Stabilization of Explicit Flow Solvers Using a Proper Orthogonal Decomposition Technique

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January 2012, Nashville, Tennessee AIAA Stabilization of Explicit Flow Solvers Using a Proper Orthogonal Decomposition Technique Kivanc Ekici 1,, Kenneth C. Hall 2, and Huang Huang 1, 1 University of Tennessee, Knoxville, Tennessee Duke University, Durham, North Carolina A new numerical technique for the stabilization of explicit computational fluid dynamic (CFD) solvers is presented. When using CFD codes to solve practical problems, one sometimes finds the solution fails to converge due to the presence of a small number of eigenvalues of the flow solver outside the unit circle. In this paper, we use the proper orthogonal decomposition technique to estimate the unstable eigenvalues and eigenmodes of the flow solver as the solution diverges for linear problems, or exhibits limit cycle oscillations for nonlinear problems. We use the resulting eigen-information to construct a preconditioner that repositions the unstable eigenvalues from outside the unit circle to a point inside the unit close to the origin, resulting in a stable flow solver. In this work, the proposed method is applied to a nonlinear steady cascade solver. However, the technique can be easily applied to external flow solvers, and to unsteady frequency-domain (time-linearized and harmonic balance) flow solvers. I. Nomenclature A i, B i i th -stage coefficients of Runge-Kutta integration D artificial dissipation vector E total energy F, G flux vectors h total enthalpy p pressure P preconditioner P r l laminar Prandtl number P r t turbulent Prandtl number Q discretized convective and viscous fluxes R residual vector S source vector S t source term for turbulence model t time U vector of conservation variables u, v Cartesian velocities V control volume x, y Cartesian coordinates X matrix of right eigenvectors γ specific heat ratio µ l dynamic laminar viscosity µ t dynamic turbulent viscosity ν kinematic viscosity ν working variable for Spalart-Allmaras turbulence model ρ density Assistant Professor, Dept. of Mechanical, Aerospace and Biomedical Engineering. Senior Member AIAA. Julian Francis Abele Professor, Dept. of Mechanical Engineering and Materials Science. Fellow AIAA. Graduate Research Assistant, Dept. of Mechanical, Aerospace and Biomedical Engineering. Student Member AIAA. Copyright c 2012 by Kivanc Ekici, Kenneth C. Hall and Huang Huang. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Copyright 2012 by Kivanc Ekici, Kenneth C. Hall and Huang Huang. Published by the, Inc., with permission. 1 of 15

