Showing papers on "Drag coefficient published in 1971"

Momentum absorption by vegetation

Abstract: Measurements were made in a wind-tunnel of the drag on elements of a simply-structured artificial crop, and of the wind profiles above and within the crop. Analysis demonstrates (i)that the drag force on an element of such an array can be calculated from the profile of the turbulent shear flow within the array, using the known (and unmodified), wind-tunnel value of the drag coefficient of the individual element; (ii)that the zero-plane displacement (d) of an aerodynamically rough surface can be identified with the level of action of the drag on its elements; and (iii)that von Karman's constant = 0.41 ± 0.03. The relation z0 = 0.36 (h – d) is suggested for the roughness parameter of vegetation of height h. Calculated values of the drag force, f, on unit column of a real stand of beans in the field, using individual-element drag coefficients (Cd) and measured wind speeds, give f = 3.5 τ0 where τ0 is the downward momentum flux derived from the shape of the wind profile above. On the evidence of conclusion (i) and the dense and complex nature of the bean canopy, the factor 3.5 is attributed to mutual sheltering of neighbouring canopy elements rather than as evidence that the Cd – values are modified by turbulent shear flow. For the artificial crop, and for the real crop, recognition of the wind-speed dependence of the individual-element drag coefficients gives values of eddy viscosity, KM, almost constant in the height range h/3 < z ⩽ h and significantly larger than those found when constant drag coefficients are assumed. Constant KM within a crop canopy is consistent with the wind profile u(z)/u(h) = {1 + α(1 – z/h)}−2: an explicit expression is given for the parameter α.

Influence of surface roughness on the cross-flow around a circular cylinder

Abstract: The influence of surface roughness on the cross-flow around a circular cylinder is the subject of the present experimental work. The investigations were carried out in a high-pressure wind tunnel, thus high Reynolds numbers up to Re = 3 × 106 could be obtained. Local pressure and skin friction distributions were measured. These quantities were evaluated to determine the total drag coefficient and the percentage of friction as functions of Reynolds number and roughness parameter. In addition the local skin friction distribution yields the angular position of boundary-layer transition from laminar to turbulent flow and the location of boundary-layer separation.

Some experimental results on sphere and disk drag

Abstract: The drag on spheres and disks moving rectilinearly through an incompressible fluid has been measured for Reynolds numbers (Re) from 5 to 100,000. Test models were mounted on a carriage which rode along a linear air bearing track system. Tests were performed by towing the models through a channel filled with glycerine-water mixtures. Forces and moments on the models were sensed by strain gage transducers; hydrogen bubble flow visualization was utilized in relating these forces to the unsteady wake flows. Steady drag results agreed with existing data except for the disk at 100 < Re < 1000, in which the drag coefficient values were up to 50% below the level of existing data; drag force unsteadiness during steady motion was always <5% for the sphere and <3% for the disk. Sphere drag measurements under constant acceleration from rest showed the apparent mass concept to be valid (at high Re) until the sphere had traveled approximately one diameter, after which the quasi-steady drag (based on instantaneous velocity) showed good agreement with the actual drag. Interference effects of the sting supports used in these tests are discussed.

An elastic sublayer model for drag reduction by dilute solutions of linear macromolecules

Abstract: Further evidence of a universal maximum drag reduction asymptote is presented. In the elastic sublayer model, inferred therefrom, the mean velocity profile during drag reduction is approximated by three zones: the usual viscous sublayer, an elastic sublayer where the mixing-length constant is derived from the maximum drag reduction asymptote, and an outermost region with Newtonian mixinglength constant. Upon integration the model yields a friction factor relation, parametric in elastic sublayer thickness, which properly reproduces the known features of turbulent dilute polymer solution flow. The dependence of elastic sublayer thickness upon flow and polymeric parameters is inferred from experimental data revealing two hitherto unknown relationships: namely that on Prandtl co-ordinates, 1/f½ vs. log Re f½ the difference in slope between a polymer solution and solvent is proportional to the square root of molar concentration and to the three-halves power of backbone chain links in the macromolecule. The proportionality constant in the preceding relationship is approximately the same for several different polymer species of carbon-carbon or similar skeletal structure in various thin solvents; there is an indication that this constant further depends upon the product of solvent viscosity times the cube of the effective bond length per chain link of the polymer species. Some recent results regarding the onset of drag reduction are also summarized.

