Large Eddy Simulation Of Flow Past A Two-dimensional Hill

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1 Large Eddy Simlation Of Flow Past A Two-dimensional Hill Sankara N.Vengadesan ) and Akihiko Nakayama ) ) Research Associate, vengades@kobe-.ac.jp, ) Professor, nakayama@kobe-.ac.jp Gradate School of Science and Technology, Kobe niversity, Kobe 67-8, Japan Trblent flow past an idealized two-dimensional hill with two different slopes is simlated by LES techniqe. Simlations are performed by both conventional and dynamic Smagorinsky models. Standard Smagorinsky case is simlated with three different types of near-wall treatments. Reslts of simlations by log-law modified to take into accont for local and instantaneos pressre gradient effects are closer to the experimental data in terms of mean velocity and wake size.. INTRODCTION Wind flow arond hill and over complex terrain is of great interest in engineering applications like transport and dispersion of polltants in the atmosphere, agro-meteorological stdy, constrction of wind mills and airport etc., Practically, an infinite nmber of sitations are possible de to varieties of hill geometry, arrangements, and approaching flow conditions. Varios experimental measrements have been reported for different configration of the hills at different Reynolds nmbers [-]. Nmerical predictions have been performed [-8], invariably all the calclations so far are of RANS type and no Large Eddy Simlation (LES) stdy has yet been reported yet. In LES method, large-scale motion is resolved by discrete comptational grid and directly compted by nmerical method and small-scales of motions are modeled. While this techniqe has been proved to be very sccessfl in simlating simple flows over smooth bondaries and is considered to be a promising tool in engineering, its application to flows in natral environment is not qite straightforward, as bondaries are generally rogh and their geometry is very complex. In the present work, we perform Large Eddy Simlation of flow past an isolated two-dimensional hill with particlar emphasis on investigation appropriate near wall bondary condition to be applied to simlation of high Re flows over natral terrain. The hill is of bell shape defined by an analytical expression. Experimental measrements are available at moderately high Reynolds nmber. Two different LES models standard and dynamic Smagorinsky and three types of near wall bondary conditions are considered for simlation. Reslts are analyzed in terms of mean velocity and trblence qantities, inflence of bondary conditions are elcidated.. DESCRIPTION OF THE HILL GEOMETRY AND FLOW CONDITION The flow configration considered is that past an isolated hill (Fig.), a smooth two-dimensional topography, defined by an analytical expression z G =, where z G is the elevation of the H + ( x / nh ) grond at horizontal position x, and H is the height of the hill. We consider this test case with two maximm vales of the slope angle viz. degree and degree, measred from the horizontal direction, determined by vales of n. Table gives details of index n, hill height H and the respective slope angle. This flow has been sbjected to a detailed experimental stdy and experimental reslts are available in Nakayama and Yokota [9]. Mean velocity and trblent stresses have been measred for the Reynolds nmber based on the oncoming reference velocity ref and H of. This is relatively a gentle topography and no flow separation is reported. z ref z G x H Fig. Flow configration Table. Hill Geometry parameters H (cm). NMERICAL METHODS The basic eqations sed in the present LES are threedimensional, time dependent, Navier-Stokes eqations, filtered in order to separate the large scale and the small-scale motions. We consider isothermal incompressible flow and solve the filtered governing eqations along with closre sbgrid-stress model. As the focs of the present work is a stdy on the inflence of bondary conditions, governing eqations are not described here and they can be fond in Nakayama and Noda [] and standard text books [,,]. LES model chosen is the conventional Smagorinsky model in which the trblent stress, R ij is modeled as Rij = k sδ ij ν G S () ij where, k s is the sbgrid trblent kinetic energy, δ ij is the Kronecker delta, ν G is the sbgrid eddy viscosity and S ij is the strain tensor. The eddy viscosity ν G is modeled by i ( ) i j ν = + G CS () x j x j xi n Maximm slope angle (degrees) where, is the grid size defined by the geometric average of the grid spacings in three directions, ( x ) x x, i is the spatially filtered velocity component in the streamwise -x i direction, in the spanwise x direction and in the cross-stream x direction, i.e (x,x,x )=(x,y,z); (,, )=(,v,w). C s is the model constant for which we se the vale of. in the case of Standard Smagorinsky model. We also se the dynamic procedre to determine the vale of C s de to Germano et al. [] with Lilly s [] modification. () Calclation domain and grid The comptational region covers the test flow shown in Fig., from abot 8.H in the pstream and H in downstream in the streamwise direction, 7H in the cross stream-wise and H in the spanwise direction. A rectanglar grid is sed, which is niformly spaced in the spanwise direction. In the streamwise direction, points are closely spaced (9 points) within H on either side from the hill smmit, stretched with a factor of.8. In the cross stream-wise direction, the first point from the grond is placed at.h near the bottom of the wall, stretched with a factor of. pto.h and then compressed with a factor of.9 pto.h and then placed nonniformly with stretching factor of. ntil the end. This grid

