Fluid Mechanics for Mechanical Engineers/Boundary Layer Approximation

>back to Chapters of Fluid Mechanics for Mechanical Engineers

Boundary Layer

edit

Boundary layer is a region where viscous force is relatively high compared to inertia, force due to pressure gradient, gravitational or electromagnetic force. Viscous force appears when there is a velocity gradient in the flow. Velocity gradients occurs usually next to the walls, where the fluid takes the velocity of the wall (no-slip condition) and in the mixing regions where flow has high velocity gradients, such as (jet flows).

When we look to the flow around an airfoil in a wind tunnel, we can clearly see the how the velocity gradient occurs on the airfoil.

 
Velocity and pressure field around an airfoil. Effect of wall on the velocity can be seen in the vicinity of the surface.

Boundary Layer Approximation

edit

Prandtl (1905) had the following hypotheses:

  • For small values of viscosity, viscous forces are only important close to the solid boundaries (within boundary layer) where no-slip condition has to be satisfied. And everywhere else they can be neglected.
  • Thickness of boundary layer approaches to zero as the viscosity becomes smaller.

Now consider the figure shown below:

 
Schematic representation of flow over an airfoil

As it is shown in the figure we are using a body-fitted coordinate system which means x direction is always tangent to the surface of the body and y direction is always normal to the surface of the body. Note that edge velocity is shown by   and the boundary layer thickness is represented by   which means they are varying along x direction. Also let   be the average boundary layer thickness along the surface of length  .

A measure for   can be obtained by an order of magnitude analysis using momentum equation in x direction.

 

Note that Cartesian form of the equation is only valid when  , where   is the local radius of curvature of the body. Let   be the characteristic magnitude of  . In other words  .   is the distance along which   changes appreciably, and it can be the length of the body. An order of magnitude analysis shows that  . Hence the first advection term in the momentum equation can be written as:

 
The second advection term is then:
 

where   is the characteristic velocity normal to the wall in the boundary layer, which occurs due to fluid displacement caused by the walls. Doing the same analysis for the viscous terms in momentum equation would give us:

  and  

for which  .

Within the boundary layer inertia (advection terms) and the viscous forces should be in the same order. In other words

 
 
 

As the final equation shows, the higher the   the thinner would be the boundary layer. Now, we can simplify the conservation equations within the boundary layer. The basic idea is to assume that the gradients across the boundary layer is much larger than the gradients along layer in the main flow direction.

 

Looking at continuity equation one can find out an estimate for   by order magnitude analysis.

 

Since   , so  

We know that   but at the same time we know that  . Hence we can say that both terms can have the same order of magnitude.In addition to that the continuity equation dictates also that both partial derivative terms should have the same value with opposite sign. Hence,

 
  since  
 
Finally, we should find an estimation for the pressure gradient. Experimental data at high   numbers show that force due to pressure gradient is in the order of inertia.
 
Moreover, it is also true that in the boundary layer pressure difference w.r.t. to ambient pressure scales with the dynamic pressure
 

Hence the order of magnitude for each term in the governing equations are as follows:

 

 

Both equations can be non-dimensionalized by   and  , respectively. Denoting the non-dimensional parameters with a *, the non-dimensional form of governing equations becomes:

 

 

Note that, since  , the second viscous term   and therefore its order of magnitude becomes 1.

For   and   and   our non-dimensional conservation equations would simplify to

 
 
 

Those are non-dimensional boundary layer equations. Going back to the dimensional variables

 
 
 

These equations are parabolic although the original Navier-Stokes equation is elliptic. Note that the second equation only says that the pressure is not dependent to y direction and it is only a function of  . Therefore we have 3 unknowns and 2 equations. Moreover, pressure at boundary layer is equal to the pressure at the edge which can be found from solving Euler equations for the outer portion of the fluid. In other words, pressure gradient along   direction can be found by using Euler equations along the edge of the boundary layer.

