Drag Formula Research Articles

Low Reynolds numbers flow past an ellipsoid of revolution of large aspect ratio

The results of Proudman & Pearson (1957) and Kaplun & Lagerstrom (1957) for a sphere and a cylinder are generalized to study an ellipsoid of revolution of large aspect ratio with its axis of revolution perpendicular to the uniform flow at infinity. The limiting case, where the Reynolds number based on the minor axis of the ellipsoid is small while the other Reynolds number based on the major axis is fixed, is studied. The following points are deduced: (1) although the body is three-dimensional the expansion is in inverse power of the logarithm of the Reynolds number as the case of a two-dimensional circular cylinder; (2) the existence of the ends and the variation of the diameter along the axis of revolution have no effect on the drag to the first order; (3) a formula for drag is obtained to higher order.

Mutual Interaction of Two Spheroids Sedimenting in a Viscous Fluid

A general solution of the Stokes equations of a viscous fluid with boundary conditions related to a spheroid is considered and the formulae for drag and moment are obtained to a certain order of approximation. The results are applied to the problem of sedimentation of two spheroids in which the spheroids have the same shape and velocity and their axes of symmetry are both in horizontal directions. The drag and torque acting on a spheroid are given in terms of mutual interaction of the spheroids. They are considered to a higher order for the special cases where the centers of the spheroids are both on a vertical line or in a horizontal plane.

The slow motion of a sphere in a rotating, viscous fluid

The uniform, slow motion of a sphere in a viscous fluid is examined in the case where the undisturbed fluid rotates with constant angular velocity Ω and the axis of rotation is taken to coincide with the line of motion. The various modifications of the classical problem for small Reynolds numbers are discussed. The main analytical result is a correction to Stokes's drag formula, valid for small values of the Reynolds number and Taylor number and tending to the classical Oseen correction as the last parameter tends to zero. The rotation of a free sphere relative to the fluid at infinity is also deduced.

Drag on an axially symmetric body vibrating slowly along its axis in a viscous fluid

Let D0 be the Stokes drag on an axially symmetric body moving parallel to its axis with velocity U0 through an unbounded fluid. The drag D experienced by the same body oscillating with velocity U = U0eiσt along its axis in the unbounded fluid is given by the expression $\frac{D}{D_0} = \left \{{1 +{\frac {d_0}{6 \surd{(2)} \pi \mu a U_0}}(1+i)M+O(M^2)}\right \}e^{i \sigma t},$ where a is any characteristic particle dimension and $M^2 = a^2 \sigma \rho |\mu$ is a dimensionaless number. The part of this drag formula which gives the energy dissipation is calculated for bodies of various shapes.

バレルメッキにおけるスクイ出シについて

It is essential to decrease the time of analysis for simplifying the control of plating bath. Because, according to the result of analysis, a suitable chemical should be added in proportion to the variation of solution composition, which is caused by three main factors, chemical, electrochemical and mechanical factors. The mechanical factor depends mainly on drawing of plating solution.From the result of the investigation of mechanism of drawing of barrel plating solution, the following empirical formula for drag out of barrel plating solution was obtained:D=DB+SS⋅DSwhere D is the total drawing amount of barrel plating solution, DB is the drawing amount from barrel only that changes depending on the porosity of barrel, DS is the drawing amount per surface area that changes depending on the ratio of total apparent volume of plating solution to the barrel volume and SS is the total surface area of plating solution.

The Diurnal Atmospheric Bulge and Its Effect on Satellite Motions.

