Kármán–Howarth equation

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In isotropic turbulence the Kármán–Howarth equation (after Theodore von Kármán and Leslie Howarth 1938[1]), which is derived from the Navier–Stokes equations, is used to describe the evolution of non-dimensional longitudinal autocorrelation[2][3][4].

Mathematical description[edit]

Consider a two-point correlation for homogeneous turbulence

For isotropic turbulence, this correlation tensor can be expressed in terms of two scalar functions, using the invariant theory of full rotation group, first derived by Howard P. Robertson in 1940[5],

where is the root mean square turbulent velocity and are turbulent velocity in all three directions. Here, is the longitudinal correlation and is the lateral correlation of velocity at two different points. From continuity equation, we have

Thus uniquely determines the two-point correlation function. Theodore von Kármán and Leslie Howarth derived the evolution equation for from Navier–Stokes equation as

where uniquely determines the triple correlation tensor


See also[edit]


  1. ^ De Karman, T., & Howarth, L. (1938). On the statistical theory of isotropic turbulence. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 164(917), 192–215.
  2. ^ Monin, A. S., & Yaglom, A. M. (2013). Statistical fluid mechanics, volume II: Mechanics of turbulence (Vol. 2). Courier Corporation.
  3. ^ Batchelor, G. K. (1953). The theory of homogeneous turbulence. Cambridge university press.
  4. ^ Spiegel, E. A. (Ed.). (2010). The Theory of Turbulence: Subrahmanyan Chandrasekhar's 1954 Lectures (Vol. 810). Springer.
  5. ^ Robertson, H. P. (1940, April). The invariant theory of isotropic turbulence. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 36, No. 2, pp. 209–223). Cambridge University Press.