Application of a Non-Linear Frequency Domain Solver to the ...

AIAA 2002-0120Application of a Non-LinearFrequency Domain Solver to theEuler and Navier-Stokes EquationsMatthew McMullen, Antony Jamesonand Juan J. Alonso

Stanford University, Stanford, CA 94305

40th AIAA Aerospace Sciences Meeting &Exhibit

January 14–17, 2002/Reno, NVFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

AIAA 2002-0120

Application of a Non-Linear Frequency DomainSolver to the Euler and Navier-Stokes

Equations

Matthew McMullen,∗ Antony Jameson†

and Juan J. Alonso‡

Stanford University, Stanford, CA 94305

This paper presents a technique used to accelerate the convergence of unsteady flowsto a periodic steady state. The basis of this procedure is to assume the time period ofthe solution’s oscillation and to transform both the solution and residual using a discreteFourier transform. However, this paper also presents a method which iteratively solvesfor the time period during the process of calculating a solution. These methods areamenable to parallel processing and convergence acceleration techniques such as multigridand implicit residual averaging. The accuracy and efficiency of the technique is verifiedby Euler and Navier-Stokes calculations for a pitching airfoil whose period of oscillationis forced. The capability to identify the natural frequency of oscillation is verified byNavier-Stokes calculations for laminar vortex shedding behind a cylinder where the timeperiod of oscillation is unknown a priori. Results show that a limited number of modescan accurately capture the major flow physics of these model cases.

IntroductionUnsteady flows still present a severe challenge to

Computational Fluid Dynamics (CFD). In generalthese flows can be subdivided into two general classes.The first are unsteady flows where the boundary con-ditions are forcing the unsteadiness at predeterminedfrequencies. Examples of this include the internal flowsof turbomachinery, the external flow fields of helicopterblades or propellers, and certain aero-elastic computa-tions. The second general class of unsteady flows arewhere instabilities in the fluid mechanical equations in-duces unsteadiness in the flow field. Examples in thisclass include (but are obviously not limited to) vortexshedding behind a cylinder, and other fluid dynamiccases involving separated flows and free shear layers.Without experimental or simplified analytic models,the temporal frequencies are difficult to determine apriori for this second class. This paper will presenta reduced order scheme capable of solving unsteadyflows for both classes of problem. The motivation fordeveloping this scheme is the need to reduce the costof unsteady CFD simulations for complex flows.

In general, time accurate solvers are designed to cap-ture any arbitrary time history in the evolution of thesolution. There are many applications, however, suchas helicopter rotors or turbomachinery where usersare typically only concerned with the data once the

∗Graduate Student, Student Member AIAA†T.V. Jones Professor of Engineering, AIAA Fellow‡Assistant Professor, AIAA MemberCopyright c© 2002 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

solution has reached a periodic steady state. Never-theless, the majority of the computational effort isexpended in resolving the decay of the initial tran-sients. Algorithmic efficiency is a function of the timeduration associated with this decay rate and the timestep permitted by the algorithm. This time step canbe selected either as a function of the CFL conditionrequired for stability, or as a function of the tempo-ral accuracy required by the user. In turbomachinerycases, any time accurate algorithm will pay a sub-stantial computational penalty. The decay rate of theinitial transients is slowed by the physical propaga-tion of waves through the length of the turbomachinerycomponent. Each wave can reflect in the opposite di-rection as it contacts a new stage, or it can propagatethrough the blade passage. As waves propagate backand forth through the machine, the length scales ofthe system are dramatically increased, and the phys-ical decay rate is substantially slowed. This situationis exacerbated in complex geometries where 100 timesteps are often needed to resolve a fundamental periodof the blade passing frequency.

A less expensive approach to the calculation of un-steady flows is to linearize the flow field about a meanflow solution and solve for the unsteadiness using a fre-quency domain approach.1 Assuming that the distur-bances are small compared to the mean flow values, theunsteady component of the solution can be expandedin a Fourier series and a decoupled equation is obtainedfor each of the fundamental modes in the expansion.This equation can be solved with small modifications

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American Institute of Aeronautics and Astronautics Paper 2002-0120

to typical steady-state solvers. The computationalcost is proportional to the cost of the steady-statesolution multiplied by the number of distinct modesrequired to attain a desired level of temporal accu-racy. However the error in the linearization procedurebecomes significant when the amplitude of unsteadi-ness in the flow is substantial, as happens in most realviscous flow examples.