2 II. Introduction Frequency-domain computational fluid dynamic (CFD) models of aerodynamic and aeroelastic phenomena have improved dramatically over the past several decades. In the turbomachinery community, investigators have used two main families of theories based on frequency-domain modeling: time-linearized methods 1 6 and nonlinear harmonic balance methods 7 14 to simulate unsteady flows. The time-linearized approach assumes that the unsteady disturbances are small compared to the mean flow, and are harmonic in time with frequency ω. These assumptions allow one to linearize the governing equations about a nonlinear steady or mean operating condition. The resulting linearized equations can be solved very efficiently using convergence acceleration techniques such as multigrid and local time stepping. In the nonlinear harmonic balance approach, the unsteady flow is also assumed to be temporally periodic with a fundamental excitation frequency ω. However, unlike the time-linearized approach, the unsteady disturbances are not assumed to be small. Using the harmonic balance technique, the dependent variables of the governing Navier-Stokes equations are stored at a number of sub-time levels spanning a single period of the unsteadiness. The various physical time level solutions are only coupled to one another through a pseudo-spectral operator that approximates the time derivatives in the Navier-Stokes equations. As in the time-linearized flow solvers, pseudo time-marching is used to drive the solution to convergence, allowing one to model dynamically nonlinear flows very efficiently. Often, nominally steady or frequency-domain unsteady flow solutions fail to converge. In the case of the timelinearized flow solvers, the solution may unexpectedly diverge exponentially. For steady or unsteady harmonic balance solvers, the solution may diverge, or more commonly, exhibit small limit cycle oscillations. The root cause of the lack of convergence may be physical, as in the case of small scale shedding behind a blunt trailing edge, or a spurious numerical artifact. In any event, the instability in the flow solution is often limited to a very small number of unstable modes. In most of the work reported in the literature, frequency-domain and steady flow solvers have not exhibited convergence problems due to numerical instabilities associated with explicit pseudo-time marching. A few exceptions are the work of Ning and He, 15 Campobasso and Giles, 16, 17 and Elliot. 18 As an example, Campobasso and Giles 16, 17 investigated a configuration in which a steady-state flow solver underwent a low-level limit cycle oscillation due to the existence of a suction side separation bubble in a turbine blade row. This limit cycle oscillation did not adversely affect the accuracy of the mean flow solution because of the small size of the oscillation. However, for the accompanying time-linearzed flow solver, the problem appeared as an exponentially diverging solution. In their work, Campobasso and Giles developed two techniques based on the GMRES algorithm and the Recursive Projection Method (RPM) to stabilize their time-linearized flow solver. In this paper, we propose a new stabilization technique based on proper orthogonal decomposition (POD). POD has been used by a number of investigators to analyze turbulence, 19 to construct reduced models of fluid dynamic systems, 20, 21 and for other purposes. Here we use the POD technique to analyze flow solver instabilities, and then use this information to precondition the flow solver to eliminate any instabilities. The current approach is similar in some ways to the RPM method developed by Campobasso and Giles in that we seek to identify the unstable eigenvalues and eigenmodes of the flow solver, and then modify the the eigenvalues to produce a stable solver. Our method is designed for use with steady nonlinear solvers, but may also be used with time-linearized or harmonic balance solvers. III. Flow Solver A. Governing Equations To demonstrate the POD stabilization method, we apply the method to the two-dimensional Reynolds-Averaged Navier-Stokes (RANS) equations, with the Spalart-Allmaras 22 turbulence model to model Reynolds stresses. In strong conservation form, these equations are given by U t + F x + G y = S (1) 2 of 15

3 where U is the vector of conservation variables, i.e., ρ ρu U = ρv ρe ρ ν The last entry ν, is the working variable in the Spalart-Allmaras turbulence model, from which the eddy viscosity is computed. The flux vectors F, G and the source term S are given by ρu ρv 0 ρuv τ yx F = ρu 2 + p τ xx ρuv τ xy ρuh τ xh ρu ν τ xν, G = ρv 2 + p τ yy, S = ρvh τ yh ρv ν τ yν The shear stresses, τ xx and τ xy, in the flux vectors are defined as τ xx = 2 ( 3 (µ l + µ t ) 2 u x v ) y ( u τ xy = (µ l + µ t ) y + v ) x In addition, the τ xh term in the energy equation and the τ xν term in the turbulence model equation are given by ( µl τ xh = uτ xx + vτ xy + + µ ) t h P r l P r t x τ xν = 3 2 (µ l + ρ ν) ν x The remaining components of the shear stresses and terms in the energy and turbulence model equations are defined similarly. For an ideal gas with a constant specific heat ratio, the pressure is related to the conserved variables through [ p = (γ 1)ρ E 1 ( u 2 + v 2)] 2 Finally, the total enthalpy is defined as h = ρe + p ρ It should be noted that the inviscid flux and source vectors depend on the conservation variables and the Cartesian coordinates. The viscous fluxes depend on the gradients of the flow velocities, temperature and the working turbulent variable. B. Numerical Integration Having presented the governing equations to be used in our analysis, we next integrate Eq. (1) on a computational grid of H-O-H topology using an explicit time-marching scheme introduced by Jameson. 23 The fact that the steady flow is periodic in the circumferential direction allows us to use a computational grid spanning a single blade, thereby reducing the computational cost. In addition, to simplify the treatment of boundary conditions, we use a vertex-based approach 24 in our discretization. Because the steady-state solution is of interest in this work, convergence acceleration techniques such as local time stepping are incorporated in the flow solver. The details of Jameson s scheme is well-documented in the literature However, its salient features are presented here for completeness. S t 3 of 15

