an introduction to turbulence modeling for cfd
TRANSCRIPT
An Introduction toTurbulence Modeling for CFD
Gerald Recktenwald∗
February 19, 2020
∗Mechanical and Materials Engineering Department Portland State University, Portland, Oregon,[email protected]
Turbulence is a Hard Problem
• Unsteady
• Many length scales
• Energy transfer between scales:→ Large eddies break up into small eddies
• Steep gradients near the wall
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Engineering Model: Flow is “Steady-in-the-Mean” (1)
0 0.5 1 1.5 2t
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
u
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Engineering Model: Flow is “Steady-in-the-Mean” (2)
Reality:
Turbulent flows are unsteady: fluctuations at a point are caused byconvection of eddies of many sizes. As eddies move through the flowthe velocity field changes in complex ways at a fixed point in space.
Turbulent flows have structures – blobs of fluid that move and thenbreak up.
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Engineering Model: Flow is “Steady-in-the-Mean” (3)
Engineering Model:
When measured with a “slow” sensor (e.g. Pitot tube) the velocityat a point is apparently steady. For basic engineering analysis wetreat flow variables (velocity components, pressure, temperature) astime averages (or ensemble averages). These averages are steady(ignorning ensemble averaging of periodic flows).
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Engineering Model: Enhanced Transport (1)
Turbulent eddies enhance mixing.
• Transport in turbulent flow is much greater than in laminar flow: e.g.pollutants spread more rapidly in a turbulent flow than a laminar flow
• As a result of enhanced local transport, mean profiles tend to be moreuniform except near walls.
• Near walls, gradients of velocity, temperature (and other scalars (e.g.,chemical concentration) are steep.
Visualize two-layer: high viscosity in core of pipe and low viscositynear the wall – Cartoon view only!
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Engineering Model: Enhanced Transport (2)
Turbulence models are a way to account for enhanced mixing whiletreating the flow as steady-in-the-mean
• Apparent effect of turbulence is to increase the effective viscosity,thermal conductivity, and diffusivity.
• Enhanced transport coefficients are properties of the flow, not realthermophysical transport coefficients.
• Most commonly used turbulence models provide a way to compute theeffective transport coefficients, e.g. the turbulence viscosity.
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Computational Models (1)
1. Direct Numerical Simulation (DNS)
• No modeling of turbulence• Resolve all time scales and spatial scales• Useful for fundamental research• Don’t scale to highest Re and hard to implement in complex
geometries
Rodi1 cites a DNS simulation in 2015
With increasing computer power, channel-flow simulations with ever-increasing R
were carried out over the years, and at the time of writing, the largest calculation
is that of Lee and Moser (2015) at R = 125, 000. This employed 242× 109 grid
points and ran for several months on the Peta-FLOP/S computer Mira.
1Journal of Hydraulic Engineering, 2017, 143(5)
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World’s Fastest computers as of November 2018
See https://www.top500.org/lists/2018/11/. Mira is now #21 with Rmax = 0.06Rmax(Summit)
Note: the average US home uses ≈1 kW.
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Trend in the Top 500
https://www.top500.org/lists/2018/11/
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Computational Models (2)
2. Large Eddy Simulation (LES)
• Model the small scale (sub-grid) turbulence• Larger, energy-containing eddies are resolved in unsteady flow• Transient data needs to be averaged to obtain engineering quantities• Useful for rigorous engineering analysis• StarCCM+ can do LES and DES (Detached Eddy Simulation)
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Computational Models (3)
3. Reynolds-Averaged Navier Stokes (RANS)
• Turbulence effects are replaced by enhanced mixing – eddy viscosity• Standard approach for basic engineering computations• Extremely limited ability to capture complex structure
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Reynolds Averaged Navier Stokes
RANS
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Reynolds Averaged Navier-Stokes (RANS)
• Average the Navier-Stokes equations in attempt to solve equations thatare steady in the mean.
• Averaging creates additional unknowns: the Reynolds stresses
• Use a turbulence model to compute the Reynolds stresses.
. Simplest models are based on the Boussinesq eddy viscosity
. Additional equations are necessary to compute the eddy viscosity
. The k − ε model is the standard engineering model
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Reynolds Decomposition (1)
Instantaneous velocity is
u(x, y, z, t) = exu(x, y, z, t) + eyv(x, y, z, t) + ezw(x, y, z, t)
Decompose each component into a mean and fluctuating component
u = U + u′ v = V + v′ w = W + w′
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Reynolds Decomposition (2)
Reynolds averaging rules
1. f + g = f + g
2. af = af (a is a constant)
3.∂f
∂s=∂f
∂s
4. fg = f g
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Reynolds Decomposition (3)
Substitute into the Navier-Stokes equations and average. Consider one ofthe convection terms in the x-direction momentum equation
∂
∂x(ρuu) =
∂
∂x
[ρ(U + u′)(U + u′)
]rule 3
=∂
∂x
[ρ(UU + 2Uu′ + u′u′
)]rule 2, with ρ = constant
=∂
∂x
(ρUU + ρu′2
)by definition of U , and Uu′ = Uu′ = 0
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Reynolds Decomposition (4)
The averaging process has the important property that although u′ = 0,
u′u′ 6= 0
Terms like u′iu′j are called Reynolds stresses, and arise from the process of
Reynolds averaging of the momentum equations.
