Turbulent diffusion

Last updated February 04, 2024

Turbulent diffusion is the transport of mass, heat, or momentum within a system due to random and chaotic time dependent motions.^[1] It occurs when turbulent fluid systems reach critical conditions in response to shear flow, which results from a combination of steep concentration gradients, density gradients, and high velocities. It occurs much more rapidly than molecular diffusion and is therefore extremely important for problems concerning mixing and transport in systems dealing with combustion, contaminants, dissolved oxygen, and solutions in industry. In these fields, turbulent diffusion acts as an excellent process for quickly reducing the concentrations of a species in a fluid or environment, in cases where this is needed for rapid mixing during processing, or rapid pollutant or contaminant reduction for safety.

However, it has been extremely difficult to develop a concrete and fully functional model that can be applied to the diffusion of a species in all turbulent systems due to the inability to characterize both an instantaneous and predicted fluid velocity simultaneously. In turbulent flow, this is a result of several characteristics such as unpredictability, rapid diffusivity, high levels of fluctuating vorticity, and dissipation of kinetic energy.^[2]

Applications

Atmospheric diffusion and pollutants

Atmospheric dispersion,^[3] or diffusion, studies how pollutants are mixed in the environment. There are many factors included in this modeling process, such as which level of atmosphere(s) the mixing is taking place, the stability of the environment and what type of contaminant and source is being mixed. The Eulerian and Lagrangian (discussed below) models have both been used to simulate atmospheric diffusion, and are important for a proper understanding of how pollutants react and mix in different environments. Both of these models take into account both vertical and horizontal wind, but additionally integrate Fickian diffusion theory to account for turbulence. While these methods have to use ideal conditions and make numerous assumptions, at this point in time, it is difficult to better calculate the effects of turbulent diffusion on pollutants. Fickian diffusion theory and further advancements in research on atmospheric diffusion can be applied to model the effects that current emission rates of pollutants from various sources have on the atmosphere.^[4]

Turbulent diffusion flames

Using planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) processes, there has been on-going research on the effects of turbulent diffusion in flames. Main areas of study include combustion systems in gas burners used for power generation and chemical reactions in jet diffusion flames involving methane (CH₄), hydrogen (H₂) and nitrogen (N₂).^[5] Additionally, double-pulse Rayleigh temperature imaging has been used to correlate extinction and ignition sites with changes in temperature and the mixing of chemicals in flames.^[6]

Modeling

Eulerian approach

The Eulerian approach to turbulent diffusion focuses on an infinitesimal volume at a specific space and time in a fixed frame of reference, at which physical properties such as mass, momentum, and temperature are measured.^[7] The model is useful because Eulerian statistics are consistently measurable and offer great application to chemical reactions. Similarly to molecular models, it must satisfy the same principles as the continuity equation below (where the advection of an element or species is balanced by its diffusion, generation by reaction, and addition from other sources or points) and the Navier–Stokes equations:

${\partial c_{i} \over \partial t}+{\partial \over \partial x_{j}}(u_{j}c_{i})=D_{i}{\partial ^{2}c_{i} \over \partial x_{j}\partial x_{j}}+R_{i}(c_{1},...,c_{N},T)+S_{i}(x,t)$

${i=1,2,...,N}$

where $c_{i}$ = species concentration of interest, $u_{j}$ = velocity t= time, $x_{j}$ = direction, $D_{i}$ = molecular diffusion constant, $R_{i}$ = rate of $c_{i}$ generated reaction, $S_{i}$ = rate of $c_{i}$ generated by source. ^[8] Note that $c_{i}$ is concentration per unit volume, and is not mixing ratio ( $kg/kg$ ) in a background fluid.

