Slender Body Theory Research Articles

Hydrodynamic Aspects on Vulnerability Criteria for Surf-Riding of Ships

For developing the International Maritime Organization (IMO) second-generation intact stability criteria regarding broaching, draft vulnerability criteria for surf-riding were agreed at the IMO in 2012. This paper describes their hydrodynamic backgrounds with captive model experiments for seven ships, a hydrodynamic theory and a random process theory. In the first level vulnerability criteria, a ship is required to reduce her Froude number of less than 0.3 in case of severe following waves. For predicting the surf-riding threshold in a global bifurcation theory, it is necessary to precisely estimate wave-induced surge force. Thus, the authors execute captive model experiments for three ships in model basins. As a result, we confirmed that the Froude-Krylov calculation overestimates the amplitude of wave-induced surge force so that an empirical formula for regulatory application is presented. For investigating the reason of this discrepancy, a slender body theory assuming low encounter frequency is applied to the situation where a ship runs with a wave. This theory suggests that change of wave-making resistance due to incident wave could reduce the amplitude of the wave-induced surge force and quantitative agreement with model experiment requires the use of CFD or an empirical formula. Thus, the authors can recommend the use of experimental correction formula for the vulnerability criteria. Based on sample calculation results of surf-riding probability of six ships in the North Atlantic, the safety level to be required in the criteria is proposed.

On the solution of exterior plane problems by the boundary element method: A physical point of view

The paper is devoted to the solution of Laplace equation by the boundary element method. The coupling between a finite element solution inside a bounded domain and a boundary integral formulation for an exterior infinite domain can be performed by producing a stiffness or matrix. It is shown in a first step that the use of classical Green's functions for plane domains can lead to impedance matrices which are not satisfying, being singular or not positive-definite. Avoiding the degenerate scale problem is classically overcome by adding to Green's function a constant which is large compared to the size of the domain. However, it is shown that this constant affects the solution of exterior problems in the case of non-null resultant of the normal gradient at the boundary. It becomes therefore important to define this constant related to a characteristic length introduced into Green's function. Using a 'slender body theory allows to show that for long cylindrical domains with a given cross section, the characteristic length is shown as being asymptotically equal to the length of the cylindrical domain. Comparing numerical or analytical 3D and 2D solutions on circular cylindrical domains confirms this result for circular cylinders.

The sedimentation of flexible filaments

AbstractThe dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. Filaments of both non-uniform and uniform cross-sectional thickness are considered. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes. These are shown to match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. The instability travels as a wave along the filament opposite the direction of gravity as it grows and the predicted growth rates are shown to compare favourably with numerical simulations. The linear eigenmodes of the governing equation are also studied, which agree well with the finite-amplitude buckled shapes arising in simulations.

Study of semidilute fibre suspension rheology with lattice-Boltzmann method

A hybrid lattice-Boltzmann numerical simulation method is undertaken to study the rheology of non-colloidal, rigid fibres in sheared, semidilute Newtonian suspension. The viscosity and the first normal stress difference are calculated with the combined use of numerical fibre orientation information and a corrected form of the slender body theory of Batchelor (J Fluid Mech Digit Arch 46(4):813–829, 1971). The corrections make the theory applicable in semidilute suspension flow (Shaqfeh and Fredrickson, Phys Fluids A: Fluid Dyn 2(1):7–24, 1990) with fibres of finite aspect ratio. The corrected theory within its framework, based on spacing among fibres in semidilute suspension, only considers hydrodynamic interactions among fibres and inferences that the effect of actual fibre–fibre mechanical contacts on rheological properties of such suspension systems remains inconsequential. To investigate this issue, the rheological properties of semidilute suspension were calculated directly from the numerical simulation. This approach accounted for both hydrodynamic and mechanical interactions among fibres. This direct measurement proved that the mechanical interactions increase both the relative shear viscosity and the first normal stress difference in the semidilute suspension to values which are larger than permissible within the framework of Batchelor’s theory.

The Changing Shape of a Liquid Drop in an Electric Field

An approximate extension of the slender body theory was used to determine the static shape of a conically ended dielectric fluid drop in an electric field. Using induced surface charge density, hydrostatic pressure and the surface tension of the liquid with interfacial tension stresses and Maxwell electric stresses, a governing equation was obtained for slender geometries for the equilibrium configuration and numerically solved for 3D. For an applied electric field, the electric energy on a spherical drop can be maximized in a weak dielectric by increasing the applied electric field. The minimum dielectric constant ratio needed to produce a conical end is 14.5 corresponding to a cone angle 31.25° .There is a sharp increment of the aspect ratio after reaching the threshold value of the applied field strength and the deformation of the fluid drop increases with the increase in dielectric constant of the fluid drop. For a particular dielectric constant ratio, the threshold electric field producing conical interface increases with the increased surface tension of the liquid. The threshold electric field for a water drop is 1.0854×104 units and the corresponding aspect ratio is 15. For the minimum dielectric ratio the cone angle of the drop decreases with applied field making the drop more stable at higher fields.

