Abstract. 1 Introduction

Transcription

1 Numerical simulation of the Humber Estuary using a non-orthogonal curvilinear coordinate system R.W. Barber, R.V. Pearson, A.P. Roberts, W.I. McKee Water Resources Research Group, Telford Institute of Environmental Systems, University of Salford, Salford M5 4WT, UK Abstract This paper describes a numerical model for simulating depth-averaged flow in an estuary. The governing hydrodynamic equations are solved using finitedifferences on a boundary-fitted non-orthogonal grid which facilitates an accurate representation of the coastline at the expense of more complex hydrodynamic equations. The continuity and momentum equations are solved explicitly by forward-time central-space differencing on a staggered grid. A flooding and drying algorithm has been implemented to allow for the changing shape of the flow domain as the tide ebbs and flows. The versatility of the numerical scheme is demonstrated by simulating tidal flow in the Humber Estuary on the east coast of England. Results from the non-orthogonal scheme indicate that the boundary-fitted approach has considerable promise for simulating the hydrodynamics of awkwardly shaped estuaries. 1 Introduction Most hydrodynamic models of tidal inlets and estuaries rely on finite-difference solutions of the depth-averaged shallow water equations expressed in a Cartesian reference frame [1,2]. In practice, a rectangular mesh with fixed grid spacing is placed over the flow domain, usually resulting in non-alignment of the physical boundaries with the edges of the computational mesh. This lack of alignment may give rise to major inaccuracies in the flow solution such as spurious vorticity generation at the sharp corners of the mesh. In addition, features such as dredged shipping channels often require local grid refinement in order to increase the accuracy of the flow predictions. Boundary-fitted curvilinear coordinate systems provide an approach which

2 30 Computer Modelling of Seas and Coastal Regions combines the computational efficiency of finite-difference procedures with the grid flexibility usually attributed to finite-element methods. The essential characteristic of a boundary-fitted grid is that the coordinate lines coincide with the flow perimeter, no matter how irregular the shape of the region. The idea of numerically-generated curvilinear coordinate systems was originally developed by Thompson et al. [3] to study flow around awkwardly shaped aerofoils. Since Thompson's pioneering work, many authors have applied the same technique to solve the shallow water equations. The earliest numerical model of the shallow water equations utilising a curvilinear boundary-fitted grid was presented by Johnson [4]. More recently, Wijbenga [5], Hauser et al. [6,7] and Borthwick & Barber [8] have proposed alternative numerical procedures for simulating watercourses of complex geometry. The purpose of this paper is to describe a shallow water equation solver for use with arbitrary non-orthogonal meshes. The model has been designed specifically for flow prediction in shallow estuaries of irregular shape. It consists essentially of two modules: the first generates the non-orthogonal grid within the prescribed flow domain, the second solves the transformed nonlinear shallow water equations in a stable and non-diffusive manner. 2 Boundary-fitted grid generation Following Thompson et al. [3], the grid generating system is taken to be a pair of elliptic Poisson equations: (i) relating the physical (x,y) coordinates to the transformed (,TJ) coordinates. (Here, the subscripts denote the usual shorthand notation for partial differentiation). The functions P and Q are the so-called 'attraction operators' or 'control functions' which may be used to alter the internal structure of the curvilinear mesh. Since all numerical calculations are performed in the transformed (,77) plane, it is first necessary to interchange the dependent and independent variables in equation (1) to give the quasi-linear elliptic system: (2) where and J is the Jacobian of the transformation, given by J = x^ - x^. The mapping expressions given in equation (2) are rewritten as finite-differences

