Page 69 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

34 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Now we remark that the above quantity K also satisfies, in the steady

state case, the equation w x v = —gradK and, correspondingly, v
-gradK = O which could be obtained from the Euler equation in the
Helmholtz form, with the same previous assumptions. Consider the
energy equation for an inviscid fluid with no heat change with its sur-
rounding (6g = 0) and with a time-free potential of the external forces,
that is

oP 2 D (p D
Pri (c+ 5) = —di iv (pv) + pgradU d -v = — PD: (2) + pPp,Y

H
or GF = 0, where = 50? +e +4 —U.
The energy equation shows that H = constant on each streamline.
From the expression of H we get, by taking the grad operator and using

the equality grad [ os + pgrad (2) = grad (2), that

grad H = grad (5 v o4 [2 - u) + grade + pgradv,

where v = 5 is the specific volume.
At the same time the first law of thermodynamics written under the
“gradient” form, .e., T grads = grade + pgradv, allows us to write
B
that grad H = T grads + grad or

grad H = T grads — wxv.

The last equality is known as the Crocco—Vazsonyi equation and it

shows that H is constant in the whole domain of the flow provided
that s =constant and w = 0. In other words, for the isentropic steady
potential fluid flows H is constant together with K.
In the absence of the external forces H = ho, where ho is the en-
thalpy at the zero velocity (stagnation) points. In this case the Crocco
— Vazsonyi equation can be written in the simplified form as gradhg =

T grads — wXv.
Generally, the values of the constants taken by K and H along a
certain streamline, in the steady case, are different. But in the case of
isentropic flows (s =constant), the constants for K and H will be the
same.
It has been shown that the modification of these constants while the
streamlines are changing (which does not occur in the case of irrotational

flows) is a direct consequence of the existence of the rotational feature
of the whole fluid flow [153].