Math 2.0

Fluid mechanics Turbulence Proposition

Kármán–Howarth equation

Builds on Navier–Stokes equationsTurbulenceViscosityTurbulence closure problemEnergy cascade

Statement

Let 𝐮(x,t)\mathbf{u}(x, t) be a statistically homogeneous and isotropic velocity field in ℝ3\R^3 that solves the incompressible Navier–Stokes equations with kinematic viscosity ν\nu and no forcing. Write ⟨⋅⟩\langle \cdot \rangle for the ensemble average, u1u_1 for the first component of 𝐮\mathbf{u}, 𝐞1\mathbf{e}_1 for the first unit vector, and u′2=⟨u12⟩u'^2 = \langle u_1^2 \rangle. Define the longitudinal correlation functions f(r,t)f(r, t) and k(r,t)k(r, t) for r>0r > 0 by u′2f(r,t)=⟨u1(x,t) u1(x+r𝐞1,t)⟩,u′3k(r,t)=⟨u1(x,t)2 u1(x+r𝐞1,t)⟩.u'^2 f(r, t) = \langle u_1(x, t)\, u_1(x + r\mathbf{e}_1, t) \rangle, \qquad u'^3 k(r, t) = \langle u_1(x, t)^2\, u_1(x + r\mathbf{e}_1, t) \rangle. Then ∂∂t(u′2f)=u′3(∂k∂r+4kr)+2νu′2(∂2f∂r2+4r∂f∂r).\frac{\partial}{\partial t}\bigl(u'^2 f\bigr) = u'^3 \left(\frac{\partial k}{\partial r} + \frac{4k}{r}\right) + 2\nu u'^2 \left(\frac{\partial^2 f}{\partial r^2} + \frac{4}{r} \frac{\partial f}{\partial r}\right).

Remarks

The proof takes the Navier–Stokes equations at the two points xx and x+r𝐞1x + r\mathbf{e}_1, multiplies them and averages. Isotropy then writes every two- and three-point tensor in terms of ff and kk. Incompressibility removes the pressure terms, because in isotropic turbulence the pressure–velocity correlation vanishes. The equation is not closed: it is one equation for the two unknowns ff and kk, an instance of the turbulence closure problem. At r=0r = 0 it gives ddtu′2=10νu′2f′′(0)=−23ε\frac{d}{dt} u'^2 = 10 \nu u'^2 f''(0) = -\frac{2}{3}\varepsilon, where ε\varepsilon is the mean dissipation rate. In the inertial range, where the viscous term and the time derivative of ff are negligible, integration gives Kolmogorov's four-fifths law ⟨δu(r)3⟩=−45εr\langle \delta u(r)^3 \rangle = -\frac{4}{5} \varepsilon r for the longitudinal increment δu(r)\delta u(r).

Going deeper

For a student

The equation is the main exact dynamical relation of homogeneous isotropic turbulence. Written for the second and third order structure functions, it yields the four-fifths law, one of the few exact results in the theory. Its Fourier transform is the equation for the energy spectrum, in which the triple correlation becomes the energy transfer term of the energy cascade.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Energy cascade, concept, no evidence levelEnergy cascadeconceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptReynolds decomposition, concept, no evidence levelReynolds decompositionconceptReynolds stress, concept, no evidence levelReynolds stressconceptReynolds-averaged Navier–Stokes equations, concept, no evidence levelReynolds-averaged Nav…conceptTurbulence, concept, no evidence levelTurbulenceconceptViscosity, concept, no evidence levelViscosityconceptTurbulence closure problem, problem, no evidence levelTurbulence closure pr…problemKármán–Howarth equation, statement, no evidence levelKármán–Howarth equati…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Energy cascade concept
  2. Navier–Stokes equations concept
  3. Reynolds decomposition concept
  4. Reynolds stress concept
  5. Reynolds-averaged Navier–Stokes equations concept
  6. Turbulence concept
  7. Viscosity concept
  8. Turbulence closure problem problem
  9. Kármán–Howarth equation statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n%E2%80%93Howarth_equation
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Kindstatement, proposition
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AboutTurbulence
UsesNavier–Stokes equations, Turbulence, Viscosity, Turbulence closure problem, Energy cascade
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Changed3 November 2026

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