Hello, welcome to my talk. Today's topic is on turbulence modeling , two-
equation turbulence models, Part 1, conventional turbulence models.
In this talk, the basic two-equation turbulence models will be introduced,
including the popular k-Epsilon and k-Omega turbulence models, as well as
the less popular k-kL and k-k TAU models. In this talk I will
introduce how these equations are derived and approximated
and the examples are given for some basic flows. In the summary the advantages
and disadvantages for k-Epsilon and k-Omega models are discussed.
In this talk I will introduce the fundamental equations for two-
equation turbulence models, including Reynolds averaged
navier-stokes equation, and the Boussinesq's hypothesis;
the turbulence transport equations for turbulence kinetic energy and
dissipation rate; the introduction of the Saffman's generalized
transport equation. And the introduction for k-Epsilon
two-equation turbulence models; the k-Omega two-equation turbulence models
and the other two-equation populism others;
applications of the table and models to the simple flows
the concluding immerse the wand ages and disadvantages are 4 KiB serum and
economic turbulence models
for the topping and the froze the dynamic equation
it's the notes average the navier-stokes equation given by this, here the capital
letters, capital UI, capital UJ and the capital P represent the averaged
velocity and the pressure in the topic and the froze
and the specific name of the stress tensor how IT a given by this this it's
a symmetric tensor introducing 6 unknowns to the Reynolds averaged
navier-stokes equation. And the question is that we have no more
laws for building the equation for solving these unknowns.
for the turbulent flows, the Reynolds stresses contain all the information
of the flow turbulence. According to Wilcox, Reynolds averaging is a
brutal simplification that losses much of the information contained in the
navier-stokes equation. So for turbulence models it is of vital importance how we
can effectively model or retrieve the flow turbulence.
so to close the Reynolds averaged Navier-Stokes equation,
the Boussinesq's hypothesis is often used, as we see in the zero-, one- and two- equation
turbulence models. Using this relation the Reynolds stress is linked with the
mean velocity, capital Ui and capital J here, and the turbulence kinetic
energy k here and NU_t here is the turbulence eddy viscosity.
for an incompressible flow, we can calculate TAU_ii as this, and we can see from the
continuity equation for the incompressible flows, these two terms will
be zero. Therefore we have the expression as this
so TAU_ii=-2k.
so substitute the Boussinesq's hypothesis to the RANS equation, we have
the Reynolds averaged navier-stokes equation, expressed as this.
so to solve this equation, the key issue is to determine the turbulence related
parameters: k and NU_t. So in this talk we will see how we can use the
two-equation turbulence models to close the dynamic equation for turbulent flows.
As we have shown the transport equation for Reynolds stresses TAU_ij can be
derived directly from the navier-stokes equation, given as this.
so in this equation you can see the production term and the viscous term
which can be calculated if the mean flow and Reynolds stresses are solved.
however we have also some other complicated terms, the pressure strain term,
given by this, which would introduce 6 unknowns; the diffusion term C_ijk, given
by this, which introduces 10 unknowns; the dissipation term, EPSILON_ij, given by
this, with 6 new unknowns. Therefore we build this transport
equation for solving the Reynolds stresses, TAU_ij of 6 unknowns, but
unfortunately, we introduce 22 new unknowns.
for simplifying the turbulence models, we are not solving the Reynolds stresses directly,
Instead, we solve for the turbulent kinetic energy k, and the
corresponding transport equation can be derived if we take i=j.
for the incompressible flow ,we have the equation as this,
in the derivation we use the expressions for these,
and consider the incompressible flow, all these four terms are zero. Now we term
the relevant terms using the Roman numerals II, III and IV. These terms
are corresponding to the regions in the figure of the energy cascading
process in the turbulent flow. In the region II, the production term
represents the interaction between the mean flow and the turbulence, which
extracts energy from the mean flow and deliver the energy to the eddies.
the region III is the diffusion region, involving some double and triple
correlations. This term represents the transport process of energy from the
large eddies to the smaller eddies; and the Region IV is the dissipation
term, including the viscous term and the dissipation rate.
based on the DNS data, the diffusion term is small for some simple flows, thus
it can be simplified using the diffusivity approximation in this
equation, which is a widely accepted in most turbulence models.
so from this equation, NU_t T is the eddy viscosity, therefore, this relation
implies that the diffusion process can be represented with the eddy viscosity.
so we can obtain the transport equation for the turbulence kinetic
energy, given in this form. this is the mostly used transport
equation for turbulence kinetic energy. So, to close this equation the
dissipation rate EPSILON and the eddy viscosity coefficient, NU_t, must be
specified or solved.