Hall et al,2,3 have proposed a harmonic balancetechnique for the fully non-linear Navier-Stokes equa-tions. McMullen et al4 have proposed an alterna-tive technique in which the equations are solved inthe frequency domain. For time-periodic flows, ei-ther method can be computationally more efficientthan time-accurate approaches without requiring theassumptions of a linearized solution. In comparisonwith time accurate solvers, which are developed forarbitrary time histories, frequency domain methodsassume the flow solution is periodic in time. Thispermits a substantial reduction in computational costwhen compared with time accurate methods. As withthe linearized method the cost is proportional to thecost of the steady-state solution multiplied by thenumber of temporal modes required. In contrast withthe linearized methods, nonlinear frequency domainmethods do not make assumptions which limit themagnitude of the unsteadiness in the flow field. Thisallows these methods to predict flow field behavior fora broader class of problems and provides solutions inwhich all the temporal modes are coupled together.

Until now, one of the major deficiencies in frequencydomain solvers was the requirement that the user spec-ify a priori the time period of the solution’s oscillation.This paper will present a method to address this prob-lem. Application of this method requires the userto provide an initial guess of the time period. Themethod then iteratively solves for the solution as theresidual of the unsteady equations is minimized.

The Non-Linear Frequency Domain method calcu-lates only a limited number of temporal modes in thesolution. Because of this, this scheme is generally clas-sified as a reduced order method.5 These schemestypically trade a reduced level of computational ac-curacy for improvements in computational efficiency.Since each flow field presents different challenges thevalidity of the method must be verified on a case bycase basis. In addition, this verification process shouldcategorize the type of results which can be accuratelypredicted for each different flow field. Capturing onlythe dominant modes in the solution field can provideaccurate estimates for global properties such as coeffi-cient of lift and drag but may produce inferior resultsfor individual unsteady pressure coefficients. In thispaper we examine results from several different testcases and compare them to experimental data.

Governing EquationsViscous unsteady fluid flows in two dimensions can

be described by the Navier Stokes equations in integralform ∫

Ω

∂W

∂tdV +

∮

∂Ω

~F · ~Nds = 0, (1)

where

W =

ρρuρvρE

~F1 = f =

ρuρu2 + p− σxx

ρuv − σxy

ρuH − uσxx − vσxy + qx

~F2 = g =

ρvρuv − σxy

ρv2 + p− σyy

ρvH − uσxy − vσyy + qy

, (2)

and ~N is the outward pointing normal on the surfaceof the control volume. The variables ρ, u, v, and E aredensity, Cartesian velocity components and specific to-tal energy respectively. The flux terms also containthe thermodynamic pressure, p, the stress tensor, σ,and the heat flux vector obtained from Fourier’s heatconduction law, ~q. Closure is provided by the follow-ing equations for the pressure, shear stresses, and heatconduction.

p = (γ − 1)ρ[e− 1

2(u2 + v2)

]

σxx = 2µux − 23µ(ux + vy)

σyy = 2µvy − 23µ(ux + vy)

σxy = σyx = µ(uy + vx)

qx = κ∂T

∂x= − γ

γ − 1µ

Pr

∂ pρ

∂x

qy = κ∂T

∂y= − γ

γ − 1µ

Pr

∂ pρ

∂y. (3)

The equations can be discretized by dividing the flowdomain into smaller cells. The volume of each cellis denoted by V . Applying Eq. 1 to each cell in themesh, we can take the time derivative operator outsideof the integral sign and the remaining integral can beapproximated by the product of the cell volume andthe current value of the flow solution at the cell cen-ter. The boundary integral is calculated by discreteintegration of the fluxes around the control volumein a manner which is equivalent to central differenc-ing. A modified JST scheme6,7 is implemented to addthird-order artificial dissipation for numerical stability.Thus Eq. 1 may be expressed as

V∂W

∂t+ R(W ) = 0. (4)

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Here

R(W ) =n∑

j=1

~Fj · ~Sj (5)

is he residual for each cell, and vecFj is the fluxthrough the cell face with area ~Sj .

Under the assumption that the solution is periodicover a given time period, we can transform the in-dependent variables and residual terms into the fre-quency domain using a discrete Fourier transform. Inthe following discussion, the variables Wk and Rk rep-resent the Fourier coefficients (for a given wavenum-ber k) of the Fourier transforms of W and R(W )respectively. Numerically this transformation is ac-complished using the Fast Fourier Transform (FFT)in order to minimize the cost of computation. Thecomputational cost of this transform is proportionalto N log N , where N is the number of time inter-vals used to describe the signal. For real solutionsthe Fourier coefficients for the negative wavenumbersare simply the complex conjugates of the coefficientsfor the positive wavenumbers. By taking advantage ofthis property we can eliminate half of the wavenum-bers from the computation. Once we have obtainedthe Fourier coefficients for the expansions of W andR(W ), we can recover these quantities using the in-verse discrete Fourier transform as follows:

W =

N2 −1∑

k=−N2

Wkeikt

R(W ) =

N2 −1∑

k=−N2

Rkeikt, (6)

wherei =

√−1.

