An investigation on the applicability of Cartesian grid approach to calculate flow over arbitrary terrain

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1 Indan Journal of Engneerng & Materals Scences Vol. 11, Deceber 2004, pp An nvestgaton on the applcablty of Cartesan grd approach to calculate flow over arbtrary terran S Vengadesan a*, A Nakayaa b & S Yokoa c a Departent of Appled Mechancs, Indan Insttute of Technology, Chenna , Inda b Departent of Scence of Regonal and Bult Envronents, Graduate School of Scence and Technology Kobe Unversty, Kobe , Japan c Envornental Flud Mechancs Laboratory, Departent of Cvl and Envronental Engneerng Stanford Unversty, Stanford, CA , USA Receved 17 Deceber 2003; accepted 9 Septeber 2004 An nvestgaton s ade on the applcablty of ethod of representng by Cartesan grd to calculate flow over natural terran. Model hll geoetry wth dfferent axu slope angles s chosen and calculatons by both Cartesan and Boundary-ftted coordnate representatons are ade. Coputatons are perfored for lanar flow at dfferent Reynolds nubers and turbulent flow condtons. Results n ters of ean velocty dstrbuton, strealne, vortcty dstrbuton and velocty vector close to the surface are analyzed and dscussed. The study ndcates that to calculate flow over undulatng natural terran, rectangular coordnate ethod of representng the geoetry s able to capture ost of the flow phenoena and t can be a vable alternatve. IPC Code: Int Cl. 7 G01F 1/00, F15D Increasng power of coputers and recent advanceents n nuercal ethods have ade coputatonal flud dynacs (CFD) as a powerful tool to calculate and predct flows of varous knds. Applcaton of CFD technque to calculate coplex three-densonal flows n natural envronent s becong possble. One of the an dffcultes n atospherc applcatons s to represent undulatng natural topography. Conventonal ethod to deal wth curved boundary s to use coordnate syste that fts the boundary curves. Once a good grd s generated, boundary condtons are set at exact locatons and the flow over t can be calculated wth relatve ease. For ore coplex boundares, fnte eleent ethods or fnte dfference ethods on unstructured grds are consdered prosng and any efforts are ade n ths drecton. However, when the boundary becoes rregular or not sooth, t becoes possble to generate a grd. In such a stuaton, usually, one has to resort to other ethods lke a ult-block approach or local refneent, but generatng a grd wth good qualty becoes dffcult. Furtherore, transforaton of the governng equatons results n a coplex syste of equatons wth any geoetrc paraeters plyng hgh coputatonal overhead. So, soe knd of approxaton s necessary and a *For correspondence: (E-al: vengades@t.ac.n) classcal ethod of usng sple rectangular Cartesan coordnates ay be seen as an alternatve choce. In Cartesan ethod, generaton of grd s trval and arbtrary shapes can be represented. Because of ths advantage, any new ethods of usng rectangular grds for coplex shapes wth varous boundary treatents are proposed 1,2. Many researchers have adopted Cartesan coordnate ethod 3-6. However, all these ethods requre geoetry defnton, and they are not avalable for natural terran or any undulatng geoetry. One dsadvantage of usng the rectangular grd for an arbtrary shape s that ether the poston of the boundary becoes approxate or the boundary condtons are appled at nterpolated ponts. In coputng flows over coplex topography or slar coplex boundary, t wll help f relatve perforance and accuraces are known for these dfferent ethods of representng the geoetry n context wth both overall results and behavour of dfferent dfferencng schees used to dscretze the non-lnear convectve ters. In the present work, test case of flow over sooth curved hll s consdered and calculaton usng both boundary-ftted coordnates and the Cartesan coordnates are copared. Frst, basc perforance of each ethod s exaned n low-reynolds nuber

