Some recent development of the numerical simulation methods for CFD Liu Ruxun Dept.of Math., USTC, Hefei Anhui 230026, China liurx@ustc.edu.cn What is Computational Fluid Dynamics (CFD) ? CFD is the systematic application of computing systems and computational solution techniques to mathematical models formulated to describe and simulate fluid dynamic phenomena. Simulation is used by engineers and physicists to forecast or reconstruct the behaviour of an engineering product or physical situation under assumed or measured boundary conditions (geometry, initial states, loads, etc.). The importance simulation techniques have great developed in recent decade years: 1.Research of models is the foundation [29] Quecedo M. et al, Comparison of two mathematical models for solving the dam break problems using the FEM method, Comput. Methods Appl. Mech. Engrg.,194(2005)3984-4005 Adopted a wrong model!! 2.Advances in solution algorithms 3.Mathematical analysis (classic and numerical analysis, discrete mathematics) 4.Computer Science (algorithms, coding, software) 5.Visualization Techniques Lab for Computational Fluid Dynamics May 28, 2004 Content 1.Introduction 2.Some classical methods ? 2-1.Donor and Acceptor ? 2-2.Harlow and Welch’s MAC (Marker and cell),PIC,FLIC ? 2-3.Leonard’s QUICK (quadratic upstream interpolation for convective kinetics) and Simple ? 2-4.van Leer’s MUSCL (monotonic upstream scheme for conservation law) ? 2-5.Collela’s PPM (piecewise parabolic method) ? 2-6.Harten’s TVD (total variation diminishing schemes) 3.Recent development of numerical simulation method ? 3-1.ENO (essentially non-oscillatory schemes) and weighted ENO ? 3-2.FVM (finite volume methods) with unstructured meshes ? 3-3.Rational approximation methods, high order compact and Pade schemes ? 3-4.CIP (cubic interpolated propagation methods) ? 3-5.VOF (volume of fluid) and Level Set methods for tracking moving- interface ? 3-6.DG (Discontinuous Galerkin finite element methods) ? 3-7.LBM (Lattice Boltzmann method ) ? 3-8.SPH (smoothed particle hydrodynamics)and meshless methods. ? 3-9.Software:Fleunt, Phoenics,Star-CD,CFX,and so on. 1.Introduction In recent years, the numerical methods of subtly simulate fluid dynamic phenomena have been advanced quickly and have succeeded in various fluid dynamics applications. In the short paper, only some important and effective new approaches will be introduced. Some methods, such as moving FEM, BEM, moving grid methods, spectral method, LES, multi-scale method and so on, isn’t able to be discussed. 2.Some classical methods we review some classical numerical methods in order to discuss recent methods and developments easily. 2-1.Donor and Acceptor methods ? Consider the numerical flux scheme (8) of the 1D shallow water equations in the cell and the neighboring cell .The numerical flux at the discontinuous (joint) point can be reconstructed by judging which is the donor- or acceptor-cell between the two cells ? (1) ? The reconstruction approach is called donor-acceptor method which has obvious mechanics character. 1 1/2 1/2 1/222 22 11 1 ( ) , ( ) , () 0 0 (), ), ii ii i hu hu FU asu or asu hu gh hu gh + ++ + + ??? ? = >< ??? ? +? ??? ? 1/2 1/2 [,] ii i Ix x ?+ = 11/23/2 [, ] iii Ixx +++ = 1/2 () i FU + 11/2 xx + = 2-2.Harlow&Welch’s MAC (Marker and cell) ? PIC (particle in cell,Evan and Harlow,1957),MAC (Marker and cell, Harlow and Welch,1965), FLIC (Fluid in cell, Gentry,Martin and Daly,1966), ALE (Arbitrary Lagrange and Euler). ? MAC method : Mmarker technique .By tracking these markers based on the velocity-field of flow, we can finely numerically simulate the free- surface of moving interface. ? Tracking markers----------Lagrange-computation. or (2) () (, ,) () (, ,) m m dx t uxyt dt dy t vxyt dt = = 1 1 11 11 ((), (),) ((), (),) n n n n t nn mm t t nn mm t xx uxtyttdt yy vxtyttdt + + ++ ++ =+ =+ ∫ ∫ 2-3.Leonard’s QUICK (quadratic upstream interpolation for convective kinetics) Leonard (1979) used a three-point upstream- weighted quadratic interpolation to construct the numerical flux at the discontinuous point (the cell interface) . (3) and Patanka and Spalding’s SIMPLE (Semi-Implicit Method for Pressure-Linked Equation,1972) 1/2 () i FU + 11/2 xx + = 1/2 1/2 ()[()] ii FU AU U ++ = ? 