Boundary Layer Fluid Mechanics

8/17/2019 Boundary Layer Fluid Mechanics

1/10

1

Boundary Layers


2/10

2

Boundary Layers• When an infinite expansion of fluid flows over a

stationary solid surface, the viscous force brings the

fluid (in the immediate contact w/ the surface) to therest (to satisfy No-slip condition)

– However, the fluid far away from the solid surface stillmaintains its velocity (stream velocity)

• The gradual change in the fluid velocity (from the

surface velocity to the stream velocity) gives rise to athin layer, known as boundary layer , BL (First

identified by Prandtl)

• Thickness () of the velocity boundary layer defined as

– The distance from the solid body at which the viscous

flow velocity is 99% of the free stream velocity

– v( ) = 0.99 . u0

• Can the boundary layer be neglected?

– No. The consideration of the physics taking place withinthe boundary layer is very crucial to accurately predict

The flow dynamics, e.g. fluid separation

The mass or heat trans ort at the solid/li uid interface

Bulk

BL

solid plate

stream velocity

(x)

v( )

x

vx(y)


3/10

3

• 2-D incompressible fluid flow over a flat plate

• The fluid domain can be divided into 2 areas in order to simplify the solution of N-S Eq

– In the bulk: The inertial term dominates; viscous effect is very minimal for high Re flow in this

domain

Viscous term can be neglected in comparison to the inertial term, without significant loss of accuracy

– In the boundary layer: The viscous term must be accounted in this domain

However, this general N-S eq. can be simplified using non-dimensionality –

– On the boundary layer: Inertial effect ~ Viscous effect

Boundary Layer Equations

Bernoulli’s Equation

(holds along a streamline)

Bulk

BL

solid plate

x

vx(y) y

Inertial term

dominates

Viscous term

dominates


4/10

4

Non-Dimensionalization of Boundary Layer Equations

• Characteristic parameters for non-dimensionalization

– x*, y*, v x* and v y* are the non-dimensional parameters

– L, , u0 and v y,0 are the characteristic parameters

v y,0 and are unknowns.

• Non-Dimensionalization of the continuity equation

– 1

st

term ~ 2nd

term

Bulk

BL

solid plate

x

vx(y) y

The characteristic velocity in the y-direction (v y,0 ) is now determined

After non-dimensionalization of C.E.


5/10

5

Non-Dimensionalization of Boundary Layer Equations

• Navier-Stokes Equation

• N-S equation within the boundary layer has been derived

• The order of magnitude of boundary layer thickness is determined

0 @ steady state 0 @ steady state0

O(~1) O(~1) O(~1) O(?)

O (~1) As Re >> 1O(~1) O(


6/10

6

Summary of the Equation of Motion in Boundary Layer

• Equation of motion inside the boundary layer (Prandtl’s B-L Equations)

Bulk

BL

solid plate

L

vx(y) y

i.e. P will be constant in the y-direction (across the thickness of the boundary layer)

Therefore, pressure gradient ( ) at the edge of the boundary layer (obtained by solving theflow Eq. in the bulk) can be used as the pressure gradient within the boundary layer.

Therefore, P * can be replaced by P bnd in this equation.

P bnd(x)

P solid (x)

Non-linear ODE but can be solved numerically to get the flow velocity in the Bundary layer

For Laminar flow

Numerical solution


7/107

Bulk

BL

solid plate

vx(y)

y=

Solution of Boundary Layer Equation

• Skin friction factor (Based on the numerical solution for the flow velocity)

• Shear stress on the plate

• Total viscous drag force on the plate

L

y

Drag force decreases with increasing Re

x

Width of the plate


8/108

Turbulent Boundary Layers

• As the flow proceeds down the solid surface, Rex and the thickness of the boundary layer ( )

increase

• Beyond a critical Rex, the flow inside the bnd layer becomes un-stable, and soon becomes turbulent

– The critical Rex for the flow over a flat-plate is 5x105 (Note: Re,cr for the flow in a pipe is ~2000)

• Empirical correlations for turbulent boundary layer

x

y

(x)

BL: Viscous force dominates

Bulk: Inertial

force dominates Re,crit

The turbulent boundary layer is thicker than laminar

The drag force is higher for the turbulent flow


9/109

Boundary Flow Separation

dp/dx =0

Flow reversal starts here

(No-shear force at this point)

A steady, incompressible, high Re flow passing over a 2-d circular cylinder

BulkBL

A

BC

D

Bulk(Inviscid approximation @ high Re)

p

A

B

D

BL

Pressure is decreased in the upstream

and increased in the downstream

Bernoulli EqBoundary layer

(Viscosity driven flow)

Pressure gradient (dp/dx) within the boundary layer follows

the same trend as in the bulkHence within the boundary layer, the pressure is

decreased in the upstream and increased in the

downstream

x

y

Bulk

upstream downstream

Favorable Adverse

A positive pressure drop in the downstream (dp/dx > 0, i.e. increasing pressure) causes boundary layerseparation in the downstream Drag force increases.


10/1010

• The onset of flow reversal can be delayed (to decrease the drag force) by

a) Streamlining the objects, e.g. smooth and elongated shape of the aerofoil

Facilitates gradual increase in the pressure drop in the downstream, andhence the flow separation gets delayed

b) Deliberately tripping the boundary layer into turbulent prior to the laminar separation, by roughening the surface or using vortex generators

As the turbulence allows a thicker boundary layer and hence delays the flow separation

It reduces the total drag force

Aerofoil wake

Gradual pressure recovery

Boundary Flow Separation

Documents

Boundary Layer Fluid Mechanics