The Kármán–Howarth Relation and the Four-Fifths Law
This appendix proves Equation (35.74) of Remark 35.86 in Fluid Dynamics:
in the inertial range of homogeneous isotropic turbulence, with no adjustable constant. The chapter calls this the one nontrivial statement about turbulence that is derived from the Navier–Stokes equations rather than assumed and tested, and sets it deliberately against the dimensional argument of Phenomenon 35.84, which is not derived. The whole weight of that claim rests on the pages below, and the section is written accordingly: everything is proved, the symmetry hypotheses are used explicitly at the four places where they are needed, and Remark A63.10 states at the end exactly how much the word exact is buying. It is buying less than it sounds, and more than anything else in the subject.
Setting
Let \(\vect{v}(\vect{x},t)\) be a random velocity field on \(\R^{3}\) obeying the incompressible Navier–Stokes equations Equation (35.32) with kinematic viscosity \(\nu\) in \(\mathrm{m}^{2}/\mathrm{s}\) and a body force \(\vect{F}\) per unit mass, and let \(\avg{\,\cdot\,}\) denote the ensemble average. The turbulence is assumed
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homogeneous: every joint moment is invariant under translations of \(\R^{3}\);
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isotropic in the strong sense: every joint moment is invariant under the full orthogonal group \(\Ogrp(3)\), rotations and reflections alike;
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statistically steady: every moment is independent of \(t\);
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forced at large scales only: \(\vect{F}\) is a random field with the same symmetries whose correlation with \(\vect{v}\) varies on a single length \(L\), the integral scale.
Write \(\vect{v}:=\vect{v}(\vect{x},t)\), \(\vect{v}':=\vect{v}(\vect{x}+\vect{r},t)\), \(\delta\vect{v}:=\vect{v}'-\vect{v}\), \(r=\abs{\vect{r}}\), \(\vect{n}=\vect{r}/r\), and \(\delta u_{\parallel}:=\delta\vect{v}\cdot\vect{n}\) for the longitudinal increment. Define
and let
be the mean dissipation rate per unit mass, in \(\mathrm{W}/\mathrm{kg}\)\(=\)\(\mathrm{m}^{2}/\mathrm{s}^{3}\). Rests on Theorem 35.37 and Definition 15.4.
Hypothesis (2) is stronger than rotational invariance alone and the difference matters: a turbulence with a mean helicity is invariant under \(\SO(3)\) but not under reflection, and the pseudo-tensor terms that reflection invariance removes below would then survive.
Isotropic tensor functions of one vector
Everything in this section rests on one algebraic lemma, which is proved here rather than quoted because the whole reduction from a field theory to two ordinary differential equations is contained in it.
Let \(T\) be a Cartesian tensor field on \(\R^{3}\setminus\set{0}\), depending on the single vector \(\vect{r}\), and covariant under \(\Ogrp(3)\): \(T_{i\ldots}(\mathsf{O}\vect{r}) =\mathsf{O}_{ii'}\cdots T_{i'\ldots}(\vect{r})\) for every orthogonal \(\mathsf{O}\). Then, with \(\vect{n}=\vect{r}/r\),
with scalar functions of \(r\) alone. A constant tensor with these symmetries and no \(\vect{r}\) dependence is zero at odd rank and a multiple of \(\delta_{ij}\) at rank two. Rests on Theorem 9.133 and Definition 18.26.
Derives Lemma A63.2. Fix \(r>0\). Covariance under the rotations that move \(\vect{r}\) shows that the components in a frame adapted to \(\vect{r}\) depend on \(r\) only, so it suffices to work at one point. Choose axes with \(\vect{n}=\hat{\vect{e}}_{3}\) and let Latin indices \(a,b,c\) run over the transverse values \(1,2\). The subgroup of \(\Ogrp(3)\) fixing \(\vect{r}\) is the group \(\Ogrp(2)\) acting on the transverse plane, and each block of components must be an invariant tensor of that group.
At rank one: \(T_{a}\) is an invariant vector of \(\Ogrp(2)\), hence zero (the element \(-\mathsf{I}\) of \(\Ogrp(2)\) reverses it), and \(T_{3}\) is free. This is Equation (A63.4).
