MOD_FreeSurf2D is an open source MATLAB code that simulates fluid velocities
and depths in rivers and streams. Although ... stand-alone representation of fluid
flow in rivers and ...... and for a plane-bedded sand, open channel (Shen and.
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Computers & Geosciences 31 (2005) 929–946 www.elsevier.com/locate/cageo
MOD_FreeSurf2D: A MATLAB surface fluid flow model for rivers and streams$ Nick Martin, Steven M. Gorelick Department of Geological and Environmental Sciences, Stanford University, Braun Hall, Bldg. 320, 450 Serra Mall, Stanford, CA 94305-2115, USA Received 6 June 2004; received in revised form 7 March 2005; accepted 7 March 2005
Abstract MOD_FreeSurf2D is an open source MATLAB code that simulates fluid velocities and depths in rivers and streams. Although this model was designed for a specific purpose, MOD_FreeSurf2D can be employed in general scenarios when the depth-averaged, shallow water equations apply. The model approximates the depth-averaged, shallow water equations with a finite volume, semi-implicit, semi-Lagrangian representation. This numerical solution method provides accuracy and stability when using model time steps that exceed the Courant–Friedrichs–Lewy (CFL) restriction. An additional benefit of the numerical representation is the ability to simulate moving land/water boundaries. Model results were shown to be accurate when compared to published data from a dam-break experiment in a 21 m flume and from velocity and depth measurements along a 400 m river reach in Idaho. Results from the dam-break experiment simulations demonstrate the model’s ability to simulate wetting and drying. Additionally, sensitivity analyses conducted on the two scenarios show model convergence and demonstrate that the model can employ time steps that exceed the CFL restriction. r 2005 Elsevier Ltd. All rights reserved. Keywords: Numerical model; Semi-Lagrangian; Semi-implicit; Depth-averaged; Shallow water
1. Introduction This paper describes the mathematical foundation, numerical implementation, and testing of an open source, MATLAB code that solves the depth-averaged, shallow-water equations. MOD_FreeSurf2D was developed to be one of a series of open source modules used to simulate hydraulic and sedimentary processes in order to model aquifer formation in fluvial environments. $ Code on server at http://www.iamg.org/CGEditor/ index.htm. Tel.: +1 5404212994; fax: +1 5404428863. E-mail address:
[email protected] (N. Martin).
Currently, his fluid flow module is the only existing component of the suite of modules (but other modules are in development). Although MOD_FreeSurf2D was designed to provide a flow velocity field to a series of coupled sediment transport and deposition modules, this model provides a stand-alone representation of fluid flow in rivers and streams. Our purpose in presenting this paper and providing this MATLAB code is to provide an accessible tool that other researchers can use and modify for their particular applications. An important design criterion was to make MOD_FreeSurf2D as generally applicable as possible. With this goal in mind, the model permits the specification of a variety of parameters that allow the model user to include or remove various
0098-3004/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2005.03.004
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components of the governing equations according to the user’s requirements. These parameters and governing equations are covered in more detail in the following sections and in the manual that is included with the MATLAB code. Additional important criteria in the design of MOD_FreeSurf2D were an open source, stable code with the ability to simulate moving land/water boundaries. With these considerations in mind, we examined many different 2D model configurations (see Table 1) before selecting a semi-implicit, semi-Lagrangian, finite volume formulation. MOD_FreeSurf2D builds on the semi-implicit, semi-Lagrangian finite-volume approximation to the depth-averaged shallow water equations of Casulli and Cheng (1992) and related work (Casulli, 1990; Casulli, 1997; Casulli, 1999; Casulli and Cattani, 1994; Casulli and Cheng, 1992; Walters and Casulli, 1998). In addition, we include our own semi-Lagrangian method in the model (Martin and Gorelick, 2005). The numerical representation employed in the model has the advantages of stability and of the ability to represent moving land/water boundaries. Although many of the numerical methods combined in the model have been used in other models, MOD_FreeSurf2D is the only open source, depthaveraged code that we know of that employs this collection of methods.
2. Governing equations The governing equations for MOD_FreeSurf2D are the depth-averaged, shallow-water equations on a regular mesh: qU qU qU þU þV qt qx qy 2 qZ q U q2 U g ðU a UÞ ¼ g þ þ T þ qx qx2 qy2 H pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 U þV U þ fV , g Cz2 qV qV qV þU þV qt qx qy 2 qZ q V q2 V þ 2 ¼ g þ qy qx2 qy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U2 þ V2 g ðV a V Þ g V fU, þ T H Cz2 qZ qðHUÞ qðHV Þ þ þ ¼ 0, qt qx qy
ð1Þ
gravitational constant, t is time, is the horizontal eddy viscosity coefficient, H is the total water depth, and f is the Coriolis parameter. Eq. (4) and Fig. 1 give the relationship among total water depth; H, undisturbed water depth; h, free surface elevation, Z: H ¼ Z þ h.
(4)
Assumptions of a well-mixed water column and of a small water depth-to-width ratio are required for simplification to these equations from the 3D primitive variable equations (Casulli and Cheng, 1992). The vertical integration of the water column, that provides the depth-averaged representation, requires top and bottom boundary representations to replace the vertical change in velocity in the viscous terms. A prescribed wind-stress coefficient, gT, and prescribed wind velocities, Va and Ua, in Eq. (5) provide the top friction boundary: u qu qz ¼ gT ðU a UÞ
u qv qz ¼ gT ðV a V Þ:
(5)
A Manning–Chezy formula given in Eq. (6) provides a bottom friction relationship (Casulli, 1999; Casulli and Cheng, 1992): pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi U 2 þV 2 U 2 þV 2 qv (6) u qu ¼ g U; u ¼ g v, 2 qz qz Cz Cz2 where Cz is the Chezy coefficient, n is the kinematic velocity coefficient, and z represents the vertical direction. Eqs. (1)–(3) provide three partial differential equations in three unknowns, Uðx; y; tÞ, V ðx; y; tÞ, and Zðx; y; tÞ. The governing equations represent fluid flow in streams, rivers, and shallow estuaries when the hydrostatic pressure assumption, the well-mixed water column assumption, and the small depth-to-width ratio assumption are valid. In spite of these limiting assumptions, the 2D shallow water equations have been successfully employed to represent fluid flow in vertically well-mixed rivers, streams, and shallow estuaries (Falconer and Chen, 1991; Garcia and Kahawita, 1986; Heniche et al., 2000; Leclerc et al., 1990; Zhao et al., 1994).
