Purely irrotational theories for the viscous effects on the oscillations of drops and bubbles

In this paper, we apply two purely irrotational theories of the motion of a viscous fluid, namely, viscous potential flow (VPF) and the dissipation method to the problem of the decay of waves on the surface of a sphere. We treat the problem of the decay of small disturbances on a viscous drop surrounded by gas of negligible density and viscosity and a bubble immersed in a viscous liquid. The instantaneous velocity field in the viscous liquid is assumed to be irrotational. In VPF, viscosity enters the problem through the viscous normal stress at the free surface. In the dissipation method, viscosity appears in the dissipation integral included in the mechanical energy equation. Comparisons of the eigenvalues from VPF and the dissipation approximation with those from the exact solution of the linearized governing equations are presented. The results show that the viscous irrotational theories exhibit most of the features of the wave dynamics described by the exact solution. In particular, VPF and DM give rise to a viscous correction for the frequency that determines the crossover from oscillatory to monotonically decaying waves. Good to reasonable quantitative agreement with the exact solution is also shown for certain ranges of modes and dimensionless viscosity: For large viscosity and short waves, VPF is a very good approximation to the exact solution. For ‘small’ viscosity and long waves, the dissipation method furnishes the best approximation.     

Neural network reconstruction of fluid flows from tracer-particle displacements

We demonstrate some of the advantages of using artificial neural networks for the post-processing of particle-tracking velocimetry (PTV) data. This study is concerned with the data obtained after particle images have been matched and the obvious outliers have been removed. We show that it is easy to produce simple back-propagation neural networks that can filter the remaining random noise and interpolate between the measurements. They do so by performing a particular form of non-linear global regression that allows them to reconstruct the fluid flow for the entire field covered by the photographs. This is obtained by training these neural networks to learn the fluid dynamics function f that maps the position x of a fluid particle at time t to its position X at time t + Δt. They can do so with a high degree of precision when provided with pairs of matching particle positions (x, X) from only about 2 to 4 pairs of PTV photographs as exemplars. We show that whether they are trained on exact or on noisy data, they learn to interpolate with such a precision that their output is within one pixel of the theoretical output. We demonstrate their accuracy by using them to draw whole streamlines or flow profiles, by iteration from a single starting point. Received: 23 November 1998/Accepted: 14 July 2000     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Algorithms for interface treatment and load computation in embedded boundary methods for fluid and fluid–structure interaction problems

Embedded boundary methods for CFD (computational fluid dynamics) simplify a number of issues. These range from meshing the fluid domain, to designing and implementing Eulerian‐based algorithms for fluid–structure applications featuring large structural motions and/or deformations. Unfortunately, embedded boundary methods also complicate other issues such as the treatment of the wall boundary conditions in general, and fluid–structure transmission conditions in particular. This paper focuses on this aspect of the problem in the context of compressible flows, the finite volume method for the fluid, and the finite element method for the structure. First, it presents a numerical method for treating simultaneously the fluid pressure and velocity conditions on static and dynamic embedded interfaces. This method is based on the exact solution of local, one‐dimensional, fluid–structure Riemann problems. Next, it describes two consistent and conservative approaches for computing the flow‐induced loads on rigid and flexible embedded structures. The first approach reconstructs the interfaces within the CFD solver. The second one represents them as zero level sets, and works instead with surrogate fluid/structure interfaces. For example, the surrogate interfaces obtained simply by joining contiguous segments of the boundary surfaces of the fluid control volumes that are the closest to the zero level sets are explored in this work. All numerical algorithms presented in this paper are applicable with any embedding CFD mesh, whether it is structured or unstructured. Their performance is illustrated by their application to the solution of three‐dimensional fluid–structure interaction problems associated with the fields of aeronautics and underwater implosion. Copyright © 2011 John Wiley & Sons, Ltd.     

Motion of two circular cylinders in an ideal fluid

A solution of the problem concerning the motion of two circular cylinders in an ideal fluid was given earlier using approximate methods; the error made in this treatment of the problem increased with approach of the cylinders to one another [1, 2]. In this paper we give an exact solution of the problem for arbitrary motion of the cylinders. We define a velocity potential, the kinetic energy, and the forces acting on the cylinders from the fluid side.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 80–84, November–December, 1970.The author wishes to thank N. S. Storozhuk for his aid in preparing this paper.     