2 distribtion gives z + =z τ/ ν of the first node abot on the top of hill and abot at = and ths viscos layer are not resolved. The total grid size is 8x6x. The crved bondary is represented by cartesian co-ordinate system with staggered mesh arrangement. The bondary conditions are applied at the mesh points closest to the real bondary, bt not exactly on it. In order to find the inflence of the approximate position of the bondary, example of calclated velocity vectors along with grid and test case geometry between two streamwise stations are shown in Fig.. This figre shows that there is no sch thing as step corners de to the approximation. Small deviation of the vectors from the direction tangent to the local bondary srface is seen, bt this does not inflence the reslts on the whole. () Nmerical schemes We solve the governing eqations by a finite difference procedre. Non-linear convective terms in the eqations are discretised by a third order pwind differencing, (TOPIA) to avoid stability problems and viscos terms are discretised by second-order accrate central differencing scheme. Inflow conditions for the streamwise velocities are adopted from experimental data. Radiation otflow condition is applied at the downstream bondary. The periodic bondary conditions are sed for the spanwise direction. In the cross flow direction, the nonslip bondary conditions are applied on the grond srface and slip conditions are applied on the top bondary. HSMAC iteration scheme is sed for calclating pressre. Time advancing of the momentm eqations is done by a secondorder accrate explicit, Adams-Bashforth method. Performance of the code had been assessed earlier for flow past a blff body and for the crved geometry by Nakayama and Noda []. All the calclations are performed with the non-dimensional time step, dt ref /H of.. All calclations have been allowed to settle down ntil nondimensional time nits, and then statistical averages over the next non-dimensional time nits are obtained that are presented below. Theoretical bondary Fig.. Enlarged plot of velocity vector near the solid bondary () Wall bondary conditions In wall-bonded flows, the only correct bondary condition at the srface is the no-slip condition, bt this reqires calclations p to the wall with sfficient grid resoltion. However, as the Reynolds nmber increases, bondary layer thickness decreases, reslting in reqirement of large nmber of grid points. In RANS type simlation, wall-fnction approach is sed as one method of way ot to meet this condition. Bt in LES the problem is severe as pointed ot by Spalart et.al. [6] and no definite soltion has been proposed yet. We perform calclation for the present test case, with non-slip bondary condition as a baseline soltion to compare and this case is referred to as. Hino and Okmra [7] have performed flow over a wavy wall by assming single-layer linear distribtion for the velocity. This approximation is good only when viscos sblayer can be resolved. When the laminar sblayer cannot be resolved by the comptational grid, artificial bondary condition may be applied at some distance from the wall. In LES, this techniqe is recently referred to as Off the wall bondary condition, (Cabot [8] ). As one method, Werner- Wengle [9] proposed instantaneos two-layer linear-power law velocity distribtion. This has been sed qite extensively in many LES calclations, reported by Rodi et.al. []. However, they cannot be sed in separated flows and non-eqilibrim flows. This two-layer model is modified into three-layer linear-loglaw version in the format given by Von Karman [], to specify the bondary conditions for the velocities in the tangential directions, at the first point from the wall. Nakayama et.al. [] have tested the validity of this bondary condition, for LES of flow over an isolated hill at Re=. The maximm slope angle of this is degrees from the horizontal direction and according to the experiment, the flow separates on the lee side and then reattaches. They fond that the prediction with log-law bondary condition was closer to the experimental data compared with calclations withot it. However, they indicate that the se of the log-law was too dissipative to show significant trblent flctations. We test once again the three-layer linear-log law in the present model and this is referred to as. In this method, an approximation to a wall law given by the following eqation is sed. + = z +, > z + > () + = ln z + -., > z + () + =. ln z + +., z + () where, z + =z τ/ ν and + =/ τ are the non-dimensionalised vertical distance and velocity respectively. The friction velocity, τ is calclated from these eqations with the velocities at the second point from the wall. Natral terrain is sbjected to wavy topography and the flow past it is sbjected local acceleration and deceleration de to pressre gradients and resistance de to roghness of bondary. In, we se the log-law modified to inclde local and instantaneos pressre gradient effects in the format given by Wilcox [], to specify velocities at the first point from the wall. The wall fnction for the velocity modified to inclde pressre gradient effects is given as: + = z +, > z + > (6) + =. ln z + +. C pc z + P +, z + (7) where, P + is the dimensionless pressre-gradient parameter defined by + ν dp P =, where s is the distance along the bondary. The above τ ds eqation is proposed for mean velocity profile. In order to apply it to the instantaneos velocity, the coefficient C pc for the pressre gradient term is set to.. Table gives the details of different comptational rns and keys sed to refer to them.. RESLTS AND DISCSSION Comptational cases to is rn for hill with maximm slope angle degrees and all the for cases are rn for hill with maximm slope angle for degrees.

