Numerical solutions of incompressible Euler and Navier-Stokes ...

ACTA MECHANICA SINICA (English Series), Vol.16, No.3, August 2000 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A.

ISSN 0567-7718

N U M E R I C A L S O L U T I O N S OF I N C O M P R E S S I B L E E U L E R AND NAVIER-STOKES EQUATIONS BY EFFICIENT DISCRETE SINGULAR CONVOLUTION METHOD* D. C. Wan

G . W . Wei +

(Shanghai Institute of Appl Math and Mech, Shanghai University, Shanghai 200072, China) +( Dept of Computational Science, National University of Singapore, Singapore 119260) A B S T R A C T : An efficient discrete singular convolution (DSC) method is introduced to the numerical solutions of incompressible Euler and Navier-Stokes equations with periodic boundary conditions. Two numerical tests of two-dimensional NavierStokes equations with periodic boundary conditions and Euler equations for doubly periodic shear layer flows are carried out by using the DSC method for spatial derivatives and fourth-order Runge-Kutta method for time advancement, respectively. The computational results show that the DSC method is efficient and robust for solving the problems of incompressible flows, and has tile potential of being extended to numerically solve much broader problems in fluid dynamics. K E Y W O R D S : incompressible flows, periodic boundary, DSC method, fourth-order Runge-Kutta method

1 INTRODUCTION Over the past decade, there has been a great deal of progress in the area of computational fluid dynamics. Along with the rapid development of computer hardware and software, more and more attention has been increasingly concentrated on the development of advanced, efficient, highly accurate numerical algorithms which can be used for the analysis and numerical solutions of highly complex flow problems. Generally speaking, there are two major approaches to construct numerical algorithms, namely, global methods and local methods. Global methods are highly localized in their spectral space, but unlocalized in the coordinate space. By contrast, local methods have excellent spatial localization, but are unlocalized in their spectral space. Therefore, global methods are much more accurate than local methods, while the major advantage of local methods is their flexibility for dealing with complex geometries and boundary conditions. The most known local methods include finite difference method[ 1], finite element method [2], finite strip [3], finite volume method[ 41 and many special high-order difference methods, such as total variation diminishing(TVD) schemes [~], essentially non-oscillatory(ENO) schemes [61, compact schemes F~9], etc. While the famous spectral approximation [1~ pseudospectral methods [12'13], fast Fourier transform [14], and differential quadrature [15] belong to the global Received 3 January 2000, revised 17 June 2000 * The project supported by the National Natural Science Foundation of China (No.19902010).

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methods. Although both global methods and local methods play important roles in the area of incompressible flow calculations, as is well known, each method has its drawbacks and merits. Hence, it is highly desirable to have a method that can offer both the accuracy of global method and the flexibility of local method. In order to combine the merits of glob~tl methods and local methods, a discrete singular convolution (DSC) method [16~1sl has been developed to accomplish tile target. The DSC method is based on the theory of distributions and the principle of singular convolutions, which were informally used by physicists and engineers, and were later illustrated in rigorous mathematical framework by Schwartz [191, Korevaar[ 2~ and others. One of distributions used in the DSC method is the delta distribution which can be approximately replaced by the continuous sequence of delta kernel [21]. The most important property of the delta kernel is that it actually forms an orthogonal basis for a reproducing kernel Hilbert space. So a function and its various order derivatives can be approximated by a linear sum of the functional values at grid points within a small bandwidth with a regularized sampling kernel. In tile DSC method, the often used delta kernel is the Shannon's delta kernel [21]. In order to improve the local and ~ y m p t o t i c behavior of Shannon's delta kernel, a regularization procedure [ls] was proposed to increase the regularity of the kernel. Therefore, discrete approximations of delta distributions can be accurately constructed by using the regularized Shannon's delta kernel [ls]. The DSC method has many attractive features: orthogonality, compact support, arbitrary regularity and simplicity, since it has the wavelet feature of time-frequency localization and avoids the difficulty and redundancy of multiresolution analysis [16,1vm'22]. It combines the advantages of both local methods and global methods: good localization and spectral accuracy. Therefore, one expects that the DSC method will be well suited to situations where classical local methods do not converge and where global methods do not apply. So far, the DSC method has been successfully applied to the treatment of the nonlinear Sine-Gordon homoctinic orbit singularity, eigenvalue problems of the Fokker-Planck equation and the SchrSdinger equation, structural analysis, formation of nanoscale morphological patterns and many other fields [22~24]. The main objective of the present paper is to apply the DSC method to numerical solutions of incompressible Euler and Navier-Stokes equations with periodic boundary conditions. The convection terms and the viscous terms in these equations are discretized by tile DSC method, and for time advancement a fourth-order Runge-Kutta method is used. A fractional time step and potential function method (FTSPFM) is adopted to overcome the difficulty occurring in treating the pressure field of the incompressible equations, and highly efficient iterative SOR solver is utilized for solving the resulting Neumann-Poisson equation. Two benchmark problems of unsteady incompressible flow, i.e., the solutions of two-dimensional Navier-Stokes equations with periodic boundary conditions and twodimensional Euler equations for doubly periodic shear layer flows are investigated, respectively. These numerical tests enable us to validate the efficiency and accuracy of the DSC method in solving the unsteady incompressible flows. 2 DISCRETE