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Diurnal Atmospheric Bulge and Its Effect on Satellite Motions. Wyatt, Stanley P. Abstract Several authors have derived expressions for the secular acceleration of a satellite moving through the high atmosphere and giving away momentum to the air. The simplest assumptions one can make are that the atmosphere is spherically symmetric, stationary, and of constant scale height. The resulting drag formula then permits one to use the observed accelerations to deduce the product of air density times the square root of scale height at the perigee level of the satellite. Analyses by a number of authors have revealed primary facts concerning the structure of the outer atmosphere. First, the scale height of the atmosphere increases with height above ground throughout all strata traveled by satellites. Second, on the daytime hemisphere the atmospheric density correlates strikingly with solar activity. Third, at heights of 500 km and more the atmosphere bulges conspicuously toward the sun. In October of 1958, when Vanguard I was passing through perigee at local time 13 or 14 hr the air density at 650 km was some 10 times greater than from mid-1959 to mid-1960 when its perigee passage occurred during the nighttime hours. At any geographic position this variation amounts to a diurnal rise and fall of the high atmosphere, with maximum expansion occurring an hour or two after noon. Jacchia has used the accelerations of four satellites to construct a model of the atmosphere in the 200-600 km range, incorporating all three of these effects (Jacchia, L. G., Smithsonian Astrophys. Obs. Spec. Rept. No. 39,1960). Because the atmosphere has neither spherical symmetry nor an exponential decrease of density, the present work is addressed to the derivation of a new and more complex drag formula relating perigee density and scale height to the observed acceleration of a satellite moving through a Jacchia-type atmosphere. The essential results are twofold. First, use of the earlier and simpler drag equation may yield atmospheric densities that are systematically incorrect by as much as 50 or 60%, the sign and amount of the error depending on the orbital elements and the orientation of the orbit relative to the bulge on any given date. Second, and provisionally, the new formula indicates that the diurnal bulge is even sharper than previously thought. This work was partly done at the Smithsonian Astrophysical Observatory. Publication: The Astronomical Journal Pub Date: September 1961 DOI: 10.1086/108429 Bibcode: 1961AJ.....66..299W full text sources ADS |

On the wall effect correction of the Stokes drag formula for axially symmetric bodies moving inside a cylindrical tube

In dieser Arbeit wird der Einfluss der Wand auf die Bewegung eines rotationssymmetrischen Korpers langs der Achse eines mit zaher Flussigkeit gefullten Rohres betrachtet. Der Widerstand des Korpers wird unter der Voraussetzung berechnet, dass die Dimensionen des Korpers gegenuber dem Radius des Rohres klein sind. Wir finden fur den WiderstandD eines Korpers, der sich mit der GeschwindigkeitU langs der Achse einer zylindrischen Rohre mit dem inneren RadiusR bewegt, woD o der Widerstand des Korpers in einer sonst den ganzen Raum fullenden Flussigkeit, κ eine Konstante etwa gleich 2.203, und μ die Viskositatskonstante ist.

Stokes flow of a conducting fluid past an axially symmetric body in the presence of a uniform magnetic field

Low Reynolds number flow of an incompressible fluid past an axially symmetric body in the presence of a uniform magnetic field is studied using a perturbation method. It is found that for small Hartmann number M an approximate drag formula is given by $ D^ \prime = D^\prime_0 \left(1 + \frac {D^\prime_0} {16\pi \rho vaU}M\right) + O(M^2),$ where D′0 is the Stokes drag for flow with no magnetic effect.

Wake of a Satellite Traversing the Ionosphere

The particle treatment is applied to a study of the structure of the wake behind a charged body moving supersonically through a low-density plasma. For the case of a body whose dimensions are considerably smaller than a Debye length, a solution is obtained which is very similar in structure to the solution obtained by using the linearized fluid dynamics equation. For the case of a disk whose radial dimensions are much larger than a Debye length, two conical regions are found in the wake. At the surface of each of these cones, over thicknesses of the order of a Debye length, the ion and electron densities are increased over their ambient values. Formulae for the electrohydrodynamic drag on a wire, and on a large disk are obtained.

