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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 "Form B" of the ideal gas equation of state to supplement the principal governing equations, namely,
<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </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>. Simultaneously, "Form A" of the ideal gas equation of state provides a relationship between the gas temperature T and the state variables <math>~P</math> and <math>~\rho</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.]
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