The equations governing the movement of the melt and the matrix of a partially molten material are obtained from the conservation of mass, momentum, and energy using expressions from the theory of mixtures. The equations define a length scale dc called the compaction length, which depends only on the material properties of the melt and matrix. A number of simple solutions to the equations show that, if the porosity is initially constant, matrix compaction only occurs within a distance ~<5C of an impermeable boundary. Elsewhere the gravitational forces are supported by the viscous stresses resulting from the movement of melt, and no compaction occurs. The velocity necessary to prevent compaction is known as the minimum fluidization velocity. In all cases the compaction rate is controlled by the.properties of the matrix. These results can only be applied to geological problems if the values of the permeability, bulk and shear viscosity of the matrix can be estimated. All three depend on the microscopic geometry of the melt, which is in turn controlled by the dihedral angle. The likely equilibrium network provides some guidance in estimating the order of magnitude of these constants, but is no substitute for good measurements, which are yet to be carried out. Partial melting by release of pressure at constant entropy is then examined as a means of produced melt within the earth. The principal results of geological interest are that a mean mantle temperature of 1350 °C is capable of producing the oceanic crustal thickness by partial melting. Local hot jets with temperatures of 1550 °C can produce aseismic ridges with crustal thicknesses of about 20 km on ridge axes, and can generate enough melt to produce the Hawaiian Ridge. Higher mantle temperatures in the Archaean can produce komatiites if these are the result of modest amounts of melting at depths of greater than 100 km, and not shallow melting of most of the rock. The compaction rate of the partially molten rock is likely to be rapid, and melt-saturated porosities in excess of perhaps 3 per cent are unlikely to persist anywhere over geological times. The movement of melt through a matrix does not transport major and trace elements with the mean velocity of the melt, but with a slower velocity whose magnitude depends on the distribution coefficient. This effect is particularly important when the melt fraction is small, and may both explain some geochemical observations and provide a means of investigating the compaction process within the earth. I N T R O D U C T I O N There is an obvious need for a simple physical model which can describe the generation of a partially molten rock, and the separation of the melt from the residual solid, which will be referred to as the matrix. If such a model is to be useful it must lead to differential equations which can be solved by standard methods. The principal aim of this paper is to propose such a model, derive the governing equations, and obtain some solutions for particularly simple cases. The model is concerned with the physics, rather than the chemistry, of the process, though the formulation is sufficiently general to allow the inclusion of complicated phase equilibria. Several effects whose importance is unclear have not been included, in order to obtain the simplest model which can describe the generation and extraction of magma. Generation of a magma containing few solid crystals requires two operations. A partially mohen rock must first be generated, either by supplying heat or by reducing the pressure and so changing the solidus temperature. Once such a rock has been formed, the melt must UounuJ of Petrology, Vol. 25, Pirt 3, pp. 713-765, 19841 at W asngton U niersity at St L ouis on M arch 5, 2013 http://petroxfordjournals.org/ D ow nladed from

The Generation and Compaction of Partially Molten Rock

Perturbed Free Shear Layers

This article reports on the International Nanofluid Property Benchmark Exercise, or INPBE, in which the thermal conductivity of identical samples of colloidally stable dispersions of nanoparticles or “nanofluids,” was measured by over 30 organizations worldwide, using a variety of experimental approaches, including the transient hot wire method, steady-state methods, and optical methods. The nanofluids tested in the exercise were comprised of aqueous and nonaqueous basefluids, metal and metal oxide particles, near-spherical and elongated particles, at low and high particle concentrations. The data analysis reveals that the data from most organizations lie within a relatively narrow band (±10% or less) about the sample average with only few outliers. The thermal conductivity of the nanofluids was found to increase with particle concentration and aspect ratio, as expected from classical theory. There are (small) systematic differences in the absolute values of the nanofluid thermal conductivity among the various experimental approaches; however, such differences tend to disappear when the data are normalized to the measured thermal conductivity of the basefluid. The effective medium theory developed for dispersed particles by Maxwell in 1881 and recently generalized by Nan et al. [J. Appl. Phys. 81, 6692 (1997)], was found to be in good agreement with the experimental data, suggesting that no anomalous enhancement of thermal conductivity was achieved in the nanofluids tested in this exercise.

A Benchmark Study on the Thermal Conductivity of Nanofluids

In the present study, it is shown that the spreading rate of a mixing layer can be greatly manipulated at very low forcing level if the mixing layer is perturbed near a subharmonic of the most-amplified frequency. The subharmonic forcing technique is able to make several vortices merge simultaneously and hence increases the spreading rate dramatically. A new mechanism, ‘collective interaction’, was found which can bypass the sequential stages of vortex merging and make a large number of vortices (ten or more) coalesce.A deeper physical insight into the evolution of the coherent structures is revealed through the investigation of a forced mixing layer. The stability and the forcing function play important roles in determining the initial formation of the vortices. The subharmonic starts to amplify at the location where the phase speed of the subharmonic matches that of the fundamental. The position where vortices are seen to align vertically coincides with the position where the measured subharmonic reaches its peak. This location is defined as the merging location, and it can be determined from the feedback equation (Ho & Nosseir 1981).The spreading rate and the velocity profiles of the forced mixing layer are distinctly different from the unforced case. The data show that the initial condition has a longlasting effect on the development of the mixing layer.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112082001438

Subharmonics and vortex merging in mixing layers

We present a new mechanism for generation of near-wall streamwise vortices – which dominate turbulence phenomena in boundary layers – using linear perturbation analysis and direct numerical simulations of turbulent channel flow. The base flow, consisting of the mean velocity profile and low-speed streaks (free from any initial vortices), is shown to be linearly unstable to sinuous normal modes only for relatively strong streaks, i.e. for wall inclination angles of streak vortex lines exceeding 50°. Analysis of streaks extracted from fully developed near-wall turbulence indicates that about 20% of streak regions in the buffer layer exceed the strength threshold for instability. More importantly, these unstable streaks exhibit only moderate (twofold) normal-mode amplification, the growth being arrested by self-annihilation of streak-flank normal vorticity due to viscous cross-diffusion. We present here an alternative, streak transient growth (STG) mechanism, capable of producing much larger (tenfold) linear ampliflcation of x-dependent disturbances. Note the distinction of STG – responsible for perturbation growth on a streak velocity distribution U(y, z) – from prior transient growth analyses of the (streakless) mean velocity U(y). We reveal that streamwise vortices are generated from the more numerous normal-mode-stable streaks, via a new STG-based scenario: (i) transient growth of perturbations leading to formation of a sheet of streamwise vorticity ωx (by a ‘shearing’ mechanism of vorticity generation), (ii) growth of sinuous streak waviness and hence ∂u/∂x as STG reaches nonlinear amplitude, and (iii) the ωx sheet’s collapse via stretching by ∂u/∂x (rather than rollup) into streamwise vortices. Significantly, the three-dimensional features of the (instantaneous) streamwise vortices of x-alternating sign generated by STG agree well with the (ensemble-averaged) coherent structures educed from fully turbulent flow. The STG-induced formation of internal shear layers, along with quadrant Reynolds stresses and other turbulence measures, also agree well with fully developed turbulence. Results indicate the prominent – possibly dominant – role of this new, transient-growth-based vortex generation scenario, and suggest interesting possibilities for robust control of drag and heat transfer.