2 466 INDIAN J. ENG. MATER. SCI., DECEMBER 2004 lanar flow, and then accuraces and stablty are exaned at hgh Reynolds nubers. Then the ethods are evaluated when appled to large-eddy sulaton (LES) of hgher Reynolds nuber turbulent flow n the sae geoetrcal regon. The test calculatons for the turbulent case are perfored for the cases n whch detaled experental data are avalable. Nuercal Strategy for Rectangular Grd (RC) Here, the nuercal ethods adopted whle usng the rectangular grd are descrbed. They are for coputng the flow of ncopressble flud of densty ρ and kneatc vscosty v. The governng equatons are the conservaton equatons for ass and oentu U J u t = 0 + { U u}= u 1 n A ν G ρ n where J G 1 n x det = ; A 1 n = J ; ν J = J U 1 p u 1 n n = A u ;. (3) (4) (5) u = 0 u u 1 p u t 2 + u = + ν ρ (1) (2) Here u s the coponent of the velocty vector n the Cartesan coordnate x. We also use the notaton (x,y,z) for (x 1,x 2,x 3 ) and (u,v,w) for (u 1,u 2,u 3 ). These equatons are dscretzed, wth varables arranged n staggered syste to apply convenently the pressurecouplng algorth. Fgure 1 shows the grd arrangeent and the ponts where the boundary condtons are appled when the sae schee s used for an arbtrary geoetry. Dscretzaton of convectve ters are done by a thrd-order upwnd dfferencng-utopia, or by conservatve secondorder central dfference schee 7. Vscous ters are dscretzed by second-order accurate central dfferencng schee. HSMAC teraton procedure s used for calculatng pressure. Te advancng of the oentu equatons s done by a second-order accurate explct, Adas-Bashforth ethod. Perforance of the code wth nuercal strategy for RC has been valdated for lanar and turbulent flow past a square cylnder aganst avalable nuercal and experental results 8. Nuercal Strategy for Boundary-Ftted Grd (BFC) In order to solve the sae ncopressble flows n boundary-ftted grd, Eqs (1) and (2) are transfored n general coordnates and they are gven as: Here, x s the Cartesan coordnate fxed n the physcal space and ξ s the general coordnate used n the coputaton. J s the Jacoban of the transforaton atrx fro x to ξ and U s the contravarant coponent of the velocty vector ultpled by J 1, whch represents the volue flux n the drecton perpendcular to the surface ξ = constant. For lanar flow, the last ter wll dsappear and for turbulent flow, whch s solved assung the eddy-vscosty forulaton for the subgrd scale stresses, the eddy vscosty ν G s added to ν as explaned n the next secton. The staggered grd s an effectve ethod of avodng the pressure oscllaton, but t s not suted for the general coordnates. The control volues n whch conservaton laws are appled are dfferent for dfferent coponents of oentu and the codes becoe excessvely coplex for three-densonal cases wth any eleents of the etrc tensor requred to be stored. Ths results n coplex codng and addtonal coputatonal loads. Recently, Fg.1 Grd arrangeent; boundary condtons are enforced at locatons arked by flled sybols

3 VENGADESAN et al.: CARTESIAN GRID APPROACH TO CALCULATE FLOW OVER ARBITRARY TERRAIN 467 collocated grd arrangeent, n whch the velocty coponents and the pressure are defned at the cell center and the fluxes are defned on the cell-boundng surfaces, has been proposed and found to be effectve 9. Therefore, the ethod based on the collocated varables s used here. Fgure 2 shows the collocated grd arrangeent of varables adopted n the present work. In the collocated grd, the basc varables u and p are defned at the cell center and the dfferencng of the conservatons of equatons of oentu s done the sae way as n the case of regular grd, whle dfferencng of the conservaton of ass s done usng the volue flux U, defned at the center of the cell surface so that pressure oscllaton s controlled the sae way as the staggered grd ethods. The ethod, we take s very close to that of Zang et al. 9 and follow the fractonal-step ethod wth the Crank-Ncolson plct dfferencng for the dagonal eleents of the vscous ters and the second-order Adas-Bashforth schee for the offdagonal vscous ters and the advectve ters. The accuracy of nterpolaton for the volue fluxes on the cell surfaces needed n the collocated grd s proved by usng the ethod of Inagak and Abe 10. The spatal dfferencng of the nonlnear advectve ters s done by ether the second-order conservatve schee or UTOPIA. All other spatal dfferencngs are done by the second-order central dfferencng schee. The above-descrbed ethods ncorporatng nuercal strategy for Boundary-ftted coordnate representaton have been coded and perforance of the code wth nuercal strategy for BFC has been valdated aganst lanar flow n a curved cavty lanar, an external flow above a curved hll and DNS of fully-developed flow n a two-densonal open channel 11. Fg. 