If we consider an inert species (no reaction) with no sources and assume molecular diffusion to be negligible, only the advection terms on the left hand side of the equation survive. The solution to this model seems trivial at first, however we have ignored the random component of the velocity plus the average velocity in u_j= ū + u_j’ that is typically associated with turbulent behavior. In turn, the concentration solution for the Eulerian model must also have a random component c_j= c+ c_j’. This results in a closure problem of infinite variables and equations and makes it impossible to solve for a definite c_i on the assumptions stated.^[9]

Fortunately there exists a closure approximation in introducing the concept of eddy diffusivity and its statistical approximations for the random concentration and velocity components from turbulent mixing:

$\langle u_{j}'c'\rangle =-K_{jj}{\partial (c) \over \partial x_{j}}$

where K_jj is the eddy diffusivity.^[8]

Substituting into the first continuity equation and ignoring reactions, sources, and molecular diffusion results in the following differential equation considering only the turbulent diffusion approximation in eddy diffusion:

${\partial c_{i} \over \partial t}+{\overline {u}}_{j}{\partial (c) \over \partial x_{j}}={\partial \over \partial x_{j}}{\bigg (}K_{jj}{\partial (c) \over \partial x_{j}}{\bigg )}$

Unlike the molecular diffusion constant D, the eddy diffusivity is a matrix expression that may vary in space, and thus may not be taken outside the outer derivative.

Lagrangian approach

The Lagrangian model to turbulent diffusion uses a moving frame of reference to follow the trajectories and displacements of the species as they move and follows the statistics of each particle individually.^[7] Initially, the particle sits at a location x’ (x₁, x₂, x₃) at time t’. The motion of the particle is described by its probability of existing in a specific volume element at time t, that is described by Ψ(x₁, x₂, x₃, t) dx₁ dx₂ dx₃ = Ψ(x,t)dx which follows the probability density function (pdf) such that:

${\boldsymbol {\psi }}(\mathbf {x} ,{\mathit {t}})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\mathit {Q}}(\mathbf {x} ,{\mathit {t}}|\mathbf {x} ',{\mathit {t}}'){\boldsymbol {\psi }}(\mathbf {x} ',{\mathit {t}}')d\mathbf {x} '$ Where function Q is the probably density for particle transition.

The concentration of particles at a location x and time t can then be calculated by summing the probabilities of the number of particles observed as follows:

$\langle c(\mathbf {x} ,{\mathit {t}})\rangle =\sum _{i=1}^{m}{\boldsymbol {\psi }}_{i}(\mathbf {x} ,{\mathit {t}})$

Which is then evaluated by returning to the pdf integral^[8]

$\langle c(\mathbf {x} ,{\mathit {t}})\rangle ==\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }{\mathit {Q}}(\mathbf {x} ,{\mathit {t}}|\mathbf {x} _{0},{\mathit {t}}_{0})\langle c(\mathbf {x} _{0},{\mathit {t}}_{0})\rangle d\mathbf {x} _{0}+\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{t_{0}}^{t}{\mathit {Q}}(\mathbf {x} ,{\mathit {t}}|\mathbf {x} ',{\mathit {t}}'){\mathit {S}}(\mathbf {x} ',{\mathit {t}}')d{\mathit {t}}d\mathbf {x} '$

Thus, this approach is used to evaluate the position and velocity of particles relative to their neighbors and environment, and approximates the random concentrations and velocities associated with turbulent diffusion in the statistics of their motion.

Solutions

The resulting solution for solving the final equations listed above for both the Eulerian and Lagrangian models for analyzing the statistics of species in turbulent flow, both result in very similar expressions for calculating the average concentration at a location from a continuous source. Both solutions develop a Gaussian Plume and are virtually identical under the assumption that the variances in the x,y,z directions are related to the eddy diffusivity:

$\langle c(x,y,z)\rangle ={\frac {q}{2\pi \sigma _{y}\sigma _{z}}}\exp {\bigg [}-{\bigg (}{\frac {y^{2}}{\sigma _{y}^{2}}}+{\frac {z^{2}}{\sigma _{z}^{2}}}{\bigg )}{\bigg ]}$

where $\sigma _{y}^{2}={\frac {2K_{yy}x}{\overline {u}}}\sigma _{z}^{2}={\frac {2K_{zz}x}{\overline {u}}}$

q= species emission rate, u = wind speed, σ_i² = variance in i direction. ^[8]

Under various external conditions such as directional flow speed (wind) and environmental conditions, the variances and diffusivities of turbulent diffusion are measured and used to calculate a good estimate of concentrations at a specific point from a source. This model is very useful in atmospheric sciences, especially when dealing with concentrations of contaminants in air pollution that emanate from sources such as combustion stacks, rivers, or strings of automobiles on a road.^[2]

