or,
or,
The first term constitutes the expression from Ackeret's linear theory.
From Linear Theory,
Airfoil lift coefficient:
, independent of thickness and camber. Camber does not cause lift in supersonic
flow.
Note the issues:
Wing aerodynamics in supersonic flow can be analyzed by a combination of:
(a) shock-expansion theory
(b) conical flow
(c) thin airfoil (2-D) linearized theory and second-order theory
(d) singularity distribution
Taylor-Maccoll Equation
(Anderson, Modern Compressible Flow, p. 296-306)
Axisymmetric: 
Properties constant along rays: 
Continuity: 
....................................................................(1)
Isentropic: 
. Then by Crocco's theorem, 
.............(2)
Using (2) in (1), for axisymmetric flow, 
.................................(3)
Euler's equation:
, where 
...........................................(4)
State equation:
..........................................................................(5)
Energy: 
...................................(6)
From (4), (5), (6), 
.....................(7)
Using 
, eq.(7) becomes:
................(8)
This is the Taylor-Maccoll equation.
Ordinary differential equation for 
. Solution : 
, and 
. In terms of 
we get
See Anderson p.301 - for numerical solution procedure.
Supersonic source: 
.
Doublet: 
axis along +z
Vortex: 
, where Q is the strength of the singularity and
hyperbolic radius 
Direct Problems (integrations with known integrands)
Lifting: Given a lifting surface, find pressure distribution on it. Need Kutta condition for subsonic trailing edges.
Nonlifting problems are conveniently solved using source
or doublet distributions: while lifting problems are usually treated using
vortex distributions.
.......................................(33)
where 
.........................................(35)
, we have:
............................................(36)
...................................................(37)
, and hence, 
.Using the convolution theorem,
........................(38)
Consider the transform inversions: Laplace tranform is essentially the same as for the 2-D case:
To perform the Fourier inversion, we write:
Since the integrand is even with respect to 
,
. The integral has been evaluated in Bateman:
for
, and = 0 for 
. Finally,
for 
, and = 0 for 
.......(39)
where 
, 
; 
; 
; 
.
If 
is known everywhere in the region of integration, then (39) is a solution.
However, often 
is unknown over some portion of the region of interest.
Recall that: 
This is generally unknown off the wing.
Exceptions:
(1) Thickness problem: 
everywhere off the wing.
(2) Supersonic Leading Edge:
Certain geometries, above a certain Mach number, will have
undisturbed flow off the wing even in the lifting case. For these supersonic
planforms, 
off the wing as well.
(3) Even in the most general case, there will be no disturbance
to the flow ahead of the rearward facing Mach lines, 
originating at the leading-most point of the lifting surface.
are sufficiently small, i.e., 
, then the regions where 
is unknown, and 
, will vanish. This is called the supersonic planform case.The mathematical problem for these planforms is essentially
the same as for a "thickness problem", whether or not lift is being produced.
This is because these is no communication between the upper and lower surfaces.
However, in the lifting problem, 
changes
sign across z=0.
The Mixed Boundary Condition Problem
Analytical solution is not possible; numerical methods
must be used. The BOX method is coinsidered here.
is approximated by
...............................................(40)
where 
and
are the dimensions of the "aerodynamic box".
= aerodynamic influence coefficients; the velocity potential at point (ij)
due to a unit "downwash" 
at point (kl).
Equation (40) can be written in matrix notation as: 
...............................(41)
The system of linear equations can be separated out as:
.............................................................(42)
where N1 = # of boxes where 
is known, 
is unknown (these are generally on the wing)
N2 = # of boxes where 
is unknown, 
is known (these are generally off the wing)
Using the last N2 equations of (42): 
.......(43)
Solving for 
,
,
In regions where 
,
..............................(44)
Using (44) in the first N1 equations,
, or
...................................(45)
Note: In evaluating the "aerodynamic influence coefficients"
it is essential to account for the singular nature of the integrand along
Mach lines. This requires analytical integration of (39) over each box
with 
assumed constant and taken outside the integral.
"Box method for generalized air loads on an oscillating flexible wing in supersonic flow".
Wing is represented by a grid of square boxes.
Potential Equation (linearized, unsteady, for supersonic flow)
, or
Wing is divided into a lattice of square boxes. Each square
is allowed to oscillate harmonically as a flat plate about it center 
. The vertical deflection is:
Vertical velocity is:
As boxes get smaller, 
and 
get smaller. So
Each box has 2 degrees of freedom: tranaslation and local angle of attack.
For a wing with supersonic leading edge and harmonically oscillating boundary condition on the z=0 plane, [NACA TR872, 194:
, and 
= downwash velocity, and the region S extends over the forward Mach cone
of any receiving point (x,y).The normal velocity is assumed constant over each box, but varies, box to box.
is the area of the box j included in the forward Mach cone from 
. Pressure at (x,y) is given by:
.
, 
Non-dimensionalize with respect to length of the sides of the square box, s.
. Then 
where 
is the pressure influence coefficient.
where 
and 
Pressure influence coefficient is the oscillating
pressure at point 
caused by an oscillating unit normal velocity of the box j. The
pressure influence coefficient is a function of M, k, and the difference
in coordinates between 
and the grid box j. This difference is in numbers. Thus, generalized influence
coefficients can be tabulated.
.
Substitute the downwash:
Work done by aerodynamic forces
. Assume that the pressure is constant over each box: Work done by the
pth mode against the aerodynamic forces of the qth mode:
. This applies for a square which lies completely within the forward Mach
cone of the receiving point. These coefficients are tabulated for given
geometries.
. The Mach line touches the tips of the square boxes. For M<1.414, the
forward Mach cone emanating from the center of a square box cuts across
the two sides as well as the front of the square box" Neighboring boxes
at the same chordwise location will influence one another.