Page 75 - Computational Fluid Dynamics for Engineers

60 2. Conservation Equations

2.4.1 Inviscid Flow

One simplification of the Navier-Stokes equations assumes all cr-stresses to be
locally negligible, which corresponds to inviscid flow. With the neglect of viscous
forces, Eq. (2.2.10) becomes

DV 1
-£ = —Vp + f (2.4.13)

which is known as the Euler equation. For a steady flow with no body forces,
the Euler equation reduces to

( t ? . V ) y = - ^ (2.4.14)
Q
If we take a dot product of the above equation with a differential length of a
streamline ds, the Euler equations integrate (see Problem 2.4) to give
V 2 [dp
+ / J? = constant (2.4.15)
~2 J Q
2
2
2
where V 2 = u +v +w . For a steady incompressible flow for which g is constant,
the integrated Euler equation, Eq. (2.4.15), becomes
P + ~QV 2 = constant (2.4.16)

which is called the Bernoulli equation. For an isentropic compressible flow [g =
1
(constant)p /^], Eq. (2.4.15) can be written as
V 2 i v
— + — — - = constant (2.4.17)
2 -y-lg
which is known as the compressible Bernoulli equation. We note that the two
forms of the Bernoulli equation are valid only along a given streamline since
the constants appearing in these equations can vary between streamlines. They
can, however, become valid everywhere in the flowfield if the flow is irrotational,
which is defined by zero vorticity
u = V x y = 0 (2.4.18)

This condition implies the existence of a scalar function 0, called the velocity
potential, defined by
V = V0 (2.4.19)

In this case the continuity equation, Eq. (2.2.1), can be combined with Eq.
(2.4.19) to obtain Laplace's equation

2
V 0 = 0 (2.4.20)