Future research

Because applying mathematical equations to turbulent flow and diffusion is so difficult, research in this area has been lacking until recently. In the past, laboratory efforts have used data from steady flow in streams or from fluids, that have a high Reynolds number, flowing through pipes, but it is difficult to obtain accurate data from these methods. This is because these methods involve ideal flow, which cannot simulate the conditions of turbulent flow necessary for developing turbulent diffusion models. With the advancement in computer-aided modeling and programming, scientists have been able to simulate turbulent flow in order to better understand turbulent diffusion in the atmosphere and in fluids.

Currently in use on research efforts are two main non-intrusive applications. The first is planar laser-induced fluorescence (PLIF), which is used to detect instantaneous concentrations at up to one million points per second. This technology can be paired with particle image velocimetry (PIV), which detects instantaneous velocity data. In addition to finding concentration and velocity data, these techniques can be used to deduce spatial correlations and changes in the environment. As technology and computer abilities are rapidly expanding, these methods will also improve greatly, and will more than likely be at the forefront of future research on modeling turbulent diffusion.^[10]

Aside from these efforts, there also have been advances in fieldwork used before computers were available. Real-time monitoring of turbulence, velocity and currents for fluid mixing is now possible. This research has proved important for studying the mixing cycles of contaminants in turbulent flows, especially for drinking water supplies.

As researching techniques and availability increase, many new areas are showing interest in utilizing these methods. Studying how robotics or computers can detect odor and contaminants in a turbulent flow is one area that will likely produce a lot of interest in research. These studies could help the advancement of recent research on placing sensors in aircraft cabins to effectively detect biological weapons and/or viruses.

Related Research Articles

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all.

<span class="mw-page-title-main">Navier–Stokes equations</span> Equations describing the motion of viscous fluid substances

The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers.

<span class="mw-page-title-main">Heat equation</span> Partial differential equation describing the evolution of temperature in a region

In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.

<span class="mw-page-title-main">Fokker–Planck equation</span> Partial differential equation

In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker-Planck equation has multiple applications in information theory, graph theory, data science, finance, economics etc.

In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

Shear stress is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

In continuum mechanics, the material derivative describes the time rate of change of some physical quantity of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.

Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. $. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature, this type of flow occurs in the swimming of microorganisms and sperm. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.$

The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, a system of partial differential equations that describe the motion of a fluid in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.

In fluid dynamics, eddy diffusion, eddy dispersion, or turbulent diffusion is a process by which fluid substances mix together due to eddy motion. These eddies can vary widely in size, from subtropical ocean gyres down to the small Kolmogorov microscales, and occur as a result of turbulence. The theory of eddy diffusion was first developed by Sir Geoffrey Ingram Taylor.

The derivation of the Navier–Stokes equations as well as its application and formulation for different families of fluids, is an important exercise in fluid dynamics with applications in mechanical engineering, physics, chemistry, heat transfer, and electrical engineering. A proof explaining the properties and bounds of the equations, such as Navier–Stokes existence and smoothness, is one of the important unsolved problems in mathematics.

Taylor dispersion or Taylor diffusion is an effect in fluid mechanics in which a shear flow can increase the effective diffusivity of a species. Essentially, the shear acts to smear out the concentration distribution in the direction of the flow, enhancing the rate at which it spreads in that direction. The effect is named after the British fluid dynamicist G. I. Taylor, who described the shear-induced dispersion for large Peclet numbers. The analysis was later generalized by Rutherford Aris for arbitrary values of the Peclet number. The dispersion process is sometimes also referred to as the Taylor-Aris dispersion.

