# Semi-Empirical Models of Turbulence: Theory and Experiment

A.T. Onufriev
Pure and Applied Mechanics institute Novosibirsk, USSR; Joint Institute of High Temperature Moscow, Russia

R.A. Safarov
Joint Institute of High Temperature Moscow, Russia

K.E. Son
Joint Institute of High Temperature Moscow, Russia

Eduard Son
Joint Institute for High Temperature RAS

## Description

This monograph presents a review of methods that exist for describing the structure of a turbulent flow and the behavior in the flow under various conditions of the average velocity, variance of the velocity pulsations, one-point and two-point correlation moments of various orders, integral scales and microscales, spectral distributions. The turbulent motion often demonstrates a lot of peculiarities because of the reason that the integral correlation scale is comparable to the characteristic size of the flow.

The book presents experimentation results on the behavior of a turbulent flow under the influence of a vorticity created by rotation of a tube with respect to its longitudinal axis; on the behavior of spectral distributions which correspond to the second, third, and fourth-order moments; on the behavior of one-dimension and joint probability densities. Peculiar features of the behavior of the joint probability densities at abnormal values of the pulsations are addressed. Results obtained in Moscow Physical and Technical Institute are used.

Bibliographic references are given for further studies on problems of choice. The book provides a list of issues which are important for the improvement of the semi-empiric theory of turbulent transport but not yet investigated to a sufficient extent.

