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AIAA 95–0048 Multigrid Unsteady Navier-Stokes Calculations with Aeroelastic Applications Juan J. Alonso, Luigi Martinelli and Antony Jameson Princeton University, Princeton, NJ 08544 33rd AIAA Aerospace Sciences Meeting and Exhibit January 9–12, 1995/Reno, NV For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344
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Page 1: Multigrid Unsteady Navier-Stokes Calculations with ...aero-comlab.stanford.edu/Papers/reno95.pdfNavier-Stokes Equations and Discretization The two-dimensional unsteady compressible

AIAA 95–0048Multigrid Unsteady Navier-StokesCalculations with AeroelasticApplicationsJuan J. Alonso, Luigi Martinelli and Antony JamesonPrinceton University, Princeton, NJ 08544

33rd AIAA Aerospace SciencesMeeting and Exhibit

January 9–12, 1995/Reno, NVFor permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

Page 2: Multigrid Unsteady Navier-Stokes Calculations with ...aero-comlab.stanford.edu/Papers/reno95.pdfNavier-Stokes Equations and Discretization The two-dimensional unsteady compressible

AIAA 95–0048

Multigrid Unsteady Navier-Stokes Calculationswith Aeroelastic Applications

Juan J. Alonso,∗ Luigi Martinelli† and Antony Jameson‡

Princeton University, Princeton, NJ 08544

An implicit approach to the solution of the unsteady two-dimensional Navier-Stokesequations is presented. After spatial discretization, the resulting set of coupled implicitnon-linear equations is solved iteratively. This is accomplished using well proven conver-gence acceleration techniques for explicit schemes such as multigrid, residual averaging,and local time-stepping in order to achieve large computational efficiency in the calcu-lation. Calculations are performed in parallel using a domain decomposition techniquewith optimized communication requirements. In addition, particular care is taken tominimize the effect of numerical dissipation with flux-limited dissipation schemes. Re-sults for the unsteady shedding flow behind a circular cylinder and for a pitching NACA64A010 airfoil are presented with experimental comparisons, showing the feasibility ofaccurate, efficient, time-dependent viscous calculations. Finally, a two-dimensional struc-tural model of the cylinder is coupled with the unsteady flow solution, and time responsesof the deflections of the structure are analyzed.

NomenclatureCl coefficient of lift

Cd coefficient of drag

Cx, Cy damping coefficients in the two coordinate di-rections

D cylinder diameter, cylinder drag

E total energy (internal plus kinetic)

E(wij) convective Euler fluxes

f ,g Euler flux vectors

H total enthalpy

Kx,Ky spring constants in the two coordinate direc-tions

L airfoil section lift (normal to free stream), pos-itive up

m cylinder mass

M∞ free stream Mach number

n frequency, 1/sec

NS(wij) viscous flux residual for cell i,j

p static pressure

qi heat flux component

R(wij) total flux residual for cell i,j

R∗ modified residual

R,S viscous flux vectors

ReD Reynolds number based on the diameter

St Strouhal frequency, St = nDU∞

T static temperature

u, v cartesian velocity components

U∞ free stream velocity

Vij volume of i,j cell

w vector of flow variables

∗Graduate Student, AIAA Member†Assistant Professor, AIAA Member‡James S. McDonnell Distinguished University Professor of

Aerospace Engineering, AIAA FellowCopyright c© 1995 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

xt, yt mesh cartesian velocity components

∆α pitching motion forcing amplitude

∆t implicit real time step

γ ratio of specific heats, γ = 1.4

ρ air density

σij viscous stress tensor components

ωf frequency of the forced oscillations

Ω, ∂Ω cell element and boundary

Introduction

UNSTEADY flow solvers are becoming a neces-sary part of the toolkit of the computational

fluid dynamicist. In order to solve problems whichare naturally unsteady (such as vortex shedding flows,moving boundary problems, fluid-structure interactionflows, etc.) it is essential to develop numerical schemeswhich provide accurate solutions at a reasonable cost.Therefore, computational efficiency is of paramountimportance for unsteady numerical solutions. As thegoverning equations increase in complexity (viscosity,turbulence, etc.), the computational efficiency of themethod determines whether the approach can be use-ful as either an engineering or a research tool.

On the other hand, the solution of both inviscid andviscous steady flows can now be easily obtained, evenfor relatively complex geometries. This progress inthe solution of steady flows has been closely coupledto the development of higher price/performance com-puting platforms, as well as to the appearance of veryefficient numerical schemes and convergence accelera-tion techniques. Therefore it seems logical to make useof some of these techniques that have proven success-ful in the solution of steady flows, and apply them tothe calculation of time dependent flows.

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Explicit time accurate methods must advance allcells in the computational domain with the same timestep to maintain consistency. The global time stepis therefore limited by the most restrictive of the al-lowable time steps in the domain. Unfortunately, dueto the Courant-Friedrichs-Levy (CFL) restriction, thiscan lead to a very large number of time steps in or-der to complete a calculation. Unless the frequency ofthe physical phenomena that we want to resolve is ofthe order of the characteristic frequencies of acousticdisturbances, such a choice of the time step is basedon the stability of the numerical scheme, and not onissues of accuracy of the solution of the problem of in-terest. Therefore, we see ourselves forced to choose atime-step that is quite a bit smaller than the one wewould require to ensure accuracy, with a consequentincrease in the time of computation.

If the problem is formulated in an implicit fash-ion, this explicit time step limitation can be bypassed.Moreover, at each time step, the implicit set of equa-tions which results from the discretization of the gov-erning equations of the flow can be cast as a modifiedsteady-state problem in pseudo-time. Then, efficienttechniques such as multigrid and residual averagingcan be applied without sacrificing time accuracy, andgreat improvements in computational performance canbe achieved.

