A Generalized Alpha Method for Integrating the Filtered NS Equations With a Stabilized FEM

A generalized-a method for integrating the ®lteredNavier±Stokes equations with a stabilized ®nite element

method

Kenneth E. Jansen a,*,1, Christian H. Whiting a,2, Gregory M. Hulbert b

a Scienti®c Computation Research Center and the Department of Mechanical Engineering, Aeronautical Engineering & Mechanics,

Rensselaer Polytechnic Institute, 110, 8th Street, Troy, NY 12180-3590, USAb Computational Mechanics Laboratory, Mechanical Engineering and Applied Mechanics, 2250 G.G. Brown,

University of Michigan, Ann Arbor, MI 48109-2125, USA

Received 26 July 1999; received in revised form 26 August 1999

Abstract

A generalized-a method is developed and analyzed for linear, ®rst-order systems. The method is then extended to the ®ltered

Navier±Stokes equations within the context of a stabilized ®nite element method. The formulation is studied through the application to

laminar ¯ow past a circular cylinder and turbulent ¯ow past a long, transverse groove. The method is formulated to obtain a second-

order accurate family of time integrators whose high frequency ampli®cation factor is the sole free parameter. Such an approach allows

the replication of midpoint rule (zero damping), Gear's method (maximal damping), or anything in between. Ó 2000 Elsevier Science

S.A. All rights reserved.

1. Introduction

As computational power increases, ¯uid dynamics researchers have been giving more attention to un-steady ¯ows. In particular, the simulation of turbulence places very strict demands on the computationalmethod due to its need to integrate a broad variation of spatial and temporal scales for very long timeperiods. The high temporal frequencies (often marginally resolved) can lead to non-physical instabilities ifthey are not controlled by high frequency damping. These poorly resolved frequencies, if left undamped,can also lead to further sti�ening of the linear algebraic system and, in some cases, to breakdown of theiterative methods used in its solution. However, the e�ect of this damping on the resolved scales present inthe system must also be monitored to ensure that the dynamics of the resolved scales are faithfully rep-resented.

While several methods have been proposed for the integration of the Navier±Stokes equations in time,there has yet to emerge one method as a clear favorite. Several have shown great promise in one type of ¯owonly to perform unsatisfactorily in another. For example, space-time ®nite element methods were proposedand analyzed by Shakib et al. [1,2] and expanded and used extensively by Tezduyar et al. [3±6]. Here, as thename implies, the weight and solution space are given a temporal dependence in addition to the usual

www.elsevier.com/locate/cmaComput. Methods Appl. Mech. Engrg. 190 (2000) 305±319

* Corresponding author. Tel.: +1-518-276-6755; fax: +1-518-276-6025.

E-mail address: [email protected] (K.E. Jansen).1 Supported by AFOSR grant F49620-97-1-0043.2 Supported by NASA LaRC grant NGT-1-52160.

0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 0 0 ) 0 0 2 0 3 - 6

spatial dependence. While these methods have yielded very accurate results, the cost has only been justi-®able on problems with a moving domain such as free surface ¯ows and/or deforming spatial domains thataccount for moving solid boundaries. In these cases, the additional cost of space-time methods is put togood use by providing a consistent tracking of the moving boundary.

In cases where the boundary is not moving, semi-discrete methods remain in favor [7]. Semi-discretemethods discretize only space with the ®nite element method, leaving a system of ordinary di�erentialequations to be integrated in time. Part of the attraction to semi-discrete methods is their long history of usein computational solid dynamics. Many algorithms have been proposed, analyzed and even designed toprovide a particular behavior needed for speci®c situations. It is often of interest to understand the behaviorof these algorithms in situations where a broad range of temporal scales are present. Here, the time step isoften chosen such that certain frequencies are marginally resolved or perhaps even completely unresolved.Given the non-linearities present in most interesting engineering systems, it is of great importance to ensurethat there is temporal damping for frequencies beyond the chosen resolution level. However, it is equallyimportant that this damping not a�ect the frequencies within the chosen resolution level. This delicatebalancing act has led to a long series of improvements to semi-discrete methods for computational soliddynamics as chronicled by Chung and Hulbert [8]. In the same reference a new method was developed andanalyzed, termed the generalized-a method. In this paper, we extend this analysis to computational ¯uiddynamics (i.e., from a second-order system to a ®rst-order system) and provide validation and large scaleapplications where we can assess its e�ciency and accuracy.

