Navier-Stokes Characteristic Boundary Conditions. for Simulations of Some Typical Flows

Transcription

1 Appled Mathematcal Scences, Vol. 4, 00, no. 8, Naver-Stokes Characterstc Boundary Condtons for Smulatons of Some Typcal Flows Chen Ln and Tang Dengbn Nanjng Unversty of Aeronautcs and Astronautcs Nanjng 006, P. R. Chna Abstract The mproved Naver-Stokes characterstc boundary condtons (NSCBC) method for drect numercal smulaton of vscous flows, combnng s order non-dsspatve compact schemes wth eght order flters,are studed n ths paper. The new boundary condtons ncludng transverse and vscous effects are appled to a comprehensve set of test problems, such as vorte-convecton, counter flow, pressure wave propagaton and boundary layer flow. The computatonal results show that these boundary condtons, whch relate to the resoluton, stablty and reflecton of numercal smulaton, have wder applcablty and hgher numercal precson. Keywords: Naver-Stokes characterstc boundary condtons (NSCBC); non-dsspatve compact schemes; drect numercal smulaton; vscous flow. INTRODUCTION The treatment of boundary condtons s one of the most recurrent ssues n computatonal flud dynamcs. Computatonal accuracy, n general, s strongly senstve to boundary soluton, whch may be spoled by spurous numercal reflectons. These spurous reflectons of physcal nformaton, adversely affect the accuracy and stablty of the solutons. Ths matter s of vtal to usng hgh order non-dsspaton schemes n drect numercal smulaton of Naver-Stokes equatons, because non-dsspaton schemes have very low numercal dsspaton,

2 880 Chen Ln and Tang Dengbn t need more accurate boundary condtons to ensure numercal stablty[]. Ths motvates the necessty for strateges to reduce reflecton and set up transparent boundary condtons. Several approaches were proposed to tackle boundary condtons. Technques based on characterstcs waves motvated much attenton. Prevous studes on the characterstc boundary condtons focused on how to suppress acoustc wave reflectons at open boundares. Intally for hyperbolc systems of Euler equatons, these approaches decompose the flow n terms of characterstc waves travelng n the drecton normal to the boundary, and reduce the boundary problem to a sutable dealng wth the ncomng waves. The dentfcaton of ncomng waves allows, n prncple, a drect control over boundary reflecton, as the boundary condton can be desgned to prevent ncomng perturbatons or to damp smoothly ther ampltude. Ths one-dmensonal appromaton of the characterstc boundary condtons was successfully appled n mult-dmensonal Euler equatons by Thompson [, 3]. An etenson to the Naver Stokes equatons was dscussed by Ponsot and Lele[], who developed a systematc approach to account for vscous terms under the locally one-dmensonal nvscd (LODI) assumptons, known as Naver Stokes characterstc boundary condtons. The mathematcal well-posedness of boundary condtons n flud dynamcs was studed by J.Olger et al [4] and L.Halpern[5]. The stablty of the vscous boundary condtons was nvestgated by P.Dutt[6]. The NSCBC approach was further etended to consder mult-component reactng flows [7-8], and subsonc nonreflectng nflow condtons were also derved as a varaton wthn the generalzed formulaton [9]. As an eample of refnement of the NSCBC approach, Sutherland and Kennedy [0] recognzed that the chemcal source terms must be consdered n the LODI formulaton n order to reproduce realstc flame propagaton through the boundary. Yoo et al [] studed the counterflow dffuson flame smulatons. It was found that transverse convecton terms must also be properly accounted for n the modfed LODI epresson to acheve correct soluton behavor. It was also observed that, smlar to the pressure relaaton used n the estng nonreflectng boundary condtons, a proper relaaton treatment for the transverse terms was also necessary n order to ensure soluton stablty. These results suggest that, for a successful applcaton to general turbulent combuston problems, the entre formulaton of the characterstc boundary condtons may have to be re-eamned for a complete resoluton of these ssues. Yoo and Im[] attempted to revst the full NSCBC formulaton n a generalzed contet, by reconsderng many terms that have been neglected before. Generalzed LODI appromatons that were applcable for a wde range of reactng flow condtons were derved. In ths paper, the mproved NSCBC method whch s compatble wth hgh order non-dsspatve compact schemes [3] s addressed. A couple of typcal