3 Computer Modelling of Seas and Coastal Regions 31 and solved using a successive-over-relaxation procedure to find a one-to-one mapping between the rectilinear transformed (,77) plane and the curvilinear physical (x,y) plane. The transformed equations are not entirely linear and therefore it is necessary for the initial (x,y) values to be close to their final positions if the solution is to converge. A reasonable initial guess is obtained by linear interpolation between the boundaries along both coordinate lines passing through each grid node. The grid generation system has the flexibility to enable islands to be created inside the flow domain and boundary-fitted grids may also be adapted to concentrate cells in regions of deep or shallow water, by altering the form of the attraction operators, P and Q, as demonstrated by Nielsen & Skovgaard [9]. 3 Governing hydrodynamic equations In the Cartesian (x,y) coordinates of the physical domain, the depth-averaged continuity equation can be written as dt dx dy and implementing the Boussinesq eddy viscosity concept to model diffusion leads to the non-conservative x-momentum equation: du,, du -rrdu /.,, dc ^wr ^hr - uu CjV uu /A\ + jj + y f y g _ + _ "* + Y (4) 8f 3;c 5y 8x pd pd * dx* dxdy dy* and the non-conservative y-momentum equation: dt dx dy dy pd pd * dx* dxdy dy* where U and V are depth-averaged velocity components in the x- and y- directions, f is the surface elevation above an arbitrary horizontal datum, D is the local water depth, g is the acceleration due to gravity, ij. is the Coriolis parameter, p is the fluid density, 7* and r^ are the wind and bed stresses respectively and ^ is the horizontal eddy viscosity coefficient. Since the numerical computations are performed on a non-orthogonal mesh, it is necessary to convert the previously presented Cartesian formulae into transformed equations written in terms of the boundary-fitted coordinates, and rj. The transformation of the Cartesian shallow water equations is accomplished using the non-conservative derivative relationships presented by Thompson et al. [3]:

4 32 Computer Modelling of Seas and Coastal Regions a%. dy -(-\ J -- 3 dr\ (6) where f denotes an arbitrary differentiable function of x and y. Substituting for all derivatives of x and y in the governing hydrodynamic equations yields the transformed shallow water expressions: dt J d(vd) _^ 3(VD) -% *{ 3_ 11 3g =o (7) dt as an + -J2L - -* + DIFFUSIVE TERMS pd pd (8) dt J dv,dy_. an,9v +Z^2-1*L +DIFFUSIVE TERMS. pd pd The transformed diffusive terms are too long to be presented here but are discussed in more detail by Pearson & Barber [10]. The wind and bed shear stresses are expressed using quadratic drag laws: (9) (10) and (11) where W% and Wy are the wind speeds in the x- and y-directions, p* is the density of air, and C* and Cf are the drag coefficients at the surface and bed respectively. In the present study, the bed friction coefficient is evaluated using Manning's equation: c - (12) 01/3 where n is the Manning roughness factor.

5 Computer Modelling of Seas and Coastal Regions 33 4 Computational procedure The transformed governing equations (7,8 & 9) are solved using an explicit, forward-time central-space finite-difference scheme. A space-staggered (,17) grid is employed for the storage of the hydrodynamic variables; the U- and V-velocity components are located at the middle of cell faces whilst the surface elevation, f, is defined at cell centres. The present layout of variables reduces the amount of spatial averaging and more importantly overcomes the troublesome 'decoupling effects' which are reported in earlier curvilinear shallow water equation solvers [4,7]. To allow for the changing shape of the estuary as the tide ebbs and flows, a 'flooding and drying' scheme has been adopted, whereby grid cells are added to, or removed from, the computational domain depending upon the local water depth. Since the scheme is explicit, the timestep is subject to the Courant- Friedrichs-Lewy stability constraint. At the land/water interface, U=V=0 noslip conditions are used and the model is driven by prescribing time varying surface elevation tidal curves at one or more 'open' boundaries. 5 Model application The model was validated by examining the tidal propagation in a 50 km long section of the Humber Estuary on the east coast of England. The Humber Estuary provides an outlet to the North Sea for the rivers Trent and Ouse, and shipping access to a number of ports including Hull, Immingham and Grimsby. Present work has concentrated on the wider lower part of the estuary where the flow patterns are of a complex two-dimensional nature and tidal effects dominate. Figure 1 shows the extent of the modelled area and the location of the intermediate field observation sites used in model verification, whilst Figure 2 shows the boundary-fitted coordinate system employed in the numerical simulation. After grid generation, the depth of the bed below a fixed 'Chart Datum' was calculated at all mesh nodes using inverse power interpolation between scattered bathymetry points. To drive the model, observed surface elevation curves were prescribed at the landward and seaward open boundaries. The model was run for three tidal cycles from a stationary initial condition with the resulting tidal elevations and velocities being recorded on the third cycle. Model calibration was accomplished by varying the bed friction coefficient. It was found that the value of Manning's n had a significant effect on the magnitude of the predicted velocities, but the computed surface elevations and velocity directions were relatively insensitive to changes in bed friction. Therefore, the model was calibrated by considering the error between the observed and predicted velocity magnitudes. A quantitative approach was adopted by minimising either the root mean square (R.M.S.) error or the goodness of fit (G.O.F.):

