Hello, welcome to my talk, on the enhanced turbulence models, Part 2. In this talk we
will introduce the popular and useful two-equation turbulence models in
both practical applications and in the research community, including the Menter's
SST model and the algebraic Reynolds stress model enhanced two-equation
turbulence models.
Two-equation turbulence models are the most popular turbulence models in the
practical applications, especially for the k-epsilon and k-Omega models due to
the simplicity and their successful applications in many different flows; and
two-equation turbulence models are the least turbulence model for the
complicated flows, according to Durbin in the reference. However, there are
limitations for the two-equation turbulence models and remedies have been
proposed for improving the capacity of the turbulence models for different and
wider applications, including: some simple enhancement, such as, Low Reynolds 
turbulence models, introduced in the first part of the talk on enhanced
turbulence models; the earlier k-Omega model has a strong
freestream dependency, as identified in the Menter's Baseline model; and the
problem in the construction of the eddy viscosity, as seen in Menter's SST k-Omega model,
the limits of the Boussinesq's hypothesis
which may be inaccurate for the simple flat plate boundary layer or shear layer;
Turbulence models with the algebraic Reynolds
stress model enhancement, via the norlinear Reynolds stress
expression; the talk includes the applications and
concluding remarks.
I would first introduce the Menter's Baseline model in this talk, which is
often omitted in the introduction of the turbulence models; the reason for this
introduction is because this baseline model serves today as the basis of the
many turbulence models, such as, SST model, the Omega-equation based Reynolds
stress model, or even the DES (detached eddy simulation) models, which
take advantages of the following turbulence models: the robust and
accurate formulation of the Wilcox k-Omega model in the near wall region; the
advantage of the freestream independence of the k-Epsilon model in the
outer part of the boundary layer. So in formulating the Menter's Baseline
model, we have two different turbulence models, the original k-Omega turbulence
model and the transformed k-Epsilon turbulence model, so the relation between the
parameters Epsilon and Omega would be given as this, here the coefficient Beta*
or C_mu are both 0.09, as we have seen in the standard k-Epsilon model.
The transport equations for the turbulences kinetic energy k are same for
both turbulence models, the only difference coefficients here
Sigma_k1 and Sigma_k2.
the transport equations for the frequency Omega are given as these,
respectively, for the original k-Omega turbulence model and transferred k-Epsilon turbulence
model. Here we can see the coefficients, Gamma 1,Gamma 2, Beta 1, Beta 2 and the
Sigma_omega1 and Sigma_omega2 would be different,
and the significant difference between the original k-Omega turbulence model
and the transformed k-Epsilon turbulence model is the term here.
this is a cross-diffusion term between the kinetic energy and the frequency
Omega, and this term is very important since the advaned k-
Omega model has included the cross-diffusion term.
So based on the two different turbulence models, the transport equations
for the Menter's Baseline k-Omega turbulence model would be given as this,
the transport equation for the kinetic energy k and the transport equation for
the frequency Omega, and here we can see a blending function F1 is employed in this.
Strictly speaking, the Menter's Baseline k-
Omega turbulence model is a zonal model, use the blending function F1. When F1 equals
(to) 1, the original k-Omega model is employed; and when F1 equals to 0, the transformed
k-Epsilon model is used, and in between the blending process is used. so
the overall closure coefficients are calculated at PHI given in this formula,
here PHI_1 is the corresponding coefficient for the original k-Omega
turbulence model, such as, Sigma_k1, Sigma_Omega1, etc; and PHI_2 is any constant in
the transformed k-Epsilon model, representing Sigma_k2, Sigma_omega2 and etc.
and the baseline model, the coefficient is given by this blended
coefficient PHI. The eddy viscosity is calculated as this formula, this is the
same as the eddy viscosity in the standard k-Omega turbulence model.
For the Menter's Baseline k-Omega turbulence model,
we have two sets of the closure coefficients for the original k-Omega
turbulence model, PHI_1 as this, and for the transformed turbulence model
PHI2 given as this.
and the blending function is defined as F1, calculated as this,
the argument is given in this formula. For different boundary layers we see
here, the solid lines, due to the different adverse pressure
gradients, the corresponding blending function F1 would be different, here we
can see that the Menter's Baseline model starts to blending before half of the
boundary layer thickness here, or even starts at the quarter of the
boundary layer thickness, the blue dashed line here.