3 Table. Comptational rns Smagorinsky Model Bondary condition Key Standard Non-slip Standard Conventional log-law Standard Log-law with pressre gradient effects Dynamic Non-slip Case D () Mean velocity Profiles of time averaged streamwise velocity component, at specified streamwise stations compted sing different bondary conditions, by two Smagorinsky models and experimental reslts are plotted in Fig. (a) for hill with degrees and Fig. (b) for hill with degrees. The inflow velocity profile for the calclations at station =- and =-6 for degrees and degrees respectively are taken from that of experiment. At =, experimental reslts show that the flow accelerates jst near the top of the hill. At the same / ref Experiment (a) Hill with slope angle degrees Fig.. Velocity profiles along selected cross-stream sections station, calclation with the nonslip bondary condition () shows the development of the bondary layer and the maximm velocity is drastically nder-predicted. Calclations sing the loglaw bondary condition () show a thinner bondary layer and reslts by the modified log-law () are seen closer to the experiment. As it can be observed, shows separation at = and = for degrees and degrees hill respectively and predicts a large re-circlation zone at frther downstream. There the reslts sing the modified log-law (), in which the effects of one of the locally changing parameters are inclded, show trends that are in better agreement with the experiment. This implies that to improve simlation reslts for a complex topography, one needs to inclde inflence of locally and temporally changing parameters () Trblence qantities Calclated shear stresses are shown in Fig. at two streamwise stations = and =6 along with the experimental data. On top of the hill, all three cases grossly nder-predict the distribtion and at farther downstream, there is a improvement in the predictions. At =, prediction with the non-slip bondary condition case is closer to the experimental vales. This can be attribted to the fact that early massive separation cased is responsible for trblent prodction. Prediction sing the conventional log-law is the worst among and this may be interpreted that in this case very small trblence is prodced. At =6, the reslts of shows a large negative shear stress which is also de to the large separation. Here the modified bondary condition appears to give reslts closer to the experiment. Reynolds normal stresses are shown in Fig. at =6 station. Case D reslts are not inclded as they are fond to be very mch ot of range. Prediction by non-slip bondary condition shows the same behavior as noticed earlier. Conventional log-law bondary condition case reslts are in between and reslts of proposed modification case nder predict the stresses. Reslts on the whole indicate that in order to improve prediction, some kind of mechanism to prodce and retain trblence is to be incorporated in the model or in the simlation procedre.. CONCLSIONS Large Eddy Simlation of flow past an idealized twodimensional hill with maximm slope angle of degrees and degrees at moderate Re nmber has been performed. Both Standard and dynamic type Smagorinsky model along with three different types of near-wall bondary conditions are considered. At first, two existing bondary conditions non-slip and conventional log-law assmption are stdied and their limiting behavior is elcidated. A modification in the conventional log-law to inclde local and instantaneos pressre gradient effects that reflects local acceleration and deceleration de to changing topography is proposed. This method improved the simlation reslts in terms of wake size and mean velocity. Bt, predictions of trblence qantities are not satisfactory. It is fond from the present stdy that for LES of the flow near grond, where the topography is never smooth and qite ndlating, by considering other parameters sch as the rate of temporal change and three-dimensionality, frther refinements may be possible.

