LINEARIZATION OF THE POTENTIAL EQUATION

The assumptions made in linearization are:
1. The time derivatives of the velocity components are all of the same order of magnitude.
2. The spatial derivatives are all of the same order of magnitude.
The full potential equation can be written as

Define perturbation quantities:

;  ; 

Small-perturbation assumption:

Assume that  and that the disturbance motions of the body are not too rapid; that is, the time rates of change are not very large.

Surface Boundary Condition

Assume that body slopes are small.
;  so that 
This leads to the linearized body surface boundary condition: 
For lifting surfaces, a more explicit form is:

Bu(x,y,z,t) = z -zu(x,y,z,t)=0

Bl(x,y,z,t) = z -zl(x,y,z,t) = 0

Planar Wing Approximation

Apply the boundary condition at z = 0, instead of at z = za.
at z =0+, and  at z = 0-
Assuming that the time and space derivatives are of the same order of magnitude,
Substituting in the potential equation,

We can replace a by its value in the undisturbed freestream without incurring 1st order errors.


 
 

Acceleration Potential

The Euler equation is
Thus the acceleration vector is the gradient of a scalar quantity which will be defined as the Acceleration Potential Y, such that
with components  ,  , 
Thus  .

Thus, Y also satisfies the linearized potential equation


 
 

Linearized Pressure Coefficient

assuming that 

Under this condition, of course, the compressible case reduces to the incompressible case. The density change due to velocity change is negligibly small.
 
 

Summary of restrictions in the linearized potential equation

1. No body forces

2. Viscous forces are negligible (regions of shear and rotation will be modeled using vortices and vortex sheets as needed).

3. Barotropic (includes isentropic flow)

4. Small body slopes.

5. Small perturbations in all flow parameters.

6. Changes with time are not too rapid.

Linearization is not valid for transonic or hypersonic flows.
 

Summary of Linearized Potential Equation Forms

Reference: Pierce, G.A., "Potential Flow Solutions". Course Notes, given in AE6030/6031.
 

The general form of th elinearized potential flow equation in stationary coordinates with uniform freestream velocity along the x-axis is:

a. Steady Incompressible Flow : we get the Laplace equation

The expression for a steady source is:

b. Unsteady incompressible flow: We still get the Laplace equation.

Note that the f here is f(x,y,z,t).

Assume that the potential can be factored into a space-dependent part and a time-dependent part. i.e., f(x,y,z,t) = g(x,y,z)h(t).

The Laplace equation reduces to:

The solution for a source is: 

c. Steady subsonic flow: 

Use the transformation:  ;  ;  ;  . This gives:

. The expression for a steady source is:

d. Unsteady Subsonic Flow:

Use a Lorentz transformation:  ;  ;  ;  ;  ;

The linearized potential equation is:

Using separation of variables as before, assume that the potential can be factored into a function of space and a function of time. 
 
 

this reduces to two ordinary differential equations:

The solution for a source is: 

where  , and 
 
 

e. Unsteady Supersonic Flow

Use the modified Lorentz transformation:

;  ;  ;  ; 

The linearized potential equation becomes:

Separating variables,  where 

Transforming back to the original coordinates,

where