Turbulent Transport Two Scale VAT Governing Equations for Obstructed and Porous Media. Introduction

Analysis of Approaches, Attempts to Average, Upscale the Turbulent Models, Governing Equations for Lower Continuum Mechanics Scale Depiction

Development of the Upper Scale Turbulent Transport Modeling Governing Equations for Porous Media

NOMENCLATURE

REFERENCES

I would like to make more assessable the structure and logic of this subsection. Getting in this direction I can modify the material from previous published version (still shortened) with the closer look at the text's parts making more comments those seems trivial for professionals in turbulent transport, still are not familiar with the VAT scaling turbulent transport concepts and mathematics.

Here I would like to remind the main unsolved difficulties in scaling description of nonlinear general and turbulent transport in fluid mechanics, thermal physics, and chemical engineering at that time - beginning of 80-th. They are unsolved for many fields up to now. Those were -

  • 1) the scaling treatment of nonlinear right hand side two operators diffusive terms;
  • 2) the basic closure understanding;
  • 3) the absence of clear understanding of communications between the scales - What to calculate and what does it mean?;
  • 4) the complete gap between the continuum mechanics, take the fluid mechanics, for example, and the scaling fluid mechanics. Meaning - Can we calculate the conventional Fluid Mechanics problem in an obstructed volume (area) and make calculated properties and conclusions used for about the Upper Scale Fluid Mechanics properties and model? And vice versa ?

    • Most of these issues are not understood in many professional communities up to now.

    The scope of these tasks is not for the web publishing, but we are trying, with the hope that the learning capabilities of internet are still to be found.

    Analysis of Approaches, Attempts to Average, Upscale the Turbulent Models, Governing Equations for Lower Continuum Mechanics Scale Depiction

    The turbulent transport processes in highly structured or porous media are of great importance due to the large variety problems of this kind in environment, earth sciences, also for the heat- and mass exchange equipment used in modern technology. These include heterogeneous media for heat exchangers and grain layers, packed columns, reactors, etc. In all cases there occurs a jet or stalled flow of fluids in channels or around the obstacles. There are, however, few theoretical developments for the flow and heat- exchange in channels of complex configuration or when flowing around nonhomogeneous bodies with randomly varied parameters. The advanced forms of laminar transport equations in porous media were developed in a paper by Crapiste et al. (1986). For turbulent transport in heterogeneous media, there are few modeling approaches and their theoretical basis and final modeling equations differ.

    The lack of a sound theoretical basis impacts the development of mathematical models for turbulent transport in the complex geometrical environments found in nuclear reactors subchannels where rod-bundle geometries are considered to be formed by subchannels. Processes in each subchannel are calculated separately (see Teyssedou et al., 1992). The equations used in these works have often been obtained from two-phase transport modeling equations (Ishii, 1975) with heterogeneity of spacial phase distributions neglected in the bulk. Three-dimensional two-fluid flow equationswere obtained by Ishii (1975) using a statistical averaging method. In his development, he essentially neglected nonlinear phenomena and took the flux forms of the diffusive terms to avoid averaging of the second power differential operators. Ishii and Mishima (1984) averaged a two-fluid momentum equation of the form MATH MATH where $\alpha _{k}$ is the local void fraction, $\tau _{i}$ is the mean interfacial shear stress, $\tau _{k}^{t}$ is the turbulent stress for the $k^{th}$ phase, MATH is the averaged viscous stress for the $k^{th}$ phase, $\Gamma _{k}$ is the mass generation and $\QTR{bf}{M}_{ik}$ is the generalized interfacial drag. Using the area average in the second time averaging procedure, Ishii and Mishima (1984) introduced a distribution of parameters to take into consideration the nonlinearity of convective term averaging. This approach cannot strictly take into account the stochastic character of various kinds of spatial phase distributions. The equations used by Lahey and Lopez de Bertodano (1991) and Lopez de Bertodano et al. (1990) are very similar with the momentum equation being MATH MATH Here the index $i$ denotes interfacial phenomena and $M_{wk}$ is the volumetric wall force on phase $k$. Additional terms in equations (1) and (2) are usually based on separate micro modeling efforts and experimental data. One of the more detailed derivations of the two-phase flow governing equations by Lahey and Drew (1988) is based on a volume averaging methodology. Among the problems was that the authors developed their own volume averaging technique without consideration of theoretical advancements developed by Whitaker and colleagues (1986a)-(1997) and Gray et al. (1993) for laminar and half-linear transport equations. The most important weaknesses are the lack of nonlinear terms (apart from the convection terms) which naturally arise and the nonexistance of interphase fluctuations.

