Page 63 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

48 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

i.e., the shock “separates” the characteristics. These characteristics are

oriented (both of them) towards the “past”, 1e., to the initial data line
= tg or towards the “future” 1.e., towards larger t.
A shock is said to obey the entropy condition if the two characteristics
which cross at each point of it are oriented backwards to the initial line
t = tg. A shock which does not observe the entropy condition is called a
rarefaction shock. In gas dynamics the rarefaction shocks are excluded

because if such shock exists, the (weak) solutions of the problem will
not be unique and, more, such a solution does not depend continuously
on the initial data (the characteristics cannot be “traced back” to the
initial line) and the basic thermodynamic principles are violated.
We shall allow only shocks which do obey the entropy condition. This
restriction will make the (weak) solution of the problem unique.

A shock is called compressive if the pressure behind the shock is
greater than the pressure in front of the shock.
One shows that for a fluid with an equation of state under the form
p = kpv” (or, more generally, whose total energy is conserved while the
specific energy is given by e = 5 pv" + =F), the fulfilment of the entropy
condition holds if and only if the shock is compressive.
It has been proved that, for a perfect gas, the so-called Weyl hypothe-

ses are satisfied, which means

Op 0p Op
Bp <9 Gra > 9 Be > 0

Then, besides the fact that the knowledge of the values of the flow pa-
rameters before the shock together with the shock displacement velocity
allows the determination of the flow parameters behind the shock, the
following properties across the shock take place:
1) There is an entropy increase which is of order 3 in 79 — 7; Or in

P1 — Po;
2) The pressure and the specific mass increase such that the shock is
compressive (pj > po and 7; < 79);
3) The normal component of the fluid velocity vs. the shock front is
supersonic before the shock, becoming subsonic after shock. Further, the
fluid flow before the shock will obviously be supersonic while after shock
it will be subsonic, the shock waves arising only within the supersonic
flows.
One can show that the Weyl hypotheses are satisfied by other gases

too.