This is the second part of the talk on zero-equation turbulence models,
the algebraic turbulence models. In this part, I will introduce some zero-equation
turbulence models, together with some variants, such as half-equation
turbulence model, and the algebraic Reynolds stress models ('ASM').
The first zero-equation turbulence model is introduced in this slide
in 1967 Cebeci-Smith proposed a two-layer model for calculating the eddy
viscosity in the boundary layer, which is given by this formula
here ym is the value of y for which the eddy viscosities of the inner and the
outer boundaries are equal, see the figure here.
In details, the eddy viscosity for the inner boundary is calculated as this.
the eddy viscosity is proportional to the mixing length squared and the
mixing length is proportional to the distance to the wall boundary y
and the modification in this square brackets. The eddy viscosity in the outer boundary
is calculated as this: it is proportional to the velocity at the edge of the layer Ue,
and the velocity thickness DELTA*_v, and it is also modified by the Klebanoff
relation F_Kleb.
The closure coefficients are given by Kappa 0.4
ALPHA 0.0168,  and A+ is given by this formula
and it can be seen the pressure gradient is included in the modification.
Generally the speeches miss model is elegant and easy to implement if the
value of ym is determined. here the estimation of y+_m is given at a
Reynolds number typically of the full developed turbulence, matching between
inner and outer layers occurs well into the log layer, thus y+_m is
estimated as this: it is proportional to the Reynolds number based on the
displacement thickness DELTA* and on velocity at the edge of the boundary
layer, Ue. For a typical Toby and the flow the
corresponding Reynolds number Re_delta* is about 10,000, therefore the
corresponding y+_m is about 420.
the second zero-equation turbulence model is the Baldwin-Lomax model
which was formulated in 1978 and it was proposed for the
removing the difficulties found in Cebeci-Smith model, when the boundary layer
thickness, the velocity thickness and the velocity at the edge of the boundary
layer if the complicated flows are studied, for instance, the separated
flows; the flows with shock waves etc.
the Baldwin-Lomax model is also a two-layer model, with the eddy viscosity
in the inner boundary calculated as this, which is proportional to the mixing-length
squared and the magnitude of the vorticity vector, OMEGA given as this.
and similarly, the mixing-length is proportional to the distance from the
boundary y and the modification in the square brackets.
For the other boundary
the eddy viscosity is calculated as this, here the wake function F_wake is given
by this, and y_max is the value of y at which l_mix times absolute OMEGA
achieves its maximum value , U_dif is the maximum value of U for the boundary
layer or the difference between the maximum velocity in the layer and the
value of U at y=y_max in the free shear layer given by this,
the closure coefficients are KAPPA 0.4; ALPHA 0.0168; A+_0= 26; C_cp=1.6; C_Kleb
0.3 and C_wk= 1.
in this slide an application of zero-equation turbulence models is made for
the flow in a horizontal pipe, the pipe flow Reynolds number based on pipe
diameter and the average velocity, Re_D=40,000,
the comparison of the velocity profile is seen in this figure, we can see Baldwin-
Lomax model gives a better prediction than Cebeci-Smith model.
and that Reynolds shear stress, both zero-equation models are very
close to the experimental data and they are very close each other.
for a comparison of the friction for the smooth pipe, Reynolds' universal law of
friction is taken as a reference as this, and  c_f can be solved using the
iteration of the equation. and we can see Baldwin-Lomax model gives a
better prediction when compared to the experimental data, while the Cebeci-Smith
model gives a slightly worse prediction. For the boundary layer, both Cebeci-
Smith and Baldwin-Lomax models are very close to the experiment data.
in summary for the Baldwin-Lomax model predicted the velocity and the Reynolds
shear stress less than 3% difference when compared to the experimental data, and
the skin friction is within 1% when compared to the Prandtl's
formula, while for the Cebeci-Smith model the predictions are slightly worse: velocity
and skin friction are both within 8%.
in this slide 1/2-equation turbulence model is introduced.
Johnson and King in 1985 and Johnson and Coakley in 1990 proposed
a non-equilibrium version of algebraic model, with the eddy viscosity calculated
as this, here MU_ti and MU_to are the calculated inner and outer
layer eddy viscosities. the eddy viscosity for inner layer is
calculated as this, all parameters can be
calculated in all these formulas or
specified, but TAU_m and u_m. here the subscript m denotes the value
at the point y=ym at which the Reynolds shear stress, TAU-xy
reaches its maximum, given by this, and that the eddy viscosity in the outer
boundary layer is given as this, this is very similar to Cebeci-Smith's
model, with the modification of the non-equilibrium parameter SIGMA(x).
In this 1/2-equation turbulence model, an ordinary differential equation for
the maximum Reynbolds stress TAU_m can be calculated in terms of for this
formula, the ordinary equation is given by this
here um is the mean velocity and (um)_eq is the value of um according to the
equilibrium algebra model when SIGMA(x)=1.
Because this differential equation is an ordinary differential equation, it's different from the partial
differential equation, therefore this turbulence model is termed as half-
equation turbulence model. And the closure coefficients are: Kappa=0.4
ALPHA 0.0168, A+=17, a_1=0.25, c_1= 0.09, and c2= 0.7, here C_dif
equals 0.5 when SIGMA(x) is larger (than)
or equal to 1, otherwise C_dif is 0.
