EFFICIENT SOLUTIONS OF THE EULER AND NAVIER-STOKES EQUATIONS FOR EXTERNAL FLOWS

Transcription

1 Scientific Bulletin f the Plitehnica University f Timisara Transactins n Mechanics Special issue The 6 th Internatinal Cnference n Hydraulic Machinery and Hydrdynamics Timisara, Rmania, Octber -, 4 EFFICIENT SOLUTIONS OF THE EULER AND NAVIER-STOKES EQUATIONS FOR EXTERNAL FLOWS Dan MATEESCU Prfessr, Aerspace Prgram Directr Department f Mechanical Engineering McGill University, Mntreal, Canada 87 Sherbrke Street West, Mntreal, Quebec, H3A K6, Canada dan.mateescu@mcgill.ca ABSTRACT This invited paper presents efficient slutins f the Euler and Navier-Stkes equatins based n finite difference and finite vlume frmulatins. They were btained with several methds recently develped fr the efficient analysis f steady and unsteady eternal flws. (i) A biased-flu methd based n an eplicit finite vlume frmulatin is first presented fr the slutin f the Euler equatins. A secnd-rder scheme is used fr the flu calculatin with an upwind r dwnwind bias fr the flw variables, accrding t the subsnic r supersnic character f the lcal flw. The methd is first validated fr cmpressible flws with epansins and shck waves in nzzles, and then used t btain slutins fr subsnic, transnic and supersnic flws past airfils. (ii) Efficient Navier- Stkes slutins are als presented fr the flws past airfils at lw Reynlds numbers, dminated by viscus effects, which are f interest fr micraircraft and unmanned air vehicle applicatins. KEYWORDS Aerdynamics, steady and unsteady flws, inviscid and viscus flws, cmputatinal fluid dynamics.. INTRODUCTION The numerical methds f slutins used in the analysis f steady and unsteady flws f engineering interest have t be characterized by a very gd cmputatinal efficiency in additin t a very gd accuracy. The requirement f a gd cmputatinal efficiency and a user-friendly implementatin is imprtant especially in the study f steady and unsteady fluid-structure interactin prblems, which requires the simultaneus slutin f the Euler r Navier-Stkes equatins in cnunctin with the equatins f the defrmatin mtin f the structures subected t fluid flws. This paper presents efficient methds based n finite difference and finite vlume frmulatins recently develped by this authr and cllabratrs fr the analysis f steady and unsteady flws with fied and scillating bundaries in subsnic, transnic and supersnic flw regimes [5,,,, 4-7]. Other efficient methds using Lagrangian and spectral frmulatins [8, 3, 8] are presented in ther papers [6, 7]. An efficient biased-flu methd, based n an eplicit finite vlume frmulatin, is first presented fr the slutin f the Euler equatins in subsnic, transnic and supersnic flws. This methd takes int accunt the physical directin f perturbatin prpagatin in cmpressible flws, aviding thus the numerical prblem f the dd-and-even-pints decupling which appears in ther methds with negative effects n the cmputatinal efficiency. A special research interest has recently been devted t the analysis f the flws at lw and very lw Reynlds numbers past airfils and wings. This interest is driven by micr-aerial-vehicles (MAV) and unmanned air vehicles (UAV) applicatins, which were made pssible by the recent advances in micr-electr-mechanical systems (MEMS). These applicatins have shwn that many questins are unanswered regarding the airfil aerdynamics at lw Reynlds numbers. The flws past airfils at lw Reynlds numbers are dminated by viscus effects, transitinal and flw separatin phenmena, which cmplicate the understanding f airfil aerdynamics in these cnditins. Accurate slutins fr the lw Reynlds number flws past airfils are als presented in this paper by efficiently slving the Navier-Stkes equatins using a finite difference methd using artificial cmpressibility.

