QUASI-3D SOLVER OF MEANDERING RIVER FLOWS BY CIP-SOROBAN SCHEME IN CYLINDRICAL COORDINATES WITH SUPPORT OF BOUNDARY FITTED COORDINATE METHOD Keisuke Yoshida, Tadaharu Ishikawa Dr. Eng., Tokyo Institute of Technology, Japan, e-mail:yoshida.k.af@m.titech.ac.jp Dr. Eng., Tokyo Institute of Technology, Japan, e-mail: ishikawa.t.ai@m.titech.ac.jp Abstract A new numerical solver is developed for the quasi-3d flow model to simulate meandering river flows. This solver can investigate the flows in an arbitrary curved river channel, by means of the adaptive CIP-Soroban (CIP-S) scheme in a cylindrical coordinate system with support of a boundary fitted coordinate (BFC) system. Time development of water velocity in an advection phase is computed by the orthogonal curvilinear coordinate system without any transformation of the governing equations, whereas a non-advection phase such as the continuity equation is solved by the BFC method, so that the mass conservation is fully satisfied and the water level is accurately calculated. From the verification of this solver on the meandering virtual river flows compared with the original BFC method, it is shown that the proposed solver accurately predicts the main flow profile in the regularly curved section of the flows, and that the horizontal velocity vectors both at the free-surface and at the bottom are reasonably predicted with the effect of the secondary currents in the curved section of the virtual river with some tributary streams. Key words adaptive CIP-Soroban scheme; cylindrical coordinate system; boundary fitted coordinate (BFC) system; meandering river flows; quasi-3d flow model 1 INTRODUCTION The flood disasters due to the abnormal climate system in recent years have frequently forced us to face the significant threat for the life of human beings and our infrastructures. This fact inevitably results in our hydraulic planning of the river management for preventing the possible natural catastrophes caused by the floods. However, it is usually in a relatively short time that we have to find the better solutions and carry out the related engineering works. Thus, the trade-off analysis with the reliable numerical simulations will be often efficient to assess the effects of such works. This is because we need much cost and consume a lot of time if we conduct the experimental measurements with huge laboratorial apparatus or carry out the field observations with a lot of collaborators, although these measures are far helpful to offer us the important information about the practical and specific solutions in the pending issues. Recently, some promising numerical codes have been established for the practical applications. Many primitive codes were developed in the early works of river engineering. These works had the academic background that the river flow is mainly governed by both the gravity force and the bottom friction, and the flow can be simply regarded as semi-steady flow. These kinds of numerical works were practically meaningful for the simple prediction of water discharge, water level and bed roughness of the river. However, many hydraulic researchers pointed out the existence of
Proceedings of 16th IAHR-APD Congress and 3rd Symposium of IAHR-ISHS the 2-D or 3-D vortex structures in the flows of the natural rivers, by their extensive measurements and observations. These structures can not be analyzed by the 1-D models, and are dominated by the flow conditions and the river geometries. For example, the natural rivers usually have the curved sections and some branches as well as the compound cross-sections. Consequently, the existing hydraulic models can treat the 2-D or 3-D nature of the flow structures. They are well established with some significant numerical techniques in a cost-conscious sense. For instance, the hydraulic researchers have developed depth-averaged type of flow models, which can be used in the shallow water basins. It is well know that the 3-D codes are disadvantageous in a practical sense, when we carry out the large-scaled simulations of the actual river flows. We often the boundary-fitted coordinate (BFC) method which is employed in the practical numerical codes. This is because, for example, if we conduct the simulations in the physical domain with curved boundaries, we can save more computer memory and remove more complexity of the grid alignment in the computational domain by using the BFC grid than by using staircase-like step grid in the orthogonal system. In the BFC method, the target equations which are to be discretized in the codes are derived from the governing equations of fluid motions by means of the mathematical transformation between the orthogonal and the non-orthogonal coordinates. It is natural that the researchers take advantage of this method in their codes. However, they ignore the negative effect of the transformation; the resulting equations contain more complicated terms. In addition, the careless numerical transformation may generate the discretization error, even if we make use of any high-order difference schemes. In the previous studies on this kind of error, Yabe et al. (1999) attempted to investigate intensively the extent of the error. They found that the numerical transformation yields such an error, if the numerical grid is not smoothly arranged; the neighboring grid spacing is not gradually changed in the physical domain. They also pointed out that if we carelessly engage the BFC method in the flow simulations, the damage to the numerical accuracy will be serious, especially in the convection terms. 