CFD Online Discussion Forums - calculation of time step

Below is my derrivation for the stability requirements of $\Delta t$ for direct solution of the navier stokes equations on a cartesian grid,

The dimensionalised navier stoke equation for the conservation of momentum, is given by
$\frac{\partial u}{\partial t}=\left(\frac{\mu}{\rho}\left(\frac{\partial^2 u}{\partial^2 x}+\frac{\partial^2 u}{\partial y^2} +\frac{\partial^2 u}{\partial z^2} \right) - \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)-\frac{1}{\rho}\frac{\partial p}{\partial x}+\frac{F_{ext}}{\rho}\right)$
for stability it is required that at all points,
$\frac{|\delta u|}{\delta t}=\left|\frac{u^{n+1}-u^n}{\Delta t}\right|<<\left|u\right|$
sub,
$\left|\frac{u^{n+1}-u^n}{\Delta t}\right|<\epsilon \left|u\right|$
where
$\epsilon$ is some suitible factor of saftey, then.
$\epsilon | u|>\Delta t\left|\frac{\mu}{\rho}\left(\frac{\partial^2 u}{\partial^2 x}+\frac{\partial^2 u}{\partial y^2} +\frac{\partial^2 u}{\partial z^2} \right) - \left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)-\frac{1}{\rho}\frac{\partial p}{\partial x}+\frac{F_{ext}}{\rho}\right|$
assuming each term has equal weight then,
$\frac{\epsilon |u|}{4}> \Delta t\frac{\mu}{\rho}\left|\frac{\partial^2 u}{\partial^2 x}+\frac{\partial^2 u}{\partial y^2} +\frac{\partial^2 u}{\partial z^2} \right|$
$\frac{\epsilon |u|}{4}>\Delta t\left|u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right|$
$\frac{\epsilon |u|}{4}>\Delta t\left|\frac{1}{\rho}\frac{\partial p}{\partial x}\right|$
$\frac{\epsilon |u|}{4}>\Delta t\left|\frac{F_{ext}}{\rho}\right|$

rearranging gives four requirements for $\Delta t$

Viscous term requirement

$\frac{\rho\epsilon |u|}{4\mu}\left|\frac{\partial^2 u}{\partial^2 x}+\frac{\partial^2 u}{\partial y^2} +\frac{\partial^2 u}{\partial z^2} \right|^{-1}> \Delta t$
the discretization of $\partial ^2 u$ is given by,
$\partial ^2 u= u_{i+1,j}+u_{i-1,j}-2u_{i,j}$
which will always have magnitude,
$|\partial ^2 u|<4|u|$
substituting gives,
$\frac{\rho\epsilon |u|}{4\mu}\left|\frac{4|u|}{\partial x^2}+\frac{4|u|}{\partial y^2} +\frac{4|u|}{\partial z^2} \right|^{-1}> \Delta t$
which simplifies to,
$\frac{\rho\epsilon }{16\mu}\left|\frac{1}{\partial x^2}+\frac{1}{\partial y^2} +\frac{1}{\partial z^2} \right|^{-1}> \Delta t$

Diffusion term requirement

$\frac{\epsilon |u|}{4}\left|u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right|^{-1}>\Delta t$
the discretization of
$\partial u$ is given by,
$|\partial u|=|u_{ij}-u_{i-1,j}|$
which will always have magnitude;
$|\partial u|<2|u|$
substituting gives,
$\frac{\epsilon |u|}{4}\left|u\frac{2 |u|}{\partial x}+v\frac{2|u |}{\partial y}+w\frac{2| u|}{\partial z}\right|^{-1}>\Delta t$
which simplifies to,
$\frac{\epsilon }{8}\left|\frac{u }{\partial x}+\frac{v}{\partial y}+\frac{w}{\partial z}\right|^{-1}>\Delta t$
$\frac{\epsilon }{8}\left|3\frac{u }{\partial x}_{max}\right|^{-1}>\Delta t$
$\frac{\epsilon }{24}\left|\frac{u }{\partial x}_{max}\right|^{-1}>\Delta t$

Pressure term Requirment

$\frac{\epsilon |u|\rho\partial x}{4|\partial p|}>\Delta t$
$\partial p$ can be discretized as,
$\partial p=p_{i,j}-p_{i+1,j}$
therfore
$|\partial p|< 2|p|$
substitution gives,
$\frac{\epsilon |u|\rho\partial x}{8|p|}>\Delta t$

External Force term Requirment

$\frac{\epsilon |u|\rho}{4|F_{ext}|}>\Delta t$

therfore if the bellow inequality is satisfied then the discretized equations will be stable,
$\Delta t < min\left( \frac{\rho\epsilon }{16\mu}\left|\frac{1}{\partial x^2}+\frac{1}{\partial y^2} +\frac{1}{\partial z^2} \right|^{-1}, \frac{\epsilon }{24}\left|\frac{u }{\partial x}_{max}\right|^{-1}, \frac{\epsilon |u|\rho\partial x}{8|p|},\frac{\epsilon |u|\rho}{4|F_{ext}|}\right)$
where

is the maximum velocity found in the system,

A slighly more relaxed stability requirement for the 2 dimensional case was given in "Griebel et al: Numerical Simulation in Fluid Dynamics: SIAM 1998"

$\Delta t = \epsilon min\left(\frac{Re}{2}\left( \frac{1}{\Delta x^2}+\frac{1}{\Delta y^2}\right)^{-1}, \frac{\Delta x}{|U_{max}|}, \frac{\Delta y}{|v_{max}|} \right)$
but there was no acompanying derrivation.

Hope this helps