383 pages, © 2014

## Table of Contents:

1 Models of Continuous Media

1.2 BALANCE EQUATIONS FOR MASS, MOMENTUM AND ENERGY

1.2.1 Continuity equation

1.2.2 General equation of transport

1.2.3 Component diffusion equations

1.3 CLOSURE OF THE EQUATION SET

1.5 VISCOUS HEAT-CONDUCTING MEDIUM

1.5.1 Vector part of entropy production

1.5.2 Tensor part of entropy production

1.6 STOKES MODEL OF VISCOUS INCOMPRESSIBLE MEDIUM (FLUID, GAS)

1.7 STRESS RELAXATION MEDIUM (MAXWELL'S MEDIUM)

2 Kinetic Models of Media

2.1.2 Maxwell distribution function

2.1.3 Boltzmann equation for a mixture of gases

2.1.4 Equations of hydromechanics

2.2 CHAPMAN–ENSKOG METHOD

2.2.1 Transport factors of indifferent gases

2.4 METHOD OF MOMENTS IN KINETIC THEORY

2.4.1 Transport properties of gas in τ-approximation

2.4.2 Heat conduction of gas in τ-approximation

2.4.3 Viscosity of gas in τ-approximation

2.4.4 Transport processes in a free-molecule flow

2.4.5 Transport processes for a finite mean free path

CONCLUSIONS FOR CHAPTERS 1 AND 2

3 Basic Notions and Methods of Description of Turbulence

3.1 GENERAL CHARACTERIZATION OF TURBULENT MOTION

3.2 REASONS FOR EMERGENCE OF TURBULENCE

3.3 POSING OF PROBLEMS IN TURBULENCE THEORY

3.3.1 Features of turbulent flows

3.4 METHODS FOR DESCRIPTION OF TURBULENT FLOWS

3.4.1 Instantaneous and mean values

3.4.2 Correlations and correlation scales

3.4.3 Moments of pulsations

3.4.4 Probability density function (PDF)

3.4.5 Examples of turbulent flows

4 Turbulence of Homogeneous and Isotropic Fields

4.1 LOCAL STRUCTURE OF TURBULENCE

4.1.1 Transport of turbulence energy across the spectrum

4.1.2 Basics of theory of locally isotropic turbulence

4.1.3 Characterization of small-scale turbulence

4.2 HOMOGENEOUS ISOTROPIC TURBULENT MOTION OF THE MEDIUM

4.2.1 Correlation functions in homogeneous isotropic turbulence

4.2.2 Von Karman–Howarth equation

4.2.3 Spectral form of correlation functions and equations

4.2.4 Unidimensional spectra for homogeneous isotropic turbulence

4.2.5 Approximate form of the spectrum for homogeneous isotropic field

4.2.6 Spectra of scalar functions

5 An Ovedrview of Some Semi-Empiric Models of Turbulence

5.1 SEMI-EMPIRIC DESCRIPTION OF TURBULENT FLOWS

5.1.1 Principle of “local similarity” in turbulence

5.2 METHOD OF STATISTICAL MOMENTS

5.2.1 Equations for first moments (SM-1 model)

5.2.2 Equations for second moments

5.2.3 Plane-parallel flow

5.2.4 Flat layer of constant friction stress

5.3 MODEL OF TURBULENCE WITH SCALE

5.3.2 k-L model of turbulence

5.3.3 Kolmogorov–Prandtl model

5.4 ε-MODELS OF TURBULENCE

5.4.1 Equation of dissipation rate

5.4.3 A modified k-ε model of turbulence

5.4.4 Drawbacks of turbulent viscosity based models

5.5 THIRD-ORDER STATISTICAL MODELS (SM-3 MODEL)

5.5.1 Differential equations for third moments

5.6 REYNOLDS STRESS MODELS (SM-2 MODELS)

5.6.1 Differential models of turbulence

5.6.2 Equations for second and third moments in the DRSM-96 model

5.6.3 Algebraic models of turbulence

5.6.4 Algebraic equations for third moments

5.7.1 Equation of turbulent diffusion

5.7.2 Diffusion in the inertial range

5.7.3 Expression of diffusivity

5.7.4 Conclusion to Section 5.7

5.8 A REVIEW OF SEMI-EMPIRIC TURBULENCE MODELS

5.8.1 Prandtl–Reichardt model for free shear flows

5.8.3 Baldwin–Lomax model

5.8.4 Prandtl–Loytsiansky–Klauser-3 model

5.8.5 Garbaruk–Strelets–Lapin model

5.8.7 Nonlinear two-parametric dissipative model

5.8.8 Differential q-ω model of turbulence

5.9.1 Kolmogorov–Prandtl model

5.9.2 Equation for turbulent friction

5.9.3 Equation for turbulent viscosity

5.10 MODELS BASED ON TWO DIFFERENTIAL EQUATIONS

5.10.1 Dissipative two-parameter model of turbulence

5.10.2 Modeling of generation, dissipation and diffusion terms in the isotropic dissipation equation