3 Naver-Stokes characterstc boundary condtons 88 flows are tested to certfcate the feasblty of the algorthm. It s concluded from computatonal results that ths method has hgh resoluton and more convenent applcablty.. CHARACTERISTIC BOUNDARY CONDITIONS The compressble Naver-Stokes equatons can be transformed nto a characterstc form, wrtten as (n -drecton) [0]: () () u (L5 L )/ ρc Vt tu du () v 3 t tv ( / ) p/ y d v L V + ρ () w+ L + V d 4 t tw + ( / ρ) p/ z = w () () () ρ L d + (L5 + L )/c ρ t ( Vt ) ρ () () p L d 5 + L t tp+γp V t V t p where (, y, z) are the spatal coordnates n a rectangular Cartesan system, t s tme, ρ s the mass densty, ( u,v,w) are the components of flow velocty, p s the () pressure, V s the velocty vector, c = ( γrt) / s the speed of sound and the subscrpt t s represents tangental (y- and z-) drectons. Ths formula can also be used for reacton flow f reacton source terms are added. The vscous terms s equaton () are gven by d u ( / ρ ) j τj d v ( / ) ρ j τjy d w = ( / ρ ) j τ jz dρ 0 d p ( γ- )[ τjk : juk - j qj] () where τ jk s the shear stress, j s the Laplace operator, q j s the heat flu, j,k=,,3. The L n equaton () are the wave-based quanttes obtaned from a k characterstc analyss of the -drecton governng equatons. These quanttes gve the temporal rate of the ampltudes of the dfferent acoustc, convectve or entropy waves at the outcome boundary, and are defned as follows: T L = ( L, L, L3, L4, L5 ) p u ρ p v w p u = [ λ ( ρc ), λ ( ), λ, λ, λ ( + ρc )] c T (3) where ( ) k λ are the characterstc veloctes:

4 88 Chen Ln and Tang Dengbn λ = u c, 3 4 λ =λ =λ = u, 5 λ = u+ c (4) Therefore, the problem of specfyng nflow/outflow condtons s now reduced to the problem of determnng the wave ampltude varatons. Accordng to characterstc system, the dentfcaton of the wave ampltude L depends on the sgn of λ. For the outward wave, L s calculated by usng one-sde dfference method from nsde the computatonal doman. For ncomng waves, however, they cannot be computed from outsde the computatonal doman, and therefore addtonal physcal consderatons must be made. An mportant problem s how to dentfy the nward wave ampltude L. For an eample, we wll ntroduce the method of calculatng L for subsonc outflow and nflow n -drecton. Subsonc outflow, nward wave ampltudes are addressed as: L p p a V = α ( ) + ( ) I + = l (5) L = α ( p p ) + ( a) I + V = 0 (6) Here, I and V denote the transverse and vscous terms, respectvely. α and a are the relaaton coeffcent of the pressure and transverse term, respectvely. The former standard NSCBC [] s gven by: L = α ( p p ) (7) The equaton (7) consders the effect of pressure relaaton only, and neglect the effect of the transverse and vscous terms. When these terms effect weakly n some flow condton, t wll not nfluence the resoluton and stablty of numercal smulaton. When the dervatve of physcal quantty n transverse terms s large enough for some flow smulatons, ths appromaton can cause numercal nstablty and numercal reflecton. We wll eplan t n detals n the followng part, and prove the scentsm and ratonalty of Naver-Stokes characterstc boundary condtons wth transverse and vscous effects. Smlar to pressure relaaton method, mproved subsonc nflow boundary condtons [] are adopted n ths paper. In the treatment of the nonreflectng nflow, the nlet values of the flow velocty vector and temperature are mposed by usng a set of relaaton terms. The modfed LODI relatons correspond to a set of lnear relaaton constrants between the nflow varables and ther prescrbed