The outline of the paper is as follows: In Section 2, we introduce the ®ltered Navier±Stokes equationswhich we discretize in space using a stabilized ®nite element method to obtain a non-linear system ofcoupled ordinary di�erential equations. Since this is a rather complicated environment to be explaining (notto mention analyzing) a new semi-discrete method, in Section 3, we introduce and analyze a linear modelproblem for these equations to allow illustration of the generalized-a method for ®rst-order systems. InSection 4, the method is extended to the system of interest and then veri®ed on a simple model problem.The method is then applied to a large scale turbulence simulation to demonstrate its e�ciency. Finally, inSection 5 some conclusions are drawn.

2. The ®nite element formulation

Consider the compressible Navier±Stokes equations (complete with continuity and total energy equa-tions) written in ®ltered form after the application of a subgrid-scale model (see [9±11] or [12,13] for details).

U ;t � Fadvi;i ÿ Fdiff

i;i �S; �1�

where

U �

U1

U2

U3

U4

U5

8>>>><>>>>:

9>>>>=>>>>; � q

1~u1

~u2

~u3

~etot

8>>>><>>>>:

9>>>>=>>>>;; Fadvi � ~uiU � p

0d1i

d2i

d3i

~ui

8>>>><>>>>:

9>>>>=>>>>;; Fdiffi �

0s1i

s2i

s3i

sij~uj ÿ qi

8>>>><>>>>:

9>>>>=>>>>;; �2�

and

sij � 2�l� lT� Sij�~u��

ÿ 1

3Skk�~u�dij

�; Sij�~u� � ~ui;j � ~uj;i

2; �3�

qi � ÿ�j� jT� ~T;i; jT � cplT

PrT

; ~etot � ~e� ~ui~ui

2; ~e � cv

~T ; �4�

where we use the overbar to denote an unweighted ®lter and a tilde to denote a density weighted ®lter. The®ltered variables are: the velocity ~ui, the pressure p, the density q, the temperature ~T and the total energy

306 K.E. Jansen et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 305±319

~etot. The constitutive laws relate the stress, sij, to the deviatoric portion of the strain, Sdij � Sij ÿ �1=3�Skkdij,

through a molecular viscosity, l, plus turbulent viscosity, lT. Similarly, the heat ¯ux, qi, is proportional tothe gradient of temperature with the proportionality constant given by the addition of a molecular con-ductivity, j, and a turbulent conductivity, jT, which is assumed proportional to the turbulent viscosity asdescribed above. While the formulation can be made to accommodate a general divariant ¯uid, the cal-culations shown within this paper assume an ideal gas, p � qR ~T , and constant speci®c heats at constantpressure, cp, and at constant volume, cv. Furthermore, since all the calculations shown in this paper are atlow Mach number where temperature variation is low we have also assumed a constant molecular viscosityand constant conductivity through a constant Prandtl number, though, this again is not a necessary sim-pli®cation. Finally, S is a body force (or source) term.

For the speci®cation of the methods that follow, it is helpful to de®ne the quasi-linear operator (withrespect to some yet to be determined variable vector Y) related to (1) as

L � A0

oot� Ai

ooxiÿ o

oxiK ij

ooxj

� �; �5�

which can be naturally decomposed into time, advective, and di�usive portions

L �Lt �Ladv �Ldiff ; �6�where Ai � Fadv

i;Y is the ith Euler Jacobian matrix, K ij is the di�usivity matrix, de®ned such thatK ijY ;j � Fdiff

i , A0 � U ;Y is the change of variables metric. For a complete description of A0, Ai and K ij, thereader is referred to [14,15]. Using this, we can write (1) as simply LY � S.

To proceed with the ®nite element discretization of the Navier±Stokes equations (1), we must de®ne our®nite element approximation spaces. First, let �X � RN represent the closure of the physical spatial domain(i.e., X [ C where C is the boundary) in N dimensions; only N � 3 is considered here. In addition, H 1�X�represents the usual Sobolev space of functions with square-integrable values and derivatives on X.

Next, X is discretized into nel ®nite elements, Xe. With this, we can de®ne the trial solution space for thesemi-discrete formulations as

Vh � fv j v��; t� 2 H 1�X�m; t 2 �0; T �; v jx2Xe2 Pk�Xe�m; v��; t� � g on Cgg; �7�and the weight function space

Wh � fw j w��; t� 2 H 1�X�m; t 2 �0; T �;w jx2Xe2 Pk�Xe�m;w��; t� � 0 on Cgg; �8�where Pk�Xe�, is the space of all polynomials de®ned on Xe, complete to order k P 1, and m is the number ofdegrees of freedom (m � 5).

