Ideal Fluid Theory Research Articles

Boundary effects on ideal fluid forces and Kelvin's minimum energy theorem

The electrostatic force on a charge above a neutral conductor is generally attractive. Surprisingly, that force becomes repulsive in certain geometries (Levin & Johnson, Am. J. Phys., vol. 79, issue 8, 2011, pp. 843–849), a result that follows from an energy theorem in electrostatics. Based on the analogous minimum energy theorem of Kelvin (Camb. Dublin Math. J., vol. 4, 1849, pp. 107–112), valid in the theory of ideal fluids, we show corresponding effects on steady and unsteady fluid forces in the presence of boundaries. Two main results are presented regarding the unsteady force. First, the added mass is proven to always increase in the presence of boundaries. Second, in a model of a body approaching a boundary, where the unsteady force is typically repulsive (Lamb, Hydrodynamics, 1975, § 137, University Press), we present a geometry where the force can be attractive. As for the steady force, there is one main result: in a model of a Bernoulli suction gripper, for which the steady force is typically attractive, we show that the force becomes repulsive in some geometries. Both the unsteady and steady forces are shown to reverse sign when boundaries approximate flow streamlines, at energy minima predicted by Kelvin's theorem.

Study on the Influence of Explosive Types on Rod Jet

In order to study the influence of explosive type on the rod jet formation of energetic composite liner, the process of rod jet formation of this liner is numerically simulated by using finite element analysis and multi-material Euler algorithm. In this paper, the rod jet formation and penetration performance of the liner are studied and analyzed by combining relevant theories and simulation calculation, and the influence law of explosive type on rod jet formation and target penetration thickness is obtained. In this paper, the stability of rod jet formed by this kind of model is verified by numerical simulation, and the influence law of explosive type on rod jet is obtained by simulating the collected data such as effective mass, tip velocity and jet length of rod jet and the equivalent target penetration thickness obtained by the quasi steady incompressible ideal fluid theory of jet penetration; it provides data support and design basis for the application of new explosives in energetic composite liner in the next stage.

Fundamentals and Knowledge Relevant to the Drag Reduction Through Air Cavitation of Ship's Hulls

Numerous experiments with ship drag reduction by air bottom cavitation in diverse countries have exhibited very different achievements. Therefore, a paper clarifying mechanics of this drag reduction and describing the proven design algorithms is appropriate. Solutions of an ideal fluid problem existing in diverse ranges of Froude number are compared and the solutions suitable for ship drag reduction are considered in more detail. It is emphasized in this paper that a cavity locker at the trailing edge of the bottom niche (recess) assigned for the cavity is necessary to reduce the necessary air supply to the cavity and to mitigate the cavity tail pulsation resulting in a drag penalty. It is also pointed out that the bottom niche depth must allow for cavity withstanding under impact of waves in seaways. Bottom cavitation may even reduce wave-induced loads on the hull. With taking into account the above-mentioned design aspects, the energy spent on the air supply can be minimized. An algorithm of bottom design based on ideal fluid theory is also explained in the paper. Comparisons with several model test results are provided to illustrate the algorithm employment.

Hamiltonian Aspects of Three-Layer Stratified Fluids

The theory of three-layer density-stratified ideal fluids is examined with a view toward its generalization to the n-layer case. The focus is on structural properties, especially for the case of a rigid upper lid constraint. We show that the long-wave dispersionless limit is a system of quasi-linear equations that do not admit Riemann invariants. We equip the layer-averaged one-dimensional model with a natural Hamiltonian structure, obtained with a suitable reduction process from the continuous density stratification structure of the full two-dimensional equations proposed by Benjamin. For a laterally unbounded fluid between horizontal rigid boundaries, the paradox about the non-conservation of horizontal total momentum is revisited, and it is shown that the pressure imbalances causing it can be intensified by three-layer setups with respect to their two-layer counterparts. The generator of the x-translational symmetry in the n-layer setup is also identified by the appropriate Hamiltonian formalism. The Boussinesq limit and a family of special solutions recently introduced by de Melo Viríssimo and Milewski are also discussed.

Complex Korteweg-de Vries equation: A deeper theory of shallow water waves.

Using Levi-Cività's theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equation, describing the complex velocity of a shallow fluid up to first order. We use perturbation theory, and the long wave, slowly varying velocity approximations for shallow water. The complex KdV equation describes the nontrivial dynamics of all water particles from the surface to the bottom of the water layer. A crucial step made in our work is the proof that a natural consequence of the complex KdV theory is that the wave elevation is described by the real KdV equation. The complex KdV approach in the theory of shallow fluids is thus more fundamental than the one based on the real KdV equation. We demonstrate how it allows direct calculation of the particle trajectories at any point of the fluid, and that these results agree well with numerical simulations of other authors.

