Dimensionless Numbers in Compressible Flow

It is helpful to normalize the equations of compressible ideal gas flow, (1.87)-(1.89), in the following manner: $\overline{\nabla} = L\,\nabla$ , $\overline{\bf v} = {\bf v}/V_0$ , $\overline{t} = (V_0/L)\,t$ , $\overline{\rho}=\rho/\rho_0$ , $\overline{{\mit\Psi}} = {\mit\Psi}/(g\,L)$ , $\overline{\chi} = (L/V_0)^2\,\chi$ , and $\overline{p} = (p-p_0)/(\rho_0\,V_0^{\,2} +\rho_0\,g\,L+ \rho_0\,\nu\,V_0/L)$ . Here, $L$ is a typical spatial variation lengthscale, $V_0$ a typical fluid velocity, $\rho_0$ a typical mass density, and $g$ a typical gravitational acceleration (assuming that ${\mit\Psi}$ represents a gravitational potential energy per unit mass). Furthermore, $p_0$ corresponds to atmospheric pressure at ground level, and is a uniform constant. It follows that $\overline{p}$ represents deviations from atmospheric pressure. All barred quantities are dimensionless, and are designed to be comparable with unity. The normalized equations of compressible ideal gas flow take the form

$$\frac{D\overline{\rho}}{D\overline{t}} = -\overline{\rho}\,\overline{\nabla}\cdot\overline{\bf v},$$ (1.102)

$$\frac{D\overline{\bf v}}{D\overline{t}} = - \left(1+\frac{1}{{\rm Fr}^{\,2}}+\frac{1}{{\rm Re}}\right)\fr... ...-\frac{1}{3}\,\overline{\nabla}(\overline{\nabla}\cdot\overline{\bf v})\right],$$ (1.103)

$$\frac{1}{\gamma-1}\left[\frac{D\overline{p}}{D\overline{t}}\right.\!\! -\!\! \left.\gamma\left(\frac{\overline{p}_0+\overline{p}}{\overl... ...bla^{\,2}\!\left( \frac{\overline{p}_0 + \overline{p}}{\overline{\rho}}\right),$$ (1.104)

$$\overline{p}_0 = \frac{1}{\gamma\,{\rm Ma}^{\,2}\,(1+1/{\rm Fr}^{\,2}+1/{\rm Re})},$$ (1.105)

where $D/D\overline{t}\equiv \partial/\partial \overline{t} + \overline{\bf v}\cdot\overline{\nabla}$ ,

$${\rm Re} = \frac{L\,V_0}{\nu},$$ (1.106)

$${\rm Fr} = \frac{V_0}{(g\,L)^{1/2}},$$ (1.107)

$${\rm Pr} = \frac{\nu}{\kappa_H},$$ (1.108)

$${\rm Ma} = \frac{V_0}{\sqrt{\gamma\,p_0/\rho_0}},$$ (1.109)

and

$$\nu =\frac{\mu}{\rho_0},$$ (1.110)

$$\kappa_H = \frac{\kappa\,M}{R\,\rho_0}.$$ (1.111)

Here, the dimensionless numbers ${\rm Re}$ , ${\rm Fr}$ , ${\rm Pr}$ , and ${\rm Ma}$ are known as the Reynolds number, Froude number, Prandtl number, and Mach number, respectively. [The latter two numbers are named after Ludwig Prandtl (1875-1953) and Ernst Mach (1838-1916), respectively.] The Reynolds number is the typical ratio of inertial to viscous forces within the gas, the square of the Froude number is the typical ratio of inertial to gravitational forces, the Prandtl number is the typical ratio of the momentum and thermal diffusion rates, and the Mach number is the typical ratio of the gas flow and sound propagation speeds. Thus, thermal diffusion is far faster than momentum diffusion when ${\rm Pr}\ll 1$ , and vice versa. Moreover, the gas flow is termed subsonic when ${\rm Ma}\ll 1$ , supersonic when ${\rm Ma}\gg 1$ , and transonic when ${\rm Ma}\sim {\cal O}(1)$ . Note that $\sqrt{\gamma\,p_0/\rho_0}$ is the speed of sound in the undisturbed gas (Reif 1965). The quantity $\kappa_H$ is called the thermal diffusivity of the gas, and has units of meters squared per second. Thus, heat typically diffuses through the gas a distance $\sqrt{\kappa_H\,t}$ meters in $t$ seconds. The thermal diffusivity of dry air at atmospheric pressure and $20^\circ\,{\rm C}$ is about $\kappa_H = 2.1\times 10^{-5}\,{\rm m^2\,s^{-1}}$ (Batchelor 2000). It follows that heat diffusion in air is a relatively slow process. The kinematic viscosity of dry air at atmospheric pressure and $20^\circ\,{\rm C}$ is about $\nu= 1.5\times 10^{-5}\,{\rm m^2\,s^{-1}}$ (Batchelor 2000). Hence, momentum diffusion in air is also a relatively slow process.