To derive the weak form of (1), the entire equation is dotted with a vector of weight functions, W 2Wh,and integrated over the spatial domain. Integration by parts is then performed to move the spatial de-rivatives onto the weight functions. This process leads to the integral equation (often referred to as the weakform): ®nd Y 2Vh such that

0 �Z

XW � A0Y ;t

ÿ ÿW ;i � Fadvi �W ;i � Fdiff

i �W �S� dXÿZ

CW � ÿÿ Fadv

i � Fdiffi

�ni dC

�Xnel

e�1

ZXeLTW � s LY� ÿS� dX: �9�

The ®rst line of (9) contains the Galerkin approximation (interior and boundary) and the second linecontains the least-squares stabilization. Streamline upwind Petrov Galerkin (SUPG, see [16] for details)stabilization is obtained by replacing LT by LT

adv. The stabilization matrix s is an important ingredient inthese methods and is well documented in Shakib [17] and in Franca and Frey [18]. Note that we havechosen to ®nd Y instead of U . As discussed in Hauke and Hughes [19], U is often not the best choice ofsolution variables, particularly when the ¯ow is nearly incompressible. For the calculations performedherein, the SUPG stabilized method was applied with linearly interpolated pressure-primitive variables, viz

K.E. Jansen et al. / Comput. Methods Appl. Mech. Engrg. 190 (2000) 305±319 307

Y �

Y1

Y2

Y3

Y4

Y5

8>>>><>>>>:

9>>>>=>>>>; �p~u1

~u2

~u3

~T

8>>>><>>>>:

9>>>>=>>>>;: �10�

By inspecting (2)±(4) it is clear that all quantities appearing in (9) may be easily calculated from (10).To develop a numerical method, the weight functions (W), the solution variable (Y), and its time de-

rivative (Y ;t) are expanded in terms of basis functions (typically piecewise polynomials; all calculationsdescribed herein were performed with a linear basis). The integrals (9) are then evaluated using Gaussquadrature resulting in a system of non-linear ordinary di�erential equations which can be written as

M _Y � N�Y�; �11�where the underbar is added to make clear that Y is the vector of solution values at discrete points (spatiallyinterpolated with the ®nite element basis functions), _Y are the time-derivative values at the same points.

3. Analysis of generalized-a method

The system of non-linear ordinary di�erential equations described in (11) is non-linear, making it unwieldyfor analysis. If we consider a linearized version with symmetric coe�cient matrices, however, the system canbe uncoupled into a series of single degree of freedom problems, resulting in the following model problem

_y � ky; �12�where k is the complex valued eigenvalue associated with the chosen mode (alternatively one can arrive at(12) by discretizing the unsteady scalar advection±di�usion problem in space in a way analogous to thatshown in Section 2).

We proceed to introduce the generalized-a method for integrating (12) from tn to tn�1 (i.e., Dt � tn�1 ÿ tn)

_yn�am � kyn�af ;

yn�1 � yn � Dt _yn � Dtc� _yn�1 ÿ _yn�;_yn�am � _yn � am� _yn�1 ÿ _yn�;yn�af � yn � af �yn�1 ÿ yn�;

�13�

where am; af and c are at this point, free parameters. These four equations can be combined to yield thefollowing system

ayn�1 � byn; or yn�1 � cyn; �14�where the solution vector at tn is given by yn � fyn;Dt _yngT

(similarly for yn�1) and the ampli®cation matrixc � aÿ1b is

c � 1

dam ÿ �af ÿ 1�cK am ÿ c

K am ÿ 1� af K�1ÿ c��

; �15�

where K � kDt, d � am ÿ af cK. It is fairly straightforward (see [20]) to show that

yn�1 � trace�c�yn ÿ det�c�ynÿ1: �16�If we further substitute a Taylor series expansion of yn�1 and ynÿ1 about yn in time, we ®nd that second-orderaccuracy can be obtained so long as

c � 1

2� am ÿ af : �17�

This is the same result found by Chung and Hulbert [8] for the second-order system.


Stability can be assessed by looking at the eigenvalues of c, fn1; n2g. To make our model problem re-¯ective of both advective and di�usive phenomena requires that K be complex. We are interested in provingstability for the left half of the complex plane since we will assume positive di�usive coe�cients in our ¯uiddynamics problems. Stability will be attained so long as the modulus of each eigenvalue is less than or equalto one. The expressions for the eigenvalues of (15) are too lengthy to be given explicitly here. Instead we willillustrate the stability constraints on am and af through the limiting values of K.

First consider the case when the time step is taken to be very small. Regardless of the value of k, Kvanishes, and the eigenvalues of c in this limit are

limDt!0

n � 1

�ÿ 1

am; 1

��18�

from which we may deduce that stability requires

am P1

2: �19�

We next consider the limit of an in®nite time step for any eigenvalue in the left complex half plane (i.e., Ktending to complex in®nity). The eigenvalues of c in this limit are

limDt!1

n � ÿ1� 2�am ÿ af �1� 2�am ÿ af � ; 1

�ÿ 1

af

��20�

from which we may deduce that stability also requires

am P af P1

2: �21�

Again, this is the same result obtained by Chung and Hulbert [8] for the second-order system. Since theyhad an additional eigenvalue (and an additional parameter in their method) they had a third constraint thatis not present here.

