Boundary layer with pressure gradient in flow direction.  

Separation & Flow induced Vibration

Unit # 5: Potter 8.6.7, 8.2, 8.3.2

Boundary layer flow with pressure gradient 

So far we neglected the pressure variation along the flow in a boundary layer

This is not valid for boundary layer over curved surface like airfoil

Owing to object’s shape the free stream velocity just outside the boundary layer varies along the length of the surface.

As per Bernoullis equation, the static pressure on the surface of the object, therefore, varies in x- direction along the surface.

There is no pressure variation in the y- direction within the boundary layer. Hence pressure in boundary layer is equal to that just outside it.

As this pressure just outside of a boundary layer varies along x axis that inside the boundary layer also varies along x axis

Separation 

In a situation where pressure increases down stream the fluid particles can move up against it  by virtue of its kinetic energy. 

Inside the boundary layer the velocity in a layer could reduce so much that the kinetic energy of the fluid particles is no longer adequate to move the particles against the pressure gradient. 

This leads to flow reversal. 

Since the fluid layer higher up still have energy to mover forward a rolling of fluid streams occurs, which is called separation

Onset of separation 

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Figure 8.27 – Influence of a strong pressure gradient on a turbulent flow: (a) a strong negative pressure gradient may re-laminarize a flow; (b) a strong positive pressure gradient causes a strong boundary layer top thicken.  (Photograph by R.E. Falco)

Bernoullis equation 

It is valid just outside Boundary Layer, where between two points (1,2) on the flow stream

       (1) 
   
Since pressure in the boundary layer is same on y axis and that just outside, the expression for pressure gradient along x is also valid inside the boundary layer.

Navier Stokes eq. is valid inside boundary layer. Eq. (8.6.45) from Potter we have 
         (2)

Substituting in Eq. (2) boundary condition at wall u=0, v=0 we get 
 
 
        (3) 

It is valid for both laminar & turbulent flows as very near the wall both flows are laminar

From the above expression we see that when pressure decreases second derivative of velocity is negative. So the velocity initially increases fast and then gently blend with the free stream velocity U

For adverse pressure gradient ( dP/dx >0) second derivative is positive at wall but must be negative at the top of boundary layer to match with U. Thus it must pass through a point of inflexion.

Separation occurs when the velocity gradient is zero at the wall and shear stress at wall is zero

Influence of the pressure gradient.

Separation  

Separation starts with zero velocity gradient at the wall  

Flow reversal takes place beyond separation point 
dP/dx >0

Adverse pressure gradient is necessary for separation 

There is no pressure change after separation So, pressure in the separated region is constant.

Fluid in turbulent boundary layer has appreciably more momentum than the flow of a laminar B.L. Thus a turbulent B.L can penetrate further into an adverse pressure gradient without separation

Smooth ball   Rough ball

Effect of a wire ring on separation

Effect of separation 

There is an increase in drag as a result of separation as it prevents pressure recovery

There is low pressure in separated region and it persists in the entire region.

Turbulent eddies formed due to separation can not convert their rotational energy back into pressure head. So there is no pressure recovery (increase).  The difference between high pressure at the front and low pressure at rear increases the drag.

This increase in drag overshadows any increase in lift due to increase in the angle of attack

Control of separation 

Streamlining reduces adverse pressure gradient beyond the maximum thickness and delays separation

Fluid particles lose kinetic energy near separation point. So these are either removed by suction or higher energy  
 
 

High energy fluid is blown near separation point

Roughening surface to force early transition to turbulent boundary layer

Separation delays by suction 

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Pressure & Velocity change in a converging diverging duct

Boundary layer growth in a nozzle-diffuser 

Area

Increasing 

Area

Constant  

Area

Decreasing 

Pressure gradient

Adverse 

Pressure gradient

Zero 

Pressure gradient

Favourable 

Pressure increases 

Pressure Constant  

Pressure decreases 

Velocity

decreasing 

Velocity Constant  

Velocity

increasing 

Diffuser 

Throat 

Nozzle

Problem (White-7.63) 

Assume that the front surface velocity on an infinitely long cylinder is given by potential theory , V = 2Usinq from which the surface pressure is computed by Bernoullis equation. In the separated flow on the rear, the pressure is assumed equal to its value at  q=90. Compute the theoretical drag coefficient and compare that with the  experimental value of 1.2 
[This problem may show the inadequacy of potential flow theory near the surface]

Flow induced vibration 
(Von Karman Vortex) 

Vortices are created on both sides of a symmetric blunt object.

However the vortices are not created simultaneously on both ends. So this leads to alternate shedding of vortices in the flow range 40<Re<10,000.

This induces a vibration, which if matched with the natural frequency of the object may be disastrous.

The frequency f is related to Strouhl number St = fD/U, where D is diameter and U is velocity. St = 0.198(1-19.7/Re) for 250<Re<2x105

Home work (Potter p-361) 

The velocity of a slow moving air (kinematic viscosity=1.6x10-5) is to be measured using a  6 cm cylinder. The velocity range expected to be 0.1<U <1 m/s. Do you expect vortex shedding to occur?

If so, what frequency would be observed by the pressure measuring device for U=1 m/s.

Drag on airfoil 

Separation is reduced by slightly bending the leading edge.

By giving air foil shape to the plate drag is further reduced

But further tilting brings back the separation