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Reynolds Decomposition (5)
There are six Reynolds stresses, which can be represented by the followingmatrix. u′u′ u′v′ u′w′
u′v′ v′v′ v′w′
u′w′ v′w′ w′w′
The matrix is symmetric because u′v′ = v′u′ and v′w′ = w′v′. TheReynolds stresses are field variables, i.e., each term in the matrix is afunction of space in the flow.
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Turbulence Kinetic Energy
Another important quantity is the turbulence kinetic energy
k =1
2u′ku
′k =
1
2
(u′u′ + v′v′ + w′w′
)(1)
If the turbulence is isotropic, then u′u′ = v′v′ = w′w′, and2
k =3
2u′u′ (2)
2Equality of normal stresses is a consequence of isotropy, not the definition of it.
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Turbulence Intensity(1)
Turbulence Intensity is a measure of the velocity associated with thefluctuations in the flow relative to the mean flow. We start by defining theturbulence intensity in each coordinate direction
TIx =
√u′u′
Ux,0, T Iy =
√v′v′
Uy,0, T Iz =
√w′w′
Uz,0, (3)
where U0 is a representative length scale
The total turbulence intensity is
TI =
√1
3(u′u′ + v′v′ + w′w′)
U0(4)
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Turbulence Intensity(2)
If the turbulence is isotropic, then from
TI =
√1
3(u′u′ + u′u′ + u′u′)
U0=
√(u′u′)
U0(5)
and from Equation (2),
TI =
√2
3k
U0(6)
thus, another interpretation of turbulence intensity is that it is a measureof (the square root of) local turbulence kinetic energy.
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Turbulence Models
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Kinds of turbulence models
1. Large eddy simulation
a. Resolve unsteady large scale motionsb. Model small scale (sub-grid scale) motions
2. Reynolds stress models
a. Treat flow as steady (no large scale motions)b. Solve model equations for the Reynolds stresses
3. Two-equation models
a. Treat flow as steady (no large scale motions)b. Use Boussinesq eddy viscosity to represent Reynolds stressesc. Solve model equations to compute eddy viscosity
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Boussinesq model (1)
The viscous stress tensor is
τij = µ
[(∂Uj∂xi
+∂Ui∂xi
)− 2
3δij
∂Uk∂xk
](7)
Assume (wish, hope) that the Reynolds stresses have the same form asthe viscous stresses.
−ρ u′iu′j = µt
[(∂Uj∂xi
+∂Ui∂xi
)− 2
3δij
∂Uk∂xk
]− 2
3ρ δijk (8)
where k is the turbulence kinetic energy
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Boussinesq model (2)
In practice, the Boussinesq model involves replacing the viscosity in themomentum equations with an effective viscosity
µeff = µ+ µt
where µ is the physical or moleclar viscosity, and µt is a turbulenceviscosity simulates the enhanced mixing due to turbulence.
General observations
• µt is a point value. µt has no direction, so therefore the Boussinesqmodel assumes that the turbulence is isotropic
• µt needs to be estimated by some other model
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Prandtl’s Mixing Length Hypothesis
Prandtl made the direct analogy with the the molecular model of viscosityfrom the kinetic theory of gases.
molecular viscosity of gases turbulence viscosity
µ =1
3ρ`fVm µt = ρ`mVt
`f = mean free path `m = mixing length
Vm = velocity of molecules Vt = velocity scale of turbulence
The mixing length is taken as an appropriate length scale for the flow.
The mixing length can vary in proportion to distance from a wall.
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Estimate of Eddy Viscosity for Pipe Flow (1)
We can estimate the magnitude of the eddy viscosity in a pipe. Prandtl’smodel is
µt ∼ ρVt`mTurbulent fluctuations are small compared to the mean velocity in thepipe, say,
0.01 ≤ u′
U≤ 0.15
To estimate µt, take u′/U ∼ 0.1 and Vt ∼ u′ so that
Vt ∼ 0.1U
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Estimate of Eddy Viscosity for Pipe Flow (2)
To make a somewhat arbitrary choice of a single length scale thatcharacterizes the turbulent mixing, take
`m ∼D
4
Combining the preceding expressions gives the following estimate ofturbulence viscosity
µt = ρ(0.1U)(0.25D) = 0.025ρUD
From the estimate of µt we can compute an effective Reynolds number
Reeff =ρUD
µt=
ρUD
0.025ρUD= 40
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The k − ε model
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How do we compute µt?
Short answer:
µt = Cµρk2
εwhere k is the turbulence kinetic energy, ε is the turbulence dissipationrate, and Cµ is a constant.
