Stanford University Department of Aeronautics & Astronautics Aerospace Computing Laboratory

Arathi Kamath Gopinath

Durand Building, Room 001
Department of Aeronautics and Astronautics
Stanford University, Stanford, CA 94305
Phone: (650) 723-9564


Doctor of Philosophy
Department of Aeronautics and Astronautics, Stanford University (2004-2007)

Master of Science
Department of Aeronautics and Astronautics, Stanford University (2002-2003)

Bachelor of Technology
Department of Naval Architecture and Ocean Enginnering, Indian Institute of Technology, Madras, India (1997-2001)

Research Interests

The Time Spectral Algorithm for Periodic Unsteady Flows

Time dependent calculations find a wide variety of applications including flutter analysis, analysis of flow around helicopter blades in forward flight and rotor-stator combinations in turbomachinery. Typically, explicit time-stepping schemes require the use of very small time steps to obtain reasonable accuracy, whereas implicit schemes allow much larger time steps but are prohibitively expensive for 3-D calculations.

The Time Spectral Method is proposed for the fast and efficient computation of periodic flows using Reynolds-Averaged Navier-Stokes calculations past two- and three- dimesnsional bodies. The efficiency of the approach derives from these attributes :
* Time discretization based on a fourier representation in time to take advantage of its periodic nature.
* Multigrid to drive a fully implicit time stepping scheme.

The accuracy and efficiency of this algorithm is verified with Euler and Navier-Stokes calculations for a pitching airfoil and pitching wing test case and compared with experimental results. Results show that engineering accuracy can be produced for the globally integrated quantities with just 4 time intervals per time-period for the pitching airfoil/wing case. The algorithm has been modified and extended to other forced response applications like rotor-stator interactions in turbomachinery, unsteady flow around vertical-axis wind turbines. For full-fledged Unsteady three-dimensional RANS calculations around these complex geometries, we have gained significant performance improvements over conventional time integration methods.

More recently, the method has been used also to predict unsteady flows where the frequency of unsteadiness is not known apriori. Typical examples include vortex shedding behind bluff bodies. We have formulated a gradient based approach in combination with the time spectral method to simulate these flows. A similar methodology has been used to predict Limit-Cycle Oscillations for model problems and various other spectral methods are used for comparison.

The Harmonic Balance Method for Turbomachinery Calculations

A full unsteady calculation through a multi-stage, multi-passage turbomachine can be prohibitively expensive, especially if such a calculation needs to be done as an everyday design tool. Even the Time Spectral method used for such a test case can be limited by computing power. We have proposed the Harmonic Balance method as a reduced order model for predicting the unsteady flow through turbomachines using Unsteady RANS equations.

This time-domain algorithm simulates the true geometry of the turbomachine (with the exact blade counts) using only one blade passage per blade row, thus leading to drastic savings in both CPU and memory requirements. Modified periodic boundary conditions are applied on the upper and lower boundaries of the single passage in order to account for the lack of a common periodic interval for each blade row. The solution algorithm allows each blade row to resolve a specified set of frequencies in order to obtain the desired computation accuracy; typically, a blade row resolves only the blade passing frequencies of its neighbors. Since every blade row is setup to resolve different frequencies the actual Harmonic Balance solution in each of these blade rows is obtained at different instances in time or time levels. The interaction between blade rows occurs through sliding mesh interfaces in physical time. Space and time interpolation are carried out at these interfaces and can, if not properly treated, introduce aliasing errors that can lead to instabilities. With appropriate resolution of the time interpolation, all instabilities are eliminated. This new procedure is demonstrated using both two-and three dimensional test cases and can be shown to significantly reduce the cost of multi-stage simulations while capturing the dominant unsteadiness in the problem.



  PDF   "Three-Dimensional Unsteady Multi-Stage Turbomachinery Simulations using the Harmonic Balance Technique", Gopinath, A., van der Weide, E., Alonso, J.J., Jameson, A., Ekici, K. and Hall, K.C., 45th AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2007-0892, Reno, Nevada, January 8-11 2007.


  PDF   "Application of the Time Spectral Method to Periodic Unsteady Vortex Shedding", Gopinath, A. and Jameson, A., 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2006-0449, Reno, Nevada, January 9-12 2006.

  PDF   "Comparative Analysis of Computational Methods for Limit-Cycle Oscillations", Gopinath, A., Beran, P.S. and Jameson, A., 47th AIAA Structures, Structural Dynamics and Materials Conference, AIAA Paper 2006-2076, Newport, Rhode Island, May 1-4 2006.


  PDF   "Turbomachinery Applications with the Time Spectral Method", van der Weide, E., Gopinath, A. and Jameson, A., 17th AIAA Computational Fluid Dynamics Conference, AIAA Paper 2005-4905, Toronto, Ontario, June 6-9 2005.

  PDF   "Time Spectral Method for Periodic Unsteady Computations over Two- and Three- Dimensional Bodies",Gopinath, A. and Jameson, A., 43rd Aerospace Sciences Meeting and Exhibit, AIAA Paper 2005-1220, Reno, Nevada, January 10-13 2005.

  PDF   "Revisiting the Vertical-Axis Wind-Turbine Design using Advanced Computational Fluid Dynamics", Vassberg, J., Gopinath, A. and Jameson, A., 43rd Aerospace Sciences Meeting and Exhibit, AIAA Paper 2005-0047, Reno, Nevada, January 10-13 2005.

  PDF   "Aerodynamics and Flight Control of Insects: A Computational Study", Sriram, Gopinath, A., Van der Weide, E., Kim, S., Tomlin, C. and Jameson, A., 43rd Aerospace Sciences Meeting and Exhibit, AIAA Paper 2005-0841, Reno, Nevada, January 10-13 2005.



Thesis (turned in on April 30, 2007)

  PDF   "Efficient Fourier-Based Algorithms for Time-Periodic Unsteady Problems"


  PDF   "Defense"
  PDF   "Reno 07"

Last Modified: Mon Jun 14 17:07:41 PST 2004