the k turbulence model - computer action...
TRANSCRIPT
The k − ε Turbulence Model
ME 448/548 Lecture Notes
Gerald Recktenwald∗
August 3, 2009
1 Overview
This article gives a very brief overview of the basic k− ε turbulence model used in CFD codes. Themajor topics discussed are
• The Reynolds Averaged Equations
• The modified Boussinesq Eddy-Viscosity Concept
• Prandtl’s Mixing Length Hypothesis
• The k and ε equations
• Boundary Conditions and Wall Functions
Good background reading for this material can be found in the books by Ferziger and Peric [1,Chapter 9] Panton [4, Chapter 23], Pope [5, Chapter 10], and Tannehill et al. [7, § 5.2, § 5.4].
1.1 What Is Turbulence?
Turbulence is a fluid flow phenomenon characterized by
• unsteadiness,
• fluid motions that appear irregular or topologically complex,
• fluid motions occurring on a wide range of physical scales,
• rapid mixing of passive contaminants (smoke, heat, concentrations of chemical species)
Unsteadiness Imagine using a probe to measure the fluid velocity at a point in a turbulent flow.To be specific, consider inserting a probe through a small hole in a duct so that the sensing elementof the probe is located near the centerline. Assume that the probe has a high frequency response,meaning that it is capable of detecting any time variation in the velocity signal. Figure 1 shows theoutput of the probe as it would be displayed on an oscilloscope. The velocity signal is characterizedby a nominal average U , and a superimposed fluctuation, u′. The average value is representativeof the mass flow rate through the duct. The fluctuating component is an essential feature of theturbulence.∗Associate Professor, Mechanical and Materials Engineering Department Portland State University, Portland,
Oregon, [email protected]
1.1 What Is Turbulence? 2
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t
u
Figure 1: Turbulent velocity signal at a point. The Reynolds decomposition separates the signalinto an average, U , and a fluctuating component, u′.
Flow Irregularity Let us continue with our imagined experiment with duct flow. Suppose therewas some way to make the flow patterns inside the duct visible. First of all, assume that the ductwalls were transparent. Then suppose that we were to inject smoke from a series of small pipeshaving outlets at a small number of vertical locations on the center plane of the duct. Finally, inorder to keep the smoke from obscuring all the details, illuminate the center plane of the duct witha sheet of laser light. The light has the effect of showing smoke that is only in the plane. All othersmoke in the duct is not illuminated and, therefore, invisible. Figure 2 is an example of such a flowvisualization.
The first thing apparent from the smoke visualization is that the flow is extremely complicated.The flow swirls and fluctuates with no apparent pattern. It would be difficult, for example, to traceout the trajectory of a particular puff of smoke.
Further complicating an identification of flow patterns is the realization that our visualizationsystem is illuminating the flow in a single plane. The smoke that we see is not confined to move inthe plane. In fact, any smoke that we see at one instant is likely to be swept out of the plane inthe next instant. Thus, although our view is two-dimensional, the flow features are fundamentallythree-dimensional as well as being unsteady.
Multiple Scales After staring at the smoke in the duct, we may begin to adjust our perspective.In particular, if we allow our eyes to follow a billow of smoke as it is carried downstream, we willsee small whorls of smoke turning and tumbling. If we were to take photographs of the smoke, wewould be able to identify whorls of many different sizes. Fluid dynamicists call the whorls “eddies”.
An exact definition of a turbulent eddy is somewhat elusive. Pope [5, p. 183] describes an eddyas “a turbulent motion, localized within a region of size `, that is at least moderately coherent overthis region”. Although the shape of an eddy is not precise, we often think of it as being roughlycircular in cross-section. Because an eddy is embedded in an unsteady, three-dimensional flow, itis in constant motion. Furthermore, the eddy will be stretched and twisted until it breaks up intoother smaller eddies. The notion that the eddy is coherent means that it exists for a period of time,
1.2 Why is Modeling Necessary? 3
Figure 2: Turbulent wake behind a cylinder at Re = 1770. White areas are oil fog illuminated by asheet of laser light. Image is from the compilation of Van Dyke [9].
and that while it exists, the fluid that constitutes the eddy, has a temporary degree of organization.For a high Reynolds number turbulent flow, there will be eddies of many different sizes1. In fact
a large scale eddy will likely be composed of many smaller scale eddies. All of these eddies, thesesomewhat ill-defined, transient structures in the flow, are in constant motion.
In our imaginary duct flow, the largest eddy will be no larger than D/2, where D is the char-acteristic size of the duct cross-section. The size of the smallest eddy depends on the Reynoldsnumber of the flow, but in general it will be much smaller than D/2. The size of the smallest eddyis determined by the mechanism of viscous dissipation.
The largest eddies tumble and interact largely under the influence of inertia, and to some extentpressure. The largest eddies are virtually immune to viscous effects. As the large eddies break upinto smaller eddies, the role of inertia continues to dominate until the eddies become small enoughthat viscous resistance converts their kinetic energy (of translation and rotation) to heat.
Rapid Mixing To complete our visualization of the turbulent duct flow, we turn off the laserlight sheet, and the streams of smoke. We illuminate the duct with a broad beam of light, andintroduce a small puff of smoke in the center of the duct. The puff of smoke is quickly dispersed asit moves downstream. For a high Reynolds number flow, the dispersal happens so quickly that thepuff might seem to disappear almost as soon as it is introduced. The smoke disappears because ithas been diluted by rapid mixing with the rest of the air in the duct. The smoke particles remain,but at such a low concentration that they are no longer visible.
The rapid mixing of the turbulent flows is due to bulk transport of fluid. A blob of fluid isstretched and reshaped by the eddy motion. This mixing is order of magnitudes more effectivethan the Brownian motion that is responsible for the transport properties that we call viscosity andthermal conductivity.