방파제 형상 연직구조물 부근을 항행하는 대형선박에 미치는 간섭력에 관한 연구

It is well known that the hydrodynamic forces and moments induced by the proximity of bank in confined waters, such as in a harbour or narrow channel affect ship`s maneuvering motion. In this paper, the calculation method based on the slender body theory for estimation of the hydrodynamic force between ship and breakwater is applied, and also, the characteristic features of hydrodynamic force acting on a large vessel in the proximity of a breakwater are described and illustrated. Furthermore, the effects of water depth and the lateral spacing between ship and breakwater are summarized and discussed.

CFD-Based Calculation of Regular Transverse Wave Force in the Maneuvering Mathematical Modeling Group (MMG) Model

The accuracy of mathematical model has a great effect on the prediction of ships maneuverability. Wave disturbing force is an important element of the Maneuvering Mathematical Modeling Group (MMG). Building regression formula of wave disturbing force acting on the ship hull can not only increase ship mathematical models accuracy but also help to predict ships maneuverability. The wave force can be calculated by the Computational Fluid Dynamics (CFD) method, which is more accurate and economical. The 3D model of M/V YUKUN was established and the cut-cell mesh was made. A numerical wave tank was developed in this work, based on the FLUENT software. The calculation results were obtained with different height, length, and period of waves. After analyzing the calculated data obtained by the CFD simulation, a regression formula was built about the above parameters. Finally, after comparing the regression formula and tradition wave force model based on the slender body theory, it can be concluded that the regular transverse wave force regression formula of the YUKUN is valid and better to make simulation for the prediction on ships maneuverability.

Nematode locomotion in unconfined and confined fluids

The millimeter-long soil-dwelling nematode Caenorhabditis elegans propels itself by producing undulations that propagate along its body and turns by assuming highly curved shapes. According to our recent study [V. Padmanabhan et al., PLoS ONE 7, e40121 (2012)10.1371/journal.pone.0040121] all these postures can be accurately described by a piecewise-harmonic-curvature model. We combine this curvature-based description with highly accurate hydrodynamic bead models to evaluate the normalized velocity and turning angles for a worm swimming in an unconfined fluid and in a parallel-wall cell. We find that the worm moves twice as fast and navigates more effectively under a strong confinement, due to the large transverse-to-longitudinal resistance-coefficient ratio resulting from the wall-mediated far-field hydrodynamic coupling between body segments. We also note that the optimal swimming gait is similar to the gait observed for nematodes swimming in high-viscosity fluids. Our bead models allow us to determine the effects of confinement and finite thickness of the body of the nematode on its locomotion. These effects are not accounted for by the classical resistive-force and slender-body theories.

Subdiffusive transport of fluctuating elastic filaments in cellular flows

The dynamics and transport properties of Brownian semiflexible filaments suspended in a two-dimensional array of counter-rotating Taylor-Green vortices are investigated using numerical simulations based on slender-body theory for low-Reynolds-number hydrodynamics. Such a flow setup has been previously proposed to capture some of the dynamics of biological polymers in motility assays. A buckling instability permits elastic filaments to migrate across such a cellular lattice in a “Brownian-like” manner even in the athermal limit. However, thermal fluctuations alter these dynamics qualitatively by driving polymers across streamlines, leading to their frequent trapping within vortical cells. As a result, thermal fluctuations, characterized here by the persistence length, are shown to lead to subdiffusive transport at long times, and this qualitative shift in behavior is substantiated by the slow decay of waiting-time distributions as a consequence of trapping events during which the filaments remain in a particular cell for extended periods of time. Velocity and mass distributions of polymers reveal statistically preferred positions within a unit cell that further corroborate this systematic shift from transport to trapping with increasing fluctuations. Comparisons to results from a continuum model for the complementary case of rigid Brownian rods in such a flow also highlight the role of elastic flexibility in dictating the nature of polymer transport.