For the case of dry air at atmospheric pressure and $20^\circ\,{\rm C}$ (Batchelor 2000),

$${\rm Re} \simeq 6.7\times 10^4\,L({\rm m})\,V_0({\rm m\,s^{-1}}),$$ (1.112)

$${\rm Fr} \simeq 3.2\times 10^{-1}\,V_0({\rm m\,s^{-1}})/[L({\rm m})]^{1/2},$$ (1.113)

$${\rm Pr} \simeq 7.2\times 10^{-1},$$ (1.114)

$${\rm Ma} \simeq 2.9\times 10^{-3}\,V_0({\rm m\,s^{-1}}).$$ (1.115)

Thus, if $L\sim 1\,{\rm m}$ and $V_0\sim 1\,{\rm m\,s^{-1}}$ , as is often the case for subsonic air dynamics close to the Earth's surface, then the previous expressions suggest that ${\rm Re}\gg 1$ , ${\rm Ma}\ll 1$ , and ${\rm Fr}, {\rm Pr} \sim {\cal O}(1)$ . It immediately follows from Equation (1.105) that $\overline{p}_0\gg 1$ . However, in this situation, Equation (1.104) is dominated by the second term in square brackets on its left-hand side. Hence, this equation can only be satisfied if the term in question is small, which implies that

$$\frac{D\overline{\rho}}{D\overline{t}} \ll 1.$$ (1.116)

Equation (1.102) then gives

$$\overline{\nabla}\cdot\overline{\bf v} \ll 1.$$ (1.117)

It is evident that subsonic (i.e., ${\rm Ma}\ll 1$ ) gas flow is essentially incompressible. The fact that ${\rm Re}\gg 1$ implies that such flow is also essentially inviscid. In the incompressible inviscid limit (in which $\overline{\nabla}\cdot\overline{\bf v}=0$ and ${\rm Re}\gg 1$ ), the (unnormalized) compressible ideal gas flow equations reduce to the previously derived, inviscid, incompressible, fluid flow equations:

$$\nabla\cdot{\bf v} = 0,$$ (1.118)

$$\frac{D{\bf v}}{Dt} =- \frac{\nabla p}{\rho} - \nabla{\mit\Psi}.$$ (1.119)

It follows that the equations which govern subsonic gas dynamics close to the surface of the Earth are essentially the same as those that govern the flow of water.

Suppose that $L\sim 1\,{\rm m}$ and $V_0\sim 300\,{\rm m\,s^{-1}}$ , as is typically the case for transonic air dynamics (e.g., air flow over the wing of a fighter jet). In this situation, Equations (1.105) and (1.112)-(1.115) yield ${\rm Re}, {\rm Fr} \gg 1$ and ${\rm Ma}, {\rm Pr}, \overline{p}_0\sim {\cal O}(1)$ . It follows that the final two terms on the right-hand sides of Equations (1.103) and (1.104) can be neglected. Thus, the (unnormalized) compressible ideal gas flow equations reduce to the following set of inviscid, adiabatic, ideal gas, flow equations:

$$\frac{D\rho}{Dt} = -\rho\,\nabla\cdot{\bf v},$$ (1.120)

$$\frac{D{\bf v}}{Dt} = -\frac{\nabla p}{\rho},$$ (1.121)

$$\frac{D}{Dt}\!\left(\frac{p}{\rho^{\,\gamma}}\right) =0.$$ (1.122)

In particular, if the initial distribution of $p/\rho^{\,\gamma}$ is uniform in space, as is often the case, then Equation (1.122) ensures that the distribution remains uniform as time progresses. In fact, it can be shown that the entropy per unit mass of an ideal gas is (Reif 1965)

$${\cal S} = \frac{c_V}{M}\ln\left(\frac{p}{\rho^{\,\gamma}}\right).$$ (1.123)

Hence, the assumption that $p/\rho^{\,\gamma}$ is uniform in space is equivalent to the assumption that the entropy per unit mass of the gas is a spatial constant. A gas for which this is the case is termed homentropic. Equation (1.122) ensures that the entropy of a co-moving gas element is a constant of the motion in transonic flow. A gas for which this is the case is termed isentropic. In the homentropic case, the previous compressible gas flow equations simplify somewhat to give

$$\frac{D\rho}{Dt} = -\rho\,\nabla\cdot{\bf v},$$ (1.124)

$$\frac{D{\bf v}}{Dt} = -\frac{\nabla p}{\rho},$$ (1.125)

$$\frac{p}{p_0} = \left(\frac{\rho}{\rho_0}\right)^\gamma.$$ (1.126)

Here, $p_0$ is atmospheric pressure, and $\rho_0$ is the density of air at atmospheric pressure. Equation (1.126) is known as the adiabatic gas law, and is a consequence of the fact that transonic gas dynamics takes place far too quickly for thermal heat conduction (which is a relatively slow process) to have any appreciable effect on the temperature distribution within the gas. Incidentally, a gas in which thermal diffusion is negligible is generally termed adiabatic.