3. Numerical approximations
ð2Þ
(3)
where U is the depth-averaged x-direction velocity component, V is the depth averaged y-direction velocity component, Z is the free surface elevation, g is the
Numerical approximations to the governing equations (Eqs. (1)–(3)) were selected to provide an accurate, stable representation of fluid flow and to simulate moving land/water boundaries. In MOD_FreeSurf2D semi-implicit and semi-Lagrangian numerical representations are combined with the finite volume method to solve the shallow water equations on a rectangular grid. Semi-implicit treatment requires that the gravitational terms in the momentum equations and the velocity divergence in the continuity equation be treated
Table 1 Comparison of a selection of 2D models for rivers, streams, and estuaries presented in literature Free Surfacec
Lagrangian representation of advectiond
Implicite
Wetting and drying
Martin and Gorelick (2005)a Akanbi and Katopodes (1988) Alcrudo and Garcia-Navarro (1993) Anastasiou and Chan (1997) Bates and Anderson (1993) Bellos et al. (1991) Benque et al. (1982) Bermudez et al. (1998) Casulli and Cheng (1992) Falconer and Chen (1991) Fennema and Chaudhry (1990) Fennema and Chaudhry (1989) Fraccarollo and Toro (1995); Toro (1992) Fujihara and Borthwick (2000) Galland et al. (1991); Hervouet and Haren (1996) Garcia and Kahawita (1986) Guillou and Nguyen (1999) Heniche et al. (2000) Hsu et al. (2000) Katopodes and Strelkoff (1978) Kawahara and Umetsu (1986) King (1977); King and Norton (1978) Layton and Panne (2002) Leclerc et al. (1990) Molls and Chaudhry (1995) Petera and Nassehi (1996) Tabuenca and Cardona (1992) Tee (1976); Tee (1977) Tseng and Chu (2000) Tucciarelli and Termini (2000) Zhao et al. (1994) Zhou and Goodwill (1997); Zhou and Stansby (1999) Zoppou and Roberts (2000)
K
K
K
K K
K K
K K
K K
K K K
K K K K
K K
K K K K
f
K K K K K
K K K K K
K
K K K
K K K K
K K
K K K K K K
K
K K K K K K K
K
K K K K K K K
K K K K
a
This article. Finite volume designation requires integral representation of the conservation equations. c Models with a free surface are those that solve an equation for free surface elevation. d Lagrangian representation of advection cateogory includes purely Lagrangian methods, semi-Lagrangian methods, and characteristics-based methods. e Implicit model include those models that are semi-implicit and those that allow designation of the degree of implicitness with the y method. f Wetting and drying designation requires that a model represent moving land-water boundaries by solving the governing equations for both wet and dry areas or by using moving or deforming calculationd grids. b
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CFL restriction:
V i,j+1/2 i-1/2,j
i,j
CFL ¼ w i+1/2,j U
i,j-1/2 Top View η
h
H
Side View
Fig. 1. Mod_FreeSurf2D variable location (top view) and variable definition (side view). Top view displays location of velocity components, total depth, and free surface elevation on computational grid. Side view shows relationship between free surface elevation, Z, total water depth, H, and undisturbed depth, h.
implicitly (Robert, 1981, 1982) while the remaining terms are treated explicitly. In semi-Lagrangian representations, advection is simulated by following the motion of the fluid. In the model, the mass- and momentum-conservative finite volume method (Fletcher, 1991) was adopted for stability and accuracy. The finite volume method is derived from the integral form of the conservation equations and is valid at both discontinuities and in smooth portions of the flow field (Zhao et al., 1994). Although our implementation is on a regular rectangular mesh, the finite volume method and not the finite difference method is employed. The combination of semi-implicit and semi-Lagrangian methods provides a stable solution method that permits relatively large model time steps without significant degradation of model accuracy. The advantage of relatively long model time steps is a smaller number of time steps, and fewer total calculations, are required to complete a simulation. A semi-implicit, semiLagrangian representation demonstrated a 6-fold increase in the maximum stable time step in applications of the atmospheric form of the primitive variable equations (Robert, 1982). In free-surface flow modeling, a semi-implicit, semi-Lagrangian portrayal permitted the relaxation of the Courant–Friedrichs–Lewy (CFL) restriction on time-step duration (Casulli, 1990; Casulli and Cattani, 1994; Casulli and Cheng, 1992). The
Dt Dxi
(7)
traditionally governs the maximum time-step duration in transient fluid flow simulations. In Eq. (7), w is the velocity component in the xi-direction, Dt is the time step size, and Dxi is the cell dimension in the xi-direction of flow. This restriction relates fluid velocity to time step size and to computational cell size and stipulates that the CFL number should be less than 1.0. 3.1. Semi-implicit representation In the semi-implicit process, a system of equations is generated where free-surface elevation, Z, is the only unknown variable at time N þ 1. This system of equations with one unknown is obtained by substituting Eqs. (9) and (10): Dt N N Nþ1 ZNþ1 H iþ1=2;j U Nþ1 ¼ ZN i;j i;j y iþ1=2;j H i1=2;j U i1=2;j Dx Dt N N Nþ1 y H i;jþ1=2 V Nþ1 i;jþ1=2 H i;j1=2 V i;j1=2 Dx Dt N N N ð1 yÞ H iþ1=2;j U N iþ1=2;j H i1=2;j U i1=2;j Dx Dt N N N H i;jþ1=2 V N ð1 yÞ i;jþ1=2 H i;j1=2 V i;j1=2 , Dx ð8Þ g Dt N N Ziþ1;j ZN U Nþ1 i;j iþ1=2;j ¼ FU iþ1=2;j ð1 yÞ Dx g Dt Nþ1 Nþ1 Ziþ1;j Zi;j y Dxrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi UN iþ1=2;j