Helical swimming in Stokes flow using a novel boundary-element method

We apply the boundary-element method to Stokes flows with helical symmetry, such as the flow driven by an immersed rotating helical flagellum. We show that the two-dimensional boundary integral method can be reduced to one dimension using the helical symmetry. The computational cost is thus much reduced while spatial resolution is maintained. We review the robustness of this method by comparing the simulation results with the experimental measurement of the motility of model helical flagella of various ratios of pitch to radius, along with predictions from resistive-force theory and slender-body theory. We also show that the modified boundary integral method provides reliable convergence if the singularities in the kernel of the integral are treated appropriately.

Robust flutter suppression for a 2-dimensional wing-store system

In this paper, a new scheme is proposed to model the flow speed uncertainties in an aeroelastic system. Compared with the traditional method, the new scheme is based on a signal transformation so as to reduce the dimension of the mathematical model of the flow speed uncertainties. Thus, the robust flutter controller can be designed easily via the reduced-order model of the aeroelastic system. To illustrate the procedure of proposed scheme, the dynamic equation of a two-dimensional wing-store aero-servo-elastic system with flow speed uncertainties is modeled by using the proposed scheme. The aerodynamic forces of the wing section and the store are computed through the use of doublet lattice approach and slender-body aerodynamic theory, respectively. To suppress the flutter instability of the wing-store aeroelastic system with flow speed uncertainties, the trailing-edge control surface is used as a control input and a single-input and multi-output (SIMO) controller is designed via the robust control theory. Numerical results indicate that, with the proposed scheme, the robust flutter control law is efficient and the flutter boundary of aeroelastic system can be greatly expanded.

Lateral Force and Yaw Moment on a Slender Body in Forward Motion at a Yaw Angle

This paper presents a solution method for obtaining the lateral hydrodynamic forces and moments on a submerged body translating at a yaw angle. The method is based on the infinite-fluid formulation of the free-surface random-vortex method (FSRVM), which is reformulated to include the use of slender-body theory. The resulting methodology is given the name: slender-body FSRVM (SB-FSRVM). It utilizes the viscous-flow capabilities of FSRVM with a slender-body theory assumption. The three-dimensional viscous-flow equations are first shown to be reducible to a sequence of two-dimensional viscous-fluid problems in the cross-flow planes with the lowest-order effects from the forward velocity included in the cross-flow plane. The theory enables one to effectively analyze the lateral forces and yaw moments on a body undergoing prescribed forward motion with the possible occurrence of cross-flow separation. Applications are made to several cases of body geometry that are in steady forward motion, but at a yawed orientation. These include the case of a long “cone-tail” body. Comparisons are made with existing data where possible.

Propulsion of microorganisms by a helical flagellum

The swimming of a bacterium or a biomimetic nanobot driven by a rotating helical flagellum is often interpreted using the resistive force theory developed by Gray and Hancock and by Lighthill, but this theory has not been tested for a range of physically relevant parameters. We test resistive force theory in experiments on macroscopic swimmers in a fluid that is highly viscous so the Reynolds number is small compared to unity, just as for swimming microorganisms. The measurements are made for the range of helical wavelengths λ, radii R, and lengths L relevant to bacterial flagella. The experiments determine thrust, torque, and drag, thus providing a complete description of swimming driven by a rotating helix at low Reynolds number. Complementary numerical simulations are conducted using the resistive force theories, the slender body theories of Lighthill and Johnson, and the regularized Stokeslet method. The experimental results differ qualitatively and quantitatively from the predictions of resistive force theory. The difference is especially large for and/or , parameter ranges common for bacteria. In contrast, the predictions of Stokeslet and slender body analyses agree with the laboratory measurements within the experimental uncertainty (a few percent) for all λ, R, and L. We present code implementing the slender body, regularized Stokeslet, and resistive force theories; thus readers can readily compute force, torque, and drag for any bacterium or nanobot driven by a rotating helical flagellum.

A fluid impulse nonlinear theory of ship motions and sea loads

A new nonlinear seakeeping theory is presented for the prediction of the large amplitude motions and wave induced loads on ships in severe seastates. The hydrodynamic forces on the ship are evaluated using a new fluid impulse theory which expresses the nonlinear Froude–Krylov, radiation and diffraction forces as time derivatives of integrals of velocity potentials over the instantaneous ship wetted surface circumventing the time consuming evaluation of the pressure from Bernoulli's equation. Analogous expressions are derived for the structural loads. The free surface problems for the ship wave disturbance are solved on vertical planes fixed relative to an earth fixed frame using a 2D + t slender body theory. The linear free surface condition is enforced on the ambient wave profile assumed locally horizontal, a nonlinear boundary condition is imposed on the ship hull and the resulting boundary value problem is solved using the two-dimensional time-domain wave source potential. Computations are presented illustrating the performance of the new theory. The application of the theory to the evaluation of the extreme statistics of the ship responses and structural loads in a stochastic seastate is also addressed.