1.2 Why is Modeling Necessary?
Suppose that to simulate a turbulent flow, we attempt to obtain a numerical solution to the unsteadyNavier-Stokes equations. This is the approach of a strategy called Direct Numerical Simulation(DNS) of turbulence. DNS is only applicable as a research tool at relatively low Reynolds numbers.The difficulty with DNS is in the spatial grid and time scale requirements to model all the importantmotions in a turbulent flow. Furthermore, any direct simulation of turbulence yields an unsteadyflow field. Extraction of mean quantities (say overall drag, or local heat transfer rate) requires thetime (or ensemble) averaging of a vast amount of data.
1The meaning of “high” is somewhat problem-dependent. Let us just assume that the velocity and length scalesof the problem are sufficient that the flow is very far from being laminar.
2 REYNOLDS AVERAGED EQUATIONS 4
Table 1: Computational cost for direct numerical simulation of isotropic turbulence. Adapted fromPope [5, Table 9.2, p. 349]. In November 2005, the Blue Gene/L computer was the fastest computerin the world. Blue Gene/L has 131072 processors and achieved a sustained 280 TFlops on theLINPACK benchmark. 1 Tflop = 1012 floating point operations per second. A high end personalcomputer has a theoretical performance of between 1 and 10 Gflop. 1 Gflop = 109 floating pointoperations per second.
CPU Time
ReL N N3 PC at 1 Gflop Supercomputerat 1 Tflop
Blue Gene/Lat 280 Tflop
100 52 1.1× 106 3 minutes 1.94 µsec 0.2 µsec
1000 291 1.0× 107 47 hours 0.05 sec 0.2 msec
10,000 1640 1.2× 108 5.3 years 47 sec 0.2 sec
100,000 9200 2.0× 109 5300 years 5.3 years 6.9 days
1,000,000 52000 3.8× 1010 53,000 centuries 5300 years 19 years
Pope [5, § 9.1.2] presents the following estimates of the computational cost of using DNS tosimulate a homogeneous turbulent flow. Homogeneous turbulence is an idealization of turbulentflow far from any bounding walls, and free of any shear or other flow structure. For a homogeneousturbulent flow in a region of size L, and having maximum mean velocity U , the Reynolds numberis Re = UL/ν. Let N be the number of points necessary to resolve all relevant flow scales in anyone coordinate direction. The total number of mesh points necessary to simulate the flow is
N3 ∼ 4.4Re9/4L
If the simulation is performed on a computer with a sustained throughput of one Gigaflop (109
floating point operations per second) the time to complete the simulate in days is
TG ∼(
ReL800
)3
Evaluating these formulas for a range of ReL gives the results in Table 1. Clearly, DNS is notapplicable for engineering design work.
2 Reynolds Averaged Equations
The governing equations for flow in Cartesian coordinates are the continuity equation
∂ρ
∂t+
∂
∂x(ρu) +
∂
∂y(ρv) +
∂
∂z(ρw) = 0 (1)
and the momentum equations (in conservative form, no body forces)
∂
∂t(ρu) +
∂
∂x(ρuu) +
∂
∂y(ρvu) +
∂
∂z(ρwu) = −∂p
∂x+∂τxx∂x
+∂τxy∂y
+∂τxz∂z
(2)
∂
∂t(ρv) +
∂
∂x(ρuv) +
∂
∂y(ρvv) +
∂
∂z(ρwv) = −∂p
∂y+∂τyx∂x
+∂τyy∂y
+∂τyz∂z
(3)
∂
∂t(ρw) +
∂
∂x(ρuw) +
∂
∂y(ρvw) +
∂
∂z(ρww) = −∂p
∂z+∂τzx∂x
+∂τzy∂y
+∂τzz∂z
(4)
2.1 Reynolds Decomposition 5
where the velocity vector is u = exu+ eyv + ezw, and the shear stresses are
τxx = µ
[2∂u
∂x− 2
3(∇ · u)
]τyy = µ
[2∂v
∂y− 2
3(∇ · u)
]τzz = µ
[2∂w
∂z− 2
3(∇ · u)
]
τxy = τyx = µ
[∂u
∂y+∂v
∂x
]τxz = τzx = µ
[∂u
∂z+∂w
∂x
]τyz = τzy = µ
[∂v
∂z+∂w
∂y
]
2.1 Reynolds Decomposition
Figure 1 shows a short sample of a turbulent signal. The Reynolds decomposition identifies anaverage value U and a fluctuating component u′ such that the instantaneous signal is
u = U + u′ (5)
The average of a signal is also designated with an overbar, i.e.
u = U
By definition of the averageu′ = 0
The governing equations are averaged with the Reynolds averaging rules which apply to any twofunctions f and g
1. f + g = f + g
2. af = af (a is a constant)
3.∂f
∂s=∂f
∂s
4. fg = f g
It is possible to define averages that do and do not satisfy these rules.A common form of averaging is time-averaging, which is defined by
U = limT→∞
(1/T )∫ t0+T
t0
u(t) dt (6)
In general both the average and fluctuating component are functions of space. For time averaging,as defined by Equation (6) the average velocity will not be a function of time, i.e., the average flowis steady. Some turbulent flows are unsteady in the mean, for example the flow inside the cylinderof an IC engine. In these cases the ensemble average (see below) is used.
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 to process of Reynolds averaging of
the momentum equations.
2.2 Ensemble Averaging 6
0 1 2 3 4 5 6 7 82
4
6
Sample 1
0 1 2 3 4 5 6 7 82
4
6
Sample 2
0 1 2 3 4 5 6 7 82
4
6
Sample 3
0 1 2 3 4 5 6 7 82
4
6
Ensemble
Figure 3: Ensemble averaging of a velocity at a point in a turbulent flow with a periodic forcingfunction.