At rank two, symmetric: \(T_{3a}\) is again an invariant transverse vector, hence zero; \(T_{ab}\) is an invariant symmetric two-dimensional tensor, hence \(\phi_{2}\delta_{ab}\), since the traceless part carries the spin-two representation of \(\SO(2)\), which has no invariant; and \(T_{33}=\phi_{1}\) is free. Writing \(\delta_{ab}=\delta_{ij}-n_{i}n_{j}\) in covariant form gives Equation (A63.5).
At rank three, symmetric in the first two indices, four blocks occur. \(T_{33a}\) is an invariant transverse vector: zero. \(T_{ab3}\) is an invariant symmetric transverse two-tensor: \(\mathcal{B}\,\delta_{ab}\). \(T_{3ab}=T_{a3b}\) is an invariant transverse two-tensor, not required symmetric, hence again \(\mathcal{C}\,\delta_{ab}\) — the antisymmetric \(\epsilon_{ab}\) is invariant under \(\SO(2)\) but changes sign under a reflection of the plane, and hypothesis (2) of Definition A63.1 excludes it. \(T_{abc}\) is an invariant transverse tensor of odd rank, and \(-\mathsf{I}\in\SO(2)\) reverses it: zero. Assembling with \(\delta_{ab}=\delta_{ij}-n_{i}n_{j}\) and renaming the free coefficient of \(n_{i}n_{j}n_{k}\) gives Equation (A63.6).
For a constant tensor the same argument applies at every \(\vect{r}\) simultaneously, so the surviving coefficients must be independent of the direction \(\vect{n}\); at odd rank every term in Equations (A63.4) and (A63.6) carries an odd number of factors \(n\), and no such expression is direction independent unless it vanishes.
∎Write \(B_{ij}(\vect{r}):=\avg{v_{i}v'_{j}}\). Then there is a single scalar \(f\), with \(f(0)=1\), such that
and \(R(r)=B_{ii}=u^{2}\left(3f+rf'\right)\); moreover \(\avg{v_{i}v_{j}v'_{k}}\) is determined by a single scalar \(k\), namely \(\avg{v_{\parallel}^{2}v'_{\parallel}}=u^{3}k(r)\), and \(\avg{v_{i}v_{j}v_{k}}=0\). Rests on Lemma A63.2 and Corollary 35.13.
Derives Corollary A63.3. \(B_{ij}\) is symmetric under the simultaneous exchange \(i\leftrightarrow j\), \(\vect{r}\rightarrow-\vect{r}\), and by Equation (A63.5) it takes the displayed form with \(f=B_{\parallel\parallel}/u^{2}\) and \(g=B_{\perp\perp}/u^{2}\), the longitudinal and transverse correlations; \(f(0)=1\) by the definition of \(u^{2}\). Incompressibility acts through \(\pp B_{ij}/\pp r_{j} =\avg{v_{i}\,\pp'_{j}v'_{j}}=0\). Writing \(h:=\left(f-g\right)/r^{2}\) so that the first term is \(u^{2}h\,r_{i}r_{j}\),
using \(\pp_{j}r=r_{j}/r\) and \(\pp_{j}\left(r_{i}r_{j}\right)=4r_{i}\). Their sum vanishes for all \(\vect{r}\), so \(h'r+4h+g'/r=0\); substituting \(h\) and clearing \(r^{2}\) leaves \(rf'+2\left(f-g\right)=0\), which is the second relation in Equation (A63.7). Contracting the first relation gives \(R=u^{2}\left(f-g+3g\right)=u^{2}\left(f+2g\right) =u^{2}\left(3f+rf'\right)\).
For the triple correlation put \(B_{ij,k}:=\avg{v_{i}v_{j}v'_{k}}\); it is symmetric in \(i,j\), so Equation (A63.6) applies with coefficients \(\mathcal{A},\mathcal{B},\mathcal{C}\). Incompressibility at the far point gives \(\pp B_{ij,k}/\pp r_{k}=\avg{v_{i}v_{j}\pp'_{k}v'_{k}}=0\); carrying out the differentiation as above,
and since \(n_{i}n_{j}\) and \(\delta_{ij}\) are independent the two coefficients must vanish separately:
Two equations for three functions leave one free function, which may be taken to be \(u^{3}k:=B_{\parallel\parallel,\parallel} =\mathcal{A}+\mathcal{B}+2\mathcal{C}\). Finally \(\avg{v_{i}v_{j}v_{k}}\) is a constant isotropic tensor of rank three, so it vanishes by the last clause of Lemma A63.2.
