A NONLINEAR HARMONIC BALANCE METHOD FOR THE CFD CODE OVERFLOW 2

IFASD-2009-050 A NONLINEAR HARMONIC BALANCE METHOD FOR THE CFD CODE OVERFLOW 2 Chad H. Custer 1, Jeffrey P. Thomas 2, Earl H. Dowell 3, and Kenneth C. Hall 4 Duke University Mechanical Engineering & Materials Science 144 Hudson Hall, Box 90300 Durham, NC 27708 1 chad.custer@duke.edu, 2 jthomas@duke.edu, 3 dowell@mail.ee.duke.edu, 4 kenneth.c.hall@duke.edu Keywords. CFD, OVERFLOW, Harmonic Balance Solution, Overset Grid. Abstract. A National Aeronautics and Space Administration computational fluid dynamics code, OVERFLOW 2, was modified to utilize a harmonic balance solution method. This modification allows for the direct calculation of the nonlinear frequency-domain solution of a periodic unsteady flow while avoiding the time consuming calculation of long physical transients that arise in aeroelastic applications. With the usual implementation of this harmonic balance method, converting an implicit flow solver from a time marching solution method to a harmonic balance solution method results in an unstable numerical scheme. However, a relatively simple and computationally inexpensive stabilization technique has been developed and is utilized in this paper. With this stabilization technique, it is possible to convert an existing implicit time-domain solver to a nonlinear frequencydomain method with minimal modifications to the existing code. This new frequencydomain version of OVERFLOW 2 utilizes the many features of the original code, such as various discretization methods and several turbulence models. The use of Chimera overset grids in OVERFLOW 2 requires care when implemented in the frequency-domain. This paper presents a harmonic balance version of OVERFLOW 2 capable of solving on overset grids for sufficiently small unsteady amplitudes. 1 INTRODUCTION Flutter and limit cycle oscillation (LCO) calculations for complex geometries of flexible, deforming structures in an aerodynamic flow are computationally expensive due to the many grid points needed to model the complex flow accurately. Flutter is the dynamic instability of the aeroelastic (fluid-structure) system and LCO is the nonlinear oscillation that may follow. Near flutter the aeroelastic damping is small and thus the physical transients can be very long. Using traditional time-domain computational fluid dynamics (CFD) methods, such transients must be modeled time-accurately, while only the periodically converged solution is typically of interest. Nonlinear frequency-domain methods offer the benefit of calculating the periodic response directly, thus avoiding the need to model long physical transients. This leads to a dramatic reduction in computational cost for lightly damped and physically unstable systems. The harmonic balance (HB) nonlinear frequency-domain solution method is highly efficient with the computational time being at least an order of magnitude faster than the time-marching solution for aeroelastic analyses [1]. Also, the HB method is capable of modeling nonlinearities [2]. These properties make the method useful for a wide variety 1

of applications such as the modeling of limit cycle oscillations in nonlinear aeroelastic systems [3, 4]. Many CFD codes exist with the majority of these codes modeling the flow field by timemarching a spatially and temporally discretized version of the conservation equations. This technique is highly versatile since any body motion can be considered. However, when studying aeroelasticity one is often primarily interested in the response of the flow field to harmonic motion of the body. While it is possible to write a nonlinear frequency-domain solver de novo, it is possible to convert a time-domain flow solver to a frequency-domain method with a modest amount of CFD code modification. Basing the HB solver on a time-domain code significantly reduces code development time. By using the time-domain flow solver to drive the frequency-domain code, we are able to utilize current state-of-the-art methods already implemented within a given time-domain solver. Specifically, we are interested in using a variety of discretization techniques and turbulence models as well as the Chimera overset grid method. 2 HARMONIC BALANCE METHOD The harmonic balance method developed by Hall et al. [5] takes advantage of the temporally periodic nature of the flow field by assuming the conservation variables can each be accurately represented by a Fourier series in time. It is then possible to recast the governing equations and solve for the Fourier coefficients of the conservation variables. One of the major features of the HB method is that the pseudo-time marching scheme used to solve for the Fourier coefficients of the conservation variables takes the same form as the scheme by which the original governing equations are solved for the steady state case. The only addition to the update step required by the HB method is the inclusion of an additional term known as the source term. The fact that the equations to be solved for the HB method mimic the original governing equations allows one to utilize computational methods already well developed in the literature. The set of partial differential equations that describe fluid flow can be expressed in the form given by Eq. 1. Here the first term clearly represents the time derivative. The second term, N(Q(x, t)), is the spatial operator representing the derivative operations in space. Q(x, t) t N(Q(x, t)) = 0 (1) Note that it is useful to begin the formulation for the harmonic balance method with this general expression for Navier-Stokes equations since many conservation laws can be expressed in this form. Consider Eq. 1 discretized in space but continuous in time. In Eq. 2, and in those to follow, l represents the index of the conservation variable and j is the nodal index. Q l,j (t) t N l,j (t) = 0 (2) The first step in deriving the HB method is to utilize the fact that the body is undergoing harmonic motion with fundamental frequency ω and to expand Q l,j (t) and N l,j (t) in 2