6 34 Computer Modelling of Seas and Coastal Regions R.M.S. = /y _y \2 ' ^ iobs ipred' N (13) G.O.F. = \ 'y _ y \2 iobs ipred' N Ev 2 'iobs (14) where the subscripts iobs and ipred refer to the observed and predicted data respectively, and N is the total number of observations. In the present study, six separate field observation sites were employed in the calibration exercise, with the closest agreement being obtained for a Manning's n of Kingston-upon-Hull North High water mark Low water mark Field site Landward boundary D '^.O ImminghaniN^ F ' -.. Grimsby Spurn Head Length scale Figure 1: Map of the Humber Estuary

7 Computer Modelling of Seas and Coastal Regions 35 Figure 2: Boundary-fitted grid representative of the Humber Estuary Time after high water at seaward boundary (hours) Figure 3: Surface elevation comparisons at Grimsby Figure 3 illustrates the spring tide surface elevation profile at Grimsby, 22 km from the seaward boundary. Although there are small discrepancies between the predicted and observed surface elevation around the time of low water, it can be seen that the numerical model successfully replicates the lengthening ebb and shortening flood tides generated by bed friction. The velocity comparisons for field observation site 'E' located approximately 5 km north of Grimsby are presented in Figures 4 and 5. It is evident that the model gives excellent agreement with the observed tidal direction but the correlation is less satisfactory for the velocity magnitude. Similar trends are exhibited at the other observation sites. Finally, Figure 6 shows the predicted velocity vectors in the eastern part of the estuary during the later stages of the ebb tide. The velocity vectors were calculated using a mesh of twice the resolution of that presented in Figure 2 and indicate that the model is generating stable and realistic flow patterns.

8 36 Computer Modelling of Seas and Coastal Regions Z 120 Q Time after high water at seaward boundary (hours) Figure 4: Velocity direction comparisons at field observation site 'E' Time after high water at seaward boundary (hours) Figure 5: Velocity magnitude comparisons at field observation site 'E' Figure 6: Predicted velocity vectors at the mouth of the estuary

9 Computer Modelling of Seas and Coastal Regions 37 6 Conclusions A technique for solving the depth-averaged shallow water equations in irregularly shaped estuaries has been presented. The procedure employs a nonorthogonal coordinate system and uses a forward-time central space finitedifference scheme to discretize the transformed curvilinear flow equations. The results for the Humber Estuary are very encouraging and indicate that the non-orthogonal boundary-fitted methodology has considerable promise for simulating the hydrodynamics of awkwardly shaped estuaries and coastal seas. References 1. Leendertse, J.J. & Gritton, E.G. A water-quality simulation model for well-mixed estuaries and coastal seas: Vol 2, Computation Procedures, The Rand Corporation, R-708-NYC, Falconer, R. A. Numerical modelling of tidal circulation in harbours, J. Waterway, Port, Coastal and Ocean Div., Proc. ASCE, 106, 31-48, Thompson, J.F., Thames, F.C. & Mastin, C.W. Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies, /. Comp. Phys., 15, , Johnson, B.H. VAHM - A vertically averaged hydrodynamic model using boundary-fitted coordinates, U.S. Army Engineer Waterways Experiment Station Hydraulics Laboratory, Vicksburg, USA, Wijbenga, J.H.A. Determination of flow patterns in rivers with curvilinear coordinates, Proc. 21st I.A.H.R. Congress, Melbourne, Australia, [reprinted as Publication No. 352, Delft Hydraulics Laboratory]. 6. Hauser, J., Paap, H.G., Eppel, D. & Mueller, A. Solution of shallow water equations for complex flow domains via boundary-fitted coordinates, Int. J. Num. Methods in Fluids, 5, , Hauser, J., Paap, H.G., Eppel, D. & Sengupta, S. Boundary conformed coordinate systems for selected two-dimensional fluid flow problems, Int. J. Num. Methods in Fluids, 6, , Borthwick, A.G.L. & Barber, R.W. River and reservoir flow modelling using the transformed shallow water equations, Int. J. Num. Methods in Fluids, 14, , Nielsen, P. & Skovgaard, O. A scheme for automatic generation of boundary-fitted depth- and depth-gradient dependent grids in arbitrary twodimensional regions, Int. J. Num. Methods in Fluids, 10, , Pearson, R.V. & Barber, R.W. Mathematical simulation of the Humber Estuary using a depth-averaged boundary-fitted tidal model, Proc. 8th Int. Conf. on Num. Methods in Laminar and Turbulent Flow, Ed. C. Taylor, Part 2, , 1993.