2 . BIASED FLUX METHOD FOR SOLVING THE EULER EQUATIONS IN SUBSONIC, TRANSONIC AND SUPERSONIC FLOWS The eplicit r implicit finite vlume methds are directly applied n structured r unstructured grids withut the need f crdinate transfrmatin. Numerus methds have been develped based n varius numerical discretizatins f the flu derivatives, such as thse develped by Jamesn et al. [8, 37] and MacCrmack et al. []. These basic methds, as well as many ther similar methds nt mentined here, were prven t be efficient and reliable in slving varius engineering prblems. Hwever, in this class f methds, the numerical prcedure des nt reflect the physical prpagatin f perturbatins, with eventual negative implicatins n the numerical stability and cnvergence, accuracy and ability t avid numerical distrtins in capturing sharp shcks. T vercme this deficiency, several numerical methds have been develped t better reflect the physical prpagatin f perturbatins. Hwever, these methds becme very invlved fr multidimensinal flws (as in the methds develped by Re [35], r based n flu splitting []), with negative implicatins n the cmputatinal efficiency; r the basic mdel develped fr quasi D flws has n mre the same physical significance fr multidimensinal flws (as in the case f Gdunv s methd [6]). A mre accurate flu calculatin using discretized frms f the Euler equatins, and taking int accunt the permissible directins f perturbatin prpagatin in cmpressible flws, has been develped by Mateescu & Lausn [3].This methd led t very accurate and cmputatinally efficient slutins; hwever, the numerical scheme became mre invlved fr multi-dimensinal flws, especially in the presence f the shck waves. The eplicit biased flu methd, develped in [4], uses a relatively simple numerical scheme t efficiently take int accunt the physical prpagatin f perturbatins, by using an upwind and dwnwind bias in the evaluatin f flues in functin f the lcal flw Mach number. This methd can be easily applied t multi-dimensinal flws... Prblem frmulatin An eplicit finite vlume frmulatin is used in this methd, in which the Euler equatins fr rtatinal cmpressible flws are epressed in the frm f + [ f ( V n) + g ] = d V V da, () t V where V and V represent the finite cntrl vlume and its bunding surface, and where f = ρ, ρv, ρ E, () { } T {, n, ( V n) } T g = p p, (3) in which ρ, p and V = iu + v + kw are the fluid density, pressure and velcity, n = in + n + kn is the utward unit vectr nrmal t the bunding surface V. Fr D flws, the cmputatinal dmain is discretized int a certain number f quadrilateral cells, N, each f these cells being characterized by an area (crrespnding t the vlume in 3D) and by its bunding cntur A i subscripts i N A i frmed by fur straight lines dented by the ( i, ) ± and ( i, ± ), characterized by their Cartesian prectins i, ±, i, ± i±,, y i±, y z and y (abslute values). In this case, the Euler equatins in matri frm can be discretized as fi = Qi ( f ), (4) t where vectr f i is the cell-averaged value f the state f and ( f ) Q { ρ, ρu, ρv, ρ E} T i is the flu peratr defined as f =, (5) i () f [ H H + H H ] = i+, i, + Ai Q (6) in which = V + ( p ρ)( γ ) E is the specific energy per unit f mass, and H = F G, (7) k, l k, l y k, l k, l k, l ρ k, l uk, l ρ k, l uk, l + pk, l F = k, l, (8a) ρ k, l uk, l vk, l ( ) ρ k, l Ek, l + pk, l uk, l ρ k, l vk, l ρ k, l vk, l uk, l G = k, l. (8b) ρ k, lvk, l + pk, l ( ) ρ k, l Ek, l + pk, l vk, l The flu vectrs F and G at the cell interfaces have t be epressed in terms f the cell-averaged values f the state vectr f, in rder t iteratively slve by time marching the eplicit Euler equatins (4). The manner in which the interface values are related by the cell-averaged values distinguishes the numerus finite vlume methds develped by