Particles accelerate the detachment of viscous liquids

During detachment of a viscous fluid extruded from a nozzle, a filament linking the droplet to the latter is formed. Under the effect of surface tension, the filament thins until pinch-off and final detachment of the droplet. In this paper, we study the effect of the presence of individual particles trapped in the filament on the detachment dynamics using granular suspensions of small volume fractions (??<?6 %). We show that even a single particle strongly modifies the detachment dynamics. The particle perturbs the thinning of the thread, and a large droplet of fluid around the particle is formed. This perturbation leads to an acceleration of the detachment of the droplet compared to the detachment observed for a pure fluid. We quantify this acceleration for single particles of different sizes and link it to similar observations for suspensions of small volume fractions. Our study also gives more insight into particulate effects on detachment of denser suspensions and allows to explain the accelerated detachment close to final pinch-off observed previously (Bonnoit et al. Phys Fluids 24(4):043304, 2012).     

Models for the deformation of a single ellipsoidal drop: a review

Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends. This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components, the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects alter the drop dynamics; and the confined non-Newtonian system.     

Propagation of weakly nonlinear disturbances of the interface between two layers of fluid

The dynamics of internal waves of small but finite amplitude in a two-layer fluid system bounded by rigid horizontal surfaces at bottom and top is investigated theoretically. For linear disturbances of the fluid interface the authors propose a polynomial approximation of the dispersion relation which has the same asymptotics as the exact formula in the limiting situations of very long and short waves. In the case of three-dimensional, weakly nonlinear disturbances of slowly varying shape (in the coordinate system moving with the wave) an equation like the wave equation is derived. This equation has Stokes solutions coinciding with the well-known results for infinitely deep layers. For fairly long disturbances solitary solutions of the model wave equation which fit the experimental data are determined. Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 125–131, January–February, 1994.     

Poroelasticity-I: Governing Equations of the Mechanics of Fluid-Saturated Porous Materials

We present here the approach to the theory of fluid-filled poroelastics based on consideration of poroelastics as a continuum of “macropoints” (representative elementary volumes), which “internal” states can be described by as a set of internal parameters, such as local relative velocity of fluid and solid, density of fluid, internal strain tensor, specific area, and position of the center of mass of porous space. We use the generalized Cauchy–Born hypothesis and suggest that there is a system of (structural) relationships between external parameters, describing the deformation of the continuum and internal parameters, characterizing the state of representative elementary volumes. We show that in nonhomogenous (and, particularly, nonlinear) poroelastics, an interaction force between solid and fluid appears. Because this force is proportional to the gradient of porosity, absent in homogeneous poroelastics, and one can neglect with dynamics of internal degrees of freedom, this force is equivalent to the interaction force, introduced earlier by Nikolaevskiy from phenomenological reasons. At last, we show that developed theory naturally incorporates three mechanisms of energy absorption: visco-inertial Darcy mechanism, “squirt flow” attenuation, and skeleton attenuation.     

Droplets splashing upon films of the same fluid of various depths

We explore the effects of fluid films of variable depths on droplets impacting into them. Corresponding to a range of fluid “film” depths, a non-dimensional parameter—H*, defined as the ratio of the film thickness to the droplet diameter—is varied in the range 0.1≤H*≤10. In general, the effect of the fluid film imposes a dramatic difference on the dynamics of the droplet–surface interaction when compared to a similar impact on a dry surface. This is illustrated by the size distribution and number of the splash products. While thin fluid films (H*≈0.1) promote splashing, thicker films (1≤H*≤10) act to inhibit it. The relative roles of surface tension and viscosity are investigated by comparison of a matrix of fluids with low and high values of these properties. Impingement conditions, as characterized by Reynolds and Weber numbers, are varied by velocity over a range from 1.34 to 4.22 m/s, maintaining a constant droplet diameter of 2.0 mm. The dependence of splashing dynamics, characterized by splash product size and number, on the fluid surface tension and viscosity and film thickness are discussed.     

