Knudsen Number and pressure

Question

When computing the Knudsen number to know if the continuum hypothesis can be applied as $\frac{k_B T}{p \sqrt{2} \pi d^2 L}$, do we use the static or total pressure of the free stream? My object is travelling at 7.6 km/s and I don't know if I should include it

Answer 1

It is the static pressure that you will need as Chet Miller correctly pointed out. Wikipedia states it should be the "total pressure" but I assume it is intended to mean the total pressure as opposite of partial pressure and should not mean the stagnation pressure (see the ambiguity for total pressure). As you correctly pointed out the mean free path - at least for the simplified gas kinetics model of solid rigid spherical particles - is completely independent of the body's velocity but that does not mean that the Knudsen number of interest is independent of the body's velocity and thus if the continuum hypothesis can be applied. I'd like to explain this further by first reasoning why the static pressure is indeed the relevant pressure using an ideal gas model and the further elaborate the last point.

The Knudsen number is a dimensionless number and as such is only a simplified concept that can be used to estimate orders of magnitude. In particular the choice of the characteristic length L is somewhat arbitrary. It should be an important dimension that defines the the physical scale of a problem and can be used to determine dynamic similitude. But which characteristic scale should one use?

The Knudsen number is defined as

$$ Kn := \frac{\lambda}{L} \phantom{spacespace} \frac{\text{mean free path}}{\text{representative physical length scale}} \tag{1}\label{1}$$

We can not yet see a connection to pressure therefore let's introduce the quotient of Mach $Ma := U / c_s$ (where $c_s := \sqrt{\left( \frac{\partial p}{\partial \rho} \right)_S } = \sqrt{\gamma R_m T}$ is the speed of sound calculated with the static temperature) and Reynolds number $Re := \frac{U L}{\nu}$

$$\frac{Ma}{Re} = \frac{\mu}{\rho L c_s} \tag{2}\label{2}$$

Let's pop \eqref{2} into \eqref{1} and we will see that

$$ Kn = \frac{Ma}{Re} \frac{\rho c_s \lambda}{\mu}. \tag{3}\label{3}$$

In order to \eqref{3} even further we will have to find some approximation to $\mu$ and $\lambda$ based on kinetic theory of gases. For this purpose one considers the Boltzmann equation with Boltzmann's Stoßzahlansatz. For the simplified model of rigid spherical particles (the easiest case) with mass $m_P$ and diameter $d$ one can find assuming a Maxwell-Boltzmann equilibrium distribution

$$\lambda = \frac{m_P}{\sqrt{2} \pi d^2 \rho} \tag{4}\label{4}$$

where $d$ is the diameter of the spherical particles and similarly for the dynamic viscosity $\mu$

$$\mu = \frac{ 5\sqrt{\pi}}{16} \frac{\sqrt{k_B m_P T}}{\pi d^2} \tag{5}\label{5}$$

Inserting \eqref{4} into \eqref{1} and furthermore using $R_m = \frac{k_B}{m_P}$ yields the formula mentioned by you

$$ Kn = \frac{m_P}{\sqrt{2} \pi d^2 \rho L} = \frac{k_B T}{\sqrt{2} \pi d^2 p L} \tag{6}\label{6}$$

while inserting \eqref{4}, \eqref{5} in \eqref{3} yields

$$ Kn = \underbrace{\frac{16}{\sqrt{2} \,\, 5 \, \sqrt{\pi}}}_{\approx 1.28} \sqrt{\gamma} \frac{Ma}{Re}. \tag{7}\label{7}$$

Similarly one can find the estimate

$$ Kn = \underbrace{\sqrt{\frac{\pi}{2}}}_{\approx 1.25} \sqrt{\gamma} \frac{Ma}{Re} \tag{8}\label{8}$$

in the literature. Large Mach numbers lead to a large mean free path and thus the continuum hypothesis $Kn \to 0$ breaks down for shock waves. The Reynolds number counters these effects. Anyways it seems as due $Kn \propto \frac{Ma}{Re}$ the Knudsen number is independent of the actual flow velocity.

But let's go back to our initial thought. What is the characteristic lengthscale of the problem? Is it really the characteristic length of the spacecraft? The continuum hypothesis can break down in several areas as the flow can be locally rarefied. Dieter Händel suggests that for hypersonic flow such as on a re-entry $Ma = \mathcal{O}(10)$ one should consider the boundary layer thickness $\delta$ as measure for hypersonic flow which scales like $\delta \propto \frac{L}{\sqrt{Re}}$ at least for low Reynolds number flows. Thus

$$Kn_{hyper} = \frac{\lambda}{\delta} \propto \frac{Ma}{\sqrt{Re}} \tag{9}\label{9}$$

holds, which now would depend on the velocity.