4 / ref Experiment References Case D Fig.. Contd., (b) Hill with slope angle degrees ) Baskaran, V., Smits, A.J., and Jobert, P.N.: A trblent flow over a crved hill, Part. Growth on an internal bondary layer, J.Flid Mech., Vol. 8, pp.7-8, 987. ) Hnt, J.C.R., Leibovich, S., and Richards, K.J.: Trblent shear flow over low hills, Royal Metrological Society, Vol., pp. -7, 988. ) Almeida, G.P., Drao, D.F.G., and Heitor, M.V.: Wake flows behind two-dimensional model hills, Experimental Thermal and Flid Science, Vol.7, pp.87-, 99. ) Ferreira, A.D., Lopes, A.M.G., Viegas, D.X., and Sosa, A.C.M.: Experimental and nmerical simlation of flow arond twodimensional hills, J. Wind Eng and Ind. Aerodyn., Vol./, pp.7-8, 99. ) Kim, H.G., Lee, C.M., Lim, H.C., and Kyong, N.H.: An experimental and nmerical stdy on the flow over two-dimensional hills, J.Wind Eng. and Ind. Aerodyn., Vol.66, pp.7-, ) Kobayashi, M.H., Pereira, J.C.F., and Siqeira, M.B.B.: Nmerical stdy of the trblent flow over and in a model forest on a D hill, J.Wind Eng. and Ind. Aerodyn., Vol., pp.7-7, 99. 7) Ying, R., and Canto, V.M.: Nmerical simlation of flow over two-dimensional hills sing a second-order trblence closre model, Bondary Layer Metrology, Vol. 8, pp.7-7, ) Peng, S., and Davidson, L.: New two-eqation eddy viscosity transport model for trblent flow comptation, AIAA J., Vol.8, n7, pp.96-,. 9) Nakayama, A., and Yokota, D.: The inflence of pressregradients and crvatre on the rogh-wall bondary layers, Proc. Meeting of JSFM, pp.89-8, (in Japanese). ) Nakayama, A., and Noda, H.: LES simlation of flow arond a blff body fitted with splitter plate, J. Wind Eng. and Ind. Aerody., Vol.8, pp.8-96,. ) Wilcox, D.C.: Trblence modeling for CFD, nd ed., DCW indstries, 998. ) Voke, P.R., Sandham, N.D., and Kleiser, L.: Direct and Large- Eddy Simlation III, Proc. of Issac Newton Institte Symposim/ERCOFTAC Workshop, Klwer Academic Pblishers, 999. ) Peyret, R., and Krase, E.: Advanced Trblent flow Comptations, CISM corses and lectres, No.9, Springer Wein, NewYork,. ) Germano, M., Piomelli,., Moin, P., and Cabot, W.H.: A dymaic sbgrid scale eddy viscosity model, Phys. Flids A, Vol., pp.76-76, 99. ) Lilly, D.K.: A proposed modification of the Germano sbgridscale closre method, Phys. Flids A, Vol., pp.6-6, 99. 6) Spalart, P.R., Jo, W.H., Strelets, M., and Allmaras, S.R.: Comments on the feasibility of LES for wings, and on a Hybrid RANS/LES approach, Advances in DNS/LES, Proc. st AFOSR Inter. Conf. on DNS and LES, Greyden Press, ) Hino, M., and Okmra, T.: Coherent strctre of trblent flow over wavy walls, Proc. 9th symp. of Trblent Shear Flows, pp.-- --, 99. 8) Cabot, W.: Wall models in Large Eddy Simlation, NASA Ames/Stanford Center for Trblence Research, Annal Research Briefs, pp.67-68, ) Werner, H., and Wengle, H: Large-Eddy simlation of trblent flow over and arond a cbe in a plate channel, Proc. 8th Trblent Shear Flows, Mnich, pp , 99. ) Rodi, W., Ferziger, H.H, Breer, M., and Porqie, M.: Stats of Large Eddy Simlation-Reslts of a Workshop, J.Flids Engg., Vol,9, pp.8-6, 997. ) Von Karman, T.: The analogy between flid friction and heat transfer, Trans. ASME, Vol.6, pp.7-7, 99. ) Nakayama, A., Miyashita, K., and Maeda, K.: Comparison of nmerical analysis of flow past topography by time-dependent RANS and LES simlation, Annal J.Hydralics Engg., JSCE, Vol., pp.-8, (in Japanese). ) Wilcox, D.C.: Wall matching, A rational alternative to wall fnctions, AIAA paper 89-6, 989.

5 (a) at = (b) at =6 Fig.. Shear stress distribtion for hill with slope angle degrees ' ref ' ref Fig.. Reynolds stress distribtion at =6 for hill with slope angle degrees

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