Calculation of shallow water ¯ows using a Cartesian cut cell approach
D.M. Causon
*, D.M. Ingram, C.G. Mingham, G. Yang, R.V. Pearson
Centre for Mathematical Modelling and Flow Analysis, The Manchester Metropolitan University, Manchester M1 5GD, UK
Received 7 May 1999; accepted 16 August 1999
Abstract
A new grid generation method for the computation of shallow water ¯ows is presented. The procedure, based on the use of cut cells on a Cartesian background mesh, can cope with shallow water problems having arbitrarily complex geometries. Although the method provides a fully boundary-®tted capability, no mesh generation in the conventional sense is required. Solid regions are simply cut out of a background Cartesian mesh with their boundaries represented by dierent types of cut cell. For the ¯ow cal-culations a multi-dimensional high resolution upwind ®nite volume scheme is used in conjunction with an ecient approximate Riemann solver to deal with complex shallow water problems involving steady or unsteady hydraulic discontinuities. The method is validated for several test problems involving unsteady shallow water ¯ows. Ó 2000 Published by Elsevier Science Ltd. All rights reserved.
Keywords:Cartesian cut cell; Shallow water equations; Riemann based schemes; Bore wave propagation; Finite volume method
1. Introduction
In the past two decades, the numerical modelling of coastal and estuarine hydrodynamics has usually been accomplished using ®nite dierence discretisations of the shallow water equations on Cartesian grids, e.g. [1,29]. However, most existing Cartesian mesh methods are restricted in their ability to ®t geometries of arbitrary complexity. This lack of generality is largely due to the representation of curved boundaries by a staircase-type approximation and because of the diculty of concen-trating mesh cells in regions of speci®c interest without the necessity to employ mesh patching or some similar approach to remesh a subdomain. An approach that has gained popularity recently is the use of boundary-®tted grids which make the boundary contour a coordinate surface [5,9,25,43]. The ¯ow equations are transformed from Cartesian to curvilinear coordinates and then ap-proximated using ®nite dierences. A grid can be gen-erated iteratively by means of an elliptic solver [37], or directly by an algebraic technique such as trans®nite interpolation [23]. However, the generation of a suitable grid is non-trivial, case speci®c, and requires signi®cant user expertise. The ¯ow equations are also more
com-plicated in the transformed coordinates and discretisa-tion of the metric transformadiscretisa-tion derivatives can give rise to geometrically induced errors [26].
Alternatively, the dierential equations may be recast in integral form with the solution variables stored at cell centres as integral averages over each cell area. The spatial discretisation then involves ¯ux balances across each cell interface. Such ®nite volume approaches [3,6,10,11,30,32,33,45,46] combine the simplicity of ®-nite dierence methods with the geometric ¯exibility of ®nite element methods. This is because, unlike methods based on the curvilinear dierential form of the ¯ow equations, the ¯ow variables remain referenced to the Cartesian frame, even when using a boundary-®tted mesh. Finite volume methods lend themselves well to techniques such as mesh patching and mesh adaptation. However, there remains the need to generate a suitable boundary-conforming mesh which is non-trivial, par-ticularly for multi-component geometries of the type illustrated in Fig. 1. The problem is that a grid topology which is optimal in one part of the ¯ow domain is generally sub-optimal in other regions. This has led to multi-block approaches [36] where grids are generated in blocks for each subdomain with coordinate line conti-guity constraints satis®ed at each block interface, and to so-called Chimera methods [7] which use a hierarchical structure of overlapping meshes but these approaches generally result in complicated computational methods. *
Corresponding author. Tel.: 3557; fax: +44-161-247-1483.
E-mail address:d.m.causon@doc.mmu.ac.uk (D.M. Causon).
An alternative is the use of a ®nite element approach which can be combined with mesh adaptation to provide a very eective calculation method. Examples of ®nite element methods abound throughout computational mechanics. However, in common with ®nite volume methods, problems which involve large scale movement of ¯ow domain boundaries inevitably require re-mesh-ing several times durre-mesh-ing calculations. This can be time-consuming and, with the attendant interpolation of data, may reduce the formal accuracy of computed so-lutions.
Cartesian cut cell methods were ®rst developed within the aerospace community speci®cally for handling multi-component geometries [8,17,40±42]. However, the basic principles have much wider application. The ap-proach is conceptually quite simple. Boundary contours are cut out of a background Cartesian mesh and cells that are partially or completely cut are singled out for special treatment. The remainder of the ¯ow cells are treated in a straightforward manner. By using cut cells the method fully retains the boundary-conforming properties of a curvilinear mesh ®nite volume or ®nite element method. Advantages of the cut cell approach over other ®nite dierence, ®nite volume or ®nite ele-ment approaches are:
1. There is no mesh generation in the conventional sense. This is replaced by relatively straightforward calculations for the boundary segment intersections with the background Cartesian mesh.
