Implementation of Preconditioned Dual-Time Procedures in ... · ShishirA.Pandya ∗...

AIAA 2003-0072

Implementation of Preconditioned Dual-Time Procedures inOVERFLOW

Shishir A. Pandya ∗

NASA Ames Research CenterMoffett Field, CA 94035

Sankaran Venkateswaran †

University of TennesseeTullahoma, TN 37388

Thomas H. Pulliam ‡

NASA Ames Research CenterMoffett Field, CA 94035

NAS-03-003

AbstractPreconditioning methods have become the method

of choice for the solution of flowfields involving thesimultaneous presence of low Mach number flow andtransonic flow regions. It is well known that thesemethods are important for insuring accurate numeri-cal discretization as well as convergence efficiency overvarious operating conditions such as low Mach num-ber, low Reynolds number and high Strouhal numbers.For unsteady problems, the preconditioning is intro-duced within a dual-time framework wherein the phys-ical time-derivatives are used to march the unsteadyequations and the preconditioned time-derivatives areused for purposes of numerical discretization and itera-tive solution. In this paper, we describe the implemen-tation of the preconditioned dual-time methodologyin the OVERFLOW code. To demonstrate the per-formance of the method, we employ both simple andpractical unsteady flowfields, including vortex prop-agation in a low Mach number flow, flowfield of animpulsively started plate (Stokes’ first problem) anda cylindrical jet in a low Mach number crossflow withground effect. All the results demonstrate that thepreconditioning algorithm improves both the numeri-cal accuracy and convergence efficiency and, thereby,

∗Aerospace Engineer, Member AIAA†Research Professor, Member AIAA‡Senior Research Scientist, Associate Fellow AIAA

This paper is declared a work of the U. S. Government and isnot subject to copyright protection in the United States.

enables low Mach number unsteady computations tobe performed at a fraction of the cost of traditionaltime-marching methods.

Introduction

Accurate and efficient numerical simulation of un-steady fluid flows is one of the remaining challenges forthe resolution of engineering problems. For high speedflows, time-marching methods developed by the aero-dynamics’ CFD community have been very successfulfor both Euler and Navier-Stokes problems [1, 2]. Like-wise, for incompressible flows, unsteady algorithmsbased on projection methods [3, 4] or artificial com-pressibility [5, 6, 7, 8] approaches have been widelyapplied. However, many engineering problems involvethe co-existence of both compressible and incompress-ible flow regimes in the same flow field. For such cases,generalized preconditioned dual-time approaches havebeen developed which allow the same computer codeto be applied in all speed regimes [9, 10].

An example of a flow field that simultaneously in-volves both low speed and high speed regions is theHarrier aircraft in near-hover (landing approach) con-dition [11]. In this situation, the aircraft’s forwardvelocity is approximately 0.04 Mach. However, at thesame time, the aircraft is kept in the air by four highspeed jet exhausts directed toward the ground. Theinteraction of the jets with the cross-wind and theground effect renders this flow field extremely time-dependent. A purely incompressible method would be

1American Institute of Aeronautics and Astronautics

appropriate for the low-speed free-stream region, butnot for the high speed jet plumes. On the other hand,a transonic flow method would be appropriate for thejets, but is inefficient for the free-stream. Thus, nu-merical procedures for the solution of such problemsmust be capable of simultaneously handling both tran-sonic and low speed flow regimes.

The difficulty that transonic algorithms face at lowspeeds arises due to the poor conditioning of the eigen-values of the time-marching system. This lack of con-ditioning leads to poor solution accuracy as well aspoor convergence properties. Over the last decade,preconditioning methods have become increasinglypopular as a means of assuring that the time-marchingalgorithm is well-conditioned in terms of both accu-racy and efficiency at all speeds [12, 13]. For steadystate problems, these methods involve the introductionof pseudo-time derivatives in lieu of the physical timederivative terms in the time-marching system. For un-steady flows, the formulation is typically cast withina dual-time-stepping strategy, wherein the physical-time derivatives are employed to follow the physicaltransients and the pseudo-time derivatives serve as aniterative device. Proper definition of the pseudo-timederivatives serves to optimize the numerical solutionprocedure, making the performance of the algorithmcommensurate with traditional transonic methods athigh speeds and with incompressible methods at lowspeeds.

In this paper, the implementation of the precon-ditioned dual-time algorithm in the OVERFLOWcode [1] is discussed. The OVERFLOW code is a com-pressible Navier-Stokes code that uses the Chimeraoverset grid approach [14] for simulating complex-bodyconfigurations such as the afore-mentioned Harrier air-craft. The main solution algorithm in the code is thediagonalized approximate-factorization procedure [15].The implementation of the preconditioned dual-timescheme in the diagonalized approximate factorizationframework has been described by Buelow et al. [16] andwe follow the same approach in this paper. Specifically,this involves a modification of the traditional diago-nalization procedure to include both the physical andpreconditioned time-derivatives, thereby avoiding theintroduction of block matrices in the implicit operator.

