Unsteady Flow Computation Using a Harmonic Balance Approach Implemented about the OVERFLOW 2 Flow Solver

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1 9th AIAA Computational Fluid Dynamics Conference // June 22-25, 2009, San Antonio, Texas Unsteady Flow Computation Using a Harmonic Balance Approach Implemented about the OVERFLOW 2 Flow Solver Jeffrey P Thomas, Chad H Custer, Earl H Dowell, and Kenneth C Hall Duke University, Durham, NC A novel approach for implementing a nonlinear unsteady frequency domain harmonic balance solution technique about existing implicit computational fluid dynamic flow solvers is presented This approach uses an explicit discretization of the harmonic balance source term, which enables the harmonic balance method to be applied to existing implicit flow solvers with minimal need for modification to the underlying implicit flow solver code The resulting harmonic balance solver can then be used for modeling nonlinear periodic unsteady flows The methodology is applied to the OVERFLOW 2 flow solver code, and results are presented for transonic viscous flow past an unsteady pitching airfoil section Unsteady aerodynamic and aeroelastic results for the F-6 fighter wing are also presented I Introduction In recent years, a novel nonlinear frequency domain harmonic balance HB technique for modeling periodic unsteady flows has been developed The methodology, sometimes also referred to as the nonlinear frequency domain NLFD approach, 2,3 enables the computational modeling of nonlinear periodic unsteady aerodynamics It has subsequently been used for modeling aeroelastic flutter-onset and limit cycle oscillation LCO response of airfoil and wing configurations 4 6 Here it is demonstrated that the harmonic balance approach can be implemented using existing computational fluid dynamic CFD solvers with minimal alteration of the underlying CFD solver computer code The HB technique then provides a method for the direct computation of a frequency domain solution for a periodic unsteady flow Previously, the harmonic balance method has been demonstrated for CFD codes that can model flows about geometric configurations such as airfoil sections and wings 4 6 However it is of great interest to use the harmonic balance method to model unsteady flows about more complex geometric configurations such as complete aircraft with external stores In order to model flows about more complex geometric configurations, there are two options The first option is to develop and then implement complex geometric modeling software into our existing harmonic balance CFD codes The second option is to apply the harmonic balance method to an existing time marching CFD code that has the capability of modeling complex geometric configurations The primary objective of this paper is to demonstrate that the harmonic balance approach can be implemented with minimal modification to existing CFD solvers that are capable of modeling flows about complex geometric configurations An example of such a code and the one considered here is OVER- FLOW 2 7,8 Based on previous experience in applying the harmonic balance method to CFD solvers, it has been found that this can be done with minimal revision of the original CFD code These CFD solvers are based on a modified form of the explicit Law-Wendroff algorithm 9 The new challenge is implementing the HB method about an implicit CFD solver such as OVERFLOW 2 It will be shown in this paper that an explicit Research Assistant Professor, Department of Mechanical Engineering and Materials Science, Senior Member AIAA Graduate Research Assistant, Department of Mechanical Engineering and Materials Science, Member AIAA William Holland Hall Professor, Department of Mechanical Engineering and Materials Science, and Dean Emeritus, School of Engineering, Honorary Fellow AIAA Julian Francis Abele Professor, Department of Mechanical Engineering and Materials Science, Associate Fellow AIAA Copyright c 2009 by Jeffrey P Thomas, Chad H Custer, Earl H Dowell, and Kenneth C Hall Published by the American Institute of Aeronautics and Astronautics, Inc with permission of 4