2.2 Ensemble Averaging
Consider an unsteady flow with a periodic forcing function (e.g. sinusoidally varying inlet pressureor inlet velocity, or a boundary that moves with simple harmonic motion). Figure 3 depicts threesamples (top three plots) and the ensemble average of the velocity measured at a point. If thevelocity vector is u = u(x, t), then the ensemble averaged velocity is
U(x, t) = limn→∞
1n
n∑i=1
ui(x, t) (7)
where n is the total number of samples and i is the index of the sample. U(x, t) is the ensembleaverage of u at a fixed location, x.
Ensemble averaging satisfies the Reynolds averaging rules. By applying the Reynolds averag-ing rules to the governing equations, a new set of equations is obtained. These equations haveensemble averaged velocities (U, V,W ) and second order turbulence correlations as dependent vari-ables. Note that if ensemble averaging is used, the average velocities will be functions of time, e.g.,U = U(x, y, z, t). The ensemble averaging has achieved a separation of the fluctuations from theaverage quantities.
2.2.1 Example of Reynolds Averaging
Each component of the velocity field is assumed to fluctuate so that we can identify three pairs ofaverage velocity (components) and fluctuating components
u = U + u′ v = V + v′ w = W + w′
2.2 Ensemble Averaging 7
Apply the Reynolds averaging rules to the left side of Equation (2). For simplicity, assume thatthere are no fluctuations of pressure or density, i.e., p′ = 0 and ρ′ = 0. Consider each term
∂
∂t(ρu) =
∂
∂t(ρu) rule 3
=∂
∂t(ρu) rule 2, with ρ = constant
=∂
∂t(ρU) by definition of U
∂
∂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 since Uu′ = Uu′ = 0
Similarly,∂
∂y(ρvu) =
∂
∂y
(ρV U + ρu′v′
)∂
∂z(ρwu) =
∂
∂z
(ρWU + ρu′w′
)Thus,
∂
∂t(ρu) +
∂
∂x(ρuu) +
∂
∂y(ρvu) +
∂
∂z(ρwu) =
∂
∂t(ρU) +
∂
∂x(ρUU) +
∂
∂y(ρV U) +
∂
∂z(ρWU)
+∂
∂x
(ρu′u′
)+
∂
∂y
(ρu′v′
)+
∂
∂z
(ρu′w′
)The second line of the preceding equation consists of second order correlations that are a direct resultof the averaging process. These second order correlations of the fluctuating velocity components arecalled the Reynolds stresses.
Applying the Reynolds averaging rules to the right hand side of Equation (2) gives
−∂p∂x
+∂τxx∂x
+∂τxy∂y
+∂τxz∂z
= −∂p∂x
+∂τxx∂x
+∂τxy∂y
+∂τxz∂z
Notice that no extra correlation terms appears due to averaging of the right hand side.Putting the averaged x-direction momentum equation back together and moving the Reynolds
stresses to the right hand side gives.
∂
∂t(ρU) +
∂
∂x(ρUU) +
∂
∂y(ρV U) +
∂
∂z(ρWU) (8)
= −∂p∂x
+∂
∂x
[τxx − ρu′u′
]+
∂
∂y
[τxy − ρu′v′
]+
∂
∂z
[τxz − ρu′w′
]Equation (8) suggests how the Reynolds stresses get their name. Averaging the governing equations(to simplify the analysis) results in additional terms that appear as stresses.
2.3 Flow That Is “Steady in the Mean” 8
Repeating the averaging process for the continuity and the y and z momentum equations gives
∂ρ
∂t+
∂
∂x(ρU) +
∂
∂y(ρV ) +
∂
∂z(ρW ) = 0 (9)
∂
∂t(ρV ) +
∂
∂x(ρUV ) +
∂
∂y(ρV V ) +
∂
∂z(ρWV ) (10)
= −∂p∂y
+∂
∂x
[τyx − ρv′u′
]+
∂
∂y
[τyy − ρv′v′
]+
∂
∂z
[τyz − ρu′w′
]∂
∂t(ρW ) +
∂
∂x(ρUW ) +
∂
∂y(ρVW ) +
∂
∂z(ρWW ) (11)
= −∂p∂z
+∂
∂x
[τzx − ρw′u′
]+
∂
∂y
[τzy − ρw′v′
]+
∂
∂z
[τzz − ρw′w′
]2.3 Flow That Is “Steady in the Mean”
In the preceding section the governing equations were transformed by performing an ensembleaverage. Now consider a flow that lacks any periodic forcing function and that does not exhibitlarge scale periodic motion of any kind. This is the case in many practical situations, for example,in turbulent flow in ducts. One expects that while the unsteady turbulent fluctuations exist, theaverage velocity field appears to be steady. Such is the case depicted by the velocity signal inFigure 1.
When time averaging of the velocity field at each point in the flow yields constant value of U ,V , W and p, then the flow is said to be steady in the mean. For such flows it is appropriate to useEquation (6) to perform Reynolds averaging of the governing equations. The result is the same asEquations (8) through (11) with the exception of the time derivatives, which vanish.
If the flow is steady in the mean then the governing equations are simplified. Although theReynolds stresses are still present, at least the average velocity field is steady. If appropriate modelsfor the Reynolds stresses can be introduced, a steady numerical solution can be obtained eventhough the flow is characterized by a high degree of fluctuation in the velocity signal. This is theheart of turbulence models used in CFD codes.
2.4 The Closure Problem
The Reynolds averaging process has created new dependent variables — the Reynolds stresses —without adding to the set of governing equations. This is called the turbulence closure problem. Forthree-dimensional, incompressible flow there are six Reynolds stresses, which can be represented bythe following matrix. 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′. The Reynolds stresses are fieldvariables, i.e., each term in the matrix is a function of space in the flow.