SINGULAR

CONVOLUTION

METHOD

For the sake of simplicity, the one-dimensional problem is chosen to demonstrate the discrete singular convolution (DSC) method. Please refer to Refs.[16--~18] for a more detailed

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discussion on the DSC method. In the DSC method, any function f(x) 6 L 2 and its derivatives with respect to a coordinate at a grid point x are approximated by a linear sum of the functional values in the narrow domain [x - x - w , x + xw] in that coordinate direction. This expression can be written as follows W

F_,

,~,~'x - xk)f(xk)

(1)

k=-W

where superscript q (q = 0, 1, 2 , . . . ) denotes the qth-order derivative with respect to x. The {xk } is an appropriate set of sampling discrete points centered around the point x. The 2 W + 1 is the computational bandwidth which is usually smaller than the whole computational domain. Therefore, the resulting approximation matrix has a banded structure, which makes the DSC method more efficient than normal global methods and is particularly valuable with respect to large scale computations. In Eq.(1), the 6a,~(x) is Gaussian regularized Shannon's delta kernel, given by (2)

where 5A(x) is the Shannon's delta kernel with the central frequency c~ = 7r/A , here A is the grid spacing 6 z ~ ( x ) - sinTrx/A (3)

7rzl A It should be pointed out that although only the Shannon's delta kernel is adopted here, there are m a n y other kernels available to replace the Shannon's delta kernel [16]. In Eq.(2), R~(x) is a Gaussian regularizer Ra(z)=exp

-

a > 0

(4)

where a determines the width of the Gaussian window and is often varied in association with the grid spacing, i.e., a = rA, here r is a parameter chosen in computations. The expression of Eq.(1) provides the highest computational efficiency both on and off a grid. In fact, it provides exact results when the sampling points are extended to an infinite series. A m a t h e m a t i c a l estimation for the choice of W, r and A is available. For example, if the L 2 error for approximating an L 2 function f(x) is set to 10 -~, the following relations are to be satisfied

r O r - BA) > ~

and

-W - >

(5)

r

where r = a / A and B is the frequency bounded for the function of interest, f(x). The first inequality states that for a given grid size A, a large r is required for the approximating high frequency component of an L 2 function. The second inequality indicates that if one chooses the ratio r = 3, then the half bandwidth W ~ 30 can be used to ensure the highest accuracy in a double precision computation (7] = 15). It is noted that Eq.(1) is very efficient since just one kernel is required for the whole computational domain [a, b] for given A and r. We refer this kernel as translationally invariant. However, there is a technical aspect which concerns Eq.(1) at a computational

