Kai Schneider

An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls

This paper is concerned with the numerical simulation of two-dimensional flame balls. We describe a Galerkin-type discretization in an adaptive basis of orthogonal wavelets. The solution is represented by means of a reduced basis being adapted in each time step. This algorithm is applied to compute the evolution of circular and elliptic thermodiffusive flames. In particular, we study the influence of the Lewis number, the strength of radiative losses and of the initial radius. The results show different scenarios. We find repeated splitting of the flame ball which is studied in some detail, extinction after excessive growth and also obtain quasi-steady flame balls.

Computation of decaying turbulence in an adaptive wavelet basis

Abstract The paper presents computations of decaying two-dimensional turbulence in an adaptive wavelet basis. At each time step the vorticity is represented by an adaptively selected set of wavelet functions which adjusts to the instantaneous distribution of vorticity. Essential features are the use of operator–adapted test functions and the adaptive evaluation of the convection term. The results of this new algorithm are compared to a classical Fourier method and a Fourier method supplemented with wavelet compression in each time step. They show that turbulent flows with a multitude of spatial scales can be computed with a reduced number of degrees of freedom. The investigation of diverse spectral and statistical criteria validates the wavelet approach.

Wavelet forcing for numerical simulation of two-dimensional turbulence

Abstract This note presents a new method to force turbulent flows computed by numerical simulations. The forcing term is defined in an inhomogeneous way, because it depends nonlinearly on the wavelet coefficients of the vorticity field. It injects energy and enstrophy as locally as possible in both physical and spectral spaces. This new forcing is defined in wavelet space in order to control the smoothness of the vortices thus excited and to model the local production of vortices by instabilities observed in turbulent flows.

Numerical simulation of the mixing of passive and reactive scalars in two-dimensional flows dominated by coherent vortices

Abstract The present paper comprises direct numerical simulation of the mixing of passive and reactive scalars in two-dimensional flows dominated by coherent vortices. By means of highly accurate pseudo-spectral methods the instationary Navier–Stokes and convection–diffusion–reaction equations are numerically integrated on the torus. We present the evolution of typical vortex arrangements and analyze their ability to mix scalars. Furthermore, we attribute the influence of vortices on the mixing to the formation of spirals and to the merging of vortices. As a way of quantifying the mixing, we consider global and local mixing time scales which describe the overall and the early variance decay, respectively. Moreover, the influence of the Schmidt number and of a chemical reaction on mixing processes are investigated.

Numerical simulation of three-dimensional instabilities of spherical flame structures

The paper is concerned with the non-stationary numerical simulation of freely propagating spherically symmetric flame structures at low Lewis number in a lean hydrogen-air mixture. The three-dimensional thermodiffusive equations with a single-step chemical reaction with Arrhenius-like temperature dependence of the reaction rate and a Stefan-Boltzmann-type radiation are integrated by means of a highly accurate Fourier-pseudo-spectral method with a semi-implicit time scheme in a parallelized version. This computational effort is necessary to fulfill the requirements resulting from the wide range of scales which are present in this reaction-diffusion problem. The algorithm is applied to compute the evolution of thermodiffusive flames, where we particularly investigate the influence of the initial flame radius and the radiative heat loss onto the development of freely evolving spherical flames. In the computations we obtain different scenarios of expanding flame structures as found in recent experimental studies under microgravity conditions, for example, the extinction because of excessive heat loss, the splitting due to cellular instability, and the observation of quasi-steady three-dimensional flame balls. Furthermore, we show the necessity of simulations in all physical dimensions to incorporate three-dimensional instabilities which in rotationally symmetric calculations cannot be included. Finally, we show the time evolution of flame structures which are initially disturbed by an artificial function to simulate natural perturbation of flame balls.

Numerical simulations on the stability of spherical flame structures

The paper presents investigations concerning the stability of spherical flames in a premixed lean hydrogen-air atmosphere and their evolution in case of instabilities. This is done by means of numerical simulations using the thermo-diffusive model with one-step finite rate chemical reaction and radiative heat loss under optically thin conditions. In the first part spherical symmetry is imposed leading to a one-dimensional problem. The results obtained in this way are compared with the asymptotic analysis and the numerical simulations from the literature. In the second part these solutions are employed as initial conditions for fully three-dimensional simulations using a high-resolution pseudo-spectral method. It allows the investigation of the nonlinear transient behavior of spherical flames with respect to three-dimensional perturbations. Different scenarios of their evolution are observed: extinction, spherical growth, and splitting. Also, for the first time, a steady flame ball is computed in a three-dimensional simulation. The different numerical and physical issues are discussed in detail and are related to available experimental observations as well as to theoretical analyses.

Numerical simulation of a mixing layer in an adaptive wavelet basis

Abstract This note presents an adaptive wavelet method to compute two-dimensional turbulent flows. The Navier–Stokes equations in vorticity–velocity form are discretized using a Petrov–Galerkin scheme. The vorticity field is developed into an orthogonal wavelet series where only the most significant coefficients are retained. The testfunctions are adapted to the linear part of the equation so that the resulting stiffness matrix turns out to be the identity. The nonlinear term is evaluated on a locally refined grid in physical space. This numerical scheme is applied to simulate a temporally developing mixing layer. A comparison with a classical pseudo-spectral method is used for validation of the new method. The results show that the formation of Kelvin–Helmholtz vortices is well captured and all scales of the flow are well represented.

Coherent Structure Eduction in Wavelet-Forced Two-Dimensional Turbulent Flows

We analyze vorticity fields obtained from direct numerical simulations (DNS) of statistically stationary two-dimensional turbulence where the forcing is done in wavelet space. We introduce a new eduction method for extracting coherent structures from two-dimensional turbulent flows. Using a nonlinear wavelet technique based on an objective universal threshold we separate the vorticity field into coherent structures and background flow. Both components are multi-scale with different scaling laws, and therefore cannot be separated by Fourier filtering. We find that the coherent structures have non-Gaussian statistics while the background flow is Gaussian, and we discuss the implications of this result for turbulence modelling.

Simulation and Analysis of Mixing in Two-Dimensional Turbulent Flows Using Fourier and Wavelet Techniques

This paper presents direct numerical simulations of mixing of passive and reactive scalars in two-dimensional flows. By means of pseudo-spectral methods the governing equations are numerically integrated. As an application we study a temporally growing mixing layer where we focus on the role of coherent vortices. Wavelet techniques are applied to obtain local spectral information about vorticity and scalars. We show that the local generation of fine scales in shear zones is strongly correlated with locally enhanced mixing. Furthermore, we examine the influence of chemical reactions.

Coherent Vortex Simulation (CVS) of 2D bluff body flows using an adaptive wavelet method with penalisation

In this paper we present an adaptive wavelet method to integrate the velocity-vorticity formulation of the two-dimensional Navier-Stokes equations coupled with a penalisation technique to handle easily solid boundaries of arbitrary shape. The validity of this method, called Coherent Vortex Simulation (CVS), is demonstrated by computing flows past different bluff bodies. Firstly, we show the computation of a flow around an impulsively started cylinder at Reynolds number 3000. The results are compared with those of a DNS using a spectral method and with others computed with two different vortex methods. Secondly, we also present simulations of a flow around a NACA air-foil profile at Reynolds number 1000.