    Zhang and Prosperetti (1994) derived averaged equations for the motion of equal sized rigid spheres suspended in a potential flow using an equation for the probability distribution. They used the small particle dilute limit approximation to "close" the momentum equations. After approximate resolution of the continuous phase fluctuation tensor $\QTR{bf}{M}_{c}$ and the vector MATH the fluctuating particle volume flux tensor, $\QTR{bf}{M}_{D}$, they recognized that (p. 199) - "Closure of the system requires an expression for the fluctuating particle volume flux tensor $\QTR{bf}{M}_{D}$ .... This missing information cannot be supplied internally by the theory without a specification of the initial conditions imposed on the particle probability distribution". They also considered the case of "finite volume fractions for the linear problem" where the problem equations were formulated for inviscid and unconvectional media. The development by Zhang and Prosperetti (1994) is a good example of the correct application of ensemble averaging. The equations they derive compare exactly with those derived from rigorous volume averaging theory (VAT) (1999b). Transport phenomena in tube bundles of nuclear reactors and heat exchangers can be modeled by treating them as porous media Khan et al. (1975). The two-dimensional momentum equations for a constant porosity distribution usually have the following form Subbotin et al. (1979), MATH MATH MATH where the physical quantities are written as averaged values and the solid phase effects are included in two coefficients of bulk resistence, $A_{x}$ and $A_{y}$ , and an effective eddy viscosity, $\nu _{eff}$, that is not equal to the turbulent eddy viscosity. These kinds of equations were not designed to deal with nonlinearities induced by the physics of the problem and the medium variable porosity or to take into account local inhomogeneities. Some of the more interesting applications of turbulent transport in heterogeneous media are to agro-meteorology, urban planning and air pollution. The first significant works on momentum and pollutant diffusion in urban environment treated as a two-phase medium were the papers by Popov (1974, 1975). In these investigations, an urban porosity function was defined based on statistical averaging of a characteristic function $\eta (x,y,z)$ for the surface roughness that is equal to zero inside of buildings and other structures and equal to unity in an outdoor space. The turbulent diffusion equation for an urban roughness porous medium after ensemble averaging is MATH MATH where MATH means porous volume ensemble averaging, and MATH porosity. Closure of the two "morphological" terms, the first and the second terms on the right hand side, were obtained using a Boussinesque analogy MATH A descriptive analysis of the deviation variables MATH, MATH and the effective diffusion coefficient $K_{ij}$ was not given. In many studies of meteorology and agronomy, the only modeling of the increase in the volume drag resistance is by addition of a nonlinear term as done by Yamada (1982), MATH MATH where MATH is fraction of the earth surface occupied by forest, $m_{s}$ is the area porosity due to a tree volume and $f_{k}$ is a Coriolis parameter. The averaging technique used by Raupach and Shaw (1982) to obtain a turbulent transport equation for a two-phase medium of agro- and forest cultures is a plain surface 2-D averaging procedure where the averaged function is defined by MATH with MATH being the area within the volume $\Delta \Omega _{p}$ occupied by air. Raupach et al. (1986) and Coppin et al. (1986) assumed that the dispersive covariances were unimportant, MATH where MATH is a fluctuation value within the canopy and MATH $u_{i}^{\prime }$. The contribution of these covariances was found by Raupach et al. (1986) to be small in the region just above the canopy from experiments with a regular rough morphology. This finding has been explained by Scherban et al. (1986), Primak et al. (1986) and Travkin and Catton (1992,1995) for regular porous (roughness) morphology in terms of VAT. Covariances are larger, however, as the result of irregular or random two-phase media. When the surface averaging used by Raupach et al. (1986) is used instead of volume averaging, especially in the case of non-isotropic media, the neglect of one of the dimensions in the averaging process results in an incorrect value. This result should be called a 2-D averaging procedure, particularly when 3-D averaging procedures are replaced by 2-D for non-isotropic urban rough layer (URL) when developing averaged transport equations. Raupach et al. (1986) later introduced a true volume averaging procedure within an air volume $\Delta \Omega _{f}$ that yielded the following averaged equation for momentum conservation MATH MATH MATH where $S_{w}$ is interfacial area. Development of this incorrect and incomplete equation (see more explanations below in the subsection -

