Separated Flow About a Slender Body in Forward and Lateral Motion

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.

*# A method for analyzing cambered bodies has been incorporated into Missile Datcom. Existing methods based on mean line theory, slender body theory, viscous crossflow theory were assessed. They were compared with wind tunnel data from subsonic to supersonic speeds for cambered bodies with axi-symmetric cross-sections. The comparison cases consisted of three configurations with discrete camber changes (deflectable nose missile) and twelve configurations with continuous camber changes. A combination of slender body and viscous crossflow theory was found to provide the best agreement with experimental data. Nomenclature

Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three‐dimensional fiber, in part due to the fact that simply prescribing data along a 1D curve does not result in a well‐posed boundary value problem for the Stokes equations in ℝ3. Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well‐posedness framework, allows us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry. © 2019 Wiley Periodicals, Inc.

Compressible flow past thin wings and slender bodies are studied. First, subsonic flow past thin wings are analyzed by means of potential flow theory. The kernel function method is introduced for arbitrary planforms undergoing simple harmonic oscillations. The spanwise polynomial approximation and the chordwise trigonometric function approximations which automatically satisfy the Kutta condition and inherit the leading edge singularity are used. The Doublet-Lattice method as a more general numerical approach is given for the analysis of the flow past additional surfaces like tail or store surfaces, which are not necessarily in plane with the wing surface or surfaces having spanwise deflections. Airfoil response to the arbitrary unsteady motion is also given for subsonic flows. A brief review of shock waves and Mach waves in a supersonic flow is given. Afterwards, unsteady supersonic potential flow is studied for a simple harmonically oscillating point source. First, unsteady flow past an airfoil is considered. Then, supersonic flow about thin wings are analyzed using the Mach box technique. Introduction to supersonic kernel function method is briefly provided. Arbitrary unsteady motion of an airfoil in a supersonic flow is presented. Slender body theory is introduced to analyze the cross flow past bodies of considerable fineness where the cross flow is shown to be approximately incompressible under certain conditions. In connection with the slender body approximation, the Munk’s airship theory is utilized to predict the stability derivatives of missile like bodies.

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.

SummaryWard’s slender-body theory is extended to derive first approximations to the external forces on slender bodies of general cross section with discontinuous profile slope. Two classes of body are considered: bodies whose profile (typified by the local radius) is continuous between the nose and base, and certain bodies whose profile is discontinuous, such as bodies with annular or side air intakes and wing-bodies on which the wing has an unswept leading edge. (Where air intakes are concerned, it is assumed that they are sharp-edged and that there is no “ spillage ” of the internal flow).The following conclusions apply to the former class of bodies. The variation of drag with Mach number is found to depend only on the discontinuities in the longitudinal rate of change of the cross-sectional area, and is thus independent of cross-sectional shape. The drag itself is unchanged if the direction of the flow is reversed. The expressions for lift and moment assume the same forms as for smooth pointed bodies, the lift depending only on conditions at the base of the body.The general theory is applied to winged bodies of revolution with an unswept wing leading edge: the results bear a marked resemblance to those obtained by Ward. The results for wings alone are seen to be applicable, with one modification, to subsonic as well as to supersonic speeds.

Immersed boundary lattice Boltzmann method has been applied to analyze the characteristics of the self-propelled fish motion swimming robot. The airfoil NACA0012 with caudal fin stroke model was considered to examine the characteristics. The foil in steady forward motion and a combination of steady-state harmonic deformation produces thrust through the formation of a flow downstream from the trailing edge. The harmonic motion of the foil causes unsteady shedding of vorticity from the trailing edge, while forming the vortices at the leading edge as well. The resultant thrust is developed by the pressure difference formed on the upper and lower surface of the airfoil. and the time averaged thrust coefficient increases as Re increase in the region of <TEX>$Re{\leqq}700$</TEX>. The suggested numerical method is suitable to develop the fish-motion model to control the swimming robot, however It would need to extend in 3D analysis to examine the higher Re and to determine the more detail mechanism of thrust production.

