Numerical Method

In this chapter we give a brief description of the numerical method underlying NaSt3DGP. For more information please refer to [1]. In several cases, different numerical methods, for example, different discretizations for time derivatives or convective terms, are possible. Details on how to control these schemes using the scene description file are given in section .

The basic mathematical model are the dimensionless time-dependent incompressible Navier-Stokes equations

$$\frac{\partial {\bf u}}{\partial t} + {\bf u}\cdot\nabla {\bf u} = \frac{{\bf g}}{Fr}-\nabla p + \frac{1}{Re} \Delta {\bf u} ~,~~{\bf x} \in \Omega,~ 0\le t \le T_{fin}$$ (3.1)

$$\nabla\cdot {\bf u} = 0 ~,$$ (3.2)

subject to appropriate initial and boundary conditions. $Re$ is the dimensionless Reynolds-number which determines the ration between inertia and viscous forces in the flow. the Reynolds-number is given by

$$Re = \frac{\rho L u_\infty}{\mu}$$

where $\mu$ is the dynamic viscosity, $\rho$ the (constant) density, $L$ a characteristic length and $u_\infty$ a characteristic velocity of the flow configuration. $Fr$ is a modified Froude-number defined as

$$Fr = \frac{u^2_\infty}{L}$$

and specifies the ration of inertia to gravitational forces.

If thermal effects or the behaviour of a scalar, driven by the flow, are of interest, () and () are completed by equations for the energy (temperature) and the transport of a scalar

$$\frac{\partial T}{\partial t} + {\bf u}\cdot\nabla T = \frac{1}{Re \, Pr} \Delta T$$ (3.3)

$$\frac{\partial C}{\partial t} + {\bf u}\cdot\nabla C = \nu_C \Delta C\quad.$$ (3.4)

$T$ is the temperature and $C$ the scalar. The dimensionless number $Pr$ is the Prandtl number and is given by

$$Pr = \frac{\nu}{\alpha}$$

where $\nu = \mu / \rho$ is the kinematic viscosity and $\alpha$ is the heat diffusion coefficient. $\nu_C$ is the diffusion constant of $C$.