∎The pressure drops out
This is the step that makes the result exact rather than a closure, and it is the one a reader should doubt until it is shown, since a correlation involving the pressure is otherwise as intractable as anything in the subject.
Under Definition A63.1, \(\avg{p\,v'_{i}}=0\) identically, and likewise \(\avg{\abs{\vect{v}}^{2}v'_{i}}=0\). Rests on Lemma A63.2 and Corollary 35.13.
Derives Lemma A63.4. Both are isotropic vector functions of \(\vect{r}\), so by Equation (A63.4) each equals \(\phi(r)n_{i}\) for some scalar \(\phi\). Both are divergence free in \(\vect{r}\): differentiating with respect to \(r_{i}\) is differentiating the far point, and
because the field is incompressible (Corollary 35.13) and the near point is held fixed. For a radial field \(\phi(r)n_{i}\) the divergence is \(r^{-2}\left(r^{2}\phi\right)'\), so \(r^{2}\phi\) is constant; the constant is zero because \(\phi\) is bounded at the origin, both quantities being finite one-point moments there. Hence \(\phi\equiv0\).
∎The two-point energy equation
Under Definition A63.1, and without any closure hypothesis whatever,
and \(\mathcal{F}(0)=2\varepsilon\) in a steady state. Rests on Theorem 35.37 and Lemma A63.4.
Derives Theorem A63.5. By homogeneity every correlation depends on \(\vect{r}=\vect{x}'-\vect{x}\) alone, so \(\pp/\pp x_{i}=-\pp/\pp r_{i}\) and \(\pp/\pp x'_{i}=+\pp/\pp r_{i}\) when applied to one. Write the Navier–Stokes equation Equation (35.32) at \(\vect{x}\) and at \(\vect{x}'\), multiply the first by \(v'_{i}\) and the second by \(v_{i}\), add and average:
The pressure bracket is \(\pp_{i}\avg{p\,v'_{i}}-\pp_{i}\avg{v_{i}p'}\) after moving the derivatives outside the averages, and both terms vanish by Lemma A63.4 — the second is the first with \(\vect{r}\) reversed. The viscous bracket is \(2\nu\nabla_{r}^{2}R\), since each Laplacian acts on one factor and becomes \(\nabla_{r}^{2}\) on the correlation.
For the nonlinear terms use incompressibility to write \(v_{j}\pp_{j}v_{i}=\pp_{j}\left(v_{j}v_{i}\right)\); then
so the nonlinear contribution is \(\pp_{r_{j}}\bigl[\avg{v_{j}v_{i}v'_{i}} -\avg{v_{i}v'_{j}v'_{i}}\bigr]\). It remains to identify the bracket. Expand
The terms \(\avg{\abs{\vect{v}'}^{2}v'_{j}}\) and \(\avg{\abs{\vect{v}}^{2}v_{j}}\) are equal by homogeneity and cancel; \(\avg{\abs{\vect{v}}^{2}v'_{j}}\) and \(\avg{\abs{\vect{v}'}^{2}v_{j}}\) vanish by Lemma A63.4; and what is left is
which is twice the bracket. This gives Equation (A63.9).
At \(\vect{r}=\vect{0}\): the divergence term vanishes, because \(\avg{\abs{\delta\vect{v}}^{2}\delta\vect{v}}=O(r^{3})\) there; the viscous term is \(2\nu\nabla_{r}^{2}R|_{0}=2\nu\avg{v_{i}\nabla^{2}v_{i}} =-2\nu\avg{\pp_{j}v_{i}\pp_{j}v_{i}}=-2\varepsilon\) by Equation (A63.3) and homogeneity; and \(\pp_{t}R(0)=0\) in a steady state. Hence \(\mathcal{F}(0)=2\varepsilon\), which is the statement that in a steady state the force injects energy at exactly the rate viscosity destroys it.
