Page 67 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

52 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

Dw gradp 1...
—— = (w- grad) v —w(divv + rot (— + Zaiv{a)
[o]
,
,
pi ~ | ) (div v)
we set [ao] = O, then we get

Dw | ; grad p
5
Det (w+ grad) v — w(divv) + rot ( } ,

For a barotropic fluid (obviously for an incompressible fluid too) be-
cause grad { = gredp and taking into account the equation of conti-

nuity, it turns out that div v = —178, so that we obtain

D [w Gs GJ
— {—]= | —-grad}v = (gradv) —
ni () (5 4 ) (9 MS

Similarly, from

DI gradp 1...
D
Di = / (- + “div(ol) - or
C

we get a = — f meee - dr such that, for a barotropic fluid, we fi-

nally have oes = 0. This result, also known as the Thompson (Lord
Kelvin) theorem, states that the circulation along a simple closed curve,

observed during its motion, is constant whenever the fluid is inviscid
(ideal), barotropic (or incompressible) and the mass (external) forces
are potential.' Correspondingly, in the above conditions, the strength
of a vortex tube is a constant too (Helmholtz).
In the case of the ideal incompressible or barotropic compressible fluid

flows, the vorticity (rotation) equation (obtained by taking the curl of
each term of the Euler equation) could be written as oe + rot (wxv) =0.
On the other hand, if we consider the flux of rotation (vorticity) across
a fluid surface %, that is ® = J { w-ndo, as divw = 0 and the formula

[153],
a= // Se + rot (woxv)| -ndo

holds, we can state the following theorem:

' The Thompson theorem requires, basically, the existence of a uniform potential of accelera-
tions. Somne recent results, which have also taken into consideration the case of nonuniform
potential of accelerations, should be mentioned [122].