The Flow Between Parallel Plates with Variable Mass Diffusivity


In Richard Feynman’s words, turbulence is ‘the most important unsolved problem of classical physics’ with  seemingly too many mathematical descriptions – each valid, more or less, under a series of restrictive conditions.


The research of Professor Fabio Gori and Dr Andrea Boghi from the University of Rome Tor Vergata  into turbulence in solutions is revealing new insights about molecular diffusion rates and correlation rules, with impacts ranging from plastic production to solar power – and even our understanding of the movement of energy from the core of the earth itself.


Read their original research at:


Image Source:  Leonid Ikan/ Shutterstock






Hello, and welcome to ResearchPod. Thanks for listening and joining us today.


In this episode, we are looking at the work of Professor Fabio Gori and Dr Andrea Boghi from the University of Rome “Tor Vergata”. Their research into turbulence in solutions is revealing new insights about molecular diffusion rates and correlation rules. The impact of their mathematical model could be as far-reaching as the realms of plastic production and harnessing nuclear and solar power – and even our understanding of the movement of energy from the core of the earth itself.


In Richard Feynman’s words, turbulence is ‘the most important unsolved problem of classical physics’. This is not because a mathematical description of the phenomenon does not exist but because there are ‘too many’ – each valid, more or less, under a series of restrictive conditions. The second law of Dynamics was adapted by Claude-Louis Navier and George Gabriel Stokes for a fluid flow. The Navier-Stokes equations describe the fluid motion very well if it flows at low speeds. However, if the speed increases or the flow occurs near an obstacle, it is impossible to describe the fluid flow exactly. For as many times as we repeat the experiment, we will never obtain the same flow pattern.


In the Navier-Stokes model, the fluid acceleration is described by a non-linear term, which is proportional to the local fluid speed and its gradient. This means that the faster the fluid flows, the bigger this term becomes, especially when the boundary of a pipe or topography changes direction. Under these conditions, small perturbations, such as micro vortices – always present in the fluid flow – are amplified. The nonlinearity of the fluid acceleration has a characteristic property. It changes the size and frequency of the vortices, causing the big, slow vortices to break up into smaller and faster vortices, a process known as ‘Energy Cascade’, which is responsible for the high mixing of cold and hot fluids and different molecules of the turbulent flow.


Over the past century, many approaches have been developed to describe turbulent flow. The first approach, and the one we will discuss here, is the statistical approach, meaning that, instead of describing instantaneous flow, we aim to describe the mean flow. This approach is known as the Reynolds-Average-Navier-Stokes approach, named after Sir Osborne Reynolds, the first to use it, but is also known as the RANS approach. It aims to derive an equation for the mean variables, such as the mean velocity, mean pressure, and so on. The problem with this approach is that, due to the nonlinearity of the fluid acceleration, averaging the Navier-Stokes equations produces an extra term. The reason for this can appear mysterious to some listeners. As an example, one can think of the square of a binomial, (a+b)^2 , which is not only equal to the square of each term, a^2+b^2, but also contains an extra term, which is the double product of each term, 2ab . In the case of the RANS approach, the extra term is the fluid variance. So, to calculate the mean flow, which is unknown, we need to know the flow variance, which is unknown as well.


Factoring in turbulence is very important for many engineering and environmental applications because turbulence enhances heat exchange and mixing. The presence of unknown terms also affects the RANS description of heat and mass transfer in a turbulent flow. This is due to the rate of change of energy or chemical concentration in the turbulent fluid flow, which is proportional to the product of the fluid velocity and, respectively, the energy or chemical concentration gradient. With a procedure similar to that described for the velocity, the RANS description of heat and mass transfer is affected by the presence of an unknown term, proportional to the cross-correlation of the fluid velocity and the energy or chemical concentration. Most fluid dynamic research aims to find a direct relationship between the mean variables, such as velocity and energy and their cross or auto-correlations. This is known as the ‘Closure Problem’, which the Scientific Community hasn’t been able to resolve so far.


As far as the transport and reaction of chemical species in turbulent flow is concerned, things get more complicated when, instead of dealing with the diffusion of simple molecules, such as salt in an aqueous solution, we want to provide a mathematical description of complex macromolecules, such as proteins in blood flow. At this point, the audience can guess where the difficulty is. Yes, the diffusion of macromolecules is a non-linear process. While for small molecules, the diffusivity of a substance is a parameter, and the diffusion is exclusively regulated by the concentration gradient of the chemical species, the diffusivity of macromolecules is itself dependent on the concentration of the macromolecule.


