Microscopic Aspects of Turbulent Transport –Conduction and Convection Unification
S. Usman§ , S. Abdallah*
And N. Katragadda*
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§Department of Mechanical Industrial and Nuclear Engineering
University of Cincinnati, Ohio
* Department of Aerospace Engineering and Engineering Mechanics
University of Cincinnati, Ohio
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Abstract: It is argued that transport of a scalar under the influence of turbulence is also influenced by the molecular properties of the scalar. This argument is supported by the observation of fluctuating velocity suppression during heavy gas cloud passage. Based on this assumption it is possible (purely on mathematical grounds) to represent a problem of simultaneous convection-diffusion by an expression identical to conduction equation using a variable conduction coefficient. The mixed convection-diffusion transport equation is linearized and rewritten in diffusion form. The new form of the transport equation combines the convection and diffusion terms in a manner similar to the heat conduction equation with a coefficient analogous to thermal conductivity. However, the coefficient developed in the mathematical manipulation is a local variable which is a function of local diffusion and local linearized velocities. It is shown that for diffusion coefficient®0 the effect of diffusion vanishes, producing a convection dominant transport mode. On the other extreme, for large (infinite diffusion coefficient) the variable coefficient reduces to “thermal conductivity”, producing pure diffusion equation.
Key-Words: - Conduction, Convection, Numerical stability, Diffusion, Turbulence
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Introduction
Analysis and modeling of transport of passive scalar in a fluid medium has been an area of research interest for the last 175 years or more. In 1827, Robert Brown – a botanist observed perpetual random motion of particles immersed in a fluid. This observation has lead to the development of the well known and generally accepted molecular kinetic theory of diffusion. Displacement or dispersion rate of a scalar under molecular transport is accepted to be caused by interaction of the diffusing particle with the surrounding fluid molecules and as a result molecular diffusion is a function of both the fluid properties and the particle characteristics (e.g. Hirschfelder et al, 1954). Significant advances have been made to represent mutual molecular diffusion coefficient of two gases which consequently becomes a function of the molecular/atomic properties of both the gas molecules. (Chapman and Cowling, 1970).
For fluid in motion, it is assumed that the molecular movement is super-imposed on the bulk transport of the fluid and if a secondary fluid (for example a dye) is introduced into a laminar flow, in addition to being transported along with the bulk flow, random molecular movements also diffuse the dye both along the flow and also in the transverse directions. Generally, it is believed that as long as the flow is strictly laminar, the transverse direction diffusion is solely due to molecular diffusion (Treylab, 1980).
However, when the bulk flow becomes turbulent a significant enhancement in diffusion rate is observed (Hinze, 1959). This turbulent diffusion is assumed to be super-imposed on the molecular diffusion. Turbulent diffusion becomes the dominant contributor to the dispersion process and in many cases molecular diffusion is considered to be negligible (Taylor, 1921). Random velocity fluctuations, which are characteristic of turbulence, are believed to be responsible for the enhanced turbulent diffusion. Scales of these fluctuating velocities are much larger than the molecular motions. Generally, in the case of turbulent diffusion modeling most of the emphasis is placed on the flow characteristics (that is the behavior of fluctuating velocities) and the turbulent diffusion rates are estimated without much consideration given to the microscopic characteristics of the molecules or particles getting diffused in the flow.
Let us pose a basic question, what is the mechanism of energy and momentum transfer from the bulk fluid to the diffusing particle? For the case of molecular diffusion, it is well established that the diffusion is a consequence of molecular collision and scattering which is caused by the random motion of the molecules (Chapman and Cowling, 1970). The question at hand is, what is the fundamental mechanism of energy and momentum exchange between the particles in turbulent motion and how the bulk fluid molecules in turbulent motion transfer their momentum and energy to the molecule and/or particle getting diffused? This is a very fundamental question with far reaching consequences.
