Page 61 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

46 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

where m = pv (the specific momentum), system which is completed by

the state equation
p= kp”.

But if instead of the equation of state p = kp7 , we consider the energy

equation

at [ery =| = 0,
x
with e = pv? + =P; , then some physical reasons show that the ac-

ceptance of the energy conservation is a much more realistic condition
than p = kp’ , k, in general, depending on entropy and so it cannot be
constant.
In fact the above system together with p = kp? does not have the
same weak solution as the same system but is completed with the energy
conservation.

There are special subjects as, for instance, the wave theory in hydro-
dynamics, where the results obtained by considering the equation of
state p = kp” are close to reality. But, generally speaking, the shock
phenomena should be treated with the system completed with the above
energy equation instead of the equation of state.
From the jump relation [F-n] = 0, across the discontinuity surface

& which moves with velocity d, we get , for any of the equations of the
above system, the jump relations

d(m] = me +r),

d{e] = [(e+p)»],

called the Rankine—Hugoniot jump relations.
If it takes a coordinate system whose displacement with uniform ve-
locity would be, at a moment ¢ = 0 , equal with the displacement velocity
of a discontinuity located at the origin of this system, then within this
new frame of coordinates, the previous relations will be rewritten

Povo = P1V1,

pove + Po = pivi tpi,