Fourier series as Q l,j (t) = Q0 l,j + N l,j (t) = N0 l,j + N H n=1 N H n=1 { QC n l,j cos(ωnt) + QS n l,j sin(ωnt) } (3) { NC n l,j cos(ωnt) + NS n l,j sin(ωnt) }. (4) Note that the number of harmonics retained in the Fourier series expansions (N H ) must be sufficient such that Q l,j (t) and N l,j (t) are represented accurately. The expansions of Q l,j (t) and N l,j (t) are then substituted into Eq. 2, resulting in a linear system where the unknowns are the Fourier coefficients of Q and N. It is then possible to relate the Fourier coefficients of Q and N to the conservation variables sampled over one period in time using the discrete Fourier transform. The result is a system of equations for each conservation variable at each node as given by ω[d]q l,j N l,j = 0 (5) where Q l,j contains sampled values of the l th conservation variable at the j th node. Q l,j (t 0 + t) N l,j (t 0 + t) Q l,j (t 0 + 2 t) N l,j (t 0 + 2 t) Q l,j =. Q l,j (t 0 + T ), N l,j =. N l,j (t 0 + T ) (6) T = 2π ω, t = 2π N T ω The airfoil locations at each of these sub-time levels is shown for a simple pitching airfoil case in Fig. 1. α max Angle of Attack, α 0 -α max 0 T/7 2T/7 3T/7 4T/7 5T/7 6T/7 T Time (a) Angle of attack for each sub-time level. (b) Airfoil location for each sub-time level. Figure 1: Sub-time level grid locations for a pitching airfoil. Equation 5 represents the governing equations recast in the frequency-domain. There are several important features of this equation. First, note that the first term in the above 3

equation is a pseudo-spectral (frequency-domain) representation of the time derivative term. This term is referred to as the source term of the equation since it contains no time derivatives per se. The second important feature of this equation is that the spatial operator is unchanged from the time-domain formulation, it is simply evaluated at multiple sub-time levels. This is critical to the goal of implementing the HB method about an existing time-domain solver. Finally, note that the vector equation for each conservation variable at each point is of size N T where N T = 2N H + 1. In order to mimic the form of the time-domain equations as closely as possible, a pseudotime (non-physical time) term is added to Eq. 5. The result is an equation that is identical in form to Eq. 2 with one extra term, i.e. the source term. Q l,j (t) τ N l,j (t) + ω[d]q l,j (t) = 0 (7) Since the frequency-domain representation of the governing equations is given by Eq. 5, the pseudo-time term can be used to iterate rapidly to convergence. Equation 7 can be discretized in pseudo-time using the same methods by which the original time-domain equations are discretized in physical time. Also, recall that the spatial operator is unchanged from the time-domain solver. This results in a finite-difference formulation of the HB method that is formally unchanged from that of the original time-domain CFD code except for the addition of the source term. When considering the finite-difference representation of Eq. 7, the source term can be included on the n pseudo-time level or the n + 1 pseudo-time level. Each of these options have serious limitations. The original time-domain linear system of equations is of size N T D N T D where N T D = N nodes N vars. If included on the n + 1 pseudo-time level, a large linear system of size N T D N T N T D N T (8) would need to be created. Not only would this increase the computational cost associated with solving the linear system, but also it would greatly deviate from the structure of the time-domain solver. This method has recently been used with success [6], however it is unsuitable for the goals of this research, i.e. to construct a harmonic balance solver for a time-domain CFD code with minimal modifications. Alternatively, if the source term is included with the n pseudo-time level, the form of the linear system is unchanged from the time-domain system. Instead of constructing a linear system that is scaled by a factor of N T, each of the sub-time levels can be solved independently. However, if implemented as described above, the scheme is numerically unstable when used in conjunction with implicit methods. Fortunately a relatively simple and computationally inexpensive stabilization technique has been developed, which allows for the source term to be evaluated at the n pseudotime level. Thomas, et al. [7] describe the stabilization method used in the present work in considerable depth, and the reader is referred this paper for this key element of the stabilized harmonic balance method. 4

