Post on 21-Dec-2015
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
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
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
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
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
(b) and (c) Stationary simulation of NS
(a) Theoretical sketch
A possibility :Non uniqueness of the Solution as Re