My research interests are, ** Reduced Order Modeling (ROM), Scientific Computing,
Lagrangian Coherent Structures (LCS), Numerical Analysis,
Computational Fluid Dynamics (CFD), Numerical Methods for PDEs.
**

ROMs attempt to replace the high rank model by low dimennsional dynamcial systems, which are valuable whenever the same problem must be solved repeatedly. ROMs reduce the dimension and computational complexity of a system, using a high resolution data set obtained by either physical experiment or numerical simulation. The primary goal is to construct surrogate models that are cheap to execute and still maintain the large scale features of the original system.

My focus is ROM for nonlinear fluid flows, i.e. Navier-Stokes Equations (NSE), and turbulence.

Standard Galerkin ROM (G-ROM) for NSE is trying to find,
$\mathbf{u}_r=U+\sum_{j=1}^r{a}_j{\phi}_j$, where $\phi_j$ is the POD basis, $U$ is centering trajectory, such that

POD basis $\phi_1$, $\phi_2$, $\phi_3$, $\phi_6$, from left to right. For more informations, look at my publications

3D turbulent cylinder flow with $Re=1\times10^4$.

Fluid particle trajectories are sensitive to changes in their initial conditions which makes the assessment of flow models and observations from individual tracer samples unreliable. There is a robust skeleton of material surfaces, Lagrangian Coherent Structures (LCSs), shaping those patterns. The LCS group at Caltech and references can be found here.

It starts with a velocity field $\mathbf{v}(x,t)$, $x=(x_1,x_2,x_3)\in U\subset \mathbf{R}^3$. Then the flow map is defined as,

$\begin{eqnarray} &\psi(x,y,t) = A\sin(\pi f(x,y))\sin(\pi y) \\ &f(x,y) = a(t)x^2+b(t)x \\ &a(t)=\epsilon\sin(\omega t)\\ &b(t)=1-2\epsilon\sin(\omega t) \end{eqnarray} $

Attracting LCSs with high resolution $1000\times500$

Repelling LCSs with high resolution $1000\times500$

Computational Fluid Dynamics emerge as in last 1950s, particularly in engineering, it is still at the stage of development where "problems involving complex geometries can be treated with simple physics and those involving simple geometry can be treated with complex physics" (Bailey 1986). The improvement in computer hardware, numerical algorithms as well as software has brought about the increase need to simulate more extreme physical conditions, higher Reynolds number, higher temperature in CFD, making it still one of the most popular research topics in the world. Many state-of-the-art open source numerical software and packages have been developed over the years.

Example:

Breaking of a dam 3D, simulated by OpenFOAM (if you can not see this movie, please try Safari browser)