The paper is organized as follows. We start by pre-senting the preconditioned dual-time algorithm and itsimplementation in the diagonalized framework of theOVERFLOW code. Following this, we present severalcomputational examples to verify the accuracy and ef-ficiency gains in various flow regimes of interest. Theexample problems include both simple test problems,such as the propagation of a Lamb vortex and Stokes’first problem, as well as more practical situations, such

as a round-jet in a low-Mach number cross-flow withground-effect. The latter problem is representative ofthe Harrier flow fields mentioned earlier. Finally, wesummarize the current status of the unsteady model-ing capability in OVERFLOW and point out the areasfor future research and development.

Preconditioned Dual-Time AlgorithmEquations of Motion

The Euler equations in generalized coordinates canbe written as follows:

∂Q

∂t+

∂E

∂ξ+

∂F

∂η+

∂G

∂ζ= 0 (1)

Here,Q = J−1Q = (ρ, ρu, ρv, ρw, e)/J,

E = J−1

ρUρuU + pρvUρwU

(e + p)U

,

F = J−1

ρVρuV

ρvV + pρwV

(e + p)V

and

G = J−1

ρWρuWρvW

ρwW + p(e + p)W

where J−1 = xξyηzζ + xζyξzη + xηyζzξ − xξyζzη −xηyξzζ − xζyηzξ and U , V , and W are the contra-variant velocities.

To carry out the iterations at each physical timestep, an artificial time(τ) term is explicitly introduced

∂Q

∂τ+

∂Q

∂t+

∂E

∂ξ+

∂F

∂η+

∂G

∂ζ= 0 (2)

Convergence of the pseudo-time(sub-iterations) ateach physical time step is important for obtaining anaccurate transient solution. To optimize the perfor-mance of the pseudo-iterations, the pseudo-time termis written in terms of the primitive variable vectorQp = (P, u, v, w, T )/J and the preconditioning matrixΓp.

Γp∂Qp

∂τ+

∂Q

∂t+

∂E

∂ξ+

∂F

∂η+

∂G

∂ζ= 0 (3)

Here Γp takes the form discussed in Ref. [9] and isgiven in the appendix.


Discretization

Discretizing Eqn. 3 with first order finite differencefor artificial time and second order backward differencefor the physical time terms results in

Γp

Qk+1p − Qk

p

∆τ+

3Qk+1 − 4Qn + Qn−1

2∆t

+δξEk+1 + δηF

k+1 + δζGk+1 = 0

(4)

Here, k is the pseudo-iteration counter, n is the timestep counter and δ represents spatial differences in thedirection indicated by the subscript. After lineariza-tion,

Γp∆Qp

∆τ+

3(Qk + Γe∆Qp) − 4Qn + Qn−1

2∆t

+ δξ(Ek + Akp∆Qp) + δη(F k + Bk

p∆Qp)

+ δζ(Gk + Ckp∆Qp) = 0

(5)

where ∆Qp = Qk+1p − Qk

p, Ap, Bp, and Cp are the fluxJacobians with respect to Qp, and Γe = ∂Q

∂Qp.

Rewriting this equation such that all terms evalu-ated at sub-iteration k or time steps n and n − 1 areon the right hand side and all terms multiplying ∆Qp

are on the left hand side[Γp + Γe

3∆τ

2∆t+ ∆τ(Ak

pδξ + Bkpδη + Ck

p δζ)]

∆Qp = Rk

(6)where

Rk = −∆τ

(3Qk − 4Qn + Qn−1

2∆t

)

−∆τ(δξEk + δηFk + δζG

k)

(7)

The key step in the derivation of a diagonalized schemerests in combining the pseudo- physical time-derivativeterms on the left hand side into a single matrix. Ac-cordingly, we define Sp = Γp + 3

2∆τ∆t Γe (see Appendix),

(Sp + ∆τAk

pδξ + ∆τBkpδη + ∆τCk

p δζ

)∆Qp = Rk

(8)Multiplying through by Γe and converting back to theconservative system, we get

(I + ∆τΓeS

−1p Akδξ + ∆τΓeS

−1p Bkδη

+ ∆τΓeS−1p Ckδζ

)∆Q = ΓeS

−1p Rk

(9)

where A, B, and C are the flux Jacobians with respectto Q.

Approximate Factorization andDiagonalization

Applying approximate factorization,(I + ∆τΓeS

−1p Akδξ

) (I + ∆τΓeS

−1p Bkδη

)(I + ∆τΓeS

−1p Ckδζ

)∆Q = ΓeS

−1p Rk

(10)

Now, ΓeS−1p A has the same eigenvalues as S−1

p Ap

and the eigenvectors of ΓeS−1p A are the columns of

ΓeXξ where Xξ are the eigenvectors of S−1p Ap. Thus,

ΓeS−1p A = ΓeS

−1p ApΓ−1

e = ΓeXξΛξX−1ξ Γ−1

e (11)

The scheme can be diagonalized using Eqn. 11 andtakes the form

ΓeXξ

(I + b∆τδξΛξ

)X−1

ξ Xη

(I + b∆τδηΛη

)X−1

η Xζ(I + b∆τδζΛζ

)X−1

ζ Γ−1e ∆Q = ΓeS

−1p Rk

(12)where 1

b = 1 + 32

∆τ∆t . The eigenvalues Λ are discussed

in the appendix.