Conservation of circulation in SPH for 2D free‐surface flows

In this paper, we study how accurately the Smoothed Particle Hydrodynamics (SPH) scheme accounts for the conservation and the generation of vorticity and circulation, in a low viscosity, weakly compressible, barotropic fluid in the context of free‐surface flows. We consider a number of simple examples to clarify the processes involved and the accuracy of the simulations. The first example is a differentially rotating fluid where the integration path for the circulation becomes progressively more complicated, whereas the structure of the velocity field remains simple. The second example is the collision of two rectangular regions of fluid. We show that SPH accurately predicts the time variation of the circulation as well as the total vorticity for selected domains advected by the fluid. Finally, a breaking wave is considered. For such a problem we show how the dynamics of the vorticity generated by the breaking process is captured by the SPH model. Copyright © 2012 John Wiley & Sons, Ltd.     

Multiresolution reproducing kernel particle method for computational fluid dynamics

Multiresolution analysis based on the reproducing kernel particle method (RKPM) is developed for computational fluid dynamics. An algorithm incorporating multiple-scale adaptive refinement is introduced. The concept of using a wavelet solution as an error indicator is also presented. A few representative numerical examples are solved to illustrate the performance of this new meshless method. Results show that the RKPM is a good candidate for tackling the widespread large-scale problems in fluid dynamics. © 1997 John Wiley & Sons, Ltd.     

Effects of Bubbles on the Hydraulic Conductivity of Porous Materials – Theoretical Results

In a porous material, both the pressure drop across a bubble and its speed are nonlinear functions of the fluid velocity. Nonlinear dynamics of bubbles in turn affect the macroscopic hydraulic conductivity, and thus the fluid velocity. We treat a porous medium as a network of tubes and combine critical path analysis with pore-scale results to predict the effects of bubble dynamics on the macroscopic hydraulic conductivity and bubble density. Critical path analysis uses percolation theory to find the dominant (approximately) one-dimensional flow paths. We find that in steady state, along percolating pathways, bubble density decreases with increasing fluid velocity, and bubble density is thus smallest in the smallest (critical) tubes. We find that the hydraulic conductivity increases monotonically with increasing capillary number up to Ca 10^–2, but may decrease for larger capillary numbers due to the relative decrease of bubble density in the critical pores. We also identify processes that can provide a positive feedback between bubble density and fluid flow along the critical paths. The feedback amplifies statistical fluctuations in the density of bubbles, producing fluctuations in the hydraulic conductivity.     

The Interface Between Fresh and Salt Groundwater in Horizontal Aquifers: The Dupuit–Forchheimer Approximation Revisited

We analyze the motion of a sharp interface between fresh and salt groundwater in horizontal, confined aquifers of infinite extend. The analysis is based on earlier results of De Josselin de Jong (Proc Euromech 143:75–82, 1981). Parameterizing the height of the interface along the horizontal base of the aquifer and assuming the validity of the Dupuit–Forchheimer approximation in both the fresh and saltwater, he derived an approximate interface motion equation. This equation is a nonlinear doubly degenerate diffusion equation in terms of the height of the interface. In that paper, he also developed a stream function-based formulation for the dynamics of a two-fluid interface. By replacing the two fluids by one hypothetical fluid, with a distribution of vortices along the interface, the exact discharge field throughout the flow domain can be determined. Starting point for our analysis is the stream function formulation. We derive an exact integro-differential equation for the movement of the interface. We show that the pointwise differential terms are identical to the approximate Dupuit–Forchheimer interface motion equation as derived by De Josselin de Jong. We analyze (mathematical) properties of the additional integral term in the exact interface motion formulation to validate the approximate Dupuit–Forchheimer interface motion equation. We also consider the case of flat interfaces, and we study the behavior of the toe of the interface. In particular, we give a criterion for finite or infinite speed of propagation.     

Exact and numerical solutions of a fully developed generalized second-grade incompressible fluid with power-law temperature-dependent viscosity

We compute exact and numerical solutions of a fully developed flow of a generalized second-grade fluid, with power-law temperature-dependent viscosity (μ=θ^-M), down an inclined plane. Analytical solutions are found for the case when M=m+1, m≠1, m being a constant that models shear thinning (m<0) or shear thickening (m>0). The exact solutions are given in terms of Bessel functions. The numerical solutions indicate that both the velocity and the temperature increase with decreasing Froude number and that there is a critical value of Fr below which temperature “overshoots” its free surface value of unity. This phenomena is not reported in the work of Massoudi and Phuoc [Fully developed flow of a modified second grade fluid with temperature dependent viscosity, Acta Mech. 150 (2001) 23-37.] for viscosity that depends exponentially on temperature.     

