Numerical Convergence Study of Nearly- Incompressible ... - CiteSeerX

Numerical Convergence Study of NearlyIncompressible, Inviscid Taylor-Green Vortex Flow Wai-Sun Don, David Gottlieb, Chi-Wang Shu Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912 E-mail: [email protected], [email protected], [email protected]

and Oleg Schilling, Leland Jameson University of California, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551 E-mail: [email protected], [email protected]

Received ; revised November 11, 2002; accepted

A spectral method and a fifth-order weighted essentially non-oscillatory method were used to examine the consequences of filtering in the numerical simulation of the three-dimensional evolution of nearly-incompressible, inviscid Taylor-Green vortex flow. It was found that numerical filtering using the high-order exponential filter and low-pass filter with sharp high mode cutoff applied in the spectral simulations significantly affects the convergence of the numerical solution. While the conservation property of the spectral method is highly desirable for fluid flows described by a system of hyperbolic conservation laws, spectral methods can yield erroneous results and conclusions at late evolution times when the flow eventually becomes under-resolved. In particular, it is demonstrated that the enstrophy and kinetic energy, which are two integral quantities often used to evaluate the quality of numerical schemes, can be misleading and should not be used unless one can assure that the solution is sufficiently well-resolved. In addition, it was shown that for the Taylor-Green vortex (for example) it is useful to compare the predictions of at least two numerical methods with different algorithmic foundations (such as a spectral and finite-difference method) in order to corroborate the conclusions from the numerical solutions when the analytical solution is not known.

D R A F T

November 11, 2002, 12:55pm

D R A F T

2

W.-S. DON ET AL.

Key Words: spectral methods, WENO method, Taylor-Green vortex, convergence

CONTENTS 1. Introduction. 2. The Euler Equations and The Taylor-Green Vortex. 3. Numerical Methods. 4. Filtering. 5. Results and Discussion. 6. Conclusion.

1.

INTRODUCTION

High-order accuracy is required in the simulation of turbulence and mixing in order to capture both the large- and small-scale structures, and accurately estimate the statistical properties and large-scale transport. The numerical methods of choice are the Fourier Galerkin method and the Fourier collocation method (which will collectively be referred to as Fourier spectral methods). In Fourier spectral methods, the solution is expanded either in a finite sum of trigonometric polynomials which serve as the basis functions in the Galerkin approach, or one requires that the solution is interpolated by a Nth-degree trigonometric polynomial at equi-spaced collocation points in the collocation approach. The Fourier collocation method will be used in this paper, as both approaches have essentially the same numerical properties and the collocation method is simpler to implement for nonlinear partial differential equations. The appeal of spectral methods is mainly based on several important properties of these methods. One property is spectral accuracy, which refers to the fact that the rate of convergence of the approximation fN depends only on the smoothness of the approximated function f. If the function f is analytic, the rate of convergence is exponential. If f is a function with p continuous derivatives, the convergence

D R A F T

November 11, 2002, 12:55pm

D R A F T

CONVERGENCE STUDY OF INVISCID TAYLOR-GREEN VORTEX FLOW

3

rate is formally O(N ¡p ). In contrast, the rate of convergence of finite-difference methods is limited by the order of the scheme. Hence, for smooth functions and for a fixed accuracy, spectral methods are more efficient than finite-difference methods. A second important property of spectral methods is that they are conservative, as well as non-dissipative and non-dispersive. For the solution of partial differential equations based on conservation laws, such as the wave equation and the Euler equations, the numerical schemes would satisfy the same conservation principles. The reader interested in the details of spectral methods is referred to the extensive literature available and references contained therein [1, 2, 3, 4, 5]. In particular, when applied to hyperbolic conservation laws such as the compressible Euler equations, spectral methods conserve quantities such as mass, momentum, and total energy discretely, which are also conserved by the continuum partial differential equations. Therefore, results computed with spectral methods are often used as a benchmark to study the numerical properties of other highorder methods such as compact finite-difference methods and the weighted essentially non-oscillatory (WENO) finite-difference method. In this work, the role and applicability of two integral quantities—the kinetic energy and enstrophy—for the assessment of numerical schemes is considered. The kinetic energy is the volume integral of ½u ¢ u=2, where ½ and u are the density and velocity fields, respectively. The enstrophy, which is the volume integral of ω ¢ ω=2, where ω = ∇ £ u is the vorticity vector field, is also computed (note that ω is a computed quantity involving the spatial derivatives of the components of the velocity vector). The inherent numerical dissipation of a finite-difference method will become an issue as the kinetic energy will be dissipated numerically over a long time integration, rendering the scheme less accurate than a spectral method. The numerical dissipation also has an important role in reducing the sharpness of the approximation of the derivative of the function. Hence the reduction of the production of the vorticity results in a smaller increase of the enstrophy than computed using the Fourier spectral method.

D R A F T

November 11, 2002, 12:55pm

D R A F T

4

W.-S. DON ET AL.

