Turbulence

In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Nobel Laureate Richard Feynman described turbulence as "the most important unsolved problem of classical physics."[1] Flow in which the kinetic energy dies out due to the action of fluid molecular viscosity is called laminar flow. While there is no theorem relating Reynolds number (Re) to turbulence, flows at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent, while those at low Reynolds numbers usually remain laminar. In pipe flow, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040[2]; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 3000. In turbulent flow, unsteady vortices appear on many scales and interact with each other. Drag due to boundary layer skin friction increases. The structure and location of boundary layer separation often changes, sometimes resulting in a reduction of overall drag. Although laminar-turbulent transition is not governed by Reynolds number, the same transition occurs if the size of the object is gradually increased, or the viscosity of the fluid is decreased, or if the density of the fluid is increased.

Contents

Features

Turbulence is highly characterized by the following features:

Irregularity: Turbulent flows are always highly irregular. This is why turbulence problems are always treated statistically rather than deterministically. Turbulent flow is always chaotic but not all chaotic flows are turbulent.

Diffusivity: Turbulence is highly associated with rapid mixing. One of the useful effects of turbulence, it tends to accelerate the homogenization of any non-uniform fluid mixture. The process which brings any non-uniform state of a system into a uniform one is called mixing and when the system is in its uniform state, the system becomes a homogeneous system. A mixing process requires sufficient input of energy which is readily available in a turbulent flow. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is regarded as diffusivity.

Turbulent diffusion is usually described by a turbulent diffusion coefficient. This turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with the molecular diffusivities, but it does not have a true physical meaning, being dependent on the flow conditions, and not a property of the fluid itself. In addition, the turbulent diffusivity concept assumes a constitutive relation between a turbulent flux and the gradient of a mean variable similar to the relation between flux and gradient that exists for molecular transport. In the best case, this assumption is only an approximation. Nevertheless, the turbulent diffusivity is the simplest approach for quantitative analysis of turbulent flows, and many models have been postulated to calculate it. For instance, in large bodies of water like oceans this coefficient can be found using Richardson's four-third power law and is governed by the random walk principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula.

Rotationality: Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as vortex stretching. In fluid dynamics, they are essentially vortices subjected to stretching associated with a corresponding increase of the component of vorticity in the stretching direction—due to the conservation of angular momentum. On the other hand, vortex stretching is the core mechanism on which the turbulence energy cascade relies to establish the structure function. In general, the stretching mechanism implies thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of fluid elements. As a result, the radial length scale of the vortices decreases and the larger flow structures break down into smaller structures. The process continues until the small scale structures are small enough to the extent where their kinetic energy is overwhelmed by the fluid's molecular viscosity and dissipated into heat. This is why turbulence is always rotational and three dimensional. For example, atmospheric cyclones are rotational but their substantially two-dimensional shapes do not allow vortex generation and so are not turbulent. On the other hand, oceanic flows are dispersive but essentially non rotational and therefore are not turbulent.

Dissipation: To sustain turbulent flow, a constant source of energy supply is required. Otherwise, turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of eddies of many different length scales. Most of the kinetic energy of the turbulent motion is contained in the large-scale structures. The energy "cascades" from these large-scale structures to smaller scale structures by an inertial and essentially inviscid mechanism. This process continues, creating smaller and smaller structures which produces a hierarchy of eddies. Eventually this process creates structures that are small enough that molecular diffusion becomes important and viscous dissipation of energy finally takes place. The scale at which this happens is the Kolmogorov length scale.

Energy cascade: Turbulent flow can be realized as a superposition of a spectrum of velocity fluctuations and eddies on an over mean flow. The eddies are loosely defined as coherent patterns of velocity, vorticity and pressure. Turbulent flows may be viewed as made of an entire hierarchy of eddies over a wide range of length scales and the hierarchy can be described by the energy spectrum that measures the energy in velocity fluctuations for each wave number. The scales in the energy cascade are generally uncontrollable and highly non-symmetric. Nevertheless, based on these length scales these eddies can be divided into three categories.

Integral length scales: Largest scales in the energy spectrum. These eddies obtain energy from the mean flow and also from each other. Thus these are the energy production eddies which contain the most of the energy. They have the large velocity fluctuation and are low in frequency. Integral scales are highly anisotropic and are defined in terms of the normalized two-point velocity correlations. The maximum length of these scales is constrained by the characteristic length of the apparatus. For example, the largest integral length scale of pipe flow is equal to the pipe diameter. In the case of atmospheric turbulence, this length can reach up to the order of several hundreds kilometers.

Kolmogorov length scales: Smallest scales in the spectrum that form the viscous sub-layer range. In this range, the energy input from nonlinear interactions and the energy drain from viscous dissipation are in exact balance. The small scales are in high frequency which is why turbulence is locally isotropic and homogeneous.

Taylor microscales: The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor micro-scales are not dissipative scale but passes down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor micro-scales as a characteristic length scale and consider the energy cascade contains only the largest and smallest scales; while the later accommodate both the inertial sub-range and the viscous-sub layer. Nevertheless, Taylor micro-scales is often used in describing the term “turbulence” more conveniently as these Taylor micro-scales play a dominant role in energy and momentum transfer in the wavenumber space.

Although it is possible to find some particular solutions of the Navier-Stokes equations governing fluid motion, all such solutions are unstable at large Reynolds numbers. Sensitive dependence on the initial and boundary conditions makes fluid flow irregular both in time and in space so that a statistical description is needed. Russian mathematician Andrey Kolmogorov proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by Richardson) and the concept of self-similarity. As a result, the Kolmogorov microscales were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified.[3] Still, a complete description of turbulence remains one of the unsolved problems in physics.

According to an apocryphal story, Werner Heisenberg was asked what he would ask God, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."[4] A similar witticism has been attributed to Horace Lamb (who had published a noted text book on Hydrodynamics)—his choice being quantum electrodynamics (instead of relativity) and turbulence. Lamb was quoted as saying in a speech to the British Association for the Advancement of Science, "I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic."[5][6]

A more detailed presentation of turbulence with emphasis on high-Reynolds number flow, intended for a general readership of physicists and applied mathematicians, is found in the Scholarpedia articles by R. Benzi and U. Frisch.[7] and by G. Falkovich.[8]

Examples of turbulence

Unsolved problems in physics
Is it possible to make a theoretical model to describe the behavior of a turbulent flow — in particular, its internal structures?

Heat and momentum transfer

When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the heat transfer and the friction coefficient.

Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual velocity v=\left( {{v}_{x}},{{v}_{y}} \right) of every particle that passed through that point at any given time. Then one would find the actual velocity fluctuating about a mean value:

{{v}_{x}}=\underbrace{\overline{{{v}_{x}}}}_{\begin{smallmatrix} \text{mean} \\ \text{value} \end{smallmatrix}}%2B\underbrace{{{{{v}'}}_{x}}}_{\text{fluctuation}}\text{ }\text{, }{{v}_{y}}=\overline{{{v}_{y}}}%2B{{{v}'}_{y}}

and similarly for temperature \left( T=\overline{T}%2B{T}' \right) and pressure \left( P=\overline{P}%2B{P}' \right), where the primed quantities denote fluctuations superposed to the mean. This decomposition of a flow variable into a mean value and a turbulent fluctuation was originally proposed by Osborne Reynolds in 1895, and is considered to be the beginning of the systematic mathematical analysis of turbulent flow, as a sub-field of fluid dynamics. While the mean values are taken as predictable variables determined by dynamics laws, the turbulent fluctuations are regarded as stochastic variables.

The heat flux and momentum transfer (represented by the shear stress \tau ) in the direction normal to the flow for a given time are

\begin{align} & q=\underbrace{{{{{v}'}}_{y}}\rho {{c}_{P}}{T}'}_{\text{experimental value}}=-{{k}_{\text{turb}}}\frac{\partial \overline{T}}{\partial y} \\ & \tau =\underbrace{-\rho \overline{{{{{v}'}}_{y}}{{{{v}'}}_{x}}}}_{\text{experimental value}}={{\mu }_{\text{turb}}}\frac{\partial \overline{{{v}_{x}}}}{\partial y} \\ \end{align}

where {{c}_{P}} is the heat capacity at constant pressure, \rho is the density of the fluid, {{\mu }_{\text{turb}}} is the coefficient of turbulent viscosity and {{k}_{\text{turb}}} is the turbulent thermal conductivity.

[11]

Kolmogorov's theory of 1941

Richardson's notion of turbulence was that a turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy.

In his original theory of 1941, Kolmogorov postulated that for very high Reynolds number, the small scale turbulent motions are statistically isotropic (i.e. no preferential spatial direction could be discerned). In general, the large scales of a flow are not isotropic, since they are determined by the particular geometrical features of the boundaries (the size characterizing the large scales will be denoted as L). Kolmogorov's idea was that in the Richardson's energy cascade this geometrical and directional information is lost, while the scale is reduced, so that the statistics of the small scales has a universal character: they are the same for all turbulent flows when the Reynolds number is sufficiently high.

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the viscosity (\nu) and the rate of energy dissipation (\varepsilon). With only these two parameters, the unique length that can be formed by dimensional analysis is

\eta = \left(\frac{\nu^3}{\varepsilon}\right)^{1/4}.

This is today known as the Kolmogorov length scale (see Kolmogorov microscales).

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length \eta, while the input of energy into the cascade comes from the decay of the large scales, of order L. These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there is a range of scales (each one with its own characteristic length r) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. \eta \ll r \ll L). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called "inertial range").

Hence, a third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range \eta \ll r \ll L are universally and uniquely determined by the scale r and the rate of energy dissipation \varepsilon.

The way in which the kinetic energy is distributed over the multiplicity of scales is a fundamental characterization of a turbulent flow. For homogeneous turbulence (i.e., statistically invariant under translations of the reference frame) this is usually done by means of the energy spectrum function E(k), where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u(x):

\mathbf{u}(\mathbf{x}) = \iiint_{\mathbb{R}^3} \widehat{\mathbf{u}}(\mathbf{k})e^{i \mathbf{k \cdot x}} \mathrm{d}^3\mathbf{k},

where û(k) is the Fourier transform of the velocity field. Thus, E(k)dk represents the contribution to the kinetic energy from all the Fourier modes with k < |k| < k + dk, and therefore,

\frac{1}{2}\langle u_i u_i \rangle = \int_{0}^{\infty}E(k)\mathrm{d}k,

where 1/2\langle u_i u_i \rangle is the mean turbulent kinetic energy of the flow. The wavenumber k corresponding to length scale r is k=2\pi/r. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

E(k) = C \varepsilon^{2/3} k^{-5/3} ,

where C would be a universal constant. This is one of the most famous results of Kolmogorov 1941 theory, and considerable experimental evidence has accumulated that supports it.[12]

In spite of this success, Kolmogorov theory is at present under revision. This theory implicitly assumes that the turbulence is statistically self-similar at different scales. This essentially means that the statistics are scale-invariant in the inertial range. A usual way of studying turbulent velocity fields is by means of velocity increments:

\delta \mathbf{u}(r) = \mathbf{u}(\mathbf{x} %2B \mathbf{r}) - \mathbf{u}(\mathbf{x});

that is, the difference in velocity between points separated by a vector r (since the turbulence is assumed isotropic, the velocity increment depends only on the modulus of r). Velocity increments are useful because they emphasize the effects of scales of the order of the separation r when statistics are computed. The statistical scale-invariance implies that the scaling of velocity increments should occur with a unique scaling exponent \beta, so that when r is scaled by a factor \lambda,

\delta \mathbf{u}(\lambda r)

should have the same statistical distribution as

\lambda^{\beta}\delta \mathbf{u}(r),

with \beta independent of the scale r. From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the velocity increments (known as structure functions in turbulence) should scale as

\langle [\delta \mathbf{u}(r)]^n \rangle = C_n \varepsilon^{n/3} r^{n/3},

where the brackets denote the statistical average, and the C_n would be universal constants.

There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the n/3 value predicted by the theory, becoming a non-linear function of the order n of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov n/3 value is very small, which explain the success of Kolmogorov theory in regards to low order statistical moments. In particular, it can be shown that when the energy spectrum follows a power law

E(k) \propto k^{-p},

with 1 < p < 3, the second order structure function has also a power law, with the form

\langle [\delta \mathbf{u}(r)]^2 \rangle \propto r^{p-1} .

Since the experimental values obtained for the second order structure function only deviate slightly from the 2/3 value predicted by Kolmogorov theory, the value for p is very near to 5/3 (differences are about 2%[13]). Thus the "Kolmogorov -5/3 spectrum" is generally observed in turbulence. However, for high order structure functions the difference with the Kolmogorov scaling is significant, and the breakdown of the statistical self-similarity is clear. This behavior, and the lack of universality of the C_n constants, are related with the phenomenon of intermittency in turbulence. This is an important area of research in this field, and a major goal of the modern theory of turbulence is to understand what is really universal in the inertial range.

See also

References and notes

  1. ^ "Turbulence theory gets a bit choppy". USA Today. September 10, 2006. http://www.usatoday.com/tech/science/columnist/vergano/2006-09-10-turbulence_x.htm. 
  2. ^ Avila, K.; D. Moxey, A. de Lozar, M. Avila, D. Barkley, B. Hof (July 2011). "The Onset of Turbulence in Pipe Flow". Science 333 (6039): 192–196. Bibcode 2011Sci...333..192A. doi:10.1126/science.1203223. http://www.sciencemag.org/content/333/6039/192. 
  3. ^ http://www.weizmann.ac.il/home/fnfal/papers/PhysToday.pdf
  4. ^ MARSHAK, ALEX (2005). 3D radiative transfer in cloudy atmospheres; pg.76. Springer. ISBN 9783540239581. http://books.google.com/books?id=wzg6wnpHyCUC. 
  5. ^ Mullin, Tom (11 November 1989). "Turbulent times for fluids". New Scientist. 
  6. ^ Davidson, P. A. (2004). Turbulence: An Introduction for Scientists and Engineers. Oxford University Press. ISBN 9780198529491. http://books.google.com/?id=rkOmKzujZB4C&pg=PA24&dq=%22when+I+die+and+go+to+Heaven+there+are+two+matters+on+which+I+hope+for+enlightenment%22#v=onepage&q=%22when%20I%20die%20and%20go%20to%20Heaven%20there%20are%20two%20matters%20on%20which%20I%20hope%20for%20enlightenment%22&f=false. 
  7. ^ http://www.scholarpedia.org/article/Turbulence R. Benzi and U. Frisch, Scholarpedia, "Turbulence".
  8. ^ http://www.scholarpedia.org/article/Cascade_and_scaling G. Falkovich, Scholarpedia, "Cascade and scaling".
  9. ^ Narasimha R, Rudra Kumar S, Prabhu A, Kailas SV (2007). "Turbulent flux events in a nearly neutral atmospheric boundary layer". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (Phil Trans R Soc Ser A, Vol. 365, pp. 841–858) 365 (1852): 841–858. Bibcode 2007RSPTA.365..841N. doi:10.1098/rsta.2006.1949. 
  10. ^ Trevethan M, Chanson H (2010). "Turbulence and Turbulent Flux Events in a Small Estuary". Environmental Fluid Mechanics (Environmental Fluid Mechanics, Vol. 10, pp. 345-368) 10 (3): 345–368. doi:10.1007/s10652-009-9134-7. http://espace.library.uq.edu.au/view/UQ:205133. 
  11. ^ H. Tennekes and J. L. Lumley, "A First Course in Turbulence", The MIT Press, (1972).
  12. ^ U. Frisch. Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press, 1995.[1]
  13. ^ J. Mathieu and J. Scott An Introduction to Turbulent Flow. Cambridge University Press, 2000.

Further reading

General

Original scientific research papers and classical monographs

External links