Showing papers in "Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences in 1991"

A Re-Examination of the Basic Postulates of Thermomechanics

Abstract: This paper is mainly concerned with a re-examination of the basic postulates and the consequent procedure for the construction of the constitutive equations of material behaviour in thermomechanics. However, the implication of the basic postulates and the significance of the related procedure for the development of the constitutive equations is also illustrated in some detail in the context of flow of heat in a rigid solid with particular reference to the propagation of thermal waves at finite speed. More specifically, after briefly examining the nature of the basic equations of motion for a system of particles within the scope of the classical newtonian mechanics, the basic postulates of the purely mechanical theory for a continuum (including its specialization for a rigid body) is re-examined. This includes some differences from the usual procedure on the subject. Next, thermal variables are introduced and after observing a useful analogy between the thermal and mechanical variables, a discussion of a theory of heat (or a purely thermal theory) is provided which differs from the usual development in the classical thermodynamics. A detailed application of the latter development is then made to the problem of heat flow in a stationary rigid solid using several different and well-motivated constitutive equations. Special cases of these include linearized theories of the classical heat flow by conduction and of heat flow transmitted as thermal waves. The remainder of the paper is concerned with thermal mechanical theory of deformable media along with discussions of a number of related issues on the subject.

On the collision of a droplet with a solid surface

Abstract: The collision dynamics of a liquid droplet on a solid metallic surface were studied using a flash photographic method. The intent was to provide clear images of the droplet structure during the deformation process. The ambient pressure (0.101 MPa), surface material (polished stainless steel), initial droplet diameter (about 1.5 mm), liquid (n-heptane) and impact Weber number (43) were fixed. The primary parameter was the surface temperature, which ranged from 24 degrees C to above the Leidenfrost temperature of the liquid. Experiments were also performed on a droplet impacting a surface on which there existed a liquid film created by deposition of a prior droplet. The evolution of wetted area and spreading rate, both of a droplet on a stainless steel surface and of a droplet spreading over a thin liquid film, were found to be independent of surface temperature during the early period of impact. This result was attributed to negligible surface tension and viscous effects, and in consequence the measurements made during the early period of the impact process were in good agreement with previously published analyses which neglected these effects. A single bubble was observed to form within the droplet during impact at low temperatures. As surface temperature was increased the population of bubbles within the droplet also increased because of progressive activation of nucleation sites on the stainless steel surface. At surface temperatures near to the boiling point of heptane, a spoke-like cellular structure in the liquid was created during the spreading process by coalescence of a ring of bubbles that had formed within the droplet. At higher temperatures, but below the Leidenfrost point, numerous bubbles appeared within the droplet, yet the overall droplet shape, particularly in the early stages of impact (< 0.8 ms), was unaffected by the presence of these bubbles. The maximum value of the diameter of liquid which spreads on the surface is shown to agree with predictions from a simplified model.

The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers

Abstract: §1. We shall denote by uα ( P ) = uα ( x 1, x 2, x 3, t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1, x 2, x 3. In considering the turbulence it is natural to assume the components of the velocity uα ( P ) at every point P = ( x 1, x 2, x 3, t ) of the considered domain G of the four-dimensional space ( x 1, x 2, x 3, t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ 2 α and (d uα /d xβ )2― are finite and bounded in every bounded subdomain of the domain G .

Dissipation of energy in the locally isotropic turbulence

Abstract: In my note (Kolmogorov 1941 a ) I defined the notion of local isotropy and introduced the quantities B d d ( r ) = [ u d ( M ′ ) − u d ( M ) ] 2 , ¯ [ u n ( M ′ ) − u n ( M ) ¯ ] 2 , where r denotes the distance between the points M and M' , ud(M) and ud(M') are the velocity components in the direction MM' ¯¯ at the points M and M' , and un(M) and un(M') are the velocity components at the points M and M' in some direction, perpendicular to MM' .