3 Basic Turbulence Modeling
Consult Tannehill [7, § 5.4] and Wilcox [11] for overviews of turbulence modeling applied to CFDcodes. Pope [5, Chapter 10] explains the fundamental deficiencies of this modeling approach.
3.1 Eddy-Viscosity Concept 9
3.1 Eddy-Viscosity Concept
The viscous stress tensor is
τij = µ
[(∂Uj∂xi
+∂Ui∂xi
)− 2
3δij
∂Uk∂xk
](12)
By direct analogy we propose (guess, assume, hope) that the Reynolds stresses have the sameform as the viscous stresses. This model introduces a turbulence viscosity µt or eddy viscosity suchthat [2, 5, 7]
−ρ u′iu′j = µt
[(∂Uj∂xi
+∂Ui∂xi
)− 2
3δij
∂Uk∂xk
]− 2
3ρ δijk (13)
where k is called the turbulence kinetic energy
k =12u′ku
′k =
12(u′u′ + v′v′ + w′w′
)(14)
Equation (13) effectively replaces the Reynolds stresses by two new unknowns, k and µt. Bothµtand k vary from point to point in the flow. It is important to realize that Equation (13) is usedbecause it is convenient, not because it is an accurate model of the Reynolds stresses.
Substituting Equation (12) and Equation (13) into the Reynolds averaged momentum equations,and simplifying for steady, incompressible flow gives
∂
∂x(ρUU) +
∂
∂y(ρV U) +
∂
∂z(ρWU)
= −∂p∂x
+∂
∂x
[µeff
∂U
∂x
]+
∂
∂y
[µeff
∂U
∂y
]+
∂
∂z
[µeff
∂U
∂z
]+ SU (15)
∂
∂x(ρUV ) +
∂
∂y(ρV V ) +
∂
∂z(ρWV )
= −∂p∂y
+∂
∂x
[µeff
∂V
∂x
]+
∂
∂y
[µeff
∂V
∂y
]+
∂
∂z
[µeff
∂V
∂z
]+ SV (16)
∂
∂x(ρUW ) +
∂
∂y(ρVW ) +
∂
∂z(ρWW )
= −∂p∂z
+∂
∂x
[µeff
∂W
∂x
]+
∂
∂y
[µeff
∂W
∂y
]+
∂
∂z
[µeff
∂W
∂z
]+ SW (17)
whereµeff = µ+ µt (18)
p = p− 13µeff ∇ · u +
23ρk (19)
and SU , SV , and SW are additional source terms due to the non-uniform viscosity. For example,
SU =∂µeff
∂y
∂V
∂x− ∂V
∂y
∂µeff
∂x+∂µeff
∂z
∂W
∂z− ∂W
∂z
∂µeff
∂x.
Use of the eddy viscosity concept in a CFD code involves replacing the true viscosity, µ, by µeff .In general µeff � µ. We now turn to the job of computing µt.
3.2 Prandtl Mixing Length Hypothesis 10
Table 2: Comparison of the molecular model of viscosity from the kinetic theory of gases withPrandtl’s mixing length hypothesis.
molecular viscosity of gases turbulence viscosity
µ =13ρ`fVm µt = ρ`mVt
`f = mean free path `m = mixing length
Vm = velocity of molecules Vt = velocity scale of turbulence
3.2 Prandtl Mixing Length Hypothesis
The Prandtl mixing length hypothesis is based on a direct analogy with the molecular theory ofviscosity from the kinetic theory of gases. In general the (true) viscosity is a macroscopic manifes-tation of the microscopic behavior of a fluid. For a gas, the viscosity arises because momentum istransferred across velocity gradient by the random motion of gas molecules (see, e.g., Panton [4,Chapter 6] and Tennekes and Lumley [8]). Prandtl hypothesized that the turbulent eddies enhancethe exchange of momentum (and passive scalars) by augmenting the molecular diffusion process.The relationship between Prandtl’s mixing length model and the viscosity model from the kinetictheory of gases is summarized in Table 2.The mixing length hypothesis has two parts:
1. A model for the turbulence viscosity
µt =ρu′iu
′j
∂U/∂y
where ∂U/∂y is the mean velocity gradient. (The mixing length was first developed for bound-ary layer flows where the only important velocity gradient is ∂U/∂y.)
2. A model for the turbulent velocity fluctuations
Vt = `m
∣∣∣∣∂U∂y∣∣∣∣
One problem with the mixing length hypothesis is that µt is not defined wherever ∂U/∂y = 0,e.g. at the centerline of a pipe. Another problem is that the mixing length hypothesis does notinclude the influence of upstream events, i.e. turbulence cannot be convected downstream. Yetanother problem is that the mixing length model requires specification of the mixing length. Forsome flows, e.g. boundary layers, and the far field of isolated turbulent jets and wakes, experimentaldata is available so that specification of a mixing length is possible. For a general flow, especiallyone involving strong recirculation, identification of a single mixing length is not possible. The ideaof the mixing length is still useful as a conceptual model, and as a point of reference for moresophisticated models.
3.3 Constant Eddy Viscosity Model
A simplistic model of turbulence involves asserting that µeff is constant throughout the flow. Thetrick, of course, is to specify a value of µeff that is meaningful. The constant eddy viscosity model
3.3 Constant Eddy Viscosity Model 11
will not allow accurate prediction of the smaller scale features of the flows. An estimate of µeff isobtained by assuming `m is uniform throughout the flow.
3.3.1 Example: Flow in a Pipe
We can easily estimate the magnitude of the eddy viscosity in a pipe. Prandtl’s model is
µt ∼ ρVt`m
In general, the magnitude of the turbulent fluctuations are small compared to the mean velocity inthe pipe.