2
þ VN iþ1=2;j
2
U Nþ1 iþ1=2;j
g Dt
Cz2iþ1=2;j H N iþ1=2;j Nþ1 gt U a U iþ1=2;j þ Dt , HN iþ1=2;j
ð9Þ
g Dt N Zi;jþ1 ZN i;j Dy g Dt Nþ1 Nþ1 Zi;jþ1; Zi;j y Dy rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N V Nþ1 i;jþ1=2 ¼ FV i;jþ1=2 ð1 yÞ
UN i;jþ1=2
g Dt
2
þ VN i;jþ1=2
Cz2i;jþ1=2 H N i;jþ1=2 gt V a V Nþ1 i;jþ1=2 þ Dt , HN i;jþ1=2
2
V Nþ1 i;jþ1=2
ð10Þ
where Dx is the distance across a computational volume in the x-direction; Dy is the distance across a volume in the y-direction, and Dt is the time step duration. Eq. (8)
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represents the discrete form of the depth-averaged continuity, Eq. (3). Eqs. (9) and (10) provide the discrete forms of the depth-averaged momentum equations, Eqs. (1) and (2). The free-surface system of equations obtained after this substitution is symmetric, positivedefinite and can be solved with a pre-conditioned conjugate gradient solver (Casulli and Cheng, 1992). After values of ZNþ1 are obtained by conjugate gradient solution of this system of equations, the ZNþ1 values are used in Eqs. (9) and (10) to explicitly update the velocity nþ1 components U nþ1 iþ1=2;j and V i;jþ1=2 . In Eqs. (8)–(10) subscripts i,j represent computational volume centers, where i 1=2 or j 1=2 denote computational volume faces. The computational mesh is the Arakawa C-grid, displayed at the top of Fig. 1,which has directional velocity, U and V, and total water depth, H, defined at the centers of volume faces, and free-surface elevation, Z, defined at volume centers. We adopt this rectangular computational grid in MOD_FreeSurf2D; however, these methods can be extended for use with unstructured grids (Casulli, 1997). The parameter y, in Eq. (8), determines the ‘‘degree of implicitness’’ of the solution and y should range between 0.5 and 1.0 for a semi-implicit method. In MOD_FreeSurf2D, the model user specifies the y-value. When y equals 0.5, the approximation is centered in time. When yequals 1.0, the approximation is fully implicit (Casulli, 1999). The term gT, in Eqs. (9) and (10), is the wind-stress coefficient. This parameter is user specified and setting this parameter to zero effectively removes the windsurface stress terms from model calculations. Va and Ua in Eqs. (9) and (10) are user prescribed wind velocities. A uniform, prescribed value for each wind velocity applies to the entire simulation domain. In Eqs. (9) and (10) the value for the Chezy coefficient, Cz, is obtained from the relationship with Manning’s roughness coefficient, Mn:
HN iþ1=2;j
Mniþ1=2;j
3.2. Semi-Lagrangian advection operator The advective operators, FU and FV in Eqs. (9) and (10), contain the advective, viscous, and Coriolis components from the governing equations. Eqs. (12) and (13) represent the semi-Lagrangian numerical approximation of these operators. U nsLbicubic represents the advective component; the Dt terms are the viscous terms, and f Dt V nsLbilinear provides the Coriolis component: n U isl þ1;j sl 2U nisl ;j sl þ U nisl 1;j sl n n FU iþ1=2;j ¼ U sLbicubic þ Dt Dx2 n n U isl ;j sl þ1 2U isl ;j sl þ U nisl ;j sl 1 þ Dt Dy2 þ f Dt V nsLbilinear ,
ð12Þ
n V isl þ1;jsl 2V nisl ;j sl þ V nisl 1;j sl FV ni;jþ1=2 ¼ V nsLbicubic þ Dt Dx2 n V isl ;j sl þ1 2V nisl ;jsl þ V nisl ;j sl 1 þ Dt Dy2 þ f Dt V nsLbilinear .
ð13Þ
A semi-Lagrangian representation of advection is a two-step process (Staniforth and Cote, 1991), and the semi-Lagrangian method employed in MOD_FreeSurf2D is presented schematically in Fig. 2. In step one, the Lagrangian step, the path of a fluid particle
destination point departure point
13/2
pathline 11/2
advective term interpolation locations Coriolis term interpolation locations
9/2
viscous term interpolation locations
7/2 5/2
.
(11)
3/2 1/2
flo
w
j =
Although it is convenient to consider Mn to be dimensionless (and we employ this convenience in the remainder of this paper), consistent dimensions for Mn in Eq. (11) are TL1=3 ðs m1=3 Þ. The equivalent value of Cz in US Customary units is obtained by multiplying H in Eq. (11) by 1.49 (Street et al., 1996). Because of this dimensional inconsistency and because Mn values are usually listed in reference sources as an acceptable range of values, we employ Mn as an estimated parameter. In MOD_FreeSurf2D, the model user supplies an Mn value for each computational volume in the simulation domain.
di re ct
io n
Cziþ1=2;j ¼
1=6
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i = 1/2
3/2
5/2
7/2
9/2
11/2
13/2
15/2
17/2
Fig. 2. Schematic of semi-Lagrangian advection representation in MOD_FreeSurf2D. Semi-Lagrangian solution for FV n8;11=2 (destination point) is shown as a two-step process; similar calculations are preformed for all FV ni;jþ1=2 . In first step, pathline tracing is performed to locate departure point. In this situation, departure point is located in computational volume (4,2). Second step, interpolation, is performed at departure point using different interpolation stencils for three different terms (advective, viscous, and Coriolis).