2 discretization treatment of the HB source term leads to unstable CFD schemes Far-field boundary conditions and finite computational domains can help to preserve numerical stability, and this has been found to be the case with explicit Law-Wendroff CFD algorithms However when implementing the HB method about implicit CFD methods for some simple test problems, numerical stability immediately became a problem It is noted that Woodgate and Badcock 3 have also recently implemented an implicit HB method They however use an implicit treatment of the HB source term which adds additional cost and complexity that the present approach avoids Fortunately, a simple modification has been identified that enables the HB source term to be treated explicitly in the overall discretization, thus enabling easy implementation of the HB method about existing implicit CFD solvers In the following, the theory of the HB method is presented, followed by a stability analysis for the HB method as applied to an implicit discretization of the scalar advection equation The numerical instability issue is demonstrated and an effective and efficient way to overcome this instability is presented The application of the HB method to existing CFD codes is discussed next, with a specific implementation to the OVER- FLOW 2 CFD solver Finally, results are presented showing a comparison between HB/OVERFLOW 2 and time-domain OVERFLOW 2 simulations for an unsteady transonic airfoil configuration in viscous flow as well as aerodynamic and aeroelastic results for the F-6 fighter aircraft wing It is noted that Custer et al 2 present aspects of using HB with the Chimera overset mesh capability of OVERFLOW 2 which allows complex geometries such as wings plus external stores to be considered II The Harmonic Balance Method The harmonic balance method development proceeds by first considering a given CFD method as representing a solver of a large system of ordinary differential equations, which may be expressed as R t = dq t dt N t = 0 where the subscript j represents a specific mesh point in the discrete computational spatial domain, and the subscript l represents a specific dependent variable at mesh point j As such, Q denotes the l th CFD solution independent variable at mesh point j, and N denotes l th component of the CFD solver spatial residual at mesh point j For the next step of the HB/CFD method development, the CFD solution Q and CFD residual N are assumed to be periodic in time with a fundamental frequency ω This enables the CFD solution and residual to be expanded in a Fourier series as Q t = ˆQ 0 + N t = ˆN 0 + ˆQcn cosωnt + ˆQ sinωnt sn n= ˆNcn cosωnt + ˆN sinωnt sn 3 n= The CFD solution and residual Fourier expansions are then truncated to a specified and sufficient number of harmonics N H Q t ˆQ N H 0 + ˆQcn cosωnt + ˆQ sinωnt sn 4 n= N t ˆN N H 0 + ˆNcn cosωnt + ˆN sinωnt sn 5 n= The next step of the HB/CFD method development consists of substituting Eqs 4 and 5 into Eq This allows Eq to also be considered as a truncated Fourier expansion ie, R t ˆR N H 0 + ˆRcn cosωnt + ˆR sinωnt sn = 0 6 n= 2 2 of 4

3 The Fourier transform of Eq 6 determines the Fourier harmonic component terms ˆR 0, ˆRcn and ˆR sn ie, ˆR 0 = ω 2π ˆR cn = ω π ˆR sn = ω π 2π/ω 0 2π/ω 0 2π/ω 0 R tdt= ˆN 0, R tcosωntdt= ωn ˆQ sn + ˆN cn, R tsinωntdt= ωn ˆQ cn + ˆN sn Since R t = 0, ˆR0, ˆRcn and ˆR sn n =, 2,,N H must also each be equal to zero This implies that ˆN 0 = 0, ωn ˆQ sn + ˆN cn = 0, 8 ωn ˆQ cn + ˆN sn = 0 for n =, 2,, N H Equation 8 constitutes a system of = 2N H + equations for the unknown Fourier harmonic terms ˆQ 0, ˆQ cn, and ˆQ sn n =, 2,, N H Equation 8 can also be written as ωa ˆQ + ˆN = ˆ0 9 where A is an matrix with row/column entries A i,j given by A i=n+,j=nh++n = n and A i=n+,j=nh++n = n for n =, 2,, N H, and all other elements of A being equal to zero ie, A = N H N H 0 0 and ˆQ 0 ˆN 0 7 ˆQ c ˆN c ˆQ = ˆQ cnh, and ˆN = ˆN cnh ˆQ c ˆN c ˆQ snh ˆN snh One could write a flow solver based on Eq 9 However depending on the degree of nonlinearity of the CFD solver, expressing the Fourier residual terms of ˆN as functions of the Fourier solution variables ˆQ could be very difficult if not impossible Hall et al realized that it is much easier to work in terms of time domain variables That is, Q t and N t stored at discrete and equally spaced subtime levels over the period T T = 2π/ω of oscillation Thus by setting Q = Q t 0 + t Q t t and N = N t 0 + t N t t 2 Q t 0 + T N t 0 + T 3 of 4

