Bottom-Up or Top-Down Problems: Lower Scale Turbulent Transport Closure Equations and Methods for the Upper Scale Closure and Solution

It is of obvious importance the issue of closure of Averaged VAT turbulent equations, primarily stemming from the one scale conventional homogeneous fluid mechanics.

The next question is - What to do with these equations when the Upper scale VAT governing equations are modeling the averaged over the REV turbulent transport process?

We do not intend here to speak, claim or study closely the lower scale turbulence phenomena. And we also not claiming to improve or to better explain the lower scale turbulent phenomena. This task belongs to the other physical discipline - and we are here not to engage ourselves into this unsolvable so far physics problem.

We need to Tell Loudly about this - as soon as many times people trying to discuss the turbulence theory as a substitute field while addressing issues, difficulties and mathematics of the Turbulent VAT.

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. However, using turbulent models to predict turbulent flow and related heat- or mass transport in porous media is still in its infancy.

Among many possible models for turbulent momentum in porous media the example of turbulence models are considered in the initial low level of hierarchical modeling. The fluctuations produced by the porous medium heterogeneities and wall roughness contribute to the overall transport processes and to the governing scalar fluctuation equations, the stress tensor equation, the kinetic energy and dissipation rate equations, must be treated along with other phenomenological dependencies.

The proper distinction should be exercised for turbulence models in the fluid volume of the REV's element, and for the REV as a whole modeling point. In one of the models 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.

In view of theoretical importance to turbulent transport in porous media, the incompressible turbulent flow within porous media can be VAT modeled using the averaging of the transport equations for the turbulent kinetic energy () and its dissipation rate ().

The Upper scale turbulent transport equations for porous media were developed by V.S.Travkin based on the generalized Volume Averaging Theory (VAT) for porous media along with the statistical and numerical methodology needed for their solution. First publications were in Russian as well as in English: Primak et al. (1986), Shcherban et al. (1986) and others; then by Travkin and Catton (1992, 1993, 1995, etc.). Simplified models based on approaches taken in meteorology and agro-science were used in these works for simulation in 80-th and beginning of 90-th. Later on the problem of the exact VAT formulation and simulation of the heterogeneous media tasks as the truly two-scale predicament opened the new windows for studies and treatment. So, this field is still in the creation stage. Any progress to be made in this field will be a significant one.

The proper averaging of the initial turbulent fluctuations equations I had carried out and published in 90-th (Travkin et al., 1999). Closure of VAT Upper scale turbulent equations with simplified turbulent kinetic energy and dissipation rate equations was obtained for few morphologies. The averaging of equations was conceded on strict HSP-VAT principles, mathematics while based on the heterogeneous WSAM type theorems for application to scaled heterogeneous and hierarchical media problems.

As one of the outputs of such an approach toward clear averaging of the very complicated non-linear differential equations one would find out that the issues of mathematical simulation (implementation) and of the physical validity outburst itself to the first row. The averaged equations became so complicated in form, that the solution of the complicated plasma physics sets of equations will be considered as a minor problem in comparison. It worth to remind that the convenient approach in plasma physics communities is to establish the teams, labs, institutions to simulate those fusion statement sets of equations.
One of related questions is regarding the real nature of the formulated averaged (upper scale model) equations. The problem itself can be formulated or sought as via either the Bottom-Up or the Top-Down statement, depending on the physics and initial conditions found for the task. The consequences might be significant as for the simulation charge.

One of the first public presentations of these complicated form equations was the first work-study on the averaging of the equations presented at the ASME/JSME Fluids Engineering Conference (1999)