1 2 1 2 1 2 63 1 1188 8 631 12888 , 0 , 0 ii i i i iii i UU Uu U UUUu +? + + ++ + + ?> ? ? = ? + ?< ? ? 2-4.van Leer’s MUSCL (monotonic upstream scheme for conservation law) van Leer (1979) uses the approximations at the time level to directly reconstruct (4) the 2nd polynomial approximation of the integrand of above integral based on characteristics property. The third order scheme has the lateral values of the cell interface ( is a third order MUSCL scheme) (5a) In order to restrain the oscillations by inserting a flux limiter is effective strategy, i.e. the scheme (5a) should be replaced as (5b) {} n ii U ? n tt= 1 1 1 (, ) i n in I UUxtdx x + + = ? ∫ 1 1/2 14 1 1/2 1 14 [(1 ) (1 ) ] [(1 ) (1 ) ] L ii i i R ii i i UU U U UU U U κκ +? + ++ =+ ?? ++? =?+?+?? 1 1/2 14 1 1/2 1 14 [(1 ) (1 ) ] [(1 ) (1 ) ] L ii i i R ii i i UU U U UU U U κκ +? + ++ =+ ?? ++? =?+?+?? null null 11/2 xx + = 1 3 κ= 2-5.Collela’s PPM (piecewise parabolic method) Collela and Woodward (1984) proposed PPM by piecewise parabolic polynomial interpolation to the definition (24). For example, the PPM scheme of a scalar equation will be (6) Fig.6 The reconstruction character of left and right limit values forMUSCL and PPM 1 2 11 22 11 22 ,6, ,, 1 6, , ,2 ( ) ( (1 )), lim ( ); lim ( ) 6( ( )) ii i Li i i ii iRiLi Ri Li xx xx n ii RiLi xx vx u u u x x x x uu u xuux uuuu ξξξ +? ? ? + ↑↓ ? ? =+?+ ? = ≤≤ ? ? ? ? ?= ? ? ==? ? ? =? + ? piecewise linear PPM Collela &Woodwar d (1984) piecewise parabolic MUSCL van eer (1979) 2-6.Harten’s TVD (total variation diminishing schemes) Harten (1983) first introduced TVD scheme including Limiter Contribution: TVD character; Reconstruction: Limiter TVD:(7) [Theory] For scalar conservation law ,(8 a scheme consistent with it can be written as (9) if the following conditions are satisfied (0) it is also a TVD one. Monotone scheme is TVD. A TVD scheme is monotonicity preserving. 1 ( ) (), () nnn n i i TV U TV U TV U x U + ≤=? ∑ ( ()) 0, () ()/ tx u f u a u f u u Jacobian matrix+ ==?? 1 1/2 1 1/2 1 ()() nn nn nn iiiii iii uuCuuDuu + ??++ =? ? + ? 1/2 1/2 1/2 1/2 0, 0 01 ii ii CD CD ++ ++ ≥≥ ? ? ≤+≤ ? 3.Recent development of the numerical simulation method for CFD ? The numerical test and study on the shallow water wave problems is one of the most active topics in computational hydraulics. Using numerical simulation and numerical analysis, taking the suitable simplified model, scientists have got a lot of significant information for various complicated shallow water wave phenomena. ? In recent years, especially, many impressive and wonderful numerical simulated results for 2D or 3D discontinuous problems are continually reported. ? In the section, several efficient simulation methods will be introduced. 3-1.ENO (essentially non-oscillatory schemes) and weighted ENO The polynomial interpolation is the foundation of most numerical methods although it can introduces spurious oscillations. Preserve the monotonic character for designing finite difference schemes is the key to avoid introducing spurious oscillatory mechanism. Harten et al (1998) developed the reconstruction methods and found the procedure to overcome spurious oscillation. ENO, especially weighted ENO basically realized the object by controlling the scale of every Newton difference-quotients and selecting suitable node-stencils during the process heightening the order of the polynomial. 