2.1 Implementation of the Harmonic Balance Method Using the present method, few changes are needed to convert OVERFLOW 2 from its time-domain solution method to a frequency-domain harmonic balance method. Most importantly, no changes are required for the handling of the temporal and spatial derivative operators. The conversion of a time-domain solver to the frequency-domain harmonic balance solution method requires modifications in three main areas of the code: initialization, iteration, and post-processing. These changes are outlined below and illustrated in Fig. 2, where the white boxes represent steps that are unchanged from the time-domain code. Shaded elements represent steps that require modification. Initialize Variables Initialization Initialize HB Variables Deform Body Grids Enter Pseudo-time Loop Iteration Enter Sub-time Loop Calculate ΔQ Exit Sub-time Loop Stabilize ΔQ Update Q Exit Pseudo-time Loop Post-processing Post-processing Figure 2: Harmonic balance method flow chart. The first set of modifications is in the initialization phase. Recall that with the harmonic balance method the solution is sampled over one period of motion, meaning that the number of sample points scales with the number of harmonics retained. This requires increased memory allocation to allow for the storage of these sub-time level solutions. Also in this phase, the computational grid is modified to correspond to the shape of the body at each of the given sub-time levels. The iteration section of the code is where the most significant changes are made. The original temporal and spatial operators are used to iterate each of the sub-time level solutions, which means that an additional sub-time loop is added. However, as noted previously, the spatial and temporal finite-difference operators are unchanged. After an update is calculated for each of the sub-time levels, the update is stabilized and the solution is advanced. Modifications are also necessary to properly account for grid motion. Since the solution being calculated is unsteady, the time metrics must be modified to 5

account for the unsteady motion. Finally, in the post-processing phase, routines are necessary to output useful information such as the Fourier coefficients of surface pressure. 3 CHIMERA OVERSET GRIDS When modeling a complex, three-dimensional body and the surrounding fluid domain, it is often difficult to create a well behaved single-block grid. The Chimera overset grid technique was developed to alleviate some of these challenges [8,9]. The general approach of the method is to create a patchwork of grids that collectively describe the surface of the body and the fluid domain. The key concept of the Chimera overset grid technique is that individual grids are created for each component of the body and the fluid domain. Since the component grids are created without regard for the other portions of the body, two steps are required after the individual grids are produced. These steps are illustrated for a two grid system where an o-grid describes the surface and near-body of a NACA 0012 airfoil and a Cartesian grid describes the far-field as shown in Fig. 3. First, it is possible that nodes describing the fluid domain of one component grid may lie within another portion of the body as is the case with the Cartesian grid points lying within the airfoil in Fig. 3(a). These points, such as the point labeled A, are blanked and a collection of these blanked points make up a hole in the grid. The result, as shown in Fig. 3(b), is a set of overlapping grids describing the body surface and fluid domain. Second, since the domain is no longer a single-block grid with clear computational directions, a method for inter-grid communication must be established. Specifically, hole boundaries and grid boundaries on the interior of the domain must be treated specially. The structured grid finite differencing stencil cannot be applied across grid boundaries. Instead, the solution at these fringe points is interpolated from donor points within the neighboring grid. For example, in Fig. 3(b) the solution on the outer most points of the airfoil grid is interpolated from the Cartesian grid. One of these airfoil fringe points is labeled B. Likewise, the Cartesian grid points nearest to the airfoil hole (such as the point labeled C ) receive data from the airfoil grid. Since each component grid or region is created independently, the gridding process is much simpler than if one were to create a single-block grid describing the entire domain. In fact, for most geometries of interest it would not be possible to create a single structured grid. Also, since component grids are created individually, it is relatively easy to modify the geometry of interest. For example, adding an under-wing store to a fighter aircraft would not require the regeneration of the grids for the fighter body or wing. Instead, a grid is created for the store and added to the set of grids describing the wing and body. There are two options for creating the domain connectivity database for use with OVER- FLOW 2. The Pegasus 5 package [10] is NASA developed software that is run as a pre-processing step where the domain connectivity does not change throughout the CFD calculation. This option is appropriate for cases without grid motion. The second option is to use the domain connectivity function (DCF) within OVERFLOW 2 [11]. The DCF performs hole cutting and determines domain connectivity at each physical time step of 6

