well the‐describe outer region the outer of velocity region of profiles velocity in pro this‐ filesrough in bed this dataset. rough Alterationbed dataset. in αAlterationand m was in minimal α and m indicating was minimal that the indicating shape of the that outer the shaperegions of of the the outer normalized regions of velocity the normalized profiles did velocity not change profiles substantially did not change with substantially bed types. with bed types. 3.2. Influence of Roughness on Velocity Profiles 3.2. InfluenceThe downstream of Roughness evolution on Velocity of velocity Profiles profiles is shown in Figure7, which illustrates changes in the velocity structure of currents in presence of roughness arrays. Herein, the The downstream evolution of velocity profiles is shown in Figure 7, which illustrates dense layer height (z) and velocity (u) were normalized with the inlet opening height (h ) changes in the velocity structure of currents in presence of roughness arrays. Herein, thein and inlet velocity (uin), respectively. In general, the velocities decreased as the currents trav- dense layer height (z) and velocity (u) were normalized with the inlet opening height (hin) eled over different bed configurations, and the peak velocities occurred at higher positions and inlet velocity (uin), respectively. In general, the velocities decreased as the currents above the bed. The driving force of these currents is the density difference between the traveledcurrents over and ambientdifferent fluid. bed configurations, The influx of ambient and the freshwater peak velocities into theoccurred underflows at higher results po‐ sitionsin reduction above of the their bed. excess The driving densities, force which of these in turn currents reduces is the the density velocity difference of these between currents whilethe currents traveling and downstream. ambient fluid. The influx of ambient freshwater into the underflows As seen in Figure7, the changes in the peak velocities and their positions were negligible for the smooth bed, while a stronger deceleration was observed in the case of the rough beds. Additionally, the distance from the bed to the point of maximum velocity

shifted further upward for the currents over the rough beds. However, the magnitude of adjustments in the peak velocities and their positions appeared to vary depending on the roughness elements configuration. A comparison of the results reveals that the smallest changes occurred for the currents over the plane surface, which was followed by the currents flowing over the rough beds with λ/Kr = 1 and 128. With increasing the spacing, the peak velocities continued to decrease further till the currents over the spacing of λ/Kr = 8 where the most significant adjustments were observed. Water 2021, 13, x FOR PEER REVIEW 9 of 16

Water 2021, 13, 1460 9 of 16 results in reduction of their excess densities, which in turn reduces the velocity of these currents while traveling downstream.

Figure 7. Velocity profiles at different locations for smooth and rough beds corresponding to Figure 7. Velocity proﬁles at different locations for smooth and rough beds corresponding to hin = 7 cm, Cin = 15 g/L, hin = 7 cm, Cin = 15 g/L, Qin = 0.5 L/s, and S = 0.25%. Qin = 0.5 L/s, and S = 0.25%.

Water 2021, 13, x FOR PEER REVIEW 10 of 16

As seen in Figure 7, the changes in the peak velocities and their positions were neg‐ ligible for the smooth bed, while a stronger deceleration was observed in the case of the rough beds. Additionally, the distance from the bed to the point of maximum velocity shifted further upward for the currents over the rough beds. However, the magnitude of adjustments in the peak velocities and their positions appeared to vary depending on the roughness elements configuration. A comparison of the results reveals that the smallest Water 2021, 13, 1460 changes occurred for the currents over the plane surface, which was followed by the10 cur of 16‐ rents flowing over the rough beds with λ/Kr = 1 and 128. With increasing the spacing, the peak velocities continued to decrease further till the currents over the spacing of λ/Kr = 8 where the most significant adjustments were observed. FollowingFollowing [35], [35], the roughened beds with λλ/Kr/Kr =1 =1 are are categorized categorized as as d d-type,‐type, and the bedsbeds with with λλ/Kr/Kr = =4, 4,8, 16, 8, 16, 32, 32,64, 64,128 128are considered are considered as k‐ astype. k-type. Figure Figure 8 illustrates8 illustrates the influ the‐ enceinfluence of different of different configurations configurations of bottom of bottom roughness roughness on the velocity on the velocity profiles profilesfor a selected for a numberselected of number experiments. of experiments. Note that Note all thatthese all currents these currents had the had same the sameinitial initial conditions, conditions, and theand adjustments the adjustments in the in profiles the profiles were were solely solely induced induced by the by bottom the bottom roughness. roughness. As seen, As seen, the primarythe primary influence influence of roughness of roughness was was decreasing decreasing the the velocity velocity of of the the currents. currents. The The currents currents lostlost their their kinetic energy as traveling over the elements due to adverse pressure gradient induced by the inclined upstream surface of the beams, which led to the deceleration of the induced by the inclined upstream surface of the beams, which led to the deceleration of currents. Also, the velocity maximum for the currents over roughened beds was located the currents. Also, the velocity maximum for the currents over roughened beds was lo‐ higher in comparison with the smooth bed case. However, the magnitude of these effects cated higher in comparison with the smooth bed case. However, the magnitude of these depended on the configuration of roughness elements, as discussed afterward. effects depended on the configuration of roughness elements, as discussed afterward.