A general recursion method for calculating diffracted intensities from crystals containing planar faults

Abstract: A general recursion algorithm is described for calculating kinematical diffraction intensities from crystals containing coherent planar faults. The method exploits the self-similar stacking sequences that occur when layers stack non-deterministically. Recursion gives a set of simple relations between average interference terms from a statistical crystal, which can be solved as a set of simultaneous equations. The diffracted intensity for a polycrystalline sample is given by the incoherent sum of scattered intensities over an ensemble of crystallites. The relations between this and previous approaches, namely the Hendricks-Teller matrix formulation, the difference equation method, the summed series formula of Cowley, and Michalski’s recurrence relations between average phase factors, are discussed. Although formally identical to these previous methods, the present recursive description has an intuitive appeal and proves easier to apply to complex crystal structure types. The method is valid for all types of planar faults, can accommodate long-range stacking correlations, and is applicable to crystals that contain only a finite number of layers. A FORTRAN program DIFFaX , based on this recursion algorithm, has been written and used to simulate powder X-ray (and neutron) diffraction patterns and single crystal electron (kinematical) diffraction patterns. Calculations for diamond-lonsdaleite and for several synthetic zeolite systems that contain high densities of stacking faults are presented as examples.

Hopkinson Techniques for Dynamic Recovery Experiments

Abstract: Novel techniques are introduced to render the classical split Hopkinson bar apparatus suitable for dynamic recovery experiments, where samples can be subjected to a single pulse of pre-assigned shape and duration, and then recovered without any additional loading, for post-test characterization; i.e., techniques for fully controlled unloading in Hopkinson bar experiments. For compression dynamic recovery tests, the new design generates a compressive pulse trailed by a tensile pulse (stress reversal), travelling toward the sample. Furthermore, all subsequent pulses which reflect off the free ends of the two bars (incident and transmission) are rendered tensile, so that the sample is subjected to a single compressive pulse whose shape and duration can also be controlled. For tension recovery experiments, the new design provides for trapping the compression pulse reflected off the sample, and the tensile pulse transmitted through the sample. In addition, a sample can be subjected to compression followed by tension, and then recovered, allowing the study of, e.g. the dynamic Bauschinger effect in materials.

On Local Isotropy of Passive Scalars in Turbulent Shear Flows

Abstract: An assessment of local isotropy and universality in high-Reynolds-number turbulent flows is presented. The emphasis is on the behaviour of passive scalar fields advected by turbulence, but a brief review of the relevant facts is given for the turbulent motion itself. Experiments suggest that local isotropy is not a natural concept for scalars in shear flows, except, perhaps, at such extreme Reynolds numbers that are of no practical relevance on Earth. Yet some type of scaling exists even at moderate Reynolds numbers. The relation between these two observations is a theme of this paper.

Latent hardening in single crystals. II. Analytical characterization and predictions

Abstract: Constitutive equations are developed that characterize the multiple-slip behaviour of crystalline materials at low temperature. A matrix of instantaneous hardening moduli that relate the rate of hardening on each slip system to all slip-rates is proposed based upon well-known observations and the latent hardening experiments reported in Part I. In general, these moduli depend on the history of slips. Simulations of various behaviours are presented for FCC single crystals (copper) that are in good agreement with observations. These include, for example, stress-strain curves in a uniaxial loading test, hardening rate variations with respect to initial orientation, latent hardening, tensile overshoot and secondary slips. Numerical calculations are facilitated using an extremum principle and a modified quadratic programming algorithm.

Hyperasymptotics for integrals with saddles

Abstract: Integrals involving exp {-kf(z)}, where $|$k$|$ is a large parameter and the contour passes through a saddle of f(z), are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles, through which the contour does not pass. These contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent. The refinement is achieved by means of an exact \`resurgence relation', expressing the original integral as its truncated saddle-point asymptotic expansion plus a remainder involving the integrals through certain \`adjacent' saddles, determined by a topological rule. Iteration of the resurgence relation, and choice of truncation near the least term of the original series, leads to a representation of the integral as a sum of contributions associated with \`multiple scattering paths among the saddles. No resummation of divergent series is involved. Each path gives a \`hyperseries', depending on the terms in the asymptotic expansions for each saddle (these depend on the particular integral being studied and so are non-universal), and certain `hyperterminant' functions defined by integrals (these are always the same and hence universal). Successive hyperseries get shorter, so the scheme naturally halts. For two saddles, the ultimate error is approximately $\varepsilon ^{2.386}$, where $\varepsilon $ (proportional to exp (-A$|$k$|$) where A is a positive constant), is the error in optimal truncation of the original series. As a numerical example, an integral with three saddles is computed hyperasymptotically.

