River, Coastal and Estuarine Morphodynamics: RCEM 2005 – Parker & García (eds)© 2006 Taylor & Francis Group, London, ISBN 0 415 39270 5

Two-dimensional morphological simulation in transcritical flow

J.A. Vasquez & R.G. MillarDepartment of Civil Engineering, University of British Columbia, Canada

P.M. StefflerDepartment of Civil and Environmental Engineering, University of Alberta, Canada

ABSTRACT: A two-dimensional Finite Element hydrodynamic model with capabilities for transcritical flowhas been coupled with a sediment transport model to simulate bed elevation changes in alluvial beds. The modelhas been tested for a knickpoint migration experiment that involved a short and very steep reach of 10% bedslope. The initial Froude was Fr = 3.5 downstream from the knickpoint with a hydraulic jump formed close tothe toe of the oversteepened reach. The results computed by the model agreed with the experimental data. Thenew upstream location of the knickpoint was correctly predicted and the computed bed profile seemed to followthe average trend of the measured profile. The model seems as a promising morphodynamic tool for alluvialstreams subject to transcritical flow.

1 INTRODUCTION

Although several river morphology models are cur-rently available, most of them are intended for appli-cations in lowland alluvial rivers with small gradients(Papanicolaou, in press). Such models usually assumesubcritical flow conditions and therefore cannot beapplied in cases where supercritical flow or hydraulicjumps are present.

A transcritical flow condition, where flow regimechanges between subcritical and supercritical, usu-ally involves the presence of a sharp shock wave orhydraulic jump, as sketched in Figure 1.

Some practical examples of transcritical flow overalluvial beds are: sediment deposition upstream ofdams (Brusnelli et al. 2001, Bellal et al. 2003); dambreak (Fraccarollo & Capart 2002, Spinewine & Zech2002); dam removal (Cantelli et al. 2004, Wong et al.2004, Cui & Wilcox, in press, Cui et al. a, b, in press);dyke breach (Spinewine et al. 2004); and knickpointmigration (Brush & Wolman 1960, Bhallamudi &Chaudhry 1991).

Figure 1. Sketch of transcritical flow after dam removal(Cui & Wilcock, in press).

Knickpoints are points along longitudinal profilesof streams where the slope increases abruptly frommild to steep (Brush & Wolman 1960). Knickpointsmay appear after meander cut-off (Brush & Wolman1960) or dam removal (Fig. 1). Their presence can leadto transcritical flow conditions and intense scour anddeposition.

Morphodynamic models with transcritical flowcapabilities are uncommon, and normally limited toone-dimensional (1D) flow (e.g. Papanicolaou, inpress, Brusnelli et al. 2001, Cui et al. a, b, in press).

The main objective of this paper is to assess thecapabilities of a new two-dimensional (2D) depth-averaged river morphology model for simulating scourand deposition in transcritical flow generated by aknickpoint. Comparisons with the experimental dataof Brush & Wolman (1960) suggested that modelis a promising tool for alluvial streams subject totranscritical flow.

2 THE NUMERICAL MODEL

2.1 River2D hydrodynamic model

River2D is a two-dimensional (2D) depth-averagedhydrodynamic model intended for use on naturalstreams and rivers and has special features for super-critical/subcritical flow transitions, ice covers, andvariable wetted area. For the spatial discretization ituses a flexible Finite Element unstructured mesh com-posed of triangular elements. River2D is based onthe 2D vertically averaged Saint Venant equations

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expressed in conservative form; which form a sys-tem of three equations representing the conservationof water mass and the two components of the momen-tum vector (Steffler and Blackburn, 2002). It has beendeveloped at the University of Alberta and it is freelyavailable at www.River2D.ca.

