fluent-adv turbulence 15.0 l01 overview

1 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential

15.0 Release

Lecture 1:

Turbulence Modeling Overview

Turbulence Modeling Using ANSYS Fluent


Outline Background Characteristics of Turbulent Flow

Scales Eliminating the small scales

Reynolds averaging Filtered equations

Turbulence modeling theory RANS turbulence models in Fluent Near wall modeling and transition modeling

Large Eddy Simulation (LES) and other Scale Resolving Simulation (SRS) models LES theory and models Hybrid RANS-LES methods

Case studies Comparison with experiments and DNS


Characteristics of Turbulent Flows Unsteady, irregular (aperiodic) motion in which transported quantities (mass,

momentum, scalar species) fluctuate in time and space

The fluctuations are responsible for enhanced mixing of transported quantities

Instantaneous fluctuations are random (unpredictable, irregular) both in space and time

Statistical averaging of fluctuations results in accountable, turbulence related transport mechanisms

Contains a wide range of eddy sizes (scales) Typical identifiable swirling patterns Large eddies carry small eddies The behavior of large eddies is different in each flow

Sensitive to upstream history

The behavior of small eddies is more universal in nature


Reynolds Number

The Reynolds number is defined as

Turbulent flows occur at large Reynolds numbers

External Flows

Internal Flows

Natural Convection

along a surface

around an obstacle

where

Other factors such as free-stream turbulence, surface conditions, blowing, suction, and other disturbances etc. may cause transition to turbulence at lower Reynolds numbers

(Rayleigh number)

(Prandtl number)

where U and L are representative velocity and length scales for a given flow. L = x, d, dh, etc.


Two Examples of Turbulence

Turbulent boundary layer on a flat plate

Homogeneous, decaying, grid-generated turbulence


Energy Cascade

Larger, higher-energy eddies, transfer energy to smaller eddies via vortex stretching

Larger eddies derive energy from mean flow Large eddy size and velocity on order of mean flow

Smallest eddies convert kinetic energy into thermal energy via viscous dissipation

Rate at which energy is dissipated is set by rate at which they receive energy from the larger eddies at start of cascade


Vortex Stretching

Existence of eddies implies vorticity

Vorticity is concentrated along vortex lines or bundles

Vortex lines/bundles become distorted from the induced velocities of the larger eddies

As the end points of a vortex line randomly move apart Vortex line increases in length but decreases in diameter Vorticity increases because angular momentum is nearly conserved

Most of the vorticity is contained within the smallest eddies

Turbulence is a highly 3D phenomenon


Smallest Scales of Turbulence

Smallest eddy --- the Kolmogorov scales: large eddy energy supply rate is balanced by the small eddy energy

dissipation rate e = -dk/dt

k ( u2+v2+w2 ) is (specific) turbulent kinetic energy [L2 / T2]

e is dissipation rate of k [L2 / T3]

Motion at smallest scales dependent upon dissipation rate, e, and kinematic viscosity, n [L2 / T]

From dimensional analysis, the Kolmogorov scales can be estimated as follows: h = (n 3 / e) 1/4; t = (n /e ) 1/2; v = (ne ) 1/4

length scale time scale velocity scale


Small scales vs Large scales Largest eddy scales:

Assume l is a characteristic size of a larger eddy Dimensional analysis is sufficient to estimate the order of large eddy supply

rate for k as: k / t turnover

The order of t turnover can be estimated as l / k 1/2 (i.e., t turnover is a time scale associated with the larger eddies)

Since e ~ k / t turnover, e ~ k 3/2 / l or l ~ k3/2 / e

Comparing l with h :

where ReT = k 1/2 l / n (turbulence Reynolds number)

4/3

4/3

4/12/3

4/13Re

)/(

)/(T

lklll

nenh1

h

l


Implication of Scales

Consider a mesh fine enough to resolve smallest eddies and large enough to capture mean flow features

Example: 2D channel flow

Ncells ~ ( 4 l / h ) 3 or

Ncells ~ ( 3Ret ) 9/4 where

Ret = ut H / 2n

ReH = 30,800 Ret = 800 Ncells = 4x107 !

H

4/13 )/( enh

ll

l

h


Direct Numerical Simulation DNS is the solution of the time-dependent Navier-Stokes equations

without recourse to modeling

Numerical time step size required, D t ~ t For 2D channel example ReH = 30,800 Number of time steps ~ 48,000

DNS is not suitable for practical industrial CFD DNS is feasible only for simple geometries and low turbulent Reynolds numbers DNS is a useful research tool

j

i

kik

ik

i

x

U

xx

p

x

UU

t

U

tt u

Ht ChannelD

Re

003.02 D


Removing the Small Scales Two methods can be used to eliminate need to resolve small scales: Reynolds Averaging

Transport equations for mean flow quantities are solved All scales of turbulence are modeled Transient solution D t is set by global unsteadiness

Filtering (LES) Transport equations for resolvable scales Resolves larger eddies; models smaller ones Inherently unsteady, D t dictated by smallest resolved eddies