FigureFigure 8. 8. ComparisonComparison of ofvelocity velocity profiles proﬁles for fordifferent different bed bed types types at X at=4 Xm =corresponding 4 m corresponding to hin = to7 cm,hin =Cin 7 cm,= 5 g/L, Cin S= = 5 1.75%, g/L, S Q =in 1.75%, = 1 L/s. Q in = 1 L/s.

ForFor the the currents currents over over λλ/Kr/Kr = 1, = 1,the the velocity velocity profiles profiles depicted depicted close close similarity similarity with with the smooththe smooth bed and bed noticeably and noticeably different different from other from roughened other roughened cases. For cases. the rough For the bed rough with λbed/Kr with= 1, theλ/Kr peak = velocity 1, the peak was velocityonly slightly was lower only slightly than that lower of the than smooth that bed. of the Neverthe smooth‐ less,bed. the Nevertheless, position of thepeak position velocity of for peak this velocity bed type for shifted this bed further type shifted upward further reaching upward h/hin =reaching 0.65 in Figure h/hin =8, 0.65as opposed in Figure to8 ,the as opposedcorresponding to the current corresponding flowing current over the flowing smooth over bed λ wherethe smooth h/hin bed= 0.4. where This h/hsuggestsin = 0.4. that This for suggests λ/Kr = 1, that the for current/Kr =lifted 1, the above current the lifted roughness above elements.the roughness The initial elements. density The difference initial density between difference the ambient between water the and ambient the dense water fluid and wasthe densesmall; fluid therefore, was small; the currents therefore, flowing the currents over the flowing dense array over of the beams dense (λ array/Kr = of 1) beams could lift(λ/Kr above = 1) the could beams lift above and showed the beams a dynamic and showed similar a dynamic to their similarcounterparts to their traveling counterparts over thetraveling smooth over bed. the The smooth front bed. of density The front currents of density flowing currents over flowing d‐type overroughness d-type was roughness exam‐ inedwas examinedby [22,34,40], by [ and22,34 a, 40similar], and mechanism a similar mechanism was found was in which found inthe which head thefloated head over floated the roughnessover the roughness elements elementsand had andonly hada slight only interaction a slight interaction with them. with In them.addition, In addition,they re‐ portedthey reported rotating rotating weak vortices weak vortices trapped trapped between between the elements, the elements, which which agrees agrees with the with find the‐ findings reported herein. ings reported herein. For the k-type rough beds, two different behaviors were observed in the collected For the k‐type rough beds, two different behaviors were observed in the collected velocity profiles of the currents. As seen in Figure8, maximum retardation of the currents velocity profiles of the currents. As seen in Figure 8, maximum retardation of the currents occurred for the rough surface with the critical spacing of λ/Kr = 8. For the spacing less than the critical value (λ/Kr =4), the retardation of the currents increased with increasing the gaps between elements. For the spacing more than the critical value (λ/Kr = 16, 32,