2. The method can be implemented very eciently such that only data relating to cells de®ning boundaries and the ¯ow domain need be held in memory. Conse-quently, a long narrow meandering ¯ow domain like a river estuary can be handled just as eciently as a region which is nearly rectangular.
3. The majority of the ¯ow domain is overlaid with a regular Cartesian mesh so that loss of solution accu-racy due to any pathological cases involving exces-sively stretched or skewed cells is avoided.
4. Moving ¯ow boundaries can be accommodated by re-computing cell-boundary intersections as boundaries move, rather than re-meshing the whole ¯ow domain or large portions of it; furthermore, the amplitude of boundary motion is unrestricted.
5. When used in conjunction with mesh adaptation adapting to irregular static or moving boundaries and/or static or moving ¯ow features (e.g. bathymet-ric data or moving fronts), the method can provide ®ne mesh resolution where required with much larger coarse spacings in regions where spatial gradients are low; thus the method oers the potential to be a high-ly ecient and versatile practical computational modelling tool.
An assessment of the accuracy of Cartesian cut cell approaches has been made by Coirier and Powell [14].
In recent years, a number of high resolution numer-ical schemes have been developed for use in computa-tional hydraulics. In [21], a Roe-type approximate Riemann solution of the two-dimensional shallow water equations was given by Glaister. Total variation di-minishing (TVD) and essentially non-oscillatory (ENO) methods have also been applied to the computation of free surface ¯ows [34,43]. Finite volume techniques based on approximate Riemann solvers were employed for the shallow water equations in [3,4,32,33]. All of these schemes require two-dimensional rectangular or transformed boundary-®tted structured meshes which lead to the diculties referred to above, particularly for complicated multi-component geometries.
In this paper, we present a new approach to mesh generation for shallow water ¯ows. The method is based on the Cartesian cut cell technique described above and previously developed for compressible ¯ow applications [41,42]. In the next section, the generation of a Cartesian cut cell mesh is described. A ®nite volume integration scheme suitable for solving the shallow water equations on cut cell meshes is presented in Section 3; the ¯ow solver is an upwind scheme of the Godunov-type based on MUSCL reconstruction [38] and a suitable approxi-mate Riemann solver. The method is then evaluated by recourse to a variety of test problems in Section 4. Ex-tensions of the method to include mesh adaptation are discussed in Section 5, and ®nally, in Section 6, some conclusions are drawn.
2. Principles of the cut cell method
2.1. Boundary de®nition
It is assumed that the physical boundaries of the ¯ow domain are de®ned by a set of data points, such as x1;y1; x2;y2;. . .; xmax;ymax, and that the data points
achieved by adding more data points. If the ¯ow domain is multiply connected, i.e. there is more than one solid region, additional sets of data points are needed. In order to cope with practical ¯ow geometries of arbitrary complexity, the approach that we adopt is the use of so-called cut cells on a background uniform or non-uni-form Cartesian grid [41,42]. The background Cartesian mesh is constructed ®rst and then any solid regions are simply cut out of it. The intersections of the boundaries of solid regions with the background grid de®ne cut cells that have one of their sides coincident with a boundary segment; the method thus produces a boundary con-forming mesh without the necessity to make the boundary a coordinate surface. In fact, there is no grid generation in the conventional sense; all that is necessary is to calculate the intersections of a series of line seg-ments with the background Cartesian mesh.
2.2. Finding the intersection points
Suppose the intersection points of a particular straight line segment, de®ned in terms of its start and end coordinates xs;ys and xe;ye, are to be found.
Before proceeding it is necessary to determine both the slopeQof the line segment and identify which grid cells contain the start and end points. The address Is;Jsof
the cell containing the start point is given by
Isint
xsÿx0
Dx
1;
Jsint
ysÿy0
Dy
1;
1
where int xreturns the integer part ofxandx0andy0are
the coordinates of the bottom left corner of the compu-tational domain. The address Ie;Jeof the end point is
found in a similar way. For convenience we identify which of the four quadrants 0°;90°, 90°;180°,
180°;270°or 270°;360°the slope of the line lies in.
Once the grid cells containing the start and end points of the line segment have been identi®ed and its slope computed the required intersection points can be found. Fig. 2 shows a line segment which cuts across a uniform background Cartesian mesh. Suppose that the intersec-tion points of the line segment with cell i;j are to be found: clearly the point a at which the line segment enters the cell is already known because it is the exit point from the previous cut cell. It thus remains neces-sary only to determine the exit point,b, for the cell i;j. Sinceais on the left side of the cell andQ2 0°;90°the
exit point must lie above and to the right ofa, locating the exit point on either the top or right-hand side of the cell. The intersection pointsb, between the line segment and the line yj1y0 j1Dy, and c, between the
line segment and the linexi1x0 i1Dx, are now
found. Since yc >yj1 the exit point must be xb;yb.
