Numerical Convergence Study of Nearly- Incompressible ......of Fourier spectral methods and the fifth-order weighted essentially non-oscillatory finite-difference method used are

Numerical Convergence Study of Nearly-Incompressible, Inviscid Taylor-Green Vortex Flow

Wai-Sun Don, David Gottlieb, Chi-Wang ShuDivision of Applied Mathematics, Brown University, 182 George Street,

Providence, RI 02912

E-mail: [email protected], [email protected], [email protected]

and

Oleg Schilling, Leland JamesonUniversity 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, in-

viscid 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 conver-

gence 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 addi-tion, 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 esti-

mate the statistical properties and large-scale transport. The numerical meth-

ods 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 polyno-

mial 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


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 com-

pressible Euler equations, spectral methods conserve quantities such as mass, mo-

mentum, 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 high-

order methods such as compact finite-difference methods and the weighted essen-

tially 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 vor-

ticity 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 nu-

merical 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.


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 pe-

riodic 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 develop-

ment 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 poly-

nomial 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 sharp-

cutoff 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



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 equa-

tions are the three-dimensional, compressible Euler equations in Cartesian coordi-


6 W.-S. DON ET AL.

nates on a [0; 2¼]£ [0; 2¼]£ [0; 2¼] periodic domain,

@½

@t+∇¢ (½u) = 0 ; (1)

@

@t(½u) +∇¢ (½u− u) = ¡∇p ; (2)

@E

@t+∇¢ [(E + p)u] = 0 ; (3)

where ½(x; t) is the density, u(x; t) = (u; v; w) is the velocity vector, p(x; t) is the

pressure, andE(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 =½u ¢ u

2+

p

° ¡ 1(4)

with the ratio of specific heats ° = 1:4.

The vorticity equation that follows from Eq. (2) is

@ω

@t+ (u ¢∇)ω = (ω ¢∇)u¡ω∇ ¢ u+

∇½£∇p½2

; (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 equa-

tion is

@

@t

³½u ¢ u2

´+∇¢

³½u ¢ u2

u´= ¡u ¢∇p ; (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

@

@t

³½u ¢ u2

´+∇¢

³½u ¢ u2

u+ pu´= 0 ; (7)



so that the kinetic energy (volume integrated)

K(t) ´Z 2¼

0

Z 2¼

0

Z 2¼

0

½u ¢ u2

dxdy dz (8)

is conserved in time.

It follows from the vorticity equation (5) that the enstrophy density equation is

@

@t

³½ω ¢ω2

´+∇¢

µ½u−ω

2ω

¶= ½ω¢

∙(ω ¢∇)u¡ω∇ ¢ u+

∇½£∇p½

¸; (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 ¢ ω2

dx dy dz : (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 propor-

tional 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


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 im-

plies 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]

limt"t¤ kω(x; t)k1 " 1 (11)

and

limt"t¤

Z t¤

0

kω(x; t)k1 dt " 1 ; (12)

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



and periodic boundary conditions on [0; 2¼]3. The initial density and pressure are

½(x; 0) = 1 ; (17)

p(x; 0) = p0 +½

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

Green 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 use-

ful 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) =

Z 2¼

0

Z 2¼

0

Z 2¼

0

½(x; 0)u(x;0) ¢ u(x; 0)2

dxdy dz (19)

=1

4

Z 2¼

0

Z 2¼

0

Z 2¼

0

[1¡ cos (2x) cos (2y)] cos2 (z) dxdy dz

= ¼3 ;

which is conserved for the inviscid Taylor-Green vortex, so that a quantitative mea-

sure 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


10 W.-S. DON ET AL.

conserve the kinetic energy very accurately to late times. As the nonlinear inter-

actions 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 compu-

tation 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 two-

dimensional, three-dimensional Taylor-Green vortex flow were compared to power-

series solutions up to t80 [11, 15]. A subsequent calculation of the symmetric Taylor-

Green 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 [bu(k; t) is the Fourier transform of the velocity field u(x; t)]

−(t) =1

2

Xk

k2 jbu(k; t)j2 (20)

=1

2

1Xk=0

A(2k) t2k

corresponding to the inviscid case was calculated numerically with 29 digits of pre-

cision up to O(t44) [18]; this involves derivatives of the velocity field with wavenum-

bers 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



the singularities using Taylor series expansions, Domb-Sykes plots, and Padé ap-

proximant techniques (see [19]). The expansion coefficients A(2k) are obtained by

calculating time derivatives of the velocity @nbu(k; 0)=@tn using the Navier-Stokesequation recursively. The convergence radius was determined by imaginary time

singularities at t ¼ p5i. The existence of a real time singularity was studied using

analytical continuation, and Padé approximants indicated a possible real singular-

ity 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 li-

brary for high-order accurate, adaptive domain simulations of the compressible

Euler equations in one, two, and three spatial dimensions. The formal spatial accu-

racy 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 accu-

racy 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 re-

gions 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-


12 W.-S. DON ET AL.

encing:

