W D McComb

Theory of turbulence

This review is concerned with modern theoretical approaches to turbulence, in which the problem can be seen as a branch of statistical field theory, and where the treatment has been strongly influenced by analogies with the quantum many-body problem. The dominant themes treated are the development (since the 1950s) of renormalized perturbation theories (RPT) and, more recently, of renormalization group (RG) methods. As fluid dynamics is rarely part of the physics curriculum, in section 1 we introduce some background concepts in fluid dynamics, followed by a skeleton treatment of the phenomenology of turbulence in section 2, taking flow through a straight pipe or a plane channel as a representative example. In section 3, the general statistical formulation of the problem is given, leading to a moment closure problem, which is analogous to the well known BBGKY hierarchy, and to the Kolmogorov -5/3 power law, which is a consequence of dimensional analysis. In section 4, we show how RPT have been used to tackle the moment closure problem, distinguishing between those which are compatible with the Kolmogorov spectrum and those which are not. In section 5, we discuss the use of RG to reduce the number of degrees of freedom in the numerical simulation of the turbulent equations of motion, while giving a clear statement of the technical problems which lie in the way of doing this. Lastly, the theories are discussed in section 6, in terms of their ability to meet the stated goals, as assessed by numerical computation and comparison with experiment.

A local energy-transfer theory of isotropic turbulence

It is shown that a nonlinear integral equation for turbulent energy transport may be reinterpreted in terms of a Heisenberg-type effective viscosity. A new equation is derived for the effective viscosity. This is found to permit general expansions of the integral kernels, in powers of wavenumber ratios, leading to local (differential) equations for the energy spectrum and effective viscosity. It is found that these equations yield the Kolmogoroff distribution as the inertial-range solution, and that the numerical predictions agree quite well with experimental results. The final equations are similar to equations recently derived by Nakano (1972), and the relationship between the two theories is discussed.

A theory of time-dependent, isotropic turbulence

Second-order equations are derived for the turbulent velocity-field correlation and propagator functions. Is is argued that the concept of the propagator may be more fully exploited as the relationship between eddies at successive times, than as the relationship between the velocity and the arbitrary stirring forces. The resulting equations differ from the direct-interaction approximation of Kraichnan by the presence of additional diffusive-type terms in the equation for the propagator. A generalisation of the diagram technique due to Wyld is used to analyse the approximation procedure to fourth order. It is shown that many higher-order terms in the perturbation series are represented in the truncated equations.

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.

Effective viscosity due to local turbulence interactions near the cutoff wavenumber in a constrained numerical simulation

If the largest resolved wavenumber in a numerical simulation of isotropic turbulence is too small, then it is well known that the energy spectrum will depart from its expected monotonic decrease with increasing wavenumber and will instead begin to increase. We show that an operational method is capable of modifying the instantaneous velocity field such that the unphysical features of the spectrum are suppressed. When the effect on the constrained simulation is interpreted in terms of an effective viscosity, this agrees well with the usual result obtained by comparison with a fully resolved direct numerical simulation, thus directly establishing the localness in wavenumber of the relevant interactions.

Renormalisation group calculation of the eddy viscosity for isotropic turbulence

Iterative averaging, which has previously been shown to be equivalent to the renormalisation group, was applied directly to the Navier-Stokes equations in k space. The better frequency separation achieved by this technique (when compared to convention RG filtering and averaging operations) was shown to be due to the order in which operations were carried out, rather than to any difference in underlying assumptions. An expression for the eddy viscosity for wavenumber scales k or=kc. For k<<kc, the eddy viscosity became constant. For k to kc the eddy viscosity had a gentle roll-off, which may be compared with the cusp at k=kc found with renormalized perturbation theory.

The inertial-range spectrum from a local energy-transfer theory of isotropic turbulence

It has been shown previously that a nonlinear integral equation for turbulent energy transport could be re-interpreted in terms of a Heisenberg-type effective viscosity. The resulting integral equations were used to derive local (differential) equations for the energy spectrum and effective viscosity. The author considers the integral formulation of the theory and restrict his attention to the inertial range of wavenumbers. It is shown that the equations yield the Kolmogoroff distribution, in the limit of infinite. Reynolds numbers. The Kolmogoroff spectrum constant is calculated and found to be alpha =2.5 which is marginally outside the experimental range. It is argued that this result is sufficient encouragement to develop a time-dependent form of the theory, which would allow a more decisive comparison with experiment.

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 Wylds formalism and based on this felt Wylds 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.

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.

Self-organization and transition to turbulence in isotropic fluid motion driven by negative damping at low wavenumbers

We observe a symmetry-breaking transition from a turbulent to a self-organized state in direct numerical simulation of the Navier–Stokes equation at very low Reynolds number. In this self-organized state the kinetic energy is contained only in modes at the lowest resolved wavenumber, the skewness vanishes, and visualization of the flows shows a lack of small-scale structure, with the vorticity and velocity vectors becoming aligned (a Beltrami flow). S Online supplementary data available from stacks.iop.org/jpa/48/25FT01/ mmedia