Relationships for Small Mach Number

Even with today's computers a simplified method can reduce the tedious work involved in computational work. In particular, the trends can be examined with analytical methods. It further will be used in the book to examine trends in derived models. It can be noticed that the Mach number involved in the above equations is in a square power. Hence, if an acceptable error is of about %1 then $M<0.1$ provides the desired range. Further, if a higher power is used, much smaller error results. First it can be noticed that the ratio of temperature to stagnation temperature, $T \over T_0$ is provided in power series. Expanding of the equations according to the binomial expansion of

$$(1+ x)^n = 1 + n x + {n(n-1) x^2\over 2!} + {n(n-1)(n-2) x^3\over 3!} + \cdots$$ (4.16)

will result in the same fashion

$${P_0\over P} = 1 + {(k -1) M^2 \over 4} + {k M^4\over 8} + {2(2-k)M^6 \over 48} \cdots$$ (4.17)

$${\rho_0\over \rho} = 1 + {(k -1) M^2 \over 4} + {k M^4\over 8} + {2(2-k)M^6 \over 48} \cdots$$ (4.18)

The pressure difference normalized by the velocity (kinetic energy) as correction factor is

$${P_0 -P \over \half \rho U^2} = 1 + \overbrace{{ M^2 \over 4} + {(2-k) M^4\over 24} + \cdots}^{compressibility\; correction}$$ (4.19)

From the above equation, it can be observed that the correction factor approaches zero when $M\longrightarrow 0$ and then equation (4.19) approaches the standard equation for incompressible flow.

The definition of the star Mach is ratio of the velocity and star speed of sound at M=1.

$$M^{*} = {U \over c^{*} } = \sqrt{k+1 \over 2} M \left( 1 - {k -1 \over 4} M^2 + \cdots \right)$$ (4.20)

$${P_0 -P \over P} = {kM^2 \over 2} \left( 1 + {M^2 \over 4} + \cdots \right)$$ (4.21)

$${\rho_0 -\rho \over \rho} = {M^2 \over 2} \left( 1 - {kM^2 \over 4} + \cdots \right)$$ (4.22)

The normalized mass rate becomes

$${\dot{m} \over A} = \sqrt{k {P_0}^2 M^2 \over RT_0} \left( 1 + {k-1 \over 4}M^2 + \cdots \right)$$ (4.23)

The ratio of the area to star area is

$${A \over A^{*}} = \left(2 \over k +1 \right)^{k +1 \over 2 (k-1)} \left( {1\over M} + {k+1 \over 4}M + {(3-k) (k+1)\over 32 } M^3 + \cdots \right)$$ (4.24)