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Spectra in the Inertial Subrange of Turbulent Flow
The largest motions, or eddies, of turbulence contain most of the kinetic energy, whereas the smallest eddies are responsible for the viscous dissipation of turbulent kinetic energy. Kolmogorov recognized that when these scales are well separated, the intermediate range of length scales would be statistically isotropic and that its characteristics would depend only on the rate at which kinetic energy is dissipated.

Dissipation is the frictional conversion of mechanical energy to thermal energy. The dissipation rate, ε, may be written down in terms of the fluctuating rates of strain in the turbulent flow and the fluid's kinematic viscosity, ν. It has dimensions of energy per unit mass per second. The production of turbulent kinetic energy at the large scales of motion is accompanied by the dissipation of this energy at the small scales.

Energy Spectrum of Turbulence
The energy spectrum of turbulence, E(k), is related to the mean turbulence kinetic energy per unit mass as


 * $$\frac{1}{2} \left( \overline{u_i u_i} \right) = \int_0^\infty E(k) dk $$

where  ui is the magnitude of the fluctuating component of the velocity, the overbar denotes an ensemble average, and k is the wavenumber. The energy spectrum, E(k), thus represents the contribution to turbulence kinetic energy by wavenumbers from k to k+dk. The largest eddies have low wavenumber, and the small eddies have high wavenumbers.

The dissipation rate may also be written in terms of the energy spectrum as:


 * $$\epsilon = 2\nu \int_0^\infty k^2E(k) dk $$

From this equation, it may again be observed that dissipation is mainly associated with high wavenumbers (small eddies) whereas kinetic energy is associated mainly with lower wavenumbers (large eddies).

Energy Spectrum in the Inertial Subrange
The transfer of energy from the low wavenumbers to the high wavenumbers is the energy cascade. This transfer brings turbulent kinetic energy from the large scales to the small scales, at which viscous friction dissipates it. In the intermediate range of scales, the so-called inertial subrange, Kolomogorov's hypotheses led to the following universal form for the energy spectrum:


 * $$E(k) = C \epsilon^{2/3} k^{-5/3} $$

An extensive body of experimental evidence supports this result, over a vast range of conditions. Experimentally, the value C=1.5 is observed.

Spectrum of Pressure Fluctuations
The pressure fluctuations in a turbulent flow may be similarly characterized. The mean-square pressure fluctuation in a turbulent flow may be represented by a pressure spectrum, π(k):


 * $$ \overline{p^2} = \int_0^\infty \pi(k) dk $$

For the case of turbulence with no mean velocity gradient (isotropic turbulence), the spectrum in the inertial subrange is given by


 * $$\pi(k) = \alpha \rho^2\epsilon^{4/3} k^{-7/3} $$

where ρ is the fluid density, and α = 1.32 C2 = 2.97. A mean-flow velocity gradient (shear flow) creates an additional, additive contribution to the inertial subrange pressure spectrum which varies as k-11/3; but the k-7/3 behavior is dominant at higher wavenumbers.

Spectrum of Capillary Disturbances at a Free Liquid Surface
Turbulent pressure fluctuations below the free surface of a liquid can drive fluctuating displacements of the liquid surface. This free-surface-turbulence interaction may also be characterized by a wavenumber spectrum. If δ is the instantaneous displacement of the surface from its average position, the mean squared displacement may be represented with a displacement spectrum G(k) as:


 * $$ \overline{\delta^2} = \int_0^\infty G(k) dk $$

A three dimensional form of the pressure spectrum may be combined with the Young-Laplace equation to show that :


 * $$G(k) \propto k^{-19/3} $$

Experimental observation of this k-19/3 law has been obtained by optical measurements of the surface of turbulent free liquid jets.