If we apply the discrete Fourier transform to thesemi-discrete form of the governing equation in Eq. 4,and we move the time derivative of the state vari-able inside the series summation, orthogonality of theFourier series ensures that the individual contributionsfrom each wavenumber are separately equal to zero

ikV Wk + Rk = 0. (7)

It follows that a periodic steady-state equation can bewritten for each independent wavenumber.

Instead of directly solving Eqs. 7 we add in a pseudo-time derivative and numerically integrate the resultingequations

VdWk

dτ+ ikV Wk + Rk = 0 (8)

to reach a steady state solution satisfying Eq. 7. Note,however, that solution of the physical problem will re-quire iteration between physical and Fourier spaces,

Fig. 1 Process Flowchart

since, due to the nonlinearity of the residual operator,R(W ), Rk cannot be computed directly from Wk. Rk

can only be computed by evaluation of the residual ata number of time locations within one periodic cycleand subsequent transformation to Fourier space usingthe discrete Fourier transform.

Figure 1 provides a flow chart of the data and trans-forms used to advance the solution through one itera-tion in pseudo time. At the beginning of this iterationwe know W at every grid point for all wavenumbers.This initial guess can simply result from the discreteFourier transform of a constant uniform flow. Usingan inverse FFT we can transform W back to the statevector W (t) in time at every grid point. At each timepoint we compute the residual R, and, using an FFTwe transform it back to the frequency domain to ob-tain R. We calculate the overall residual by adding Rto the source term ikV W . This overall residual, spec-ified in the frequency domain, is used to compute thenew approximate W , and the process returns to thebeginning of the cycle.

Solution TechniquesThe modified Eq. 8 represents the solution of a

steady system of equations in the frequency domain.This facilitates established convergence acceleration(originally developed for steady systems) techniquesto improve computational efficiency. A V or W cyclefull multigrid scheme with variable local pseudo-timesteps and implicit residual averaging has been imple-mented. The solution at each grid point is advanced inpseudo time using a modified multi-step RK scheme.Except for the residual averaging operations, the ex-plicit nature of the time advancement scheme facili-tates parallelization. The current solver utilizes multi-ple blocks distributed on different processors. A dualhalo scheme serves to locally retrieve the state vec-tor from neighboring grids. MPI libraries implementthe actual communication between processors. Theamount of inter-block communication scales linearlywith the number of wavenumbers the user specifies apriori.

Gradient Based Methods forDetermining the Time Period

For the class of flows where the time period of os-cillation is not known a priori we propose a methodto iteratively determine this parameter. This methodis based on forming a gradient of the residual withrespect to the time period. This gradient informa-tion is then used to iteratively update the time period

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as convergence is obtained for the unsteady flow solu-tion. Thomas et al8 have proposed a harmonic balancemethod that solves the unsteady equations in the timedomain. They argue that working in the time domainis easier, and facilitates reuse of existing codes. How-ever, solving the equations in the frequency domainprovide obvious approaches to forming gradients thatare the basis of this method.

To begin the derivation, the wavenumber k is calcu-lated by normalizing the sinusoidal period of oscilla-tion 2π by the time period of interest T .

k =2πn

T(9)

The unsteady residual in Eq. 8 can be can then bewritten as a function of the time period T .

−VdWn

dτ=

i2πnV

TWn + Rn (10)

The process of finding a solution to the unsteady flowequations is analogous to an optimization problemwhere the magnitude of the unsteady residual is min-imized. We can calculate a gradient of this unsteadyresidual with respect to the time period. This gradi-ent may then be used to iteratively modify the timeperiod until the unsteady residual is minimized.

In order to implement this concept, we rewrite ourunsteady residual as a figure of merit In defined foreach wavenumber.

In = −VdWn

dτ(11)

Because In is a complex quantity we minimize thesquare of the magnitude of this quantity. We can forma gradient of this cost function with respect to the timeperiod as

12

∂∣∣∣In

∣∣∣2

∂T= Inr

∂Inr

∂T+ Ini

∂Ini

∂T(12)

The quantity In is already calculated while monitoringthe convergence of the solution (note that the real andimaginary parts of In are Inr and Ini respectively).The partial derivative terms can be expanded in thefollowing equations.

∂Inr

∂T=

2πnV Wni

T 2(13)

∂Ini

∂T= −2πnV Wnr

T 2(14)

The formulas can be further simplified by introducingcross product notation. We can write the fourier co-efficient of the solution and residual in terms of twovectors.

