Annular liquid jets and other axisymmetric free-surface flows at high Reynolds numbers

A perturbation method based on a long wavelength approximation is used to obtain the leading order equations governing the fluid dynamics of laminar, annular, round and compound liquid jets and liquid films on convex and concave cylindrical surfaces. An approximate, integral balance method is also used to determine the inviscid core and the thickness of the boundary layers of annular liquid jets near the nozzle exit. The steady state equations are transformed into parabolic ones by means of the von Mises transformation and solved in an adaptive, staggered grid to determine the axial velocity distribution and the location of the free surfaces. It is shown that, for free surface flows subject to inertia, gravity and surface tension, there is a contraction near the nozzle which increases as the Reynolds and Froude numbers are decreased, and is nearly independent of the Weber number for Weber numbers larger than about one hundred. It is also shown that this contraction depends on the flow considered, and is larger for films on convex surfaces. It is also shown that, for round jets, the acceleration of the jet's free surface is larger than that of the jet's centerline, although, sufficiently far from the nozzle exit, the axial velocity is uniform across the jet.     

Asymptotic Analysis and Stability of Inviscid Liquid Sheets

Asymptotic methods are employed to determine the leading-order equations that govern the fluid dynamics of slender, and thin and slender, inviscid, irrotational, planar liquid sheets subject to pressure differences and gravity. Two flow regimes have been identified depending on the Weber number, and analytical solutions to the steady state equations are provided. Linear stability studies indicate that the sinuous mode corresponds to Weber numbers on the order of unity, while the varicose mode is associated with small Weber numbers. For small Weber numbers, the nonlinear stability of liquid sheets is determined analytically in terms of elliptic integrals of the first and second kinds. It is also shown that the sinuous mode of thin and slender liquid sheets is identical to the same mode for slender sheets.     

Compressible fluids and active potentials

We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.     

The Cauchy-Poisson problem in an inviscid stratified liquid

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface response of a nonrotating inviscid stratified liquid of finite or infinite depth to the initial displacement of the free surface. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and distance from the origin of disturbance. It is shown that the asymptotic solution consists of the classical free surface gravity waves and the internal waves.     

Slender body expansions in potential theory along a finite straight line

Consider a distribution of singularities in a potential field along a finite straight line such that the potential satisfies the Laplace equation. An example is a distribution of sources representing a ship or missile moving with forward velocity in a potential inviscid flow field. Such bodies are often truncated or bluff at the ends, and so the strength of the resulting distributions may not gradually tend to zero close to these ends and may instead be non-zero finite. A near-field expansion is obtained which accounts for this using the slender body theory integral splitting method. All terms in the expansion are obtained, and the coefficient of each term in the infinite sequence is given in terms of differentials of the distribution strength. Hence an exact separation of variables solution (separating the axial distance from the cross-sectional distances) is obtained for the potential. This is different from previous representations in that it represents a distribution over a finite length, and the resulting expansion is a simple single summation expression that is straightforward to apply. The resulting numerical scheme is discussed, in particular the evaluation close to the ends and also a comparison between the presented slender body theory and existing numerical methods.     

Nonsimilarity solutions of a class of free convection problems

The nonsimilarity solutions of laminar boundary layer free convection flows along a rotating isothermal plate under nonuniform gravity and along a non isothermal vertical plate have been obtained. The governing nonlinear partial differential equations are reduced to a set of ordinary differential equations and then solved numerically using the method of ‘shooting’ with least square convergence criterion. The results for large values of axial distance are found to be more accurate than those obtained by series and integral methods, but for small values of axial distance they are in good agreement.     

Some Potential Problems for Slender Tori

An integral equation approach is given for solving potentialproblems involving Dirichlet boundary conditions applied onthe surface of a slender torus of general cross-section. Inthis context the word "slender" implies that the ratio of atypical cross-section dimension to a dimension characterizingthe hole in the torus is small. Some applications are givento electrostatics and fluid mechanics, and the method is comparedwith that recently given by Cade (1978).     