Brenneis and Eberle1 solved the unsteady two- andthree-dimensional implicit unfactored Euler equationsby using a non-linear Newton method and a Gauss-Seidel algorithm for relaxation purposes. They used afirst order accurate discretization of the time derivativeoperator which allowed them to use on the order offifty to one hundred time steps per cycle of oscillationof the physical phenomena in question.

Jameson2 used a second order backwards differenceoperator for the time derivative term, and appliedmultigrid and residual averaging to the solution of theimplicit system. With this approach he was able tosolve unsteady flows about two- and three-dimensionalconfigurations in twenty to thirty time steps per cycleof oscillation. Later, this method was shown to workwell for a third order accurate stiffly stable backwardsdifference formula, as well as for aeroelastic calcula-tions.3 With this discretization, only fifteen steps perperiod of oscillation are necessary to ensure an accu-rate solution.

It has become clear that inviscid models yield a costeffective approximation to the solution of unsteadyproblems, but when strong shocks and separated flowsare involved, it is necessary to incorporate viscosityand turbulence effects into the model.2, 5 Especiallyin transonic flows where shocks are present, there is atendency of inviscid models to overpredict the strengthof the shock, and to place the location of these shocksslightly aft of the experimental location.1 With thisperspective in mind it is only natural to think of ap-

plying a similar approach to that of Jameson2 to thesolution of the unsteady Navier-Stokes equations. Inthis case, the explicit time-step restrictions are evenless tolerable, since the ratio of the maximum to theminimum size of the cells in a viscous mesh can spanseveral orders of magnitude. This means that the gainsto be obtained from an implicit method are even moresubstantial than those reported for the Euler equa-tions. Melson et al.4 applied Jameson’s approach tothe solution of the unsteady Thin Layer Navier-Stokesequations and pointed out the possible gain in robust-ness of the algorithm when some terms are treatedimplicitly in the Runge-Kutta time-stepping. Arnoneet al.5, 6 followed the same approach and computedflows over rows of cylinders, bicircular arc airfoils, andturbomachinery flows at subsonic and transonic Machnumbers with very promising results.

In this work we present a similar approach appliedto the solution of the full Navier-Stokes equations,8, 9

where all these improvements to the convergence ofthe method have been incorporated to the scheme,and various flux-limited dissipation schemes are eval-uated.7 Vortex shedding flows over stationary andoscillating cylinders in an infinite free stream are cal-culated, as well as the unsteady flow over a pitchingNACA 64A010 airfoil. Experimental comparison ofthe unsteady lift coefficient for the NACA 64A010airfoil is shown, and preliminary results of the aeroe-lastic coupling of the flow over a cylinder with a two-dimensional model of the structure are also presented.

Navier-Stokes Equations andDiscretization

The two-dimensional unsteady compressibleNavier-Stokes equations, after the appropriate non-dimensionalizations, can be written in a cartesiancoordinate system (x, y) as:

∂w∂t

+∂f∂x

+∂g∂y

=(∂R∂x

+∂S∂y

), (1)

where w is the vector of flow variables, f and g arethe convective flux vectors, and R and S are the vis-cous flux vectors in each of the coordinate directions.Consider a control volume Ω with boundary ∂Ω whichmoves with cartesian velocity components xt and yt.The equations of motion of the fluid can then be writ-ten in integral form as

d

dt

∫∫

Ω

w dx dy +∮

∂Ω

(f dy − g dx) =

=∮

∂Ω

(R dy − S dx), (2)

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where w is the vector of flow variables

w =

ρρuρvρE

,

f , g are the Euler flux vectors

f =

ρ(u− xt)ρu(u− xt) + pρv(u− xt)

ρE(u− xt) + pu

, g =

ρ(v − yt)ρu(v − yt)

ρv(v − yt) + pρE(v − yt) + pv

,

and

R =

0σxx

σxy

uσxx + vσxy − qx

,

S =

0σxy

σyy

uσxy + vσyy + qy

are the viscous flux vectors, where the stress tensorcomponents and the heat flux vector are given by

σxx = 2µux − 23µ(ux + vy)

σyy = 2µvy − 23µ(ux + vy)

σxy = σyx = µ(uy + vx)

qx = −κ∂T∂x

qy = −κ∂T∂y.

The coefficient of viscosity, µ, is calculated accordingto Sutherland’s law. Also, for an ideal gas, the equa-tion of state may be written as

p = (γ − 1) ρ[E − 1

2(u2 + v2)

].

Turbulence effects can be taken into account by use ofa suitable turbulence model. In this work, a Baldwin-Lomax algebraic turbulence model11 has been usedwhen necessary. Details of the implementation can befound in9 and.10 When the integral governing equa-tions (2) are applied independently to each cell in thedomain, we obtain a set of coupled ordinary differen-tial equations of the form

d

dt(wij Vij) + E(wij) + NS(wij) + D(wij) = 0, (3)

where E(wij) are the convective Euler fluxes, NS(wij)are the Navier-Stokes viscous fluxes, and D(wij) arethe artificial dissipation fluxes added for numerical sta-bility reasons. This equation (3) can be discretized

implicitly as follows (drop the i, j subscripts for clar-ity):

d

dt[wn+1 V n+1] + R(wn+1) = 0, (4)

where R is the sum of the three flux contributions,and the superscripts denote the time step of the calcu-lation. If we discretize the time derivative term with,say, a backwards difference second order accurate op-erator, we obtain

32∆t

[wn+1V n+1] − 2∆t

[wnV n] +1

2∆t[wn−1V n−1]

+ R(wn+1) = 0. (5)

This equation for wn+1 is non-linear due to the pres-ence of the R(wn+1) term and cannot be solved di-rectly. One must therefore resort to iterative methodsin order to obtain the solution. The time integrationof the Navier-Stokes equations at each time step canthen be seen as a modified pseudo-time steady-stateproblem with a slightly altered residual

R∗(w) =3

2∆t[w V n+1] − 2

∆t[wnV n]

+1

2∆t[wn−1V n−1] + R(w).

