Open AccessOn the theory of statistical and isotropic turbulence
Author(s) -
W. Heisenberg
Publication year - 1948
Publication title -
proceedings of the royal society of london. series a, mathematical and physical sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.814
H-Index - 135
eISSN - 2053-9169
pISSN - 0080-4630
DOI - 10.1098/rspa.1948.0127
Subject(s) - turbulence , physics , isotropy , reynolds number , mathematical physics , statistical theory , energy (signal processing) , dissipation , distribution (mathematics) , statistical physics , quantum mechanics , mathematics , mathematical analysis , statistics , mechanics
The statistical theory of turbulence, initiated by Taylor (1935) and v. Kármán & Howarth (1938), has recently been developed so far that a satisfactory explanation of the spectral distribution of energy among the turbulent eddies can be given. In fact Kolmogoroff (1941a, b ) and independently Onsager (1945) and v. Weizsaecker (1948) have introduced a similarity hypothesis, which allows a determination of the spectrum for eddies with large Reynolds numbers, and the author (Heisenberg 1948) has extended these calculations to include those frequency components which have small Reynolds numbers. Since the distribution of energy among the largest eddies must be a geometrical and not a statistical problem, one may say that the statistical part of the spectrum is now well understood. Recently Batchelor & Townsend (1947, 1948a, b ) have studied the decay of turbulent motion caused by a mesh grating in a wind tunnel, and the following discussions will apply the statistical theory to this problem. For the calculations the notation of Heisenberg (1948) will be used. IfpF(k)dk denotes the energy contained between the wave numbersk andk +dk , the following equation for the dissipation of energy was given (Heisenberg 1948, equation (13)):Sk = {μ + pk ∫(F (k '') /k ''3 )dk " } ∫k 0 2F (k ')k '2 dk '.