5.10.3 Model form of the isotropic dissipation equation. Constants of dissipative model

5.10.4 Family of two-parameter dissipative k-ε models of turbulence

5.10.5 Low-Reynolds effects in k-ε models

5.10.6 Suffman–Wilcox k-ω model

5.10.7 Other two-equation models

5.10.8 Menter's two-layer k-ω model

5.10.9 Analysis of turbulence models

5.11 ANALYSIS OF THE STATISTICAL MOMENT METHODS

6 Probability Density in the Turbulent Transport Theory

6.1 DEFINITION OF A PROBABILITY DENSITY

6.2 SEMI-EMPIRIC EQUATION FOR PDF

6.4 DERIVING TRANSPORT EQUATIONS FROM PDF EQUATIONS

6.4.1 Allowing for viscous diffusion

6.5 APPROXIMATE EQUATIONS FOR SECOND AND THIRD ORDER MOMENTS

6.6 A SUBSIDING HOMOGENEOUS ISOTROPIC TURBULENCE

6.6.1 Approximations of relaxation times

6.7 APPROXIMATIONS FOR THIRD-ORDER MOMENTS

6.8 JOINT PROBABILITY DENSITIES IN A TURBULENT FLOW

7 Anisotropy of Turbulent Transport

7.1 ANISOTROPY OF PULSATIONS IN THE FLOWWITH A CONSTANT SHEAR RATE

7.1.2 A PDF model that allows for Zhukovsky lift

7.2 TURBULENTWAKE IN A STRATIFIED MEDIUM

7.3 A WAKE BEHIND A BODY WITH ZERO EXCESS MOMENTUM

7.3.1 Effect of stratification

7.3.2 Effect of the streamline curvature on the turbulence properties (Isayev)

8 Turbulent Motion in the Ground Layer of the Atmosphere

8.1 TURBULENT MOTION OF A STRATIFIED MEDIUM

8.2 TRANSPORT EQUATIONS IN A STRATIFIED TURBULENT FLOW

8.2.1 Turbulent transport of heat

8.3 ALGEBRAIC MODEL FOR SECOND-ORDER MOMENTS. JAUGASHTIN'S MODEL

8.3.2 Allowing for vorticity in the velocity field

8.3.4 Expressions for diffusive fluxes

8.3.5 Two-parameter dissipative model that takes buoyancy into account (by Isayev)

9 'Negative' Viscosity Phenomenon

9.1 PECULIARITIES OF TURBULENCE IN PLANARWAKES AND SHEAR FLOWS

9.2 PLANARWAKE BEHIND A CYLINDER

10 Intermittency, Viscous Sublayer, Wall Functions

10.2 ALLOWING FOR INTERMITTENCY IN TURBULENT TRANSPORT MODELS

10.2.1 Comparison with the models by Dali, Harlow and Sheng

10.3 AN HIERARCHICAL MODEL OF TURBULENT TRANSPORT

10.3.1 A role played by the intermittency effect in the viscous sublayer

10.4 APPROXIMATION OF THE PROBABILITY DENSITY FUNCTION CONSIDERING THE INTERMITTENCY

10.5 A MODEL OF TRANSPORT THAT MAKES USE OF THE INTERMITTENCY COEFFICIENT

10.5.1 Continuity equation

10.5.5 Mean velocity of the flow

10.5.6 The method of wall functions (Isayev)

CONCLUSION TO THE CHAPTER

11 Turbulent Motion in Tubes and Rotating Flows

11.1 TURBULENT FLOWIN A ROUND RECTILINEAR TUBE

11.2 REACHING THE STEADY TURBULENT STATE IN A TUBE

11.3 TURBULENT TRANSPORT IN A VORTEX CORE

11.4 EXPERIMENTAL INVESTIGATIONS OF TURBULENT FLOWS WITH ROTATION

11.4.1 Experimental bench and measurement methods

11.4.2 A flow with the rotating exit part of the channel

11.4.3 Non-isothermal flow

12 Computational Modeling of Turbulent Flows

12.1 NUMERICAL MODELING USING MODELS OF TURBULENCE BY BELOTSERKOVSKY [15, 16]

12.1.1 Turbulent transport in swirling flows

12.1.2 Polar coordinate system

12.1.3 Cylindrical coordinate system

12.1.4 Swirling turbulent flows in channels

12.1.7 The ARSM model in the boundary layer approximation

12.1.8 Third-moment equations in the boundary layer approximation

12.1.9 The ARSM model for cylindric coordinates

12.2 TURBULENT FLOWIN A ROTATING TUBE

12.2.1 A model of transport equations for turbulent stresses

12.2.2 Algebraic model for Reynolds stresses

12.2.3 Numerical solution of the problem of mixing in a cylindrical chamber

12.2.4 Computational algorithms

12.3 A BOUNDARY LAYER APPROXIMATION FOR A MIXING CHAMBER FLOW

12.4 SWIRLING FLOWIN A CONICAL DIFFUSER

12.4.1 Formulation of the problem

12.4.2 Computational modeling of the flow in the diffuser

12.5 TURBULENT FLOWIN A CURVED CHANNEL

12.5.1 Straight channel areas (A and C)

12.5.2 The mean velocity vector's turn area (B)

12.5.3 Numerical modeling of a channel flow

CONCLUSION TO THE CHAPTER