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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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