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A telescopic local grid refinement technique for wind flow simulation over complex terrain

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A telescopic local grid refinement technique for wind flow simulation over complex terrain
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  WIND ENERGY Wind Energ.  2001;  4 :77–98 (DOI: 10.1002/we.48) Research  Article  ATelescopicLocalGridRefinement TechniqueforWindFlowSimulationoverComplexTerrain A. Theodorakakos and G. Bergeles ∗ Department of Mechanical Engineering, Laboratory of Aerodynamics, National Technical University of Athens, 9 Heroon Polytehneiou, 15773 Athens,Greece Key words: local gridrefinement; flowover complexterrain;numericalsimulation  A telescopic local grid refinement technique is developed in order to enhance the accuracy level of wind field predictions in a subregion of a complex topography. A 3D simulation of thewindflowfieldovercomplexterrainshasbeencarriedout.ThegoverningNavier–Stokesconservation equations of the flow field are solved numerically in a three-dimensional generalizedcurvilinearnon-orthogonalgrid,usingCartesianvelocitycomponents,followingthefinitevolumeapproximationandapressurecorrectionmethod.Turbulenceissimulated by a two-equation transport model. The reliability of the general flow solver is first tested by simulating the flow past a cube. The second test case simulated is the flow over the Askervein hill, with a detailed comparison of predicted and measured velocities. The third case presented concerns flow field simulation over a complete island. Comparison withmeasurementsrevealsthesignificanceoftheaccuratediscretizationofthetopographyand theuseoftelescopicmeshesontheresults. Copyright  © 2001JohnWiley&Sons,Ltd. Introduction Predictions of the effects on near-ground mean wind velocity and turbulence quantities caused by complexterrain features (i.e. hills, mountains or other surface obstructions) have received considerable attention inrecent years because of increased interest in the siting of wind generators, in tall building construction, inpollutant dispersion, etc.Jackson and Hunt 1 developed a linear two-dimensional theory for flow perturbations induced by hills of low aspect ratio. Mason and Sykes 2 extended this theory to three dimensions. The main feature of this theoryis the distinction between the outer layer where the flow is pressure-driven and the inner layer where theturbulent stresses play a role. Linear equations are used for the velocity perturbations and solved analytically inwavenumber space. Since then, many studies based on these theories have been conducted giving satisfactoryresults. For example, Beljaars  et al . 3 developed a mixed spectral finite difference model where the equationsare linearized around the upstream flow and Fourier-transformed in the two horizontal co-ordinates. Theresulting set of ordinary differential equations in the vertical co-ordinate is solved by means of a finitedifference technique.On the other hand, in order to simulate the wind flow over an extended complex area, solution of thegoverning Navier–Stokes equations using finite volume techniques seems to be preferable. Some typicalexamples of such studies will be presented hereafter. Bergeles 4 has presented predictions of turbulent flowaround two-dimensional hills, solving the full Navier–Stokes equations, using a body-fitted orthogonal Ł Correspondence to: G. Bergeles, Department of Mechanical Engineering, Laboratory of Aerodynamics, NationalTechnical University of Athens, 9 Heroon Polytehneiou, 15773 Athens, Greece.Contract/grant sponsor: DGXII of the European Union; Contract/grant number: JOR3-CT96-0033. CCC 1095–4244/99/020113–11 $17.50  Received 23 January 2001 Copyright  󰂩  2001 John Wiley & Sons, Ltd.  Revised 7 September 2001 Accepted 26 September 2001  78 A. Theodorakakos and G. Bergeles curvilinear co-ordinate system and the two-equation  k  – ε  turbulence model. Comparison of the predictionsrevealed excellent agreement with the JH (Jackson and Hunt) theory 1 for low hills and with measurementsfor high hills. Mouzakis and Bergeles 5 have presented predictions of the two-dimensional turbulent flow overa triangular rib using similar methods to Bergeles. 4 Despite the complexity of the flow, with an extendedflow separation region present behind the rib, the predictions are reasonably accurate compared with themeasurements, although the inevitable non-orthogonality of the grid at the apex of the triangle may beresponsible for the small differences. Apsley and Castro 6 used a finite volume solver for the numericalinvestigation of the flow over two-dimensional and three-dimensional hills for adiabatic and stably stratifiedflows. The results indicated the significance of the non-isotropy of eddy diffusivity in predicting dispersion aswell as the need for  k  – ε  turbulence model modifications to account for streamline curvature and streamwisestrains on dissipation of turbulence. Trifonopoulos  et al . 