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Date Created: 10/21/14

1 MECE 6353 CFD Notes MECE 353 CSMPUTATISNAL FLUID DYNAMICS Basic eqns of Ineompressible Fluid Flow amp Heat Transfer NAVHERST IS uoVuE lVp1Vzu t 0 VuO E u uXyzti VXyztj wXyztk p pXyzt VZi j k V2V V coVgtlt u x y z Well posed problem gt Appropriate Boundary Initial Conditions SIPECHAL CASES 2D Flow UL Stokes ow uVu ltlt vV2u Re ltlt 1 Re T Inviscid ow VV2u ltlt uVu Re gtgt 1 Irrotational ow o O gt u Vcl Boundary layer ow Thin regions near solid walls viscosity amp vorticity important Steady ow Z 0 8t FSRCES C NVE TI N EUATI gtN D9 Ft ocV2G in D 2 MECE 6353 CFD Notes 9 9Xyzt is temperature concentration passive scalar Boundary Condition on 8D 9 Dirichlet 1ltVG n heat flux Neumann Initial Cond Gxt O G0x SPECEAL CASES Pure conduction uV0 ltlt ocV2G Pure convection ocV2G ltlt u V0 TUULENT FLCDWS E NUMERHCAL APPlIiAClEIES Chaotic unsteady motion wide spectrum of scales 1 Turbulence Models Challenge How to accurately predict physics with realistic computational effort a Algebraic models based on mixing length or eddy Viscosity U time mean of turbulent ow smooth regular Simple example Boussinesq turbulent eddy Viscosity l UVU E Vp vtV2U V u 0 Has smooth solutions Model frequently Very inaccurate b One equation models include eqn for k turbulent kinetic energy Generally doesn t capture enough physics c Twoequation models 2 3 MECE 6353 CFD Notes Add additional eqn for 8 dissipation or a dissipation rate gt k8 or ka model Most popular and extensively used turbulence model despite well documented shortcomings d Secondorder closure models Typically 7 additional eqns for Reynolds stresses and length scale Complexity of these models tax present computers 2 Direct Numerical Simulations Solve full NS eqns Wo model Large range of scales gt need many modes for accurate soln eg For RA 50 need Ol06 grid pts 9 pts T w RK4 3 Subgrid Scale SGS or Large Eddy Simulation LES Model only the subgrid scales a compromise Simulations are hardware limited CMIPUTER HAW CltoNsHDET1oNs 1 Serial processor IBM 650 early microcomputers SISD Single Instruction Single Datastream 2 Early pipeline machines CDC 7600 510 mega ops 3 Vector pipeline machines Cray l SIMD 4 Multiprocessors Cray XMP4 Cray YMP 16 MECE 6353 CFD Notes Multiple units can directly increase speed in parallel computations 5 Massively parallel computers Jaguar Oak Ridge National Laboratory 23 peta op computer 1015 oating point operationssec 18688 dualcore AMD opteron chips 224256 Processing cores 300 Terabytes of memory Beowulf Clusters 10000 microprocessors Googleplex uses 450000 servers 1 EN Performance Development PF39 3 3241940 71 1 1quot 10 PFlop3 1 a 5 ul 1759ooIF sum 1 PFlop3 133 Jquot 11quot I II I 100 TF0p3 1 8 39 g 24570 EFU Jquot Zg quot E 10 T ops 4 13 I B I SIquot quot I 3 I g I TFIop339 3 r I E A I I I391 II 5 I 1oocHops W Bquotquot I E E 1UGF1ops Ian39s lt In 2 a 1 GFIops 03 39 I 1UUMFlOp3TIrrI IIIIIIIIIIIIIIIIIIIIIIII 0quot V In 9 quot oo 039 G v quotKl 397 ID 9 l 07 Q 07 67 CD 039 639 O7 O D D Q 3 3 Q Q Q Q 39 07 039 07 Oquot 039 G 039 Q Q Q Q 3 Q Q Q C Q F 39 39 J J GI quot394 OJ OJ 0 J J OJ 0 MECE 6353 CFD Notes c I A 0 Proected