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The Governing Equations for Inviscid, Compressible Quasi 1D Flow

    Quasi 1D flow represents a situation in which the fluid is restricted to motion along one coordinate in space, but for which the effective cross-sectional area of the fluid domain is allowed to vary along that coordinate direction.  The governing equations for this fluid motion are derived beginning from the integral form of the Navier-Stokes equations.  The flow is considered to be inviscid and without body forces, and the formulation requires a total of three equations: continuity, x-direction momentum, and energy.  A typical fluid element is pictured in Figure 1.  

typical quasi 1-D Cell
Figure 1. A Typical Quasi-1D Fluid Element

    For this situation, the integral form of the governing equations are given as follows:

continuity eqn
(1)
  momentum eqn.
(2)
  energy eqn.
(3)

    These equations are presented here without derivation.  The derivations from basic principles can be found in Anderson[].  Note that this formulation can be written in vector form by combining the three above relations into one.

governing eqns.
(4)

The three equation set can be written in a single form for brevity.  The application of the finite volume formulation applies to each equation.  The prime appears in the flux term because the sign of the flux must be accounted for in taking the integral (as the dot product has been removed from the previous equation).

Governing eqn. - vector form
(5)

This form is called a "conservative" form.  Using the conservative form of the Euler equations is desirable because discontinuities like shocks are admitted automatically as part of the solution with no special action necessary.  The variables making up the vector Q are called the conservative variables.  All solution is done in terms of these three conservative variables, so primitive variables which might be of more interest (things like velocity, pressure, etc.) must be calculated as functions of the conservative variables.  Fortunately, the three conservative variables completely define the one dimensional flow field, and any other flow variable can be calculated form them as shown in the next section.

conservative variable vector
(6)


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