~Wn = Wnr i + Wnij~In = Inr i + Inij (15)

Description Variable ValueAGARD CaseNumber CT6 DI 55Airfoil 64A010Mean Angleof Attack αm 0.00Angle of AttackVariation α0 ±1.01

Reynolds Number Re∞ 12.56x106

Mach Number M∞ 0.796Reduced Frequency kc 0.202

Table 1 AGARD Test Case Descriptions

Using this notation the gradient can be expressed asthe magnitude of the cross product of the above vec-tors.

12

∂∣∣∣ ˆIn

∣∣∣2

∂T=

2πnV

T 2| ~In × ~Wn| (16)

The time period can be updated using the gradientinformation by selecting a stable step ∆T .

Tn+1 = Tn −∆T∂

∣∣∣In

∣∣∣2

∂T(17)

Typically one can start with an initial guess in thevicinity of the final answer for the time period. Anunsteady flow solution can be obtained by solving theunsteady equations to some residual level. The abovegradient can then be used to adjust the time step ateach iteration in the solution. The solution and gra-dients are hence simultaneously updated, and a finalsolution can be calculated to any arbitrary residuallevel.

Results - Pitching AirfoilIn the following subsections we present results from

numerical simulations of the Euler and Navier-Stokesequations for a two dimensional airfoil undergoing aperiodic pitching motion. The motion of the airfoil isidentical to the experiments performed by Davis.9 Thecharacteristics of the test are summarized in Table 1.

Euler Equations

Two separate grid configurations were used to gener-ate the Euler results. The first grid configuration wasan ”O-mesh” generated by a conformal mapping pro-cedure. The second topology was a ”C-mesh” whichwas generated using a hyperbolic grid generation tool.Table 2 provides a list of the different grids used inthe Euler studies. Figures 2 and 3 depict the nearfield resolution of both the ”O-mesh” and ”C-Mesh”grids respectively.

It should be noted that the measured coordinates ofthe experimental cross-section do not exactly matchthe 64A010 theoretical profile. The deviation between

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Topology Dimensions Mean Mean GridBoundary SpacingDistance at Wall(Chords) (Chords)

O-mesh 81x33 128 0.0096O-mesh 161x33 128 0.0097C-mesh 129x33 22 0.0087C-mesh 193x49 22 0.0087

Table 2 Euler Mesh Descriptions

theoretical12,13 and measured coordinates is plotted inFigure 5. The impact this deviation had on the solu-tion was investigated using the higher fidelity Navier-Stokes solutions presented in upcoming sections. Forthe Euler simulations, the theoretical coordinates forthe 64A010 airfoil were used in all cases.

In general, the Euler solutions were relatively in-sensitive to mesh variations. Steady state simulationswere run and compared to the subsonic experimentalresults of Peterson.14 The results of all the numericalsimulations reproduce the experimental results withsufficient accuracy; verifying the spatial resolution ofall the grids.

On each mesh listed in Table 2 we have computedsolutions which use 1,2 and 3 temporal modes for a to-tal of 12 different solutions. Coefficient of lift (Cl) dataas a function of angle of attack, for all solution permu-tations, is provided in Figure 6. It is noteworthy, thatlittle deviation in the global coefficients is achievedwith additional temporal resolution. One can concludethat Euler solutions can be obtained with only onemode; which incurs the lowest possible computationalcost. The magnitude of the Cl ellipse is larger thanthe experimental results for every case. Having veri-fied the spatial accuracy of the grid, and investigatedthe effect of increasing temporal resolution; it seemsthat the over prediction of the Cl experimental resultsmay be attributed to the lack of viscous damping inthe governing equations, or to errors in the experimen-tal data. Comparisons with the viscous results in thefollowing sections tend to support this assertion.

Figure 7 provides coefficient of moment (Cm) datafor the ”C-mesh” cases listed in Table 2. The Cm datadeviates badly from the experimental results. Thetrend is not surprising, considering that previous inves-tigators, using similar spatial discretizations,10 showedsimilar discrepancies between the data and time accu-rate computations of this flow field.

Navier Stokes Equations

A ”C-mesh” grid configuration was used exclusivelyto generate the Navier-Stokes results. The grids weregenerated with the same hyperbolic mesh generationtool used in the previous section. Parameters definingthe three different grids used in this survey are pro-vided in Table 3. The highest density grid, 257x65

Topology Dimensions Boundary Mean GridDistance Spacing(Chords) at Wall

(y+)C-mesh 129x33 15 11.6C-mesh 193x49 12 6.9C-mesh 257x65 12 3.8

Table 3 Viscous Mesh Descriptions

points, is displayed in Figure 4.Each set of parameters defined in Table 3 were used

to generate grids around both the 64A010 airfoil andthe measured airfoil of the CT6 experiment. All ofthese grids were used to provide solutions employing1,2 and 3 temporal modes. Nine different solutionswere created for grids based on the 64A010 coordi-nates, and another nine solutions were created for themeasured coordinates of the experimental CT6 airfoil.