Structure and dynamics of homogeneous turbulence: models and simulations

Abstract: This paper presents a review of recent results on homogeneous turbulence. We discuss results obtained by direct numerical simulation as well as phenomenological models for the interpretation and understanding of these flows. In particular, we show that homogeneous turbulence can be well described in terms of a weakly correlated, random background field that is generally consistent with the classical Kolmogorov theory of turbulence, and strongly correlated, highly localized structures, that are largely responsible for intermittency effects and deviations from Kolmogorov scaling. These results give a unified dynamical picture of turbulence that describes both the energetics and intermittency of homogeneous turbulence, and allows us to develop a quantitative model for the description of the statistics of turbulence at small scales.

Analysis of radiative scattering for multiple sphere configurations

Abstract: An analysis of radiative scattering for an arbitrary configuration of neighbouring spheres is presented. The analysis builds upon the previously developed superposition solution, in which the scattered field is expressed as a superposition of vector spherical harmonic expansions written about each sphere in the ensemble. The addition theorems for vector spherical harmonics, which transform harmonics from one coordinate system into another, are rederived, and simple recurrence relations for the addition coefficients are developed. The relations allow for a very efficient implementation of the 'order of scattering' solution technique for determining the scattered field coefficients for each sphere. Prediction of the radiative absorption and scattering characteristics of small particles is important to researchers in a number of fields, e.g. atmospheric modelling, analysis of radiative transfer from flames, and development of nonintrusive laser-based optical diagnostic methods. Computation of the radiative characteristics of spherical particles from Lorenz-Mie theory is practically a trivial matter due to the existence of efficient computer codes (Bohren & Huffman 1983). However, it is not unusual to encounter situations in which the individual particles, while spherical in shape, are so close together that the 'isolated sphere' assumption inherent in Lorenz-Mie theory is questionable. A common example is soot formed in combustion processes. Electron micrographs of the individual soot particles reveal them to be agglomerates of a large number of primary, spherical particles (Dobbins

Chaotic Phenomena Triggering the Escape from a Potential Well

Abstract: This paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold. The aim is to show how the well-known atoms of dissipative dynamics (saddle-node folds, period-doubling flips, cascades to chaos, boundary crises, etc.) assemble to form molecules of overall response (hierarchies of cusps, incomplete Feigenbaum trees, etc.). Particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle, the latter being accurately predicted by a Melnikov analysis. After escape, chaotic transients are shown to conform to recent scaling laws. Analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible.

A uniqueness proof for the Wulff Theorem

Abstract: The Wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of (small) elastic crystals. It consists in minimising the (generally anisotropic) surface energy among sets of given volume. A solution of this problem is given by a geometric construction due to Wulff. In the class of sets of finite perimeter this was first shown by J. E. Taylor who, using methods of geometric measure theory, also proved uniqueness. Here a more analytic uniqueness proof is presented. The main ingredient is a sharpened version of the Brunn–Minkowski inequality.

Latent hardening in single crystals. I: Theory and experiments

Abstract: Accurate measurements of the initial yield stress on previously latent slip systems as well as a reinterpretation of widely reported experimental observations have led to a new description of single crystal hardening within the framework of the incremental (flow) theory of plasticity. Slip interactions and the history of slips are essential in explaining well-known physical phenomena such as stage II deformation and latent hardening. Guidelines for deriving the set of instantaneous hardening moduli are given in terms of inequality restrictions. Although time-independent behaviour is assumed throughout the present study, these restrictions are expected to apply as well to time-dependent creep behaviour at low to intermediate temperatures. In Part II, a complete constitutive theory is developed with analytical forms given for the instantaneous hardening moduli.