7 compare predictions of the wind field over complexterrain obtained from a simple model of mass conservation type with the results obtained by solution of thefull 3D Reynolds equations using a zero-equation, mixing length turbulent closure. The comparison showsthat the simple model should be used with caution in regions where flow retardation or separation exists,while the latter method is free from limitations and can be used in any kind of complex terrain. Glekasand Bergeles 8 have presented a numerical method for the prediction of recirculating flows, based on thesolution of the time-averaged Navier–Stokes equations, using generalized non-orthogonal co-ordinates, viaa finite volume technique. Several two-dimensional test cases have been simulated successfully in orderto prove the capability of the method to work without difficulty with highly complicated geometries, thuseliminating many of the problems of using orthogonal grids. Kim  et al . 9 used a similar method based on thefinite volume method and the SIMPLEC algorithm for the prediction of the flow over two-dimensional hillterrain. They concluded that the use of the  k  – ε  turbulence model on a non-orthogonal grid seems preferablein predicting the attached flow field, because of the significant saving in computational time compared withthe low-Reynolds-number model. Kadja  et al . 10 follow an alternative method for the simulation of windflow over complex terrains. They use a Cartesian grid, but instead of following the simpler but somewhatolder method of approximating the ground using a stairstep approximation, the ground is treated as aporous medium. The accuracy of the predictions is further improved by applying a local grid refinementmethod.In all previous works the inadequacy of a single numerical grid to account for local topographiccomplexities or to achieve a desired grid fineness near the ground while at the same time avoidingexcessive CPU memory requirements was evident, and this is the motivation for the present work. Gridembedding or grid local refinement has often been applied for the solution of the compressible 11 , 12 or incompressible 13–15 Navier–Stokes equations. However, the use of similar techniques, using multiplegrid refinement levels, in wind simulation over large-scale complex terrains has received much lessattention.In the work presented in this article, the governing Navier–Stokes conservation equations of the flow fieldare solved numerically in a three-dimensional generalized curvilinear non-orthogonal grid, using Cartesianvelocity components, following the finite volume approximation and a pressure correction method. A two-equation transport model simulates turbulence. The novelty of the present work compared with previous work is the use of local grid refinement in a telescopic manner within a collocated finite volume approach, whileretaining the simplicity of a structured grid approach.The first case simulated is the flow field past a cube, and predictions for the velocity field are comparedwith the measurements of Castro and Robins. 16 The second case presented is the simulation of the flow overthe Askervein hill, for which detailed and accurate measurements are available. 17 The third case presentedconcerns flow field simulation over an island, using the local grid refinement technique in order to enhancethe accuracy level of the predictions inside a small region of the topography, while the total region simulatedextends beyond the coasts of the island for reliable boundary conditions.The comparison of predictions with measurements proves the accuracy of the method and illustrates theadvantages that the use of local grid refinement in the numerical simulation of large-scale terrains can offerwhile increasing the level of accuracy in selected locations of interest. Copyright  󰂩  2001 John Wiley & Sons, Ltd.  Wind Energ.  2001;  4 :77–98  A Telescopic Local Grid Refinement Technique 79 SolutionMethod  The equations in conservation formulation for the mass, momentum and enthalpy coupled with the two-equation  k  – ε  model of turbulence, 18 for the steady state, are written in an arbitrary co-ordinate system andfor Cartesian velocity components  u i . 19 The general forms of these transport equations are presented in theAppendix. These transport equations are integrated and discretized over a common control volume followingthe finite volume method.The grid employed is structured, non-orthogonal curvilinear and is generated using geometrical interpolationmethods. The grid arrangement is collocated, where all unknown variables are stored in the centre of thecomputational cell. In order to avoid pressure–velocity decoupling problems, arising from the fact that pressureand velocities are calculated in the same location, the convective flux through each cell face is calculatedusing the modification first proposed by Rhie and Chow 20 for a Cartesian grid and extended here for the 3Dproblem in generalized curvilinear co-ordinates. The key feature of this approach is that the velocity whichis used to calculate the convective