Number Concentration of Cylindrical Particles in a Fluidized Bed

In this study, a three-dimensional model based on RANS, slender-body theory and Newton-Euler dynamics is established to study the number concentration, one of the most important fluidization characteristics of cylindrical particles. Also, the effects of interaction between cylindrical particles are taken into account by introducing the rigid collision dynamics. To validate the model, the fluidization experiments of cylindrical particles in a cold-state fluidized bed are carried out. The number concentration characteristics of cylindrical particles are obtained from computational fluid dynamics (CFD) simulation. It is found that cylindrical particles arriving at the exit of the riser the earliest come from the near-wall regions, the horizontal transfer of so many cylindrical particles from the radial centre regions to the near-wall regions is evident. Meanwhile, there is no distinct relationship between the number concentration and inlet wind velocity.

On thin or slender bodies

This is a review of thin-body and slender-body theories, with indications of some new applications. Topics discussed include bodies with near-constant surface pressure, subsonic and supersonic aerodynamics, ship hydrodynamics, slender bodies in Stokes flow, slender footings in elastic media, and slender moonpools. Mathematical features of the thin- and slender-body approximations are also discussed, especially nonlocal convolution terms modelling three-dimensionality in the otherwise two-dimensional near field, end effects, and the role of the logarithm of the slenderness ratio. This review was presented by the first author as the IMA Lighthill Memorial Lecture at the British Applied Mathematics Colloquium (BAMC) 2004. doi:10.1017/S1446181112000120

A Wave Resistance Formulation for Slender Bodies at Moderate to High Speeds

T. Francis Ogilvie (1972) developed a Green's function method for calculating the wave profile of slender ships with fine bows. He recognized that near a slender ship's bow, rates of change of flow variables axially should be greater than those typically assumed in slender body theory. Ogilvie's result is still a slender body theory in that the rates of change in the near field are different transversely (a half-order different) than axially; however, the difference in order of magnitude between them is less than in the usual slender body theory. Typical of slender body theory, this formulation results in a downstream stepping solution (along the ship's length) in which downstream effects are not reflected upstream. Ogilvie, however, developed a solution only for wedge-shaped bodies. Taravella, Vorus, and Givan (2010) developed a general solution to Ogilvie's formulation for arbitrary slender ships. In this article, the general solution has been expanded for use on moderate to high-speed ships. The wake trench has been accounted for. The results for wave resistance have been calculated and are compared with previously published model test data.

The Changing Shape of a Liquid Drop in an Electric Field

An approximate extension of the slender body theory was used to determine the static shape of a conically ended dielectric fluid drop in an electric field. Using induced surface charge density, hydrostatic pressure and the surface tension of the liquid with interfacial tension stresses and Maxwell electric stresses, a governing equation was obtained for slender geometries for the equilibrium configuration and numerically solved for 3D. For an applied electric field, the electric energy on a spherical drop can be maximized in a weak dielectric by increasing the applied electric field. The minimum dielectric constant ratio needed to produce a conical end is 14.5 corresponding to a cone angle 31.25° .There is a sharp increment of the aspect ratio after reaching the threshold value of the applied field strength and the deformation of the fluid drop increases with the increase in dielectric constant of the fluid drop. For a particular dielectric constant ratio, the threshold electric field producing conical interface increases with the increased surface tension of the liquid. The threshold electric field for a water drop is 1.0854×104 units and the corresponding aspect ratio is 15. For the minimum dielectric ratio the cone angle of the drop decreases with applied field making the drop more stable at higher fields.

The Aerodynamic Characterization of Generic Tail Fin Shapes

This paper describes the experimental characterization of the yaw response of a range of generic tail fin shapes: delta wings, right triangles, rectangles, and ellipses. The fins were tested in a wind tunnel without the complication of blades and nacelle. They were released from an initial angle, usually 45°, and the subsequent response recorded using an accurate potentiometer. Unsteady tail fin performance can be analysed using unsteady slender body theory and the quasi-steady analysis originally developed for wind vanes. Both predict a linear, second order response so the measured responses are quantified in terms of the natural frequency and damping ratio. It is shown that unsteady slender body theory is slightly more accurate, but both fail to describe the full complexity of the response.

Triform Wing-Body Loading

This paper presents the application of the slender-body theory to triform-finned missiles to derive detailed spanload distributions and the overall normal and side forces generated by the fin panels in the presence of the body. A comparison is made between the results of a panel-based aeroprediction code and the distributions and integrated loads given by the theory to moderate angles of attack at several Mach numbers.