RESEARCH INTERESTS

• Computational Fluid Dynamics
• Wavelet Methods for Modeling and Simulation of Complex Multi-Scale Phenomena
• Theoretical and Numerical Studies in Turbulence
• Large Eddy Simulations of Turbulent Flows
• Turbulence Modeling

CURRENT RESEARCH TOPICS

Parallel adaptive wavelet Environment for Multiscale Modeling (PawEMM++)

Today there are a number of problems in engineering and science, which share a single common computational challenge: the ability to solve and/or model accurately and efficiently a wide range of spatial and temporal scales. Numerical simulation of such problems requires either the use of highly adaptive physics based numerical algorithms, the use of reduced models that capture “important” physics of the problem at a lower cost, or the combination of both approaches. In addition, with the rapidly increasing ability to model large problems and the constant demand to extract and visualize the information relatively quickly or even interactively, the scientific visualization of very large data sets has become a challenge in itself. Currently we are working on development of multi-scale modeling and simulation environment capable of performing different fidelity simulations for single/multi-phase, inert/reactive, compressible/incompressible, transitional and turbulent flows in complex geometries. At the core of the problem solving environment is an integrated adaptive multi-scale/multi-form modeling and simulation framework that on-the-fly identifies regions of the flow with a suitable model-form, differentiates the most dominant (energetic) structures that control the overall dynamics of the flow; and resolves and "tracks" on a space-time adaptive mesh these dynamically-dominant flow structures, while modeling the effect of the unresolved motions using the compatible multi-level model form. The unique feature of the problem-solving environment is a unified, dynamically adaptive, wavelet multi-resolution (multi-scale), and multi-form approach to numerical algorithms and solvers, modeling and visualization.

Hierarchical Wavelet-based Modeling of Turbulent Flows

Since the inception of Computational Fluid Dynamics, turbulence modeling and numerical methods evolved as two separate fields of research with the perception that once a turbulence model is developed, any suitable computational approach can be used for the numerical simulations of the model. Latest advancements in wavelet-based adaptive multi-resolution methodologies for the solution of partial dierential equations, combined with the unique properties of wavelet analysis to unambiguously identify and isolate localized dynamically dominant flow structures, make it feasible to develop a cardinally different framework for hierarchical modeling and simulation of turbulent flows that fully utilizes spatial/temporal turbulent flow intermittency and tightly integrates numerics and physics-based modeling. The integration is achieved by combining spatially and temporally varying wavelet thresholding with hierarchical wavelet-based turbulence modeling. The resulting approach provides automatic smooth transition from directly resolving all flow physics to capturing only the energetic/coherent structures, leading to a dynamically adaptive variable fidelity approach.  Our current efforts are focused on the development of  the unified framework  that will allow for synergistic transition among models of different hierarchy, namely, the adaptive Wavelet-based DNS, the Adaptive wall-resolved LES, Adaptive wall-modeled LES, and adaptive wavelet-based Unsteady RANS and application of the approach to modeling and simulation of industrially relevant flows.

Parallel Adaptive Wavelet Collocation Method (P-AWCM) for Solution of Multi-Scale Problems

Four general classes of methods for solving nonlinear partial differential equations on adaptive computational meshes have been developed by our group. Each method uses the adaptive wavelet collocation method (AWCM) based on bi-orthogonal lifted interpolating wavelets to construct a computational grid adapted to the solution. The wavelet decomposition naturally provides a set of nested multi-scale grids adapted to the solution, and we take advantage of this property in developing our methods. In the first two methods we implement a traditional time marching scheme for parabolic and hyperbolic partial differential equations, but use AWCM to adapt the computational grid to the solution at each time step. When hyperbolic equations are solved an additional wavelet-based procedure for shock capturing is used. With this procedure the mesh is refined in the vicinity of the shock up to a-priori specified resolution and the shock is smoothed out using localized numerical viscosity. The third method simply uses the multi-scale wavelet decomposition as the basis for an adaptive multilevel method for nonlinear elliptic equations. Recently, we have begun to investigate a combination of the first three approaches to produce an adaptive simultaneous space–time method. In this case, both the space-time grid adapts locally to the solution, and the final solution is obtained simultaneously in the entire space–time domain of interest. Our current efforts are focused on further development of the parallel wavelet-based methods with mesh and anisotropy adaptation.