An improved slender-body theory for Stokes flow

1980 ◽  
Vol 99 (2) ◽  
pp. 411-431 ◽  
Author(s):  
Robert E. Johnson
Keyword(s):  
Boundary Condition ◽  
Cross Sections ◽  
Unit Length ◽  
The Body ◽  
Entire Body ◽  
Slender Body Theory ◽  
Inertia Effects ◽  
Force Coefficients ◽  
Centre Line ◽  
Slender Bodies

The present study examines the flow past slender bodies possessing finite centre-line curvature in a viscous, incompressible fluid without any appreciable inertia effects. We consider slender bodies having arbitrary centre-line configurations, circular transverse cross-sections, and longitudinal cross-sections which are approximately elliptic close to the body ends (i.e. prolate-spheroidal body ends). The no-slip boundary condition on the body surface is satisfied, using a convenient stepwise procedure, to higher orders in the slenderness parameter (ε) than has previously been possible. In fact, the boundary condition is satisfied up to an error term of O(ε2) by distributing appropriate stokeslets, potential doublets, rotlets, sources, stresslets and quadrupoles on the body centre-line. The methods used here produce an integral equation valid along the entire body length, including the ends, whose solution determines the stokeslet strength or equivalently the force per unit length up to a term of O(ε2). The O(ε2) correction to the stokeslet strength is also found. The theory is used to examine the motion of a partial torus and a helix of finite length. For helical bodies comparisons are made between the present theory and the resistive-force theory using the force coefficients of Gray & Hancock and Lighthill. For the motion considered the Gray & Hancock force coefficients generally underestimate the force per unit length, whereas Lighthill's coefficients provide good agreement except in the vicinity of the body ends.

Inertia effects on the motion of long slender bodies

1989 ◽  
Vol 209 ◽  
pp. 435-462 ◽  
Author(s):  
R. E. Khayat ◽  
R. G. Cox
Keyword(s):  
Reynolds Number ◽  
Asymptotic Expansion ◽  
Body Length ◽  
Unit Length ◽  
The Body ◽  
Total Force ◽  
Cross Sectional ◽  
Inertia Effects ◽  
Slender Bodies ◽  
Valid Solution

A solid long slender body with curved centreline is held at rest in a fluid undergoing a uniform flow. Assuming that the Reynolds number Re based on body length is fixed, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio κ of the cross-sectional radius to body length. In the limit of large Re, this result is no longer valid and an asymptotic expansion in κRe is necessary. A uniformly valid solution is obtained from these two expansions. The total force and torque acting on a body with a straight centreline are explicitly determined. The limiting cases of small and large Re are studied in detail.

Slender-body theory for slow viscous flow

1976 ◽  
Vol 75 (4) ◽  
pp. 705-714 ◽  
Author(s):  
Joseph B. Keller ◽  
Sol I. Rubinow
Keyword(s):  
Integral Equation ◽  
Incompressible Fluid ◽  
Viscous Incompressible Fluid ◽  
Unit Length ◽  
The Body ◽  
Slender Body ◽  
Slow Flow ◽  
Slender Body Theory ◽  
Slow Viscous Flow ◽  
Body Theory

Slow flow of a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions. The main result is an integral equation for the force per unit length exerted on the body by the fluid. The novelty is that the body is permitted to twist and dilate in addition to undergoing the translating, bending and stretching, which have been considered by others. The method of derivation is relatively simple, and the resulting integral equation does not involve the limiting processes which occur in the previous work.

scholarly journals Slender body theory for particles with non-circular cross-sections with application to particle dynamics in shear flows

2019 ◽  
Vol 877 ◽  
pp. 1098-1133 ◽  
Author(s):  
Neeraj S. Borker ◽  
Donald L. Koch
Keyword(s):  
Cross Section ◽  
Relative Motion ◽  
Cross Sections ◽  
Stokes Flow ◽  
Unit Length ◽  
Fluid Velocity ◽  
Simple Shear Flow ◽  
Slender Body Theory ◽  
Aspect Ratios ◽  
Cross Sectional Dimension

