fluent-adv turbulence 15.0 l01 overview
DESCRIPTION
Ansys Fluent Turbulence OverviewTRANSCRIPT
-
1 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
15.0 Release
Lecture 1:
Turbulence Modeling Overview
Turbulence Modeling Using ANSYS Fluent
-
2 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
3 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
4 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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.
-
5 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
Two Examples of Turbulence
Turbulent boundary layer on a flat plate
Homogeneous, decaying, grid-generated turbulence
-
6 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
7 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
8 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
9 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
10 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
11 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
12 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
13 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
Prediction Methods
-
14 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
RANS Modeling - Velocity Decomposition Consider a point in the given flow field:
( ) ( ) ( )txutxUtxu iii ,,,
u'i
Ui ui
time
u
-
15 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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,
-
16 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
17 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
18 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
19 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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|
-
20 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
21 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
22 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
23 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
24 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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
-
25 2014 ANSYS, Inc. April 22, 2014 ANSYS Confidential
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.