Turbulence Training Manual
Chapter 9 Turbulence
Introduction to CFX
Turbulence
What is Turbulence?
Training Manual
• Unsteady, irregular (non-periodic) motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space – Identifiable swirling patterns characterize turbulent eddies – Enhanced mixing (matter, momentum, energy, etc.) results
• Fluid properties and velocity exhibit random variations – Statistical averaging results in accountable, turbulence related transport
mechanisms – This characteristic allows for turbulence modeling
• Contains a wide range of turbulent eddy sizes (scales spectrum) – The size/velocity of large eddies is on the order of the mean flow • Large eddies derive energy from the mean flow – Energy is transferred from larger eddies to smaller eddies • In the smallest eddies, turbulent energy is converted to internal energy by viscous dissipation
Turbulence
Is the Flow Turbulent?
Training Manual
• Flows can be characterized by the Reynolds Number, Re External Flows Re x
≥ 500,000
Re d ≥ 20,000
where along a surface around an obstacle
Internal Flows Re d h ≥ 2,300
ρU L Re L = µ L = x, d , d h , etc.
Other factors such as free-stream turbulence, surface conditions, and disturbances may cause transition to turbulence at lower Reynolds numbers
Natural Convection Ra Pr
≥ 10
9
where
β g L3 ∆T ρ 2 C p β g L3 ∆T Ra = = is the Rayleigh number να µ k ν µ C p Pr = = is the Prandtl number α k
Turbulence
Observation by O. Reynolds
Training Manual
Laminar (Low Reynolds Number)
Transition (Increasing Reynolds Number)
Turbulent (Higher Reynolds Number)
Turbulence
Turbulent Flow Structures
Small structures
Training Manual
Large structures
Energy Cascade Richardson (1922)
Turbulence
Governing Equations
Training Manual
Conservation Equations Continuity
∂ ρ ∂ + ( ρ ui ) = 0 ∂t ∂ xi
Momentum
∂ ∂ ∂P ∂τ ij + ( ρ ui ) + ( ρ ui u j ) = − ∂t ∂ x j ∂ xi ∂ x j
Energy
∂ ∂ ∂P ∂ ∂T + ( ρ htot ) + ( ρ htot u j ) = (uiτ ij + λ ) ∂t ∂ x j ∂t ∂ x j ∂ x j
where
∂ui ∂u j 2 ∂ui + + τ ij = µ δ ij ∂ x j ∂ xi 3 ∂ x j
htot = h +
1 2
2
ui
Note that there is no turbulence equation in the governing Navier-Stokes equations!
Turbulence
Overview of Computational Approaches
Training Manual
• Direct Numerical Simulation (DNS) – Theoretically, all turbulent (and laminar / transition) flows can be simulated by
numerically solving the full Navier-Stokes equations – Resolves the whole spectrum of scales. No modeling is required – But the cost is too prohibitive! Not practical for industrial flows
• Large Eddy Simulation (LES) type models – Solves the spatially averaged N-S equations – Large eddies are directly resolved, but eddies smaller than the mesh are modeled – Less expensive than DNS, but the amount of computational resources and efforts
are still too large for most practical applications
• Reynolds-Averaged Navier-Stokes (RANS) models – Solve time-averaged Navier-Stokes equations – All turbulent length scales are modeled in RANS • Various different models are available
– This is the most widely used approach for calculating industrial flows
• There is not yet a single, practical turbulence model that can reliably predict all turbulent flows with sufficient accuracy
Turbulence
RANS Modeling – Time Averaging
Training Manual
• Ensemble (time) averaging may be used to extract the mean flow properties from the instantaneous ones – The instantaneous velocity,