For the 64A010 coordinates, the coefficient of liftresults are provided in Figure 8. The variation in Cl

is minimal throughout the range of grid and modecombinations. The coarsest mesh using only a sin-gle mode provides a reasonable approximation to thehigher resolution cases. Although the viscous termshave mitigated the effect, there is still a tendency forthe numerical data to overshoot the experimental re-sults.

Coefficient of lift results for the CT6 experimentalairfoil are provided in Figure 9. Since the measuredcoordinates are not perfectly symmetric, there is aslight amount of asymmetry in the ellipses. Compari-son of Figures 8 and 9 shows that there is a significantamount of variation in the Cl data for a slight variationin the airfoil geometry.

The coefficient of moment data is more sensitive tovariations in the solution than the Cl data. Figures 10-12 provide Cm data as function of angle of attack foreach mesh used in this survey. The coarsest mesh pro-vides small variations in Cm over the range of temporalmodes employed in the solution. The finer meshes, ca-pable of resolving a larger number of temporal modes,show some variation in Cm with temporal accuracy.In both cases, the magnitude of the ellipse (formed bythe numerical results) is lessened with an increasingnumber of modes. This effect slightly moves the nu-merical results closer to the experiment data, but thedifference between the two sets is still significant.

Due to the variation in the numerical results, Cm

data for both the CT6 experimental airfoil and 64A010airfoil are provided in Figure 13. The differences notedabove (between the CT6 and 64A010 solutions) in theCl data are further magnified by the sensitivity of theCm data. The figure shows that there is a signifi-cant variation in the moment coefficient over the entirerange of solution permutations.

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Mesh 1 Mode 2 Modes 3 Modes(secs/cycle) (secs/cycle) (secs/cycle)

129x33 0.59 1.0 1.5193x49 1.9 2.9 4.5

Table 4 Execution Time

Computational Cost

This subsection provides data on the computationalcost of the nonlinear frequency domain solver. Wewill provide data on the convergence rates of the forcecoefficients and the execution speed of the code. Forthe Cl and Cm convergence surveys, we will narrowour focus to solutions using only one temporal mode.In addition, we will only provide data for two differentgrid configurations. The 129x33 ”C-mesh” will be usedfor all the Euler results and the 193x49 ”C-mesh” willbe used in the compilation of the viscous results.

Figures 14- 15 provide coefficient of lift data (forboth Euler and Navier-Stokes cases) as a function ofthe number of multigrid cycles used in the solver. TheEuler solver provides a converged Cl solution within10 to 15 cycles. Due to the increased condition num-ber, the convergence rate of the Navier-Stokes solver isslower. This latter case takes approximately 60 cyclesto provide a reasonably converged result. Figures 16-17 provide coefficient of moment data as a function ofthe number of multigrid cycles. A trend similar to theCl data is observed, with 20 to 30 cycles required toconverge the Euler case, and 120 cycles for the viscouscase.

The speed at which the code executes is providedin Table 4. The matrix provides execution times(in terms of seconds per multigrid cycle) as a func-tion of the number of temporal modes used in thesolution. The results were collected using a desktopcomputer equipped with an AMD 1.4Ghz Athlon pro-cessor. Note that the number of cycles required toconverge the global coefficients is roughly independentof the number of temporal modes used in the solution.The required execution time can be approximated bythe product of the number of required multigrid cyclesand the amount of time needed per cycle. All the tim-ing runs use 64 bit floating point arithmetic. 32 bitmath can be utilized and will reduce execution timesby approximately 50 percent. The timing results con-firm that the computational cost approximately growslike 2N +1, where N is the number of unsteady modes(not including the time average) used in the solution.

Results - Cylinder FlowOur second test case is the laminar flow around a cir-

cular cylinder. At Reynolds numbers between 49 and19415 vortices alternately shed off behind the cylinderinto a two-dimensional wake known as a Karman vor-tex street. The unsteady perturbations in this wake

are substantial and represent a good test of the non-linear frequency domain method. In contrast with theforced oscillations of the pitching airfoil, the vortexshedding is a function of instabilities in the shear layerof the flow, and the shedding frequency is not knowna priori. Experimental methods or simplified analytictechniques can be used to provide initial guesses atthis parameter. But the exact shedding frequency pre-dicted by the discretized equations is unknown at thestart of any numerical investigation.

