Turbulent Flows and their Numerical Modeling

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1 REVIEW Lecture 4: Numercal Flud Mechancs Sprng 015 Lecture 5 Fnte Volume on Complex geometres Computaton of convectve fluxes: x y For md-pont rule: Fe ( vn. ) ds fs e e ( ( vn. ) e Se e m e e e ( Se ue Se ve) S e y η NW N n ne η n ξ nw ξ e E n P n n w W S se sw n SE S x Image by MIT OpenCourseWare. Computaton of dffusve fluxes: md-pont rule for complex geometres often used d x y Ether use shape functon (x, y), wth md-pont rule: Fe ( k. n) ) e Se kese Se x e y e Or compute dervatves at CV centers frst, then nterpolate to cell faces. Opton nclude ether: x Gauss Theorem: dv dv cs c dv x x x P P CV 4 c faces old d E P Deferred-correcton approach: Fe ks e e ks e e e n re rp old old Comments on 3D x where s nterpolated from,e.g. e P cs c x P 4 c faces dv Numercal Flud Mechancs PFJL Lecture 5, 1

2 Numercal Flud Mechancs Sprng 015 Lecture 5 REVIEW Lecture 4, Cont d: Turbulent Flows and ther Numercal Modelng Propertes of Turbulent Flows Strrng and Mxng Energy Cascade and Scales Numercal Methods for Turbulent Flows: Classfcaton Drect Numercal Smulatons (DNS) for Turbulent Flows Reynolds-averaged Naver-Stokes (RANS) Mean and fluctuatons Reynolds Stresses Turbulence closures: Eddy vscosty and dffusvty, Mxng-length Models, k-ε Models Reynolds-Stress Equaton Models Large-Eddy Smulatons (LES) Spatal flterng LES subgrd-scale stresses Examples η (Kolmogorov mcroscale) Image by MIT OpenCourseWare. 3/4 / ~ O(Re L ) Numercal Flud Mechancs PFJL Lecture 5, L Re L UL /3 5/3 1 Turbulent Wavenumber Spectrum and Scales S SK (, ) A K L 1 K 1 ε = turbulent energy dsspaton

3 References and Readng Assgnments Chapter 9 on Turbulent Flows of J. H. Ferzger and M. Perc, Computatonal Methods for Flud Dynamcs. Sprnger, NY, 3rd ed., 00 Chapter 3 on Turbulence and ts Modellng of H. Versteeg, W. Malalasekra, An Introducton to Computatonal Flud Dynamcs: The Fnte Volume Method. Prentce Hall, Second Edton. Chapter 4 of I. M. Cohen and P. K. Kundu. Flud Mechancs. Academc Press, Fourth Edton, 008 Chapter 3 on Turbulence Models of T. Cebec, J. P. Shao, F. Kafyeke and E. Laurendeau, Computatonal Flud Dynamcs for Engneers. Sprnger, 005 Durbn, Paul A., and Gorazd Medc. Flud dynamcs wth a computatonal perspectve. Cambrdge Unversty Press, 007. Numercal Flud Mechancs PFJL Lecture 5, 3

4 Drect Numercal Smulatons (DNS) for Turbulent Flows Most accurate approach Solve NS wth no averagng or approxmaton other than numercal dscretzatons whose errors can be estmated/controlled Smplest conceptually, all s resolved: Sze of doman must be at least a few tmes the dstance L over whch fluctuatons are correlated (L= largest eddy scale) Resoluton must capture all knetc energy dsspaton,.e. grd sze must be smaller than vscous scale, the Kolmogorov scale, For homogenous sotropc turbulence, unform grd s adequate, hence number of grd ponts (DOFs) n each drecton s (Tennekes and Lumley, 1976): L O UL 3/4 / ~ (Re L ) ReL In 3D, total cost (f tme-step scales as grd sze, for stablty and/or accuracy): CPU and RAM lmt the sze of the problem: 6 10 Peta-Flops/s for 1 hour: Re ~ (extremely optmstc) L Largest DNS ever performed up to 013: x 1536 x 1150 mesh, Numercal Flud Mechancs 3/4 3/4 3 3 L L O L ~ O Re Re (Re ) Re L 500 PFJL Lecture 5, 4

