High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics 4 and DNS problems. This is made possible for spatial discretization by optimizing 5 explicit and compact differencing procedures that minimize numerical error in the 6 spectral plane. While zero-diffusion nine point explicit scheme has been proposed 7 for the interior, additional highaccuracyone-sided stencils have alsobeendeveloped 8 for ghost cells near the boundary. A new compact scheme has also been proposed 9 fornon-periodicproblems—obtainedbyusingmultivariateoptimization technique. 10 Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise 11 and that rules out dissipation that is otherwise introduced via spatial and tempo12 ral discretizations. Acoustics problems are wave propagation problems and hence 13 require Dispersion Relation Preservation (DRP) schemes that simultaneously meet 14 high accuracy requirements and keeping numerical and physical dispersion relation 15 identical. Emphasis is on high accuracy than high order for both DNS and acous16 tics.While higher order implies higher accuracy for spatial discretization, it is shown 17 here not to be the same for time discretization. Specifically it is shown that the 2nd 18 order accurate Adams-Bashforth (AB)—scheme produces unphysical results com19 pared tofirst order accurateEuler scheme.This occurs, as theAB-scheme introduces 20 a spurious computational mode in addition to the physical mode that apportions to 21 itself a significant part of the initial condition that is subsequently heavily damped. 22 Additionally, AB-scheme has poor DRP property making it a poor method for 23 DNS and acoustics. These issues are highlighted here with the help of a solution 24 for (a) Navier–Stokes equation for the temporal instability problem of flow past a 25 rotating cylinder and (b) the inviscid response of a fluid dynamical system excited 26 by simultaneous application of acoustic, vortical and entropic pulses in an uniform 27 flow. The last problem admits analytic solution for small amplitude pulses and can 28 be used to calibrate different methods for the treatment of non-reflecting boundary 29 conditions as well. 30