®
COMSOL Multiphysics

Turbulent Flow ModelingTurbulent Flow

Determining the Flow Regime, Re At low 𝑅𝑅𝑅𝑅 (≪ 1): creeping (Stokes) flow

Viscous forces tend to damp out all flow disturbances
Reversible smooth flow pattern At intermediate 𝑅𝑅𝑅𝑅 ~1 − 2000∗ : laminar flow
Inertial forces become increasingly important

Viscous forces are confined to boundary layers, shear layers, and wakes
Regular, smooth flow pattern At high 𝑅𝑅𝑅𝑅 (> 4000∗ ): turbulent flow

Flow disturbances grow by nonlinear interactions producing a cascade of eddies (vortices)
Viscous damping is active everywhere but only has a significant effect on thesmallest eddies
Disordered (chaotic) flow pattern*Critical Re of 2000 has been mentioned above as an example considering an internal channel flow

Reynolds Number Examples𝑥𝑥

𝑈𝑈0 𝑥𝑥𝑅𝑅𝑅𝑅𝑥𝑥 =𝜈𝜈
𝑈𝑈0 𝐿𝐿0𝑅𝑅𝑅𝑅𝐿𝐿 =𝜈𝜈

𝑈𝑈0 𝐷𝐷𝐻𝐻4𝐴𝐴, 𝐷𝐷𝐻𝐻 ≡𝑅𝑅𝑅𝑅𝐷𝐷ℎ =𝜈𝜈𝑃𝑃
𝑈𝑈0 𝐷𝐷𝑅𝑅𝑅𝑅𝐷𝐷 =𝜈𝜈

𝐿𝐿0𝐷𝐷

Determining the Flow Regime, Ma
 For 𝑀𝑀𝑀𝑀 = 0: formally incompressible flow

Speed of sound is infinite (hypothetical case), instantaneous spread of pressure disturbances For 0 < 𝑀𝑀𝑀𝑀 < 0.3: weakly compressible flow
Density changes due to pressure by max. 5%

Density changes can also occur due to the dependency on temperature For 𝑀𝑀𝑀𝑀 > 0.3: compressible flow

Thermodynamic effects not negligible
Transonic, supersonic flow with shock waves

Choosing the Right Fluid Flow Interface
𝑴𝑴𝑴𝑴 < 𝟎𝟎. 𝟑𝟑
𝑅𝑅𝑅𝑅 ≪ 1
𝑅𝑅𝑅𝑅~1 − 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅 > 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅 > 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅 > 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐

𝑴𝑴𝑴𝑴 > 𝟎𝟎. 𝟑𝟑
𝑅𝑅𝑅𝑅 < 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑅𝑅 > 𝑅𝑅𝑒𝑒𝑐𝑐𝑐𝑐
𝑅𝑅𝑒𝑒 → ∞Laminar, Transitional, and Turbulent Flowu(y)
u(y)

u∞

Log layer
Buffer layer
Viscous sublayer

Laminar

TransitionTurbulent

Introduce turbulent viscosity term (𝜇𝜇 𝑇𝑇 )

 Reynolds Averaged Navier-Stokes (RANS) equations:ρ()∂uT+ ρ u ⋅ ∇u + ∇ ⋅ (ρ u′ ⊗ u′) = −∇P + µ ∇u + (∇u ) + F∂tρ (∇ ⋅ u ) = 0

 Model the turbulent fluctuations using turbulent viscosity µT :(− µT ∇u + (∇u )

ρT)

()

∂uT+ ρ u ⋅ ∇u = −∇P + (µ + µT ) ∇u + (∇u ) + F∂tρ (∇ ⋅ u ) = 0

In COMSOL: 9 different turbulence models to solve for 𝜇𝜇 𝑇𝑇 = spf.muTComputational Cost forTurbulence Models
 Algebraic yPlus and L-VEL Spalart–Allmaras
 k-epsilon
 k-omega, SST and
low Re k-epsilon
 Realizable k-epsilon
 v2-fIncreasing cost

Algebraic Turbulence Models
 Algebraic turbulence models are faster and more robust but, generally less accurate thantransport-equation turbulence models
 Reynolds number based on local velocity magnitude and wall distance𝜌𝜌𝑈𝑈𝑈𝑈̅
𝑈𝑈 𝜌𝜌𝑢𝑢̅ 𝜏𝜏 𝑦𝑦== 𝑢𝑢+ 𝑦𝑦 + ,𝑅𝑅𝑅𝑅 =𝜇𝜇̅𝑢𝑢𝜏𝜏 𝜇𝜇̅𝜇𝜇̅𝜇𝜇̅ + 𝜇𝜇 𝑇𝑇 = +𝜕𝜕𝑢𝑢 ⁄𝜕𝜕𝑦𝑦 + Algebraic yPlus model prescribes 𝑢𝑢+ 𝑦𝑦 + from an extension of the logarithmic wall law andsolves for 𝑦𝑦 +
 L-VEL model prescribes 𝑦𝑦 + 𝑢𝑢+ from a different extension of the logarithmic wall law andsolves for 𝑢𝑢+
NOTE: No turbulent transport

no wake interference and no in/outflow of turbulenceTransport-Equation Turbulence Models (2-Eqn.)
Transport-equation turbulence models solve for the transport of turbulence quantities.
 k- ε : Transport equations for 𝑘𝑘 and ε. General purpose model with wall functions forsmooth and rough walls.
 Realizable k- ε : Realizability constraints are built into the model parameters.

 k- ω : Transport equations for 𝑘𝑘 and ω. More accurate than the k-epsilon model close towalls and in recirculation regions. Wall functions for smooth and rough walls but can also
resolve the flow close to walls.
 SST: Combines the k- ω model close to walls with the k- ε model in the outer flow region.
Can resolve the wall layer.
 Low Re k- ε : Wall resolved k- ε model.

