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Finite Volume Formulation for Inviscid, Compressible Quasi 1D Flow

    From eqn. (5), the flux can be integrated over the element surface as shown in Figure 1.

(6)

The flux for each face is shown split into the convective and pressure contributions for clarity.  These surface integrals can be eliminated by replacing them with the appropriate section averages of the flux times the associated section area.  For sections 1 and 2, both the convective and pressure components are retained, but section 3 retains only the pressure term since velocity is zero by definition on the outer surface.  The surface 1 term is negative as a result of the dot product between velocity and the area normal.  The wall flux term will be evaluated more specifically below.  The reulst of the integration is given by eqn. (7).

(7)

The wall pressure force term shown in eqn. (7) affects only the momentum equation.  The continuity and energy components of the vector are zero.  This momentum force contribution is the component of the pressure in the x-direction.  This component is defined as the average wall pressure times the sine of the inclination angle.  The sine of the angle can be approximated by the derivative of area for small cell sizes.

(8)

An approximation is still required for the averaged wall pressure in terms of the stored variables.  The wall pressure must be approximated by the section average pressure of the cell.  With this definition and eqn. (8), the wall pressure force vector becomes:

(9)

Equation (9) is equal to the wall force variable in the final term of eqn. (7).  The area at the two flow boundaries are known in this case since these boundaries are grid point locations.  Equation (10) gives the resulting equation.

(10)

The terms in front of the time derivative can be re-arranged to give eqn. (11).

(11)

Equation (11) gives the final finite volume expression, accurate to the order of the flux calculation method.  The flux terms at the left and right boundaries must be evaluated, which will be accomplished in future sections.  The finite volume formulation establishes an algebraic form for the spatial terms of the governing equations, while the transient term is left in differential form.  It will be dealt with in the next section.  The spatial terms consist of a flux derivative representation and a source term representing the area variation.  These spatial terms are referred to as a "residual," shown in eqn. (12).  The residual has a significance in the time integration routines and as a monitor of steady state convergence.

(12)

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