chapter 24 - intro to turbulence modeling in cfd

7/29/2019 Chapter 24 - Intro to Turbulence Modeling in CFD

1/23

1

Introduction to Turbulence Modeling in CFD

-------------------------------------------------------------------------------------------------------

Copyright 2012 by R.M. Barron. All rights reserved. No part of these notes may be reproduced or distributed

in any form or by any means, mechanical or electronic, including but not limited to photocopying, recording,storage or retrieval system, without prior written permission from the author.

What is Turbulence?

Turbulent flowmay be defined as a flow which contains self-

sustaining fluctuations of flow properties, such as velocity, pressure,

temperature, imposed on the main flow.

In laminar flow, fluid layers transfer some of their momentum to adjacent

layers through the motion of molecules (Microscopic scale)

In turbulent flow, the momentum transfer between the layers is due to the

migration of chunks of fluid (Macroscopic scale)

The Macroscopic motion in turbulence results in instantaneous fluctuations

of fluid velocity components

CHAPTER 24


2/23

2

Turbulent flows can be broadly categorized into 3 groups:

o Boundary layers and wall-bounded flows

o Shear layers

o Grid-generated turbulent flows

Boundary layers and wall-bounded flows:

- majority of turbulent kinetic energy is produced near the walls

- small scale eddies dominate in the near wall region

- small scale eddies are more organized and have similar structure- can be turbulent boundary layer (bounded by wall and free stream), or fully

developed turbulent flow (bounded by walls)


3/23

3

Shear layer:

- flow expands in the streamwise

direction

- typically develops universal characteristics,

considered self-preserving

- subcategorized as free shear layers,

jets and wakes

Fig. 1: 3 categories of shear flows


4/23

4

Grid-generated turbulence:

- produced by passing an initially uniform and irrotational flow through a

grid composed of rods

- vortices generated by the rods interact with each other and degenerate

into turbulence

- typically isotropic, i.e., fluctuations have no preferred direction


5/23

5

Fig. 2: Transition to turbulence in a jet flow


6/23

6Fig. 3: Transition to turbulence for flow over a flat plate


7/23

7

Remarks:

o Small-scale eddies are more organized and have similar structure and

characteristics in all turbulent flows

o Large-scale eddies in shear layers are fairly regular and have a coherent

structure

o The large-scale coherent structures are quasi-2D, but have 3D small-

scale eddies imposed on themo The growth of large-scale eddies if mainly due to inertia and pressure

effects

o The growth of small-scale eddies is mainly due to viscosity; viscosity is

the dominant factor limiting the size of small-scale eddies

o Turbulence is an unsteady phenomena; it is inherently 3D


8/23

8

Remarks continued:

o Eddies overlap in space and larger eddies carry smaller eddies

o The majority of turbulence production in a boundary layer takes place in the

near-wall buffer zone, where bursting occurs

o Kinetic energy contained in large eddies is transferred to smaller eddies

through a process called cascading

o The kinetic energy ultimately dissipates into heat at the level of smallesteddies due to molecular viscosity

o Wide range of length scales and time scales in any turbulent flow

o

Kolmogorov length scale is smallest length scale of turbulence; 0.1 1mm

o An increase in flow Reynolds number results in longer energy cascade, a

wider range of eddy sizes and the smallest eddies getting smaller


9/23

9

The macroscopic motion in turbulence results in instantaneous fluctuation

of fluid velocity components:

Fig. 4: Average and fluctuating quantities for steady (left) and

unsteady (right) turbulent flows

Turbulence Modeling


10/23

10

Three approaches are most commonly used in the numerical

simulation of turbulent flows:

o Direct Numerical Simulation (DNS)

o Large-Eddy Simulation (LES)

o Reynolds Averaged Navier-Stokes (RANS)

A fourth approach, Favre Averaged Navier-Stokes (FANS) is less

commonly used, but is preferrable to RANS in the numerical

simulation of compressible turbulent flows


11/23

11

LES:

- the smallest scales of the flow are filtered out

- remaining (larger) scales are directly computed by the filtered Navier-

Stokes equations

- the filtered small scales are determined with subgrid-scale models

DNS:

- directly solves the governing eqns. of fluid mechanics, i.e., continuity,

momentum and energy equations

- all relevant scales of turbulence must be accommodated

- number of grid points needed scales as Re9/4 for 3D and as Re2 for 2D

- extremely computationally expensive; impractical for most applications


12/23

12

RANS:

- the instantaneous flow quantities are represented as the sum of a mean

value and a time-dependent fluctuating value (see Fig. 4):

where

and

lim1 ,

lim1

, 0.

classical time-averaged

+

fluctuating


13/23

13

FANS:

