Crocco’s Theorem

Consider two-dimensional, steady, inviscid flow in natural coordinates (l, n) such that l is along the streamline direction and n is perpendicular to the direction of the streamline. The advantage of using natural coordinate system – a coordinate system in which one coordinate is along the streamline direction and other normal to it – is that the flow velocity is always along the streamline direction and the velocity normal to streamline is zero.

Though this is a two-dimensional flow, we can apply one-dimensional analysis, by considering the portion between the two streamlines 1 and 2 (as shown in Figure 9.3) as a streamtube and taking the third dimension to be to.

Let us consider unit width in the third direction, for the present study. For this flow, the equation of continuity is:

The /-momentum equation can also be expressed as: dV dp pV— = —-. f dl dl The n-momentum equation is: dV = 0. But there will be centrifugal force acting in the n-direction. Therefore: pV2 _ dp R dn

(9.27)

(9.28)

The energy equation is:

V2 h + — = h0.

(9.29)

The relation between the entropy and enthalpy can be expressed as [1]:

Differentiation of Equation (9.29) gives dh + VdV = dh0. Therefore, the entropy equation becomes:

Tds = — V dV + — ) + dh0. p

This equation can be split as follows: (i)

ds! dV 1 dp dl dl p dl I

ds f dV 1 dp dh0 dn у dn p dn J dn Introducing dp/dl from Equation (9.27) and dp/dn from Equation (9.28) into the above two equations, we get:

ds T — = 0 dl ds (dV V dh0 T — = – V—————– + —-. dn V dn R ) dn

But, (R — = z is the vorticity of the flow. Therefore:

This is known as Crocco’s theorem for two-dimensional flows. From this, it is seen that the rotation depends on the rate of change of entropy and stagnation enthalpy normal to the streamlines.

Crocco’s theorem essentially relates entropy gradients to vorticity, in steady, frictionless, nonconduct­ing, adiabatic flows. In this form, Crocco’s equation shows that if entropy (s) is a constant, the vorticity (Z) must be zero. Likewise, if vorticity Z is zero, the entropy gradient in the direction normal to the streamline (ds/dn) must be zero, implying that the entropy (s) is a constant. That is, isentropic flows are irrotational andirrotational flows are isentropic. This result is true, in general, only for steady flows of inviscid fluids in which there are no body forces acting and the stagnation enthalpy is a constant.

From Equation (9.30a) it is seen that the entropy does not change along a streamline. Also, Equation (9.30Z>) shows how entropy varies normal to the streamlines.

The circulation is:

By Stokes theorem, the vorticity Z is given by:

(9.33)

dV^_ 9Vy dy dz J dVx_dVz dz dx J dVy _dVx dx dy J

where Zx, Zy, Zz are the vorticity components. The two conditions that are necessary for a frictionless flow to be isentropic throughout are:

1. h0 = constant, throughout the flow.

2. Z = 0, throughout the flow.

From Equation (9.33), Z = 0 for irrotational flow. That is, if a frictionless flow is to be isentropic, the total enthalpy should be constant throughout and the flow should be irrotational.

It is usual to write Equation (9.33) as follows:

Z = (V x V)

i j к

_ d d d

dx dy dz

( V V V (

Vx Vy Vz

When Z = 0

Since h0 = constant, T0 = constant (perfect gas). For this type of flow, we can show that:

T ds RT0 dp 0

V dn V p0 dn

From Equation (9.34), it is seen that in an irrotational flow (that is, with Z = 0), stagnation pressure does not change normal to the streamlines. Even when there is a shock in the flow field, p0 changes along the streamlines at the shock, but does not change normal to the streamlines.

Let h0 = constant (isoenergic flow). Then Equation (9.31) can be written in vector form as:

T grad s + V x curl V = grad h0	(9.35a)

where grad s stands for increase of entropy s in the n-direction. For a steady, inviscid and isoenergic flow:

T grad s + V x curl V = 0

V x curl V =-Tgrads. (9.35b)

If s = constant, V x curl V = 0. This implies that (a) the flow is irrotational, that is, curl V = 0, or (b) V is parallel to curl V.

Irrotational flow

For irrotational flows (curl V = 0), a potential function ф exists such that:

(9.36)

On expanding Equation (9.36), we have:

дф дф

iVx + jVy + kVz = ідф + jJL +

dx dy

Therefore, the velocity components are given by:

дф	V= дф y dy’
dx

The advantage of introducing ф is that the three unknowns Vx, Vy and Vz in a general three-dimensional flow are reduced to a single unknown ф. With ф, the irrotationality conditions defined by Equation (9.33) may be expressed as follows:

дVz

Zx = IT дy

д дф д

д^ д^ J дz

Also, the incompressible continuity equation v • V = 0 becomes:

д2ф д2 ф д2ф dx2 + dy2 + dz2

or

v2ф = 0 •

This is Laplace’s equation. With the introduction of ф, the three equations of motion can be replaced, at least for incompressible flow, by one Laplace equation, which is a linear equation.

9.6.1 Basic Solutions of Laplace’s Equation

We know from our basic studies on fluid flows [2] that:

1. For uniform flow (towards positive x-direction), the potential function is:

ф = V» x.

2. For a source of strength Q, the potential function is:

, Q ,

ф = — ln r.

2n

3. For a doublet of strength p. (issuing in negative x-direction), the potential function is:

p. cos в

ф = ——– •

r

4. For a potential (free) vortex (counterclockwise) with circulation Г, the potential function is:

Г

ф = ^в.

2п