64, 128), the retardation of the currents diminished with increasing the gap between the elements. For the critical spacing, the currents depicted the smallest velocity maxima and the highest position of velocity maxima. As an example, according to Figure8, the peak velocity for the current over λ/Kr = 8 decreased by 32.3%, and its height rose by 149.8% compared to the plane bed case. Flow separation occurs above each beam as the current propagates over the roughness elements. The ﬂow reattaches on the bottom wall in between the beams and separates Water 2021, 13, 1460 11 of 16

again as the next element is approached. This creates a vortex below the separated flow at the downstream side of the beams which ejects to the overlying flow [41,42]. For the spacing between the roughness elements that were large enough, the reattachment length was not affected by the presence of other elements [41,42]; this critical spacing appeared to be λ/Kr = 8 herein. The critical value represents the spacing in which the maximum strength of ejection occurred, which could have significant effects on the density currents propagation and entrainment. For single-phase flows, the strength of the ejections from between the roughness elements got more till the critical spacing of λ/Kr ≈ 7 was achieved [41,42]. Herein, the evidence was shown that this rule applied to density currents as well. For the spacing less than the critical value (λ/Kr = 4), further deceleration of the currents was attributed to the larger spacing between the beams, and the reattachment length was expected to shorten due to adverse pressure gradient induced by the next element. The larger gaps between the beams required more energy to maintain a vortex in motion in that area. Also, the volume of released fluid from between the elements to the current was more significant making more disruption leading to deceleration of the currents. For the rough beds, increasing the spacing between elements increased the distance above the bed where the maximum velocity attains. As observed in the velocity profiles, for the spacing above the critical one (λ/Kr = 16, 32, 64, 128), increasing the distance between the elements had little impact on reducing the velocity of the currents. In other words, increasing the cavities between the beams decreased the retardation of the currents. For λ/Kr = 128, when the elements were too far apart, the current dynamics reverted to that over a smooth surface with the highly dispersed elements representing very small individual elements in the path of the current. Regarding the spacings more than the critical value, the reattachment length was expected to increase, and the overlying fluid dynamics were once again similar to those encountered above the smooth surface.

3.3. Shear Stress The measurement of the velocity profiles allowed for a backward calculation of the bottom shear stresses. This study employed the mean flow method to estimate the shear velocities [13,14]. The velocity profiles within the inner region vary logarithmically with depth, and the Von Karman–Prandtl Law of the Wall applies to them as [43]:

u∗ z u = ln (5) k z0 where u is the velocity at height z above the bed, u* is shear velocity, K = 0.41 is Von Karman constant, and z0 is zero velocity roughness height. The Equation (5) can be rewritten as:

u∗ u∗ u = ln z − ln z = mx + c (6) k k 0 As mentioned in Section 2.2, the velocity (u) was collected at several points (z) above the bed within the body of density currents. To determine the distance above the bed in which the logarithmic law was valid, all the data points were used to regress ln(z) versus 2 u. This was observed that the best ﬁt could be achieved at Z/hm = 0.88 with R = 0.86. Therefore, only the data points which fell within this distance were used to calculate the shear velocity, and slope (m) of the regression line was used to compute the shear velocity as u* = Km. The intercept (c) of the regression line was equivalent to z0 which is theoretically the height where the current velocity becomes zero. The bottom stress (τ) was related to the near-bed density (ϕ) of the currents and shear velocity (u*) as τ = ϕ u*2 [14]. For each bed type, the shear velocities were calculated for all the velocity proﬁles at the data collection stations along the channel. The resulted bed shear stresses were then averaged at each location and normalized by the inlet velocity 2 (uin) and initial density (ϕin) as τin = ϕin uin , as illustrated in Figure9. Additionally, the distance (X) of data collection stations from the inlet gate was normalized by the total Water 2021, 13, x FOR PEER REVIEW 12 of 16

