Energy cascade

The energy spectrum of turbulence, E(k), is related to the mean turbulence kinetic energy per unit mass as[2]

${\displaystyle {\frac {1}{2}}\left({\overline {u_{i}u_{i}}}\right)=\int _{0}^{\infty }E(k)\;dk,}$

where u_i are the components of the fluctuating velocity, the overbar denotes an ensemble average, summation over i is implied, 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.

Since diffusion goes as the Laplacian of velocity, the dissipation rate may be written in terms of the energy spectrum as:

${\displaystyle \varepsilon =2\nu \int _{0}^{\infty }k^{2}E(k)\;dk,}$

with ν the kinematic viscosity of the fluid. From this equation, it may again be observed that dissipation is mainly associated with high wavenumbers (small eddies) even though kinetic energy is associated mainly with lower wavenumbers (large eddies).

The transfer of energy from the low wavenumbers to the high wavenumbers is the energy cascade. This transfer brings turbulence 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, Kolmogorov's hypotheses lead to the following universal form for the energy spectrum:

${\displaystyle E(k)=C\varepsilon ^{2/3}k^{-5/3}.}$

An extensive body of experimental evidence supports this result, over a vast range of conditions. Experimentally, the value C = 1.5 is observed.[2]

The result ${\displaystyle E(k)\sim k^{-5/3}}$ was first stated by independently by Alexander Obukhov in 1941.[3] Obukhov's result is equivalent to a Fourier transform of Kolmogorov's 1941 result[4] for the turbulent structure function.[5]

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):

${\displaystyle {\overline {p^{2}}}=\int _{0}^{\infty }\pi (k)\;dk.}$

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

${\displaystyle \pi (k)=\alpha \rho ^{2}\varepsilon ^{4/3}k^{-7/3},}$

where ρ is the fluid density, and α = 1.32 C² = 2.97.[6] 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.[7]

Pressure fluctuations below the free surface of a liquid can drive fluctuating displacements of the liquid surface, which at small wavelengths are modulated by surface tension. 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:

${\displaystyle {\overline {\delta ^{2}}}=\int _{0}^{\infty }G(k)\;dk.}$

A three dimensional form of the pressure spectrum may be combined with the Young–Laplace equation to show that:[8]

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

Experimental observation of this k^−19/3 law has been obtained by optical measurements of the surface of turbulent free liquid jets.[8]