Time-domain implementation of an impedance boundary condition with boundary layer correction

A time-domain boundary condition is derived that accounts for the acoustic impedance of a thin boundary layer over an impedance boundary, based on the asymptotic frequency-domain boundary condition of Brambley [2011, AIAA J. 49(6), pp. 1272–1282]. A finite-difference reference implementation of this condition is presented and carefully validated against both an analytic solution and a discrete dispersion analysis for a simple test case. The discrete dispersion analysis enables the distinction between real physical instabilities and artificial numerical instabilities. The cause of the latter is suggested to be a combination of the real physical instabilities present and the aliasing and artificial zero group velocity of finite-difference schemes. It is suggested that these are general properties of any numerical discretization of an unstable system. Existing numerical filters are found to be inadequate to remove these artificial instabilities as they have a too wide pass band. The properties of numerical filters required to address this issue are discussed and a number of selective filters are presented that may prove useful in general. These filters are capable of removing only the artificial numerical instabilities, allowing the reference implementation to correctly reproduce the stability properties of the analytic solution.E.J. Brambley gratefully acknowledges the support of the Royal Society through a University Research Fellowship, and Fellowship of Gonville & Caius College, Cambridge.This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jcp.2016.05.06

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