School of Mechanical Aerospace and Civil Engineering Unseady Flow Problems T. J. Craf George Begg Building, C41 TPFE MSc CFD-1 Reading: J. Ferziger, M. Peric, Compuaional Mehods for Fluid Dynamics H.K. Verseeg, W. Malalaseara, An Inroducion o Compuaional Fluid Dynamics: The Finie Volume Mehod S.V. Paanar, Numerical Hea Transfer and Fluid Flow Noes: hp://cfd.mace.mancheser.ac.u/mcfd - People - T. Craf - Online Teaching Maerial Overview In sudying he discreizaion of fluid flow problems so far we have considered only seady-sae siuaions. This has mean we have only discreized spaial derivaives. Some examples of unseady flows: Flow Through a Tigh Pipe Bend Very high Re number and rough pipe wall lead o large-scale unseady srucures in he flow. Separaed Flow wih Periodic Forcing Bacward-facing sep, wih periodic injecion/sucion a he sep corner. Unseady Flow Problems 2010/11 2 / 30 In his lecure we consider how o handle unseady flow problems, and any addiional complicaions his migh inroduce. The Navier-Soes momenum equaions for incompressible flow) now become U i + U iu j) = 1 P + ν U ) i 1) x j ρ x i x j x j When considering how o discreize he ime derivaives, we need o recognise a major difference beween space and ime coordinaes. Mos real fluid flow problems are a leas o some exen) ellipic in naure meaning ha any forcing inroduced has an effec in all spaial direcions. However, he ime dependence is purely parabolic any forcing affecs only he fuure, never he pas. To reflec his, soluion mehods end o advance in ime in a marching manner. Iniial values are provided a ime = 0 and he soluion mehod proceeds by advancing in ime in a sep-by-sep manner. Unseady Flow Problems 2010/11 3 / 30 Explici vs. Implici Treamens We consider he generic ime-dependen problem = f,)) 2) Inegraing wih respec o ime, over one ime sep, gives or + + d = f,))d 3) + n+1) = n) + f,))d 4) where he superscrip n) denoes a quaniy evaluaed a ime, and n+1) a ime +. The problem is hen how o approximae he inegral on he righ hand side. Unseady Flow Problems 2010/11 4 / 30
If we approximae he inegral using he value of f a he iniial ime,, hen we obain f) Anoher alernaive is o approximae he inegral using a weighed average of he iniial and final values of f. f) n+1) = n) + f, n) ) 5) This is a fully explici mehod. Once we have values a ime sep n, equaion 5) can be applied in a simple explici fashion o obain he value of a ime sep n+1. + In he case of an equally weighed average he inegral is evaluaed using he rapezium rule: n+1) = n) + 2 [ ] f, n) )+f +, n+1) ) 7) + If he inegral is approximaed using he value of he inegrand a he final ime, +, hen we have n+1) = n) + f +, n+1) ) 6) Again, his is an implici scheme, and will ypically require an ieraive procedure o obain n+1). This is a fully implici mehod. Values of n+1) appear on boh he lef and righ hand sides of he equaion. In a general case i will ofen require an ieraive soluion procedure o obain n+1). Unseady Flow Problems 2010/11 5 / 30 Unseady Flow Problems 2010/11 6 / 30 Accuracy of Temporal Schemes By expanding as a Taylor series expansion abou ime we can examine he accuracy of he fully explici scheme: ) + ) = )+ + )2 2 ) 2! 2 + O ) 3 ) Hence we have = )+f,)) + O ) 2 ) n+1) = n) + f, n) ) + O ) 2 ) 8) From his we can see ha equaion 5) shows a runcaion error proporional o ) 2 over a single ime sep. However, he number of ime seps ha mus be performed for a given simulaed ime is proporional o 1/. Hence he oal error associaed wih use of he scheme is proporional o, and he scheme is referred o as firs order accurae. Unseady Flow Problems 2010/11 7 / 30 The fully implici scheme of equaion 6) can also be shown o be firs order accurae. The rapezium rule approximaion of equaion 7) can be shown o be second order accurae, since + ) = + /2)+ ) + )2 /4 2 ) 2 + /2 2! 2 + O ) 3 ) + /2 9) ) = + /2) ) + )2 /4 2 ) 2 + /2 2! 2 + O ) 3 ) + /2 10) Hence n+1) = n+1/2) + 2 f + /2,n+1/2) )+ )2 /4 2 ) 2! 2 + O ) 3 ) + /2 11) n) = n+1/2) 2 f + /2,n+1/2) )+ )2 /4 2 ) 2! 2 + O ) 3 ) + /2 12) Unseady Flow Problems 2010/11 8 / 30
