Week 9

THE ENERGY SPECTRUM FUNCTION
1
2 ϕii (k)
is a function dependent on a 3D vector argument. Can we
simplify this dependence, by first asking about the magnitude of k,
then worry about the remaining details?
A surface integral within a spherical shell of radius k ≡ |k|:

¨
1
E(k) = 2 ϕii (k) dS(k)
(see., e.g. T&L Eq. 8.15; Eq. 6.188 of Pope)

This is often simply called the (3D) energy spectrum. If the tur-
bulence is isotropic, since energy is a scalar, energy distribution in
wavenumber space should be independent of the direction of k: hence
in terms of k would suffice.
We get the entire wavenumber space by covering 0 < k < ∞

ˆ ∞ ˆ ∞ ¨
1 1
E(k) dk = 2 ϕii (k) dS(k) dk = 2 ui ui
0 0
E(k) has dimensions of energy per unit mass per wavenumber: [L3][T ]−2.
Since most of the TKE is in the large scales, in general we expect

E(k) to to be high at low k, and low at high k. The shape of the
spectrum is thus of great interest.
In experiments, and in flows that are not isotropic, often only a 1D

spectrum (function of one wavenumber component) is available.
Formal relations between 1D and 3D spectra are known (T&L Eq.

8.1.13, Pope Eq. 6.217). Useful for comparisons of theory and DNS
with experiment, if local isotropy is likely.
1
KOLMOGOROV SIMILARITY HYPOTHESES
A question out of both curiosity and practicality is: are there any
similarities between E(k) in different turbulent flows? And likewise
for spatial structure of fluctuating velocity field in physical space.
Kolmogorov 1941 (in two papers written in 1941, English reprint

1991) provided two important hypotheses.
First Hypothesis: Dissipation range
At sufficiently high Reynolds number, the small scales are (locally)

isotropic, and have an universal structure independent of the details
of the large scales, characterized only by viscosity (ν) and the energy
dissipation rate (ϵ).
Consider the relative velocity between two points in place, at a dis-

tance r ≪ ℓ apart. The hypothesis is saying

(∆r u)2 r
= f only (r ≪ ℓ)
ν aϵb η
Dimensional analysis gives a = 1/2, b = 1/2. Here f (·) is to be an
universal function, with two (longitudinal and transverse) versions.
The functional form of f (·) is not precisely known, except in the

“ballistic” limit (r ≪ η), where a Taylor-series expansion shows:
∆r u ∼ r(∂u/∂x) ⇒ (∆r u)2 ∝ r2
in case of a longitudinal velocity increment. For transverse incre-
ments we also have r2 behavior but a different proportionality factor.
2
For the spectrum, the same hypothesis leads to
E(k)
= f (kη) only (k ≫ 1/ℓ)
ν 5/4ϵ1/4
kη is the “Kolmogorov-scaled wavenumber”.
The supposed universality of the small scales has great implications.
• it may be easier to work towards developing a model for the small

scales, that is hopefully applicable to various high-Re flows.
• then we only need to compute the large scales directly.
Hence large-eddy simulations (LES), with sub-grid scale (SGS) mod-

eling. Presumably, not anticipated by Kolmogorov in 1941.
Second Hypothesis: Inertial range
If the Reynolds number is high enough (higher than for the first
hypothesis) we can find a range of scales such that
η≪r≪ℓ; or 1/ℓ ≪ k ≪ 1/η
In physical space, this means both r/η ≫ 1 and r/ℓ ≪ 1. Then, at
scale size r, neither viscosity nor large-scales geometry have a signif-
icant effect. But ϵ still relevant since it is also linked to interaction
between large scales and small scales.
(∆r u)2 = f (r, ϵ) only, ⇒ ∝ r2/3
Similar arguments for the spectrum give
E(k) ∝ ϵ2/3k −5/3
which is a famous result, known as the “five-thirds law”.
3
Collection of data on 1D energy spectra scaled by Kolmogorov vari-
ables (Pope, p. 235), up to the mid 1990s.
4
“Compensated” 3D energy spectrum in Kolmogorov scaling, from
DNS of isotropic turbulence. Increasing times: lines A to K.
5
K41: THE EVIDENCE (EXPT. + DNS)
A lot of research (thousands of papers) has been performed to test

the various predictions of the K41 similarity hypotheses.
Generally, the first hypothesis is well-verified for most practical pur-

poses, at least at the second-moment level.
Second hypothesis is harder, because most experiments and DNS

face the challenge of obtaining a sufficiently wide range of scales.
In experiments, reliance on local isotropy relations for the mean dis-

sipation rate contributes to increased uncertainty.
We have not considered (say) the following aspects here:
Intermittency: in fact, ϵ fluctuates. At the simplest level, ⟨ϵn⟩ =

̸ ⟨ϵ⟩n
for any n ̸= 1. Especially problematic at high orders.
Extreme events: they are reflected in higher-order moments. They

also show, to say (ν 3/ϵ)1/4 is the “smallest scale” is misleading.
Can the theory be applied to fluctuations of temperature fields?
Turbulent dispersion: a wide range of time scales is important. But

T /τη increases with Re slower than ℓ/η does.
To look up the literature: one good way is to find out (re: the Web
of Science) from citations to the following review article
Sreenivasan & Antonia, “The phenomenology of small-scale turbu-

lence”. Annu. Rev. Fluid Mech. 29, 435-472 (1997).
6

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THE ENERGY SPECTRUM FUNCTION

A surface integral within a spherical shell of radius k ≡ |k|:

(see., e.g. T&L Eq. 8.15; Eq. 6.188 of Pope)

We get the entire wavenumber space by covering 0 < k < ∞

Since most of the TKE is in the large scales, in general we expect

In experiments, and in flows that are not isotropic, often only a 1D

Formal relations between 1D and 3D spectra are known (T&L Eq.

Kolmogorov 1941 (in two papers written in 1941, English reprint

First Hypothesis: Dissipation range

At sufficiently high Reynolds number, the small scales are (locally)

Consider the relative velocity between two points in place, at a dis-

The functional form of f (·) is not precisely known, except in the

The supposed universality of the small scales has great implications.

• it may be easier to work towards developing a model for the small

Hence large-eddy simulations (LES), with sub-grid scale (SGS) mod-

Second Hypothesis: Inertial range

A lot of research (thousands of papers) has been performed to test

Generally, the first hypothesis is well-verified for most practical pur-

Second hypothesis is harder, because most experiments and DNS

In experiments, reliance on local isotropy relations for the mean dis-

We have not considered (say) the following aspects here:

Intermittency: in fact, ϵ fluctuates. At the simplest level, ⟨ϵn⟩ =

Extreme events: they are reflected in higher-order moments. They

Can the theory be applied to fluctuations of temperature fields?

Turbulent dispersion: a wide range of time scales is important. But

Sreenivasan & Antonia, “The phenomenology of small-scale turbu-

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