4 where t = 2π/ ω, one can then relate ˆQ to Q, and ˆN to N, via a constant Fourier transform matrix E ie, ˆQ = EQ and ˆN = E N 3 where E = 2 cos cos cos sin Equation 9 can then be rewritten as /2 cos 2π 0 2π 2π NH 2π sin 2π NH cos cos sin /2 cos 2π 0 2 2π 2 2π NH 2 2π 2 sin 2π NH 2 Subsequently, multiplying Eq 5 by the inverse of E results in or equivalently 2π 0 NT /2 cos 2π NT cos 2π NH NT sin 2π NT sin 2π NH NT 4 ωaeq + E N = ˆ0 5 ωe AEQ + E E N = ˆ0 6 ωdq + N = ˆ0 7 where D = E AE Equations 9 and 7 are equivalent, however Eq 7 is much easier to work with in terms of an existing CFD solver The N terms can be computed directly from the CFD solver of interest Equation 7 is a steady state equation, however one can pseudo time march Eq 7 to solve for the solution Q ie, δq + ωdq + N = ˆ0 8 δτ For example, if one were using an explicit time marching technique for their nominal CFD method, one could iterate the following equation to obtain the HB solution Q n+ = Q n t ωdq n + N n 9 However it will be shown in the next section that a careful treatment of the HB source term ωdq is necessary to insure numerical stability III Dispersion Analysis To demonstrate the numerical instability issue when pseudo time marching Eq 7, a dispersion analysis is performed for an implicit discretization of Eq 8 with two different discretization strategies for the HB source term ωdq For the stability analysis, the HB method is applied to the scalar advection equation The HB system of equations for Eq 20 when considering one harmonic is where τ is the pseudo time marching variable and u x, τ Q = u 2 x, τ u 3 x, τ u t + a u = 0 20 x Q τ + a[i] Q + ω[d]q = 0 2 x = ut 0 + tx, τ ut tx, τ ut 0 + Tx, τ 22 4 of 4

5 where T = 2π/ω and t = 2π/3ω For one harmonic, 3 [D] = For the dispersion analysis, consider a wave of the form [ Qτ, x = ˆQexp j φτ τ θx ] x 24 The amplification factor g is defined as g = Qτ + τ, x Qτ +, x = expjφ 25 The exact dispersion relations for Eq 2 are φ = νθ, φ 2 = νθ + ω τ, and φ 3 = νθ ω τ 26 The magnitude of the amplification factor for each of these roots is equal to one g = e jφ = e jνθ = 27 g 2 = e jφ 2 e = jνθ+ω τ = 28 g 3 = e jφ3 = e jνθ ω τ = 29 This means a sine wave will neither grow nor decay in pseudo time for the exact HB system of equations The first discretization considered for Eq 2 is the implicit Euler technique with an implicit treatment of the HB source term Q n+ i Q n i + a[i] Qn+ i+ Qn+ i τ 2 x For this discretization, the magnitude of the dispersion analysis amplification factors are: g = ν 2 sin θ 2 + g 2 = ν sin θ + ω τ 2 + g 3 = ν sin θ ω τ ω[d]q n+ i = 0 30 As such, this scheme is unconditionally stable However the drawback is that implicit treatment of the HB source term will require extensive modification of an existing CFD solver One will need to construct, compute, and invert a large and sparse Jacobian matrix Explicit treatment of the HB source term is far simpler However, explicit treatment of the HB source term, ie, Q n+ i Q n i τ 2 x results in the following amplification factor magnitudes: + a[i] Qn+ i+ Qn+ i 3 + ω[d]q n i = 0 32 g = ν 2 sin θ of 4