  • - "Modeling and Averaging in Meteorology of Heterogeneous Domains - Follow-up the NATO PST.ASI.980064"

    of this website) was based on intrinsic averaged values of MATH or MATH, whereas averages of vector field variables over the entire REV are more correct (see, for example, Kheifets, Neimark, 1982, etc.). Raupach et al. (1986) next simplified all the closure requirements by developing a bulk overall drag coefficient. The second, third and fifth terms on the right hand side of equation (Raupach et al., 1986) are represented by a common drag resistance term. For a stationary fully developed boundary layer, they write MATH

    where $C_{de}$ is an element drag coefficient and $S_{pe}$ is an element area density - frontal area per unit volume. A wide range of flow regimes is reported in papers by Fand et al. (1987) and Dybbs and Edwards (1982). The latter work revealed that there were four regimes for regular spherical packing, and that only when the Reynolds number based on pore diameter, $Re_{ch}$, exceeded 350 could the flow regime be considered to be turbulent flow. The Fand et al. (1987) investigation of a randomly packed porous medium made up of single size spheres showed that the fully developed turbulent regime occurs when $Re_{p}$ > 120 where $Re_{p}$ is particle Reynolds number.

    Some words need to be said on the paper by Antohe and Lage (1997), which we even commented in the letter to editor of "International Journal of Heat and Mass Transfer":

    ".... We believe the derived equations are deficient and will attempt to explain why in the following paragraphs by analysis of the paper.

    The approach taken by the authors demonstrates a misunderstanding of the basic physical principles of hierarchical modeling to obtain a phenomenological description of a transport process in porous media. The authors choose a set of phenomenological equations that are themselves the result of assumptions and simplifications and treat them as if they have a pedigree. The development of a set of equations that are rigorous does not allow one to use correlation based models developed by others that are themselves based on approximate conceptions of what the physical processes are dependent on. These models or terms in the equations already include many observed effects. After all that was their purpose. It is inadmissable for one to include such correlations in the Navier Stokes equations as done by Antohe and Lage because this results in the effects being included in the governing equations twice. Doing so incorporates the Darcy term and other factors (porosity function, for example) in the Navier-Stokes equations with the goal of further developing $k$-$\varepsilon $ equations.

    A number of serious deficiencies found in the paper are the following: 1) The authors initial set of equations are based on the assumption that the turbulent fluctuations and fluctuations caused by the porous medium are of the same nature. They are not and serious error can result if they are assumed to be the same. 2) The initial set of equations (1)-(3) is incorrect MATH MATH MATH MATH MATH MATH Darcy, Forchheimer and Brinkman terms are arbitrarily included in this set of equations that are assumed to be valid at some undefined scale. Porosity is arbitrarily included in the energy equation as a multiplicative factor in the convective term, again assuming that this is the correct equation for scale level heat transport description. 3) Given these deficiencies, the derived equations, ((7), (8) MATH MATH MATH MATH MATH as well as (11), (12), (17), (19), (32), (33), (42), (45) and others) are incorrect. 4) Use and manipulation of equations obtained from different levels of approximation and different scales, using equations (9) and (18) in (17) for example, leads to an equations set that has little rigor or generality. Given the above observations, the conclusions presented in the abstract of the paper that "Among them, this conclusion supports the hypothesis of having microscopic turbulence, known to exist at high speed flow, damped by the volume averaging process. Therefore, turbulence models derived directly from the general (macroscopic) equations will inevitably fail to characterize accurately turbulence induced by the porous matrix in a microscopic sense," are not correct. Before one can reach such conclusions, the derivations of the equations upon which it is based must be valid...."

    Volume averaging procedures were used by Masuoka and Takatsu (1996) to derive their volume averaged turbulent transport equations. Like numerous other studies of multiphase transport, the major difficulties of averaging the terms on the right hand were overcome by using assumed artificial closure models for the stress components terms. As a result, the averaged turbulent momentum equation, for example, has conventional additional resistance terms like the averaged momentum equation developed by Vafai and Tien (1981) for laminar regime transport in porous medium. A major assumption is the linearity of the fluctuation terms obtained, for example, by neglect of additional terms in the momentum equation.

    I would like to discuss here in more detail the main arguments following the papers by Masuoka and Takatsu (1996) and Takatsu and Masuoka (1997), including as much elements as it was done in those years. The momentum equation in Masuoka, and Takatsu (1996) work is MATH where
    MATH MATH MATH where $k$ is the turbulent kinetic energy. The averaging procedure for this vectorial equation provided using the intrinsic phase average MATH MATH Drawbacks of this kind of averaging for a vector variable equation was addressed by Kheifets and Neimark (1982). In further development the closure of fluctuations stress terms of the right hand side momentum equation tems for both the momentum and one-temperature energy equation were announced neglegable. The closure of right hand side very complicated expressions was obtained like MATH and MATH MATH "where $\sigma $ is the correction factor which is introduced to extend the concept of the hydrodynamic conductance defined by Darcy's law to the turbulent flow." ?! So, this is the clear intention to stay within the area of laminar phenomena parameters. Also, the biggest assumption in this work is that the linearity of equation's terms related to fluctuations induced by porosity is presented as MATH Further, after that kind of closure the averaged momentum equation became MATH

    which is the turbulent momentum equation with additional "Darcy like" term. Evidently, this equation - its derivation and final form based on the same type "heuristic" voluntary approach when the final form of equation was established in mind even before beginning of the derivation issues considered.

