Rankine Body Research Articles

Drag of Rankine bodies embedded in porous media at small Darcy numbers

This paper studies the flow over a solid inclusion embedded in a porous medium of low permeability. We study Rankine bodies which represent circular cylinders and spheres when the aspect ratio is one, and thin plates and almost uniform rods when the aspect ratio is small. Using boundary layer theory, the flow field and pressure distribution are determined. It is found that the pressure drag dominates the shear drag except for small aspect ratios. Thus, geometry plays a decisive role in the choice of using the Darcy equation or the Darcy-Brinkman equation. The results are important for the determination of flow resistance of a non-homogeneous porous medium which has solid inclusions.

적응제어기법을 이용한 수면근처에서 운항하는 몰수체의 심도제어기 설계

A submerged body moving near the free surface needs to maintain its attitude and position to accomplish missions. It is necessary to validate the performance of a designed controller before a sea trial. The hydrodynamic coefficients of maneuvering are generally obtained by experiments or computational fluid dynamics, but these coefficients have uncertainty. Environmental loads such as the wave exciting force and suction force act on the submerged body when it moves near the free surface. Thus, a controller for the submerged body should be robust to parameter uncertainty and environmental loads. In this paper, the six-degree-of-freedom equations of motions for the submerged body are constructed. The suction force is calculated using the double Rankine body method. An adaptive control method based on an artificial neural network and proportional-integral-derivative control are used for the depth controller. Simulations are performed under various depth and speed conditions, and the results show the effectiveness of the designed controller.

Low-order modeling of wind farm aerodynamics using leaky Rankine bodies

We develop and characterize a low-order model of the mean flow through an array of vertical-axis wind turbines (VAWTs), consisting of a uniform flow and pairs of potential sources and sinks to represent each VAWT. The source and sink in each pair are of unequal strength, thereby forming a “leaky Rankine body” (LRB). In contrast to a classical Rankine body, which forms closed streamlines around a bluff body in potential flow, the LRB streamlines have a qualitatively similar appearance to a separated bluff body wake; hence, the LRB concept is used presently to model the VAWT wake. The relative strengths of the source and sink are determined from first principles analysis of an actuator disk model of the VAWTs. The LRB model is compared with field measurements of various VAWT array configurations measured over a 3-yr campaign. It is found that the LRB model correctly predicts the ranking of array performances to within statistical certainty. Furthermore, by using the LRB model to predict the flow around two-turbine and three-turbine arrays, we show that there are two competing fluid dynamic mechanisms that contribute to the overall array performance: turbine blockage, which locally accelerates the flow; and turbine wake formation, which locally decelerates the flow as energy is extracted. A key advantage of the LRB model is that optimal turbine array configurations can be found with significantly less computational expense than higher fidelity numerical simulations of the flow and much more rapidly than in experiments.

On the Calculation of Two-Dimensional Added Mass Coefficients by the Taylor Theorem and the Method of Fundamental Solutions

ABSTRACTThis work integrates the Taylor theorem and the method of fundamental solutions to develop a numerical tool for estimating the added mass coefficient tensor for a solid object of any convex shape moving in potential flow. In potential flow theory, the Taylor theorem calculates the added mass coefficient tensor for a Rankine body with algebraic manipulations of the properties of the internal singularities employed to generate the corresponding flow. To apply this theorem for objects in other shapes, the singularity strength and locations are required information which is facilitated numerically in this work by the method of fundamental solutions (MFS). The developed scheme is tested on a circle, an ellipse, a square, and a rhombus and the numerical results are in good agreement with the corresponding analytical values. A final example of a Cassini oval is also considered to show the potential applications on bio-engineering problems.

Analysis of the unstable Tollmien–Schlichting mode on bodies with a rounded leading edge using the parabolized stability equation

The interaction between free-stream disturbances and the boundary layer on a body with a rounded leading edge is considered in this paper. A method which incorporates calculations using the parabolized stability equation in the Orr–Sommerfeld region, along with an upstream boundary condition derived from asymptotic theory in the vicinity of the leading edge, is generalized to bodies with an inviscid slip velocity which tends to a constant far downstream. We present results for the position of the lower branch neutral stability point and the magnitude of the unstable Tollmien–Schlichting (T-S) mode at this point for both a parabolic body and the Rankine body. For the Rankine body, which has an adverse pressure gradient along its surface far from the nose, we find a double maximum in the T-S wave amplitude for sufficiently large Reynolds numbers.