A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids

Abstract: This paper is the first in a series of investigations having the overall objective of developing a new technique for treating the slow viscous motion past finite assemblages of particles of arbitrary shape. The new method, termed the multi-pole representation technique, is based on the theory that any object conforming to a natural co-ordinate system in a particle assemblage can be approximated by a truncated series of multi-lobular disturbances in which the accuracy of the representation is systematically improved by the addition of higher order multipoles. The essential elements of this theory are illustrated by examining the flows past finite line arrays of axisymmetric bodies such as spheres and spheroids which conform to special natural co-ordinate systems. It is demonstrated that this new procedure converges more rapidly and is simpler to use than the method of reflexions and represents the desired boundaries more precisely than the point-force approximation even when the objects are touching one another. Comparison of these solutions with the exact solutions of Stimson & Jeffery (1926) for the two sphere problem demonstrates the rapidity of convergence of this multipole procedure even when the spheres are touching. Drag results are also presented for flows past chains containing up to 101 spheres as well as for chains containing up to 15 prolate or oblate spheroids. The potential value of the technique is suggested by the rapidity with which the drag calculations were made, the 101 sphere problem requiring about 10 seconds on an IBM 360–65 computer to determine both the fluid flow and the drag coefficient.

Free-Fall Behavior of Planar Snow Crystals, Conical Graupel and Small Hail

Abstract: Drag coefficients and Best numbers of models of six planar snow crystals, two conical graupel and two conical small-hail particles were determined experimentally in glycerin-water mixtures and salt solutions. The Reynolds number (Re) range covered for the crystals was 0.1 to 200 and for the conical models 10 to 2000. It was found that the drag coefficients of dendritic shapes differed by factors of up to 4 from that of a disc of equal thickness and at an equal Reynolds number. The drag ratio is roughly constant with Re and linearly related to the ratio of the respective surface areas. The drag coefficients of the conical models assumed values between 0.5 and 2.0. During steady fall they decreased with increasing Re; however, as soon as oscillations started this trend reversed. Since tumbling does occur for larger graupel and small hail at Re > 300–1000 their main characteristic motions and frequencies are also discussed. Values for oscillation frequencies are given and the motions are described i...

Drag reduction in rough pipes

Abstract: Drag reduction caused by dilute, distilled water solutions of four polyethylene-oxides and one polyacrylamide, molecular weights respectively 0·1 × 106 to 8 × 106 and 13 × 106, was studied experimentally in one smooth and three sand-roughened pipes, relative roughnesses (R/k) 15, 23, 35, all of about 0·34, in inside diameter. The onset of drag reduction in the rough pipes occurred at the same wall shear stress as in smooth; the onset wall shear stress was essentially independent of polymer concentration, varied inversely as the square of polymer radius of gyration and was unaffected by the flow regime, hydraulically smooth, transitional or fully rough, during which onset occurred. Following onset a flow regime was observed wherein the fractional slip, i.e. fractional increase in mean velocity relative to solvent at a given friction velocity, obtained with a given polymer solution in a rough pipe was the same as the fractional slip in the smooth pipe despite marked differences in the respective rough and smooth friction factors. This ‘effectively smooth’ regime prevailed for values of non-dimensional roughness k+* < k+ < k+es from onset, k+*, to an upper limit given by k+es ∼ 50 for all of the present experiments. For k+ < k+es, the fractional slip in the rough pipes was always less than that corresponding to smooth and was a function of relative roughness as well as flow and polymeric parameters. The maximum drag reduction possible in the rough pipes was limited by an asymptote which was independent of polymeric parameters. Under asymptotic conditions, friction factors in all the rough pipes identically obeyed the smooth pipe friction factor relation for k+ < 12; the onset of roughness at k+ ∼ 12 indicated that the maximum viscous sublayer thickness attained during drag reduction was approximately 2½ times Newtonian.

The response of a tropical cyclone model to variations in boundary layer parameters, initial conditions, lateral boundary conditions, and domain size

Abstract: Tropical cyclone model experiments are summarized in which the drag coefficient and the analogous exchange coefficients for sensible and latent heat are varied. During the early portions of the immature stage, the response of the model storm follows linear theory and growth is more rapid with larger drag coefficients. However, the ultimate intensity reached by model storms varies inversely with the drag coefficient. The experiments indicate that air-sea exchanges of latent heat are crucial for the development and maintenance of the model storm. The air-sea exchange of sensible heat appears to be far less important. Experiments conducted with open lateral boundary conditions revealed that the structure and intensity of the mature stage of the model cyclone is relatively insensitive to the initial perturbation and to the size of the computational domain. The time required to reach the mature stage is, however, quite sensitive to these influences. Comparisons between experiments with open and mechan...