The above numerical issues are studied here by considering the evolution of a nonlinear fluid flow: the Taylor-Green vortex. The spatial and temporal evolution of the incompressible, inviscid Taylor-Green vortex flow in a three-dimensional periodic domain is perhaps the simplest model for the investigation of the nonlinear transfer of kinetic energy among eddies with a range of spatial scales. When the flow has a finite Reynolds number, the kinetic energy generated by velocity shear is dissipated by the smallest scales, which provides a simple model for the development of a turbulent flow and the cascade of energy from larger to smaller scales. The initial condition is smooth and consists of a first-degree trigonometric polynomial in all three directions. For a long-time integration, the numerical scheme should capture the behavior of the flow field accurately as long as possible. Thus, spectral methods are ideal for this application. However, as the flow evolves according to the nonlinear Euler equations, the flow rotates about the vertical axis z. As the velocity increases with radius, the flow begins to twist about the center of the domain forming a vortex core with a diminishing radius. Apparently, the radius of vortex core tends to zero and forms a flow singularity at a finite time. While the putative existence of a finite-time singularity remains a controversial issue, the purpose of the present study is to examine some of the subtle points regarding the properties of different numerical schemes and the conclusions drawn from them. Prior to the appearance of the singularity in the flow, the solution is smooth and well-resolved by the WENO and spectral methods. As the core of the vortex flow increasingly twists up in time, spectral methods will become unstable as the high modes can no longer be supported by the fixed number of terms in the Fourier series expansion. Various low-pass filters, such as the exponential filter and sharpcutoff filter, are often implemented to suppress the nonlinear instability allowing the integration to proceed further in time. The exponential filter attenuates the amplitude of the high modes to zero exponentially smoothly, while the sharp-cutoff

D R A F T

November 11, 2002, 12:55pm

D R A F T


5

filter set all of the amplitudes of the high modes with mode number greater than a specified cutoff mode Nc , to zero. The common choice of the cutoff mode is Nc = 2N=3, where N is the total number of Fourier modes used. This is also often used as the dealiasing technique [3] for the solution of nonlinear partial differential equations. In contrast, high resolution finite-difference methods (such as WENO methods) are often stable and no additional numerical techniques are needed. Using the sharp-cutoff filter, it was found that the form of the filter used in spectral methods leads to a stable, yet diverging in time, solution. Although the solution did not converge, the kinetic energy was well-conserved in time and the enstrophy growth was in qualitatively better agreement with the expected physics. In contrast, the exponential filter, when properly tuned, yields a stable, accurate and converging solution that is superior to that given by most finite-difference methods. This paper consists of the following sections. The equations and initial conditions for the Taylor-Green vortex flow are formulated in Section 2. A brief description of Fourier spectral methods and the fifth-order weighted essentially non-oscillatory finite-difference method used are given in Section 3. The applications of filtering in spectral methods are discussed in Section 4. The results of the simulation of the Taylor-Green vortex flow are presented and discussed in Section 5. Finally, conclusions are given in Section 6. 2.

THE EULER EQUATIONS AND THE TAYLOR-GREEN VORTEX

The compressible Euler equations and their global conservation properties are reviewed here, together with the Taylor-Green vortex flow and its applications.

2.1.

The Euler Equations

In the absence of external forces and molecular dissipation, the governing equations are the three-dimensional, compressible Euler equations in Cartesian coordi-

D R A F T

November 11, 2002, 12:55pm

D R A F T

6

W.-S. DON ET AL.

nates on a [0; 2¼] £ [0; 2¼] £ [0; 2¼] periodic domain, @½ + ∇¢ (½ u) = 0 ; @t

(1)

@ (½ u) + ∇¢ (½ u − u) = ¡∇p ; @t

(2)

@E + ∇¢ [(E + p) u] = 0 ; @t

(3)

where ½(x; t) is the density, u(x; t) = (u; v; w) is the velocity vector, p(x; t) is the pressure, and E(x; t) is the total (kinetic plus internal) energy. With the assumption of an ideal gas law, the total energy can be written in terms of the velocity and pressure as E=

p ½u ¢ u + 2 °¡1

(4)

with the ratio of specific heats ° = 1:4. The vorticity equation that follows from Eq. (2) is ∇½ £ ∇p @ω + (u ¢ ∇) ω = (ω ¢ ∇) u ¡ ω∇ ¢ u + ; @t ½2

(5)

where the terms on the right side correspond to vortex stretching, dilatation, and baroclinic vorticity production. In a strictly-incompressible flow, the last two terms on the right side of Eq. (5) vanish. It follows from the momentum equation (2) that the kinetic energy density equation is ³½ u ¢ u ´ @ ³½ u ¢ u´ + ∇¢ u = ¡u ¢ ∇p ; @t 2 2

(6)

which is not conserved in a compressible flow. However, integration over all space of Eq. (3) immediately shows that the total energy is conserved. For an incompressible flow in which ½ = constant and ∇ ¢ u = 0, Eq. (6) can be rewritten in the form ³½ u ¢ u ´ @ ³½ u ¢ u´ + ∇¢ u + pu = 0; @t 2 2

D R A F T

November 11, 2002, 12:55pm

(7)

D R A F T


7

so that the kinetic energy (volume integrated) K(t) ´

Z

Z

2¼ 0

2¼ 0

Z

2¼ 0

½u¢ u dx dy dz 2

(8)

is conserved in time. It follows from the vorticity equation (5) that the enstrophy density equation is @ ³½ ω ¢ ω ´ + ∇¢ @t 2

µ

½u − ω ω 2

¶

∙ ¸ ∇½ £ ∇p = ½ ω¢ (ω ¢ ∇) u ¡ ω∇ ¢ u + ; ½

(9)

which is also not conserved. Hence, there is no conservation in time of the enstrophy (volume integrated) −(t) ´

Z

2¼ 0

Z

2¼ 0

Z

2¼ 0

ω¢ω dx dy dz : 2

(10)

In a strictly-incompressible flow, the last two terms on the right side of Eq. (9) vanish. 2.2.