0.01 ≤ u′
U≤ 0.15
To estimate µt, take u′/U ∼ 0.1 and Vt ∼ u′ so that
Vt ∼ 0.1U
The flow in the pipe contains eddies ranging from half the diameter of the pipe down to the scaleat which kinetic energy is dissipated by viscosity. To make a somewhat arbitrary choice of a singlelength scale that characterizes the turbulent mixing, take
`m ∼D
4
Combining the preceding expressions gives the following estimate of turbulence 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
Therefore, the constant viscosity model has the effect of modeling the fluid as if it were very viscous.At first this may seem counter-intuitive, but deeper examination shows that the large apparentviscosity is consistent with Prandtl’s mixing length hypothesis.
Recall that the mixing length hypothesis is based on an analogy with the model of viscosityobtained from the kinetic theory of gases. There, it is assumed that the small scale random motionof individual molecules gives rise to a transport of momentum whenever there is a velocity gradient.In the mixing length model, fluid blobs transported by turbulent fluctuations take the role of theindividual molecules in the kinetic theory. Thus, turbulent fluctuations enhance the transport ofmomentum. The increase in momentum transport is reflected by the increase in the apparentviscosity.
Though this model is intuitively compelling, the reality of turbulent mixing is not so simple.Pope [5, § 10.1] uses experimental data and scaling arguments to show that the mechanics of molec-ular diffusion and turbulent mixing happen on vastly different time scales. Molecular mixing happensvery quickly because the velocity of molecules on the microscopic scale is very high2. Turbulenteddies rotate on a time scale comparable to the mean flow velocities. As a consequence, models ofturbulent mixing that rely on mean velocity gradients only work when the mean flow field changesrelatively slowly in the flow direction.
2From the kinetic theory of gases (see, e.g. Wark [10]), the RMS velocity of a molecule of gas is vrms =p
3kT/m,where k = 1.38×10−23 J/K/molecule is Boltzmann’s constant, T is the absolute temperature of the gas, m =M/NA
is the mass per molecule,M is the molar mass of the gas, and NA = 6.024×1026 atoms/k-mol is Avogadro’s number.For Nitrogen (M = 28 kg/k-mol) at 294 K, vrms = 511 m/s.
4 THE K − ε MODEL 12
3.3.2 Constant Turbulence Viscosity Models in CFD Codes
The constant viscosity model attempts to simulate the effect of turbulent fluctuations by increasingthe apparent viscosity of the fluid. This is a crude interpretation of the mixing length model, for itassumes that the mixing length is uniform throughout the flow. In fact, turbulent transport occursdue to eddies of many different sizes. Near walls, the eddies are smaller. Downstream protrudingobjects such as cylinders or other bluff bodies, the eddy size is strongly influenced by the size of theobject. Shear layers, e.g. where two streams of different velocity mix, create high levels of turbulentfluctuations. All of these flow features are not accounted for by a model that assumes a uniformturbulence viscosity.
Though the constant turbulence viscosity model cannot be expected to yield accurate simulationof many flow features, it is still useful in CFD codes. The constant viscosity model is especially help-ful for preliminary analysis, and for developing an initial guess at the flow field. More sophisticatedturbulence models require significantly greater computational effort. In addition, more complexmodels may also introduce convergence problems for the calculations. Therefore, when beginninga new CFD simulation, it is helpful to use the constant turbulence viscosity model to generate afirst guess at the flow field. The computations can then be restarted with a more sophisticatedturbulence model after the large scale features of the flow have been established.
4 The k − ε Model
The k − ε model was first proposed by Jones and Launder [3]. It is now consider the standardturbulence model for engineering simulation of flows.
The modified Boussinesq eddy viscosity model overcomes the first problem of the mixing lengthhypothesis, viz. that µt is not defined in regions of zero shear (∂U/∂y = 0). To relate µt to theReynolds stresses, and assume that
Vt ∝√k
so thatµt = C ρ`mk
1/2 (20)
where C is a constant. Using this model µt is nonzero everywhere in the flow that k is nonzero.The new independent variables of the turbulence model are `m and k.
4.1 The k Equation
An exact equation for k is obtained by taking the inner product of the velocity vector and themomentum equation (in vector form). The result after some algebra is a conservation equation fork (see [2]).
Dk
Dt=− ∂
∂xi
[u′i
(p
ρ+ k
)]︸ ︷︷ ︸
convective diffusion
− u′iu′j
∂Uj∂xi︸ ︷︷ ︸
production by deformation
+ ν∂
∂xi
[u′j
(∂u′i∂xj
+∂uj∂xi
)]︸ ︷︷ ︸
production by viscous shear
− ν(∂u′i∂xj
+∂uj∂xi
)∂u′j∂xi︸ ︷︷ ︸
dissipation
4.2 The ε Equation 13
Like the Reynolds averaged equations this equation also has higher order correlations. The solutionis to model these correlations. The standard form of the model is (see [7])
Dk
Dt=
∂
∂xi
[µeff
σk
∂k
∂xi
]︸ ︷︷ ︸
diffusion
+[µt
(∂Ui∂xj
+∂Uj∂xi
)− 2
3ρ δijk
]∂Uj∂xi︸ ︷︷ ︸
production
− cDρ k3/2
`m︸ ︷︷ ︸dissipation
(21)
Note that the model equation for k includes the mixing length, `m in the dissipation term.Equation (21) is a conservation equation for k. The left hand side has terms for storage and
convection. These terms are balanced on the right hand side by diffusion, and source terms due toturbulence production (source) and turbulence dissipation (sink).
If `m were known, then we could obtain a numerical solution to Equation (21) using the sameapproximations applied to the momentum equations. The result would be a spatial distribution ofk on the computational mesh. From the discrete k(x, y, z) field, Equation (20) would be used tocompute the local µt, which in turn would be used in the discrete form of the momentum equations.The net effect is that solution of the momentum equations would require simultaneous solution ofthe k equation. This would be an improvement over the constant viscosity model.
