Holger Foysi

An explicit filtering method for large eddy simulation of compressible flows

A method for large eddy simulation (LES) is presented in which the sub-grid-scale modeling is achieved by filtering procedures alone. The procedure derives from a deconvolution model, and provides a mathematically consistent approximation of unresolved terms arising from any type of nonlinearity. The formal steps of primary filtering to obtain LES equations, approximate deconvolution to construct the subgrid model term and regularization are combined into an equivalent filter. This filter should be an almost perfect low pass filter below a cut-off wavenumber and then fall off smoothly. The procedure has been applied to a pressure-velocity-entropy formulation of the Navier–Stokes equations for compressible flow to perform LES of two fully developed, turbulent, supersonic channel flows and has been assessed by comparison against direct numerical simulation (DNS) data. Mach numbers are 1.5 and 3.0, and Reynolds numbers are 3000 and 6000, respectively. Effects of filter cut-off location, choice of differentiation scheme (a fifth-order compact upwind formula and a symmetric sixth-order compact formula were used), and grid refinement are examined. The effects are consistent with, and are readily understood by reference to, filtering characteristics of the differentiation and the LES filter. All simulations demonstrate a uniform convergence towards their respective DNS solutions.

Compressibility effects and turbulence scalings in supersonic channel flow

Turbulence in supersonic channel flow is studied using direct numerical simulation. The ability of outer and inner scalings to collapse profiles of turbulent stresses onto their incompressible counterparts is investigated. Such collapse is adequate with outer scaling when sufficiently far from the wall, but not with inner scaling. Compressibility effects on the turbulent stresses, their anisotropy, and their balance equations are identified. A reduction in the near-wall pressure-strain, found responsible for the changed Reynolds-stress profiles, is explained using a Greens-function-based analysis of the pressure field

Compressible turbulent channel and pipe flow: similarities and differences

Direct numerical simulation (DNS) is used to explore similarities and differences between fully developed supersonic turbulent plane channel and axisymmetric non-swirling pipe flow bounded by isothermal walls. The comparison is based on equal friction Mach number, friction Reynolds number, Prandtl number, ratio of specific heats and viscosity exponent. The channel half-width and pipe radius are chosen to define the Reynolds numbers. To what extent and why mean flow quantities, second-order turbulence statistics and terms in the Reynolds stress equations coincide or diverge in both flows are investigated. The role of the fluctuating pressure in causing characteristic differences among correlations involving pressure fluctuations is identified via a Green-function-based analysis of the pressure field.

On the turbulence structure in inert and reacting compressible mixing layers

Direct numerical simulation is used to investigate effects of heat release and compressibility on mixing-layer turbulence during a period of self-similarity. Temporally evolving mixing layers are analysed at convective Mach numbers between 0.15 and 1.1 and in a Reynolds number range of 15000 to 35000 based on vorticity thickness. The turbulence inhibiting effects of heat release are traced back to mean density variations using an analysis of the fluctuating pressure field based on a Greens function.

Comment on "Statistical symmetries of the Lundgren-Monin-Novikov hierarchy".

The article by M. Wacławczyk et al. [Phys. Rev. E 90, 013022 (2014)] proposes two new statistical symmetries in the classical theory for turbulent hydrodynamic flows. In this Comment, however, we show that both symmetries are unphysical due to violating the principle of causality. In addition, they must get broken in order to be consistent with all physical constraints naturally arising in the statistical Lundgren-Monin-Novikov (LMN) description of turbulence. As a result, we state that besides the well-known classical symmetries of the LMN equations no new statistical symmetries exist. Finally, we criticize the relation between intermittency and global symmetries as it is presented throughout that study.We present a critical examination of the recent article by Wac lawczyk et al. (2014) which proposes two new statistical symmetries in the classical theory for turbulent hydrodynamic flows. We first show that both symmetries are unphysical in that they induce inconsistencies due to violating the principle of causality. In addition, they must get broken in order to be consistent with all physical constraints naturally arising in the statistical Lundgren-Monin-Novikov (LMN) description of turbulence. As a result, we state that besides the well-known classical symmetries of the LMN equations no new statistical symmetries exist. Yet, aside from this particular issue, the article by Wac lawczyk et al. (2014) is flawed in more than one respect, ranging from an incomplete proof, to a self-contradicting statement up to an incorrect claim. All these aspects will be listed, discussed and corrected, thus obtaining a completely opposite conclusion in our study than the article by Wac lawczyk et al. (2014) is proposing.

