Boundary layer with pressure gradient in flow direction.

1
Boundary layer with pressure gradient in flow
direction.

• Separation Flow induced Vibration
• Unit 5 Potter 8.6.7, 8.2, 8.3.2

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 objects 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

3
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

4
Onset of separation
5
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)
6
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)

7

• Substituting in Eq. (2) boundary condition at
  wall u0, v0 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 gt0) 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

8
Influence of the pressure gradient.
9
Separation

• Separation starts with zero velocity gradient at
  the wall
• Flow reversal takes place beyond separation
  point dP/dx gt0
• 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

10
Smooth ball Rough ball
11
Effect of a wire ring on separation
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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

13
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

14
Separation delays by suction
15
Pressure Velocity change in a converging
diverging duct
16
Boundary layer growth in a nozzle-diffuser
Nozzle Throat Diffuser
Area Decreasing Area Constant Area Increasing
Velocity increasing Velocity Constant Velocity decreasing
Pressure decreases Pressure Constant Pressure increases
Pressure gradient Favourable Pressure gradient Zero Pressure gradient Adverse
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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 q90. Compute the
  theoretical drag coefficient and compare that
  with the experimental value of 1.2This problem
  may show the inadequacy of potential flow theory
  near the surface

18
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
  40ltRelt10,000.
• 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 250ltRelt2x105

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Home work (Potter p-361)

• The velocity of a slow moving air (kinematic
  viscosity1.6x10-5) is to be measured using a 6
  cm cylinder. The velocity range expected to be
  0.1ltU lt1 m/s. Do you expect vortex shedding to
  occur?
• If so, what frequency would be observed by the
  pressure measuring device for U1 m/s.

20
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