Mechanics of Dynamic Debonding

Abstract: Singular fields around a crack running dynamically along the interface between two anisotropic substrates are examined Emphasis is placed on extending an established framework for interface fracture mechanics to include rapidly applied loads, fast crack propagation and strain rate dependent material response For a crack running at non-uniform speed, the crack tip behaviour is governed by an instantaneous steady-state, two-dimensional singularity This simplifies the problem, rendering the Stroh techniques applicable In general, the singularity oscillates, similar to quasistatic cracks The oscillation index is infinite when the crack runs at the Rayleigh wave speed of the more compliant material, suggesting a large contact zone may exist behind the crack tip at high speeds In contrast to a crack in homogeneous materials, an interface crack has a finite energy factor at the lower Rayleigh wave speed Singular fields are presented for isotropic bimaterials, so are the key quantities for orthotropic bimaterials Implications on crack branching and substrate cracking are discussed Dynamic stress intensity factors for anisotropic bimaterials are solved for several basic steady state configurations, including the Yoffe, Gol'dshtein and Dugdale problems Under time-independent loading, the dynamic stress intensity factor can be factorized into its equilibrium counterpart and the universal functions of crack speed

Some applications of Kolmogorov's turbulence research in the field of combustion

Abstract: This paper reviews recent developments in understanding of ways in which various characteristic scales in the Kolmogorov energy cascade control the wrinkling and stretching of thin laminar premixed flames in turbulent flows. Some unresolved problems are identified. Results of recent direct numerical simulations of turbulent combustion are discussed. Distributions of flame strain and curvature, obtained from constant density simulations, are presented and a parametrization of these results is suggested. This parametrization is then used to derive a new theoretical model to allow for effects of detailed finite chemical reaction rate mechanisms in engineering calculations of turbulent combustion.

On the non-radial oscillations of a star

Abstract: A complete theory of the non-radial oscillations of a static spherically symmetric distribution of matter, described in terms of an energy density and an isotropic pressure, is developed, ab initio, on the premise that the oscillations are excited by incident gravitational waves. The equations, as formulated, enable the decoupling of the equations governing the perturbations in the metric of the space-time from the equations governing the hydrodynamical variables. This decoupling of the equations reduces the problem of determining the complex characteristic frequencies of the quasi-normal modes of the non-radial oscillations to a problem in the scattering of incident gravitational waves by the curvature of the space-time and the matter content of the source acting as a potential. The present paper is restricted (for the sake of simplicity) to the case when the underlying equation of state is barotropic. The algorism developed for the determination of the quasi-normal modes is directly confirmed by comparison with an independent evaluation by the extant alternative algorism. Both polar and axial perturbations are considered. Dipole oscillations (which do not emit gravitational waves), are also treated as a particularly simple special case. Thus, all aspects of the theory of the non-radial oscillations of stars find a unified treatment in the present approach. The reduction achieved in this paper, besides providing a fresh understanding of known physical problems when formulated in the spirit of general relativity, provides also a basis for an understanding, at a deeper level, of Newtonian theory itself.

Continuum Damage Mechanics Modelling of High Temperature Deformation and Failure in a Pipe Weldment

Abstract: Predictions have been made of the deformation and failure history of a pipe weldment using a finite element creep continuum damage mechanics model, which incorporates the characteristic material properties of the parent metal, the weld metal and heat-affected zone microstructures of the weldment. It is shown that the computer predictions are in close agreement with the results of large-scale pressure vessel tests, provided that the material characterization is carried out correctly, and that the constitutive equations which control the evolution of creep strain and damage, represent the dominant physical mechanisms present.