2 IDEAS AND OBJECTIVES 2.1 NEW IDEA OF FLOW SOLVER The hydraulic engineers usually pay attention to the appropriate grid spacing in their computation, so that they can clearly capture the regions of strong variable variation. Recently, the BFC method has been widely employed in the practical simulations because we can easily adjust the physical mesh size, according to our issues. However, as is described before, the numerical error caused by the method often cancels its advantage. Therefore, Yoshida and Ishikawa (2007) developed a new type of numerical scheme for the shallow water model, which is referred to as cylindrical CIP-Soroban scheme. In this scheme, some ideas are offered as follows; (1) semi-structured grid allocation named as Soroban-grid was used in order to realize the new category of boundary-fitted grid system, (2) the advection term was calculated by the high-accuracy CIP scheme in the cylindrical coordinates with the rules fitted for the new grid system and (3) the numerical transformation was not conducted. In the study by Yoshida and Ishikawa (2007), the geometry of the river bank-line was so simple that the previous method was not just practical. Thus, in this paper we propose a new method in which we can consider the actual river flows bounded with the irregular bank-lines. This cylindrical CIP-Soroban scheme effectively incorporates the BFC method when we couple the continuity equation with the momentum equations.. 952
October 20-23 2008, Hohai University, Nanjing, China 2.2 OBJECTIVE OF THIS STUDY The objective of this study is to propose the new numerical method for the river flows with the arbitrary irregular bank-lines by the new method to verify the accuracy of this method in the virtual meandering shallow flows, and to investigate the performance of the model in the virtual river flows with some tributaries. Fig. 1 Two kinds of orthogonal curvilinear coordinate system (, r θ,), z (, s n,) z and horizontal river geometry 3 NUMERICAL PROCEDURES 3.1 GOVERNING EQUATIONS 3.1.1 BASIC EQUTIONS The following assumptions are made in the present model: 1) incompressible Newtonian fluid; 2) constant viscosity; 3) hydrostatic pressure profile along the water depth; 4) negligible wind shear over the water surface. With these assumptions, the 2-D governing equations of the open-channel flows are obtained by integrating the Reynolds-averaged continuity and Navier-Stokes equations from the channel bottom to the water surface with the Galerkin method, as follows; h 1 1 + [ u] + {( 1+ σ n)[ v] } = 0 (1) t 1+ σn s 1+ σn 1 [ pu] + [ puu] + [ puv] t 1+ σ n s n 2σ + [ puv] 1 + σ n g H Dp u = [ p] + u + p κ 1+ σ n s Dt z z [ prey s] + (2) 1 [ pv] + [ puv] + [ pvv] t 1+ σ n s n σ puu ( vv) 1+ σ n g H Dp v = [ p] + v + p κ 1+ σ n n Dt z z [ prey n] + (3) where t = time; s, n and z = orthogonal curvilinear coordinates defined in Fig.1, in which r, θ and z = cylindrical coordinates; h=local flow depth; u and v=s and n components of horizontal velocity; ρ = fluid density; g = gravity acceleration; σ = local curvature of the channel; H = water level; Reys and Reyn = s and n components of the Reynolds stress; κ = eddy viscosity; p = weighted or shape function used in the Galerkin method; D/ Dt= substantial derivative and [α ] = mathematical symbol which denotes the integration of any physical valueα from the river bed to the free-surface. 3.1.2 QUASI-3D FLOW MODEL It is not by the conventional shallow water flow model, but by the Galerkin method that we can take account of the so-called dispersion stresses in the 2-D modeling. This stress plays great role of the momentum transfer in the curved sections of open-channel flows, and is induced by the gap between the depth-averaged velocity and the actual velocity. In the Galerkin method, the weighted function p is the same as the shape function f. In this study, two kinds of vertical mode are considered for the easy implementation; the depth-averaged component and deviation components. Thus, the vertical profile of the streamwise velocity u is simply expressed as follows; u = u0 + fu1, where the subscript 0 means the depth-averaged term and the 1 means the deviation one. In this model, any empirical formulas are not used for the prediction of this kind 953
Proceedings of 16th IAHR-APD Congress and 3rd Symposium of IAHR-ISHS of secondary currents in the curved channel. 3.2 NUMERICAL SOLUTION In this section, we shall show the outline of the main characteristics of the new method used in our numerical simulation. The key point is the effective combination of the previous numerical schemes; we applied the schemes collaterally to improve the accuracy. In the original CIP-Soroban scheme proposed by Yabe et al. (1999), the governing equations are split into two phases; one is the advection phase and the other is the non-advection phase. As for the advection phase, this study follows almost of all the numerical rules adopted in the original scheme. For example, the adaptive M-type CIP scheme is applied to compute the terms. This type of CIP scheme consists of the 1-D original CIP scheme along the Soroban line and the linear interpolation scheme across the lines. The complete explanations are found in their paper. On the other hand, the non-advection terms are discretized by the existing BFC method in order to satisfy the mass balance and to offer an easy implementation. In the original CIP-Soroban method, the physical variables are located on the non-staggered grid. In this grid system, an unphysical oscillation sometimes breaks down the simulation due to both the variable location and the numerical schemes. In order to couple the continuity equation (1) with the momentum equations (2) and (3) without any unphysical oscillations and numerical errors, our study adopts the momentum interpolation method on collocation grids (Rhie and Chow s (1983)). 