4 In the vertex-based Jameson scheme, 24 a control volume is formed by the union of the cells that meet at the vertex. In the two-dimensional case the control volume is made of four cells surrounding the vertex. Written in a semi-discrete form, the Navier-Stokes are given by d (VU) + Q (U) D (U) = 0 dt where V is the volume of union cells, Q (U) denotes to the discretized convective and viscous flux terms, and D (U) denotes to the artificial dissipative term. The convective flux at the cell surface center is evaluated by averaging the conservation variables at the corresponding cell vertex. The flux balance at each vertex can be ensured by summing all the fluxes from each constituting cell. To eliminate odd-even decoupling and oscillations around shock waves, a blend of second and fourth-order dissipative terms, D (U), are included in Jameson s formulation. The system of differential equations are marched temporally toward a steady state solution with a modified multi-stage Runge-Kutta scheme. The conserved variables at the (i + 1)-st stage of each iteration are computed from U (i+1) = U (0) A (i+1) t V [ ( Q U (i)) B (i+1) D (U (i)) ( ) 1 B (i+1) D (U (0))] where U (0) are the conserved variables from the previous iteration. For the five-stage Runge-Kutta integration used in this work, coefficients are given by A 1 = 1 4, A 2 = 1 6, A 3 = 3 8, A 4 = 1 2, A 5 = 1 B 1 = 1, B 2 = 0, B 3 = 14 25, B 4 = 0, B 5 = To reduce the computational cost and maintain a good high frequency error damping property, the viscous fluxes are frozen at the first stage and the artificial dissipative terms are evaluated only at the first, third and fifth stages. IV. Stabilization In general, an explicit CFD scheme can be thought as an iterative process that drives a vector of residuals R to convergence, that is, U n+1 = U n + R (U n ) (2) where n is the iteration number and U is the vector of conserved variables. The residual operator, R, is nonlinear for a steady flow solver. However, to motivate the development of our method (and to simplify the discussion), assume for the moment that R can be written in the linear form given by, R (U n ) = AU n b (3) where the matrix A and the vector b are functions of the mean flow. Note that, in this work we use an explicit Jameson scheme, but the method described in this paper can be applied to any scheme having the generic form given by Eq. (2). Inserting Eq. (3) into Eq. (2), one obtains, U n+1 = (I + A) U n b (4) This scheme is unstable whenever the magnitude of any eigenvalue z of (I + A) is greater than or equal to unity. Therefore, if one can identify those unstable eigenmodes and modify the eigenvalues of the scheme so that all the eigenvalues lie inside the unit circle, then the numerical scheme will be stable ensuring convergence of the explicit scheme. Consider for the moment the homogeneous problem (b = 0). In this case, the residual operator given in Eq. (4) can be rewritten as, U n+1 = U n + AU n (5) Further suppose the ith eigenvalue z i is the only eigenvalue that lies outside the unit circle. To stabilize the homogeneous solver in this simple case, we propose a modification to the explicit operator of the form U n+1 = U n + ( I + β i x i y T ) i AU n (6) 4 of 15