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boundary. It is obvious that the functions, f(xk), need to be located outside tile computational domain [a, b], where their values are usually undefined. Therefore, it is necessary to make assumptions about those function values that are beyond the computational domain. In the DSC algorithm, such functions of f(xk) are obtained according to their corresponding boundary conditions. For example, in Dirichlet boundary conditions, such f(xk) can be taken to be f(a) or f(b); in periodic boundary conditions, such f(x~.) can be obtained by periodic extension from their corresponding values inside the computational domain [(~,b]; in Neumann boundary conditions, such f(xk) should be determined by f' (a) (or f' (b)). When the regularized Shannon's delta kernel is used, the detailed expressions for ~))~(z), 5~!o(x) and 6a,o_(X) (2) 9 can be easily given out

(--) exp

X2

sin ~ -

~(o)

(~ r o)

(6)

A (z = 0)

1

COS

sin ~ -

( - ~ ) exp ( - 2@2)

exp

-~--5a2

7"fX2

A sin ~-

~

exp /TO-2

(x r o)

-2-g

A (z = 0)

0

.

~sin

(..) -~

exp

(..) - ~-ffa2

(.x) (..)

- 2 cos -~- exp

X

cos ~ 2 '

=

X2

exp a2

-~-a2

sin -~- exp + 2

()()

G2___~X 7f

A

,./rx 3

~

()

sin ~-- exp ~x

-~a 2

x sin -~- exp -~-~

+

7r(/4

+ (s)

(.') (:~ # o)

A

71-20-2

3 + A---5-3o-2

(z = O)

Once r is chosen, the coefficients ~(~!r (x), 3~!o (x) and 6~Ir (x) are only grid-dependent. So when the grid point distributions are given, the coefficients can be computed once and stored during the compution.

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3 FORMULATIONS

227

AND SOLUTION PROCEDURES

3.1 G o v e r n i n g E q u a t i o n s The two-dimensional incompressible Navier-Stokes or Euler equations in the Cartesian co-ordinates in the dimensionless form are given as follows Continuity equation

Ou

Ov

(9)

o-~+N =0 Momentum equation in the horizontal direction

Ou Ot --

Op 1 (02u 02u'~ _ ( Ou Ou'~ Ox + -~e \ Ox2 + ~y2 ] \ U-~x + V-~y]

(10)

Momentum equation in the vertical direction

Ov_

Op + 1 (02v + 02v~ - ( u Ov ~ + V ~ yOv)

Ot

Oy

-~e \ Ox2

(11)

Oy2 ]

In these equations, the characteristic length is L, the characteristic velocity is U0, and the characteristic time is L/Uo. The u and v are the dimensionless velocity components in the x and y directions respectively, p is the dimensionless pressure, t is the dimensionless time, defined by u = (t/Uo, v = ~/Uo, p = p~ (pU02), t = tUo/L respectively, in which ~, ~, f and t are the corresponding dimensional ones, and p is the fluid density (constant). In the momentum equations, the parameter Re = UoL/v is the Reynolds number in which v is the kinematic viscosity of the fluid. Then, Re > 0 is for the Navier-Stokes equations, and Re ~ oz (or u = 0) for the Euler equations. For the easy design of the solution algorithm, define Du

(~v

D(U) = ~ + -Oy

(12)

L(U) = F(U) - Vp

(13)

F(U) = [f, gjT

U = [u, v]T

1 (o2u f = ~

02v~

(14)

0u + V ~ y

\Ox 2 + ay2) -

1 (02v

cOP]T Vp = [~-~ , ~yyj

( U-~x Ov + V~y Ov )

(15)

g = ~ \ ax 2 + Oy~/ So the system of Eqs.(9)~.,(ll) can be simplified as follows

D(U) = 0

(16)

OU Ot = L(U) = F(U) - Vp

(17)

These equations are the starting point for spatial and temporal discretizations described in the following subsection.