flux through a cell face is not calculated by a linear interpolation of theadjacent cell velocities, but is modified to be directly linked to the two adjacent pressure nodes. Followingthis procedure, a pressure prediction–correction method (similar to the well-known SIMPLE algorithm 21 ) isused in order to derive the pressure correction equation from the continuity equation.The convective and normal diffusion terms are discretized using the hybrid scheme, which behaves as acentral difference scheme at low Peclet numbers and as a first-order upwind scheme at higher Peclet numbers.Although this differencing scheme is known to introduce artificial diffusivity, its stability and robustness stillmake it suitable for simulations of technical interest, with complex geometries, such as simulations of the windflow over an extended region. For comparison reasons, a bounded second-order upwind (BSOU) scheme 22 wasused in one of the test cases presented in this work. The cross-diffusion terms and the second-order derivativesare discretized using a standard central difference scheme. These terms are moved to the right-hand side of the conservation equation and are treated explicitly in the iterative procedure.At the inlet boundaries of the domain, all variables have fixed values. At the ground surface the standard wallfunctions that arise from the use of the  k  – ε  turbulence model are employed. In cases where surface roughnessshould be incorporated, this was introduced through modification of the wall function parameter  E  accordingto Wilcox. 23 The other boundaries of the solution domain are set far from the regions of interest in order tobe able to use symmetry conditions without affecting the results. The use of entrainment conditions at theseboundaries has been avoided where possible, as these can cause convergence problems. The computationaldomain is also extended far enough in height (e.g. for the Askervein simulation the maximum hill height isapproximately 4% of the height of the computational domain) in order to reduce blockage effects. The set of linear equations that results after the discretization of the conservation equations is solved iteratively using aTDMA (tridiagonal matrix algorithm) solver. TelescopicLocalGridRefinementTechnique The physical domain is discretized using a curvilinear non-orthogonal grid. In regions of high gradients orin regions of special interest, finer mesh spacing is required and embedded cells are used. Dividing the cellsof the ‘primary’ mesh in each direction generates these embedded cells. The division factor can be set toany integer value, not necessarily 2, i.e. halving the cells. This factor can be different in each of the threedirections (e.g. a 3D cell can be divided into six cells with no division in the  X -direction, division by a factorof 2 in the  Y -direction and division by a factor of 3 in the  Z -direction). The ‘secondary’ finer mesh generatedis then considered to be totally independent of the ‘primary’ mesh; it can then be rearranged in order tofollow more precisely the geometry of the region it occupies (such as ground level in a wind simulation, e.g.as shown in Figure 1). The only restriction applied at the moment concerns the interfacial surface, where thegrid lines of the two grids should coincide.The solution, concerning the grids that are used for the calculation, proceeds in steps. In the beginning,only the coarse (first) grid is solved. After convergence has been achieved, the first level of grid refinement(second grid) is added into the calculation (thus solving both the ‘outer’ coarse grid and the ‘inner’ finer grid). Copyright  󰂩  2001 John Wiley & Sons, Ltd.  Wind Energ.  2001;  4 :77–98  80 A. Theodorakakos and G. Bergeles Coarse grid.Dividing the coarse grid.Rearrange the fine grid to follow thegeometry more accurately. Figure 1. Steps for fine mesh generation At this stage, when the second grid is included in the simulation, the first coarse grid is modified in orderto occupy only the region around the second finer grid. The starting values of the unknown values for thesecond grid are obtained by interpolation from the already calculated solution of the first grid in this region.When the iterative procedure for the solution of the flow field in the set of two grids converges, the sameprocedure can be repeated in order to embed a second level of grid refinement (third grid) inside the secondgrid. This process can be repeated, leading to as many grid refinement levels as desired.As mentioned above, each grid is totally independent of any other. Different three-dimensional arraysare used to store all the variables (including the grid co-ordinates) in each of these meshes. Connectivityinformation is required for the interface surfaces between two neighbouring meshes. Each mesh is solvedseparately, explicit from each other, using a standard TDMA solver, within each iteration of the solutionprocedure. It would be possible to solve all the grids implicitly and simultaneously by using a solver that doesnot require diagonal