5 where x i and y i are the right and left eigenvectors associated with the unstable eigenvalue and β i is a constant to be determined. Note that if the numerical solution of the steady flow field is stable, β i is taken to be zero and the original form of the operator is recovered. However, if the solution is unstable (including the scenario where the residual value goes into a limit cycle) the value of β i will be non-zero. In general, since the linear homogeneous problem is considered, the solution can be thought of as a linear combination of the eigenvectors of the system, so that U = Xα (7) where X is the matrix of right eigenvalues and α is a vector that represents the weighting coefficients of each eigenvector. Substituting Eq. (7) into Eq. (6), one gets Xα n+1 = Xα n + ( I + β i x i y T ) i AXα n (8) Next, pre-multiplying Eq. (8) by the transpose of the left eigenvector (y T i ) and making use of the orthogonality of the right and the left eigenvectors, i.e., { y T 0 i j i x j = 1 i = j and one gets y T i (I + A) x j = { 0 i j i = j z i α n+1 i = α n i + [z i + β i (z i 1)] α n i (9) where α i is the generalized coordinate associated with the unstable eigenvalue z i. The effect of the preconditioning, then, is to change the eigenvalue of the unstable mode from z i to z i + β i (z i 1), leaving the stable eigenvalues and eigenmodes unchanged. To completely eliminate the unstable error mode in one iteration, we set z i β i = z i 1 (10) so that the modified eigenvalue is zero. Recall that to simplify the discussion for the development of the technique, we have considered that only one eigenvalue was unstable and that the problem was homogeneous. For the more general case where more than one eigenvalue is unstable, and b 0, the preconditioned scheme can be written as [ ] U n+1 = U n + I + i β i x i y T i [AU n b] (11) where the sum is over the set of unstable eigenmodes. Similarly, for nonlinear CFD schemes, the preconditioning technique can be expressed as [ ] or U n+1 = U n + I + i β i x i y T i R (U n ) (12) U n+1 = U n + PR (U n ) (13) where the eigenvalues and the eigenvectors are those of the nonlinear problem linearized about some mean flow and P is the preconditioning matrix. Therefore, to stabilize a diverging or limit cycling CFD solver, we use eigen-information associated with the unstable or limit cycling modes to precondition the solver. In the next section, we present a proper orthogonal decomposition (POD) technique used to compute approximately the unstable eigenvalues and eigenmodes. 5 of 15

6 V. Determination of POD Vectors For the case where the steady solution residual is diverging or exhibiting limit cycle, the unsteadiness in the solution will be dominated by just a few modes of motion. One way to identify these modes is to use proper orthogonal decomposition (POD). The process is as follows. Every few iterations, the entire flow solution is stored. Each of these several stored solutions or snapshots is loaded as a column vector into a tall, narrow matrix (S), so that S = { U 1 Ū} { U 2 Ū} {... U K Ū} (14) where U 1, U 2, are the solution snapshots and Ū = 1 K K U k (15) is the average of the solution snapshots. We seek to construct optimal basis vectors that are linear combinations of the columns of the snapshot matrix, that is Φ = Sv (16) The so-called proper orthogonal decomposition vectors are found by first solving the small eigenvalue problem defined by k=1 S T Sv k = λ k v k, k = 1,..., K (17) where λ k and v k are the eigenvalues and eigenvectors of the matrix S T S. Those eigenmodes with the largest eigenvalues are dominant. Next, we use several of the dominant POD vectors to estimate the eigenvalues and eigenmodes of the flow solver, that is, we assume that, x i Φγ i (18) where γ i is a vector whose length is equal to the number of dominate POD vectors retained in Φ. Thus the eigenvalue problem of the flow solver, (I + A) x i = z i x i (19) can be rewritten as (I + A) Φγ i = z i Φγ i (20) Finally, because this system is over constrained, we project the error against the basis vectors to obtain the reduced order eigenvalue problem, Φ T (I + A) Φγ i = z i Φ T Φγ i ; i = 1,, I (21) The above equation approximates the eigenvalues of the explicit solution scheme using a reduced order model, which is small (on the order of 12 12) and very inexpensive to calculate. Note that, although A may not be easily determined, the product AΦ can be easily obtained using numerical differentiation eliminating the need to explicitly linearize the code. In this work, we use second-order finite differencing to calculate AΦ in a matrix-free approach. VI. Numerical Results To demonstrate the effectiveness of the novel stabilization technique proposed in this work, we consider the Eleventh Standard Configuration, 27 which is a two-dimensional viscous turbine cascade (see Fig. 1). The turbine row is comprised of airfoils with stagger angle of degrees and a chord-to-pitch ratio (solidity) of Experimental measurements for this configuration were reported by Fransson et al. 27 In addition, a number of investigators presented unsteady computations 16, 17, for this test case to validate their flow solvers. Experimental data is available for a number of different flow conditions for this turbine. However, we consider only the transonic off-design condition in this paper because this (nominally steady) case demonstrates a numerical 6 of 15

7 limit cycle oscillation caused by a separation bubble that forms on the suction side of the blades near the leading edge. Note that, the same test case was also used by Campobasso and Giles 16, 17 to validate their numerical stabilization techniques based on the GMRES and Recursive Projection Methods (RPM) applied to unsteady time-linearized flow solvers. Before presenting the results of the POD stabilization scheme, we present results of the unmodified CFD scheme to illustrate this limit cycle. A. Unmodified Flow Solver Solutions (No POD Stabilization) Generally speaking, because of the use of local time-stepping, the solution of nonlinear steady flows is not timeaccurate. However, the pseudo time-marching of the governing equations reflects some physical properties of the flow field causing a limit cycle behavior in convergence. 16 Figure 1 shows the H-O-H viscous computational grid we used to compute the flow field; the grid contains nodes in the O-grid block, and nodes in each of the two H-grid blocks. The flow enters the blade Figure 1. H-O-H grid used for flow computations. row with a Mach number of 0.38 and a flow angle of 34 deg with respect to the axial direction. The Reynolds number, based on inlet flow conditions and the chord length, is approximately 1,100,000. The isentropic Mach number at the exit of the blade row is 0.99 causing a weak shock formation on the suction side of the blades at approximately 80% of the chord. As mentioned earlier, a separation bubble forms between the leading edge and 30% chord of the suction side [see Fig. 2] that introduces a numerical instability associated with the bubble, and hence, the steady flow solver fails to converge. As can be seen from Fig. 3, the steady solution residual drops one order of magnitude within the first 750 iterations, but then grows, and eventually goes into a periodic limit cycle around 3000 iterations. A fully converged solution cannot be achieved for this case without the POD stabilization. Nevertheless, the (unconverged) contour plots from our nominal flow solver, shown in Fig. 4 at the end of 5000 iterations, 16, 17 agree very well with the computations of Campobasso and Giles. The pressure and absolute Mach number contours within the passage clearly show the shock formation as well as the separation bubble on the suction side. As reported by Campobasso and Giles, 16 a small-amplitude limit-cycle behavior about a mean flow does not necessarily prevent the steady solution from converging to a level acceptable for engineering purposes. Next, we compare the results obtained from our nominal solver (no POD-stabilization applied) and available experimental data. Shown in Fig. 5 are the isentropic Mach number values on the surface of the blade. Overall, the predicted values are in good agreement with the experimental data except for a small region upstream of the 7 of 15

8 Figure 2. Flow field near the leading edge. Left: Streamlines. Right: Velocity vectors Log(Global Residual) Iteration Number Figure 3. Convergence history. Note the limit-cycle behavior instability due to shedding. 8 of 15

9 Figure 4. Contour plots for the Eleventh Standard Configuration. Left: Mach number. Right: Static pressure. Note that a single blade passage is used in computations. Multiple passages are shown here for clarity. weak shock. However, as pointed out by Fransson,27 fairly small changes in the inlet flow conditions may result in significantly different flow structures near the shock, which was also observed by other investigators. Figure 5 also reveals small oscillations in surface isentropic Mach number values near the leading edge of the blades. These oscillations are mainly due to the instability caused by a shedding phenomenon near the separation bubble. 2 Surface isentropic Mach number Present solver Experiment Surface location, x/c Figure 5. Surface isentropic Mach number distribution. Note small oscillations caused by the numerical instability near the leading edge. 9 of 15

