Delayed-time domain impedance boundary conditions (D-TDIBC)

Defining acoustically well-posed boundary conditions is one of the major numerical and theoretical challenges in compressible Navier-Stokes simulations. We present the novel Delayed-Time Domain Impedance Boundary Condition (D-TDIBC) technique developed to impose a time delay to acoustic wave reflection. Unlike previous similar TDIBC derivations (Fung and Ju, 2001-2004 [1], [2], Scalo et al., 2015 [3] and Lin et al., 2016 [4]), D-TDIBC relies on the modeling of the reflection coefficient. An iterative fit is used to determine the model constants along with a low-pass filtering strategy to limit the model to the frequency range of interest. D-TDIBC can be used to truncate portions of the domain by introducing a time delay in the acoustic response of the boundary accounting for the travel time of inviscid planar acoustic waves in the truncated sections: it gives the opportunity to save computational resources and to study several geometries without the need to regenerate computational grids. The D-TDIBC method is applied here to time-delayed fully reflective conditions. D-TDIBC simulations of inviscid planar acoustic-wave propagating in truncated ducts demonstrate that the time delay is correctly reproduced, preserving wave amplitude and phase. A 2D thermoacoustically unstable combustion setup is used as a final test case: Direct Numerical Simulation (DNS) of an unstable laminar flame is performed using a reduced domain along with D-TDIBC to model the truncated portion. Results are in excellent agreement with the same calculation performed over the full domain. The unstable modes frequencies, amplitudes and shapes are accurately predicted. The results demonstrate that D-TDIBC offers a flexible and cost-effective approach for numerical investigations of problems in aeroacoustics and thermoacoustics.