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located at each U iþ1=2;j and V ijþ1=2 location on the Eulerian computational grid (e.g. the destination point in Fig. 2) is traced back through Dt of travel time to the particle’s departure point (isL,jsL). Pathline tracing, is accomplished in MOD_FreeSurf2D with choice of either a semi-analytical particle path-line tracing method (Martin and Gorelick, 2005) or a classical four-step, explicit Runge–Kutta method for path line tracing (Zheng and Bennett, 2002). In step two, the Eulerian step, the velocity components in the advective operator are interpolated from the known U niþ1=2;j and V ni;jþ1=2 velocity components on the Eulerian grid around (isL,jsL). Because pathline tracing follows the representative fluid particles back Dt in time, the velocity values on the Eulerian grid are known from the calculations for the previous time step (i.e. time step n). Different interpolation stencils are employed for the three components of advective operators as shown in Fig. 2. The advective component is expected to be the dominant term in our applications and a bicubic Lagrange polynomial interpolation is employed in MOD_FreeSurf2D to calculate U nsLbicubic . The Coriolis and viscous terms in Eqs. (12) and (13) can be included or removed from calculations depending on the requirements of the model user. The option to retain these terms is included in the model to make the model as generally applicable as possible. If the Coriolis terms are retained, bilinear interpolation is employed to calculate V nsLbilinear in the Coriolis term (see Fig. 2), and the Coriolis parameter, f, is calculated with the f-plane model (Kundu, 1990) given by f ¼ 2O sin y0 ,
(14)
where y0 is the central latitude of the simulation region and O is the rotation rate of the earth. To remove the Coriolis terms from Eqs. (12) and (13), the model user would set O to zero. For the viscous terms, a constant horizontal eddy viscosity coefficient, e, is specified by the user. Because the eddy viscosity terms are generally small relative to the other terms in Eqs. (1) and (2), is often set equal to zero (Casulli and Cheng, 1992) effectively removing these terms from numerical representation. If the viscous terms are included in model calculations, a centered difference stencil is employed in the calculation of these terms as shown in Fig. 2. 3.3. Representation of wetting and drying An important feature of the model is that channel boundaries within the simulation domain do not require specification, as the model automatically determines wetting or drying in response to any change in flow conditions. The Arakawa C-grid layout (see Fig. 1) and total depth variables, employed by Casulli and Cheng (1992) and adopted here, ensure that the normal velocity
across this face is zero when total depth at a volume face is zero. Because the free-surface representation of total water depth (see Fig. 1 and Eqs. (15) and (16)) allows volume faces to naturally wet or dry in a conservative manner as flow conditions warrant, the correct closed boundary conditions are enforced at every dry face. This treatment enables the model to determine automatically the location of water/land boundaries: Nþ1 Nþ1 H Nþ1 ¼ max 0; h þ Z ; h þ Z , (15) iþ1=2;j iþ1=2;j i;j iþ1;j iþ1=2;j Nþ1 Nþ1 ¼ max 0; h þ Z ; h þ Z H Nþ1 i;jþ1=2 i;jþ1=2 i;j i;jþ1 . i;jþ1=2
(16)
3.4. Boundary conditions Two general types of domain boundaries, open and closed, exist for any simulation domain involving a free surface. Closed boundaries are simply boundaries across which no flow occurs, and generally represent a coast, a shoreline, or a riverbank. Open boundaries are virtual non-physical boundaries across which water may flow (Agoshkov et al., 1994), such as a river inflow boundary. In general, open boundaries may be inflow only, outflow only, or both inflow and outflow. For both open and closed boundaries, three dependent variables, the total water depth, the normal velocity component, and the transverse velocity component, require specification or other treatment. Open boundaries can be treated with either Dirichlet (fixed value) or radiation boundary conditions in MOD_FreeSurf2D. Depth-averaged velocity, total water depth, and total water flux can be fixed at open boundaries. Fixed total water depth and water flux boundaries should only be applied to inflow boundaries. Dirichlet velocity conditions may be applied to both inflow and outflow boundaries (Agoshkov et al., 1994). Two types of radiation boundary conditions are available. Both radiation conditions should only be applied to outflow open boundaries in the model. The first type of radiation boundary is a projection of velocity normal to the domain boundary: qU qU þ U upw ¼ 0, qt qn
(17)
where the drift velocity term, U upw , is simply the upwinded, normal directional velocity component, and n represents the direction normal to the domain boundary (Tseng, 2003). This simple condition does an adequate job of allowing information to propagate out of the simulation domain. The other radiation boundary condition is applied to free-surface elevation, Z, Eq. (18). Orlanski (1976) developed this boundary condition to limit, or absorb, wave reflections at open boundaries. The core of the method is the calculation of a propagation velocity, C n ,
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from grid points surrounding the boundary with the leap-frog finite-difference representation: qZ qZ þ C n ¼ 0. qt qn
(18)