1. ENO method Consider the initial value problem of the scalar conservation law (11) Numerical-flux scheme: . (12) ENO method adopts a Newton polynomial to reconstruct the numerical flux : 1). Choose a initial two-cell stencil 2). Extend the stencil to left and right side to construct new stencils 3). Compute the two quotients corresponding two cell-stencils and compare them And choose the one having min-absolute-value and expand Newton polynomial 4). Substitute and calculate the value 0 (() 0, , 0; () ()/ (.0) () tx ufu xR t aau fuu ux u x + =∈>==?? = 11 22 1 [( ) ( )] nn ii ii uurfu fu + +? =? ? 1 1 {, } ii SII + = 1 2 () i f u + 22 11 12 {}{, }; { , }{ } Li ii Rii i SI IISII I ?+ ++ =∪ = ∪ 11 1 1 22 22 11 12 1 11 21 (, ) ( ,) ( , ) (, ) () ; () ii i i i i ii LL RR ii i i Nxx Nx x Nx x Nxx NS NS xx xx +? ++ + +? ++ ?? == 12 11 1 ( ) ( , )( ) ( )( ) iiii i i i i p xuNxx xx Nxxxx + ?+ =+ ? + ? ? 1/ 2i xx + = 1/2 1/2 1/2 1/2 (), ()(() nn ii i i uPx fu fPx ++ + + == 2. Weighted ENO methods Weighted ENO saves all of possible cell-stencil and combines their corresponding effects to construct a weighted approximation polynomial. For example: for 3 three-cell stencil case (13) 1) Construct the three Newton polynomials by means of consistency conditions (14) 2) Calculate the values (15) 3) Weighted plus to get the numerical flux and (16) 222 (1) 2 1 (2) 1 1 (3) 1 2 { , , }, { , , }, { , , } iii iii iii SIIISIIISIII ?? ? + ++ === () () 2 () 0 1 2 1 () , ( ) , j rjr I i r i Pxdxu jS x x x Paaa x ξξξξ =∈ ? ? =+ + = ? ∫ () r Px 1 2 1 2 1 2 2() (1) 2 1 2 1 (2) 1 1 1 1 2(3) (3) 1 2 1 2 171 {,,}, 366 151 {,,}, 663 15 1 {, , }, 36 6 iii i i i i iii i i i i ii i i i i i SIIIu u u u SIII u u uu SIII u uu u ?? ? ? + ? +?+ + ++ + + + == ==? 11 22 3 () 1 nr r ii r uwu + + = = ∑ 1/2 n i u + 1 2 () i f u + Fig.1 The numerical results of circle dam-breaking by WENO (Z.L.Xu&R.X.Liu) 3-2.FVM (finite volume methods) with unstructured meshes Consider 2D shallow water equations (in fact, depth-averaging 3D inviscid incompressible flow considering Coriolis force, and the change of river-bed ) (17a) or their compact form (17b) Integrating above equations over a control volume of unstructured mesh (Fig.1) and using Green Theorem, we can obtain (18) 22 1 2 22 1 2 ( ( )) ( ( )) 0 ,() ,() , b b txy z x z y UFU GU S hhu hv U hu F U hu gh G U hvu S gh fvh hv huv hv gh gh fuh ? ? ? ? ++= ?? ???? ? ? ?? ???? ? ? ==+ = =?+ ?? ? ? ?? ? ? ??+ ?? ?? ? ? ?? ?? () , () ((),()) t UFUSFUFUGU ?? +?? = = 1 ()|| || k i ki i kl k V lV V i dU Fn l Sdv dt V ? ∈? =? ? + ∑ ∫ (0, 200) (100, 200) (200, 200) (100, 170) (100, 95) (0, 0) (100, 0) (200, 0) Flow 0 50 100 150 200 0 25 50 75 100 125 150 175 200 6 8 10 0 50 100 150 200 0 50 100 150 200 0501015020 0 25 50 75 100 125 150 175 200 9 .2 1 8 .8 5 8.49 8 .1 4 8 .1 4 7.78 7 .4 2 7. 06 6.7 6. 34 5. 27 6. 3 4 5.99 5 .6 3 5 .2 7 5.63 5. 27 1 .25 1.5 .75 2 0 15 30 45 60 75 90 0 10 20 30 40 0 102030405060708090 0 10 20 30 40 1 . 0 6 1 .1 2 1 .3 6 1 . 7 9 1 . 9 2 1 . 6 7 1.55 1 .5 5 1.55 1.61 1.67 1.67 1.61 1 .6 7 1 .6 7 1.36 1 .4 2 1. 42 1.36 1 .4 2 1 .6 1 Fig.2 The numerical simulation results by FVM (J.W.Wang&R.X.Liu) 3-3.Rational approximation methods, high order compact and Pad schemes The Runge (gibbs) phenomeno of polynomial interpolation 2 1 () , [5,5] 1 fx x x =∈? + S.K.Lele (1992, J. Comput.Phys.,Vol.103,pp16-42) High order compact- or Pade finite difference scheme. Let (19) By Taylor expansion one can obtain the relations between the coefficient relations (20) or (21) 11 2 2 33 11 2 2 33 222 22 11 246 22 11 49 ()() ()() ii i i ii iii i ii iii ff f f ff ii ii i xxx f ff f ff f ff ii ii i xxx ff ff fa b c ff ff fa b c βα +? +? +? + ?+?+? ?? +? +? ??? ?+ ?+ ?+ +? +? ??? ′′ ′′′ ++ += + + ′ ′ ′′ ′′ ′′ ++ += + + (1)! ! 