The Finite Element Method used in River2D’shydrodynamic model is based on the StreamlineUpwind Petrov-Galerkin (SUPG) weighted residualformulation (Hicks & Steffler 1992, Ghanem et al.1995). In this technique, upstream biased test functionsare used to ensure solution stability under the full rangeof flow conditions, including subcritical, supercritical,and transcritical flow. A fully conservative discretiza-tion is implemented which ensures that no fluid massis lost or gained over the modeled domain. This alsoallows implementation of boundary conditions as nat-ural flow or forced conditions. The equations solvedby the River2D are (Steffler and Blackburn 2002):

The water continuity equation:

The vertically averaged momentum equation in thex-direction:

The vertically averaged momentum equation in they-direction:

where h = water depth; (u, v) = depth-averaged veloc-ities in the (x,y) directions; qx = uh = flow discharge inx-direction per unit width; qy = vh = flow discharge inthe y-direction per unit width; (Sox,Soy) = bed slopesin the (x,y) directions; Sfx and Sfy = correspondingfriction slopes; τxx, τxy, τyx, τyy = components of thehorizontal turbulent stress tensor; ρ = water density;and g = gravitation acceleration.

The basic assumptions in equations (1) through (3)are (Steffler and Blackburn 2002):

– The pressure distribution is hydrostatic, which lim-its the accuracy in areas of steep slopes and rapidchanges in bed slopes.

– The horizontal velocities are constant over the depth.Information on secondary flows and circulations isnot available.

– Coriolis and wind forces are assumed negligible.

The friction slope terms depend on the bed shearstresses which are assumed to depend on the magni-tude and direction of the vertically averaged velocities.For example, in the x-direction:

τbx is the bed shear stress in the x direction and C∗ isthe dimensionless Chezy coefficient, which is relatedto the effective roughness height ks through

e = 2.7182; C∗ is related to Chezy’s C coefficientthrough

The vertically-averaged turbulent shear stresses aremodeled with a Boussinesq type eddy viscosity. Forexample:

where υt = eddy viscosity coefficient assumed bydefault as

Unlike other popular 2D hydrodynamic modelsbased of Finite Elements (e.g. RMA2, FESWMS),River2D does not require artificial eddy viscosity forconvergence. In fact, River2D is rather insensitive tothe value of eddy viscosity.

2.2 River2D-Morphology

River2D was extended by including a solver for thebed load sediment continuity (Exner) equation

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where zb = bed elevation; λ = porosity of the bedmaterial; t = time; and (qsx, qsy) = components of thevolumetric rate of bedload transport per unit width qs(ignoring the effects of secondary flow and bed slope):

Equation (9) is discretized using a conventionalGalerkin Finite Element Method (GFEM). The timeadvance is performed using a Runge-Kutta secondorder scheme. Since GFEM lacks upwinding prop-erties (equivalent to a central difference scheme) itis known to be adequate for diffusive problems, butnot for advection-dominated problems (Hirsch 1987,Chung 2002, Donea & Huerta 2003).

This extended version is known as River2D-Morphology and has been successfully applied tosimulate aggradation, degradation and bend scour inalluvial channels (Vasquez et al. 2005a, b). However,the model should not be applied to problems whereadvection is important, such as migrating bedforms orsediment waves.

3 EXPERIMENTAL TEST CASE

3.1 Experimental data

Brush & Wolman (1960) carried out a laboratorystudy of the behavior of knickpoints in noncohesivesediment. The experiments were conducted at the Uni-versity of Maryland in a flume 15.85 m (52 ft) longand 1.22 m (4 ft) wide. Before starting each experi-mental run a trapezoidal channel 0.21 m (0.7 ft) wideand 0.03 m (0.1 ft) deep with rounded corners wasmolded in noncohesive sand. A short over-steepenedreach with a length of 0.30 m (1 ft) and a fall of 0.03 m(0.1 ft), was located 10.8 m from the flume entrance.This oversteepened reach immediately downstream ofthe knickpoint had a 10% slope which was signifi-cantly steeper than the adjacent reaches. A total of5 runs were performed with different slopes in theupstream and downstream reaches (between 0.125 and0.88%). In every run, the slope of both water sur-face and bed below the knickpoint decreased withtime, causing the position of the knickpoint to migrateupstream.As the knickpoint moved upstream the chan-nel directly above it narrowed. At the lower end of thesteep reach, sediment eroded from above deposited asa dune, which moved downstream causing the channelto locally widen.