The question remains, “How can `m be specified?”
4.2 The ε Equation
In a turbulent flow, some mechanism or device is responsible for adding energy to the fluid. In a ductflow, a fan converts electrical energy to work, and imparts momentum to the fluid. In atmosphericflows, e.g. the weather in the troposphere, solar heating causes large columns of air to rise, whileprecipitation induces downdrafts. These external energy inputs cause large scale motions in thefluid. Motions on a large scale correspond to large Reynolds numbers. The organization of theselarge scale motions is not stable, i.e. a large swirling vortex tends to break up into smaller vortices.
High Reynolds number turbulent flows have motions that range over many length scales. Thelargest scales correspond to the scale of the energy input mechanism or to the size of the boundingwalls. At the large scales, viscosity is unimportant. In other words, the large vortex structures existand breakup into smaller scale vortices under the influence of pressure and inertia alone. Viscouseffects are important predominantly at the smallest scales of motion. At the smallest scales thevelocity gradients are smoothed out by the dissipative effect of viscosity.
In a typical turbulent flow, the kinetic energy of the largest scale motions is transferred tosuccessively smaller scale motions without loss. It is only at the small scales that energy dissipationoccurs.
We can use a simple scaling argument to estimate the dissipation rate (see Panton [4, § 22.10]).Consider a flow where u0 is linear velocity associated with the largest eddy. The kinetic energyof this eddy is proportional to u2
0. If the diameter of the eddy is L, then the eddy completes arevolution in time L/u0. Let ε be the rate at which energy is dissipated. An upper estimate of ε is
ε ∼ kinetic energy of an eddytime for one rotation
=u2
0
L/u0=u3
0
L(22)
Ultimately all of this energy is dissipated. In a steady flow the amount of energy dissipated is equalto the amount of energy that must be continually supplied to the flow.
In a turbulent flow the energy is dissipated at the smallest scales. Thus, although Equation (22)provides an estimate for the magnitude of the dissipation rate, it does not correspond to the actualdissipation mechanism. The turbulence kinetic energy, k is a measure of the energy in the velocity
4.3 Calculation of Effective Viscosity 14
fluctuations. If the turbulence is isotropic, i.e. if |u′| = |v′| = |w′| then3
k =32u′u′ isotropic turbulence
and we can estimate the velocity of the fluctuations with |u′| ∼√k. Furthermore if we define `ε as
the dissipation length scale, i.e. the length scale on which the viscous dissipation mechanism occurs,then we can write the dissipation rate as
ε ∼ CDk3/2
`ε(23)
where CD is to be a constant that is assumed adjust the magnitude of the right hand side to the truemagnitude of ε. Equation 23 is a scaling relationship, not the equation that allows us to computethe distribution of ε.
We can develop a relationship between µt and ε as follows. Solve Equation (23) for `ε.
`ε ∼ CDk3/2
ε(24)
Use `ε as an indication of the scale for `m. We do not assert that `ε and `m are equal. Ratherassume that their ratio is constant. Thus, if `m = C ′`ε, where C ′ is a constant, then we use thefollowing estimates in Prandtl’s mixing length model
µt ∼ ρVt`m Prandtl
Vt ∼√k isotropic turbulence
`ε ∼ CDk3/2
εscale estimate
to get
µt = Cρ
(CD
k3/2
ε
)(k1/2
)or
µt = Cµρk2
ε(25)
The standard form of the modeled conservation equation for ε is (check this?)
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
(26)
4.3 Calculation of Effective Viscosity
A typical turbulent flow would involve the following iterative loop:
1. Set-up and solve equations for the velocities and pressure, using the current guess at theeffective viscosity.
2. Solve the k and ε equations3|u′| = |v′| = |w′| is a consequence of isotropy, not the definition of it.
4.4 Wall Functions 15
ue
δ
~ 0.1δ
~ 0.1δ
viscous sublayer
0 ² y+ 8<~
buffer layer
8 y+ 30
overlap layer
(a.k.a. log region,
inerial sublayer)
30 y+
<~
<~
<~
Figure 4: Velocity profile in a turbulent boundary layer.
3. At each point in the flow compute
µt = Cµρk2
ε
4. Use µt as appropriate to compute the effective diffusion coefficient for each dependent variable.
5. Return to step 1 until convergence.
4.4 Wall Functions
In the basic CFD model of turbulent flow, the boundary conditions for the dependent variables (u, v,w, T , k, ε) are implemented with wall functions. Wall functions are derived from a semi-empiricalmodel of turbulent boundary layer flow called the law-of-the-wall. More sophisticated near-walltreatments are available.
As depicted in Figure 5 the k − ε model applies to the central part of the calculation domain.Because velocity and temperature gradients are very steep near the wall, it is often impractical toresolve all the details of the flow in the near-wall region. Wall functions are an economical (in termsof mesh points and CPU effort) way to bridge the region between the true wall boundary values andthe turbulent core flow. Before discussing the implementation of wall functions we will first reviewthe features of turbulent boundary layers and introduce the law-of-the-wall.
4.4.1 Law of the Wall
See Panton [4] for a clear and concise introduction to the scaling laws for turbulent boundary layers.Panton also gives a nice introduction to turbulent flow in general. More detailed discussions areprovided by Hinze [2] and Pope [5].
Figure 4 depicts the profile of the mean velocity in a turbulent boundary layer. The boundarylayer is a region of thickness δ over which the velocity varies from 0 to ue, where ue is the localvelocity of the external flow. The external velocity is assumed to vary slowly with position, i.e.ue is not expected to change much over distances of the order of, say, 10δ. When this is true,the boundary layer will have a universal structure that is the same in many different flows. Thisassumption of universal structure is at the heart of the near wall model of turbulence used in thecommon CFD models of turbulent flow.
