Sphere Drag Research Articles

Diapirism in the earth's mantle: Experiments on the motion of a hot sphere in a fluid with temperature-dependent viscosity

The ascent of magma diapirs through the earth's mantle is modelled experimentally by the motion of a hot metal sphere through a fluid whose viscosity varies strongly with temperature. The dimensionless drag on the sphere (drag number D ) and the heat transfer from it (Nusselt number Nu ) are found as functions of the dimensionless velocity of the sphere (Peclet number Pe ) and the viscosity contrast μ ∞ / μ 0 = 10 γ , where μ ∞ and μ 0 are the viscosities of the fluid far from the sphere and at its surface. The drag D = D ( Pe , γ ) has two limits. For large Pe and small γ (“Stokes” limit), the drag approaches the Stokes' Law result. For small Pe and large γ (“lubrication” limit), the drag is orders of magnitude less than that predicted by Stokes' Law. Nu is a function of Pe alone. For reasonable values of the diapir radius and the viscosity contrast, the dimensionless scale height Pe /3 Nu may exceed a critical value, resulting in progressive melting during ascent. This suggests that diapirs may ascend great distances through the mantle while remaining largely molten. Lamont-Doherty Geological Observatory Contribution No. 3414.

On the Motion of Spheres in Fluid at Low Reynolds Numbers : Part.1 Flow past a Solid Sphere

A solid -sphere drag is measured at low Reynolds numbers, of which the estimated uncertainty is ± 1.2 %. The measurements show that the Stokes formula for the sphere drag is applicable only at Re=0 and that the Oseen formula for the sphere drag represents the actual drag most accurately at the Reynolds numbers below 0.4. Here the Reynolds number is related to the sphere diameter. New method to visualize streamlines for the steady flow past a body is developed by exploiting a naturally occurring phenomenon in glycerol. An application of this method to visualization of streamlines around a solid sphere reveals that the first -order solution for the Oseen equation represents the flow field around a sphere most accurately at the Reynolds numbers below 0.4.

A similarity analysis of the droplet trajectory equation

A procedure is established to reduce the number of similarity parameters in the trajectory equation for particles in a moving fluid. This is accomplished by the use of an approximate sphere drag law to derive a new trajectory scaling parameter. The modified inertia parameter proposed by Langmuir specifically for the aircraft icing problem is analyzed and for the first time a closed-form solution is obtained. Both experimental and analytical results are presented to verify this new trajectory scaling parameter.

The effects of turbulence and surface roughness on the drag of spheres in thin jets

AbstractAn extension is made to previous work on spheres in thin jets to cover the cases of high stream turbulence and sphere surface roughness. Limited results for the latter agree with past work by Achenbach. Results for cases of increasing stream turbulence line up not only with other high turbulence data but also with Dryden's long-established results for turbulence levels up to a few percent and thereby present some problems, both in the interpretation of the physical flow processes and in nomenclature for the criticality conditions involved.

Asymptotic models in the theory of a diffusion screening layer

23. K. S. Hadjimichalis and C. L. Brundin, "The effect of wall temperature on sphere drag in hypersonic transition flow," in: Rarefied Gas Dynamics~ Vol. 2, Porz--Wahn (1974), DI3 11-D13/9. 24. W. H. Sims, "Experimental sphere drag results in the near-free molecule regime," in: Rarefied Gas Dynamics, Vol. i, Academic Press, New York (1969), pp. 751-756. 25. J. Aroesty, "Sphere drag in low-density supersonic flow," in: Rarefied Gas Dynamics, Vol. 2, Academic Press, New York (1963), pp. 261-277.

Sphere drag at Mach numbers from 0·3 to 2·0 at Reynolds numbers approaching 107

Our analysis of 18th and 19th century cannon firings shows that knowledge of sphere drag can be substantially extended into the region of 0·3 ≤ M∞ ≤ 2·0 and Re∞d up to 107. Bashforth’s chronographic measurements (1868) are of a quality comparable to modern measurements. The data of Mayevski (chronograph, 1868), Hutton (ballistic pendulum, 1787-1791), and Didion (ballistic pendulum, 1839-1840) are of lesser accuracy but in agreement with Bashforth’s. These cannon data are combined with modern data to provide the most extensive curves available of CD vs. Re∞d in this region. Interesting features of these curves for M∞ ≤ 1·0 are briefly described.