It is argued that in the absence of any other mechanism, the exchange of momentum and energy between the bulk fluid molecules (even in turbulent motion) and the diffusing particle is via molecular collision and scattering. There are two very significant aspects of this argument;
1) If both the molecular and turbulent diffusions are the result of the same fundamental process, that is molecular collisions and scattering, are we justified in treating them as independent and additive processes as the commonly used approach (see any standard text on turbulence and viscous flow, e.g., White 1991)?
2) Is the use of an “eddy viscosity” and “turbulent diffusivity” terms which depend only on the molecular viscosity of the bulk fluid and the flow characteristics justified, without due consideration given to the microscopic nature of particles or molecules getting diffused?
Saffman (1960) introduced the concept of interaction between molecular diffusion and turbulence and concluded that the interaction can potentially reduce dispersion. There is lot to be gained from this classical work. Saffman’s analysis, and almost all the other existing literature, is based on the assumption that turbulent and molecular diffusion are "independent and additive" processes. His analysis did not offer any insight into the dependence of turbulent transport properties on the molecular characteristics.
Heavy gas dispersion
Recently, a great deal of interest has been placed on the atmospheric dispersion of heavy gases (Hartwig, 1984). Primary motivation for these studies is for environmental impact analysis of pollutant spills etc. Several experimental and theoretical studies have been conducted pertaining to various aspects of heavy gas dispersion in atmosphere (Davies, 1999). In general, it has been observed that heavy gases do not follow the expected rate of diffusion in atmospheric air flow. Heavy gas diffusion is much slower than expected and a density-based stratification is observed (Hunt et. al 1983 and Ooms & Tennekes, 1984). These aspects of heavy gas diffusion have made modeling of heavy gas dispersion both a challenging and interesting problem.
One aspect of heavy gas that has been reported in the literature is the suppression of fluctuating velocity during the passage of the heavy gas cloud (Koopman et. al 1989). Heavy gas clouds can be classified into two groups; ones that are made up of a gas with heavy molecules and others where the gas molecular weight is not significantly larger than the bulk fluid but is at a temperature/pressure that the density of the spill gas is much larger than the bulk of the fluid in motion. Or in other words, the spill is not in thermodynamic equilibrium with the bulk fluid. For both of these cases the fluctuating velocity damping can be explained using the argument that the mechanism of energy and momentum transfer between molecules in a turbulent flow field and the spill gas is via molecular collision and scattering.
Consider the case of atmospheric dispersion of a gas with molecular mass significantly larger than the average mass of air (say N2) molecule. For the simplifying assumption of elastic scattering where both energy and momentum are conserved, as the mass of the target particle increases the average particle velocity after elastic scattering interaction reduces. In fact for the case of isentropic scattering (that is, the angular probability of scattering is uniformly distributed) the average velocity becomes proportional to the ratio m/M where m is the mass of the incident particle and M is the mass of the heavy target. Therefore, when a puff of heavy gas is introduced in the bulk flow, there is a suppression of fluctuating velocity during the passage of the heavy gas cloud. The extent of suppression depends on the ratio of the molecular masses and the initial velocity of the heavy gas.
Likewise, in the other case when the spill gas molecular mass is not significantly different than the bulk fluid but the spill gas is at a much lower temperature, most of the K.E. transferred to the spill gas will be used to achieve the thermodynamic equilibrium and as a result the average fluid velocity will reduce. As Mohan et. al (1995) discussed, a transition to a true passive transport will take place after a duration of time which depends on the atmospheric flow condition and the density of the spill gas.
The phenomenon of fluctuating velocity suppression is observed and reported by Koopman et. al (1989).
In the present analysis unification of conduction and convection phenomena is brought about by writing the well known vorticity transport equation in a simple diffusion equation form with a spatially and temporally varying diffusion coefficient.