Fractal dimensions and spectra of interfaces with application to turbulence

Abstract: This paper is concerned with the analysis of any convoluted surface in two or three dimensions which has a self-similar structure, and which may simply be defined as a mathematical surface or as an interface where there is a sharp change in the value of a scalar field F(x) (say from 0 to 1). The different methods of analysis that are related to each other here are based on spectra of the scalar which have the form $\Gamma $(k) $\propto $ k$^{-p}$, (1) the Kolmogorov capacity D$\_{\text{K}}$ of the interface or D$\_{\text{K}}^{\prime}$ of the intersections of the interface with a plane or a line (both being defined by algorithms for counting the minimum number of boxes of sizes $\epsilon $ either covering the surface or its intersection), and the Hausdorff dimensions D$\_{\text{H}}$, D$\_{\text{H}}^{\prime}$ (which are defined differently). It is demonstrated that interfaces with a localized self-similar structure around accumulation points, such as spirals, may have non-integer capacities D$\_{\text{K}}$ and D$\_{\text{K}}^{\prime}$ even though their Hausdorff dimension is integer and equal to the topological dimension of the surface. It is explained how the same surface can have different values of D$\_{\text{K}}$ and D$\_{\text{K}}^{\prime}$ over different asymptotic ranges of $\epsilon $. There are other (fractal) surfaces where both D$\_{\text{H}}$ and D$\_{\text{K}}$ are non-integer which are convoluted on a wide range of scales with the same form of self-similarity everywhere on the surface. Distinctions are drawn between these two kinds of interface which have local and global self-similarity respectively. If the intersections of the interface with any line form a set of points that is statistically homogeneous and independent of the location and orientation of the line and also that is self-similar over a sufficiently wide range of spacing that a capacity D$\_{\text{K}}^{\prime}$ can be defined, it is shown that the scalar F has a power spectrum of the form of (1) and that the exponent p is related to D$\_{\text{K}}^{\prime}$ by p+D$\_{\text{K}}^{\prime}$ = 2. (2) This quite general result for interfaces is varified analytically and computationally for spirals. In experiments with scalar interfaces in different turbulent flows at high Reynolds and Prandtl number, K. R. Sreenivasan, R. Ramshankar and C. Meneveau's measurements showed that D$\_{\text{K}}^{\prime}$ = 0.33 for values of $\epsilon $ within the inertial range and D$\_{\text{K}}^{\prime}$ = 0 for smaller values (in the microscale range). The values of p derived from (2) are consistent with the theory of G. K. Batchelor and many measurements of scalar spectra. For fractal interfaces with global self-similarity, the values of D$\_{\text{H}}$ of an interface and D$\_{\text{H}}^{\prime}$ of the intersections of the interface with a line, have been shown previously to be related simply to each other by the topological dimension E of the interface so that D$\_{\text{H}}$ = D$\_{\text{H}}^{\prime}$ + E. No such theorem exists in general for the Kolmogorov capacities D$\_{\text{K}}$ and D$\_{\text{K}}^{\prime}$. But it is shown analytically and computationally that for the case of spirals, over a certain range of resolutions $\epsilon $, D$\_{\text{K}}$ = D$\_{\text{K}}^{\prime}$ + E. (3) This corresponds in practice to the measurable range of typical experimental spirals with fewer than about five turns. Over a range of smaller length scales $\epsilon $, where a larger number of turns is resolved and which is experimentally difficult to measure, (3) is not correct. The result (3) has been previously suggested based on experimental results. Finally we demonstrate that interfaces for which there is a well-defined value of capacity (which is indeed the case for spirals of three turns) are only found to have self-similar spectra if there is a much wider range of length scales (e.g. more than 50 turns of the spiral) than is needed for the capacity D$\_{\text{K}}$ to be measurable. As well as demonstrating this computationally, this is proved mathematically for interfaces having a particular class of accumulation points whose intersections with straight lines form a self-similar sequence of points x$\_{n}\propto $ n$^{-\alpha}$; the power spectrum of F only tends to the self-similar form (1) if $(\epsilon \_{\min}/\epsilon \_{\max})^{1-D\_{\text{K}}^{\prime}}\ll $ 1, whereas the capacity measure simply requires that $\epsilon \_{\min}/\epsilon \_{\max}\ll $ 1. So when D$\_{\text{K}}^{\prime}$ > 0, the criteria for the spectra requires a winder range of scales in the convolutions of the interface. This is consistent with the finding that reliable measurements of D$\_{\text{K}}$ can be computed from measurements of interfaces in laboratory experiments, but in the same experiments computations of spectra are often not of the form (1). Therefore, despite this apparent discrepancy with the general result (2) (which implies a value of p given a value of D$\_{\text{K}}^{\prime}$), the above theoretical argument supports the deduction from such experiments that if a non-integer value of D$\_{\text{K}}$ is measured, the interface does indeed have a self-similar structure; but it is not self-similar over a wide enough range of length scales to satisfy (2). Consequently, for turbulent flows, stating that the capacity should have its asymptotic value rather than (as is usual) the spectrum should be equal to its asymptotic form (e.g. in (1) p = ${\textstyle\frac{5}{3}}$) may be the correct necessary condition for deciding whether, in a given flow, the interfaces have the characteristic structure found at very high Reynolds number. Many of the results here may be of value in other scientific fields, where convoluted interfaces are also studied.