Free-Flight Measurements of Sphere Drag at Subsonic, Transonic, Supersonic, and Hypersonic Speeds for Continuum, Transition, and Near-Free- Molecular Flow Conditions

Abstract: : A comprehensive set of measurements was made in a ballistic range which permits the sphere drag coefficient to be derived with an uncertainty of approximately +2 or -2 percent in the flight regime 0.1 < M < 6.0 and 20 < Re < 100,000 for Tw/T approx = 1.0. Sufficient information is also presented to predict the effect of wall temperature on sphere drag coefficient when Tw/T is not = 1.0 for 2 < M < 6. This investigation represents the most comprehensive experimental program to date to define sphere drag in the velocity-altitude regime applicable to the falling sphere technique for defining upper air density.

Windbreak Drag as Influenced by Porosity

Abstract: W width, porosity, and flexibility all influence windspeed reduction lee of windbreaks. Atmospheric stability and surface roughness of the surounding area also affect windspeed reduction (5)*. But windbreak porosity is the major factor (11). Classifying plant windbreaks on the basis of porosity would facilitate comparisons of effectiveness among plant windbreaks. Also, experimental results from nonliving windbreaks usually are presented in terms of windbreak porosity; hence, the results would be easier to apply to plant windbreaks if they too were classified by porosity (5, 11, 13). Investigators have attempted to determine the porosity of plant windbreaks. Nokkentved (7) and Jensen (5) tried to use pictures, while Grundmann and Niemann (4) attempted to use a ratio of windspeed in the open field to windspeed at a leeward position to indicate porosity. Neither method proved satisfactory (11). George et al (3) used enlarged photographs and an overlayed dotted grid, and reported the method reliable when checked against windbreaks of known porosity. Fryrear (2) used shielded photocells to measure reflected sunlight that penetrated annual-crop windbreaks and then compared photocell output with and without the windbreaks to calculate relative porosities. Because the photocells must be shielded from direct sunlight, his method would be difficult to apply to large windbreaks. Another approach to the porosity problem is to measure drag coefficients of windbreaks with known porosities and compare them with drag coefficients of plant windbreaks with unknown porosities. Drag coefficients can be computed using momentum transfer principles, which are well established for airfoil drag measurements (10). They recently have been used successfully to determine windbreak drag in the surface boundary layer (12). This investigation determined relationships between windbreak drag co-

Motion of a Dislocation Acted on by a Viscous Drag through an Array of Discrete Obstacles

Abstract: A computer model was used to calculate the shape and average velocity v of a dislocation acted on by a viscous drag, and moving through a field of point obstacles. As a special case, the operation of a Frank‐Read source was computed also. At sufficiently low velocities (and stress levels), the behavior is governed entirely by the obstacle spacing and strength; conventional tensile testing probably falls in this regime. At sufficiently high velocities (and stress levels), the behavior is governed entirely by the viscous drag; shock deformation falls in this regime. In between lies a wide range in which the two effects superimpose, each contributing measurably to the flow stress. The strain rate sensitivity m = (∂lnv/∂lnσ)T, where σ is the stress, reflects this behavior. At low and high velocities, respectively, m characterizes the obstacles and the drag. But in between m is characteristic of neither, and in fact reflects the change in the contributions of the two strengthening mechanisms when the applied s...

The drag on oscillating flat plates in liquids at low Reynolds numbers

Abstract: Liquid flow about oscillating flat plates, determining drag coefficient relationships to low Reynolds number and period parameter from graphical representation

Mass transfer and drag coefficients of bubbles rising in dilute aqueous solutions

Abstract: The effects of aliphatic alcohols on mass transfer and drag coefficients of a carbon dioxide bubble were investigated in very dilute aqueous solutions. A high speed photography technique provided an almost continuous record of the bubble volume, shape, and oscillation, as related to the height of the bubble in the liquid and the time elapsed from the moment of its release. The results showed that the concentration, molecular size, and structure of the investigated alcohols have a pronounced effect on mass transfer as well as drag coefficients of the rising bubble. The significance of these effects with respect to idealized models of a highly circulating bubble and a solid sphere was considered, and a functional relationship was found to exist. This relationship may be applied to systems more complex than a single bubble.