Diffusiophoresis is the spontaneous motion of colloidal particles or molecules in a fluid, induced by a concentration gradient of a different substance. In other words, it is motion of one species, A, in response to a concentration gradient in another species, B. Typically, A is colloidal particles which are in aqueous solution in which B is a dissolved salt such as sodium chloride, and so the particles of A are much larger than the ions of B. But both A and B could be polymer molecules, and B could be a small molecule. For example, concentration gradients in ethanol solutions in water move 1 μm diameter colloidal particles with diffusiophoretic velocities $of order 0.1 to 1 μm/s, the movement is towards regions of the solution with lower ethanol concentration. Both species A and B will typically be diffusing but diffusiophoresis is distinct from simple diffusion: in simple diffusion a species A moves down a gradient in its own concentration.$

For a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of water waves, experiences a net Stokes drift velocity in the direction of wave propagation.

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<span class="mw-page-title-main">Diffusion</span> Transport of dissolved species from the highest to the lowest concentration region

Diffusion is the net movement of anything generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics, probability theory, information theory, neural networks, finance, and marketing.

The multiphase particle-in-cell method (MP-PIC) is a numerical method for modeling particle-fluid and particle-particle interactions in a computational fluid dynamics (CFD) calculation. The MP-PIC method achieves greater stability than its particle-in-cell predecessor by simultaneously treating the solid particles as computational particles and as a continuum. In the MP-PIC approach, the particle properties are mapped from the Lagrangian coordinates to an Eulerian grid through the use of interpolation functions. After evaluation of the continuum derivative terms, the particle properties are mapped back to the individual particles. This method has proven to be stable in dense particle flows, computationally efficient, and physically accurate. This has allowed the MP-PIC method to be used as particle-flow solver for the simulation of industrial-scale chemical processes involving particle-fluid flows.

In the science of fluid flow, Stokes' paradox is the phenomenon that there can be no creeping flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial steady-state solution for the Stokes equations around an infinitely long cylinder. This is opposed to the 3-dimensional case, where Stokes' method provides a solution to the problem of flow around a sphere.

Lagrangian ocean analysis is a way of analysing ocean dynamics by computing the trajectories of virtual fluid particles, following the Lagrangian perspective of fluid flow, from a specified velocity field. Often, the Eulerian velocity field used as an input for Lagrangian ocean analysis has been computed using an ocean general circulation model (OGCM). Lagrangian techniques can be employed on a range of scales, from modelling the dispersal of biological matter within the Great Barrier Reef to global scales. Lagrangian ocean analysis has numerous applications, from modelling the diffusion of tracers, through the dispersal of aircraft debris and plastics, to determining the biological connectivity of ocean regions.

References

↑ Hideto Yoshida; Masuda, Hiroaki; Higashitani, Kō (2006). Powder Technology Handbook (3rd ed.). Boca Raton: CRC. ISBN 978-1-57444-782-8. OCLC 64679080.
1 2 Roberts, P.J.W., & Webster, D.R. (2002). "Turbulent Diffusion". In Shen, Hayley H. (ed.). Environmental fluid mechanics: theories and applications. New York: American Society of Civil Engineers. ISBN 978-0-7844-0629-8. OCLC 50090138.{{cite book}}: CS1 maint: multiple names: authors list (link)
↑ Beychok, M.R. (2005). Fundamentals Of Stack Gas Dispersion (4th ed.). ISBN 978-0-9644588-0-2.
↑ Walcek, C. J. (2002). "Effects of wind shear on pollution dispersion". Atmospheric Environment. 36 (3): 511–7. Bibcode:2002AtmEn..36..511W. doi:10.1016/S1352-2310(01)00383-1.
↑ Su, L. K., Sun, O. S., & Mungal, M. G. (2006). "Experimental investigation of stabilization mechanisms in turbulent, lifted jet diffusion flames". Combustion and Flame. 144 (3): 494–512. doi:10.1016/j.combustflame.2005.08.010.{{cite journal}}: CS1 maint: multiple names: authors list (link)
↑ Hult, J., Meier, U., Meier, W., Harvey, A., & Kaminski, C. F. (2005). "Experimental analysis of local flame extinction in a turbulent jet diffusion flame by high repetition 2-D laser techniques and multi-scalar measurements". Proceedings of the Combustion Institute. 30 (1): 701–9. doi:10.1016/j.proci.2004.08.069.{{cite journal}}: CS1 maint: multiple names: authors list (link)
1 2 Csanady, G.T. (1973). Turbulent diffusion in the environment. Geophysics and astrophysics monographs. Vol. 3. Boston, MA: D.Reidel Publishing Company. ISBN 978-90-277-0260-9. OCLC 629866.
1 2 3 4 Pandis, Spyros N.; Seinfeld, John H. (2006). Atmospheric chemistry and physics: from air pollution to climate change. J. Wiley. ISBN 978-0-471-72017-1. OCLC 62493628.
↑ AMS Glossary, Initials. (2010, March 1). Closure problem
↑ Arima, T., Matsuura, Y., & Oharu, S. (2007). "Computation of air flows and motion of environmental pollutants over complex geographical topographies". Journal of Computational and Applied Mathematics. 204 (1): 187–196. doi:10.1016/j.cam.2006.04.036.{{cite journal}}: CS1 maint: multiple names: authors list (link)