6 g 2 = g 3 = + ω 2 τ 2 ν 2 sin θ 2 + This discretization is only stable if ν 2 sin 2 θ ω 2 τ 2 As such very short and/or long wavelengths will be unstable In fact, it turns out that techniques such as the first-order upwind method and the Lax-Wendroff method are also unstable with an explicit treatment of the HB source term However a simple stabilization technique exists which enables the HB source term to be treated explicitly while providing numerical stability and this is discussed next IV Stabilization Technique As demonstrated in the previous section, explicit treatment of the HB source term leads to an unstable discretization However numerical stability can be recovered as follows Starting from Eq 8, first make a variable transformation of the form Q = e ωτd V 34 where e ωτd is the matrix exponential In this case, Eq 8 becomes or e ωτdδv δτ Then discretize the time derivative term and revert back to Q variables via Equation 36 becomes or ωde ωτd V + N + ωde ωτd V = ˆ0 V n+ The next step is to approximate e ω τd as δv δτ V n τ + e ωτd N = ˆ e ωτd N n = ˆ0 36 V = e ωτd Q 37 τ eωτd e ω τd Q n+ Q n + e ωτd N n = ˆ0 e ω τd Q n+ Q n + N n τ = ˆ0 38 e ω τd I + ω τd 39 Equation 38 then becomes [ ] I + ω τd Q n+ Q n + N n = ˆ0 40 τ Then subtract and add ωdq n to the left hand side of Eq 40 ie, or τ Equation 40 then becomes [ ] I + ω τd Q n+ Q n ωdq n + N n + ωdq n = ˆ0 τ [ ] I + ω τd Q n+ Q n + N n + ωdq n = ˆ0 Q n+ Q n τ + I + ω τd N n + ωdq n = ˆ0 4 6 of 4

7 Note, this equation looks like the original discretization with the exception of the matrix I + ω τd multiplying the HB residual N n + ωdq n At steady state, Eq, 7 N + ωdq = ˆ0 is still being solved The cost of computing I + ω τd is negligible since it only requires the inversion of a small matrix Next, we consider the proposed stabilization technique Q n+ i Q n i + [I] + ω τ[d] τ a[i] Qn+ i+ Qn+ i + ω[d]q n i 2 x The magnitude of the dispersion analysis amplification factors in this case are: g = ν 2 sin θ 2 + g 2 = ν sin θ + ω τ 2 + g 3 = ν sin θ ω τ 2 + = 0 42 As can be seen, this scheme utilizing the HB source term stabilization technique is unconditionally stable In fact, the amplification factors are identical to the implicit Euler discretization technique with the implicit treatment of the HB source term It can also be demonstrated that the HB source term stabilization matrix I + ω τd can be used stabilize explicit schemes such as the first-order upwind method and the Lax- Wendroff method when these methods are applied to the HB system of equations Eq 2 V Implementing the HB Method About an Existing CFD Solver Implementing the HB method Eq 7 within an existing flow solver is straightforward to do All that is required is a redimensioning of the primary arrays by a factor of and the implementation of a do-loop about the residual computation section of code If the nominal flow solver can be expressed in pseudo code as DO CALL RESIDUALQ:,X:,XD:,DQ: Q: = Q: + DQ: CALL CHECK_CONVERGENCEDQ:,L2_RESIDUAL IF L2_RESIDUAL < CONVERGENCE_TOLERANCE EXIT ENDDO then the HB/CFD variant of that same solver can be written as DO DO N =,2*N_HARMONICS+ CALL RESIDUALQHBN,:,XHBN,:,DQHBN,: ENDDO CALL SOURCE_HBQHB:,:,SOURCE_TERM_HB:,: QHB:,: = QHB:,: + DQHB:,: + SOURCE_TERM_HB:,: CALL CHECK_CONVERGENCE_HBDQHB:,:,L2_RESIDUAL IF L2_RESIDUAL < CONVERGENCE_TOLERANCE EXIT ENDDO where QHB:,: and DQHB:,: are harmonic balance solver arrays, which are similar to the arrays Q: and DQ: that store the CFD solution and CFD solution residual in the nominal code The arrays QHB:,: and DQHB:,: however have an extra dimension of in order to store each HB solution and solution residual at the subtime levels over a period of oscillation The array SOURCE TERM HB:,: represents the HB source term with the stabilization matrix ie, I + ω τd ωdq 43 7 of 4

