On Fourier's law of heat conduction

A linear theory of fluid is considered in which the gradients of density, internal energy and velocity are among the constitutive variables. Thus the heat flux may be a linear combination of the gradients of density and internal energy. It is proved that this linear combination may be written as the gradient of temperature so that Fourier's law of heat conduction holds.     

On the Stress Field of a Nonlinear Elastic Solid Torus with a Toroidal Inclusion

Standard measures of local deformation such as deformation gradient, strain, elastic deformation, and plastic deformation are dimensionless. However, many macroscopic and submacroscopic geometrical changes observed in continuous bodies result in the formation of zones across whose boundaries significant changes in geometry can occur. In order to predict the sizes of such zones and their influence on material response, theories of elasticity and plasticity have been employed in which second gradients of deformation, gradients of strain, as well as gradients of elastic or of plastic deformation are taken into account. The theory of structured deformations provides additive decompositions of first deformation gradient and of second deformation gradient, valid for large deformations of any material, in which each term has a multiscale geometrical interpretation corresponding to the presence or absence of submacroscopic disarrangements (non-smooth geometrical changes such as slips and void formation). This article provides a field theory that broadens the earlier field theory, elasticity with disarrangements, by including energetic contributions from submacroscopic “gradient-disarrangements” (limits of averages of jumps in gradients of approximating deformations) and by treating particular kinematical conditions as internal constraints. An explicit formula is obtained showing the manner in which submacroscopic gradient-disarrangements determine a defectiveness density analogous to the dislocation density in theories of plasticity. A version of the new field theory incorporates this defectiveness density to obtain a counterpart of strain-gradient plasticity, while another instance of elasticity with gradient-disarrangements recovers an instance of strain-gradient elasticity with symmetric Cauchy stress. All versions of the new theory included here are compatible with the Second Law of Thermodynamics.     

Turbulent Transport Coefficients with Allowance for Density Fluctuations

A model of turbulent transfer processes modified to take into account the effect of density fluctuations is proposed. The model is used for finding expressions for the turbulent transfer coefficients (viscosity, thermal conductivity, and diffusion of admixtures) in flows of the boundary-layer type in the presence of small density gradients. The turbulent Prandtl number distribution in the far wake behind a heated cylinder is compared with the results of measurements.     

Constitutive equations for multiphase mixtures of fluids

Constitutive equations for a multiphase mixture of fluids are presented. The mixture is assumed to consist of a single non-uniform temperature and no change is allowed. The theory is based on the conservation and balance equations of multiphase mixtures proposed by Dobran, and the constitutive assumption allows for the effects of temperature gradient, density gradients, velocity gradients, velocities and accelerations. A linearized form of the constitutive equations is presented for an arbitrary number of phases, and restrictions on the constitutive assumption are investigated by the second law of thermodynamics. The theory yielded a significant number of results and they are compared with previous investigations.     

Change in the wind flow regime in stratified lakes

Wind flows in meromictic saline lakes in which the water column is not mixed to the bottom for at least one year are studied. This leads to the formation of upper and depth layers with small density gradients, between which there is a water layer with a large density gradient. It has been shown that, depending on the density stratification and the wind speed, wind flows (in the vertical plane) of two types are possible: with one or two circulation zones. For a two-layer lake model, a criterion for the change in the wind flow regime is proposed.     

Statistics of velocity gradients in turbulence at moderate reynolds numbers

A direct numerical simulation of the Navier-Stokes equation is performed in order to investigate the small scale structure of turbulence at moderately large microscale Reynolds numbers 40–140, using the spectral method with 340³ modes starting from a high-symmetric flow. It is shown that the small scale motion is statistically isotropic. The probability density distribution of the velocity is Gaussian, while those of the velocity gradients and the vorticity are not Gaussian but have long exponential tails. The moments of the velocity gradients are expressed in terms of the gamma function, and the ratio of the moments of the velocity gradients of successive orders increases linearly with the order. A comparison is made with a laboratory experiment.     

SEMI-IMPLICIT FINITE VOLUME SHALLOW-WATER FLOW AND SOLUTE TRANSPORT SOLVER WITH k–ε TURBULENCE MODEL

A 3D semi-implicit finite volume scheme for shallow- water flow with the hydrostatic pressure assumption has been developed using the σ-co-ordinate system, incorporating a standard k–ε turbulence transport model and variable density solute transport with the Boussinesq approximation for the resulting horizontal pressure gradients. The mesh spacing in the vertical direction varies parabolically to give fine resolution near the bed and free surface to resolve high gradients of velocity, k and ε. In this study, wall functions are used at the bed (defined by the bed roughness) and wind stress at the surface is not considered. Surface elevation gradient terms and vertical diffusion terms are handled implicitly and horizontal diffusion and source terms explicitly, including the Boussinesq pressure gradient term due to the horizontal density gradient. The advection terms are handled in explicit (conservative) form using linear upwind interpolation giving second-order accuracy. A fully coupled solution for the flow field is obtained by substi- tuting for velocity in the depth-integrated continuity equation and solving for surface elevation using a conjugate gradient equation solver. Evaluation of horizontal gradients in the σ-co-ordinate system requires high-order derivatives which can cause spurious flows and this is avoided by obtaining these gradients in real space. In this paper the method is applied to parallel oscillatory (tidal) flow in deep and shallow water and compared with field measurements. It is then applied to current flow about a conical island of small side slope where vortex shedding occurs and velocities are compared with data from the laboratory. Computed concentration distributions are also compared with dye visualization and an example of the influence of temperature on plume dispersion is presented. © 1997 John Wiley & Sons, Ltd.     

A New dynamic model for study of dislocation pattern formation

Based on the principle given in nonlinear diffusion-reaction dynamics, a new dynamic model for dislocation patterning is proposed by introducing a relaxation time to the relation between dislocation density and dislocation flux. The so-called chemical potential like quantities, which appear in the model can be derived from variation principle for free energy functional of dislocated media, where the free energy density function is expessed in terms of not only the dislocation density itself but also their spatial gradients. The linear stability analysis on the governing equations of a simple dislocation density shows that there exists an intrinsic wave number leading to bifurcation of space structure of dislocation density. At the same time, the numerical results also demonstrate the coexistence and transition between different dislocation patterns. The project supported by the National Natural Science Foundation of China, Grant No.19392300     

On the approximation of horizontal gradients in sigma co-ordinates for bathymetry with steep bottom slopes

Nowadays the simulation of free surface flow and transport in rivers, estuaries and seas is often based upon three-dimensional modelling systems. Most of these three-dimensional modelling systems use sigma co-ordinates in the vertical. By the use of the sigma transformation the water column can be divided into the same number of layers independently of the water depth. Especially for steep bottom slopes combined with vertical stratification of the density, sigma-transformed grids impose numerical problems for the accurate approximation of horizontal gradients. This paper deals with algorithms for the approximation in sigma co-ordinates of the horizontal diffusive fluxes of temperature and salinity and for the approximation of the horizontal pressure gradients. The approximation of the horizontal diffusive fluxes is based upon a finite volume method. The approximation of the pressure gradients is directly related to the approximation of the diffusive fluxes. Artificial vertical diffusion and artificial flow due to truncation errors are minimized. The method described in this paper is not hampered by the so-called ‘hydrostatic consistency condition’. This will be illustrated by numerical experiments.     

On configurational forces in multiplicative elastoplasticity

The main goal of this work consists in the elaboration of the material or rather configurational mechanics in the context of multiplicative elastoplasticity. This nowadays well-established approach, which is inherently related to the concept of a material isomorphism or in other words to a local rearrangement, is adopted as a paradigm for the general modelling of finite inelasticity. The overall motion in space is throughout assumed to be compatible and sufficiently smooth. According to the underlying configurations, namely the material and the spatial configuration as well as what we call the intermediate configuration, different representations of balance of linear momentum are set up for the static case. The underlying flux terms are thereby identified as stress tensors of Piola and Cauchy type and are assumed to derive from a free energy density function, thus taking hyperelastic formats. Moreover, the incorporated source terms, namely the configurational volume forces, are identified by comparison arguments. These quantities include gradients of distortions as well as dislocation density tensors. In particular those dislocation density tensors related to the elastic or plastic distortion do not vanish due to the general incompatibility of the intermediate configuration. As a result, configurational volume forces which are settled in the intermediate configuration embody non-vanishing dislocation density tensors while their material counterparts directly incorporate non-vanishing gradients of distortions. This fundamental property enables us to recover the celebrated Peach–Koehler force for finite inelasticity, acting on a single dislocation, from the intermediate configuration volume forces.     

Structure of the fluid interface in an external magnetic field in the presence of a magnetizable surfactant

The structure of the flat interface between two conventional fluids in an external magnetic field in the presence of a magnetizable surfactant is investigated with account for the dependence of the free energy of the system on the surfactant concentration gradients and the bearing phase density. The dependence of the surface tension tensor components on the magnetic field strength is determined.     

On the uncertainty of digital PIV and PTV near walls

The reliable measurement of mean flow properties near walls and interfaces between different fluids or fluid and gas phases is a very important task, as well as a challenging problem, in many fields of science and technology. Due to the decreasing concentration of tracer particles and the strong flow gradients, these velocity measurements are usually biased. To investigate the reason and the effect of the bias errors systematically, a detailed theoretical analysis was performed using window-correlation, singe-pixel ensemble-correlation and particle tracking evaluation methods. The different findings were validated experimentally for microscopic, long-range microscopic and large field imaging conditions. It is shown that for constant flow gradients and homogeneous particle image density, the bias errors are usually averaged out. This legitimates the use of these techniques far away from walls or interfaces. However, for inhomogeneous seeding and/or nonconstant flow gradients, only PTV image analysis techniques give reliable results. This implies that for wall distances below half an interrogation window dimension, the singe-pixel ensemble-correlation or PTV evaluation should always be applied. For distances smaller than the particle image diameter, only PTV yields reliable results.     

