1. Introduction
Modeling of macroscopic transport for laminar flows in permeable media can be based on the volume-average methodology [
1]. When properties of the fluid vary with time in addition to spatial deviations, two possible methodologies can be followed in order to obtain macroscopic equations. One can first apply time averaging [
2], or else, one can start with volume averaging [
3]. These two approaches lead to distinct sets of equations. However, the order of the averaging is immaterial if the variables are split using the double-decomposition theory, which is fully described in [
4] and briefly revisited here. This new concept sheds some light on the existing controversy about which order mathematical operators should be applied to governing equations when double averaging the equation set.
The double-decomposition idea was initially developed for the flow variables in porous media and extended to several flows and cases. A review of all these cases is beyond the scope of the present paper and, to the interested reader, the fundamentals of the double-decomposition concept can be found in [
5] in addition to its extension to moving media, which is detailed in [
6]. Examples including double diffusion [
7] and turbulent combustion in porous media [
8] are mentioned here as applications of the ideas compiled in [
4]. Other studies in the literature dealing with modeling turbulence in porous media include a review on methodologies [
9], Direct Numerical Simulations (DNS) [
10], two-scale models [
11] and flow in channels with permeable walls [
12].
Further, the study of turbulence though porous media can find application in several areas, including engineering and environmental research. Examples of the former are studies in porous combustors [
13], moving bed reactors [
14], thermal energy systems [
15], volumetric solar collectors [
16], impinging jets for cooling or heating [
17], among others. Applications of the study of turbulence in porous media can also be useful to environmental research such as simulation of forest fires [
18] and flows over vegetations [
19].
Here, a brief review of this new double-decomposition concept is revisited, but with no much of details, which can be found in [
4,
5,
6,
7,
8].
2. Flow Equations
The local instantaneous continuity and momentum equations are given as [
4,
5,
6]:
where
u,
ρ,
p,
μ and
g are the velocity vector, density, pressure, fluid viscosity and gravity acceleration vector, respectively.
2.1. Averaging Operators
Any general quantity
φ can be averaged in time as:
giving then
where
is the time fluctuation of
around its average value,
, with
.
Likewise, the volume average of
over a Representative Elementary Volume (REV) in a porous medium is given by [
20]
being,
where
=
/Δ
V is the porosity of the medium,
is the volume occupied by the fluid and
is the total volume (fluid plus solid). Further,
where
is the spatial deviation of
with respect to the volume average,
.
Gradient, divergent and time rate are also volume averaged as [
1,
20,
21]:
where
is the interfacial area,
is the velocity of phase
i,
n is the unit vector normal to
,
is a scalar and
a vector.
2.2. Averaged Balance Equations
By applying time averaging to Equations (1) and (2), one gets:
with
where
,
k,
and
I are the mean deformation tensor, the turbulent kinetic energy per unit mass, the turbulent viscosity and the unity tensor, respectively, and
is the production of
k due to the gradients of
u.
3. Double Decomposition
Using the operators defined above, applied simultaneously, any variable can be split into the four components, as (see [
4,
5,
6] for details):
or
Equation (15) entails the double-decomposition concept and the significance of the four terms on the right-hand-side of Equation (14) can be reviewed as: (1) is the intrinsic average of the time mean value of , (2) is the volume average of the fluctuating component, (3) is the deviation of the time-averaged value and (4) is the spatial deviation of the time fluctuation.
4. Momentum Equation
Applying both the volume and time averaging gives for the continuity Equation (11),
irrespective of the order of integration [
5,
6].
Also, regardless of the order of application of the average operations over the momentum equation, one has:
where
is the time-averaged total drag force per unit volume.
Applying the double-decomposition idea seen before to the inertia term
will result in
The meaning of the four terms on the right-hand side of Equation (20) can be seen as: I—convective term of , II—turbulent Reynolds stresses over ρ due to , III—dispersion due to , a term also present in laminar flow and IV—turbulent dispersion in a porous medium due to .
5. Macroscopic Two-Equation Models
The turbulence kinetic energy can be defined in two ways depending on the order of integration in the time and volume domains. In Ref. [
3], macroscopic
k was defined as
=
. On the other hand, if one proceeds with time averaging first, one ends up with
=
[
2]. The derivation of transport equations for
and
can be found in [
4].
