352:. In fluid dynamics, they are essentially vortices subjected to stretching associated with a corresponding increase of the component of vorticity in the stretching directionâdue 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 and maintain identifiable 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 that their kinetic energy can be transformed by the fluid's molecular viscosity into heat. Turbulent flow 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.
287:
1526:
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.
1661:). 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").
1471:
775:
109:
78:, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is commonly realized in low viscosity fluids. In general terms, in turbulent flow, unsteady
120:
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length scale and consider the energy cascade to contain only the largest and smallest scales; while the latter accommodate both the inertial subrange and the viscous sublayer. Nevertheless, Taylor microscales are often used in describing the term "turbulence" more conveniently as these Taylor microscales play a dominant role in energy and momentum transfer in the wavenumber space.
799:
Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, as more kinetic energy is absorbed by a more viscous fluid. The
Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation.
326:. 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
1466:{\displaystyle {\begin{aligned}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{aligned}}}
389:. The eddies are loosely defined as coherent patterns of flow 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 flow velocity fluctuations for each length scale (
1164:
1537:, 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
1541:). 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.
1843:
233:
In many geophysical flows (rivers, atmospheric boundary layer), the flow turbulence is dominated by the coherent structures and turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence. The turbulent events are associated with
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flow over the front of the sphere would be laminar at typical conditions. However, the boundary layer would separate early, as the pressure gradient switched from favorable (pressure decreasing in the flow direction) to unfavorable (pressure increasing in the flow direction), creating a large region
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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 flow velocity scales and time scales (turnover time) dependent on the length scale. The large eddies
709:
The intermediate scales between the largest and the smallest scales which make the inertial subrange. Taylor microscales are not dissipative scales, but pass down the energy from the largest to the smallest without dissipation. Some literatures do not consider Taylor microscales as a characteristic
216:
A jet exhausting from a nozzle into a quiescent fluid. As the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created. These layers separate the fast moving jet from the external fluid, and at a certain critical
Reynolds number they become unstable and
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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 and non-intermittent in the inertial range. A usual way of
372:
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
330:
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
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and are defined in terms of the normalized two-point flow 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
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would be a universal constant. This is one of the most famous results of
Kolmogorov 1941 theory , describing transport of energy through scale space without any loss or gain. The Kolmogorov five-thirds law was first observed in a tidal channel , and considerable experimental evidence has since
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in the case of a bounding surface such as the interior of a pipe. A similar effect is created by the introduction of a stream of higher velocity fluid, such as the hot gases from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow.
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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.
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517:
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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
674:
2997:. 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 universal in the inertial range, and how to deduce intermittency properties from the Navier-Stokes equations, i.e. from first principles.
696:
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 have high frequency, causing turbulence to be locally
314:
The readily available supply of energy in turbulent flows tends to accelerate the homogenization (mixing) of fluid mixtures. The characteristic which is responsible for the enhanced mixing and increased rates of mass, momentum and energy transports in a flow is called
807:
between two different cases of fluid flow, such as between a model aircraft, and its full size version. Such scaling is not always linear and the application of
Reynolds numbers to both situations allows scaling factors to be developed. A flow situation in which the
718:
governing fluid motion, all such solutions are unstable to finite perturbations 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. The
2981:
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
229:
Bridge supports (piers) in water. When river flow is slow, water flows smoothly around the support legs. When the flow is faster, a higher
Reynolds number is associated with the flow. The flow may start off laminar but is quickly separated from the leg and becomes
267:, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced
1996:
2450:
553:
Large eddies obtain energy from the mean flow and also from each other. Thus, these are the energy production eddies which contain most of the energy. They have the large flow velocity fluctuation and are low in frequency. Integral scales are highly
2924:
1021:
959:
While there is no theorem directly relating the non-dimensional
Reynolds number 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
234:
coherent flow structures such as eddies and turbulent bursting, and they play a critical role in terms of sediment scour, accretion and transport in rivers as well as contaminant mixing and dispersion in rivers and estuaries, and in the atmosphere.
766:: "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 more optimistic."
367:
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
802:
This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the
Reynolds number is also used in scaling of fluid dynamics problems, and is used to determine
964:, for example, turbulence can first be sustained if the Reynolds number is larger than a critical value of about 2040; moreover, the turbulence is generally interspersed with laminar flow until a larger Reynolds number of about 4000.
1725:
362:
To sustain turbulent flow, a persistent source of energy supply is required because turbulence dissipates rapidly as the kinetic energy is converted into internal energy by viscous shear stress. Turbulence causes the formation of
2359:
164:. To prevent this, the surface is dimpled to perturb the boundary layer and promote turbulence. This results in higher skin friction, but it moves the point of boundary layer separation further along, resulting in lower drag.
1623:
2145:
93:, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence create a very complex phenomenon. Physicist
2598:
305:
Turbulent flows are always highly irregular. For this reason, turbulence problems are normally treated statistically rather than deterministically. Turbulent flow is chaotic. However, not all chaotic flows are
2575:
1643:. 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
1647:) 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.
905:
3727:
Jin, Y.; Uth, M.-F.; Kuznetsov, A. V.; Herwig, H. (2 February 2015). "Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study".
2768:
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
1227:
4119:
Sommerfeld, Arnold (1908). "Ein Beitrag zur hydrodynamischen ErklĂ€erung der turbulenten FlĂŒssigkeitsbewegĂŒngen" [A Contribution to Hydrodynamic Explanation of Turbulent Fluid Motions].
1635:
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
4056:
2521:
407:
1201:), 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
3027:
393:). 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.
2471:
when statistics are computed. The statistical scale-invariance without intermittency implies that the scaling of flow velocity increments should occur with a unique scaling exponent
4236:
2818:
1899:
564:
2371:
1159:{\displaystyle v_{x}=\underbrace {{\overline {v}}_{x}} _{\text{mean value}}+\underbrace {v'_{x}} _{\text{fluctuation}}\quad {\text{and}}\quad v_{y}={\overline {v}}_{y}+v'_{y}\,;}
2837:
2181:
763:
3290:
544:
4703:
3441:"Turbulent effects on the microphysics and initiation of warm rain in deep convective clouds: 2-D simulations by a spectral mixed-phase microphysics cloud model"
1838:{\displaystyle \mathbf {u} (\mathbf {x} )=\iiint _{\mathbb {R} ^{3}}{\hat {\mathbf {u} }}(\mathbf {k} )e^{i\mathbf {k\cdot x} }\,\mathrm {d} ^{3}\mathbf {k} \,,}
983:
When flow is turbulent, particles exhibit additional transverse motion which enhances the rate of energy and momentum exchange between them thus increasing the
4629:
1544:
Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the
4688:
4614:
2193:
3089:
2063:. Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is
1561:
2588:. From this fact, and other results of Kolmogorov 1941 theory, it follows that the statistical moments of the flow velocity increments (known as
2069:
1015:
of every particle that passed through that point at any given time. Then one would find the actual flow velocity fluctuating about a mean value:
778:
The plume from this candle flame goes from laminar to turbulent. The Reynolds number can be used to predict where this transition will take place
727:
proposed the first statistical theory of turbulence, based on the aforementioned notion of the energy cascade (an idea originally introduced by
2704:{\displaystyle {\Big \langle }{\big (}\delta \mathbf {u} (r){\big )}^{n}{\Big \rangle }=C_{n}\langle (\varepsilon r)^{\frac {n}{3}}\rangle \,,}
196:
286:
4590:
4540:
4335:
4310:
4095:
4070:
4022:
3947:
3922:
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2748:
of the structure function. The universality of the constants have also been questioned. For low orders the discrepancy with the Kolmogorov
994:
Assume for a two-dimensional turbulent flow that one was able to locate a specific point in the fluid and measure the actual flow velocity
248:
155:. (This can be best understood by considering the golf ball to be stationary, with air flowing over it.) If the golf ball were smooth, the
331:
many models have been postulated to calculate it. For instance, in large bodies of water like oceans this coefficient can be found using
4711:
4637:
4502:
839:
laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
277:
Strategies used by animals for olfactory navigation, and their success, are heavily influenced by turbulence affecting the odor plume.