Artificial DissipationThe diagonalized approximate factorization scheme

employed in OVERFLOW uses central differences todiscretize the spacial directions. It has been shownby many researchers that, to maintain uniform accu-racy across different flow regimes, it is important toformulate the artificial dissipation terms for the pre-conditioned system. In the dual-time context, thismeans that the artificial dissipation terms must be con-structed using the pseudo-time derivatives rather thanthe physical time-derivatives. Accordingly, a typicalsecond order dissipation term can be written as

εe∂

∂ξ

(Γpσ(Γ−1

p Ap)∂Qp

∂ξ

)(13)

where σ(A) denotes the spectral radius of the matrixA.

In the implicit scheme, the second-order dissipationterms appear in the implicit LHS operator as well. Forthe purposes of diagonalization, however, it is neces-sary to modify the spectral radius term σ(Γ−1

p Ap) withσ(S−1

p Ap). This approximation appears only on theLHS and has been shown by Buelow et al. [16] to havelittle or no effecct on the convergence efficiency. Wefurther point out that the modification does not im-pact the accuracy of the discrete formulation.

Preconditioner DefinitionThe definition of the preconditioning matrix (Γp)

is determined by the parameter Mp (see Eqn. 17 in


the appendix). In the present case, this parameter isdefined as

M2p = MIN(MAX(M2

i ,M2u ,M

2min), 1) (14)

where Mi is the local Mach number scale convention-ally used in steady preconditioning [17] and M2

min isa cutoff value usually set to 3 × M2

∞. Mu is a newunsteady scale given by

Mu =Lu

π∆ta(15)

where Lu is a user specified unsteady length scale [16],and a is the speed of sound. Note that when Mp = 1,preconditioning is not applied.

Note that the unsteady preconditioning varies thechoice of the preconditioning parameter between theinviscid choice and the no preconditioning choice. Foracoustic problems, the physical time-step size is smallsince it is based upon the acoustic CFL number. Theunsteady preconditioning parameter then approachesunity (equivalent to the no preconditioning choice).On the other hand, for vortex propagation problems,the physical time step is large as it is selected basedupon the particle CFL number. The corresponding un-steady preconditioning parameter then approaches theMach number (equivalent to the steady precondition-ing choice). For intermediate physical time step sizes,the unsteady choice provides preconditioning valuesbetween these two limits. As discussed in Ref. [10],the unsteady preconditioning choice optimizes conver-gence efficiency of the pseudo-time iterations regard-less of the choice of physical time step. In most cases,it also insures accurate scaling of the artificial dissi-pation terms. The exceptions to this statement arediscussed further in the Results section.

ResultsVortex propagation

Vortex propagation is used as the first example of anunsteady flow field. Trends for this problem have beenpreviously established in Ref. [10] based on stabilityanalysis. Specifically, a Lamb vortex is defined withthe following velocity distribution.

Vr(r, θ) = 0, Vθ = Γ

(1 − exp −r2

φ2

r

)(16)

where Γ is the vortex strength and φ is a character-istic radius. The initial conditions are specified usingthis velocity distribution and the vortex is propagatedusing the dual-time-stepping scheme described in theprevious section. The present study is conducted atfree stream Mach numbers of 0.1 and 0.001. To study

the performance of the scheme with and without pre-conditioning, the sub-iteration convergence is studiedfor several time steps. The time step is characterizedby CFLu = u∆t

∆x , where u is the speed of propagation,∆t is the physical time step and ∆x is the grid size.Ideally, for efficient time-marching of vortex propaga-tion problems, this CFL number should be of orderunity.

Here as well as in the other test cases presentedin this paper, we compare the convergence of thesub-iterations at each physical time-step for three dif-ferent choices of the preconditioning matrix, namely,no preconditioning, steady preconditioning and un-steady preconditioning. The first choice correspondsto the traditional time-marching algorithm with dualtime-stepping, which is the base scheme in the OVER-FLOW code. The second choice adopts the standardsteady state preonditioning choice used in steady lowMach computations. Finally, the third choice concernsthe present implementation and is given in Eqn. 14.We note that the length scale, Lu, is typically selectedto be the height of the domain. which is in accordancewith the suggested value in Ref. [10].

Figure 1 shows the results for M = 0.1 for four val-ues of CFLu. For the smaller values of CFLu (0.1 and1), the no preconditioning and unsteady precondition-ing cases converge rapidly, taking about 200 iterationsto reach machine zero. On the other hand, the steadypreconditioning choice is observed to yield poor per-formance. At the higher CFL numbers (10 and 100),the steady and unsteady preconditioning choices con-verge well, taking about 300 to 400 iterations to reachmachine zero, while the no preconditioning choice nowyields poorer performance. It is clear from these re-sults that the unsteady preconditioning choice essen-tially behaves like the no preconditioning choice forsmall time steps and like the steady preconditioningchoice for large time steps. Importantly, the unsteadychoice yields the best performance for all physical timestep sizes. We point out however that for efficientand accurate solutions, the optimal physical time stepchoice would correspond to CFLu = 1.

