Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/14

"As is known from similarity theory and hydrodynamics of an incompressible fluid, the Reynolds number characterizes the relation of inertia and viscosity. The Peclet number plays an analogous role in that it characterizes the relation of molar heat transfer of a flowing fluid and the heat flows transferred by molecular thermal conductivity.

Thus, a high Reynolds number means that one may disregard viscosity in gas dynamics equations. A high Peclet number means that thermal conductivity may be ignored; it means that along the flow line motion takes place virtually adiabatically.

From the molecular-kinetic theory it follows that in gases the ratio of thermal conduction to volume thermal capacity (known as thermal diffusivity) is approximately equal to the viscosity to density ratio (known as kinematic viscosity). For this reason in a gas flow the Reynolds number is quite close to the Peclet number, and both conditions (namely, a high Reynolds number and a high Peclet number) coincide.

Following Karman we can give a different formulation to the condition of a high Reynolds number. We use the molecular expression for the viscosity coefficient

$\eta =\nu \rho ={\frac {1}{2}}\rho c'l,$

(I-9)

where $l$ is the length of the free path of the molecules in the gas, $c'$ is the velocity of molecules, a quantity equal in magnitude to the speed of sound, and <math\nu</math> is kinematic viscosity (cm$^{2}/$sec).

If we substitute the expression for viscosity into the Reynolds number formula, we get

$Re={\frac {U\rho d}{\eta }}={\frac {Ud}{\nu }}=3{\frac {Ud}{c'l}}\approx M{\frac {d}{l}}$

(I-10)

where $d$ is the characteristic size, $U$ is the characteristic velocity of the motion investigated.

The relation between the speed of motion and the speed of sound is known as the Mach number

${\frac {u}{c}}=M.$

(I-11)

In the field of gas dynamics interesting us, where the speed of motion is of the