While having two parameters free in the method has a certain appeal we recall that our goal was to ®nd amethod with strict control of high frequency damping. Therefore it is enlightening to express the twoparameters am and af in terms of the spectral radius for an in®nite time step. Chung and Hulbert referred tothis parameter as q1

q1 � limDt!1

max�n1; n2�: �22�

By requiring the q1 from each eigenvalue in (20) to take on the same value we can express am and af interms of q1, viz

am � 1

2

3ÿ q11� q1

� �; af � 1

1� q1�23�

thereby de®ning a second-order accurate, one-parameter family of methods with a speci®ed high frequencydamping.

The importance of casting the parameters in this way is that one has precise control over the damping offrequencies that are high relative to the resolution level. If q1 is chosen to be zero the method is said toannihilate the highest frequency in one step (only for a linear problem). This method has the same spectralstability as Gear's two-step backward di�erence method [21]. If q1 is chosen to be one, then the highestfrequency (as well as all others) are preserved (in the linear problem). This method corresponds to themidpoint rule which is equivalent to the trapezoidal rule for linear problems.

For linear problems with all frequencies resolved, the midpoint rule has the very nice property of in-troducing no damping regardless of the time step. However, if k is purely imaginary, while the modulusremains equal to one, the eigenvalue limK!1 n1 � ÿ1. This has the e�ect of causing the solution to switchsign on each step. Clearly this behavior is unacceptable. In these cases, it is important to have q1 strictly


less than one so that high frequencies do not spoil long term integrations. The example of vortex sheddingfrom a circular cylinder (presented below) will illustrate this e�ect.

The problem is made worse by non-linearities. In many cases of interest q1 must be reduced to besubstantially less than one to obtain an acceptable solution. The particularly attractive feature of thismethod is that a whole family of time integrators can be tested with precise control of q1. In this way, q1may be increased or decreased according to the needs of the problem of interest. Examples are given inSections 5 and 6 to illustrate this process.

4. Generalized-a method for the Navier±Stokes equations

The primary di�culty in extending the work from the previous section to the full Navier±Stokesequations is the non-linearity that is introduced. Straightforward extension of (13) to (11) leads to a non-linear system of algebraic equations to be solved at each step. For ease of presentation we also drop theunderbar from the vector of nodal time derivatives of the solution _Y ! _Y and the vector of nodal values ofthe solution Y ! Y , viz.

G� _Yn�am ;Yn�af � �Mn�am_Yn�am ÿN�Yn�af � � 0; �24�

Yn�1 � Yn � Dt_Yn � cDt

_Yn�1

�ÿ _Yn

�; �25�

_Yn�am � _Yn � am_Yn�1

�ÿ _Yn

�; �26�

Yn�af � Yn � af Yn�1� ÿ Yn�: �27�

Here, we have introduced G to be the vector of nodal values of the non-linear residual. The non-linearitiesare best handled by introducing a predictor±multicorrector algorithm similar to those proposed by Brooksand Hughes [16]. By making a prediction of the solution and its time derivative at time tn�1 we start thealgorithm. Since we will be making multiple corrections, we introduce a superscript (inside parentheses) torepresent the iteration number. In this notation, our predictor is given an iteration count of zero and isgiven by

Y�0�n�1 � Yn; �28�

_Y�0�n�1 �

cÿ 1

c_Yn; �29�

where (28) predicts that the solution will be the same as it was at the previous time step and (29) is the timederivative at tn�1 that is consistent with (25) (i.e., the predictor that preserves second-order accuracy). Otherchoices of predictor are possible but discussion is left until later for clarity.

After making the prediction, the algorithm enters a loop of multi-corrector passes with i initialized tozero. The ®rst operation within the loop is the calculation of the solution at tn�af and the time derivative ofthe solution at tn�am

Y �i�n�af� Yn � af Y

�i�n�1

�ÿ Yn

�; �30�

_Y �i�n�am� _Yn � am

_Y�i�n�1

�ÿ _Yn

�: �31�

These quantities enable the evaluation of G �i�� _Y �i�n�am;Y �i�n�af

� which, for small i can be expected to be far fromits desired value of 0. To ®nd an improvement to the current values of (30) and (31) we utilize a Newton'slinearization of G �i� with respect to the solution variable, viz.


