Approximate Solutions of Boundary Layer Equations

(8.139) | ||

(8.140) |

subject to the boundary conditions

Furthermore, it follows from Equations (8.140), (8.142), and (8.143) that

The previous expression can be thought of as an alternative form of Equation (8.143). As we saw in Section 8.4, the boundary layer equations can be solved exactly when takes the special form . However, in the general case, we must resort to approximation methods.

Following Pohlhausen (Schlichting 1987), let us assume that

(8.145) |

where , and . In particular, suppose that

where , , , , and are constants. This expression automatically satisfies the boundary condition (8.141). Moreover, the boundary conditions (8.142) and (8.144) imply that , and

where , and

Finally, let us assume that , , and are continuous at : that is,

These constraints corresponds to the reasonable requirements that the velocity, vorticity, and viscous stress tensor, respectively, be continuous across the layer. Given that , Equations (8.146), (8.147), and (8.149)-(8.151) yield

for , where

(8.153) | ||

(8.154) |

Thus, the tangential velocity profile across the layer is a function of a single parameter, , which is termed the

It follows from Equations (8.115), (8.116), and (8.152)-(8.154) that

(8.155) | ||

(8.156) |

Furthermore,

(8.157) |

The von Kármán momentum integral, (8.117), can be rearranged to give

Defining

we obtain

where

It is generally necessary to integrate Equation (8.158) from the stagnation point at the front of the obstacle, through the point of maximum tangential velocity, to the separation point on the back side of the obstacle. At the stagnation point we have and , which implies that . Furthermore, at the point of maximum tangential velocity we have and , which implies that . Finally, as we have already seen, at the separation point, which implies, from Equation (8.161), that .

As was first pointed out by Walz (Schlichting 1987), and is illustrated in Figure 8.14, it is a fairly good approximation to replace by the linear function for in the physically relevant range. The approximation is particularly accurate on the front side of the obstacle (where ). Making use of this approximation, Equations (8.159) and (8.160) reduce to the linear differential equation

(8.165) |

which can be integrated to give

assuming that the stagnation point corresponds to . It follows that

Recall that the separation point corresponds to , where .

Suppose that , which corresponds to uniform flow over a flat plate. (See Section 8.5.) It follows from Equations (8.166) and (8.167) that

(8.168) |

where , and . Moreover, according to Equations (8.148) and (8.162), and . Hence, the displacement width of the boundary layer becomes

(8.169) |

This approximate result compares very favorably with the exact result, (8.73).

Suppose that and , which corresponds to uniform transverse flow around a circular cylinder of radius . (See Section 8.8.) Equation (8.167) yields

(8.170) |

Figure 8.15 shows determined from the previous formula. It can be seen that when . In other words, the separation point is located from the stagnation point at the front of the cylinder. This suggests that the low pressure wake behind the cylinder is almost as wide as the cylinder itself, and that the associated form drag is comparatively large.

Suppose, finally, that . If is negative then, as illustrated in Figure 8.16, this corresponds to uniform flow over the back surface of a semi-infinite wedge whose angle of dip is

(8.171) |

(See Section 8.4.) It follows from Equation (8.167) that

(8.172) |

We expect boundary layer separation on the back surface of the wedge when . This corresponds to , where

(8.173) |

Hence, boundary layer separation can be prevented by making the wedge's angle of dip sufficiently shallow: that is, by streamlining the wedge, which has the effect of reducing the deceleration of the flow on the wedge's back surface. The critical value of (i.e., ) at which separation occurs in our approximate solution is very similar to the critical value of (i.e., ) at which the exact self-similar solutions described in Section 8.4 can no longer be found. This suggests that the absence of self-similar solutions for is related to boundary layer separation.