5 Drect Numercal Smulatons (DNS) for Turbulent Flows: Numercs DNS lkely gves more nformaton that many engneers need (closer to expermental data) But, t can be used for turbulence studes, e.g. coherent structures dynamcs and other fundamental research Allow to construct better RANS models or even correlaton models Numercal Methods for DNS All NS solvers we have seen are useful Small tme-steps requred for bounded errors: Explct methods are fne n smple geometres (stablty satsfed due to small tme-step needed for accuracy) Implct methods near boundares or complex geometres (large dervatves n vscous terms normal to the walls can lead to numercal nstabltes treated mplctly) DNS, Backward facng step Le and Mon (008) source unknown. All rghts reserved. Ths content s excluded from our Creatve Commons lcense. For more nformaton, see Numercal Flud Mechancs PFJL Lecture 5, 5

6 Drect Numercal Smulatons (DNS) for Turbulent Flows: Numercs, Cont d Tme-marchng methods commonly used Explct nd to 4 th order accurate (Runge-Kutta, Adams-Bashforth, Leapfrog): R-K s often more accurate for same cost For same order of accuracy, R-K s allow larger tme-steps Crank-Ncolson often used for mplct schemes Must be conservatve, ncludng knetc energy Spatal dscretzaton schemes should have low dsspaton Upwnd schemes often too dffusve: error larger than molecular dffuson! Hgh-order fnte dfference Spectral methods (use Fourer seres to estmate dervatves) Manly useful for smple (perodc) geometres (FFT) Use spectral elements nstead (Patera, Karnadaks, etc) (Hybrdzable)-(Dscontnuous)-Galerkn (FE Methods): Cockburn et al. Numercal Flud Mechancs PFJL Lecture 5, 6

7 Drect Numercal Smulatons (DNS) for Turbulent Flows: Numercs, Cont d Challenges: Storage for states at ntermedate tme steps ( R-K s of low storage) Total dscretzaton error and turbulence spectrum Total error: both order of dscretzaton and values of dervatves (spectrum) Measure of total error: ntegrate over whole turbulent spectrum Dffcult to measure accuracy due to (unstable) nature of turbulent flow Due to predctablty lmt of turbulence Hence, statstcal propertes of two solutons are often compared Smplest measure: turbulent spectrum Generatng ntal condtons: as much art as scence Intal condtons remembered over sgnfcant eddy-turnover tme Data assmlaton, smoothng schemes to obtan ICs Generatng boundary condtons Perodc for smple problems, Radatng/Sponge condtons for realstc cases Numercal Flud Mechancs PFJL Lecture 5, 7

8 Example: Spatal Decay of Turbulence Created by an Oscllatng Boundary Brggs et al (1996) Oscllatng grd on top of quescent flud creates turbulence Decays n ntensty away from grd by turbulent dffuson : strrng + mxng Used spectral method, perodc, 3 rd order R-K DNS results agree wth data Used to test turbulence closure models Dd not work well because not derved for that type of turbulence Sprnger. All rghts reserved. Ths content s excluded from our Creatve Commons lcense. For more nformaton, see Source: Ferzger, J. and M. Perc. Computatonal Methods for Flud Dynamcs. 3rd ed. Sprnger, 001. Note: DNS have at tmes found out when laboratory set-up was not proper Numercal Flud Mechancs Sprnger. All rghts reserved. Ths content s excluded from our Creatve Commons lcense. For more nformaton, see Source: Ferzger, J. and M. Perc. Computatonal Methods for Flud Dynamcs. 3rd ed. Sprnger, 001. PFJL Lecture 5, 8

9 Reynolds-averaged Naver-Stokes (RANS) Many scence and engneerng applcatons focus on averages RANS models: based on deas of Osborne Reynolds All unsteadness regarded as part of turbulence and averaged out By averagng, nonlnear terms n NS eqns. lead to new product terms that must be modeled Separaton nto mean and fluctuatons Movng tme-average: Ensemble average: N l r ( x,) t ( x,) t N r1 Reynolds-averagng: ether of the above two averages u u u'; ( x,) t ( x,) t ( x,) t Numercal Flud Mechancs PFJL Lecture 5, 9 tt / 0 l where ( x, t) ( x, t) dt ;.e. ( x, t) 0 T0 tt / u 0 T (A) u' u T 0 t T Graphs showng (A) the tme average for a statstcally steady flow and (B) the ensemble average for an unsteady flow. u u u' (B) Image by MIT OpenCourseWare. t