ui (x, t ) = lim
N → ∞
1
N
u i,
N
∑
ui
(n )
(
, t )
x
n =1
ui (x, t ) = ui (x, t ) + ui′(x, t ) Instantaneous component
is split into average and fluctuating components
Time-average Fluctuating component component
ui′(x, t ) ui (x, t )
ui (x, t )
Example: Fully-Developed Turbulent Pipe Flow Velocity Profile
• The Reynolds-averaged momentum equations are as follows
∂ui ∂ui ∂ p ∂ ∂ui ∂ Rij ρ + uk =− + µ + ∂ xk ∂ xi ∂ x j ∂ x j ∂ x j ∂t
Rij = −ρui′u ′ j (Reynolds stress tensor)
– The Reynolds stresses are additional unknowns introduced by the averaging
procedure, hence they must be modeled (related to the averaged flow quantities) in order to close the system of governing equations
Turbulence
RANS Modeling – The Closure Problem
Training Manual
• Closure problem: Relate the unknown Reynolds Stresses to the known mean flow variables through new equations – The new equations are the turbulence model
• Equations can be: – Algebraic – Transport equations
• All turbulence models contain empiricism – Equations cannot be derived from fundamental principles – Some calibrating to observed solutions and “intelligent
guessing” is contained in the models
Turbulence
RANS Modeling – The Closure Problem
Training Manual
• The RANS models can be closed in one of the following ways (1) Eddy Viscosity Models (via the Boussinesq hypothesis)
Rij
∂ui ∂u j 2 ∂uk 2 − µT = −ρ ui′u′ j = µT + δ − ρ k δij ij 3 ∂ x j ∂ xi 3 ∂ xk
– Boussinesq hypothesis – Reynolds stresses are modeled using an eddy (or turbulent) viscosity, μ T. The hypothesis is reasonable for simple turbulent shear
flows: boundary layers, round jets, mixing layers, channel flows, etc.
(2) Reynolds-Stress Models (via transport equations for Reynolds stresses) – Modeling is still required for many terms in the transport equations – RSM is more advantageous in complex 3D turbulent flows with large streamline
curvature and swirl, but the model is more complex, computationally intensive, more difficult to converge than eddy viscosity models
Turbulence
Available Turbulence Models
Training Manual
• A large number of turbulence models are available in CFX, some have very specific applications while others can be applied to a wider class of flows with a reasonable degree of confidence RANS Eddy-viscosity Models:
RANS Reynolds-Stress Models:
1) Zero Equation model.
1) LRR Reynolds Stress
2) Standard k- ε model.
2) QI Reynolds Stress
3) RNG k-ε model.
3) Speziale, Sarkar and Gatski Reynolds Stress
4) Standard k- ω model.
4) SMC-ω model
5) Baseline (BSL) zonal k- ω based model.
5) Baseline (BSL) Reynolds' Stress model
6) SST zonal k-ω based model. 7) (k-ε)1E model.
Eddy Simulation Models: 1) Large Eddy Simulation (LES) [transient] 2) Detached Eddy Simulation (DES)* [transient] 3) Scale Adaptive Simulation SST (SAS)* [transient]
*
Not available in the ANSYS CFD-Flo product
Turbulence
Turbulence Near the Wall
Training Manual
• The velocity profile near the wall is important: – Pressure Drop – Separation – Shear Effects – Recirculation
• Turbulence models are generally suited to model the flow outside the boundary layer • Examination of experimental data yields a wide variety of results in the boundary layer
The above graph shows non- dimensional velocity versus non- dimensional distance from the wall. Different flows show different boundary layer profiles.