The Taylor-Green Vortex

The global existence of solutions (and the possible existence of singularities) of the initial-value problem for the incompressible, three-dimensional Navier-Stokes equation and Euler equation have been investigated analytically for short-times and numerically for short and longer times using the viscous and inviscid Taylor-Green vortex, respectively. For the case of the Euler equation, it has been conjectured that the vorticity may become singular if the nonlinear process of vortex stretching is faster than exponential, so that some vortex surfaces become infinitely long in a finite time. A consequence of the classical Kelvin and Helmholtz theorems of fluid dynamics [6] is that vortex lines move with a fluid and the growth of the vorticity is proportional to the stretching of a vortex filament in an inviscid, incompressible, constant density fluid. These observations have been applied to the dynamics of very large Reynolds number turbulent flows (which are nearly-inviscid). In an unbounded domain (e.g., a periodic domain), these theorems also imply that an inviscid flow

D R A F T

November 11, 2002, 12:55pm

D R A F T

8

W.-S. DON ET AL.

that is initially-smooth remains smooth for all later times if the stretching of vortex filaments is bounded. While the absence of vortex stretching in two dimensions implies the global regularity of weak solutions to the two-dimensional, incompressible Euler equation, the existence of vortex stretching, tangling, and rotation in three dimensions may result in some vortex filaments becoming topologically wound such that the end points of the filament are separated by a finite distance: in this case, a spontaneous singularity can occur after a finite evolution time. In particular, it was shown that if there exists a critical time t¤ at which a solution loses its regularity, then [7, 8, 9] lim t"t¤

kω(x; t)k1 " 1

(11)

Z

kω(x; t)k1 dt " 1 ;

(12)

and lim t"t¤

t¤ 0

which is consistent with the divergence of kω(x; t)k as (t ¡ t¤ )¡n with n > 1. Thus, the development of a singularity is directly related to the production of vorticity. The Taylor-Green vortex flow is the three-dimensional, incompressible flow that evolves from an initially two-dimensional, single-mode initial velocity field of the form u(x; 0) = sin (kx) cos (ky) cos (kz) ;

(13)

v(x; 0) = ¡ cos (kx) sin (ky) cos (kz) ;

(14)

w(x; 0) = 0

(15)

with wavenumber k=

2¼ ¸

(16)

=1

D R A F T

November 11, 2002, 12:55pm

D R A F T

9


and periodic boundary conditions on [0; 2¼]3 . The initial density and pressure are ½(x; 0) = 1 ;

(17)

½ [cos (2z) + 2 cos (2x) + cos (2y) ¡ 2] 16

p(x; 0) = p0 +

(18)

with p0 = 100 chosen to limit the Mach number to approximately 0:08 and render the flow nearly-incompressible for all time in the simulations; note that Eq. (18) is a solution of the pressure Poisson equation. The symmetries of the TaylorGreen vortex [10, 11] are not directly relevant to the present investigation, and are not discussed further. The initial conditions (13)—(15) specify two-dimensional streamlines. The flow with an initially two-dimensional velocity field with w = 0 becomes three-dimensional for t > 0 by the existence of a pressure gradient. As the time-evolution of the Taylor-Green vortex entails a kinetic energy cascade, the evolution of this simple flow has been used to study the effects of both viscous dissipation in Navier-Stokes dynamics and numerical dissipation in the solution of the Euler equation. Thus, the evolution of Taylor-Green vortex flow provides a useful quantitative diagnostic of the intrinsic numerical dissipation and flow symmetry preservation in a discretization scheme for the Navier-Stokes equation [12, 13]. The initial kinetic energy is K(0) =

=

1 4

Z

2¼ 0

Z

2¼ 0

Z

2¼ 0

Z

2¼ 0

Z

2¼ 0

Z

2¼ 0

½(x; 0) u(x; 0) ¢ u(x; 0) dx dy dz 2

(19)

[1 ¡ cos (2x) cos (2y)] cos2 (z) dx dy dz = ¼3 ;

which is conserved for the inviscid Taylor-Green vortex, so that a quantitative measure of numerical dissipation inherent in a numerical algorithm can be assessed by observing the rate of decrease of the kinetic energy from its initial value during the time-evolution of the flow. In particular, a dealised pseudospectral method should

D R A F T

November 11, 2002, 12:55pm

D R A F T

10

W.-S. DON ET AL.

conserve the kinetic energy very accurately to late times. As the nonlinear interactions generate successively smaller scales, numerical simulations are limited to times during which all of the scales can be resolved with sufficient accuracy. The enstrophy involves derivatives of the velocity field components, so that its computation is sensitive to the accuracy with which the small scales can be represented numerically. Direct numerical simulations (DNS) of incompressible Taylor-Green vortex flow governed by the Navier-Stokes equations have been performed. Exponential growth of the enstrophy was observed in 8002 DNS of a two-dimensional Taylor-Green vortex for general periodic flows and at 20482 for flows with large-scale symmetries [14]. Dealised pseudospectral 323 , 643 , 1283 , and 2563 DNS of an initially twodimensional, three-dimensional Taylor-Green vortex flow were compared to powerseries solutions up to t80 [11, 15]. A subsequent calculation of the symmetric TaylorGreen vortex was performed at a resolution of 8643 [16, 17]. In the inviscid case, the exponential growth regime is characterized by colliding, oppositely-oriented vortex structures, which deform into sheets at large evolution times. The enstrophy [b u(k; t) is the Fourier transform of the velocity field u(x; t)] −(t) =

1X 2 k jb u(k; t)j2 2

(20)

k

=

1

1 X (2k) 2k A t 2 k=0

corresponding to the inviscid case was calculated numerically with 29 digits of precision up to O(t44 ) [18]; this involves derivatives of the velocity field with wavenumbers up to kmax = 23. The initial condition (13)—(15) is smooth in x, so that − is analytic at t = 0. Given a finite number of its Taylor series coefficients at the origin, the objectives of the analysis are to analytically-continue − in t 2 C and determine if there are any singularities on the positive real axis. While this problem is actually ill-posed, it may still be possible to estimate the locations and other properties of

