Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840

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Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier .com/locate /cma

Improving stability of stabilized and multiscale formulations in flow simulationsat small time steps

M.-C. Hsu a,*, Y. Bazilevs a, V.M. Calo b,c, T.E. Tezduyar d, T.J.R. Hughes e

a Department of Structural Engineering, University of California, San Diego, 9500 Gilman Drive, Mail Code 0085, La Jolla, CA 92093, USAb Earth and Environmental Science and Engineering, King Abdullah University of Science and Technology, P.O. Box 55455, Jeddah 21534, Saudi Arabiac Applied Mathematics and Computational Science, King Abdullah University of Science and Technology, P.O. Box 55455, Jeddah 21534, Saudi Arabiad Mechanical Engineering, Rice University – MS 321, 6100 Main Street, Houston, TX 77005, USAe Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th Street, 1 University Station C0200, Austin, TX 78712, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 June 2008Received in revised form 2 May 2009Accepted 27 June 2009Available online 4 July 2009

Keywords:Variational multiscale methodsStabilized methodsAdvection–diffusion equationElement-vector-based sIncompressible Navier–Stokes equationsTurbulence modelingTurbulent channel flow

0045-7825/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cma.2009.06.019

* Corresponding author.E-mail address: m5hsu@ucsd.edu (M.-C. Hsu).

The objective of this paper is to show that use of the element-vector-based definition of stabilizationparameters, introduced in [T.E. Tezduyar, Computation of moving boundaries and interfaces and stabil-ization parameters, Int. J. Numer. Methods Fluids 43 (2003) 555–575; T.E. Tezduyar, Y. Osawa, Finite ele-ment stabilization parameters computed from element matrices and vectors, Comput. Methods Appl.Mech. Engrg. 190 (2000) 411–430], circumvents the well-known instability associated with conventionalstabilized formulations at small time steps. We describe formulations for linear advection–diffusion andincompressible Navier–Stokes equations and test them on three benchmark problems: advection of an L-shaped discontinuity, laminar flow in a square domain at low Reynolds number, and turbulent channelflow at friction-velocity Reynolds number of 395.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Residual-based variational multiscale methods were recentlyadvocated and successfully used for turbulence modeling in theLES regime in [3]. The main idea of the variational multiscale ap-proach is the a-priori decomposition of the underlying functionalweighting and solution spaces into coarse and fine scales. Thecoarse scales are associated with the numerical approximation,while the fine scales are associated with the subgrid scalesand, as a result, require modeling. Residual-based models wereproposed in [3], where the fine-scale field was assumed to beproportional to the residuals of the coarse scale. The proportion-ality factors, the so-called s’s, are obtained through asymptoticscaling arguments that were developed in stabilized methodstheory over the last three decades (see, e.g., [9,23,35,37,29,40,38,21,10,41]). Recently s’s were identified with low-ordermoments of the fine-scale Green’s function, a fundamental objectin the design and analysis of variational multiscale methods (see[27]).

Stabilization parameters play a critical role in the success of thestabilized and variational multiscale methodologies. Despite dec-

ll rights reserved.

ades of research devoted to them, several outstanding issues re-main. One such issue, that to this day generates much debate inthe community, is the dependence of s’s on the time step fortime-dependent simulations. In particular, conventional defini-tions of s’s that include the time step are not well-behaved forsmall time steps in both time-dependent and steady-state regimes.On the contrary, conventional definitions of s’s that do not includethe time step size are not sufficiently robust for complex flow sit-uations. Evidence of the latter fact will be shown in the numericalexample portion of this paper.

In recent works, Harari [17] and Harari and Hauke [18] exam-ined the small time step problem in the context of diffusion andadvection–diffusion problems, respectively, and introduced simplemodifications to s’s to account for the small time step limit.

The small time step deficiency was also addressed in Codinaet al. [15] in the context of variational multiscale methods forincompressible flow by means of so-called ‘‘dynamic subgridscales.” In [15] an ordinary differential equation and asymptoticscaling arguments are used to advance the fine-scale field in time.The fine-scale field becomes a ‘‘history variable” that needs to bestored at each integration point, leading to a computational struc-ture that is similar to that for inelastic constitutive equations incomputational solid mechanics (see, e.g., Simo and Hughes [36]).Calculations and an analytical stability analysis confirm the good

M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840 829

behavior of the dynamic subgrid scales approach [15]. A simplifiedimplementation that only requires an additional vector of the sizeof the global unknowns may be found in the recent article by Hou-zeaux and Principe [20].