(a) Near-body and far-fields grid prior to hole cutting. (b) Fringe points for a NACA 0012 two-grid system. Figure 3: Set of Chimera overset grids describing the NACA 0012 airfoil. the CFD calculation. DCF is the appropriate choice when performing calculations in the time-domain on moving grids. 3.1 Time Domain Aerodynamics of the AIM-120 Missile The flutter and LCO response of a fighter aircraft is highly dependent on the store configuration. The underwing and wingtip positioning of fuel tanks, bombs and missiles dramatically influences the aeroelastic response of the fighter. One configuration of particular interest consists of the AIM-120 missile, shown in Fig. 4, mounted on the wingtips of the F-16. Although the AIM-120 is small in size relative to the F-16, modeling the missile requires a large number of grid points due to the many fins. The grid system describing the isolated AIM-120 used for inviscid calculations consists of 19 near-body grids and one far-field grid. In all, the set of grids total three million points. The desire to calculate the aerodynamics of the AIM-120 highlights two important points. First, the fact that this relatively simple geometry requires three million points for an inviscid calculation shows that computational requirements are very large when performing aeroelastic calculations on complex bodies. Iterating this system through long physical 7

Figure 4: Surface grids describing the AIM-120 missile. transients would be computationally expensive. Second, the AIM-120 calculation shows the need for overset grid methods. It would be virtually impossible to construct a singleblock, structured grid that would accurately describe this geometry. These two points illustrate the need for an HB overset grid solver. With this capability it will be possible to avoid calculating long physical transients in the time-domain by instead solving directly for the periodic response using the HB method. Even with the capability of solving in the frequency-domain, it is desirable to reduce the size of the computational domain when possible. Here we will consider three models of the isolated AIM-120 at a steady angle of attack. The first and most expensive model solves the Euler equations on the full three million node grid system described above. The simplest model employs slender body theory [12, 13] as described by Eq. 9 [ dl dx = ρ da U 2 z a dx x + U z ] [ ] a + ρ A U 2 2 z a t x + 2U 2 z a 2 x t + 2 z a t 2 ) A = π (s 2 R 2 + R4 where s the semi-span of the fins, R is the radius of the missile body, and z a is the deflection of the body. It will be shown that slender body theory (SBT) performs well away from the fins, however overpredicts the lift due to the fins. As such, the final model is a SBD/CFD hybrid model. Slender body theory is used to model the forces on the body of the missile, while the Euler equations are solved on grids that describe a single forward and single aft fin. Each fin is modeled as being cantilevered from a wall and isolated from the other. Clearly, each of the latter two models includes substantial assumptions. Slender body theory assumes that the body (including the fins) is sufficiently slender such that the flow is effectively two-dimensional in the cross-flow plane. This assumption is violated in the region of the fins. The hybrid model assumes that the aerodynamic influence of each component on the others is small. 8 s 2 (9)

Figure 5(a) shows the coefficient of lift versus angle of attack for each of the three models. If the three million node calculation is to be considered the benchmark, slender body theory overpredicts the lift by nearly a factor of two while the hybrid model agrees more closely with the three million node calculation. Figure 5(b) displays the coefficient of lift per chord-inch along the AIM-120 missile. By studying this figure, it becomes clear 2.5 2 Slender Body Theory Full Geometry Hybrid Model Coefficient of Lift, C L 1.5 1 0.5 Coefficient of Lift per Chord-inch per degree AoA 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Angle of Attack, α [degrees] (a) Coefficient of lift as a function of angle of attack. 0.05 0.04 0.03 0.02 0.01 0-0.01 Slender Body Theory 1.0 o 2.0 o 4.0 o -0.02 0 20 40 60 80 100 120 140 Chord Position from Nose [inches] (b) Coefficient of lift per chord-inch per degree angle of attack along the AIM-120 missile. Figure 5: Steady aerodynamic response of the AIM-120. In Fig. 5(b) solid lines correspond to calculations performed on the full geometry and symbols correspond to hybrid model calculations. that the slender body theory model of the AIM-120 performs well away from the fins, however the model drastically overpredicts the lift due to the fins. The hybrid model more accurately predicts the lift due to the fins, however it neglects the lift due to the aerodynamic influence of the fins on the body. The hybrid model produces acceptable preliminary results for steady flow trials while dramatically reducing computational cost. The entire AIM-120 is modeled by three million 9