Performance Development 10CIPFlop3 I 1 10 PFlop3 a 5 I Sum 1 PHD 1 Trend Line 1OUTFops ant 8 2457 Llne M 3 10 TFop3 Sum Tlend Line 5 9 1 maps 1 100 I3Flop3 10 GFlop3 9quot lt Apr 1 GFIopsT9 lUUMF0p3Inr1v Tl I III I I I I III I I I O In quot39 67 039 LO 1 67 v 00 L0 N 639 O7 639 O7 O D O O O 0 v v v 39 22228888888888 3939ww w to p500 oIg39 Challenge for CFD Full DNS of ow past complete aircraft will require an exa op 1018 ops computer Constructive interaction required among computer designers optimal machine architecture numerical analysts parallel algorithm development theorists subgrid scale models experiments elucidate fundamental physics NUMEC APPR AC1EI 6 MECE 6353 CFD Notes Want to solve probs of following type 56 E L6 L is a linear spatial operator eg L u V OLV2 E 3 basic components of soln procedure 1 Temporal discretization 8t gt At 2 Spatial discretization 8 gt AX 3 Operator inversionsoln 1 TEMMDL DISCTIZATKDN Consider ODE IVP Analytically 9 90 qt39dt39 Denote discrete time step by At amp let 9 9nAt 9t Then D E can be approximated by Jim39 I Choice of different oci gives schemes of Varying complexity amp accuracy eg OL2 1 oci O i 75 2 gt 6111 6 Atq quotEuler39s methodquot an explicit scheme 7 MECE 6353 CFD Notes error cumulative 4 911 9n Atqn At t 20 SPATEAJL DIS RETJ1ZATJI N Frnrra EDHHEJERJENCES Consider ODE BVP L9qonO x 1 BC39s Disoretize gt 1 Gj 9jAX GXj O N AX E L gt L Lij a nite dimensional matrix 330 PPETR INVERSIN To solve Lij j qi ltgt LG q need to quotinvertquot Lij by some algorithm EVA LUAT1 gtN on A NUMECAL SCHEME Truncation Error d1 zu t r Try FTCS Forward Time Centered Space scheme uI11 uI1 JTJ uf1 uI11 8 MECE 6353 CFD Notes Determine truncation error TE by Taylor series expansion a1 0quot2u u 1 u 0c 1JLuv a 038 At AX2 1 I I I I 2ujJT 1 PDE FDE 2u At 4u AX2 t Haj 2 Ba 12 TE OAX means TE S KAX as AX gt O How Well does FDE approx PDE DefinitionA With a if AX v OTE 039 e g scheme for heat eqn uI3911 uI391 1 a n n n n J 2AtJ AT2uj1 uj 1 uj 1uHi has TE given by g u 2u Atjz 1 3u I A 2 Z 12o3lt4nj X aaznaj AX 6amp3 nj A02 At Suppose At AX gt 0 but 3 remains constant AX Then FDE gt 9 MECE 6353 CFD Notes a z a at tz kz instead of the heat eqn e g y Cy exact soln yX yOeCX Euler39s method yn1 yn chyn or yn1 yn1Ch h AX Thus yn yo1ch 1r1 ch gt 1 FDE T with X since yn1 lynl I1 Chl gt yn Unstable Stabl Rech Imch 2 oNVERlteENoE Soln to FDE gt soln of PDE With same IC39s amp BC39s as AX At gt O Lax39s Equivalence Theorem For properly posed IVP amp consistent FDE stability COnVergenCeConVergence rate or order How fast does uf gt u as At AX gt O Accurac Measure diff u u in L2 or LOO norms J Storage How much computer memory T with N 10 MECE 6353 CFD Notes Operation count on CPU time How does accuracy of scheme affect N GENEL APPRA EI Begin by studying 3 important basic physical processes convection di usion amp quotequilibriumquot General advectiondiffusion eqn uoV 9aV2 9q A B C S Convection A B gt Hyperbolic equations Di usions A C gt Parabolic equations quotEquilibrium quot39 C S gt Elliptic equations S Source Term Begin with 1D examples gt higher dimensions gt combine gt NS eqns 10 11 MECE 6353 CFD Notes THME STEP DES TIZATI N Ordinary