In this article, we present another approach that is based on theelement-vector-based (EVB) definition of s’s proposed in [39,37].In this methodology, s’s are built by examining relative magni-tudes of the terms appearing in the variational equations, whichis yet another form of a scaling argument. Our calculations confirmthat this methodology is robust for all time step sizes. Furthermore,an appropriate definition of the stabilization parameter for thesteady flow regime is recovered.

In Section 2 we present the continuous and discrete versions ofthe variational formulation of the linear time-dependent advec-tion–diffusion equation, show the details of the formulation ofthe conventional and EVB stabilization parameters, and comparetheir performance on the benchmark example of advection of anL-shaped discontinuity. In Section 3 we present the residual-basedvariational multiscale formulation of the incompressible Navier–Stokes equations and compare the performance of the conven-tional and EVB stabilization parameters on two benchmark prob-lems: laminar flow in a square domain at low Reynolds numberand turbulent channel flow at friction-velocity Reynolds numberof 395.

In all numerical calculations we use the generalized-a methodfor time integration with the high frequency damping parameterq1 set to 0.5 (see, e.g., [14,31]). We use B-splines of maximal con-tinuity for the spatial discretization as we have demonstratedthese provide superior accuracy and robustness compared withC0-continuous finite elements in flow simulations (see, e.g., [3–6,22]). Nevertheless, we feel the main conclusions have more gen-eral applicability, and, in particular, are applicable to low-order fi-nite elements. These are presented in Section 4.

2. Advection–diffusion equation

2.1. Continuous problem

Consider the following time-dependent advection–diffusionequation for /:

L/ ¼ f in X; ð1Þ

where

L/ ¼ @/@tþ a � $/� $ � ðj$/Þ; ð2Þ

in which f is the given source, a is the solenoidal velocity field, j isthe diffusivity, and X is the spatial domain of the problem. Theessential and natural boundary conditions associated are:

/ ¼ g on CD; ð3Þj$/ � n ¼ h on CN; ð4Þ

where CD and CN are the Dirichlet and Neumann parts of theboundary C of the domain X; g and h are the prescribed data, andn is the unit outward boundary normal.

Let Vg and V0 denote the trial solution and weighting functionspaces, respectively, where subscripts g and 0 refer to the essentialboundary conditions. The variational counterpart of Eq. (1) reads:find / 2Vg such that 8w 2V0,

w;@/@tþ a � $/

� �X

þ ð$w;j$/ÞX ¼ ðw; f ÞX þ ðw;hÞCN; ð5Þ

where ð�; �ÞX and ð�; �ÞCNdenote the L2-inner products on X and CN ,

respectively.

2.2. Discrete formulation

Let Vhg and Vh

0 be finite-dimensional subspaces of Vg and V0,respectively. We state the semi-discrete formulation of the advec-tion–diffusion problem as follows: find /h 2Vh

g such that8wh 2Vh

0,

wh;@/h

@tþ a � $/h

!X

þ $wh;j$/h� �

X� ðwh; f ÞX � ðwh; hÞCN

þXnel

e¼1

ðsa � $wh;L/h � f ÞXe¼ 0: ð6Þ

Variational equation (6) is the SUPG formulation of the advec-tion–diffusion problem [9]. In the sequel we first provide the con-ventional definitions of the stabilization parameter s appearing in(6), followed by the EVB definition, first proposed in [39,37].

2.2.1. Conventional definition of s’sThe following definition of s is often employed in practice (see,

e.g., [28,34,3])

s ¼ 4Dt2 þ a � Gaþ CIj2G : G� ��1=2

; ð7Þ

where Dt is the time step size, CI is a positive constant, independentof the mesh size, derived from the element-wise inverse estimate(see, e.g., [32]),

Gij ¼X3

k¼1

@nk

@xi

@nk

@xj; ð8Þ

G : G ¼X3

i;j¼1

GijGij; ð9Þ

a � Ga ¼X3

i;j¼1

aiGijaj; ð10Þ

and @n=@x is the inverse Jacobian of the element mapping betweenthe parametric and physical domains.

Although often employed in practice, s defined by Eq. (7) is sub-ject to the following criticisms:

(1) Definition (7) is not robust for small time steps. Namely, inthe limit of Dt ! 0, the Dt term becomes dominant ands! 0. As a result, formulation (6) reverts to the Galerkin’smethod, which, in turn, is known to be unstable for theadvection-dominated case. As a result, the practitioner isforced to select a time step that is not ‘‘too small” to avoidnumerical instability associated with the small time steplimit.

(2) Definition (7) may not be suitable for steady solutionsobtained in time-dependent computations because thesteady state depends on Dt through (7).

In view of these remarks, the Dt dependence in s is sometimesomitted and s takes on the following definition:

s ¼ ða � Gaþ CIj2G : GÞ�1=2: ð11Þ

Although favorable in some situations, such as steady solutions,simply omitting Dt from s may lead to divergent results in practicalcomputations. We will demonstrate this in the numerical exam-ples section of this paper.

