Free Streamline Theory Separated Flows Wakes and Cavities
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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
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