The modeled equations read,
where
Dm represents the dispersion of
, which reads [
4]:
and involves correlations between deviations of velocity,
and
,
is the production rate of
due to the gradient of the macroscopic time-mean velocity
and,
where
are the production rate of
due to the mean gradients of
and the generation rate of intrinsic
k due the presence of the porous matrix, respectively,
is a constant,
K is the medium permeability and
is a constant to be commented below. The term
represents extra production of
due to the presence of solid material inside the integration volume and it is of null value for the limiting case of clear fluid flow when
.
5.1. Constant for the Macroscopic Model
The constant
introduced in Equation (24) was determined for the closure of the mathematical model. For macroscopic fully developed unidimensional flow in isotropic and homogeneous media, the limiting values for
and
are given the values
and
, respectively. In this limiting condition, transport terms are neglected and production and dissipation terms balance each other reducing the equations to:
or
Or say, for a steady-state fully developed one-dimensional macroscopic flow through an infinite porous medium (), transient and transport terms in Equations (23) and (31) vanish, in addition to . In this particular case, production due to the porous matrix, , balances the dissipation of k, giving rise to Equation (25). Renaming k and ε for this particular condition, we have Equation (26). Therefore, and are the values for and , respectively, when turbulence attains the equilibrium state, that is, when production and dissipation mechanisms solely dictate the levels of k and ε.
Using now the limiting cases
and
, both Equations (26) can be combined into the non-dimensional forms:
or
In order to determine
from Equation (28), microscopic computations in unit cells for different porosity and permeabilities were used to calculate the corresponding limiting values
and
(see Ref. [
4] for details). Once these intrinsic values were obtained, they were plugged into Equation (27). The value of
equal to 0.28 was found by noting the collapse of all data into a straight line.
5.2. Proposals for Macroscopic k
If we compare the definitions of
and
considering the above, we get [
4]:
where “vol. int.” means volume integration. We see that not all the turbulent kinetic energy of the flow is accounted for by
. Further, if we expand the term
by means of Equation (7), a connection between the two generation rates can also be written as follows:
We also note here that the production rate of
accounts for only part of the general production rate responsible for the overall balance of
.
5.3. Macroscopic Equations for ε
For the dissipation rate
, the modeled transport equation reads (see details in [
4]),
where
and
are constants and the terms from the left-hand-side of Equation (31) are the accumulation, convection, diffusion and source/sink terms.
For
, a proposal for a final modeled equation reads (see [
3,
22]):
The dissipation rates also carry a correspondence if we expand this as follows:
If the porosity
is considered to be constant,
indicating that an additional dissipation rate is necessary to fully account for the energy decay process inside the REV.
6. Concluding Remarks
In this paper we have revisited a methodology for the analysis of turbulent flow in permeable media, which was first published in the early 2000’s. A novel concept, called the double-decomposition idea, is revisited, showing how a variable can be decomposed in both time and volume in order to simultaneously account for fluctuations (in time) and deviations (in space) around mean values. Transport equations for the mean and turbulence flow have been presented.
Since the introduction of the double-decomposition concept, several authors have worked on similar treatments for turbulence in porous media and at different levels of complexity, sometimes combining what was already understood and detailed, some other instances dividing the turbulence spectrum into bands, each of which was handled by its own transport equation. In most works, however, the time–volume or volume–time sequence of integration has always played a role in setting up the overall modeling strategy.
Further areas of study for upcoming researchers might include the numerical and experimental evaluations of individual terms in the double-averaged equations above. The evaluation of individual terms and comparison with experiments would shed more light on the proposals herein, ultimately helping to refine a general tool for the analysis of several import engineering and environmental flows.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Acknowledgments
The author would like to express his appreciation to CNPq, CAPES and FAPESP, Brazil, for their invaluable support over the last two decades. Thanks are also due to the author’s former and current graduate students who conducted their graduate programs under the topic discussed here. This work was written during the author’s tenure at Purdue University under a Fulbright Chair in Interdisciplinary studies. Support from both Fulbright Foundation and Purdue University is greatly appreciated.
Conflicts of Interest
The author declares no conflict of interest.
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