This process is repeated for all subsequent grid cells intersected by the line segment until the cell Ie;Je is
reached. The cases where Qlies in the other quadrants can be treated analogously.
In cases where a non-uniform Cartesian mesh is re-quired, the grid is de®ned using two one-dimensional arraysx iandy jand a slightly modi®ed version of the algorithm is employed.
Initially, all cells in the background mesh are ¯agged as ¯ow cells or solid cells. Once all the cut cell inter-sections have been established, the cells which intersect the boundary of any solid region are registered as cut cells. Sweeps across the background mesh are then performed to identify which cells or rows of cells are bounded by solid or partially cut cells: these are regis-tered as solid cells. Three primary cell types are thus formed in the computational domain. These are the cut cell and two types of uncut cell, namely the ¯ow cell and solid cell (see Fig. 3). In the case of cut cells, these may be further categorised into four sub-types depending upon the slope Q of the boundary intersec-tion with the cell. Fig. 4 shows all possibilities for the four dierent sub-types of cut cell. Using these cell types, the geometry of a solid region can be accurately represented.
Fig. 2. Finding the intersection points of a line segment.
3. The ¯ow solver on a cut cell mesh
3.1. The shallow water equations
The shallow water equations (SWE) are a reduced form of the depth averaged Navier±Stokes equations which represent the conservation of mass and momen-tum. Admissible solutions may admit ¯ow discontinu-ities such as bore waves.
In dierential conservative form the equations are oU
whereFandGare the convective ¯ux vectors,Ais the Coriolis force term,Bthe wind shear stress term,Cthe
bed shear stress term, D the bed slope term and the viscous stress ¯ux vectorsFv andGv are
Fv
In the above,fis the Coriolis force,gthe acceleration due to gravity,qthe water density,u; vthe velocities in the x andy directions, respectively, /the geopotential (gh, h is the water depth), sxw; syw the wind shear stresses and sxf; syf the bed shear stresses in the x;y directions, rxx; ryy; sxy; syx the normal and shear stress terms, respectively, and bx; by are the bed slopes (mea-sured downwards) in the x,ydirections.
The integral form of the equations is o closed by the control surface S.
The homogeneous part of (3) are the convective ¯ow equations that describe the time evolution of the depth and velocity of the water over the physical region of interest in the absence of source terms. These equations are of hyperbolic type and admit discontinuous solu-tions (e.g. bore waves) which are dicult to resolve numerically. Any numerical scheme applied to the full ¯ow equations (3) must be able to deal satisfactorily with their behaviour. In the present work the viscous transport and source terms, Qs, are ignored and the
equation set solved is
o
Details of how the source terms are discretised in the present ®nite volume method may be found in [27].
3.2. Finite volume discretisation of the ¯ow equations
Since the inviscid shallow water Eq. (4) constitute a genuinely hyperbolic system, the numerical schemes developed for systems of conservation laws can be ap-plied to them. Here, we apply the MUSCL-Hancock ®nite volume scheme [39] with appropriate modi®ca-tions for Cartesian cut cell meshes.
This is a two-step, high resolution, upwind scheme of the Godunov type. The predictor step uses a non-con-servative approach, that de®nes an intermediate cell centre value over a time interval Dt=2:
AUnij 1=2 AU
whereAis the cell area, cell side vectorSnk the length of sidek multiplied by the outward-pointing unit normal vectornandmis the maximum number of cell sides. For a ¯ow (or uncut) cell,m4; for a cut cell,m3±5. The ¯ux vector functionH Ukis evaluated at each cell side
kfollowing a linear reconstruction of the ¯ow solution within each cell, via
Uk Unijrk rUnij; 7
where rk is the normal distance vector from the cell centroid to sidekandrUnijis a limited gradient vector in space (to be de®ned in Section 3.4).
The corrector step of the scheme is fully conservative. The intermediate solution from the predictor step is used to de®ne a set of left- and right-hand states for a series of Riemann problems. The solution of these Riemann problems provides a set of upwind cell interface ¯uxes which are used to update the ¯ow solution over the time intervalDt, i.e.
AUnij1 AU a local Riemann problem normal to the cell interfacek
and the left- and right-hand states at the interface are calculated by
wherelandmrefer to the right neighbouring cell of the celli;j.
To solve the Riemann problem, the approximate Riemann solver of Harten, Lax and van Leer (HLL) [24] is used. Other approximate Riemann solvers could be employed but we have found the HLL solver to be ac-curate and robust in practice.
3.3. HLL Riemann solver at cell interfaces
The HLL Riemann solver provides the Riemann so-lution at each cell interface based on the following as-sumptions:
1. The star region between the two acoustic waves is simpli®ed by a single intermediate wave state. 2. The only waves present are the left and right acoustic
waves which depend on a priori wave speed estimates sL andsR, see Fig. 5.
