中国科学技术大学：数值模拟新发展（PPT讲稿）Some recent development of the numerical simulation methods for CFD

Some recent development of the numerical simulation methods for CFd Liu ruxun Dept of Math. USTC, Hefei anhui 230026, China liunx@ustc.edu.cn

Some recent development of the numerical simulation methods for CFD Liu Ruxun Dept.of Math., UST C, 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.comPuterScience(algorithms,codingsoftware 5. Visualization techniques Lab for Computational Fluid Dynamics May28,2004

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, C omparison of two mathematical m o d els for solving the dam break probl ems using the FEM method, C o m put. Methods Appl. Mech. Engrg.,194(2005)39 8 4-40 05 Adopted a wrong m o d el!! 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. Dome classical methods 2-1. Donor and Acceptor 2-2. Harlow and Welch's MAC Marker and cell), PIC, FLIC 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

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

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 uss recent methods and developments easily 1. Donor and Acceptor methods Consider the numerical flux scheme(8) of the Id shallow water equations in the cell 1;=[x-1/2,x, +/2 ]and the neighboring cell I +=[x +/2,x +3/2].The numerical flux F(ua)at the discontinuous joint) point x=x,+/2 can be reconstructed by judging which is the donor-or acceptor-cell between the two cells (h) F(1+2) as u m>05or i+1 (h2+gh2), -(h2+8gh2)1 asl12<0(1) The reconstruction approach is called donor-acceptor method which has obvious mechanics character

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 2 2 1 1 1/ 2 2 2 1/ 2 2 2 1 ( ) , ( ) , ( ) 0 0 ( ) , ( ) , i i i i i i i hu hu F U a s u o r a s u hu gh hu gh + + + + + ⎧ ⎫ ⎧ ⎫ = ⎨ ⎬ > < ⎨ ⎬ + − + ⎩ ⎭ ⎩ ⎭ 1/ 2 1/ 2 [ , ] i i i I x x = − + 1 1/ 2 3/ 2 [ , ] i i i I x x + + = + 1/ 2 ( ) F Ui+ 1 1/ 2 x x = +

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: Marker 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 ax u(x, y, t) n+1 n+1 +1 +.(x(,y(O),)t dy,(t) or m=v(x,y, t) ymt=ymt+l. v(x(t),y(t), t)dt

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 u x y t dt dy t v x y t dt = = 1 1 1 1 1 1 ( ( ), ( ), ) ( ( ), ( ), ) n n n n t n n m m t t n n m m t x x u x t y t t dt y y v x t y t t dt + + + + + + = + = + ∫ ∫

B Leonard's QUICK(quadratic upstream interpolation for convective kinetics) Leonard (1979)used a three-point upstream weighted quadratic interpolation to construct the numerical flux F(Uu) at the discontinuous point (the cell interface )x=x+1/2 gU7+3U1一U1,L1>0 U,+3U-1U,L.,<0 (3) F(U1+12)=[A(U)Ul+1a2 Patanka and Spalding s SIMPLE (Semi-Implicit Method for Pressure-Linked Equation, 1972)

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 ( ) F Ui+ 1 1/ 2 x x = + 1/ 2 1/ 2 ( ) [ ( ) ] F Ui i + + = A U ⋅U 1 2 1 2 1 2 6 3 1 8 8 1 1 8 6 3 1 8 8 1 2 8 , 0 , 0 i i i i i i i i i U U U u U U U U u + − + + + + + ⎧ + − > ⎪ = ⎨ + − < ⎪⎩

4. van Leer's MUSCL(monotonic upstream scheme for conservation law) van Leer ( 1979)uses the approximations Uiy at the time level t=t,to directly reconstruct U(, tn+ndx 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 x=xu12(K=3 is a third order muscl scheme) 2=U1+[(1-x)△U1+(1+x)△U (5a) U/1a2=Ul1-4(1+x)△U1+(1-x)△U In order to restrain the oscillations by inserting a flux limiter is effective strategy i.e. the scheme(5a should be replaced as (5b)

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 Ui i ∀ n t t = 1 1 1 ( , ) i n i n I U U x t dx x + = + ∆ ∫ 1 1/ 2 4 1 1 1/ 2 1 4 1 [(1 ) (1 ) ] [(1 ) (1 ) ] L i i i i R i i i i U U U U U U U U κ κ κ κ + − + + + = + − ∆ + + ∆ = − + ∆ + − ∆ 1 1/ 2 4 1 1 1/ 2 1 4 1 [(1 ) (1 ) ] [(1 ) (1 ) ] L i i i i R i i i i U U U U U U U U κ κ κ κ + − + + + = + − ∆ + + ∆ = − + ∆ + − ∆   1 1/ 2 x x = + 13 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 v(x)=l11+5(41+l6(1-5) x △l1=lR;-1L lim (x) lin Im (l+l21) MUSCL van eer 1979) piecewise linear Collet Woodwar piecewise parabolic (1984) Fig. 6 The reconstruction character of left and right limit values forMUSCL and PF

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 1 1 2 2 1 1 2 2 , 6 , , , , , 1 6 , 2 , , ( ) ( (1 )), lim ( ); lim ( ) 6( ( )) i i i L i i i i i i R i L i R i x x L i x x n i i R i L i x x v x u u u x x x x u u u u u x u u x u u u u ξ ξ ξ + − − − + ↑ ↓ ⎧ − ⎪ = + ∆ + − = ≤ ≤ ⎪ ∆ ⎪ ∆ = − ⎨⎪ = = ⎪⎪ = − + ⎩ 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 TVDT(Um)sTW(",T(")=△∑An TTheory For scalar conservation law u,+(f(u)=0, a(u)=af(u)/au Jacobian matrix (8) a scheme consistent with it can be written as -1/2 (l2-121)+D+12(l+1-l1) 9 if the following conditions are satisfied 1+1/20. D12≥0 0≤Cm2+D12s1 it is also a TVd one Monotone scheme iS TVD A TVD Scheme is monotonicity preserving

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 ( ) ( ), ( ) n n n ni i TV U TV U TV U x U + ≤ = ∆ ∑ ∆ ( ( )) 0, ( ) ( )/ t x u + f u = = a u ∂f u ∂u Jacobian matrix 1 1/ 2 1 1/ 2 1 ( ) ( ) n n n n n n i i i i i i i i u u C u u D u u + = − − − − + + + − 1/ 2 1/ 2 1/ 2 1/ 2 0, 0 0 1 i i i i C D C D + + + + ⎧ ≥ ≥ ⎨⎩ ≤ + ≤