4. Model applications MOD_FreeSurf2D was applied to two test applications for which there were published fluid-flow data sets. The first application employed published data from a dam-break flume experiment to demonstrate the model’s simulation capability and to highlight the model’s success in treating rapidly wetting and drying conditions. The second case used detailed river-flow data to check the model’s capability to reproduce velocities and water depths on the reach scale. 4.1. Application I: Flume experiment dam-break case A variety of 2D fluid flow models have been tested on dam-break style conceptual problems (Alcrudo and Garcia-Navarro, 1993; Anastasiou and Chan, 1997; Bermudez et al., 1998; Fennema and Chaudhry, 1989; Fennema and Chaudhry, 1990; Galland et al., 1991; Garcia and Kahawita, 1986; Zhao et al., 1994; Zoppou and Roberts, 2000), have been used to model dam-break style experimental setups (Bellos et al., 1991; Fraccarollo and Toro, 1995; Hsu et al., 2000; Jovanovic’ and Djordjevic, 1995; Molls and Chaudhry, 1995; Tseng and Chu, 2000), and have been employed to model dambreak induced inundation (Hervouet, 2000; Tucciarelli and Termini, 2000). Although the high gradient in the free surface across the dam produces substantial deviations from the assumptions underlying the shallow water equations, the dam break problem is often considered as a 2D flow (Fraccarollo and Toro, 1995). A dam-break experiment, the initially dry bed case of Bellos et al. (1992) with an initial water depth of 0.15 m behind the dam, and a flume slope of 0.002, was chosen for testing of MOD_FreeSurf2D. This particular case has been employed to test other 2D models (Bellos et al.,
935
1991; Hsu et al., 2000; Jovanovic’ and Djordjevic, 1995; Tseng and Chu, 2000). It involves a flume 21.2 m in length with an open downstream end and a closed upstream end (Fig. 3). A dam is located 8.5 m from the upstream end (12.7 m from the downstream end). The flume contains a curved constriction, beginning at 5.0 m from the closed end and terminating at 4.7 m from the open end. The narrowest part of the flume is 0.6 m and occurs at the dam location. In the areas outside of the constriction, the flume is 1.4 m wide. In the experiment of interest, the slope of the flume is 0.002, and the water depth behind the dam is 0.15 m. Water depth during the experiment was measured at eight different locations along the flume midline. Wave meters recorded measurements at areas of critical to subcritical flow regime. Specifically, wave meters were located at two locations upstream of the dam (x ¼ 8:5 and 4:5 m) and at locations directly adjacent to the dam on each side (x ¼ 0:0 and þ0:0 m). Pressure transducers were employed to measure water depth, assuming a hydrostatic pressure distribution, at four locations downstream of that dam characterized by super-critical flow conditions (x ¼ þ2:5, þ5:0, þ7:5, and þ10:0 m). The removal of the dam, or the water release time, is the starting point for measurements. The experiment lasted for approximately 70 s with water depth measurements published for about 62 s after dam removal. Water depth data for the experiment are provided for each measurement location by approximately 50 plotted points. These points were taken from the published data with digitization software. Each point corresponds to the average signal value for a time interval of 1.4 s (Bellos et al., 1992). The model was used to simulate this experiment given the flume topography, the initial water height behind the dam, and a homogeneous value of Manning’s roughness coefficient. A radiation velocity boundary condition, Eq. (17), was specified for the outflow end of the flume. Bellos et al. (1991) suggest employing a Manning’s roughness of 0.012 for the glass and smooth-steel flume configuration. The experiment was simulated with Mn ¼ 0:012, but a slightly better match to the measured depth values downstream of the dam is
Fig. 3. Plan view flume layout for dam-break experiment of Bellos et al. (1992). Wave meter locations are shown with circles. Pressure transducer locations are displayed with stars. At the beginning of the experiment, dam is removed and water behind dam travels from left to right and out of the open end of flume. Flume slope is 0.002. Flume length is 21.2 m and width is 1.4 m. Location 1 is at x ¼ 8:50 m; Location 2 is at x ¼ 4:00 m, Location 3 is at x ¼ 0:00 m, Location 4 is at x ¼ þ0:0 m, Location 5 is at x ¼ þ2:50 m, Location 6 is at x ¼ þ5:00 m, Location 7 is at x ¼ þ7:50 m, and Location 8 is at x ¼ þ10:00 m.
ARTICLE IN PRESS N. Martin, S.M. Gorelick / Computers & Geosciences 31 (2005) 929–946
0
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Hydrograph x = -0.00 m Simulated Depth [m] Measured Depth [m]
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20
30
40
50
60
70
Time [s]
Hydrograph x = +10.00 m Simulated Depth [m] Measured Depth [m]
0.06 0.04 0.02 0.00
(H)
30
Hydrograph x = +5.00 m
0.08
0.08
70
Time [s]
30
Time [s]
0.00
Hydrograph x = +7.50 m
0
0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0
(F)
0.06
0.00
20
Time [s]
0.10
70
Time [s] 0.08
10
(D)
Simulated Depth [m] Measured Depth [m]
(E)
Simulated Depth [m] Measured Depth [m]
Hydrograph x = +0.00 m
Hydrograph x = +2.50 m
0.08
0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0
(B)
Water Depth [m]
Water Depth [m] Water Depth [m]
50
Time [s] 0.10
Water Depth [m]
40
Time [s]
(C)
(G)
30
Water Depth [m]
Simulated Depth [m] Measured Depth [m]
(A) 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
Hydrograph x = -4.00 m
Hydrograph x = -8.50 m
Water Depth [m]
0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
Water Depth [m]
Water Depth [m]
936
0
10
20
30
40
50
60
70
Time [s]
Fig. 4. Dam-break flume data compared to simulated depth values. Simulated depths, for Mn ¼ 0:010, y ¼ 0:8, Dx ¼ 0:1250 m, Dy ¼ 0:5000 m, and Dt ¼ 0:103 s, and measured depths are displayed for each experimental measurement location. No data are provided for Fig. 4G. Crce ¼ 1:0 for this simulation.