4.4 Wall Functions 16
In the left half of Figure 4 the velocity profile is divided into two regions: an inner region betweenthe wall and roughly 0.1δ, and an outer region between approximately 0.1δ and the free stream.The inner region is further divided into three layers as shown in the right half of Figure 4. Wallfunctions are computational models used to bridge the numerical approximation in the turbulentcore flow (the edge of which corresponds to the outer region), and the wall without resolving thedetails of the inner region. In other words, wall functions are models of the turbulent momentumtransport in the region 0 ≤ y ≤ 0.1δ.
The extent of the sublayers of the inner region are characterized by a dimensionless variable
y+ =yu∗ν
(27)
where y is the distance normal to the wall, u∗ is called the friction velocity and ν is the kinematicviscosity of the fluid. Note that y+ has the same form as a Reynolds number where the characteristicvelocity is u∗ instead of ue.
The delineation of an inner region (y < 0.1δ) and outer region (y > 0.1δ) reflects a fundamentaltruth about turbulent boundary layers. The outer region is characterized by the flow outside theboundary layer, and the inner region is characterized by the conditions at the wall. Near the wallthe velocity profile has the functional form (see, e.g., [4])
u = fin(y, ρ, ν, τw, εw) (28)
where τw is the wall shear stress, and εw is a length scale characterizing the wall roughness4. In theouter region the velocity profile has the functional form
u− ue = fout
(y, δ, ρ,
dp
dx, τw
)(29)
Comparing Equation (28) and (29) we see that the near wall region is insensitive to the externalpressure gradient, the boundary layer thickness, δ, and the external velocity, ue. Also note that thewall shear stress, τw, affects both the inner and outer profiles.
The dimensionless form of the inner part of the profile (Equation (28)) is
u+ = Fin(y+, ε+w) (30)
whereu+ =
u
u∗(31)
and the friction velocity is defined by
u∗ =√τwρ
(32)
Note that τw has the units of pressure (stress), and (by definition of u∗) τw = ρu2∗. The dimensionless
wall roughness isε+w =
εwu∗ν
(33)
Equation (30) is called the law-of-the-wall. This so-called law is a semi-empirical model whichdescribes the near-wall behavior of boundary layers and flow in ducts. The law-of-the-wall coversthe three sublayers depicted in the right half of Figure 4.
4Do not confuse the roughness scale, εw, with the turbulence dissipation rate, ε, used in the k − ε model.
4.4 Wall Functions 17
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.
Figure 5: Wall functions apply to thin layers between solid walls and the central part of the flowdomain.
In the viscous sublayer the shear stress is dominated by viscous forces5. The velocity profile inthe viscous sublayer is
u+ = y+ or u = y (34)
i.e., the mean velocity profile in the viscous sublayer is nominally linear. Equation (34) is hard toverify experimentally because the region y+ < 5 is very close to the wall. It is difficult to build aprobe that is small enough to accurately measure the velocity in this region. We do know, however,that all velocity fluctuations are zero at the wall so the importance of the Reynolds stresses vanishas y → 0.
The middle part of the inner layer is called the buffer layer. It has no simple correlating equation.The outer part of the inner layer is called the overlap layer or the log-law layer. The log-law layeris correlated by an equation of the form
u+ = c1 ln y+ + c2 (35)
where c1 and c2 are constants.The dimensionless form of the outer part of the profile (Equation (29)) is
u− ueu∗
= Fout
(y
δ,δ
ρu2∗
dp
dx
)(36)
In a CFD code the outer part of the velocity profile is not used.
4.4.2 Wall Function Boundary Conditions
Wall functions provide a computational glue layer that allow the turbulent core flow to be linkedto the physical boundary conditions at the wall of the domain. Wall functions are used because itis often not economical to resolve the structure of the turbulent boundary layer, especially in thesublayers close to the wall. Simulation of the full turbulent boundary layer would require many
5Some older texts refer to this as the “laminar” sublayer. It is misleading to consider this a laminar layer becausethe flow in the sublayer is characterized by slower motions driven by the turbulent eddies outside of the viscoussublayer. Although viscous forces are dominant in this sublayer, the flow in the sublayer is not necessarily smoothand sheet-like — the defining characteristic of laminar flow.
4.4 Wall Functions 18
I
B
yI
ures
Figure 6: Wall functions model the variation of dependent variables between nodes adjacent to asolid boundaries. Node B is on the boundary. Node I is the nearest interior node close to node B.
nodes (cells) to be placed very close to the wall. This consumes memory (to store element and nodedata) and increases the CPU time to achieve a converged solution6. Instead of resolving the detailsof the wall boundary layers, wall functions attempt to impose the boundary condition informationon the core flow in such a way that the correct flux of momentum (drag) and heat at the wall isobtained. One way to look at wall functions is to consider them as a computational device forspecifying µeff for nodes on and near the wall. The µeff near the wall needs to be modified by thepresence of the wall, and wall functions achieve approximate values of µeff from the law-of-the-wall.
Figure 5 depicts the flow very close to a solid wall. The circles labeled “B” and “I” representa node on the boundary, and the nearest internal node on the computational mesh. In the wallfunction model, node I is assumed to lie outside of the log-law region, i.e. y+
I > 30. The purposeof the wall function is to model the wall shear stress from the value of the mean velocity at node I.The model enables a prediction of the wall shear stress (and wall heat flux if there energy transportis being simulated) without resolving the detailed structure of the boundary layer with many nodesbetween node B and node I.