Stripes or Spots? Nonlinear Effects in Bifurcation of Reaction-Diffusion Equations on the Square

Abstract: Bifurcation to spatial patterns in a two-dimensional reaction-diffusion medium is considered. The selection of stripes versus spots is shown to depend on the nonlinear terms and cannot be discerned from the linearized model. The absence of quadratic terms leads to stripes but in most common models quadratic terms will lead to spot patterns. Examples that include neural nets and more general pattern formation equations are considered.

The Wulff Theorem Revisited

Abstract: The parametrized indicator measures and the Brunn-Minkowski inequality are used to prove that the Wulff set W ‪‪‪‪‪г‪ is a minimizer for the surface energy where the density is the support function of W ‪‪‪‪‪г . The support of the indicator measures associated to minimizing sequences is characterized. It is shown that if W ‪‪‪‪‪г is polyhedral then minimizing sequences cannot oscillate.

The Effective Conductivity of Random Suspensions of Spherical Particles

Abstract: The effective conductivity of an infinite, random, mono-disperse, hard-sphere suspension is reported for particle to matrix conductivity ratios of ∞, 10 and 0.01 for sphere volume fractions, c, up to 0.6. The conductivities are computed with a method previously described by the authors, which includes both far- and near-field interactions, and the particle configurations are generated via a Monte Carlo method. The results are consistent with the previous theoretical work of D. J. Jeffrey to O(c^2) and the bounds computed by S. Torquato and F. Lado. It is also found that the Clausius-Mosotti equation is reasonably accurate for conductivity ratios of 10 or less all the way up to 60 % (by volume). The calculated conductivities compare very well with those of experiments. In addition, percolation-like numerical experiments are performed on periodically replicated cubic lattices of N nearly touching spheres with an infinite particle to matrix conductivity ratio where the conductivity is computed as spheres are removed one by one from the lattice. Under suitable normalization of the conductivity and volume fraction, it is found that the initial volume fraction must be extremely close to maximum packing in order to observe a percolation transition, indicating that the near-field effects must be very large relative to far-field effects. These percolation transitions occur at the accepted values for simple (sc), bodycentred (BCC) and face-centred (FCC) cubic lattices. Also, the vulnerability of the lattices computed here are exactly those of previous investigators. Due to limited data above the percolation threshold, we could not correlate the conductivity with a power law near the threshold; however, it can be correlated with a power law for large normalized volume fractions. In this case the exponents are found to be 1.70, 1.75 and 1.79 for sc, BCC and FCC lattices respectively.

On the non-radial oscillations of a star III. A reconsideration of the axial modes

Abstract: It is shown that for stars with radii in the range 2.25 GM/c$^{2}$ < R < ca. 3GM/c$^{2}$, quasinormal axial modes of oscillation are possible. These modes are explicitly evaluated for stellar models of uniform energy density.

From Global Scaling, a la Kolmogorov, to Local Multifractal Scaling in Fully Developed Turbulence

Abstract: In the first part of the paper a modern presentation of scaling ideas is made. It includes a reformulation of Kolmogorov’s 1941 theory bypassing the universality problem pointed out by Landau and a presentation of the multifractal theory with emphasis on scaling rather than on cascades. In the second part, various historical aspects are discussed. The importance of Kolmogorov’s rigorous derivation of the -4/5 ϵl law for the third order structure function in his last 1941 turbulence paper is stressed; this paper also contains evidence that he was aware of universality not being essential to the 1941 theory. An inequality is established relating the exponents ζ 2 p of the structure functions of order 2 p and the maximum velocity excursion. It follows that models (such as the Obukhov-Kolmogorov 1962 log-normal model), in which ζ 2 p does not increase monotonically, are inconsistent with the basic physics of incompressible flow. This result is independent of Novikov’s 1971 inequality; in particular, the proof presented here does not rely on the (questionable) relation, proposed by Obukhov and Kolmogorov, between instantaneous velocity increments and local averages of the dissipation.