In this case, the vector of flow variables w which sat-isfies the equation R∗(w) = 0 is the w(n+1) vectorwe are looking for. In order to obtain this solutionvector, we can reformulate the problem at each timestep as the following modified steady-state problem ina fictitious time, t∗:

dwdt∗

+ R∗(w) = 0, (6)

to which one can apply the fast convergence techniquesused for steady-state calculations. Applying this pro-cess repeatedly, one can advance the flow field solutionforward in time in a very efficient fashion. The readeris referred to2 for a more complete description of theapproach.

In this work, a third order accurate discretizationof the time derivative operator has been used in allcases. When applied to the model one-dimensionalproblem, this discretization is stiffly stable, but dueto the presence of viscous fluxes, this slight restric-tion does not present practical problems. When aFourier analysis for the model one-dimensional equa-tion is performed, it is seen that the Fourier symbolof the residual shifts in the negative direction of thereal axis by an amount which is proportional to thepseudo-time step in each cell, ∆t∗, with a proportion-ality constant that depends on the order of accuracyof the time discretization. There exists the possibility,for the large cells in the domain (where ∆t∗ is large),that the numerical scheme will become unstable. Inthe original code, the time step was cut off at a given

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level in order to restrict the shifting of the Fourier sym-bol of the residual out of the stability region of theRunge-Kutta time-stepping scheme. The proportion-ality constant becomes larger as the order of accuracyof the time discretization increases, and curiously, asthe real time step ∆t decreases. Melson et al.4 treatedthe 3

2∆t [w V n+1] term implicitly in the Runge-Kuttatime-stepping. In this case, the one-dimensional analy-sis indicates the disappearance of the stability problemin the larger cells of the domain, and a possible de-crease in the amplification factor for all frequencies.In a sense, their approach is equivalent to a rescalingof the pseudo-time step for every cell in the domain,and, in practice, does not lead to noticeable improve-ments in convergence. Nevertheless, this change in thealgorithm increases considerably the robustness of themethod, and eliminates the need of an extra parame-ter. Therefore, it has been adopted in our code.

Numerical Dissipation

The calculation of compressible flows at transonicand supersonic Mach numbers requires the implemen-tation of non-oscillatory discrete schemes which com-bine high accuracy with high resolution of shock wavesand contact discontinuities. These schemes must alsobe formulated in such a way that they facilitate thetreatment of complex geometric shapes. When deal-ing with viscous flows, care must be taken in ordernot to spoil the accurate resolution of the viscous phe-nomena. The difficulty rests on the fact that shockcapturing requires the construction of schemes whichare numerically dissipative, a requirement which couldaffect the global accuracy of the solution of the physi-cal viscous problem.

In a recent paper,7 starting from the local extremumdiminishing (LED) principle proposed by Jameson,12

the accuracy of a large family of schemes, comprisingboth high resolution switched and flux-limited dissipa-tion schemes, has been evaluated for viscous solutions.In particular, both symmetric limited positive (SLIP),and upstream limited positive (USLIP) schemes, werefound to yield very accurate resolution of laminarboundary layers on a wide range of Mach and Reynoldsnumbers.

More recently, a new approach to the formulation ofthe artificial dissipation which combines sharp resolu-tion of discontinuities with good accuracy in boundarylayers and wakes has been proposed.13 This new ap-proach has been denominated H-CUSP, and was usedin this paper. Validation for steady viscous flows anda description of the scheme can be found in an accom-panying paper.25

One of the objectives of the present work is to inves-tigate the validity of the H-CUSP scheme for unsteadyviscous flows with moving shock waves.

Aeroelastic ModelThe model used in these initial calculations is sim-

ilar to the typical section wing model widely used inaeroelastic calculations.14, 15 In this case, instead ofhaving a torsional spring to model the pitching de-flection of the wing, the cylinder is connected to twolinear springs along each of the coordinate axes. Sinceboth Cl and Cd are functions of time, displacementsin both coordinate directions will be excited by theunsteadiness in the flow.

The governing equations of the structural model aresimply:

mx+ Cxx+Kxx = L

my + Cy y +Kyy = D,

where L and D are the lift and the drag of the cylinderrespectively, m is the mass of the cylinder, Cx and Cy

are the damping coefficients of the dampers, and Kx

and Ky are the spring constants of the horizontal andvertical springs (all of which can be seen in Figure 1).These equations are already in modal form (since theyare fully uncoupled), and can be solved implicitly usingthe approach in.3 This approach involves the decom-position of each of these modal equations into a systemof first order differential equations which is later diag-onalized and integrated in time with a Runge-Kuttaapproach. The reader is referred to3 for more detailson this procedure.

The structural model is coupled to the Navier-Stokesequations in the pseudo-time steady-state calculationfor each time step. This coupling is fully implicit, andtherefore, at the end of each pseudo-time iteration,both systems of equations are in full agreement witheach other. The cylinder is held in place until thevortex shedding flow pattern is fully developed, andis then released and left to evolve due to its own self-induced forces.