2 Varables n collocated arrangeent Sub-grd Stress Model for LES When the above ethods of flow soluton are used n large eddy sulaton (LES) of turbulent flows, all varables n the equatons are the respectve fltered quanttes of the nstantaneous flow. The addtonal stress called subgrd-scale stress R appears n the oentu equatons. If t s odeled by conventonal eddy-vscosty odel, whch s gven as, 2 R = ksδ 2νG S (6) 3 where, k s s the sub-grd turbulent knetc energy, δ s the Kronecker delta, ν G s the sub-grd eddy vscosty and S s the stran tensor, then ν G needs to be added ν n the oentu equatons. ν G s odeled by the Sagornsky odel u ( ) u u ν = Δ + G CS (7) where, Δ s the grd sze defned by the geoetrc average of the grd spacngs n three drectons, ( Δ x ) 1 3 1Δx2Δx, u 3 s now the spatally fltered velocty. C S s Sagornsky constant, whch s chosen as In the case of BFC, coordnate transfored equatons of (6) and (7) are used. Test Flow Past a Model Curved Hll The flow confguraton consdered shown n Fg. 6 s that past a odel solated hll. It s a sooth twodensonal topography, defned by an analytcal z expresson G 1 =,where z H 1 + ( x / nh ) 4 G s the elevaton of the ground at horzontal poston x, and H s the heght of the hll and x s the horzontal dstance fro the center of the hll. Reynolds nuber Re for the present flow confguraton s defned by the oncong velocty U ref and axu hll heght H. The steepness of the hll s deterned by the value of n. For n=2.8, the largest slope angle s 20 degrees and the flow n ths case contans a sall separate bubble at low Reynolds nubers. For n=2.3, the largest slope s 25 degrees and the flow s consderably dfferent wth larger flow separaton. For n=2.0, the largest slope s 15 degrees. These are the test cases used for the present coparatve test runs. Calculatons are perfored at two Reynolds nubers of 100 and 500

4 468 INDIAN J. ENG. MATER. SCI., DECEMBER 2004 and results at Re=100 are copared wth avalable nuercal results 12. Coputatonal Doan and Grd The coputatonal regon covers the test flow shown n Fg. 3. In RC, the geoetry s approxated and the boundary condton s appled at the grd pont nearest to the theoretcal boundary. The coputatonal doan extends fro about 8.5H upstrea and 14H downstrea of the hll of 25 degrees. For the hll wth axu slope angle of 20 degrees, t covers the regon fro -10.5H to 17H. The calculaton doan extends 7H n the cross-streawse and 4H n the span-wse drectons. In the strea-wse drecton, ponts are closely spaced wthn 4H on ether sde fro the hll sut. In the cross-strea-wse drecton, the frst pont fro the ground s placed at 0.03H near the botto of the wall, stretched up to 0.5H and then copressed up to 1.2H and then placed non-unforly untl the top boundary. In the span-wse drecton n ether grd syste, grds are unforly spaced. The total grd sze s and respectvely for 25- degree and 20-degree slope hlls. The coputatonal doan for hll wth axu slope angle of 15 degrees covers the regon of 12H and 19H on the upstrea and downstrea respectvely. In the other drectons, the doan s kept the sae as that used for 25 degrees case. The total grd sze used s In the case of BFC, the grd s generated by transforng the physcal space on the rectangular coputatonal space by an ellptc equaton and Fg. 3 Geoetry of flow over a curved hll generated coordnates ft the actual boundary. The esh s ade orthogonal near the boundary snce n that stuaton, boundary condton for pressure s not requred, f staggered or collocated grd s used. In order to copare the results wth those by RC, calculaton doan, dstance of the frst grd pont fro the ground and grd sze s the kept as that of n RC. Fg. 4 shows the typcal coputatonal grd used for hll wth axu slope angle of 20 degrees. Boundary Condtons Boundary condtons are enforced at nflow, downstrea, top and botto boundares. Perodc boundary condtons are enforced for the span-wse drecton. Non-slp boundary condtons are appled on the ground surface and slp condtons on the top boundary. At the downstrea plane, the radaton boundary condtons are used. At the nflow plane, unfor flow s specfed for low Reynolds nuber test cases. Results and Dscusson Calculatons at low Reynolds nuber flow Table 1 gves the lst of calculaton cases and keys used for lanar flow calculaton. Calculatons are run wth non-densonal te ncreent of Δt=0.001H/U ref and the results are copared at the sae non-densonal te T=40H/U ref. In order to assess the nuercal ethods wth UTOPIA schee, whch s known to ntroduce nuercal vscosty and conservatve central schee, calculatons are perfored for the hll wth axu slope angle of 20 degrees at Re=100, and copared wth nuercal calculaton. Fgure 5a presents a coparson of strealnes, and good agreeent s seen. Contours of transverse vortcty obtaned by RC and present BFC grds are copared n Fg. 5b wth those by Myashta 12. In Ref.12, calculaton s two-densonal Fg. 4 BFC grd used for flow over odel hll geoetry

