A Simple Strategy for Capturing the Unstable Steady Solution of an Unsteady Flow
ABSTRACT
A simple strategy for capturing unstable steady solution is proposed in this paper. By enlarging the real time step, the unstable high frequency modes of the unsteady flow can be effectively suppressed because of the damping action on the real time level. The steady component will not be changed during real time marching. When the unsteady flow solution converges to a steady state, the partial derivatives of real time in unsteady governing equations will disappear automatically. The steady solution is the exacted unstable steady solution. The method is validated by the laminar flow past a circular cylinder and a transonic buffet of NACA0012 airfoil. This approach can acquire high-precision unstable steady solutions quickly and effectively without reducing the spatial discretization accuracy. This work provides the basis for unstable flow modeling and analysis.
A simple strategy for capturing unstable steady solution is proposed in this paper. By enlarging the real time step, the unstable high frequency modes of the unsteady flow can be effectively suppressed because of the damping action on the real time level. The steady component will not be changed during real time marching. When the unsteady flow solution converges to a steady state, the partial derivatives of real time in unsteady governing equations will disappear automatically. The steady solution is the exacted unstable steady solution. The method is validated by the laminar flow past a circular cylinder and a transonic buffet of NACA0012 airfoil. This approach can acquire high-precision unstable steady solutions quickly and effectively without reducing the spatial discretization accuracy. This work provides the basis for unstable flow modeling and analysis.
Cite this paper
Zhang, W. , Liu, Y. and Li, J. (2015) A Simple Strategy for Capturing the Unstable Steady Solution of an Unsteady Flow. Open Journal of Fluid Dynamics, 5, 188-198. doi: 10.4236/ojfd.2015.52021.
Zhang, W. , Liu, Y. and Li, J. (2015) A Simple Strategy for Capturing the Unstable Steady Solution of an Unsteady Flow. Open Journal of Fluid Dynamics, 5, 188-198. doi: 10.4236/ojfd.2015.52021.
References
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http://dx.doi.org/10.1016/j.jfluidstructs.2011.03.021 [11] Sengupta, T.K., Bhole, A. and Sreejith, N.A. (2013) Direct Numerical Simulation of 2D Transonic Flows around Airfoils. Computers & Fluids, 88, 19-37.
http://dx.doi.org/10.1016/j.compfluid.2013.08.007 [12] Alshabu, A. and Olivier, H. (2008) Unsteady Wave Phenomena on a Supercritical Airfoil. AIAA Journal, 46, 2066-2073.
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http://dx.doi.org/10.2514/1.J051329 [14] Crouch, J.D., Garbaruk, A. and Magidov, D. (2007) Predicting the Onset of Flow Unsteadiness Based on Global Instability. Journal of Computational Physics, 224, 924-940.
http://dx.doi.org/10.1016/j.jcp.2006.10.035 [15] Crouch, J.D., Garbaruk, A., Magidov, D. and Travin, A. (2009) Origin of Transonic Buffet on Aerofoils. Journal of Fluid Mechanics, 628, 357-369.
http://dx.doi.org/10.1017/S0022112009006673 [16] Woodgate, M.A. and Badcock, K.J. (2007) Fast Prediction of Transonic Aeroelastic Stability and Limit Cycles. AIAA Journal, 45, 1370-1381.
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http://dx.doi.org/10.1017/jfm.2012.330 [18] Ahuja, S. and Rowley, C.W. (2010) Feedback Control of Unstable Steady States of Flow past a Flat Plate Using Reduced-Order Estimators. Journal of Fluid Mechanics, 645, 447-478.
http://dx.doi.org/10.1017/S0022112009992655 [19] Barkley, D. (2006) Linear Analysis of the Cylinder Wake Mean Flow. Europhysics Letters, 75, 750-756.
http://dx.doi.org/10.1209/epl/i2006-10168-7 [20] Jessie, W., Simone, C. and Angelo, I. (2009) Feedback Control by Low-Order Modeling of the Laminar Flow past a Bluff Body. Journal of Fluid Mechanics, 634, 405-418.
http://dx.doi.org/10.1017/S0022112009990590 [21] He, N., Tourlidakis, A. and Elder, R.L. (2007) Comparisons of Steady and Time-Averaged Unsteady Flow Predictions for Impeller-Diffuser Interactions in a Centrifugal Compressor Stage. Proceedings of GT2007 ASME Turbo Expo 2007: Power for Land, Sea and Air, Montreal, 14-17 May 2007.
