Self-Similar Boundary Layers

Let us search for a self-similar solution to Equation (8.35) of the general form

where

This implies that , and . Here, and are constants. Moreover, has dimensions of velocity, whereas , , and , are dimensionless. Transforming variables from , to , , we find that

(8.43) | ||

(8.44) |

Hence,

(8.45) | ||

(8.46) | ||

(8.47) | ||

(8.48) | ||

(8.49) |

where . Thus, Equation (8.35) becomes

(8.50) |

Because the left-hand side of the previous equation is a (non-constant) function of , while the right-hand side is a function of (and as and are independent variables), the equation can only be satisfied if its right-hand side takes a constant value. In fact, if

(8.51) |

then

(which is consistent with our initial guess), and

where

(8.54) |

Expression (8.53) is known as the

and

(The final condition corresponds to the requirement that the vorticity tend to zero at the edge of the layer.) Note, from Equations (8.39), (8.42), (8.47), (8.52), (8.55), and (8.56), that the normally integrated vorticity within the boundary layer is

(8.58) |

Furthermore, from Equations (8.40), (8.47), and (8.52), the -component of the viscous force per unit area acting on the surface of the obstacle is

It is convenient to parameterize this quantity in terms of a

(8.60) |

It follows that

(8.61) |

where

(8.62) |

is the effective Reynolds number of the flow on the outer edge of the layer at position . Hence, . Finally, according to Equation (8.41), the width of the boundary layer is approximately

(8.63) |

which implies that .

If then the external tangential velocity profile, , corresponds to that of irrotational inviscid flow incident, in a symmetric fashion, on a semi-infinite wedge whose apex subtends an angle , where . (See Section 5.10, and Figure 5.10.) In this case, can be interpreted as the tangential velocity a distance along the surface of the wedge from its apex (in the direction of the flow). By analogy, if then the external velocity profile corresponds to that of irrotational inviscid flow parallel to a semi-infinite flat plate (which can be thought of as a wedge whose apex subtends zero angle). In this case, can be interpreted as the tangential velocity a distance along the surface of the plate from its leading edge (in the direction of the flow). (See Section 8.5.) Finally, if then the external velocity profile is that of symmetric irrotational inviscid flow over the back surface of a semi-infinite wedge whose apex subtends an angle , where . (See Section 5.11, and Figure 5.11.) In this case, can be interpreted as the tangential velocity a distance along the surface of the wedge from its apex (in the direction of the flow).

Unfortunately, the Falkner-Skan equation, (8.53), possesses no general analytic solutions. However, this equation is relatively straightforward to solve via numerical methods.
Figure 8.2 shows
, calculated numerically as a function of
, for the solutions of Equation (8.53) that
satisfy the boundary conditions (8.55)-(8.57). In addition, Figure 8.3 shows
versus
, calculated numerically
for various different values of
.
Because
as
, solutions of the Falkner-Skan equation with
have no physical significance. For
, it can be seen, from Figures 8.2 and 8.3, that there is a single
solution branch characterized by
and
. This branch is termed the *forward flow* branch, because it
is such that the tangential velocity,
, is in the same direction as the
external tangential velocity [i.e.,
] across the whole layer (i.e.,
).
The forward flow branch is characterized by a positive skin friction coefficient,
.
It can also be seen that for
there exists a second solution branch, which is termed the *reversed flow*
branch, because it is such that the tangential velocity is in the opposite direction to the
external tangential velocity in the region of the layer immediately adjacent to the surface of the obstacle (which corresponds to
).
The reversed flow branch is characterized by a negative skin friction coefficient. The reversed flow
solutions are probably unphysical, because reversed flow close to the wall is generally associated
with a phenomenon known as boundary layer
separation (see Section 8.10) that invalidates the boundary layer orderings.
It can be seen that the two
solution branches merge together at
,
which corresponds to
. Moreover, there are no solutions to the Falkner-Skan equation with
or
.
The disappearance of solutions when
becomes too negative (i.e., when the deceleration of the
external flow becomes too large) is also related to boundary layer separation.