Turbulence
Turbulence Near the Wall
Training Manual
• By scaling the variables near the wall the velocity profile data takes on a predictable form (transitioning from linear to logarithmic behavior) Scaling the non-dimensional velocity and non- dimensional distance from the wall results in a predictable boundary layer profile for a wide range of flows
Linear Logarithmic
• Since near wall conditions are often predictable, functions can be used to determine the near wall profiles rather than using a fine mesh to actually resolve the profile – These functions are called wall functions
Turbulence
Turbulence Near the Wall
Training Manual
• Fewer nodes are needed normal to the wall when wall functions are used y
y u
Wall functions used to resolve boundary layer Boundary layer
u
Wall functions not used to resolve boundary layer
Turbulence
Turbulence Near The Wall
Training Manual
• y+ is the non-dimensional distance from the wall – It is used to measure the distance of the first node away from the wall
y u
Boundary layer
y+
• Wall functions are only valid within specific y + values • If y+ is too high the first node is outside the boundary layer and wall functions will be imposed too far into the domain • If y+ is too low the first node will lie in the laminar (viscous) part of the boundary layer where wall functions are not valid
Turbulence
Turbulence Near the Wall
Training Manual
• In some situations, such as boundary layer separation, wall functions do not correctly predict the boundary layer profile
Wall functions applicable
Wall functions not applicable
• In these cases wall functions should not be used • Instead, directly resolving the boundary layer can provide accurate results • Not all turbulence models allow the wall functions to be turned off
Turbulence
k-epsilon Model
Training Manual
• Standard k- Model – The “industrial CFD” standard since it offer a good compromise between
numerical effort and computational accuracy – Wall functions are always used – y+ should typically be < 300 for the wall functions to be valid – There is no lower limit on y+ • CFX uses Scalable wall functions • If your mesh results in y+ values below the valid range of the wall functions, the nodes nearest the wall are effectively ignored • This ensures valid results, within the model limitations, but is a waste of mesh – Known limitations: • Separation generally under predicted since wall functions are used • Inaccuracies with swirling flows and flows with strong streamline curvature
Turbulence
k-omega Model • k-
Training Manual
Model
– One of the advantages of the k-
formulation is the near wall treatment for low-Reynolds number computations • Here “low-Reynolds” refers to the turbulent Reynolds number, which is low in the viscous sub-layer, not the device Reynolds number • In other words “low-Reynolds number computations” means the near wall mesh is fine enough to resolve the laminar (viscous) part of the boundary layer which is very close to the wall
– A low-Reynolds number k- model only requires y+ <= 2 • If a low-Re k- model were available, it would require a much small y+ – In industrial flows, even y+ <= 2 cannot be guaranteed in most
applications and for this reason, a new automatic near wall treatment was developed for the k- models
Turbulence
k-omega Model • k-
Training Manual
Model (continued)
– The Automatic wall treatment for the k-
models switches between a lowReynolds number formulation (i.e. direct resolution of the boundary layer) at low y+ values and a wall function approach at higher y+ values
– This lets you take advantage of a fine near-wall mesh when present
Airfoil at Mach 0.5 showing the mesh and y+ values. y+ values are >2. A finer near wall mesh is required to achieve y+ < 2.
Turbulence
SST Model
Training Manual
• Shear Stress Transport (SST) Model – The SST model is based on the k-
model and has the same automatic
wall treatment – It accounts for the transport of the turbulent shear stress and gives highly accurate predictions of the onset and the amount of flow separation – This is a good default choice
k-ε fails to predict separation
SST result and experiment
Experiment Gersten et al.
Turbulence
y+ for the SST and k-omega Models
Training Manual
• When using the SST or k- models y+ should be < 300 so that the wall function approach is valid • This will not take advantage of the low-Reynolds formulation, which is necessary for accurate separation prediction • However, the model can still be used on these coarser near-wall mesh and produce valid results, within the limitations of the wall functions
• To take full advantage of the low-Reynolds formulation y+ should be < 2
Turbulence
Estimating y+
Training Manual
• It is useful to estimate y + before obtaining a solution – Saves time!
