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Fluid Mechanics. Lecture 6 The boundary-layer equations. The need for the boundary-layer model.

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fluid mechanics

Fluid Mechanics

Lecture 6

The boundary-layer equations

the need for the boundary layer model
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.
objectives of this lecture
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.
the governing equations
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,δ, is very thin , i.e. d << L and ReL >>1.

Continuity:

x-momentum:

y-momentum:

non dimensionalized form of n s equations
Non-dimensionalized form of N-S Equations

L

L

  • Non-dimensional-ize equations using V, a constant (approach) velocity, L ,an overall dimension i.e.
  • U*= U/ V; V*=V/V; x*=x/L; y*=y/L
  • P*=???

( A question for you)

non dimensionalized n s equations
Non-dimensionalized N-S equations

L

L

x has a magnitude comparable to L

  • Since , .
  • Hence we write

x* has an order of magnitude of 1.

non dimensionalized n s equations1
Non-dimensionalized N-S equations

L

L

  • Since and , .
  • Hence we writey*= O(d).
  • Also we have

y* is at least an order of magnitude smaller than 1.

continuity equation
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.

x momentum equation1
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:

y momentum equation
y-momentum equation

To do an order of magnitude analysis for each term and estimate the order of magnitude for

y momentum equation1
y-momentum equation

 Hence at most

y momentum equation2
y-momentum equation

L

L

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

two qualifiers
Two qualifiers
  • If the surface has substantial longitudinal curvature (/R >0.1)

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 .
summary of assumptions
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.
boundary layer equations
Boundary layer equations

L

Continuity

x-momentum

  • 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:

At wall :

Free stream:

Inlet:

L

y = 0; U = V = 0

y = d; U = V U

U(x0,y), V(x0,y)

boundary layer over a curved surface
Boundary layer over a curved surface

Pressure gradient across boundary layer:

Assume a linear velocity distribution, i.e. integrating from y=0 to d gives

Hence pressure variations across the boundary layer are negligible when

limitations
Limitations
  • Large Reynolds number, typically Re >1000
  • Boundary-layer approximations inaccurate beyond the point of separation.
  • The flow becomes turbulent when Re > 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.