Discontinuous solutions of the Navier-Stokes equations for compressible flow

We prove local existence and study properties of discontinuous solutions of the Navier-Stokes equations for one-dimensional, compressible, nonisentropic flow. We assume that, modulo a step function, the initial data is in L² and the initial velocity and density are in the space BV. We show that the velocity and the temperature become smoothed out in positive time, and that discontinuities in the density, pressure, and gradients of the velocity and temperature persist for all time. We also show that for stable gases these discontinuities decay exponentially in time, more rapidly for smaller viscosities.     

圆筒容器内起旋和降旋过程中分层液体混合现象

针对旋转圆筒容器内的两种分层流体,应用平面激光诱导荧光与高速摄影技术对互溶与不互溶液体进行了实验研究.结果表明,上下层液体密度梯度与黏度梯度方向是产生界面不稳定的关键因素.当二者的梯度方向相同时,起旋过程不会发生剧烈混合,降旋过程中轻流体会冲击重流体;当二者的梯度方向相反时,起旋过程中形成抽吸效应,其后期发生界面破碎,降旋过程中重流体冲击轻流体.在旋转的3个过程中,降旋过程对混合的作用最大,无论梯度方向是否相同,都会发生液体间的界面不稳定.     

Completeness,conservation and error in SPH for fluids

Smoothed particle hydrodynamics (SPH) is becoming increasingly common in the numerical simulation of complex fluid flows and an understanding of the errors is necessary. Recent advances have established techniques for ensuring completeness conditions (low‐order polynomials are interpolated exactly) are enforced when estimating property gradients, but the consequences on errors have not been investigated. Here, we present an expression for the error in an SPH estimate, accounting for completeness, an expression that applies to SPH generally. We revisit the derivation of the SPH equations for fluids, paying particular attention to the conservation principles. We find that a common method for enforcing completeness violates a property required of the kernel gradients, namely that gradients with respect the two position variables be equal and opposite. In such models this means conservation principles are not enforced and we present results that show this. As an aside we show the summation interpolant for density is a solution of, and may be used in the place of, the discretized, symmetrized continuity equation. Finally, we examine two examples of discretization errors, namely numerical boundary layers and the existence of crystallized states. Copyright © 2007 John Wiley & Sons, Ltd.     

Investigation of scalar measurement error in diffusion and mixing processes

In variable density, multi-fluid and reacting flows, the degree of molecular mixing is a critical component of turbulent transfer and mixing models. Also, in many microflows and low Reynolds number flows, scalar diffusion length- and time-scales play a significant role in the mixing dynamics. Characterization of such molecular mixing processes requires scalar measurement devices with a small probe volume size. Spatial averaging, which occurs due to finite probe volume size, can lead to errors in resolving the density or scalar gradients between pockets of unmixed fluids. Given a probe volume size and a priori knowledge of the functional profile of the diffusion layer being measured, we obtain an estimate for the measurement error due to spatial averaging and make the corrections accordingly. An analytical model for the measure of scalar mixing is developed as a predictor for the growth of scalar gradients in a variable scalar flow. The model is applied to a buoyancy-driven mixing layer with a Prandtl number of 7. Measurements within the mixing layer have shown that initial entrainment of unmixed fluid causes a decrease in the measured amount of molecular mixing at the centerplane. Following this period of initial entrainment, the fluids within the mixing layer exhibit an increase in the degree of molecular mixing.     

Laser beam deviation as a local density probe

In axisymmetric flows, we show that the laser beam deviation technique can be associated with an Abel inversion calculation in order to give the local density and its gradients. We report density profile measurements in the different regions (zone of silence and barrel shock) of nitrogen and argon supersonic free jets near the nozzle exit. Three different nozzle geometries have been tested at different stagnation pressures. Results are compared to calculations obtained by the method of characteristics.     

Experimental analysis of high-speed helium jet flow using four-dimensional digital speckle tomography

A high-speed and initial helium jet flow has been analyzed by a developed four-dimensional digital speckle tomography. Multiple high-speed cameras have been used to capture movements of speckles in multiple angles of view simultaneously because a shape of a nozzle for the jet flow is asymmetric and the initial jet flow is fast and unsteady. The speckle movements between no flow and helium jet flow from the asymmetric nozzle controlled by a solenoid valve have been obtained by a cross-correlation tracking method so that those distances can be transferred to deflection angles of laser rays for density gradients. The four-dimensional density fields for the high-speed helium jet flow have been reconstructed from the deflection angles by a developed real-time tomography method.