Direct numerical simulation of the turbulent flow in an elliptical pipe

The turbulent flow in a pipe with an elliptical cross section is directly simulated at $\operatorname{Re}=4000$ (where the Reynolds number $\operatorname{Re}$ is calculated in terms of the mean velocity and the hydraulic diameter). The incompressible Navier–Stokes equations are solved in curvilinear orthogonal coordinates by using a central-difference approximation in space and a third-order accurate semi-implicit Runge–Kutta method for time integration. The discrete equations inherit some properties of the original differential equations, in particular, the neutrality of the convective terms and of the pressure gradient in the kinetic energy production. The distributions of the mean and fluctuation characteristics of the turbulent motion over the pipe's cross section are computed.