10 B. Modified Flow Solver Solutions (With POD Stabilization) In this section, we demonstrate how our proposed technique can be used to identify and eliminate the nonlinear limit cycle oscillation found in the example of the previous section. As shown in the previous section, the steady flow computation for the Eleventh Standard test case fails to converge due to a numerical instability caused by a physical phenomenon, that is, the shedding near the separation bubble is clearly seen in the contours of ρe shown in Fig. 6. Here contour plots are generated at every three iterations from the 5001 st to the 5031 st iteration and it is apparent that the limit-cycle behavior has a period spanning approximately 30 iterations. Evidence of shedding can also be seen from the convergence history plot presented in Fig. 7. Figure 6. Contours of ρe for different iteration numbers. Top left to right: Iteration numbers from 5001 to Bottom left to right: Iteration numbers from 5016 to The first step in the POD stabilization algorithm is to assemble the solution snapshots of the POD basis vectors. After the solution goes into limit-cycle at the end of 5000 iterations, we store the computed CFD computations every three iterations for the next 30 iterations. This results in a total of eleven snapshots. We choose the snapshots to span a single cycle of the limit cycle. Shorter periods will not contain all the relevant solution behavior to obtain good POD vectors; longer periods provide no additional useful information. Although not presented here, we also tried fewer or more snapshots and found that ten to 15 snapshots are needed to stabilize the solver considered in this work. Having collected the solution snapshots, the next step is to compute the POD modes using Eq. (17). Next, these POD modes are used to compute estimates of the unstable eigenvalues and eigenvectors using Eq (21). Figure 8 shows the eleven eigenvalues computed in this way. As can be seen, the eigenanalysis reveals two unstable modes, which lie just outside the unit circle and are complex conjugates of one another. Note that we perform the eigenanalyses to determine the unstable modes just once after the stabilization is turned on, and in any event is a very inexpensive operation because of the small size of the problem. Having identified the offending eigenmodes, at each subsequent iteration of the flow solver, we precondition the flow solver using Eq. (12) where the values of β i are computed from Eq. (10) for the two unstable modes. Figure 9 shows the convergence history for the stabilized solver. It can be seen that once the POD stabilization is turned on, the residual drops three orders of magnitude in a few hundred iterations eliminating the numerical instability and fully converges to machine accuracy in around 7500 iterations. Note that for the preconditioning here, we used a total of eleven solution snapshots to compute the five most dominant POD vectors, which in 10 of 15

11 -4.70 Log(Global Residual) Iteration Number Figure 7. Convergence history. Note the limit-cycle behavior instability due to shedding Unstable modes Im(λ) Re(λ) Figure 8. Eigenvalues of the CFD solution. Note that eleven snapshots were used and no order reduction was performed. 11 of 15

12 turn were used to find the two least stable flow solver eigenmodes used to construct the preconditioning. When the total number of POD vectors used was four or fewer, the computations diverged, presumably because the computed flow solve eigen-information was not sufficiently accurate to construct an effective preconditioner. 0.0 POD Stabilization enabled after 5000 iterations -5.0 Log(Global Residual) Iteration Number Figure 9. Convergence history using the POD stabilization technique. Next, we plot the surface isentropic Mach number values when the POD stabilization is applied. As seen in Fig. 10, the stabilization technique eliminates the spatial oscillations near the leading edge of the blades. To further investigate the effectiveness of the proposed method, we turned off the preconditioning at the end of iterations, restarted the solver from the fully-converged solution, and performed computations for an additional iterations. Figure 11 shows the global residual for this case. It is apparent that the numerical instability associated with the shedding phenomenon causes the solution to diverge slowly when the POD technique is disabled. Finally, to re-stabilize the flow solver, we revert back to the preconditioning technique and restart the solution after iterations. Figure 12 shows the convergence histories for the preconditioning constructed using two different sets of solution snapshots. When the snapshot are taken every ten iterations as the solution is diverging (between and iterations), the convergence rate is the slowest. This is not surprising, because the period of the instability is around 15 iterations. When the solution snapshots are taken at more closely spaced intervals, the convergence rate increases because the POD vectors provide a better basis for estimating the eigenvalues and eigenmodes of the system that are used to construct the preconditioner. VII. Conclusions A numerical stabilization method based on a novel POD technique was presented for explicit CFD solvers. Using this scheme, we first take snapshots of the non-converging solution. Using these snapshots, we form POD basis vectors, which we then use to compute approximate eigenvalues and eigenmodes of the unstable CFD modes. Finally, the estimates of the unstable eigenvalues and eigenmodes are used to precondition the explicit scheme, resulting in a stable flow solver. The method was applied to a representative flow solver, and was found to produce fully converged solutions for cases that would otherwise not converge. 12 of 15