Transport-Equation Turbulence Models (1 and 3 Eqn.)
 Spalart–Allmaras: Transport equation for 𝜐𝜐�𝑇𝑇 . Wall resolved turbulence model developed
for aerodynamic applications.

 v2-f: An extended low-Re k-ε model. In addition to 𝑘𝑘 and ε, it also solves for the wall normal
fluctuations (v2) and an elliptic relaxation function (f). Good predictive capabilities for flow
over curved surfaces. Can resolve the wall layer.

Wall Treatment
𝑢𝑢+ 𝑦𝑦 +Wall offset

𝑢𝑢+ =1log 𝑦𝑦 + + B
𝜅𝜅

𝑢𝑢+ = 𝑦𝑦 +

 Wall functions: A wall offset is applied. The computational domain always start in the loglayer.
 Low-Reynolds-number treatment: The viscous sublayer must be resolved.
 Automatic wall treatment: A wall offset is applied. The computational domain can start
anywhere in the boundary layer.

Resolution for Low-Re Wall Treatment

Algebraic yPlus
L-VEL
Accurate wall stressk-omega
Low-Re k-epsilon
SSTMay still converge

Algebraic Turbulence Models
 Turbulent viscosity evaluated from theReynolds number based on local speed andwall distance
Algebraic yPlus
L-VEL Advantages:
RobustComputationally inexpensive

 Disadvantage:
Less accurate

Surface temperature and streamlines in a benchmark for electronic cooling.

Transport Equations for Turbulence: Two Equation Models
k-ε models
The standard k-ε model withrealizability constraints
The realizable k-ε modelk-ω model
The revised Wilcox k-ω model (1998) withrealizability constraints

SST model
Combines the k-ε model in the free streamwith the k-ω model close to walls

Low Re k-ε model
The AKN k-ε modelBenchmark model of a NACA0012 airfoil
using the SST turbulence model.

Transport Equation Turbulence Models
Spalart-Allmaras
One-equation model with rotationalcorrection, developed for
aerodynamic applications

v2-f model
An extension of the k-ε model which accounts
for turbulence anisotropy by solving for thewall-normal turbulence velocity fluctuations

Flow in a hydrocylone. A typical application where v2-f gives superior
results over two-equation models such as k-ε or SST.

Wall Treatment
Wall functions for smooth and rough walls
Robust, applicable for coarse meshes, but limited
accuracy
Supported by k-ε, Realizable k-ε and k-ω

Low-Reynolds-number treatment
Resolves the flow all the way down towalls, accurate
Supported by all turbulence models exceptk-ε and realizable k-ε

Low ReWall functions

Automatic wall treatment
Switches between low-Re treatment and wall functions,accuracy according to local
mesh resolution
Inherits the robustness provided bywall functions
Default for all turbulence models except k-ε and realizablek-ε

Flow in a pipe elbow simulated with the k-ω model.

Large Eddy Simulation
 The larger three-dimensional, unsteadyeddies are resolved, whereas the effect ofthe smaller eddies is modeled.
 Simulations must be three-dimensional andtime dependent.
 Computationally demanding The three current interfaces are based on
variational multiscale methods
RBVM
RBVMWV
Smagorinsky

LES of turbulent channel flow at 𝑅𝑅𝑅𝑅𝑇𝑇 = 395

Detached Eddy Simulation
 Hybrid method between RANS and large
eddy simulation (LES)Simulations must be
three-dimensional and time dependent.
 RANS is used in the boundary layer and LES
is used elsewhere.
 Requires a less dense boundary layer meshcompared with a pure LES which
substantially reduces the memory
requirement and computation
 Spalart–Allmaras turbulence model with thefollowing LES models
RBVM
RBVMWV
Smagorinsky

Flow over an Obstacle with DES

DemoOutlet

Model Definition Laminar flow in water Fully developed flow at the inlet
 Pressure condition at the outlet
 No-slip conditions at walls
 Symmetry conditions at the twolateral surfaces
 Why?
Typical expansion found in manysystems, e.g. in medtech
Benchmark with flow separation

InletWalls

Symmetry

SymmetryDue to symmetry, we only have to model one eight of the model domain, provided that the flow is steady andthat the inlet flow is perpendicular to the inlet boundary.

WallsSymmetry

OutletModel Definition
 Extension of previous laminar flow
example:Elongate outlet section to avoid
recirculation zone close to outlet

 Turbulent flow in waterk-ε turbulence model

Inlet

Symmetry

 Fully developed flow at the inlet
 Pressure condition at the outlet
 Wall functions at walls
 Symmetry conditions at the twolateral surfaces

It is possible to change the model settings from laminar flow to turbulent flow and also chose turbulence model.

Results
 Flow and pressure fields
 Length of recirculation zone Note:Recirculation reaches the outlet-> elongate the outlet section

u0 = 0.6[m/s]