Reply by Author to M. J. Walsh

Henderson, C. B., Coefficients of Spheres in Continuum and Rarefied Flows, AIAA Journal, Vol. 14, June 1966, pp. 707708. Bailey, A. B. and Hiatt, J., Free-Flight Measurements of Sphere Drag at Subsonic, Transonic, Supersonic, and Hypersonic Speeds for Continuum, Transition, and Near-Free-Molecular Flow Conditions, Arnold Engineering Development Center, Tullahoma, Tenn., AEDCTR-70-291, March 1971. Walsh, M. J., Coefficient Equations for Small Particles in High Speed Flows, AIAA Journal, Vol. 13, Nov. 1975, pp. 15261528. Walsh, M. J., Influence of Particle Drag Coefficient on Particle Motion in High-Speed Flow with Typical Laser Velocimetry Applications, NASATN D-8120, Feb. 1976. Zarin, N. A., Measurement of Non-Continuum and Turbulence Effects on Subsonic Sphere Drag, NASA CR-1585, June 1970. Emmons, H. W., Ed., Fundamentals of Gas Dynamics, Vol. HI. High Speed Aerodynamics and Jet Propulsion, Princeton University Press, Princeton, N.J., 1958, p. 689.

Comment on 'Drag coefficient of spheres in continuum and rarefied flows'

A paper by Henderson (1976) provides a method of predicting experimental sphere drag data. This approach uses two equations for the drag coefficient, one for relative Mach number less than one, one for relative Mach number greater than 1.75. For relative Mach numbers between these limits a linear interpolation procedure is followed. In a comment on this paper, it is claimed, on the basis of comparing predictions with experimental results, that a method proposed by Walsh (1975) gives better predictions of the drag coefficient for relative Mach numbers less than 1.75, provided that a modification of the procedure is made for relative Mach numbers less than 0.1. For values over 1.75, both methods are considered equally accurate. In a reply to this comment, it is agreed that the Walsh method is more accurate when Reynolds numbers are within a range between 20 and 200, and Mach numbers are between 0.5 and 1.25. Presumed errors and possible limitations in the Walsh procedure for predicting drag coefficients are discussed.

Sphere drag at transonic speeds and high Reynolds numbers

M of sphere drag have been made in the AEDC aeroballistics range G over the Maeh number range 0.9 A/o, <1.4 at a Reynolds number of approximately 10. These values of sphere drag were found to be larger than the values derived from the experimental summary curves presented in Ref. 1. The change in sphere drag with Mach number near Mach 1, as well as with Reynolds number in excess of 10, is significant and care is required in establishing summary curves. On the basis of these more recent measurements and those contained in Refs. 2 and .3, the summary curves presented in Ref. 1 have been reevaluated. A plot of the revised values of sphere drag f or 5 x 10 </?£«, <10 and 0.1 <M^ < 1.75 is presented in Fig. 1. The difference between the previous and present values of sphere drag coefficient at M^ =0.955 (Fig. 1) is representative of the changes that have been made, in the Mach number range of 0.8 to 1.5. Sphere drag coefficient is shown to increase with increasing Reynolds number for 10 <Rex ^ 10 for Mach numbers ranging from 0.9 to 1.75 (Fig. 1).

Flow around and heat and mass transfer of a sphere with blowing at medium reynolds numbers

The Navier-Stokes and heat- and mass-transfer equations are solved numerically for a sphere with uniform blowing over the surface in the Reynolds number range up to 20. A method of refining the boundary conditions far from the sphere is proposed in both problems. A difference scheme from other authors is used to solve the hydrodynamic problem, and an explicit difference scheme with a second order of approximation is used for the heat problem. It is shown that blowing diminishes the aerodynamic drag of the sphere and the temperature or concentration gradient at its surface, i.e., the heat- and mass-transfer intensity.

Drag coefficient equations for small particles in high speed flows

The paper determines the effect of various available drag coefficient equations on particle velocity calculations for typical two phase flows encountered in supersonic and turbulent laser velocimeter applications. The predictions of the particle drag coefficient equations are compared with experimental sphere drag data. For the laser velocimeter applications, the relative Mach number less than 2 and the relative Reynolds number less than 200 are of particular importance.

Sphere drag coefficient for subsonic speeds in continuum and free-molecule flows

An extensive series of measurements of sphere drag coefficients has been made in an aeroballistic range for a broad range of Reynolds and Mach numbers. These measurements have been compared with those obtained in other test facilities. As a result of this comparison it has been possible to suggest reasons for many of the inconsistencies in the earlier measurements and to establish more accurate values of the sphere drag coefficient for M∞ [lsim ] 0·2 and 10−2 [lsim ] Re∞ [lsim ] 107.

Flow around a sphere with material cross flow at low reynolds numbers

An analytical solution is carried out for the problem of the flow around a sphere with material cross flow at Reynolds numbers less than 1 and a blowing velocity less than the free stream velocity. The method of asymptotic expansions of Pearson and Proudman is used for the solution. Expressions are obtained for the distribution of the current and velocity component functions as well as for the aerodynamic drag coefficient of the sphere. It is shown that blowing diminishes the sphere drag, where its influence will increase as the Reynolds number grows.