coefficient matrices (e.g. a preconditioned biconjugate gradient method), thus achievingfaster convergence of the solution. On the other hand, the added complexity in doing this (ideally all thearrays of the computer code should be transformed to one-dimensional) is quite high, and it is questionablewhether one should use such an approach (grid refinement with implicit simultaneous solution of all grids)or instead adopt a completely unstructured grid approach (where local grid refinement does not require anyspecial treatment from a computational point of view).The data structure is illustrated in Figure 2. Figure 2(a) shows the way that information from the fine gridis transferred to the coarse grid. On the east side of centre cell ‘P’ a pseudo-cell is considered on the otherside of the interface (denoted ‘E’, broken line). The values of this pseudo-cell are calculated from the morerecent values of cells ‘E1’ and ‘E2’ of the fine mesh which this cell overlaps. The analogous treatment for thefine grid is shown in Figure 2(b). For each cell ‘P’ adjacent to the interface a pseudo-cell ‘W’ is consideredon the other side of the interface in order to close the set of linear equations for this region. The values for thispseudo-cell are calculated by interpolation of the most recent values of the surrounding cell of both meshes.This procedure of adding an extra layer of pseudo-cells beyond the interface for each grid is automated andis not part of the mesh generation procedure. The values of the flow field in these pseudo-cells are used asboundary conditions for the solution of the corresponding region. Continuity of the variables is not imposedin any other way. The results show that this method of transferring information from one grid to anotherresults in a continuous flow field and full convergence of the solution. Flux conservation across the interfacesis automatically ensured if the fluxes are calculated in the same way for grid nodes on both sides of theinterface. Copyright  󰂩  2001 John Wiley & Sons, Ltd.  Wind Energ.  2001;  4 :77–98  A Telescopic Local Grid Refinement Technique 81 sEData from coarse to fine gridS1SWNCoarse gridInterface surface between 2 gridsP1WwWN1NPwneFine gridData from fine to coarse gridSsInterface surface between 2 gridsCoarse gridnwPNE2E1EeEeFine grid(a) (b) Figure 2. Representation of the grid refinement technique The technique used aims to be as general as possible while keeping the simplicity of using three-dimensionalarrays for storage of the variables. ResultsandDiscussion FlowFieldpastaCube The first case simulated is the flow field past a cube. The cube has an edge length of   H D 200 mm andis positioned on the floor of the wind tunnel. The flow upstream of the cube has the profile of a turbulentboundary layer over a flat plate. The thickness of the boundary layer equals 10 times the cube’s edge length, U  z D1  D 0 Ð 5 m s  1 and the Reynolds number of the flow based on  U  z D1  and the cube’s edge length equals4000. Castro and Robins 16 have published extensive measurements of this configuration.Five different numerical grids were used (shown in Figure 3). The first one (‘case 1’) does not include alocally refined region and consists of 45 ð 41 ð 37 cells in the  x  -,  y  - and  z -directions respectively (7 ð 7 ð 7cells on the cube). The second case (‘case 2’) also does not include a locally refined region and consists of 90 ð 82 ð 74 cells in the  x  -,  y  - and  z -directions respectively, which is the same as the case 1 grid refined by afactor of 2 in each direction for the whole computational domain (14 ð 14 ð 14 cells on the cube). The thirdcase (‘case 3’) consists of the grid of case 1 with a locally refined region around the cube, which consists of 52 ð 38 ð 38 cells (2 ð 2 ð 2 refinement ratio in each direction). The fourth case (‘case 4’) consists of thegrid of case 1 with the same locally refined region around the cube as in the previous case, but which this timeconsists of 104 ð 76 ð 76 cells (4 ð 4 ð 4 refinement ratio in each direction). Finally, the fifth case (‘case5’) consists of the grid of case 3 with an added level of grid refinement around the cube (44 ð 44 ð 36). Thegrid arrangements for all cases are shown in Figure 3.Figure 4 presents the comparison of predicted and measured axial velocities at two different locations(along the line  x   /   H D 1,  y   /   H D 0 and the line  x   /   H D 0 Ð 75,  z  /   H D 0 Ð 5) and the pressure coefficient  c p  (alongthe centreline of the cube), where the point  x   D 0,  y   D 0,  z D 0 is considered to be at the centre of the baseof the cube. It must be noted that the pressure coefficient is calculated as c p  D  P  P  z D1 12 U 2  z D  H The results for all five cases examined are very close to each other. The differences between predictions andexperiment, especially inside the wake of the cube, can be attributed to the use of the  k  – ε  turbulence model, Copyright  󰂩  2001 John Wiley & Sons, Ltd.  Wind Energ.  2001;  4 :77–98
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