4 RESULT AND DISCUSSIONS 4.1 NUMERICAL VERIFICATION To test the M-type CIP scheme and verify the present method, a simple numerical simulation were carried out for a virtual meandering open-channel flows. In this test, the secondary currents were not taken into account in the numerical model, and the numerical results show the steady-state of flow fields because of the constant boundary conditions. The present method was compared with the conventional BFC method. In the BFC method, the advection term is discretized by the 5th-order upwind difference scheme. Fig.2 shows a schematic representation of the virtual meandering open-channel which was equipped with both straight sections and regularly sine-generated curves ones. Fig.3 shows the comparison of the numerical results of the cross-sectional profile of mass flux M = Uh at the test section A-A ; h = water depth, U = x -component of the depthaveraged velocity. The notation ( N, N ) denotes the 2-D grid numbers in the numerical domain. The Manning s roughness coefficient is set to be 0.01, and the time increment is 0.1s. We can expect that in general the numerical results with more grids show good performance than that with fewer grids because we solve the problem by the finite difference method. The numerical results in the figure show us that the present method is superior to the 5 th -order upwind BFC method if we use the same numerical grids. x y 954 Fig. 2 Virtual meandering channel with regularly cross-sectional boundary in flat bed open circle: distinct Soroban grid; A-A : test section; B :channel width; l, l :longitudinal length; a :amplitude
October 20-23 2008, Hohai University, Nanjing, China Fig. 3 Cross-sectional distribution of mass flux M (at the test cross-section A-A ) 4.2 APPLICATION TO RIVER FLOWS WITH SOME BRANCHES In actual rivers, the flows are much influenced by the geometry of the channel and the boundary conditions. For example, the branch is one of the typical geographical patterns that we will observe in natural rivers, and the flows around the branch point are interactively affected by the tributary streams. In this section, the application of the present model is demonstrated in the virtual river with tributaries. Fig.4 shows a sketch of the virtual river with the rectangular flat cross-sections, which is connected with two tributaries. The boundary conditions such as a water discharge at an upstream end and a surface level at a downstream one are constant. The roughness coefficient is set as 0.03, and time increment is set as 0.5s. In this calculation, the secondary currents are taken into account. Fig. 5 show the numerical result of the cross-sectional profile of the horizontal velocity vectors at each node in the test cross-sections (a) B-B and (b) C-C. The three kinds of velocity vectors are plotted in the figure, and they are calculated by the quasi-3d model. It can be found that the present method reasonably predict the horizontal velocity vectors due to the secondary currents in the curved sections and near the branches. Fig. 5 Cross-sectional distribution of velocity vectors at each node in the test cross-sections (a) B-B and (b) C-C ( n = cross-sectional axis from the right bank to the left one; u = surface flow; u = bottom flow and u = depth-averaged flow) 5 CONCLUSIONS Fig. 4 Schematic representation of numerical grids of virtual river with two tributaries ( Q = constant water discharge of the main river at upstream end; H = constant surface level of the main river at downstream end and Q, Q = constant water discharge of the tributaries at upstream end) In the present paper, a quasi-3d flow model was established to simulate the meandering river flows. The numerical method employed in the model is the cylindrical CIP-Soroban scheme with support of the boundary-fitted coordinate (BFC) method. The model was tested by comparing the present data with the data by the existing pure BFC. The numerical results showed that the flow fields 955
Proceedings of 16th IAHR-APD Congress and 3rd Symposium of IAHR-ISHS were reasonably predicted by the proposed method. The future works will be required to compare our results with the observation data. REFERENCES Rhie, C. M and Chow, W. L. (1983) Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J., Vol.21, pp.1525-1532. Yabe, T., Mizoe, H. Takizawa, K., Moriki, H., IM, H. and Ogata, Y (2004) Higher-order schemes with CIP method and adaptive Soroban grid towards mesh-free scheme, J. Comput. Phys., Vol.194, pp57-77. Yoshida, K. and Ishikawa, T. (2007) Numerical prediction of meandering river flows by CIP-Soroban grids in cylindrical coordinate system, C1.c, Proc. 32 nd IAHR Congress, Italy. 956