∎For unforced (decaying) homogeneous isotropic turbulence, Equation (A63.9) is equivalent to
with \(f\) and \(k\) the scalars of Corollary A63.3. Rests on Theorem A63.5 and Corollary A63.3.
Derives Proposition A63.6. Write \(T(r)\) for the scalar in \(\avg{\abs{\delta\vect{v}}^{2}\delta v_{j}}=T(r)n_{j}\), legitimate by Equation (A63.4); then \(\vect{\nabla}_{r}\cdot\avg{\abs{\delta\vect{v}}^{2}\delta\vect{v}} =r^{-2}\left(r^{2}T\right)'\) and \(\nabla_{r}^{2}R=r^{-2}\left(r^{2}R'\right)'\), so Equation (A63.9) unforced reads
By Corollary A63.3, \(R=u^{2}\left(3f+rf'\right)=u^{2}\left(r^{3}f\right)'/r^{2}\), and Structure functions below gives \(T=u^{3}\left(8k+2rk'\right)\). Apply the operator \(\mathcal{L}[\Phi]:=r^{-2}\pp_{r}\left(r^{3}\Phi\right)\) to each term of Equation (A63.10). On the left, \(\mathcal{L}\bigl[\pp_{t}(u^{2}f)\bigr]=\pp_{t}R\). On the right,
the last step by direct differentiation of \(r^{2}T=u^{3}\left(8r^{2}k+2r^{3}k'\right)\); and
since \(r^{2}R'=u^{2}\left(4r^{2}f'+r^{3}f''\right)\). So Equation (A63.10) maps term by term onto Equation (A63.11); and \(\mathcal{L}\) is injective on functions regular at the origin, since \(\mathcal{L}[\Phi]=0\) forces \(r^{3}\Phi\) constant, hence \(\Phi\propto r^{-3}\), hence \(\Phi=0\).
∎Equation (A63.10) is the Kármán–Howarth relation, obtained by von Kármán and Howarth in 1938 (Proceedings of the Royal Society of London A 164, 192–215). That paper is not in the bibliography of this treatise, and the statement is not being taken from it: everything above is derived here from Equation (35.32) and Definition A63.1, so the attribution is historical and nothing rests on it. The equation is exact and it does not close: one equation relates the two unknown functions \(f\) and \(k\), and nothing determines \(k\). What Kolmogorov saw is that in one limit the undetermined function drops out.
Structure functions
With \(S_{ijk}:=\avg{\delta v_{i}\delta v_{j}\delta v_{k}}\) and \(T(r)\) as above,
Rests on Corollary A63.3 and Lemma A63.2.
Derives Lemma A63.7. Expand \(S_{ijk}\) in the eight products of \(v\) and \(v'\). The two single-point terms \(\avg{v_{i}v_{j}v_{k}}\) and \(\avg{v'_{i}v'_{j}v'_{k}}\) vanish by Corollary A63.3. Of the remaining six, three have two factors at \(\vect{x}\) and three have two at \(\vect{x}+\vect{r}\); using homogeneity to write the latter as \(B\) evaluated at \(-\vect{r}\), and the fact that \(B_{ij,k}\) is odd — every term of Equation (A63.6) carries an odd number of factors \(\vect{n}\) — the two groups add rather than cancel:
the second equality by adding the three copies of Equation (A63.6) with indices permuted. Write \(W:=\mathcal{A}+\mathcal{B}+2\mathcal{C}=u^{3}k\) and \(V:=\mathcal{B}+2\mathcal{C}\). Contracting with \(n_{i}n_{j}n_{k}\) gives \(S_{3}=6\mathcal{A}+6V=6W\), the first relation in Equation (A63.12); contracting on \(i=j\) and then with \(n_{k}\) gives \(T=S_{iik}n_{k}=6\mathcal{A}+10V=S_{3}+4V\).