3 varius authrs. Thus, in the methd develped by Jamesn and his cllabratrs [8, 37], the flues at an interface are calculated by using an algebraic mean f the cell-averaged values f the tw neighbring cells, such as f i ±, = ( fi ±, + f ), r ( F F ) F + i±, = i±,. This very simple apprach led t rbust algrithms, which have been successfully used t slve varius aerdynamic prblems. Hwever, this flu calculatin leads t the dd-and-even-pints decupling (mst bvius in the case f rectangular cells, where the flw variable changes fr the ddnumbered cells depend nly n the flw variables in the even-numbered cells, and vice-versa). T avid this prblem, an artificially-added dissipatin, invlving secnd and furth rder terms, has been ingeniusly develped by Jamesn [8, 37]. This artificially-added dissipatin implies, hwever, a heavy cmputatinal effrt, especially t calculate the furth-rder dissipatin terms, undermining thus the efficiency f the simple interface-flues algrithm. Als, due t this added dissipatin, the captured shcks are less sharp. In additin, the flu variables calculated at an interface in this manner always depend n bth sets f flw variables in the upwind and dwnwind neighbring cells, and this des nt crrespnd t the actual directins f perturbatin prpagatin in cmpressible flws. Fr eample, the perturbatins prpagate in quasi D flw with the characteristic velcities u a, u and u + a (where a is the speed f sund), which are all psitive fr supersnic flws ( u > a ); in this case, the flu values depend nly n the upstream flw variables, and bviusly nt n the dwnstream nes, as btained in the flu calculatin based n algebraic averaging... Methd f slutin In the biased-flu develped in [4], the flu values are calculated using simple weighted epressins invlving the upwind and dwnwind cells, such as ρ ρ ρ = ρ +, (9a) ( ) i u ( ) ui v ( ) ui p ( ) pi i +, ρ +, u u i+, u + +, =, (9b) v v i+, u + +, =, (9c) p p i+, p + +, =, (9d) ρ where the values f the weighting parameters, u v p, and are taken between and in functin f the subsnic r supersnic character f the lcal flw. A subsnic quasi D nzzle flw is physically and mathematically well defined by specifying the stagnatin pressure and temperature (r speed f sund) at the inlet and the static pressure at the utlet. In cntrast, a supersnic quasi D nzzle flw is well defined by specifying all flw variables at the inlet (and n cnditin at the utlet). By analgy with the nzzle flw, at any interface between tw cells, the static pressure has t be characterized by a dwnwind bias in a lcally subsnic flw, which can be mdeled by cnsidering lwer values fr the weighting parameter = (such as. t.4), and by an p upwind bias in a lcally supersnic flw, mdeled by cnsidering higher values fr =, (such as p.6 t.9). At the same time, the ther fluid dynamic variables, ρ, u and v, have an upwind bias at the interface in bth subsnic and supersnic lcal flws, mdeled by higher values f the weighting parameters ρ u = = =. The v ptimum values fr the upwind- and dwnwindbiased parameters, and, were fund by numerical eperimentatins t be in the range.75.8 and, respectively, The numerical results presented in this paper have been btained with =.75 and =. 5. As a result f this upwind-dwnwind biased flu calculatin, this methd des nt present the ddand-even-pints decupling, and hence there is n need fr a furth-rder artificially-added dissipatin fr subsnic and supersnic flws; this enhances cnsiderably the cmputatinal efficiency f the biased-flu methd. Fr the flws invlving shck waves, a secndrder dissipatin is added nly in the vicinity f the shck, which is detected using a sensr based ν i n a nrmalized secnd-rder difference f the pressure, as that used by Jamesn et al. [8]; fr eample, in the case f quasi D flws, ν = p p + p p + p p. () i ( i+ i i ) i ( f ) () f i+ i i + Thus the flu peratr Q secnd-rder dissipatin term Q i A D i is augmented by a in the frm () f = [ H i+, H i, + H + H i D i ()] f, () where D d f + d f + d f d,() ( ) f i f = i+, i, + + in which, fr eample, () f ( f ) d i, = i±, i± +, f ± ε, (3)