- for compressible flow, must account for density

(and temperature) variations

- classical Reynolds averaging leads to triple correlations involving ,

and - the instantaneous flow quantities are represented as the sum of a mean

(density-weighted) value and a time-dependent fluctuating value

1 lim 1 ,,

where and 0.

density-weighted time-averaged

+

fluctuating

NOTE: density and pressure are Reynolds averaged, all other

flow variables are Favre averaged


14/23

14

Substitute these into the instantaneous Navier-Stokes equations

(eg., conservative form):

To derive the RANS equations for incompressible flow, take


15/23

15

Applying the appropriate averaging rules gives the RANS eqns:


16/23

16

These equations contain 6 additional stresses

These stresses are referred to as the Reynolds stresses. In turbulent flow,

o normal stresses , and are never zero.

o shear stresses , and are associated

with correlations between different velocity components. They

could be zero, if the velocity fluctuations are statistically

independent, the time average would be zero.

o shear stresses are usually very large compared to the viscous

stresses.


17/23

17

CLOSURE PROBLEM

o RANS eqns contain 6 extra unknowns, the Reynolds stresses

o simple formulae for these extra stresses are not available due

to the complexity of turbulence

The main task of turbulence modeling is to develop computational

procedures to predict the Reynolds stresses

o must be sufficiently accurate

o must be sufficiently general

o for most engineering applications it is not necessary to resolve the

details of the fluctuations

o more important to study the effects of the turbulence on the mean flow,

i.e., determine the Reynolds stresses

o for most applications, model should be simple and economical to run


18/23

18

TURBULENCE MODELS

Classical models:

zero-equation models (e.g., mixing length; uses algebraic equation)

1-equation models (e.g., Spalart-Allmaras; uses 1 PDE)

2-equation models (e.g., k-, k-; uses 2 PDEs)

Reynolds stress equation model (RSM; uses 6 PDEs)

algebraic stress model (ASM)

A turbulence model is a semi-empirical equation relating the Reynolds

stresses to the mean flow variables, with various constants in the equation

provided from experimental studies

Large eddy Simulation:

based on space-filtered equations (LES)


19/23

19

Boussinesq Assumption (1877)

In 2D, the turbulent shear stress -r represents the macroscopicmomentum exchange due to turbulence. In thex-momentum eqn there are

terms involving the derivatives of the turbulent stresses, i.e., .

There are similar terms in the viscous dissipation terms, i.e.,

.

In the Boussinesq assumption, these terms are combined together by

expressing the Reynolds stresses as

- = t(

+

and - = t(

+

The viscous stresses and turbulent stresses are then combined as

The quantity t is called the eddy or turbulent viscosity.

[

(

(

( + t)[+


20/23

20

Prandtl Hypothesis

o In laminar flow, it is well-established that the mixing of fluid is at the

molecular level. The viscous stresses and heat fluxes are due to

momentum and energy transfer by molecules travelling a distance of the

mean-free path before collision.

o In turbulent flow, it is assumed that lumps of fluid travel a finite distance

before collision and before losing their identity. This finite distance is

referred to as the Prandtl mixing length, lm.

o Prandtl hypothesis:

= ()2

Comparing with the Boussinesq assumption gives an expression for the

eddy viscosity = ()


21/23

21

For 2D RANS: = [()2

+ ()2

]

1/2

Note: Closure problem not yet solved. We have replaced the unknowns (in

2D) , and with the unknown mixing length

.

Table 1: Mixing lengths for 2D turbulent flows

In this Table: y+ = uy/ (wall-distance)

where u

= (w

/)1/2

(friction velocity)


22/23

22

ONE & TWO EQUATION TURBULENCE MODELS

The major kinetic energy of the turbulence exists in the large eddies due to

their intense velocity fluctuation. On the other hand, the energy associated

with smaller eddies is small due to their smaller velocity fluctuations. The

turbulent kinetic energy is obtained from its definition:

One equation models involve a PDE for the velocity scale (transport of

turbulence is accounted for). The velocity scale is characterized by turbulent

kinetic energy (k). The length scale is specified algebraically.

Two equation models involve 2 PDEs, one for velocity scale and the other

for turbulent dissipation (caused by work done by the smallest eddies

against viscous stress diffusion). Turbulent dissipation is defined by:

k- and k- are the most common two-equation turbulence models.


23/23

23

k- TURBULENCE MODEL

Transport equations of both kinetic and dissipation can be derived. These

equations have the following form in 2-D:

wherePk is the production of turbulence, defined as

Pk= ij(Ui/xj)

Typical (standard) constants are

k= 1.0, = 1.3, c1 = 1.44, c2 = 1.92

and the turbulent viscosity is related to by = ck2/ ,where c = 0.09.

chapter 24 - intro to turbulence modeling in cfd

Documents