Under these assumptions, the vector of conserved variables U and the corresponding one-dimensional ¯ux vectorFin the star region can be determined from the conservation laws in the integral form
R
Hence, the HLL approximate Riemann solutions are de®ned as
In the general two-dimensional case, the upwind ¯ux corresponding to (10) and (11) for use in (8) is
H Uÿ Lk;URk
k is the outward pointing unit normal side vector to cell sidekand suitable wave speed estimates sL; sR (see Fig. 5) are given by Davis [16]
For shallow water problems with a dry bed, the wave speed estimates that are used are [20]
The time step employed is
with an analogous de®nition for Dty. The Courant numbermwas taken in our calculations to be 0.5 which is close to the stability bound based on a linear von Neumann analysis.
3.4. Calculation of gradients and reconstruction technique
In order to achieve second-order spatial accuracy, reconstruction of the ¯ow data within each cell is nec-essary, requiring appropriate gradient information. The gradient calculation is straightforward on ¯ow cells away from a solid boundary. First, a central dierence gradient of conserved variables is estimated using the known values at neighbouring cell centres. Then, a slope limiter function is applied to prevent spurious over- or under-shoots. For cells near a solid boundary, however, a dierent gradient calculation is needed. In [17], a linear reconstruction technique has been proposed for dierent types of cut cell, based on cells in the neighbourhood of the cut cell interface and a suitable path integral. However, since we wish to accommodate moving boundaries in future work, we have modi®ed the gra-dient calculation appropriately.
If re¯ection boundary conditions are used on a solid boundary, the variables in ®ctional cell R can be ob-tained by (see Fig. 6)
/R /ij;
vRvijÿ2 vijn
ÿ
n: 18
The gradients on cut cell i;jmay be of two types: ¯uid and solid. We calculate the ¯uid gradients and solid gradients separately, i.e.
limiter function that is used to prevent over- or under-shoots. The limiter function may take one of the fol-lowing forms, e.g.:
I. The Superbee limiter
G a;b smax0;min 2jbj;sa;min jbj;2sa;
ssign b: 23
II. The van Leer limiter
G a;b ajbj jajb
jaj jbj : 24
Once the two types of gradients are calculated, a length average technique is used to obtain unique gra-dients in the cut cell
UxD thexandy directions, respectively.
Since DxfDxsDx and Dyf DysDy, we note that if Dyf Dy, Dys0, so Ux Ufx; otherwise, if
DysDy, Dyf 0, so UxUsx. Ux andUy are compo-nents of a gradient vector in the cut cell, i.e.
rUij
Ux
Uy
: 27
Given the gradient vector rUij, a reconstructed so-lution vectorU x;ycan be found anywhere within the cut cell from
U x;y Uijr rUij; 28
where r is the normal distance vector from the cell centroid to any speci®c interface or solid boundary.
3.5. Merging of cells
force an unrealistically small time step. To overcome this problem, a cell merging technique [12,13] is imple-mented that is simple and can be easily extended later to moving boundary problems (that are not considered here). The basic idea is to combine several neighbouring cells together so that the interfaces between merged cells are ignored and waves can travel in a newly combined larger cell without reducing the global time step. To implement this technique, we need ®rst to determine which cells to merge. We de®ne a minimum areaAminfor
a cut cell and then proceed as follows:
1. Check each cut cell area against theAmin criterion. If
its area is larger than Amin, no further processing is
needed so go to step 3; if its area is smaller than Amin, a check is made to see whether the cell has
al-ready appeared in the cell merging list. If so, no fur-ther processing is needed so go to step 3; if not, a neighbouring cell is needed for merging so go to step 2.
2. The choice of which neighbouring cell to merge with depends on the slope of the solid side of the cut cell. For example, suppose cut cell A in Fig. 7 is smaller thanAmin, and is of sub-type one. By comparing the
right hand and bottom ¯ow interfaces and merging in the direction of the longer one we ®nd that a suit-able cell for cut cell A to merge with is cut cell C. For other sub-types of cut cell, an analogous approach is used. Once a neighbouring cell has been identi®ed it is saved to the cell merging list.
3. Proceed to the next cut cell and then repeat steps 1 and 2 until all cut cells have been processed.
The choice for the minimum area criterion Amin is
based in practice on a trade-o between the time-step constraint and resolution accuracy. In our calculations, we chose to setAminto one half of the smallest ¯ow cell
size.
