Scaleless algebraic energy spectra for the incompressible Navier–Stokes equation; relation to other nonlinear problems

Abstract

We use the Wiener expansion, a.k.a. the Wiener–Hermite (W–H) expansion, which is based on weighted integrals of the white noise process (also called Feynman path integrals) to represent turbulence, near to Gaussian. Fully developed turbulence-fluctuation Reynolds (Re) numbers greater than 10 4 are treated. The energy transfer function T( k, q) (transfer from wavenumber q to k) is shown, and for the equilibrium 5/3 law energy spectrum, it is seen that the dependence on k 0, the energy range wavenumber, drops out so that dimensionally the energy spectrum must be algebraic, 5/3 in the inertial subrange (ISR) at equilibrium, for driven turbulence. An interesting consequence of the k 0 independence is the loss of the last evidence of the large scale drive of the turbulence. The form of the driving force has no influence. The same conclusion follows for decaying turbulence, with the slight modification that rate of energy loss (viscous decay) is a (slow) function of time. The integral over the transfer (the net energy gain/loss) converges independently of k v, the viscous cutoff so that the turbulence is scaleless (see Barenblatt and Zel'dovich [Barenblatt, G.I., Zel'dovich, Ya.B., 1972. Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech. 4, 285–312.] for a full discussion of the influence of this property). The scaleless character is a necessary ingredient of fractals. Fractals of course have the property that they preserve their qualitative character at any magnification, and this characteristic would be upset by the presence of a scale external to the process. The transfer from wavenumbers of order k dominates that from wave numbers of order k 0 because of the following. First, the volume of phase space available for transfer, of order the large wavenumber k, is enormous compared with the small region in the vicinity of the energy spectrum peak, k 0. Second, the pressure term in conjunction with incompressibility yields geometric coefficients (in the transfer) which are proportional to the square of the (small) wavenumber, q, in the region k 0. The effect of moderate compressibility (which will be discussed) is to destroy this second small behavior of the transfer in the energy range at a threshold Mach number; then a different ISR spectrum can develop, as is known to be the case. This opens the way for processes with other phenomena, e.g., buoyancy effects, to yield other spectra (than the 5/3). We introduce a compressible flow model by relieving incompressibility and calculate the resulting transfer. There is a critical (fluctuation) Reynolds number, Re c′=2.5(Ma′) −8. If Re′<Re c′, there is only the compressible spectrum, −6/3. If Re′ is larger, then there is a critical wavenumber k c=2 k 0(Ma′) −6, forming a knee. For k< k c, the spectrum is −5/3, and for k> k c, it will be −6/3. There are geophysical (and of course astrophysical) processes where this condition is fulfilled. The discussion here gives, to the author's knowledge, the first explanation of the 5/3 spectrum which goes beyond dimensional analysis (with additional ad-hoc assumptions concerning appropriate parameters, specifically an assumed independence of k 0).