D R A F T

November 11, 2002, 12:55pm

D R A F T


11

the singularities using Taylor series expansions, Domb-Sykes plots, and Padé approximant techniques (see [19]). The expansion coefficients A(2k) are obtained by b (k; 0)=@tn using the Navier-Stokes calculating time derivatives of the velocity @ n u

equation recursively. The convergence radius was determined by imaginary time p singularities at t ¼ 5i. The existence of a real time singularity was studied using

analytical continuation, and Padé approximants indicated a possible real singularity at the critical time tc ¼ §5:2. An analysis of dn −=dtn using series expansions and Padé approximants suggested that the time derivatives become singular before the maximum enstrophy is attained (see [18, 20]). However, it was subsequently concluded [11, 15, 16, 17] that neither numerical simulations nor series expansions conclusively showed evidence for an inviscid finite-time singularity.

3.

NUMERICAL METHODS

The time-evolution scheme, spectral method, and WENO method used in the present study are briefly described here. The results presented in this paper were obtained using the code library WS-Adaptive, which is a WENO and spectral library for high-order accurate, adaptive domain simulations of the compressible Euler equations in one, two, and three spatial dimensions. The formal spatial accuracy of the WENO module is fifth-order, and the accuracy of the spectral module depends on the order of the exponential filter that is applied. The high-order accuracy allows high Fourier wavenumber modes to be evolved efficiently for long-time integrations. Additionally, WS-Adaptive has an adaptive domain capability, so that the computational grid adjusts dynamically to optimize the spatial resolution in regions of the flow field which contain small-scale structure (however, this capability is not used in the present simulations). 3.1.

The Third-Order TVD Runge-Kutta Time-Evolution Scheme

The Shu-Osher [34] third-order TVD Runge-Kutta time-evolution scheme is used to solve the system of ordinary differential equations resulting from spatial differ-

D R A F T

November 11, 2002, 12:55pm

D R A F T

12

W.-S. DON ET AL.

encing: φ(1) = φn + L(φn ) ¢t ; φ(2) = φn+1 =

(21)

3 φn + φ(1) + L(φ(1) ) ¢t ; 4

(22)

φn + 2 φ(2) + 2 L(φ(2) ) ¢t ; 3

(23)

where L(¢) is the spatial operator, φn and φn+1 are the data arrays at the nth and (n + 1) th timestep, respectively, and φ(1) and φ(2) are two temporary arrays at the intermediate Runge-Kutta steps. The scheme is stable for both the Fourier spectral and WENO methods used in this paper with ¢t ¼ O(N ¡1 ) and N the number of collocation points. 3.2.

The Spectral Method

The Taylor-Green Vortex flow is periodic and has anti-symmetry in all three coordinate directions; hence, the Fourier collocation method is used in this study. To reduce the computation required, the (anti)-symmetry property of the flow is also taken into account in the simulation, and results in a reduction of computation by a factor of eight. It is also known that this flow possesses other symmetries (see [10, 11]), but these symmetries were not utilized in this study. In spectral methods the approximation error depends only on the regularity of the approximated function. A typical error estimate is of the form jf (x) ¡ fN (x)j ∙ C N

1=2¡p

"Z

2¼ 0

#1=2 ¯ p ¯ ¯ @ f (k) ¯2 ¯ ¯ dk ; ¯ @kp ¯

(24)

where fN (x) is the Fourier approximation to the function f(x), with similar expressions for other spectral and collocation approximations. This estimate shows that smoother functions yield better approximations, and the error depends on the pth derivative of the function. In fact, if the function is analytic then the convergence is exponential, jf (x) ¡ fN (x)j ∙ C e¡®N :

D R A F T

November 11, 2002, 12:55pm

(25)

D R A F T


13

A related result is that in order to resolve a wave using Fourier collocation methods one needs a classical resolution of two points per wavelength according to the Nyquist sampling theorem [1], while many more points per wavelength are needed for low-order finite-difference methods depending on the modal content of the flow and the computation time [24, 25]. The accuracy of spectral methods allows numerical simulations that use a minimal number of grid points to obtain the same results obtained using lower-order methods with many more grid points. Thus, for limited computational resources, spectral methods are ideal for developing deeper insights into the physics of flows with small-scale structure by providing the maximum resolution.1

3.3.

The Fifth-Order Weighted Essentially Non-Oscillatory Method

The weighted essentially non-oscillatory (WENO) finite-difference method (see [26, 27, 28, 29, 30]) is an improvement over the essentially non-oscillatory (ENO) methods (see [21, 29, 30, 31, 32, 33, 34]) for discontinuous problems. ENO and WENO methods are high-order accurate finite-difference methods optimized for piecewise-smooth flows with discontinuities between the smooth regions: a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil so as to avoid crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO methods have been quite successful in applications, especially for problems containing both shocks and complicated smooth flow structures, such as compressible turbulence and aeroacoustics. In the present study, a fifth-order WENO method with global Lax-Friedrichs flux splitting [27] is used to solve the Euler equations. The right and left eigenvectors, 1 The

software library PseudoPack was used to compute derivatives based on the Fourier method

(see http://www.cfm.brown.edu/people/wsdon/home.html); derivatives can also be computed using Chebyshev and Legendre collocation methods with PseudoPack. The PseudoPack library is contained within the WS-Adaptive library.

