Page 65 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

50 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

can be numerically evaluated by considering, usually, some time steps.

Therefore using this approach one calculates numerically the elements
of U instead of genuine variables v1,v2,v3 and e. Of course once the
numerical values for the U components are determined, the numerical
values for the genuine parameters are immediately obtained by p = p,
4)?
e+ >
_ pei a r) Se

Vi ,e= p . In the case of the inviscid fluids we
2
wil follow the same procedure with the simplification oj; = 0.
In the case of the steady flow, we have ou = Q. That is why for such
problems one frequently uses numerical techniques of marching type.
For instance if the solution of the problem is obtained via a marching

procedure in the direction of the « axis, then our equation could be
written in the form oF = J-— on a7 oH

Here F becomes the “solution” vector while the dependent variables

are pu,, pu? + p, puiv2, pv,v3 and lov (c + ) + pri]. From these
variables it would be possible to get again the genuine variables even
if this time the calculations are more complicated than in the previous
case.

Let us notice now that the generic form considered for our equations
contains only the first order derivatives with respect to x; and ¢ and all
these derivatives are on the left side, which makes it a strong conservative
form. This is in opposition with the previous forms of our equations (for
instance the energy equation) where the spatial coordinates derivatives
could occur on the right side too. That is why these last equations are
considered to be in a weak conservative form.

The strong conservative form is the most used in CFD. To understand
“why’’, it would be sufficient to make an analysis of the fluid flows which
involve some “shock waves’. We will see later, that such flows imply
discontinuities in variable p, p, u;,T etc. Iffor determining of such flows

we would use, for instance, the so-called “shock capturing” method,
the strong conservative form leads to such numerical results that the
corresponding fluid is smooth and stable, while the other forms of these
equations lead to unrealistic oscillations, to an incorrect location of the
discontinuities (the shock) and to unstable solutions. The main reason
for this situation consists in the remark that whereas the “genuine”
variables are discontinuous, the dependent variables like pv and p+ pv?
are continuous across the shock wave (Rankine—Hugoniot relations).