The advantage of using the unsteady preconditioneris observed to be more significant at even lower speedsas Fig. 2 shows for a free stream Mach number of0.001. At this speed, for low values of CFLu thesteady preconditioner does not converge in the sub-iteration. The unsteady preconditioner is observed tofollow the nonpreconditioned case as expected (see Fig.2(a)). At CFLu = 0.1, however, the unsteady pre-conditioner outperforms the non-preconditioned case.At CFLu = 1, the steady preconditioner is conver-gent, but like the nonpreconditioned case it displayspoor performance. Again the unsteady preconditioner


0 100 200 300 400 500Subiteration Count

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Nor

m o

f ∆Q

No preconditonerSteady PreconditionerUnsteady Preconditioner

M=0.1, CFLu=0.1

a) CFLu = 0.1


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b) CFLu = 1.0


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Nor

m o

f ∆Q


M=0.1, CFLu=100

d) CFLu = 100.0

Fig. 1 Comparisons of residual convergence with and without preconditioner for a Lamb vortex at M = 0.1

outperforms the other choices. At an even higherCFLu = 10 the steady preconditioner performs as wellas its unsteady counterpart. It is thus clear that in allcases the unsteady preconditioning provides optimalperformance of the pseudo-time convergence. Notethat these trends are similar to those presented in Ref.[10].

The subiteration convergence is important to studybecause it has an impact on the accuracy of the solu-tion. If the subiterations are not properly converged,the solution will be inaccurate. Of course, practicalcomputations do not require that the sub-iterations beconverged to machine zero. Therefore, to establish theamount of work required to obtain a solution, the num-ber of sub-iterations required for 4-order convergenceis plotted in Fig. 3 against CFLu. For smaller valuesof CFLu where the unsteady preconditioner performsas well as the no preconditioning choice, a relatively

small number of sub-iterations are required. However,at these small CFLu, a large number of time stepsis needed to march the solution a specified amount oftime. Accordingly, the CPU time required for march-ing the vortex one rotation is plotted in Fig. 4. Hereat low values of CFLu, the CPU time required is pro-hibitively high. For CFLu = O(1), where the vortexis marching approximately 1 grid point per time step,the unsteady preconditioner requires the least numberof sub-iterations and makes one rotation in the leastamount of CPU time. At larger values of CFLu, thesolution is no longer correct as the time step is simplytoo high to accurately capture the vortex convection.From these results, it is clear that the preconditioneddual-time scheme provides substantial savings in CPUtime for low Mach flowfields.

We next turn our attention to computational ac-curacy. Vortex computations are notoriously difficult



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m o

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a) CFLu = 0.01


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Fig. 2 Comparisons of residual convergence with and without preconditioner for a Lamb vortex atM = 0.001

and the second-order schemes considered here are espe-cially prone to excessive numerical damping. However,we are primarily concerned with a comparative assess-ment of the different preconditioning choices. Figure 5shows vorticity contours of the initial condition (Fig.5a) and the solution after one rotation through the (pe-riodic) domain for the various preconditioning choices(Figs. 5b, c and d). The non-preconditioned result(Fig. 5b) is wrong, with the correct vortex locationbut strength significantly damped out. Note that thisis true even though the inner iterations are fully con-verged. Thus, the discrepancy is due to the pooraccuracy of the numerical discretization (i.e. excessiveartificial dissipation) and not due to lack of conver-gence. In contrast, the steady (Fig. 5c) and unsteadypreconditioning (Fig. 5d) results predict the correctlocation of the vortex and a more respectable vortex

strength. These results clearly point to the bene-fit of preconditioning in limiting excessive numericaldissipation and hence preserving the accuracy of thenumerical formulation as well as computational effi-ciency.

Stokes’ first problem

For the next test case, we consider the solution of animpulsively started plate, which involves the unsteadydevelopment of a viscous boundary layer. Commonlyreferred to as Stokes’ first problem, there is a closedform self-similar solution, which makes this an idealtest case for verifying both convergence and accuracyproperties of the preconditioned scheme consideredhere.

To systematically study the computational results,we use an isotropic mesh with a fixed aspect ratio of


0.001 0.01 0.1 1 10 100

CFLu

0

100

200

300

400N

umbe

r of

sub

itera

tions

to c

onve

rge

4 or

ders

No Preconditionersteady preconditionerUnsteady Preconditioner

Lamb Vortex, M=0.001

Fig. 3 Number of sub-iterations required for 4order convergence for a Lamb vortex at M = 0.001

0.001 0.01 0.1 1 10 100

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3000

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tim

e to

Adv

ance

1 R

otat

ion

No PreconditionerSteady PreconditionerUnsteady Preconditioner

Lamb Vortex, M=0.001

Fig. 4 CPU time required for 4 orderconverged(sub-iteration) Lamb vortex to make onerotation at M = 0.001

unity. We use 11 grid points in the self-similar direc-tion and 101 grid points in the wall-normal direction.The domain height is 10−4 for a ∆y = 10−6. On thisuniform Cartesian grid, the initial boundary layer isselected to extend to approximately 51 points off thesurface. The initial condition is obtained from theanalytical solution and the computations are carriedout to follow the continued evolution of the boundarylayer.

