Standing and travelling waves in the shallow-water circular hydraulic jump

A wave equation for a time-dependent perturbation about the steady shallow-water solution emulates the metric an acoustic white hole, even upon the incorporation of nonlinearity in the lowest order. A standing wave in the sub-critical region of the flow is stabilised by viscosity, and the resulting time scale for the amplitude decay helps in providing a scaling argument for the formation of the hydraulic jump. A standing wave in the super-critical region, on the other hand, displays an unstable character, which, although somewhat mitigated by viscosity, needs nonlinear effects to be saturated. A travelling wave moving upstream from the sub-critical region, destabilises the flow in the vicinity of the jump, for which experimental support has been given.     

A level-set continuum method for two-phase flows with insoluble surfactant

A level-set continuum surface force method is presented to compute two-phase flows with insoluble surfactant. Our method recasts the Navier–Stokes equations for a two-phase flow with insoluble surfactant as “one-fluid” formulation. Interfacial transport and interfacial jump conditions are treated using the level-set method and the discrete Dirac function. Based on the density-weighted projection method, a stable semi-implicit scheme is used to decouple the velocity components in solving the regularized Navier–Stokes equations. It allows numerical simulations for a wide range of viscosity ratios and density ratios.Numerical simulations on single drop deformation in a 2D shear flow are presented. Simulations on two drop interaction shows that surfactants can play a critical role in preventing drop coalescence. A fully 3D simulation demonstrating the physical interactions of multiple surfactant-laden drops is presented.     

Variable Fluid Property Effect on Heat Transfer and Frictional Flow Characteristics of Water Flowing through Microchannel

The effects of temperature-dependent viscosity and thermal conductivity on heat transfer and frictional flow characteristics of water flowing through a microchannel are numerically investigated in this work. The hydrodynamically and thermally developing flow with no-slip, notemperature jump, and constant wall heat flux boundary condition is numerically studied using 2D continuum-based conservation equations. A significant deviation in Nusselt number from conventional theory is observed due to flattening of axial velocity profile due to temperaturedependent viscosity variation. The Nusselt number shows a significant deviation from conventional theory due to flattening of the radial temperature profile due to temperature-dependent thermal conductivity variation. It is noted that the deviation in Nusselt number from conventional theory is maximum for combined temperature-dependent viscosity and thermal conductivity variations. The effects of temperature-dependent viscosity and thermal conductivity on the Fanning friction factor are also investigated. Additionally, the effects of variable fluid properties on Poiseuille number, Prandtl number, and Peclet number are also investigated.     

Relativistic Mechanics of Continuous Media

In this work we study the relativistic mechanics of continuous media on a fundamental level using a manifestly covariant proper time procedure. We formulate equations of motion and continuity (and constitutive equations) that are the starting point for any calculations regarding continuous media. In the force free limit, the standard relativistic equations are regained, so that these equations can be regarded as a generalization of the standard procedure. In the case of an inviscid fluid we derive an analogue of the Bernoulli equation. For irrotational flow we prove that the velocity field can be derived from a potential. If in addition, the fluid is incompressible, the potential must obey the d'Alembert equation, and thus the problem is reduced to solving the d'Alembert equation with specific boundary conditions (in both space and time). The solutions indicate the existence of light velocity sound waves in an incompressible fluid (a result known in previous literature ⁽¹⁹⁾ ). Relaxing the constraints and allowing the fluid to become linearly compressible one can derive a wave equation, from which the sound velocity can again be computed. For a stationary background flow, it has been demonstrated that the sound velocity attains its correct values for the incompressible and nonrelativistic limits. Finally viscosity is introduced, bulk and shear viscosity constants are defined, and we formulate equations for the motion of a viscous fluid.     

基于第二粘性的Navier-Stokes方程组求解正激波结构

为考察气体第二粘性（体积粘性）对正激波内部流动的影响机制,数值求解含第二粘性的一维Navier-Stokes方程组.结果表明：第二粘性对激波内部的密度、热流和能量分布等物理量具有抹平效应,导致热流和熵流的峰值减小、激波厚度增加,体积粘性耗散的增加使得一部分机械能转化为内能;考虑第二粘性所计算的密度分布和激波厚度大为改善,与实验数据吻合较好;当马赫数为1.2≤Ma≤10,激波内部的Knudsen数满足0.12≤Kn≤0.4,对于马赫数Ma≤4.0的激波内部流动,考虑第二粘性的连续流Navier-Stokes方程组能够准确地模拟正激波结构.     