    In their experimental work Takatsu and Masuoka, "Turbulent phenomena in flow through porous media", Journal of Porous Media, Vol. 1, No. 3, pp.243-251, (1997), studied by visualization the flow in the flat vertical channel with very narrow width of the channel 1 cm and circular cylinders crossing the channel in staggered regular pattern as a bank of short tubes. In fact the experiments were set up in a 3D environment. They write in the abstract: "The present experiments support the fact that the Forchheimer flow resistance and dispersion have a close relevance to the turbulent mixing". To come to this conclusion they used the modeling equations, which are derived by voluntarilly methods and are closed to be as the Forchheimer pressure resistance model. So, while reading this conclusion one needs understand that it is obtained due to the fact that the model used in this work contains the Forchheimer type resisitance term which in turn points out to this conclusion. They used the following correlations for experimental data reduction - with the hydraulic diameter MATH also they used the flow resistance factor which is MATH and $Re_{tak}$ MATH where MATH a) the resistance factor byKnudsen and Katz (1958) for laminar regime across a bank of tubes MATH this correlation after substitution of the $f_{br}$ and $Re_{tak}$ through the $f_{f}$ and $Re_{por}$ gives MATH and for turbulent flow across a bank of tubes MATH MATH b) and by Bird et al. (1960) for packed beds MATH or MATH which is very close (actually almost equal) to the Ergun's correlation obtained for $Re_{por}$ in Travkin and Catton (1995) MATH

    Also they arrived to an "analysis" of "averaged" kinetic energy equation as

    MATH where the production term should be like this tensorial relation MATH They did accounting for the additional term in "averaged" kinetic energy equation MATH MATH then they did the convenient thing declaring that according to their definition this term should be as MATH which is unexplainable.

    Conclusion can be made that the procedures used in this work were constructed not on the consistent scientific approach ( justifiable or not ) - but using heuristic judgements to bring the final form of governing equations close to conventional one phase, one scale fluid mechanics equations.

    The paper by Wang and Takle (1995) is devoted to the same noble aim - to develop the turbulent filtration governing equation using averaging methods. Reading this paper: p. 75: "In this paper, we derive high-wavenumber-averaged equations and high-frequency turbulence budget equations which hold in the entire space rather than only in the air-space of a porous medium". The latter statement is the correct note. Authors state incorrectly that: ''Time averaging followed by spatial averaging implies that obstacle elements interact only with time-averaged flow and turbulence and do not interact with turbulent properties of the bounday-layer shear flow. The turbulence energy-cascade process is precluded under these assumptions.'' ''Therefore, schemes that first use the conventional time-averaging method do not allow for detailed description of interactions between the wake turbulence and shear flow.''

    To these statements needs to be said:

    Firstly: the assumptions mentioned in the text are not assumptions but merely applied averaging procedures.

    Secondly: When writing these statements authors forget to admit that the whole idea of the VAT is to perform the modeling and mathematical description on both scales ( levels) of a medium. This implies that the corect modeling on the first level of hierarchy gives the correct physical picture - including also the boundary-layer features modeling within the fluid region of the REV and in vicinity to interface surface.

    Thirdly: It is the situation when results and performance should be in clear correspondence with the admitted or expected physical and mathematical initial constructions. Meaning, that expectation of the results should not be evaluated on the basis of idealistical desires of improving the basics of turbulence theory using the VAT turbulent filtration governing equations and models, but strictly in agreement with undertaken or admitted just at the beginning the turbulence theory models themselves and their simplifications which are independently accepted in science many years ago for that or another purposes. The limitations, at this case, are directly inhereted from the initial turbulence assumptions - but not from the averaging procedures peculiarities.