The characteristic length (Lu) that appears in thedefinition of the unsteady preconditioning parameter(Eqn. 15) is typically chosen to be the length orwidth of the computational domain. Here, the rele-vant length scale is the height of the domain, which isgiven by 10−4. Figure 6 shows the sensitivity of thesub-iteration convergence to the selection of this pa-rameter for a von Neumann number of V NN = 1.67

(V NN = ν∆t∆y2 ). It is clear that values of 10−4 (and

less) yield the best convergence rates. Figure 6 alsoshows the convergence rates with no preconditioningand the steady preconditioning choices. It is clear thatthese choices yield substantially poorer sub-iterationconvergence.

To accurately capture the boundary layer growth,an appropriate physical time step for the numericalmarching process must be chosen. The viscous timestep or Von Neumann number (V NN) is the char-acteristic time-step of interest in this problem. It isdesirable to use a von Neumann number of 1 which cor-responds approximately to a boundary layer growth of1 grid point per time step. To assess the convergencebehavior of the low Mach preconditioner with respectto V NN , the number of sub-iterations required to con-verge the problem 4 orders of magnitude is plotted fora given time step in Fig. 7. For small values of V NN ,the no preconditioning choice shows good convergencebehavior. However, for higher values of V NN , theconvergence is extremely slow and the solution can notbe obtained in a reasonable number of sub-iterations.Using steady preconditioning, the convergence hangsbefore getting to 4 orders for V NN < 0.5. For highervalues of V NN proper convergence can be obtained,but very large time steps can lead to poor resolution ofthe transient evolution of the solution. The unsteadypreconditioner exhibits good sub-iteration convergencebehavior regardless of the choice of V NN .

The CPU time required to get four orders of sub-iteration convergence is shown in Fig. 8. It is clearthat the unsteady preconditioning formulation yieldsthe best CPU times for all choices of VNN. As men-tioned earlier, optimal efficiency and accuracy is ob-tained for VNN of order unity. Under these conditions,the no preconditioning case does not provide good con-vergence, while the steady and unsteady precondition-ing choices provide good sub-iteration convergence.Indeed, Figure 7 indicates that the dual time precon-ditioned formulation provides significant CPU savingsover the traditional (non-preconditioned) formulation.

As mentioned earlier, the choice of precondition-ing also influences the accuracy of the computationsthrough the definition of the artificial dissipationterms. Table 1 compares the solutions obtained at asingle point in the flowfield with the exact solution forseveral choices of VNN and preconditionings. It canbe observed that the no preconditioning choice is notconvergent for large VNN, while the steady precondi-tioning choice is not convergent for small VNN. Theunsteady preconditioning, in contrast, is always con-vergent. The most accurate solutions are obtained bythe steady and unsteady preconditioning choices whenthe VNN is order unity. It can also be observed that


0.01

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0.05

a) Exact Solution

-0.009

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-0.020.040.04

0.00

0.04 0.06

0.02 0.06

0.080.10

0.02

0.00

0.04 0.040.12

0.06

0.080.02

0.00

0.02

-0.0

20.060.08

0.08

0.06

0.040.08

0.06 0.06

0.04

0.00

-0.02

-0.0

2

d) Unsteady Preconditioner

Fig. 5 Comparison of solution with and without preconditioner for a Lamb vortex at M = 0.001

the no preconditioning result for small VNN is some-what inaccurate even though the solution is converged.This is because the poor scaling of the artificial dissi-pation terms yields a more diffusive formulation. Fora more detailed explanantion, the reader is referred toRef. [9]. Interestingly, the unsteady preconditioningresult for small VNN is also somewhat in error. Thisis because the unsteady preconditioning approachesthe no preconditioning formulation at small VNN andthe corresponding artificial dissipation terms becomepoorly scaled. This difficulty may be readily circum-vented by separating the preconditioning formulationused for the time-derivative (which is responsible forconvergence effficiency) and that used for the artifi-cial dissipation terms (responsible for accuracy). Such

Table 1 Solution comparison for Stokes’ first prob-lem with respect to various values of V NN

No Steady Unsteady ExactVNN Precond. Precond. Precond. Solution0.017 0.37151 0.37617∗ 0.37465 0.376220.167 0.37177∗ 0.37617∗ 0.37606 0.376221.67 0.37645∗ 0.37618 0.37619 0.37622

∗ Not converged

a formulation may be implemented within a multiplepseudo-time framework as discussed in Ref. [18], butis beyond the scope of the present article.