Simulation of unsaturated water flow around obstructions: three-dimensional Rankine bodies

Sink and source combinations were used to simulate flow around barriers such as might exist within a stony, unsaturated soil system. Although more complicated than two-dimensional cases which were previously presented, the three-dimensional “Rankine bodies” are more appropriate for simulating flow around real objects. Generally, the appropriate potentials and velocity fields are easy to write down and evaluate, in fact, much easier than the alternative solutions based on scattering theory. Also, arrays of bodies can be arranged to simulate a repeated pattern rather than deal with only single barriers, cavities, etc. However, the shapes of the bodies are not known ahead of time and can only be roughly controlled. For the same Darcian flow, a simulated stony layer was shown to be wetter than that without stones and the effective saturated conductivity reduced somewhat less than 1 - f s where f s is the fraction of stones. Overall there is a qualitative agreement with previous theoretical and experimental results for both saturated and unsaturated flow.

Unsaturated water flow around obstructions simulated by two-dimensional Rankine bodies : A. W. Warrick & G. G. Fennemore, Advances in Water Resources, 18(6), 1995, pp 375–382

Unsaturated water flow around obstructions simulated by two-dimensional Rankine bodies : A. W. Warrick & G. G. Fennemore, Advances in Water Resources, 18(6), 1995, pp 375–382

First-order transient free convection about a horizontal cylinder embedded in a porous medium

An extension is given of the recent work by Pop et al. (Fluid Dynamics Research 12 (1993) 295) concerning transient convection in a porous medium surrounding a horizontal circular cylinder with a sudden temperature change. The horizontal cylindrical boundary may be a smooth closed curve of arbitrary shape. Only the first order of the matched asymptotic expansion is considered. In analogy with the classical theory of Rankine bodies, we demonstrate how various choices of harmonic functions generate different cylinder contours a posteriori.

Unsaturated water flow around obstructions simulated by two-dimensional Rankine bodies

Unsaturated flow systems were studied which consisted of two-dimensional doublets embedded in a porous medium. The isolated flow regions formed around the doublets (source-sink combinations) were called Rankine bodies as a generalization of the classical geometries used in fluid mechanics for solutions to Laplace's equation. The motivation was to simulate unsaturated flow around subterranean objects, such as cavities and rock inclusions. The results were generally successful and patterns for single inclusions qualitatively matched more elaborate solutions based on scattering theory. A complex system (of five inclusions) was presented to simulate a layered case of rock inclusions within a profile.

Three-dimensional flow around a submerged body in finite-depth water

The three-dimensional fluid flow around an axisymmetric body submerged in a finite-depth fluid is calculated by an analytical/numerical method based on a Green's function formulation. The flow around a submerged axisymmetric body, such as the infinite-fluid analogue of a Rankine body, can be constructed by superposition of a source and a sink along the axis of symmetry. Analytical evaluation is complicated because of the singular Cauchy-type principal-value integrals with infinite and semi-infinite limits. In this study these integrals are evaluated by using a set of adaptive numerical quadratures. This approach is direct, and it does not require an asymptotic expansion. The results for the three-dimensional flow calculations are applied for the evaluation of pressure signatures of several Rankine-type bodies with different Froude numbers. Streamlines were determined by a second-order finite-difference algorithm that follows a fluid particle by solving an appropriate initial-value problem. As expected, the shape distortion from the infinite-fluid Rankine body geometry was significant when the slenderness and the linearity (small-wave elevation and slope) approximations became inappropriate.

A Geometrical Acoustics Approach for Calculating the Effects of Flow on Acoustics Scattering

A ray acoustics approach to flow effects on acoustics scattering is considered, based on a combination of geometrical acoustics and the paraxial ray approximation. The approach allows for scattering by objects of arbitrary shapes, inhomogeneous and moving media. Applications to flow effects on scattering include the effect of a wall shear layer on scattering from a cylindrical shaped fuselage and the effects of an incompressible inviscid flow on the directivity pattern of the scattered field from a cylinder and a Rankine body. It has been found that the flow causes a modification and displacement of the lobes of the directivity pattern and the shadow zone of the scattered field, which can be important at Mach numbers greater than 0·2.

Numerical calculation for the flow of submerged bodies under a free surface

The flow field generated by a Rankine body moving under a free surface in afinite-depth water is calculated by potential theory. Velocity field generated by a source located at the origin is calculated first by using highly efficient and adaptive quadratures of the QUADPACK library. This solution is used for generating the flow around a Rankine body by locating a source and an equal strength sink along the body axis. Results agree well with the existing literature.