The log-law part (cf. Equation (35)) of the law-of-the-wall can be written (see e.g. Rodi [6])
ures
u∗=
1κ
ln(y+E) (37)
where ures is the resultant velocity parallel to the wall, κ = 0.4 is von Karman’s constant, and Eis the roughness parameter (E = 9.0 for smooth walls). Use a finite-difference model to relate thewall shear stress to the velocity gradient in the y direction
τw = µeff∂u
∂y
∣∣∣∣w
≈ µeffuI − uByI
(38)
where µeff is the effective viscosity. Note that the velocity profile between node B and node I isassumed to look like the right half of Figure 4. The role of the wall function is to provide the estimateof µeff that compensates for the crude estimate of the velocity gradient given by Equation (38).
Since the no-slip condition requires uB = 0, Equation (38) can be rearranged as
µeff = τwyIuI
Using the definitions of y+, u+, and u∗ the preceding equation becomes
µeff = τw(y+I ν/u∗)u+I u∗
=τwu2∗
y+
u+= µ
y+I
u+I
(39)
6Because the solution is a superlinear function of the total number of cells in the mesh.
5 SUMMARY 19
where y+I and u+
I are evaluated at the interior point, “I”.Evaluation of µeff from Equation (39) depends on the value of y+
I . Ideally the first interior pointis far enough from the wall so that y+
I > 30. In that case, Equation (37) is used to relate y+I to u+
I ,i.e.
u+I =
1κ
ln(y+I E) =⇒
y+I
u+I
=κy+
I
ln(y+I E)
If y+I < 30 the wall function model breaks down because there is no universal functional relationship
for the buffer layer (8 ≤ y+ ≤ 30). When y+I < 30 most CFD codes print a warning message. Some
codes apply ad-hoc analytical corrections to the wall function boundary condition. If y+I < 5
Equation (34) may be used, i.e,µeff = µ if y+
I < 5
Otherwise
µeff =κµy+
ln(Ey+)if y+
I > 30
5 Summary
1. Reynolds averaging leads to equations governing the mean or ensemble averaged flow field
2. Reynolds averaging also introduces the Reynolds stresses and creates the closure problem
3. Closure in the k − ε model is achieved by
a. relating −ρ u′iu′j to µtb. using model equations to evaluate µt
For the constant eddy-viscosity model, µt is assumed to be uniform throughout the flow field.For the k − ε model µt is a local function of the k and ε fields. Wall functions are used tocompute the value of µt near solid boundaries for the k − ε model.
4. Eddy-viscosity models are not accurate for flows with anisotropic Reynolds stresses, and flowsfor which the local production and dissipation of turbulence kinetic energy is not in balance.
5. k is the turbulence kinetic energy
• k has a direct physical significance
• there is an exact equation for k, but it has higher order correlations that cannot bedirectly computed
• In the k − ε model, the “k” is an approximation to the true k
. turbulence is assumed to be isotropic
. k represents energy contained in eddies of many sizes
. k is governed by a transport equation with terms that are models of the exact terms
6. ε is the turbulence dissipation rate
• ε represents the action of the small eddies that are responsible for dissipating the kineticenergy of turbulence into heat.
• an exact differential equation for ε can be derived, but it contains higher order correlationsthat cannot be directly computed
6 ADVICE 20
• In the k − ε model, the “ε” equation is an approximation to the true ε equation
7. Boundary conditions in a k − ε model are often implemented by wall functions
• The turbulent boundary layer model is used to avoid resolving steep gradients near walls
6 Advice
Turbulence modeling is an inexact art. Most commercial codes offer a variety of turbulence models.Here we have described the standard k − ε model. We have barely scratched the surface of thisimportant topic.
It is important that you do additional research on your application to determine which turbulencemodel should be used. You should not expect the k − ε model to give good results in flow withrapid streamwise changes in the mean flow variables, or in flows with strong swirling components.Other models to investigate are the k − ω model and Reynolds stress models. Many commercialCFD codes include these models as options. The k−ω model has some advantage over the standardk− ε model for simulating near-wall flows and flows with strong streamwise pressure gradients. TheReynolds stress models are theoretically more sound because they do not rely on the eddy viscositymodel to relate the Reynolds stresses to the mean flow gradients.
Research in this area is continuing, and you should expect to be faced with more choices inturbulence modeling in the future. Above all, remember this: just because you have obtained asimulation of a turbulent flow with a turbulence model, in no way guarantees that this simulationis accurate.
References
[1] Joel H. Ferziger and Milovan Peric. Computational Methods for Fluid Dynamics. Springer-Verlag, Berlin, third edition, 2001.
[2] J.O. Hinze. Turbulence. McGraw-Hill, New York, second edition, 1975.
[3] W.P. Jones and B. E. Launder. The prediction of laminarization with a two-equation model ofturbulence. International Journal of Heat and Mass Transfer, 15:301–314, 1972.
[4] Ronald L. Panton. Incompressible Flow. Wiley, New York, second edition, 1996.
[5] Stephen B. Pope. Turbulent Flows. Cambridge University Press, Cambridge, UK, 2000.
[6] Wolfgang Rodi. Turbulence Models and Their Application in Hydraulics. International Associ-ation for Hydraulic Research, Delft, Netherlands, second edition, 1984.
[7] John C. Tannehill, Dale A. Anderson, and Richard A. Pletcher. Computational Fluid Mechanicsand Heat Transfer. Taylor and Francis, Washington, D.C., second edition, 1997.
[8] H. Tennekes and J.L. Lumley. A First Course in Turbulence. MIT Press, Cambridge, MA,1972.
[9] Milton Van Dyke, editor. An Album of Fluid Motion. The Parabolic Press, Stanford, CA, 1982.
[10] Kenneth Wark. Thermodynamics. McGraw-Hill, New York, third edition, 1977.
[11] David C Wilcox. Turbulence Modeling for CFD. DCW Industries, Inc., La Canada, CA, 1993.