The complete model is supposed to represent thefirst bending mode in both coordinate directions of along cable in water or air, which is a relevant prob-lem for both marine structures and civil engineeringproblems.17

Parallelization StrategyIncreasingly complex fluid flow models require high

performance computing facilities. A cost effective solu-tion for problems of this type requiring fast CPU’s andlarge internal memory is the use of a parallel comput-ing paradigm. For numerical efficiency, one typicallyincorporates convergence acceleration techniques suchas multigrid and implicit residual smoothing. Messagepassing becomes necessary in this new environment,and severely limits the performance of processes thatare inherently communication intensive.

UFLO82NSP is parallelized using a domain decom-position model, a SPMD (Single Program Multiple

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Data) strategy, and the MPI (Message Passing Inter-face Standard) Library for message passing.19 Flowswere typically computed on an O-mesh of size ni×nj =384 × 96. This domain was decomposed into sub-domains containing ni

Np× nj points, where Np is the

number of subdomains used. Communication betweensubdomains was performed through halo cells sur-rounding each subdomain boundary. A two-level halowas sufficient to calculate the convective, viscous, anddissipative fluxes in the finest mesh of the cycle. Inthe coarser levels of the multigrid cycle, a single levelhalo suffices.

For problems with a low task granularity (ratio ofthe number of bytes received by a processor to thenumber of floating point operations it performs), largeparallel efficiencies can be obtained. Unfortunately,convergence acceleration techniques developed in the1980s base their success on global communication inthe computational domain. Thus, current multigridand especially implicit residual smoothing techniqueshinder parallel performance. Nevertheless, one caneffectively deal with these issues by modifying the al-gorithm to minimize the amount of messages passed.Several techniques to deal with the parallelization ofmultigrid and implicit residual smoothing were de-scribed in.20 In particular, the usage of an iterativeapproach for the implicit residual averaging, restoresparallel efficiencies to a level around 90% for 8 proces-sors.

With this and other improvements incorporated tothe parallel version of the code, the unsteady com-putations in this work can be carried out in a smallamount of time, which allows for almost interactiveparameter studies, as well as continuous feedback onthe computations.

Parallel benchmarks were performed on an IBM SP2platform with up to 8 processors. In Figure 2, we cansee the parallel speedup curves for the full code, includ-ing multigrid and residual averaging, for two differentmessage passing implementations on the IBM SP1 andSP2 distributed memory machines. For these struc-tured meshes, the domain decomposition is straightfor-ward, and nearly perfect load balance is achieved. Forthe size of the meshes used in this study, parallel per-formance starts to drop off for 12 or more processors,since the amount of computations that each processorperforms becomes offset by the increasing amount ofcommunication required by a larger number of proces-sors.

For illustration purposes, Figure 3 shows the Machnumber contours on a NACA 64A010 airfoil at 2 angleof attack, and Re∞ = 5000, with the processor bound-aries being the nearly radial coordinate emanatingfrom the airfoil. This calculation used 12 processors.The continuity of the Mach number contour lines is anindicative of the quality of the parallel solution.

Results and Discussion

This section is divided according to the type of com-putations performed. First, we will show the resultsfor the simulation of the vortex shedding flow over acircular cylinder at a Reynolds number ReD = 500.Experimental investigations21–23 show that a laminar,two-dimensional wake exists in the Reynolds numberrange 40 − 180. After that, the wake becomes three-dimensional. In the first section we are only interestedin the properties of the numerical scheme. Therefore,the fact that the real flow is three-dimensional is ofno consequence. Secondly, we will show the resultsfor the unsteady transonic flow over a pitching NACA64A010 airfoil and compare it to experimental data.Finally, aeroelastic calculations for a structural modelof a circular cylinder are carried out, and comparisonsof the maximum oscillation amplitude with experimen-tal data are presented.

Vortex shedding over a circular cylinder atReD = 500

In order to test the accuracy of the baseline code,we chose a problem that is well documented both ex-perimentally and numerically, such that meaningfulcomparisons of the computed solutions were possible.Initially, we investigated the problem of vortex shed-ding behind a circular cylinder. It is well known16 thatin the Reynolds number range from 40 to 5000, a reg-ular Karman vortex street is observed. In this range,the Strouhal frequency of vortex shedding, St = n D

U∞,

where n is the real frequency, D is the diameter of thecylinder, and U∞ is the free stream velocity, is only afunction of the Reynolds number.

We set out to perform a refinement study in bothspace and time in order to determine the minimumresolutions that must be used if one wants to havefully resolved calculations. During these studies, theReynolds number was fixed at ReD = 500, the Machnumber for the computation was M∞ = 0.20 (whichis very nearly incompressible flow), and the far fieldboundary of the mesh was fixed at 50 cylinder diame-ters. In Figure 4, we can see a detail of the inner partof the grid used in this calculation. In the 384 × 96mesh, about 32 cells were placed in the boundary layerin order to fully resolve the effects of the presenceof the solid wall. The mesh stretches geometricallyfrom the surface of the cylinder in order to cluster thepoints near the solid surface. At the solid wall, theno-slip kinematic condition is imposed. The wall isalso considered to be adiabatic. In the far field, in-viscid, non-reflecting boundary conditions are used.27

Due to the symmetry of the code and the initial condi-tions, vortex shedding can only be caused by roundofferror in the simulation. This was proven to be truein one of the initial simulations. However, in orderto avoid unnecessary waste of computational time, thecylinder was forced in pitch for one cycle of amplitude

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∆α = 1, at a frequency, ωf , close to the vortex shed-ding frequency. After this forced cycle, the cylinderremains stationary for the rest of the calculation. Theflow slowly develops an asymmetric instability, and al-ternate shedding proceeds in a periodic fashion. InFigure 5, we can see the typical development of theperiodic response in the coefficients of lift and drag.Notice the small notch produced by the sudden ceasein the forced motion of the cylinder. After the flow isforced, the strength of the shed vortices increases untilit reaches a limit amplitude in both Cl and Cd. Noticeas well that the frequency of the coefficient of dragis twice that of the coefficient of lift due to symme-try considerations. Figure 10 shows the instantaneousstreamlines at three different times in the shedding cy-cle once periodicity has fully developed, and one canclearly see the process by which the separating bound-ary layer feeds vorticity into the vortex that is aboutto be shed. Finally, in Figure 6, we can see the firstfour vortices that are captured before the grid coarsensbeyond the point where an accurate solution can beobtained.