Fluid mechanics of flapping wings

This article presents the unsteady aerodynamic performance of crosswind stability obtained numerically for the ATM train. Results of numerical investigations of airflow past a train under different yawing conditions are summarized. Variations of occurrence flow angle from parallel to normal with respect to the direction of forward train motion resulted in the development of different flow patterns. The numerical simulation addresses the ability to resolve the flow field around the train subjected to relatively large yaw angles with three-dimensional Reynolds-averaged Navier-Stokes equations (RANS). <TEX>${\kappa}-{\varepsilon}$</TEX> turbulence model solved on a multi-block structured grid using a finite volume method. The massively separated flow for the higher yaw angles on the leeward side of the train justifies the use of RANS, where the results show good agreement with verification results. A method of solution is presented that can predict all aerodynamic coefficients and the wind characteristic curve at variety of angles at different speed.

Hydroelasticity and the field radiated by a slender elastic body into an unbounded fluid

Numerical simulation of ultra-high speed supercavitating flows considering the effects of the water compressibility

In this paper, the first experimental results of tests with a new model support designed for simulations of high frequency and high amplitude pitching maneuvers are presented. A generic missile model with a blunt nose was used for which static test data from earlier experiments with a conventional model support were existing. At the beginning, static tests were done for comparison purposes to judge the influence of the new setup (model support, sting for the model, pivot arm) on the measured forces and moments. Afterwards, dynamic tests with sine oscillations at frequencies of $$f_{\rm a}=0.05\ldots 4\,\hbox{Hz}$$ and angles of attack varying between $$\alpha = 0^{\circ} \ldots 45^{\circ}$$ were performed. Beside the qualification of this new test rig, the tests were used to study the “Phantom Yaw Effect” and to prove an interactive method of its control. This phenomenon is characterized by unwanted yawing moments resulting from asymmetric vortices which can occur on slender bodies at high angles of attack. In the tests, a lee-vortex control device with symmetrically arranged longitudinal slot nozzles producing “air-jet strakes” was demonstrated to decrease the yawing moments both under static and dynamic conditions.

Wash and Wave Resistance of Ships in Finite Water Depth

Occurrence of flow separation within the boundary-layer is primarily attributed to the growth of adverse pres-sure gradient within the boundary-layer. Such a separating boundary-layer can, however, be made to stay attached to the bounding surface due to various remedies where the surface transverse curvature (TVC) is one of them while the body contour is another. Assistive role of the surface transverse curvature is utilized in delaying the boundary-layer separation in a retarded flow past the slender bodies by assuming different body configurations. Axially-symmetric bodies of revolution of different shapes having a varying cross section in x, following the power-law form, have been considered in this study. It is shown that the point of separation is strongly dependent upon the surface curvature parameter k and the body-shape index n, which in fact control the flow separation. This fact ensures that an increase in the surface curvature parameter k and the body-shape index n leads to an increase in the wall shear stress which consequently delays the flow separation. Significant delay in flow separation is observed on cylinders having sufficiently large transverse curvature. Percent increase in the separation length corresponding to various values of the curvature parameter and the body-shape index has been calculated and reported in the form of Tables.

In case of offshore support vessels, it is quite important for their behaviours in low speed manoeuvrability. It is not easy to operate offshore support vessels and it raises an importance to train crew by a ship handling simulator. Therefore it is quite hot issue for ship handling simulator vendors to provide mathematical model of support vessels, but from the users’ point-of-view the model used for the simulator is not clear. Up to now some mathematical models are proposed in the literature, but the validation is not yet done carefully. For example, cross flow model is the main part for hydrodynamic force in low speed ship manoeuvring. From model ship experiments of various types of ships and from some comparison study, it is found that further discussion is necessary. In this paper, sway force and yaw moment as well as surge force will be treated in low speed condition with certain sway and yaw motions. There are many experiment results for such hydrodynamic forces and moment, especially for sway force and yaw moment, for sway motion, but those for yaw motion is quite limited. In this paper, such hydrodynamic forces and moment for both sway and yaw motions are calculated and validated with experiment results. Some manoeuvring simulation will also demonstrate which terms and/or expressions in the mathematical model are affecting the motion.