Calculate the characteristic derivatives for outgoing characteristics using one-sided derivatives and for incoming characteristics from the physical boundary conditions. Then use the boundary equations (49-52 or 53-56) to determine the time derivatives of the variables. Assume subsonic flow. Then there is always one outgoing wave, one incoming wave and the advection (entropy) wave which switches between incoming and outgoing depending on the normal fluid velocity.

- Outgoing characteristic wave

There is one outgoing characteristic waveEvaluate using one-sided derivatives,

- Entropy:

The entropy characteristic wave equation isand the equation for the entropy is

For inflows, to keep the entropy constant, set . To make the entropy tend toward some reference value, , set

where the constant for some time constant , which might be taken as or for instance, and the other variables have their mean values or initial boundary values.

For outflows, evaluate using one-sided derivatives,

- Incoming characteristic wave

The incoming characteristic wave equation isThe no-reflection condition is . However, additional constraints should be imposed to ensure constant bottom pressure or zero net mass flux. Thus take

Since,

this should make the net mass flux . Also,

so this should make the pressure .

- Horizontal velocities

The characteristic equation for the horizontal velocities is

For outflows evaluate using one-sided derivatives,

For inflows, set

Then calculate the time-derivative of the horizontal velocities from the boundary equations.

- Viscous stresses and normal energy flux

Set the normal derivative of the tangential viscous stresses and energy flux to zero,

Finally, solve equations (53-56) for the time-derivatives of the conserved variables or equations (49-52) for the time-derivatives of the primitive variables.

Wed Sep 19 18:16:24 MET DST 2001