u. This was observed that the best fit could be achieved at Z/hm = 0.88 with R2 = 0.86. Therefore, only the data points which fell within this distance were used to calculate the shear velocity, and slope (m) of the regression line was used to compute the shear velocity as u* = Km. The intercept (c) of the regression line was equivalent to z0 which is theoreti‐ cally the height where the current velocity becomes zero. The bottom stress (τ) was related to the near‐bed density (φ) of the currents and shear velocity (u*) as τ = φ u*2 [14]. For each bed type, the shear velocities were calculated for Water 2021, 13, 1460 all the velocity profiles at the data collection stations along the channel. The resulted12 bed of 16 shear stresses were then averaged at each location and normalized by the inlet velocity (uin) and initial density (φin) as τin = φin uin2, as illustrated in Figure 9. Additionally, the distance (X) of data collection stations from the inlet gate was normalized by the total lengthlength L L of of the the channel channel downstream downstream of ofthe the gate. gate. As seen, As seen, the shear the shear stress stress was almost was almost con‐ stantconstant along along the thechannel, channel, which which agrees agrees with with the the findings ﬁndings of of Nourmohammadi Nourmohammadi et et al. al. [13]. [13]. However,However, the the bottom bottom stresses stresses for for the the currents over over the rough beds varied among the roughnessroughness configurations. conﬁgurations.

FigureFigure 9. 9. BottomBottom shear shear stress stress for for different different bed bed configurations. conﬁgurations.

ForFor the the λλ/Kr = = 1, 1, the the bottom bottom stresses stresses remained remained almost almost unchanged unchanged and and similar similar to to the casecase of of the the smooth smooth bed. bed. This This can can be be attributed attributed to to the the flow ﬂow of of these these currents currents from from above above the the elementselements in in the the case case of of this bed type, which minimizes the interaction of these currents withwith the the bottom roughness. With With increasing increasing the the gaps between the elements, the bottom stressstress levels levels decreased decreased until until the the spacing spacing of of λ/Krλ/Kr = 8 = where 8 where its minimum its minimum occurred. occurred. This This can becan deduced be deduced that thathigher higher maximum maximum velocity velocity positions positions for these for these currents currents resulted resulted in the in there‐ ductionreduction of ofbottom bottom stresses. stresses. For For the the spacing spacing above above the the critical critical one one (roughened (roughened beds beds with λλ/Kr/Kr = = 16, 16, 32, 32, 64, 64, 128), 128), increasing increasing the the cavities cavities between between the the beams beams rose rose the the bottom shear stress.stress. The The bottom bottom stresses stresses for for the currents over λλ/Kr = = 128 128 resembled resembled those those of of the the plane surfacesurface and and remained remained roughly roughly unchanged along the channel. TheThe shear stress atat thethe bottom bottom of of a a density density current current can can induce induce bed bed erosion. erosion. The The erosion ero‐ sionof a softof a soft mobile mobile bed bed occurs occurs when when the bottomthe bottom shear shear stress stress exerted exerted by theby the density density current cur‐ rentexceeds exceeds the particle’sthe particle’s threshold threshold of motion of motion [44]. [44]. The The eroded eroded sediments sediments are entrained are entrained into intothe densitythe density currents currents creating creating self-sustaining self‐sustaining currents currents which which are highlyare highly unpredictable unpredictable and destructive for marine engineering installations. These currents can maintain their forward and destructive for marine engineering installations. These currents can maintain their motion for long distances if there is sufﬁcient sediment exchange with the bed. According forward motion for long distances if there is sufficient sediment exchange with the bed. to the results, this can be implied that the existence of roughness arrays could inhibit According to the results, this can be implied that the existence of roughness arrays could erosion through decreasing bottom shear stress. Roughness-induced changes in the bed inhibit erosion through decreasing bottom shear stress. Roughness‐induced changes in erosion-siltation balance could deteriorate the capacity of these currents to preserve their the bed erosion‐siltation balance could deteriorate the capacity of these currents to pre‐ density and hence driving buoyancy forces. serve their density and hence driving buoyancy forces. 3.4. Dimensional Analysis for Velocity Estimation Dimensional analysis using Buckingham Pi theorem showed that the velocity of the currents was a function of Froude number (Frin), bed slope (S), and roughness parameter (λ/Kr) as: u λ = f(Frin, S, ) (7) uin Kr