Nose Drag in Free-Molecule Flow and Its Minimization

The superaerodynamic nose drag of a body in a free-molecule flow involves two parameters: the speed ratio 5 between ordered and random molecular motions (modified Mach Number), and the temperature ratio Tr/T between the solid surface and undisturbed gas. Simplifications of the drag formula are obtained at hypersonic as well as low-subsonic extremes. To minimize the drag on a nose of specified length and base radius, the ordinary method of calculus of variation was found inadequate. A generalized approach has, accordingly, been developed, and the specification of end conditions is discussed at length. Results of the present investigation indicate that in all cases an optimum nose requires a flat tip. The optimum nose curve for the hypersonic extreme does not depend on the temperature ratio, but that for the low-subsonic extreme varies in the following manner: for a hot body the curve is convex; for a cold body, concave. An optimum solution exists in a restricted range of specification only. With prescribed tip and base radii the admissible nose length is bounded below for the cases of hypersonic and low-subsonic hot body and bounded above for the case of low-subsonic cold body. A vanishing tip radius leads to an infinitely long nose in the former and a vanishing nose in the latter case. Optimum nose curves for several temperature ratios at the lowsubsonic extreme, as well as the one for hypersonic extreme, are presented. I t is observed that at the low-subsonic extreme, with Tr/T —> co, the hot-body solution asymptotically approaches the hypersonic solution—i.e., a slender conventional warhead with a flat t ip; whereas with Tr/T —*0, the cold-body solution asymptotically approaches the minimal-surface solution—i.e., (a) with prescribed tip radius, a catenoid; (b) with unspecified tip radius, a flat disc.

Optimum Thickness and Lift Distribution for a Fuselage in the Presence of a Prescribed Wing System

THE RULE, in supersonic flow, for the optimization of a fuselage-thickness distribution in the presence of a prescribed wing-thickness distribution has been described by various authors—e.g., Ward 1 and Jones. I t is interesting to see how easily this rule is derived in terms of singularity distributions from the drag formula developed by Hayes, and how, from Hayes ' formula, the generalization of the rule to include lift and sideforce as well as source distributions is immediately evident. Such a generalization, in terms of area distributions and forces, has already been presented by Lomax and Heaslet. However, the subject seems of sufficient interest to be reconsidered from a slightly different viewpoint. Since the drag formula of Hayes, which is valid within linear theory, is expressed in terms of singularities, it seems natural to determine optimum configurations in terms of singularities and then later obtain the geometry of the configuration. (The analysis presented here appears also in a slightly different form and in somewhat more detail in reference 5.) For midwing monoplanes, the rule in its usual form (without forces) is adequate to prescribe the opt imum fuselage-thickness distribution. However, for configurations such as high-wing monoplanes and ring wings, the thickness distributions and the lift and sideforce distributions do not act independently in producing drag. Thus, in general, the optimization of the fuselage-thickness distribution is affected by the lift and side-force distributions on the wing as well as by the wing thickness. In addition, there is the possibility of optimizing the fuselage lift distribution. This generalized rule prescribes what lineal distribution of sources, lift, and side force should be used to represent a fuselage in order to minimize the wave drag of the fuselage in combination with a given wing (the wing being represented by a spatial distribution

VII. The Kármán street of vortices in a channel of finite breadth

1. The following investigations deal with a Kármán street of vortices, or unsymmetrical double row, in a channel of finite width. A discussion on the symmetrical double row has also been incorporated. For the sake of convenience the work has been divided into two distinct parts—(1) Systems of Line Vortices in a Channel of Finite Breadth, and (II) The Kármán Drag Formula for a Channel of Finite Breadth. Part I has itself been divided into two sections headed Problem I and Problem II, and these sections deal with the stream lines and stability of the unsymmetrical and symmetrical double row respectively. Part II discusses Problem I from the point of view of hydrodynamics and contains a section dealing with the hydrodynamical significance of the stability investigations. Summaries of the investigations have been kept distinct and can be found in paragraphs (1.5) and (1.3) of Part I and Part II respectively.