2.2.2. Element-vector-based (EVB) sLet Na; a ¼ 1; . . . ;nsh, be the element-level basis functions, and

nsh be the number of element-level basis functions. Let ð�; �ÞXede-

note the L2-inner product over the element domain Xe correspond-ing to element e. We define the following element-level vectors:

= 0

= 0

= 0= 1

= 1

= 0

a = cos ,sin( )= 10 6, L = 1

= 45°

1

2L

1

4L

Fig. 1. Advection of an L-shaped front. Problem description.

Min = 0.0000, Max = 1.0000

830 M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840

cV ¼ fcag;ca ¼ ðNa;a � $/hÞXe

; ð12Þ~kV ¼ f~kag;~ka ¼ ða � $Na;a � $/hÞXe

; ð13Þ~cV ¼ f~cag;

~ca ¼ a � $Na;@/h

@t

!Xe

: ð14Þ

The components of the EVB s are:

s�11 ¼

k~kVkkcVk

; ð15Þ

s�12 ¼

k~cVkkcVk

; ð16Þ

s�13 ¼ s�1

1 Pe�1; ð17Þ

where Pe is an element-level Peclet number,

Pe ¼ kak2

jkcVkk~kVk

: ð18Þ

In Eqs. (15)–(18), k � k denotes the l2-norm. For example, for thevector cV ,

kcVk ¼X

a

c2a

!1=2

: ð19Þ

We define s as

s ¼ s�21 þ s�2

2 þ s�23

� ��1=2: ð20Þ

Remarks

(1) The s2 term defined in Eq. (16) directly brings in the depen-dence of s on the time step size. In the case when the solu-tion is changing in time, this term is active, while in thesteady regime, it is identically zero. As a result, this con-struction automatically selects a s appropriate for differentsolution regimes, making it attractive for practicalapplications.

(2) The definition of s given by Eqs. (15)–(20) can be seen as anonlinear definition because it depends on the solution evenin linear problems. However, in marching from time level nto nþ 1, the element vectors can be evaluated at level n. Thiseliminates the need of solving a nonlinear discrete system ineach time step when the continuous problem is linear.

(3) The definition of s given by Eqs. (15)–(20) makes s an ele-ment-level constant. One may also define an EVB s for everyglobal basis function in the formulation by taking a to be aglobal basis function index and avoiding summation overa’s (this option, as well as other variants, were pointed outin [37,39]). In fact, we may go as far as recommending thistechnique for computations that employ truly global basisfunctions, such as spectral methods, where conventionaldefinitions of s (i.e., (7) and (11)) are not directly applicabledue to the global support of the basis functions. However,we note that no experience has yet been gained with meth-ods using global basis functions.

(4) Note that, in contrast to the original references [39,37], theelement Peclet number Pe (see Eq. (18)) is defined usingthe element-vector rather than element matrix arrays. Thisis done in the interest of simplicity. It is believed that bothdefinitions will give similar results.

(5) In Refs. [11–13] the authors successfully employed theelement-matrix-based variant of s for compressible flowcomputations.

(6) Note that the EVB s was designed using scaling arguments todeliver accuracy comparable to the conventional s.

2.3. Tests with an L-shaped discontinuity advected skew to mesh

We test the proposed methodology on the advection of anL-shaped front benchmark originally presented in [4]. The problemset-up is given in Fig. 1. The diffusivity j ¼ 10�6, the advection an-gle is chosen to be 45� and the magnitude of the advective speed isset to unity. The domain is a unit square subdivided into 20 � 20square elements. At time t ¼ 0 the value of the scalar field is setto unity in the interior of the L-shaped block located in the lowerleft-hand corner of the domain. Elsewhere in the domain /h isset to zero, creating an interior layer with an L-shaped concavefront as shown in Fig. 2. The solution is advanced in time untilt ¼ 0:25 as well as to the steady state. Given that the diffusion coef-ficient is very small compared to the advection velocity and themesh size, for all practical purposes the problem correspondsnumerically to pure advection. We will refer to the interior layeras the discontinuity front, and its location and shape at t ¼ 0:25are illustrated in Fig. 1 with a dashed line.

Fig. 2. Advection of an L-shaped front. Elevation plot of the initial condition.

Min = -0.0054, Max = 1.0320

(a) Conventional τ with Δt

Min = -0.0054, Max = 1.0299

(b) Conventional τ without Δt

Min = -0.0070, Max = 1.0285

(c) EVB τ

Fig. 3. Advection of an L-shaped front. Results using C5-continuous splines of order six. Elevation plot of the transient solution (interpolated with 40 � 40 bilinear elements)at t ¼ 0:25 with Dt ¼ 0:025.

