POTENTIAL FLOW

Objective:

Get a method to describe flow velocity fields and relate them to surface shapes consistently. Once the velocity field is known, the pressures and hence the loads can be calculated.
 

Strategy:

Describe the flowfield as the effect of variations in one scalar quantity.
 

Consider the vector identity:

, i.e., The curl of the gradient of a scalar function is zero.

Thus, if we have a vector relation of the form

it should be possible to express the vector u as the gradient of the scalar function f.

We defined "vorticity" as

. This is a measure of rotation in the flow; in fact the vorticity is half the angular velocity.

Thus,  implies "irrotational flow".

Thus if we have irrotational flow, the vector u can describe the gradient of some scalar function f. This function is called the "velocity potential. Thus,

or,  ;  ; 
 

Unsteady Bernoulli Equation

From Euler's equation,

Use the vector identity:

If the flow is irrotational,

. With this, the Euler equation above becomes

From the definition of a barotropic fluid,

, so that 

or 

 

Substituting in the Euler equation,

This is the unsteady Bernoulli equation.
 

 

If we replace f by  , the velocity field is not modified. Also, if

then 

POTENTIAL EQUATION

Continuity equation in differential form is:
, or,

Use the definition of the velocity potential:  , and the definition

The continuity equation reduces to:


 

Special Case of Constant-Density Flow:

If the density r is constant, this reduces to: . This is the Laplace equation.

Note:

1. The Laplace equation describes unsteady potential flow is r is constant. In most low-speed aerodynamics problems, assuming incompressible flow also ensures constant density.

2. This does NOT mean that the flowfield solution is the same for an airfoil at a degrees steady angle of attack in steady flow, and an airfoil passing through a degrees during an unsteady maneuver or during a change in flow conditions.

General case of compressible flow

Differentiate the unsteady Bernoulli equation with respect to time:

For barotropic flow,

This can be written as 

For the case of isentropic flow,

Substituting,

Momentum equation for barotropic flow is:

Multiply by

Substituting, we get the full potential equation:

Notes:

1. Incorporates the continuity and momentum equations for barotropic flow.

2. One scalar equation for f replaces 4 scalar equations for u,v,w and r.
 

Boundary Conditions:

1. Disturbances due to the body cannot grow as distance from the body increases.

Wake goes to infinity in potential flow, and waves go to infinity in supersonic flow, since there is no viscosity to dissipate them.

2. The flow at the body surface must follow the body motion.

A 3-dimensional, deforming body shape can be described by

B(x,y,z,t) = 0

The the surface boundary condition is

or, 

Note: For steady flow, this reduces to 

Describing the surface by B(x,y,z,t) = z-za(x,y,z,t)=0, we see that

, and