Hydrodynamic Fluctuations in Laminar Fluid Flow. I. Fluctuating Orr-Sommerfeld Equation

In recent years it has become evident that fluctuating hydrodynamics predicts that fluctuations in nonequilibrium states are always spatially long ranged. In this paper we consider the application of fluctuating hydrodynamics to laminar fluid flow, using plane Couette flow as a representative example. Specifically, fluctuating hydrodynamics yields a stochastic Orr-Sommerfeld equation for the wall-normal velocity fluctuations, where spontaneous thermal noise acts as a random source.This stochastic equation needs to be solved subject to appropriate boundary conditions. We show how an exact solution can be obtained from an expansion in terms of the eigenfunctions of the Orr-Sommerfeld hydrodynamic operator. We demonstrate the presence of a flow-induced enhancement of the wall-normal velocity fluctuations and a resulting flow-induced energy amplification and provide a quantitative analysis how these quantities depend on wave number and Reynolds number.     

Direct numerical simulation of the turbulent flows of power-law fluids in a circular pipe

Fully developed turbulent pipe flows of power-law fluids are studied by means of direct numerical simulation. Two series of calculations at generalised Reynolds numbers of approximately 10000 and 20000 were carried out. Five different power law indexes n from 0.4 to 1 were considered. The distributions of components of Reynolds stress tensor, averaged viscosity, viscosity fluctuations, and measures of turbulent anisotropy are presented. The friction coefficient predicted by the simulations is in a good agreement with the correlation obtained from experiment. Flows of power-law fluids exhibit stronger anisotropy of the Reynolds stress tensor compared with the flow of Newtonian fluid. The turbulence anisotropy becomes more significant with the decreasing flow index n. An increase in apparent viscosity away from the wall leads to the damping of the wall-normal velocity pulsations. The suppression of the turbulent energy redistribution between the Reynolds stress tensor components observed in the simulations leads to a strong domination of the axial velocity pulsations. The damping of wall-normal velocity pulsations leads to a reduction of the fluctuating transport of momentum from the core toward the wall, which explains the effect of drag reduction.     

Fluid dynamics of a flow excited resonance,Part II: Flow acoustic interaction

This is the second of two companion papers in which the physics and detailed fluid dynamics of a flow excited resonance are examined. The approach is rather different from those previously used, in which stability theory has been applied to small wavelike disturbances in a linearly unstable shear layer, with an equivalent source driving the sound field which provides the feedback. In the approach used here, the physics of the flow acoustic interaction is explained in terms of the detailed momentum and energy exchanges occurring inside the fluid. Gross properties of the flow and resonance are described in terms of the parameters necessary to determine the behaviour of the feedback system. In this second paper it is shown that two relatively distinct momentum balances can be considered in the resonator neck region. One can be identified with the vortically induced pressure and velocity fluctuations and the other with the reciprocating potential flow. The fluctuating Coriolis force caused by the interaction of the potential and vortical flows is shown to be the only term in the linearized momentum equation which is not directly balanced by a fluctuating pressure gradient. This force provides the mechanism for the exchange of the mean energies associated with the mean and fluctuating momenta, respectively. A source and sink of energy are identified in which mean energy associated with fluctuating momentum is extracted from and returned to the mean flow, respectively. The imbalance between the source and sink is responsible for both the radiated acoustic power and the power carried away by the vortices as they convect downstream. This radiated acoustic power and vortically convected power, and the source and sink powers, are all of the same order of magnitude. With the vortex shedding and reciprocating potential flow “phase locked” the amplitude of the steady state oscillations is determined by the condition that the net power produced in the resonator neck (the source power less the sink power) is equal to the sum of the radiated acoustic power and that carried by the vortices.     

Viscoelastic effects on asymmetric two‐dimensional vortex patterns in a strongly coupled dusty plasma

Strongly coupled dusty plasma medium is often described as a viscoelastic fluid that retains its memory. In a flowing dusty plasma medium, vortices of different sizes appear when the flow does not remain laminar. The vortices also merge to transfer energy between different scales. In the present work, we study the effect of viscoelasticity and compressibility over a localized vortex structure and multiple rotational vortices in a strongly coupled viscoelastic dusty plasma medium. In case of single rotating vortex flow, a transverse wave is generated from the localized vortex source and the evolution time of generated waves is found to be reduced due to finite viscoelasticity and compressibility of the medium. It is found that the viscoelasticity suppresses the dispersion of vorticity. In the presence of multiple vortices, we find, the vortex mergers get highly affected in the presence of memory effect of the fluid, and thus the dynamics of the medium gets completely altered compared to a non‐viscoelastic fluid. For a compressible fluid, viscoelasticity dampens the energy in the sonic waves generated in the medium. Thus a highly viscoelastic and compressible fluid, in some cases, behaves similarly to an incompressible viscoelastic fluid. The wave‐front like rings propagate in elliptical orbits keeping the footprint of the earlier position of the point‐vortex. The rings collide with each other even within the patch vortex region forming regions of high vorticity at the point of intersection and pass through each other.     

Immersed boundary method for the simulation of 2D viscous flow based on vorticity–velocity formulations

A new immersed boundary method based on vorticity–velocity formulations for the simulation of 2D incompressible viscous flow is proposed in present paper. The velocity and vorticity are respectively divided into two parts: one is the velocity and vorticity without the influence of the immersed boundary, and the other is the corrected velocity and the corrected vorticity derived from the influence of the immersed boundary. The corrected velocity is obtained from the multi-direct forcing to ensure the well satisfaction of the no-slip boundary condition at the immersed boundary. The corrected vorticity is derived from the vorticity transport equation. The third-order Runge–Kutta for time stepping, the fourth-order finite difference scheme for spatial derivatives and the fourth-order discretized Poisson for solving velocity are applied in present flow solver. Three cases including decaying vortices, flow past a stationary circular cylinder and an in-line oscillating cylinder in a fluid at rest are conducted to validate the method proposed in this paper. And the results of the simulations show good agreements with previous numerical and experimental results. This indicates the validity and the accuracy of present immersed boundary method based on vorticity–velocity formulations.     

Turbulent structures in an optimal Taylor–Couette flow between concentric counter-rotating cylinders

The turbulent structures formed in a Taylor–Couette (TC) flow established between two concentric counter-rotating cylinders were explored numerically. The shear Reynolds number was set to Re_shear = 8000 and the radius ratio was set to r_i/r_o = 0.5. An optimal flow corresponding to the maximal angular velocity transport between the cylinders was selected for the TC flow. The mean velocity profile reached its steepest value near the cylinders in the optimal TC flow. The streamwise velocity correlations at the outer cylinder in the gap exceeded those at the inner cylinder. The large convective transport of angular velocity in the gap generated a maximal angular velocity flux to achieve the optimal flow. The angular velocity flux generated by the momentum source exceeded that generated by the momentum sink. The vorticity dispersion was larger near the inner cylinder than near the outer cylinder, but vorticity stretching near the outer cylinder exceeded than that near the inner cylinder. The skin friction coefficient budgets were plotted using the velocity–vorticity correlation. The vortex stretching contributions dominated the skin friction budgets. The area near the inner cylinder was populated by stronger vortices, but their population density was smaller than the population density of the vortices near the outer cylinder. The probability density functions of the wall-normal and streamwise velocity fluctuations delineated the presence of the large wall-normal velocity fluctuations near the outer cylinder.     

Influence of structural and doping parameter variations on Si and $$\hbox {Si}_{1-x}$$$$\hbox {Ge}_{x}$$Gex double gate tunnel FETs: An analysis for RF performance enhancement

In this paper, we propose a stochastic evolution of the early Universe which can lead to a fractal correlation in galactic distribution in the Universe. The stochastic equation of state, due to fluctuating creation rates of various components in a many-component fluid, leads to a fluctuating expansion rate for the Universe in the early epochs. It provides persistent fluctuations in the number count vs. apparent magnitude relation, as expected from the observation of a fractal distribution of the galaxies. We also present a stochastic evolution of density perturbations in the early Universe.     

Physical mechanism of the two-dimensional enstrophy cascade

In two-dimensional turbulence, irreversible forward transfer of enstrophy requires anticorrelation of the turbulent vorticity transport vector and the inertial-range vorticity gradient. We investigate the basic mechanism by numerical simulation of the forced Navier-Stokes equation. In particular, we obtain the probability distributions of the local enstrophy flux and of the alignment angle between vorticity gradient and transport vector. These are surprisingly symmetric and cannot be explained by a local eddy-viscosity approximation. The vorticity transport tends to be directed along streamlines of the flow and only weakly aligned down the fluctuating vorticity gradient. All these features are well explained by a local nonlinear model. The physical origin of the cascade lies in steepening of inertial-range vorticity gradients due to compression of vorticity level sets by the large-scale strain field.     