Min = -0.0052, Max = 1.0148

(a) Conventional τ with Δt

Min = -0.0043, Max = 1.0054

(b) Conventional τ without Δt

Min = -0.0051, Max = 1.0128

(c) EVB τ

Fig. 4. Advection of an L-shaped front. Results using C5-continuous splines of order six. Elevation plot of the steady-state solution (interpolated with 40 � 40 bilinearelements) with Dt ¼ 0:025.

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We compare solutions produced with conventional s’s definedaccording to Eqs. (7) and (11) and the EVB version defined accord-ing to Eqs. (15)–(20). Both transient and steady-state solutions areconsidered. C5-continuous B-splines of degree six are employed forall test cases. Very high-order B-splines have been shown to beespecially effective for solutions of the advection–diffusionproblem with thin layers (see, e.g., [22,4]). Time stepsDt ¼ 0:025;Dt ¼ 0:00625, and Dt ¼ 0:003125 are used, thelast one being much smaller than what is required for time-accuracy.

Results for all three s’s computed at Dt ¼ 0:025 are shown inFigs. 3 and 4. At this time step the results are similar for all s’sfor both transient and steady-state solutions. A small but notice-able ‘‘wiggle” is present at the steady state for both the conven-tional s with Dt and EVB s. Nevertheless, the maximumovershoot does not exceed 1.5% and the solution quality is compa-rable in all cases.

Results for Dt ¼ 0:00625 are presented in Figs. 5 and 6. Tran-sient responses for all three cases are similar. Comparison of thesteady-state results for this time step reveals the deficiency forthe conventional s with small Dt: s is dominated by the Dt termand it is too small to guarantee a stable steady-state solution(see Fig. 6a). In contrast, both the conventional s without Dt and

the EVB s deliver nearly identical, stable, and accurate solutionsat the steady state.

Results for Dt ¼ 0:003125 are shown in Figs. 7 and 8. At thistime step, at t ¼ 0:25, the conventional s with Dt produces smallbut noticeable wiggles that propagate from the interior layer in-ward (see Fig. 7a). At the steady state, at this small time step, theeffect of the Dt term in the conventional definition of s leads to sig-nificant oscillations (see Fig. 8a), where stability is achieved for theconventional s without Dt and the EVB s. There are, once again,small oscillations for the EVB s.

For this problem, the conventional s without Dt generates solu-tions of somewhat better quality than the EVB s. However, as willbe shown in the sequel, the conventional definition of s without Dtis not sufficiently robust in more complicated situations.

3. Variational multiscale residual-based turbulence modeling

In this section we adapt the definition of the EVB s’s to theincompressible Navier–Stokes equations and make use of themin the context of the variational multiscale residual-based turbu-lence modeling (see [3]). EVB s’s for stabilized formulations ofthe incompressible Navier–Stokes equations were first proposedin [39,37].

Min = -0.0048, Max = 1.0097

(a) Conventional τ with Δt

Min = -0.0082, Max = 1.0044

(b) Conventional τ without Δt

Min = -0.0096, Max = 1.0039

(c) EVB τ

Fig. 5. Advection of an L-shaped front. Results using C5-continuous splines of order six. Elevation plot of the transient solution (interpolated with 40 � 40 bilinear elements)at t ¼ 0:25 with Dt ¼ 0:00625.

Min = -0.1734, Max = 1.2642

(a) Conventional τ with Δt

Min = -0.0043, Max = 1.0054

(b) Conventional τ without Δt

Min = -0.0051, Max = 1.0128

(c) EVB τ

Fig. 6. Advection of an L-shaped front. Results using C5-continuous splines of order six. Elevation plot of the steady-state solution (interpolated with 40 � 40 bilinearelements) with Dt ¼ 0:00625.

832 M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840

3.1. Continuous problem

We begin by considering a weak formulation of the incompress-ible Navier–Stokes equations. Let V represent both the trial solu-tion and weighting function spaces, which are assumed to be thesame (we assume for simplicity of exposition that u = 0 on C).We also assume that

RX pðtÞdX ¼ 0 for all t 2�0; T½. As before,

ð�; �ÞX denotes the L2-inner product with respect to the domain X.The variational formulation is stated as follows: findU ¼ fu; pg 2V such that 8W ¼ fw; qg 2V,

BðW;UÞ ¼ LðWÞ; ð21Þwhere

BðW;UÞ ¼ w;@u@t

� �X

� ð$w;u� uÞX � ð$ �w; pÞX þ ðq;$ � uÞX

þ ð$sw;2m$suÞX; ð22ÞLðWÞ ¼ ðw; f ÞX; ð23Þ

and

$su ¼ 12ð$uþ $uTÞ: ð24Þ

Here, f is the given body force (per unit mass), m is the kinematic vis-cosity and p is the pressure divided by the density.