    Analysis of the authors mathematical developments:

    Their averaged NS equation (15) is almost correct (excluding the second left hand term MATH instead of correct MATH) by form when comparing with the VAT equation MATH MATH MATH but actually this equation is incorrect because of the different definitions for their fluctuation variables $\ddot{u}_{i}$. (In this above equation dropped two terms irrelavent to our analysis - influences of Coriolis and buoyancy forces). The fluctuations of velocity $\ddot{u}_{i}$ and other functions $\ddot{f}$ are not equal to intrinsic fluctuations $\widehat{v}_{i},$ nevertheless that both are the spacial fluctuations and according to their definition (10) MATH but in VAT MATH and as the consequence the averaging of this fluctuation gives MATH which contradicts with their (11) making (11) incorrect ! Also their equation (17) is the equation which is defined as the kind of closure equation for the laminar fluctuations in the fluid region field MATH and there is no trust to this equation - it has no integral terms, etc.? Assumption (18) which states - that the four terms in the equation (17) are the resistance terms - is just the pure arbitrary speculation. Authors used to represent the averaged velocity field as the sum of time averaged MATH and instant averaged velocities MATH MATH and at the same time going from their previous definitions one can deduce that MATH so MATH or the full instant velocity is MATH

    Their mean motion equation (24) was developed on an easy basis because it obtains the form MATH MATH MATH after simple time averaging of the previously averaged laminar regime equation $($WT1$)$. This is really simple. The last term in this equation is the convenient resistance term commonly used in meteorology - which is the reflection of two last integral terms in the $($WT1$)$ and had undertaken no change during the time averaging procedure - which is, of coarse, inappropriate for transient statements, but as a first approximation is acceptable.

    Apart of the previous remark that this equation is incorrect due to incorrect treatment of fluctuation variable $\ddot{u}_{j}$ (plus incorrect writing of the second left hand term), still there are following questions:

    1) The velocity field MATH is not the turbulent velocity by definition, so what is it?

    2) What is the relation of this velocity to the real turbulent velocity $\overline{U}_{i}$ field happened to be in the porous media?

    3) For the steady state regime the laminar velocity field $\widetilde{U}_{i}$ that plays so important role in this equation development is the field without time dependent fluctuations $\widetilde{U}_{i}=$ MATH so, where from the time fluctuations does appear in the further developed part of this work the turbulent form of equation ? But we know that there are the steady state mean turbulent flows exist ! And they have field's time fluctuations as the inseparable part of their physics.

    4) The same kind of questions can be adressed to the issue of fluctuation variables MATH , $\ddot{u}_{j}$ sense (meaning) ?

    5) The impression is that in this development the porous medium itself is the major cause of the fluid turbulent motions which is not. Yes, it is the essential part of the turbulence characterized by porous medium, but not the major cause of the turbulent motion appearance.

    6) The $Closure$ of the momentum equation (24) is not understandable at the moment - a) the fluctuation variable $\ddot{u}_{j}$ is the laminar regime fluctuation by definition (10); b) the fluctuation variable MATH is not defined as usual instant point related velocity fluctuation, so it should be determined after laminar regime solution MATH would be known.

    When this equation compared with the VAT turbulent filtration (with the molecular viscosity terms neglected for simplicity) equation MATH MATH MATH MATH one would observe that in the latter equation for each variable exists the definition which is recognizable on each scale of hierarchy.

    We will stop here to analyze the various attempts to derive the averaged turbulent transport equations,

    referring visitors to more observations which can be found in - Fundamentals of Hierarchical Scaled Description in Physics and Technology/ Are there any other Methods and Theories available?
    http://travkin-hspt.com/fundament/04.htm

    and return to the HSP-VAT, which is based on the WSAM kind of theorems - heterogeneous media analogs to the only tool used for the derivation of mathematical physics governing equations in homogeneous medium.

    Development of the Upper Scale Turbulent Transport Modeling Governing Equations for Porous Media

    Fluid flow in a porous layer or medium can be characterized by several modes. Let us single out from among them the three modes found in a highly porous media. The first is flow around isolated obstacle elements, or inside an isolated pore. The second is interaction of traces or a hyper-turbulent mode. The third is fluid flow between obstacles or inside a blocked interconnected swarm of channels (filtration mode). The models developed by Scherban et al. (1986), Primak et al. (1986), and Travkin and Catton (1998) were primarily for laminar and turbulent nonlinear filtration and hyper-turbulent modes transport. Specific features of flows in the channels of filtered media include the following: 1) increased drag due to microroughness on the channel boundary surfaces, 2) gravity effects for meteorological problems, 3) free convection effects, 4) the effects of secondary flows of the second kind and curved stream-lines, 5) large-scale vortex effects, and 6) the anisotropic nature of turbulent transfer and resulting anisotropy of turbulent viscosity.

    It is well-known that in spacial boundary flows, an important role is played by the gradients of normal Reynolds stresses and that this is the case for flows in porous medium channels as well. As a rule, flow symmetry is not observed in these channels. Therefore, in channel turbulence models, the shear components of the Reynolds stress tensors have a decisive effect on the flow characteristics. At present, however turbulence models that are less than second order can not be successfully employed for simulating such flows (Rodi, 1980; Lumley 1978; and Shvab and Bezprozvannykh, 1984). Derivation of the equations of turbulent flow and diffusion in a highly porous medium during the filtration mode is based on the theory of averaging of the turbulent transfer equation in the liquid phase and the transfer equations in the solid phase of a heterogeneous medium (Primak et al., (1986); and Shcherban et al., 1986) over a specified REV .

