Boundary layer with pressure gradient in flow direction. - PowerPoint PPT Presentation

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Boundary layer with pressure gradient in flow direction.

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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 ... – PowerPoint PPT presentation

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Title: 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
12
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
17
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

19
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
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