4559:
4521:
4483:
4285:
3268:
291:
4795:
3067:
4014:
348:
Turbulent flows have non-zero vorticity and are characterized by a strong three-dimensional vortex generation mechanism known as
2532:
3012:
739:
were named after him. It is now known that the self-similarity is broken so the statistical description is presently modified.
559:
turbulence, this length can reach up to the order of several hundreds kilometers.: The integral length scale can be defined as
245:
Is it possible to make a theoretical model to describe the behavior of a turbulent flowâin particular, its internal structures?
860:
794:
forces within a fluid which is subject to relative internal movement due to different fluid velocities, in what is known as a
3062:
100:
The turbulence intensity affects many fields, for examples fish ecology, air pollution, precipitation, and climate change.
4683:
4609:
3684:
3072:
758:? And why turbulence? I really believe he will have an answer for the first." A similar witticism has been attributed to
115:
and turbulent water flow over the hull of a submarine. As the relative velocity of the water increases turbulence occurs.
4866:
4851:
4744:
1708:
is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field
743:
3771:
3052:
3042:
2724:
There is considerable evidence that turbulent flows deviate from this behavior. The scaling exponents deviate from the
715:
512:{\displaystyle T=\left({\frac {1}{\langle u'u'\rangle }}\right)\int _{0}^{\infty }\langle u'u'(\tau )\rangle \,d\tau }
35:
339:
principle. In rivers and large ocean currents, the diffusion coefficient is given by variations of Elder's formula.
4670:
2489:
1664:
Hence, a third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range
271:, bruits (and therefore turbulent flow) can be heard in some vessels that have been narrowed by the disease process.
4832:
TurBase public database with experimental data from European High Performance Infrastructures in Turbulence (EuHIT)
4816:
2467:). Flow velocity increments are useful because they emphasize the effects of scales of the order of the separation
200:
148:
becomes turbulent as its Reynolds number increases with increases in flow velocity and characteristic length scale.
3392:"Influence of Intermittent Turbulence on Air Pollution and Its Dispersion in Winter 2016/2017 over Beijing, China"
842:
turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic
385:, turbulent flow can be realized as a superposition of a spectrum of flow velocity fluctuations and eddies upon a
3077:
2993:
in turbulence and can be related to the non-trivial scaling behavior of the dissipation rate averaged over scale
984:
961:
4786:
4780:
4404:"A critical analysis of turbulence modulation in particulate flow systems: a review of the experimental studies"
3173:
2929:
Since the experimental values obtained for the second order structure function only deviate slightly from the
1991:{\displaystyle {\tfrac {1}{2}}\left\langle u_{i}u_{i}\right\rangle =\int _{0}^{\infty }E(k)\,\mathrm {d} k\,,}
669:{\displaystyle L=\left({\frac {1}{\langle u'u'\rangle }}\right)\int _{0}^{\infty }\langle u'u'(r)\rangle \,dr}
4612:(1941). "The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers".
3807:
4871:
4062:
2445:{\displaystyle \delta \mathbf {u} (r)=\mathbf {u} (\mathbf {x} +\mathbf {r} )-\mathbf {u} (\mathbf {x} )\,;}
178:
2774:
2919:{\displaystyle {\Big \langle }{\big (}\delta \mathbf {u} (r){\big )}^{2}{\Big \rangle }\propto r^{p-1}\,,}
1629:
736:
691:
374:
145:
3390:
Wei, Wei; Zhang, Hongsheng; Cai, Xuhui; Song, Yu; Bian, Yuxuan; Xiao, Kaitao; Zhang, He (February 2020).
4876:
4861:
4856:
3057:
3047:
843:
364:
323:
4434:
2461:(since the turbulence is assumed isotropic, the flow velocity increment depends only on the modulus of
2153:
4720:
4646:
4451:
4360:
4155:
3981:
3819:
3736:
3693:
3632:
3566:
3553:
Kunze, Eric; Dower, John F.; Beveridge, Ian; Dewey, Richard; Bartlett, Kevin P. (22 September 2006).
3511:
3452:
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3348:
3299:
3224:
3022:
1514:
755:
728:
332:
167:
31:
4402:
Hoque, Mohammad Mainul; Joshi, Jyeshtharaj B.; Evans, Geoffrey M.; Mitra, Subhasish (31 July 2023).
4199:"The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds' Numbers"
4846:
4805:
4790:
4630:"The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers"
4351:
Meneveau, C.; Sreenivasan, K.R. (1991). "The multifractal nature of turbulent energy dissipation".
3104:
3007:
1545:
754:, given the opportunity. His reply was: "When I meet God, I am going to ask him two questions: Why
171:
3625:
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
1555:. With only these two parameters, the unique length that can be formed by dimensional analysis is
4736:
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4376:
4179:
3843:
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804:
294:. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows.
226:
work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence.
1639:, while the input of energy into the cascade comes from the decay of the large scales, of order
3215:
Ting, F. C. K.; Kirby, J. T. (1996). "Dynamics of surf-zone turbulence in a spilling breaker".
774:
4821:
4586:
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948:
747:
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704:
349:
71:, which occurs when a fluid flows in parallel layers with no disruption between those layers.
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4728:
4654:
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4368:
4248:
4163:
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3964:
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3640:
3574:
3519:
3460:
3411:
3356:
3307:
3232:
3165:
3139:
3124:
1202:
210:
The external flow over all kinds of vehicles such as cars, airplanes, ships, and submarines.
124:
529:
191:
The flow conditions in many industrial equipment (such as pipes, ducts, precipitators, gas
3891:
3177:
3134:
3129:
3032:
1534:
821:
783:
732:
268:
108:
94:
90:
3254:
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4198:
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3352:
3303:
3228:
4139:
3675:
3114:
3037:
809:
795:
382:
220:
Biologically generated turbulence resulting from swimming animals affects ocean mixing.
156:
83:
44:
4143:
3554:
3523:
3360:
3337:"Turbulence, larval fish ecology and fisheries recruitment: a review of field studies"
17:
4840:
4740:
4666:
4442:
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3539:
3236:
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2990:
2354:{\displaystyle E(k)=K_{0}\varepsilon ^{\frac {2}{3}}k^{-{\frac {5}{3}}}\exp \left\,,}
1484:
369:
343:
75:
64:
4183:
3602:
3376:
119:
4471:
4273:
3713:
3660:
3336:
3017:
1210:
915:
832:
817:
260:
141:
112:
68:
56:
4570:
3489:
1873:
represents the contribution to the kinetic energy from all the Fourier modes with
97:
described turbulence as the most important unsolved problem in classical physics.
4008:
3679:
2365:
studying turbulent flow velocity fields is by means of flow velocity increments:
3144:
3109:
3084:
943:
759:
555:
357:
336:
204:
185:
967:
The transition occurs if the size of the object is gradually increased, or the
4810:
4372:
4252:
3784:
3705:
3617:
3416:
3391:
2455:
that is, the difference in flow velocity between points separated by a vector
1530:
390:
256:
223:
4214:
4105:
3887:
3839:
3586:
3531:
3474:
3425:
3368:
4416:
4403:
4167:
3895:
3578:
3286:"New developments in understanding interfacial processes in turbulent flows"
3260:
1618:{\displaystyle \eta =\left({\frac {\nu ^{3}}{\varepsilon }}\right)^{1/4}\,.}
1501:
968:
813:
698:
386:
161:
152:
137:
4732:
4658:
4175:
3652:
3644:
3594:
3321:
3312:
2744:
value predicted by the theory, becoming a non-linear function of the order
2140:{\displaystyle E(k)=K_{0}\varepsilon ^{\frac {2}{3}}k^{-{\frac {5}{3}}}\,,}
4571:"Introductory Lectures on Turbulence â Physics, Mathematics, and Modeling"
3806:
Reddy, Gautam; Murthy, Venkatesh N.; Vergassola, Massimo (10 March 2022).
2831:, the second order structure function has also a power law, with the form
933:
is a characteristic velocity of the fluid with respect to the object (m/s)
3465:
3440:
988:
924:
192:
128:
79:
60:
4583:
Turbulence â Introduction to Theory and Applications of Turbulent Flows
3748:
972:
920:
791:
4463:
3993:
3618:"Turbulent flux events in a nearly neutral atmospheric boundary layer"
3555:"Observations of Biologically Generated Turbulence in a Coastal Inlet"
3119:
2187:
Outside of the inertial area, one can find the formula below :
847:
720:
274:
Recently, turbulence in porous media became a highly debated subject.
264:
27:
Motion characterized by chaotic changes in pressure and flow velocity
4800:
3439:
Benmoshe, N.; Pinsky, M.; Pokrovsky, A.; Khain, A. (27 March 2012).
4827:
Johns Hopkins public database with direct numerical simulation data
3506:
237:
3616:
Narasimha, R.; Rudra Kumar, S.; Prabhu, A.; Kailas, S. V. (2007).
952:
787:
773:
285:
118:
107:
82:
appear of many sizes which interact with each other, consequently
4686:(1941). "Dissipation of Energy in Locally Isotropic Turbulence".
2031:
is the mean turbulent kinetic energy of the flow. The wavenumber
782:
The onset of turbulence can be, to some extent, predicted by the
714:
Although it is possible to find some particular solutions of the
401:
The integral time scale for a Lagrangian flow can be defined as:
4826:
3909:
3907:
3905:
327:
812:
is significantly absorbed due to the action of fluid molecular
4038:
Mullin, Tom (11 November 1989). "Turbulent times for fluids".
751:
89:
The onset of turbulence can be predicted by the dimensionless
74:
Turbulence is commonly observed in everyday phenomena such as
4581:
Nieuwstadt, F. T. M.; Boersma, B. J.; Westerweel, J. (2016).
4395:
3808:"Olfactory Sensing and Navigation in Turbulent Environments"
4831:
4704:"Dissipation of energy in the locally isotropic turbulence"
4550:
Bohr, T.; Jensen, M. H.; Paladin, G.; Vulpiani, A. (1998).
3913:
Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R. (2012).
3872:. Springer Science & Business Media. pp. 265â307.
2714:
where the brackets denote the statistical average, and the
1859:
is the Fourier transform of the flow velocity field. Thus,
1216:) in the direction normal to the flow for a given time are
688:′ is the velocity fluctuation in that same direction.
4603:
Original scientific research papers and classic monographs
2570:{\displaystyle \lambda ^{\beta }\delta \mathbf {u} (r)\,,}
4495:
Turbulence â An Introduction for Scientists and Engineers
4330:. Cambridge ; New York: Cambridge University Press.
4090:. Cambridge ; New York: Cambridge University Press.
3680:"Turbulence and Turbulent Flux Events in a Small Estuary"
4058:
Turbulence: An Introduction for Scientists and Engineers
3028:
Different types of boundary conditions in fluid dynamics
1628:
This is today known as the Kolmogorov length scale (see
1209:
The heat flux and momentum transfer (represented by the
900:{\displaystyle \mathrm {Re} ={\frac {\rho vL}{\mu }}\,,}
684:
is the distance between two measurement locations, and
298:
Turbulence is characterized by the following features:
1904:
4433:
Falkovich, Gregory; Sreenivasan, K. R. (April 2006).
3963:
Falkovich, Gregory; Sreenivasan, K. R. (April 2006).
3868:
Ferziger, Joel H.; Peric, Milovan (6 December 2012).
2840:
2777:
2601:
2535:
2492:
2374:
2196:
2156:
2072:
1902:
1728:
1678:
are universally and uniquely determined by the scale
1564:
1225:
1024:
863:
567:
532:
410:
174:(the blurring of images seen through the atmosphere).
4514:
Non-equilibrium Statistical Mechanics and Turbulence
4305:. Oxford ; New York: Oxford University Press.
2945:
value predicted by Kolmogorov theory, the value for
170:
experienced during airplane flight, as well as poor
4533:
Statistical Theory and Modeling for Turbulent Flows
4235:Grant, H. L.; Stewart, R. W.; Moilliet, A. (1962).
742:A complete description of turbulence is one of the
2965:(differences are about 2%). Thus the "Kolmogorov â
2918:
2812:
2703:
2569:
2515:
2444:
2353:
2175:
2139:
1990:
1837:
1617:
1465:
1158:
899:
764:British Association for the Advancement of Science
668:
538:
511:
160:of low pressure behind the ball that creates high
4822:Fluid Mechanics website with movies, Q&A, etc
3291:Philosophical Transactions of the Royal Society A
2888:
2843:
2649:
2604:
2526:should have the same statistical distribution as
3933:
3931:
820:regime. For this the dimensionless quantity the
4512:Cardy, J.; Falkovich, G.; Gawedzki, K. (2008).
4138:Avila, K.; Moxey, D.; de Lozar, A.; Avila, M.;
335:'s four-third power law and is governed by the
290:Flow visualization of a turbulent jet, made by
4702:Kolmogorov, Andrey Nikolaevich (8 July 1991).
