Boundary conditions

Fan face boundary conditions. At the fan face, outflow boundary condition is applied by imposing an incoming spinning mode (m, n):

dp , y&P = dPfan ydpfan dt dx dt dx

with V = uo + sqrtc2 — u^.

Pfan = Pom, n) (AJm (kr R) + BYm (kr R))

(kr R)+BYm (kr R)) -ip0c0(k—M°kx)

krPlmn)(AJ’m(krR)+BY^(krR))

—ip0c0(k—M°kx)

___ ™P°{m, n)(AJ’m (kr R)+BYI (kr R))

Wfan — r(—ip0c0 (k—M° kx))

At the far-field boundary, radiation or outflow boundary condition of Tam is applied. For 2D problems, radiation boundary conditions can be applied also, but non-refhctive boundary conditions from Giles [12] are more suitable.

Hard and soft walls. Non-penetration boundary condition on the hard wall is expressed as:

u. n = 0

In the previous expression, the grid lines are assumed to be orthogonal to the wall,

Following Myers impedance condition defined in the frequncy domain, acous­tic impedance boundary condition is applied on the soft wall boundary using the following relation:

dp dp ^

гир + u0 ——b v0 — = —iu)Zvn dx dr

Extension to non-linear propagation

The non-linear solver is based on the linear solver, each harmonics consid­ered in the computational domain is calculated seperately as described above. The non-linear term of the Euler equations are added to the linearized Euler equations 1, 2, 3 and 4 after the solution and spatial derivatives are transformed in the time domain via Fourier transform. Residuals of the Euler equations are calculated and are transformed back to the frequency domain. The boundary conditions at the far-field, fan face and hard wall are imposed for each har­monic in the same manner, as well as for the soft wall boundary condition. Thanks to the frequency domain approach, the impedance boundary condition for each harmonic is satisfied with the frequency dependency.