Page 307 - Computational Fluid Dynamics for Engineers

10.2 Shock Jump Relations 297

reversible, that is expansion waves are unphysical as they would violate the
second law of thermodynamics, which states that entropy must increase.
Physical shock waves are continuous across a very narrow viscous region,
and the numerical shock wave computed with an inviscid model is represented
as a discontinuous jump. Continuous flow solutions are referred as "genuine"
solutions, whereas discontinuous flow solutions are referred as "weak" solutions.
Inviscid shock jump relations can be derived for each set of compressible gov-
erning equations. For the TSD and full-potential equations, they are referred as
the isentropic jump relations whereas, in the Euler equations, they are called
the Rankine-Hugoniot relations after their founders. The isentropic relations
imply that the entropy rise through the shock wave is assumed negligible. This
has direct consequences on the TSD and full-potential solutions, as they allow
incorrect expansion shocks as their weak solutions. This is clearly inadmissible
and means of correcting for this deficiency have been devised over the years.
The shock jump relations can be derived starting from a generic conservation
equation for steady state
dE OF
0 (10.2.1)
dx dy
where E and F are the fluxes. For the TSD equation, they are [Eq. (P2.6.2)]

7 + l C 2
E=(l-M* Q)<p x (10.2.2a)

F = <p y (10.2.2b)
For the full-potential equation, they are [Eqs. (2.1.12b) and (2.3.19)]

(10.2.3a)
Q<Pz
F = gcp y (10.2.3b)
where in both cases, V = V</?
For the Euler equations, the fluxes are [Eq. (2.2.30)]

gu gv
gu 2 + p guv
E and F (10.2.4)
guv gv 2 + p
+ P) + P)\
lu(E t v{E t
The conservation equation (10.2.1) can be written in divergence form

V - F = 0 (10.2.5)

Applying the divergence theorem, as in a finite volume representation of the
discretized equation (10.2.5) over the discontinuous control volume representing
a shock wave (Fig. 10.1), one obtains