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. - FEDSM99-7275.

Due to restriction on the size of presented for the 3-rd 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 non-linear 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.

Turbulent Kinetic Energy and Dissipation Rate Equation Models for Momentum Transport in Porous Media

V. S. Travkin, K. Hu and I. Catton

Department of Mechanical & Aerospace Engineering

University of California, Los Angeles, CA 90095

Abstract

Among the many possible models for turbulent momentum in porous media, the so-called k-$\varepsilon $ 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 ($k$) and its dissipation rate ($\varepsilon $).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.

Nomenclature

$\widetilde{c}_{d}$ - mean skin friction coefficient over the turbulent area of $\partial S_{w}$ [-]
$c_{d}$ - mean drag resistance coefficient in the REV [-]
$c_{dp}$ - mean form resistance coefficient in the REV [-]
$c_{fL}$ - mean skin friction coefficient over the laminar region inside of the REV [-]
MATH - dimensionless coefficients

$ds$ - interface differential area in the REV [$m^{2}$]
$F$ - form drag
$f$ - friction factor
$\widetilde{f}$ - averaged value $f$ over $\Delta \Omega _{f}$
$<f>_{f}$ - value $f$, averaged over $\Delta \Omega _{f}$ in a REV $\Delta \Omega $
$\widehat{f}$ - value f morpho-fluctuation in a $\Omega _{f}$

$g$ - gravity $[m/s^{2}]$
$G$ - buoyant production / destruction
$k$ - turbulent kinetic energy $[m^{2}/s^{2}]$
$k_{f}$ - fluid thermal conductivity [$W/(mK)$]
$K_{m}$ - averaged turbulent eddy viscosity [$m^{2}/s$]
$L$ - integral length scale of turbulence
$m$ - porosity [-]

$p$ - pressure [$Pa$]
$S_{w}$ - specific surface of a porous medium MATH
$S_{wp}$ - MATH
$S_{\perp }$ - cross flow projected area of obstacles [$m^{2}$]
$\partial S_{w}$ - internal surface in the REV [$m^{2}$]
$T$ - temperature $[K]$
$u$ - velocity in x-direction $[m/s]$
$u_{\ast rk}^{2}$ - square friction velocity at the interface surface $[m^{2}/s^{2}]$
$U,V,W$ - mean velocity $[m/s]$
$w$ - velocity in z-direction $[m/s]$
$x$ - space coordinate $[m]$

Subscripts

$f$ - fluid phase
$i,j$ - component of turbulent vector variable
$L$ - laminar
$s$ - solid phase
$t$ - turbulent

Superscripts

$\thicksim $ - value in fluid phase averaged over the REV
$-$ - mean turbulent quantity
$\prime $ - fluctuation turbulent quantity

Greek letters

MATH - averaged heat transfer coefficient over $\partial S_{w}$ $[W/(m^{2}K)]$
$\beta $ - thermal coefficient of volume expansion $[K^{-1}]$
$\Delta \Omega $ - representative elementary volume (REV) $[m^{3}]$
$\Delta \Omega _{f}$ - pore volume in a REV $[m^{3}]$
$\Delta \Omega _{s}$ - solid phase volume in a REV $[m^{3}]$
$\sigma _{k}$ - turbulent coefficient exchange ratio MATH [-]
$\sigma _{T}$ - turbulent coefficient exchange ratio MATH [-]
$\nu $ - kinematic viscosity $[m^{2}/s]$
$\varrho $ - density $[kg/m^{3}]$
$\varepsilon $ - dissipation rate of turbulent kinetic energy
$\tau _{w}$ - wall shear stress $[N/m^{2}]$

1. Introduction

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 $k-\varepsilon $ 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 $k-L$ models based on approaches taken in meteorology and agro-science were used in these works. Antohe and Lage (1997) presented a two - equation $k-\varepsilon $ 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 $k-\varepsilon $ equations in porous media based on the VAT methods (see Travkin and Catton, 1998; and Whitaker, 1997).

2. Incompressible flow $k$- $\varepsilon $ model in Porous Media

Three types of turbulence models can be identified as approximate engineering methods or models. They are the two-equation Eddy-Viscosity Models (EVMs), the differential Re-stress 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 $k$-$\varepsilon $ 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 $k$-$\varepsilon $ equations will be our starting point. For steady state incompressible flow, the transport equation determining the distribution of turbulent kinetic energy (TKE) MATHderived by Rodi (1984) reads
MATH

It can also be written in the form
MATH


MATH

MATH
where MATH 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
MATH

MATH


MATH

MATH


MATH

The term with the triple factor-operator product is
MATH
which is the kinetic energy production term. In order to derive the averaged form of this term, first simplify the product of tensors MATH and MATH
MATH
where
MATH
As long as the partial derivatives MATH can be presented as
MATH
which in turn transforms the partial derivatives
MATH
where MATH is the dyadic. Similarly, the other two vectorial variables take the form as
MATH

MATH
then Eqn.( ) can be expressed as MATH
MATH
The second term $A_{2}$ in Eqn.() is a product of tensors MATH and will be of the form
MATH

Averaging of the above operators involves averaging the triple product-operator terms . The general triple product decomposition averaging is formulated as
MATH
With this as guidance, the needed averaged terms can be obtained. For example, for the additive term
MATH
is derived by assigning
MATH
For the term $A_{1i},$ one obtains
MATH
where
MATH
with averaged and fluctuation variables
MATH

