Boundary Layer Equations

Suppose that the equations of irrotational flow have already been solved to determine the fluid velocity outside the boundary layer. This velocity must be such that its normal component is zero at the outer edge of the layer (i.e., at ). On the other hand, the tangential component of the fluid velocity at the outer edge of the layer, (say), is generally non-zero. Here, we are assuming, for the sake of simplicity, that there is no spatial variation in the -direction, so that both the irrotational flow and the boundary layer are effectively two-dimensional. Likewise, we are also assuming that all flows are steady, so that any time variation can be neglected. The motion of the fluid within the boundary layer is governed by the equations of steady-state, incompressible, two-dimensional, viscous flow, which take the form (see Section 1.14)

where is the (constant) density, and the kinematic viscosity. Here, Equation (8.1) is the equation of continuity, whereas Equations (8.2) and (8.3) are the - and -components of the fluid equation of motion, respectively. The boundary conditions at the outer edge of the layer, where it interfaces with the irrotational fluid, are

(8.4) | ||

(8.5) |

as . Here, is the fluid pressure at the outer edge of the layer, and

(because , and viscosity is negligible, just outside the layer). The boundary conditions at the inner edge of the layer, where it interfaces with the impenetrable surface, are

(8.7) | ||

(8.8) |

Of course, the first of these constraints corresponds to the no slip condition.

Let be a typical value of the external tangential velocity, , and let be the typical variation length-scale of this quantity. It is reasonable to suppose that and are also the characteristic tangential flow velocity and variation length-scale in the -direction, respectively, of the boundary layer. Of course, is the typical variation length-scale of the layer in the -direction. Moreover, , because the layer is assumed to be thin. It is helpful to define the normalized variables

(8.9) | ||

(8.10) | ||

(8.11) | ||

(8.12) | ||

(8.13) |

where and are constants. All of these variables are designed to be inside the layer. Equation (8.1) yields

(8.14) |

In order for the terms in this equation to balance one another, we need

(8.15) |

In other words, within the layer, continuity requires the typical flow velocity in the -direction, , to be much smaller than that in the -direction, .

Equation (8.2) gives

(8.16) |

In order for the pressure term on the right-hand side of the previous equation to be of similar magnitude to the advective terms on the left-hand side, we require that

(8.17) |

Furthermore, in order for the viscous term on the right-hand side to balance the other terms, we need

where

(8.19) |

is the Reynolds number of the flow external to the layer. (See Section 1.16.) The assumption that can be seen to imply that . In other words, the normal thickness of the boundary layer separating an irrotational flow pattern from a rigid surface is only much less than the typical variation length-scale of the pattern when the Reynolds number of the flow is much greater than unity.

Equation (8.3) yields

(8.20) |

In the limit , this reduces to

(8.21) |

Hence, , where

, and use has been made of Equation (8.6). In other words, the pressure is uniform across the layer, in the direction normal to the surface of the obstacle, and is thus the same as that on the outer edge of the layer.

Retaining only terms, our final set of normalized layer equations becomes

subject to the boundary conditions

(8.25) |

and

(8.26) | ||

(8.27) |

In unnormalized form, the previous set of layer equations are written

subject to the boundary conditions

(8.30) |

(note that really means ), and

(8.31) | ||

(8.32) |

Equation (8.28) can be automatically satisfied by expressing the flow velocity in terms of a stream function: that is,

(8.33) | ||

(8.34) |

In this case, Equation (8.29) reduces to

subject to the boundary conditions

and

(8.37) | ||

(8.38) |

To lowest order, the vorticity internal to the layer,

whereas the -component of the viscous force per unit area acting on the surface of the obstacle is written (see Section 1.18)