Run 1, which has more detailed information, wasselected for the numerical simulation. For Run 1 the

Figure 2. Resolution of Finite Element mesh around knick-point and steep 10% slope reach (parallel gray lines are bedcontours every 0.001 m).

longitudinal bed slope changed at a knickpoint from0.125% to 10% and back to 0.125% again (Fig. 3).The channel was molded in sand 0.67 mm in diameterwith initial water depth ho = 0.0137 m and constantflow rate Q = 0.59 L/s. During Run 1 the width of thechannel increased downstream of the knickpoint bybank erosion to about 0.25 m; while it got deeper andnarrower upstream.

At the beginning of Run 1, no sediment was mov-ing in the reaches above and below the steep reach.The erosion and deposition resulted solely from thepresence of the knickpoint. Sediment eroded abovethe knickpoint was deposited below. The head andtoe of the steep reach moved upstream and down-stream respectively. Sediment transport rate was notmeasured.

3.2 Numerical model

In the numerical model the channel was assumed asrectangular with a fixed width of 0.21 m. An irregularmesh was used; the average size of the elements wasabout 0.20 m away from the steep reach and about0.04 m around the knickpoint (Fig. 2). The mesh wasrefined close to the knickpoint to capture the strongflow gradients and the hydraulic jump. The mesh had869 triangular elements and 563 nodes. The roughnessheight was set to ks = 0.3 mm and porosity to λ = 0.4.

The boundary conditions for the hydrodynamicmodel were constant discharge in the upstream inflowsection, and constant water surface elevation in thedownstream outflow section.

4 RESULTS

Figure 3 shows the initial water surface profile com-puted before the beginning of the morphological sim-ulation. Away from the steep reach, the flow remainedsubcritical with a Froude around 0.5; which corre-sponds to a velocity of about 0.2 m/s and a depth of0.014 m, in agreement with observed values (Table 3in Brush & Wolman 1960). As the flow approachesthe knickpoint, it accelerates and becomes supercriti-cal. The maximum velocity of about 0.7 m/s is reached

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Figure 3. Initial profiles of bed elevation and water surfaceelevation (WSE) around knickpoint and steep reach (t = 0).

over the steep reach, the Froude number being as highas 3.5 with a water depth of only 0.004 m.

Downstream from the toe of the steep reach, theflow returns to subcritical through a hydraulic jump.The sharp front of the jump was captured by the model,as shown in Figure 3, without any noticeable spuriousoscillation.

Since most sediment transport equations are derivedfor subcritical conditions, there is considerable uncer-tainty in the sediment transport equation for thisexperiment. Some preliminary runs were made testing3 different sediment transport equations: Engelund-Hansen, Van Rijn and the empirical equation usedby Bhallamudi & Chaudhry (1991) for this particularexperiment:

None of the equations provided a perfect fit withthe measured data; but Engelund and Hansen andVan Rijn equations notoriously underpredicted theupstream migration of the knickpoint (results notshown). Therefore, equation (11) was adopted.

A very small time step was used �t = 0.01 s to pre-vent large bed changes during the initial period of thesimulation. The results showed that both the computedbed and water surface profiles were rather smooth,without any noticeable numerical oscillation.

Figure 4 shows the computed bed and water sur-face profiles at the early stages of the simulation. Itcan be noticed that the most dramatic changes hap-pen during the first minute. The profiles seem to pivotaround the middle point of the steep reach. The inten-sity of the hydraulic jump decreases rapidly. The toe ofthe steep reach and the hydraulic jump moved rapidlydownstream; while the knickpoint migrated upstream.

A comparison between the computed profiles andthe bed profiles measured by Brush & Wolman (1960)at t = 160 min is shown in Figure 5. By this time,the intensity of the hydraulic jump has notoriously

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Figure 4. Computed profiles of bed (black lines) and watersurface (gray lines) at the beginning of the simulation: t = 0,1 and 6 min.