叶栅内定常及非定常粘性流动数值模拟

本文给出了一个模拟叶栅内准三维定常和非定常粘性流动的数值方法。对于定常流动,采用ＴＶＤ Ｌａｘ－Ｗｅｎｄｒｏｆｆ格式和代数湍流模型求解雷诺平均Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程,使用当地时间步长和多网格技术使计算加速收敛到定常状态;对于非定常流动,使用双时间步长和全隐式离散,采用与求解定常流动相似的多网格方法求解隐式离散方程。文中给出了ＶＫＩ透平叶栅内的定常流结果和１．５级透平叶栅内的非定常数值结果。     

Rapid communication: Transverse spin with coupled plasmons

The steady two-dimensional flow and heat transfer of a non-Newtonian power-law nanofluid over a stretching surface under convective boundary conditions and temperature-dependent fluid viscosity has been numerically investigated. The power-law rheology is adopted to describe non-Newtonian characteristics of the flow. Four different types of nanoparticles, namely copper (Cu), silver (Ag), alumina (Al ₂ O ₃) and titanium oxide (TiO ₂) are considered by using sodium alginate (SA) as the base non-Newtonian fluid. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The transformed equations are solved numerically by using a shooting method with fourth-order Runge–Kutta integration scheme. The results show that the effect of viscosity on the heat transfer rate is remarkable only for relatively strong convective heating. Moreover, the skin friction coefficient and the rate of heat transfer increase with an increase in Biot number.     

Shear viscosity for inelastic hard spheres

The hydrodynamic equations of the Enskog theory for inelastic hard spheres is considered as a model for rapid flow granular fluids at finite densities. A detailed analysis of the shear viscosity of the granular fluid has been done using homogenous cooling state (HCS) and uniform shear flow (USF) models. It is found that shear viscosity is sensitive to the coefficient of restitution α and pair correlation function at contact. The collisional part of the Newtonian shear viscosity is found to be dominant than its kinetic part.     

Rheological Analysis of CNT Suspended Nanofluid with Variable Viscosity: Numerical Solution

In this article, we discuss the two-dimensional stagnation-point flow of carbon nanotubes towards a stretching sheet with water as the base fluid under the influence temperature dependent viscosity. Similarity transformations are used to simplify the governing boundary layer equations for nanofluid. This is the first article on the stagnation point flow of CNTs over a stretching sheet with variable viscosity. A well known Reynold's model of viscosity is used. Single wall CNTs are used with water as a base fluid. The resulting nonlinear coupled equations with the relevant boundary conditions are solved numerically using shooting method. The influence of the flow parameters on the dimensionless velocity, temperature, skin friction, and Nusselt numbers are explored and presented in forms of graphs and interpreted physically.     

A negative biasing experiment to test numerical calculations basedon a neoclassical L-H transition model in stellarators

Numerical calculations based on Shaing's L-H transition theory in stellarators and performed by Dahi et al. determine viscosity as a function of ion flow speed in the interchangeable module stellarator (IMS). The calculations predict local maximums in viscosity at flows corresponding to Mach numbers -2 and -10, but not elsewhere. The local peaks manifest themselves as jumps in flow speed, and as regions of high radial electric field (E_r) shear. By inducting flows swept between Mach numbers ±5, an electron injection biasing probe revealed a jump at -2, but none at +2, in agreement with the numerical results. A series of flow profiles at constant bias confirmed this result. Altogether, these data agree well with the numerical calculations and provide support for Shaing's L-H transition model as applied to stellarators     

二维卡门涡街的格子Boltzmann仿真

格子气自动机和格子Boltzmann方法的迅速发展提供了一类求解流体力学问题的新的方法。本文中，我们介绍了Boltzmann方法，解决了格子气方法中的缺点，通过选择适当平衡分布及其参数，导出了Navier－Stokers方程，并得到了声速和粘性系数。最后在微机上模拟了在无限长平板流动问题及绕单分离板的流动问题，得到了卡门涡街。结果表该模型有格子气方法及其它的数值方法所没有的优点，计算更精确、更直观、更有效。     

Mechanics of the turbulent-nonturbulent interface of a jet

We report the results of an experimental investigation of the mechanics and transport processes at the bounding interface between the turbulent and nonturbulent regions of flow in a turbulent jet, which shows the existence of a finite jump in the tangential velocity at the interface. This is associated with small-scale eddying motion at the outward propagating interface (nibbling) by which irrotational fluid becomes turbulent, and this implies that large-scale engulfment is not the dominant entrainment process. Interpretation of the jump as a singular structure yields an essential and significant contribution to the mean shear in the jet mixing region. Finally, our observations provide a justification for Prandtl's original hypothesis of a constant eddy viscosity in the nonturbulent outer jet region.     