http://dx.doi.org/10.1115/gt2007-27985 [22] Chen, Z.L. and Merkle, C.L. (2011) Contrast between Steady and Time-Averaged Unsteady Combustion Simulations. Computers & Fluids, 44, 328-338.
http://dx.doi.org/10.1016/j.compfluid.2011.01.032 [23] Mittal, S. (2008) Global Linear Stability Analysis of Time-Averaged Flows. International Journal for Numerical Methods in Fluids, 58, 111-118.
http://dx.doi.org/10.1002/fld.1714 [24] Langhorne, P.J., Dowling, A.P. and Hooper N. (1990) Practical Active Control System for Combustion Oscillations. Journal of Propulsion and Power, 6, 324-333.
http://dx.doi.org/10.2514/3.25437 [25] Tieron, J.E. and Doyle, J.C. (1992) Multi Mode Active Stabilization of a Rijke Tube. Proceedings of the ASME Winter Annual Meeting, 38, 65-68. [26] Roe, P.L. (1981) Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes. Journal of Computational Physics, 43, 357-372.
http://dx.doi.org/10.1016/0021-9991(81)90128-5 [27] Jameson, A. (1991) Time Dependent Calculations Using Multigrid, with Applications to Unsteady Flows past Airfoils and Wings. Proceedings of the AIAA 10th Computational Fluid Dynamics Conference, Honolulu, 24-26 June 1991.
[1] [1] Dowell, E.H. and Hall, K.C. (2001) Modeling of Fluid-Structure Interaction. Annual Reviews of Fluid Mechanics, 33, 445-490.
http://dx.doi.org/10.1146/annurev.fluid.33.1.445 [2] Lucia, D.J., Beran, P.S. and Silva, W.A. (2004) Reduced-Order Modeling: New Approaches for Computational Physics. Progress in Aerospace Sciences, 40, 51-117.
http://dx.doi.org/10.1016/j.paerosci.2003.12.001 [3] Silva, W. (2005) Identification of Nonlinear Aeroelastic Systems Based on the Volterra Theory: Progress and Opportunity. Nonlinear Dynamics, 39, 25-62.
http://dx.doi.org/10.1007/s11071-005-1907-z [4] Amsallem, D. and Farhat, C. (2008) Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity. AIAA Journal, 46, 1803-1813.
http://dx.doi.org/10.2514/1.35374 [5] Marzocca, P., Silva, W.A. and Librescu, L. (2004) Nonlinear Open/Closed-Loop Aeroelastic Analysis of Airfoils via Volterra Series. AIAA Journal, 42, 673-686.
http://dx.doi.org/10.2514/1.9552 [6] Zhang, W.W., Ye, Z.Y. and Zhang, C.A. (2009) Aeroservoelastic Analysis for Transonic Missile Based on Computational Fluid Dynamics. Journal of Aircraft, 46, 2178-2183.
http://dx.doi.org/10.2514/1.45249 [7] Zhang, W.W. and Ye, Z.Y. (2010) Effect of Control Surface on Airfoil Flutter in Transonic Flow. Acta Astronautica, 66, 999-1007.
http://dx.doi.org/10.1016/j.actaastro.2009.09.016 [8] Zhang, W.W., Wang, B.B. and Ye, Z.Y. (2012) Efficient Method for Limit Cycle Flutter Analysis by Nonlinear Aerodynamic Reduced-Order Models. AIAA Journal, 50, 1019-1028.
http://dx.doi.org/10.2514/1.J050581 [9] Williamson, C.H.K. and Govardhan, R. (2004) Vortex-Induced Vibrations. Annual Review of Fluid Mechanics, 36, 431-455.
http://dx.doi.org/10.1146/annurev.fluid.36.050802.122128 [10] Bearman, P.W. (2011) Circular Cylinder Wakes and Vortex-Induced Vibrations. Journal of Fluids and Structures, 27, 648-658.
http://dx.doi.org/10.1016/j.jfluidstructs.2011.03.021 [11] Sengupta, T.K., Bhole, A. and Sreejith, N.A. (2013) Direct Numerical Simulation of 2D Transonic Flows around Airfoils. Computers & Fluids, 88, 19-37.