Wave action by arrays of vertical cylinders with arbitrary smooth cross-section

This paper presents an analytical model to solve the linear wave diffraction problem by arrays of bottom-mounted cylinders with arbitrary smooth cross-section. Based on the assumption of ideal fluid and potential theory, the unknown coefficients of total velocity potential can be solved by system of linear equations, which are obtained from the boundary conditions. The accuracy of the present method is verified by comparing it with the numerical tool in terms of the wave force and wave run-up. Multiple cylinders with different configurations are tested, the cross-section of which is circular with cosine perturbation. The results show that the proposed method could obtain an accurate prediction of the wave action with multiple cylinder problems. Finally, the diffraction wave is investigated on arrays of bottom-mounted cylinders with different cross-section and layout. The near-trapping problem with effects of the multi-body interaction are also investigated.

Challenging Problems on Ventilated Cavitation and Paths to Their Computational Solutions

Ventilated cavitation has been successfully employed as ship drag reduction technology and potentially can mitigate flow-induced vibration. The obtained successes were based on solutions of design problems considered in the framework of ideal fluid theory with their following validation by towing tank tests. However, various aspects of the interaction of ventilated cavities with the viscous flows around the ship hulls remain unclear, whereas there is usually no possibility to simultaneously keep the full-scale Froude number and cavitation number in the test facilities. So, the further progress of the application of ventilated cavitation substantially depends on the ability of computational tools to predict this interaction. This paper briefly describes the state-of-the-art computation of ventilated cavitation and points out the most challenging unsolved problems that appeared in the model tests (prediction of air demand by cavities, ventilation effect on ship drag, on hydrofoil lift, and on the propagation of shock waves in cavities).

Capillary Waves Joint to Super-Hydrophobic Bars

Super-hydrophobic coating reduces hydrodynamic drag of rigid surfaces due to separation of a part of the surface from the flow by air. The drag reduction ratio is proportional to the ratio of the surface area covered by air to the whole surface area. The maximum ratio may be achieved for coating with a regular spanwise super-hydrophobic bar. The air–water boundary over such bar would be a capillary wave with wavelength equal the distance between bar apexes. The numerical analysis of such waves was carried out by solving a two-dimensional nonlinear free-boundary problem of ideal fluid theory. Besides several wave shapes, the main computational results include dependencies of wavelengths and dimensionless pressure coefficient necessary for wave maintenance on Weber number. These dependencies make it possible to select the bar size and inflow speed allowing for existence of such waves and the highest drag reduction ratios.

An affine model of the dynamics of astrophysical discs

Thin astrophysical discs are very often modelled using the equations of two-dimensional hydrodynamics. We derive an extension of this model that describes more accurately the behaviour of a thin disc in the absence of self-gravity, magnetic fields and complex internal motions. The ideal fluid theory is derived directly from Hamilton's Principle for a three-dimensional fluid after making a specific approximation to the deformation gradient tensor. We express the equations in Eulerian form after projection on to a reference plane. The disc is thought of as a set of fluid columns, each of which is capable of a time-dependent affine transformation, consisting of a translation together with a linear transformation in three dimensions. Therefore, in addition to the usual two-dimensional hydrodynamics in the reference plane, the theory allows for a deformation of the midplane (as occurs in warped discs) and for the internal shearing motions that accompany such deformations. It also allows for the vertical expansions driven in non-circular discs by a variation of the vertical gravitational field around the horizontal streamlines, or by a divergence of the horizontal velocity. The equations of the affine model embody conservation laws for energy and potential vorticity, even for non-planar discs. We verify that they reproduce exactly the linear theories of three-dimensional warped and eccentric discs in a secular approximation. However, the affine model does not rely on any secular or small-amplitude assumptions and should be useful in more general circumstances.

Analysis of interaction between ship bottom air cavity and boundary layer

The successful designs of hulls for ships employing drag reduction by air bottom cavitation have been based on solutions of inverse problems of the theory of ideal incompressible fluid. However, prediction of the drag reduction ratio, the air demand by ventilated cavities and the cavity impact on the hull–propeller interaction is impossible in the framework of this theory because all mentioned characteristics depend on interaction of air cavities with the ship boundary layers. Because the known CFD tools are not fitted to ventilated cavitation at low Froude numbers, an analysis of this interaction requires a novel flow model. This model includes the incompressible air flow in the ventilated cavity, the compressible flow of a water–air mixture in the boundary layer on cavities and downstream of them and the curl-free incompressible outer water flow. The provided 2D computations employing this model allows for explanations of the earlier observed effects and for prediction of the air demand by ventilated cavities. The computed velocity profiles downstream of cavities are in the accordance with the available experimental data.