Correlation and interpretation of zeolitic diffusion coefficients

Abstract: Kinetic data for the sorption of gases in zeolites are generally correlated in terms of Fickian diffusion coefficients. The diffusivities are, however, in general concentration dependent, indicating that the simple Fickian equation gives an inadequate description of the zeolitic transport process. A more convenient parameter for the correlation of kinetic data is the hydrodynamic drag coefficient k, which is related to the diffusivity by the expression : D=(RT/k)(∂ ln p/∂ ln c). Experimental data for the sorption of methane, ethane, propane and n-butane in 5A zeolite are presented. For these systems, the drag coefficients are effectively independent of concentration and the strong concentration dependence of the diffusivity can be entirely accounted for by the non-linearity of the sorption equilibrium isotherms. Correlation of the temperature dependence of the diffusivity in terms of an Eyring equation is unsatisfactory since the apparent activation energy is concentration dependent. A more satisfactory correlation may be obtained by considering separately the temperature dependence of the drag coefficients and the equilibrium isotherms. For the four sorbates studied, the drag coefficients may be expressed as a single function of reduced temperature. Comments on the mechanistic interpretation of the data are also included.

Control of gliding angle in Rüppell's Griffon Vulture Gyps rüppellii

Abstract: 1. The drag of the frozen, wingless body of a Ruppell9s griffon vulture was measured in a wind tunnel with a simple drag balance. The drag coefficient with feet and neck retracted was 0.43, based on the greatest cross-sectional area of the body. 2. The drag of the body was trebled by fully lowering the feet, and more than quadrupled when the tail was lowered as well, apparently owing to separation of the flow over the back. The drag coefficient of the legs and feet, based on their frontal area, varied from 0.89 to 1.08 in different positions. 3. At low speeds the use of the feet alone should reduce the glide ratio from about 15 to 10, but the airbrake effect becomes progressively more marked at higher speeds. At lower speeds reduction of the wing area produces a greater steepening of the gliding angle, but at the expense of increasing the minimum speed. Increase of induced drag would provide a highly effective gliding angle control at very low speeds, and it is suggested that this is achieved by raising the secondary feathers, which would alter the spanwise lift distribution by transferring a greater proportion of the lift to the primaries.

The influence of domain wall bowing on eddy current drag

Abstract: A method for calculating accurately the profile and eddy current drag of a moving ferromagnetic boundary when it is severely bowed by the inhomogeneous eddy current field is described. This technique is applied to the simplest case of a single domain wall in a steady state of motion under constant driving field. The eddy current drag coefficient is found to decrease with increasing wall velocity, slowly at first, but ultimately as the inverse square root of the velocity. As the motion is in the steady state this reduction in drag coefficient is entirely due to a change in wall profile. The calculations of both drag and profile are consistent with the experiments of Helmiss [5]. Es wird eine Methode zur genauen Berechnung des Profils und der Wirbelstromdampfung einer sich bewegenden ferromagnetischen Wand beschrieben, welche unter dem Einflus eines inhomogenen Wirbelstromfeldes erheblich durchgebogen ist. Die Methode wird auf den einfachsten Fall einer einzelnen Blochwand in gleichformiger Bewegung bei konstantem Feld angewandt. Es wird gefunden, das der Wirbelstrom-Dampfungskoeffizient mit steigender Wandgeschwindigkeit abnimmt, und zwar zunachst langsam und dann umgekehrt proportional der Quadratwurzel aus der Geschwindigkeit. Da die Bewegung gleichformig ist, mus diese Verkleinerung des Dampfungskoeffizienten ausschlieslich mit der Veranderung des Wandprofils zusammenhangen. Die Berechnungen der Dampfung und des Wandprofils stehen beide in Einklang mit den Experimenten von Helmiss [5].