Variational equations (21)–(23) imply weak satisfaction of thelinear momentum equations and incompressibility constraint,namely@u@tþ $ � ðu� uÞ þ $p� $ � ð2m$suÞ � f ¼ 0 in X; ð25Þ

$ � u ¼ 0 in X: ð26Þ

3.2. Discrete formulation

Below, we recall the discrete variational formulation of theincompressible Navier–Stokes equations (see [3]). We approximateEqs. (21)–(23) by the following variational problem over the finite-element spaces: find Uh ¼ fuh; phg 2Vh such that 8Wh ¼fwh; qhg 2Vh,

BðWh;UhÞ þ uh � $wh þ $qh; sMrM� �

X þ $ �wh; sCrC� �

X

þ ð$whÞT uh; sMrM

� �X� $wh; sMrM � sMrM� �

X

¼ LðWhÞ; ð27Þwhere

rMðuh; phÞ ¼ @uh

@tþ uh � $uh þ $ph � mDuh � f ; ð28Þ

rCðuhÞ ¼ $ � uh: ð29Þ

Min = -0.5710, Max = 1.5893

(a) Conventional τ with Δt

Min = -0.0043, Max = 1.0054

(b) Conventional τ without Δt

Min = -0.0051, Max = 1.0128

(c) EVB τ

Fig. 8. Advection of an L-shaped front. Results using C5-continuous splines of order six. Elevation plot of the steady-state solution (interpolated with 40 � 40 bilinear

Min = -0.0110, Max = 1.0125

(a) Conventional τ with Δt

Min = -0.0082, Max = 1.0045

(b) Conventional τ without Δt

Min = -0.0087, Max = 1.0040

(c) EVB τ

Fig. 7. Advection of an L-shaped front. Results using C5-continuous splines of order six. Elevation plot of the transient solution (interpolated with 40 � 40 bilinear elements)at t ¼ 0:25 with Dt ¼ 0:003125.

M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840 833

Remarks

(1) This is a residual-based variational multiscale method forincompressible Navier–Stokes equations (see [3]). For back-ground, see, e.g., [19,24–26,28,30]. The first term on the left-hand side of Eq. (27) is the Galerkin term; the next twoterms are classical stabilization terms; and the last twoterms are the additional terms produced by the variationalmultiscale method.

(2) Note that in the definition of the momentum residual givenby Eq. (28), we applied the incompressibility constraint tothe advective and the viscous stress terms. As a result, theadvective term appears in the convection form, while theviscous stress term appears in the Laplacian form (i.e., com-pare Eqs. (25) and (28)). Computational experience indicatesthat this form is more favorable for the stability of the dis-crete formulation.

elements) with Dt ¼ 0:003125.

3.2.1. Conventional definition of s’sWe recall the conventional definition of sM that is commonly

employed in practice and is form-identical to Eq. (7):

sM ¼4

Dt2 þ uh � Guh þ CIm2G : G� ��1=2

: ð30Þ

In Eq. (27), sC is given as

sC ¼ ðsMg � gÞ�1: ð31Þ

This definition of sC comes from the small-scale Shur complementoperator for the pressure (see [2]). In Eq. (31), g is a vector obtainedby summing @n=@x on its first index as

gi ¼X3

j¼1

@nj

@xi; ð32Þ

g � g ¼X3

i¼1

gigi: ð33Þ

As in the previous section on the time-dependent advection–diffusion equation, we also consider sM in which the dependenceon the time step size is omitted, that is,

sM ¼ uh � Guh þ CIm2G : G� ��1=2

: ð34Þ

3.2.2. EVB s’s for the incompressible Navier–Stokes equationsWe use the notation of the previous section. In addition, let ei be

the ith Cartesian basis vector. We define the following element-le-vel vectors:

u = u

u =sin x 0.7( )sin y + 0.2( )cos x 0.7( )cos y + 0.2( )

L = 1

cos x( )cos y( ) + sin x 0.7( )cos x 0.7( )+2 2 sin x 0.7( )sin y + 0.2( )

sin x( )sin y( ) sin y + 0.2( )cos y + 0.2( )+2 2 cos x 0.7( )cos y + 0.2( )

p = sin x( )cos y( ) + cos(1) 1( )sin(1)

= 1.0

f =

u = u

u = u

u = u

Fig. 9. Laminar flow in a square domain at low Reynolds number. Problemdescription.