Cylindrical jet in cross flow with ground effectA complex unsteady flow field akin to the problem

of a Harrier in ground effect is a jet in cross flow with


0 100 200 300 400 5001e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07 no Presteady Preuns. Pre. CHLEN=1e-4uns. Pre. CHLEN=1e-3uns. Pre. CHLEN=1e-2uns. Pre. CHLEN=1e-1uns. Pre. CHLEN=1e-6uns. Pre. CHLEN=1e-5

Comparison of subiteration performance with respect to changes in Lu

M=0.001

Fig. 6 Parameter study to determine best Lu forM = 0.001 on a grid of AR = 1 based on sub-iterationresidual convergence of the Stokes’ first problem

0.001 0.01 0.1 1 10 100

von Neumann Number

0

100

200

300

400

Num

ber

of s

ubit

erat

ions

to c

onve

rge

4 or

ders

No Preconditionersteady preconditionerUnsteady Preconditioner

Stokes’ first problem, M=0.001, AR=1 grid

Fig. 7 Number of sub-iterations required 4 orderconvergence for Stokes’ first problem at M = 0.001on a grid of AR = 1

ground effect. This problem has been studied exper-imentally by Cimbala et. al. [19] with a jet at Mach0.13 issuing out of a 3 in. cylindrical tube in the cen-ter of a wind tunnel. The crosswind Mach numberis 0.013. The jet subsequently impinges on the windtunnel floor 9 in. away from the tube exit.

For the present simulation a single grid with cylin-drical topology is used with stretching to assureproper viscous mesh spacing near the tube walls andground. The mesh consists of approximately 250,000grid points.

The experimental results of Cimbala et al. [19] re-veal that the jet impinges on the ground and splays inall directions. The cross flow pushes the jet creatinga horseshoe shaped vortex which surrounds the jet on

0.001 0.01 0.1 1 10 100

von Neumann Number

-100

-50

0

50

100

150

200

250

300

350

400

450

500

CP

U tim

e to

co

nve

rge

4 o

rde

rs

No PreconditionerSteady PreconditionerUnsteady Preconditioner

Stokes’ first problem, M=0.001, AR=1 grid

Fig. 8 CPU time required for 4 orderconverged(sub-iteration) Stokes’ first problemboundary layer to grow 10 grid points at M = 0.001on a grid of AR = 1

3 sides. The ground vortex in turn is observed to pul-sate with a frequency of about 1-5 Hz, a result of theinteraction between the jet with the low speed cross-flow. The general features of the flowfield are observedin Fig. 9, which shows a particle trace issued from thejet at a fixed point in time. The jet is seen to impingethe ground and a horseshoe vortex is seen in front ofthe jet.

The computation of this flowfield with a traditionaldual time scheme (without preconditioning) is stableonly for small time steps. This stability restrictionmay be attributed to the poor convergence of thesub-iterations at large time step values. Our stud-ies indicated that the maximum physical time stepthat could be reliably used was 5.5 × 10−5 s. Us-ing this time step, more than 18,000 time steps arerequired to resolve a frequency of 1 Hz (that is char-acteristic of the ground vortex unsteadiness). Clearly,this would render the computation prohibitively ex-pensive. Moreover, the non-preconditioned choice alsoleads to a more dissipative numerical solution, whichwould also compromise the accuracy of the simula-tions. In contrast, when the unsteady preconditioneddual time scheme is employed, much larger time-stepsmay be stably employed and reliable inner iterationconvergence is obtained. The computations presentedin this paper were obtained with a physical time stepsize of 5.5 × 10−3 s., which corresponds to about 180time steps per 1 Hz cycle. We point out however thatthe larger physical time steps typically necessitate theuse of more inner iterations (compared with five inneriterations used in the traditional algorithm). In thepresent calculations, 50 inner iterations were employedat each physical time step. Thus, the preconditioned


Fig. 9 The particle trace at an instance in time for a jet at M = 0.13 in a M = 0.013 cross flow

dual time scheme yields a potential CPU savings ofabout an order of magnitude. Figure 10 shows a plotof the L2 norm of the residual at each physical timestep. The periodic unsteady nature of the flowfield isevident. The lowest frequency indicated in these re-sults is approximately 1.8Hz. We point out that these

16400 16600 16800 17000 17200 17400Time Step Number

−5.8

−5.6

−5.4

−5.2

−5

−4.8

−4.6

Log1

0(L2

nor

m o

f del

ta Q

)

Residual HistoryLog10(L2 norm of delta Q)

Fig. 10 The residual with respect to time step forthe jet in cross flow

results represent a preliminary effort of computing thiscomplex flowfield. Detailed evaluations are presentlyunderway and will be the subject of a future article.Here, it is sufficient to state that the preconditioneddual-time formulation shows potential in computingcomplex multiple-timescale, 3-D problems with signif-icantly improved accuracy and efficiency.

Summary

A preconditioned dual-time algorithm has been im-plemented in the OVERFLOW code. The approachfollows the method of Buelow et al. to tailor the formu-lation within the diagonalized solution procedure usedin the OVERFLOW code. Modifications of the pre-conditioning selection to account for unsteady scaleshave also been included. The method significantly en-hances the capability of the code, enabling it to handleunsteady flows over a wide range of Mach numbers andtime scales. Specifically, the preconditioned dual-timemethod enhances accuracy by improving the scalingof the artifical dissipation terms used in the numer-ical discretization procedure and improves efficiencyby optimizing the number of sub-iterations requiredat each physical time step. For low Mach number, low


frequency problems, in particular, significant savingsin CPU time are possible.

