Introduction to the Turbulence Models Seyhan Uygur Onbaşıoğlu

Not in order but interacting Physical domain Computational domain Boundary Conditions Turbulence Model Numerical Method (discretization etc.) Grid type and layout

History of the dependent variable φ

If you deal with CFD you should know about turbulence! Flow around/in cars, aeroplanes, buildings. Boundary layers and wakes around/after cars, aeroplanes, buildings. Flow and combustion in engines, gasturbines, combustors. Air movements in rooms, enclosures…

Mean and fluctuation If the mean flow is steady we divide turbulence into mean and fluctuation parts.

Irregularity, randomness, dissipation Turbulent flow is irregular, random and chaotic. Random but deterministic. There is a spectrum of scales from the largest to the smallest.

Scales Largest scale is the order of the flow geometry, boundary layer thickness, jet width Smallest scale is the smallest eddy scale where the viscous stresses are dissipated into internal energy.

Eddies… An eddy is a turbulent motion localized within a region of size l. Sizes of the eddy differ from the flow length L, to the smallest eddy size. For each eddy size there is a velocity and time scale

From Big to the Smaller Eddies… Each eddy has Reynolds number. Large eddies are unstable and break up into smaller eddies and so on…

The Energy Cascade

The Energy Cascade

Energy cascade… This energy cascade continues until the Reynolds number is sufficiently small that energy is dissipated by viscous effects: the eddy motion is stable, and molecular viscosity is responsible for dissipation.

Vorticity = 0 “vortex stretching” will stop. Ul/ν=1

Turbulence is random contains different scales 3D

Scales and the dissipation Since the KE is destroyed by the viscous forces, the larger viscosity means larger scales. The amount of energy is to be dissipated is ε. More dissipation means the larger velocity gradient.

Scales The dimensions on LHS =RHS Similarly, Called Kolmogorov scale.

Energy Spectrum I Range for large, energy containing eddies, II the inertial subrange, III Range for small, isotropic scales

Energy Spectrum

Energy Spectrum

Kolmogorov Spectrum Law or -5/3 Law Explains the difficulty of LES and DNS

The Kinetic Energy of Turbulence

A statistical measure is an average of some kind: over the symmetry coordinates, if any are available (e.g, a time average for stationary flows); over multiple realizations (e.g, an ensemble); or over the phase space of solutions if the dynamics are homogeneous. In a statistically homogenous turbulent flow, measurable statistical quantities such as the mean velocity or the turbulent kinetic energy are the same at every point in the flow.

Probability density functions and moments A complete description of a turbulent variable v at a given location and instant in time is given by the probability density function (PDF), P(v), where P(v)dv is the probability of the variable v taking a value between v and v+ dv, and

Navier-Stokes Denklemleri

Structure of the Transport Equation

Reynolds Averaging and RANS

Reynolds averaging (1)

Reynolds Averaging (2)

Phase averaging

Some Rules

Obtaining the turbulent transport N-S for fluctuations

Obtaining the turbulent transport Energy for temperature fluctuations

Obtaining the turbulent transport averaging fluctuations

Obtaining the turbulent transport

Transport equation for the turbulent stress

Transport equation for the turbulent stress…

The terms of transport equation for the turbulent stress The first 2 terms are turbulent diffusion of Reynolds stresses The 3 rd term is molecular d,ffus,on of Reynolds stresses 4 th and 5 th are production of the Reynolds stresses

The turbulent stress(es)

The terms of transport equation for the turbulent stress 6 th term is the dissipation and the 7 th term is the pressure strain term (making the flow isotropic.)

The turbulent kinetic energy

The dissipation of the turbulent kinetic energy We need a transport equation!

Obtaining the transport equation for the dissipation of the turbulent kinetic energy

The transport equation for the dissipation of the kinetic energy

The transport equation for the dissipation of the kinetic energy…

The terms of the dissipation equation The first two terms are the turbulent diffusion of the dissipation The third one is the molecular diffusion of the dissipation The 4 th and the 5 th terms are the production of the dissipation The last two terms are the destruction of the dissipation.

Equation for turbulent heat flux

Closure Problem

Closure Problem…

Method for closing

Method for closing

Without solving RS transport

Without solving RS transport…

Without solving RS transport…

Without solving RS transport…

Without solving RS transport…

Without solving RS transport…

Without solving RS transport…

Exact and Modelled Equations

For non-isotropic diffusivity

Modelled RS equation

Modelled RS equation…

Modelled k equation

Modelled dissipation equation

Free shear flows

Flow

Round jets