Having identi®ed suitable neighbouring cells for merging, we now describe the adjustments made to the
¯ow solver. The following description relates to the predictor step of the MUSCL-Hancock solver (6); changes to the corrector step (8) at merged cut cells follow analogously. Consider ®rst the standard un-modi®ed procedure for updating two cellsAandB, see Fig. 8
where the ¯ux vector function H Uk is evaluated at each cell side kfollowing a linear reconstruction of the ¯ow solution within the cell. In the case of merged cells, we proceed instead by ignoring the interface between cells A and B, and then update the merged cell C simply by combining the volume updates of cells A and B:
D AUCD AUAD AUB 31
with the ¯ux vector function H Uknow based on vol-ume averaged cell centre data
AUC AUA AUB: 33
In this case, it should be noted that the ¯uxes on the interface between cells A and B cancel out automatically via cell side vectors of opposite sense and as a conse-quence of the ¯ux calculation being conservative. Ef-fectively, the ¯uxes balance is now being carried out over each of the (generally more than four) sides of the enlarged cell. The conserved variableUfor cell C at time tn1 is now
Although the cell merging process may in principle reduce the integration accuracy on solid boundaries, it has been found not to aect global solution accuracy [15]. Since the update for the merged cell is based on a volume-weighted average, the process is fully conserva-tive. The cell merging procedures described above and associated scheme formulation accommodate cases (not
considered in this paper) where cell boundaries move during calculations.
4. Results and discussion
Numerical experiments for some external and inter-nal bore wave diraction problems are presented in this section to illustrate the favourable properties of the present calculation method for studying two-dimen-sional shallow water ¯ows with strong hydraulic jumps. In the following calculations, two types of boundary conditions are encountered: those corresponding to solid and transmissive boundaries. No ¯ow is permitted through a solid boundary and therefore, at a solid in-terface,qS 0 which means that the corresponding ¯ux vectorHSdepends only on the geopotential. The required value for the geopotential is taken from the adjacent ¯uid cell. At a transmissive boundary, the data necessary to compute the ¯ux vectorH at the interface are taken to be that at the adjacent interior cell centre.
4.1. Two-dimensional dam-break problem
The two-dimensional dam-break problem has been used previously as a benchmark test problem by Fen-nema and Chaudhry [19]. The problem models a partial dam break or rapid opening of a sluice gate. The com-putational basin has dimensions of 200200 m2 with
the reservoir headwater level H110 m and the
tail-water levelH25 m. A schematic representation of the
geometry is shown in Fig. 9. At the instant of failure of the dam wall, water is released through a 75 m wide breach, and a leading bore and abrupt depression are formed in the basin travelling in opposite directions. In the computations, a uniform Cartesian mesh with 4040 cells was used on a computational domain of 200200 m2. Fig. 10(a) shows a 3D view of water
surface elevation and water depth contours at time
t7:2 s. These results compare favourably with those presented in [3,4,19,21].
To demonstrate that the present method can deal with shallow water ¯ows with a dry bed, a dam-break problem with initial heightsH110 m and H20:0 m
has been simulated. The geometry and mesh are the same as the above. In the dry bed case, the wave speed estimates (14) are replaced by (15). Fig. 10(b) shows the water surface elevation and water depth contours at time t7:2 s for 0 tailwater height (dry bed). Fig. 9. Reservoir geometry for the partial dam-break problem.
Comparing the two cases, we can see the leading bore in the dry bed case is lower than that in the wet bed case but travels much faster. At the same time stage, the leading bore in the dry bed case reaches the computa-tional boundary.
4.2. Bore re¯ection at an inclined boundary
Mach re¯ection of a bore wave is known to occur at an inclined bank in shallow water [28]. Practical exam-ples may arise during bore wave propagation in an es-tuary or wave interaction with a structure, e.g. a pier. This phenomenon is equivalent to shock wave re¯ection at a wedge in gas dynamics [22]. The problem can be stated as follows: a planar incident bore wave at Froude number FS moving from right to left encounters a
boundary inclined at a ®xed anglehwinto the direction
of ¯ow and is subsequently re¯ected. Two primary types of bore re¯ection occur depending on the Froude number of the incident boreFSand the bank inclination
angle hw; these are: regular re¯ection (RR) and Mach
re¯ection (MR). For the Mach re¯ection case, several sub-types exist, namely: single Mach re¯ection (SMR), transitional (or, complex) Mach re¯ection (TMR) and double Mach re¯ection (DMR). The initial position of the incoming bore can be arbitrarily placed a certain distance to the right of the corner point on the bound-ary. The conditions between the pre-bore (L) and post-bore (R) states are related by
/Rr/L; number of the bore and the Froude number in the pre-bore state, respectively. Initial conditions for the four cases are given in Table 1. A uniform Cartesian mesh with 270270 cells was used on a computational domain of 9:09:0 m2. Fig. 11 illustrates the grid
layout. For illustrative purposes, the depicted grid resolution is less than that actually used in the compu-tations.
AtFS2 andhw27, a single Mach re¯ection
oc-curs (see Fig. 12(a)). In the SMR case, the incident and re¯ected shocks meet at atriple pointand aMach stemis formed, normal to the inclined boundary. The triple point moves along a straight line trajectory originating at the corner point on the boundary. A slip line extends from the triple point into the region between the Mach stem and the re¯ected shock.
