Fluid Mechanics

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# Fluid Mechanics - PowerPoint PPT Presentation

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

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## Fluid Mechanics

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### Fluid Mechanics

Lecture 6

The boundary-layer equations

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
• 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
• 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

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

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 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

[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 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

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

y-momentum equation

 Hence at most

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
• 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
• 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

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