Menter furthered his development on the combined turbulence model, named as the
SST model (short for Shear Stress Transport model). The transport equations
for the SST turbulence model is exactly same as those in the Baseline model, but
with slightly different closure coefficients,
The overall closure coefficients calculated as PHI, given as this, here the blending
function is F2, and PHI_1, PHI_2, represent the closure coefficients for
the original and the transformed turbulence models, respectively.
the significant difference between the Baseline model and the SST model is the
eddy viscosity formulation: eddy viscosity is calculated as this formula,
here if a1*Omega is larger than capital Omega*F2, then the
eddy viscosity is the same as that in the Baseline model.
So the closure coefficients and the relations for the Menter's SST k-Omega turbulence
models are given for this: Set 1, PHI1 as this, if we compare the SST k-Omega
turbulence model to the Baseline model, we can see the coefficient for Sigma_k1
and Sigma_omega1 are different, and the closure coefficients for the transformed
k-Epsilon model are exactly same as that used in the Baseline model.
so the blending function is defined as F2, if we compare to f1, we can see
the difference of the formulations for F2 and F1, the argument for the blending
function F2 is calculated in this formula.
so we draw out the boundary layers and blending functions, we have the
drawing as this, and for different boundary layers, see the solid lines, we
have the slightly different blending function F2, and we can see from this
blending function, the Menter's SST k-Omega model starts to blend at nearly 3/4 of
the boundary layer thickness. The another difference for the Menter's SST k-Omega
model is the blending function is not very sensitive to the different
boundary layers, this is a very important feature for the Menter's SST k-Omega
turbulence model.
The example for comparing different turbulence models is the Driver's flow, see
in this picture, in which the horizontal circular cylinder is placed in the
tunnel, and the experiment setup allows different adverse pressure
gradients acting on the flow past the horizontal cylinder. This experiment has
been widely used for calibrating the new turbulence models, and I referred to this
experiment many times in my talks on turbulence models.
The first example is the boundary layer profile on the horizontal cylinder at
five different locations,
the turbulence models include the Menter's SST, Menter's Baseline, Wilcox k-
Omega, and Jones-Launder k-Epsilon model. And from the comparison with
the experimental data, we can see Menter's SST model predicts the boundary layer
profile very accurately, best in all five locations, and the Jones-
Launder k-Epsilon model is the worst in all the cases.
The 2nd comparison is for the predictions of the turbulent shear
stresses. Again the comparison shows the Menter's
SST model gives the best answers for most locations, although it may not be so
good as that for the boundary layer prediction.
In this slide, a discussion is made for the Boussinesq's hypothesis. As we all
know in the turbulence models, such as, zero-, one-, two- and some other turbulence models,
the Boussinesq's hypothesis is used for linking the Reynolds stresses with the
averaged velocity gradient as this. This hypothesis has obvious shortcomings,
such as, no historic effect on the Reynolds stress, tensor; it cannot
well represents the flow over the curved surface; the flow in ducts with
the second motions; and the flow in rotating frame; And also for some
complicated 3d flows. For example, if we take the flat plate boundary layer
or the simple shear layer, both are driven by U(y). From the relevant 
references, we can see for the simple flat plate boundary layer, the normal Reynolds
stress ratio for three different directions: 4 compared to 2
and 3. And in the simple shear layer, these three normal Reynolds stresses
are: 1.07k to 0.23k and to 0.7k. So if we make an analysis to the Boussinesq's
formulation, we can see: TAU_11, given as this, TAU_22 given as
this, and TAU_33. Based on the assumption of the mean flow, capital U is
only dependent on y, so all these mean flow velocity gradients are zero, therefore, we
have all these Reynolds stress components as these, these will lead  to the
ratio of the normal Reynolds stresses: those are equal. So if we compare
these theoretical values to the experiment data, we can see the
difference. Therefore, the Boussinesq's hypothesis is very inaccurate, even in
such simple turbulent flows.
Basically algebraic Reynolds stress models have been proposed for avoiding
solving the full Reynolds stresses so to reduce the computations in the
modeling of the turbulent flow. However in reality the algebraic Reynolds
stress model, including the traditional algebraic Reynolds stress model, and
the explicit algebraic Reynolds stress models, are not independent or
self-contained, they can be applied only together with
two-equation turbulence models, for which the turbulent kinetic energy k
and the dissipation rate, Epsilon, are available. As such, in the turbulent flow
modeling the computation would be slightly more than the conventional two-
equation models, but would be significantly less than the full Reynolds
stress model (the 7-equation turbulence model). And this approach could
remove some shortcomings of the Boussinesq's hypothesis, such as, the
anisotropy of the Reynolds stresses. So as early as in 1970s,
the problem has been recognized, and the methods have been developed for the problem.
Rodi in 1972 and 1977 proposed the
transport of Reynolds stresses would be proportional to that of the turbulent
kinetic energy k, which means the anisotropy is constant along the streamlines,
with the mathematical equation for the approximation, given as this,
such an approximation would lead to a nonlinear algebraic equation for
calculating the Reynolds stresses TAU_ij as this. Superficially with the
suitable approximations to the dissipation rate, Epsilon_ij, and the
pressure-strain rate correlation, capital PI_ij, this nonlinear equation can
be used for solving the Reynolds stresses. However, to solve this nonlinear
Reynolds stress tensor, we must specify the turbulence kinetic energy k and the
turbulence dissipation rate, Epsilon. Therefore, this equation can be used for
calculating the Reynolds stress component,
like many other explicit algebraic Reynolds stress models, which could
remove some shortcomings of the simple Boussinesq's hypothesis.
In 1980, Wilcox and Rubesin proposed a Reynolds stress tensor,
expressed in a formula as this, here S_ij is the standard strain rate and capital Omega_ij
is the vorticity tensor, and the coefficient 8/9 is used to
ensure the normal Reynolds stresses would be correct in this form 
for the flat plate boundary layer. However, when we use this, we must have all the
variables k and NU_t, this means the Reynolds stress tensor must be used
combining with other turbulence models. And in 1992, Gatski and Speziale
proposed a Reynolds stress tensor as this formula, and the parameters
XI and ETA are calculated as this, and again we can see for calculating the
Reynolds stress tensor, the parameter k and Epsilon must be available,
and in this expression, to ensure the fraction to be free of singularity,
an approximation, called Pade approximation, can be used as this.
So in this slide, another explicit algebraic Reynolds stress model is
introduced, which was proposed by Wallin and Johansson in 2000, this model works
on the Reynolds stress anisotropy tensor a_ij, defined as this, and in
constructing the anisotropy tensor a_ij, the non-dimensionless strain rate,
S hat_ij, and capital OMEGA_hat_ij, defined in this form.
here we can see the turbulence kinetic energy and the dissipation rate must be
known, and the anisotropy tensor is given by this
complicated formula, and the coefficient Beta depends on
the scalar invariants, S_hat_ij and capital Omega_hat_ij, and also
some other independent invariants are given in this form.
In this slide, an application of the explicit algebraic Reynolds
stress model enhanced turbulence models to a high lift airfoil, the
National High Lift Program (NHLP), this airfoil is a 3-element airfoil
and the comparison is made for the angle of attack 20.18 deg
at the Mach number 0.197, and the Reynolds number 3.5 million.
the pressure coefficients at the station 1 and station 2 are compared. Here
the turbulence models include the new k-Omega model, with the explicit algebraic
Reynolds stress model enhancement; the Menter's Baseline k-Omega model, with the
explicit algebraic Reynolds stress model enhancement; and the Menter's
SST k-Omega model. So from the comparison with the experimental data
for the pressure coefficients, we can see all 3 turbulence models predict the
coefficients very well for this very complicated flow.
so from this table, we can see 3 k-Omega SST models with the enhancement of
explicit algebraic Reynolds stress model, it can predict the lift very
accurately, with the error about 1% or even less. Here for a comparison the Wilcox
k-Omega model is used, its prediction for the lift is slightly over 2.0%.
And for the drag prediction, the
Rumsey k-Omega model, with the explicit algebraic Reynolds stress model, gave
the very accurate prediction. So in this example, we can see even with the
enhancement of the explicit algebraic Reynolds stress models,
it cannot guarantee a good prediction.
and in this example, we can see compared to the prediction of the lift
the drag prediction is much more complicated.
According to Menter, tubrulence modelling, especially, the RANS models would
remain the optimal choice for many years to come, with the modeling of laminar-
turbulent transition in the focus. In the Author's opinion, the most popular
turbulence models in the industry applications would be the Menter's SST k-
Omega turbulence model, followed with Spalart-Allmaras one-equation turbulence model,
the latter is especially developed and tuned for aerodynamics applications.
Relatively, the k-Omega model could provide a good feature for the laminar-
turbulent transition modelling, but they are yet too sensitive to the freestream
parameters, therefore, Menter combined the advantages of the k-Epsilon and k-
Omega models to formulate the Baseline turbulence model, which can be used to
serve as the basis of many turbulence models, such as, the SST model, Omega-
equation based Reynolds stress model, or detached eddy simulation models; and
provide a blending fucntion or method in developing new turbulence models
or zonal methods. The explicit algebraic RSM are
basically the nonlinear extension to the Boussinesq's hypothesis, which
accounts for the effects of anisotropy; and allows the inclusion of
curvature and the rotation effects; In application, the turbulence models
enhanced with an explicit algebraic Reynolds stress model could
significantly improve the prediction of the flow, as the prediction of multi-
element NHLP airfoil.