4628:Kolmogorov, Andrey Nikolaevich (8 July 1991).
4531:Durbin, P. A.; Pettersson Reif, B. A. (2001).
2989:constants, are related with the phenomenon of
2516:{\displaystyle \delta \mathbf {u} (\lambda r)}
140:. For the first few centimeters, the smoke is
2875:
2850:
2636:
2611:
213:The motions of matter in stellar atmospheres.
8:
3445:Journal of Geophysical Research: Atmospheres
2694:
2667:
656:
628:
603:
584:
499:
471:
446:
427:
30:For the turbulence felt on an airplane, see
4783:, Scientific papers and books on turbulence
4689:Proceedings of the USSR Academy of Sciences
4615:Proceedings of the USSR Academy of Sciences
4010:3D radiative transfer in cloudy atmospheres
3938:Tennekes, Hendrik; Lumley, John L. (1972).
4476:Turbulence: The Legacy of A. N. Kolmogorov
4278:Turbulence: The Legacy of A. N. Kolmogorov
3917:. Netherlands: Elsevier Inc. pp. 537â601.
3284:Eames, I.; Flor, J. B. (17 January 2011).
3248:
3246:
4415:
4237:"Turbulence spectra from a tidal channel"
3812:Annual Review of Condensed Matter Physics
3783:
3505:
3464:
3415:
3311:
2912:
2900:
2887:
2886:
2880:
2874:
2873:
2858:
2849:
2848:
2842:
2841:
2839:
2806:
2797:
2776:
2697:
2683:
2661:
2648:
2647:
2641:
2635:
2634:
2619:
2610:
2609:
2603:
2602:
2600:
2563:
2549:
2540:
2534:
2496:
2491:
2438:
2430:
2422:
2411:
2403:
2395:
2378:
2373:
2347:
2331:
2315:
2305:
2298:
2281:
2271:
2245:
2241:
2226:
2216:
2195:
2161:
2155:
2133:
2121:
2117:
2102:
2092:
2071:
1984:
1976:
1975:
1957:
1952:
1934:
1924:
1903:
1901:
1831:
1826:
1820:
1815:
1813:
1800:
1796:
1784:
1770:
1768:
1767:
1759:
1755:
1754:
1752:
1737:
1729:
1727:
1611:
1601:
1597:
1582:
1576:
1563:
1455:
1438:
1428:
1421:
1415:
1402:
1381:
1368:
1361:
1352:
1333:
1312:
1306:
1300:
1284:
1264:
1248:
1241:
1226:
1224:
1152:
1143:
1130:
1120:
1110:
1100:
1093:
1079:
1073:
1063:
1052:
1042:
1039:
1029:
1023:
939:is a characteristic linear dimension (m)
893:
875:
864:
862:
659:
622:
617:
578:
566:
531:
526:′ is the velocity fluctuation, and
502:
465:
460:
421:
409:
4552:Dynamical Systems Approach to Turbulence
4303:Developments in the Theory of Turbulence
4121:International Congress of Mathematicians
3870:Computational Methods for Fluid Dynamics
3832:10.1146/annurev-conmatphys-031720-032754
4806:international CFD database iCFDdatabase
4645:(1991). Translated by Levin, V.: 9â13.
3192:
3157:
197:dynamic scraped surface heat exchangers
4435:"Lessons from hydrodynamic turbulence"
4144:"The Onset of Turbulence in Pipe Flow"
3965:"Lessons from hydrodynamic turbulence"
3490:"The power spectrum of climate change"
3164:The story has also been attributed to
3090:NavierâStokes existence and smoothness
546:is the time lag between measurements.
4326:Mathieu, Jean; Scott, Julian (2000).
2813:{\displaystyle E(k)\propto k^{-p}\,,}
971:of the fluid is decreased, or if the
7:
4759:The Theory of Homogeneous Turbulence
3253:Tennekes, H.; Lumley, J. L. (1972).
746:. According to an apocryphal story,
322:is usually described by a turbulent
199:, etc.) and machines (for instance,
131:wing passing through coloured smoke
1682:and the rate of energy dissipation
1551:and the rate of energy dissipation
249:(more unsolved problems in physics)
86:due to friction effects increases.
4712:Proceedings of the Royal Society A
4638:Proceedings of the Royal Society A
3494:The European Physical Journal Plus
3396:Journal of Meteorological Research
3335:MacKENZIE, Brian R (August 2000).
1977:
1958:
1816:
1446:
1424:
1324:
1309:
868:
865:
854:The Reynolds number is defined as
623:
466:
259:, a stethoscope is used to detect
25:
4328:An Introduction to Turbulent Flow
55:is fluid motion characterized by
3770:Mackenzie, Dana (6 March 2023).
2859:
2620:
2550:
2497:
2431:
2423:
2412:
2404:
2396:
2379:
2176:{\displaystyle K_{0}\approx 1.5}
1827:
1807:
1801:
1785:
1771:
1738:
1730:
1529:In his original theory of 1941,
1500:is the coefficient of turbulent
4408:Reviews in Chemical Engineering
3942:. Cambridge, Mass.: MIT Press.
3772:"How animals follow their nose"
3524:10.1140/epjp/s13360-022-02773-w
3202:Introduction to Fluid Mechanics
3013:Atmospheric dispersion modeling
2592:in turbulence) should scale as
1169:and similarly for temperature (
1105:
1099:
4787:Center for Turbulence Research
4781:Center for Turbulence Research
4684:Kolmogorov, Andrey Nikolaevich
4610:Kolmogorov, Andrey Nikolaevich
4554:. Cambridge University Press.
4516:. Cambridge University Press.
4478:. Cambridge University Press.
4301:Leslie, David Clement (1983).
4280:. Cambridge University Press.
3488:Sneppen, Albert (5 May 2022).
2869:
2863:
2787:
2781:
2721:would be universal constants.
2680:
2670:
2630:
2624:
2560:
2554:
2510:
2501:
2435:
2427:
2416:
2400:
2389:
2383:
2206:
2200:
2184:accumulated that supports it.
2082:
2076:
2035:corresponding to length scale
1972:
1966:
1789:
1781:
1775:
1742:
1734:
1533:postulated that for very high
1520:
653:
647:
496:
490:
1:
4761:. Cambridge University Press.
4585:(Online ed.). Springer.
3685:Environmental Fluid Mechanics
3361:10.1016/s0399-1784(00)00142-0
3073:Lagrangian coherent structure
1491:is the density of the fluid,
850:and other flow instabilities.
188:and intense oceanic currents.
3940:A First Course in Turbulence
3256:A First Course in Turbulence
3237:10.1016/0378-3839(95)00037-2
3068:KelvinâHelmholtz instability
1433:
1391:
1317:
1125:
1047:
955:(Pa·s or N·s/m or kg/(m·s)).
835:and turbulent flow regimes:
750:was asked what he would ask
744:unsolved problems in physics
184:The oceanic and atmospheric
4796:Scientific American article
4497:. Oxford University Press.
4203:Akademiia Nauk SSSR Doklady
4086:Falkovich, Gregory (2011).
3996:– via weizmann.ac.il.
1521:Kolmogorov's theory of 1941
975:of the fluid is increased.
240:Unsolved problem in physics
201:internal combustion engines
36:Turbulence (disambiguation)
4893:
4241:Journal of Fluid Mechanics
3729:Journal of Fluid Mechanics
979:Heat and momentum transfer
292:laser-induced fluorescence
29:
4757:Batchelor, G. K. (1953).
4700:Translated into English:
4626:Translated into English:
4576:. University of Kentucky.
4569:McDonough, J. M. (2007).
4535:. John Wiley & Sons.
4373:10.1017/S0022112091001830
4253:10.1017/S002211206200018X
3785:10.1146/knowable-030623-4
3706:10.1007/s10652-009-9134-7
3417:10.1007/s13351-020-9128-4
3078:Turbulence kinetic energy
3063:HagenâPoiseuille equation
2584:independent of the scale
217:break down to turbulence.
4812:Turbulent flow in a pipe
4493:Davidson, P. A. (2004).
4055:Davidson, P. A. (2004).
1692:energy spectrum function
692:Kolmogorov length scales
255:In the medical field of
177:Most of the terrestrial
67:. It is in contrast to
4801:Air Turbulence Forecast
4417:10.1515/revce-2022-0068
4197:Kolmogorov, A. (1941).
4168:10.1126/science.1203223
4063:Oxford University Press
3579:10.1126/science.1129378
3053:NavierâStokes equations
3043:DarcyâWeisbach equation
716:NavierâStokes equations
375:Kolmogorov length scale
179:atmospheric circulation
4733:10.1098/rspa.1991.0076
4659:10.1098/rspa.1991.0075
4142:; B. Hof (July 2011).
4007:Marshak, Alex (2005).
3645:10.1098/rsta.2006.1949
3313:10.1098/rsta.2010.0332
3200:Batchelor, G. (2000).
2920:
2814:
2705:
2571:
2517:
2479:is scaled by a factor
2446:
2355:
2177:
2141:
1992:
1839:
1630:Kolmogorov microscales
1619:
1487:at constant pressure,
1467:
1160:
901:
828:) is used as a guide.
790:of inertial forces to
779:
737:Kolmogorov microscales
670:
550:Integral length scales
540:
513:
295:
132:
116:
104:Examples of turbulence
34:. For other uses, see
18:Atmospheric turbulence
4676:on 23 September 2015.
3058:Large eddy simulation
2921:
2815:
2706:
2572:
2518:
2447:
2356:
2178:
2142:
1993:
1840:
1620:
1468:
1161:
902:
777:
731:) and the concept of
671:
541:
539:{\displaystyle \tau }
514:
324:diffusion coefficient
289:
122:
111:
3466:10.1029/2011jd016603
3023:Clear-air turbulence
2838:
2775:
2599:
2533:
2490:
2372:
2194:
2154:
2070:
1900:
1726:
1562:
1515:thermal conductivity
1223:
1022:
861:
735:. As a result, the
565:
530:
408:
373:this happens is the
168:Clear-air turbulence
136:Smoke rising from a
32:Clear-air turbulence
4867:Transport phenomena
4852:Concepts in physics
4791:Stanford University
4725:1991RSPSA.434...15K
4651:1991RSPSA.434....9K
4456:2006PhT....59d..43F
4365:1991JFM...224..429M
4160:2011Sci...333..192A
3986:2006PhT....59d..43F
3824:2022ARCMP..13..191R
3741:2015JFM...766...76J
3698:2010EFM....10..345T
3637:2007RSPTA.365..841N
3571:2006Sci...313.1768K
3565:(5794): 1768â1770.
3516:2022EPJP..137..555S
3457:2012JGRD..117.6220B
3408:2020JMetR..34..176W
3353:2000AcOc...23..357M
3304:2011RSPTA.369..702E
3229:1996CoasE..27..131T
3217:Coastal Engineering
3174:Theodore von KĂĄrmĂĄn
3105:Turbulence modeling
3008:Astronomical seeing
2590:structure functions
1962:
1546:kinematic viscosity
1389:
1376:
1256:
1151:
1087:
770:Onset of turbulence
762:in a speech to the
627:
470:
397:Integral time scale
320:Turbulent diffusion
172:astronomical seeing
3778:. Annual Reviews.
3749:10.1017/jfm.2015.9
2916:
2810:
2701:
2567:
2513:
2442:
2351:
2173:
2137:
1988:
1948:
1913:
1835:
1615:
1463:
1461:
1407:
1404:experimental value
1400:
1377:
1364:
1289:
1286:experimental value
1282:
1244:
1156:
1139:
1098:
1091:
1075:
1068:
1061:
897:
805:dynamic similitude
780:
705:Taylor microscales
666:
613:
536:
509:
456:
296:
133:
123:Turbulence in the
117:
4771:
4770:
4592:978-3-319-31599-7
4542:978-0-470-68931-8
4464:10.1063/1.2207037
4337:978-0-521-57066-4
4312:978-0-19-856161-3
4154:(6039): 192â196.
4097:978-1-107-00575-4
4072:978-0-19-852949-1
4024:978-3-540-23958-1
3994:10.1063/1.2207037
3949:978-0-262-20019-6
3923:978-0-12-382100-3
3879:978-3-642-56026-2
3776:Knowable Magazine
3631:(1852): 841â858.
3341:Oceanologica Acta
3298:(1937): 702â705.
3170:Arnold Sommerfeld
3100:Taylor microscale
2691:
2339:
2325:
2291:
2253:
2234:
2129:
2110:
1912:
1893:, and therefore,
1778:
1591:
1513:is the turbulent
1453:
1436:
1418:
1405:
1394:
1353:
1351:
1331:
1320:
1303:
1287:
1242:
1240:
1128:
1103:
1096:
1074:
1072:
1066:
1050:
1040:
1038:
949:dynamic viscosity
891:
748:Werner Heisenberg
725:Andrey Kolmogorov
607:
450:
350:vortex stretching
16:(Redirected from
4884:
4813:
4762:
4751:
4749:
4743:. Archived from
4708:
4697:
4677:
4675:
4669:. Archived from
4634:
4623:
4596:
4577:
4575:
4565:
4546:
4527:
4508:
4489:
4467:
4439:
4429:
4419:
4396:
4385:
4384:
4348:
4342:
4341:
4323:
4317:
4316:
4298:
4292:
4291:
4270:
4264:
4263:
4261:
4259:
4232:
4226:
4225:
4223:
4221:
4194:
4188:
4187:
4135:
4129:
4128:
4116:
4110:
4109:
4083:
4077:
4076:
4052:
4046:
4045:
4035:
4029:
4028:
4004:
3998:
3997:
3969:
3960:
3954:
3953:
3935:
3926:
3911:
3900:
3899:
3865:
3859:
3858:
3856:
3854:
3803:
3797:
3796:
3794:
3792:
3787:
3767:
3761:
3760:
3724:
3718:
3717:
3671:
3665:
3664:
3622:
3613:
3607:
3606:
3550:
3544:
3543:
3509:
3485:
3479:
3478:
3468:
3436:
3430:
3429:
3419:
3387:
3381:
3380:
3332:
3326:
3325:
3315:
3281:
3275:
3274:
3250:
3241:
3240:
3223:(3â4): 131â160.
3212:
3206:
3205:
3197:
3181:
3166:John von Neumann
3162:
3140:Wingtip vortices
3125:Vortex generator
2996:
2988:
2980:
2978:
2977:
2974:
2971:
2964:
2962:
2961:
2958:
2955:
2949:is very near to
2948:
2944:
2942:
2941:
2938:
2935:
2925:
2923:
2922:
2917:
2911:
2910:
2892:
2891:
2885:
2884:
2879:
2878:
2862:
2854:
2853:
2847:
2846:
2830:
2819:
2817:
2816:
2811:
2805:
2804:
2767:
2766:
2764:
2763:
2760:
2757:
2747:
2743:
2742:
2740:
2739:
2736:
2733:
2720:
2710:
2708:
2707:
2702:
2693:
2692:
2684:
2666:
2665:
2653:
2652:
2646:
2645:
2640:
2639:
2623:
2615:
2614:
2608:
2607:
2587:
2583:
2576:
2574:
2573:
2568:
2553:
2545:
2544:
2522:
2520:
2519:
2514:
2500:
2482:
2478:
2474:
2470:
2466:
2460:
2451:
2449:
2448:
2443:
2434:
2426:
2415:
2407:
2399:
2382:
2360:
2358:
2357:
2352:
2346:
2342:
2341:
2340:
2332:
2330:
2326:
2321:
2320:
2319:
2310:
2309:
2299:
2292:
2287:
2286:
2285:
2272:
2256:
2255:
2254:
2246:
2236:
2235:
2227:
2221:
2220:
2182:
2180:
2179:
2174:
2166:
2165:
2146:
2144:
2143:
2138:
2132:
2131:
2130:
2122:
2112:
2111:
2103:
2097:
2096:
2062:
2061:
2059:
2058:
2053:
2050:
2038:
2034:
2030:
2017:
2015:
2014:
2011:
2008:
1997:
1995:
1994:
1989:
1980:
1961:
1956:
1944:
1940:
1939:
1938:
1929:
1928:
1914:
1905:
1892:
1883:
1872:
1858:
1844:
1842:
1841:
1836:
1830:
1825:
1824:
1819:
1812:
1811:
1810:
1788:
1780:
1779:
1774:
1769:
1766:
1765:
1764:
1763:
1758:
1741:
1733:
1718:
1707:
1703:
1685:
1681:
1677:
1660:
1646:
1642:
1638:
1624:
1622:
1621:
1616:
1610:
1609:
1605:
1596:
1592:
1587:
1586:
1577:
1554:
1550:
1540:
1535:Reynolds numbers
1512:
1499:
1490:
1482:
1472:
1470:
1469:
1464:
1462:
1454:
1452:
1444:
1443:
1442:
1437:
1429:
1422:
1420:
1419:
1416:
1406:
1403:
1401:
1396:
1395:
1390:
1385:
1372:
1362:
1332:
1330:
1322:
1321:
1313:
1307:
1305:
1304:
1301:
1288:
1285:
1283:
1278:
1277:
1269:
1268:
1252:
1215:
1203:Osborne Reynolds
1200:
1195:
1185:) and pressure (
1184:
1179:
1165:
1163:
1162:
1157:
1147:
1135:
1134:
1129:
1121:
1115:
1114:
1104:
1101:
1097:
1094:
1092:
1083:
1067:
1064:
1062:
1057:
1056:
1051:
1043:
1034:
1033:
1014:
946:
938:
932:
918:
906:
904:
903:
898:
892:
887:
876:
871:
831:With respect to
827:
816:gives rise to a
701:and homogeneous.
675:
673:
672:
667:
646:
638:
626:
621:
612:
608:
606:
602:
594:
579:
545:
543:
542:
537:
518:
516:
515:
510:
489:
481:
469:
464:
455:
451:
449:
445:
437:
422:
241:
21:
4892:
4891:
4887:
4886:
4885:
4883:
4882:
4881:
4837:
4836:
4811:
4777:
4772:
4756:
4750:on 6 July 2011.
4747:
4719:(1980): 15â17.
4706:
4701:
4682:
4673:
4632:
4627:
4608:
4593:
4580:
4573:
4568:
4562:
4549:
4543:
4530:
4524:
4511:
4505:
4492:
4486:
4470:
4437:
4432:
4401:
4393:
4391:Further reading
4388:
4350:
4349:
4345:
4338:
4325:
4324:
4320:
4313:
4300:
4299:
4295:
4288:
4272:
4271:
4267:
4257:
4255:
4234:
4233:
4229:
4219:
4217:
4196:
4195:
4191:
4137:
4136:
4132:
4118:
4117:
4113:
4098:
4088:Fluid Mechanics
4085:
4084:
4080:
4073:
4054:
4053:
4049:
4037:
4036:
4032:
4025:
4006:
4005:
4001:
3967:
3962:
3961:
3957:
3950:
3937:
3936:
3929:
3915:Fluid Mechanics
3912:
3903:
3880:
3867:
3866:
3862:
3852:
3850:
3805:
3804:
3800:
3790:
3788:
3769:
3768:
3764:
3726:
3725:
3721:
3674:Trevethan, M.;
3673:
3672:
3668:
3620:
3615:
3614:
3610:
3552:
3551:
3547:
3487:
3486:
3482:
3438:
3437:
3433:
3389:
3388:
3384:
3334:
3333:
3329:
3283:
3282:
3278:
3271:
3252:
3251:
3244:
3214:
3213:
3209:
3199:
3198:
3194:
3190:
3185:
3184:
3178:Albert Einstein
3163:
3159:
3154:
3149:
3135:Wave turbulence
3130:Wake turbulence
3033:Eddy covariance
3003:
2994:
2987:
2983:
2975:
2972:
2969:
2968:
2966:
2959:
2956:
2953:
2952:
2950:
2946:
2939:
2936:
2933:
2932:
2930:
2896:
2872:
2836:
2835:
2824:
2793:
2773:
2772:
2761:
2758:
2753:
2752:
2750:
2749:
2745:
2737:
2734:
2729:
2728:
2726:
2725:
2719:
2715:
2679:
2657:
2633:
2597:
2596:
2585:
2581:
2536:
2531:
2530:
2488:
2487:
2480:
2476:
2475:, so that when
2472:
2468:
2462:
2456:
2370:
2369:
2311:
2301:
2300:
2294:
2293:
2277:
2273:
2267:
2263:
2237:
2222:
2212:
2192:
2191:
2157:
2152:
2151:
2113:
2098:
2088:
2068:
2067:
2054:
2051:
2048:
2047:
2045:
2040:
2036:
2032:
2027:
2023:
2012:
2009:
2006:
2005:
2003:
2002:
1930:
1920:
1919:
1915:
1898:
1897:
1879:
1874:
1860:
1849:
1814:
1792:
1753:
1748:
1724:
1723:
1709:
1705:
1694:
1683:
1679:
1665:
1648:
1644:
1640:
1636:
1578:
1572:
1571:
1560:
1559:
1552:
1548:
1538:
1523:
1511:
1505:
1498:
1492:
1488:
1481:
1477:
1460:
1459:
1445:
1427:
1423:
1411:
1363:
1354:
1344:
1338:
1337:
1323:
1308:
1296:
1270:
1260:
1243:
1233:
1221:
1220:
1213:
1191:
1186:
1175:
1170:
1119:
1106:
1041:
1025:
1020:
1019:
1011:
1004:
995:
981:
962:Poiseuille flow
942:
936:
930:
914:
877:
859:
858:
825:
822:Reynolds number
786:, which is the
784:Reynolds number
772:
733:self-similarity
639:
631:
595:
587:
583:
574:
563:
562:
528:
527:
482:
474:
438:
430:
426:
417:
406:
405:
284:
269:atherosclerosis
252:
251:
246:
243:
239:
106:
95:Richard Feynman
91:Reynolds number
39:
28:
23:
22:
15:
12:
11:
5:
4890:
4888:
4880:
4879:
4874:
4872:Fluid dynamics
4869:
4864:
4859:
4854:
4849:
4839:
4838:
4835:
4834:
4829:
4824:
4819:
4808:
4803:
4798:
4793:
4784:
4776:
4775:External links
4773:
4769:
4768:
4764:
4763:
4754:
4753:
4752:
4692:(in Russian).
4680:
4679:
4678:
4618:(in Russian).
4605:
4604:
4599:
4598:
4597:
4591:
4578:
4566:
4560:
4547:
4541:
4528:
4522:
4509:
4504:978-0198529491
4503:
4490:
4484:
4468:
4430:
4394:
4392:
4389:
4387:
4386:
4343:
4336:
4318:
4311:
4293:
4286:
4265:
4247:(2): 241â268.
4227:
4189:
4130:
4111:
4096:
4078:
4071:
4047:
4030:
4023:
4017:. p. 76.
3999:
3955:
3948:
3927:
3901:
3878:
3860:
3818:(1): 191â213.
3798:
3762:
3719:
3692:(3): 345â368.
3666:
3608:
3545:
3480:
3431:
3402:(1): 176â188.
3382:
3347:(4): 357â375.
3327:
3276:
3269:
3242:
3207:
3191:
3189:
3186:
3183:
3182:
3156:
3155:
3153:
3150:
3148:
3147:
3142:
3137:
3132:
3127:
3122:
3117:
3115:Vertical draft
3112:
3107:
3102:
3097:
3092:
3087:
3082:
3081:
3080:
3075:
3070:
3065:
3060:
3055:
3050:
3045:
3038:Fluid dynamics
3035:
3030:
3025:
3020:
3015:
3010:
3004:
3002:
2999:
2985:
2927:
2926:
2915:
2909:
2906:
2903:
2899:
2895:
2890:
2883:
2877:
2871:
2868:
2865:
2861:
2857:
2852:
2845:
2821:
2820:
2809:
2803:
2800:
2796:
2792:
2789:
2786:
2783:
2780:
2717:
2712:
2711:
2700:
2696:
2690:
2687:
2682:
2678:
2675:
2672:
2669:
2664:
2660:
2656:
2651:
2644:
2638:
2632:
2629:
2626:
2622:
2618:
2613:
2606:
2578:
2577:
2566:
2562:
2559:
2556:
2552:
2548:
2543:
2539:
2524:
2523:
2512:
2509:
2506:
2503:
2499:
2495:
2453:
2452:
2441:
2437:
2433:
2429:
2425:
2421:
2418:
2414:
2410:
2406:
2402:
2398:
2394:
2391:
2388:
2385:
2381:
2377:
2362:
2361:
2350:
2345:
2338:
2335:
2329:
2324:
2318:
2314:
2308:
2304:
2297:
2290:
2284:
2280:
2276:
2270:
2266:
2262:
2259:
2252:
2249:
2244:
2240:
2233:
2230:
2225:
2219:
2215:
2211:
2208:
2205:
2202:
2199:
2172:
2169:
2164:
2160:
2148:
2147:
2136:
2128:
2125:
2120:
2116:
2109:
2106:
2101:
2095:
2091:
2087:
2084:
2081:
2078:
2075:
2025:
2021:
1999:
1998:
1987:
1983:
1979:
1974:
1971:
1968:
1965:
1960:
1955:
1951:
1947:
1943:
1937:
1933:
1927:
1923:
1918:
1911:
1908:
1846:
1845:
1834:
1829:
1823:
1818:
1809:
1806:
1803:
1799:
1795:
1791:
1787:
1783:
1777:
1773:
1762:
1757:
1751:
1747:
1744:
1740:
1736:
1732:
1626:
1625:
1614:
1608:
1604:
1600:
1595:
1590:
1585:
1581:
1575:
1570:
1567:
1522:
1519:
1509:
1496:
1479:
1474:
1473:
1458:
1451:
1448:
1441:
1435:
1432:
1426:
1414:
1410:
1399:
1393:
1388:
1384:
1380:
1375:
1371:
1367:
1360:
1357:
1350:
1347:
1345:
1343:
1340:
1339:
1336:
1329:
1326:
1319:
1316:
1311:
1299:
1295:
1292:
1281:
1276:
1273:
1267:
1263:
1259:
1255:
1251:
1247:
1239:
1236:
1234:
1232:
1229:
1228:
1167:
1166:
1155:
1150:
1146:
1142:
1138:
1133:
1127:
1124:
1118:
1113:
1109:
1090:
1086:
1082:
1078:
1071:
1060:
1055:
1049:
1046:
1037:
1032:
1028:
1009:
1002:
980:
977:
957:
956:
940:
934:
928:
923:of the fluid (
908:
907:
896:
890:
886:
883:
880:
874:
870:
867:
852:
851:
840:
810:kinetic energy
796:boundary layer
771:
768:
723:mathematician
712:
711:
707:
702:
694:
689:
678:
677:
676:
665:
662:
658:
655:
652:
649:
645:
642:
637:
634:
630:
625:
620:
616:
611:
605:
601:
598:
593:
590:
586:
582:
577:
573:
570:
551:
535:
520:
519:
508:
505:
501:
498:
495:
492:
488:
485:
480:
477:
473:
468:
463:
459:
454:
448:
444:
441:
436:
433:
429:
425:
420:
416:
413:
399:
398:
383:energy cascade
379:
378:
360:
354:
353:
346:
317:
316:
315:"diffusivity".