Differential Eqns ODE39s Initial Value Problem IVP t a9t090 gt990 flteegtde O Classi cation of timestepping schemes Let 9n 911A9 Then in general I 1jlee scheme isj Otherwise it is If k gt 1 scheme is EX 1 quot05quot methods d CjI X191iCit 0116St p 1 order implicit oneSt p 1 order imp1iltgtit oneSt pe 2nd order EX 2 F 11 12 MECE 6353 CFD Notes EX 3 II Accuracy EX ocmethods Taylorexpand as before 9 9 1 fen d9 m 1d29 At2 i At Ati Cit 2 dt2 P PA dt Combine terms with common powers of At n1 n n i Ot fn1 1 Ot P1 fn At A in E A2 ti2dt2 X dt O t TE 2 SetRHSOanduseg Lfto gettheEDE 2 J 2 dt fAot 2dt2oAt Note First term of TE gt 0 When oc 12 gt Crank Nicolson scheme is 2nd order Error eqn Define a G 9 Then from ODE amp FDE en must satisfy OtfiCn1 e1 1 Ot fo in TE 12 13 MECE 6353 CFD Notes For stability need to have Lh T l5n1 Consider on methods for model problem With f9t 9 q Then ocke 1 1 oc e gt t En an 1 1 ocxAt 1 OCAt Case 1 on O For absolute stability need 1 tAt 1gt 1 1 iAt 1gtiAt 2 gtAts3AtC1 1 50 this Sch m Case 2 0c 12 1 l 2 lei an Case 3 ocl 8I11 811 8n1 Ln 1 kAt gt Recall Lax39s Equivalence Theorem For properly posed IVP amp consistent FDE stability convergence 13 14 MECE 6353 CFD Notes Local vs Global errors Global errors given by error eqn in form 8nl 8n At However note that at each time step actual local error is only 8 8 OAt2 Paomc PDEVS Unsteady diffusion heat eqn Mixed IVP BVP 19 329 6t 032 9Ot 91t O BC39s 9XO sin TEX IC39s G e75 sin TEX Analytical soln TemporalSpatial Discretization In general 6n1 an 0 26n1 0 26n 0 26n 1 m 2 F 2 2 9 2 9 At 6X 6X 6X 9 Choice of F determines timestepping scheme need to couple this With FD representation of spatial derivatives e g W 1 1 0 90 9 O 91 sin TCXi Examine computational molecules 14 15 MECE 6353 CFD Notes 0 O O O I p7 9 Ii 0 O J 2 X0 0 gI 1 X0 0 gI 1 aO al Explicit Implicit Accuracy Taylor expand FDE as before n1 n n n jgi gi 591 1At 2 ii At2 At 5t 2 5t n 2 n AXX6 azgzi AX 641 OAx4 6X 12 k n i AXX6 AXX6 AtAXX 5 OAt2 Now use the PDE to relate spatial amp temporal derivs 0quot 329 i tz 3t 3t 3t K 032 032 t 034 so that the Taylorexpanded difference eqn becomes OR 6 Ata 1 L46 0Ax2 Atz 0 t 0395 2 03 TE Since TE gt O as At AX gt O scheme is consistent For on 01 TE OAt AX2 1st order time 2nd order in space For on 12 TE OAt2 AX2 2nd order time amp space 15 16 MECE 6353 CFD Notes Stability Prediction methods 1 Matrix method Consider Euler scheme n1 n AXXG 0139 At 0 0 1G 1 20 031 where r A Ax r031 1 2rG rO1 It follows that max 6 67 Thus a su cient condition for absolute stability is 1 2r2r 1gt1 2rS1 2r gt Axz rlt1 ieAtS 2 2 Time stepping constraints like this determine stz ness of the eqn More generally a system is stz if El widely disparate time scales in the problem Generally the matrix method involves determining the eigenvalues of A where e A e AP M It can be shown that error in FDE can be written as an Zlck t xk a linear combination of the eigenvectors xk of A Xk Wk Xk amp that all will remain bounded if tk S 1 Vk Gershgorin39s Theorem The modulus of the largest eigenvalue of a square matrix cannot exceed the Sr 1 2rrmax 16 17 MECE 6353 CFD Notes largest sum of the moduli of the terms along any row or column Matrix