D R A F T

November 11, 2002, 12:55pm

D R A F T

14

W.-S. DON ET AL.

and the eigenvalues of the Jacobian of the Roe-averaged Euler flux are computed at each grid point. The WENO reconstruction procedure is then applied on characteristics to reconstruct the positive and negative components of the flux based on its wind directions. The resulting fluxes are then recombined to form the total flux at the cell boundary.

4.

FILTERING

Most finite-difference methods are stabilized by the inherent numerical dissipation of the high-mode component of the resolved frequency spectrum. In contrast, spectral methods do not possess any numerical dissipation: all modes are captured exactly, which is a significant advantage when the flow is adequately resolved. For highly-nonlinear problems, the nonlinear interaction of the modes generates additional modes that spread over the entire solution spectrum. This leads to the well-known nonlinear instability as the high frequency error grows without bound in time. In spectral methods, a dissipative term known as the spectral vanishing viscosity in the form of a high, even-order term is added to the partial differential equations. With the appropriate choice of parameters for the dissipative term, spectral accuracy can be assured [35, 36, 37]. It has been shown that with a suitable addition of (spectrally small) artificial dissipation to the high modes only, the method converges. The spectral vanishing viscosity (SVV) method [35, 36] was used in the present study, which can be illustrated as follows. Consider the nonlinear scalar hyperbolic equation @Á @f (Á) + = 0: @t @x

(26)

A spectral approximation involves seeking a polynomial of degree N, ÁN (x; t), such that @fN (ÁN ) @ÁN + = 0: @t @x

D R A F T

November 11, 2002, 12:55pm

(27)

D R A F T


15

In the SVV method, the equation is regularized by an additional dissipative term @fN (ÁN ) @ÁN @ 2s ÁN + = ² (¡1)s+1 : @t @x @x2s

(28)

In [38] it was shown that if ² = ®N 1¡2s , where s » log N , then the solution of Eq. (28) converges to the correct entropy solution. The SVV method can be easily applied if one uses the time-splitting technique and solves in the first step @fN (ÁN ) @ÁN + =0 @t @x

(29)

@ÁN @ 2s ÁN = ² (¡1)s+1 : @t @x2s

(30)

and in the second step

Note that the Fourier polynomials ei¼kx are the eigenfunctions of the operator @ 2 =@x2 with eigenvalues k2 . Thus if

ÁN (x; t) =

N X

Ák (t) ei¼kx ;

(31)

k=0

then solving the second step analytically over one timestep is equivalent to modifying the Fourier coefficients Ák (t) as "

Ák (t + ¢t) = Ák (t) exp ¡® N ¢t

µ

k N

¶2s #

:

(32)

This is a low-pass filter that can be performed without any additional work. This high-order dissipative term can be recast in the form of filtering (see [22]). Given the Fourier approximation

fN (x) =

N X

fk ei¼kx ;

(33)

¾(k=N ) fk ei¼kx :

(34)

k=¡N

construct a filtered sum ¾ fN (x) =

N X

k=¡N

D R A F T

November 11, 2002, 12:55pm

D R A F T

16

W.-S. DON ET AL.

Following Vandeven [23], define a p > 1 order filter function ¾(!) 2 C 1 [¡1; 1] satisfying ¾(0) = 1 ; ¯ ¯ @ j ¾(!) ¯ = 0; @!j ¯ !=0

¾(§1) = 0 ; ¯ ¯ @ j ¾(!) ¯ = 0; @!j ¯ !=§1

:

(35)

j∙p

It can be shown that the filtered sum (34) approximates the original function quite well away from discontinuities. A commonly used filter function is the exponential filter, ¾(k=N ) =

8 > : exp ¡® ¯¯ k¡Nc ¯¯ jkj > N c N¡Nc

(36)

where k = ¡N; : : : ; N , Nc is the cutoff mode, ® = ¡ ln ² (² is the machine zero), and p is the order of the filter. The exponential filter offers the flexibility of changing the order of the filter simply by specifying a different p. Thus, varying p with N yields exponential accuracy according to [23]. Alternatively, rather than allowing the high modes to decay to zero smoothly, all modes greater than a fixed cutoff mode Nc can be set to zero, i.e., the sharp-cutoff filter ¾(k=N ) =

8 > < 1 jkj ∙ Nc > : 0 jkj > Nc

;

(37)

where k = ¡N; : : : ; N , and Nc is the cutoff mode. 5.

RESULTS AND DISCUSSION

As discussed in the Introduction, the enstrophy is often used in the numerical solution of the Euler and Navier-Stokes equation to measure the growth of vorticity in time and to predict the appearance of finite-time singularities in Taylor-Green vortex flow. Since no numerical scheme can resolve the apparent singularity of the vortex core with a given finite grid, the enstrophy can be used to quantify the resolving power of a given scheme. Similarly, the kinetic energy is often used to measure the loss of conservation of a numerical scheme. It is generally accepted that

D R A F T

November 11, 2002, 12:55pm

D R A F T


17

a numerical scheme should exactly or approximately satisfy the same conservation properties of the partial differential equations being solved. Thus, the numerical dissipation of the scheme can be quantified by measuring the loss of kinetic energy during the time-evolution. For convenience, the enstrophy and kinetic energy are normalized by their respective initial values at time t = 0, −(0) and K(0). For the Taylor-Green vortex flow, it is conjectured that the normalized enstrophy −(t)=−(0) increases from unity to infinity by a finite time t ¼ 5, and the normalized kinetic energy K(t)=K(0) should maintain a constant value of nearly unity, as there is no dissipative mechanism in the system of partial differential equations describing the flow. In this study, the following three schemes are used: 1. the Fourier collocation method with the sharp-cutoff filter (F-SF); 2. the Fourier collocation method with the exponential filter (F-EF), and; 3. the fifth-order WENO (WENO-5) finite-difference method. To distinguish the different sub-cases considered here, the following notation is used. The case of the Fourier collocation method with sharp-cutoff filter (F-SF) with the cutoff parameter Nc =

