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.

scholarly journals Remarks on Regularized Stokeslets in Slender Body Theory

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 283
Author(s):  
Laurel Ohm
Keyword(s):  
Stokes Flow ◽  
Regularization Parameter ◽  
Slender Body ◽  
Slender Body Theory ◽  
Classical Counterpart ◽  
Regularized Stokeslets ◽  
The Difference ◽  
Body Theory ◽  
Force Data ◽  
Well Posed

We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius ϵ>0. Denoting the regularization parameter by δ, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive L∞ bounds for the difference between regularized SBT and its classical counterpart in terms of δ, ϵ, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to log(δ/ϵ); in particular, δ=ϵ is necessary to avoid an O(1) discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to log(1+δ2/ϵ2), and any choice of δ∝ϵ will result in an O(1) discrepancy as ϵ→0. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed slender body PDE which classical SBT is known to approximate. Numerics verify this O(1) discrepancy but also indicate that the difference may have little impact in practice.

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.

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.

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

2012 ◽  
Vol 56 (04) ◽  
pp. 207-214
Author(s):  
Brandon M. Taravella ◽  
William S. Vorus
Keyword(s):  
General Solution ◽  
High Speed ◽  
Near Field ◽  
Wave Resistance ◽  
Slender Body ◽  
Slender Body Theory ◽  
Rates Of Change ◽  
Order Of Magnitude ◽  
Flow Variables ◽  
Body Theory

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.

scholarly journals Dropping slender-body theory into the mud

2019 ◽  
Vol 862 ◽  
pp. 1-4
Author(s):  
S. E. Spagnolie
Keyword(s):  
Fluid Flow ◽  
Aspect Ratio ◽  
Slender Body ◽  
Viscoplastic Fluid ◽  
Slender Body Theory ◽  
Viscous Fluid Flow ◽  
Physical Systems ◽  
Surrounding Environment ◽  
Slender Bodies ◽  
Body Theory

The equations describing classical viscous fluid flow are notoriously challenging to solve, even approximately, when the flow is host to one or many immersed bodies. When an immersed body is slender, the smallness of its aspect ratio can sometimes be used as a basis for a ‘slender-body theory’ describing its interaction with the surrounding environment. If the fluid is complex, however, such theories are generally invalid and efforts to understand the dynamics of immersed bodies are almost entirely numerical in nature. In a valiant effort, Hewitt & Balmforth (J. Fluid Mech., vol. 856, 2018, pp. 870–897) have unearthed a theory to describe the motion of slender bodies in a viscoplastic fluid, ‘fluids’ such as mud or toothpaste which can be coaxed to flow, but only with a sufficiently large amount of forcing. Mathematical theories for some tremendously complicated physical systems may now be within reach.

scholarly journals Viscoplastic slender-body theory

2018 ◽  
Vol 856 ◽  
pp. 870-897 ◽  
Author(s):  
D. R. Hewitt ◽  
N. J. Balmforth
Keyword(s):  
Yield Stress ◽  
Numerical Approach ◽  
The Body ◽  
Constitutive Law ◽  
Slender Body ◽  
Viscoplastic Fluid ◽  
Slender Body Theory ◽  
Local Flow ◽  
Flow Around A Cylinder ◽  
Body Theory

The theory of slow viscous flow around a slender body is generalized to the situation where the ambient fluid has a yield stress. The local flow around a cylinder that is moving along or perpendicular to its axis, and rotating, provides a first step in this theory. Unlike for a Newtonian fluid, the nonlinearity associated with the viscoplastic constitutive law precludes one from linearly superposing solutions corresponding to each independent component of motion, and instead demands a full numerical approach to the problem. This is accomplished for the case of a Bingham fluid, along with a consideration of some asymptotic limits in which analytical progress is possible. Since the yield stress of the fluid strongly localizes the flow around the body, the leading-order slender-body approximation is rendered significantly more accurate than the equivalent Newtonian problem. The theory is applied to the sedimentation of inclined cylinders, bent rods and helices, and compared with some experimental data. Finally, the theory is applied to the locomotion of a cylindrical filament driven by helical waves through a viscoplastic fluid.