Conduction convection unification
The steady state transport equation for two dimensions (X,Y) can be written as;
(1)
where is the transport scalar, (,) are the velocities in (X,Y) directions respectively, D is the diffusion coefficient. This non-linear partial differential equation is linearized and rewritten in a diffusion form, the linearized form is;
(2)
where and are the velocity components at the location (x,y) in the sub-domain. Both and can be considered local constants. Equation (2) is written in the following integrating factor form in (X and Y) directions;
(3)
Multiplying and dividing the first term in equation (3) by and the second term by,
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(4)
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Or
(5)
Where
(6)
Equation (5) is similar to the heat conduction equation, where the dependent variable,and the factor, represent the temperature and the thermal conductivity respectively. It should be noted that here, in equation (6) is variable with respect to (X,Y) the space variables. It may also be noted that can be estimated for every field point in a flow field.
To obtain a numerical scheme Equation (5) is integrated over the control volume defined by 5 point domain shown in Fig. (1).
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(7)
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If and are taken to be constants on the control volume surfaces and equal to the intermediate values between the grid points, Eq. (7) results into the 5-point scheme
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(8)
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Where (8a)
(8b)
(8c)
(8d)
The rest of the coefficients in Eq. (8) can be obtained by permutation. and are the space increments in X and Y directions respectively. Equation (8) was previously derived, consequently no further testing for the 5- point scheme is considered in the present work.
Numerical Results
Equation (8), the finite difference form of Eq. (1) is solved at each grid point of the square domain using the Successive Over-Relaxation method (SOR). The relaxation factor is taken as unity. The numerical solutions are computed using VAX/VMS. The following two cases are considered
First case:
Second case: and .
Boundary conditions:
Exact Solution is given by:
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The numerical results for both the cases are obtained using equal space increments in X and Y directions. The results of the first case are computed in about 3 CPU seconds, Fig. (2). This case is considered here in order to check the validity of the present method at low cell Reynolds numbers. The results for the second case are given in Fig. (3) in comparison with the exact solution. These results are obtained in lesser iterations than the first case.
Discussion and conclusions
Based on the observation made by Koopman et. al (1989) and the argument presented here it seems logical to assume that the fundamental mechanism of energy and momentum transfer between the bulk fluid molecules and the diffusing gas/particles is molecular collision and scattering. This assumption raises several basic questions very pertinent to turbulent transport phenomenon.
It is shown here that a convection-diffusion equation can be represented as a pure diffusion equation with a variable conductivity. That variable conductivity coefficient is a function of the convection velocity and the molecular diffusivity. In other words, the dispersion of a scalar quantity is a diffusion process with diffusion coefficient an exponential function of the molecular diffusion coefficient and the convection velocities. It is important to point out that the exponent in the variable conductivity in equation (6) is a function of the local properties versus the global property, the diffusion coefficient. The analysis presented by equations (5) and (6) is applicable in general for flow momentum equations, the energy equations and the vorticity equations. The implication for momentum and vorticity equations is that the exponential conductivity,is the cell Reynolds number (Peclet number). This scheme leads to a positive conductivity coefficient and a diagonal dominance of the discrete formulation for numerical solution of equation (5) for all Peclet numbers versus direct solution for equation (1). Diagonal dominance has the obvious advantage of numerical stability and elimination of oscillations resulting from non-physical phenomenon. In other words, equation (5) filters non-physical oscillations in equation (1) of the cases of large Peclet numbers.
The proposed approach of unifying diffusion and convection is coherent with Yakhot’s (2003) suggestion that the dissipative scale is much smaller than Kolmogorov’s (1941) scale. In other words, there can potentially be continuity between molecular scales and the turbulent scales. It is only that these very high Reynolds numbers capable of producing extremely small length and time scales are impractical to produce using conventional mechanical means.
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References
Chapman, S., & Cowling T.G., (1970), The Mathematical Theory of Nonuniform Gases, 3rd Ed., Cambridge Press, London.
Hartwig S., (1984) Heavy Gas and Risk Assessment III : proceedings of the Third Symposium on Heavy Gas and Risk Assessment, Bonn, Wissenschaftszentrum, November 12-13.