8 Note, two other modifications have to be made to the nominal CFD solver code in order to implement the HB method One is to add the necessary code to treat the effect of unsteady CFD mesh motion, and the other is the additional code to model the unsteady solid wall boundary conditions For the mesh motion effect, one simply modifies the flux computation routines For instance, if F represents the l th element of the x component flux at mesh point j, F would be modified to be F t n = F t n ẋt n Q t n 44 in the HB solver Let X j be a vector whose elements are the x component of mesh point j at the subtime levels over a period of oscillation x j t 0 + t x j t t X j = 45 x j t 0 + T If we let Ẋ j be a vector whose elements are the time derivative of the x component of mesh point j at the subtime levels over a period of oscillation, ie ẋt n, Ẋ j = ẋ j t 0 + t ẋ j t t, 46 ẋ j t 0 + T it can be shown that Ẋ j is related to X j via the following equation Ẋ j = ωdx j 47 The vector Ẋ j can be determined before the HB solution process begins The y and z flux components are modified in a similar manner ie G t n = G t n ẏt n Q t n 48 H t n = H t n żt n Q t n 49 Next, in order to represent viscous solid wall boundary conditions with the HB method, all that is necessary is to modify the subroutine that treats the solid wall boundary conditions such that the velocity components at the wall are set to u j t n = ẋt n v j t n = ẏt n w j t n = żt n 50 An attractive feature of OVERFLOW 2 is that the mesh motion flux and moving wall boundary condition terms are all calculated and stored in arrays for time-domain simulations It has been found that one can simply use the same arrays for the HB mesh motion flux and moving wall boundary conditions terms As a result, none of the flux or boundary condition routines in OVERFLOW 2 needs to be modified VI NACA 002 Airfoil Section Computations In this section, results are presented for viscous transonic unsteady flow about a NACA 002 airfoil section using the HB method implemented in the OVERFLOW 2 CFD code These results are then compared to the results from the OVERFLOW 2 Newton time accurate time-stepping solution method for the same configuration Figure a shows the computational mesh used for the NACA 002 airfoil section A c-mesh 8 of 4

9 a Computational Grid b Steady Flow Mach Contours Figure NACA 002 Airfoil Computational Mesh and Steady Flow Mach Number Contours topology is used with 40 mesh points circumferentially about and 75 mesh points radially to the airfoil An unsteady oscillating amplitude α of one degree in pitch about the airfoil quarter chord is considered for an oscillation reduced frequency ω = ωc/u of 05 where c is the airfoil chord and U is the freestream velocity The freestream Mach number is 08, the Reynolds number is three million, and the mean angleof-attack is zero degrees Figure b shows computed steady flow Mach contours A strong shock wave near mid-chord is evident The computational method is the HLLC 3 scheme together with the Spalart-Almaras turbulence model 4 Next, Fig 2 shows computed the real and imaginary parts of the first harmonic unsteady surface pressure distributions when using different numbers of harmonics in the harmonic balance method As can be seen, well converged solutions are obtained when using as few as two harmonics Figure 3 shows a comparison OVERFLOW 2 time-domain and HB/OVERFLOW 2 solutions for the real and imaginary parts of the unsteady first harmonic airfoil surface pressure distributions Newton timestepping is used with five Newton subiterations and 80 steps per cycle for the OVERFLOW 2 time-domain solution, and three harmonics are used for the HB/OVERFLOW 2 solution As can be seen, the OVER- FLOW 2 time domain and HB/OVERFLOW 2 results are nearly identical Finally, Fig 4 shows the real and imaginary parts of the unsteady first harmonic airfoil lift coefficient as a function of total iteration count for both the OVERFLOW 2 time domain and HB/OVERFLOW 2 solutions As can be seen, the convergence trends are very similar for both the time-domain and HB methods It should be noted that for the time-domain method, one must specify the number of Newton subiterations, steps per period, and total number of periods The time-domain solution also has to be Fourier transformed to determine the frequency domain content An attractive feature of the HB method is that it is not necessary to consider Newton subiterations, steps per period, and the number of periods needed to converge a time marching transient solution With the HB method, one solves directly for the frequency domain solution Because of these differences in the time marching and HB methods, it is difficult to say whether one approach is faster than the other for computing the unsteady aerodynamic solution for prescribed airfoil motion A time-domain solution may have to be run for several cycles of airfoil motion in order to achieve sufficient frequency domain harmonic convergence, and many Newton subiterations and/or steps per period may be necessary for sufficient time accuracy Even so, the great advantage of the HB method is when the CFD code is combined with a structural model for aeroelastic computations where the structural motion is unknown a priori Because of the low levels of damping typical in the critical aeroelastic modes, the computer run times for a time marching method 9 of 4