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Figure 5. Comparison between computed and measured bedprofiles at t = 160 min.

decreased since the maximum value of the Froudenumber was only slightly larger than 1.0.The steep bedslope has reduced to only 0.44% from its original valueof 10%. The computed bed profile was smoother thanthe observed profile and tended to follow the averagetrend of the measurements. The new upstream locationof the knickpoint agreed well with the observed value.

Bhallamudi & Chaudhry (1991) also simulated thisexperiment using a 1D model. However, they did notobserve a hydraulic jump in their numerical simula-tions (Bhallamudi, pers. comm.); likely because thespace discretization used �x = 0.30 m was too large tocapture the sharp front of the jump, and also becausetheir downstream water level was higher (ho = 0.03 m).

5 DISCUSSION

The morphodynamic simulation of the knickpointmigration experiment is a challenging problembecause of the steep 10% bed slope that gives riseto a hydraulic jump and transcritical flow conditions.

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1D morphological model with transcritical flow capa-bilities exist (e.g. Busnelli et al. 2001, Fraccarollo &Capart 2002, Papanicolaou et al., in press, Cui et al.a,b, in press); but they are still rare or inexistent in2D or 3D models (Spinewine & Zech 2002). Theshock-capturing techniques incorporated in River2D,added to the flexibility of the Finite Element methodto increase mesh resolution around areas with stronggradients (Fig. 2), made possible to simulate the sharphydraulic jump front (Fig. 3), without any numeri-cal oscillation. This test case would be intractable formost 2D commercial models currently available (e.g.RMA2, MIKE 21C, FESWMS).

Busnelli et al. (2001) argue that hydraulic jumpstend to rapidly disappear in alluvial beds underconstant water discharge. The experiments reportedby Bellal et al. (2003) show that amplitude of thehydraulic jump is drastically reduced by a movablesediment front. At least qualitatively, the model seemsto show a rapid reduction in the intensity of the jumpas bed evolves (Fig. 4). In the final profiles shown inFigure 5, the jump is barely noticeable.

The agreement between measured and computedbed profiles is reasonable (Fig. 5), considering thelarge uncertainty in the sediment transport equationand the fact that both bank erosion and depositionhave been completely ignored by assuming a fixedwidth. Brush & Wolman (1960) reported that channelbecame wider downstream of the knickpoint and nar-rower upstream; the maxima changes in width wereabout ±20% of the original width. Recently, Cantelliet al. (2004) and Wong et al. (2004) have demonstratedthe importance of width changes in the morphologicalevolution of steep channels after dam removal; whichcan be considered as a knickpoint-like problem.

A very interesting and potential application ofRiver2D-Morphology is the simulation of sedimenttransport after dam removal, which is a topic of highcurrent interest (Cantelli et al. 2004, Wong et al. 2004,Cui & Wilcox, in press, Cui et al. a,b, in press). Thehydraulic conditions after dam removal shown in Fig-ure 1 (from Cui & Wilcox, in press) are practicallyidentical to those depicted in Figure 3, with the over-steepened reach having a slope close the angle ofrepose of sediment. The computed bed profiles shownin Figure 4 also have some similarities with the initialbed profiles after dam removal computed by Cui &Wilcox (in press) for the Marmot Dam.

6 SUMMARY AND CONCLUSIONS

The recently developed River2D-Morphology model(Vasquez et al. 2005a, b) has been applied to themorphodynamic simulation of a knickpoint migra-tion experiment in transcritical flow with successfulresults.

The main features of the model are:

– It uses a two-dimensional unstructured mesh basedon the Finite Element method.

– It incorporates shock capturing techniques to simu-late the sharp fronts of hydraulic jumps.

– It can dynamically update the bed elevation as scourand deposition progresses, in both subcritical andsupercritical regimes.

Currently, the main limitations of the sedimenttransport model are:

– Suspended sediment is ignored.– Sediment size is assumed uniform.– Sediment waves (advection-dominated problems)

can not be properly simulated.

However, the present limitations listed above onlyreflect the early stage of development of this model,which is probably one of the first Finite Element depth-averaged models with capabilities for transcritical flowmorphodynamics.

The potential of River2D-Morphology to simulatealluvial rivers in both lowland and mountain reacheslooks promising.

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