Momentum flux and wave spectra observations from an ocean tower

Abstract: Three hundred seventy-seven eddy correlation estimates of momentum flux were obtained with a ‘flux meter’ system and cross-spectrum analysis at an elevation of 7.5 meters above mean sea level at Argus Island Tower during 1964, 1967, and 1969. Argus Island, a former research tower which stands in 60 meters of water, is located 45 km southwest of Bermuda with an unrestricted fetch in most directions. The influence of the pre-existing ocean waves in the open ocean (January to May) is manifest in a decreasing drag coefficient as mean horizontal wind velocities increase up to 6.5 m sec−1. As the wind velocity increases up to 14.5 m sec−1, the differential interplay of wave background and atmospheric stability is associated with a nearly linear increase of drag coefficient under unstable conditions, and stability effects appear to be more dominant than the surface waves. However, the influence of the ocean-surface waves is clearly evident when drag coefficients are observed at neutral and stable conditions. Under these conditions the observations indicated an oscillation of the drag coefficient as function of mean horizontal wind velocities between 7 and 12 m sec−1. There is some evidence of the dependence of the drag coefficient on dimensionless wave-phase velocity and dimensionless wave height. Wave-induced peaks in vertical velocity and quadrature spectra, corresponding to ocean-wave spectral peaks, are illustrated for light winds (4–6 m sec−1) and moderate winds (9.5–10.5 m sec−1). The ratio of the standard deviation of the vertical wind velocity to the friction velocity increased linearly as a function of the ratio of wave-phase velocity of the spectral peak to the friction velocity at all observed stabilities.

Cavity-flow wall effects and correction rules

Abstract: This paper is intended to evaluate the wall effects in the pure-drag case of plane cavity flow past an arbitrary body held in a closed tunnel, and to establish an accurate correction rule. The three theoretical models in common use, namely, the open-wake, Riabouchinsky and re-entrant-jet models, are employed to provide solutions in the form of some functional equations. From these theoretical solutions several different rules for the correction of wall effects are derived for symmetric wedges. These simple correction rules are found to be accurate, as compared with their corresponding exact numerical solutions, for all wedge angles and for small to moderate 'tunnel-spacing ratio' (the ratio of body frontal width to tunnel spacing). According to these correction rules, conversion of a drag coefficient, measured experimentally in a closed tunnel, to the corresponding unbounded flow case requires only the data of the conventional cavitation number and the tunnel-spacing ratio if based on the open-wake model, though using the Riabouchinsky model it requires an additional measurement of the minimum pressure along the tunnel wall. The numerical results for symmetric wedges show that the wall effects invariably result in a lower drag coefficient than in an unbounded flow at the same cavitation number, and that this percentage drag reduction increases with decreasing wedge angle and/or with decreasing tunnel spacing relative to the body frontal width. This indicates that the wall effects are generally more significant for thinner bodies in cavity flows, and they become exceedingly small for sufficiently blunt bodies. Physical explanations for these remarkable features of cavity-flow wall effects are sought; they are supported by the present experimental investigation of the pressure distribution on the wetted body surface as the flow parameters are varied. It is also found that the theoretical drag coefficient based on the Riabouchinsky model is smaller than that predicted by the open-wake model, all the flow parameters being equal, except when the flow approaches the choked state (with the cavity becoming infinitely long in a closed tunnel), which is the limiting case common to all theoretical models. This difference between the two flow models becomes especially pronounced for smaller wedge angles, shorter cavities, and with tunnel walls farther apart. In order to gauge the degree of accuracy of these theoretical models in approximating the real flows, and to ascertain the validity of the correction rules, a series of definitive experiments was carefully designed to complement the theory, and then carried out in a high-speed water tunnel. The measurements on a series of fully cavitating wedges at zero incidence suggest that, of the theoretical models, that due to Riabouchinsky is superior throughout the range tested. The accuracy of the correction rule based on that model has also been firmly established. Although the experimental investigation has been limited to symmetric wedges only, this correction rule (equations (85), (86) of the text) is expected to possess a general validity, at least for symmetric bodies without too large curvatures, since the geometry of the body profile is only implicitly involved in the correction formula. This experimental study is perhaps one of a very few with the particular objective of scrutinizing various theoretical cavity-flow models.

The response of a middle-latitude model atmosphere to forcing by topography and stationary heat sources1,,2

Abstract: The middle-latitude standing wave problem is investigated by means of a quasi-geostrophic, linear, steady-state model in which the zonal current is perturbed by the lower boundary topography and by a distribution of heat sources and sinks. All the perturbations are assumed to have a single meridional wavelength and the dissipation is considered to take place in the surface boundary layer using, as a first approach, a horizontally uniform drag coefficient. After investigating some basic properties of the model atmosphere, some computations are made to determine its response to the combined forcing by topography and by diabatic heating for January 1962. The resulting perturbations are found to be in rather good agreement with the observed standing waves. The results also indicate that the standing waves forced by the topography are in about the same position as those forced by the diabatic heating and that the former have somewhat larger amplitudes than the latter. The effect of allowing the drag c...