    The initial turbulent transport equation set for the first level of the hierarchy, microelement or pore, was taken to be of the form ( see, for example Rodi, 1980; and Patel et al., 1985; among many others) MATH MATH MATH Here MATH and its fluctuation represent any scalar field that might be transported into either of the porous medium phases, and the last terms on the right hand side of (14) and (15) are source terms. Next we introduce free stream turbulence into the hierarchy. Let us represent the turbulent values as follows: MATH where the index $'k'$ stands for the turbulent components independent of inhomogeneities of dimensions and properties of the multitude of porous media channels (pores), and $'r'$ stands for contributions due to the porous medium inhomogeneity. Being independent of the dimensions and properties of the inhomogeneities of the porous media configurations, sections and boundary surfaces does not mean that the distribution of values of MATH, and $u_{k}^{\prime }$ are altogether independent of the distance to the wall, pressure distribution, etc. Thus, the values MATH , or $u_{k}^{\prime }$ stand for the values generally accepted in the turbulence theory, i.e. when a plane surface is referred to, these values are those of a classical turbulent boundary layer. When a round-section channel is involved, and even if the cross-section of this channel is not round, but without disturbing nonhomogeneities in the section, then the characteristics of this regular section (and flow ) may be considered to be those that could be marked with index $k$. Hence, if a channel in a porous medium can be approximated by superposition of smooth regular (of regular shape ) channels, then it is possible to give such a flow its characteristics and designated them with the index $k$, which stands for the basic (canonical) values of the turbulent quantities. Triple decomposition techniques have been used in papers by Brereton and Kodal (1992) and Bisset et al. (1991) among others. The latter utilized triple decomposition, conditional averaging and double averaging to analyze the structure of large-scale organized motion over the rough plate.

    It should be noted that there are problems where MATH, and $u_{k}^{\prime }$ can be found from known theoretical or experimental expressions (correlations) where the definitions of MATH, and $u_{k}^{\prime }$ are equivalent to the solution of an independent problem ( for example - turbulent flow in a curved channel). The same thing can be said about flow around a separate obstacle located on a plain surface. In this case one can write MATH The term MATH appears if the flow is through a nonuniform array of obstacles. If all the obstacles are the same and ordered, then MATH can be taken equal to 0. Naturally, the term $u_{k}^{\prime }$ in this particular case doesn't equal the fluctuation vector $u_{ks}^{\prime }$ over a smooth, plain surface. The following hypothes about the additive components is developed to correct the above deficiencies MATH

    It should be noted that solutions to the equations for the turbulent characteristics may be influenced by external parameters of the problem, namely, by the coefficients and boundary conditions which themselves can carry information about porous medium morphological features. The adoption of a hypothesis about the additive components of functions representing turbulent filtration facilitates the problem of averaging the equations for the Reynolds stresses and covariations of fluctuations (flows) in pores over the REV.

    After averaging the basic initial set of turbulent transport equations over the REV and using the averaging formalism developed in the works by Primak et al. (1986), Shcherban et al. (1986), Primak and Travkin (1989), one obtains the following equations for mass conservation MATH for turbulent filtration (with molecular viscosity terms neglected for simplicity) MATH MATH MATH

    MATH
    and for scalar diffusion (with molecular diffusivity terms neglected) MATH MATH
    MATH MATH

    Many details and possible variants of the above equations with tensorial terms are found in Primak et al. (1986), Shcherban et al. (1986), Travkin and Catton (1995, 1998). Using an approximation to K - theory in an elementary channel (pore), the equation for turbulent diffusion of n-th species takes the following more complex form after being averaged MATH MATH MATH MATH

    In the more general case, the momentum flux integrals on the right-hand sides of equations (16) through (17) do not equal zero, since there could be penetration through the phase transition boundary changing the boundary conditions in the microelement to allow for heat -and mass exchange through the interface surface as the values of velocity, concentrations and temperature at $\partial S_{w}$ do not equal zero (see also Crapiste et al., 1986). The first term on the right-hand side of equation (18) is the divergence of the REV averaged product of velocity fluctuations and admixture concentration caused by random morphological properties of the medium being penetrated and is responsible for morphoconvectional dispersion of admixture in this particular porous medium. The third term on the right-hand side of equation (18) can be associated with the notion of morphodiffusive dispersion of a substance or heat in a randomly nonhomogeneous medium. The term with MATH may also reflect, specifically, the impact of microroughness from the previous level of the simulation hierarchy. The importance of accounting for this roughness has been demonstrated by many studies. The remaining step is to account for the microroughness characteristics of the previous level.

