Theory of viscous and thermal attenuation of sound by small spheres

The problem is to calculate the attenuation of plane sound waves passing through a viscous, heat-conducting fluid containing small spherical inhomogeneities. The attenuation is calculated by evaluating the rate of increase of entropy caused by two irreversible processes: (1) the mechanical work done by the viscous stresses in the presence of velocity gradients, and (2) the flow of heat down the thermal gradients. The method is first applied to a homogeneous fluid with no spheres and shown to give the classical Stokes-Kirchhoff expressions. The method is then used to calculate the additional viscous and thermal attenuation when small spheres are present. The viscous attenuation agrees with Epstein's result obtained in 1941 for a non-heat-conducting fluid. The thermal attenuation is found to be similar in form to the viscous attenuation and, for gases, of comparable magnitude. The general results are applied to the case of water drops in air and air bubbles in water.

For water drops in air the viscous and thermal attenuations are camparable; the thermal losses occur almost entirely in the air, the thermal dissipation in the water being negligible. The theoretical values are compared with Knudsen's experimental data for fogs and found to agree in order of magnitude and dependence on frequency. For air bubbles in water the viscous losses are negligible and the calculated attenuation is almost completely due to thermal losses occurring in the air inside the bubbles, the thermal dissipation in the water being relatively small. (These results apply only to non-resonant bubbles whose radius changes but slightly during the acoustic cycle.)

On the transport properties of fluid-particle flow

The hydrodynamic forces acting on a solid particle in a viscous, incompressible fluid medium at low Reynolds number flow is investigated mathematically as a prerequisite to the understanding of transport processes in two-phase flow involving solid particles and fluid. Viscous interaction between a small number of spherical particles and continuous solid boundaries as well as fluid interface are analyzed under a “point-force” approximation. Non-spherical and elastic spherical particles in a simple shear flow area are then considered. Non-steady motion of a spherical particle is briefly touched upon to illustrate the transient effect of particle motion.

A macroscopic continuum description of particle-fluid flow is formulated in terms of spatial averages yielding a set of particle continuum and bulk fluid equations. Phenomenological formulas describing the transport processes in a fluid medium are extended to cases where the volume concentration of solid particles is sufficiently high to exert an important influence. Hydrodynamic forces acting on a spherical solid particle in such a system, e.g. drag, torque, rotational coupling force, and viscous collision force between streams of different sized particles moving relative to each other are obtained. Phenomenological constants, such as the shear viscosity coefficient, and the diffusion coefficient of the bulk fluid, are found as a function of the material properties of the constituents of the two-phase system and the volume concentration of solid. For transient heat conduction phenomena, it is found that the introduction of a complex conductivity for the bulk fluid permits a simple mathematical description of this otherwise complicated process. The rate of heat transfer between particle continuum and bulk fluid is also investigated by means of an Oseen-type approximation to the energy equation.

Particle fluid mechanics in shear flows, acoustic waves, and shock waves

Three different categories of flow problems of a fluid containing small particles are being considered here. They are: (i) a fluid containing small, non-reacting particles (Parts I and II); (ii) a fluid containing reacting particles (Parts III and IV); and (iii) a fluid containing particles of two distinct sizes with collisions between two groups of particles (Part V).

Part I

A numerical solution is obtained for a fluid containing small particles flowing over an infinite disc rotating at a constant angular velocity. It is a boundary layer type flow, and the boundary layer thickness for the mixture is estimated. For large Reynolds number, the solution suggests the boundary layer approximation of a fluid-particle mixture by assuming W = W_p. The error introduced is consistent with the Prandtl’s boundary layer approximation. Outside the boundary layer, the flow field has to satisfy the “inviscid equation” in which the viscous stress terms are absent while the drag force between the particle cloud and the fluid is still important. Increase of particle concentration reduces the boundary layer thickness and the amount of mixture being transported outwardly is reduced. A new parameter, β = 1/Ω τ_v, is introduced which is also proportional to μ. The secondary flow of the particle cloud depends very much on β. For small values of β, the particle cloud velocity attains its maximum value on the surface of the disc, and for infinitely large values of β, both the radial and axial particle velocity components vanish on the surface of the disc.

Part II

The “inviscid” equation for a gas-particle mixture is linearized to describe the flow over a wavy wall. Corresponding to the Prandtl-Glauert equation for pure gas, a fourth order partial differential equation in terms of the velocity potential ϕ is obtained for the mixture. The solution is obtained for the flow over a periodic wavy wall. For equilibrium flows where λ_v and λ_T approach zero and frozen flows in which λ_v and λ_T become infinitely large, the flow problem is basically similar to that obtained by Ackeret for a pure gas. For finite values of λ_v and λ_T, all quantities except v are not in phase with the wavy wall. Thus the drag coefficient C_D is present even in the subsonic case, and similarly, all quantities decay exponentially for supersonic flows. The phase shift and the attenuation factor increase for increasing particle concentration.

Part III

Using the boundary layer approximation, the initial development of the combustion zone between the laminar mixing of two parallel streams of oxidizing agent and small, solid, combustible particles suspended in an inert gas is investigated. For the special case when the two streams are moving at the same speed, a Green’s function exists for the differential equations describing first order gas temperature and oxidizer concentration. Solutions in terms of error functions and exponential integrals are obtained. Reactions occur within a relatively thin region of the order of λ_D. Thus, it seems advantageous in the general study of two-dimensional laminar flame problems to introduce a chemical boundary layer of thickness λ_D within which reactions take place. Outside this chemical boundary layer, the flow field corresponds to the ordinary fluid dynamics without chemical reaction.

Part IV

The shock wave structure in a condensing medium of small liquid droplets suspended in a homogeneous gas-vapor mixture consists of the conventional compressive wave followed by a relaxation region in which the particle cloud and gas mixture attain momentum and thermal equilibrium. Immediately following the compressive wave, the partial pressure corresponding to the vapor concentration in the gas mixture is higher than the vapor pressure of the liquid droplets and condensation sets in. Farther downstream of the shock, evaporation appears when the particle temperature is raised by the hot surrounding gas mixture. The thickness of the condensation region depends very much on the latent heat. For relatively high latent heat, the condensation zone is small compared with Ʌ_D.

For solid particles suspended initially in an inert gas, the relaxation zone immediately following the compression wave consists of a region where the particle temperature is first being raised to its melting point. When the particles are totally melted as the particle temperature is further increased, evaporation of the particles also plays a role.

The equilibrium condition downstream of the shock can be calculated and is independent of the model of the particle-gas mixture interaction.

Part V

For a gas containing particles of two distinct sizes and satisfying certain conditions, momentum transfer due to collisions between the two groups of particles can be taken into consideration using the classical elastic spherical ball model. Both in the relatively simple problem of normal shock wave and the perturbation solutions for the nozzle flow, the transfer of momentum due to collisions which decreases the velocity difference between the two groups of particles is clearly demonstrated. The difference in temperature as compared with the collisionless case is quite negligible.

Fast Lattice Green's Function Methods for Viscous Incompressible Flows on Unbounded Domains

In this thesis, a collection of novel numerical techniques culminating in a fast, parallel method for the direct numerical simulation of incompressible viscous flows around surfaces immersed in unbounded fluid domains is presented. At the core of all these techniques is the use of the fundamental solutions, or lattice Green’s functions, of discrete operators to solve inhomogeneous elliptic difference equations arising in the discretization of the three-dimensional incompressible Navier-Stokes equations on unbounded regular grids. In addition to automatically enforcing the natural free-space boundary conditions, these new lattice Green’s function techniques facilitate the implementation of robust staggered-Cartesian-grid flow solvers with efficient nodal distributions and fast multipole methods. The provable conservation and stability properties of the appropriately combined discretization and solution techniques ensure robust numerical solutions. Numerical experiments on thin vortex rings, low-aspect-ratio flat plates, and spheres are used verify the accuracy, physical fidelity, and computational efficiency of the present formulations.

Turbomachinery computational fluid dynamics: asymptotes and paradigm shifts.

This paper reviews the development of computational fluid dynamics (CFD) specifically for turbomachinery simulations and with a particular focus on application to problems with complex geometry. The review is structured by considering this development as a series of paradigm shifts, followed by asymptotes. The original S1-S2 blade-blade-throughflow model is briefly described, followed by the development of two-dimensional then three-dimensional blade-blade analysis. This in turn evolved from inviscid to viscous analysis and then from steady to unsteady flow simulations. This development trajectory led over a surprisingly small number of years to an accepted approach-a 'CFD orthodoxy'. A very important current area of intense interest and activity in turbomachinery simulation is in accounting for real geometry effects, not just in the secondary air and turbine cooling systems but also associated with the primary path. The requirements here are threefold: capturing and representing these geometries in a computer model; making rapid design changes to these complex geometries; and managing the very large associated computational models on PC clusters. Accordingly, the challenges in the application of the current CFD orthodoxy to complex geometries are described in some detail. The main aim of this paper is to argue that the current CFD orthodoxy is on a new asymptote and is not in fact suited for application to complex geometries and that a paradigm shift must be sought. In particular, the new paradigm must be geometry centric and inherently parallel without serial bottlenecks. The main contribution of this paper is to describe such a potential paradigm shift, inspired by the animation industry, based on a fundamental shift in perspective from explicit to implicit geometry and then illustrate this with a number of applications to turbomachinery.

Global linear and nonlinear stability of viscous confined plane wakes with co-flow

The global stability of confined uniform density wakes is studied numerically, using two-dimensional linear global modes and nonlinear direct numerical simulations. The wake inflow velocity is varied between different amounts of co-flow (base bleed). In accordance with previous studies, we find that the frequencies of both the most unstable linear and the saturated nonlinear global mode increase with confinement. For wake Reynolds number Re = 100 we find the confinement to be stabilising, decreasing the growth rate of the linear and the saturation amplitude of the nonlinear modes. The dampening effect is connected to the streamwise development of the base flow, and decreases for more parallel flows at higher Re. The linear analysis reveals that the critical wake velocities are almost identical for unconfined and confined wakes at Re ≈ 400. Further, the results are compared with literature data for an inviscid parallel wake. The confined wake is found to be more stable than its inviscid counterpart, whereas the unconfined wake is more unstable than the inviscid wake. The main reason for both is the base flow development. A detailed comparison of the linear and nonlinear results reveals that the most unstable linear global mode gives in all cases an excellent prediction of the initial nonlinear behaviour and therefore the stability boundary. However, the nonlinear saturated state is different, mainly for higher Re. For Re = 100, the saturated frequency differs less than 5% from the linear frequency, and trends regarding confinement observed in the linear analysis are confirmed.

Eccentricity effects on acoustic radiation from a spherical source suspended within a thermoviscous fluid sphere.

Acoustic radiation from a spherical source undergoing angularly periodic axisymmetric harmonic surface vibrations while eccentrically suspended within a thermoviscous fluid sphere, which is immersed in a viscous thermally conducting unbounded fluid medium, is analyzed in an exact fashion. The formulation uses the appropriate wave-harmonic field expansions along with the translational addition theorem for spherical wave functions and the relevant boundary conditions to develop a closed-form solution in form of infinite series. The analytical results are illustrated with a numerical example in which the vibrating source is eccentrically positioned within a chemical fluid sphere submerged in water. The modal acoustic radiation impedance load on the source and the radiated far-field pressure are evaluated and discussed for representative values of the parameters characterizing the system. The proposed model can lead to a better understanding of dynamic response of an underwater acoustic lens. It is equally applicable in miniature transducer analysis and design with applications in medical ultrasonics.

Magnetically actuated artificial cilia: the effect of fluid inertia.

Natural cilia are hairlike microtubule-based structures that are able to move fluid on the micrometer scale using asymmetric motion. In this article, we follow a biomimetic approach to design artificial cilia lining the inner surfaces of microfluidic channels with the goal of propelling fluid. The artificial cilia consist of polymer films filled with superparamagnetic nanoparticles, which can mimic the motion of natural cilia when subjected to a rotating magnetic field. To obtain the magnetic field and associated magnetization local to the cilia, we solve the Maxwell equations, from which the magnetic body moments and forces can be deduced. To obtain the ciliary motion, we solve the dynamic equations of motion, which are then fully coupled to the Navier-Stokes equations that describe the fluid flow around the cilia, thus taking full account of fluid inertial forces. The dimensionless parameters that govern the deformation behavior of the cilia and the associated fluid flow are arrived at using the principle of virtual work. The physical response of the cilia and the fluid flow for different combinations of elastic, fluid viscous, and inertia forces are identified.