Problem Statement The aerodynamics weak interaction theory potential flow assumption u º -Ñf significantly simplifies the Navier-Stokes equation system. Substituting this definition into DM yields DM: L(f) = -Ñ2f = 0, on W Ì Ân
f(xb) = 0 on ¶Wo The GWSh process leads to the template pseudo-code GWSN Þ GWSh = Se{WSe} º 0
{WS}e = ( ) ( ) { } (-1) [M2KK]{PHI} + ( ) (
) { } (1) [M200] {
For quasi-one dimensional potential flow in a duct of variable cross-sectional area A(x), DM takes the form
DM: L(f)
=
The GWSh
process produces the template pseudo-code {WS}e = ( ) ( ) {AREA} (-1) [A3011] {PHI} + ( ) ( ) {AREA} (-1) [A3101] {PHI} + (-1) ( ) {AREA} (-1) [A3101] {PHI} + (UDOTN)
( ) {AREA} ( ) [ONE] { } and note the theory has
generated cancellation of the area derivative term in DM.
The potential function is only a computational variable, to be
manipulated to generate the desired aerodynamics pressure distribution.
Bernoulli’s equation is a streamline integral on DP, equivalently DE,
hence pressure can be post-processed via the GWSh on DE DE: p(x) = p¥ - (½)ru×u
L(p)
= p - p¥
+ (½)rÑf×Ñf
= 0 where r
is the density (assumed uniform) of the fluid. The resultant template
pseudo-code is {WS(ph)}e = ( ) ( ) { } (1) [A200] {P} + (-1) ( ) { } (1) [A200] {Pinf} + (rho,
1/2) ( ) {PHI} (-1) [A3101] {PHI} One can also predict the
velocity field via
kinematics:
L(u)
= u(x) + Ñf×
The GWSh
template pseudo-code is {WS(uh)}e= ( ) ( ) { } (1) [A200] {U} + ( ) ( )
{ } ( ) [A201] {PHI}
The available theory for asymptotic convergence in the energy norm
remains valid for all variables as
The duct cross-sectional area distribution is assumed of the form
Figure 1. Duct cross-sectional area distribution. |