Combining hese expansions gives n+1) n) = f + /2, n+1/2) )+O ) 3 ) 13) The approximaion ) f + /2, n+1/2) ) 0.5 f, n) )+f +, n+1) ) 14) Sabiliy of Temporal Schemes Anoher imporan issue wih emporal discreizaion schemes is ha of sabiliy. Explici schemes in paricular are prone o insabiliies if oo large a ime sep is employed. is a second order accurae cenred approximaion. The runcaion error over a single ime sep is hus proporional o ) 3, so over a fixed ime inegraion he scheme is second order. To see why his should be, consider a 1-D spaial, ime dependen problem, dominaed by convecive effecs. Zone of Influence The flow condiions a some poin xi influence an increasingly wide region of he flow as ime progresses. x i x Equally, we can idenify a region of he flow ha has an influence on condiions a he poin x i a any ime. Zone of dependence x i x Unseady Flow Problems 2010/11 9 / 30 Unseady Flow Problems 2010/11 10 / 30 In a fully explici scheme, since he value of a ime n) is deermined purely from is values a ime n 1), we need o ensure ha he spaial sencil does conain all he relevan flow informaion a ime n 1). For example, if a 3-poin spaial sencil is employed, hen for he problem shown i covers he zone of dependence a ime, so he scheme could be expeced o be sable. If, however, a ime sep of 2 were aen, no all he relevan flow informaion a ime 2 would be included in he discreizaion, and he scheme migh be unsable. 2 Hence sabiliy ofen imposes a limi on he ime sep. The sabiliy of differen schemes will be examined in he following secions when applying hem o unseady convecion/diffusion problems. Unseady Flow Problems 2010/11 11 / 30 x Time Sepping for Seady Problems In some cases he soluion o a seady flow problem is obained by inroducing a ime derivaive erm and solving he unseady problem, marching he soluion o a seady sae. In such cases he accuracy of he ime discreizaion is no crucial since he ime derivaive will be zero a he end of he calculaion), and one ypically wans o ae raher large ime seps. The sabiliy of he scheme is herefore paricularly imporan in hese cases, since he ime-sep size ha can be applied will be limied by his, raher han by accuracy consideraions. In a soluion procedure such as he above, he ime-sepping is effecively playing a similar role o under-relaxaion, which is normally used in solving coupled sysems in a segregaed fashion. In fac, when used in his conex, one does no even have o use he same ime-sep a each spaial locaion, and local ime-sepping can be employed. Unseady Flow Problems 2010/11 12 / 30
The Explici Euler Scheme For simpliciy we consider he ime dependen convecion/diffusion problem for some variable : + ) Uj = Γ ) x j x j x j 15) The explici Euler mehod is obained by moving all erms excep / o he righ hand side and applying he fully explici formulaion: = ) Uj + Γ ) f,)) 16) x j x j x j The ime discreizaion hen resuls in he approximaion + ) = )+f,)) 17) As we shall see, sabiliy of he scheme depends boh on he ime sep size and how we approximae he spaial derivaives in he funcion f. Unseady Flow Problems 2010/11 13 / 30 If we consider he 1-D problem wih consan velociy U and diffusiviy Γ hen he differenial equaion becomes = U x + Γ 2 x 2 18) Using cenral differences o approximae spaial derivaives hen leads o [ n) n+1) i = n) i+1 i + U ) n) n) i+1 + Γ 2n) i + n) )] x) 2 19) The compuaional sencil involves hree poins a ime level n and one a level n+1. Given iniial condiions he calculaion can hen proceed, evaluaing n+1) i in a simple poinwise manner. i i+1 Unseady Flow Problems 2010/11 14 / 30 + Sabiliy Analysis As indicaed earlier, he sabiliy of a emporal discreizaion is ofen an imporan issue, paricularly wih explici schemes. Broadly, we inerpre a sable scheme o be one ha leads o a bounded soluion if he original differenial equaion would have done so. A mehod ha is frequenly used o explore sabiliy issues for linear problems is he Von-Neumann sabiliy analysis. If one assumes here are no forcing erms, hen he discreized equaion for can be wrien in he general form n+1) = A n) 20) If we also assume ha he eigenvecors of A form a basis, hen can be wrien in erms of hese: n) = ξ n) a 21) where a are he eigenvecors of A and ξ n) he associaed coefficiens. Unseady Flow Problems 2010/11 15 / 30 The difference beween soluions for a successive ime seps using some appropriae norm) is hen ) n+1) n) = ξ n+1) ξ n) a 22) Using he discreized equaion o relae his difference o ha a he previous imesep leads o n+1) n) = A n) n 1) ) = A ξ n) = α ξ n) where α is he eigenvalue corresponding o eigenvecor a ξ n 1) )a ξ n 1) )a Hence he difference beween he soluions a successive imeseps ges smaller corresponding o a sable soluion) if all he eigenvalues of he marix A are less han uniy. Unseady Flow Problems 2010/11 16 / 30