10 Reynolds-averaged Naver-Stokes (RANS) Varance, r.m.s. and hgher-moments: Correlatons: In tme: In space: tt0 / l ( x,) t ( x,) t dt u ( u u )( ) u u T 0 tt / rms p 1 k u' v' w' Note: some arbtrarness n the decomposton Turbulent knetc energy: 0 tt 0 / 0 l p ( x,) t ( x, t) dt wth p 3,4, etc T 0 tt / R(t 1,t ; x) u'(x t, 1 ) u'(x t 1 1, ) for a statonary process : R(; x) u'(x, t) u'(x, t ) R( x, x ; t) u'( x, t) u'( x, t) for a homogeneous process : R( ; t ) u '( x, t ) u '( x, t ) uu u'; and n the defnton that fluctuatons = turbulence Numercal Flud Mechancs PFJL Lecture 5, 10

11 Reynolds-averaged Naver-Stokes (RANS) Contnuty and Momentum Equatons, ncompressble: u x 0 u ( uu ) p + t x x x Applyng ether the ensemble or tme-averages to these equatons leads to the RANS eqns. In both cases, averagng any lnear term n conservaton equaton gves the dentcal term, but for the average quantty Average the equatons, nsertng the decomposton: u u u the tme and space dervatves commute wth the averagng operator u x u t u x u t Numercal Flud Mechancs PFJL Lecture 5, 11

12 Reynolds-averaged Naver-Stokes (RANS) Averaged contnuty and momentum equatons: u x where 0 u ( u u u u ) p + t x x x u u x x For a scalar conservaton equaton e.g. for ctmean nternal energy Terms that are products of fluctuatons reman: Reynolds stresses: p Turbulent scalar flux: uu u Equatons are thus not closed (more unknown varables than equatons) Closure requres specfyng u u and u n terms of the mean quanttes and/or ther dervatves (any Taylor seres decomposton of mean quanttes) Numercal Flud Mechancs (ncompressble, no body forces) ( u u) t x x x PFJL Lecture 5, 1

13 Reynolds Stresses: uu Re Total stress actng on mean flow: If turbulent fluctuatons are sotropc: u u x x Off dagonal elements of Dagonal elements equal: cancel Average of product of fluctuatons not zero Consder mean shear flow: If parcel s gong up (v >0 ), t slows down neghbors, hence u <0 (opposte for v <0) Hence: and u uv 0 for 0 y u u v u w v v w w Re Other meanngs of Reynolds stress: (acts as turb. dffus. ) Rate of mean momentum transfer by turb. fluctatons Average flux of -momentum along -drecton Re u v w u y 0 Numercal Flud Mechancs uu Re u ( y) Academc Press. All rghts reserved. Ths content s excluded from our Creatve Commons lcense. For more nformaton, see PFJL Lecture 5, 13

14 Smplest Turbulence Closure Model Eddy Vscosty and Eddy Dffusvty Models Effect of turbulence s to ncrease strrng/mxng on the mean-felds, hence ncrease effectve vscosty or effectve dffusvty Hence, Eddy-vscosty Model and Eddy-dffusvty Model Re u u u u t k x x 3 Last term n Reynolds stress s requred to ensure correct results for the sum of normal stresses: u 1 u u t k 3 0 u ' v ' w' u ' v ' w' x 3 The use of scalar naccurate u t x Re k, t t assumpton of sotropc turbulence, whch s often Snce turbulent transports (momentum or scalars, e.g. nternal energy) are due to average strrng or eddy mxng, we expect smlar values for t and t. Ths s the so-called Reynolds analogy,.e. Turbulent Prandtl number ~ 1: t t 1 Numercal Flud Mechancs t PFJL Lecture 5, 14

15 Turbulence Closures: Mxng-Length Models Mxng length models attempt to vary unknown μ t as a functon of poston Man parameters avalable: turbulent knetc energy k [m /s ] or velocty u *, large eddy length scale L => Dmensonal analyss: Observatons and assumptons: Most k s contaned n largest eddes of mxng-length L u Largest eddes nteract most wth mean flow Hence, L. Ths s Prandtl s mxng length model. t u y Smlar to mean-free path n thermodyn: dstance before parcel mxes wth others * u u 1 For a plate flow, Prandtl assumed: Ly u y * ln y const. y u Mxng-length turbulent Reynolds stress: * u Mxng length model can also be used for scalars: Numercal Flud Mechancs PFJL Lecture 5, 15 k C t / * u L * C dmensonless constant u u f( u,,, ) x x Re * u n D, mostly xy uv u f( ) cl y u y Re u u xy uv L y y t L u u x y y t t