obtained by a modest adjustment in Mn. Fig. 4 displays simulated water depths at each measurement location for a simulation with Dx ¼ 0:1250 m, Dy ¼ 0:0500 m, Dt ¼ 0:103 sec, and Mn ¼ 0:010. A roughness coefficient of 0.010 is not unreasonable for a glass and smooth-steel flume since a minimum value of 0.011 is given for a smooth-steel, lined, open channel (Street et al. 1996) and for a plane-bedded sand, open channel (Shen and Julien, 1993). We decreased Mn below 0.010 and found
that the simulated versus measured depth comparison deteriorated. In Fig. 4, simulated depth values match the measured values well for the three measurement locations above the dam. Simulated values below the dam do not match measured values as well as above the dam because the downstream locations are characterized by critical to super-critical flow. The model does not simulate these downstream locations as well, because the important
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vertical velocity information in the transition from subto super-critical flow is not reproduced by the model as a result of depth-averaging. However, the model is stable and robust enough to simulate this experiment moderately well despite the super-critical flow regions. In addition to the results displayed in Fig. 4, 40 simulations were completed to explore the effects of temporal and spatial discretizations. The four discretizations employed were Dx ¼ 0:0625 and Dy ¼ 0:0250 m, Dx ¼ 0:1250 and Dy ¼ 0:0500 m, Dx ¼ 0:2500 and Dy ¼ 0:1000 m, and Dx ¼ 0:5000 and Dy ¼ 0:2000 m. Ten different time steps, corresponding to Crce of 0.3–2.0 (see Eq. (19)), were used with each spatial discretization: Crce ¼
pffiffiffiffiffiffiffi Dt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dt ¼ 9:81 0:15 . gH Dx Dx
(19)
Crce is employed to represent the CFL criterion in this case because the celerity provided by the initial dambreak is expected to be the dominant model speed. Multiple spatial layouts and time steps provide further evidence of model consistency and convergence, and show that the model can provide accurate results when the CFL criterion is exceeded. Figs. 5–7 display results from a suite of simulations employing the four different discretizations, different time steps to provide the various Crce numbers, and an Mn equal to 0.010. These figures demonstrate model convergence, and display the relative accuracy of simulations at relatively high values of Crce. Fig. 5 displays the difference in simulated water mass remaining in the flume. Essentially the same amount of water is left in the simulation domain after 70 s when the value of Crce is less than or equal to 1.5. When the Crce value is
Fig. 6. Convergence and accuracy for simulated depths at four sub-critical to critical locations. Sum of Normalized Errors, SNE, is provided by Eq. (20A). For sub- to critical locations, Crce less than or equal to 1.4 are providing roughly the same degree of accuracy.
Fig. 7. Convergence and accuracy for simulated depths at three super-critical locations. Sum of Normalized Errors, SNE, is provided by Eq. (20A). For super-critical locations, Crce less than or equal to 1.4 are providing roughly the same degree of accuracy.
Fig. 5. Percentage of the initial water mass remaining in flume after 70 s. Simulations with Crce larger than 1.5 are not providing results of the same degree of accuracy as lower Crce value simulations.
greater than 1.5, results show that water depth is no longer accurately represented. The Sum of Normalized root mean square Errors (SNE) is employed as a measure of relative accuracy and is displayed in Figs. 6 and 7. Here it provides a statistic that is used to compare simulated water depths at each location to the measured values from Bellos et al. (1992). The SNE given in Eq. (20A), is obtained by computing the root mean square error, RMSE, at a location (Eq. (20B)), dividing it by the maximum measured water depth at that location, and summing the values over all
ARTICLE IN PRESS N. Martin, S.M. Gorelick / Computers & Geosciences 31 (2005) 929–946
locations (S total locations) at which observations exist: S X RMSE k , max H k k¼1
(20A)
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u M u1 X ðxf xv Þ2 . RMSE ¼ t M 1
(20B)
Water Depth [m]
In Eqs. (20A) and (20B) xf is the simulated value; xv is the measured value, and M is the number of measurements. The SNE decreases as the spatial representation is refined, as shown in Figs. 6 and 7. Convergence occurs at both the sub-critical and critical locations. Accurate simulated water depths are generated for the four subcritical to critical measurement locations when Crce does not exceed 1.5. Similar results were obtained for the super-critical locations but at a lower Crce; accurate simulated depths are generated at super-critical locations when Crce does not exceed 1.4. Wetting of the flume below the location of the dam is accurately simulated as shown clearly in Fig. 4. In addition, Fig. 8, shows the simulation results when the time frame in the dam-break case time was extended by 5 min. At the end of 6 min, all measurement locations are effectively dry. Taken together, these three figures demonstrate the ability of the model to simulate wetting and drying in response to rapidly changing flow conditions.
0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
Hydrograph x = -8.50 m Simulated Depth [m]
0
100 200 300 400 500 600
(A)
Water Depth [m]
0.10
Time [s]
Simulated Depth [m] 0.06 0.04 0.02 0.00
0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
0
100 200 300 400 500 600
Time [s]
Hydrograph x = +0.00 m Simulated Depth [m]
0
(B)
Hydrograph x = +5.00 m
0.08
(C)
The second test case for the model comes from data provided by a US Geological Survey study of flow in the Kootenai River, Idaho (Lipscomb et al., 1998). As part of this study, velocity and water depth data were collected along 41 transects approximately 10 m apart with an Acoustic Doppler Current Profiler (ADCP) for three reaches of the Kootenai River. The transects, taken together, provide a 3D profile of stream velocity in each reach. To test the model’s predictive capability on the reach scale, simulated water depth and water velocities for Reach 1 of the Kootenai River were compared to the data collected by the USGS. The ADCP data provide a set of 3D velocities. Because the model solves the depth-averaged shallowwater equations, the velocity values collected for each water column must be transformed into a single value for each water column location. Depth averaging was done using numerical integration and yielded 1591 depth-averaged values. Specifically, Simpson’s Rule was used to integrate the velocity values in each water column. Then, the average value was obtained by dividing by the total water depth at each measurement location. The depth-averaged velocity and depth data are presented in the contour plots of Figs. 9 and 10. Topography, water depth, and a Manning’s roughness coefficient were used in the model to simulate Reach 1. The topography data from the study are depths
Water Depth [m]
SNE ¼
4.2. Application II: Flow in a 400 m reach of the Kootenai River, Idaho
100 200 300 400 500 600
Time [s]
Hydrograph x = +10.00 m 0.08
Water Depth [m]
938
Simulated Depth [m] 0.06 0.04 0.02 0.00 0
(D)
100 200 300 400 500 600
Time [s]
Fig. 8. Simulated dam-break flume experiment water depths at four measurement locations over 6 min. In this simulation, Dx ¼ 0:1250 m, Dy ¼ 0:0500 m, Dt ¼ 0:103 s, y ¼ 0:8, and Mn ¼ 0:010. At end of 6 min, all measurement locations are effectively dry.
ARTICLE IN PRESS N. Martin, S.M. Gorelick / Computers & Geosciences 31 (2005) 929–946
0
Northing in Meters
5403900
0.8
0.4 0.7 0.6
0.2
0.8 0.9
0.9
1.0
0.9
0.7
0.6
939
0.8 0.7 0.6
0.8 5403800
0.7 0.2
0.4
0.7
0
0.4
0.6 0.2
0 5403700 543400
543500
543600
543700
543800
Easting in Meters Fig. 9. Contours of velocity magnitudes obtained by depth-averaging velocity data for Reach 1 of the Kootenai River. Horizontal axis is meters east-west. Vertical axis is meters north-south. Contours are velocity magnitudes in meters per second.