MATH

MATH

MATH

MATH
The same process is applied to the averaged operators MATHand MATH. Applying this technique to the averaging of term $A_{2}$ yields the following formulae
MATH
where similar averaging procedures give three more scalar triple operand terms in the averaged $k$ equation
MATH
where
MATH
with averaged and fluctuation variables
MATH

MATH

MATH
Substitution of $\overline{V}$ and $\overline{W}$ for $\overline{U}$ when averaging $A_{2j}$ and $A_{2k}$ 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
MATH
The averaged turbulent kinetic energy transport equation will be


MATH
or
MATH
A similar transformation can be justified for the steady state $\varepsilon $ equation
MATH
or
MATH
Following the same averaging procedures outlined above leads to the final form of the averaged dissipation rate equation,
MATH
or
MATH
A reasonable assumption to take the equality for the turbulent viscosity as for the averaging procedure
MATH
which after averaging becomes
MATH

3. Turbulent $k$ - $L$ Model for Porous Media

Closure is needed for practical application of the equations presented in the previous section. The closure scheme given below for the $k-L$ model equations may serve as guidance for finding closure for the $k-\varepsilon $ model equations.

According to Monin and Yaglom (1975), the transport equation for the kinetic energy of the mean motion MATH is


MATH
where
MATH
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, MATH 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 $k$ because there are no other sources or sinks for the fluctuation motion kinetic energy. Thus


MATH
The one-dimensional equation for momentum flux in a flat channel filled with regular porous medium, under steady-state conditions and certain other simplifications, with no flow penetration through $\partial S_{w}$, has the following form for a horizontally homogeneous stream (Travkin and Catton, 1992, 1995)
MATH
The integral friction resistance terms in momentum equation are
MATH
where
MATH

MATH
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,
MATH
The result is
MATH
The general volume force is represented by


MATH
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
MATH
A term like this in the kinetic energy equation accounts for the influence of additional volume momentum resistance forces.

4. Conclusions

Mathematical models for modeling momentum transport in a porous media are developed using volume averaging theory (VAT) methods. These models use $k$ - $\varepsilon $ and $k$ - $L$ 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.

Acknowledgment

This work was partly sponsored by the Department of Energy, Office of Basic Energy Sciences through the grant DE-FG03-89ER14033 A002.

References

Antohe, B. V. and Lage, J. L., (1997), A General Two-Equation Macroscopic Turbulence Model for Incompressible Flow in Porous Media, Int. J. Heat Mass Transfer, Vol. 40, No. 13, pp. 3013-3024

Lee, K. and Howell, J. R. (1987), Forced Convective and Radiative Transfer within a Highly Porous Layer Exposed to a Turbulent External Flow Field, Proc. 2nd ASME/JSME Thermal Engineering Joint Conf ., Vol. 2, pp.377-386.

Masuoka, T. and Takatsu, Y. (1996), Turbulence Model for Flow through Porous Media, Int. J. Heat Mass Transfer, Vol. 39, No. 13, pp. 2803-2809.

Menzhulin, G. V., (1970), The Method of Calculating the Meteorological Regimes in the Vegetation Cover, Meteorology and Hydrology, No. 2, pp. 92-99.

Monin, A. S. and Yaglom, A. M. (1975), Statistical Fluid Mechanics, J. Lumley (ed.). MIT Press, Cambridge, MA.

Primak, A.V., Shcherban, A.N. and Travkin, V.S., (1986), Turbulent Transfer in Urban Agglomerations on the Basis of Experimental Statistical Models of Roughness Layer Morphological Properties, Trans. World Meteorological Organization Conf. Air Pollution Modelling Application, Geneva, Vol. 2, pp. 259-266.

Rodi, W. (1984). Turbulence Models and Their Applications in Hydraulics - a State of the Art Review, International Association for Hydraulic Research.

Shcherban, A.N., Primak, A.V., and Travkin, V.S. (1986), Mathematical Models of Flow and Mass Transfer in Urban Roughness Layer,'' Problemy Kontrolya i Zaschita Atmosfery ot Zagryazneniya, No. 12, pp. 3-10 (in Russian).

Travkin V. S. and Catton, I., (1992), Models of Turbulent Thermal Diffusivity and Transfer Coefficients for a Regular Packed Bed of Spheres, Procs., 28th National Heat Transfer Conference, San Diego, CA, ASME, HTD-Vol. 193, pp. 15-23.

Travkin, V. S., Catton, I., and Gratton, L., (1993), Single Phase Turbulent Transport in Prescribed Non- isotropic and Stochastic Porous Media, Heat Transfer In Porous Media, ASME, HTD Vol. 240, pp. 43- 48.

Travkin V. S. and Catton, I., (1995), A Two Temperature Model for Turbulent Flow and Heat Transfer in a Porous Layer, Journal of Fluid Engineering, Vol. 117, pp. 181-188.

Travkin, V.S. and Catton, I., (1998), ''Porous Media Transport Descriptions-Non-Local, Linear and Non-Linear against Effective Thermal/Fluid Properties'', Advances in Colloid and Interface Science, Vol. 76-77, pp. 389-443.

Whitaker, S. (1997), ''Volume Averaging of Transport Equations'', Chap. 1, in Fluid Transport in Porous Media, Computational Mechanics Publications, Southampton, UK, 1997.


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