For he simplified problem being considered, on a uniform grid, he eigenvecors can be expressed as combinaions of Fourier modes wih sines and cosines. Hence o analyse sabiliy we consider as being represened by a combinaion of Fourier modes of he form n) j = ξ n expij x) 23) where i = 1, he index j denoes he spaial posiion and superscrip n) he ime level. We hen examine wheher he complex) coefficiens ξ associaed wih he Fourier modes are less han or greaer han uniy in magniude. For sabiliy heir magniudes should be less han uniy. To examine he sabiliy of he explici scheme oulined earlier, we hus subsiue he form ξ n expij x) ino he discreized equaion 19), leading o [ ] expi x) exp i x) ξ n+1 = ξ {1 n U [ ]} expi x) 2+exp i x) +Γ x) 2 24) which gives ξ = 1 U Γ [expi x) exp i x)]+ [expi x)+exp i x) 2] x) 2 = 1+2Γ / x) 2 )[cos x) 1] iu / x)sin x) maing use of he ideniies for complex numbers: i sinx)=0.5expix) exp ix)) and cosx)=0.5expix)+exp ix)) Since ξ is complex, is magniude is given by ξ 2 = 1+ 2Γ ) 2 ) U 2 [cos x) 1] + x) 2 x sin x) 25) Unseady Flow Problems 2010/11 17 / 30 Unseady Flow Problems 2010/11 18 / 30 Two limiing cases can be considered: In he absence of diffusion, we have ) U 2 ξ 2 = 1+ x sin x) 26) so ξ is always greaer han uniy and he scheme is uncondiionally unsable! In he absence of convecion we have ξ 2 = 1+ 2Γ ) 2 [cos x) 1] 27) x) 2 The maximum value of ξ hus occurs when cos x) = 1, when we have ξ 2 = 1 4 Γ ) 2 28) x) 2 Thus ξ < 1 provided ha Γ / x) 2 < 1/2, and he scheme is condiionally sable. Unseady Flow Problems 2010/11 19 / 30 Anoher way of examining sabiliy and boundedness of his problem is o wrie he discreized equaion in he form n+1) i = 1 2Γ ) x) 2 n) Γ i + x) 2 U ) n) Γ i+1 + x) 2 + U ) n) 29) Suppose ha represens some scalar quaniy ha canno physically be negaive. If any of he coefficiens associaed wih n) i, n) i+1 or n) ae a negaive value hen i is possible ha n+1) i could become negaive, leading o an unbounded soluion. To ensure all coefficiens are posiive requires and 1 2Γ x) 2 > 0 giving Γ x) 2 < 1/2 Γ x) 2 U > 0 giving U x Γ < 2 Unseady Flow Problems 2010/11 20 / 30
The firs of hese condiions is he same as found by he Von-Neumann analysis. The second is ha he cell Pecle number should be less han 2. This is, in fac, a condiion ha can be found for he cenred convecion scheme o be bounded in seady convecion/diffusion problems. The resricion Γ / x) 2 < 1/2 can impose a raher serious consrain on he ime sep for pracical calculaions. If he grid spacing is halved, for example, he ime sep has o be reduced by a facor of four. Upwinding Convecion Terms If we reain explici Euler ime discreizaion, bu now use he firs order upwind scheme o discreize convecive erms, he discreized form of he above unseady convecion/diffusion problem assuming U > 0) becomes n+1) i = n) i U n) i n) x ) + Γ n) i+1 2n) i + n) x) 2 ) 30) or n+1) i = 1 2Γ x) 2 U ) n) x i + Γ Γ x) 2 n) i+1 + x) 2 + U ) n) x 31) To ensure he coefficiens remain posiive, we now need 1 2Γ x) 2 U x > 0 giving < 1 2Γ)/ x) 2 + U/ x For he case of negligible diffusion his limi becomes U x < 1 or < x U Unseady Flow Problems 2010/11 21 / 30 Unseady Flow Problems 2010/11 22 / 30 The Von-Neumann analysis can also be applied o his case, leading o he same limi on. The quaniy U / x is nown as he Couran Number, and represens he raio of he ime sep o he ime required for flow informaion o be conveced across he cell. Thus, for a purely convecive flow, he inroducion of upwinding for he convecive flux leads o condiional sabiliy of he explici Euler mehod. Sabiliy is achieved provided he Couran number is less han uniy. Noe ha his condiion implies ha should be sufficienly small so ha a fluid elemen canno ravel more han one grid cell lengh in each ime sep. I is worh noing ha he exac sabiliy condiions obained above relae o he raher simplified problem currenly considered. In real fluid flow problems he precise crieria may be differen, alhough he limis found above are generally good guidelines. Unseady Flow Problems 2010/11 23 / 30 Unseady Convecion/Diffusion Example The 1-D unseady convecion/diffusion problem T + U T x = T Γ 2 x 2 on he inerval 0 < x < 1 wih U = 0.2 and Γ = 0.2. Iniial condiions are { 1 for 0.3 < x < 0.7 T = 0 oherwise Boundary condiions are T/ x = 0 a x = 0 and 1. Cenral spaial differencing. Compued profiles of T shown a = 0.0, 0.18, 0.36, 0.54 and 0.72. Unseady Flow Problems 2010/11 24 / 30
