Analysis of artificial pressure equations in numerical simulations of a turbulent channel flow

Abstract

Recently, several methods have been proposed to simulate incompressible fluid flows using an artificial pressure evolution equation, avoiding the resolution of a Poisson equation. These methods can be seen as various levels of approximation of the compressible Navier-Stokes equation in the low Mach number limit. We study the simulation of incompressible wall-bounded flows using several artificial pressure equations in order to determine the most relevant approximations. The simulations are stable using a finite difference method in a staggered grid system, even without diffusive term, and converge to the incompressible solution, both in direct numerical simulations and for coarser meshes, to be used in large-eddy simulations. A pressure equation with a convective and a diffusive term produces a more accurate solution than a compressible solver or methods involving more approximations. This suggests that it is near to an optimal level of approximation. The presence of a convective term in the pressure evolution equation is in particular crucial for the accuracy of the method. The rate of convergence of the solution in terms of artificial Mach number is studied numerically and validates the theoretical quadratic convergence rate. We demonstrate that this property can be used to accelerate the rate of convergence using an extrapolation in terms of artificial Mach number. Since the approach is based on an explicit and local system of equations, the numerical procedure is massively parallelisable and has low memory requirements.

Artificial compressibility method for strongly anisothermal low Mach number flows

“…The validity of this assumption will be discussed in more details in the following. The convective term in equation 7 has been found crucial for incompressible flows in a previous paper [18]. The diffusive term in equation 7 physically represents thermal conduction and may thus not be neglected in the case of strongly anisothermal flows that are the target of this study.…”

“…Although thermoacoustic waves [30] are produced by the system of equations, the method is not expected to be relevant to their study as their velocity has been reduced artificially. Compared to the artificial methods devised for incompressible flows [11,13,14,18], the proposed methodology includes two pressures in the final set of equations ( 5)-( 9), the thermodynamical pressure and the mechanical pressure. This decomposition is useful to take into account anisothermal effects because the two pressures are affected differently by a reduction of the speed of sound.…”

“…The latter approach includes the α-transformation of O'Rourke and Bracco [7], the pressure gradient scaling (PGS) method of Ramshaw et al [8], the acoustic speed reduction (ASR) method of Wang and Trouvé [9], the kinetically reduced local Navier-Stokes (KRLNS) equations of Ansumali et al [10] and Karlin et al [11], the artificial acoustic stiffness reduction method (AASCM) of Salinas-Vázquez et al [12], the entropically damped artificial compressibility (EDAC) method of Clausen [13] and the general pressure (GP) equation of Toutant [14]. These methods provide successive improvements to the numerical simulation of incompressible flows using artificial pressure equations and have been validated extensively for both laminar and turbulent viscous flows in the literature [11,15,13,16,17,18]. To the best knowledge of the authors, low Mach number flows with large temperature variations have been to date relatively ignored by these developments despite their ubiquitousness in a large variety of industrial applications, including heat exchangers, propulsion systems or nuclear or concentrated solar power plants [19,20,21,22].…”

An Artificial Compressibility Method for 1D Simulation of Open-Channel and Pressurized-Pipe Flow

“…More than 200 papers were retrieved by database searches for 2015-2020 using the terms artificial compressibility, pseudo compressibility, and dual time stepping. These studies include analysis of the relationships between AC and fractional step methods [86,87], studies of turbulent flow [88][89][90], development of discontinuous Galerkin methods [91][92][93], parallel solution of large-scale problems [94][95][96], multi-fluid simulations [97], further advances to entropically-damped AC methods [98][99][100], implementation high-order numerical schemes [101][102][103], use of characteristic methods [104][105][106], application for non-hydrostatic effects [107,108], magneto-hydrodynamic simulations [109][110][111], solution with lattice Boltzmann methods [112], and solution by smoothed particle hydrodynamics [113,114]. Note the above are examples and should not be considered an exhaustive list of recent work.…”

The QUICK scheme is a third‐order finite‐volume scheme with point‐valued numerical solutions

“…As such, there is no doubt that the QUICK interpolation scheme is third-order accurate since it is designed to be exact for quadratic functions. However, a controversy arose quickly and it still exists even today about the order of accuracy of the resulting convection scheme: third-order accurate [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] or second-order accurate. [26][27][28][29][30][31][32][33][34] Several authors including Leonard himself have attempted to resolve the confusion in the 1990s, [35][36][37][38][39] but the resolution does not seem to have been achieved as we can find recent references stating that the QUICK scheme is second-order accurate.…”