Fluid-magnetic helicity in axisymmetric stationary relativistic magnetohydrodynamics

The present work is intended to gain a fruitful insight into the understanding of the formations of magneto-vortex configurations and their role in the physical processes of mutual exchange of energies associated with fluid’s motion and the magnetic fields in an axisymmetric stationary hydromagnetic system subject to strong gravitational field (e.g., neutron star/magnetar). It is found that the vorticity flux vector field associated with vorticity 2-form is a linear combination of fluid’s vorticity vector and of magnetic vorticity vector. The vorticity flux vector obeys Helmholtz’s flux conservation. The energy equation associated with the vorticity flux vector field is deduced. It is shown that the mechanical rotation of vorticity flux surfaces contributes to the formation of vorticity flux vector field. The dynamo action for the generation of toroidal components of vorticity flux vector field is described in the presence of meridional circulations. It is shown that the stretching of twisting magnetic lines due to differential rotation leads to the breakdown of gravitational isorotation in the absence of meridional circulations. An explicit expression consists of rotation of vorticity flux surface, energy and angular momentum per baryon for the fluid-magnetic helicity current vector is obtained. The conservation of fluid-magnetic helicity is demonstrated. It is found that the fluid-magnetic helicity displays the energy spectrum arising due to the interaction between the mechanical rotation of vorticity flux surfaces and the fluid’s motion obeying Euler’s equations. The dissipation of a linear combination of modified fluid helicity and magnetic twist is shown to occur due to coupled effect of frame dragging and meridional circulation. It is found that the growing twist of magnetic lines causes the dissipation of modified fluid helicity in the absence of meridional circulations.     

MC-Smooth: A mass-conserving,smooth vorticity–velocity formulation for multi-dimensional flows

A novel vorticity–velocity formulation of the Navier–Stokes equations – the Mass-Conserving, Smooth (MC-Smooth) vorticity–velocity formulation – is developed in this work. The governing equations of the MC-Smooth formulation include a new second-order Poisson-like elliptic velocity equation, along with the vorticity transport equation, the energy conservation equation, and N_spec species mass balance equations. In this study, the MC-Smooth formulation is compared to two pre-existing vorticity–velocity formulations by applying each formulation to confined and unconfined axisymmetric laminar diffusion flame problems. For both applications, very good to excellent agreement for the simulation results of the three formulations has been obtained. The MC-Smooth formulation requires the least CPU time and can overcome the limitations of the other two pre-existing vorticity–velocity formulations by ensuring mass conservation and solution smoothness over a broader range of flow conditions. In addition to these benefits, other important features of the MC-Smooth formulation include: (1) it does not require the use of a staggered grid, and (2) it does not require excessive grid refinement to ensure mass conservation. The MC-Smooth formulation is a computationally attractive approach that can effectively extend the applicability of the vorticity–velocity formulation.     

On the Langevin formalism for nonlinear and nonequilibrium hydrodynamic fluctuations

The Landau-Lifshitz method of fluctuating hydrodynamics is generalized to the cases of nonlinear and nonequilibrium fluctuations. For a simple one-component fluid, the multiplicative random fluxes are constructed by using universal Gaussian variables with variances independent of the specific parameters of a fluid. It is shown that the nonlinear Langevin formalism proposed is equivalent to the approach based on the hydrodynamic Fokker-Planck equation derived earlier by statistical-mechanical methods. Then, the scheme is extended to the case of two-component fluids, where cross effects must be taken into account. In conclusion, the connection of the present formalism with the Keizer approach to nonequilibrium fluctuations is discussed.     

An exactly solvable model for coherent and incoherent exciton motion

We treat the motion of a Frenkel exciton using a Hamiltonian which comprises a completely coherent part and a fluctuating part which describes both fluctuations of the energy of a localized exciton and fluctuations of the transition matrix elements between different lattice sites. Under the assumption that the fluctuating forces are Markoffian and Gaussian we derive exactly a density matrix equation which can be solved by a Green's function method.     

Gravity,energy conservation,and parameter values in collapse models

We interpret the probability rule of the CSL collapse theory to mean to mean that the scalar field which causes collapse is the gravitational curvature scalar with two sources, the expectation value of the mass density (smeared over the GRW scale a) and a white noise fluctuating source. We examine two models of the fluctuating source, monopole fluctuations and dipole fluctuations, and show that these correspond to two well-known CSL models. We relate the two GRW parameters of CSL to fundamental constants, and we explain the energy increase of particles due to collapse as arising from the loss of vacuum gravitational energy.     

The effects of porosity and permeability on fluid flow and heat transfer of multi walled carbon nano-tubes suspended in oil (MWCNT/Oil nano-fluid) in a microchannel filled with a porous medium

The forced convection heat transfer and laminar flow in a two-dimensional microchannel filled with a porous medium is numerically investigated. The nano-particles which have been used are multi walled carbon nano-tubes (MWCNT) suspended in oil as the based fluid. The assumption of no-slip condition between the base fluid and nano-particles as well as the thermal equilibrium between them allows us to study the nanofluid in a single phase. The nanofluid flow through the microchannel has been modeled using the Darcy–Forchheimer equation. It is also assumed that there is a thermal equilibrium between the solid phase and the nanofluid for energy transfer. The walls of the microchannel are under the influence of a fluctuating heat flux. Also, the slip velocity boundary condition has been assumed along the walls. The effects of Darcy number, porosity and slip coefficients and Reynolds number on the velocity and temperature profiles and Nusselt number will be studied in this research.     