It remains to express \(V\) through \(W\). Adding the two relations Equation (A63.8) and using \(\mathcal{A}+\mathcal{B}=W-2\mathcal{C}\),
and substituting \(\mathcal{B}=V-2\mathcal{C}\) into the second of Equation (A63.8) gives \(V'+2V/r=2\mathcal{C}'+2\mathcal{C}/r\). With the expression just obtained for \(\mathcal{C}\), the right-hand side is \(\left[rW''+4W'+2W/r\right]/2\), so
the last step because \(\left(r^{3}W'\right)'=3r^{2}W'+r^{3}W''\) and \(\left(r^{2}W\right)'=2rW+r^{2}W'\). Integrating and discarding the \(r^{-2}\) homogeneous solution by regularity at the origin, \(V=\tfrac{1}{2}\left(rW\right)'\). Hence \(T=S_{3}+2\left(rW\right)'=S_{3}+\tfrac{1}{3}\left(rS_{3}\right)'\), and in terms of \(k\), \(T=6u^{3}k+2u^{3}\left(k+rk'\right) =u^{3}\left(8k+2rk'\right)\).
∎\(D(r):=\avg{\abs{\delta\vect{v}}^{2}}=2\left[R(0)-R(r)\right] =3S_{2}+rS_{2}'\). Rests on Corollary A63.3.
Derives Lemma A63.8. \(\avg{\abs{\delta\vect{v}}^{2}} =\avg{\abs{\vect{v}}^{2}}+\avg{\abs{\vect{v}'}^{2}} -2\avg{\vect{v}\cdot\vect{v}'}\), and the first two are \(R(0)=3u^{2}\) each by homogeneity. For the second equality, Equation (A63.7) gives \(S_{2}=2u^{2}\left(1-f\right)\) and the transverse structure function \(\avg{\left(\delta u_{\perp}\right)^{2}}=2u^{2}\left(1-g\right) =S_{2}+\tfrac{1}{2}rS_{2}'\), using \(g=f+\tfrac{1}{2}rf'\); summing one longitudinal and two transverse contributions gives \(D=3S_{2}+rS_{2}'\).
∎The inertial range and the four-fifths law
Under Definition A63.1, for every separation \(r\ll L\),
and consequently, in the inertial range \(\eta\ll r\ll L\) where the viscous term is negligible,
which is Equation (35.74). Here \(\eta=\left(\nu^{3}/\varepsilon\right)^{1/4}\) is the Kolmogorov length Equation (35.72). Rests on Theorem A63.5, Lemma A63.7 and Lemma A63.8.
Derives Theorem A63.9. In a steady state the left-hand side of Equation (A63.9) vanishes, and in radial form
Multiply by \(r^{2}\) and integrate from \(0\) to \(r\). The first term gives \(\tfrac{1}{2}r^{2}T\), the boundary term at the origin vanishing because \(T=O(r^{3})\). The second gives \(2\nu r^{2}R'(r)\), likewise. For the third, \(\mathcal{F}\) varies on the scale \(L\) by hypothesis (4), so for \(r\ll L\) it may be replaced by \(\mathcal{F}(0)=2\varepsilon\), giving \(\tfrac{2}{3}\varepsilon r^{3}\). Hence
By Lemma A63.8, \(R=R(0)-\tfrac{1}{2}D\), so \(R'=-\tfrac{1}{2}D'\) and
This is already exact and already the whole content; the rest is translation. Substituting \(T=\left(r^{4}S_{3}\right)'/3r^{3}\) — which is Equation (A63.12) rewritten, since \(\left(r^{4}S_{3}\right)'=r^{3}\left(4S_{3}+rS_{3}'\right) =3r^{3}\left[S_{3}+\tfrac{1}{3}\left(rS_{3}\right)'\right]\) — and \(D=\left(r^{3}S_{2}\right)'/r^{2}\) from Lemma A63.8, one verifies that Equation (A63.13) solves Equation (A63.15): with \(S_{3}=6\nu S_{2}'-\tfrac{4}{5}\varepsilon r\),
because \(D'=\left(3S_{2}+rS_{2}'\right)'=4S_{2}'+rS_{2}''\). Since Equation (A63.15) determines \(S_{3}\) given \(S_{2}\) — it is a first-order linear equation with the regularity condition \(S_{3}=O(r^{3})\) at the origin fixing the integration constant — Equation (A63.13) is the solution.
For Equation (A63.14): the viscous term \(6\nu S_{2}'\) is of order \(\nu S_{2}/r\), and in the inertial range \(S_{2}\sim\left(\varepsilon r\right)^{2/3}\), so the ratio of the viscous to the inertial term is of order \(\nu\varepsilon^{2/3}r^{-4/3}/\varepsilon r =\left(\eta/r\right)^{4/3}\), which is negligible for \(r\gg\eta\); and \(r\ll L\) was used already. Dropping it leaves Equation (A63.14).