method can give too strong a condition ie su cient but not necessary amp doesn39t Work V eqns But it is simple amp extends to noneonst eoef eient eases 2 Represent FD soln by Fourier examine behavior of indi grows scheme is potential Z a sinflltxltor Z a The all are the Fourier eoeffs of determined from the IC39 OIIC Applying this to our ocmethod FD 9 9 mlfjf It 1 ocgte1 29 91 Substitute in the Fourier expansion of 9 gt N l N l Za 1 a1 sin 7ZlXJ raZ 0ka 1 sin 7ZlXJ k1 k1 N 1 rl 05 c7ka1 sin nkxj Gk 21 cos 7zlltAx k1 Thus afg a1 rocoka1 r 1 odokafg or allyl aE1 7 005k 1 rocok 17 18 MECE 6353 CFD Notes Case 1 Euler forward a a l rok So we have absolute stability if l 1 Gk lt l gt r S 2Gk V k From de nition of Gk 1 2 maXo39k4gtrS gtAtS k 2 2 Euler forward difference scheme DT 0025 DX 1 R 25 O t 000 E 39 t 100 quot39 D t 200 t 0U 18 19 MECE 6353 CFD Notes Euler forward dilference scheme DT 007 DX 1 R 7 O t 000 0 t 210 0 t 230 00 02 04 06 08 10 Euler for r 1 h This Varies With k like k Gk lGk O 0 1 N 3 1 O 2N 3 3 2 N 4 3 Frequently occur in unstable num sim Case 2 CrankNicolson E afg a j2 unconditionally stable 1 2 Case 3 Full Implicit 81 an1 T 1 rok unconditionally stable 19 20 MECE 6353 CFD Notes Full implicit differencing scheme DT 1DX 1R 10 Ot0 I I 9 t1 quot U t2 t3 63910 012 04 06 08 fio X Implicit scheme stable even for large R but error now much larger Note severity of time stepping constraint for Euler forward Atcr AX2 Doubling spatial resolution requires At gt At4 gt Work goes up by factor 8 in 1D 32 in 3D Von Neumann Method More General Context Consider L9 L a discrete spatial operator From linear algebra L S A S14 20 21 MECE 6353 CFD Notes where A is the diagonal matrix of eigenvalues of L kk amp S is the eigenvector matrix Consider expanding 9 in these eigenvectors gt S a 0 Then from the DB gt L161 usiai s1A1sHs1a1 2 Aa aka So stability is essentially governed by most unstable equation For this example a3 Sjk Sill kTCXj A 91 amp Ax In general the kk are estimated numerically ExplicitImplicit Which is best n1 n 9 9 A o Explicit t C BN1 L 911 Implicit At Explicit Mult by A usually sparse efficient calc Very stiff conditionally stable gt small At Implicit Unconditionally stable Need to invert C altho sparse can be expensive 21 22 MECE 6353 CFD Notes Also consider Accuracy If Atacc lt Atcr El no penalty for explicit scheme Machine architecture Explicit methods easier to Vectorize Discretization schemes How sparse are A amp C Geometry Implicit can be harder in complex geometry HYPEQDLEC PDEVS Convection dominated processes Model Problem g 0 9xO 90x 9Ot 91t which has general soln of the form 9 90x t Periodic waveform travelling to the right at velocity 1 Consider as example 90 sin 27tx 22 23 MECE 6353 CFD Notes Propagating Wave Soln 00 02 04 06 08 10 Dispersion amp Dissipation For model eqn examine solns of the form 0 Reei2 kX ei2TlZD eiznkix 39 o ei2TlZlltlX Ct Here k wavenumber 270 7 wavelength o frequency C phase speed Plug into eqn gt 36 36 i27m 9 i27zlltc9 gt akorc1 il 039 Thus 0 is undamped ie nondissipative and eqn is nondispersive gt C const 75 ck Compare this with heat eqn Gt GXX amp examine solns of form 9 ReeGt ei2TlZlX Plug into eqn gt 