G �i� _Y �i�n�am;Y �i�n�af

� ��

oG �i� _Y �i�n�am;Y �i�n�af

� �oY �i�n�af

DY �i�n�af� 0: �32�

The ®rst term of (32) is simply the non-linear residual. The second is the tangent matrix multiplying theincrement of the solution. This yields a linear matrix problem to be solved for each corrector step

K �i�DY �i�n�af� ÿG �i�; �33�

where K �i� is the tangent matrix. Once this system is solved for DY �i�n�af, the solution is updated

Y �i�1�n�af� Y �i�n�af

� DY �i�n�af; �34�

_Y �i�1�n�am� 1

�ÿ am

c

�_Yn � am

cDtafY �i�1�

n�af

�ÿ Yn

�; �35�

and i is incremented. If i < imax the algorithm returns to solve (32), then initiating the next corrector pass.Otherwise, the solution at time step tn�1 is determined as follows:

Yn�1 � Yn �Y �imax�

n�afÿ Yn

af; �36�

_Yn�1 � _Yn �_Y �imax�

n�amÿ _Yn

am: �37�

This completes the step from tn ! tn�1. If more time steps are required, n is incremented and the algorithmreturns to the prediction phase for the next step (i.e. (30) and (31)).

We have glossed over a few important aspects of the algorithm in the interest of a compact presentation.We address these issues in the remarks that follow.

Remarks.

1. There are at least three other predictors of interest. In the base algorithm we described a Same _Y pre-dictor. This predictor is the most robust but does little to accelerate convergence within a given time stepfor unsteady ¯ows since it presumes the ¯ow to be steady. For the same reason it is the best predictorwhen a steady ¯ow is desired. However, in this case one should not choose q1 based on (17) and (23) butshould instead set am � af � c � 1 to obtain the ®rst-order accurate backward Euler method. The threealternative predictors are given below

�a� Zero _Y

_Y�0�n�1 � 0; �38�

Y�0�n�1 � Yn � Dt�1ÿ c�af

_Yn: �39��b� Same _Y

_Y�0�n�1 � _Yn; �40�

Y�0�n�1 � Yn � Dt

_Yn: �41��c� Same DYn�1 � DYn � Yn ÿ Ynÿ1;

Y�0�n�1 � Yn � �Yn ÿ Ynÿ1�; �42�

_Y�0�n�1 �

Yn ÿ Ynÿ1

Dtc� cÿ 1

c_Yn: �43�

These alternative predictors have shown superiority on some problems. However, they have also led toinstabilities in some problems when employing a ®xed number of corrector passes (leading to di�erent


levels of satisfaction of (24) before moving on to the next step). It seems that the non-linearity cansometimes make minor changes in the predicted values lead to wild excursions from the solution of (24).Of the three, the Same DYn�1 o�ered the most improvement, often allowing imax to be reduced by onerelative to the Same Y . The downside of this predictor is that it cannot be used to start the algorithm(Ynÿ1 is not available). However, it is possible to start with one of the other three predictors and switchto this predictor after the ®rst step. Furthermore, after some manipulation of the formulas it is possibleto update Y �0�n�af

and _Y �0�n�amin place making this predictor not require more memory than the others as

might have been suggested by (42) and (43). If the number of corrector passes is taken su�ciently largethat (24) is satis®ed on every step, then the choice of predictor has no e�ect on the ®nal solution butpractical concerns usually make this practice unattractive.

2. The implementation given in (24) uses a ``consistent'' mass matrix, Mn�am , evaluated at n� am. For thesake of computational convenience we often evaluate this term at n� af thereby avoiding the evaluationof the variables, Y at a second temporal point for each quadrature point. The results have, in all casescompared, been insensitive to this shortcut motivating the albeit slight computational savings. This ®nd-ing may be more re¯ective of the very mild non-linearities present in the mass term when using a pressureprimitive variable formulation (for incompressible ¯ow M is independent of Y making the point mute).Other, more highly non-linear variables (such as entropy variables) may be more sensitive to the treat-ment of this term.

3. The tangent matrix K �i� is very large and very expensive to compute if done in the most straightfor-ward way. There are a number ways in which the storage and the cost can be reduced due to the factthat iterative methods are usually employed to ®nd a solution to (33). In particular, GMRES methodswhich were ®rst proposed by Saad and Schultz [22] perform very well in conjunction with stabilized®nite element methods applied to a variety of ¯uid dynamics problems. Shakib et al. [2]demonstratedthe e�ciency of element-by-element GMRES solution techniques. Johan et al. [23] proposed a matrix-free extension to this work. We currently employ both solvers. Furthermore, we have recently linkedour code with the PETSc solver library to enable testing of sparse storage implementations ofGMRES as well as other solvers and a variety of pre-conditioners. Publications of this work are forth-coming.

