Module 5

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Module 5:

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Boundary Layer Equations for

laminar flow over a flat plate

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Blasius Exact Solution

• Blasius performed a transformation technique to change the set of two partial differential equations into a single ordinary differential equation.

• Continuity and Momentum equations

Boundary conditions, y = 0, u = v = 0, y  , u = U.

• The exact solution of boundary layer equations is known as

similarity solution. For the flow over a flat plate, the similarity solution is often referred to as Blasius solution.

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• Blasius transformed the above partial differential equations into single, higher-order, nonlinear, ordinary differential equation.

• Boundary conditions

• where is Blasius similarity variable

• Blasius solved this third order non-linear differentia equation through analytical techniques.

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Turbulent Boundary Layer Over A Flat Plate

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Turbulent Boundary Layer Over A Flat Plate



Generally, the boundary layer may be divided into an internal

zone covering about 0.2δ from the wall and the external zone

outside it.



In the external zone the inertia forces dominate the flow.

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Boundary Layer Separation

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Boundary layer flow with pressure gradient

• In the case of a flat plate we neglect the pressure variation along the flow in a boundary layer.

• This is not valid for boundary layer over curved surface like airfoil

• Owing to object’s shape the free stream velocity just outside the boundary layer varies along the length of the surface.

• As per Bernoulli's 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.

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Boundary Layer Separation

2 2 1 y u dx dp y u v x u u e          

_



dx dp y u _e y  1 0 2 2          

 Momentum Equations to flat plates:

 Close to the wall (y=0) , u=v=0, therefore

 Outside Boundary layer: ₀

2 2    y u

At the body surface, therefore ∂2u/∂y2 must always have the same sign as the pressure gradient dp_e/dx.

 The external pressure gradient can be:

o dp_e/dx=0 <–> U_∞ constant (Paralell outer streamlines):

o dp_e/dx>0 <–> U_∞decreases (Divergent outer streamlines):

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

Zero pressure gradient:

dp_e/dx=0 <–> U₀ constant (Paralell outer streamlines):

y

u

Inflection point at the wall

No separation of boundary layer 0 2 2           y y u 0 0 2 2           y y u

Boundary Layer Separation

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

Favourable pressure gradient:

dp_e/dx<0 <–> U₀ increases (Convergent outer streamlines):

0 2 2           y y u y 0 0 2 2           y y u

Curvature of velocity profile remains constant

No boundary layer separation

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 Adverse pressure gradient:

dp_e/dx>0 <–> U₀ decreases (Divergent outer streamlines):

Boundary Layer Separation

Curvature of velocity profile can change 0 2 2           y y u 0 0 2 2           y y u Boundary layer

Separation can occur

y

P.I.

Separated Boundary Layer

For adverse pressure gradient ( dP/dx >0) 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.

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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.

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Control of Boundary Layer Separation

• The total drag on a body is attributed to form drag and skin friction drag. In some flow configurations, the contribution of form drag becomes significant.

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 There are some popular methods for this purpose which are stated as follows.

By giving the profile of the body a streamlined shape

Streamlining reduces adverse pressure gradient beyond the maximum thickness and delays separation

• This has an elongated shape in the rear part to reduce the magnitude of the pressure gradient.

• The optimum contour for a streamlined body is the one for which the wake zone is very narrow and the form drag is minimum.

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Control of Boundary Layer Separation

 The injection of fluid through porous wall can also control the boundary layer separation.

 This is generally accomplished by blowing high energy fluid particles tangentially from the location where separation would have taken place otherwise.

 The injection of fluid promotes turbulence.

 This increases skin friction. But the form drag is reduced considerably due to suppression of flow separation.

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Control of Boundary Layer Separation

 Fluid particles lose kinetic energy near separation point. So these are either removed by suction or higher energy.