13 2 Surface isentropic Mach number Present solver (No POD stabilization) Present solver (With POD stabilization) Experiment Oscillations eliminated with POD stabilization Surface location, x/c Figure 10. Surface isentropic Mach number distribution. Note how POD technique eliminates oscillations caused by the numerical instability near the leading edge. 0.0 POD Stabilization enabled after 5000 iterations Log(Global Residual) POD Stabilization disabled after iters Log(Global Residual) Iteration Number Iteration Number Figure 11. Convergence histories. Left: Solution restarted at the end of iterations and without the POD stabilization technique. Right: Close-up view. 13 of 15

14 0.0 New snapshots (Every 10 iters from ) New snapshots (Every iteration from ) Log(Global Residual) POD stabilization restarted Iteration Number Figure 12. Convergence histories for restarted POD solver References 1 Hall, K. C. and Crawley, E. F., Calculation of Unsteady Flows in Turbomachinery Using the Linearized Euler Equations, AIAA Journal, Vol. 27, No. 6, June 1989, pp Hall, K. C. and Lorence, C. B., Calculation of Three-Dimensional Unsteady Flows in Turbomachinery Using the Linearized Harmonic Euler Equations, Journal of Turbomachinery, Vol. 15, No. 4, Oct. 1993, pp Hall, K. C., Clark, W. S., and Lorence, C. B., A Linearized Euler Analysis of Unsteady Transonic Flows in Turbomachinery, Journal of Turbomachinery, Vol. 116, July 1994, pp Clark, W. S. and Hall, K. C., A Time-Linearized Navier-Stokes Analysis of Stall Flutter, Journal of Turbomachinery, Vol. 122, July 2000, pp Sbardella, L. and Imregun, M., Linearized unsteady viscous turbomachinery flows using hybrid grids, Journal of Turbomachinery, Vol. 123, 2001, pp Hall, K. C. and Ekici, K., Multistage Coupling for Unsteady Flows in Turbomachinery, AIAA Journal, Vol. 43, No. 3, March 2005, pp He, L. and Ning, W., Efficient Approach for Analysis of Unsteady Viscous Flows in Turbomachines, AIAA Journal, Vol. 36, No. 11, 1998, pp Hall, K. C., Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique, presented at the Kerrebrock Symposium, A Symposium in Honor of Professor Jack L. Kerrebrock s 70th Birthday, Massachusetts Inst. of Technology, Cambridge, MA, Jan 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, May 2002, pp Ekici, K. and Hall, K. C., Nonlinear Analysis of Unsteady Flows in Multistage Turbomachines Using Harmonic Balance, AIAA Journal, Vol. 45, No. 5, May 2007, pp Ekici, K., Hall, K. C., and Dowell, E. H., Computationally Fast Harmonic Balance Methods for Unsteady Aerodynamic Predictions of Helicopter Rotors, Journal of Computational Physics, Vol. 227, No. 12, 2008, pp Ekici, K. and Hall, K. C., Nonlinear Frequency-Domain Analysis of Unsteady Flows in Turbomachinery with Multiple Excitation Frequencies, AIAA Journal, Vol. 46, No. 8, Aug. 2008, pp Ekici, K. and Hall, K. C., Harmonic Balance Analysis of Limit Cycle Oscillations in Turbomachinery, AIAA Journal, Vol. 49, No. 7, July Huang, H. and Ekici, K., A Harmonic Balance Method for the Analysis of Unsteady Flows in Cascades, AIAA Paper , Ning, W. and He, L., Some Modeling Issues on Trailing-Edge Vortex Shedding, AIAA Journal, Vol. 39, No. 5, 2001, pp Campobasso, M. S. and Giles, M. B., Effects of Flow Instabilities on the Linear Analysis of Turbomachinery Aeroelasticity, Journal of Propulsion and Power, Vol. 19, No. 2, 2003, pp of 15

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