Since 1895, when Sir Osbourne Reynolds first proposed the RANS methodology in ‘On the dynamic theory of incompressible viscous fluids’, this problem was never addressed. That is until 2010, when two researchers from the University of Rome “Tor Vergata”, Professor Fabio Gori and Dr Andrea Boghi, published in the ASME Journal of Fluids Engineering a paper titled: ‘On a New Passive Scalar Equation With Variable Mass Diffusivity: Flow Between Parallel Plates’.


In this paper, Gori and Boghi first applied the RANS description of turbulent flow to the diffusion of macromolecules in solution, discovering the presence of two unknown correlations instead of one. The first is the well-known correlation between the velocity fluctuations and the concentration of the chemical species. The second term introduced in this work is the correlation between the chemical diffusivity of the species and the concentration gradient of the chemical species.


Gori and Boghi derived two transport equations: the first governs the diffusion of the mean concentration of the chemical species, and the second, its variance. This approach is common in turbulence modelling. Transport equations for higher order statistical moments, such as the variance that is a second order moment, are derived and are used to simplify the solution of lower order moments, such as the mean, the first order moment. However, this procedure has the drawback of increasing the number of unknowns in the system.


By deriving the two transport equations, the authors discovered something new. In analogy with the correlation between velocity and concentration, the new additional term behaves as an additional diffusive term. However, unlike the first term, which increases the concentration variance, the new term reduces it, acting as a diffusion regulariser. Nevertheless, more than just this physical consideration is needed to help resolve the closure problem for the new unknown correlation.


To eliminate the unknown terms in the system, Gori and Boghi made the following hypothesis: the mean concentration is always bigger than its mean root square. This condition is not true for the fluid velocity since the mean velocity can be zero. At the same time, its fluctuation can be significant. But it is a reasonable assumption for complex solutions that is the object of this study. The authors linearised the molecular diffusivity around the mean concentration value. This hypothesis has the following consequences: the mean molecular diffusivity is equal in first approximation to the molecular diffusivity calculated at the mean concentration, but the diffusivity fluctuations are directly proportional to the concentration fluctuations. This hypothesis is sufficient to eliminate the additional unknown in the transport equation of the mean concentration. The authors conclude that the additional diffusion term is proportional to the gradient of the concentration variance. The consequence of this result is that if one wants to investigate the diffusion of complex molecules in turbulent flow, one must calculate the concentration variance at each point of the flow.


This can be achieved by using the concentration variance transport equation introduced by Gori and Boghi in this work. Nevertheless, three additional unknown terms remained in this equation. The pair were able to eliminate two of these terms using an order-of-magnitude analysis. At the same time, the last one could be simplified using a boundary approximation close to the container wall. The authors developed a computer code to solve their mathematical model and applied the model to the case of turbulent diffusion in a fluid flowing between two parallel plates. The authors chose this configuration to validate their result with the literature, showing a good agreement in case of constant diffusivity.


Gori and Boghi analysed the solution at different values of the molecular diffusivity at the centre of the channel to understand in which situation the extra terms can prove to be important compared to the classical RANS approach to model the transport of chemical species. The results show that the present model has significant potential for applications involving the heat transfer of fluids with low thermal conductivity or the chemical diffusion of molecules with low molecular diffusivity.


Potential applications of the model in the heat transfer field include molten salt heat exchange in nuclear power plants and solar thermal power stations, engine oils (for decreasing frictional heat), and simulations of the heat exchange from the Earth’s mantle to the outer core.


Further applications of the model could be in the mass transfer field, including the diffusion of polymers, such as polyvinyl-pyrrolidone, ethanol, propanol, polyethylene glycol, alkanolamines, acetone, and benzene in aqueous solutions, the diffusion of Lanthanides and actinides, the diffusion of globular proteins, such as albumin or lactoferrin in aqueous solutions, and perhaps even for describing drug delivery in turbulent blood flow.


That’s all for this episode, thanks for listening, and be sure to click through to the paper linked in the show notes for this episode to read more about the research from Professor Gori and Dr Boghi. And as always, stay subscribed to ResearchPod for more of the latest science. See you again soon.

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