Page 62 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 47

(€9 + po) vg = (ey +1) v1,
where the subscripts identify the state “O” before the jump and the state
“1” after the jump. Ifm = povo = p11 = 0, the respective discontinuity

is of contact type because vg = vy = O show that these discontinuities
move with the fluid.
If m # 0 the discontinuity will be called a shock wave or, shorter,
a Shock. As v9 # 0, v3 #0, the fluid is passing through shock or,
equivalently, the shock is moving through fluid.
That part of the gas (fluid) which does not cross the shock is called
the shock front (the state “O’”) while the part after the shock is called
the back of the shock (the state “1”’).

From the Rankine—Hugoniot relations we could get simple algebraic
relations which allow the determination of the parameters after shock
(state “1”) by using their values before shock (state “O”’’).
If co = ve and cy = yo are the sound speed in front and, respec-

tively, behind the shock, then denoting by Mp = 2 and M,; = “a
(vf and v} being the projections of the fluid velocity on the shock nor-

mal, at the origin of the system) and by 79 = a5 and 7; = a , we easily
get the relations

T™1 — To 2 1
5-1},
=

Pi-~Po _2y_
—
1),
= (Mé
Po y+1
which determine 7; and p; with the data before the shock.
Analogously, we have

je
1— M? =

ee
and from the perfect gases law p = pRT' we obtain for the new” tem-
perature 7; the evaluation

npr, _ 2y—1) (yMB +1) (MB 1)
T, = To ——
ToPo (y +1) Mg

relation which, together with the above ones, solves completely the pro-
posed problem.
In what follows we will see what type of conditions should be imposed
to ensure the uniqueness of the (weak) physically correct solution.
It is easy to check that through every point of a shock in the (z,t)

plane one can draw two characteristics, one of each side of the shock,