The whole field of treating the lower scale turbulent closure equations when the Upper scale VAT averaged fields are sought is made of the issues of averaging the additional expressions and closure equations used to be in turbulent science to support a variety of homogeneous fluid turbulent flow theories. One of them is  "To Average or Not to Average? And What are the consequences?" And how to do "if to make" the averaging of those highly nonlinear equations? We mentioned in few places of this website, that the very critical, important methodology of nonlinear terms VAT averaging is the complicated one.
Here is the text with the paper presented at the 3rd ASME/JSME Fluids Engineering Conference, San Francisco  Travkin, V. S., Hu, K., and Catton, I. (1999). Turbulent kinetic energy and dissipation rate equation models for momentum transport in porous media. In Proc. 3rd ASME/JSME Fluids Engineering Conf.  FEDSM997275.
Due to restriction on the size of presented for the 3rd ASME/JSME(1999) conference papers this text was presented only using the 6 printed pages. It is really short, and I can'not extend it here with the all supportive, research and technical questions described.
This paper should be viewed as the consistent study to seek the upper (second) scale kinetic energy and dissipation rate equations averaged mathematical formulations, those are the result of strict application of the theorems developed in VAT for linear and nonlinear mathematical expressions.
And at the same time, this paper is of no means suggesting that I am in a favor of using these exactly forms of said above equations. I have some work and studies, also used the simplified forms of these equations, and can here to make some sort of ascertaining that this problem is standing alone and up for future investigations.
Among the many possible models for turbulent momentum in porous media, the socalled k turbulence models are considered to be appropriate for the lowest level of hierarchical modeling. The fluctuations produced by the porous medium heterogeneity and wall roughness contribute to the overall transport processes and to the governing scalar fluctuation equations, the Reynolds stress equation, the turbulent kinetic energy (TKE) and the dissipation rate of TKE. These equations must be treated along with other phenomenological dependencies. In view of the theoretical importance to turbulent transport in porous media, incompressible turbulent flow within porous media is modeled using the proper averaging of the transport equations for the turbulent kinetic energy () and its dissipation rate ().The closures of these model equations are obtained for a few morphologies. The averaging of these equations is based on strict principles developed for application of volume averaging theory (VAT). A variety of closure alternatives are considered for application to the VAT turbulent kinetic energy and VAT dissipation rate of TKE resulting in a set of simulation equations. An analysis of other equations used in heterogeneous transport theories is given for comparison.
  mean skin friction coefficient over the turbulent area of []  
  mean drag resistance coefficient in the REV []  
  mean form resistance coefficient in the REV []  
  mean skin friction coefficient over the laminar region inside of the REV []  
  dimensionless coefficients 
  interface differential area in the REV []  
  form drag  
  friction factor  
  averaged value over  
  value , averaged over in a REV  
  value f morphofluctuation in a 
  gravity  
  buoyant production / destruction  
  turbulent kinetic energy  
  fluid thermal conductivity []  
  averaged turbulent eddy viscosity []  
  integral length scale of turbulence  
  porosity [] 
  pressure []  
  specific surface of a porous medium  
  
  cross flow projected area of obstacles []  
  internal surface in the REV []  
  temperature  
  velocity in xdirection  
  square friction velocity at the interface surface  
  mean velocity  
  velocity in zdirection  
  space coordinate 
  fluid phase  
  component of turbulent vector variable  
  laminar  
  solid phase  
  turbulent 
  value in fluid phase averaged over the REV  
  mean turbulent quantity  
  fluctuation turbulent quantity 
  averaged heat transfer coefficient over  
  thermal coefficient of volume expansion  
  representative elementary volume (REV)  
  pore volume in a REV  
  solid phase volume in a REV  
  turbulent coefficient exchange ratio []  
  turbulent coefficient exchange ratio []  
  kinematic viscosity  
  density  
  dissipation rate of turbulent kinetic energy  
  wall shear stress 
Turbulence modeling, a part of computational fluid dynamics, emerged as a distinct discipline and in conjunction with suitable numerical methods, became one of the more widely employed predictive tools in fluid mechanics and in heat and mass transfer. Using turbulent models to predict turbulent flow and related heat transfer in porous media is still in its infancy. Lee and Howell (1987) proposed a model for flow through porous media with high porosity using the same eddy viscosity for the porous media as one which is commonly used for a pure fluid. Turbulent transport equations for porous media were developed by Primak et al. (1986), Shcherban et al. (1986) and Travkin and Catton (1992, 1993, 1995) based on the generalized Volume Averaging Theory (VAT) for highly porous media along with the statistical and numerical methodology needed for their solution. Simplified models based on approaches taken in meteorology and agroscience were used in these works. Antohe and Lage (1997) presented a two  equation turbulence model for incompressible flow within a fluid saturated and rigid porous medium that is the result of incorrect procedures. Masuoka and Takatsu (1996) proposed a 0  equation model for the turbulent flow through porous media consisting of packed spheres.
In spite of many achievements in mathematical modeling of turbulent flow in porous media, there are still some physical effects that are not well modeled leaving the porous media research community without satisfactory tools. The motivation of this paper is to develop a more complete and rigorous model for the equations in porous media based on the VAT methods (see Travkin and Catton, 1998; and Whitaker, 1997).
Three types of turbulence models can be identified as approximate engineering
methods or models. They are the twoequation EddyViscosity Models (EVMs), the
differential Restress equation model (DSM) and intermediate (truncated) and
''hybrid'' models. Among the above possible models for turbulent momentum in
porous media, two  equation EVMs in their rudimentary forms have a major
advantage in their simplicity and practical usability. We will try to maintain
the blend of simplicity and acceptable predictive ability of popular models
such as the