834 M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840

cV ¼ ½ca;i�;ca;i ¼ Naei;uh � $uh

� �Xe; ð35Þ

~kV ¼ ½~ka;i�;~ka;i ¼ uh � rNaei;uh � $uh

� �Xe; ð36Þ

~krV ¼ ~kr

a;i

h i;

~kra;i ¼ r � $Naei; r � $uh

� �Xe; ð37Þ

~cV ¼ ½~ca;i�;

~ca;i ¼ uh � $Naei;@uh

@t

� �Xe

: ð38Þ

Fig. 10. Laminar flow in a square domain at low Reynolds number. Pressure contoursvelocity in the domain interior and zero pressure everywhere.

Fig. 11. Laminar flow in a square domain at low Reynolds number

In Eq. (36), r is the unit vector in the direction of the solutionabsolute value gradient,

r ¼ $kuhkk$kuhkk : ð39Þ

The three components of the EVB s are:

s�11 ¼

k~kVkkcVk

; ð40Þ

s�12 ¼

k~cVkkcVk

; ð41Þ

s�13 ¼ s�1

1mk~kr

VkkcVk

: ð42Þ

We define sM as

sM ¼ s�21 þ s�2

2 þ s�23

� ��1=2: ð43Þ

Remarks

(1) In Eqs.(40)–(42), as before, k � k is used to denote the vectorl2-norm, that is, for the vector cV ,

of the s

. Contou

kcVk ¼X

a;i

c2a;i

!1=2

: ð44Þ

(2) In Eq. (36), ~krV is introduced for boundary layer flows. Near

solid boundaries, the vector r points in the direction orthog-onal to the solid wall, and, as a result, for boundary layer ele-ments with high aspect ratios, the EVB s switches to theviscous limit with the mesh size h corresponding to the ele-ment wall-normal dimension.

olution at the first time step for Dt ¼ 0:01. The initial conditions are: zero

rs of the steady-state pressure solution for Dt ¼ 0:01.

Fig. 12. Laminar flow in a square domain at low Reynolds number. Pressure contours of the first- and second-step solutions for Dt ¼ 0:000001. The initial conditions are set tobe the steady-state solution from Dt ¼ 0:01.

Fig. 13. Laminar flow in a square domain at low Reynolds number. Contours of the steady-state pressure solution for Dt ¼ 0:000001.

Fig. 14. Turbulent channel flow. Problem set-up.

M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840 835

(3) The equation for sC remains unchanged (see Eq. (31)),although sM employed in its definition now comes fromEq. (43).

(4) Unlike in the original references [39,37], we make use of apurely vector-based definition for s3 in Eq. (42). This is donein the interest of simplicity and is well suited for matrix-freecomputations (not employed in this work).

(5) The sM does not depend on the pressure and is not the sameas the EVB sM for the Navier–Stokes equations proposed byTezduyar and Osawa [37,39]. As such, the present EVB sM

will have a different Stokes limit. This is an issue that isnot studied here, but warrants further investigation.

3.3. Laminar flow in a square domain at low Reynolds number

The small time step problem also appears when stabilizedrather than BB-stable formulations are employed for the time-dependent Stokes (see e.g., [8,7]) as well as low Reynolds number

836 M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840

Navier–Stokes (see [15]) problems. To examine the performance ofthe different s’s in this regime, we consider a test case originallyproposed in [7] in the context of the Stokes equations, and solvedin [15] using the Navier–Stokes equations. We consider the lattercase. The problem’s set-up and its analytical solution are presentedin Fig. 9.

We solve the problem using Dirichlet boundary conditions onthe velocity field. The velocity boundary condition is obtained asfollows: we perform an L2 projection of the analytical velocity solu-tion onto our discrete space, which consists of C1-continuous qua-dratic B-splines. The control variables at the domain boundary areassigned Dirichlet boundary conditions with values that come fromthe L2 projection. To assign the initial conditions, we set the veloc-

U+

100 1010

5

10

15

20

25

Δ t = 0.1

Quadratic

(a) Mean stream-w

w+

0

0.5

1

1.5

v+

0

0.5

1

τM, conv w/o ΔtDNS

u+

0 100 20

0.5

1

1.5

2

2.5

3

3.5 τM, conv w/o Δt

τM, EVB τM, co

DNS

(b) Velocity flu

Fig. 15. Turbulent channel flow at Res ¼ 395 using quadratic NURBS. Results compu

ity to zero in the domain interior and the pressure to zero every-where in the domain, including the boundary. We use a uniformmesh of 80 � 80 elements.

