# Research

## Research Interests

My research interests are primarily in the field of fluid dynamics. As a computational fluid dynamist, I'm most interested in:

- high-order numerical methods
- massively parallel flow solver development
- unstructured moving & deforming grids
- high performance parallel computing
- rotating flows, vortex dynamics and turbulence

## Research Activities

My research so far can be very generally summarized as numerically solving the Navier-Stokes equations (and their variants) that govern the motion of fluids. The Navier-Stokes equations, when written in conservative form, are:
$$
\begin{aligned}
&\frac{\partial\rho}{\partial t} + \frac{\partial\rho u_j}{\partial x_j} = 0, \\[1.2em]
&\frac{\partial\rho u_i}{\partial t} + \frac{\partial}{\partial x_j}(\rho u_i u_j + p \delta_{ij}) = \frac{\partial\tau_{ij}}{\partial x_j}, \\[1.2em]
&\frac{\partial \rho E}{\partial t} + \frac{\partial}{\partial x_j}\Big((\rho E+p)u_j\Big) = \frac{\partial}{\partial x_j}(u_i \tau_{ij}) - \frac{\partial q_j}{\partial x_j},
\end{aligned}
$$
where $\rho$ is density, $u$ is velocity, $p$ is pressure, $\delta_{ij}$ is the the Kronecker delta, $\tau_{ij} = 2\mu e_{ij} + \lambda e_{kk} \delta_{ij}$ is the shear stress tensor, $\mu$ is the dynamic viscosity, $\lambda=-(2/3)\mu$ for Newtonian fluids based on Stokes assumption, $e_{ij} = (u_{i,j} + u_{j,i})/2$ is the strain rate tensor, $E=e+u_k u_k /2$ is the total energy per unit mass, $e$ is the internal energy per unit mass, $q_j = -\kappa \partial T/\partial x_j$ is the heat flux vector, $\kappa$ is the thermal conductivity, $T$ is temperature. Ideal gas has $p = \rho R T$, $e=p/(\rho(\gamma-1))$, where $R$ is the specific gas constant.

More specifically, I am working or have worked on the following research topics: