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2015 | OriginalPaper | Buchkapitel

18. Transport Phenomena in Turbulent Flow

verfasst von : Roberto Mauri

Erschienen in: Transport Phenomena in Multiphase Flows

Verlag: Springer International Publishing

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Abstract

In this chapter we intend to provide a brief introduction to the study of transport phenomena in the presence of turbulent flows. This is a very large and quite specialized topic, and here we will confine our treatment to some basic features, above all in the context of pipe flow, referring the interested readers to more specialized texts (See, for example, Turbulent Flows by S.B. Pope, Cambridge University Press.). Most of the discussion on convective transport phenomena in the preceding chapters assumes that the flow is laminar, while the case of turbulent flows has been treated so far using empirical correlations, that are often subjected to stringent geometric constraints. The reason for concentrating on laminar flows is quite obvious: transport phenomena involving laminar flows can be treated using appropriate simplifying hypothesis, such as the fact that the process is stationary or uni-directional, so that the problem can be treated systematically, often even finding an analytical solution. No such simplifying features can be assumed to apply to turbulent flows, which are always irregular, transient and three-dimensional, and therefore very complex to treat. These difficulties notwithstanding, turbulent flows pop up everywhere in engineering and the natural world, regardless of whether we are optimizing a heat exchanger or watching a cloud moving in the sky. The importance of this subject has stimulated the interest of many investigators, leading to an impressive amount of experimental and theoretical studies. Unfortunately, while the experiments have produced many useful empirical relationships (albeit often with narrow applicability conditions), a complete theory of turbulence is still lacking, although its fundamental characteristics are well understood. After a brief introduction (Sect. 18.1), in Sect. 18.2 we describe the Kolmogorov scaling arguments, thus determining the characteristic time- and length-scales of turbulence. Then, in Sect. 18.3, we present the Reynolds decomposition and the associated governing equations, involving the turbulent fluxes of mass, momentum and energy. Next, Sect. 18.4, a closure model for the diffusive turbulent fluxes is described, based on the Reynolds analogy and Prandtl’s mixing length model. This approach is applied in Sect. 18.5 to determine the logarithmic velocity profile near a wall, and then to describe turbulent pipe flow. Finally, in Sect. 18.6, more complex models are briefly sketched.

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Fußnoten
1
This, so called, principle of local isotropy is the basic assumption of the most commonly used turbulence model.
 
2
Very near the walls the velocity field is laminar and therefore there is no turbulence.
 
3
The entrance length is the distance from the tube entrance that is required for the velocity or temperature profile to reach their respective fully developed profile.
 
4
This result can be obtained by substituting the units of ν (m2/s) and ε (m2/s3), and solving the algebraic equation in the unknowns α and β that result from equating the exponents of m (meter) and of s (second). Note that we would obtain the same result had we assumed that δ depends separately on the dynamic viscosity, μ, and on the density, ρ.
 
5
This condition is always satisfied when turbulence is statistically stationary, which is the case that is considered here. However, in the presence of shock waves or other discontinuities, this condition is not satisfied any more, and therefore the definition (18.1.2) ceases to be valid.
 
6
We assume that the characteristic time of the variation of \(\langle f\rangle\) is much longer than τ δ .
 
7
Here V is the mean volume flow rate per a unit cross sectional area, as distinct from the mean velocity at the centerline.
 
8
For a turbulent flow past a flat plate it is found that A = 25.
 
9
Naturally, once we go back to the dimensional variables v and y, the resulting velocity profile strongly depends on Re.
 
10
In fact, the maximum velocity is only 1.22 times larger than the mean velocity, instead of twice larger, as in laminar flow.
 
11
See W.M. Deen, Analysis of Transport Phenomena, Oxford University Press, pp. 541–544.
 
12
An accurate treatment of the K-ε model can be found in C.G. Speziale, Annu. Rev. Fluid Mech. 23, 107–157 (1991).
 
Metadaten
Titel
Transport Phenomena in Turbulent Flow
verfasst von
Roberto Mauri
Copyright-Jahr
2015
DOI
https://doi.org/10.1007/978-3-319-15793-1_18

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