312:
308:
307:
303:
283:
280:
279:
278:
275:
272:
247:
244:
238:
236:
235:
231:
227:
221:
218:
214:
211:
208:
189:
182:
175:
165:
157:boundary layer
149:
105:
102:
53:turbulent flow
45:fluid dynamics
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4889:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4844:
4842:
4833:
4830:
4828:
4825:
4823:
4820:
4818:
4814:
4809:
4807:
4804:
4802:
4799:
4797:
4794:
4792:
4788:
4785:
4782:
4779:
4778:
4774:
4767:
4760:
4755:
4746:
4742:
4738:
4734:
4730:
4726:
4722:
4718:
4714:
4713:
4705:
4699:
4698:
4695:
4691:
4690:
4685:
4681:
4672:
4668:
4664:
4660:
4656:
4652:
4648:
4644:
4640:
4639:
4631:
4625:
4624:
4621:
4617:
4616:
4611:
4607:
4606:
4602:
4601:
4600:
4594:
4588:
4584:
4579:
4572:
4567:
4563:
4561:9780521475143
4557:
4553:
4548:
4544:
4538:
4534:
4529:
4525:
4523:9780521715140
4519:
4515:
4510:
4506:
4500:
4496:
4491:
4487:
4485:9780521457132
4481:
4477:
4473:
4469:
4465:
4461:
4457:
4453:
4449:
4445:
4444:
4443:Physics Today
4436:
4431:
4427:
4423:
4418:
4413:
4409:
4405:
4400:
4399:
4398:
4397:
4390:
4382:
4378:
4374:
4370:
4366:
4362:
4358:
4354:
4353:J. Fluid Mech
4347:
4344:
4339:
4333:
4329:
4322:
4319:
4314:
4308:
4304:
4297:
4294:
4289:
4287:9780521457132
4283:
4279:
4275:
4269:
4266:
4254:
4250:
4246:
4242:
4238:
4231:
4228:
4216:
4212:
4208:
4204:
4200:
4193:
4190:
4185:
4181:
4177:
4173:
4169:
4165:
4161:
4157:
4153:
4149:
4145:
4141:
4134:
4131:
4126:
4122:
4115:
4112:
4107:
4103:
4099:
4093:
4089:
4082:
4079:
4074:
4068:
4064:
4060:
4059:
4051:
4048:
4043:
4042:
4041:New Scientist
4034:
4031:
4026:
4020:
4016:
4012:
4011:
4003:
4000:
3995:
3991:
3987:
3983:
3979:
3975:
3974:
3973:Physics Today
3966:
3959:
3956:
3951:
3945:
3941:
3934:
3932:
3928:
3924:
3920:
3916:
3910:
3908:
3906:
3902:
3897:
3893:
3889:
3885:
3881:
3875:
3871:
3864:
3861:
3849:
3845:
3841:
3837:
3833:
3829:
3825:
3821:
3817:
3813:
3809:
3802:
3799:
3786:
3781:
3777:
3773:
3766:
3763:
3758:
3754:
3750:
3746:
3742:
3738:
3734:
3730:
3723:
3720:
3715:
3711:
3707:
3703:
3699:
3695:
3691:
3687:
3686:
3681:
3677:
3670:
3667:
3662:
3658:
3654:
3650:
3646:
3642:
3638:
3634:
3630:
3626:
3619:
3612:
3609:
3604:
3600:
3596:
3592:
3588:
3584:
3580:
3576:
3572:
3568:
3564:
3560:
3556:
3549:
3546:
3541:
3537:
3533:
3529:
3525:
3521:
3517:
3513:
3508:
3503:
3499:
3495:
3491:
3484:
3481:
3476:
3472:
3467:
3462:
3458:
3454:
3450:
3446:
3442:
3435:
3432:
3427:
3423:
3418:
3413:
3409:
3405:
3401:
3397:
3393:
3386:
3383:
3378:
3374:
3370:
3366:
3362:
3358:
3354:
3350:
3346:
3342:
3338:
3331:
3328:
3323:
3319:
3314:
3309:
3305:
3301:
3297:
3293:
3292:
3287:
3280:
3277:
3272:
3270:9780262200196
3266:
3262:
3258:
3257:
3249:
3247:
3243:
3238:
3234:
3230:
3226:
3222:
3218:
3211:
3208:
3203:
3196:
3193:
3187:
3179:
3175:
3171:
3167:
3161:
3158:
3151:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3095:Swing bowling
3093:
3091:
3088:
3086:
3083:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3040:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3006:
3005:
3000:
2998:
2992:
2991:intermittency
2913:
2907:
2904:
2901:
2897:
2893:
2881:
2866:
2855:
2834:
2833:
2832:
2828:
2807:
2801:
2798:
2794:
2790:
2784:
2778:
2771:
2770:
2769:
2756:
2732:
2722:
2698:
2688:
2685:
2676:
2673:
2662:
2658:
2654:
2642:
2627:
2616:
2595:
2594:
2593:
2591:
2564:
2557:
2546:
2541:
2537:
2529:
2528:
2527:
2507:
2504:
2493:
2486:
2485:
2484:
2465:
2459:
2439:
2419:
2408:
2392:
2386:
2375:
2368:
2367:
2366:
2348:
2343:
2336:
2333:
2327:
2322:
2316:
2312:
2306:
2302:
2295:
2288:
2282:
2278:
2274:
2268:
2264:
2260:
2257:
2250:
2247:
2242:
2238:
2231:
2228:
2223:
2217:
2213:
2209:
2203:
2197:
2190:
2189:
2188:
2185:
2170:
2167:
2162:
2158:
2134:
2126:
2123:
2118:
2114:
2107:
2104:
2099:
2093:
2089:
2085:
2079:
2073:
2066:
2065:
2064:
2057:
2043:
2028:
1985:
1981:
1969:
1963:
1953:
1949:
1945:
1941:
1935:
1931:
1925:
1921:
1916:
1909:
1906:
1896:
1895:
1894:
1891:
1887:
1882:
1877:
1871:
1867:
1863:
1856:
1852:
1832:
1821:
1804:
1797:
1793:
1760:
1749:
1745:
1722:
1721:
1720:
1716:
1712:
1701:
1697:
1693:
1687:
1676:
1672:
1668:
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4862:Chaos theory
4857:Aerodynamics
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4209:: 301â305.
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3676:Chanson, H.
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3110:Velocimetry
1878:< |
1095:fluctuation
760:Horace Lamb
556:anisotropic
358:Dissipation
337:random walk
311:Diffusivity
224:Snow fences
59:changes in
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4841:Categories
4622:: 299â303.
4472:Frisch, U.
4274:Frisch, U.
4127:: 116â124.
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3188:References
1531:Kolmogorov
1065:mean value
756:relativity
729:Richardson
391:wavenumber
333:Richardson
306:turbulent.
257:cardiology
230:turbulent.
125:tip vortex
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3369:0399-1784
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428:⟨
387:mean flow
381:Via this
193:scrubbers
162:form drag
153:golf ball
138:cigarette
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3853:13 March
3791:13 March
3678:(2010).
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370:inviscid
282:Features
129:airplane
127:from an
80:vortices
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4148:Science
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3820:Bibcode
3737:Bibcode
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3661:1975604
3633:Bibcode
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365:eddies
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3373:S2CID
3152:Notes
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