12 2 232(2), (2) mmm m m abc a b c m order approximation error α β αβ + ++=+ + ++= + +? () 1 11 , , , T Nx AF BF F f f ?? ′==null () 1 11 , , , T Nx AF BF F f f ?? ′′ ==null Fig.3 Steady-state stream and vorticity distributions for lid-driver cavity problem at Re=100(left) and 1000 (right) simulated by Compact method 3-4.CIP (cubic interpolated propagation methods) CIP-method is developed by Yabe and Aoki (1991). It is a semi-Lagrange type method using Hermite interpolation reconstruction for convective problems or conservation laws. Consider the following convective equation and it’s gradient equation (22) Take the two-node stencil and give the data at the time level, the Hermite interpolation on the cell (23) Obviously, the unknown value at thetime level can be got based on the characteristics relationship (24) 0; , f fffuf uuf tx t x x x ′ ′?? ?? ? ? ′′+= +=? = ?? ?? ? ? 1 23 01 2 3 011 12 1 1 3 1 1 () () , , , 3 (2 + ), ( + )-2 i xx x iiiii iii Px H a a a a a f a xma u xmm a xmm u ξξξξξ ? ? ? ???? ?? ==+++ = ==? =?? =? ? 1 ()(1), / n ii f Px u t H c c u t x + = ??= ? =? ? Fig.11 The evolution of distribution f Fig.4 The evolution of distribution of Vlasov equation and the logarithm of the density by CIP method Fig.12 CIP simulation results for Zelasak problems (R.Zhang & R.X.Liu) 3-5.VOF (volume of fluid) and Level Set methods for Tracking Moving-Interface VOF (Volume of Fluids) method (Hirt and Nichols,1981) and Level Set method (Osher and Sethian,1988) To numerically simulate the rising-out and breaking-up of gas bulb in fluid, wave or dam breaking, etc, is great significance to comprehension and research of many physical phenomena. Especially, to numerically research of developing process for the interior, microcosmic structure and character of these phenomena is a more practical approach. The mathematical models are composed by two parts 1). Main-field governing model (considering the tension due to the moving interface) (25) () () 1 2|| ()0 , 2 , ( ), = , T C C V t V VV g f f C t pI D D u u n n σσ ρ ρ ρ ρτρ σκ τμ κ ? ? ? +?? = ? ? +?? ? =? + =? ? ? =+ =?+? ?? = 2). The interface governing equation will be either the fluid volume function equation (26) or the level set function equation (27) [Comment] Moving-interface problems in hydrodynamics usually are some kind of multi-value problems, therefore SWE-model isn’t appropriate. ( ) 0, cell Caimgfluid in cell VC C t τ τ ? ? +??= = ? ( , (0)), ; 0, ( ,0) 0, (0); ( , (0)), . dx x Vx t dx x ? ?? + ? ? Γ∈? ? ? +??= = ∈Γ ? ? ? ?Γ ∈? ? nullnull nullnull nullnull ? The numerical simulation procedure is composed by alternate computing processes of both main-field computing and moving-interface computing with reconstruction- or re- initialization. Fig.13 Time evolution of rising bubble problems (Pan D. and Chang C.H.) Fig.15 Breaking waves by particle level set method (D.Enright et al, Using the particle level set method and a second order accurate presssue boundary condition for free surface flows) Fig.16 Drop impact onto liquid pool (D.Enright et al, Using the particle level set method and a second order accurate presssue boundary condition for free surface flows) Fig.16 Pouring water into a cylindrical glass using the particale level set method (D.Enright,D.Fedkiv,et al,A hybrid particle level set method for improved interface capturing, JCP,183-1(2002)83-116 ) Fig.17 Numerical results of 3D wave breaking by coupling by a VOF method and BEF method at x-profile (B.Biausser et al., Proceedings of the Thirteenth Inter. Offshore and Polar Engineering Conference,2003) 3-6.Lattice Boltzmann method ? Lattice Boltzmann method(LBM) has caused considerable attention recently (Chen S.and Doolen G. D.,1998). ? This method can be either considered as an extension of the lattice gas automaton or as a special discrete form of the Boltzmann equation for kinetic theory. The LBM is based on the statistical physics and describes the microscopic