Both methods introduce additional terms that must be modeled for closure


Prediction Methods


RANS Modeling - Velocity Decomposition Consider a point in the given flow field:

( ) ( ) ( )txutxUtxu iii ,,,

u'i

Ui ui

time

u


RANS Modeling - Ensemble Averaging

Ensemble (Phase) average:

Applicable to non-stationary flows such as periodic or quasi-periodic flows involving deterministic structures

( ) ( )( )

N

n

n

iN

i txuN

txU1

,1

lim,


Deriving RANS Equations Substitute mean and fluctuating velocities in instantaneous Navier-

Stokes equations and average:

Some averaging rules: Given f = F + f and y = + y

Mass-weighted (Favre) averaging used for compressible flows

( )

j

ii

jik

iikk

ii

x

uU

xx

pp

x

uUuU

t

uU )()()(

)(

.,0;0;;0; etcFFF yfyyffyff


RANS Equations Reynolds Averaged Navier-Stokes equations:

New equations are identical to original except : The transported variables, U, , etc., now represent the mean flow quantities Additional terms appear:

Rij are called the Reynolds Stresses

Effectively a stress

These are the terms to be modeled

(prime notation dropped) ( )

j

ji

j

i

jik

ik

i

x

uu

x

U

xx

p

x

UU

t

U

jiij uuR

ji

j

i

j

uux

U

x


Turbulence Modeling Approaches Boussinesq approach

isotropic relies on dimensional analysis

Reynolds stress transport models

no assumption of isotropy contains more physics most complex and computationally expensive


The Boussinesq Approach

Relates the Reynolds stresses to the mean flow by a turbulent (eddy) viscosity, t

Relation is drawn from analogy with molecular transport of momentum

Assumptions valid at molecular level, not necessarily valid at macroscopic level t is a scalar (Rij aligned with strain-rate tensor, Sij) Taylor series expansion valid if lmfp | 2U/ y 2|


Modeling t Basic approach made through dimensional arguments Units of nt = t / are [L2/T] Typically one needs 2 out of the 3 scales:

velocity length time

Models classified in terms of number of transport equations solved, e.g., zero-equation one-equation two-equation


Zero Equation Model

Prandtl mixing length model:

Relation is drawn from same analogy with molecular transport of momentum:

The mixing length model:

assumes that vmix is proportional to lmix & strain rate: requires lmix to be prescribed lmix must be calibrated for each problem

Very crude approach, but economical Not suitable for general purpose CFD though can be useful where a very crude

estimate of turbulence is required

i

j

j

iijijijt

x

U

x

USSSl

2

1;22mix

mfpth2

1lv

mixmix 2

1lt v

ijijSSl 2mixmix v


One-Equation Models Traditionally, one-equation models were based on transport equation for k

(turbulent kinetic energy) to calculate velocity scale, v = k

Circumvents assumed relationship between v and turbulence length scale (mixing) Use of transport equation allows history effects to be accounted for

Length scale still specified algebraically based on the mean flow

Very dependent on problem type Approach not suited to general purpose CFD


Turbulence Kinetic Energy Equation Exact k equation derived from sum of products of Navier-Stokes equations with

fluctuating velocities

(Trace of the Reynolds Stress transport equations)

where (incompressible form)

jjii

jjj

iij

jj upuuu

x

k

xx

UR

x

kU

t

k'

2

1e

unsteady &

convective production dissipation molecular

diffusion

turbulent

transport

pressure

diffusion

k

i

k

i

x

u

x

u

ne


Modeled Equation for k The production, dissipation, turbulent transport, and pressure diffusion terms

must be modeled

Rij in production term is calculated from the Boussinesq formula Turbulent transport and pressure diffusion:

e = CD k3/2/ l from dimensional arguments t = CD k2/ e ( recall t k1/2l ) CD , sk and l are model parameters to be specified

Necessity to specify l limits usefulness of this model

Advanced one-equation models use a different approach (without having to specify a length scale):

solving eddy viscosity directly

jk

tjjii

x

kupuuu

s

'

2

1 Using t / s k assumes k can be transported by turbulence similarly to U


Summary Turbulent flows are inherently unsteady, three-dimensional and irregular.

A broad range of time and length scales exist in turbulent flows.

Turbulent flows are governed by the Navier-Stokes equations, but the need to resolve all scales from the dissipative (Kolmogorov) scales to the mean flow scales makes direct simulation too expensive to be feasible for industrial applications.

Reynolds averaging is one of the approaches used to eliminate the small scales. The application of this approach leads to the Reynolds Averaged Navier-Stokes (RANS) equations.

The Reynolds stress terms in the RANS equation require modeling in order to obtain a closed system of equations.

Two branches of RANS modeling are eddy viscosity models (EVM) and Reynolds stress models (RSM).

EVM are based on the Boussinesq approximation. They can be classified in terms of the number of equations that are solved to provide the turbulent viscosity.

fluent-adv turbulence 15.0 l01 overview

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