4. Shakib et al. [2] and many others have pointed out the importance of freezing the coe�cient matrices(A0; Ai, and K ij) when calculating the tangent. Johan et al. [23] found that the true tangent could accel-erate convergence once the solution became ``close'' to converged, though they still advocated freezingthe coe�cients when outside this region. This situation remains the same in our experience. For timesteps of practical interest, the initial guess of the solution is often not su�ciently close, requiring useof the frozen coe�cient tangent. Further simpli®cations are also possible. Semi-implicit methods canbe created by neglecting completely various terms in the tangent. ``Explicit'' methods can be createdby neglecting all terms but the time-derivative term. The quotes around explicit re¯ect that consistenttreatment of the time-derivative terms will still result in a matrix problem unless the resulting matrixis lumped. Thus far, we have only studied the performance of this algorithm under the fully implicit case.The other cases are left for future work.

5. An important step in the linearization process is determining the dependence of _Y �i�n�amon Y �i�n�af

. The de-

pendence is apparent from (35). This expression is obtained by solving for _Y �i�1�n�am

in (25)±(27). Recall thatthese are only nodal solution and time-derivative vectors making the tangent

o _Y�i�A;n�am

oY�i�B;n�af

� I5�5

amdAB

cDtaf; �44�

where A and B are global node numbers considered to be selected from 1 to the number of nodal points,dAB is the Kronecker delta function which is one only when A � B and is otherwise zero, and I5�5 is a 5 by5 identity matrix. By including this term we implicitly account for the dependence of the nodal values ofthe time derivative of the solution on the nodal values of the solution that is built into the generalized-amethod.

6. We have also omitted the discussion of boundary conditions. After each prediction or correction it isimportant to con®rm that the solution satis®es the boundary conditions.


5. Validation on ¯ow past a circular cylinder

The ®rst example will provide an application of the generalized-a method to the relatively well under-stood ¯ow around a circular cylinder at a Reynolds number of 100 (based on the cylinder diameter andin¯ow velocity). We have simulated this ¯ow with compressible and incompressible stabilized ®nite elementcodes. Here, we have chosen to present the incompressible code results since they are much more sensitiveto the choice of time integration parameters. This incompressible formulation is completely described in[24], and the details will not be given here.

The problem geometry and boundary conditions are depicted in Fig. 1. In addition to the boundaryconditions shown, we have imposed zero x3 velocity and no tangential traction on the two x3 planes tosimulate two-dimensional conditions with our three-dimensional code. A complete description of thisproblem can be found in a variety of refs.: Shakib et al. [2], Hauke and Hughes [15], and Behr et al. [7]. Wechoose boundary conditions consistent with those of Shakib et al. [2]. At a Reynolds number of 100, thesalient feature of this ¯ow is the periodic shedding of vortices from the cylinder creating time varying liftand drag forces determined by integrating the forces on the cylinder surface. This single, dominant fre-quency allows a simple study of the new time integrator.

In this context, we will study the e�ect of the high-frequency damping parameter (q1) on the lift anddrag pro®les. The ¯ow is solved in time assuming the cylinder is initially at rest, and is immediately ac-celerated to a velocity of u0 (often referred to as an impulsive start). We have taken a time step of 0:1(normalized by the cylinder diameter) and for each time step, three Newton corrector passes are performed,insuring that the normalized change in the velocity increment is less than 5� 10ÿ4. This time step a�ords 60time steps per period of the lift force and 30 time steps per period of the drag force.

The ®rst four plots in Fig. 2 show the lift and drag forces (fl and fd, respectively) plotted against the non-dimensional time, t�, for q1 � 0:0; 0:25; 0:5, and 0:75 (q1 � 1:0 is left o� the ®rst four plots for reasonsthat will be clear shortly) for two di�erent time windows (the one on the right being a zoomed view). Fromthese plots we make the following observations: (i) the period and amplitude of both the lift and the dragare very weak functions of q1 (which might be expected from the observation that 30 points per wavelength is adequate for second-order accurate methods), (ii) a small amplitude undulation is present in theq1 � 0:75 case, (iii) the elimination of this undulation for the cases of q16 0:5 re¯ects the increasingdi�culty the method faces as q1 approaches 1, (iv) the q1 � 0 case appears to start its transition from asteady separation to a saturated unsteady ¯ow much earlier than the other methods.