    One dimensional mathematical statements will be used in what follows for simplicity. Admission of specific types of medium irregularity or randomness requires that complicated additional expressions be included in the generalized governing equations. Treatment of these additional terms becomes a crucial step once the governing averaged equations are written. An attempt to implement some basic departures from a porous medium with strictly regular morphology descriptions into a method for evaluation of some of the less tractable, additional terms is explained below.

    The 1-D momentum equation with terms representing a detailed description of the medium morphology is depicted as follows MATH MATH MATH where $K_{_{m}}$ is the turbulent eddy viscosity, $u_{_{\ast rk}}^{2}$ is the square friction velocity at the upper boundary of surface roughness layer $h_{r}$ averaged over interface surface $S_{w}$. General statements for energy transport in a porous medium require two-temperature treatments. Travkin et al. (1994), Gratton et al. (1995) showed that the proper form for the turbulent heat transfer equation in the fluid phase using one-equation K-theory closure with primarily 1-D convective heat transfer is MATH MATH MATH while in the neighboring solid phase, the corresponding equation is MATH MATH or MATH MATH The generalized longitudinal 1-D mass transport equation in the fluid phase, including description of potential morpho-fluctuation influences, for a medium morphology with only 1-D fluctuations is written MATH MATH MATH while the corresponding nonlinear equation for the solid phase is MATH MATH or MATH MATH

    It is very important to remind here that the VAT and its techniques seems the only tool, and I understand why, which allowed to solve few canonical problems for porous and heterogeneous media exactly - seen in few places of this website. No other theory, method so far was able to do the solutions on both scales, in spite of the claims.


    We recognize that the theory and the VAT mathematics those have been given in this subsection are rather short for this formidable task, and hoping that the some additional materials shown in the following sections of this website -

  • - "Fundamentals of Hierarchical Scaled (VAT) Description of Transport in Heterogeneous Media"
  • - "Fluid Mechanics"
  • - "Thermal Physics"
  • - "Electrodynamics"

    and others, will fill some gaps in the exposition, while speaking on many more VAT turbulent and nonlinear governing equations those scattered throughout the website describing the specific problems.