Free vibrations and stability of hydraulic cylinder fixed elastically on both ends

The boundary value problem of the stability and free vibration of a hydraulic cylinder has been formulated and solved in this paper. The considered hydraulic cylinder has been elastically fixed at both ends and loaded by Euler force. An elastical fixation has been modelled by rotational springs. The mentioned above hydraulic cylinder consists of a piston rod and cylinder replaced by rods. There are conditions of continuity between the rods. The boundary value problem has been formulated on the basis of minimum potential energy (static problem) and on the basis of Hamilton's principle (free vibration problem). In this paper example results of numerical calculations of the stability and free vibration have been presented. Experimental research has been performed in order to verify the correctness of the assumed mathematical model. Professional measuring apparatus and special stand for research into the slender systems have been used in experiment. Natural frequencies have been measured in dependence on the values of an external load. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Annular liquid jets in zero gravity

The equations describing the steady-state behavior of long annular liquid jets and liquid membranes in zero gravity are solved analytically as a function of the pressure difference across the jet or membrane, Weber number, and nozzle exit angle. The ranges of the parameters for which the analytical solutions are valid are determined, and analytical solutions of the mass absorption rate are obtained as a function of the Peclet and Weber numbers, nozzle exit angle, pressure difference, and thickness of the annular liquid jet. It is shown that the convergence length of annular liquid jets and liquid membranes increases as the Weber number, nozzle exit angle, and pressure coefficient are increased. It is also shown that the mass absoption rate increases as the nozzle exit angle, pressure coefficient, and Weber number are increased; however, the mass absorption rate decreases as the Peclet number and annular jet initial thickness-to-radius ratio are increased.     

Aerodynamic noise from wave-turbulence interaction

In order to estimate the acoustic energy scattered when a unit volume of free turbulence, such as in free jets, interacts with a plane steady sound wave, theoretical expressions are derived for two simple models of turbulence: eddy model and isotropic model. The effect of convection by mean motion of the energy-bearing eddies on the incident sound wave and on the sound generated from wave-turbulence interaction is taken into account. Finally, by means of a representative calculation, the directionality pattern and Mach number dependence of the noise so generated is discussed.     

Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

The conservation laws for Prandtl’s boundary layer equations for an incompressible fluid governing the flow in radial and two-dimensional jets are investigated. For both radial and two-dimensional jets the partial Lagrangian method is used to derive conservation laws for the system of two differential equations for the velocity components. The Lie point symmetries are calculated for both cases and a symmetry is associated with the conserved vector that is used to establish the conserved quantity for the jet. This associated symmetry is then used to derive the group invariant solution for the system governing the flow in the free jet.     

Abelian groups that are small with respect to different classes of groups

In this paper, we study Abelian groups that are small with respect to different classes of groups. Completely decomposable torsion free groups that are small with respect to an arbitrary class of torsion free groups are described completely. Direct products of groups small with respect to the class of slender groups are derived.     

Simulation of a long slender structure in turbulent flow

The dynamic behaviour of a long slender structure in turbulent flow can vary considerably, depending on the particular excitation. Regimes from simple static deformation, over periodic to irregular and chaotic behaviour are observed. An example is the self-excited periodic motion of a long slender structure in cross flow. In such cases, the fluid flow and the structure movement are strongly coupled. The paper presents a robust partitioned coupling approach based on the Immersed Boundary Method which integrates the geometrically exact Cosserat rod model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of mold filling of Al gravity casting and validation with X-ray in-situ observation

In the mold filling simulation, element parameters including volume filled ratio, surface dimensionless distance, and surface filled ratio, were proposed to describe the shape and location of free surfaces in DFDM (Direct Finite Difference Method) elements. A model of the filling process was established, specially taking into account the mass, momentum and heat transfer in the vicinity of free surfaces. It was applied to an experimental AC4C (Al–7Si–0.4Mg) gravity casting. With a special X-ray apparatus, in-situ observation and record of actual mold filling process of the casting were carried out. The simulation results were validated and analyzed by comparing with the observation. The liquid flowing in the casting runner and ingate as well as the evolution of free surfaces were analyzed and discussed.     