 

The boundary conditions of the problem are as follows

 
 
 
 

Solution of this problem would be in an iterative manner in which one should guess the thickness of the boundary layer and then solve the Euler equation to find the pressure distribution over length  , then by finding the pressure distribution one can find the velocity distribution within boundary layer and by having velocity distribution one can find the thickness of the boundary layer to compare it with the guessed one. If they match we have reached to the final solution if not the new thickness should be taken as next guess and the procedure would repeat the same way as before.


Different Measures of Boundary Layer Thickness

edit

There are 3 different definitions for boundary layer listed below:

1.   where  . Based on this definition the boundary layer is going to be defined where the velocity of the fluid is 0.99 times of the edge velocity (upstream velocity).
2. Displacement thickness ( ):The name of this definition comes from the displacement of the streamlines at the presence of walls contacting with the fluid.
 
 
 

Solving for  

 

As  

There are some practical usages of using  

  • Design of ducts and wind tunnels
    1. Assume frictionless flow and dimension the tunnel
    2. Estimate   along the tunnel
    3. Calculate  
    4. Enlarge the duct by   to obtain the same mass flow rate
  • Finding   at the edge of the boundary layer
    1. Neglect boundary layer (frictionless flow assumption) and calculate   over the body surface
    2. Solve boundary layer by  
    3. Calculate  
    4. Displace the body by  
    5. Calculate   again by frictionless flow assumption
    6. Repeat steps from 2 to 5 until a convergence criterion is satisfied
3. Momentum thickness ( ):

It is defined such that   is the momentum loss due to presence of boundary layer. Hence,

 
where
 
 
 

From which

 

For  

 

Flat Plate Boundary Layer

edit

Flat plate is a special case where the edge velocity is constant ( ). Hence, as a consequence of Euler equation   along the flat plate. The boundary layer equations reduces to

 
 

Boundary conditions are as follows:

 
 
 
 

Now we have two equations and two unknowns which means that our system of equations is closed and can be solved. One way to solve it is through using the similarity solution. We consider the non-dimensional velocity profile as a function of one variable ( ).

  where  

Hence,

 

Also based on our earlier findings ( ) we conclude that  

Since the problem is two dimensional, it is easier to work with stream function ( ). It is a function whose isolines represents the streamlines and they are perpendicular to the velocity potential ( ). So

 

Note that, inserting those equalities into the continuity equation will result in a zero sum. In other words, stream function satisfies the continuity equation per definition.

At   location   represents the flow rate. Using the similarity variable   would give us

 

Non-dimensional form or the similarity form of   function is

 

After inserting the velocities written in terms of stream function into the momentum equation, momentum equations would become

 

Each term can be written as a function of  . From here onwards, for simplicity   and   are used


Since  

 

Note that to find   we should use chain rule in differentiation.

  where  
using the quotient rule of differentiation:
 

Combining derived relations for   and   we would obtain

 

Similarly

 

where   and

 

since  

 

Combining derived relations for   and we would obtain

 

Using this relation, the relationships for   and   can be written as:

  ,
 

  should be derived also:

 

 

 

Using the equalities found already   and   and the relation  

 


Finally substituting partial derivatives of   into the momentum equation would give us

 

It shows that if we are looking for a x-independent solution for the similarity variable   then

 

Note that if we choose that constant as   then  

That way our equation would become

 

Noting that  ,   and  

Then the boundary conditions are as follows

 
  (since   at the wall)

In order to solve the above differential equation, one need to know  , which is not known. However, instead

  is known.

Therefore, it is possible to solve this equation by shooting method. Briefly, with an initial guess for   the equation will be solved and the condition   can be be checked. Depending on the result  

will be modified iteratively in the further steps till solution satisfies  .

The exact solution delivers:

 

There are approximate solutions for the laminar velocity profile, such as:

 

Utilizing this relation and the integral form of mass and momentum equations, one can show:

 

For turbulent flow, the following power law can be used for the non-dimensional velocity profile: