Second-order solutions for random interfacial waves in N-layer density-stratified fluid with steady uniform currents

In the present paper, the random inter facial waves in N-layer density-stratified fluids moving at different steady uniform speeds are researched by using an expansion technique, and the second-order a symptotic solutions of the random displacements of the density interfaces and the associated velocity potentials in N-layer fluid are presented based on the small amplitude wave theory. The obtained results indicate that the wave-wave second-order nonlinear interactions of the wave components and the second-order nonlinear interactions between the waves and currents are described. As expected, the solutions include those derived by Chen (2006) as a special case where the steady uniform currents of the N-layer fluids are taken as zero, and the solutions also reduce to those obtained by Song (2005) for second-order solutions for random interfacial waves with steady uniform currents if N=2.

Second-order Stokes wave solutions for intefacial waves in three-layer stratified fluid with background current

In this paper, interfacial waves in three-layer stratified fluid with background current are investigated using a perturbation method, and the second-order asymptotic solutions of the velocity potentials and the second-order Stokes wave solutions of the associated elevations of the interfacial waves are presented based on the small amplitude wave theory, and the Kelvin-Helmholtz instability of interfacial waves is studied. As expected, for three-layer stratified fluid with background current, the first-order asymptotic solutions (linear wave solutions), dispersion relation and the second-order asymptotic solutions derived depend on not only the depths and densities of the three-layer fluid but also the background current of the fluids, and the second-order Stokes wave solutions of the associated elevations of the interfacial waves describe not only the second-order nonlinear wave-wave interactions between the interfacial waves but also the second-order nonlinear interactions between the interfacial waves and currents. It is also noted that the solutions obtained from the present work include the theoretical results derived by Chen et al (2005) as a special case. It also shows that with the given wave number k (real number) the interfacial waves may show Kelvin-Helmholtz instability.

Investigation into the performance of turbulence models for fluid flow and heat transfer phenomena in electronic applications

Computational Fluid Dynamics (CFD) is gradually becoming a powerful and almost essential tool for the design, development and optimization of engineering applications. However the mathematical modelling of the erratic turbulent motion remains the key issue when tackling such flow phenomena. The reliability of CFD analysis depends heavily on the turbulence model employed together with the wall functions implemented. In order to resolve the abrupt changes in the turbulent energy and other parameters situated at near wall regions a particularly fine mesh is necessary which inevitably increases the computer storage and run-time requirements. Turbulence modelling can be considered to be one of the three key elements in CFD. Precise mathematical theories have evolved for the other two key elements, grid generation and algorithm development. The principal objective of turbulence modelling is to enhance computational procedures of efficient accuracy to reproduce the main structures of three dimensional fluid flows. The flow within an electronic system can be characterized as being in a transitional state due to the low velocities and relatively small dimensions encountered. This paper presents simulated CFD results for an investigation into the predictive capability of turbulence models when considering both fluid flow and heat transfer phenomena. Also a new two-layer hybrid kε / kl turbulence model for electronic application areas will be presented which holds the advantages of being cheap in terms of the computational mesh required and is also economical with regards to run-time.

Nonlinear saturation of baroclinic instability: part II: continuously stratified fluid

Rigorous upper bounds are derived that limit the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow in a continuously stratified, quasi-geostrophic, semi-infinite fluid. Bounds are obtained bath on the depth-integrated eddy potential enstrophy and on the eddy available potential energy (APE) at the ground. The method used to derive the bounds is essentially analogous to that used in Part I of this study for the two-layer model: it relies on the existence of a nonlinear Liapunov (normed) stability theorem, which is a finite-amplitude generalization of the Charney-Stern theorem. As in Part I, the bounds are valid both for conservative (unforced, inviscid) flow, as well as for forced-dissipative flow when the dissipation is proportional to the potential vorticity in the interior, and to the potential temperature at the ground. The character of the results depends on the dimensionless external parameter γ = f02ξ/β0N2H, where ξ is the maximum vertical shear of the zonal wind, H is the density scale height, and the other symbols have their usual meaning. When γ ≫ 1, corresponding to “deep” unstable modes (vertical scale ≈H), the bound on the eddy potential enstrophy is just the total potential enstrophy in the system; but when γ≪1, corresponding to ‘shallow’ unstable modes (vertical scale ≈γH), the eddy potential enstrophy can be bounded well below the total amount available in the system. In neither case can the bound on the eddy APE prevent a complete neutralization of the surface temperature gradient which is in accord with numerical experience. For the special case of the Charney model of baroclinic instability, and in the limit of infinitesimal initial eddy disturbance amplitude, the bound states that the dimensionless eddy potential enstrophy cannot exceed (γ + 1)2/24&gamma2h when γ ≥ 1, or 1/6;&gammah when γ ≤ 1; here h = HN/f0L is the dimensionless scale height and L is the width of the channel. These bounds are very similar to (though of course generally larger than) ad hoc estimates based on baroclinic-adjustment arguments. The possibility of using these kinds of bounds for eddy-amplitude closure in a transient-eddy parameterization scheme is also discussed.

Instabilities in two-liquid layers subject to a horizontal temperature gradient

We study theoretically the stability of two superposed fluid layers heated laterally. The fluids are supposed to be immiscible, the interface undeformable and of infinite horizontal extension. Combined thermocapillary and buoyancy forces give rise to a basic flow when a temperature difference is applied. The calculations are performed for a melt of GaAs under a layer of molten B2 O3 , a configuration of considerable technological importance. Four dif- ferent flow patterns and five temperature configurations are found for the basic state in this system. A linear stability analysis shows that the basic state may be destabilized by oscilla- tory motions leading to the so-called hydrothermal waves. Depending on the relative height of the two layers these hydrothermal waves propagate parallel or perpendicular to the temperature gradient. This analysis reveals that these perturbations can alter significantly the liquid flow in the liquid-encapsulated crystal growth techniques.

A two-fluid model for tissue growth within a dynamic flow environment

We study the growth of a tissue construct in a perfusion bioreactor, focussing on its response to the mechanical environment. The bioreactor system is modelled as a two-dimensional channel containing a tissue construct through which a flow of culture medium is driven. We employ a multiphase formulation of the type presented by G. Lemon, J. King, H. Byrne, O. Jensen and K. Shakesheff in their study (Multiphase modelling of tissue growth using the theory of mixtures. J. Math. Biol. 52(2), 2006, 571–594) restricted to two interacting fluid phases, representing a cell population (and attendant extracellular matrix) and a culture medium, and employ the simplifying limit of large interphase viscous drag after S. Franks in her study (Mathematical Modelling of Tumour Growth and Stability. Ph.D. Thesis, University of Nottingham, UK, 2002) and S. Franks and J. King in their study Interactions between a uniformly proliferating tumour and its surrounding: Uniform material properties. Math. Med. Biol. 20, 2003, 47–89). The novel aspects of this study are: (i) the investigation of the effect of an imposed flow on the growth of the tissue construct, and (ii) the inclusion of a chanotransduction mechanism regulating the response of the cells to the local mechanical environment. Specifically, we consider the response of the cells to their local density and the culture medium pressure. As such, this study forms the first step towards a general multiphase formulation that incorporates the effect of mechanotransduction on the growth and morphology of a tissue construct. The model is analysed using analytic and numerical techniques, the results of which illustrate the potential use of the model to predict the dominant regulatory stimuli in a cell population.

Air Entrainment in the Turbulent Ship Hull Boundary Layer

Turbulent fluctuations in the vicinity of the water free surface along a flat, vertically oriented surface-piercing plate are studied experimentally using a laboratory-scale experiment. In this experiment, a meter-wide stainless steel belt travels horizontally in a loop around two rollers with vertically oriented axes, which are separated by 7.5 meters. This belt device is mounted inside a large water tank with the water level set just below the top edge of the belt. The belt, rollers, and supporting frame are contained within a sheet metal box to keep the device dry except for one 6-meter-long straight test section between rollers. The belt is launched from rest with an acceleration of up to 3-g in order to quickly reach steady state velocity. This creates a temporally evolving boundary layer analogous to the spatially evolving boundary layer created along a flat-sided ship moving at the same velocity, with a length equivalent to the length of belt that has passed the measurement region since the belt motion began. Surface profile measurements in planes normal to the belt surface are conducted using cinematic Laser Induced Fluorescence and quantitative surface profiles are extracted at each instant in time. Using these measurements, free surface fluctuations are examined and the propagation behavior of these free surface ripples is studied. It is found that free surface fluctuations are generated in a region close to the belt surface, where sub-surface velocity fluctuations influence the behavior of these free surface features. These rapidly-changing surface features close to the belt appear to lead to the generation of freely-propagating waves far from the belt, outside the influence of the boundary layer. Sub-surface PIV measurements are performed in order to study the modification of the boundary layer flow field due to the effects of the water free surface. Cinematic planar PIV measurements are performed in horizontal planes parallel to the free surface by imaging the flow from underneath the tank, providing streamwise and wall-normal velocity fields. Additional planar PIV experiments are performed in vertical planes parallel to the belt surface in order to study the bahvior of streamwise and vertical velocity fields. It is found that the boundary layer grows rapidly near the free surface, leading to an overall thicker boundary layer close to the surface. This rapid boundary layer growth appears to be linked to a process of free surface bursting, the sudden onset of free surface fluctuations. Cinematic white light movies are recorded from beneath the water surface in order to determine the onset location of air entrainment. In addition, qualitative observations of these processes are made in order to determine the mechanisms leading to air entrainment present in this flow.

Diagnostics and two-dimensional simulation of low-frequency inductively coupled plasmas with neutral gas heating and electron heat fluxes

This article presents the results on the diagnostics and numerical modeling of low-frequency (∼460 KHz) inductively coupled plasmas generated in a cylindrical metal chamber by an external flat spiral coil. Experimental data on the electron number densities and temperatures, electron energy distribution functions, and optical emission intensities of the abundant plasma species in low/intermediate pressure argon discharges are included. The spatial profiles of the plasma density, electron temperature, and excited argon species are computed, for different rf powers and working gas pressures, using the two-dimensional fluid approach. The model allows one to achieve a reasonable agreement between the computed and experimental data. The effect of the neutral gas temperature on the plasma parameters is also investigated. It is shown that neutral gas heating (at rf powers≥0.55kW) is one of the key factors that control the electron number density and temperature. The dependence of the average rf power loss, per electron-ion pair created, on the working gas pressure shows that the electron heat flux to the walls appears to be a critical factor in the total power loss in the discharge.

Reactive species in Ar+H2 plasma-aided nanofabrication : two-dimensional discharge modelling

Understanding the generation of reactive species in a plasma is an important step towards creating reliable and robust plasma-aided nanofabrication processes. A two-dimensional fluid simulation of the number densities of surface preparation species in a low-temperature, low-pressure, non-equilibrium Ar+H2 plasma is conducted. The operating pressure and H2 partial pressure have been varied between 70-200 mTorr and 0.1-50%, respectively. An emphasis is placed on the application of these results to nanofabrication. A reasonable balance between operating pressures and H 2 partial pressures that would optimize the number densities of the two working units largely responsible for activation and passivation of surface dangling bonds (Ar+ and H respectively) in order to achieve acceptable rates of surface activation and passivation is obtained. It is found that higher operating pressures (150-200 mTorr) and lower H2 partial pressures (∼5%) are required in order to ensure high number densities of Ar+ and H species. This paper contributes to the improvement of the controllability and predictability of plasma-based nanoassembly processes.

Two-dimensional simulation of nanoassembly precursor species in Ar+H2+C2H2 reactive plasmas

The results of two-dimensional fluid simulation of number densities and fluxes of the main building blocks and surface preparation species involved in nanoassembly of carbon-based nanopatterns in Ar+H2+C2H2 reactive plasmas are reported. It is shown that the process parameters and non-uniformity of surface fluxes of each particular species may affect the targeted nanopattern quality. The results can be used to improve predictability of plasma-aided nanofabrication processes and optimize the parameters of plasma nanotools.KGaA, Weinheim.

A sufficient criterion for Rayleigh-Taylor instability of incompressible viscous three-layer flow

Using normal mode analysis Rayleigh-Taylor instability is investigated for three-layer viscous stratified incompressible steady flow, when the top 3rd and bottom 1st layers extend up to infinity, the middle layer has a small thickness δ. The wave Reynolds number in the middle layer is assumed to be sufficiently small. A dispersion relation (a seventh degree polynomial in wave frequency ω) valid up to the order of the maximal value of all possible Kj (j less-than-or-equals, slant 0, K is the wave number) in each coefficient of the polynomial is obtained. A sufficient condition for instability is found out for the first time, pursuing a medium wavelength analysis. It depends on ratios (α and β) of the coefficients of viscosity, the thickness of the middle layer δ, surface tension ratio T and wave number K. This is a new analytical criterion for Rayleigh-Taylor instability of three-layer fluids. It recovers the results of the corresponding problem for two-layer fluids. Among the results obtained, it is observed that taking the coefficients of viscosity of 2nd and 3rd layers same can inhibit the effect of surface tension completely. For large wave number K, the thickness of the middle layer should be correspondingly small to keep the domain of dependence of the threshold wave number Kc constant for fixed α, β and T.

A sufficient criterion for Kelvin–Helmholtz instability in the magnetopause boundary‐layer region

The Kelvin–Helmholtz instability has been investigated for the magnetopause boundary‐layer region by the linearized method. The plasma in magnetosheath and magnetopause is assumed to be semi‐infinitely extended homogeneous, nondissipative, and incompressible. It is observed that, if one relation of two plasma speeds on the two sides of the magnetopause, wave number, and boundary‐layer thickness exceeds a certain threshold, the instability sets in. This new analytically sufficient criterion for excitation of instability in the three‐layer plasma flow generalizes the corresponding Chandrasekhar’s instability criterion for two‐layer plasma flow. The known results have been recovered and modified, the new results have been discovered. It is proved that the velocity threshold for the onset of instability is low when the magnitude of the magnetosheath and boundary‐layer region magnetic field and the angle between them are small. Also the threshold depends on the direction of plasma flow. The following results are observed numerically. The growth of the instability is sensitive to the magnetic field direction in the magnetosheath. A slight variation in the magnetic field direction in the second region can substantially change the relative velocity threshold for instability. When the ratio of the density of the second and third layer (magnetosphere) increases or that of the first and third layer decreases, the threshold decreases. Apart from this a necessary criterion for instability is obtained for a particular case.

Role of the cesium antimonide layer in the Na2KSb/Cs3Sb photocathode

When radiation of sufficiently high energy is incident on the surface of a semiconductor photocathode, electrons are excited from the valence band to the conduction band and these may contribute to the photocurrent. The photocurrent in a single-layer cathode is found to be small, because of collisions within the cathode material, the electron affinity condition, etc. It is observed that when a thin layer of n-type cesium antimonide (Cs3Sb) is deposited over a p-type layer of sodium potassium antimonide (Na2KSb), there occurs a sharp rise in the photocurrent. The causes for the dramatic increase in the photocurrent obtainable from a sodium potassium antimonide cathode, by depositing a thin layer of cesium antimonide are analyzed in this article. It has been shown that the interface between sodium potassium antimonide and cesium antimonide can result in lowering of the electron affinity to a level below the bottom of the conduction band of sodium potassium antimonide. The drift field that arises at the heterointerface enables the electrons to reach the surface, leading to the emission of almost all the photogenerated electrons within the cathode. The processes involved in photoemission from such a double-layer cathode are examined from a theoretical point of view. The spectral response of the two-layer cathode is also found to be better than that of a single-layer cathode.

Parameter estimation of a two-horizon soil profile by combining crop canopy and surface soil moisture observations using GLUE

Estimation of soil parameters by inverse modeling using observations on either surface soil moisture or crop variables has been successfully attempted in many studies, but difficulties to estimate root zone properties arise when heterogeneous layered soils are considered. The objective of this study was to explore the potential of combining observations on surface soil moisture and crop variables - leaf area index (LAI) and above-ground biomass for estimating soil parameters (water holding capacity and soil depth) in a two-layered soil system using inversion of the crop model STICS. This was performed using GLUE method on a synthetic data set on varying soil types and on a data set from a field experiment carried out in two maize plots in South India. The main results were (i) combination of surface soil moisture and above-ground biomass provided consistently good estimates with small uncertainity of soil properties for the two soil layers, for a wide range of soil paramater values, both in the synthetic and the field experiment, (ii) above-ground biomass was found to give relatively better estimates and lower uncertainty than LAI when combined with surface soil moisture, especially for estimation of soil depth, (iii) surface soil moisture data, either alone or combined with crop variables, provided a very good estimate of the water holding capacity of the upper soil layer with very small uncertainty whereas using the surface soil moisture alone gave very poor estimates of the soil properties of the deeper layer, and (iv) using crop variables alone (else above-ground biomass or LAI) provided reasonable estimates of the deeper layer properties depending on the soil type but provided poor estimates of the first layer properties. The robustness of combining observations of the surface soil moisture and the above-ground biomass for estimating two layer soil properties, which was demonstrated using both synthetic and field experiments in this study, needs now to be tested for a broader range of climatic conditions and crop types, to assess its potential for spatial applications. (C) 2012 Elsevier B.V. All rights reserved.

The influence of stacks on flow patterns and stratification associated with natural ventilation

We investigate the steady state natural ventilation of an enclosed space in which vent A, located at height hA above the floor, is connected to a vertical stack with a termination at height H, while the second vent, B, at height hB above the floor, connects directly to the exterior. We first examine the flow regimes which develop with a distributed source of heating at the base of the space. If hBhB>hA, then two different flow regimes may develop. Either (i) there is inflow through vent B and outflow through vent A, or (ii) the flow reverses, with inflow down the stack into vent A and outflow through vent B. With inflow through vent A, the internal temperature and ventilation rate depend on the relative height of the two vents, A and B, while with inflow through vent B, they depend on the height of vent B relative to the height of the termination of the stack H. With a point source of heating, a similar transition occurs, with a unique flow regime when vent B is lower than vent A, and two possible regimes with vent B higher than vent A. In general, with a point source of buoyancy, each steady state is characterised by a two-layer density stratification. Depending on the relative heights of the two vents, in the case of outflow through vent A connected to the stack, the interface between these layers may lie above, at the same level as or below vent A, leading to discharge of either pure upper layer, a mixture of upper and lower layer, or pure lower layer fluid. In the case of inflow through vent A connected to the stack, the interface always lies below the outflow vent B. Also, in this case, if the inflow vent A lies above the interface, then the lower layer becomes of intermediate density between the upper layer and the external fluid, whereas if the interface lies above the inflow vent A, then the lower layer is composed purely of external fluid. We develop expressions to predict the transitions between these flow regimes, in terms of the heights and areas of the two vents and the stack, and we successfully test these with new laboratory experiments. We conclude with a discussion of the implications of our results for real buildings.