0
Northing in Meters
5403900
2 4 6
8
10
12
4 6
14 12 10
15
14 14 5403800
10
8 10
12
14
14
8
12 10 8 6 4
8 6
2
4
6 2
0
543600
543700
4 0 5403700 543400
543500
543800
Easting in Meters Fig. 10. Contours of measured water depths in Reach 1 of the Kootenai River. Horizontal (east-west) and vertical (north-south) axes in meters. Contours are total water depth in meters.
to the bottom measurements along each transect. These data were interpolated to square grids of 30, 15, 10, 6, 5, and 3 m using a quadratic radial basis function (Golden Software, 1999). The grids are aligned in such a way that all the defined larger grid points are represented in every higher resolution grid. A complementary initial waterdepth grid was generated in the same manner for each different grid size. A uniform value of Manning’s roughness coefficient, Mn ¼ 0:055, was applied to every computational volume in the simulation domain. This roughness coefficient value is from the higher end of the expected range of roughness coefficients in rivers (Barnes, 1967; Street et al., 1996) but provided a better fit for simulated velocities than obtained with lower coefficient values. The mean discharge on the day of
measurement, 1240 m3/s (Lipscomb et al., 1998), was used for the inflow boundary condition and the freesurface radiation condition from Eq. (18) provided the outflow boundary condition. Flow was distributed by multiplying the vertically integrated velocities from the ADCP by the local water depths. Given the Dirichlet inflow boundary, steady state was achieved by marching through time until all velocity and depth values were constant over time. A number of simulations were completed for Reach 1 using the initial and boundary conditions stipulated above to investigate the effects of time step size. The simulated velocity and depth values were compared to the depth-averaged measurements. Bilinear interpolation of the simulated values provided velocity and depth
ARTICLE IN PRESS N. Martin, S.M. Gorelick / Computers & Geosciences 31 (2005) 929–946
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0 0.4 0.6 0.7 0.8 0.9
Northing [m]
5403900
0.2
0.4 0.6 0.7
0.8
0.9 1.0 1.0
0.9 0.8 0.7
0.9 0.8
5403800
0.6
0.7 0.8
0.7
0.6 0.4 0.2
0.4 0.2
0
0.6 0
5403700 543400
543500
543600
543700
543800
Easting [m] Fig. 11. Contours of simulated velocity values for Reach 1 of the Kootenai River. In this simulation, Dx ¼ Dy ¼ 5:0 m, Dt ¼ 2:4 s, and y ¼ 0:8. Horizontal axis is meters east-west. Vertical axis is meters north-south. Contours are velocity magnitudes in meters per second.
4
Northing [m]
5403900
2 4 6 8 10
8 6
2
5403700 543400
14 14 12
14 12 10
14 12 10
5403800
12
12 15
14
0 6 8 10
10 8 6
8 6 4 2
4 2 0
4 0
543500
543600
543700
543800
Easting [m] Fig. 12. Contours of simulated water depths for Reach 1 of the Kootenai River. In this simulation, Dx ¼ Dy ¼ 5:0 m, Dt ¼ 2:4 s, and y ¼ 0:8. Horizontal axis is meters east-west. Vertical axis is meters north-south. Contours are total water depth in meters.
magnitudes at each water column measurement location. Fig. 11 displays contours of the simulated velocity values at the measurement locations. Fig. 12 shows the contours of simulated water depth at the measurement locations. The simulation that generated Figs. 11 and 12 employed Dx ¼ Dy ¼ 5:0 m, Dt ¼ 2:4 s, with y ¼ 0:8. Simulations with spatial discretizations of 10 m or smaller provide similar results at steady state. Visual comparison of velocities in Figs. 9 and 11 and water depths in Figs. 10 and 12 indicates the simulation ability of the model. The measured and simulated velocity contour plots and depth contour plots have similar contour shapes and magnitudes. Quantitative comparison shows a median absolute residual of 0.065 m/s and a RMSE (Eq. (20B)) of velocity of
0.108 m/s. These values can be viewed in the context of the depth-averaged maximum measured velocity for the reach, which is 1.175 m/s, and the median measured depth-averaged velocity, which is 0.752 m/s. The median absolute residual of depth is 0.088 m and the RMSE of depth is 0.337 m. These values can be viewed in the context of the maximum measured water depth of 16.340 m, and the median measured depth of 7.580 m. Fig. 13 contains a contour map of velocity residuals, ðxf 2xv Þ from Eq. (20B), and shows that the location of the larger magnitude velocity residuals is along the bank or close to the outflow boundary. One reason that the larger magnitude velocity residuals are located adjacent to the riverbank is that the rectangular finite-volume discretization does not fully capture the irregular
ARTICLE IN PRESS N. Martin, S.M. Gorelick / Computers & Geosciences 31 (2005) 929–946
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Fig. 13. Filled contour plot of velocity residuals calculated from the simulation displayed in Fig. 11. Large magnitude residuals are primarily located near riverbank and near outflow boundary. Velocity residuals are in meters per second.
Fig. 14. Histogram of velocity residual values obtained from the simulation displayed in Fig. 11. These residual values were employed to generate Fig. 13. Residual values are roughly centered on zero with a median residual value of 0.025, maximum residual value of 0.352, and minimum residual value of –0.546.
geometry of the riverbank. Although the model permits the use of varying roughness coefficients across the simulation domain, a uniform Manning’s roughness coefficient was used in this simulation. Improvement in simulation results relative to data might be achieved by varying the roughness coefficient across the channel or reach. In Figs. 14 and 15, histograms of velocity and depth residual values are displayed. Fig. 14 demonstrates that
the velocity residual values are clustered about zero with a slight positive bias of median residual of 0.025 m/s, or 3.4% of the median velocity. Fig. 15 shows that the histogram of depth residuals is also centered about zero. We also investigated the accuracy of the simulations when the CFLo1 condition was greatly exceeded. Fig. 16 displays the change in the RMSE of velocity as the grid Courant number increases. The grid Courant
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Fig. 15. Histogram of water depth residuals obtained from the simulation displayed in Fig. 12. Residual values are roughly centered on zero with a median residual value of 0.016, maximum residual value of 2.029, and minimum residual value of –2.278.