The last observation is somewhat counter intuitive. One might expect the method with the highestdamping would be the last to leave a steady ¯ow in favor of an unsteady ¯ow. The last two plots shed somelight on this mystery. They indicate that the impulsive start can introduce a rather large, highest-frequencyunsteadiness to the ¯ow. Furthermore, these plots indicate the severe errors that may occur when usingq1 � 1:0 (trapezoidal rule) for this ¯ow. The ®rst of these two plots shows the time window including theimpulsive start. This plot clearly shows the strong damping characteristics of the q1 � 0 case, where theinitial disturbance due to the impulsive start is damped out within a couple of time steps. Also clear from

Fig. 1. Cylinder geometry and boundary conditions.


this plot is that the initial disturbance is almost completely preserved in the q1 � 1 case, polluting the entiresolution. Intermediate values of q1 annihilate this unresolvable frequency in a manor predictable by theirproximity to the two extreme cases. It is conjectured that the physical instability leading to limit cyclevortex shedding is in the well resolved and almost completely undamped range for all chosen values of06 q16 1. However, by using a shorter time to annihilate the highest, unresolvable frequency, the lowervalues of q1 actually pick up the physical instability (which starts at extremely low values and grows ex-ponentially before saturating on the limit cycle) sooner. The higher values of q1 are unable to pick up thisvery low amplitude physical instability until they bring the unresolvable frequency below its level. Thisshould not be misconstrued as an advocation of q1 � 0:0 in all cases but it does demonstrate the bene®t inat least one case of having the ability to annihilate a large, unresolvable frequency rapidly. In simulations ofpractical interest, this ability must be balanced by the fact that there often exist a continuous range offrequencies that one is interested in resolving rather than two, widely separated ones as shown here. Inthose cases, higher values of q1 are desirable to maintain the ability to accurately integrate waves withsigni®cantly less than 30±60 time steps per period.

Fig. 2. Lift, fl, and drag, fd, forces on the cylinder for di�erent time windows: - - - - q1 � 0:00; ±�± q1 � 0:25; � � � � � � q1 � 0:50; ±±±±

q1 � 0:75; ±±±± � q1 � 1:0.


The ®nal plot in Fig. 2 shows that the highest frequency can be excited even without an impulsive start.In this case, the ¯ow was restarted from the case of q1 � 0:0 (at t� � 200, where no high frequencies werevisibly present). However, the non-linearities inherent to the Navier±Stokes equations needed little time tobuild up energy in this highest frequency when the time integrator was switched to q1 � 1:0, a time in-tegrator that is powerless to control these frequencies. This highest frequency mode does saturate in am-plitude though leaving a surprisingly accurate signal that can be recovered by ®ltering this signal in time.This is again fortuitous to this case though, owing to the wide separation of the frequencies causing little ifany interaction. Again it must be stressed that in the problems of interest, the continuous range of scaleswill su�er much greater contamination due to the stronger interaction of waves of close proximity infrequency space. In these cases, the energy in the highest frequencies may also fail to saturate leading to abreakdown of the solution technique. Clearly, some capacity to insure the annihilation of unresolvablewaves is critical to maintain the ®delity of the well and marginally resolved waves, thus the motivation forthe design of the method with careful control of q1 and the advocacy for the availability and use of in-termediate values.

6. Application to turbulence simulation

The method described above has been applied to a number of turbulence simulations [12,13,25,26] in-cluding fundamental ¯ows such as decay of isotropic turbulence and channel ¯ow and very complex ¯owslike a supersonic jet and ¯ow over an airfoil. These were large-eddy simulations (LES). As the namesuggests, this turbulence simulation technique seeks to resolve the large eddies of the turbulent ¯ow both inspace and in time while modeling the turbulent eddies which fall below a certain cuto� of resolution. Theneed for high temporal accuracy with this technique was the driving motivation behind development of themethod described above. In this section, we will describe a recent application to ¯ow over a very longcavity. A detailed study of this ¯ow awaits results of ongoing calculations but results obtained thus farillustrate the fundamental character of the solutions obtained with this method.

6.1. Flow past a very long groove

Consider a turbulent boundary layer (Reh � Uh=m � 1300 where h is the momentum thickness [27], andU is the speed of ¯uid in the free stream) encountering a very long groove, transverse to the ¯ow direction.Since the groove is very long, the ¯ow can be considered two-dimensional in the mean. Indeed, Reynolds-averaged Navier±Stokes simulations of this ¯ow would be carried out using the two-dimensional geometry.However, to do the LES correctly requires resolution of the three-dimensional ¯uctuations. In this case, thesimulation geometry is an extrusion of the two-dimensional cross-section of the groove into the transversedirection. The transverse length must be chosen long enough that the turbulent ¯uctuations are de-cor-related (i.e., their two point correlation becomes very small) between the two lateral planes, the conditionnecessary for application of periodic boundary conditions.

