Research

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• Large Eddy Simulation of Turbulent Flows
• Geophysical Flows (Gravity Currents)
• ViTLES - A Parallel Finite Element CFD Platform
• Mesh Adaptation
• Aluminum Reduction

Most of my recent research has been centered around Large Eddy Simulation (LES). LES is one of the most successful techniques for the numerical simulation of turbulent flows. Introduced more than 30 years ago for geophysical applications, LES has been developed mainly in the engineering community. LES aims at approximating the large structures (eddies) in the turbulent flow. The effect of the small structures on the large ones is modeled, usually by using physical insight. Due to its low computational cost, LES is a popular approach for numerical simulations of complex turbulent flows.

In my Ph.D. thesis I started a focused effort on providing a sound mathematical support for the derivation and discretization of LES models. Here are links to some of my papers on mathematical modeling and analysis, numerical validation and testing, numerical analysis, and boundary conditions for some new LES models.

For the last couple of years, Bill Layton, Luigi Berselli, and I have been writing a BOOK on Mathematical Analysis of Large Eddy Simulation. The book is in its final editing stages, and we are under contract with Springer Verlag to submit it by the end of the year.

Mathematical Modeling and Analysis for LES

• Large Eddy Simulation for Turbulent Flows,
  Ph.D. thesis, PDF file, University of Pittsburgh, 2000.

• Approximating the Larger Eddies in Fluid Motion III: the Boussinesq Model for Turbulent Fluctuations,
  (with W. J. Layton), An. St. Univ. "Al. I. Cuza, vol. 44, 1998, pp. 245-261.

• Mathematical Analysis for the Rational Large Eddy Simulation Model,
  (with L. C. Berselli, G. P. Galdi, and W. J. Layton), Math. Models Meth. Appl. Sc., vol. 12 (8), 2002, pp. 1131--1152.

• A Higher Order Subfilter-Scale Model for Large Eddy Simulation,
  (with L. C. Berselli), J. Comp. Appl. Math., vol. 159, 2003, pp. 411-430.

Numerical Validation and Testing for LES

• A numerical study of a class of LES models,
  (with V. John, W. J. Layton, G. Matthies and L. Tobiska), Int. J. Comput. Fluid Dyn., vol. 17 (1), 2003, pp. 75-85.

• A 3D Channel Flow Simulation at Re_{\tau}=180 Using a Rational LES Model,
  (with P. Fischer), Proceedings of Third AFOSR International Conference on DNS/LES, eds. C. Liu, L. Sakell, and T. Beutner,
  Greyden Press, 2001, pp. 283--290.

• Large Eddy Simulation of Turbulent Channel Flows by the Rational LES Model,
  (with P. Fischer), Physics of Fluids, vol. 15(10), 2003, pp. 3036--3047.

• Backscatter in the Rational LES Model,
  (with P. Fischer), Computers and Fluids, vol. 33(5-6), 2004, pp. 783-790.

Numerical Analysis for LES

• Genuinely Nonlinear Models for Convection-Dominated Problems,
  PDF file, Comput. Math. Appl.

• Convergence of Finite Element Approximations of Large Eddy Motion,
  (with V. John and W. J. Layton), Num. Meth. P.D.E.s, vol. 18(6), 2002, pp. 689-710.

Boundary Conditions for LES

• Approximate Deconvolution Boundary Conditions for Large Eddy Simulation,
  (with J. Borggaard), compressed postscript file, submitted for publication, August 2004.

Gravity Currents

Much of my current research is on gravity currents, cold masses of water protruding into warm ones. The mixing between the cold and warm water takes place through the Kelvin-Helmholtz instability (see figure above). Gravity currents are believed to have a significant role in general ocean circulation models. However, their numerical simulation is very challenging due to the large (geophysical scale) parameters involved. My main interest in the numerical investigation of gravity currents is the ability of LES to perform at parameters prohibitive for a Direct Numerical Simulation (DNS) approach.
Here are some of my papers on gravity currents.

• Three-dimensional turbulent bottom density currents from a high-order nonhydrostatic spectral element model,
  (with T. Ozgokmen, P. Fischer, and J. Duan), J. Phys. Oceanogr., vol. 34(9), 2004, pp. 2006-2026.

• Entrainment in bottom gravity currents over complex topography from three-dimensional nonydrostatic simulations,
  (with T. Ozgokmen, P. Fischer, and J. Duan), Geophys. Res. Letters vol. 31(13), L13212, 2004.

• Enstrophy and ergodicity of gravity currents,
  (with V.P. Bongolan-Walsh, J. Duan, H. Go, T. Ozgokmen, and P.F. Fischer), IMA volume on SPDEs, in press, 2004.

ViTLES - The Virginia Tech Large Eddy Simulator

Two years ago, my colleague Jeff Borggaard and I, together with a group of enthusiastic graduate students at Virginia Tech, started building a computational platform for LES. The scientific and engineering applications targeted yielded a couple of requirements for our code:
(1) geometric flexibility for being able to treat complex computational domains, such as those encountered in realistic geophysical and engineering applications;
(2) parallelism for being able to tackle realistic parameters in our numerical simulations.
Thus, ViTLES is a parallel, finite element computational platform. For more details, please visit our working web page.

Some of my earlier research focused on mesh adaptation. Specifically, I investigated the use of edge swapping in improving the solution quality in convection-dominated problems. Here are a couple of papers on this subject:

• A Flow-Aligning Algorithm for Convection-Dominated Problems,
  Int. J. Num. Meth. Eng., vol. 46, No. 7, 1999, pp. 993-1000.

• A 3D Flow-Aligning Algorithm for Convection-Diffusion Problems,
  Appl. Math. Letters, vol. 4, No. 12, 1999, pp. 67-70.

One project I worked on was Aluminum Reduction. This international collaboration with the University of Puerto La Cruz, Venezuela, aimed at efficient numerical simulation of aluminum reduction. The ultimate goal was to optimize the process through a careful parameter choice.
Here are some papers on this subject:

• Distribucion de Temperatura en las Esquinas de una Celda de Reduccion de Aluminio Tipo Hall-Heroult,
  (with E. Gutierrez, N. Troyani and W. J. Layton), IV Congreso Iberoamericano de Ingenieria Mecanica, Santiago de Chile, Chile, 1999.

• Corner Distribution of Voltage in Hall-Heroult Aluminum Reduction Cells,
  (with E. Gutierrez, N. Troyani and W. J. Layton), IV International Congress: Energy, Environment and Technological Innovation, Rome, 1999, pp. 1445-1451.

• The Center Section Temperature Distribution in Hall-Heroult Aluminum Reduction Cells from a Three-Dimensional Finite Element Simulation,
  (with E. Gutierrez, N. Troyani and W. J. Layton), International Thermal Energy Congress, Cesme, Izmir, Turkey, July 2001.

• Algoritmos Computacionales para Resolver el Problema Termoelectrico en Tres Dimensiones en Celdas de Reduccion de Aluminio del Tipo Hall-Heroult,
  (with E. Gutierrez, N. Troyani and W. J. Layton), Universidad Ciencia y Tecnologia, Vol. 3, No.12, pp. 17-24, 1999.

• Formulacion Variacional del Problema Termoelectrico de una Celda de Reduccion Hall-Heroult en Tres Dimensiones,
  (with E. Gutierrez, N. Troyani and W. J. Layton), Universidad Ciencia y Tecnologia, Vol. 3, No. 9, pp. 25-29, 1999.