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Supplemental Relations
Apart from the independent variables <math>~t</math> and <math>~\vec{x}</math>, our principal governing equations involve the vector velocity <math>~\vec{v}</math>, and the four scalar variables, <math>~\Phi</math>, <math>~P</math>, <math>~\rho</math>, and <math>~\epsilon</math>. Because the variables outnumber the equations by one, one (additional) supplemental relationship between the physical variables must be specified in order to close the set of equations.
Also, in order to complete the unique specification of a particular physical problem, either a steady-state flow field or initial conditions must be specified, depending on whether one is studying a time-independent (structure) or time-dependent (stability or dynamics) problem, respectively. Throughout this H_Book, the following strategy will be adopted in order to complete the physical specification of each examined system:
- For time-independent problems, we will ...
- adopt a structural relationship between <math>~P</math> and <math>~\rho</math>, and
- specify a steady-state flow-field.
- For time-dependent problems, we will ...
- adopt an equation of state, and
- specify initial conditions.
Time-Dependent Problems
Equation of State
The equation of state that generally will be adopted for time-dependent problems is one that describes an ideal gas. As the accompanying discussion illustrates, the ideal gas equation of state can assume a variety of different forms. Throughout this H_Book, we frequently will use either "Form A" or "Form B" of the ideal gas equation of state, as displayed in the following Table, to supplement the principal governing equation.
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Ideal Gas Equation of State |
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| Form A | Form B | ||
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<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math> |
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"Form A" provides a relationship between the gas temperature <math>~T</math> and the state variables <math>~P</math> and <math>~\rho</math>, where <math>~\Re</math> is the gas constant and <math>\mu</math> is the mean molecular weight of the gas. "Form B" relates the state variables <math>~P</math> and <math>~\rho</math> to <math>~\epsilon</math>, where the ratio of specific heats <math>~\gamma_\mathrm{g}</math> is assumed to be independent of both <math>~\vec{x}</math> and <math>~t</math>. [See Tassoul (1978) — specifically the discussion associated with Chapter 4, Eq. 13 — for a more general statement related to the proper specification of the supplemental, equation of state relationship.]
Initial Conditions
For time-dependent problems, the principal governing equations must be supplemented further through the specification of initial conditions. Frequently throughout this H_Book, we will select as initial conditions a specification of <math>~\rho</math>( <math>~\vec{x}</math>, <math>~t</math> = 0), <math>~P</math>( <math>~\vec{x}</math>, <math>~t</math> = 0), and <math>~\vec{v}</math>( <math>~\vec{x}</math>, <math>~t</math> = 0) that, as a group themselves, define a static or steady-state equilibrium structure. Perturbation or computational fluid dynamic (CFD) techniques can be used to test the stability of such structures.
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