2N 3 2,

where N is the total number of the collocation

points, is denoted as (F-SF-23N). For the Fourier collocation method with exponential filter (F-EF), the order of the exponential filter (16th-order) is appended to (F-EF), i.e., (F-EF-16). If the cutoff parameter is non-zero, its value Nc (20, for example) is appended to (F-EF-16) as well. Various numbers of collocation points N are used for each of the three methods and subcases: N = 64, 128, 256, and 384. The Taylor-Green vortex flow is evolved up to a final time t = 5, just before the flow apparently becomes a singular. The time-evolution of the normalized enstrophy and kinetic energy are shown in Figures 1 and 2, respectively, for the four cases studied here: the Fourier collocation method with the 16th-order exponential filter (F-EF-16) and with the 10th-order

D R A F T

November 11, 2002, 12:55pm

D R A F T

18

W.-S. DON ET AL.

exponential filter and cutoff mode Nc =

3N 8 2

(F-EF-10-38N), the sharp-cutoff filter

with Nc = 2=3N (F-SF-23N), and the fifth-order WENO method (WENO-5) for various numbers of collocation points N = 64; 128 and 256. As observed from these figures, at an early time of t ¼ 3:5 the rate of increase of the enstrophy − computed with all of the schemes—(F-EF-16), (F-EF-10-38N), (F-SF-23N) and (WENO-5)—are virtually indistinguishable. This indicates that the vorticity field produced by the vortical flow at early times is well-captured by all of the schemes. However, at later times when the vortex core steepens up into a high gradient, large differences between the computed values of the enstrophy appear. The largest value of the enstrophy is that computed using the Fourier collocation method with the sharp-cutoff filter (F-SF-23N), followed closely by that computed using the exponential filter with mode cutoff (F-EF-16-38N). The Fourier collocation method with exponential filter with no cutoff (F-EF-16) gives an enstrophy with a slightly smaller growth rate than the other two Fourier methods. The latetime behavior of the (WENO-5) scheme shows that the production of enstrophy by this scheme begins to slow down and does not produce as much vorticity as the other schemes: this is due to the dissipative nature of the finite-difference scheme at the developing sharp fronts. Figure 3 shows plane cut-away views of the enstrophy field computed using the Fourier collocation method with the 16th-order exponential filter (F-EF-16) at times t = 3 and t = 5, respectively, at a spatial resolution of 2563 . Figure 4 shows plane cut-away views of the enstrophy field computed using the WENO method at times t = 3 and t = 5, respectively, at a spatial resolution of 2563 . Figure 5 shows plane cut-away views of the kinetic energy field computed using the Fourier collocation method with the 16th-order exponential filter (F-EF-16) at times t = 3 and t = 5, respectively, at a spatial resolution of 2563 . Figure 6 shows plane cut-away views of the kinetic energy field computed using the WENO method at times t = 3 and t = 5, respectively, at a spatial resolution of 2563 .

D R A F T

November 11, 2002, 12:55pm

D R A F T


19

With respect to the kinetic energy K(t), all schemes are conservative until time t ¼ 3:0 as shown in Figure 2. In particular, the (F-SF-23N) scheme conserves K(t) and remains constant for time up to t = 6 (and longer). The Fourier collocation method with the exponential filter (F-EF-16) and (F-EF-10-38N) lose < 5% of the initial kinetic energy. At late times, the (WENO-5) scheme dissipates K(t) the most, losing 25% of the initial value at the lowest resolution. The numerical dissipation of K(t) can be reduced dramatically as the resolution of the simulation increases. However, the dissipation of the kinetic energy can only be delayed and cannot be prevented for each of the schemes, except for the Fourier collocation method with sharp-cutoff filter (F-SF-23N). Based on an initial consideration of these numerical results, it seems apparent that the spectral method with sharp-cutoff filter (F-SF-23N) is the best scheme among those tested here: it simultaneously produces the largest amount of enstrophy in qualitative agreement with theory, and conserves the kinetic energy as stipulated by the conservation principles of the Euler equations in the nearlyincompressible limit. However, this observation and conclusion can be misleading if they are based solely on these two integral quantities. They are insufficient to assess the performance of numerical schemes, especially for simulations of highlynonlinear turbulent flows which are characterized by strong vortex stretching and deformation. To illustrate this, the normalized enstrophy −(t)=−(0) and kinetic energy K(t)=K(0) computed using the Fourier collocation method with the sharpcutoff filter is shown in Figure 7. Here, Nc = 4 was used instead of Nc = 2N=3, i.e., only the first four modes contribute to the flow field. With only four modes retained, the solution remains very smooth: no high modes can survive the sharpcutoff filter, and no large gradients develop in the flow despite the highly nonlinear nature of the equations. Hence, the production of the enstrophy −(t) is sharply curtailed and small, as depicted in the figure. A peculiarity of this case is that the kinetic energy K(t) is conserved, even when only four Fourier modes are kept: this

D R A F T

November 11, 2002, 12:55pm

D R A F T

20

W.-S. DON ET AL.