10 Real Part Unsteady Pressure, Rep /q Harmonic 2 Harmonics 3 Harmonics 4 Harmonics Airfoil Surface Location, x/c Imaginary Part Unsteady Pressure, Imp /q Harmonic 2 Harmonics 3 Harmonics 4 Harmonics Airfoil Surface Location, x/c a Real Part b Imaginary Part Figure 2 A Comparison of the Number of Harmonics Used in Harmonic Balance Method for the NACA 002 Airfoil Unsteady Surface Pressure Distributions may be much greater than for a simulation of a prescribed structural motion By contrast the computation time for the HB method is little changed VII F-6 Fighter Wing Unsteady Aerodynamic Modal Force Computations Time-domain and HB computational comparisons have also been obtained for the F-6 fighter wing In this example, M = 05, α 0 = 5 degree, and an altitude of h = 2000 is considered For the unsteady wing motion, the first anti-symmetric bending mode for an F-6 fighter weapons and stores structural configuration referred to as configuration number five See Thomas et al 5 is considered The unsteady frequency is eight Hertz, and the modal coordinate amplitude is ξ 2 = 00 inch Figure 5 shows a comparison of the computed modal force as a function of the total number of solver iterations for both OVERFLOW 2 time-domain and HB calculations The computational method is OVERFLOW 2 s central differencing option together with the Spalart-Almaras turbulence mode One harmonic is used for the HB/OVERFLOW 2 method, and as with the airfoil case in the previous section, five Newton subiterations and 80 steps per cycle are used for the OVERFLOW 2 time-domain solution As can be see, the convergence rates are similar to those for the case of the transonic airfoil configuration VIII F-6 Fighter Wing Unsteady Aeroelastic Computations OVERFLOW 2 time-domain and HB computations for prescribed motion appear to have about the same computational cost We believe however that the HB method has great potential for reducing the computational cost of aeroelastic computations To illustrate, we consider the flutter onset altitude versus Mach number trend for an F-6 fighter weapons and stores structural configuration known as configuration number one See Thomas et al 5 The solid line in Fig 6 shows the computed flutter onset altitude versus Mach number boundary based using the flutter-onset/lco solution technique of Thomas et al 5,6,6 that is based on HB calculations using HB/OVERFLOW 2 and which allows for the solution of the precise matched point flutter onset altitude for a given Mach number By contrast, in order to determine the flutter-onset boundary using the time-domain approach, one must use a divide and conquer strategy For instance, the inset figures in Fig 6 show time-domain aeroelastic computations for dynamic pressures corresponding to altitudes of 0, -5000, and feet at a Mach number of M = 095 As can be seen, the time domain 0 of 4

11 Real Part Unsteady Pressure, Rep /q Time Domain Harmonic Balance Airfoil Surface Location, x/c Imaginary Part Unsteady Pressure, Imp /q Time Domain Harmonic Balance Airfoil Surface Location, x/c a Real Part b Imaginary Part Figure 3 A Comparison of OVERFLOW 2 Time-Domain and Harmonic Balance NACA 002 Airfoil Surface Pressure Distributions solutions have to be run for several cycles of aeroelastic response in order to determine aeroelastic stability or divergence for a given altitude and Mach number combination This is due to low aeroelastic damping which is typical of many aeroelastic problems Each point on the flutter-onset boundary determined by using flutter-onset/lco solution technique of Thomas et al in conjunction with HB/OVERFLOW 2 was computed in a day or two, whereas the OVERFLOW 2 time-domain aeroelastic solutions each took on the order of a week for the time intervals shown in the inset figures of the Fig 6 IX Conclusions Using a newly created stabilization procedure, the harmonic balance solution method has been implemented in the CFD flow solver OVERFLOW 2 with only a modest modification to the underlying OVER- FLOW 2 code The harmonic balance method can also be applied to other well known implicit CFD solvers using the technique presented in this paper The harmonic balance method provides for the direct computation of a frequency domain solution for a periodic unsteady flow, and such solutions are particularly valuable for aeroelastic calculations Although not necessarily faster than time-domain solutions for unsteady flows due to prescribed airfoil or wing motions where the transient oscillations are relatively short, the HB method is expected to offer significant computational cost savings for aeroelastic problems where low levels of damping exist in the critical aeroelastic modes and thus the transient oscillations are much longer References Hall, K C, Thomas, J P, and Clark, W S, Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique, AIAA Journal, Vol 40, No 5, 2002, pp McMullen, M S and Jameson, A, The Computational Efficiency of Nonlinear Frequency Domain Methods, Journal of Computational Physics, Vol 22, No 2, March 2006, pp Cagnone, J-S and Nadarajah, S K, Implicit Nonlinear Frequency-Domain Spectral-Difference Scheme for Periodic Euler Flow, AIAA JOURNAL, Vol 47, No 2, FEB 2009, pp Thomas, J P, Dowell, E H, and Hall, K C, Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter and Limit Cycle Oscillations, AIAA Journal, Vol 40, No 4, April 2002, pp Thomas, J P, Dowell, E H, and Hall, K C, Modeling Viscous Transonic Limit Cycle Oscillation Behavior Using a Harmonic Balance Approach, Journal of Aircraft, Vol 4, No 6, November-December 2004, pp Thomas, J P, Dowell, E H, Hall, K C, and Jr, C M D, Further Invesitgation of Modeling Limit Cycle Oscillation Behavior of the F-6 Fighter Using a Harmonic Balance Approach, AIAA Paper , Presented at the 46th of 4