Four meshes were used in the grid refinement studyconducted for this flow. The sizes of these mesheswere: 96× 24, 192× 48, 384× 96, and 384× 192. Thefirst two meshes (as we can see in Figure 7) do notprovide the appropriate resolution of the viscous phe-nomena. The added viscosity (in the form of artificialdissipation introduced by the numerical scheme) re-tards the development of the shedding, and limits thepeak to peak variation of the coefficient of lift, whileincreasing the coefficient of drag. The flows in the fol-lowing two meshes exhibit a slight difference in thetime to develop the full vortex shedding motion, but,as we can see in Table 1, the differences in the Strouhalfrequency of shedding, and the Cl amplitude are on theorder of one half of a percent. Apart from the smalldifference in phase, it was considered that the 384×96mesh provided the necessary accuracy for the solutionof this type of flows, and therefore, all the subsequentcalculations in this paper are performed in meshes ofthese dimensions. All calculations in this mesh refine-ment study were done with 20 time steps per periodof oscillation of the lift coefficient. Due to the bunch-ing of the cells close to the surface of the cylinder,Courant numbers of the order of 50, 000 and higher canbe achieved in these calculations with the implicit timestepping scheme. Thus, the implicit scheme can be ofthe order of one thousand times more efficient than anexplicit scheme, which could only achieve compara-ble accuracy at prohibitive computational costs. Thecalculated Strouhal frequency was St = 0.231, whichagrees quite well with the experimental value of 0.235extrapolated from the curves for incompressible flowin the laminar shedding regime.

In order to verify the properties of the discretizationof the time derivative operator, we conducted a study

Mesh Size StClClmax Cdavg StCd

96 × 24 0.13384 0.05624 0.68857 0.27292192 × 48 0.22466 0.87248 1.10768 0.44981384 × 96 0.23313 1.14946 1.31523 0.46735384 × 192 0.23103 1.14671 1.35923 0.46868

Table 1 Relevant Quantities in the Spatial GridRefinement Study

Time stepsper period StCl

Clmax Cdavg StCd

10 0.22812 0.92011 1.24888 0.4569820 0.23313 1.14946 1.31523 0.4673530 0.22320 1.06801 1.22030 0.4466940 0.22469 1.04562 1.21759 0.4487850 0.22470 1.04560 1.21723 0.44923

Table 2 Relevant Quantities in the Time Step Re-finement Study.

which used 10, 20, 30, 40, and 50 time steps per cycleof oscillation of the lift coefficient for the same param-eters in the problem above. The size of the mesh usedin this study was 384 × 96. The same procedure wasfollowed to force the flow into its periodic solution.Figure 8 shows the time history of Cl for the calcula-tions with 10, 20, 30, and 40 time steps per period afterthe shedding motion is fully developed. The results for50 time steps per period are omitted since they coin-cided exactly with those for 40 time steps per period,and would make the figure less readable. The resultsfor 30 and 40 time steps per period are in very goodagreement except for a slight difference in phase. InTable 2, we see that the Strouhal frequency of sheddingdiffers by 0.6% for these last two cases. It was there-fore estimated that a number of time steps between 30and 40 is sufficient to ensure a very accurate solution.With this in mind, the simulations in the followingsections used 36 time steps per period of oscillation ofthe meaningful quantities. All these calculations havebeen made with a third order accurate discretizationof the time derivative term. While the third order dis-cretization requires the storage of an extra level of thegrid and flow variables, the number of time steps perperiod that are necessary is smaller, thus making itmore computationally efficient. It is believed that thetradeoff between computational efficiency and mem-ory storage might favor the second order scheme forviscous three-dimensional calculations.

Pitching NACA 64A010 Airfoil

One of the advantages of the current formulationis that it can be applied to the whole Mach numberrange. The algorithm discussed in this paper cov-ers the compressible regime (subsonic, transonic, andsupersonic), whereas an accompanying paper18 dis-cusses the implementation of an artificial compressibil-ity method for truly incompressible flows. Moreover,the algorithm is valid for complex configurations, and

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one needs not be limited to the solution of flows overa cylinder. Our area of interest is mainly in transonicflows, and therefore, we wanted to compute the un-steady flow over a pitching airfoil, in this case a NACA64A010 airfoil. In the case of turbulent calculations,explicit methods become unaffordably expensive, andthe current method can really be of help.