indicates that energy conservation is not necessarily a good measure and can be misleading, as the solution is clearly very poorly resolved with only four modes. Thus, a conservative scheme is not necessarily an accurate scheme. Next, it will be argued that the computed enstrophy is also a poor measure of the production of vorticity when the flow becomes under-resolved at late times. In the Taylor-Green vortex flow, the flow develops a singularity at the center of a tightly twisted vortex core. The production of vorticity is proportional to the derivative of the velocity vector. When the flow becomes singular or nearly singular, numerical oscillations appear as the grid can no longer support the development of the steep gradients at the center of the vortex core. To demonstrate this, the kinetic energy and enstrophy field are shown at time t = 3 (when the flow is still smooth) and at time t = 5 (when the vorticity is largest) in Figures 8 and 9, respectively, with a spatial resolution of 1283 . The numerical solution develops small oscillations at early time (t < 4 at resolution 1283 and t < 2 at resolution 643 ), even when the solution is apparently smooth and well-behaved. At later times, the numerical solution using the sharp-cutoff filter (F-SF-23N) becomes highly-oscillatory and non-converging, but remains stable for a long-time integration, while all others schemes yield a stable and converging solution. As shown previously in Figure 2, the normalized total kinetic energy K(t)=K(0) is conserved and maintains a constant value of nearly unity. Coupled with the seemingly reasonable growth of the enstrophy and the conservation of the kinetic energy, this oscillatory state can be misinterpreted as a ‘transition to turbulence’, as a similar oscillatory solution can be obtained as the resolution is increased. The three-dimensional volume ‘streamrod’ of the enstrophy field computed using the Fourier collocation method with the 16th-order exponential filter (F-EF-16) and the WENO scheme (WENO-5) are shown in Figures 10 and 11 at times t = 3 and t = 5, respectively, at a spatial resolution of 2563 . The streamrod is a threedimensional-volume streamline trace with a defined thickness and a polygonal cross-

D R A F T

November 11, 2002, 12:55pm

D R A F T


21

section. The cross-section of a streamrod rotates around a volume streamline in accordance with the local stream-wise vorticity. The center of the streamrod is a regular, three-dimensional volume streamline. Streamrods have an orientation at each timestep, and the cross-section of the rod is a regular pentagon. These figures exhibit the small-scale structure of the evolving Taylor-Green vortex, and show the intense vorticity (enstrophy) produced in the vortex cores located near the corners of the computational domain.

6.

CONCLUSION

The present investigation considered the evolution of the nearly-incompressible, inviscid Taylor-Green vortex in three dimensions using spectral methods and a fifth-order WENO finite-difference method. The conservation properties and convergence of the numerical solutions were compared using two volume-integrated quantities–the kinetic energy and enstrophy. The spectral simulations were performed using an exponential and sharp-cutoff filter to stabilize the numerical computations. It was shown that the choice of filter strongly affects the computed kinetic energy and enstrophy: for the Taylor Green flow, the exponential filter yields a converged solution, but not the sharp-cutoff filter. It was demonstrated that extreme care is needed to use and interpret the computed kinetic energy and enstrophy in numerical assessments of numerical schemes at late evolution times, as the solution eventually becomes under-resolved. In particular, the computed enstrophy is not a physically meaningful quantity if the computation at late times is under-resolved and is not converged. Another important conclusion is that it is extremely useful to compare the results computed with one class of numerical schemes with the results of a method from a different class of numerical schemes, especially when an analytical solution is unavailable (as in the case of most nonlinear problems).

D R A F T

November 11, 2002, 12:55pm

D R A F T

22

W.-S. DON ET AL.

ACKNOWLEDGMENT This work was performed under AFOSR Grant No. F49620-02-1-0113, DOE Grant No. DEFG02-96ER25346, and under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

REFERENCES 1. D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Application, CBMS-NSF Regional Conference Series in Applied Mathematics Vol. 26 (SIAM, Philadelphia, 1977).

2. D. Gottlieb, M. Y. Hussaini, and S. A. Orszag, Spectral Methods for Partial Differential Equations (SIAM, Philadelphia, 1984).

3. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics (Springer-Verlag, New York, 1990).

4. C. Canuto, Spectral Methods, in Computational Fluid Dynamics, Proceedings of the Les Houches Summer School on Theoretical Physics 1993 Session LVIX, M. Lesieur, P. Comte and J. Zinn-Justin eds. (North-Holland, Amsterdam, 1996).

5. C. Bernardi and Y. Maday, Spectral Methods, in Handbook of Numerical Analysis Vol. 5, P. G. Ciarlet and J. L. Lions eds. (North-Holland, Amsterdam, 1997).

6. G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Mathematical Library (Cambridge University Press, Cambridge, 2000).

7. J. T. Beale, T. Kato, and A. Majda, Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations, Comm. Math. Phys. 94, 61 (1984).

