Free Streamline Theory

Separated FlowsWakes and Cavities

Flow approximation

Viscosity is necessary to provoke separation, but if we introduce the separation "by hand", viscosity is not relevant anymore.

Solves the D'Alambert Paradoxe : Drag on bodies with zero viscosity

3.1 Flow over a plate

The pressure (and then the velocity modulus) is constant along the separation

streamline

=

The separation streamline is a free streamline

is the cavity

parameter

3.1 Flow over a plate

3.1 Flow over a plate

Separation has to be smooth otherwise U=0 at separation is not consistent with the velocity on the free stream line

Form of the potential near separation

3.1 Flow over a plate

Cases study with k

3.1 Flow over a plate

Villat condition US=U : the cavity pressure is the lowest

Subcritical flow Supercritical flow

1. Separation angle deduced from Villat condition (k= 0 at separation)

2. Pressure cavity is prescribed to p

3.1 Flow over a plate

Subcritical flow

Supercritical flow

1. Separation angle is prescribed and k>0

2. Pressure cavity is prescribed to p

3.1 Flow over a plate

3.1 Flow over a plate

Flow boundaries in the z-plane (physical space)

Represent the flow in the -planeand then apply the SC theorem

(W=0)

3.1 Flow over a plate

Show that

+1-1

3.1 Flow over a plate

Represent the flow in the W-planeand then in the W1/2 plane

(W=0)

3.1 Flow over a plate

Show that :

+1-1

3.1 Flow over a plate

Correspondance between two half planes gives :

Extract

and show that :

3.1 Flow over a plate

Compute z0 and k = d/(4+) and the shape of the free streamline

3.1 Flow over a plate

From the pressure distribution around the plate, the drag is:

In experiments, CD 2

Similar problem with circular cylinder :CD0=0.5 while in experiments CD 1.2

The pressure in the cavity is not p, but lower !

1. Separation angle is prescribed and k>0

2. Pressure cavity is prescribed to pb

It is a fit of the experimental data !

Improvment of the theory

3.1 Flow over a plate

Work only if the separation position is similar to that of the theory at pc=p ( i.e. C=0, is called the Helmholtz flow that gives CD0)

3.1 Flow over a plate

A cavity cannot close freely in the fluid (if no gravity effect) Closure models

L/d ~ (-Cpb)-n

Limiting of the stationary NS solution as Re ∞

Academic case

L ~ d Re

Imagine the flow stays stationary as Re∞ free streamline theory solution

(b) and (c) Stationary simulation of NS

(a) Theoretical sketch

A candidate solution of NS as Re ∞ ?

Cpb0

Cx0.5

L = O(Re) : infinite length

Kirchoff helmholtz flow :

Limiting stationary solution as Re ∞

Academic case

Cpb>0 !!!

CD0 ?

Numerical simulation

Limiting stationary solution as Re ∞

Academic case

(b) and (c) Stationary simulation of NS

(a) Theoretical sketch

A possibility :Non uniqueness of the Solution as Re

Super cavitating wakes

Kirchoff helmholtz flow ? :

vapor

liquid

Super cavitating wakes