∎The direction of the inequality is the physics. \(S_{3}<0\) says that the odd moment of the longitudinal increment is negative: the fluid at \(\vect{x}+\vect{r}\) is, on average, moving towards the fluid at \(\vect{x}\) more strongly than away from it, in the cubic sense. That asymmetry is the cascade, and Equation (A63.14) measures it against \(\varepsilon\) with no free constant. Reversed sign would be an inverse cascade, which is what two-dimensional turbulence does and three-dimensional turbulence does not.
The chapter calls Equation (35.74) the one exact nontrivial result about turbulence, and the reader is owed the bill.
Homogeneity was used at every step where \(\pp/\pp x_{i}=-\pp/\pp r_{i}\) was applied, and in Lemma A63.8. No real flow is homogeneous — there is always a boundary and always a forcing region — so the statement is about an idealized ensemble that laboratory turbulence approximates locally.
Isotropy, including reflection, was used three times: in Lemma A63.2 to reduce tensor fields to scalars, in Lemma A63.4 to kill the pressure term, and in Lemma A63.7. Anisotropy at the large scales survives into the inertial range more persistently than K41 supposes, and the four-fifths law is correspondingly harder to observe cleanly than its exactness suggests.
Stationarity set \(\pp_{t}R=0\). For decaying turbulence the same calculation retains a term in \(\pp_{t}\) and Equation (A63.14) acquires a correction that is small only when the decay time exceeds the eddy turnover time at separation \(r\).
The ordering of the limits. Two neglects were made, \(r\ll L\) and \(r\gg\eta\), and they are compatible only when \(L/\eta\gg1\), that is at large Reynolds number by Equation (35.73). The four-fifths law is a statement about an asymptotic regime, and the inertial range at attainable Reynolds numbers is short.
Finite dissipation as \(\nu\rightarrow0\). The whole argument treats \(\varepsilon\) as a fixed quantity while the viscous term is dropped. That \(\varepsilon\) tends to a non-zero limit as \(\nu\rightarrow0\) — the dissipation anomaly — is an assumption, not a theorem; it is strongly supported by measurement and it is what makes the cascade picture consistent, but it is precisely the kind of statement whose proof would require knowing that the equations have the solutions Remark 35.48 says are not known to exist.
What survives all five qualifications is still remarkable, and the comparison with Phenomenon 35.84 is the point of stating it: the exponent \(-5/3\) follows from hypotheses that measurement has falsified in detail — the intermittency corrections are real — while the coefficient \(-4/5\) follows from the equations themselves, and measurement has found no correction to it. Remark 35.87 sets the whole position out, and Experiment: Fluid Flow and Turbulence collects the measurements a future theory will be asked to reproduce.
Equation (A63.14) is Kolmogorov's, and the paper that contains it is the companion of [Kolmogorov:1941] — the dissipation paper, Doklady Akademii Nauk SSSR 32 (1941) 16–18, recorded in the bibliographic note of that entry. The entry [Kolmogorov:1941] itself is the local-structure paper, which states the similarity hypotheses used for Equation (35.71). Remark 35.86 already warns that the K41 attributions must be read carefully, the spectral form being Obukhov's [Obukhov:1941]; the same care applies within Kolmogorov's own output, and the two 1941 papers are not interchangeable.
The Kármán–Howarth Relation and the Four-Fifths Law discharges Equation (35.74) inside Remark 35.86 of Fluid Dynamics, and with it the chapter's claim that exactly one nontrivial consequence of Equation (35.32) about turbulence is derived rather than assumed. The reader returning there should carry back two things this section supplies and the chapter cannot: that the Kármán–Howarth relation Equation (A63.10) is exact but does not close, one equation in two unknown functions, which is the closure problem of Remark 35.83 in its sharpest form; and that the four-fifths law survives the non-closure only because the undetermined function is exactly the one the inertial-range limit eliminates. Everything else in Section 35.9 is dimensional analysis and measurement, as Remark 35.87 says.