23 24 MECE 6353 CFD Notes 09 27Z39l29 32 gt 0 27zllt2 a ak gt 9 is damped or dissipative What about higher order spatial derivatives Consider 9 6 with solns of form e12 kquote 12 t 03919 i27ra i27zk39 53 gt c 27zllt2 a ex XXX Which is the dispersion relation Thus 9 is undamped but eqn is dispersive In general odd order higher derivs are undamped dispersive higher order even derivs are damped dissipative iLHG 93 2 9 Gj S111 2TCXj I Taylor eXpand gt 61 61 d6 d6 6261 4 AX6 J J g 3 OAt2AX2 At 3 at ax 2 at I TE I TE gt O as At AX gt O gt consistent From PDE g 2 OAt2AX2 t 5X 2 5X EDE is convectiondiffusion eqn With negative diffusion ie backwards heat eqn EXpeot severe instability 24 25 MECE 6353 CFD Notes Von Neumann Stability Analysis Examine soln expansion of form 00 2 39kx lt9 Za e 7 J ltassume con symm a1 afk k OO Use trig identities as before to get expressions like f1 13 Z2i sin 27zlltAxbke27 kXj k J Scheme can be Written as I1 I1 C I1 I1 9j 1 gr 2 6j1 9j 1 C 2 E Here C 72 c is the Courcmt number Plug Fourier expansions into eqn amp equating factors of like terms 3 a a1 iC sin 27tlltAxl all 3 a1 all 1 iC sin 2n1ltAx gt la1l a1 1 C2sin2 27tlltAxl1 2 Thus la1lgta V Ck Note that the VN analysis enables us to get a more precise error estimate S n1 ak 1 C212 C22 n1 ak I1 sake n 2 At 6 C 2 a1E6Xp2AX2jI1AtJ So errors are bounded under the constraint that AtAx2 lt const When Ax At gt O n gt oo nAt xed This is a very restrictive condition for problems of this type 3 this scheme not useful S a1 n1 ak 25 26 MECE 6353 CFD Notes Complex stability diagrams Useful to analyze the ODE Q dt la X complex as a model of our PDE39s for stability purposes The FD discretization can be represented as afg 211 At where kk represents the spatial discretization operation In general we have stability if aI11 g Since a1 a1 1 ZAt must have 1 ZAt 3 1 kkafg I1 21 quotconvection axisquot AA t plane quotdilfusion axisquot Recall that for FTCS for the heat eqn k E AX2 where on 21 cos 27zkAX From this it follows immediately that for the heat eqn 26 27 MECE 6353 CFD Notes 2 1maXAt L gt 2 3 At lt AX 2 For the convection eqn M i sin 21tlltAX AX 311Atgt1V AtgtO 3 unconditional instablity Note Euler forward time stepping gives circle of stability above Other schemes have different stability zones E11161 f01 Wa1 d Basie scheme ellH H 0 At AX Taylor series expans 3 EDE g lAX At OAX2At2 at 03 2 k TE gt O 3 consistency Von Neumann stability Plug Fourier expansion 6 Z a ezmkxj into FDE 3 k oo a111 allgll Cl e211i1ltAX or la121lagl 2C2 Cl cos 2TClAXT 2 a11y need y S 1 Since 1 eos 21tkAX 2 0 must have 2C2 2C S O 3 C S13 Compare this condition At S AX With heat eqn At S AX2 27 28 MECE 6353 CFD Notes quotconve ction axisquot unstabl 2At plane c 1 quotdffu H 1 s10n axis c5 Damping moves km off the imaginary axis into the stability region Numerical diffusion The EDE for was S0 for nite AX signi cant can be Very 28 29 MECE 6353 CFD Notes Euler forward upwind Convection Eqn DX 05 DT 025 C 5 i gt r O t 3900quot 0 t 1000 39 D t 2000 r I t 3000 3 A t 4000 00 02 04 06 08 10 CrankNicolson centered differences The 59 term in the TE appears to cause either instability FTCS or damping Upwind What about CN em1 