“Every mathematician believes that he is ahead of the others. The reason none state this belief in public is because they are intelligent people.”
A.N. Kolmogorov
Introduction
The Kolmogorov theory of turbulence, which was published in a series of three small publications in 1941, is one of the most widely mentioned but least understood turbulence theories. The original publications are difficult to understand due to unclear assumptions and non-trivial outcomes. On the other hand, these results have stood the “test of time,” and should be regarded as correct within the limits of the assumptions used to acquire them.
We will begin by stating Kolmogorov’s “universality assumptions,” and then introduce three related hypotheses employed by Frisch in: “Turbulence: The Legacy of A. N. Kolmogorov” to prove the celebrated 4/5 law. We then present the three main results provided by Kolmogorov in 1941 (known as the K41 theory), and discuss some of the consequences of these.
Kolmogorov’s “universality” assumptions
Before presenting individual claims based on these assumptions, it is useful to quickly discuss the broader context in which they were produced. Solutions to the Navier-Stokes equations have symmetries at low Reynolds numbers, which break as the Reynolds number increases. These symmetries are eventually recovered statistically as the flow becomes turbulent. Within this paradigm:
Kolmogorov’s first universality assumption (sometimes dubbed his “first similarity hypothesis”) is as follows:
At very high, but not infinite, Reynolds number, all of the small-scale statistical properties are uniquely and universally determined by the length scale ℓ, the mean dissipation rate (per unit mass) ε and the viscosity ν.
Similarly, the second universality assumption is as follows:
In the limit of infinite Reynolds number, all small-scale statistical properties are uniquely and universally determined by the length scale ℓ and the mean dissipation rate ε.
Before proceeding, it is important to define terminology and notation. First, it is vital to highlight that both of these assertions refer to “small-scale” statistical features. Thus, we must first define what we mean by “small scale.” This is not the same as the fluctuating quantities in a Reynolds decomposition (though classical turbulence theorists sometimes equate them).
To begin, keep in mind that in a Reynolds decomposition, the lowest moments of variable quantities, such as average u′ is zero. In this scenario, we should associate small-scale values with the high-pass filtered portion of an N.-S. solution, i.e., the high-wavenumber/high-frequency components. Thus, in the context of our Hilbert-space decomposition (which I shall not go into at this stage).
A slightly different, but related (essentially equivalent) interpretation of small scale is to associate this with length scales that are very much smaller than the integral scale.
We may now examine the specifics of Kolmogorov’s two similarity assumptions. The first concerns finite-Reynolds number behavior, which indicates that statistics will be uniquely (and universally) established by length and dissipation scales, as well as viscosity. The second hypothesis addresses the Re → ∞ limit. In this instance, we anticipate v → 0, Thus, universal statistical behaviors should no longer be dependent on v. We point out that the predicted universality is what generates the majority of issues with these assumptions. It should also be noted that the flow attributes homogeneity and local isotropy are widely viewed as critical to the application of the Kolmogorov theory, yet neither is expressly needed in the similarity hypothesis. However, as we’ll see later, these are required to show the 4/5 law.
Hypotheses put forward by Uriel Frisch
Frisch’s treatment of the Kolmogorov theory is the result of many meticulous and extensive reviews and analyses by a number of scholars between 1980 and 1990, culminating in a far more easily understandable portrayal of this theory. We’ll include a shortened and significantly restructured version of this in the post. The beginning point is Frisch’s three hypotheses, which may be summarized as follows:
Hypothesis 1:
At small scales and away from limits, the Re → ∞ limit statistically restores all conceivable symmetries of NSE, which are often violated by physical turbulence processes.
Hypothesis 2:
Under the same assumptions as before, turbulent flow is self-similar at small scales; that is, it has a unique scaling exponent h such that:
The increments ℓ and λℓ are modest compared to the integral scale.
Hypothesis 3:
Similarly to Hypothesis 1, turbulent flow has a limited, non vanishing mean rate of dissipation per unit mass (ε).
It is crucial to analyze some of the specifics and consequences of these hypotheses, as well as to compare them to Kolmogorov’s two original universality assumptions. We should reiterate that tiny scales are connected with length scales that are far smaller than the integral scale.