4 () where ε ( ν ) i±, = k ma i, ν i±,, with k having a typical value f /4. This secnd-rder dissipatin des nt essentially affect the cmputatinal efficiency, since its applicatin is restricted nly t the flws with shck waves, and in this case nly t very narrw regins (several cells wide) in the vicinity f the shcks detected by the sensr ν. Equatin (6) is then slved by eplicit iteratins with a furth-rder Runge-Kutta scheme, similar t that used in [8], which can be epressed fr the n n+ n pseud-time levels t and t = t + t in the frm n ( ) Qi ( ) () ( t ) Q ( f ) f =, (4a) f () n i fi t f n = f, (4b) () i i i (3) n () fi = fi t Q i ( f ( n ) n n () f fi ( 6) [ Qi ( f ) Q i ( + i = t + f ) i ), (4c) () (3) ( f ) Q ( )] + Q f, (4d) i +.3. Methd validatin The methd is first validated fr quasi D flw in a circular-arc-bump nzzle. The present slutin, which did nt require any artificial dissipatin, was fund in very gd agreement with the eact slutin btained fr this quasi D flw as shwn in Figure ; als shwn is the slutin btained with Jamesn s methd [8] using artificially-added dissipatin, which required 33% mre cmputing time per iteratin than the present biased-flu methd (CFL number was. in bth methds). i.4. Slutins fr D subsnic, transnic and supersnic cnfined flws The slutins btained with the biased-flu methd fr the D subsnic, transnic and supersnic flws in the same circular-arc-bump nzzle are shwn in Figures, 3 and 4. Figure. Subsnic nzzle flw. Mach number distributins n the lwer and upper walls and is-mach lines. Cmparisn with [33]. Figure. Mach number variatin in a circular-arcbump nzzle (quasi D flw). Figure 3. Transnic nzzle flw. Mach number distributins n the lwer and upper walls and is-mach lines. Cmparisn with [5].

5 -Cp.5. Present Slutin Pulliam's Cmpt. Results[4] /chrd Figure 6. Pressure cefficient distributin n the NACA airfil at M =.8 and = 6.5. Cmparisn with eperiments [39]. Figure 4. Supersnic nzzle flw. Mach number distributins n the lwer and upper walls and is-mach lines. Cmparisn with [5]..5. Subsnic, transnic and supersnic airfil flws Slutins fr the subsnic, transnic and supersnic flws have been als btained with the biased flu methds. The present slutins cmputed fr the pressure distributins n the NACA airfil are cmpared in Figures 5 and 6 with the eperimental results btained in [39] fr subsnic flw (at Mach number =.53 and M angle f attack =.5 ) and fr transnic flw (at Mach number =.8 and angle f attack M = 6.5 ). The present slutins were fund in gd agreement with the eperimental results..5 -Cp..5. Present slutin Eperiment[3] Figure 7. Mach number distributin n the NACA airfil and its symmetry-ais etensin at =. and =, and the isbar lines. M /chrd Figure 5. Pressure cefficient distributin n the NACA airfil at M =.53 and =.5. Cmparisn with eperiments [39].

