Separated Flows
Wakes and Cavities
Free Streamline Theory
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
s is the cavity
parameter
The pressure
(and then the velocity modulus)
is constant along the separation
streamline
=
The separation streamline is a
free streamline
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
Supercritical flow
Subcritical 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)
(W=0)
Represent the flow in the -plane
and then apply the SC theorem
3.1 Flow over a plate
Show that
-1
+1
3.1 Flow over a plate
(W=0)
Represent the flow in the W-plane
and then in the W1/2 plane
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 !
Improvment of the theory
1. Separation angle is prescribed and k>0
2. Pressure cavity is prescribed to pb
It is a fit of the experimental data !
3.1 Flow over a plate
Work only if the separation position is similar
to that of the theory at pc=p ( i.e. sC=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
Imagine the flow stays stationary as Re∞
free streamline theory solution
(a) Theoretical sketch
(b) and (c) Stationary simulation of NS
Kirchoff helmholtz flow :
A candidate solution of NS as Re ∞ ?
Cpb0
L ~ d Re
Cx0.5
L = O(Re) : infinite length
Limiting stationary solution as Re ∞
Academic case
Numerical
simulation
Cpb>0 !!!
CD0 ?
Limiting stationary solution as Re ∞
Academic case
(a) Theoretical sketch
(b) and (c) Stationary simulation of NS
A possibility :Non uniqueness of
the Solution as Re 
Super cavitating wakes
Kirchoff helmholtz flow ? :
liquid
vapor
Super cavitating wakes