where u is the average velocity of the current and uin is the velocity of the current at the inlet. Note that the Reynolds numbers at the inlet were almost more than 2000 for all the experiments and the currents were turbulent. Hence, the viscous effects could be neglected [39], and the Reynolds number was excluded from the analysis. The relationships between the dependent and independent parameters were determined by performing a regression analysis on the experimental data using the statistical package SPSS. Sixty Water 2021, 13, 1460 13 of 16

percent of the experimental data were used for the regression analysis. The coefﬁcient of determination (R2) was used here to display how well our datasets ﬁt the proposed relations. To verify the accuracy of the proposed relations, the amount of error (E) was calculated. Forty percent of the datasets were used herein to investigate the accuracy of the driven equations. The error measures differences between values predicted by the n ∑i=1|uO−uP| equations and the observed values from the experimentation as E = n , where ∑i=1 uO uO is the velocities driven from the observed data at the experiments, uP is the velocities predicted by the proposed relationships, and n is the number of velocities. For the smooth bed, the following equation was drawn:

u 0.245 0.026 = 2.263 (Frin) S R2 = 0.80, E = 10.3% (8) uin With including the roughness parameter, the relationships for the roughened beds with 1 < λ/Kr ≤ 8 and 8 < λ/Kr ≤ 128 are shown in Equations (9) and (10), respectively.

−0.128 u 0.245 λ 0.026 = 2.250 (Frin) S R2 = 0.81, E = 11.6% (9) uin Kr

0.085 u 0.245 λ 0.026 = 1.452 (Frin) S R2 = 0.80, E = 11.1% (10) uin Kr Herein, the values of the coefﬁcient of determination were close to unity showing our data ﬁt the suggested relationships well. The errors were also near zero indicating minimal differences between the values predicted by the suggested relationships and the ones measured in the laboratory. As mentioned before, forty percent of the dataset was used for error analysis of the proposed equations. To illustrate the accuracy of the equations, the measured values of mean velocities for this dataset were drawn against their predicted values by the equations, as seen in Figure 10. This demonstrates that the

Water 2021, 13, x FOR PEER REVIEWsuggested relationships could estimate the mean velocities almost within the range14 of of 16 ±15% error.

FigureFigure 10. 10.Comparison Comparison of of the the measured measured and and predicted predicted values values of of the the mean mean velocities velocities (cm/s). (cm/s).

4.4. ConclusionsConclusions ThisThis study study aimed aimed to to explain explain the the velocity velocity structure structure of density of density currents currents over roughover rough beds. Thebeds. coefficients The coefficients of the of well-known the well‐known relationships relationships for wall for and wall jet and regions jet regions of the of velocity the ve‐ profileslocity profiles were modified were modified for different for different bed roughness bed roughness configurations. configurations. While the bed While roughness the bed wasroughness found to was notably found affect to notably the shape affect of velocity the shape profiles of velocity in the innerprofiles region, in the the inner alterations region, inthe the alterations outer regions in the were outer minimal. regions Thewere primary minimal. influence The primary of roughness influence was of decreasingroughness was the decreasing the velocity of density currents. However, the magnitude of this deceleration depended on the spacing of the roughness elements. For the rough bed with λ/Kr = 1, the velocity profiles resembled those of the smooth surface and markedly differed from other roughened cases. There was also a critical spacing of the roughness elements (λ/Kr = 8) over which the currents demonstrated the lowest velocities and the highest location of peak velocities. For the spacing more than the critical value, the influence of bottom roughness was diminished. In other words, increasing the distance between elements had little impact on reducing the velocity of the currents. The rough bed with λ/Kr = 128 demonstrated very little influence on the currents, and the flow dynamics reverted to that of the smooth bed. Based on the dimensional analysis, new equations were developed in which the av‐ erage velocity of the currents was related to inlet Froude number, bed roughness, and bed slope. It was also observed that roughness‐induced changes in the velocity of currents can alter shear stress along the bed. The bottom stresses declined as the gaps between the elements increased and reached their minimum at the spacing of λ/Kr = 8. In dam reser‐ voirs, the deceleration of currents allows more time for sedimentation, and this decreases the sediment concentration which is their driving force. Additionally, roughness‐induced changes in the velocity profiles inhibit erosion by decreasing the bed shear stress. There‐ fore, bed roughness can be effective in the control of density‐driven currents in the reser‐ voirs and can prolong their lifespan.