5 Naver-Stokes characterstc boundary condtons 883 upstream values. Comparng wth drect subsonc nflow condtons [], ths method has wder applcablty, and superorty. In -drecton, subsonc nflow, nward wave ampltudes are addressed as: = 0 : L = ( T T ) +I + V L = ( v v ) +I + V ( ) ( ) ( ) ( β ) ( ) ( ) 0 3 β L = ( w w ) +I + V L = β ( u u ) +I + V (8) ( ) ( ) ( ) ( 4 β ) ( ) ( ) = l : L u u V = β ( l ) +I + L = β ( T Tl ) +I + V L v v V 3 = β 3( l ) +I L4 = 4( w wl ) +I 4 + V4 β (9) where T 0, u0, v0, w0 and T, u, v, w s the accurate value of physcal varables l l l l at the nflow boundary, β s the correspondng relaaton coeffcents. It has pad many attentons that control quanttes of Naver-Stokes equatons n the conservaton form are conservaton varables, but the physcal boundary condtons are usually epressed by prmtve varables and ther dervatves. In ths paper, equatons () are drectly solved usng prmtve varables epressed by Sutherland s method [0]. For the corner pont, the characterstc method s appled smultaneously n both drectons, and the boundary condton s gven by: () () y u (L5 L )/ ρc M T du () (y) (y) v T L 3 (M5 M )/ ρc d v () w+ L + T 4 M = d w () () () (y) (y) (y) ρ L T + (L5 + L )/c M 3 + (M5 + M )/c 4 dρ () () (y) (y) p L T 5 + L M5 + M 5 d p (0) M = ( M, M, M, M, M ) ( y) ( y) ( y) ( y) ( y) ( y) T p v u ρ p w p v = [ λ ( ρc ), λ, λ ( ), λ, λ ( + ρc )] y y y y c y y y y ( y) ( y) ( y) ( y) ( y) T u v w p ( ρw) p w T = [ w, w, w +,, w +γp ] z z z ρ z z z z T () where ( y ) k λ are the characterstc veloctes ( y) ( y) ( y) ( y) ( y) λ = v c, λ =λ =λ = v, λ = v+ c ()

6 884 Chen Ln and Tang Dengbn 3. NUMERICAL TEST AND ANALYSIS In the numercal smulaton of Naver-Stokes equatons, s order compact schemes are adopted for transverse terms and vscous terms, and three order and four order compact schemes are appled n the boundary ponts and near boundary ponts, respectvely. Four step Runge-Kutta method s used n the tme-ntegarton. The artfcal dsspaton [4] s replaced by eght order flters. Net, numercal test of some typcal flows s studed and analyzed. 3. Vorte convecton n the free flow The two-dmensonal vorte convects through a non-reflectng boundary. Ths s a typcal test used to evaluate boundary condtons. The confguraton corresponds to a sngle vorte n a unform flow feld along the -drecton. The ntal flow feld s prescrbed by: Ψ u u y ( 0) + ( y y0) = + / ρ, Ψ= C ep( ) (3) v 0 Ψ R c where Ψ s a stream functon for an ncompressble vorte, C s the vorte strength, R c s the vorte radus,and ( 0,y 0 ) s the locaton of the vorte center[] The ntal pressure feld correspondng to the vorte s gven by: C ( ) + ( y y ) p = p ρ ep( ) (4) R 0 0 c Rc The flow condtons are specfed as Ma = u / c = 0. where Ma s Mach number, u s the free flow velocty, c s sound velocty, R / l = 0., C/( cl ) = The doman sze s.0mm. 0mm wth 00 grds n each drecton. The reference temperature and pressure are 300K and atm, respectvely, and the flud s ar. The mamum velocty nduced by the vorte s0.5cm / s. At the nflow boundary = 0, standard non-reflectng subsonc nflow boundary condton [] s appled. At the outflow boundary = l, three dfferent non-reflectng boundary condtons are tested by usng A, A and A3, respectvely, n order to llustrate the effect of transverse term and vscous terms n boundary condtons. A: Standard non-reflectng boundary condton ( L ) = α ( p p ) s adopted. From c

7 Naver-Stokes characterstc boundary condtons 885 equaton (), we can obtan the effectve boundary condton: p u ( ρc ) = α ( p p ) + ( I + V ) (5) A: L = ( p p ) +I + V α s appled, whch s referred as Naver-Stokes characterstc boundary condtons wth transverse terms and vscous terms. So the effectve boundary condton has only pressure relaaton term n RHS: p u ( ρc ) = α ( p p ) (6) A3: The modfed characterstc boundary condton (5) s adopted. The transverse term n ( L ) has dfferences wth the accurate vales of physcal flows n some cases. Smlar to the pressure relaaton method, relaaton of transverse term s adopted. In followng standard asymptotc analyss at low Mach number flows [], an estmate of an approprate relaaton coeffcent a= Ma s derved, and effectve boundary condtons become: p u ( ρc ) = α ( p p ) MaS (7) In the y -drecton, the standard non-reflectng boundary condton [] s appled n test A-A3.The addtonal vscous condtons are: n y -drecton: q / y = 0, τ y / y = 0; y n -drecton: τ / = 0 ( = 0), q / = 0 and τ y / = 0 ( = l ). A A A3 Fg. Vortcty socontours ( / s ) at t = 8.8μ sec The vortcty socontours passng through the outflow boundary are shown n Fg. for A-A3. No sgnfcant dfference can be found, even f dfferent boundary boundary condtons are used. The center regon of the vorte for A and A are slghtly dstorted, showng an upward and downward moton at the