φ(1) = φn + L(φn)¢t ; (21)

φ(2) =3φn +φ(1) + L(φ(1))¢t

4; (22)

φn+1 =φn + 2φ(2) + 2L(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 ∙ CN1=2¡p

"Z 2¼

0

¯@pf(k)

@kp

¯2dk

#1=2

; (24)

where fN(x) is the Fourier approximation to the function f(x), with similar expres-

sions 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 : (25)



A related result is that in order to resolve a wave using Fourier collocation meth-

ods 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 nu-

merical 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 maxi-

mum 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 nonlin-

ear adaptive procedure is used to automatically choose the locally smoothest stencil

so as to avoid crossing discontinuities in the interpolation procedure as much as pos-

sible. ENO and WENO methods have been quite successful in applications, espe-

cially 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,

1The 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 us-

ing Chebyshev and Legendre collocation methods with PseudoPack. The PseudoPack library is

contained within the WS-Adaptive library.


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 char-

acteristics 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 dissipa-

tion 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 ad-

ditional 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 accu-

racy 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 con-

verges. 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

@t+

@f(Á)

@x= 0 : (26)

A spectral approximation involves seeking a polynomial of degree N , ÁN(x; t), such

that

@ÁN

@t+

@fN(ÁN)

@x= 0 : (27)



In the SVV method, the equation is regularized by an additional dissipative term

@ÁN

@t+

@fN(ÁN)

@x= ² (¡1)

s+1 @2sÁN

@x2s: (28)

In [38] it was shown that if ² = ®N1¡2s, where s » logN , 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

@ÁN

@t+

@fN(ÁN)

@x= 0 (29)

and in the second step

@ÁN

@t= ² (¡1)

s+1 @2sÁN

@x2s: (30)

Note that the Fourier polynomials ei¼kx are the eigenfunctions of the operator

@2=@x2 with eigenvalues k2. Thus if

ÁN(x; t) =NX

k=0

Ák(t) ei¼kx ; (31)

then solving the second step analytically over one timestep is equivalent to modi-

fying the Fourier coefficients Ák(t) as

Ák(t+¢t) = Ák(t) exp

"¡®N ¢t

µk

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) =NX

k=¡N

fk ei¼kx ; (33)

construct a filtered sum

f¾N(x) =

NXk=¡N

¾(k=N) fk ei¼kx : (34)


16 W.-S. DON ET AL.

Following Vandeven [23], define a p > 1 order filter function ¾(!) 2 C1[¡1; 1]

satisfying

¾(0) = 1 ; ¾(§1) = 0 ;

@j¾(!)@!j

¯!=0

= 0 ; @j¾(!)@!j

¯!=§1

= 0 ; j ∙ p: (35)

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><>: 1 jkj ∙ Nc

exp³¡®

¯k¡Nc

N¡Nc

¯p´jkj > Nc

; (36)

where k = ¡N; : : : ;N , Nc is the cutoffmode, ® = ¡ 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



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 respec-

tive 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 =23N2 , where N is the total number of the collocation

points, is denoted as (F-SF-23N). For the Fourier collocation method with expo-

nential 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


18 W.-S. DON ET AL.

exponential filter and cutoff mode Nc =38N2 (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 colloca-

tion 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 late-

time 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.



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 en-

strophy in qualitative agreement with theory, and conserves the kinetic energy

as stipulated by the conservation principles of the Euler equations in the nearly-

incompressible 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 highly-

nonlinear 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 sharp-

cutoff 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 sharp-

cutoff 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


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 three-

dimensional-volume streamline trace with a defined thickness and a polygonal cross-



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 con-

vergence of the numerical solutions were compared using two volume-integrated

quantities–the kinetic energy and enstrophy. The spectral simulations were per-

formed using an exponential and sharp-cutoff filter to stabilize the numerical com-

putations. 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, es-

pecially when an analytical solution is unavailable (as in the case of most nonlinear

problems).


22 W.-S. DON ET AL.

ACKNOWLEDGMENT

This work was performed under AFOSR Grant No. F49620-02-1-0113, DOE Grant No. DE-

FG02-96ER25346, 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.

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26 W.-S. DON ET AL.

Time

En

stro

ph

y

0 1 2 3 4 5 6

5

10

15

20

25

30

Time

En

stro

ph

y

0 1 2 3 4 5 6

5

10

15

20

25

30

Time

En

stro

ph

y

0 1 2 3 4 5 6

5

10

15

20

25

30

Time

En

stro

ph

y

0 1 2 3 4 5 6

5

10

15

20

25

30

FIG. 1. 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 =38N2(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.


Documents

Numerical Convergence Study of Nearly- Incompressible ......of Fourier spectral methods and the fifth-order weighted essentially non-oscillatory finite-difference method used are