If the bore Froude numberFSis increased to 4 at the
same bank inclination anglehw27, a TMR is formed
(see Fig. 12(b)). The TMR case is similar to the SMR case except that a kink occurs in the re¯ected shock, while the slip line leading down to the inclined boundary rolls up into a vortex.
AtFS 4 andhw50, a DMR is obtained (see Fig.
12(c)). The DMR is similar to the TMR case except that a second Mach stem appears between the kink in the re¯ected shock and the slip line. Meanwhile, a second slip line emanates from the second triple point.
Finally, increasinghw to 60at FS4, the Mach
re-¯ection becomes a regular rere-¯ection (see Fig. 12(d)). In the case of RR, only two bores exist: the incident bore and re¯ected bore. The two bores meet at a re¯ection point on the surface of the bank. Transition from RR to SMR occurs when acoustic disturbances originating at the corner point on the bank catch up with the re¯ection point.
Although experimental data are not available for comparison purposes, the present results compare very favourably with those obtained previously by the au-thors using a similar ¯ow solver on a conventional transformed boundary ®tted grid [32]. Experimental results corresponding to the analogous cases in gas dy-namics have been given by Deschambault and Glass [18]. In the present calculations, all of the primary ¯ow features such as the re¯ected bore, diracted bore, vor-tex and Mach stem are well resolved thus demonstrating the ecacy of the cut cell approach.
Table 1
Initial conditions for bore re¯ection at an inclined boundary
Type FS hw hL uL vL
SMR 2.03 27° 1.0 0.0 0.0
TMR 4.0 27° 1.0 0.0 0.0
DMR 4.0 50° 1.0 0.0 0.0
RR 4.0 60° 1.0 0.0 0.0
4.3. Bore re¯ection at a circular pier
The case of unsteady bore wave re¯ection at a cir-cular pier has been studied numerically by Yang and Hsu [44] who used a high resolution ®nite dierence scheme on a curvilinear boundary conforming grid. Initially, a plane bore wave at Froude number 2.81 is located at a certain distance upstream of the pier and travels towards it. In our calculations, a Cartesian mesh with 260346 cells was used on a computational do-main of 3:95:19 m2, where the diameter of the pier
was 1 m and its centre was located at the point 2;2:5. The pre-bore and post-bore conditions were calculated using the bore wave relations (36). Fig. 13 illustrates the grid layout. Again, for clarity, a coarser grid than that
actually employed in the calculations is shown. For comparison purposes, a sequence of frames depicting the bore/pier diraction process were obtained at ap-proximately the same time stages as the ones in [44]. Fig. 14 shows ®ve sequential depth contour plots. As one can see, regular re¯ection occurs initially due to the large inclination angle at the pier surface. As the bore wave diracts around the pier the inclination angle at the surface decreases until transition occurs from regular re¯ection to Mach re¯ection. As the incident bore wave passes the crown of the pier the Mach stem starts to diract around the surface. Later, further downstream, a Mach bore interaction takes place in the wake region and a Mach disc gradually forms. The computed results compare well with those in [44].
Finally, Fig. 15 shows a comparison of depth con-tours at time t0:3 s (time t0 corresponds to the arrival of the incident bore at the front face of the pier) for solutions obtained on four meshes each having half the mesh interval in each coordinate di-rection of its predecessor. These results have been used to derive a measure of grid convergence through the use of a grid convergence index (GCI), as proposed by
Roache [35]. The idea of a GCI is to provide for the uniform reporting of grid re®nement studies in com-putational ¯uid dynamics [2,31]. The method provides an objective asymptotic approach to the quanti®cation of uncertainty of grid convergence. The basic idea is to approximately relate the results from any grid re®ne-ment test to the expected results from a grid doubling using a second-order method. The GCI is based upon Fig. 14. Bore re¯ection at a circular pier ± depth contours atFb2:81: (a)t0:03 s; (b)t0:05 s; (c)t0:1 s; (d)t0:2 s; (e)t0:25 s.
a grid re®nement error estimator derived from the theory of generalised Richardson extrapolation. The simple formulae are independent of the equations be-ing discretised and the dimensionality of the problem, and can be applied a posteriori to solutions on two grids with no reference to the codes, algorithms or governing equations which produced the solutions as long as the original solutions are second-order accurate.
Given the water depth at the same location on a ®ne grid and a coarse grid denoted byh1andh2, respectively,
the GCI is given by
GCI213
21
j j
rpÿ1; 38
wherepis the formal order of accuracy of the numerical method (2 in the present case),ris the mesh re®nement factor
rDx2
Dx1
and 21 is a measure of the percentage relative error
between the two solutions at the given location, i.e.