8 886 Chen Ln and Tang Dengbn boundary, respectvely. On the other hand,, y velocty s shown n Fg.. It reveals that the numercal artfacts arse because of the mplementatons of the ncorrect boundary condton. From the socontours, contrast to A3, A yelds acceleraton n the u velocty and a deceleraton n the v velocty, whle A leads to an opposte behavor. A A A3 Fg., y velocty socontours ( cm / s ) at t = 8.8μ sec

9 Naver-Stokes characterstc boundary condtons 887 A A A3 Fg.3 Temporal varatons of pressure socontours (atm). (from top to bottom, t=3.0; 8.8; μ sec) The temporal varatons of pressure socontours for A-A3 are shown n Fg.3. The results ndcate that a sgnfcant amount of spurous pressure feld generates at the boundary employng A condton wthout the transverse term and vscous term. The vsble numercal reflectng appears from the velocty and pressure feld, as the vorte travels out of the boundary and the shape changes. As the vorte approaches the boundary, the v γ p term n equaton (5) becomes domnant. y Therefore, the effectve boundary condtons cause the numercal reflectng, and these terms can not be gnored. The transverse term and vscous term are ncluded n the boundary condtons for A; hence the pressure varaton s more confned near the boundary wth smaller ampltude than those for A. Snce the leadng u order n equaton (6) s ρ c = 0, ts net effect s to damp out the -drecton velocty near the boundary. All the neglected terms are reconsdered for A3, and the transverse term relaaton s also adopted. The results show that there s not numercal reflectng n the pressure and velocty feld, and the vorte shape does not change as t travels out of the boundary. 3. Counter flow The counter flow s used n the boundary condton test. A true challenge occurs n a counterflow confguraton where velocty needs to be mposed at the two opposng nflow boundares. It s hard to control the nflow velocty and the absolute value of pressure. The modfed nflow characterstc boundary condton s appled. Intal flow condton s same as C.S.YOO [].Three dfferent boundares are tested and compared n ths paper. B:Modfed nflow condtons (8) (9) are used n -drecton, but transverse term and vscous terms are not ncluded. In the y -drecton, standard characterstc boundary condton A s appled.

10 888 Chen Ln and Tang Dengbn B:Modfed nflow condtons (8) (9) are used n -drecton, and transverse term and vscous term are consdered. In the characterstc boundary condton A s appled. y -drecton, standard B3:Modfed nflow condtons (8) (9) are used n -drecton, and transverse term and vscous term are consdered. In the y -drecton, modfed characterstc boundary condton A s appled. The results of B are shown n Fg.4, the u velocty socounters s dstorted, nlet velocty value changes, and the background pressure values can not mantan. The value n the center reaches.37 atm. The results of B are shown n Fg.5, the nlet velocty value s preserved, but the background pressure value become even large, reaches.49 atm n the center. Consderng transverse terms n both drectons, the results of B3 are shown n Fg.6. The nflow velocty and background pressure are both preserved. The center value of pressure s.00 atm, whch s as same as the reference s value []. Our numercal algorthms, whch combnes mproved NSCBC method wth hgh order non-dsspatve compact schemes, has good relablty. Fg.4 Pressure and -drecton velocty at t = 0ms for condton B Fg.5 Pressure and -drecton velocty at t = 0ms for condton B

11 Naver-Stokes characterstc boundary condtons 889 Fg.6 Pressure and -drecton velocty at t = 0ms for condton B3 3.3 Pressure wave propagaton n statonary flow The pressure feld s ntalzed wth a pressure pulse: p = B( 0) ( y y0) A( + 0 ) e sn( ωt) p = B( 0) ( y y0) A( y + y0 ) e sn( ωt) y (8) The reference varables for scale, densty, velocty, temperature, pressure, tme, vscosty are λ (wave length), ρ, c, T, ρ c, λ / c and μ respectvely. The computatonal doman s a square ( l yl = ).The pressure wave s nduced at center of the square. The parameters are A= 0.005, B = 400, ω = π.reynolds number s Modfed NSCBC s adopted at the boundares. Pressure counters at tme 4 and 6 are presented n Fg 7(a) (b) and (c). As wave pass through the boundary, the pressure counter shows s dsplaed n Fg.7 (d). The dstrbuton of pressure of -drecton centerlne s shown n Fg.7(e) at tme 0. Once the pressure front meets the boundary, the boundary condton s remarkably capable of preservng the correct physcal nformaton at the boundary edges and corners.