In the past,2 the inviscid flow over this airfoil hasbeen calculated, and the unsteady Cl vs. α curvewas found to agree quite well with the experiment.The experimental results lie on a periodic curve inthe shape of an oval, which is slightly broader thanthe inviscid calculations. In Figure 9 we can see thatthe inclusion of viscous effects tilts the inviscid ovalin the direction of the experimental results. The veryslight difference still existing can be due to inaccu-racies in the turbulence model. For attached flows,we can conclude that inviscid solutions are the bestcompromise for the prediction of lift related proper-ties. However, if in addition one wants to obtain dragand pitching moment information, the only option isto use the full Navier-Stokes equations with a suitableturbulence model. Figure 11 shows the motion of theshocks in the upper and lower surfaces of the airfoil fora little more than half a pitching cycle. The snapshotsare arranged by rows. As the airfoil pitches up, theshock in the upper surface moves aft at the same timeas it becomes stronger. The shock in the lower sur-face moves forward, weakens, and disappears. As theairfoil pitches down, the opposite begins to happen,with a small phase lag. By virtue of the usage of theimproved dissipation constructions (H-CUSP in thiscase) in Section , shocks can be resolved very crisply,with the additional advantage that the actual viscos-ity in the flow field is contaminated to a much lesserdegree. Finally, Figure 12 shows the coefficient of pres-sure along a coordinate line that circles the airfoil rightoutside of the boundary layer. One can clearly seethat these the results, which were obtained with theH-CUSP scheme, are by all means superior to the onesthat could be obtained with a scalar dissipation modelwhich spreads the shock over 4 or 5 cells instead of 1or 2.

Preliminary Aeroelastic Calculations for aCircular Cylinder

In this section we present preliminary calculationsof an aeroelastic model of a circular cylinder in a freestream with the motion induced by its own aerody-namic forces. The details of the model, which has verysimple governing equations of motion, are shown inFigure 1. This model has been previously tested witha Galerkin spectral formulation by Blackburn et al.17

As the computation proceeds, vortex shedding occursand the motion of the structure is dictated by the vari-ation in time of the force coefficients. The Reynoldsnumber of the flow is once more ReD = 500, the di-

mensions of the mesh are 384 × 96, and the time stepis taken such that it corresponds to 1/36 of the pe-riod dictated by a Strouhal number St = 0.2. Thecylinder is given an initial displacement and velocityin the y coordinate direction, and after 5 time steps, itis allowed to evolve due to its own self-induced forces.The cylinder mass ratio is taken to be µ = 5.0 for allcases, and the natural frequency of the springs is cho-sen to match the same Strouhal number St = 0.2. Thedamping coefficient of the springs is varied to comparewith experimental data. It is clear that if the motionof the cylinder did not considerably affect the forcecoefficients, the amplitude of the oscillations would in-crease without bound. In reality this is not so, and, asthe cylinder moves, the lift force is limited, preventingthe vertical displacement from becoming infinite. Nev-ertheless, comparisons between the force coefficientsobtained during free and forced vibrations can stillprovide very beneficial insights. First of all, in Fig-ure 13, we present the results of a typical calculation inwhich the reduced damping17 ( ζ

µ = 8π2St2mζ/ρ∞D)is 5.0 (the reduced damping is a parameter that coa-lesces the spring damping coefficient and the cylindermass ratio). This figure clearly resembles the evolutionof Cl and Cd for the stationary cylinder in the previ-ous section. The vertical motion is centered about thestarting position, whereas the horizontal motion hasan average positive displacement due to the existenceof a net drag. If the reduced damping is decreased to1.0, the amplitude of the vertical motion increases (asone would expect, since more of the energy transferredfrom the fluid into the structure is used in storing po-tential energy in the spring, and a smaller part of it isdissipated in the dampers). Figure 14 shows a phaseplot of the time accurate trajectory of the center of ourcylinder. It can be seen that the motion slowly devel-ops into a periodic orbit centered about the verticalorigin and displaced to the right by an amount pro-portional to the drag coefficient. Notice the fact thatthe periodic orbit is in the shape of a figure-eight, sincethe frequency of the horizontal forcing (drag) is twicethat of the vertical one (lift).

When one varies the reduced damping for the sameconditions, the maximum vertical displacement of thecylinder varies accordingly. Griffin24 has compiled aset of experimental data for this type of flows, forwidely varying Reynolds Numbers (300 < ReD < 106).Blackburn and Karniadakis17 have performed simi-lar computations, and all these results, together withthose obtained by the current method, are presentedin Figure 15. One can see that although the Reynoldsnumber of the data is, in general, not matched withthat of the computation, the trend is captured quitereasonably, and the maximum amplitude of the mo-tion can be predicted quite accurately in the mediumand low reduced damping range. Both numerical cal-culations are in fairly good agreement, which was to

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be expected since they are both two-dimensional mod-els, and have the same deficiencies for the predictionof physics which are clearly three-dimensional.

The flow over a cylinder undergoing forced oscil-lations has also been studied in comparison to theaeroelastic solutions. Similarly to other studies17, 26

the phenomenon of “lock in” is observed for a rangeof forcing frequencies ωf slightly above and below thenatural vortex shedding frequency of the stationarycylinder. Differences in the topology of the wake werealso observed, and will be the object of a further study.

Computer Requirements

Most of the calculations in this paper used a 384×96structured mesh, divided into a number of subdomainsthat ranged from 12 to 16. Each subdomain was as-signed to a different processor of an IBM SP2 Parallelcomputer, all of which collaborated to speed up thesolution of the problem via explicit message passingaccomplished with the MPI standard. In general, 40multigrid cycles per time step were used in order toget a well converged solution at the end of each timestep, and 36 steps per oscillation period were taken.With these parameters, a complete oscillation periodrequires about 14 minutes on 12 processors.

In contrast, when calculations were performed ona single node of an IBM SP1 System (equivalent toan RS6000/580) workstation, a complete cycle can becomputed in 100 minutes.

It must be noted that the parallelization of the codeis a key enabling technology required to obtain almostimmediate feedback on the behavior of the code. In thefuture, improvements to the parallel program will bemade, in order to be able to use larger computationalmeshes and produce solutions in a similar amount oftime.

The implicit time stepping method, together withthe parallel implementation of the algorithm makethis method an extremely efficient tool to performunsteady viscous calculations. In fact, without theseimprovements to the method, computation of fully re-solved three-dimensional unsteady flows would not bea possibility in the near future.