8. G. Ponce, Remarks on a paper by J. T. Beale, T. Kato, and A. Majda, Comm. Math. Phys. 98, 349 (1985).

9. A. Majda, Vorticity and the Mathematical Theory of Incompressible Fluid Flow, Comm. Pure Appl. Math. 39, S187 (1986).

D R A F T

November 11, 2002, 12:55pm

D R A F T


23

10. S. A. Orszag, in Proceedings of the Symposium on Computer Methods in Applied Sciences and Engineering, R. Glowinski and J. L. Lions, eds., part II (Springer-Verlag, New York, 1974). 11. M. E. Brachet, D. I. Meiron, S. A. Orszag, B. G. Nickel, R. H. Morf, and U. Frisch, Small-scale structure of the Taylor-Green vortex, J. Fluid Mech. 130, 411 (1983). 12. M. Mossi, Simulation of Benchmark and Industrial Unsteady Compressible Turbulent Fluid Flows, Ph.D. thesis, École Polytechnique Fédérale de Lausanne (1999). 13. S. Benhamadouche and D. Laurence, Global kinetic energy conservation with unstructured meshes, Int. J. Num. Meth. Fluids 40, 561 (2002). 14. M. E. Brachet, M. Meneguzzi, H. Politano, and P. L. Sulem, The dynamics of freely decaying two-dimensional turbulence, J. Fluid Mech. 194, 333 (1988). 15. M. E. Brachet, D. I. Meiron, S. A. Orszag, B. G. Nickel, R. H. Morf, and U. Frisch, The Taylor-Green Vortex and Fully-Developed Turbulence, J. Stat. Phys. 34, 1049 (1984). 16. M. E. Brachet, Direct simulation of three-dimensional turbulence in the Taylor-Green vortex, Fluid Dyn. Res. 8, 1 (1991). 17. M. E. Brachet, M. Meneguzzi, A. Vincent, H. Politano, and P. L. Sulem, Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional Euler flows, Phys. Fluids A 4, 2845 (1992). 18. R. H. Morf, S. A. Orszag, and U. Frisch, Spontaneous Singularity in Three-Dimensional Inviscid Incompressible Flow, Phys. Rev. Lett. 44, 572 (1980). 19. K. Ohkitani and J. D. Gibbon, Numerical study of singularity formation in a class of Euler and Navier-Stokes flows, Phys. Fluids 12, 3181 (2000). 20. P. G. Saffman, Dynamics of vorticity, J. Fluid Mech. 106, 49 (1981). 21. C.-W. Shu and S. Osher, Efficient Implementation of Essentially Non-oscillatory ShockCapturing Schemes, J. Comp. Phys. 77, 439 (1988). 22. D. Gottlieb and J. S. Hesthaven, Spectral methods for hyperbolic problems, J. Comp. Appl. Math. 128, 83 (2001).

D R A F T

November 11, 2002, 12:55pm

D R A F T

24

W.-S. DON ET AL.

23. H. Vandeven, Family of Spectral Filters for Discontinuous Problems, J. Sci. Comp. 24, 37 (1992). 24. H. O. Kreiss and J. Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus 24, 199 (1972). 25. L. Jameson, High Order Schemes for Resolving Waves: Number of Points per Wavelength, J. Sci. Comp. 15, 417 (2000). 26. X.-D. Liu, S. Osher, and T. Chan, Weighted Essentially Non-oscillatory Schemes, J. Comp. Phys. 115, 200 (1994). 27. G.-S. Jiang and C.-W. Shu, Efficient Implementation of Weighted ENO Schemes, J. Comp. Phys. 126, 202 (1996). 28. D. Balsara and C.-W. Shu, Monotonicity Preserving Weighted Essentially Non-Oscillatory Schemes with Increasingly High Order of Accuracy, J. Comp. Phys. 160, 405 (2000). 29. C.-W. Shu, Essentially Non-oscillatory and Weighted Essentially Non-oscillatory Schemes for Hyperbolic Conservation Laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics Vol. 1697 (Springer-Verlag, New York, 1998). 30. C.-W. Shu, High Order ENO and WENO Schemes for Computational Fluid Dynamics, in High-Order Methods for Computational Physics, T. J. Barth and H. Deconinck eds. (SpringerVerlag, New York, 1999). 31. A. Harten, S. Osher, B. Engquist, and S. R. Chakravarthy, Some Results on Uniformly HighOrder Accurate Essentially Nonoscillatory Schemes, Appl. Num. Math. 2, 347 (1986). 32. A. Harten and S. Osher, Uniformly High-Order Accurate Non-Oscillatory Schemes, SIAM J. Num. Anal. 24, 279 (1987). 33. A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy, Uniformly High Order Essentially Non-Oscillatory Systems, III, J. Comp. Phys. 71, 231 (1987). 34. C.-W. Shu and S. Osher, Efficient Implementation of Essentially Non-oscillatory ShockCapturing Schemes, II, J. Comp. Phys. 83, 32 (1989).

D R A F T

November 11, 2002, 12:55pm

D R A F T


25

35. E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM J. Num. Anal. 26, 30 (1989). 36. E. Tadmor, Shock Capturing by the Spectral Viscosity Method, in Proceedings of ICOSAHOM 89, IMACS (North-Holland, Amsterdam, 1989). 37. Y. Maday, S. Ould Kaber, and E. Tadmor, Legendre Pseudospectral Viscosity Method for Nonlinear Conservation Laws, SIAM J. Num. Anal. 30, 321 (1993). 38. H. Ma, Chebyshev-Legendre Super Spectral Viscosity Method for Nonlinear Conservation Laws, SIAM J. Num. Anal. 35, 869 (1998).

D R A F T

November 11, 2002, 12:55pm

D R A F T

26

30

30

25

25

20

20

Enstrophy

Enstrophy

W.-S. DON ET AL.

15

15

10

10

5

5

0

1

2

3

4

5

6

0

1

2

30

30

25

25

20

20

15

10

5

5

1

2

3

4

5

6

0

Time

FIG. 1.

4

5

6

4

5

6

15

10

0

3

Time

Enstrophy

Enstrophy

Time

1

2

3

Time

Time-evolution of the normalized enstrophy Ω(t)/Ω(0) using: the Fourier collocation

method with the 16th-order exponential filter (F-EF-16) (top left), the Fourier collocation method with the 10th-order exponential filter and Nc =

3N 8 2

(F-EF-10-38N) (top right), the Fourier

collocation method with the sharp-cutoff filter with Nc = 2/3N (F-SF-23N) (bottom left), and the fifth-order WENO method (WENO-5) (bottom right). The solid line, the dashed line, and the dot-dashed line correspond to N = 64, 128 and 256 collocation points, respectively.

D R A F T

November 11, 2002, 12:55pm

D R A F T