6 .6. Unsteady flw slutins fr scillating airfils The biased-flu methd presented in Sectin. has been etended t slve the unsteady flws past airfils by using a three-pints-backward scheme fr the Euler equatins n mving grids. In the case f the mving grids, the Euler equatins () becme ~ f + [ f ([ V V] n) + g ] = d V V da, (5) t V where V ~ represents the velcity f the mving grid bunding surface. After implementing the finite vlume frmulatin, the resulting equatin is discretized in real time by using a three-pintsbackward scheme. After the real-time discretizatin, the slutin is btained at each new real-time step by using a pseud-time iterative prcedure similar t the biased-flu methd presented in Sectin. (details f the mathematical derivatins are presented in [3] ). As a sample f results, Figure 8 illustrates a typical slutin btained fr the unsteady lift cefficient f an airfil eecuting pitching scillatins in the presence f the grund, which is situated at varius distances frm the airfil, such as.5, and times the airfil chrd.. h/chrd >.8 h/chrd = h/chrd = angle f attack( ) Figure 8. Unsteady lift cefficient f an airfil NACA eecuting pitching scillatins, () t = sin ωt, at. M =.6 in the presence f the grund, situated at three different distances. 3. AIRFOIL AERODYNAMICS AT LOW REYNOLDS NUMBERS A special interest has recently been devted t the aerdynamics f airfils at lw and very lw Reynlds numbers. This interest is driven by a h variety f applicatins ranging frm dmestic windmills t special military aircraft and unmanned air vehicles (UAV), which were made pssible by the recent advances in the micr-electrmechanical-systems (MEMS). Very small aircrafts called micr-aerial-vehicles (MAV) can perate in varius envirnments including tunnels, desert and ungle (fr eamples see references included in [5]). These applicatins have shwn that many questins are unanswered regarding the airfil aerdynamics at lw and very lw Reynlds numbers. The flws past airfils at lw Reynlds numbers are dminated by viscus effects, transitinal and flw separatin phenmena, which cmplicate the understanding f airfil aerdynamics in these cnditins. Recently, research n MAV has been initiated at Stanfrd University by Kunz and Kr [9], wh used in their study the INSD cde frm NASA Ames, develped by Rgers & Kwak [36], after eperimenting FLO3 develped by Jamesn [8] and MSES by Drela & Giles [3,4] with limited success. Mre recently, Lentink and Gerritsma [] fund ut that the incmpressible cde ISNAS develped by Segal et al. [38] did nt cnverge well fr this challenging flw regime. The present analysis f airfils in incmpressible flws at lw Reynlds numbers is based n an etensin f the methd presented in Sectin. f [6] fr the slutin f the Navier-Stkes equatins, which reduces the flw prblem t several scalar tridiagnal systems f equatins, enhancing thus the cmputatinal efficiency f the methd. 3.. Methd f slutin Cnsider an airfil f chrd c placed at an incidence in a unifrm stream f velcity U and defined, as shwn in Figure 9, by the equatins f the upper and lwer surfaces y = e ( ) = h( ) + e(), (6a) y = e ( ) = h( ) e(), (6b) where the subscripts and refer t the upper and lwer surfaces, and where h ( ) and e( ) define, respectively, the camberline and airfil thickness variatin alng the airfil chrd. The flw past the airfil at lw Reynlds number is slved in a rectangular cmputatinal dmain btained frm the physical flw dmain, illustrated in Figure 7, by the crdinate transfrmatin X = cs + y sin, Y = f (, y), (7) where

7 sin + y cs fr < and H < y < H [ y e ( ) ] cs H fr < < and y > e ( ) H [ sin + e ( ) cs] [ y + e ( ) ] cs H fr < < and y < e ( ) f (, y) = H + [ sin e ( ) cs], (8) H 3 + ( sin + y cs) H fr > and y > H 3 H + H 3 H 3 + ( sin + y cs) H fr > and y < H 3 H + H 3 in which H 3 = sin and H, H are defined in Figure 9. The upstream inflw and dwnstream utflw bundaries f the cmputatinal dmain are defined by X = L and X = L + cs. The flw past the airfil at lw Reynlds number is Upper far-field bundary slved further using the methd presented in Sectin. f ur paper [6], nting that in this case the length f reference is c instead f H, the reference velcity is U instead f U, and the crdinate transfrmatin (8) replaces the general transfrmatin (4) frm [6]. cy Out-flw bundary U c y U v c H V In-flw bundary c h( ) c e( ) U u c e ( ) c X U c e( ) c e ( ) c H 3 c H c c c L c L Lwer far-field bundary Figure 9. Gemetry f a cambered airfil in a unifrm flw at incidence.

8 3.. Methd validatin fr symmetric airfils at zer angle f attack The pressure cefficient distributins cmputed with this methd fr NACA and 8 airfils are cmpared in Figure with the results btained by Kunz & Kr [9] using the INSD cde frm NASA Ames, based n an upwind scheme develped by Rgers & Kwak [36]. The present slutins were fund in ecellent agreement with the results btained by Kunz & Kr, [9] as shwn in Figure. The influence f the lw Reynlds number n the pressure cefficient distributins is shwn in Figure, which als shws fr cmparisn the inviscid flw slutins calculated frm [4, 9]. One can ntice imprtant changes in the pressure distributin at the lw Reynlds numbers. -. NACA, Re= NACA 8, Re= Kunz & Kr...3 Kunz & Kr NACA, Re=6 Inviscid Kunz & Kr Kunz & Kr NACA 8, Re=6 Inviscid Figure. Pressure cefficient distributins n NACA and NACA 8 airfils at zer incidence and Reynlds numbers Re =, and Re = 6 Re = 6 Re = NACA Re = Inviscid Inviscid Re = 6 Re = 6 Re = NACA 8 Re = Figure. Influence f the Reynlds number n the pressure cefficient distributin n NACA and NACA 8 airfils at zer incidence.