    NOMENCLATURE

    $a$- thermal diffusivity [$m^{2}/s$]
    $c_{d}$ - mean drag resistance coefficient in the REV [-]
    $\widetilde{c}_{d}$ - mean skin friction coefficient over the turbulent area of $\partial S_{w}$ [-]
    $c_{dp}$ - mean form resistance coefficient in the REV [-]
    $c_{d,sph}$ - drag resistance coefficient upon single sphere [-]
    $c_{fL}$ - mean skin friction coefficient over the laminar region inside of the REV [-]
    $c_p$ - specific heat [$J/(kg\cdot K)$]
    $C_1$ - constant coefficient in Kolmogorov turbulent exchange coefficient correlation [-]
    $d_{ch}$ - character pore size in the cross section [$m$]
    $d_{i}$ - diameter [m] of i-th pore [$m$]
    $d_{p}$ - particle diameter [$m$]
    $ds$ - interphase differential area in porous medium [$m^{2}$]
    $D_{f}$ - molecular diffusion coefficient $[m^{2}/s$], also - tube or pore diameter [$m$]
    $D_{h}$ - flat channel hydraulic diameter [$m$]
    $D_{s}$ -diffusion coefficient in solid[$m^{2}/s$]
    $\partial S_w$ - internal surface in the REV [$m^2$]
    MATH - averaged over $\Delta \Omega _{f}$ value $f$ - intrinsic averaged variable
    MATH - value f, averaged over $\Delta \Omega _{f}$ in a REV - phase averaged variable
    $<f>_f$ - value $f$, averaged over $\Delta \Omega _f$ in a REV
    MATH - morpho-fluctuation value of $f$ in a $\Omega _{f}$
    $g$ - gravitational constant [$1/m^{2}$]
    $H$ - width of the channel [$m$]
    $h$ - averaged heat transfer coefficient over $\partial S_{w}$ [$W/(m^{2}K)$], half-width of the channel [$m$]
    $h_{r}$ - pore scale microroughness layer thickness [$m$]
    $\partial S_{w}$ - internal surface in the REV [$m^{2}$]
    $k_{f}$ - fluid thermal conductivity [$W/(mK)$]
    $k_{s}$ - solid phase thermal conductivity [$W/(mK)$]
    $K$ - permeability [$m^{2}$]
    $K_{b}$ - turbulent kinetic energy exchange coefficient [$m^{2}/s$]
    $K_{c}$ - turbulent diffusion coefficient [$m^{2}/s$]
    $K_{m}$ - turbulent eddy viscosity [$m^{2}/s$]
    $K_{sT}$ - effective thermal conductivity of solid phase [$W/(mK)$]
    $K_{T}$ - turbulent eddy thermal conductivity [$W/(mK)$]
    $l$ - turbulence mixing length [$m$]
    $L$ - scale [$m$]
    MATH - averaged porosity [-]
    $m_{s}$ - surface porosity [-]
    $n$ - number of pores [-]
    $n_{i}$ - number of pores with diameter of type i [-]
    $Nu_{_{por}}$ - =MATH, interface surface Nusselt number [-]
    $p$ - pressure [$Pa$]; or pitch in regular porous 2D and 3D medium [$m$]; or phase function [-]
    $Pe_{h}$ - =$Re_{_{h}}Pr$, Darcy velocity pore scale Peclet number [-]
    $Pe_{p}$ - =$Re_{_{p}}Pr,\ $particle radius Peclet number [-]
    $Pr$- = $\frac{\nu }{a_{f}}$, Prandtl number [-]
    $Q_{0}$ - outward heat flux [$W/m^{2}$]
    $Re_{ch}$ - Reynolds number of pore hydraulic diameter [-]
    $Re_{_{h}}$- =MATH Darcy velocity Reynolds number of pore hydraulic diameter [-]
    $Re_{p}$- =MATH, particle Reynolds number [-]
    $Re_{_{por}}$- =MATH Reynolds number of general scale pore hydraulic diameter [-]
    $S_{cr}$ - total cross sectional area available to flow [$m^{2}$]
    $S_{w}$ - specific surface of a porous medium MATH [$1/m$]
    $S_{wp}$ MATH
    $S_{\perp }=S_{pr}$ - cross flow projected area of obstacles [$m^{2}$]
    $T$ - temperature $[K]$
    $T_a$ - characteristic temperature for given temperature range $[K] $
    $T_s$ - solid phase temperature $[K]$
    $T_w$ - wall temperature $[K]$
    $T_0$ - reference temperature $[K]$
    $U,$ $u$ - velocity in x-direction $[m/s]$
    $u_{\ast rk}^{2}$ - square friction velocity at the upper boundary of hr averaged over surface $\partial S_{w}$ [$m^{2}/s^{2}$]
    $V$ - velocity $[m/s]$
    $V_{D}$ - =MATHDarcy velocity [$m/s$]
    $W$ - velocity in z-direction $[m/s]$

    Subscripts


    $e$ - effective
    $f$ - fluid phase
    $i$ - component of turbulent vector variable; or species or pore type
    $k$- component of turbulent variable that designates turbulent ''microeffects'' on a pore level
    $L$ - laminar
    $m$ - scale value or medium
    $r$ - roughness
    $s$ - solid phase
    $T$ - turbulent
    $w$ - wall

    Superscripts


    $\thicksim $- value in fluid phase averaged over the REV
    $\smallfrown \ $- value in solid phase averaged over the REV
    $-$- mean turbulent quantity
    $\prime $ -turbulent fluctuation value
    $\ast $ -equilibrium values at the assigned surface or complex conjugate variable

    Greek letters


    MATH- averaged heat transfer coefficient over $\partial S_w$ $[W/(m^2K)]$
    $\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 $[m_{3}]$
    $\varepsilon _{d},$ $\varepsilon _{m}-$ electric permittivity [Fr/m]
    $\mu $ - dynamic viscosity [$kg/(ms)$] or [$Pas$]
    $\mu _{m}$ - magnetic permeability [H/m]
    $\nu $ - kinematic viscosity [$m^{2}/s$]; also $\nu $ - frequency [Hz]
    $\varrho $ - density [$kg/m^{3}$]; also $\rho $ - electric charge density [C/m$^{3}$]
    $\sigma _{e}$ - medium specific electric conductivity [A/V/m]
    $\Phi $ - electric scalar potential [V]
    $\psi $ - particle intensity per unit energy (frequency)
    MATH - ensemble-averaged value of $\psi $
    MATH - interface ensemble-averaged value of $\psi ,$ with phase $j$ being to the left
    $\omega $ - angular frequency [rad/s]
    $\chi $ - magnetic susceptibility [-]
    $\varkappa _{\nu a}$ = $\varkappa _{a}$ - absorption coefficient [1/m]
    $\varkappa _{\nu s}$ = $\varkappa _{s}$ - scattering coefficient [1/m]

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