5 VENGADESAN et al.: CARTESIAN GRID APPROACH TO CALCULATE FLOW OVER ARBITRARY TERRAIN 469 Maxu slope angle Re Table 1 Detals of calculaton cases Grd arrangeent Schee Key used RC conservatve central (CC2) H201RC RC upwnd (UB3) H201RU BFC conservatve central H201BC BFC upwnd H201BU RC conservatve central H251R BFC conservatve central H251B RC conservatve central H255R 500 BFC conservatve central H255B RC upwnd - BFC upwnd RC upwnd - BFC upwnd - Fg. 5b Coparson of calculaton results for hll wth axu slope angle of 20 degrees contours of transverse vortcty Fg. 5a Coparson of calculaton results for hll wth axu slope angle of 20 degrees strealnes and nuercal ethods adopted n that study to perfor URANS calculatons are reported n Nakayaa and Myashta 13. Much denser grd of s used and the nuercal accuracy s consdered to be better. Results agree very well although the recrculaton flow s calculated slghtly weaker n both RC and BFC. Fgure 5c presents a coparson of the profles of strea-wse velocty coponent u/u ref by RC and BFC along several strea-wse statons. Sall dfference between results by upwnd and central schee n RC s notced. But, the agreeent s generally good as expected fro the coparson of the strealnes. These ean that RC and BFC calculatons wth an approprate upwnd schee and conservatve central schee produce good results, coparable wth each other and thus the coputatonal ethods ay be sad to have been verfed. Lanar calculatons are then perfored by both RC and BFC for flow past the hll of axu slope angle of 25 degrees at low Reynolds nuber of 100 n order to verfy agan the basc nuercal procedures. Then calculatons at hgher Reynolds nuber of 500

6 470 INDIAN J. ENG. MATER. SCI., DECEMBER 2004 Fg. 5c Coparson of calculaton results for hll wth axu slope angle of 20 degrees strea-wse velocty along specfed statons [( ) Myashta 12, ( ) UB3, ( ) CC2 Fg. 6b Coparson of calculaton results for hll wth axu slope angle of 25 degrees strealne Fg. 6a Coparson of calculaton results for hll wth axu slope angle of 25 degrees contours of transverse vortcty are perfored. The calculated span-wse vortcty dstrbuton and strealnes by RC and BFC are shown n Fg. 6. At low Re=100, results calculated by both grd systes for all the cases agree well wth each other. For hgher Reynolds nuber of 500, we see dfferences n the calculated results. Sall oscllatons appear n the plot of span-wse vortcty calculated by RC grd, as notced whle perforng calculaton of flow past square cylnder 8. It has also been ponted out Poell and Balaras 5 n the calculaton of flow past a bluff body slar oscllaton appears, when the cell Reynolds nuber becoes large and the central dfferencng s used for convectve ters. Ths phenoenon can be suppressed by usng an upwnd dfferencng whch appears to be necessary for stable calculaton of turbulent flows at uch hgher Reynolds nubers when perfored on Cartesan coordnates. The results of these lanar flow calculatons perfored for flow over curved body wth unfxed separaton wth ether sall separaton or large separaton by two dfferent coordnate representatons of the geoetry can be suarzed as follows. Basc nuercal ethods eployng conservatve central schee and upwnd schee for convectve ters are valdated for both BFC and RC for the case of the odel hll geoetry wth axu slope angle of 20 degrees. At low Reynolds nuber for ths test case, flow experences sall separaton, and results by the both schees on both coordnate representaton yeld closer results. At hgher slope angle and at slghtly hgher Re, whle the calculaton usng the central dfference schee for the convectve ters leads to pressure oscllaton when the geoetry s approxated by rectangular grd, results are stable when the geoetry s represented by a boundaryftted grd or by usng upwnd schee. Calculatons at hgh Reynolds nuber flow Coparson for low and oderate Reynolds nuber flow was dscussed. The sae nvestgaton s