9 3.3. Slutins fr airfils at incidence Samples f the present slutins cmputed fr several symmetric and cambered airfils at varius lw Reynlds numbers and angles f attack are shwn in Figures - fr the lift, drag and pressure cefficients. These slutins were fund in gd agreement with available previus results [9]. A detailed study f the flw separatin n airfils at lw Reynlds numbers has als been perfrmed fr varius values f the airfil relative thickness and camber, maimum camber psitin, incidence and Reynlds number. Sample f results are shwn in Figures, 3 and Table NACA, Re= NACA 44 NACA 444 NACA Kunz & Kr Kunz & Kr NACA, Re= C D s, Re= NACA 44 NACA 444 NACA 644 s, Re= NACA 8, Re = Kunz & Kr Figure. Lift cefficient variatin fr NACA and 8 airfils at Re= and. C.5 L NACA 44 NACA 444 NACA 644 s, Re=8..3 C D.5.7 Figure 3. Lift, drag and drag plar diagrams fr NACA 44, NACA 444 and NACA 644 airfils at Re=8.

10 .8.7 NACA 44, Re =.8.7 NACA 444, Re = Kunz & Kr Kunz & Kr... C D Figure 4. Lift cefficient variatin fr NACA 44 airfil at Re=. Figure 6. Drag plar fr NACA 444 airfil at Re= and..8.7 NACA 44, Re =.. Re=4 Re=6 Re=8 Re= Kunz & Kr... D.3.4 NACA 44, Re = C.8 C D s NACA 444 NACA Kunz & Kr.7.8 C D Re=4 Re=6.. s Re=8 Re=...4 C D Figure 5. Drag plar fr NACA 44 airfil at Re= and. Figure 7. Drag variatin and drag plars fr NACA 444 airfil at Reynlds numbers Re=4, 6, 8 and.

11 -. NACA, Re= NACA, Re = & = Kunz & Kr NACA, Re = & = NACA, Re = & = Figure 8. Pressure cefficient distributins n NACA airfil at Re = and =,, 4 and NACA 8, Re= NACA 8, Re = & = Kunz & Kr NACA 8, Re = & = NACA 8, Re = & = Figure 9. Pressure cefficient distributins n NACA 8 airfil at Re = and,, = 4 and 6.

12 NACA 444, Re = 4 & = NACA 444, Re = 4 & = NACA 444, Re = 4 & = Figure. Pressure cefficient distributins n NACA 444 airfil at Re =4 and =,, 4 and NACA 444, Re = 4 & = NACA 444, Re = 8 & = NACA 444, Re = 8 & = NACA 444, Re = 8 & = NACA 444, Re = 8 & = Figure. Pressure cefficient distributins n NACA 444 airfil at Re =8 and =,, 4 and 6.

13 Table. Flw separatin cmparisn fr symmetric NACA airfils at Re= and = 6. Airfil type Separatin psitin, s Reattachment psitin, r Separatin length, l s NACA NACA NACA NACA y y y Figure. Streamline cnturs fr NACA 4 airfil at Re= and = 4, 6 and. 7.

14 .. y y y y Figure 3. Influence f Reynlds number: Streamline cnturs fr NACA 444 airfil at = 8 Reynlds numbers Re=4, 6, 8 and, respectively. and