Author Contributions: Conceptualization, R.N. and M.H.J.; data curation, R.N. and M.H.J.; formal analysis, R.N. and M.H.J.; funding acquisition, R.N., M.H.J., Z.Is., Z.Ib., M.J., M.R.M.H. and D.S.M.I.; investigation, R.N. and M.H.J.; methodology, R.N. and M.H.J.; project administration, R.N. and M.H.J.; resources, R.N., M.H.J. and Z.Is.; software, R.N.; supervision, M.H.J. and Z.Is.; validation, R.N., M.H.J. and Z.Is.; visualization, R.N.; writing—original draft, R.N. and M.H.J.; writing—review and editing, R.N., M.H.J., Z.Is., Z.Ib., M.J., M.R.M.H. and D.S.M.I. All authors have read and agreed to the published version of the manuscript. Funding: The funding for this study was provided by the Fundamental Research Grant Scheme (FRGS/1/2015/STWN01/UTM/02/2) under the Ministry of Higher Education Malaysia , and Research University Grant (Q.J130000.2522.09H06) by Universiti Teknologi Malaysia. The authors would also

Water 2021, 13, 1460 14 of 16

velocity of density currents. However, the magnitude of this deceleration depended on the spacing of the roughness elements. For the rough bed with λ/Kr = 1, the velocity profiles resembled those of the smooth surface and markedly differed from other roughened cases. There was also a critical spacing of the roughness elements (λ/Kr = 8) over which the currents demonstrated the lowest velocities and the highest location of peak velocities. For the spacing more than the critical value, the influence of bottom roughness was diminished. In other words, increasing the distance between elements had little impact on reducing the velocity of the currents. The rough bed with λ/Kr = 128 demonstrated very little influence on the currents, and the flow dynamics reverted to that of the smooth bed. Based on the dimensional analysis, new equations were developed in which the average velocity of the currents was related to inlet Froude number, bed roughness, and bed slope. It was also observed that roughness-induced changes in the velocity of currents can alter shear stress along the bed. The bottom stresses declined as the gaps between the elements increased and reached their minimum at the spacing of λ/Kr = 8. In dam reservoirs, the deceleration of currents allows more time for sedimentation, and this decreases the sediment concentration which is their driving force. Additionally, roughness- induced changes in the velocity profiles inhibit erosion by decreasing the bed shear stress. Therefore, bed roughness can be effective in the control of density-driven currents in the reservoirs and can prolong their lifespan.

Author Contributions: Conceptualization, R.N. and M.H.J.; data curation, R.N. and M.H.J.; formal analysis, R.N. and M.H.J.; funding acquisition, R.N., M.H.J., Z.I. (Zulhilmi Ismail), Z.I. (Zulkiflee Ibrahim), M.J., M.R.M.H. and D.S.M.I.; investigation, R.N. and M.H.J.; methodology, R.N. and M.H.J.; project administration, R.N. and M.H.J.; resources, R.N., M.H.J. and Z.I. (Zulhilmi Ismail); software, R.N.; supervision, M.H.J. and Z.I. (Zulhilmi Ismail); validation, R.N., M.H.J. and Z.I. (Zulhilmi Ismail); visualization, R.N.; writing—original draft, R.N. and M.H.J.; writing—review and editing, R.N., M.H.J., Z.I. (Zulhilmi Ismail), Z.I. (Zulkiflee Ibrahim), M.J., M.R.M.H. and D.S.M.I. All authors have read and agreed to the published version of the manuscript. Funding: The funding for this study was provided by the Fundamental Research Grant Scheme (FRGS/1/2015/STWN01/UTM/02/2) under the Ministry of Higher Education Malaysia, and Re- search University Grant (Q.J130000.2522.09H06) by Universiti Teknologi Malaysia. The authors would also like to acknowledge the financial support under contract research from Bahtera Offshore (M) Sdn. Bhd. (R.J130000.7651.4C246). Conflicts of Interest: The authors declare no conflict of interests.

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