The Stream Function by 6rlEDhL


									Ghosh - 550                                Page 1                                 6/22/2012

                           Boundary Layer Theory

In the context of the course Transport Phenomena we will discuss the velocity, thermal
and concentration boundary layers. For chemical engineers, the concentration boundary
layers are also of importance. However for mechanical engineers the emphasis is more on
the former two. All three kinds of boundary layers are very similar in terms of their
physical characteristics. Studying one in depth would almost guarantee the understanding
of the other two. Therefore we shall focus mostly on the velocity boundary layer and
introduce all the relevant topics in complete details. Later the other two types of
boundary layers will be discussed in relation to their similarities with the velocity
boundary layer. Since this area is the focal point of all of transport theory, we shall
introduce some physical concepts first and follow up with the mathematical rigor.

                          The Boundary Layer Concept
Let us consider a thermal analogy. We shall conduct a thought experiment. Imagine that
you are interested in recording a fluid property such as temperature in a flow field. We
heat up a metal sphere in an oven and probe the temperature in the fluid around it using a

If there is no flow around the body we may record the same fluid temperature at points
equally distant from the body. However the temperature will drop as we move further
away (as shown)

                                           T = T1
                                                            T0 > T1 > T2 > T3
                                      To       T = T2        here

                                             T = T3

In other words, if we allow the natural convection to occur, the temperature field
T(x,y,z) may be equally felt at all points equidistant from the origin, following the
symmetrical propagation of temperature as shown above.

Now let us place a fan in the flow and introduce a little flow from left towards right.
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                                                        Temperature decreases

              V  0
              But small

                               T3 T2 T1   To              Wake Region

In this case, you would agree that the points to the left of the sphere would record
lower temperatures than the region behind the body (called the wake region) where
the fluid particles would be hotter than the particles ahead of the body. Also, due to
symmetry, the constant temperature lines would be symmetric around the top and bottom
regions. If you extend this thought and work with a very high-speed flow, the
temperature increase in the flow field may be confined in a very narrow region near
the body (because the fluid does not flow in the direction perpendicular to the body).
The picture would appear as:

                          High                      A

                                               To            Wake


For points A & B marked in the figure you will hardly record any increase in
temperature due to the hot object because the temperature field will quickly
disappear in the directions perpendicular to the flow (resulting in ambient
temperatures at those points). However, higher temperatures would still be felt in the
wake region. This shows that for high-speed flows, a fluid property such as
temperature can be confined in a very narrow region, called the thermal boundary
layer. For velocity and concentration boundary layers, similar analogies hold for
the fluid velocity and concentration properties. For example, fluid rotation (or,
vorticity) that occur due to the no slip condition at a solid wall will deteriorate
rapidly as we move away from the wall in the direction perpendicular to the flow,
confining the viscous effects in a very narrow region around the body called the
velocity boundary layer.
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Prandtl was the first to derive the equations that hold inside the boundary layers. Before
we introduce them, let us review some flow equations first.

                    Non-Dimensional Fluid Flow Equations
Let us non-dimensionalize the continuity and x-momentum equation in incompressible
fluid flows:
                                u v w
Continuity Equation:                   0
                                x y z
                                   u   u u  u 
                                      u v  w 
                                   t   x y  z 
x-momentum equation:
                                            2u  2u  2u 
                                         2  2  2 
                                             x
                                      x          y   z  

Note that the body force from x was dropped for discussions.

                                x       y      z         t
                       x        , y  , z  , t  
                                L       L      L        L
                                                         
        Let:                                            U 
                                u        v       w          p
                           u   , v   , w  , p  
                                U       U       U          U 2

In the above expressions, the primed quantities represent non-dimensional variables,
which are all expressed as the non-primed variable over a characteristic variable. For
example, if an object such as an airfoil is placed in a free stream U, the chord of the
airfoil may be chosen as the characteristic length, and the characteristic velocity = U.



At this moment, there is no characteristic time or pressure variable. Thus those
values will be derived by the use of L,  and U. For example,    will have units of
Ghosh - 550                               Page 4                                  6/22/2012

time. Similarly U2 is twice the dynamic pressure of the free stream. We choose these
as the characteristic time and pressure. Using the above variables, the equations may be
written as:

                        u v w
         Continuity:              0
                        x y z

                        u       u       u       u 
                              u       v       w
                        t       x       y       z 
                           p   1   2 u  2 u  2 u 
                                                     
                           x  Re L  x  2 y 2 z  2 
                                                          
Similarly the y and z momentum equations may be written. Thus the continuity and
momentum equations may be written in vector form as:
                  V   0                                                (A)
                DV               1       
                         p        2V                              (B)
                 Dt              Re L

                       ˆ  ˆ  ˆ
                      i      j      k
                     x  y z 
                D                              
where:                    u       v      w
                Dt  t       x       v      z 
                       2      2       2
                 
                                    
                      x  2 y 2 z  2

and,            Re L           = Reynolds Number (based upon length, L)
It is interesting to note that when all terms are non-dimensionalized, we recover a
non-dimensional number such as Reynolds Number. Let us quickly remind
ourselves that the Reynolds number is a ratio of inertia and viscous forces in a fluid
flow. This non-dimensional quantity is extremely important for understanding fluid
flows. When flow speeds are very low (or, alternatively the viscosity is very high) we get
the case of ReL  0. This flow region is called the Stokes Flow region. We may extend
the Stokes Flow applications to engineering problems for ReL < 10 at most. (Typically
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ReL < 1). In this flow range, fluid viscous effects are so large that fluid inertia forces
are negligible compared to the pressure and viscous forces. As the ReL increases
from this range, we get the Oseen’s flow range where ReL  1. Again for engineering
approximations, we can extend Oseen’s flow all the way up to about ReL 100. In the
Oseen’s range, all fluid forces are equally important. Thus inertia force is a result
of external pressure and viscous forces on an object. However, as the flow ReL  ,
which is the regime of boundary layer flows, equation (B) above implies that viscous
force term is negligibly small. This means in the high-speed flows, the viscous forces
are negligibly small compared to the pressure and inertia forces. This is a true
statement for the majority of the fluid flows outside the boundary layer. However, this
was the root cause of the D’Alembert’s Paradox also. For centuries, people knew that
dropping the viscous term in the boundary layer analysis yielded absurd results.
However, there was no solution (other than Euler’s equation for potential flows) for fluid
flow equations inside the boundary layers. Prandtl first resolved this difficulty.


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