Here an example of a mild flow separation and recirculation is given for
illustrating the zero-equation and the half-equation turbulence models. the example
is a horizontal cylinder of a length about 500 mm and a diameter of
140 mm, the uniform flow comes from the left-hand side as shown in the figure,
and it should be mentioned that the wind tunnel
test section in this part is not uniform, but expends along the length of
the cylinder, not shown here, details can be found in this reference.
the comparisons are made for the
friction coefficient Cf, here it can be seen that the half-equation turbulence
model predicts the friction coefficient much better than the zero-equation
Baldwin-Lomax model. For the pressure coefficient Cp, and again we can see that
the half-equation turbulence model predicts the pressure coefficient better
than Baldwin-Lomax model. In fact for this specific example, the half-equation
model predicts the pressure coefficient better than many one- and two-
equation turbulence models.
Here in this talk I put the algebraic Reynolds stress model ('ASM'),
it is for a purpose for clarifying the confusions, at least this is a
confusion to me before.
Basically algebraic Reynolds stress models
have been proposed for avoiding solving the full Reynolds stresses, so to reduce the
computation in the modelling of the turbulent flows.
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, like
those real zero-equation model: they can be applied only together with the two
equation covers mother for which the
kinetic energy k and the dissipation
rate EPSILON are available. As such, in the turbulence flow modeling the
computation would be slightly more than convention two-equation turbulence model
but it would be significantly less then the full Reynolds stress model with seven
equations, and this approach could remove some shortcomings in the Boussinesq's
hypothesis, such as the anisotropy of the Reynolds stresses.
in 1972 and 76, Rodi assumed the transport of the Reynolds stress is
proportional to that of kinetic energy k, meaning that the anisotropy is
constant along the streamlines, and the mathematical equation for this
approximation is given as this.
this approximation could lead to a nonlinear
algebraic equation that can be used for determining the Reynolds stress tensor,
given by this. Superficially, with suitable approximations of the
dissipation rate EPSILON_ij and the pressure-strain correlation tensor PI_ij,
so this approximation equation provides a nonlinear equation for solving the
Reynolds stresses, so it is possible to avoid solving the differential transport
equation for Reynolds stresses. Therefore, this approach is called algebraic Reynolds stress model,
'ASM'. However this equation must be used
together with other turbulence models, since the kinetic energy k and the
dissipation rate EPSILON must be approximated first. Normally these
parameters are only available from other
turbulence models, such as the standard
k_EPSILON  model. So far I haven't seen the modelling method for the
dissipation rate, Epsilon.
so we will see in this slide why we cannot solve the algebraic Reynolds stress
equation as the conventional turbulence models.
The first point is that
the algebraic Reynolds stress model cannot solve for the kinetic energy,
which is unfortunately needed in the equation if we solve for other Reynolds
stress component.
if we take the indices i=j,
then we can obtain the following equation based on the
Rodi's approximation in the previous slide, and if we apply these relations, we
can see for the incompressible flows we
have this and this terms are zero
and we also following the definition for the kinetic energy and its relation with
the trace of the Reynolds stress tensor, TAU_ii and the dissipation rate
EPSILON_ii for this, here EPSILON is defined as this. Therefore we can deduce
the relation as this, this equation is equivalent to this equation. therefore
the Rodi's approximation cannot provide any information for the kinetic energy.
In addition, as pointed out that by Speziale
in 1997, such a nonlinear equation may have mathematical difficulties since we
may have multi solutions or singularities in some cases, which could
cause numerical difficulties for solving the equation. therefore Speziale
recommended that traditional algebraic stress model should be abandoned in the
future application, in favour of the regularised, explicit algebraic
(Reynolds) stress models.
it is very natural that the explicit algebraic Reynolds stress
model could be used together with and thus enhance the two equation models,
since the EASM could more accurately represent the anisotropy of the
Reynolds stress than the linear Boussinesq's model which is employed in the
convention two-equation models. An advantage of such an approach is
computationally less expensive than the full second-order Reynolds stress models,
for instance, Gatski and Speziale in 1992 proposed an explicit Reynolds
stress model, given by this expression, here S_ij and OMEGA_ij are the strain-
rate and vorticity tensor, respectively, and ETA and XI are given as this.
here an example of a full-developed turbulent channel flow subject to
a spanwise rotation with the constant angular velocity OMEGA. This is a
turbulent flow we can see here the asymmetric flow which is specially
difficult for the conventional turbulence models based on the Boussinesq's hypothesis.
The explicit algebraic Reynolds stress model
with the standard k-EPSILON model could predict such a flow pretty well,
as seen in this figure.
so in summary, the advantages for the zero-equation models: zero-equation
models are simple and easy-to-implement in the numerical simulation; zero-equation
models are ready numerical stable, which might be a good choice for a pre-
calculation of the flows. Zero-equation models are quite success in
some simple flows, especially for the Johnson-King half-equation model.
This model has been successfully applied in many transonic flows and some
results may be even better than the modern advanced turbulence models.
The algebraic Reynolds stress model could provide a better representation
of the anisotropy of the Reynolds stresses than the Boussinesq's hypothesis, but
I must say here the algebraic Reynolds stress model or the explicit algebraic
Reynolds stress model are not real zero-equation models, they must be used
together with other turbulence models.
For the downside of the zero-equation model,
Generally the zero-equation models are incomplete models, that is, this type
of the models would work well only for the flows for which the models
have been fine-tuned. For instance, the five free shears flows, we have to use the
five different mixing lengths; the extrapolation of the zero-equation model to the
complete flows or the new area would be dangerous, no assurance,
nor guarantee for the result.  As a result of the limits of zero-equation models,
they are not the options for the advanced research and applications.
For instance, no options of zero-equation model in ANSYS Fluent; no entries in the
NASA Langley Research Center, Turbulence Modeling Resource, although there are many
different types of turbulence models are available in this commercial software or
the turbulence modeling resource.