12 890 Chen Ln and Tang Dengbn (a) (b) (c) (d) (e) Fg.7 Evoluton of the pressure dstrbuton

13 Naver-Stokes characterstc boundary condtons Subsonc boundary layer flow over the flat plate Net,we study the boundary layer flow over the flat plate at M a =0.5 and Re =00 whch s based on the nflow dsplacement thckness. The doman length and heghts are both 30. The number of grd ponts s The modfed NSCBC condton s used at upper free stream flow and outflow boundary. The dstrbuton of computatonal velocty and temperature normalzed by the edge values at = 55.7 s shown n Fg.8. In spte of the qute low Reynolds number and coarse grd, comparng computatonal results wth Blasus soluton, the agreement s qute good. Fg.9 gves the pressure at the wall, whch almost keeps the same, and there s nearly no non-physcal pressure reflectng at the outflow boundary Computatonal result smlarty soluton Computatonal result smlarty soluton U/U e 0.4 T/T e y/δ y/δ Fg.8 dstrbuton of velocty and temperature at = p/p e /δ Fg.9 pressure at the wall 4 CONCLUSIONS A number of lngerng ssues of spurous soluton behavor has encountered n some flow smulatons n the past usng standard NSCBC. In these flows the

14 89 Chen Ln and Tang Dengbn transverse terms and vscous terms affect the boundary condtons. Improved Naver-Stokes characterstc boundary condtons usng hgh order schemes were adpoted n ths paper. A couple of typcal cases were tested to certfcate the feasblty of the algorthm. Ths method has more convenent applcablty, hgh resoluton, and smaller reflectng wave ampltude at the boundary. Obvously, t s concluded that ths algorthm s qute sutable for drect smulaton of Naver-Stokes equatons. REFERENCES [] T.J.Ponsot and S.K.Lele, Boundary condtons for drect smulatons of compressble vscous flow, Journal of Computatonal Physcs, 0(99), [] K.W.Thompson,Tme-dependent boundary condtons for hyperbolc systems, Journal of Computatonal Physcs, 68(987),-4. [3] K.W.Thompson,Tme-dependent boundary condtons for hyperbolc systems, Journal of Computatonal Physcs,89(990), [4] J.Olger and A.Sundström, Theoretcal and practcal aspects of some ntal boundary-value problems n flud-dynamcs, SIAM Journal on Appled Mathematcs,35(978): [5] L.Halpern,Artfcal boundary condtons for ncompletely parabolc perturbatons of hyperbolc systems, SIAM Journal on Mathematcal Analyss,(99), [6] P.Dutt,Stable boundary condtons and dfference schemes for Naver-Stokes equatons, SIAM Journal on Numercal Analyss, 5(988), [7] M.Baum, T.J.Ponsot and D.Thévenn, Accurate boundary condtons for multcomponent reactve flows, Journal of Computatonal Physcs, 76(994),47-6. [8] N Okong'o and J Bellan,Consstent boundary condtons for multcomponent real gas mtures based on characterstc waves, Journal of Computatonal Physcs,76(994),

15 Naver-Stokes characterstc boundary condtons 893 [9] T.J.Ponsot and D.Veynante,Theoretcal and Numercal Combuston, Second Edton. R.T.Edwards Inc, Phladelpha.005. [0] J.C.Sutherland and C.A.Kennedy,Improved boundary condtons for vscous reactng compressble flows, Journal of Computatonal Physcs, 9(003), [] C.S.Yoo,Y.Wang, A.Trouvé,and H.G. Im, Characterstc boundary condtons for drect numercal smulatons of turbulent counterflow flames, Combuston Theory and Modellng, 9(005), [] C.S.Yoo and H.G.Im, Characterstc boundary condtons for smulatons of compressble reactng flows wth mult-dmensonal, vscous, and reacton effects, Combuston Theory and Modelng () (007), [3] S.K.LeLe, Compact Fnte Dfference Schemes wth Spectral-lke Resoluton, Journal of Computatonal Physcs, 03(99), 6-4. [4] V.G.Datta and R.V.Mguel,Hgh-order schemes for Naver-Stokes equatons: algorthm and mplementaton nto FDL3DI,AFRL-VA-WP-TR Receved: August, 009