ConclusionsA fast, accurate solver for the unsteady Navier-

Stokes equations has been developed, and preliminaryresults have been presented. The computational ef-ficiency is two orders of magnitude higher than theequivalent explicit method for the same problem. Themethod can compute the wake and vortex sheddingflows behind bluff bodies with great accuracy, and canresolve the details of the unsteady flow field. Theimproved artificial dissipation constructions of12 areshown to yield more accurate shock resolution for tran-sonic viscous flows. Initial aeroelastic results suggestthat this solver can be coupled with a two-dimensional

aeroelastic model as in,3 in order to obtain flutterboundaries for airfoils representing an outboard sec-tion of a transonic swept wing. Furthermore, theparallel implementation of the algorithm scales ratherwell for a given mesh, and facilitates (due to reducedfeedback cycle times) the understanding of unsteadyviscous flows.

AcknowledgmentThis work has benefited from the generous support

of ARPA under Grant No. N00014-92-J-1796 andAFOSR under Grant No. AFOSR-91-0391. Compu-tational time on the NAS SP2 system was provided asa part of the IBM-CRA initiative.

References1Brenneis, A. and Eberle, A., “Application of an Implicit

Relaxation Method Solving the Euler Equations for Time-Accurate Unsteady Problems,” Journal of Fluids Engineering,Vol. 112, pp. 510-520, December 1990.

2Jameson, A., “Time Dependent Calculations Using Multi-grid, with Applications to Unsteady Flows Past Airfoils andWings,” AIAA Paper 91-1596, June 1991.

3Alonso, J. J. and Jameson, A., “Fully-Implicit Time-Marching Aeroelastic Solutions”, AIAA Paper 94-0056, AIAA32nd Aerospace Sciences Meeting, Reno, NV, January 1994.

4Melson, N. D., Sanetrik, M. D., and Atkins, H. L.,“Time-Accurate Navier-Stokes Calculations with Multigrid Acceler-ation”, Procs. of the Sixth Copper Mountain Conference onMultigrid Methods, Copper Mountain, Colorado, April, 1993.

5Arnone, A., Liou, M. S., and Povinelli, L. A., “MultigridTime-Accurate Integrations of Navier-Stokes Equations,” 24thFluid Dynamics Conference, July 1993, Orlando, Florida, AIAAPaper 93-3361.

6Arnone, A., “On the Use of Multigrid in Turbomachin-ery Calculations,” Proceedings of the International Workshopon Solution Techniques for Large-Scale CFD Problems, W.G.Habashi, Editor, Montreal, September, 1994.

7Tatsumi, S., Martinelli, L., and Jameson, A., “Design, Im-plementation, and Validation of Flux Limited Schemes for theSolution of the Compressible Navier-Stokes Equations”, AIAAPaper 94-0647, AIAA 32nd Aerospace Sciences Meeting, Reno,NV, January 1994.

8Martinelli, L., “Calculation of Viscous Flows with a Multi-grid Method,” Ph.D. Dissertation, Mechanical & Aerospace En-gineering Department, Princeton University, October, 1987.

9Martinelli, L. and Jameson, A., “Validation of a MultigridMethod for the Reynolds Averaged Equations,” AIAA Paper88-0414, AIAA 26th Aerospace Sciences Meeting, Reno, NV,January, 1988.

10Liu, F. and Jameson, A., “Multigrid Navier-Stokes Calcu-lations for Three-Dimensional Cascades,” AIAA Paper 92-0190,AIAA 30th Aerospace Sciences Meeting, Reno, NV, January,1992.

11Baldwin, B. S. and Lomax, H., “Thin Layer Approximationand Algebraic Model for Separated Turbulent Flows,” AIAAPaper 78-0257, 1978.

12Jameson, A., “Artificial Diffusion, Upwind Biasing, Lim-iters, and their Effect on Accuracy and Multigrid Convergencein Transonic and Hypersonic Flow,” presented at the AIAA 11thComputational Fluid Dynamics Conference, Orlando, July 1993.

13Jameson, A., “Analysis and design of numerical schemes forgas dynamics 2, artificial diffusion and discrete shock structure,”Int. J. of Comp. Fluid Dyn., To Appear.

14Bisplinghoff, R. L., Ashley, H. and Halfman, R. L., Aeroe-lasticity, Addison-Wesley, Reading, Massachusetts, 1955.

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15Dowell, E. H., Curtiss, H. C., Scanlan, R. H. and Sisto,F., A Modern Course in Aeroelasticity, Sijthoff and Noordhoff,Alphen ann den Rijn, The Netherlands, 1978.

16Schlichting, H., Boundary Layer Theory, 7th Edition,McGraw-Hill, 1979.

17Blackburn, H. M., and Karniadakis, G. E., “Two- andThree- Dimensional Vortex-Induced Vibration of a Circular”,ISOPE-93 Conference, Singapore, 1993.

18Belov, A., Martinelli, L., and Jameson, A., “A New ImplicitAlgorithm with Multigrid for Unsteady Incompressible FlowCalculations,” AIAA Paper 95-0049, AIAA 33rd Aerospace Sci-ences Meeting, Reno, NV, January, 1995.

19“MPI: A Message-Passing Interface Standard,” MessagePassing Interface Forum, March, 1994.

20Alonso, J. J., Mitty, T. J., Martinelli, L., and Jameson,A.,“A Two-Dimensional Multigrid-Driven Navier-Stokes Solverfor Multiprocessor Architectures,” Presented at the ParallelCFD ’94 Conference, Kyoto, Japan, May 1994.

