Marta Wacławczyk

Probability density function computation of turbulent flows with a new near-wall model

The modeling and computation of near-wall turbulent flows is addressed with the probability density function (PDF) method for velocity and the turbulent frequency. Near-wall extensions are considered in detail and a new model for viscous transport is proposed. A method of elliptic relaxation for a blending function is applied to model the pressure–strain term. A numerical integration scheme is developed to deal with the near-wall singularity of coefficients that appears in the discrete formulation. The PDF equation is solved by a Monte Carlo method and the whole approach appears as a self-contained Lagrangian simulation using stochastic particles. For the sake of a numerical example, the fully developed channel flow case is computed; results are compared with the available direct numerical simulation data.

Spray penetration in a turbulent flow

Analytical expressions for mass concentration of liquid fuel in a spray are derived taking into account the effects of gas turbulence, and assuming that the influence of droplets on gas is small (intitial stage of spray development). Beyond a certain distance the spray is expected to be fully dispersed. This distance is identified with the maximum spray penetration. Then the influence of turbulence on the spray stopping distance is discussed and the rms spray penetration is computed from a trajectory (Lagrangian) approach. Finally, the problem of spray penetration is investigated in a homogeneous two-phase flow regime taking into account the dispersion of spray away from its axis. It is predicted that for realistic values of spray parameters the spray penetration at large distances from the nozzle is expected to be proportional to t2/3 (in the case when this dispersion is not taken into account this distance is proportional to t1/2). The t2/3 law is supported by experimental observations for a high pressure injector.

Application of the extended Lie group analysis to the Hopf functional formulation of the Burgers equation

A study concerning the Lie group analysis of the functional differential equations has been performed. This is a continuation of the previous common work of Oberlack and Waclawczyk [“On the extension of Lie group analysis to functional differential equations,” Arch. Mech. 58, 597 (2006)] where Lie group theory has been extended to functional differential equations and a special solution of the functional formulation of the Burgers equation was derived based on the calculated set of infinitesimals. Here we derive an infinite set of symmetry transformations of this equation and find new and more general invariant solutions. With this we get a step closer to solutions of functional differential equations, which, e.g., give a complete statistical description of turbulence in case of the famous Hopf-Novikov-Stokes functional differential equations.

Full velocity-scalar probability density function computation of heated channel flow with wall function approach

A joint velocity-scalar probability density function (PDF) method is presented to model and simulate turbulent flows with passive inert scalars (here temperature). The full PDF approach is applied for wall-bounded flows. In the present work, the boundary conditions are imposed in the logarithmic region and the modeling is therefore performed in the wall-function spirit. The PDF equation is solved by a Monte Carlo method and the whole approach appears as a Lagrangian simulation using stochastic particles. The purpose of the work is to analyze the behavior of classical PDF models in the near-wall region and to develop new particle boundary conditions for the velocity and scalars attached to each particle. First of all, the logarithmic region is described as an equilibrium zone and resulting analytical formulas for second-order temperature–velocity statistics 〈θ2〉, 〈uθ〉, 〈vθ〉 are derived. Boundary conditions for scalars are then developed and formulated in terms of instantaneous particle variables. These res...

Probability density function computation of heated turbulent channel flow with the bounded Langevin model

The paper addresses the modelling and computation of heated turbulent flowswith temperature treated as a dynamically passive scalar variable. The probability density function (PDF) method is appliedto wall-bounded turbulence. In a one-point PDF statistical approach with additional scalars, turbulent mixing models remain an open issue.The bounded Langevin model for scalar mixing is presentedin the form of a stochastic diffusion process in continuous time and is shown to successfully predict the turbulent mixing of initially bimodal scalar distribution. In the paper, two variants of the PDF approach are formulated and solved in the Lagrangian setting. First, the stand alone joint velocity-scalar PDF is considered with the assumptions of log-layer for both velocity and temperature that result in the Lagrangian wall-function approach. Corresponding formulae are derived for the equilibrium values of the second order temperature-velocity statistics. The Lagrangian stochastic evolution equations are solved for v...

Lie Symmetry Analysis of the Hopf Functional-Differential Equation

In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Waclawczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extended Lie symmetry analysis in the physical space where we have to deal with the transformation of the integration variable in the appearing integral terms. The method is based on the transformation of the product y(x)dx appearing in the integral terms and applied to the functional formulation of the viscous Burgers equation. The extended Lie symmetry analysis furnishes all known symmetries of the viscous Burgers equation and is able to provide new symmetries associated with the Hopf formulation of the viscous Burgers equation. Hence, it can be employed as an important tool for applications in continuum mechanics.

Modelling of turbulence-interface interactions in stratified two-phase flows

In this work we describe interactions between turbulence and water-air surface in the ensemble-averaged picture where, instead of a sharp interface between the phases we deal with a surface layer where the probability of the surface position is nonzero. Changes of the turbulent kinetic energy and the characteristic size of the eddy influences the width of the surface layer. We present a numerical solution of the equation for the intermittency function which describes the probability of finding the water phase at the given point and time.

Lie Symmetries of the Lundgren−Monin−Novikov Hierarchy

In this work we consider the statistical approach to turbulence represented by the Lundgren-Monin-Novikov (LMN) hierarchy of equations for the probability density functions (PDFs). After a review of the properties that the PDFs have to satisfy, we first show the basic Galilean invariance of the LMN equations; then we discuss the extended Galilean one and finally we present a transformation of the PDFs and examine the conditions which have to be satisfied so that this transformation represents a symmetry of the LMN hierarchy corresponding in the Multi-Point Correlation (MPC) approach to one of the so called statistical symmetries found using the Lie symmetry machinery in [6] for the infinite hierarchy of equations satisfied by the correlation functions from which some decay exponents of turbulent scaling law could be worked out.

Invariant closure proposals for the interface tracking in two-phase turbulent flows

The present work focuses on the formulation of new modelling approaches to ensemble-averaged equations describing multiphase flows, based on the symmetries admitted by these equations. Particular attention will be given to the proper treatment of the unclosed terms in the equation for the interface tracking which represent the influence of unresolved part of the surface. Modelling of those terms is crucial in flows with heat and mass transfer. In the work, two approaches to track the interface will be considered: the level-set function and sharp-step indicator function method. The differences between the two approaches in terms of turbulence-interface interactions modelling will be outlined.

Fractal reconstruction of sub-grid scales for large eddy simulation of atmospheric turbulence

We present a fractal sub-grid scale model for large eddy simulation (LES) of atmospheric flows. The fractal model is based on the fractality assumption of turbulent velocity field with a dynamical hypothesis based on energy dissipation. The fractal model reconstruct sub-grid velocity field from the knowledge of its filtered values on LES grid, by means of fractal interpolation, proposed by Scotti and Meneveau (1999). The characteristics of the reconstructed signal depends on the (free) stretching parameters, which is related to the fractal dimension of the signal. In previous studies, the stretching parameters was assumed to be constant in space and time and are obtained from experimental velocity signals of homogeneous and isotropic turbulence. To improve this method and account for the stretching parameter variability, we calculate the probability distribution function of the stretching parameter from direct numerical simulation (DNS) data of stratocumulus-top boundary layer (STBL) (courtesy of Prof. J.-P. Mellado from the Max Planck Institute of Meteorology) using the geometric method proposed by Mazel and Hayes. We perform 1D a priori test and compare statistics of the constructed velocity increment with DNS velocity increments.