Page 58 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 43

We will see that if the viscosity coefficients tend to zero, the solution

of a viscous fluid problem does not converge to the solution of the same
problem considered for an inviscid fluid. More precisely, we will establish
that this convergence is non-uniform in an immediate vicinity of the
body surface (where the condition v_ = 0 is also lost) where another
approximation (than that given by the model of inviscid fluid) should
be considered.

Concerning the boundary conditions they should be completed, in
the case of unbounded domains, with a given behaviour at infinity (far
distance) for the flow parameters.
All these features analyzed above are associated with the physical na-
ture of the fluid flow. Within the CFD we must take care to use the
most appropriate and accurate numerical implementation of the bound-
ary conditions, a problem of great interest in CFD. We will return to

this subject later in this book.

3.5 Shock Waves

In a fluid, besides the surfaces (curves) loci of weak discontinuities
there could also occur some strong discontinuities surfaces (curves) or
shock waves where the unknowns themselves have such discontinuities
in passing from one side to the other side of the surface (curve). To de-
termine the relations which connect the limiting values of the unknowns
from each side of the shock wave (the shock relations), we should use

again the general principles but under the integral form which accepts
lower regularity requirements on these unknowns. Once these relations
are established, we will see that if we know the state of the fluid in
front of the wave (the state “OQ”) and the discontinuities displacement
velocity d, it will be always possible to determine the state of the fluid
“behind” the shock wave (the state “1”). We will deal only with the

case of perfect gases where the internal specific energy is e = E (<4)

and the total specific energy is £ pv? + pe, the fluid being considered
in adiabatic (isothermic) evolution. This entails total energy conserva-
tion, a requirement which prevails in the equation of state in the form
_— y
p = kp’.
Now we introduce the concept of “weak” solution which allows the
consideration of unknowns with discontinuities. Let us take, for instance,

'3Tt is shown that the entropy increase, required by the second law of thermodynamics,
associated with a shock raise, does not agree with an equation of state in the form p = kp?
where kis constant.