21Williamson, C. H. K.,“Defining a Universal and ContiguousStrouhal-Reynolds Number Relationship for the Laminar VortexShedding of a Circular Cylinder,” Physics of Fluids, 31:2742–2744, October, 1988.

22Konig, M., Eisenlohr, H., and Eckelmann, H. “The FineStructure in the Strouhal-Reynolds Number Relationship ofthe Laminar Wake of a Circular Cylinder,” Physics of Fluids,2:1607–1614, September, 1990.

23Hammache, M. and Gharib, M., “An Experimental Studyof the Parallel and Oblique Vortex Shedding from CircularCylinders,” Fluid. Mech., 232:567–590, 1991.

24Griffin, O. M., “Vortex-Induced Vibrations of MarineStructures in Uniform and Sheared Currents,” NSF Workshopon Riser Dynamics, University of Michigan, 1992.

25Tatsumi, S, Martinelli, L., and Jameson, A.,“A NewHigh Resolution Scheme for Compressible Viscous Flows withShocks,” AIAA Paper 95-0466, AIAA 33rd Aerospace SciencesMeeting, Reno, NV, January, 1995.

26Mittal, S. and Tezduyar,T. E., “A Finite Element Study ofIncompressible Flows Past Oscillating Cylinders and Aerofoils,”International Journal for Numerical Methods in Fluids, vol. 15,1073-1118, 1992.

27Venkatakrishnan, V.,“Computation of Unsteady TransonicFlows over Moving Airfoils,” Ph.D. Dissertation, Dept. of Me-chanical and Aerospace Engineering, Princeton University, Oc-tober 1986.

K

K

K

K

Uinf

C

C

C

C

x x

yy

yy

xx

Fig. 1 Aeroelastic Model for Cylinder Self-Induced Vibrations.

1 2 3 4 5 6 7 81

2

3

4

5

6

7

8

Number of processors

Spe

edup

Parallel Speedup vs. Ideal for Full Multigrid Algorithm

−−− Ideal

−o− SP1 − PVM

−*− SP2 − PVMe

−+− SP2 − MPI

Fig. 2 Parallel Speedup for UFLO82NSP for Dif-ferent Message Passing Standards on Two DifferentPlatforms.

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MACH NUMBER

Fig. 3 NACA 64A010 airfoil at 2 angle of at-tack, Re∞ = 5000. Radial lines show interprocessorboundaries.

Fig. 4 Detail of the O-mesh for Unsteady Calcu-lations.

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Cl and Cd Time Histories

Non-Dimensional Time

Cl a

nd C

d

Fig. 5 Time History of the Coefficients of Lift andDrag.

Fig. 6 Entropy Contours in the Flow Field.

20 40 60 80 100 120 140 160

−1.5

−1

−0.5

0

0.5

1

1.5

Non−Dimensional Time

Cl a

nd C

d

Global View of the Effect of Grid Refinement in the Cl and Cd Time Histories

−o− 96x24 Mesh−*− 192x48 Mesh−+− 384x96 Mesh−x− 384x192 Mesh

Fig. 7 Time Histories of Cl and Cd for the GridRefinement Study.

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190 195 200 205 210 215 220 225 230

−1

−0.5

0

0.5

1

Non−Dimensional Time

Cl

Cl Time Histories for the Time Step Refinement Study

−o− 10 Time Steps per Period−*− 20 Time Steps per Period−+− 30 Time Steps per Period−x− 40 Time Steps per Period

Fig. 8 Time Histories of Cl for the Time Step Refinement Study.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Angle of Attack (degrees)

Coe

ffici

ent o

f Lift

Cl vs. Alpha Time History

o Experimental Data−−− Navier−Stokes Results−.− Euler Results

Fig. 9 Coefficient of Lift Time History.

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1

1

1

t = 233.7

t=236.8

t = 240.0

Fig. 10 Instantaneous Streamlines at Three Different Times in the Vortex Shedding Process.

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Fig. 11 Mach Number Contours. Pitching Airfoil Case. Re = 1.0 × 106, M∞ = 0.796, Kc = 0.202. Readfigures by lines.

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−0.2 0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

1.4

x/c

−C

p

Coefficient of Pressure on Upper and Lower Surfaces of the NACA 64A010 Airfoil

Fig. 12 Coefficient of Pressure on a Grid Line Just Outside of the Boundary Layer.

0 10 20 30 40 50 60 70 80 90−0.6

−0.4

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Non−Dimensional Structural Time

x an

d y

(hal

f dia

met

ers)

Time Histories of the Horizontal and Vertical Displacements of the Center of the Cylinder

−o− x−displacement

−*− y−displacement

Fig. 13 Time Evolution of the Coordinate Displacements for ζµ= 5.0.

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0 0.2 0.4 0.6 0.8 1 1.2−1.5

−1

−0.5

0

0.5

1

1.5

x displacement (half diameters)

y di

spla

cem

ent (

half

diam

eter

s)

Phase Plot for Aeroelastic Motion of a Cylinder

Fig. 14 Phase Plot for the Aeroelastic Motion of a Circular Cylinder.

2ρm/ D = 1

Various experiments

MARINE STRUCTURES

MARINE CABLES

0.1

0.2

0.4

0.6

1.0

2.0

4.0

6.0

10.0

0.005 0.01 0.02 0.05 0.1 0.2 0.5 1.0 2.0 5.0 10.0

Alonso et al., 1994

Blackburn et al., 1993

Reduced Damping

Cro

ss-F

low

Dis

plac

emen

t (in

cyl

inde

r di

amet

ers)

Fig. 15 Maximum Cross Flow Displacement (in Cylinder Diameters) for Several Experimental Results,Blackburn et al., and the present method for µ = 5.0.

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