picture of particle-movement in an extremely simplified way, but on the macroscopic level it gives a correct average description of a fluid. It is parallel in nature due to the locality of particle interaction and the transport of particle information, so it is well suited to massively parallel computing. The single-relaxation-time or BGK Boltzmann model (28) : distribution function per particle, : micro-velocity and Boltzmann-Maxwellian distribution function (29) : number of space dimension, the macro-density, velocity and temperature. Integrating (56) along the characteristics we have (30) By the 2 nd order Chapman-Enskog expansion and neglecting 2 nd order remainders one can obtain common Navier-Stokes equations (31) , Df f f g f relaxation time Dt t ξλ λ ? ? ≡+??=? ? ? (,,)ffxtξ= null , gξ 2 /2 () exp (2 ) 2 D V g RT RT ρξ π ? ?? =? ? ? ? ? ,,,DVTρ 1 (,,)(,)[(,)(,)], / tt t fx t fx t fx t gx t τ ξδξ δ ξ ξ ξ τ λδ++? =? ? ≡ nullnullnullnull 2 ()0V t V VV p V t ρ ρ ν ? +?? = ? ? + ?? = ?? + ? ? The general procedure for LBM: (1) Given the initial conditions for density and velocity (2) Compute the equilibrium-distribution (31) (3) Choose suitable relaxation time and compute new distribution function by (31) (4) Update the density and velocity Fig.17 9-bit and 7-bit lattice models for lattice Boltzmann methods , Vρ 2 2 24 ()() 1; 2, 9 - 9, 9 - 2, 9 - , , 4, 7- 8, 7- 1, 7- eq le V ke V lV f cmcmc bit bit bit lkm bit bit bit αα αα ωρ ???? =+ + ? ?? ?? ??? === ??? ??? Fig.18 The streamline and vorticity for lid-driver cavity problem by LBM (Zhang and Liu ) X.M.Wei,et al,The Lattice Boltzmann method for gaseous phenomena Fig.20 Hot steam rising up from a teapot and its spout ( X.M.Wei,et al,The Lattice Boltzmann method for gaseous phenomena) 3-7.Discontinuous Finite Element Methods Runge-Kutta discontinuous Galerkin (RKDG) finite element method Consider the following governing equations (32a where (32b) or in a compact form (32c) 0 [0, ] 0, [0, ] ( , ,0) ( , ) ( , ) ( , , ) | ( ) T UFG T txy uxy u xy xy uxyt tγ ??× ? ?? ++= ?× ??? =∈? = 22 1 2 22 1 2 , , hhu hv U hu F hu gh G huv hv huv hv gh ???? ? ? ???? ? ? ==+ = ?? ? ? ?? ? ? ?? + ?? ? ? ?? 0, U F t ? ? +?? = ? The Runge-Kutta DG method can be implemented as follows (1) Space discretization Make the triangulation for the domain: K: triangular element with three edges (33) (2) Construct finite element approximation for (61c) (34) : smooth test function : outward unit normal to the edge : numerical flux h K??? = ∪ lK∈? { (): | (), } k hh hK h VvL v PKK ∞ =∈? ∈ ?∈? , ()0, hlKh h h KlK lK d Uv dxdy h v ds F v dxdy v V dt ? ∈? + ??? =?∈ ∑ ∫∫∫ null (, ) h vxy V∈ ,lK n , () lK hFn ? =? null (3) Tine discretization Based on above finite element approximation we can obtain a system of ODEs (35) best instable solution approach is TVD-Runge-Kutta time-marching algorithm. The 3 rd order TVD Runge-Kutta scheme is (36) () h hh dU LU dt = (1) (2) (1) (1) 1(2)(2) () 31 {(} 44 12 {(} 33 nn n nn UU tLU UUUtLU UUUtLU + =+? =+ +? =+ +? Fig.22 The numerical results of the contaminant transport scenario in Galveston Bay off the coast of Texas. (Aizinger and Dawson) 3-8.SPH (smoothed particle hydrodynamics) Lucy(1977),Gingold and Monaghan(1977) [22] L.Cueto-Felgueroso,et al, On the Galerkin formulation of SPH, Inter.J. Numer.Methods Fluids, 60(2004)1475-1512 The flood simulationed by 100000 fluid particles ( S.Premoze et al, Eurographics, 22(2003)) Perigaud G. and Saurel R., A compressible flow model with capillary effects, J. Comput.Phys., 209(2005)139-178 3-9.Software:Fleunt, Phoenics,Star-CD,CFX,and so on. References [1] Liu R.X. and C.-W. Shu, Some new methods for Computational Fluid Dynamics, Chinese Science Publisher,2002 [2] Jing X.K., Liu R.X. and Jian B.C. 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Engrg.,194(2005)3984- 4005 Thank you!