http://dx.doi.org/10.1016/j.compfluid.2013.08.007 [12] Alshabu, A. and Olivier, H. (2008) Unsteady Wave Phenomena on a Supercritical Airfoil. AIAA Journal, 46, 2066-2073.
http://dx.doi.org/10.2514/1.35516 [13] Iovnovich, M. and Raveh, D.E. (2012) Reynolds-Averaged Navier-Stokes Study of the Shock-Buffet Instability Mechanism. AIAA Journal, 50, 880-890.
http://dx.doi.org/10.2514/1.J051329 [14] Crouch, J.D., Garbaruk, A. and Magidov, D. (2007) Predicting the Onset of Flow Unsteadiness Based on Global Instability. Journal of Computational Physics, 224, 924-940.
http://dx.doi.org/10.1016/j.jcp.2006.10.035 [15] Crouch, J.D., Garbaruk, A., Magidov, D. and Travin, A. (2009) Origin of Transonic Buffet on Aerofoils. Journal of Fluid Mechanics, 628, 357-369.
http://dx.doi.org/10.1017/S0022112009006673 [16] Woodgate, M.A. and Badcock, K.J. (2007) Fast Prediction of Transonic Aeroelastic Stability and Limit Cycles. AIAA Journal, 45, 1370-1381.
http://dx.doi.org/10.2514/1.25604 [17] Illingworth, S.J., Morgans, A.S. and Rowley, C.W. (2012) Feedback Control of Cavity Flow Oscillations Using Simple Linear Models. Journal of Fluid Mechanics, 709, 223-248.
http://dx.doi.org/10.1017/jfm.2012.330 [18] Ahuja, S. and Rowley, C.W. (2010) Feedback Control of Unstable Steady States of Flow past a Flat Plate Using Reduced-Order Estimators. Journal of Fluid Mechanics, 645, 447-478.
http://dx.doi.org/10.1017/S0022112009992655 [19] Barkley, D. (2006) Linear Analysis of the Cylinder Wake Mean Flow. Europhysics Letters, 75, 750-756.
http://dx.doi.org/10.1209/epl/i2006-10168-7 [20] Jessie, W., Simone, C. and Angelo, I. (2009) Feedback Control by Low-Order Modeling of the Laminar Flow past a Bluff Body. Journal of Fluid Mechanics, 634, 405-418.
http://dx.doi.org/10.1017/S0022112009990590 [21] He, N., Tourlidakis, A. and Elder, R.L. (2007) Comparisons of Steady and Time-Averaged Unsteady Flow Predictions for Impeller-Diffuser Interactions in a Centrifugal Compressor Stage. Proceedings of GT2007 ASME Turbo Expo 2007: Power for Land, Sea and Air, Montreal, 14-17 May 2007.
http://dx.doi.org/10.1115/gt2007-27985 [22] Chen, Z.L. and Merkle, C.L. (2011) Contrast between Steady and Time-Averaged Unsteady Combustion Simulations. Computers & Fluids, 44, 328-338.
http://dx.doi.org/10.1016/j.compfluid.2011.01.032 [23] Mittal, S. (2008) Global Linear Stability Analysis of Time-Averaged Flows. International Journal for Numerical Methods in Fluids, 58, 111-118.
http://dx.doi.org/10.1002/fld.1714 [24] Langhorne, P.J., Dowling, A.P. and Hooper N. (1990) Practical Active Control System for Combustion Oscillations. Journal of Propulsion and Power, 6, 324-333.
http://dx.doi.org/10.2514/3.25437 [25] Tieron, J.E. and Doyle, J.C. (1992) Multi Mode Active Stabilization of a Rijke Tube. Proceedings of the ASME Winter Annual Meeting, 38, 65-68. [26] Roe, P.L. (1981) Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes. Journal of Computational Physics, 43, 357-372.
http://dx.doi.org/10.1016/0021-9991(81)90128-5 [27] Jameson, A. (1991) Time Dependent Calculations Using Multigrid, with Applications to Unsteady Flows past Airfoils and Wings. Proceedings of the AIAA 10th Computational Fluid Dynamics Conference, Honolulu, 24-26 June 1991.