Convective variational approach to relativistic thermodynamics of dissipative fluids

Abstract: Using guidelines provided by Noether identities arising from a generalized variation procedure of convective type, a new (nonlinear and exactly self-consistent) category of relativistic thermodynamic models is developed for the systematic representation of viscous conducting fluid media (allowing for several independent charged or neutral chemical constituents). Apart from the provision of a set of dissipation coefficients of the usual (reactivity, resistivity, and viscosity) type, the specification of a particular model is determined just by giving the algebraic dependence a single ‘master function’, Λ say, on the relevant dynamical variables, which are supposed here to consist of an entropy current 4-vector and a set of particle current 4-vectors corresponding to the various chemical constituents, together with a set of symmetric (rank 3) viscosity tensors, which are considered as being dynamically independent of the corresponding current vectors except in the degenerate limit of linear viscosity. The master function is set up as a generalization of an ordinary lagrangian function, to which it reduces in the relevant non - dissipative limit, and, as in the conservative case, it is used for the construction of derived quantities in such a way that appropriate self-consistency conditions are satisfied as identities. In particular the relevant stress-momentum-energy tensor is obtained directly in terms of the independent variables and of their dynamical conjugates (whose role is hidden in the traditional approach as developed by Israel & Stewart), which are set of ordinary 4-momentum (not 4-momentum density) covectors associated with the independent currents, and a set of generalised Cauchy type strain (not strain - rate) tensors associated with the independent viscous stress contributions. The range of application of the category obtained in this way is intended to include that of the standard (Israel-Stewart) formalism to which it is expected to be effectively equivalent in the limit of sufficiently small deviations from thermodynamic equilibrium.

Comparative lagrangian formulations for localized buckling

Abstract: An energy functional for a strut on a nonlinear softening foundation is worked into two different lagrangian forms, in fast and slow space respectively. The developments originate independently of the underlying differential equation, and carry some quite general features. In each case, the kinetic energy is an indefinite quadratic form. In fast space, this leads to an escape phenomenon with fractal properties. In slow space, kinetic energy is added to a potential contribution that is familiar from modal formulations. Together, and in conjunction with a recent set of numerical experiments, they illustrate the extra complexities of localized, as opposed to distributed periodic, buckling.

Bubble collapse and the initiation of explosion

Abstract: Experiments were conducted to investigate the initiation of an emulsion explosive containing cavities. Cylindrical cavities were created in thin sheets of either gelatine or an ammonium nitrate/sodium nitrate emulsion confined between transparent blocks. Shocks were launched into the sheets with either a flier-plate or an explosive plane-wave generator so as to collapse the cavities asymmetrically. The closure of the cavities and subsequent reaction in the explosive was photographed by using high- speed framing cameras. The collapse of the cavity proceeded in several stages. First, a high-speed jet was formed which crossed the cavity and hit the downstream wall sending out a shock wave into the surrounding material. Secondly, gas within the cavity was heated by rapid compression achieving temperatures sufficient to lead to gas luminescence. Finally, the jet penetrated the downstream wall to form a pair of vortices which travelled downstream with the flow. When such a cavity collapsed in an explosive, a reaction was observed to start in the vapour contained within the cavity and in the material around the heated gas. The ignition of material at the point at which the jet hit was found to be the principal ignition mechanism.