We solve the problem at the large time step of Dt ¼ 0:01using all three definitions of s. The fluid pressure is shown inFigs. 10 and 11, which correspond to the solution at the endof the first time step and the steady-state solution, respectively.In all cases the solutions are stable. At the steady state, in theinterior of the domain, the discrete pressure solution is veryclose to its analytical counterpart. However, at the domainboundary, the pressure solution develops a boundary layer,which is a well-known artifact that occurs in stabilized formula-tions of Stokes equations as well as Navier–Stokes equations in

y+102

τM, EVB

τM, conv w/o Δt

τM, conv w/ Δt

DNS

ise velocity

τM, EVBτM, conv w/ Δt

y+

00 300 400

nv w/ Δt

ctuations

ted at Dt ¼ 0:1 using the conventional sM with and without Dt, and the EVB sM .

M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840 837

the low Reynolds number regime. (See [16] for an explanation ofthis phenomenon and a simple remedy, which was not imple-mented in this work.)

The sensitivity of the three formulations to small time steps isstudied as follows. We take the steady-state solution from thelarge time step of Dt ¼ 0:01 as the initial condition for the simu-lation at Dt ¼ 0:000001. This is the smallest time step employedin [7,15]. This procedure of starting with the initial conditionsthat are close to the steady-state solution of the underlying equa-tions is similar to that employed in [15]. The expected result isthat the solution does not change, that is, the steady-state resultsare not sensitive to the definition of the stabilization parameter.

U+

100 1010

5

10

15

20

25

Conventional τM with ΔtQuadratic

Δt =

(a) Mean strea

w+

0

0.5

1

1.5

v+

0

0.5

1

u+

0 100 20

0.5

1

1.5

2

2.5

3

3.5 Δt = 0.00625

Δt = 0.00156

DNS

Δt = 0.025

Δt = 0.000390625 Δt = 0.0

(b) Velocit

Fig. 16. Turbulent channel flow at Res ¼ 395 using quadratic NURBS. Results usiDt ¼ 0:000390625.

Fig. 12 shows the pressure solution for the first two time stepsfor all methods considered. The solution remains nearly identicalto the initial condition for the cases of the EVB s and conventionals without Dt. On the other hand, for the conventional s with Dt,the pressure solution rapidly departs from the initial condition,which means that the steady-state solution is sensitive to thisdefinition of the stabilization parameter. Also note that the pres-sure changes sign from the first to the second time step, indicat-ing oscillatory-in-time behavior. Despite the rapid transient in thecase of the conventional s with Dt, the solution reaches a steadystate that is very similar to that of the other s definitions (seeFig. 13).

y+

102

Δt = 0.000390625

0.0015625

Δt = 0.025

Δt = 0.00625

DNS

m-wise velocity

y+

00 300 400

Δt = 0.000390625

25

Δt = 0.00625 Δt = 0.025015625

y fluctuations

ng conventional sM with Dt for Dt ¼ 0:025;Dt ¼ 0:00625;Dt ¼ 0:0015625 and

y+

U+

100 101102

0

5

10

15

20

25

EVB τM

Quadratic

DNS

Δt = 0.025

Δt = 0.0015625

Δt = 0.000390625

Δt = 0.00625

(a) Mean stream-wise velocity

w+

0

0.5

1

1.5

v+

0

0.5

1

Δt = 0.000390625Δt = 0.00625Δt = 0.025 Δt = 0.0015625

y+

u+

0 100 200 300 4000

0.5

1

1.5

2

2.5

3

3.5

Δt = 0.00625

Δt = 0.000390625

DNS

Δt = 0.025

Δt = 0.0015625

(b) Velocity fluctuations

Fig. 17. Turbulent channel flow at Res ¼ 395 using quadratic NURBS. Results using EVB sM for Dt ¼ 0:025;Dt ¼ 0:00625;Dt ¼ 0:0015625 and Dt ¼ 0:000390625.

838 M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840

Remarks

(1) Despite the sensitivity to the time step size, no spatial pres-sure instability was observed for any of the three s defini-tions. Even in the case of the small time step and theconventional s with Dt, the transient pressure solutionremained smooth. This is in contrast to the results of[7,15], and may be attributable to the differences in the spa-tial discretization used here and in the aforementionedreferences.

(2) We would also like to point out that in [39] the authors usedsM defined in Eq. (43) on the test problem of vortex sheddingabout a circular cylinder at low Reynolds number Re ¼ 100.Because of the non-uniformity of the mesh used, the local

Courant number (i.e. the normalized time step size) variedfrom approximately 1.0 to as small as 0.04. The resultsshowed that even at such small time steps, the element-vec-tor-based s was stable and yielded an accurate solution.

