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
Figure 8.27 –
Influence of a
strong pressure
gradient on a
turbulent flow:
(a) a strong
negative pressure
gradient may relaminarize 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
•
P1 V12 P2 V22

 
(1)
 2
 2
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
u
u
1 dP   2u
(2)
u v


x
y
 dx  y 2
• Substituting in Eq. (2) boundary condition at wall u=0, v=0 we
get
 2u
1 dP
y
•
•
•
•

2
wall
 dx
(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
Pressure & Velocity change in a
converging diverging duct
Boundary layer growth in a nozzlediffuser
Nozzle
Throat
Diffuser
Area
Area
Decreasing
Velocity
increasing
Constant
Velocity
Constant
Area
Increasing
Pressure
decreases
Pressure
Constant
Pressure gradient
Favourable
Pressure gradient
Zero
Velocity
decreasing
Pressure
increases
Pressure gradient
Adverse
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(119.7/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