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Discrete Formulations

Because the continuity Eq.(16) contains only velocity components, and there is no obvious link with the pressure, so the restriction to incompressible equations introduces the computational difficulty. To overcome the diffi(:ulty in sovling the pressure in incompressible equations, there are many specially designed approaches available. For example, semi-implicit method for pressure linked equation (SIMPLE), artificial compressibility method, marker and cell (MAC) method, fractional-step projection method and its many variations are widely used in the field. In this work, we adopt a fractional-time-step and potential-function method (FTSPFM), which is a variant of the MAC method, for solving the governing Eqs.(16) and (17). Details of this FTSPFM are outlined below. All spatial derivatives in Eqs.(16) and (17) are discretized by using the DSC method. Uniform grids in both x- and y-directions are employed with Az and Ay denoting the grid sizes in x- and y-directions respectively. We denote a grid point by xi = i A x and yj = j a y and the differences of two grid points by xi - xi+~. = - k A x and yj - Yj+k = - k A y . The values of u, v and p at a grid point (xi, y j) a r e denoted by ~ti,j, l)i,j and Pi,j, respectively. All equations are approximated at the point (xi, y j). By using Eq.(1), the discretized forms of Eqs.(12),-~(15) can be expressed as follows

W Dh(U) :

Z

W 5~).

,

Z

k=- W

~ 5(la)(kAz)pi+k " kk= W ~

Vhp =

fh = -~e L*:=-W

5(')a,.(kAy)vi'j+*:

(18)

IT

(19)

k=- W

W

' Z

(5~,)~(kAy)pi'j+k

k=-W

W k=-w

k'~(kAy)ui'j+k

W

W

k=-W

k=-W

] (20)

1 -% = R e

I ~

k=-W

W k=-W

W k=-W L h ( U ) = F h ( U ) -- V h p

W k=-W F h ( U ) = [fh, gh] T

1

(21)

(22)

where 5(al!o and 5 (2),~ are coefficients of the regularized Shannon's delta kernel given in Eqs.(7) and (8) respectively. By substituting Eqs.(18),-,(22) into Eqs.(16) and (17), the following semi-discretized approximation for Eqs.(16) and (17) can be obtained Dh(U) = 0

(23)

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dt = Lh(U) = Fh(U) - Vhp

229 (24)

A fourth-order Runge-Kutta (R-K) method[ 2~] is used to discretize the resulting ordinary differential Eq.(24)

(22)

V(1) =OllVn-b31 [/kt(rh(Vn)--~hp(1))]

(26) v (3) =

~ + 33

U n+l = a4 ( - U n +

(27)

-

(2S)

U (1) -b 2U (:')

where (al, a2, a3, a4) = (1, 1, 1, 1/3) and (31,32,33,34) = (1/2, 1/2, 1, 1/6). The At is time increment and the U '~ and pn are the velocity and pressure at time tn respectively. The U O) and p(1) U(2) and p(2) as well as U (3) and p(3) are their corresponding values at the first, the second and the third steps of the fourth-order R-K method, respectively. While U •+1 and pn+l are the velocity and pressure at time tn+l. At each step of the R-K method, the FTSPFM is adopted to solve Eqs.(23) and (24). In the F T S P F M method, an intermediate velocity field and a potential flmction are employed to correct the values of the velocity and pressure fields at a new time sVep. As a sample solution algorithm, consider the first step of the R-K method. Assume that at time t~, the velocity U '~ and pressure p~ are known, while U (1) and p(1) at the first step of R-K method are unknown. Let us introduce a first step intermediate velocity field U *(1) that satisfies the same boundary condition as that for the physical velocity fields U n+l at time tn+l, which is defined by U *(1) = oQ U n q- 31 [At (Fh(U n) - Vhp'~)] (29) Because Eq.(29) is explicit in terms of U '~ and p'~, integration stability requires the time increment satisfying the Courant-Friedrich-Lewy (CFL) condition max(At)