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

                                    on ¶Wb,         on ¶Wi

                        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× = 0 

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
A(x) = Ao + Bsin(2px/L). Figure 1 illustrates the geometry for Ao = 0.1m2, B = 0.05 m2 and L = 0.5 m. The remaining data specifications are onset speed U¥ = 5m/s, and the fluid density, r = 480 kg/m3

Figure 1. Duct cross-sectional area distribution.