Infrared properties of the energy spectrum in freely decaying isotropic turbulence.

The low wave number expansion of the energy spectrum takes the well known form E(k,t)=E_{2}(t)k^{2}+E_{4}(t)k^{4}+⋯, where the coefficients are weighted integrals against the correlation function C(r,t). We show that expressing E(k,t) in terms of the longitudinal correlation function f(r,t) immediately yields E_{2}(t)=0 by cancellation. We verify that the same result is obtained using the correlation function C(r,t), provided only that f(r,t) falls off faster than r^{-3} at large values of r. As power-law forms are widely studied for the purpose of establishing bounds, we consider the family of model correlations f(r,t)=α_{n}(t)r^{-n}, for positive integer n, at large values of the separation r. We find that for the special case n=3, the relationship connecting f(r,t) and C(r,t) becomes indeterminate, and (exceptionally) E_{2}≠0, but that this solution is unphysical in that the viscous term in the Kármán-Howarth equation vanishes. Lastly, we show that E_{4}(t) is independent of time, without needing to assume the exponential decrease of correlation functions at large distances.

Nonuniversality and Finite Dissipation in Decaying Magnetohydrodynamic Turbulence.

A model equation for the Reynolds number dependence of the dimensionless dissipation rate in freely decaying homogeneous magnetohydrodynamic turbulence in the absence of a mean magnetic field is derived from the real-space energy balance equation, leading to Cϵ=Cϵ,∞+C/R-+O(1/R-(2)), where R- is a generalized Reynolds number. The constant Cϵ,∞ describes the total energy transfer flux. This flux depends on magnetic and cross helicities, because these affect the nonlinear transfer of energy, suggesting that the value of Cϵ,∞ is not universal. Direct numerical simulations were conducted on up to 2048(3) grid points, showing good agreement between data and the model. The model suggests that the magnitude of cosmological-scale magnetic fields is controlled by the values of the vector field correlations. The ideas introduced here can be used to derive similar model equations for other turbulent systems.

Energy transfer and dissipation in forced isotropic turbulence.

A model for the Reynolds-number dependence of the dimensionless dissipation rate C(ɛ) was derived from the dimensionless Kármán-Howarth equation, resulting in C(ɛ)=C(ɛ,∞)+C/R(L)+O(1/R(L)(2)), where R(L) is the integral scale Reynolds number. The coefficients C and C(ɛ,∞) arise from asymptotic expansions of the dimensionless second- and third-order structure functions. This theoretical work was supplemented by direct numerical simulations (DNSs) of forced isotropic turbulence for integral scale Reynolds numbers up to R(L)=5875 (R(λ)=435), which were used to establish that the decay of dimensionless dissipation with increasing Reynolds number took the form of a power law R(L)(n) with exponent value n=-1.000±0.009 and that this decay of C(ɛ) was actually due to the increase in the Taylor surrogate U(3)/L. The model equation was fitted to data from the DNS, which resulted in the value C=18.9±1.3 and in an asymptotic value for C(ɛ) in the infinite Reynolds-number limit of C(ɛ,∞)=0.468±0.006.

Spectral analysis of structure functions and their scaling exponents in forced isotropic turbulence.

The pseudospectral method, in conjunction with a technique for obtaining scaling exponents ζ_{n} from the structure functions S_{n}(r), is presented as an alternative to the extended self-similarity (ESS) method and the use of generalized structure functions. We propose plotting the ratio |S_{n}(r)/S_{3}(r)| against the separation r in accordance with a standard technique for analyzing experimental data. This method differs from the ESS technique, which plots S_{n}(r) against S_{3}(r), with the assumption S_{3}(r)∼r. Using our method for the particular case of S_{2}(r) we obtain the result that the exponent ζ_{2} decreases as the Taylor-Reynolds number increases, with ζ_{2}→0.679±0.013 as R_{λ}→∞. This supports the idea of finite-viscosity corrections to the K41 prediction for S_{2}, and is the opposite of the result obtained by ESS. The pseudospectral method also permits the forcing to be taken into account exactly through the calculation of the energy input in real space from the work spectrum of the stirring forces.

Eulerian field-theoretic closure formalisms for fluid turbulence.

The formalisms of Wyld [Ann. Phys. 14, 143 (1961)] and Martin, Siggia, and Rose (MSR) [Phys. Rev. A 8, 423 (1973)] address the closure problem of a statistical treatment of homogeneous isotropic turbulence (HIT) based on techniques primarily developed for quantum field theory. In the Wyld formalism, there is a well-known double-counting problem, for which an ad hoc solution was suggested by Lee [Ann. Phys. 32, 292 (1965)]. We show how to implement this correction in a more natural way from the basic equations of the formalism. This leads to what we call the Improved Wyld-Lee Renormalized Perturbation Theory. MSR had noted that their formalism had more vertex functions than Wyld's formalism and based on this felt Wyld's formalism was incorrect. However a careful comparison of both formalisms here shows that the Wyld formalism follows a different procedure to that of the MSR formalism and so the treatment of vertex corrections appears in different ways in the two formalisms. Taking that into account, along with clarifications made to both formalisms, we find that they are equivalent and we demonstrate this up to fourth order.

Asymptotic freedom, non-Gaussian perturbation theory, and the application of renormalization group theory to isotropic turbulence.