grid points and each fin grid consists of about 50,000 nodes. This means that the hybrid model can produce results about 30 times faster than calculations performed on the full geometry. When considering the aeroelastic calculation, the reduced frequencies of interest based on fin chord of the motion are small. Since for steady trials it has been shown that the fins are primarily responsible for the lift produced, it is believed that the hybrid model will also work well for preliminary aeroelastic calculation. However, as illustrated, in order to accurately calculate the aerodynamic forces on a complex body such as the AIM-120, the full geometry must be represented accurately. 3.2 Overset Grids with the Harmonic Balance Method With the need for overset grid capabilities with the HB version of OVERFLOW 2 demonstrated, we will consider the modifications necessary for the frequency-domain implementation of overset grids. Since the sub-time levels of the harmonic balance method correspond to the time-domain solution sampled within one period of motion, the body grid is in physically different positions for each sub-time level. For example, consider the unsteady motion to be a pitching airfoil. With three harmonics retained, there are seven sub-time level grids on which the solution is calculated, as illustrated by Fig. 1. For large amplitude body motion, each of the sub-time level grid configurations need to be treated separately. Holes must be cut in the far-field grid and the connectivity database must be created for each configuration. It would be possible to modify the domain connectivity function within OVERFLOW 2 to perform these tasks and is the subject of continuing work. However, the scenario is greatly simplified if only small amplitude motions are considered. This paper presents a frequency-domain implementation of overset grids that is valid for small amplitude motion. This assumption of small amplitudes allows for the grid holes and connectivity to remain unchanged between sub-time levels. Thus, Pegasus 5 can be used as a pre-processing step to blank hole points and create a single connectivity database to be used for all sub-time levels. Under these assumptions very few further modifications need to be made to the harmonic balance capable version of OVERFLOW 2. The details of how the near-body grid is deformed for each sub-time level and the post-processing of data is slightly modified, however the concepts are unchanged. Time domain OVERFLOW 2, and therefore the HB version of OVERFLOW 2, stores all solution variables for all grids in a single array using pointers to access the proper block of data for a given grid. As long as the HB additions to the code follow the same principles, overset grids are handled in the same manner as a single-block grid. 4 HARMONIC BALANCE OVERSET GRID RESULTS A goal of this research program is to perform flutter and LCO calculations in the frequencydomain on complex geometries modeled by overset grids. Toward that end, the implementation of overset grids with the frequency-domain HB version of OVERFLOW 2 is first validated using a simple 2-D pitching airfoil case. The overset grid trial will be validated against calculations performed on a standard c-grid. 10

For consistency between the overset and single-block grid trials, the overset grid is created from the c-grid and consists of three grid blocks. The first grid block is the near-body grid. Since this grid is simply the inner portion of the original c-grid, it too is a c-grid topology. The other two grid blocks make up the far-field. One grid consists of the outer layers of the c-grid while the final grid block describes the far-field aft of the airfoil. The overset grids are shown in Fig. 6. Figure 6: Set of overset grids describing the NACA 0012 airfoil. The validation trials consist of the NACA 0012 airfoil pitching about the quarter-chord at a reduced frequency based on airfoil chord of 0.5. Pitch amplitudes vary from 0.5 to 4.0. The RANS model employs the Spalart-Allmaras one-equation turbulence model and is run at Mach 0.5. The pentadiagonal Beam and Warming scheme is used to discretize the pseudo-time derivative terms while the spatial operator is discretized using the central difference method. The original c-grid consists of 401 nodes in the circumferential direction and 75 nodes in the radial direction; the overset version of the grid contains slightly more nodes due to the overlapping regions as shown in Fig. 6. Three harmonics are retained in the Fourier series. The major assumption of this implementation of overset grids for use with the harmonic balance method is that the relative grid motion is small. This assumption allows for the grid holes and connectivity to remain unchanged between sub-time levels. This assumption, and its limitations, are illustrated below. Consider the case with the unsteady pitch amplitude of 2.0. It is possible to study the converged Mach contours for each of the sub-time levels of the HB solution method to determine the effects of the small amplitude assumption. As illustrated in Fig. 1, for the three harmonic case, the final sub-time level location corresponds to the largest displacement from the mean location. Since the intergrid communication is established for the mean position, Fig. 7(a) shows that the Mach contours transition smoothly from the near-body grid to the far-field grid when in this configuration. However, with the near-body grid in the physical location of the final subtime level there is a discontinuity in the solution when transitioning from the near-body to the far-field grid, which is most apparent at the wake. Clearly this discontinuity is not physical and illustrates the need for the unsteady motion to be sufficiently small. However, it is important to note that this technique is appropriate for many applications such 11