Forces acting on a small particle in an acoustical field in a thermoviscous fluid.

We present a theoretical analysis of the acoustic radiation force on a single small spherical particle, either a thermoviscous fluid droplet or a thermoelastic solid particle, suspended in a viscous and heat-conducting fluid medium. Within the perturbation assumptions, our analysis places no restrictions on the length scales of the viscous and thermal boundary-layer thicknesses δ(s) and δ(t) relative to the particle radius a, but it assumes the particle to be small in comparison to the acoustic wavelength λ. This is the limit relevant to scattering of ultrasound waves from nanometer- and micrometer-sized particles. For particles of size comparable to or smaller than the boundary layers, the thermoviscous theory leads to profound consequences for the acoustic radiation force. Not only do we predict forces orders of magnitude larger than expected from ideal-fluid theory, but for certain relevant choices of materials, we also find a sign change in the acoustic radiation force on different-sized but otherwise identical particles. These findings lead to the concept of a particle-size-dependent acoustophoretic contrast factor, highly relevant to acoustic separation of microparticles in gases, as well as to handling of nanoparticles in lab-on-a-chip systems.

The “Berkeley Wedge”: an asymmetrical energy-capturing floating breakwater of high performance

This research demonstrates the physical feasibility of constructing a single degree of freedom ocean-wave energy capturing device of extremely high efficiency, which serves also as an almost-perfect floating breakwater. The device consists of an asymmetric heaving floater and a power-take-off (PTO) system. A special geometry was first designed to have minimal effects from viscosity. As such, potential-flow analysis proved adequate for predicting the behavior of the device even in a real-fluid environment. Per ideal-fluid theory, at the heave resonance frequency, when the absorption damping of the PTO is matched with the radiation damping of the floater, an energy extraction efficiency of 96.34% can be achieved. At this high-efficiency condition, the transmitted waves beyond the floater are 12.5% of the incident-wave amplitude (or a mere 1.56% of the incident-wave energy). A working two-dimensional physical unit of 0.227m beam and 0.8m draft was designed and fabricated to work with the permanent magnet linear generator developed in (Yeung et al., 2012). The floater-PTO system was tested at The Model-Testing Facility of the University of California (UC), Berkeley. Experiments showed that when the PTO operates at even just 48% of the optimal damping, the device yields already an energy-capturing efficiency of 82%, with only 2% of the incident-wave energy transmitted. These latter values are in agreement with the theoretical predictions. Restraining lateral force and pitching moment are shown, as well as how wave energy is directed to the PTO.

On Noether’s Theorem for the Euler–Poincaré Equation on the Diffeomorphism Group with Advected Quantities

We show how Noether conservation laws can be obtained from the particle relabelling symmetries in the Euler-Poincar\'e theory of ideal fluids with advected quantities. All calculations can be performed without Lagrangian variables, by using the Eulerian vector fields that generate the symmetries, and we identify the time-evolution equation that these vector fields satisfy. When advected quantities (such as advected scalars or densities) are present, there is an additional constraint that the vector fields must leave the advected quantities invariant. We show that if this constraint is satisfied initially then it will be satisfied for all times. We then show how to solve these constraint equations in various examples to obtain evolution equations from the conservation laws. We also discuss some fluid conservation laws in the Euler-Poincar\'e theory that do not arise from Noether symmetries, and explain the relationship between the conservation laws obtained here, and the Kelvin-Noether theorem given in Section 4 of Holm, Marsden and Ratiu, {\it Adv. in Math.}, 1998.

Ship Drag Reduction by Air Bottom Ventilated Cavitation in Calm Water and in Waves

The goal herein is ship drag reduction by air bottom cavitation in the moderate range of Froude number Fr (0.4 < Fr < 0.65) in both calm water and in waves. A ship hull with a bottom niche terminating in a cavity locker/seal (suppressing cavity tail oscillations and reducing the air escape from the cavity) was designed using nonlinear ideal fluid theory. The wave impact on the cavity shape and drag reduction was estimated with a novel analytical approach that takes into account the air compressibility in the cavity and air entrainment by the water. The model drag was measured in the Naval Surface Warfare Center linear tow tank at different drafts in calm water and in waves. The baseline configuration was with the niche closed by a flat cover. The attained total drag reduction at 0.45 < Fr < 0.63 was up to 25%, whereas the air supply power was under 4% of the gain in the required propulsion power. The air cavity was stable in waves (up to sea state 5 for a 90 meter ship) and the effectiveness of drag reduction by cavitation in seaway was greater than in calm water due to smaller wave-induced additional drag of the ship with air bottom cavity. Two identical models were built and tested also as a seatrain. However, the percentage drag reduction due to cavity ventilation in the seatrain configuration was less than for a single hull. The need for fine tuning the air supply distribution between the hulls was found.