The Implici Euler Scheme One drawbac of explici schemes is ha sabiliy consrains can ofen resul in very small ime seps having o be used. This is a paricular problem in flows where he ime scales are fairly large, or where ime sepping is used o march o a seady sae soluion and an accurae resoluion of he ransien sage is no required). If he convecion and diffusion erms and oher sources) are evaluaed a he new ime level n+1, we arrive a he implici Euler mehod. For he unseady convecion/diffusion problem, wih cenral differences for convecive fluxes, he discreized equaion now becomes [ n+1) n+1) i = n) i+1 n+1) ] [ n+1) i+1 2 n+1) i + n+1) ] i U + Γ x) 2 32) or, on rearranging: [ 1 + 2Γ ] x) 2 n+1) i [ Γ x) 2 U ] [ n+1) Γ i+1 x) 2 + U ] n+1) = n) i 33) Unseady Flow Problems 2010/11 25 / 30 Noe ha his resuls in a ri-diagonal sysem, similar o ha for he equivalen seady sae problem, bu wih an addiional posiive) erm in he diagonal elemen and he addiional source erm n) i /. Since he inroducion of he ime discreizaion does no aler he sign of he coefficiens, he soluion is uncondiionally sable alhough, as in he seady case, he use of cenral differencing for convecion can lead o oscillaions if he cell Pecle number is oo large). Von-Neumann analysis can be used o confirm he scheme s sabiliy. One drawbac of he scheme is ha one has o solve a large coupled sysem of equaions a each ime sep. Since here will ypically be a large number of ime seps, employing an efficien solver is crucial. An advanage of he scheme is ha if marching o a seady sae soluion, very large ime seps can be used in a sable fashion. However, if a ime accurae soluion is required, accuracy consideraions may sill lead o a raher small ime sep because of he firs order runcaion error. Unseady Flow Problems 2010/11 26 / 30 The Cran-Nicolson Scheme When considering generic emporal discreizaions we noed ha approximaing he ime inegral beween and + via he rapezium rule resuled in a second order scheme. Such a scheme applied o parial differenial equaions is nown as he Cran-Nicolson Scheme. For he unseady convecion/diffusion problem, wih cenral differencing for convecive erms, we now ge n+1) i = n) i U n+1) i+1 n+1) ) + Γ n+1) i+1 2 n+1) i + n+1) ) 2 2 x) 2 U n) i+1 n) ) + Γ n) i+1 2n) i + n) ) 2 2 x) 2 or 1 + Γ ) x) 2 n+1) i U 4 x + Γ ) ) 2 n+1) U i+1 4 x + Γ ) ) 2 n+1) 1 = Γ ) x) 2 n) i + U 4 x + Γ ) ) 2 n) U i+1 + 4 x + Γ ) ) 2 n) Unseady Flow Problems 2010/11 27 / 30 The scheme is again implici, so requires a marix sysem soluion a each ime sep. Von-Neumann analysis shows he scheme o be uncondiionally sable, alhough in pracice i can produce oscillaions or become unsable for raher large ime seps. However, he second order accuracy means ha larger ime seps can usually be aen han wih he firs order schemes, whils reaining accepable emporal accuracy. Unseady Flow Problems 2010/11 28 / 30
Unseady Convecion/Diffusion Example Considering he same problem as in he earlier example, bu now using Implici Euler or Cran-Nicolson ime schemes. Boh ime schemes applied wih cenral differencing for spaial derivaives. Recall ha he explici scheme was unsable by = 0.065. Implici Euler Cran-Nicolson Unseady Flow Problems 2010/11 29 / 30 Oher Time Discreizaion Schemes The schemes presened above are no he only ones available in many CFD codes, alhough hey are some of he more widely-used ones. They do also illusrae many of he consideraions regarding accuracy and sabiliy ha should be borne in mind when choosing which scheme o use for a paricular problem. Higher order schemes can be consruced by employing a larger ime sencil for example by evaluaing erms a ime levels n+1, n and n 1. In some cases addiional sabiliy can be gained by including influences from more han wo ime levels. However, he requiremen o sore more ime levels does increase he overall sorage and hence memory) requiremens. In mos general purpose finie volume schemes he underlying approximaions are second order, so here may be lile poin in using a emporal discreizaion of higher order han his. However, for some applicaions or purposes i can be appropriae. Unseady Flow Problems 2010/11 30 / 30