(a) Grid configuration used for connectivity. (b) Physical grid configuration. Figure 7: Mach contours of the seventh sub-time level solution for a three-harmonic calculation. as flutter point and limit cycle oscillation calculations where the motion is substantially less than 2.0. Figure 8 displays the zeroth and first harmonic of the unsteady pressure on the airfoil for various unsteady pitch amplitudes. The symbols correspond to the single-grid solution while the solid lines correspond to the solution calculated on the overset grid. Clearly the agreement is quite good, even for the large amplitude cases. Next, consider a transonic case with a free stream Mach number of 0.8. For this case the pseudo-time term is discretized using the SSOR algorithm [14] and the spatial derivatives are discretized using the HLLC method [15]. For a steady flow at this Mach number, the NACA 0012 experiences a shock near the mid-chord. It might be expected that due to the large gradients in the vicinity of the shock, the single connection method outlined above would break down. However, Fig. 9 shows that for unsteady amplitudes of up to 1.0, the method in fact performs very well. The agreement of the overset grid results with the single-block trial may be better than expected given the discontinuity in the flow field shown in Fig. 7(b). However, it is important to recall that the flow solver is unaware of this discontinuity since the grids 12

Zeroth Harmonic of Unsteady Pressure, p 0 / q 7 6.5 6 5.5 α 1 = 0.5 α 1 = 1.0 α 1 = 2.0 α 1 = 4.0 5 0 0.2 0.4 0.6 0.8 1 Airfoil Surface Location, x/c (a) Zeroth harmonic of unsteady pressure. Real Part of Normalized Unsteady Pressure, Re(p 1 ) / (q α 1 ) 0.2 0.15 0.1 0.05 0-0.05-0.1-0.15-0.2 0 0.2 0.4 0.6 0.8 1 Airfoil Surface Location, x/c α 1 = 0.5 α 1 = 1.0 α 1 = 2.0 α 1 = 4.0 (b) Real part of the unsteady pressure. Imaginary Part of Normalized Unsteady Pressure, Im(p 1 ) / (q α 1 ) 0.08 0.06 0.04 0.02 0-0.02-0.04-0.06-0.08 0 0.2 0.4 0.6 0.8 1 Airfoil Surface Location, x/c α 1 = 0.5 α 1 = 1.0 α 1 = 2.0 α 1 = 4.0 (c) Imaginary part of the unsteady pressure. Figure 8: Comparison of single- and multi-block grid solutions for a pitching airfoil at M = 0.5. Symbols correspond to single-block grid calculations and lines correspond to multi-block grid calculations. communicate as shown in Fig. 7(a). 5 DISCUSSION A method has been presented that allows for the use of overset grids with a new nonlinear frequency-domain harmonic balance solver for the NASA CFD code OVERFLOW 2. This overset grid method currently assumes small amplitude motion, which allows for hole cutting and grid connectivity to be performed using Pegasus 5 as a pre-processing step. The computational results validate this technique using a single- and multi-block airfoil grid simulating a pitching airfoil. The results show very good agreement for sufficiently modest, but realistic, unsteady amplitudes for both subsonic and transonic flow regimes. The next phase of this research will modify the domain connectivity function within OVERFLOW 2 to perform hole cutting and determine grid connectivity for each of the sub-time levels separately. This will remove the sufficiently small amplitude requirement and allow for the study of larger motions. 13