3.4. Turbulent channel flow at Res ¼ 395

Our last numerical example is an equilibrium turbulent channelflow at Reynolds number 395 based on the friction velocity andchannel half-width. The problem set-up is shown in Fig. 14. Thecomputational domain is a rectangular box of size 2p� 2� 2

3 p inthe stream-wise, wall-normal, and span-wise directions, respec-

M.-C. Hsu et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 828–840 839

tively. A no-slip Dirichlet boundary condition is set at the wallðy ¼ �1Þ, while the stream-wise and the span-wise directions areassigned periodic boundary conditions. The flow is driven by a con-stant pressure gradient, fx, acting in the stream-wise direction. Thevalues of the kinematic viscosity m and the forcing fx are set to1:472� 10�4 and 3:37204� 10�3, respectively, which results in abulk stream-wise velocity of unity.

The computations are performed on a mesh of 323 elements andthe mesh is graded in the wall-normal direction to better capturethe boundary layer. For the spatial discretization we employ iso-geometric analysis (see, e.g., [22,3,5,1]) and C1-continuous qua-dratic B-splines are considered for this test case. Numericalresults are reported in the form of statistics of the mean velocityand root-mean-square velocity fluctuations. Statistics are obtainedby sampling the solution field at the mesh knots and averaging inthe stream-wise and span-wise directions as well as in time. Com-parison of the statistical quantities of interest with the DNS data ofMoser et al. [33] is made in order to assess the accuracy androbustness of the proposed procedure. All the results are presentedin non-dimensional wall units.

We compare the numerical results obtained using the conven-tional definition of sM , with and without the Dt term, and theEVB definition of sM . We consider five time step sizes: Dt ¼ 0:1;Dt ¼ 0:025;Dt ¼ 0:00625;Dt ¼ 0:0015625 and Dt ¼ 0:000390625.Fig. 15 corresponds to the simulation at Dt ¼ 0:1. At this large timestep the solutions are very similar for all sM ’s, while the conven-tional sM with the Dt term showing a slightly more accurate meanflow result.

We note that Dt ¼ 0:1 is the only time step out of the five con-sidered for which we were able to generate a numerical solutionusing the conventional sM without Dt. For all other time steps weobserved rapid divergence with this sM . It is now well known fromnumerous test computations carried out by different researchersthat some problems require the inclusion of the transient term inthe expression for sM and some problems require the exclusionof this term. The turbulent channel flow is a problem that requiresthe inclusion of the Dt term, and hence formulation (34) fails toproduce a convergent solution for time step sizes smaller thanDt ¼ 0:1.

In Fig. 16 we compare the solutions obtained with the conven-tional sM with Dt for the four time step sizes. Stable solutions are ob-tained for all time step sizes, and for larger time steps are verysimilar, but the accuracy of the results deteriorates for smaller timesteps. This is especially evident for the mean flow and stream-wisefluctuation. The results for the EVB sM are presented in Fig. 17. Theresults for all time steps are nearly identical and, in particular, nodeterioration of accuracy with decreasing time step size is observed.

4. Conclusions

We have numerically studied the behavior of element-vector-based (EVB) stabilization parameters in stabilized/variational mul-tiscale formulations of the time-dependent linear advection–diffu-sion and incompressible Navier–Stokes equations. In comparisonwith conventional definitions of stabilization parameters, whenthe conventional definitions yield stable results, the EVB parame-ters yielded comparable accuracy, sometimes slightly worse,sometimes slightly better. However, the conventional definitionsthat depend on the time step yield instabilities at small time steps,and the conventional definitions that do not depend on the timestep yield instabilities in complex, time-dependent flows, such asthe turbulent channel flow, whereas the EVB definitions behavedrobustly in all cases studied. At large time steps, all approachesseem to be more or less equivalent. These findings are consistentwith earlier cases in [37,39].

The shortcomings of the EVB procedure are that no stabilityanalysis is available to validate it theoretically and it introducesa nonlinearity in otherwise linear problems. With respect to theseshortcomings, the dynamic subgrid scales approach of Codina et al.[15] is superior. However, the EVB procedure has some attributes,namely, it is simply implemented in existing stabilized/variationalmultiscale computer programs, it does not entail the solution of anevolution equation for the small scales and this avoids the storageof small scales as history variables, and it provides a fair combina-tion of accuracy and robustness in contrast with conventional def-initions. We understand the approach is ad hoc and we do notbelieve it is the ultimate solution to the definition of small scalesin stabilized and variational multiscale methods, but based onour results and others we know of [37,39], it appears to be a sim-ple, effective and practically useful option. When things work, theyusually work for a reason. Is there a fundamental basis of the EVBprocedure? This is an intriguing question that can only be an-swered by future research.

Acknowledgements

We wish to thank the Texas Advanced Computing Center(TACC) at the University of Texas at Austin for providing HPC re-sources that have contributed to the research results reportedwithin this paper. Support of Teragrid Grant No. MCAD7S032 isalso gratefully acknowledged.

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