Boundary Layer

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  • Topic: Boundary layer, Viscosity, Aerodynamics
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  • Published : March 29, 2013
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BOUNDARY LAYER THEORY
INTRODUCTION


The concept of boundary layer was 1st introduced by L.Prandtl in 1904.

Figure 7-1. Viscous flow around airfoil



A structure having a shape
that provides lift,
propulsion, stability, or
directional control in a
flying object.

Boundary layer is formed whenever there is a relative motion between the boundary and the fluid.
Boundary layer thickness:
1. Standard thickness
- signified by ,” it is define as the distance from the boundary layer of solid body measured in y-direction to the point where the velocity of the fluid is approximately equal to 0.99 times the free stream (U) velocity of fluid”.

- Because the boundary layer thickness is defined in terms of the velocity distribution, it is sometimes called the velocity thickness or the velocity boundary layer thickness. Figure 7-2 illustrates the boundary layer thickness.

Figure 7-2 Standard boundary layer(http://aerojockey.com/papers/bl/node2.html) 2. Displacement thickness: “it is the distance measured perpendicular to the boundary where the free/main stream is displaced on relation of formation of boundary layer”. 


0

 *   1 

u
dy
U

It represents the amount of that the thickness of the body must be increased so that the invented uniform inviscid flow of the model has the same flow rate properties as the actual viscous flow.

Figure 7-3 Displacement thickness
3. Momentum thickness: “it is the distance measured perpendicular to the boundary of the solid body where the boundary should be displaced to compensate for reduction in momentum of the flowing fluid on relation of formation of boundary layer”. 

u u
1  dy
U U
0

 

**See Munson,2010 textbook for Prandtl/Blasius Boundary layer solution**

Example 1:
The velocity distribution in the boundary layer at a plate is given by,

u 3 y 
y2 

  0.5 2 

U 2    


Calculate the ratio of displacement thickness to standard boundary layer thickness ( */) and ratio of momentum thickness to standard boundary layer thickness (/). Solution:
For (*/):




3 y 

y2 

0









 *   1  
  2     0.5  2  dy




3 y2
y3 
3


 y 
 0.5 2        
2 2
4
6
3d  0 


And for (/):


 3 y  

y 2  

3 y 

y2 


 


   
 2     0.5  2  1   2     0.5  2  dy 

  

0 


 3 y 11 y 2 3 y 3 1 y 4 
 



 dy
2 4 2 23 44
0
11
3
1  19
 19
3
       
 
12
8
20  120
 120
4

Solve the following question:
The velocity distribution in the boundary layer at a plate is given by, u
 y  y
 2    
U
   

Calculate the following:
1. Displacement thickness (ans: /3)
2. Momentum thickness (ans: 2/15)
Solution:

2

MOMENTUM EQUATION FOR BOUNDARY LAYER
 From the Munson textbook (page 478) it is stated from the momentum integral equation for boundary layer flow on a flat plate, the wall/plate shear stress become: d
 w  U 2
dx or also known as 0
0
d
 Whereby,
is known as Von Karman momentum equation,

U 2 dx
This eq. can be used to find the frictional drag on smooth plate for laminar and turbulent boundary layer.
 For local coefficient of drag or coefficient of skin friction , C * D 

 As for average coefficient of drag, C D 

w
1
U 2
2

FD

1
AU 2
2
where, force because of drag is FD = shear stress (w) x area of surface or plate (A)  For a small distance, dx of the plate, FD = 0 x (B x dx) [B = width of plate] so, total drag force on the plate of length, L, become;

L

FD   FD   0  B  dx
0

(in Munson, the symbol of FD = Ð)
 The following boundary conditions must be satisfied for any assumed velocity distribution: du
y  0, u  0,
 finite value
1. At the surface of the plate:
dy
2. At the outer edge of...
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