Wavelet analysis of Navier-Stokes flow

Three-dimensional turbulence is analyzed by wavelet transform. To keep the number of degrees of freedom in appropriate bounds, a reduced set of wavelets is used. Integrating the equation of motion, the following results are obtained: We find strong intermittent fluctuations of the energy dissipation rate and of the vorticity in the viscous range. The vorticity shows the tendency of alignment with the direction of least shear and is organized within elongated tubes. Inertial range properties of the flow are addressed by means of an eddy viscosity or by strongly reducing the spatial resolution of the wavelet basis. In the high Reynolds number limit there seem to be hardly any scaling corrections to the ?5/3-law in accordance with other recent results.     

Direct numerical simulation of stochastically forced laminar plane couette flow: peculiarities of hydrodynamic fluctuations

The background of three-dimensional hydrodynamic (vortical) fluctuations in a stochastically forced, laminar, incompressible, plane Couette flow is simulated numerically. The fluctuating field is anisotropic and has well pronounced peculiarities: (i) the hydrodynamic fluctuations exhibit nonexponential, transient growth; (ii) fluctuations with the streamwise characteristic length scale about 2 times larger than the channel width are predominant in the fluctuating spectrum instead of streamwise constant ones; (iii) nonzero cross correlations of velocity (even streamwise-spanwise) components appear; (iv) stochastic forcing destroys the spanwise reflection symmetry (inherent to the linear and full Navier-Stokes equations in a case of the Couette flow) and causes an asymmetry of the dynamical processes.     

Fluctuation effects in disordered Peierls systems

We review the density of states (DOS) and related quantities of quasi one‐dimensional disordered Peierls systems in which fluctuation effects of a backscattering potential play a crucial role. The low‐energy behavior of non‐interacting fermions which are subject to a static random backscattering potential will be described by the strictly one‐dimensional fluctuating gap model (FGM). Recently, the FGM has also been used to explain the pseudogap phenomenon in high‐T_c superconductors. We develop a non‐perturbative method which allows for a simultaneous calculation of the DOS and inverse localization length for an arbitrary given disorder potential by solving a simple initial value problem. In the white noise limit, we recover all known results by solving a Fokker‐Planck equation. For the physically interesting case of finite correlation lengths, we use analytical and numerical methods to show that a complex order parameter leads to a suppression of the DOS, i.e. a pseudogap, and that for a real order parameter this pseudogap is overshadowed by a singularity in the DOS. We will also consider the case of classical phase fluctuations which applies to low temperatures where amplitude fluctuations are frozen out. For this regime we present analytic results for the DOS, the inverse localization length, the specific heat, and the Pauli susceptibility.     

Fluctuating initial conditions and fluctuations in elliptic and triangular flow

In heavy ion collisions, event-by-event fluctuations in participating nucleon positions can lead to triangular flow. With fluctuating initial conditions, flow coefficients will also fluctuate. In a hydrodynamic model, we study the fluctuations in elliptic and triangular flow, due to fluctuating initial conditions. Both elliptic and triangular flow fluctuates strongly, triangular flow more strongly than the elliptic flow. Strong fluctuations greatly reduce the sensitivity of elliptic and triangular flow to viscosity.     

On gas-dynamics and heat exchange of flows in cylindrical channels in the presence of heat sources bounded along a longitudinal coordinate

The analysis of energy balance equation for viscous laminar flow of fluid or gas in the cylindrical channel in the area (zone) of warm-up bounded along the longitudinal coordinate is made. It was found that at laminar flow of fluid or gas in a round pipe, in each warm-up area bounded along the longitudinal coordinate there are the areas of direct and reverse flows separated by a plane that is a locus of points where temperature is maximal for each fixed value of radial coordinate r.     

Jet engine combustion noise: Pressure,entropy and vorticity perturbations produced by unsteady combustion or heat addition

By idealizing combustion or heat addition processes to occur over a short distance in the flow direction it is possible to calculate the amplitude and phase of the disturbances corresponding to small amplitude fluctuations in the heat addition. The fluctuating heat input is assumed to vary sinusoidally with time and with distance along the direction normal to the flow. Pressure waves propagate away from the heat input region upstream and downstream, whilst on the downstream side waves of vorticity and entropy are convected away. Strong resonant peaks in the pressure and vorticity waves are present close to the cut-off condition of the pressure waves in two dimensions. Generally the wave amplitudes tend to be higher when the mean flow velocity into the region is close to sonic and to become smaller as the steady heat input is increased. For a simplified calculation in which the combustion chamber discharges directly into a multi-stage turbine the downstream noise was predominantly due to the interaction of the entropy with the turbine (i.e., “indirect” rather than “direct” combustion noise).