15 4. CONCLUSIONS The paper presents efficient slutins f the Euler and Navier-Stkes equatins based n finite difference and finite vlume frmulatins. A biased-flu methd using an eplicit finite vlume frmulatin is first presented fr the slutin f the Euler equatins. A secnd-rder scheme is used fr the flu calculatin, with an upwind r dwnwind bias fr the flw variables, accrding t the subsnic r supersnic character f the lcal flw. This methd avids thus the ddand-even-pints decupling, which is present in ther methds, and as a result, des nt require an added artificial-dissipatin, displaying a very gd cmputatinal efficiency in all prblems studied. Validated fr cmpressible flws with epansins and shck waves in nzzles, this methd is then used t btain slutins fr subsnic, transnic and supersnic flws past airfils. Efficient Navier-Stkes slutins are als presented fr the viscus flws past airfils at lw Reynlds numbers. These slutins are btained in a rectangular cmputatinal dmain, defined by a crdinate transfrmatin, with a methd using artificial cmpressibility and a finite difference frmulatin n a stretched staggered grid. As a result f the crdinate transfrmatin, there is n need fr a cmple grid generatin prcedure. The cmputatinal efficiency f this methd is very gd due t a special decupling prcedure f the mmentum equatins, with the aid f the cntinuity equatin augmented by artificial cmpressibility, which finally reduces the flw prblem t the efficient slutin f scalar tridiagnal systems. This methd has been successfully validated by cmparisn with previus available results, btained by Kunz and Kr, fr the lift, drag and pressure cefficients fr Reynlds numbers between and 6. Then the methd has been used t btain slutins at lwer Reynlds numbers between 4 and 8, fr which there are n previus results. A detailed analysis f the flw separatins n airfils at lw Reynlds number has als been perfrmed with this methd. REFERENCES. Andersn W.K., Thmas J.L., VanLeer B. (986) Cmparisn f finite vlume flu vectr splittings fr the Euler equatins. AIAA Jurnal 4: Chrin A. (967) A numerical methd fr slving incmpressible viscus flw prblems. Jurnal f Cmputatinal Physics : Drela M., Giles M.B. (987) Viscus-Inviscid analysis f transnic and lw Reynlds number airfils. AIAA Jurnal 5: Drela M., Giles, M.B. (987) ISES- A Tw- Dimensinal Viscus Aerdynamic Design and Analysis Cde. AIAA Paper Eidelman S., Clella P., Shreeve R.P. (984) Applicatin f the Gdunv methd and its secnd-rder etensin t cascade flw mdeling. AIAA Jurnal : Gdunv S.K. (959) Finite difference methd fr numerical cmputatin f discntinuus slutins f the equatins f fluid dynamics. Math Sb 47: Huteau T., Lee T., Mateescu D. () Flw past a -D backward-facing step with an scillating wall. Jurnal f Fluids and Structures 4: Jamesn A., Schmidt W., Turkel E. (98) Numerical Slutin f the Euler Equatins by Finite Vlume Methds Using Runge-Kutta Time Stepping Schemes. AIAA Paper Kunz P., Kr I. () Analysis and design f airfils fr use at ultra-lw Reynlds numbers. Prceedings f the AIAA Fied, Flapping and Rtating Wing Aerdynamics at Very Lw Reynlds Numbers Cnference. Ntre Dame, June 5-7, pp Lee T., Mateescu D. (998) Eperimental and numerical investigatins f -D backwardfacing step flw. Jurnal f Fluids and Structures : Lentink D., Gerritsma M. (3) Influence f airfil shape n perfrmance in insect flight. AIAA Paper , June 3-6. MacCrmack R.W., Rizzi A.W., Inuye M. (976) Steady supersnic flwfields with embedded subsnic regins. Cmputatinal methds and prblems in aernautical fluid dynamics, Academic Press, pp Mateescu D., Abd M. (5) Lw-Reynlds number aerdynamics f airfils at incidence. Applied Aerdynamics Cnference, 43rd AIAA Aerspace Sciences Meeting, Ren, Nevada, AIAA Paper 5-38, pp Mateescu D., Abd M. (4) Efficient secndrder analytical slutins fr airfils in subsnic flws. Aerspace Science and Technlgy Jurnal (accepted fr publicatin) 5. Mateescu D., Abd M. (4) Aerdynamic analysis f airfils at very lw Reynlds numbers. Applied Aerdynamics Cnference, 4nd AIAA Aerspace Sciences Meeting, Ren, Nevada, AIAA Paper 4-53, pp. -

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