Wedge dislocations and three-dimensional gravity

The expression for the free energy of arbitrary static distribution of wedge dislocations in elastic media is proposed. In the framework of geometric theory of defects, the free energy is given by the Euclidean action for (1+2)-dimensional gravity interacting with N point particles. Relative movement of particles in gravity corresponds to bending of dislocations. The equations of equilibrium are obtained and analyzed. For two dislocations, the solution is found explicitly through hypergeometric functions.     

Conservation laws and conserved quantities for laminar two-dimensional and radial jets

A systematic way to derive the conserved quantities for the liquid jet, free jet and wall jet using conservation laws is presented. Both two-dimensional and radial jets are considered. The jet flows are described by Prandtl’s momentum boundary layer equation and the continuity equation. The multiplier approach (also know as variational derivative approach) is first applied to construct a basis of conserved vectors for the system. The basis consists of two conserved vectors. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived for the liquid jet and the free jet. The multiplier approach is then applied to construct a basis of conserved vectors for the third-order partial differential equation for the stream function. The basis consists of two local conserved vectors one of which is a non-local conserved vector for the system. The conserved quantities for the free jet and the wall jet are derived from the corresponding conservation laws and boundary conditions. The approach gives a unified treatment to the derivation of conserved quantities for jet flows and may lead to a new classification of jets through conserved vectors and their multipliers.     

Hamiltonian modelling and buckling analysis of a nonlinear flexible beam with actuation at the bottom

ABSTRACT

The use of beams and similar structural elements is finding increasing application in many areas including micro and nanotechnology devices. For the purpose of buckling analysis and control, it is essential to account for nonlinear terms in the strains while modelling these flexible structures. Further, the Poisson’s effect can be accounted in modelling by the use of a two-dimensional stress–strain relationship. This paper studies the buckling effect for a slender, vertical beam (in the clamped-free configuration) with horizontal actuation at the fixed end and a tip-mass at the free end. Including also the inextensibility constraint of the beam, the equations of motion are derived. A preliminary modal analysis of the system has been carried out to describe candidate post-buckling configurations and study the stability properties of these equilibria. The vertical configuration of the beam under the action of gravity is without loss of generality, since the objective is to model a potential field that determines the equilibria. Neglecting the inextensibility constraint, the equations of motion are then casted in port-Hamiltonian form with appropriately defined flows and efforts as a basis for structure-preserving discretization and simulation. Finally, the finite-dimensional model is simulated to obtain the time response of the tip-mass for different loading conditions.     

Flow in a waterfall with large Froude number

An approximate method is developed to solve the full nonlinear equations governing two-dimensional irrotational flow in a free waterfall, falling under the influence of gravity, at high Froude number based on conditions far upstream. Schwarz—Christoffel transformation is used to map the region, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique, for large Froude number, are used as a basis for the approximate solution of the problem. A complete solution, up to second-order approximation, for the downstream free-surfaces profiles, for different Froude number, is discussed and illustrated. The obtained approximate solutions are compared with those of other authors. Favourable agreement with other results suggests that this method is effective in dealing with flow problems strongly influenced by gravity and high Froude number. The results obtained by this method are sufficiently accurate for practical purposes.     

Oscillatory flow about a cylinder pair

Oscillatory flow about a pair of circular cylinders is considered.The distance between the cylinders can be varied as can theangle that the undisturbed oscillatory flow makes with the linejoining the cylinder centres. In common with other fluid flowsdominated by oscillatory flow, a time-independent, or steadystreaming, motion develops. Attention is focused on the caseof high streaming Reynolds numbers and the resulting jets thaterupt from the surfaces of the cylinders.     

Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle <90° admit locked chains, which is precisely the threshold beyond which the slender property no longer holds.