A model equation with an extended range of asymptotic freedom is used as the basis of a non-Gaussian perturbation expansion in powers of the control parameter of the conditional average. Re-expansion in the local Reynolds number allows the systematic rederivation of an earlier, heuristic theory [W. D. McComb and A. G. Watt, Phys. Rev. Lett. 65, 3281 (1990)].

Eulerian spectral closures for isotropic turbulence using a time-ordered fluctuation-dissipation relation.

Procedures for time-ordering the covariance function, as given in a previous paper [K. Kiyani and W. D. McComb, Phys. Rev. E 70, 066303 (2004)], are extended and used to show that the response function associated at second order with the Kraichnan-Wyld perturbation series can be determined by a local (in wave number) energy balance. These time-ordering procedures also allow the two-time formulation to be reduced to time-independent form by means of exponential approximations and it is verified that the response equation does not have an infrared divergence at infinite Reynolds number. Last, single-time Markovianized closure equations (stated in our previous paper) are derived and shown to be compatible with the Kolmogorov distribution without the need to introduce an ad hoc constant.

Galilean invariance and vertex renormalization in turbulence theory.

The Navier-Stokes equation is invariant under Galilean transformation of the instantaneous velocity field. However, the total velocity transformation is effected by transformation of the mean velocity alone. For a constant mean velocity, the equation of motion for the fluctuating velocity is automatically Galilean invariant in the comoving frame, and vertex renormalization is not constrained by this symmetry.

Time-ordered fluctuation-dissipation relation for incompressible isotropic turbulence.

The Kraichnan-Wyld perturbation expansion is used to justify the introduction of a renormalized response function connecting two-point covariances at different times. The resulting relationship was specialized by a suitable choice of initial conditions to the form of a fluctuation-dissipation relation (FDR). This was further developed to reconcile the time symmetry of the covariance with the causality of the response by the introduction of time ordering along with a counterterm. This formulation provides a solution to an old problem, that of representing the time dependence of the covariance and response by exponential forms. We show that the derivative (with respect to difference time) of the covariance now vanishes at the origin. This allows one to study the relationships between two-time spectral closures and time-independent theories such as the self-consistent field theory of Edwards or the more recent renormalization group approaches. We also show that the renormalized response function is transitive with respect to intermediate times and report a different Langevin equation model for turbulence. We note the potential value of this time-ordering procedure in all applications of the FDR.

Spectral intermode coupling in a model of isotropic turbulence.

We investigate the nonlinear coupling between the so-called explicit modes, identified with wave numbers k such that 0< or =k< or =k(c), and implicit modes, defined such that k(c)< or =k< or =k(max). Here k(c) is an arbitrarily chosen cutoff wave number and k(max) is the ultraviolet cutoff as determined by viscous damping. The stresses arising from the nonlinearity in the Navier-Stokes equations are categorized as "implicit-implicit" (or "Reynolds") and "explicit-implicit" (or "cross"). These arise from dynamic coupling between different regions of wave number space. Their respective effects on momentum, kinetic energy, and energy flux are assessed. The analysis is based on a model system comprising the Navier-Stokes equations and the Edwards-Fokker-Planck energy equation [S. F. Edwards, J. Fluid Mech. 18, 239 (1964)] which is known to retain all the symmetries of homogeneous, isotropic turbulence. The Reynolds stress is found to be responsible for long-range energy transfers. It can be represented by an effective viscosity and is mainly determined by dynamical friction. The cross term is more complicated, involving both diffusive and frictional effects. For long-range coupling it can be expressed as a modification of the effective viscosity, while for short-range coupling it may be modeled on the assumption that implicit scales are slaved to explicit scales. Thus, both the random and coherent aspects of intermode coupling in turbulent flows are relevant in the cross term. The imposition of a continuity requirement on energy transfer leads to a new parametrization that represents the effect of absent modes in a truncated spectral simulation, and takes into account the phase-coupling (coherent) effects, as well as the usual viscositylike (random) effects.

Solution of functional equations and reduction of dimension in the local energy transfer theory of incompressible, three-dimensional turbulence.

It is shown that the set of integrodifferential and algebraic functional equations of the local energy transfer theory may be considerably reduced in dimension for the case of isotropic turbulence. This is achieved without restricting the solution space. The basis for this is a complete analytical solution to the functional equations Q(k;t,t('))=H(k;t,t('))Q(k;t('),t(')) and H(k;t,s)H(k;s,t('))=H(k;t,t(')). The solution is proved to depend only on a single function straight phi(k;t) solely determining Q and H. Hence the dimension of both the dependent and the independent variables is reduced by one. From the latter, the corresponding two integrodifferential equations are lowered to a single integrodifferential equation for straight phi(k;t), extended by an integral side condition on the k dependence of straight phi(k;t). In the limit nu-->0, a partial solution to the reduced set of equations is presented in the Appendix.

Renormalized expression for the turbulent energy dissipation rate.

Conditional elimination of degrees of freedom is shown to lead to an exact expression for the rate of turbulent energy dissipation in terms of a renormalized viscosity and a correction. The correction is neglected on the basis of a previous hypothesis [W. D. McComb and C. Johnston, J. Phys. A 33, L15 (2000)] that there is a range of parameters for which a quasistochastic estimate is a good approximation to the exact conditional average. This hypothesis was tested by a perturbative calculation to second order in the local Reynolds number, and the Kolmogorov prefactor (taken as a measure of the renormalized dissipation rate) was found to reach a fixed point which was insensitive to initial values of the kinematic viscosity and to values of the spatial rescaling factor h in the range 0.4< or =h< or =0.8.