Incompressible Irrotational Axisymmetric Flow about a Body of Revolution: The Inverse Problem

The flow of an incompressible fluid past an axisymmetric shape is considered in the form of the inverse problem in hydrodynamics. For a given pressure or velocity distributon, the appropriate body shape is determined iteratively. The method of line sources and sinks is used to represent the body shape. The procedure is simple and accurate and convergence is more rapid than that obtained by other investigators using surface-source distributions. Examples chosen to represent the method include spheres, Rankine bodies, a constant velocity body, and an airfoil-shape axisymmetric body. Results compare very well with exact solutions and with calculated results of others when comparison is possible.

Bottom Depth Effects on Regular Surface Waves due to a Submerged Rankine Body with Attached Vertical Column

The role of finite-depth bottom on the amplitudes and slopes of the regular surface waves resulting from passage of a body at uniform velocity beneath an otherwise smooth surface has been considered. The velocity is restricted by the requirement that the Froude number based on bottom depth be less than unity. The first-order integral representation for these regular waves amplitudes are well known, but combinations of these integrals to produce a body with attached vertical column lead to interactions which significantly alter the character of the finite-depth bottom effect. Consideration of the simplified case of the sphere allows qualitative conclusions regarding wavelength and amplitude alterations, and also facilitates definition of nearand far-field regions. A typical first-order bodyplus-vertical-column calculation shows wavelength effects similar to the sphere, but a significant interaction phenomenon at infinite bottom depth is progressively cancelled as depth decreases, implying large amplitude magnifications due to the finite depth. A check of the critical wake angle as a function of bottom depth yields the expected first-order equations, but the classical table of wake angle at various Froude numbers based on bottom depth is correctec}.

Velocity measurements in regions of upstream influence of a body in aligned-fields MHD flow

Experiments are described in which velocities were measured ahead of a semi-infinite Rankine body moving parallel to a uniform magnetic field in a conducting fluid. The flow disturbance in front of the body is found to increase in length asN½, whereNis the interaction parameter. In most of the experiments this parameter was varied from 4 to about 50. Measurements made along the axis of symmetry in the flow show that there is a relatively short region of stagnant fluid directly ahead of the body. The major part of the disturbance is found to consist of a much longer region in which the flow undergoes transition from conditions in the free stream to conditions near the body. Velocity profiles across the flow in this region show that for increasedN, at a fixed distance ahead of the body, the velocity defect increases and the disturbance becomes more confined radially. Although the radial gradients in the flow increase withN, they are found to be much smaller than would be expected in a flow containing thin current layers. A physical model of the flow which has currents and pressures consistent with these results is discussed.

Experiments on the upstream wake in magneto-fluid dynamics

Measurements have been made of the perturbation magnetic field in front of a semi-infinite Rankine body, moving parallel to a uniform impressed magnetic field in a conducting fluid. The purpose of these experiments was to investigate the so-called upstream wake effect which has been predicted by theory. It is believed that these are the first experiments in which the upstream wake has been observed. Although the wake was found to exist as predicted when the Alfvén number is greater than one, its decay behaviour was remarkably different from that which was predicted. The solutions for an infinite medium predicted that in the wake the perturbations should decay inversely as the distance from the body. However, the experiments showed that the perturbations decayed exponentially. It was finally shown that this change in the decay behaviour was an effect of the walls and the conducting material surrounding the fluid.

Forces, moments, and added masses for Rankine bodies

The dynamical theory of the motion of a body through an inviscid and incompressible fluid has yielded three relations: a first, due to Kirchhoff, which expresses the force and moment acting on the body in terms of added masses; a second, initiated by Taylor, which expresses added masses in terms of singularities within the bòdy; and a third, initiated by Lagally, which expresses the forces and moments in terms of these singularities. The present investigation is concerned with generalizations of the Taylor and Lagally theorems to include unsteady flow and arbitrary translational and rotational motion of the body, to present new and simple derivations of these theorems, and to compare the Kirchhoff and Lagally methods for obtaining forces and moments. In contrast with previous generalizations, the Taylor theorem is derived when other boundaries are present; for the added-mass coefficients due to rotation alone, for which no relations were known, it is shown that these relations do not exist in general, although approximate ones are found for elongated bodies. The derivation of the Lagally theorem leads to new terms, compact expressions for the force and moment, and the complete expressions of the forces and moments in terms of singularities for elongated bodies.

On a generalization of Taylor’s virtual mass relation for Rankine bodies

On a generalization of Taylor’s virtual mass relation for Rankine bodies