7 VENGADESAN et al.: CARTESIAN GRID APPROACH TO CALCULATE FLOW OVER ARBITRARY TERRAIN 471 now extended to hgh Reynolds nuber flow. For the sae odel hll geoetry, ean velocty and turbulent stresses have been easured for the Reynolds nuber based on the oncong reference velocty U ref and H of (ref. 14). Model hlls of 25 degrees and 15 degrees are consdered. LES ethodology s adopted. Coputaton doan, esh sze, and dstance of the frst grd pont were kept sae those were used n rectangular coordnate calculaton. Ths grd dstrbuton for the test Reynolds nuber of gves vertcal dstance n vscous unts, z + =zu τ/ ν of the frst node about 20 and 15 on the top of hll and at x/h=4 respectvely for the hll odel wth axu slope of 25 degrees and thus vscous layer are not resolved. So, the esh s not of hgh resoluton to resolve the lanar sub-layer, a key proble encountered n LES of hgh Reynolds nuber practcal flows when perfored wth there s a ltaton on coputer resource. Inflow condton for velocty at staton x/h=-4 and x/h=-6 for 25 degrees and 15 degrees respectvely are taken fro that of experent. Calculatons are ade wth thrdorder upwnd-based schee for convectve ters, and non-slp condton as wall boundary condton. Calculated velocty vectors close to the boundary by both grd systes n the regon where the boundary curvature s the largest s presented n Fg. 7 for both hll wth 15 degrees and 25 degrees slope angle. Ths fgure shows that there s no such thng as step corners due to the approxaton n the rectangular grd. In both cases, vectors are seen to be close to tangent to local boundary. Sall devaton of the Fg. 7 Coparson of velocty vector near the sold boundary

8 472 INDIAN J. ENG. MATER. SCI., DECEMBER 2004 natural terran. Calculatons are perfored at low and hgh Reynolds nuber lanar flows to valdate the nuercal ethods. Whle the calculaton usng central dfference schee for the convectve ters leads to pressure oscllaton when the geoetry s approxated by rectangular grd, results are stable when the geoetry s represented by a boundaryftted grd or by usng upwnd schee. At hgher Re for turbulent flow condton, LES calculatons results by wth UTOPIA schee for the convectve ters show slar trend n both grd systes. Hence, when calculaton of flow past natural topography or coplex geoetry or open-channel flows wth deforng free surfaces or flow doan for whch t s dffcult to generate BFC, s to be perfored, rectangular coordnate approxaton of the geoetry wth stablty assured by upwnd schee for the convectve ters s a good alternatve and sulaton by ths ethod captures the flow features. Fg. 8 Coparson of ean strea-wse velocty (U/U ref ) vectors fro the drecton tangent to the boundary s observed n the case of calculaton by rectangular grd. However, ths does not nfluence the results on the whole, wth respect to ths partcular study s concerned, as seen n the results of the ean streawse velocty (U/U ref ), along specfed strea-wse statons shown n Fg. 8. In both test cases of the hll, both geoetry representatons predct separaton, recrculaton and they are qualtatvely close to each other. Conclusons Flow past odel hll geoetry s coputed by both rectangular and boundary-ftted coordnate representaton to nvestgate the applcablty of rectangular coordnate ethod to calculate flow over References 1 Forrer H & Jeltch R, J Cop Phys, 140 (1998) Goldsten D, Handler R & Srovch L, J Cop Phys, 105 (1993) Ye T, Mttal R, Udayakuar H S & Shyy W, J Cop Phys, 156 (1999) Verzcco R, Mohd Yusof J, Orland P & Haworth D, AIAA J, 38 (2000) Poell U & Balaras E, 3rd AFOSR Int Conf on DNS and LES, Dallas, USA, 2001, Maudar S, Iaccarno G & Durbn P, Annual Research Brefs, Centre for Turbulence Research, Stanford Unversty, 2001, Mornsh S, Lund T, Vaslyev O V & Mon P, J Cop Phys, 143 (1998) Nakayaa A & Vengadesan S N, Int J Nuer Methods Fluds, 38 (2002) Zang Y, Street R L & Koseff J R, J Cop Phys, 114 (1994) Inagak M & Abe K, Trans JSME B, 64 (1998) Vengadesan S, Yokoa S & Nakayaa A, J Aero Soc Inda, (2003) (councated). 12 Myashta K, Ph.D Thess, Kobe Unversty, Japan, (2001). 13 Nakayaa A & Myashta K, Int J Nuer Methods Heat Flud Flow, 11 (2001) Nakayaa A & Yokota D, Ann J Hydraul Eng, JSCE, 45 (2001) 43.