39 Q n1 n lAX0 lAX0 0 gt At 2 J 2 J 9 e f 0 At 4AX Thus tridiagonal matrix will be diag dom only if L lt L gt C lt 2 ZAX At So have almost as strong restriction as with explicit scheme EDE gt 29 30 MECE 6353 CFD Notes p iAt2 lAX2j t 5x 12 6 So expect OAt2 Ax2 vN stability analysis gt 1 Esin 27tlltAx ai1 2 at 2 lai1lIai2I Vczk 1 ijcsin 27tlltAx unconditional stability and no ampli cation errors Analyze error from behavior of soln of the form 6 e2TC1lX e2TC10lt in the EDEgt 2mm 21tik 1 11 2At2 Ax22n1lt3 gt o L 2 1 2 k 1 mm 6Ax22n1ltgt 1 Ad Where Ad is the phase error This is numerical dispersion fxn of k Max error occurs When k N2 gt c 0 El major probs With this CNCD scheme For nonlinear probs matrix inversion very expensive What about higher order explicit AdamsBashforth centered differences For ET fz general AB schemes are of the form 11 1 n M 395quotF For model Wave eqn prob gt 30 31 MECE 6353 CFD Notes 9 9 H D L AX9j lAX9j 1 0 At 2 2 Using Taylor expansions get EDE 0igg imz lAX2ji at 03 12 6 03 So have dispersive errors leading in time lagging in space Stability Von Neumann analysis 00 27rilltX 6 Za e J k OO method gt second order difference eqn a121 a 1 gicsin 2TClAX a11 1sin ZTCKAX O FD eqns have solns of the form all Ll Solving for Lt givesj 2 H 1 icsin 27tlltAX sin327tlltAX iic3sin327IlltAX c4sin427tkAX Q icsin 27tlltAX c2sin227IlltAX 39 the desired consistent With the soln of the PDE bu1 This c1riS6S in For small c J lt l gt stability However 31 32 MECE 6353 CFD Notes n1 ak W n k unstable But growth is very slow for C 1 a factor 10 growth in unstable mode takes Euler 11 C2 gt 460 time steps AB2 1 C34 gt 92000 time steps Thus Whi0h is 0f as long as C is small 9 ugnuhgnagar has the corresponding EDE 346 lAX2 6 OAt3 4 OAtAX2 at 03 6 k 03 dispersion dissipation This Thus almost as stable as upwinding but With small damping 1 4 O 4 12 1 4 g1 c S111 27Z39lAX 3 1 c gt NNPEEM lDHC UNlDARY CNDl1TIl NS For 1D Wave eqn 0 tgt00SXS1 t x 9XO sin 27zX 9Ot sin27zt exact soln is 0Xt sin 27tXt Problem is well posed only With no BC given at X l CrankNicolson Centered Differences 32 33 MECE 6353 CFD Notes Same scheme as before but now problem at x 1 can39t apply scheme amp no BC gt need spurious or numerical BC to use this scheme when order of FDE gt order of PDE eg GN1 9N AX b 2 GN GN1 Ax gt 9N1 9N 1 9N 39 9N 1 2Ax Ax based on linear extrapolation Thus at j N 9 9 i W1 9 9 9amp4 At 2 Ax Ax a onesided first order approximation Soln is forced at x 0 so no longer get phase errors Explicit schemes Von Neumann method cannot be directly applied to determine BC stability AdamsBashforth Can use at j N T 0 At 2 Ax 2 Ax This is stable altho Ol errors at boundary they do not tend to contaminate interior soln 33 34 MECE 6353 CFD Notes AB2 CD Wave Eqn Errors Due to Out ow BC 02 01 39 3 o 5 as 00 A t 5 E 0 t 10 39539 G4 01 02 00 02 04 06 08 10 LeapFrog n 1 n1 n n ej 9j191O j1 N 2At 1 21AX as as 0000 0 2At AX This scheme is clearly unstable Leap Frog CD Wave Eqn OutFlow BC 0 t 000 0 t 2004 D t 4004 34 35 MECE 6353 CFD Notes These out ow BC39s introduce damping which doesn39t affect AB2 stability but destabilizes leapfrog 35

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