12 Unsteady Lift Coefficient, c l /α Real Part of Unsteady Lift, Rec l /α Time Domain Harmonic Balance Imaginary Part of Unsteady Lift, Imc l /α Total Number of Iterations, N Figure 4 OVERFLOW 2 Time Domain and HB/OVERFLOW 2 Unsteady Lift Convergence Histories AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials SDM Conference, Austin, TX 7 Nichols, R H, Tramel, R W, and Buning, P G, Solver and Turbulence Model Upgrades to OVERFLOW 2 for Unsteady and High-Speed Applications, AIAA Paper , 25th Applied Aerodynamics Conference, San Francisco, CA 8 Nichols, R H and Heikkinen, B D, Validation of Implicit Algorithms for Unsteady Flows Including Moving and Deforming Grids, Journal of Aircraft, Vol 43, No 5, September-October 2006, pp Ni, R-H, A Multiple Grid Scheme for Solving the Euler Equations, AIAA Journal, Vol 20, No, November 982, pp Davis, R L, Ni, R H, and Bowley, W W, Prediction of Compressible, Laminar Viscous Flows Using a Time-Marching Control Volume and Multiple-Grid Technique, AIAA Journal, Vol 22, No, November 984, pp Saxor, A P, A Numerical Analysis of 3-D Inviscid Stator/Rotor Interactions Using Non-Reflecting Boundary Conditions, Tech Rep 209, MIT, March 992, Gas Turbine Laboratory Report Custer, C H, Thomas, J P, Dowell, E H, and Hall, K C, A Nonlinear Harmonic Balance Method for the CFD Code OVERFLOW 2, Interational Forum on Aeroelasticity and Structural Dynamics IFASD Paper , Seattle, WA 3 Toro, E F, Spruce, M, and Speares, W, Restoration of the Contact Surface in the HLL Riemann Solver, Shock Waves, Vol 4, 994, pp Spalart, P R and Allmaras, S R, A One Equation Turbulence Model for Aerodynamic Flows, AIAA Paper Thomas, J P, Dowell, E H, Hall, K C, and Denegri, C M, Virtual Aeroelastic Flight Testing for the F-6 Fighter with Stores, AIAA Paper , US Air Force T&E Days, Destin, FL 6 Thomas, J P, Dowell, E H, and Hall, K C, Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter and Limit Cycle Oscillations, AIAA Journal, Vol 40, No 4, April 2002, pp of 4

13 5 ξ 2 *00 4 Real Part of Unsteady Modal Force Unsteady Modal Force, Q 2 /ρ a Time Domain Harmonic Balance Imaginary Part of Unsteady Modal Force Total Number of Iterations, N Figure 5 OVERFLOW 2 Time Domain and HB/OVERFLOW 2 Unsteady Modal Force Convergence Histories for the F-6 Fighter Wing 3 of 4

14 Figure 6 OVERFLOW2 F-6 Fighter Wing Aeroelastic Flutter Onset Altitude Versus Mach Number 4 of 4