MenuComputational Fluid Mechanics. Incompressible Navier Stokes Equations
Introduction
The incompressible Navier-Stokes equations is a system of partial differential equations that describe the motion of a viscous, incompressible, Newtonian fluid. If $\Omega \subset \mathbb{R}^d$ with $d=2,3$ is the domain that the fluid occupies, these equations are $$ \left\{ \begin{array}{r r} \rho_t + \nabla\cdot( \rho u ) = 0, & \texttt{in~} \Omega, \\ \rho u_t + ( \rho u \cdot \nabla) u -\mu \Delta u + \nabla p = f, &\texttt{in~} \Omega, \\ \nabla\cdot u = 0, &\texttt{in~} \Omega. \end{array} \right. $$ Here, $f$ is a given internal dragging force (for instance gravity), and the unknowns are the density $\rho$, the velocity $u$ and the pressure $p.$ This system must be complemented with suitable boundary and initial conditions.
The effective approximation of the solutions to the Navier-Stokes equations presents several challenges. To begin with, the system is nonlinear. Moreover, the velocity and the pressure are coupled through the constraint $\nabla \cdot u = 0,$ which gives the problem a saddle-point structure and, thus, requires some computational effort to approximate the solutions.
Constant Density Flows
Let us consider the case when the density is constant. Since one of the difficulties in the approximation of the solutions to the Navier-Stokes equations come from the fact that they are coupled through the incompressibility constraint, one of the most popular approaches to approximate the solutions is to decouple this constraint from the diffusion operator. This is the main idea behind fractional time-stepping techniques.
A grosso modo, fractional time-stepping techniques for constant density flows work as follows: First we update the velocity using a diffusion equation, then update the pressure using a Poisson equation. One of the research directions we have is the development of fractional time-stepping techniques that are accurate and efficient.
Numerical Illustrations
Here are examples of two-dimensional simulations. These films show the time evolution of vorticity generated around a NACA12 foil impulsively started from rest at zero angle of attack at three different Reynolds numbers. (Authors: J.-L. Guermond, A. Marra (Politecnico Milano), L. Quatapelle (Politecnico Milano))Here are other examples of flows in a two-dimensional channel. (Author: A. Salgado)
Variable Density Flows
In the case when the density is variable it was, until recently, the common viewpoint in the literature that fractional time-stepping techniques lead to a time-dependent variable-coefficient second order partial differential equation for the determination of the pressure. This makes the determination of the pressure computer intensive, and the main bottleneck for solving variable density flows.
We are conducting a research program aiming at simplifying this approach. Our new algorithms show that the pressure can actually be computed by simply solving a Poisson problem at each time step. This substantially simplifies the method, since Poisson problems can be solved very fast.