Anomalous transport in fluid fleld with random waiting time depending on the preceding jump length

Anomalous(or non-Fickian) transport behaviors of particles have been widely observed in complex porous media.To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields,in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced,and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived.As examples,two generalized advection-dispersion equations for Gaussian distribution and levy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

Anomalous transport in fluid field with random waiting time depending on the preceding jump length

Anomalous (or non-Fickian) transport behaviors of particles have been widely observed in complex porous media. To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields, in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced, and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived. As examples, two generalized advection-dispersion equations for Gaussian distribution and lévy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

MHD free convection flow over an isothermal vertical cone with temperature dependent viscosity

Free convection flow over an isothermal vertical cone immersed in a fluid with variable viscosity and MHD is studied in this paper. Using appropriate variables, the basic equations are transformed into the non-dimensional boundary-layer equations. These equations are then solved numerically using a very efficient implicit finite-difference method known as Crankl-Nicolson scheme. Detailed results for the velocity, temperature, skin friction, and heat transfer rates for a selection of parameter sets consisting of the viscosity parameter, magnetic field parameter, and Prandtl number are discussed. In order to validate our numerical results, the present results are compared with the available work in the literature and are found to be in an excellent agreement.     

Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions

Several theories for weakly damped free-surface flows have been formulated. In this Letter we use the linear approximation to the Navier-Stokes equations to derive a new set of equations for potential flow which include dissipation due to viscosity. A viscous correction is added not only to the irrotational pressure (Bernoulli's equation), but also to the kinematic boundary condition. The nonlinear Schrödinger (NLS) equation that one can derive from the new set of equations to describe the modulations of weakly nonlinear, weakly damped deep-water gravity waves turns out to be the classical damped version of the NLS equation that has been used by many authors without rigorous justification.     

The effect of turbulence damping on acoustic wave propagation in tubes

The attenuation of sound due to the interaction between a low Mach number turbulent boundary layer and acoustic waves can be significant at low frequencies or in narrow tubes. In a recent publication by the present authors the acoustics of charge air coolers for passenger cars has been identified as an interesting application where turbulence attenuation can be of importance. Favourable low-frequency damping has been observed that could be used for control of the in-duct sound that is created by the engine gas exchange process. Analytical frequency-dependent models for the eddy viscosity that controls the momentum and thermal boundary layers are available but are restricted to thin acoustic boundary layers. For cases with cross-sections of a few millimetres a model based on thin acoustic boundary layers will not be applicable in the frequency range of interest.In the present paper a frequency-dependent axis-symmetric numerical model for interaction between turbulence and acoustic waves is proposed. A finite element scheme is used to formulate the time harmonic linearized convective equations for conservation of mass, momentum and energy into one coupled system of equations. The turbulence is introduced with a linear model for the eddy viscosity that is added to the shear viscosity. The proposed model is validated by comparison with experimental data from the literature.     

Impact of Internal Heat Source on Mixed Convective Transverse Transport of Viscoplastic Material under Viscosity Variation

This communication addresses the impact of heat source/sink along with mixed convection on oblique flow of Casson fluid having variable viscosity. Similarity analysis has been utilized to model governing equations, which are simplified to set of nonlinear differential equations. Computational procedure of shooting algorithm along with 4 th order Range-Kutta-Fehlberg scheme is opted to attain the velocity and temperature distributions. Impact of imperative parameters on Casson fluid flow, temperature, significant physical quantities such as skin friction, local heat flux and streamlines are displayed via graphs.     

Impact of Internal Heat Source on Mixed Convective Transverse Transport of Viscoplastic Material under Viscosity Variation

This communication addresses the impact of heat source/sink along with mixed convection on oblique flow of Casson fluid having variable viscosity. Similarity analysis has been utilized to model governing equations, which are simplified to set of nonlinear differential equations. Computational procedure of shooting algorithm along with 4th order Range-Kutta-Fehlberg scheme is opted to attain the velocity and temperature distributions. Impact of imperative parameters on Casson fluid flow, temperature, significant physical quantities such as skin friction, local heat flux and streamlines are displayed via graphs.