Analytical solutions for non-Newtonian fluid flows in pipe-like domains

This paper deals with some unsteady unidirectional transient flows of an Oldroyd-B fluid in unbounded domains which geometrically are axisymmetric pipe-like. An expansion theorem of Steklov is used to obtain exact solutions for flows satisfying no-slip boundary conditions. The well known solutions for a Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. The steady state solutions are also obtained for t→∞.     

On variational features of vortex flows

Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones, the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997; Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property. This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion is discussed.      

Dynamics of a rigid body with an ellipsoidal cavity filled with viscous fluid

We revisit the approach proposed by F.L.  Chernousko to modeling the dynamics of a rigid body with a cavity entirely filled with a highly viscous fluid. Within the approach, a finite-dimensional model of the body+fluid system is offered and the influence of the fluid is represented as a special torque acting upon the body with solidified fluid. Our aim is to develop further and expand a few technical aspects of the Chernousko model. In particular, we offer a coordinate-free form for some essential formulas and consider the case of constrained dynamics. To illustrate the results obtained we explore the motion of a physical pendulum with a fluid-filled cavity on a rotating platform.     

An embedded boundary framework for compressible turbulent flow and fluid–structure computations on structured and unstructured grids

The finite volume method with exact two‐phase Riemann problems (FIVER) is a two‐faceted computational method for compressible multi‐material (fluid–fluid, fluid–structure, and multi‐fluid–structure) problems characterized by large density jumps, and/or highly nonlinear structural motions and deformations. For compressible multi‐phase flow problems, FIVER is a Godunov‐type discretization scheme characterized by the construction and solution at the material interfaces of local, exact, two‐phase Riemann problems. For compressible fluid–structure interaction (FSI) problems, it is an embedded boundary method for computational fluid dynamics (CFD) capable of handling large structural deformations and topological changes. Originally developed for inviscid multi‐material computations on nonbody‐fitted structured and unstructured grids, FIVER is extended in this paper to laminar and turbulent viscous flow and FSI problems. To this effect, it is equipped with carefully designed extrapolation schemes for populating the ghost fluid values needed for the construction, in the vicinity of the fluid–structure interface, of second‐order spatial approximations of the viscous fluxes and source terms associated with Reynolds averaged Navier–Stokes (RANS)‐based turbulence models and large eddy simulation (LES). Two support algorithms, which pertain to the application of any embedded boundary method for CFD to the robust, accurate, and fast solution of FSI problems, are also presented in this paper. The first one focuses on the fast computation of the time‐dependent distance to the wall because it is required by many RANS‐based turbulence models. The second algorithm addresses the robust and accurate computation of the flow‐induced forces and moments on embedded discrete surfaces, and their finite element representations when these surfaces are flexible. Equipped with these two auxiliary algorithms, the extension of FIVER to viscous flow and FSI problems is first verified with the LES of a turbulent flow past an immobile prolate spheroid, and the computation of a series of unsteady laminar flows past two counter‐rotating cylinders. Then, its potential for the solution of complex, turbulent, and flexible FSI problems is also demonstrated with the simulation, using the Spalart–Allmaras turbulence model, of the vertical tail buffeting of an F/A‐18 aircraft configuration and the comparison of the obtained numerical results with flight test data. Copyright © 2014 John Wiley & Sons, Ltd.     

Macroscopic Behavior of Microscopic Oscillations in Harmonic Lattices via Wigner-Husimi Transforms

We consider the dynamics of infinite harmonic lattices in the limit of the lattice distance ɛ tending to 0. We allow for general polyatomic crystals, but assume exact periodicity such that the system can be solved, in principle, by Fourier-transform and linear-algebra methods. Our aim is to derive macroscopic continuum limit equations for ɛ →0. For the weak limit of displacements and velocities we obtain the equation of linear elastodynamics, where the elasticity tensor is obtained as a Γ-limit. The weak limit of the local energy density can be described by generalizations of the Wigner-Husimi measure, which satisfies a transport equation on the product of physical space and Fourier space. The concepts are illustrated via several examples and a comparison to Whitham's modulation equation.