Mach Number

In an ideal gas, the local Mach number of the flow is defined (see Section 1.17)

$${\rm Ma} = \frac{v}{c},$$ (14.57)

where $c=\sqrt{\gamma\,{\cal R}\,T}$ is the local sound speed. [See Equation (14.45).] Setting $v^{\,2}= {\rm Ma}^{\,2}\,c^{\,2}$ in Equation (14.55), we obtain

$$\frac{T}{T_0}=\left[1+\frac{1}{2}\,(\gamma-1)\,{\rm Ma}^{\,2}\right]^{-1},$$ (14.58)

where use has been made of Equations (14.15) and (14.24), which imply that

$${\cal C}_V =\left(\frac{1}{\gamma-1}\right){\cal R},$$ (14.59)

$${\cal C}_P =\left(\frac{\gamma}{\gamma-1}\right){\cal R}.$$ (14.60)

Incidentally, the relation (14.58) is valid for any streamline, because the stagnation temperature, $T_0$ , can be defined, by means of Equation (14.55), even when the streamline in question does not pass through a stagnation point. We can combine Equation (14.58) with the isentropic relation, $p\,/\rho^{\,\gamma}={\rm constant}$ along a streamline (see Section 14.3), as well as the ideal gas law, (14.1), to give

$$\frac{p}{p_0} =\left(\frac{T}{T_0}\right)^{\gamma/(\gamma-1)}=\left[1+\frac{1}{2}\,(\gamma-1)\,{\rm Ma}^{\,2}\right]^{-\gamma/(\gamma-1)} ,$$ (14.61)

$$\frac{\rho}{\rho_0} =\left(\frac{T}{T_0}\right)^{1/(\gamma-1)}=\left[1+\frac{1}{2}\,(\gamma-1)\,{\rm Ma}^{\,2}\right]^{-1/(\gamma-1)}.$$ (14.62)

Here, $p_0$ and $\rho_0$ are the pressure and density, respectively, at the stagnation point. In principle, the stagnation values, $T_0$ , $p_0$ , and $\rho_0$ , can be different on different streamlines. However, if a solid object moves through a homogeneous ideal gas that is asymptotically at rest then the stagnation parameters become true constants, independent of the streamline. Such flow is said to be homentropic.

A point where the speed of a steadily flowing ideal gas equals the local speed of sound, $v=c$ , is termed a sonic point. The sonic temperature, $T_1$ , pressure, $p_1$ , and density, $\rho_1$ , are simply related to the stagnation values. In fact, setting ${\rm Ma}=1$ in Equations (14.58), (14.61), and (14.62), we obtain

$$\frac{T_1}{T_0} = \left(\frac{2}{\gamma+1}\right),$$ (14.63)

$$\frac{p_1}{p_0} = \left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)},$$ (14.64)

$$\frac{\rho_1}{\rho_0} = \left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)}.$$ (14.65)

Finally, if we combine Equations (14.58), and (14.61)-(14.65), then we get

$$\frac{T}{T_1} =\left[1+\left(\frac{\gamma-1}{\gamma+1}\right)\left({\rm Ma}^{\,2}-1\right)\right]^{-1},$$ (14.66)

$$\frac{p}{p_1} =\left[1+\left(\frac{\gamma-1}{\gamma+1}\right)\left({\rm Ma}^{\,2}-1\right)\right]^{-\gamma/(\gamma-1)},$$ (14.67)

$$\frac{\rho}{\rho_1} =\left[1+\left(\frac{\gamma-1}{\gamma+1}\right)\left({\rm Ma}^{\,2}-1\right)\right]^{-1/(\gamma-1)}.$$ (14.68)