The enhancements made possible by the method aredemonstrated for both simple and practical flowfields.The simple test problems include the propagation ofa Lamb vortex in an straight channel and the flow-field of an impulsively started plate. Computationalresults reveal that the dual-time scheme with unsteadypreconditioning provides optimal sub-iteration conver-gence for all choices of time step size and Mach num-ber. Moreover, improved accuracy is also obtainedwith the method.

Preliminary computations of a round jet in a lowMach crossflow with ground effect have also been per-formed. Here, the preconditioned dual-time schemeenables the use of much larger physical time step sizes,which is important for efficiently resolving the low fre-quency pulsations of the horshoe-shaped ground vor-tex. More detailed computations are currently beingperformed and will be compared with experimentaltrends. The eventual goal is to apply the validatedcode to the computation of Harrier flowfields.

The limitation of the present implementation isthat the same preconditioning is employed for boththe artificial dissipation formulation and for the time-derivative. The disadvantage of this is that the finalsolution is dependent upon the choice of time-stepsize. This difficulty may be effectively circumventedthrough using separate preconditioners for the dissi-pation and time-derivative terms. The former thencontrols the accuracy of the discrete formulation, whilethe latter controls convergence efficiency of the sub-iterations. Implementation of such a multi-level pre-conditioning formulation will be the subject of futurework.

Acknowledgments

The authors wish to thank Mike Olsen of NASAAmes Research Center and Scott Murman of ELORETfor many useful discussions. Dr. Venkateswaran wassupported in part by NASA Ames Research Center(cooperative agreement No. NCC 2-5493).

References[1]P. G. Buning, D. C. Jespersen, T. H. Pulliam, W. M.

Chan, J. P. Slotnick, S. E. Krist, and K. J. Renze. OVERFLOWUsers Manual. NASA.

[2]J. P. Slotnick, M. Kandula, and P. G. Buning. Navier-stokes simulation of the space shuttle launch vehicle flight tran-sonic flowfield using a large scale chimera grid system. AIAAPaper 94-1860, June 1994.

[3]J. Kim and P. Moin. Application of a Fraction-StepMethod to Incompressible Navier-Stokes Equations. J. ofComputational Physics, 59:308–23, 1985.

[4]C. Kiris and D. Kwak. Numerical Solution of Incompress-

ible Navier-Stokes Equations using a Fractional-Step Approach.Computers & Fluids, 30:829–51, 2001.

[5]A. J. Chorin. A numerical Method for Solving Incom-pressible Viscous Flow Problems. J. of Computational Physics,2(12):12–26, 1967.

[6]C. L. Merkle and M. Athavale. A Time Accurate Un-steady Incompressible Algorithm based on Artificial Compress-ibility. In AIAA 8th Computational Fluid Dynamics Conference,Honolulu, HI, June 1987. AIAA paper 87-1137.

[7]D. Kwak, J. L. C. Chang, S. P. Shanks, andS. Chakravarthy. A Three-Dimensional Incompressible Navier-Stokes Flow Solver using Primitive Variables. AIAA Journal,24(3):390–6, 1977.

[8]S. E. Rogers, D. Kwak, and C. Kiris. Numerical Solutionof the Incompressible Navier-Stokes Equations for Steady-Stateand Time-Dependent Problems. AIAA paper 89-0463.

[9]S. Venkateswaran and C. L. Merkle. Efficiency and ac-curacy issues in contemporary cfd algorithms. Fluids 2000Conference, Denver, CO, June 2000. AIAA Paper 2000-2251.

[10]S. Venkateswaran and C. L. Merkle. Dual Time Steppingand Preconditioning for unsteady Computation. AIAA paper95-0078, January 1995.

[11]N. Chaderjian, S. A. Pandya, J. Ahmad, and S. Murman.Progress Toward Generation of a Navier-Stokes Database for aHarrier in Ground Effect. AIAA paper 2002-5966, November2002.

[12]J. M. Weiss and W. A. Smith. Preconditioning Appliedto Variable and Constant Density Time-Accurate Flows on Un-structured Meshes. AIAA paper 94-2209, 1994.

[13]E. Turkel. Preconditioning Techniques in ComputationalFluid Dynamics. Annual Review of Fluid Mechanics, 31:385–416, 1999.

[14]J. A. Benek, P. G. Buning, and J. L. Steger. A 3-dchimera grd embedding technique. AIAA Paper 85-1523-CP,July 1985.

[15]T. H. Pulliam and D. S. Chaussee. A Diagonal form ofan Implicit Approximate Factorization Algorithm. Journal ofComputational Physics, 39:347–363, 1981.

[16]Philip E. O. Buelow, Douglas A. Schwer, Jinzhang Feng,and Charles L. Merkle. A Preconditioned Dual-Time, Diago-nalized ADI scheme for Unsteady Computations. AIAA paper97-2101, 1997.

[17]D. Jespersen, T. Pulliam, and P. Buning. Recent En-hancements to OVERFLOW. AIAA paper 97-0644.