Last year, McMullen et al4 presented results for thiscylinder flow case at Reynolds number of 180. Theseresults were produced without the ability of varyingthe time period. Instead, the Strouhal number wasfixed a priori based on experimental results. Coeffi-cient of drag and pressure results from this previouspaper are included in Table 5. The experimentalresults cited by this paper are included in Table 6.Counter intuitively, the predicted base suction coef-ficient deviates away from the experimental data astemporal resolution is increased.

Temporal Modes −Cpb Cd

1 0.832 1.2573 0.895 1.3065 0.903 1.3117 0.903 1.311

Table 5 Time Averaged Data versus TemporalModes Source: McMullen et al4

Experiment −Cpb Cd

Williamson and Roshko16 0.8265Henderson17 0.83 1.34

Table 6 Time Averaged Experimental Data

Here we present a new set of results generated bythe Gradient Based Variable Time Period (GBVTP)algorithm. These results are compiled from a sweepof simulations over a range of Reynolds numbers andtemporal accuracies. This sweep encompassed 40 dif-ferent numerical simulations. Ten different Reynoldsnumbers were simulated between 60 and 150. Foreach Reynolds number, 4 separate simulations wererun with 1,3,5 and 7 time varying harmonics. Thisbreadth of simulations will statistically give a betterestimate in the accuracy of these solutions; eliminat-ing any random results that can occur with a singlepoint comparison.

The grid used in these simulations is identical tothe grid previously employed by McMullen et al.4 Fig-ure 18 displays this grid directly near the cylinder wall.The dimensions are 256 by 128 points in the circum-ferential and radial directions respectively. The meshboundary is 200 chords from the center of the cylin-der. An exponential function stretches the grid in theradial direction with the smallest spacing of 3.54e−03

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chords occurring at the wall. At the top of the cylinderroughly 15 grid points capture the boundary layer.

Figures 19- 21 show the Strouhal frequency, meanbase suction coefficient, and mean drag coefficientas functions of Reynolds number and temporal ac-curacy. For the purpose of validation, experimentalresults from Williamson et al16,18–21 and numericalresults from Henderson et al17 are plotted on thesefigures. The latter numerical experiments are consid-ered highly accurate. They employ a spectral elementmethod based on 8th order accurate polynomials.

Unlike our previous results, our new predictions con-verge toward the experimental results as temporal ac-curacy is increased. For the cases where more than onetemporal mode was used, the percentage difference inthe Strouhal data between the experiment and numer-ical results is approximately 3.5 percent. This erroris consistent throughout the range of Reynolds num-bers, with the numerical simulations constantly underpredicting the experimental values. When a single har-monic is used, we observe another under predictingtrend. In this case, the numerical Strouhal data be-comes a better approximate of the experimental resultsas the Reynolds number decreases.

Unlike the Strouhal data, the mean base pressure co-efficient converges to experimental values at the lowerReynolds number for all different temporal resolutions.As the Reynolds number increases, only the least ac-curate solution drastically diverges. Slightly differentobservations can be made for the mean drag coefficientdata. At the higher temporal resolutions, the numericresults consistently under predict the mean drag coef-ficient data of Henderson. When the components ofdrag are separated into viscous and pressure compo-nents the same trend holds true.

The convergence patterns of the global coefficientssuggest that the magnitude of the higher harmonics inthe exact solution continues to increase with increasingReynolds number. This observation is supported byFigure 22 which shows the L2 norm of the magnitudeof the solution’s Fourier coefficient. As the Reynoldsnumber decreases the decay rate in the solutions higherharmonic increases. For the lower temporal accuracysimulations, the error incurred by the unresolved fre-quencies in the solution decays as the Reynolds num-ber decreases.

We now examine this equation: If one does notuse a variable time period algorithm, how accurate isthe solution based on any close guess of the Strouhalnumber? This question is partially answered in Fig-ures 23 and 24 which quantify the minimum residual asa function of the distance between the exact Strouhalnumber and one used in fixed Strouhal number sim-ulation. The data was generated by first obtainingthe exact Strouhal value using a gradient based tech-nique. Subsequent fixed Strouhal number simulationswere run at Strouhal number perturbed a given dis-

tance from the exact value. Each simulation wouldconverge to a residual level greater than the machinezero value. This is the residual plotted in semi-logformat in Figure 23 and log-log format in Figure 24.With these figures one can estimate the convergenceerror in the solution as a function of Strouhal number;since the residual scales with the distance between anapproximate solution and the exact one. This sup-ports the assertion that there is a unique time periodthat provides an exact (machine level representation)solution to the discretized equations.