16 Mxng Length Models: What s L ( ) m? In smple D flows, mxng-length models agree well wth data In these flows, mxng length L proportonal to physcal sze (D, etc) Here are some examples: The McGraw-Hll Companes. All rghts reserved. Ths content s excluded from our Creatve Commons lcense. For more nformaton, see Source: Schlchtng, H. Boundary Layer Theory. 7th ed. The McGraw-Hll Companes, But, n general turbulence, more than one space and tme scale! Numercal Flud Mechancs PFJL Lecture 5, 16

17 Turbulence Closures: k - ε Models Mxng-length = zero-equaton Model One mght fnd a PDE to compute of k and other turbulent quanttes uu and u Re as a functon Turbulence model requres at least a length scale and a velocty scale, hence two PDEs? Knetc energy equatons (ncompressble flows) 1 1 Defne Total KE u u, K 1 and u Mean KE: Take mean mom. eqn., multply by to obtan: u u u e x x Rate of change of K p u e u u u u K ( Ku) u + e e u u t x x x x + Advecton of K = Transport of K by pressure ( p work) + Transport of K by mean vscous stresses + Transport of K by Reynolds stresses - Rate of vscous dsspaton of K - Rate of decay of K due turbulence producton Numercal Flud Mechancs PFJL Lecture 5, 17

18 Turbulence Closures: k - ε Models, Cont d Turbulent knetc energy equaton Obtan momentum eq. for the turbulent velocty u (total eq. - mean eq.) Defne the fluctuatng stran rate: Multply by u (sum over ) and average to obtan the eqn. for Rate of change of k 1. ( ) pu ' e u u u u k ku u + e. e u u t x x x x + Advecton of k = Transport of k by turbulent pressure Ths equaton s smlar than that of K, but wth prme quanttes Last term s now opposte n sgn: s the rate of shear producton of k : P Next to last term = rate of vscous dsspaton of k : These two terms often of the same order (ths s how Kolmogorov mcroscale s defned) e.g. consder steady state turbulence (steady k) + e Transport of k by vscous stresses (dffuson of k) If Boussnesq flud, the KE eqs. also contan buoyant loss/producton terms 1 u u x x Numercal Flud Mechancs + Transport of k by Reynolds stresses - k Rate of vscous dsspaton of k + 1 uu Rate of producton of k uu k e. e ( e e p. u. m) u x PFJL Lecture 5, 18

19 Turbulence Closures: k - ε Models, Cont d Parameterzatons for the standard k equaton: For ncompressble flows, the vscous transport term s: The other two turbulent energy transport terms are thus modeled usng: 1 t p ' u ( u u. u ) k k x t ( t 1) Ths s analogous to an eddy-dffuson of a scalar model, recall: In some models, eddy-dffusons are tensors The producton term: usng agan the eddy vscosty model for the Rey. Stresses u u u u u u u Pk u u t k t x x x 3 x x x x All together, we have all unknown terms for the k equaton parameterzed, as long as e. e the rate of vscous dsspaton of k s known: k ( k u ) k k u u u + t t x x x x x x x x t k t t eu k x x x u t x Numercal Flud Mechancs PFJL Lecture 5, 19

20 Turbulence Closures: k - ε Models, Cont d The standard k - ε model equatons (Launder and Spaldng, 1974) 3 There are several choces for e. e [ ] m / s. The standard popular one s based on the equlbrum turbulent flows hypothess: In equlbrum turbulent flows, ε the rate of vscous dsspaton of k s n balance wth P k the rate of producton of k (.e. the energy cascade): u * u u u Pk u u t O t u / L e. e x x x x Recall the scalngs: * * k u / t C u L Ths gves the length scale and the turbulent vscosty scalngs: L k k 3/ t C As a result, one can obtan an equaton for ε (wth a lot of assumptons): ( u ) t C1Pk C t x x x k k k where and are constants. The producton and destructon terms t C, C 1, C of ε are assumed proportonal to those of k (the ratos ε / k s for dmensons) Numercal Flud Mechancs PFJL Lecture 5, 0