Ambient Supercavities of Slender Bodies of Revolution

1986 ◽  
Vol 30 (03) ◽  
pp. 215-219
Author(s):  
William S. Vorus
Keyword(s):  
Cavitation Number ◽  
Slender Body ◽  
Surface Velocity ◽  
Body Of Revolution ◽  
Cavity Pressure ◽  
Slender Body Theory ◽  
Bodies Of Revolution ◽  
Specific Analysis ◽  
Slender Bodies ◽  
Body Theory

Slender-body theory is applied in an analysis of the flow about the general supercavitating streamlined body of revolution. The formulation is specialized to the case of ambient cavity pressure (zero cavitation number) for the specific analysis conducted. Numerical procedures are outlined. The methodology is demonstrated in calculating the cavity shapes, surface velocity distributions, and cavity form drag coefficients for three idealized bodies. These are the convex paraboloid, the conical frustrum, and the concave paraboloid. Characteristic differences in the flows in each of the cases are discussed.

The wave drag at zero lift of slender delta wings and similar configurations

1956 ◽  
Vol 1 (3) ◽  
pp. 337-348 ◽  
Author(s):  
M. J. Lighthill
Keyword(s):  
Uniform Distribution ◽  
Present Method ◽  
Wave Drag ◽  
The Body ◽  
Slender Body ◽  
Trailing Edge ◽  
Slender Body Theory ◽  
Edge Angle ◽  
Finite Wing ◽  
Body Theory

Ward's slender-body theory of supersonic flow is applied to bodies terminating in either (i) a single trailing edge at right angles to the oncoming supersonic stream, or (ii) two trailing edges at right angles to one another as well as to the oncoming stream, or (iii) a cylindrical section with two or four identical fins equally spaced round it. The wave drag at zero lift, D, is given by the expression $\frac {D}{\frac {1}{2}\rho U^2} &=& \frac {1}{2\pi}\int^l_0 \int^l_0 log\frac{1}{|s-z|}S^{\prime \prime}(s)S^{\prime \prime}(z)dsdz - \\ &-& \frac{S^\prime (l)}{\pi}\int^l_0 log \frac {l}{l-z}S^{\prime \prime}(z)dz + \frac{S^{\prime 2}(l)}{2\pi} \{ log \frac{l}{(M^2-1)^{1|2}b}+k \} $ where l is the length of the body, b the semi-span of the trailing edge (or length of trailing edge of a single fin), and S(z) is the cross-sectional area of the body at a distance z behind the apex. The constant k depends on the distribution of trailing-edge angle along the span for each trailing-edge configuration. In case (i) it is 1·5 for a uniform distribution of trailing-edge angle and 1·64 for an elliptic distribution. In case (ii) it is 1·28 for a uniform distribution and 1·44 for an elliptic distribution. Study of case (iii) indicates that interference effects due to the presence of the body reduce the drag of the fins. For example, with a uniform distribution of trailing-edge angle, k for two fins falls from 1·5 in the absence of a body to 1·06 when the body radius equals the trailing-edge semi-span, while k for four fins falls from 1·28 to 0·45 under the same conditions.Where ordinary finite-wing theory is applicable, the present method must agree with it for small $(M^2-1)^{1|2}b|l$, and this is confirmed by two examples (§3), but within the limit imposed by slenderness the present method is of course more widely applicable, as well as simpler, than finite-wing theory.It is not known experimentally whether slender-body theory gives accurate predictions of drag at zero lift, for the shapes here discussed, under the conditions for which on theoretical grounds it might be expected to do so. It should be noted that, although tests have not yet been made on ideally suitable bodies, no clear the drag is therefore twice that of a wing made up of two of them. The final stages of the process cannot be represented by slender-body theory, but the initial trend may well be indicated fairly accurately.

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.

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