As previously noted, we can interpret small-scale homogeneity (the criterion for which is not expressly specified) in the context of velocity increments:
δu(x, ℓ) = u(x + ℓ) − u(x)
In particular, for stationary flows, the statistics of these increments must be invariant under arbitrary translations r.
<δu(x + r, ℓ)> = <δu(x, ℓ)> (<…> is some generic averaging procedure)
Similarly, isotropy of velocity increases requires invariance under arbitrary rotations of ℓ and δu.
Hypotheses Explained
Most importantly, when put together, they are utilized to acquire the implications of the 2/3 law as well as proof of the 4/5 law. Furthermore, we observe that they do not coincide with the Kolmogorov universality assumptions. We see, as Frisch observed, that Frisch does not apply the first of the Kolmogorov assumptions; specifically, all of the hypotheses employed by Frisch include the Re → ∞ limit.
Another issue to note is that all three of Frisch’s hypotheses assume statistical recovery of Navier-Stokes symmetries, but the Kolmogorov assumptions do not explicitly indicate this. It is, however, required in the proofs of the Kolmogorov findings, but it is unclear if Kolmogorov meant this merely because experimental evidence of this feature was not very strong at the time of his K41 studies.
The third part of Frisch’s first hypothesis is the need to be “away from boundaries”. Again, this was not clearly stated in the Kolmogorov assumptions, and it is the first indication that homogeneity or isotropy would be invoked – neither of which can be anticipated to hold in the vicinity of solid borders.
Finally, Hypothesis 3 agrees with experimental findings and has substantial mathematical implications. To understand the nature of them, consider the definition of dissipation rate:
ε = 2νIISII2
where S is the strain rate tensor, with components of the type:
Sij = 1/2 { ∂ui/∂xj + ∂uj/∂xi)
Clearly, as Re → ∞, as required by all of Frisch’s assumptions (and Kolmogorov’s second assumption),
v → 0. If ε remains finite, some derivatives of U must become unbounded. However, we underline that ε is a mean dissipation rate. The average is produced spatially in the framework of Frisch’s arguments.
From a mathematical perspective, finite ε in the Re → ∞ limit is compatible with recent NSE theories, regardless of the notion of ε. In particular, such theories allow for unboundedness of U on sets of zero measure, and an averaging procedure (essentially integration) can disregard such sets without altering the average value.
Principal results of the K41 theory
As previously stated, the K41 theory yields three primary results: the limited dissipation rate, the 2/3 law, and the 4/5 law. The first two are based on dimensional analysis and experimental data to establish indeterminate constants, with the second reflects a modification of Kolmogorov’s K62 theory to align with current experimental findings. The third one, on the other hand, is a mathematically proven result of the Navier-Stokes equation.
Finite Dissipation Rate as Re → ∞
Frisch uses the physical example of drag caused by flow past a bluff object to support his argument for limited dissipation rate as Re → ∞. Wind tunnel experiments typically produce plots of drag coefficient, CD (defined as FD/(1/2ρU2)A), where FD is drag force and A is area over which it acts, versus Reynolds number, as shown in the figure below.
Plots like these highlight numerous key themes. The first is that for low Reynolds number, CD relies quite significantly on Re, and up to Re ∼ 10, it is approximately inversely proportional. It should be observed that while such flows are laminar and stable, some of the broken symmetries that occur on the route to turbulence will have already occurred. For Re > 102, CD does not rely on Re until the “drag crisis” hits at 105-106. However, once Re grows beyond this point, CD remains virtually constant. We may recall that the Moody diagram of skin friction coefficient vs. Re for pipe and duct flows illustrates a comparable pattern.
At this point, it is useful to link the drag coefficient to the energy dissipation rate since the drag coefficient in general relies on Re, resulting in nonzero dissipation as Re → ∞.
This will be accomplished simply by dimensional analysis. The power (W˙) required to convey a flow past an object is proportional to its speed (U) and size of the object – a length scale (L):
where F is force; M is mass, and T is time. W˙ is proportional to mass times rate of viscous dissipation and may be connected to drag coefficient as follows:
remember that by definition:
Equally:
This leads directly to:
The dissipation rate is often expressed in terms of unit mass, with ε ∼ L2/T3. Therefore, we write:
ε equals power per unit mass. Remember from the preceding figure (CD Vs. Re) that at high Re, when a flow should be turbulent, CD is independent of Re, and so ε is independent of viscosity as Re → ∞. This gives a heuristic demonstration of the following:
Finite dissipation: When all control parameters in a turbulent flow experiment are maintained constant except for viscosity, the energy dissipation per unit mass (de/dt ∼ ε) follows a finite positive limit.
The 2/3 Law
We begin by expressing Kolmogorov’s 2/3 law, as provided by Frisch. We provide a form of a specific formula that corresponds to the assertion and demonstrate how the well-known k−5/3 energy spectrum is directly derived from it.
Kolmogorov’s 2/3 law:
In turbulent flow with a high Reynolds number, the mean-square velocity increment <(δu(ℓ))2> between two places separated by a distance ℓ behaves essentially as the two-thirds power of the distance.
Several considerations are appropriate for interpreting this outcome. First, (δu)2 is a (squared) magnitude of the vector variable δu, which may not make sense unless the turbulence is isotropic. However, it appears that the same scaling applies to individual components of velocity increases, regardless of isotropy.
The length (ℓ) is more important. The theories stated in previous subsections suggest that ℓ should be significantly less than the integral scale. Indeed, the 2/3 law holds precisely in the inertial subrange of the energy spectrum, which corresponds roughly to Taylor microscale lengths.
Finally, the “very-high Reynolds number” part of the wording of this law is critical. For low Re (but still high enough to be turbulent), the range of ℓ where the 2/3 rule holds is quite limited; nevertheless, as Re grows, this range expands.
We may now offer a calculation based on the 2/3 rule. To do this, we first understand that <(δu(ℓ))> is the second-order structure function of u specified in general.
In doing so, we will aid in defining the structure function. Structure functions of order p are defined, somewhat heuristically, by:
where r is a vector pointing between two close places of “measurement” of the quantity u; r is the magnitude of r; and < · > signifies any appropriate average, conducted across all samples with the same value of r. This is the common structural function for scalar numbers. We suppressed temporal notation, but observed that, in general, Sp will depend on time, unless < · > incorporates temporal averaging.
The second-order structure function of u, established in my earlier mathematical explanation of a Structure Function, is represented as <(δu(ℓ))2>, meaning:
As previously stated, S2 is closely connected to kinetic energy and “energy” in general. As a reminder, the inertial subrange is defined in Def. 1.77 as the range of scales over which viscosity is insignificant; viscous forces are dominated by inertial forces. In in particular, this relates to Re being sufficiently high that for low to moderate wavenumbers, advective effects entirely outweigh diffusive effects in the Navier-Stokes equations.
This is consistent with Kolmogorov’s second universality assumption, which states that turbulence statistics are solely determined by the length scale ℓ and the dissipation rate ε. Because S2 is related with kinetic energy, its general dimensions are L2/T2. We know that ε ∼ L2/T3, and our goal is to identify a combination of ℓ and ε with energy dimensions. The only possible combination of such variables to construct S2 is then:
Where C serves as a universal constant.
The 2/3 law has two obvious ramifications. The first is the form of the turbulence kinetic energy spectrum, and the second is the exponent h in Frisch’s Hypothesis 2 on the self-similarity of Navier-Stokes turbulence. Each of these will be discussed in the next two subsections:
The k−5/3 energy spectrum
We utilize S2(ℓ) to obtain the well-known Kolmogorov k−5/3 inertial-range scaling of the turbulent energy spectrum. To achieve this, we first observe that S2 in the equation above is stated in physical space, which we shall denote as E to remind us that it represents energy. We write then:
But what is required is a function of wavenumber k. Clearly, we should be able to define an incremental component of E in Fourier space using dE = E(k)dk. The small-scale total (cumulative) energy for all wavenumbers above an arbitrary (but chosen in the inertial subrange) one k must be:
Frisch shows, under relatively generic circumstances, that E(k) must fulfill:
Then substitute this into the integral for E(k):
Now, note that the range of n provided above assures that:
Since k is arbitrary, we may write E(k) = CnE(k)k for any k. Finally, we notice that up to a scaling constant (depending on the basis functions used for a Fourier representation), k = 1/ℓ, this is based on our former result :
So, we may be assured to obtain:
And rearrangement yields Kolmogorov’s well-known finding for the inertial range energy spectrum:
where CK represents the Kolmogorov constant. Experimental measurements indicate that this constant is not genuinely constant, with values ranging from unity to two. Kolmogorov did not present the finding in his 1941 papers. Rather, it was originally delivered by Obukhov in 1941.
Furthermore, because the Russian technical literature was not generally available in the West until much later, this identical finding was independently uncovered by numerous other scholars throughout the mid to late 1940s.
The scaling exponent h
Another consequence can be simply obtained from the 2/3 law. Remember that Frisch’s second hypothesis stated that velocity increments are self-similar on small scales and that there exists a unique scaling exponent h, which was not established at the time. The 2/3 law allows us to calculate the value of this exponent.
In Frisch second hypothesis we saw that under the some assumptions turbulent flow is self-similar at small scales; that is, it has a unique scaling exponent h such that:
which we write in terms of the second-order structural function S2:
And, based on the unique scaling exponent h, it must be:
As a result, the unique scaling exponent is h = 1/3. We note that, while this derivation appears to need isotropy, Frisch points out that an alternate derivation does not.
The 4/5 Law
The 4/5 law, while less frequently mentioned and potentially less immediately practical than the 2/3 law, is, in many ways, the most significant of the K41 discoveries. This is true for two primary reasons.
First, it is obtained directly from the N.-S. equations (although with a number of restrictive assumptions), and unlike the 2/3 rule, it contains no adjustable constants—it is the only accurate answer for the Navier-Stokes equations at high Re. Second, and just as important, it has been validated in countless laboratory tests. The 4/5 law states that third-order structure functions of velocity increments grow linearly with separation distance ℓ.
Kolmogorov’s 4/5 law
In the limit of infinite Reynolds number, the third-order longitudinal structure function of homogeneous isotropic turbulence, evaluated for increments ℓ small compared with the integral scale, is given in terms of the mean energy dissipation rate per unit mass as:
Derivation of this particular solution is long and nontrivial, hence we will not present it here (you may find it in: Turbulence, the Legacy of A. N. Kolmogorov, by Frisch – chapter 6 – good luck with that). We should highlight, however, that the derivation offered in the Frisch essay requires three conditions: homogeneity, isotropy, and fully developed turbulence. We have not previously highlighted the last of them, so we will make a few extra points here. We will begin by using the definition that states that fully developed turbulence occurs at high Re, which may be interpreted simply as implying that statistical quantifications of the flow do not change with flow direction. According to this definition, the terminology high-Re turbulence, in which statistics no longer change in the flow direction. In comparison to the other two conditions, needing fully-developed turbulence simply adds the assertion of high Re (as stated in the 4/5 law).
As a result, all of our previous ideas have entailed this in some way, and the homogeneity and isotropy criteria have essentially covered it up. Frisch used particular information to prove the above:
are the following:
i) Energy is only input on large scales.
ii) N.-S. solutions tend to a stationary state at large times.
iii) The mean dissipation rate, ε, stays finite as Re → ∞.
Furthermore, the self likeness of Frisch’s Hypothesis 2 mentioned before is not applied.
By combining assumptions and their consequences, the following third-order ODE for S3 driven by ε may be obtained:
For the longitudinal situation, the following equation has a unique, exact solution:
Conclusions
This essay was a humble and simple attempt at providing a quick overview of the Kolmogorov K41 theory. This statistical method of the N.-S. equations differs from Reynolds averaging by using a Hilbert-space decomposition for flow variables. Furthermore, the Kolmogorov results do not involve any modeling; nonetheless, relatively stringent assumptions must be made in order to achieve them.
There are three main results:
i) finite dissipation rate as Re → ∞
ii) the 2/3 law
iii) the 4/5 law
The first two are primarily empirical, although they may also be determined via dimensional analysis. The second step yields two results: the inertial subrange’s k−5/3 energy spectrum and a universal scaling constant (h = 1/3) related with self-similarity on small scales.
The 4/5 law, the final crucial conclusion, may be simply derived from the N.-S. equations under the high-Re homogeneous isotropic turbulence hypothesis. It is precise (no variable constants), nontrivial, and consistent with experimental evidence. It is the only such result known at this time.
Comments on “The Kolmogorov Theory of Turbulence”