The use of the boundary element method in solving the problems of sound scattering by an elastic noncircular cylinder

A semianalytic solution of the problem of sound scattering by an elastic cylinder with a noncircular cross section is proposed. The problem is solved by the analytic-numerical boundary element method in terms of the ideal fluid theory and the linear theory of elasticity.

Simulation of transverse self-oscillations for a free circular cylinder mounted with a narrow gap in a plane channel

The experimentally observed self-oscillations of a cylinder mounted with a narrow gap in a plane channel are simulated. The added masses of the cylinder are calculated in the framework of ideal fluid theory by a generalized image method. In order to describe the self-oscillations in a real fluid, some dissipative factors and an impulsive impact force exerted on the cylinder are introduced.

Second-gradient plane deformations of ideal fibre-reinforced materials II: forming flows of fibre–resin systems when fibres resist bending

The continuum theory of ideal fibre-reinforced fluids, namely incompressible viscous fluids exhibiting some direction of inextensibility is extended to account for the fibre bending stiffness; namely a property that prevents discontinuity of the fibre slope under normal loading conditions. The principal kinematics of this new theoretical development is consistent with three-dimensional forming flows of fibre–resin systems though, for simplicity, formulation of relevant constitutive equations is confined within the framework of relevant plane flows. The theory adopts the macroscopic view that the resin matrix behaves as a viscous fluid but the resin and fibres form a homogeneous composite material. Consideration of the fibre bending resistance requires the inclusion of couple-stress and, hence, non-symmetric stress. The outlined theoretical developments are therefore relevant to polar-media behaviour; in this context, the anisotropic viscous fluids of interest become part of the material class of the so-called polar fluids. For plane flows of this type of fluids, a manner is also outlined in which the non-symmetric stress distributions sought can be determined by solving two simultaneous, first-order linear differential equations. Moreover, a relevant stress-resultants technique is adopted and extended appropriately to make possible complete determination of the kinematics dictating the creeping forming plane flow of the composite fluids of interest. Details of the mechanisms that capture fibre bending resistance are revealed and illustrated through a relatively simple example application. This considers and resolves the forming flow process of an ideal fibre-reinforced composite, moulded into a sharp corner under the action of an external line force.

SCALING OF THE ANOMALOUS BOOST IN RELATIVISTIC JET BOUNDARY LAYER

We investigate the one-dimensional interaction of a relativistic jet and an external medium. Relativistic magnetohydrodynamic simulations show an anomalous boost of the jet fluid in the boundary layer, as previously reported. We describe the boost mechanism using an ideal relativistic fluid and magnetohydrodynamic theory. The kinetic model is also examined for further understanding. Simple scaling laws for the maximum Lorentz factor are derived, and verified by the simulations.

Warm relativistic fluid description of the laser wake field accelerator

A warm relativistic ideal fluid theory is proposed to study the dynamics of laser-induced high amplitude longitudinal waves in rarefied plasmas. It is demonstrated that fluid equations may be closed using an adiabatic approximation in the local rest frame of the plasma fluid. It is shown that considering three-dimensional thermal fluctuations of plasma for an interacting laser pulse that is invariant in transverse directions, considerably changes the features of describing fluid equations compared to that of previously considered one-dimensional thermal-fluctuations models. For a given shape of laser pulse these equations are solved numerically and their results are compared with those obtained from cold fluid theories.

Exact density functional theory for ideal polymer fluids with nearest neighbor bonding constraints

We present a new density functional theory of ideal polymer fluids, assuming nearest-neighbor bonding constraints. The free energy functional is expressed in terms of end site densities of chain segments and thus has a simpler mathematical structure than previously used expressions using multipoint distributions. This work is based on a formalism proposed by Tripathi and Chapman [Phys. Rev. Lett. 94, 087801 (2005)]. Those authors obtain an approximate free energy functional for ideal polymers in terms of monomer site densities. Calculations on both repulsive and attractive surfaces show that their theory is reasonably accurate in some cases, but does differ significantly from the exact result for longer polymers with attractive surfaces. We suggest that segment end site densities, rather than monomer site densities, are the preferred choice of "site functions" for expressing the free energy functional of polymer fluids. We illustrate the application of our theory to derive an expression for the free energy of an ideal fluid of infinitely long polymers.