ConclusionsThe numerical results confirm the accuracy and

computational efficiency of the nonlinear frequency do-main method which we have implemented for severalchallenging problems of significant practical impor-tance.

For the cylinder shedding test case, relatively ac-curate global coefficients were obtained using threetemporal modes. For the forced pitching airfoil caseonly one temporal mode was needed to accurately pre-dict the coefficient of lift. Experimental data for thecoefficient of moment is poorly predicted. The limitednumber of modes needed for each case confirms thepromise that this method can provide a computation-ally efficient algorithm for complex problems.

We have also shown that in problems like vortexshedding from the cylinder, the dominant natural fre-quency can be predicted by the Gradient Based Vari-able Time Period (GBVTP) method.

AcknowledgmentsThis work was sponsored by the Department of

Energy under contract number LLNL B341491 aspart of the Accelerated Strategic Computing Initiative(ASCI) program. The authors would like to acknowl-edge the help extended by Charles Williamson andSanford Davis for their personal communication oftheir experimental results.

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8J.P. Thomas, E.H. Dowell, and K.C. Hall. Nonlin-ear Inviscid Aerodynamic Effects on Transonic Divergence,Flutter and Limit Cycle Oscillations. AIAA paper 01-1209,42nd AIAA/ASME/ASCE/AHS/ASC Structures, StructuralDynamics, and Materials Conference & Exhibit, Seattle, WA,April 2001.

9S.S. Davis. NACA 64A010 (NASA Ames Model) Oscilla-tory Pitching. AGARD Report 702, AGARD, January 1982.

10N.A. Pierce and J.J. Alonso. Efficient Computation of Un-steady Viscous Flow by an Implicit Preconditioned MultigridMethod. AIAA Journal, 36:401–408, 1998.

11J. Hsu and A. Jameson. Personal communications on un-steady flow solvers, April 2001.

12Ira Abbott and Albert Von Doenhoff. Theory of WingSections. Dover Publications Inc, New York,, 1959.

13Michael Selig. UIUC Airfoil Coordinates Database. Tech-nical report, University of Illinois, February 2000.

14Robert Peterson. The Boundary-Layer and Stalling Char-acteristics of the NACA 64A010 Airfoil Section. NACA Tech-nical Report 2235, NACA, November 1950.

15C.H.K. Williamson. Vortex Dynamics in the CylinderWake. Annual Review Fluid Mech., 28:477–539, 1996.

16C.H.K. Williamson. Defining a Universal and ContinuousStrouhal-Reynolds Number Relationship for the Laminar VortexShedding of a Circular Cylinder. Phys. Fluids, 31:2742, 1988a.

17R.D. Henderson. Details of the Drag Curve Near the Onsetof Vortex Shedding. Physics of Fluids, 1995.

18C.H.K. Williamson and A. Roshko. Measurements of BasePressure in the Wake of a Cylinder at Low Reynolds Numbers.Z. Flugwiss. Weltraumforsch, 1990.

19C.H.K. Williamson and G.L. Brown. A Series in(1/sqrt(Re)) to Represent the Strouhal-Reynolds Number Re-lationship of the Cylinder Wake. J. Fluids and Struc., 1998.

20C.H.K. Williamson. The Existence of Two Stages in theTransition to Three-Dimensionality of a Cylinder Wake. Phys.Fluids, 31:3165, 1988b.

21C.H.K Williamson. Three-Dimensional Wake Transition.J. Fluid Mech., 328:345–407, 1996.

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−0.2 0 0.2 0.4 0.6 0.8 1 1.2

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Fig. 2 Nearfield of 81x33 Euler ”O-mesh”

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Fig. 3 Nearfield of 129x33 Euler ”C-mesh”

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Fig. 4 Nearfield of 257x65 Viscous ”C-mesh”

9 of 16


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

Y C

oord

inat

e of

Airf

oil

X Coordinate of Airfoil

Experiment Upper−Experiment LowerAbott Von−DoenhoffUIUC Database

Fig. 5 Deviation Between Measured Coordinates of Experimental Foil and Theoretical 64A010 Airfoil

−1.5 −1 −0.5 0 0.5 1 1.5−0.125

−0.1

−0.075

−0.05

−0.025

0

0.025

0.05

0.075

0.1

0.125

Angle of Attack (degrees)

Coe

ffici

ent o

f Lift

ExperimentEuler Numeric Solutions

Fig. 6 Coefficient of Lift versus Angle of Attack - All Euler Solutions

−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent

ExperimentEuler Numeric Solutions

Fig. 7 Coefficient of Moment versus Angle of Attack - All Euler Solutions

10 of 16


−1.5 −1 −0.5 0 0.5 1 1.5−0.125

−0.1

−0.075

−0.05

−0.025

0

0.025

0.05

0.075

0.1

0.125


Coe

ffici

ent o

f Lift

ExperimentN−S Numeric Solutions

Fig. 8 Coefficient of Lift versus Angle of Attack - All Viscous Solutions of 64A010 Airfoil