In turbulence flows, since the dissipation happens in the very small
length scales, thus it is generally accepted an
isotropic relation for the dissipation rate, given as this.
and if we take i=j,  we have this and the dissipation rate
EPSILON is given by this formula, and the exact transport equation
for the dissipation rate EPSILON is derived as this,
from this equation we can see the transport equation for the dissipation
rate is very complicated, especially in this large ellipse which involves various
complicated double and triple correlations, and these terms must be
modeled. One big question is how we can best model these terms?
so it should be noted this transport equation for the dissipation rate is
solved only for the k-Epsilon two-equation model,
while in other two-equation turbulence models, for instance, the k-Omega (not Epsilon) model,
the transport equation for the frequency Omega would be proposed based on the
physical reasoning, not derived directly from the navier-stokes equation.
the transport equation for the turbulent kinetic energy is
given as this. and the corresponding transport equation
is given as this. Now we can look at these two equations, we can see the
similarities between these two transport equations: Term I represents the local
change of the turbulent kinetic energy and the dissipation rate;
Term II is the convective term for these two parameters; and Term III is the
production terms for these two parameters; Term IV is the viscous terms,
and the other terms in the green ellipses,  these two terms are very complicated, which must be
modeled in the turbulence modeling.
in this slide, the Saffman's generalized transport equation is
introduced. The Saffman's generalized transport equation is given in
this form, here Q is the turbulence related parameter, and A is the function
of turbulence and the mean velocity. NU_d is the eddy
diffusivity or called the effective turbulence diffusivity. If we look at
this equation, in the left hand side, are those local change and the convective term.
these are consistent with the exact transport equation we have seen in the
previous slides, and the first term in the right hand side is corresponding
to the production term and the last term corresponding to the eddy viscosity term.
and the diffusivity coefficient can be regarded as the sum of the molecular
viscosity and the eddy viscosity. Based on the generalized transport
equation, Saffman could formulated his k-Omega model.
It should be noted Saffman's transport equation is for Omega squared,
and this k-Omega turbulence model has evolved to a very successful k-Omega
model of modern form. In fact, Kolmogorov formulated a similar
k-Omega model earlier in 1941, but his work in Russian was not unknown outside the
former Soviet Union until much later time, please see the information in the
reference [2].
Launder and Sharma in 1974 re-fined and tuned the k-Epsilon model, which was
generally accepted as the standard k-Epsilon model. The transport equation
for the kinetic energy k is given by the standard transport equation,
and the transport equation for the dissipation rate EPSILON is formulated
as this. if we compare these 2 transport
equations, we can see the transport equation for the dissipation rate can
be basically derived from the transport equation for the kinetic energy by multiplying
EPSILON/k, and adjusting the corresponding coefficients
C_epsilon1 and C_epsilon2 etc, so this analog form is employed in most two-
equation turbulence models. One question the author has: in the exact
transport equation for the dissipation rate, we have the exact production term
given in this, but the constructed transport equation for the dissipation rate
EPSILON is different, WHY?
The kinetic eddy viscosity is calculated as this formula,
so this relation is basically established on the dimensional analysis,
a question may be asked: why the turbulence eddy viscosity NU_t is only
related to k and EPSILON. So based on the dimensional analysis, we
can see the unit for the eddy viscosity the unit is meter squared per
second and the kinetic energy the unit is meters squared per second squared and
the dissipation rate the unit is meter squared per second cubed, so if we
make the analysis on the unit we can see k squared divided by epsilon, we have
the unit, same as the eddy viscosity, meters squared per second.
so for the standard k-Epsilon, the closure coefficients and we also
have two auxiliary relations, given as this.
the auxiliary relations are not necessary for the k-Epsilon model
itself, but it could provide the links of the model with the k-Omega
model, and the Prandtl's one-equation model.
Saffman independently developed a k-Omega model in 1970, which was inherited by
Wilcox and had been constantly improved by Wilcox and many others.
following is the Wilcox 2006 k-Omega model:
the transport equation for the turbulent kinetic energy k is given by
this form, and here we can see the dissipation rate EPSILON is replaced by the
expression as this; and the transport equation for the turbulence
frequency Omega is given as this, so from these two transport equations,
again we can see this transport equation for the turbulence frequency Omega can
be derived from the transport equation for k, by multiplying Omega/K
and the adjusted corresponding coefficients ALPHA, BETA and etc.
here in the transport equation for the turbulence frequency, the cross-
diffusion term is added, this term is used to solve the problem of the freestream
dependency and this cross-diffusion term can be understood to resemble the Menter's
blending function in his SST k-Omega model.
the kinematic viscosity is calculated as this, NU_t equals K divided by
Omega tilde, the parameter Omega is given by this limiter,
So this limiter could limit the
magnitude of the eddy viscosity, hence, it could greatly improves incompressible
and transonic flow predictions. The closure coefficients and the auxiliary
relations are given in here and here Sigma_d is a function, similar to the Menter's
blending function.
so as we know in understanding of the fluid turbulence, two important parameters:
the turbulence length scale L and the turbulence time
scale, TAU, are frequently referred to, therefore it would be a natural choice
that the three equation fabulous models are constructed based on these two
parameters. so these include the Rotta's k-kl model,
which was proposed in 1968. This model basically extended the Pramdtl's one-
equation model, the transport equation for the turbulent kinetic
energy k is the same as the transport equation in the Prandtl's one-equation
model, here we can see the dissipation rate EPSILON is replaced by the Prandtl's
hypothesis as this, and the transport equation for the
parameter kl is given as this, so using this equation the turbulent
length scale l can be solved from this transport equation, rather than
specifying the length scale, as in the Prandtl's one-equation turbulence model,
because specifying the turbulence length scalar could be sometimes an impossible
task, even for those experienced researchers. so the kinematic eddy
viscosity is calculated in this formula, and the closure coefficients are given here.