Bilger, R.W. (1989). "Turbulent diffusion flames". Annu. Rev. Fluid Mech. 21: 101–135. Bibcode:1989AnRFM..21..101B. doi:10.1146/annurev.fl.21.010189.000533.

External links

https://share.sandia.gov/crf/article.php?id=144 -Turbulent Diffusion Flame Research
http://www.shodor.org/master/environmental/air/plume/index.html -Gaussian Plume Model Calculator
http://courses.washington.edu/cewa567/Plumes.PDF - Turbulent Diffusion and Gaussian Plume Model from University of Washington

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[1] Hideto Yoshida; Masuda, Hiroaki; Higashitani, Kō (2006). Powder Technology Handbook (3rd ed.). Boca Raton: CRC. ISBN 978-1-57444-782-8. OCLC 64679080.

[Roberts02-2] 1 2 Roberts, P.J.W., & Webster, D.R. (2002). "Turbulent Diffusion". In Shen, Hayley H. (ed.). Environmental fluid mechanics: theories and applications. New York: American Society of Civil Engineers. ISBN 978-0-7844-0629-8. OCLC 50090138.{{cite book}}: CS1 maint: multiple names: authors list (link)

[3] Beychok, M.R. (2005). Fundamentals Of Stack Gas Dispersion (4th ed.). ISBN 978-0-9644588-0-2.

[4] Walcek, C. J. (2002). "Effects of wind shear on pollution dispersion". Atmospheric Environment. 36 (3): 511–7. Bibcode:2002AtmEn..36..511W. doi:10.1016/S1352-2310(01)00383-1.

[5] Su, L. K., Sun, O. S., & Mungal, M. G. (2006). "Experimental investigation of stabilization mechanisms in turbulent, lifted jet diffusion flames". Combustion and Flame. 144 (3): 494–512. doi:10.1016/j.combustflame.2005.08.010.{{cite journal}}: CS1 maint: multiple names: authors list (link)

[6] Hult, J., Meier, U., Meier, W., Harvey, A., & Kaminski, C. F. (2005). "Experimental analysis of local flame extinction in a turbulent jet diffusion flame by high repetition 2-D laser techniques and multi-scalar measurements". Proceedings of the Combustion Institute. 30 (1): 701–9. doi:10.1016/j.proci.2004.08.069.{{cite journal}}: CS1 maint: multiple names: authors list (link)

[Csanady73-7] 1 2 Csanady, G.T. (1973). Turbulent diffusion in the environment. Geophysics and astrophysics monographs. Vol. 3. Boston, MA: D.Reidel Publishing Company. ISBN 978-90-277-0260-9. OCLC 629866.

[Pandis06-8] 1 2 3 4 Pandis, Spyros N.; Seinfeld, John H. (2006). Atmospheric chemistry and physics: from air pollution to climate change. J. Wiley. ISBN 978-0-471-72017-1. OCLC 62493628.

[9] AMS Glossary, Initials. (2010, March 1). Closure problem

[10] Arima, T., Matsuura, Y., & Oharu, S. (2007). "Computation of air flows and motion of environmental pollutants over complex geographical topographies". Journal of Computational and Applied Mathematics. 204 (1): 187–196. doi:10.1016/j.cam.2006.04.036.{{cite journal}}: CS1 maint: multiple names: authors list (link)

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