−1.5 −1 −0.5 0 0.5 1 1.5−0.125

−0.1

−0.075

−0.05

−0.025

0

0.025

0.05

0.075

0.1

0.125


Coe

ffici

ent o

f Lift

ExperimentN−S Numeric Solutions

Fig. 9 Coefficient of Lift versus Angle of Attack - All Viscous Solutions of CT6 Experimental Airfoil

−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent

Experiment1 Mode2 Mode3 Mode

Fig. 10 Coefficient of Moment versus Angle of Attack - Viscous Solution - 129x33 Grid of 64A010 Airfoil

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−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent



−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent



−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent


Fig. 13 Coefficient of Moment versus Angle of Attack - Viscous Solution - All Grids of CT6 ExperimentalAirfoil

12 of 16


−1.5 −1 −0.5 0 0.5 1 1.5−0.125

−0.1

−0.075

−0.05

−0.025

0

0.025

0.05

0.075

0.1

0.125


Coe

ffici

ent o

f Lift

1 Cycle5 Cycle10 Cycle15 Cycle

Fig. 14 Convergence of Coefficient of Lift versus Angle of Attack - Euler - 129x33 Grid

−1.5 −1 −0.5 0 0.5 1 1.5−0.125

−0.1

−0.075

−0.05

−0.025

0

0.025

0.05

0.075

0.1

0.125


Coe

ffici

ent o

f Lift


Fig. 15 Convergence of Coefficient of Lift versus Angle of Attack - Viscous - 193x49 Grid

−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent


Fig. 16 Convergence of Coefficient of Moment versus Angle of Attack - Euler - 129x33 Grid

13 of 16


−1.5 −1 −0.5 0 0.5 1 1.5−0.015

−0.0125

−0.01

−0.0075

−0.005

−0.0025

0

0.0025

0.005

0.0075

0.01

0.0125

0.015


Coe

ffici

ent o

f Mom

ent


Fig. 17 Convergence of Coefficient of Moment versus Angle of Attack - Viscous - 193x49 Grid

−2 −1 0 1 2 3 4 5 6−4

−3

−2

−1

0

1

2

3

4

Fig. 18 Computational Grid for Cylinder Flow

14 of 16


60 70 80 90 100 110 120 130 140 1500.12

0.13

0.14

0.15

0.16

0.17

0.18

0.19

Reynolds Number

Str

ouha

l Num

ber

1 Harmonic3 Harmonic5 Harmonic7 HarmonicWilliamson 1988

Fig. 19 Strouhal Numbers versus Reynolds Number For Laminar Vortex Shedding

60 70 80 90 100 110 120 130 140 1500.5

0.6

0.7

0.8

0.9

Reynolds Number

Neg

ativ

e B

ase

Pre

ssur

e C

oeffi

cien

t

1 Harmonic3 Harmonic5 Harmonic7 HarmonicHenderson 1995Williamson 1990

Fig. 20 Mean Base Pressure Coefficient versus Reynolds Number For Laminar Vortex Shedding

60 70 80 90 100 110 120 130 140 1500.2

0.4

0.6

0.8

1

1.2

1.4

Reynolds Number

Mea

n C

oeffi

cien

t of D

rag

Viscous Component of Cd

Pressure Component of Cd

Total Cd

1 Harmonic3 Harmonic5 Harmonic7 HarmonicHenderson 1995

Fig. 21 Mean Coefficient of Drag versus Reynolds Number For Laminar Vortex Shedding

15 of 16


60 70 80 90 100 110 120 130 140 15010

−7

10−6

10−5

10−4

10−3

10−2

L2 N

orm

Of D

ensi

ty*E

nerg

y

Reynolds Number

1st Harmonic2nd Harmonic3rd Harmonic4th Harmonic5th Harmonic6th Harmonic7th Harmonic

Fig. 22 L2 Norm of Solution Energy versus Reynolds For Laminar Vortex Shedding

0.1 0.15 0.210

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Min

imum

Res

idua

l

Stroudhal Number

Fig. 23 Minimum Residual versus Strouhal Frequency

10−15

10−10

10−5

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

ABS(Stroudhal Number−Machine Zero Stroudhal Number)/ Machine Zero Stroudhal Number

Min

imum

Res

idua

l

Average ComponentFirst Harmonic

Fig. 24 Minimum Residual versus Strouhal Frequency

16 of 16


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