21100
h1ÿh2
h1
:
Table 2
Bore re¯ection at a circular pier: water depth (in metres) at selected points on four grids with the GCI at each location
Point h4 h3 h2 h1 GCI43 GCI32 GCI21
0:8;2:5 4.955 5.465 5.442 5.446 10.3 0.4 0.1
3:4;2:5 4.039 3.807 3.775 3.750 5.7 0.8 0.7
3:8;4:5 3.638 3.812 3.794 3.779 4.8 0.5 0.4
3:8;4:3 3.600 3.694 3.648 3.630 2.6 1.2 0.5
3:8;4:0 3.516 3.476 3.445 3.434 1.1 0.9 0.3
3:4;4:7 3.749 3.928 3.910 3.893 4.8 0.5 0.4
2:4;4:7 3.926 4.237 4.299 4.296 7.9 1.5 0.1
2:1;4:5 4.178 4.412 4.424 4.412 5.6 0.3 0.3
1:7;4:3 4.429 4.623 4.651 4.631 4.4 0.6 0.4
1:5;4:1 4.273 4.739 4.755 4.757 10.9 0.3 0.0
2:7;3:0 1.297 1.094 1.143 1.139 15.7 4.5 0.4
Dx m 0.08 0.04 0.02 0.01
If three or more grids are used in a mesh re®nement study the solution is in the asymptotic region of grid convergence if GCI216rpGCI32 [35].
Table 2 shows predicted depths (in metres) and values of the GCI at various points in the domain for the four grids considered (see also Fig. 16). Since the present method is second order (p2) and the number of mesh points is doubled each time (r2) we require CGI2164 CGI32to demonstrate a converged solution at
a particular point. Comparing CGI21 and CGI32in this
way shows that the solution is convergent at the points listed and the GCI provides a measure of the percentage error in the solution at that point. An interested reader may check other data points in the domain in a similar way. A comparison between the CGI43 and CGI32
val-ues, however, shows that the grid convergence was not uniformly achieved on the coarse 6060 grid when compared with the ®ner 120120 grid.
4.4. Bore diraction in a contraction±expansion channel
Finally, the case of an incident bore wave at Froude number 3 travelling through a symmetric contraction± expansion channel has been simulated. This case is rel-evant to bore propagation in an estuary. The channel is formed by two cosine curves linked together at the throat position. The describing equations for the left-hand (or upper) bank are as follows:
y
1:0 ÿ26x6ÿ1;
ÿ0:375 cos px 0:625 ÿ1<x60;
ÿ0:625 cos p
2x 0:875 0<x62;
1:5 2<x64:
8 > > <
> > :
39
The pre- and post-bore conditions were determined by the relations (36). The incident bore wave was posi-tioned initially at the entrance to the curved portion of the channel. A Cartesian mesh with 360200 cells was used on a computational domain of 5:43:0 m2, where
the width of the throat was 0.5 m. Fig. 17 illustrates the grid layout using a coarser grid than that actually em-ployed in the calculations. Fig. 18 shows a sequence of depth contours of bore diraction in the channel and Fig. 19 illustrates depth contours and Froude number contours at t0:45 s. Initially the incident bore is re-¯ected at the walls of the contraction section of the channel where regular re¯ection occurs. As the incident bore reaches the throat section of the channel transition to Mach Re¯ection occurs. Mach stems are formed in the throat region which subsequently diract to form a secondary bore wave in the expansion section. Mean-while, the re¯ected bore waves are re¯ected repeatedly at the channel walls. At a later time the leading bore wave continues to evolve with the secondary bore formed by the interaction of the two Mach discs following behind while the re¯ected bore waves develop into a planar bore wave travelling upstream. Once again, the computed results compare well with the calculations in [44].
5. Mesh adaptation
In the cut cell approach, data input for meshing a speci®c domain only requires one ®le containing the x-and y-coordinates of the boundary nodes and descrip-tors of the boundary type. Required user inputs are simply the initial size of the Cartesian cells in thex- and y-directions, denoted by Dx and Dy, respectively. The grid generator then proceeds to create an initial mesh. This is a completely automatic procedure controlled by the validity of the boundary subsequently cut from the mesh whereby a grid is created such that each boundary cell is cut a maximum of twice, implying that a cut cell may be a three-, four-, or ®ve-sided polygon. When using the initialDxandDyvalues prescribed by the user, it is usually the case that in certain areas of the boundary domain cell intersections exceed 2 per cell, implying that the mesh resolution is too coarse to re-solve the geometrical features in this region. In this case, mesh adaptation is called for and oending grid cells are automatically re®ned until an initial grid is created where each boundary cell has at most two cuts and each cut boundary forms a closed polygon. After the
gener-Fig. 16. Bore re¯ection at a circular pier: depth distribution (m) along the line of symmetry for four levels of mesh re®nement.
