Product: ABAQUS/Standard
In this problem the acoustic field in a duct with mean flow at high subsonic speed is analyzed using the direct-solution steady-state dynamic procedure. Analytical solutions for this problem also exist and are provided for comparison with the numerical results obtained. Real and complex frequency analysis results for the reverberant case are also examined.
The model is a simple column of elements oriented along the x-axis. The units used in this case are consistent with air: 
, 
× 106, the column length is 4, and the Mach number is 0.5. The frequency range of interest is 50 to 300 cycles per second.
Two physical cases are examined: a reverberant end condition and an open condition. In both cases the real and imaginary parts of the acoustic pressure are prescribed at one end of the duct using the *BOUNDARY option. Default nonreflective impedance conditions are applied on the opposite end of the duct to simulate the open case; no loads, boundary conditions, or impedance conditions are required for the reverberant case.
The general analytical solution of the steady-state sound pressure along the length of the duct with uniform flow at (subsonic) Mach number 
is given by
, 
, and the constants 
and 
are defined by the prescribed load and end conditions. At the 
end, we set 
for both the reverberant and open cases. At 
, the reverberant and open conditions depend on the Mach number. To see this, recall the variational form of the acoustic equation with flow, as used in the derivation of the finite elements for these problems in ABAQUS: 
reduces to
, to establish the following constants:
For the open-end case with the same boundary condition at , only right-traveling waves exist; that is,
to the boundary traction integral and simplifying, we obtain the boundary term as
The real and complex frequency analyses are performed separately for the reverberant physical case.
The responses for the reverberant and open-ended cases are obtained by conducting direct-solution steady-state dynamic analysis steps. The analysis frequencies are chosen between 50 and 300 cycles per second. Analytic and computed results agree as expected.
The results for the real and complex frequency analysis also agree with the expected results.
Direct-solution steady-state dynamic analysis, all two-dimensional elements tested.
Direct-solution steady-state dynamic analysis, all three-dimensional elements tested.
Real and complex eigenanalysis, all two-dimensional elements tested.
Real and complex eigenanalysis, all three-dimensional elements tested.