[18]S. Venkateswaran, C. L. Merkle, X. Zeng, and D. Li.Influence of large-scale pressure changes on preconditioned so-lutions. AIAA Paper 2002-2957, June 2002.

[19]J. M. Cimbala, M. L. Billet, D. P. Gaublomme, and J. C.Oefelein. Experiments on the unsteadiness associated with aground vortex. J. of Aircraft, 28(4):261–267, April 1991.

APPENDIX

The Preconditioning MatrixThe preconditioning matrix Γp has the form defined

in [16].

Γp =

ρ′p 0 0 0 ρTuρ′p ρ 0 0 uρTvρ′p 0 ρ 0 vρTwρ′p 0 0 ρ wρT

(ρhp + h0ρ′p − 1) ρu ρv ρw ρhT + h0ρT

(17)


where ρ′p = 1εpa2 and a is the speed of sound. Here εp =

M2p

1+(γ−1)M2p

. However, in practice ρ′p is never computed

directly. Instead, computing the eigenvalues of S−1p Ap

(Eqn. 22) involves the evaluation of the terms ρhT

d′ anddd′ . These terms are defined as follows.

ρhTd′

=ρhT

ρhT ρ′′p + (1 − ρhp)ρT

=ρcp

ρcp(bρ′p + (1 − b)ρp) − ρT

=cpa

2

cp( bεp

+ γ − γb) − γR

=a2

bεp

− γb + 1

=M2

pa2

b−M2p (b− 1)

(18)

Defining β′ = M2p

b−M2p (b−1)

ρhTd′

= β′a2 (19)

Also,d

d′=

ρhT

d′ρhT

d

= β′ (20)

The values of M2p and b now controls the behavior of

the preconditioner. The parameter b switches the be-havior of the preconditioner from unsteady to steady.For steady flows, b = 1 and β′ = M2

p .For unsteady flows, M2

p is defined by Eqn. 14.

Definition of Sp

Sp =1b

ρ′′p 0 0 0 ρTuρ′′p ρ 0 0 uρTvρ′′p 0 ρ 0 vρTwρ′′p 0 0 ρ wρT

h0ρ′′p − (1 − ρhp) ρu ρv ρw ρhT + h0ρT

(21)where ρ′′p = bρ′p − (b− 1)ρp. and 1

b = 1 + 32

∆τ∆t

Eigenvalues of S−1p Ap

λ1,2,3 = bU

λ4,5 =b

2

U (

1 +d

d′

)±

√U2

(1 − d

d′

)2

+ 4ρhTd′

(22)Eigenvalues for S−1

P Bp and S−1P Cp are similar.

Eigenvectors of S−1p Ap

For the repeated eigenvalues(λ = bU), many choicesof the eigenvectors are possible. The following choicekeeps the eigenvectors associated with the linear eigen-values from becoming degenerate by assuring that theeigenvectors are non-vanishing independent of the ge-ometric orientation [17].

00ξz−ξyξx

0−ξz

0ξxξy

0ξy−ξx

0ξz

(23)

where ξx,y,z = ξx,y,z

|∇ξ|The eigenvectors corresponding to the eigenvalues

of λ+ and λ− are

−λ−−bU dd′

λ+−λ−bξx

ρ(λ+−λ−)bξy

ρ(λ+−λ−)bξz

ρ(λ+−λ−)

− 1−ρhp

ρhT

λ−−bU dd′

λ+−λ−

;

λ+−bU dd′

λ+−λ−

− bξx

ρ(λ+−λ−)

− bξy

ρ(λ+−λ−)

− bξz

ρ(λ+−λ−)

1−ρhp

ρhT

λ+−bU dd′

λ+−λ−

(24)

For a perfect gas, the left and right eigenvector ma-trices can be written as follows.

Xξ =

0 0 0 λ+−bUλ+−λ−

λ−−bUλ+−λ−

0 −bξz bξybξx

ρ(λ+−λ−) − bξx

ρ(λ+−λ−)

bξz 0 −bξxbξy

ρ(λ+−λ−) − bξy

ρ(λ+−λ−)

−bξy bξx 0 bξz

ρ(λ+−λ−) − bξz

ρ(λ+−λ−)

bξx bξy bξzγ−1ργ

λ+−bUλ+−λ− −γ−1

ργλ−−bUλ+−λ−

(25)where

λ+ − λ− = b

√U2(1 − d

d′)2 + 4

ρhTd′

|∇ξ|2

= b√U2(1 − β′)2 + 4a2β′|∇ξ|2

(26)

X−1ξ =

1b

−γ−1γρ ξx 0 ξz −ξy ξx

−γ−1γρ ξy −ξz 0 ξx ξy

−γ−1γρ ξz ξy −ξx 0 ξz

1 A1ξx A1ξy A1ξz 01 A2ξx A2ξy A2ξz 0

(27)

where A1 = −ρ(λ− − bU), A2 = −ρ(λ+ − bU), andξ can be replaced with η or ζ to obtain the eigen-

vector matrices for S−1P Bp, and S−1

P Cp.


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