In 1986, Zeieman and Wolfshtein proposed the k-k TAU model,
the transport equation for the turbulence kinetic energy is given by this and
the transport equation for the parameter k TAU is given by this.
if we compare these two transport equations, we can see the transport
equation for the parameter k TAU is very similar for the transport equation for
the turbulent kinetic energy, multiplying the parameter TAU and
adjusted the coefficients here C_tau1, C_tau2 etc.
the kinematic eddy viscosity is calculated in this formula, and the
closure coefficients are given here, and also the auxiliary relations given here
which link the k-k TAU model to the k-Epsilon, k-Omega and the Prandtl's
turbulence length scale.
The first example is for turbulent boundary layer. From the theoretical
derivation and the experiment data, it has been shown the log layer can be
expressed as in this formula, here Kappa 0.4 is von Karman constant,
and C is a constant. and for the sublayer, y approaching to
zero, we should have the asymptotic solutions to the parameter k or for the
expression this (would be a constant). Based on the solution of the boundary layer,
using different turbulence models, it can be seen that the constant C
can be well predicted using the k-Omega models,
but no turbulence model could predict the component n correctly, when y
approaches to zero, and no model could reproduce the expression correctly,
since all the predictions are far from the measurements or the
theoretical values.
the second example is for the prediction of the spreading rate of the free shear
flows. the definitions of the spreading rates
for different flows are given in here. so in this table we can see the
predictions with the k-Omega models, k-Epsilon and the RNG k-Episode
model, which is an enhanced k-Epsilon model. all these spreading rates are
compared with the measurement value, so from the prediction we can see the k-
Omega model gives the best answer to all these spreading rates and in this
table we can also see that it is not necessary the enhanced RNG k-Epsilon
model could predict the flows better than the standard k-Epsilon model.
This is understandable since if the enhanced RNG k-Epsilon model can predict
all the flows better than the (standard) k-Epsilon model, then the standard k_Epsilon
model would disappear. and in this table more different
turbulence models are compared and again the Wilcox's 2006 k-Omega model gives the
best prediction for the free shear flows.
The third example is the backward facing step flow, the comparison is made
for two different top wall angles: ALPHA=0 deg, and ALPHA=6 deg,
so the comparison using different
turbulence models for the reattachment of the flow
we can see the one-equation Spalart-Allmaras model, k-Omega
model and k-Omega/k-Epsilon model, the Menter's SST model, predict the
reattachment close to the experiment data.
so if we calculate the error of the prediction, compared to the
measurement data, we can see for the top wall angle
zero degree, k-Epsilon model under-predicts the attachment by 17% and for the top
wall angle, 6deg,  it under-predicts the reattachment by 32%, while for the k-Omega
model, the error is 7.7% and 4.3%.
So from this example we can see k-Omega model is much better than the
standard k-Epsilon model.
so in this slide the summaries will be given for the conventional two-equation
turbulence models, especially for k-Epsilon and k-Omega models.
Basically, the two equation turbulence models are complete in principle, but
turbulence models will never be perfect in the author's
opinion: they can only be used to solve some problems in the flow dynamics, not
all the problems. Therefore the applicability to wider range flows
is a key factor for all turbulence models. According to Menter, how to
construct the kinematic eddy viscosity is a critical factor for a successful
turbulence modeling.
The k-Epsilon model was the most widely
used two-equation turbulence model before 1990s. The reason why the k-Epsilon model
had been the most widely used two-equation model, it might be for the
transport equation of the dissipstion rate, because it is derived from the
navier-stokes equation directly, but it is very complicated,
thus many terms of it must be modelled, or approximated; The k-Epsilon model is
extremely difficult to integrate through the viscous sublayer and requires
viscous correction; the application successfully to some
flows, but it is difficult for the flows with the adverse pressure gradient and
for the separated flows; the k-Epsilon model might need to be
fine-tuned before a given application.
For the k-Omega model, now it has become the most widely used two-equation
turbulence model, according to Wilcox.
the transport equation for the turbulence frequency Omega is not
derived from the Navier-Stokes equation, which is purely based on the physical
reasoning and from the Saffman's generalized transport equation;
the k-Omega model is significantly more accurate for 2-D boundary layer, with both
adverse and favorable pressure gradients;
the model can be easily integrated through the viscous sublayer;
the k-Omega model accurately reproduces the subtle features of turbulence
kinetic energy behavior close to the solid boundary condition; the early k-Omega
model has the strong freestream dependency.