Turbulent momentum and passive scalar transport in supersonic channel flow

Direct numerical simulations of compressible turbulent channel flow including passive scalar transport have been performed at five Mach numbers, M, ranging from 0.3 to 3.5 and Reynolds numbers, Re, ranging from 181 to 1030. The Prandtl and Schmidt numbers are 0.7 and 1.0, respectively, in all cases. The passive scalar is added to the flow through one channel wall and removed through the other, leading to an S-shaped mean scalar profile with non-zero gradient in the channel centre. The paper describes the set of compressible flow equations, which is integrated using high-order numerical schemes in space and time. Statistical equations are presented for fully developed flow, including budgets for the Reynolds stresses, the turbulent scalar fluxes and the scalar variance. Results are presented for second order moments and the terms in the mentioned balance equations. Outer scalings are found suitable to collapse incompressible and compressible data. The reduction in the near-wall pressure-strain and pressure-scalar gradient correlations due to compressibility is explained using a Green-function-based analysis of the fluctuating pressure field.

DNS of Passive Scalar Transport in Turbulent Supersonic Channel Flow

Direct numerical simulations (DNS) of compressible supersonic channel flow of air at Reynolds numbers ranging from Re τ = 180 to Re τ = 560 and Mach numbers ranging from M = 0.3 to M = 3.0 have been performed. A Navier-Stokes solver of high order accuracy has been vectorized and parallelized to run efficiently on the Hitachi SR8000-F1. Budgets of the Reynolds stresses and the passive scalar fluxes are presented, as well as explanations concerning the reduction of the pressure-correlation terms, using a Greens function approach.

Comment on “Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation” [J. Math. Phys. 54, 072901 (2013)]

The quest to find new statistical symmetries in the theory of turbulence is an ongoing research endeavor which is still in its beginning and exploratory stage. In our comment we show that the recently performed study of Waclawczyk and Oberlack [J. Math. Phys. 54, 072901 (2013)] failed to present such new statistical symmetries. Despite their existence within a functional Fourier space of the statistical Burgers equation, they all can be reduced to the classical and well-known symmetries of the underlying deterministic Burgers equation itself, except for one symmetry, but which, as we will demonstrate, is only a mathematical artefact without any physical meaning. Moreover, we show that the proposed connection between the translation invariance of the multi-point moments and a symmetry transformation associated to a certain invariant solution of the inviscid functional Burgers equation is invalid. In general, their study constructs and discusses new particular solutions of the functional Burgers equation without referring them to the well-established general solution. Finally, we also see a shortcoming in the presented methodology as being too restricted to construct a complete set of Lie point symmetries for functional equations. In particular, for the considered Burgers equation essential symmetries are not captured.

Optimal Control of a Plane Jet Using the Adjoint Method

The optimal control of noise emitted by a 2D plane jet and first results for 3D plane jets are presented. The so-called adjoint-equations provide sensitivity information used in a gradient-based minimization of aerodynamic sound even for high dimensional control problems. The goal is to reduce the noise in the farfield of a plane jet by applying cooling and heating in small control volumes located in the jet shear layers. The farfield-sound was directly calculated by solving the compressible Navier-Stokes-equations and non-reflecting boundary conditions for the adjoint-equations were derived and implemented along with sponge zones to minimize reflections. Using optimal control a sound pressure level (SPL) reduction of up to 7dB could be reached. Furthermore 3D DNS results indicate large differences in the size and coherence of structures in the adjoint field.

On Sound Generated by a Globally Unstable Round Jet

Direct numerical (DNS) and large-eddy simulations (LES) of a strongly heated globally unstable round jet are juxtaposed with respect to aerodynamical mean characteristics and the sound being generated. The sound field is computed by a hybrid approach using the acoustic perturbation equations (APE). All used codes have been adopted to massive-parallel supercomputers. This way results can be obtained in a reasonable time frame. When compared to the DNS results, the LES is capable to capture the major characteristics of the emitted sound field in the forward direction. The sideline and backward direction that are dominated by small scale noise reveal larger discrepancies that are due to the inherent restrictions of large eddy simulations.