Fig. 18. Bore diraction in a contraction±expansion channel ± sequence of depth contours: (a)t0:067 s; (b)t0:11 s; (c)t0:125 s; (d)t0:15 s; (e)t0:19 s; (f)t0:31 s; (g)t0:36 s; (h)t0:4 s.
ation of the initial mesh, user interaction is possible to realise the required mesh for a particular study. This provides re®nement to all, internal only, or particular individual boundaries as required, or re®nement to speci®c interior zones of the mesh. At this point the grid may also be re®ned to data such as a bathymetric depth ®eld.
The use of pointer structures in the adaptive grid generation technique is fundamental to the hierarchical nature of the data structures employed. Pointer struc-tures allow storage allocation and deallocation at run-time resulting in very ecient memory management. A high-level programming language such as Fortran 90 is suitable for these purposes. Mesh generation com-mences by checking each cell in the background mesh for intersection with the ¯ow domain whereupon the cell is added to a linked list of cells forming the initial mesh. By only adding cells which intersect the domain, storage is only required for cells interior to the ¯ow domain. This is particularly ecient in cases where the ¯ow
do-main is a long, thin, meandering shape such as a river estuary. Next, the cut cell data are calculated from the boundary data and the background cells and if any cell is intersected more than twice this cell is subdivided into four cells. It follows that each cell created has a parent and four children which forms the basis of a quadtree data structure and that each node in the linked list of background cells has a quadtree associated with it.
A key feature of this data structure is that cells may be created and removed with relative simplicity. The hierarchical grid structure may be illustrated by refer-ence to the following example. Fig. 20(a) shows an ini-tial background mesh covering an arbitrary, slender, headland type feature together with a representative linked list. It can be seen that cells 6 and 8 are cut four times and so these cells are targeted for subdivision. Quadtree re®nement results in each cell being subdi-vided into four as shown in Fig. 20(b). To avoid ex-cessive cell size variation locally, a nearest neighbour strategy is taken with re®ned cells. Thus, upon
ment, no neighbouring cell is allowed to be more than one level of subdivision dierent. This test is performed by searching through the linked list/quadtree data structure to identify oending cells and is continued until no invalid cells exist anywhere in the mesh. The pointer structure ensures that the searching process is highly ecient and allows the mixed linked list/quadtree data structure to be navigated in a very general way. Fig. 20(b) shows that after re®nement the nearest neighbour constraint is satis®ed. After the cell level in-tegrity checking, the boundary is recut from the result-ing grid. However, the fourth child of node eight is still intersected four times and so, in a similar way, this cell is subdivided into four children, as shown in Fig. 20(c). After cell integrity checking is complete, the resulting mesh is as shown in Fig. 20(d). At this stage, the cut boundary satis®es the two cuts per cell criterion and initial grid generation halts. Of course, the user may then customise the grid as necessary by adapting to particular interior zones and/or local bathymetry. The ®nal result is a high resolution boundary ®tted mesh.
The practical potential of the meshing technique is demonstrated by application of the grid generator to an area of the South±West Netherlands coast. This region is multiply connected and Fig. 21(a) shows the result of meshing the whole region using square background cells
with DxDy 5000 m and performing re®nement to boundaries. Figs. 21(b)±(d) show enhanced detail of the local mesh around the coastline. It can be seen how the grid cell sizes vary from being small in coastal areas and gradually increase as the mesh progresses out into the open sea. Thus, a ®ne grid need only be used where strictly necessary in terms of accuracy or topographical consid-erations thereby reducing computational overheads.
6. Conclusions
A new mesh generation technique for the computa-tion of shallow water ¯ows has been presented. Based on a Cartesian cut cell approach and multi-dimensional high resolution upwind scheme, the present method has the following attractive features:
1. By using cut cells on a background Cartesian mesh, solid regions are simply cut out of a background grid; hence the present method oers a fully boundary-®t-ted gridding capability which can cope with complex shallow water ¯ows around arbitrarily complicated topography. No grid generation is required in the conventional sense.
2. A high-order MUSCL-Hancock ®nite volume scheme used in conjunction with slope limiting and an HLL
proximate Riemann solver provides high resolution solutions to the shallow water equations. This enables the method to deal with complex shallow water ¯ows such as dam-breaks, bore waves and general subcritical or supercritical ¯ows involving strong discontinuities. The method has been validated for a range of test problems involving unsteady shallow water ¯ows. When used in conjunction with local mesh adaptation, the method provides high resolution of local geometric features such as irregular coastline and variable bathy-metry. The results indicate the promise of the new method for solving a wide range of shallow water ¯ows for arbitrarily complicated con®gurations. The cut cell approach can also be applied very eciently to problems with moving boundaries. In such cases there is no re-quirement for local or global re-meshing as is the case with other ®nite volume or ®nite element approaches; all that is necessary is to update the cut cell information as the boundary moves, which only involves a small per-centage of the total number of computational cells in the mesh. A detailed description of extensions of the cut cell method to moving boundary problems is not given in the present paper, but will be reported in future articles.
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