Fractal Crushing of Ice and Brittle Solids

Abstract: A significant `scale effect' is observed when sea ice forces on structures are measured at field scale: the force per unit contact area is not independent of area, but decreases with increasing area Fragments of broken materials are found to have a fractal size distribution, with a fractal dimension close to 25 over a remarkably wide range of fragment size The research described in this paper brings these two observations together, and shows that they can be explained by a simple model of crushing, which incorporates the relation between fragment size and splitting force predicted by linear elastic fracture mechanics The model indicates a special role for the fractal dimension of 25, and predicts a relation between force and area, consistent with field observations

Kolmogorov's contributions to the physical and geometrical understanding of small-scale turbulence and recent developments

Abstract: This paper reviews how Kolmogorov postulated for the first time the existence of a steady statistical state for small-scale turbulence, and its defining parameters of dissipation rate and kinematic viscosity. Thence he made quantitative predictions of the statistics by extending previous methods of dimensional scaling to multiscale random processes. We present theoretical arguments and experimental evidence to indicate when the small-scale motions might tend to a universal form (paradoxically not necessarily in uniform flows when the large scales are gaussian and isotropic), and discuss the implications for the kinematics and dynamics of the fact that there must be singularities in the velocity field associated with the - inertial range spectrum. These may be particular forms of eddy or 'eigenstructure' such as spiral vortices, which may not be unique to turbulent flows. Also, they tend to lead to the notable spiral contours of scalars in turbulence, whose self-similar structure enables the 'boxcounting' technique to be used to measure the 'capacity' DK of the contours themselves or of their intersections with lines, DK. Although the capacity, a term invented by Kolmogorov (and studied thoroughly by Kolmogorov & Tikhomirov), is like the exponent 2p of a spectrum in being a measure of the distribution of length scales (D' being related to 2p in the limit of very high Reynolds numbers), the capacity is also different in that experimentally it can be evaluated at local regions within a flow and at lower values of the Reynolds number. Thus Kolmogorov & Tikhomirov provide the basis for a more widely applicable measure of the self-similar structure of turbulence. Finally, we also review how Kolmogorov's concept of the universal spatial structure of the small scales, together with appropriate additional physical hypotheses, enables other aspects of turbulence to be understood at these scales; in particular the general forms of the temporal statistics such as the highfrequency (inertial range) spectra in eulerian and lagrangian frames of reference, and the perturbations to the small scales caused by non-isotropic, non-gaussian and inhomogeneous large-scale motions. 1. Kolmogorov's papers: review and comments (a) Introduction In this review we join with the other contributors to this special publication in celebrating some of Kolmogorov's great contributions to fluid mechanics and mathematics, and showing in some small way how his genius has inspired further

Kolmogorov Similarity Hypotheses for Scalar Fields: Sampling Intermittent Turbulent Mixing in the Ocean and Galaxy

Abstract: Kolmogorov's three universal similarity hypotheses are extrapolated to describe scalar fields like temperature mixed by turbulence. The analogous first and second hypotheses for scalars include the effects of Prandtl number and rate-of-strain mixing. Application of velocity and scalar similarity hypotheses to the ocean must take into account the damping of active turbulence by density stratification and the Earth's rotation to form fossil turbulence. By the analogous Kolmogorov third hypothesis for scalars, temperature dissipation rates $\chi $ averaged over lengths r > L$\_{\text{K}}$ should be lognormally distributed with intermittency factors $\sigma ^{2}$ that increase with increasing turbulence energy length scales L$\_{\text{O}}$ as $\sigma \_{\ln \langle \chi \rangle \_{r}}^{2}\approx \mu \_{\theta}$ ln (L$\_{\text{O}}$/r). Tests of kolmogorovian velocity and scalar universal similarity hypotheses for very large ranges of turbulence length and timescales are provided by data from the ocean and the galactic interstellar medium. These ranges are from 1 to 9 decades in the ocean, and over 12 decades in the interstellar medium. The universal constant for turbulent mixing intermittency $\mu _{\theta}$ is estimated from oceanic data to be 0.44 $\pm $ 0.01, which is remarkably close to estimates for Kolmogorov's turbulence intermittency constant $\mu $ of 0.45 $\pm $ 0.05 from galactic as well as atmospheric data. Extreme intermittency complicates the oceanic sampling problem, and may lead to quantitative and qualitative undersampling errors in estimates of mean oceanic dissipation rates and fluxes. Intermittency of turbulence and mixing in the interstellar medium may be a factor in the formation of stars.