21 Turbulence Closures: k - ε Models, Cont d The standard k - ε model (RANS) equatons are thus: k ( ku) t k k + t e. e t x x k x x x ( u ) t C1Pk C t x x x k k wth t C k The Reynolds stresses are obtaned from: The most commonly used values for the constants are: Two new PDEs are relatvely smple to mplement (same form as NS) Rate of change of k or ε + Advecton of k or ε = Transport of k or ε by eddy-dffuson stresses C 0.09, C 1.44, C 1.9, 1.0, k But, tme-scales for k - ε are much shorter than for the mean flow Other k ε models: Spalart-Allmaras ν L; Wlcox or Menter k ω; ansotropc k ε s; etc. Numercal Flud Mechancs + Rate of producton of k or ε - Rate of vscous dsspaton of k or ε Re u u u u t k x x 3 PFJL Lecture 5, 1

22 Turbulence Closures: k - ε Models, Cont d Numercs for standard k - ε models Snce tme-scales for k - ε are much shorter than for the mean flow, ther equatons are treated separately Mean-flow NS outer teraton can be frst performed usng old k - ε Strongly non-lnear equatons for k - ε are then ntegrated (outer-teraton) wth smaller tme-step and under-relaxaton Smaller space scales requres fner-grds near walls for k ε eqns Otherwse, too low resoluton can lead to wggles and negatve k - ε If grds are the same, need to use schemes that reduce oscllatons Boundary condtons for k - ε models Smlar than for other scalar eqns., except at sold walls Inlet: k, ε gven (from data or from lterature) Outlet or symmetry axs: normal dervatves set to zero (or other OBCs) Free stream: k, ε gven or zero-dervatves Sold walls: depends on Re Numercal Flud Mechancs PFJL Lecture 5,

23 Turbulence Closures: k - ε Models, Cont d Sold-walls boundary condtons for k - ε models No-slp BC would be standard: Hence, approprate to set k = 0 at the wall But, dsspaton not zero at the wall use : At hgh-reynolds numbers: One can avod the need to solve k ε rght at the wall by usng an analytcal shape wall functon : At hgh-re, n logarthmc layer : u u 1 L y u y u ln y const. * * y u If dsspaton balances turbulence k producton, recall: L 3/ ; t C Combnng, one obtans: L y wthout resolvng the vscous sub-layer and one can match: For more detals, ncludng low-re cases, see references k 0 u / k k or n n wall wall u+ = 1 κ lnn+ +B u + = n + κ = 0.41 (von Karman's const.) Logarthmc regon n+ Image by MIT OpenCourseWare. 3/ k u * 3 * wall u Numercal Flud Mechancs PFJL Lecture 5, 3

24 Turbulence Closures: k - ε Models, Cont d Example: Flow around an engne Valve (Llek et al, 1991) k ε model, D ax-symmetrc Boundary-ftted, structured grd nd order CDS, 3-grds refnement BCs: wall functons at the walls Physcs: separaton at valve throat Comparsons wth data not bad Such CFD study can reduce number of experments/tests requred Sprnger. All rghts reserved. Ths content s excluded from our Creatve Commons lcense. For more nformaton, see Please also see fgures 9.13 and 9.14 from Fgs 9.13 and 9.14 from Ferzger, J., and M. Perc. Computatonal Methods for Flud Dynamcs. 3rd ed. Sprnger, 001. Numercal Flud Mechancs PFJL Lecture 5, 4

25 Reynolds-Stress Equaton Models (RSMs) Underlyng assumpton of Eddy vscosty/dffusvty models and of k - ε models s that of sotropc turbulence, whch fals n many flows Some have used ansotropc eddy-terms, but not common Instead, one can drectly solve transport equatons for the Reynolds stresses themselves: uu and u These are among the most complex RANS used today. Ther equatons can be derved from NS For momentum, the sx transport eqs., one for each Reynolds stress, contan: dffuson, pressure-stran and dsspaton/producton terms whch are unknown In these nd order models, assumptons are made on these terms and resultng PDEs are solved, as well as an equaton for ε Extra = 7 PDEs to be solved ncrease cost. Mostly used for academc research (assumptons on unknown terms stll beng compared to data) Numercal Flud Mechancs PFJL Lecture 5, 5