Zeroth Harmonic of Unsteady Pressure, p 0 / q 3.5 3 2.5 2 1.5 α 1 = 0.5 α 1 = 1.0 1 0 0.2 0.4 0.6 0.8 1 Airfoil Surface Location, x/c (a) Zeroth harmonic of unsteady pressure. Real Part of Normalized Unsteady Pressure, Re(p 1 ) / (q α 1 ) 0.3 0.2 0.1 0-0.1-0.2-0.3 0 0.2 0.4 0.6 0.8 1 Airfoil Surface Location, x/c α 1 = 0.5 α 1 = 1.0 (b) Real part of the unsteady pressure. Imaginary Part of Normalized Unsteady Pressure, Im(p 1 ) / (q α 1 ) 0.4 0.2 0-0.2-0.4 0 0.2 0.4 0.6 0.8 1 Airfoil Surface Location, x/c α 1 = 0.5 α 1 = 1.0 (c) Imaginary part of the unsteady pressure. Figure 9: Comparison of single- and multi-block grid solutions for a pitching airfoil at M = 0.8. Symbols correspond to single-block grid calculations and lines correspond to multi-block grid calculations. 6 ACKNOWLEDGMENTS Chad Custer is funded by a NASA Graduate Student Research Program (GSRP) fellowship. He would like to acknowledge the generous support and very helpful advice of his NASA mentor, Russ Rausch, of the Aeroelasticity Branch at the NASA Langley Research Center. 7 REFERENCES [1] Hall, K. C., Thomas, J. P., Ekici, K., et al. (2003). Frequency domain techniques for complex and nonlinear flows in turbomachinery. In 33rd AIAA Fluid Dynamics Conference and Exhibit. Orlando, FL: AIAA Paper 2003-3998. [2] Thomas, J. P., Dowell, E. H., and Hall, K. C. (2002). A harmonic balance approach for modeling three-dimensional nonlinear unsteady aerodynamics and aeroelasticity. In ASME International Mechanical Engineering Conference. New Orleans, LA: ASME Paper IMECE-2002-32532. [3] Thomas, J. P., Dowell, E. H., and Hall, K. C. (2002). Modeling viscous transonic limit cycle oscillation behavior using a harmonic balance approach. In 43rd 14

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit. Denver, CO: AIAA Paper 2002-1414. [4] Thomas, J. P., Dowell, E. H., and Hall, K. C. (2003). Modeling limit cycle oscillations of an NLR 7301 airfoil aeroelastic configuration including correlation with experiment. In 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit. Norfolk, VA: AIAA Paper 2003-1429. [5] Hall, K. C., Thomas, J. P., and Clark, W. S. (2002). Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA Journal, 40(5), 879 886. [6] Woodgate, M. A. and Badcock, K. J. (2009). Implicit harmonic balance solver for transonic flow with forced motions. AIAA Journal, 47(4), 893 901. [7] Thomas, J. P., Custer, C. H., Dowell, E. H., et al. (2009). Unsteady flow computation using a harmonic balance approach implemented about the OVERFLOW 2 flow solver. In 19th AIAA Computational Fluid Dynamics Conference. San Antonio, TX: AIAA Paper 2009-4270. [8] Benek, J. A., Buning, P. G., and Steger, J. L. (1985). A 3-D Chimera grid embedding technique. In 7th Computational Fluid Dynamics Conference. Cincinnati, OH: AIAA Paper 1985-1523, pp. 322 331. [9] Benek, J. A., Donegan, T. L., and Suhs, N. E. (1987). Extended Chimera grid embedding scheme with application to viscous flows. In 8th Computational Fluid Dynamics Conference. Honolulu, HI: AIAA Paper 1987-1126, pp. 283 291. [10] Rogers, S. E., Suhs, N. E., and Dietz, W. E. (2003). PEGASUS 5: An automated preprocessor for overset-grid computational fluid dynamics. AIAA Journal, 41(6), 1037 1045. [11] Nichols, R. H. and Buning, P. G. (2008). User s Manual for OVERFLOW 2.1, 2.1t ed. [12] Bisplinghoff, R. L., Ashley, H., and Halfman, R. L. (1996). Aeroelasticity. Mineola, NY: Dover Publications. [13] Ashley, H. and Landahl, M. (1985). Aerodynamics of Wings and Bodies. Reading, MA: Dover Publications. [14] Nichols, R. H., Tramel, R. W., and Buning, P. G. (2006). Solver and turbulence model upgrades to OVERFLOW 2 for unsteady and high-speed applications. In 25th Applied Aerodynamics Conference. San Francisco, CA: AIAA Paper 2006-2824. [15] Toro, E. F., Spruce, M., and Speares, W. (1994). Restoration of the contact surface in the HLL-Riemann solver. Shock Waves, 4(1), 25 34. 15