Freestream has velocity components along the u,v and w directions, in general. However, for the steady case we can always transform coordinates so that the lateral component v is zero, so lets assume this for simplicity. The angle of attack can be defined as:
.........(1)Boundary Conditions are:
......................................(2)
.........................................(3)
Approach
Find a singularity distribution satisfying (3).
Find the velocity vector q at each point of interest.
Find the pressure from the Bernoulli equation.
Represent f' as:
. The wing surface is z = h(x,y) or, the equation to the surface can be
written as:
. Here the surface normal is 
on the upper surface, and is -n on the lower surface for a thin flat wing.
from (3).
. The unknown becomes f,
satisfying 
.
The boundary condition becomes
, applied on z = h.
Use the small-perturbation assumption:
; so that 
; 
; 
Assumes thin wing, small perturbation, small angle of attack. These become invalid near the leading edge.
Using
; and 
, the boundary condition becomes:
Also, 
. This is the planar
wing approximation.
We distribute singularities in some manner to represent the configuration
and its wake. Then we compute the velocity 
induced at each point 
on the surface as:
The quantity within the integral on the lhs is the induced
velocity at 
due to the singularity distribution
over the volume dx0dy0dz0. Lets call this quanity K. Thus the general integral
equation is:
The types of singularities used vary from one "computor"
to another.
Example: Vortex Sheet Distribution
A particularly useful singularity distribution is a vortex sheet distribution. The sheet might consist of unknown distributions of vorticity aligned along two axes, i.e., gx(x,y) and gy(x,y), placed on the wing's projected area at the z=0 plane as shown below.
The integral equation then would be:
Subject to:
(Kutta condition) and 
(Kelvin's theorem).
Note: Biot-Savart Law is: 
3. Proper accounting of the wake.
Problem Formulation
Inertial coordinates: (X,Y,Z)
Body-fixed coordinates: (x,y,z); origin at 
at orientation Q(t) relative to the inertial coordinates.
Example of 
for constant
velocity
is: 
Velocity potential in the inertial frame of reference obeys the Laplace equation:
which describes the full potential,
which is also the perturbation due to the body.
Zero-normal-velocity boundary condition
in (X,Y,Z) coordinates, where 
is the surface velocity and 
is the normal
vector as viewed from the inertial reference frame.
Note: Time dependence is introduced through this boundary
condition. The normal vector varies with time.
Far-field boundary condition
where 
By Kelvin's theorem, angular momentum cannot change in
the potential flow region. Thus the circulation G around a contour enclosing
the wing and its wake is conserved. 
for
any t.
Transform to body-fixed coordinates:
Kinematic velocity of the surface: 
where 
, 
is the position vector 
in the body coordinates. 
is the rate of rotation of the body frame of reference.
If additional relative motion occurs within the (x,y,z) system due to flap deflection, for example,
and 
Transforming derivatives
The continuity equation is invariant to coordinate transformation:
in (x,y,z) coordinates.
The boundary condition in the body frame of reference is: 
i.e., 
If there is non-zero velocity 
along the surface, the r.h.s. above would be 
.
The surface is defined in the body frame of reference as
, or 
because fluid velocity is 
in the (X,Y,Z)
system.
. In the body-fixed reference
system, 
; 
. Hence,
. Replacing 
with a 3-D relative motion 
,
The integral equation says that the velocity induced at a surface point by the singularity distribution, integrated over the wing and wake, must satisfy the surface normal velocity condition. The surface boundary condition is often specified in one of two ways:
a) Neumann: surface normal velocity is zero.
b) Dirichlet: potential inside the body is constant;
surface is a streamline. This is useful for steady 2-D flow over
thick bodies.
For unique solutions, we need some physical constraints:
Wake strength: 2-D Kuta condition at trailing edge: 
. If the trailing edge has a finite angle, it is sufficient to assume that
the wake leaves the trailing edge at a median angle 
.
Kelvin condition: 
. This is used to calculate change in wake circulation.
Wake Shape: Force-free wake requires 
according to the Kutta-Joukowsky theorem, or, the velocity is parallel
to the wake circulation vector.
Computation of Pressure
in X,Y,Z coordinates.
Converting to (x,y,z),
In the case of 3-D panel methods, it is simpler to use
. Here Q and p are the local
flow speed and pressure, vref is the magnitude of the kinematic velocity:
.
Conclusions
1. For incompressible flow, the instantaneous solution is independent of time derivatives.
Speed of sound is assumed infinite: influence of a momentary boundary condition is immediately radiated across the whole flowfield.
Hence steady-state techniques can be used to treat the
time-dependent problem by substituting theinstantaneous boundary condition
at each instant. Wake shape, however, does depend on the history of the
motion, hence an appropriate vortex wake model has to be developed. For
lifting problems, the wake separationline must be prescribed. The Kutta
condition is assumed valid for reduced frequencies <1. i.e., 
. Along the trailing edges, 
The body geometry is built up using surface panel elements. Panels may be flat, with uniform singularity strength distributed over them. The panel need coincide, and be tangential to, the actual body surface only at the control point (or collocation point) of each panel. The wake is also represented by panels.
Kinematics:
The source strengths are used to represent the local kinematic velocity.
(see p.428 of K&P),
where the solid surface boundary condition is:
The Kutta condition of zero vorticity at the trailing edge relates the latest wake panel directly to the wing's unknown doublets. Once shed, the wake convects with strength unchanged.
Choice of singularities:
Constant-strength source + doublet distribution on each panel. Note that a constant-strength doublet panel = vortex loop around the panel. In the wake, the panels are vortex loops.
The Dirichlet boundary condition is applied at a surface point on a thick body as follows:
At the collocation point of the ith panel, 
at each time t,
where 
is normal velocity
induced at ith collocation point by unit doublet distribution on panel
k;
is the normal velocity at i
due to unit strength at wake panel panel l,
is the normal velocity at i
due unit source strength at k.
are doublet, vortex and source
strengths, respectively.
Time-Stepping
Start at t=0. During each step, the strength of the latest
wake row is computed using the Kutta condition, and previously shed wake
vortex srengths remain unchanged. At time 
,
, where
if no wake is shed from this panel, or
if it is shedding a wake panel.
The source strengths 
are known from kinematic
velocity.
This system gives N equations for the N values of 
at time t = 
. For time t>
,
Note that 
does not include
the latest wake row: the influence of the latest wake row is included in 
. At each time step, the 2nd and 3rd terms are known, and can be moved
to the right hand side.
So we are to solve a set of linear equations at each time step:
. If the body geometry is not
changing with time, it is enough to compute 
only once.
Thus, 
Computation of velocity, pressures and loads
Perturbation velocity on panel surface:
; 
; 
where l and m are the local tangential
coordinates.
Example: Perturbation velocity in l direction is:
Total velocity at collocation point k = kinematic + perturbation.
;
Aerodynamic Load due to an element of area 
is:
Vortex Wake Rollup
A Flow chart of the computation procedure is given below:
