Fluid Mechanics

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Fluid Mechanics

• Lecture 6
• The boundary-layer equations

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The need for the boundary-layer model

• While the flow past a streamlined body may be
  well described by the inviscid flow (and even the
  potential flow) equations over almost all the
  flow region, those equations do not satisfy the
  fact that because of finite viscosity of real
  fluids the flow velocity at the wall itself
  must vanish.
• So, we need a flow model that uses the simplest
  possible form of the Navier Stokes equation but
  which does enable the no-slip condition to be
  satisfied.
• Such a model was first developed by Ludwig
  Prandtl in 1904.

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Objectives of this lecture

• To explore the simplification of the Navier
  Stokes equations to obtain the boundary layer
  equations for steady 2D laminar flow.
• To understand the assumptions used in deriving
  these equations.
• To understand the conditions in which the
  boundary-layer equations can be used reliably.

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The governing equations

• Navier-Stokes equations
• We seek to simplify these equations by neglecting
  terms which are less important under particular
  circumstances.
• Key assumptions the thickness of the region
  where viscous effects are significant,d, is very
  thin , i.e. d ltlt L and ReL gtgt1.

Continuity x-momentum y-momentum
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Non-dimensionalized form of N-S Equations

• Non-dimensional-ize equations using V?, a
  constant (approach) velocity, L ,an overall
  dimension i.e.
• U U/ V? VV/V? xx/L yy/L
• P???
• ( A question for you)

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Non-dimensionalized N-S equations
x has a magnitude comparable to L

• Since , .
• Hence we write

x has an order of magnitude of 1.
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Non-dimensionalized N-S equations

• Since and ,
  .
• Hence we write y O(d).
• Also we have

y is at least an order of magnitude smaller than
1.
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Continuity equation
V has to be of order O(d) to satisfy
continuity, i.e..
?No term can be omitted hence the continuity
equation remains as it is, i.e.
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x-momentum equation


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x-momentum equation
To make the above equation valid, we must have
ReL has to be large and x-gradients in the
viscous term can be dropped in comparison with
y-gradients. The dimensional form of the equation
thus becomes
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y-momentum equation
To do an order of magnitude analysis for each
term and estimate the order of magnitude for
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y-momentum equation
? Hence at most

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y-momentum equation
?The pressure can be assumed to be constant
across the boundary layer over a flat plate.
Hence the pressure only varies in the x-direction
and the pressure at the wall is equal to that at
the edge of the layer, i.e.
U?
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Two qualifiers

• If the surface has substantial longitudinal
  curvature (?/R gt0.1)
• it may not be adequate to
• assume constant pressure across boundary layer.
  Then one needs to apply radial equilibrium to
  compute P (see Slide 16)
• In 3D boundary layers (not covered in this course
  but very important in the industrial world) one
  needs to be able to work out the presssure
  variations in the y-z plane (normal to the mean
  flow) to compute the secondary velocities .

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Summary of assumptions

• Basic assumption
• Derived results
• V is small, i.e.
• Re must be large
• and then only velocity gradients normal to the
  wall are significant in the viscous term
• The pressure is constant across the boundary
  layer (for 2D nearly straight) flows, i.e.

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Boundary layer equations

• Since disappears, the equations become
  of parabolic type which can be solved by knowing
  only the inlet and boundary conditions... i.e. no
  feedback from downstream back upstream.
• Unknowns U and V (P? may be assumed known)
• Boundary conditions

Continuity x-momentum
At wall Free stream Inlet
y 0 U V 0 y d U V? ?U? U(x0,y),
V(x0,y)
L
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Boundary layer over a curved surface
Pressure gradient across boundary layer
Assume a linear velocity distribution, i.e.
integrating from y0 to d gives
Hence pressure variations across the boundary
layer are negligible when
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Limitations

• Large Reynolds number, typically Re gt1000
• Boundary-layer approximations inaccurate beyond
  the point of separation.
• The flow becomes turbulent when Re gt 500,000. In
  that case the averaged equations may be
  describable by an adapted for of momentum
  equation to be treated later.
• Applies to boundary layers over surfaces with
  large radius of curvature.