Fig. 3 illustrates the mesh on the boundary of the computational domain, with annotation of thephysical dimensions and boundary conditions. The dimensions are normalized by the depth of the groove(H). From this we see that the groove length (2H ) is twice the groove depth and that boundary layerthickness is about 50% of the groove depth (0:5H ). A second Reynolds number is needed to characterizethis problem, ReH � 27; 000. As the groove height and length are varied, the fundamental character of the¯ow can be changed dramatically. In particular, for certain combinations, a resonance condition can occurwherein the fundamental frequency of the groove (and perhaps a few sub-harmonics) become quite strong(in contrast to the usual turbulence frequency spectrum which is rather broad and more equally distributed)turning an otherwise nearly silent device into a whistle. In some engineering devices, this phenomenon isquite deleterious and therefore worthy of design effort to avoid.

Before showing solutions to this problem we turn our attention to Fig. 4 where we have zoomed in onthe groove and removed the front and side walls to allow viewing of the grid resolution on all surfaces. Notein particular, the very ®ne near-wall resolution on the incoming turbulent boundary layer before thegroove. This ®ne grid is necessary to resolve the turbulent eddies in the near wall region as shown in Fig. 5


Fig. 3. Computational domain and boundary conditions. Out¯ow planes include the plane opposite the in¯ow and the top plane.

Front periodic plane (shown) matches with back periodic plane (not shown). No slip, isothermal walls assumed (not shown) on the

plate, the groove bottom, and the two vertical groove walls.

Fig. 4. Mesh on boundary with front plane, in¯ow plane and out¯ow plane cut away to show interior.


through isosurfaces of the streamwise vorticity. These small vortices which are created by the wall shear arefreed from the wall at the groove inception creating a free shear layer. The growth of these structures ismatched by a growth in the grid as can be seen by referring back to Fig. 4. This shear layer is unstable,causing the streamwise vortices to roll up in much the same way as in the laminar Kelvin±Helmholtz in-stability.

The free shear layer is again interrupted by the end of the groove, where it is seen to impact upon thegroove far wall. At this point, the ¯ow is divided, some going over the groove, some going down the farwall. This downward ¯ow, combined with the shear across the top of the groove, causes a complicateddriven ``cavity'' ¯ow within the groove. The ¯uctuations from this cavity ¯ow feed back to the very base ofthe shear layer at the groove inception.

This ¯ow has a number of complex features that provide challenges for accurate simulation. As theprimary topic of this paper is the description of a new time integrator we will con®ne our discussion to thoserelated to temporal integration. The simulation was performed with q1 � 0:5 and a time step of 0:002H=Uresulting in an average of 10 GMRES Krylov vectors per solve. Simulations performed at a higher value ofq1 required signi®cantly more Krylov vectors (due to equation sti�ening associated with unresolvable highfrequencies), and based on the cylinder results, additional damping was judged to be acceptable. The

Fig. 5. Four isosurfaces of streamwise vorticity (red, large positive; dark blue, large negative value) with the streamwise velocity

superposed on the back plane.


impinging ¯ow on the groove far wall is the most likely source of the high frequencies in this ¯ow, ne-cessitating the reduction of q1. Although the results are preliminary, Fig. 6 shows the time history of thevertical velocity according to the LES solution at various points within the ¯ow. One can see the changingcharacter from a turbulent boundary layer (where vertical velocity ¯uctuations are suppressed by the solidwall) to a free shear layer (where the instabilities grow and appear to be locking onto the fundamentalfrequency) then, after the groove, the recovery back to a turbulent boundary layer. Clearly, simulations ofthis type require not only high spatial accuracy [28] but also, accurate, well understood, temporal accuracy.

7. Conclusions

A generalized-a method was developed and analyzed for linear, ®rst-order systems. A key desirableattribute of the method is that it allows the high frequency ampli®cation factor to be set directly, therebydeveloping a range of time integrators from zero damping midpoint rule to maximal damping Gear'smethod (or anything in between). The method was then extended to the ®ltered Navier±Stokes equationswithin the context of a stabilized ®nite element method. Two applications were given: one quite simple(laminar vortex shedding behind a circular cylinder), where the behavior of the high frequency dampingfactor could be observed, and another quite complex (turbulent ¯ow over a very long transverse groove),where the need for such an integrator is made evident due to the broad range of frequencies present in thesimulation.

Acknowledgements

The authors also acknowledge Francois Bastin and Farzin Shakib whose early discussions aided thealgorithmic development of the generalized-a method for the Navier±Stokes equations. This work has been

Fig. 6. Time series of vertical velocity collected at various positions at each time step.


supported by the AFOSR (F49620-97-1-0043), by NASA LaRC (NGT-1-52160), and by the Center forTurbulence Research (CTR) (a joint NASA±Stanford University program). It has also enjoyed generouscomputer resources from the Army High Performance Computing Research Center (AHPCRC) and theNaval Research Lab (NRL).

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