Displacement Effects Displacement Effects

of Boundary Layer on Potential Flow

1  Potential Flow Perturbation

Asymptotic analysis of the Viscous Aerodynamic Problem indicates that the boundary layer perturbs the potential flow via the displacement effect by an amount which is O(Re-1/2). This perturbation may be important when high prediction accuracy is required for aerodynamic quantities such as lift and drag. For separated flow the perturbation to the potential flow can be O(1), and accounting for the perturbation is essential.

Two common methods are formualted here to modify the potential flow problem to account for the boundary layer's presence. Each method is constructed so as to match the vertical velocity ve of the actual flow just outside the viscous region.

2  Actual Flow

The actual flow situation is shown in Figure .

Figure 1: Actual flow showing displaced streamlines

The vertical velocity at some location ye is computed in terms of the mass defect gradient via the continuity equation.

 v(x,ye)    º   ve(x)
 =
 óõ ye 0 ¶v ¶y dy
 =
 óõ ye 0 - ¶u ¶x dy
 =
 óõ ye 0 ¶ ¶x ( ue - u )  dy  -  ye due dx
 =
 d dx éê ë ue óõ ye 0 æç è 1 - u ue ö÷ ø dy ùú û - ye due dx
 ve
 =
 d dx ( ue d* )  - ye due dx

3  Displacement Body Model

This model employ the concept of a fictitious displacement body, which is offset from the actual body by a distance D(x).

Figure 2: Flow over fictitious displacement body

The displacement body is assumed to have a purely potential flow tangent to it. The resulting ve in this situation is again computed using the continuity equation.

 ve(x)
 =
 ue d D dx + óõ ye D ¶v ¶y dy
 =
 ue d D dx - óõ ye D ¶u ¶x dy
 =
 ue d D dx -  ( ye - D) due dx
 ve
 =
 d dx ( ue D)  - ye due dx
Requiring that this be equal to the ve for the actual flow, gives
 D(x)    =   d*(x)
so that the necessary offset distance for the displacement body is just the displacement thickness.

4  Wall Blowing Model

This model employs the actual body shape, but defines a fictitious wall blowing distribution vwall(x).

Figure 3: Flow over body with fictitious blowing velocity

The flow is assumed to be purely potential, but it is not tangent to the actual body because of vwall(x) ¹ 0. The resulting ve at some distance above the wall follows from continuity.

 ve(x)
 =
 vwall   + óõ ye 0 ¶v ¶y dy
 =
 vwall   - óõ ye 0 ¶u ¶x dy
 ve
 =
 vwall  - ye due dx
Again requiring that this be equal to the ve for the actual flow, gives
 vwall    = d dx ( ue d* )
The increasing mass defect is effectively canceled by the wall blowing velocity to allow the fictitious flow to remain potential.