"... the complexities of things can so easily & dramatically escape the simplicity of the equations which describe them. Unaware of the scope of simple equations, man has often concluded that nothing short of God, not mere equations, is required to explain the complexities of the world."
- Richard Feynmann

The above comment aptly sums up the field I work in. I specialize in Computational fluid dynamics (CFD), pursuing my doctoral degree, under the advisory of Dr.Gary Cheng. As part of the Computational Simulation Laboratory (CSL), I am currently involved  in a research work that involves the development and Validation of a Transient Viscous flow solver based on a Space-Time CE/SE framework framework.

"Space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" - Hermann Minkowski

The method of space-time Conservation Element and Solution Element (CE/SE) is a new numerical framework for solving conservation laws with several attractive features. It is being developed by Dr. S.C. Chang and coworkers at NASA Glenn Research Centre. This high-resolution, multi-dimensional numerical framework has been built from scratch with extensive consideration on physics and rigorous mathematical proof, thereby doing away with some of the limitations of traditional numerical simulation methods. The promise of extremely low numerical dissipation and simplicity of construction makes the CE/SE paradigm very promising for CFD simulations. Since its inception, the CE/SE framework has been successfully adapted to model model flow phenomena such as acoustic waves, traveling and interacting shocks, detonation waves, cavitation, shedding vortices etc.

Unique features of CE/SE

• Unified treatment of space and time.
• Enforcement of local and global space-time flux conservation.
• Use of a space-time staggered mesh that allows for evaluation of fluxes at the cell interfaces without solving the Riemann problem (no use of MOC).
• Schemes built from a non-dissipative core scheme, allowing for control of numerical dissipation effectively and with mathematical justification (not ad-hoc).
• Both the flow variable and its spatial derivatives are treated as independent unknowns.
• For flows in multiple spatial dimensions, no directional splitting is employed. Truly multi-dimensional.

Our primary focus is simulation of flow instability problems, such as spray-combustion instability and aeroacoustics. To achieve this capability, as part of my Masters thesis, a 2-D transient viscous flow solver was successfully developed and validated against standard benchmark testcases such as driven cavity, laminar flow over a circular & square cylinder and flow over a thin splitter plate. A 2-D Euler solver,written in Fortran 90, obtained from NASA Glenn Research Center was used as a starting point and the Navier-Stokes solver was built upon it. It can currently handle supersonic/ subsonic flows of perfect gases ( single species). The code has been parallelized using MPICH and uses METIS for domain decomposition. It uses unstructured grids ( triangular and quadrilateral meshes) for flow simulations. Click on the image below to know more details on the CE/SE methodology and for some preliminary results.

In the future , we plan to extend this flow solver to three dimensions in space (i.e. 4-D problems in space and time) along with capability to handle multi-species chemically reacting flows.

Conference Publications:

1. Venkatachari, B., Cheng, G.C. and Chang, S.C, "Development of a Transient Viscous Flow Solver Based on Conservation Element- Solution Element Framework," AIAA-2004-3413, 40th AIAA Joint Propulsion Conference,Fort Lauderdale,FL, July 2004.
2. Venkatachari, B., Cheng, G.C. and Chang, S.C, "Courant Number Insensitive Transient Viscous Flow Solver Based on CE/SE Framework," AIAA- 2005-0093, 43rd AIAA Aerospace Sciences Meeting, Reno, NV, Jan 2005.
3. Venkatachari, B., Cheng, G.C. and Chang, S.C, "Validation and Verification of Courant Number Insensitive CE/SE Method for Transient Viscous Flow Simulations ," AIAA-2005-4356, 40th AIAA Joint Propulsion Conference and Exhibit, Tucson, Az, July 2005
4.B.S. Venkatachari, G.C. Cheng, S.C. Chang,B.K. Soni "Validation and verification of Courant number insensitive CE/SE method for transient viscous flow simulations," presented at MASCOT05 (5th Meeting on Applied Scientific Computing and Tools)- a guest event in TCNCAE2005,Lecce, Italy October 3-7, 2005

Masters Thesis

Venkatachari, B.,"Development and Validation of a Transient Viscous Flow Solver based on Space-Time CE/SE Framework", August 2005, UAB. (Pdf file)