Fig. 16. RMSE of velocity, from Eq. (20B), for each Crmax value, from Eq. (21). All of the velocity RMSE values are close to 0.10 demonstrating that the program maintains accuracy for high values of Crmax . The decrease in RMSE with increase in Crmax is result of smoothing effect shown in Fig. 18 and 19.
number for these simulations is given by Crmax ¼ U max
Dt , Dx
(21)
where Umax is the maximum simulated directional velocity component. As Crmax increases, the RMSE of velocity remains close to 0.10 and even decreases slightly. At Crmax values above the largest value given
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Fig. 17. RMSE of simulated water depth values, from Eq. (20B), for each Crmax value, from Eq. (21). Consistent decrease in RMSE with decrease in Dx demonstrates model convergence. RMSE decreases with Dx because resolution of irregular riverbank improves.
Northing [m]
5403900
0 0.4 0.6 0.7 0.8 0.2
0.4
0.7 0.8 0.9
0.6
1.0
0.9 0.8
0.9
0.8 0.2
0.6
5403700 543400
0.9 0.8 0.7
0.7
0.6
0.6 0.4
0.4 0
5403800
0.7
1.0
0.2
0
543500
543600
543700
543800
Easting [m] Fig. 18. Contours of velocity magnitudes, simulated with Dx ¼ Dy ¼ 5:0 m, Dt ¼ 10:0 s and y ¼ 0:8, for Reach 1 of the Kootenai River. Velocity contours are in meters per second. This simulation corresponds to Crmax ¼ 2:0. Size of 1.0 m/s contour has decreased relative to Fig. 11.
for each Dx, the model did not converge to a solution. Fig. 17 displays the change in the RMSE of simulated depth values as Crmax increases. The RMSE depth values do not change with Crmax. However, the depth RMSE values become smaller as Dx decreases and better resolution is obtained of the irregular water/land boundary. Examination of Figs. 16 and 17 suggests that the model provides a reasonable steady-state solution for Crmax exceeding 10. To examine the slight decrease in the RMSE of velocity as Crmax increases, Figs. 18 and 19, contour
plots of velocity magnitude from two additional simulations, are presented. These two figures represent simulations employing identical conditions and parameter values to those that produced Fig. 11 except that the model time step and Crmax are different for each simulation. In Fig. 11, (Dt ¼ 2:4; Dt ¼ 10:0 in Fig. 18, and (Dt ¼ 50:0 in Fig. 19). Using Eq. (21) and U max equal to the maximum simulated component velocity for each time step, Crmax ¼ 0:5 when Dt ¼ 2:4; Crmax ¼ 2:0 when Dt ¼ 10:0, and Crmax ¼ 10:0 when Dt ¼ 50:0. In these three figures, the size of the 1.0 m/s contour
ARTICLE IN PRESS N. Martin, S.M. Gorelick / Computers & Geosciences 31 (2005) 929–946
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0
Northing [m]
5403900
0.7
0.6 0.2
0.4 0.6 0.7
0.9
0.8
0.8 0.9
0.9
0.8
0.9
0.9
0.7
0.8
5403800
0.7 0.6
5403700 543400
0.6
0.7 0.8 0.6 0.4
0.4 0.2
0.2
0
0
543500
543600
543700
543800
Easting [m] Fig. 19. Contours of velocity magnitudes, simulated with Dx ¼ Dy ¼ 5:0 m, Dt ¼ 50:0 s and y ¼ 0.8, for Reach 1 of the Kootenai River. Velocity contours are in meters per second. This simulation corresponds to Crmax ¼ 10:0. Size of 1.0-m/s contour has decreased relative to Figs. 11 and 18.
decreases as Dt and Crmax increase while the other contours maintain shape and location in results from all three simulations. This suggests that employing a larger time step smoothes the values with the largest magnitudes. The 1.0 m/s contours of the measured velocity values in Fig. 9 are similar in size and location to the equivalent contour in Fig. 19. The improvement in the fit of the 1.0 m/s contour as Crmax increases accounts for the slight improvement in RMSE of velocity.
5. Conclusions MOD_FreeSurf2D solves the depth-averaged, shallow-water equations using a semi-implicit, semi-Lagrangian numerical approximation. This open source code primarily employs the numerical methods proposed by Casulli and Cheng (1992) and has the advantages of stability, representation of moving land/water boundaries, and the ability to employ time step sizes that exceed the CFL criterion. Although the model was designed for a specific purpose, it can be employed to simulate water depth and water velocity in general situations where the governing shallow water equations apply. Comparisons to data at the laboratory and field scales show that the model accurately simulated both a transient dam-break flume experiment and steady-state flow in the Kootenai River. Sensitivity analyses conducted on the two cases demonstrate that the model yields accurate results when the CFL condition is exceeded and demonstrate model convergence. Accuracy was maintained when the CFL value based on celerity, Crce, was 1.4 in a transient solution and when the CFL value based on the maximum velocity, Crmax,
exceeds 10 in a steady-state solution. The dam-break flume simulations also demonstrate that the model’s ability to simulate wetting and drying of the domain.
Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. EAR0207177. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. This material is also based upon work supported under a Stanford Graduate Fellowship. In addition, we would like to thank S. Lipscomb and C. Berenbrock for providing the Kootenai River data. We would also like to thank R.L. Street for his insight into and his suggestions on free surface flow modeling. We want to acknowledge the boundary condition suggestions received from Y.-H. Tseng. Finally, we would like to thank two anonymous reviewers for insightful suggestions that improved the quality of this paper.
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