models and employ them for the computation and predictions of turbulent flows,
heat and mass transfer and associated transport phenomena in porous media. The
existing closure equations for the

equations will be our starting point. For steady state incompressible flow,
the transport equation determining the distribution of turbulent kinetic
energy (TKE)
derived
by Rodi (1984) reads
It can also be written in the form
where
is the transposed dyadic.
Following the averaging scheme presented by Travkin and Catton (1992), the
averaged forms of the two factor terms in the turbulent kinetic energy
equation are
The term with the triple factoroperator product is
which is the kinetic energy production term. In order to derive the averaged
form of this term, first simplify the product of tensors
and
where
As long as the partial derivatives
can be presented as
which in turn transforms the partial derivatives
where
is the dyadic. Similarly, the other two vectorial variables take the form as
then Eqn.( ) can be expressed as
The second term
in Eqn.() is a product of tensors
and will be of the form
Averaging of the above operators involves averaging the triple
productoperator terms . The general triple product decomposition averaging is
formulated as
With this as guidance, the needed averaged terms can be obtained. For example,
for the additive term
is derived by assigning
For the term
one obtains
where
with averaged and fluctuation variables
The same process is applied to the averaged operators
and
.
Applying this technique to the averaging of term
yields the following formulae
where similar averaging procedures give three more scalar triple operand terms
in the averaged
equation
where
with averaged and fluctuation variables
Substitution of
and
for
when averaging
and
in the above formulae substitution is the only difference between these terms.
By combining all six terms that describe averaged production terms in the
equation for turbulent kinetic energy transport in porous media, one can write
The averaged turbulent kinetic energy transport equation will be
or
A similar transformation can be justified for the steady state
equation
or
Following the same averaging procedures outlined above leads to the final form
of the averaged dissipation rate equation,
or
A reasonable assumption to take the equality for the turbulent viscosity as
for the averaging procedure
which after averaging becomes
Closure is needed for practical application of the equations presented in the previous section. The closure scheme given below for the model equations may serve as guidance for finding closure for the model equations.
According to Monin and Yaglom (1975), the transport equation for the kinetic energy of the mean motion is
where
denotes the rate of dissipation of energy of the mean motion under the action
of molecular viscosity. The first term on the right hand side,
the loss of the kinetic energy due to interaction with the obstacles in the
mean stream, can be accounted for by complete transformation to fluctuation
kinetic energy
because there are no other sources or sinks for the fluctuation motion kinetic
energy. Thus
The onedimensional equation for momentum flux in a flat channel filled with
regular porous medium, under steadystate conditions and certain other
simplifications, with no flow penetration through
,
has the following form for a horizontally homogeneous stream (Travkin and
Catton, 1992, 1995)
The integral friction resistance terms in momentum equation are
where
The pressure drag resistance integral term is closed in a manner similar to
that for a one component pressure resistance coefficient over a single
obstacle,
The result is
The general volume force is represented by
The influence of the volume drag resistance forces within the porous media on
the turbulent fluctuation energy balance is taken into consideration by
considering the contribution of pulse type drag forces. Assuming that most of
the mean motion kinetic energy lost due to interaction of the flow with the
porous medium solid obstacles translates into increasing of the turbulent
fluctuation energy ( Monin and Yaglom, 1975; Menzhulin, 1970) brings one to
the conclusion that
A term like this in the kinetic energy equation accounts for the influence of
additional volume momentum resistance forces.
Mathematical models for modeling momentum transport in a porous media are developed using volume averaging theory (VAT) methods. These models use  and  fluctuation equations at the lowest level for closure of the subdomain homogeneous turbulent transport description. Averaging procedures performed over highly nonlinear components of those equations leads to the complicated final forms of the equations. Comparison with previous work and closure examples are given. Not withstanding the complicated appearance of kinetic energy and dissipation rate equations in porous media, the variables involved are completely described with known averaged variables and their variations. Subsequent work will use the derived equations in porous media momentum transport modeling.
This work was partly sponsored by the Department of Energy, Office of Basic Energy Sciences through the grant DEFG0389ER14033 A002.
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