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9.6.2 Physics of Compressible Flows

Compressible flows are typically characterized by the total pressure $p_0$ and total temperature $T_0$ of the flow. For an ideal gas, these quantities can be related to the static pressure and temperature by the following:


 \frac{p_0}{p} = exp (\frac{\int_{T}^{T_0} \frac{C_p}{T} dT}{R}) (9.6-3)

For constant C $_p$, Equation  9.6-3 reduces to


$\displaystyle \frac{p_0}{p}$ $\textstyle =$ $\displaystyle \left ( 1 + \frac{\gamma-1}{2} {\rm M}^2 \right )^{\gamma/(\gamma - 1)}$ (9.6-4)
$\displaystyle \;$      
$\displaystyle \frac{T_0}{T}$ $\textstyle =$ $\displaystyle 1 + \frac{\gamma - 1}{2} {\rm M}^2$ (9.6-5)

These relationships describe the variation of the static pressure and temperature in the flow as the velocity (Mach number) changes under isentropic conditions. For example, given a pressure ratio from inlet to exit (total to static), Equation  9.6-4 can be used to estimate the exit Mach number which would exist in a one-dimensional isentropic flow. For air, Equation  9.6-4 predicts a choked flow (Mach number of 1.0) at an isentropic pressure ratio, $p/p_0$, of 0.5283. This choked flow condition will be established at the point of minimum flow area (e.g., in the throat of a nozzle). In the subsequent area expansion the flow may either accelerate to a supersonic flow in which the pressure will continue to drop, or return to subsonic flow conditions, decelerating with a pressure rise. If a supersonic flow is exposed to an imposed pressure increase, a shock will occur, with a sudden pressure rise and deceleration accomplished across the shock.



Basic Equations for Compressible Flows


Compressible flows are described by the standard continuity and momentum equations solved by FLUENT, and you do not need to activate any special physical models (other than the compressible treatment of density as detailed below). The energy equation solved by FLUENT correctly incorporates the coupling between the flow velocity and the static temperature, and should be activated whenever you are solving a compressible flow. In addition, if you are using the pressure-based solver, you should activate the viscous dissipation terms in Equation  13.2-1, which become important in high-Mach-number flows.



The Compressible Form of the Gas Law


For compressible flows, the ideal gas law is written in the following form:


 \rho = \frac{p_{\rm op} + p}{\frac{R}{M_w}T} (9.6-6)

where $p_{\rm op}$ is the operating pressure defined in the Operating Conditions panel, $p$ is the local static pressure relative to the operating pressure, $R$ is the universal gas constant, and $M_w$ is the molecular weight. The temperature, $T$, will be computed from the energy equation.


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