Mean Free Path of Emitted Molecules and Correlation of Sphere Drag Data

The mean free path between molecules emitted from and incident to a surface in rarefied flow is considered. The problem is formulated for the case of two classes of hard sphere molecules with different Maxwellian velocity distribution functions. Approximate analytical solutions are obtained to the integral describing the collision frequency between emitted and incident molecules. These analytical solutions are compared with numerical solutions, and the domain of validity of the analytical results is established. The analytical solutions provide a simple expression for the mean free path which is used to define a correlation parameter. This parameter is used to correlate a great deal of low-density supersonic and hypersonic sphere drag data which have a wide range of freestream and body surface conditions.

Sphere Drag Coefficients for a Broad Range of Mach and Reynolds Numbers

The purpose of the present investigation was to establish accurate values of sphere drag coefficient in the flight regime 0.1 < Mx < 6.0 and 2 x 10 < Re^ < 10 for TJT^ % 1.0. To this end, an extensive series of measurements was made in a ballistic range. These measurements, together with other published data, permit the derivation of sphere drag confidents with an uncertainty of + 2 % in this flight regime. In addition, sifficient information is presented such that reasonable estimates of sphere drag confident can be made for TJT^ ^ 1.0, 0.05 < M^ < 20.0, and 2 x 10~ < Re*, < W

Sting-free measurements of sphere drag in laminar flow

An aerodynamic investigation was conducted to determine the laminar-flow drag coefficient of spheres of various sizes in a subsonic wind tunnel. The tests were conducted using the M.I.T.-N.A.S.A. prototype magnetic-balance system. By measuring the drag of different sized spheres without model support interference the tunnel wall effect can be deduced. The present results indicate that the classical wind tunnel correction does not completely account for the effects of model size and wall interference. That is, the corrected drag coefficient data for the different sphere sizes differ among themselves in the region of Reynolds number overlap.A comparison of the present sphere drag results with those of numerous other investigations including free-flight and ballistic-range data is given. The drag coefficients presented here are slightly lower than those of other workers for Reynolds numbers ranging from 20 000 to 150 000, but fall between the limits of experimental scatter for Reynolds numbers from 150 000 to 260 000.An analysis of the estimated error in the present data indicates the primary source to be measurement of the wind tunnel parameters rather than errors resulting from the balance system.

On the response of a sphere to an acoustic pulse

The motion of a rigid sphere responding to the passage of an acoustic pulse is considered by means of a simple approximate model which neglects the diffraction of the pulse front. The model is based on a solution of the equivalent inviscid problem and assumes that, initially, the flow field around the sphere corresponds to the steady incompressible flow of an inviscid fluid over a sphere at rest. For t &gt; 0, the motion is studied by means of the unsteady Stokes equations. Results for the sphere's velocity, displacement and drag are obtained in closed form in terms of tabulated functions and compared with results obtained by using the Stokes drag. It is found that, when the ratio of gas density to sphere material density is finite, the initial response of the sphere differs considerably from that predicted by the use of the Stokes drag. However, when the ratio of gas density to sphere material density is infinitesimal, the differences disappear. These results may be of some importance in the study of shock-induced droplet collisions in aerosol clouds.

The Hydrodynamic Drag on a Small Sphere in an Ionized Gas

A theoretical and experimental study was made of the hydrodynamic drag on a small sphere moving relative to a plasma in local thermodynamic equilibrium. The results have application to velocity measurement in ionized gases using the technique of injecting small particles and following their motion as a function of time. The previously available sphere drag results for fluids with properties which vary linearly with temperature are shown to be in error by as much as 40 percent for an argon plasma at 13,000 deg K, for example, because large temperature gradients occur in the vicinity of the sphere where fluid properties vary nonlinearly with temperature. A new drag calculation for a sphere Reynolds number of zero has been made for an argon plasma taking into account nonlinear variations of transport properties with temperature. Sphere deflection measurements in an argon plasma have been made at Reynolds numbers between 0.3 and 1.5. The interpretation of these measurements in terms of sphere drag is subject to confirmation of the plasma transport properties used in the data reduction, but the difference between the measured drag and the drag calculated for zero Reynolds number appears to be approximately the same as in the classical case for invariant fluid properties.

Drag Measurement in a Low-Density Gas Stream

Two methods of measuring the drag of bodies in a low-density gas stream are described. The supersonic stream had a mean free path as great as 4 cm, and the drag forces were small. A drag balance was adapted from an electrical analytical balance to measure forces up to 70 dynes with less than 1-percent error. With this balance, the drag of spheres was measured for the Knudsen number range from 1.0 to 5. The freedrop trajectory method yielded drag coefficients in the range of Knudsen numbers from 2.5 to 30. These tests demonstrated the feasibility of making accurate drag measurements under the free-molecule conditions that can be generated by exhausting a gas into a cryogenically pumped vacuum chamber.

Sphere Drag in Solid Rockets—Non-Continuum and Turbulence Effects

Momentum transfer between gas and condensed phase in metallized solid propellant rocket motors, measuring noncontinuum and turbulence effects on sphere drag