We need to estimate k and ε
In the k − ε model, we solve two transport equations: one for the k fieldand one for the ε field.
Then, the effective viscosity in the Boussinesq model is
µeff = µ+ µt
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k − ε model
Standard equations. See [3].
Transport equation for k
Dk
Dt=
∂
∂xi
[µeff
σk
∂k
∂xi
]︸ ︷︷ ︸
diffusion
+
[µt
(∂Ui∂xj
+∂Uj∂xi
)− 2
3ρ δijk
]∂Uj∂xi︸ ︷︷ ︸
production
− cDρ k3/2
`m︸ ︷︷ ︸dissipation
Transport equation for ε
Dε
Dt=
∂
∂xi
[µeff
σε
∂ε
∂xi
]︸ ︷︷ ︸
diffusion
+ cε,1
[µt
(∂Ui∂xj
+∂Uj∂xi
)− 2
3ρ δijk
]∂Uj∂xi︸ ︷︷ ︸
production
− cε,1ρ ε2
k︸ ︷︷ ︸dissipation
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What is ε?
Use a simple scaling argument to set an upper bound on the dissipationrate (see Panton [1, § 22.10]).
velocity scale of largest eddy ∼ u0
kinetic energy of the eddy ∼ u20
length scale of the eddy ∼ L
turn-over time of the eddy ∼ L
u0
An upper estimate of the dissipation rate, ε is
ε ∼ kinetic energy of an eddy
time for one rotation=
u20
L/u0=u3
0
L(9)
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What is ε?
But . . . dissipation happens on the smallest scale.
Thus, ε ∼ u30
Lgives an upper bound, but not a good scale
At the smallest scales the turbulence is (tends to be) isotropic. Theturbulence kinetic energy of isotropic turbulence is
k =3
2u′u′ isotropic turbulence
Thus, we can estimate the velocity scale as
|u′| ∼√k so that ε ∼ k3/2
L
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What is ε?
Introduce a dissipation length scale, `ε such that the dissipation rate is
ε ∼ CDk3/2
`ε(10)
where CD corrects (,) any error in the scale estimate.
But. . . now we need to estimate `ε
Turn Equation (10) around
`ε ∼ CDk3/2
ε(11)
Use Prandtl’s mixing length hypothesis to estimate ε from the mixinglength
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Prandtl’s mixing length hypothesis
`m is the length scale for the eddies responsible for mixing, i.e., the mostenergetic eddies. `m > `ε.
`m = C ′`ε
Prandtl’s estimate of the turbulence viscosity (eddy viscosity) is
µt ∼ ρVt`m
where Vt is the velocity scale
At the dissipation scale, which is isotropic
Vt ∼√k
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Prandtl’s mixing length hypothesis
Combine estimates of Vt, `ε in Prandtl’s expression for eddy viscosity
µt = Cρ
(CD
k3/2
ε
)(k1/2
)or
µt = Cµρk2
ε(12)
where Cµ is a constant adjusted from experimental values
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Calculation of the eddy viscosity, µt, in a CFD code
1. Solve equations for velocities and pressure, using current guess at µeff.
2. Solve the k and ε equations
3. At each point in the flow compute
µt = Cµρk2
ε
4. Update µeff = µ+ µt.
5. Return to step 1 until convergence.
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Wall Functions
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Wall Functions
In many engineering flows, we don’t have the computational power toresolve the velocity profile near the wall. One solution is to fudge it withwall functions
Expandthe y scale
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Wall variables
Scaling arguments lead to the inner (near-wall) coordinates
y+ =yu∗ν
(13)
u∗ =
√τwρ
friction velocity (14)
u+ =u
u∗(15)
In the viscous sublayeru+ = y+
In the log-law (inertial) sublayer
u+ = c1 ln y+ + c2 (16)
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Wall treatment
k-ε model is used to compute μeff
in the central region of the flow
Wall functions are used to compute μeff
near solid surfaces.
Remember the prism layer cells in StarCCM+?
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Wall treatment
Adjust the near-wall mesh spacing so that near-boundary cells have goodvalues of y+, where good depends on the wall treatment.
In StarCCM+, the value of y+I is called the “wall y+”.
I
B
yI
ures
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Wall treatment models in StarCCM+
The main wall treatment models in StarCCM+ are
• High y+: Classic wall treatment
• Low y+: Extend cells into the viscous sublayer
• All y+: Recommended model that blends low y+ and high y+
Wall treatment is considered in another slide deck.
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References
[1] Ronald L. Panton. Incompressible Flow. Wiley, New York, secondedition, 1996.
[2] Stephen B. Pope. Turbulent Flows. Cambridge University Press,Cambridge, UK, 2000.
[3] John C. Tannehill, Dale A. Anderson, and Richard A. Pletcher.Computational Fluid Mechanics and Heat Transfer. Taylor andFrancis, Washington, D.C., second edition, 1997.
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