This paper presents a theory to obtain the force per unit length acting on a slender filament with a non-circular cross-section moving in a fluid at low Reynolds number. Using a regular perturbation of the inner solution, we show that the force per unit length has $O(1/\ln (2A))+O(\unicode[STIX]{x1D6FC}/\ln ^{2}(2A))$ contributions driven by the relative motion of the particle and the local fluid velocity and an $O(\unicode[STIX]{x1D6FC}/(\ln (2A)A))$ contribution driven by the gradient in the imposed fluid velocity. Here, the aspect ratio ($A=l/a_{0}$) is defined as the ratio of the particle size ($l$) to the cross-sectional dimension ($a_{0}$) and $\unicode[STIX]{x1D6FC}$ is the amplitude of the non-circular perturbation. Using thought experiments, we show that two-lobed and three-lobed cross-sections affect the response to relative motion and velocity gradients, respectively. A two-dimensional Stokes flow calculation is used to extend the perturbation analysis to cross-sections that deviate significantly from a circle (i.e. $\unicode[STIX]{x1D6FC}\sim O(1)$). We demonstrate the ability of our method to accurately compute the resistance to translation and rotation of a slender triaxial ellipsoid. Furthermore, we illustrate novel dynamics of straight rods in a simple shear flow that translate and rotate quasi-periodically if they have two-lobed cross-section, and rotate chaotically and translate diffusively if they have a combination of two- and three-lobed cross-sections. Finally, we show the remarkable ability of our theory to accurately predict the motion of rings, retaining great accuracy for moderate aspect ratios (${\sim}10$) and cross-sections that deviate significantly from a circle, thereby making our theory a computationally inexpensive alternative to other Stokes flow solvers.

Variational properties of steady fall in Stokes flow

1972 ◽  
Vol 52 (2) ◽  
pp. 321-344 ◽  
Author(s):  
H. F. Weinberger
Keyword(s):  
Stokes Flow ◽  
Variational Principles ◽  
Vertical Axis ◽  
The Body ◽  
Centre Of Mass ◽  
Slender Body Theory ◽  
First Approximation ◽  
Slender Bodies ◽  
Settling Speed ◽  
Body Theory

It is shown that for a given body and a given orientationgthere is always a position of the centre of mass which produces a stable falling motion in a very viscous fluid withgvertical and, in general, with a spin about the vertical axis. The corresponding terminal settling speed is bounded by means of several variational principles.Relations between the terminal speeds for falls with different downward directions and between the terminal speed and the geometry of the body are deduced. In particular, it is proved that for a large class of slender bodies the first approximation to the drag obtained from the slender-body theory of Burgers (1938) is correct. It follows that the ratio of the terminal speeds for falls with the long axis vertical and horizontal is near two.

Supersonic Flow Past Slender Bodies With Discontinuous Profile Slope

1955 ◽  
Vol 6 (2) ◽  
pp. 114-124 ◽  
Author(s):  
L. E. Fraenkel ◽  
H. Portnoy
Keyword(s):  
Internal Flow ◽  
Leading Edge ◽  
Rate Of Change ◽  
The Body ◽  
Cross Sectional ◽  
Slender Body Theory ◽  
Bodies Of Revolution ◽  
Slender Bodies ◽  
External Forces ◽  
Sectional Shape

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.

Calculation of Subsonic Normal Force and Centre of Pressure Position of Bodies of Revolution Using a Slender Body Theory – Boundary Layer Method

1980 ◽  
Vol 31 (1) ◽  
pp. 1-25
Author(s):  
K.D. Thomson
Keyword(s):  
Boundary Layer ◽  
Velocity Distribution ◽  
Normal Force ◽  
The Body ◽  
Slender Body ◽  
Centre Of Pressure ◽  
Slender Body Theory ◽  
Bodies Of Revolution ◽  
Slender Bodies ◽  
Body Theory

SummaryThe aim of this paper is to present a method for predicting the aerodynamic characteristics of slender bodies of revolution at small incidence, under flow conditions such that the boundary layer is turbulent. Firstly a panel method based on slender body theory is developed and used to calculate the surface velocity distribution on the body at zero incidence. Secondly this velocity distribution is used in conjunction with an existing boundary layer estimation method to calculate the growth of boundary layer displacement thickness which is added to the body to produce the effective aerodynamic profile. Finally, recourse is again made to slender body theory to calculate the normal force curve slope and centre of pressure position of the effective aerodynamic profile. Comparisons made between predictions and experiment for a number of slender bodies extending from highly boattailed configurations to ogive-cylinders, and covering a large range of boundary layer growth rates, indicate that the method is useful for missile design purposes.