• Use the following formula based on flow over a flat plate:
∆ y = L y
+
−13 / 14
74 Re L
is the actual distance between the wall and first node – L is a flow length scale – y+ is the desired y + value – Re L is the Reynolds Number based on the length scale L –
∆ y
• See the documentation for a derivation of this formula – ANSYS CFX-Solver Modeling Guide >> Turbulence and Near-Wall
Modeling >> Modeling Flow Near the Wall >> Guidelines for Mesh Generation
Turbulence
Other Turbulence Models
Training Manual
• When RANS models are not adequate, Eddy Simulation models can be used – As already mentioned, these are more computationally expensive
• Large Eddy Simulation (LES) – Resolves the large eddies, models the small eddies – Problem: Requires a very fine boundary layer mesh, making it
impractical for most flows
• Detached Eddy Simulation (DES) – Uses a RANS model in the boundary layer, switches over to LES in the
bulk flow – A “standard” boundary layer mesh can be used – Problem: the RANS to LES switch depends on the mesh, which can give unphysical results on the “wrong” mesh
• Scale-Adaptive Simulation (SAS) – Like DES, but without the mesh dependency problems
Turbulence
Inlet Turbulence Conditions
Training Manual
• Unless turbulence is being directly simulated, it is accounted for by modeling the transport of turbulence properties, for example k and ε • Similar to mass and momentum, turbulence variables require boundary condition specifications – Several options exist for the specification of turbulence quantities at inlets
(details on next slide)
• Unless you have absolutely no idea of the turbulence levels in your simulation (in which case, you can use the Medium (Intensity = 5%) option), you should use well chosen values of turbulence intensities and length scales – Nominal turbulence intensities range from 1% to 5% but will depend on
your specific application
• The default turbulence intensity value of 0.037 (that is, 3.7%) is sufficient for nominal turbulence through a circular inlet, and is a good estimate in the absence of experimental data
Turbulence
Inlet Turbulence Conditions
Training Manual
• Default Intensity and Autocompute Length Scale –
The default turbulence intensity of 0.037 (3.7%) is used together w ith a computed length scale to approximate inlet values of k and . The length scale is calculated to take into account varying levels of turbulence. In general, the autocomputed length scale is not suitable for external flows
• Intensity and Autocompute Length Scale –
This option allows you to specify a value of turbulence intensity but the length scale is still automatically computed. The allowable range of turbulence intensities is restricted to 0.1%-10.0% to corre spond to very low and very high levels of turbulence accordingly. In general, the autocomputed length scale is not suitable for external flows
• Intensity and Length Scale –
You can specify the turbulence intensity and length scale directly, from which values of k and ε are calculated
• Low (Intensity = 1%) –
This defines a 1% intensity and a viscosity ratio equal to 1
• Medium (Intensity = 5%) – –
This defines a 5% intensity and a viscosity ratio equal to 10 This is the recommended option if you do not have any information about the inlet turbulence
• High (Intensity = 10%) –
This defines a 10% intensity and a viscosity ratio equal to 1 00
• Specified Intensity and Eddy Viscosity Ratio –
Use this feature if you wish to enter your own values for intensity and viscosity ratio
• k and Epsilon –
Specify the values of k and ε directly
• Zero Gradient –
Use this setting for fully developed turbulence conditions
Turbulence
Example: Pipe Expansion with Heat Transfer
Training Manual
• Reynolds Number ReD= 40750 • Fully Developed Turbulent Flow at Inlet • Experiments by Baughn et al. (1984)
.
q=const
.
H
q=0 D d
Inlet
axis H
40 x H
Outlet
Turbulence
Example: Pipe Expansion with Heat Transfer
Training Manual
• Plot shows dimensionless distance versus Nusselt Number • Best agreement is with SST and k-omega models which do a better job of capturing flow recirculation zones accurately
Turbulence
Summary: Turbulence Modeling Guidelines
Training Manual
• Successful turbulence modeling requires engineering judgment of: – Flow physics – Computer resources available – Project requirements • Accuracy • Turnaround time – Near-wall treatments
• Modeling procedure – – – –
Calculate characteristic Re and determine whether the flow is turbulent Estimate y+ before generating the mesh The SST model is good choice for most flows Use the Reynolds Stress Model or the SST model with Curvature Correction (see documentation) for highly swirling, 3-D, rotating flows