26 Reynolds-Stress Equaton Models (RSMs), Cont d Equatons for uu Re Rate of change of Reynolds Stress Re Re Re ( u ) u u Re u Re u p m m Cm t x x x xm x m x x where + Advecton of Reynolds Stress The dsspaton (as ε but now a tensor) s : The 3 rd order turbulence dffusons are: Smplest and most common 3 rd order closures: 4 Isotropc dsspaton: e. e one ε PDE must be solved 3 3 Several models for pressure-stran used (attempt to make t more sotropc), see Launder et al) The 3 rd order turbulence dffusons: usually modeled usng an eddy-flux model, but nonlnear models also used Actve research = Pressure-stran redstrbuton of Reynolds stresses + Rate of producton of Reynolds stress Numercal Flud Mechancs + Rate of tensor dsspaton of Reynolds stress e. e k k - Rates of vscous and turbulent dffusons (dsspatons) of Reynolds stress C uu. u p' u p' u m m k k PFJL Lecture 5, 6

27 Large Eddy Smulaton (LES) Turbulent Flows contan large range of tme/space scales However, larger-scale motons often much more energetc than small scale ones Smaller scales often provde less transport smulaton that treats larger eddes more accurately than smaller ones makes sense LES: Instead of tme-averagng, LES uses spatal flterng to separate large and small eddes Models smaller eddes as a unversal behavor 3D, tme-dependent and expensve, but much less than DNS Preferred method at very hgh Re or very complex geometry u DNS (A) LES t (A) The tme dependence of a component of velocty at one pont; (B) Representaton of turbulent moton. DNS (B) LES Image by MIT OpenCourseWare. Numercal Flud Mechancs PFJL Lecture 5, 7

28 Large Eddy Smulaton (LES), Cont d Spatal Flterng of quanttes The larger-scale (the ones to be resolved) are essentally a local spatal average of the full feld For example, the fltered velocty s: where G( xx, ; ) wdth Δ s the flter kernel, a localzaton functon of support/cutoff Example of Flters: Gaussan, box, top-hat and spectral-cutoff (Fourer) flters When NS, ncompressble flows, constant densty s averaged, one obtans u x 0 u ( u u ) p u u + t x x x x x Contnuty s lnear, thus flterng does not change ts shape Smplfcatons occur f flter does not depend on postons: u ( x, t) G( x, x; ) u ( x, t) dv V G( x, x; ) G( x x; ) Numercal Flud Mechancs PFJL Lecture 5, 8

29 Large Eddy Smulaton (LES), Cont d LES sub-grd-scale stresses It s mportant to note that Ths quantty s hard to compute One ntroduces the sub-grd-scale Reynolds Stresses, whch s the dfference between the two: It represents the large scale momentum flux caused by the acton of the small or unresolved scales (SG s somewhat a msnomer) Example of models: u u u u More advanced models: mxed models, dynamc models, deconvoluton models, etc. u u SG Mxed eqns, e.g. Partally-averaged Naver-Stokes (PANS): RANS LES Numercal Flud Mechancs u u Smagornsky: t s an eddy vscosty model SG 1 3 SG u u e x x Hgher-order SGS models kk t t PFJL Lecture 5, 9

30 Examples (see Durbn and Medc, 009) Fgures removed due to copyrght restrctons. Please see fgures 6.1, 6., 6., 6.3, 6.6, and 6.7 n Durbn, P. and G. Medc. Flud Dynamcs wth a Computatonal Perspectve. Vol. 10. Cambrdge Unversty Press, 007. [Prevew wth Google Books]. Numercal Flud Mechancs PFJL Lecture 5, 30

31 Examples (see Durbn and Medc, 009) Fgures removed due to copyrght restrctons. Please see fgures 6.1, 6., 6., 6.3, 6.6, and 6.7 n Durbn, P. and G. Medc. Flud Dynamcs wth a Computatonal Perspectve. Vol. 10. Cambrdge Unversty Press, 007. [Prevew wth Google Books]. Numercal Flud Mechancs PFJL Lecture 5, 31

32 Examples (see Durbn and Medc, 009) Fgures removed due to copyrght restrctons. Please see fgures 6.1, 6., 6., 6.3, 6.6, and 6.7 n Durbn, P. and G. Medc. Flud Dynamcs wth a Computatonal Perspectve. Vol. 10. Cambrdge Unversty Press, 007. [Prevew wth Google Books]. Numercal Flud Mechancs PFJL Lecture 5, 3

33 Examples (see Durbn and Medc, 009) Fgures removed due to copyrght restrctons. Please see fgures 6.1, 6., 6., 6.3, 6.6, and 6.7 n Durbn, P. and G. Medc. Flud Dynamcs wth a Computatonal Perspectve. Vol. 10. Cambrdge Unversty Press, 007. [Prevew wth Google Books]. Numercal Flud Mechancs PFJL Lecture 5, 33

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