A Slender-Body Theory in Acoustics

1995 ◽  
Author(s):  
Michel Tran Van Nhieu
Keyword(s):  
Sound Field ◽  
Practical Method ◽  
The Body ◽  
Slender Body ◽  
Slender Body Theory ◽  
New Approach ◽  
Singular Perturbation Methods ◽  
Slender Bodies ◽  
Computation Load ◽  
Body Theory

Abstract A new approach has been proposed to calculate the sound pressure field radiated or scattered from slender bodies. The theoretical analysis is based upon mathematical singular perturbation methods leading to the so-called slender-body approximation. The theory provides a practical method to compute the sound field as it allows a great simplification in the geometrical representation of the body surface and a reduced computation load. This paper summarizes the main results obtained with this approximation at the present state of investigation.

scholarly journals Hydromechanics of low-Reynolds-number flow. Part 5. Motion of a slender torus

1979 ◽  
Vol 95 (2) ◽  
pp. 263-277 ◽  
Author(s):  
Robert E. Johnson ◽  
Theodore Y. Wu
Keyword(s):  
Stokes Flow ◽  
Unit Length ◽  
Flow Characteristics ◽  
Slip Boundary Condition ◽  
The Body ◽  
Accurate Analysis ◽  
Cross Sectional ◽  
Reynolds Number Flow ◽  
Singularity Method ◽  
Centre Line

In order to elucidate the general Stokes flow characteristics present for slender bodies of finite centre-line curvature the singularity method for Stokes flow has been employed to construct solutions to the flow past a slender torus. The symmetry of the geometry and absence of ends has made a highly accurate analysis possible. The no-slip boundary condition on the body surface is satisfied up to an error term of O(ε2 In ε), where ε is the slenderness parameter (ratio of cross-sectional radius to centre-line radius). This degree of accuracy makes it possible to determine the force per unit length experienced by the torus up to a term of O(ε2). A comparison is made between the force coefficients of the slender torus to those of a straight slender body to illustrate the large differences that may occur as a result of the finite centre-line curvature.

scholarly journals Extensions of d'Alembert's paradox for elongated bodies

2015 ◽  
Vol 471 (2181) ◽  
pp. 20150106
Author(s):  
Philippe R. Spalart
Keyword(s):  
Cross Section ◽  
Unit Length ◽  
Angle Of Attack ◽  
Three Dimensional ◽  
Added Mass ◽  
The Body ◽  
Pitching Moment ◽  
Constant Cross Section ◽  
Slender Body Theory ◽  
The Cross

The steady incompressible irrotational flow past a three-dimensional body of any shape generates no forces. The historic paradox refers only to drag, but lift is also zero, which has been known but not emphasized. The new material concerns a body with a long constant cross section, such as a train. The final results for forces and moments are very simple. With zero angle of attack, we show that the force vectors on the front and rear parts of the body are each (asymptotically) equal to zero, if the pressure is referred to the freestream pressure. The lift and drag coefficients, based on frontal area, vanish proportionally to d / l and ( d / l ) 2 , respectively, where d / l is the diameter-to-length ratio. This applies to any shape of the cross section, and of the ends. With an angle of attack, the nose and tail forces are non-zero but depend only on the angle of attack and the cross section's added mass per unit length. The pitching moment is proportional to the total added mass and the sine of twice the angle of attack. The present results clarify slender-body theory results. The practical consequence is that, for a long body with constant cross section, the shape of the nose or the tail is irrelevant to its own ‘partial’ drag and lift, and to the pitching moment.

scholarly journals Regularized Stokeslets Lines Suitable for Slender Bodies in Viscous Flow

Fluids ◽  
2021 ◽  
Vol 6 (9) ◽  
pp. 335
Author(s):  
Boan Zhao ◽  
Lyndon Koens
Keyword(s):  
Regularization Parameter ◽  
Near Field ◽  
The Body ◽  
Slender Body ◽  
Slender Body Theory ◽  
Slip Boundary ◽  
Slender Bodies ◽  
Angular Dependency ◽  
Regularized Stokeslets ◽  
Body Theory

Slender-body approximations have been successfully used to explain many phenomena in low-Reynolds number fluid mechanics. These approximations typically use a line of singularity solutions to represent flow. These singularities can be difficult to implement numerically because they diverge at their origin. Hence, people have regularized these singularities to overcome this issue. This regularization blurs the force over a small blob and thereby removing divergent behaviour. However, it is unclear how best to regularize the singularities to minimize errors. In this paper, we investigate if a line of regularized Stokeslets can describe the flow around a slender body. This is achieved by comparing the asymptotic behaviour of the flow from the line of regularized Stokeslets with the results from slender-body theory. We find that the flow far from the body can be captured if the regularization parameter is proportional to the radius of the slender body. This is consistent with what is assumed in numerical simulations and provides a choice for the proportionality constant. However, more stringent requirements must be placed on the regularization blob to capture the near field flow outside a slender body. This inability to replicate the local behaviour indicates that many regularizations cannot satisfy the no-slip boundary conditions on the body’s surface to leading order, with one of the most commonly used blobs showing an angular dependency of velocity along any cross section. This problem can be overcome with compactly supported blobs, and we construct one such example blob, which can be effectively used to simulate the flow around a slender body.

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