/pdf/coherent-structure-generation-in-near-wall-turbulence-36o0srxpc5.pdf

Coherent structure generation in near-wall turbulence

In this paper we study the development of large-scale wavelike eddies, or instability waves, in a turbulent free shear flow. The model is based on splitting the flow into three components: the mean flow, the instability wave and the fine-scale turbulence. The wave is considered to be sufficiently weak so that it is developing in a pre-existing, known turbulent mean shear flow. The basis for the wave development is its time-averaged kinetic energy flux equation in integral form and the wave description is obtained through a shape assumption: the amplitude is determined by the energy equation; the shape function and local characteristics are determined by the local linear stability theory. The wave energy changes as it is convected into a different streamwise position where its instability properties change. The energy balancing mechanisms are production, work done by the wave pressure gradients and the energy transfer between the wave and the fine-scale turbulence via the wave-induced Reynolds stresses. The latter is taken to be dissipative via an eddy-viscosity model, inertia—elastic effects not being considered. According to forceful evidence from observations in turbulent free shear flows, the wave development is taken as being upstream controlled and begins from a distinct origin rather than being the result of local forcing by variations of the fine-scale turbulent Reynolds stresses. The wave energy flux initially grows via energy supplied by the inflexional mean flow when the shear layer is relatively thin but eventually decays through action of the finescale turbulence, directly via the dissipative energy transfer and indirectly via the turbulence-diffused, rapidly thickened mean shear flow, which renders the production mechanism less available. Numerical calculations are carried out for a turbulent mean shear flow, with speed Tie on one side and zero in the ambient region, its distribution being approximated by a sine profile in the Howarth-Dorodnitsyn co-ordinate. The flow develops from an initial boundary layer of finite thickness δ0 to a similar free-mixing layer far downstream. The wave is characterized by a dimensionless frequency parameter β0 formed from the wave frequency β*, ue, and δ0. Convection speeds, in general, increase in the downstream direction. They are subsonic initially for Mach numbers Me 2 peaking in the local intensity levels occurs when convection speeds are supersonic and this may explain the observed supersonic far-field radiation a t the higher jet speeds. Induced wave patterns in the ambient region are determined by the complex instability-wave speed rather than the real convection speed alone, consequently ambient wave patterns exist even a t subsonic convection speeds, but are more heavily damped near the origin and fan out laterally downstream for a given β0. According to the present model, if the waves are given an upstream excitation level about 10−3–10−2 times that of Ue, resembling, for instance, the levels of the upstream wall turbulent boundary-layer fluctuations over a wide, low-frequency, spectrum or other possible disturbances a t the nozzle exit, the development of the calculated near noise field as a function of downstream distance bears striking resemblances to the observed near jet noise field and is thus fully sufficient to explain such observations. This comparison leads to the suggestion that the essential energetics of the large-scale wavelike eddies are that they are formed at the origin, amplified and subsequently decay in a developing mean turbulent free shear flow. Therefore this leads also to a most important indication of shear-layer instabilities and noise control. If their historical evolution can be controlled so can the noise from the damaging wavelike eddies. Methods of control on the basis of this model are discussed.

Developing large-scale wavelike eddies and the near jet noise field

The development of a quantitative understanding of large-scale coherent structures in shear flows has led to the recognition of their evolutionary features and 'rules' of nonlinear interaction among each other. These determinations, in conjunction with the study of fine-grained turbulence and mean motion, constitute a general view of hydrodynamical instabilities that has resulted in useful concepts for the achievement of shear-flow control. The dynamical role of longitudinal structures in free shear flows is presently exemplified by the axisymmetric and helical modes in a round jet, which resemble the two-dimensional and spanwise-periodic three-dimensional modes in a two-dimensional mean flow.

Coherent Structures in Transitional and Turbulent Free Shear Flows

The theoretical capability of identifying the source of turbulent jet noise is assessed in comparison with experimental data. Account is taken of axisymmetric and spiral turbulence modes in Lighthill's (1952) formulation of turbulent noise. Coherent structures interacting with the mean flow and the fine-grained turbulence are the primary noise sources, modeled as an oscillating streamwise distribution. Low-frequencies arise farther downstream while high-frequencies congregate close to the nozzle lip. Previous measurements at various exit velocities, angles with respect to the nozzle axis, the Strouhal number and downstream distance are discussed. The model successfully predicted the angular distribution of noise frequency due to coherent structures. Further work is indicated on compressibility effects.

Sound generated aerodynamically revisited: large-scale structures in a turbulent jet as a source of sound

Our recent studies of the interactions between large-scale structures and fine-grained turbulence in plane mixing layers have shown that many physical features of the problem may easily be obtained from an approximate energy integral description (Liu & Merkine 1976; Alper & Liu 1978), and this is confirmed by computational models (Gatski & Liu 1980). In this work, therefore, the approximate description is used to study at length the development of large-scale coherent structures in the technologically important problem of the round turbulent jet. The analysis begins from the radially integrated form of the kinetic energy equations of the mean flow, the large scale structure and the fine-grained turbulence, which are obtained through the use of the usual Reynolds time average and a conditional average with reference to the frequency of the idealized monochromatic component of the large-scale wavelike structure. This forms the basis for obtaining the ‘amplitude equations’ for the three components of the flow in terms of the mean flow momentum thickness, the large-scale structure kinetic energy and the fine-grained turbulence kinetic energy across the jet. These are obtained via the accompanying shape assumptions which also implicitly address the closure problems. The large-scale structure is also characterized by the Strouhal number St = fd/U e , where f is the frequency, d is the jet diameter and U e is the jet exit velocity, and by the azimuthal wave number n . The calculations are compared with the well controlled forced-jet observations of Binder & Favre-Marinet (1973), Favre-Marinet (1975), Favre-Marinet & Binder (1979) and Moore (1977). Although the present approximate considerations are not directed at structural details, the comparisons with observations on this aspect are most encouraging. Further theoretical work is presented that addresses the fundamental understanding of the mechanisms leading to the development of large-scale structures in turbulent jets. In general, large-scale structures in the range 0.02 ^ St St is increased, the streamwise location of the peak signal moves upstream, and the streamwise lifespan shortens. Consequently, high-frequency components of the large-scale structure dominate upstream while low-frequency components prevail further downstream. The Strouhal number that gives rise to maximum amplification is about 0.70 for weak initial levels of the large-scale structure and decreases to about 0.35 for very strong initial levels. As the initial energy level of the large-scale structure increases, its maximum relative amplification decreases until a level is reached beyond which the large-scale structure decays immediately downstream. This is explained in terms of the modification of the mean flow by the increasingly high energy levels of the large-scale structure in such a manner that it chokes off its own energy supply from the mean flow. At low Strouhal numbers the n = 1 helical component amplifies initially more than the n = 0 axisymmetric component for the same initial energy level. However, the n = 1 mode decays subsequently much faster than the n = 0 mode for all Strouhal numbers. This is attributable to the azimuthally-related wave-induced turbulent shear stresses in the n = 1 mode which give rise to additional mechanisms for energy transfer to the fine-grained turbulence. The possible control of the large-scale structure is fully explored through considering adjustments of the energy levels of the fine-grained turbulence and changes in initial mean velocity profile through changes in the momentum thickness at the nozzle exit. Increasing the initial turbulence levels and smoothing the mean nozzle exit velocity profile places restraints upon the downstream amplification of the large-scale structure. The non-equilibrium development of the large-scale structure is sensitive to its own initial conditions and spectral content, the initial condition of the fine-grained turbulence and the mean flow. Physical and quantitative studies of the large-scale structure in turbulent shear flows thus necessitate that the nature of such an initial environment be established and understood.

A study of the interactions between large-scale coherent structures and fine-grained turbulence in a round jet

The present study begins with the solution of the linearized partial differential equations for viscous, secondary instabilities of nonlinearly developed primary Gortler rolls. A global energy balance verifies the amplification rates obtained from secondary instability analysis and confirms that the sinuous mode would dominate. This therefore explains the strong resemblance between the rms‐streamwise u‐velocity structure obtained by hot‐wire measurements [Swearingen and Blackwelder, J. Fluid Mech. 182, 255 (1987)] in the cross‐sectional yz plane and that of the sinuous mode. The u contours of the secondary disturbance have peaks at locations corresponding to the shoulder regions of the mushroom‐like primary streamwise U‐velocity structure and even more intense regions surrounding the mushroom stem near the wall. This structure is explained by detailed analysis of the energy‐balancing mechanisms for u2/2 in yz plane: the Reynolds stress‐conversion mechanism predominantly affects the primary spanwise rate o...

Joseph T. C. Liu

Papers

Developing large-scale wavelike eddies and the near jet noise field

Coherent Structures in Transitional and Turbulent Free Shear Flows

Sound generated aerodynamically revisited: large-scale structures in a turbulent jet as a source of sound

A study of the interactions between large-scale coherent structures and fine-grained turbulence in a round jet

On the mechanism of sinuous and varicose modes in three‐dimensional viscous secondary instability of nonlinear Görtler rolls