The drag reduction of dilute polymer solutions as a function of solvent power, viscosity, and temperature

Abstract: : The frictional drag reduction of high-molecular-weight polyethylene oxide and polystyrene solutions under turbulent flow conditions was studied as a function of temperature, solvent power, and solvent viscosity. A rotating disk apparatus was used to make the drag reduction measurements. For aqueous polyethylene oxide solutions, at concentrations well above that needed to produce maximum drag reduction, all drag reduction data reduced to a common curve when percent drag reduction was plotted against the Reynolds number for the flow. However, for polyethylene oxide solutions below this optimum concentration the drag reduction-versus-Reynolds number curves showed decreasing drag reduction with increasing temperature. The data are explained primarily in terms of the inverse temperature solubility characteristics of polyethylene oxide in water. The percent drag reduction of polystyrene in nonaqueous liquids was found to be greater in good solvents than in poor ones. It was also found that increases in solvent viscosity and decreases in temperature increased the percent drag reduction. The results are discussed in relation to the current drag reduction theories and are shown to be in opposition to Virk's theory. It is concluded from the data that drag reduction is very likely a function of a relaxation time phenomenon involving the polymer molecules and the flow system. The results also emphasize the importance of considering solvent power, viscosity, and temperature in the design of an efficient drag reduction system. (Author)

Approximate Bridging Relations in the Transitional Regime between Continuum and Free-Molecule Flows

Abstract: Approximate bridging relations for aerodynamic quantities in the transitional regime between free-molecule and continuum flows are developed. Equations are presented for heat transfer, surface shear, and drag, and are proposed for lift and pressure distribution. The equations can be considered as semiempirical. They are derived from models based on simplified kinetic theory; the structure of the equations is such that they are asymptotically correct at the free-molecule and continuum extremes, and this tends to place bounds on errors resulting from a simplified analysis. The equations presented are essentially for blunt bodies; that may be used as engineering approximations in many cases for which more rigorously developed specific relations do not exist.

Aerodynamic decelerators - An engineering review

Abstract: Nomenclature CD = drag coefficient based on S CDS — drag area of fully inflated parachute, ft2 (CiwS)r == drag area of reefed parachute, ft2 d = constructed parachute diameter across base of conical parachute, ft Di = inflated diameter, f d, ft hr — release altitude, msl, ft j£t = outflow coefficient (approximately 0.6) Ki = inflow coefficient (a value o^f 0.7 is used for a first approximation, and adjusted as determined by experimental data) Lr = circumferential length of reefing line, ft AP = pressure across canopy, lb/ft2 q = dynamic pressure JpF2, lb/ft2 R = radius of canopy during inflation normal to mean local canopy contour, ft Rm = maximum inflated radius of canopy, ft (approximately | of constructed radius) RQ = initial radius of canopy at start of inflation, ft (radius of suspension lugs or pack radius) s = tensile stress, lb/ft2 S = area of base of conical chute, 7rd2/4, ft2 t = thickness of ribbon, ft if = filling time, sec T — ribbon tensile load, Ib V = vehicle velocity, fps Fo = vehicle velocity at start of parachute filling, fps w = ribbon width, ft W = vehicle weight, Ib XG = geometric porosity of canopy p = air density, slugs/ft3

Motion of an Isolated Buoyant Thermal

Abstract: Using a simplified model, the governing equations of the motion of an isolated buoyant thermal are formulated. The general solutions which include both large and small density differences and both the initial accelerating motion and the final decelerating motion, are obtained. The asymptotic behavior for large time is given and compared with experimental results. The effect of various parameters, such as density difference, mass entrainment, and effective drag coefficient are also discussed.

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

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

Matched asymptotic solutions for optimum lift controlled atmosphericentry

Abstract: This paper considers the problem of optimal lift control of a hypersonic lifting body during atmospheric entry for the drag coefficient, a function of the angle of attack, and the atmospheric density, an arbitrary function of altitude. The solution obtained is valid for entering the planetary atmosphere from the Keplerian region, as well as from low altitudes. The method of matched asymptotic expansions was employed, and separate expansions were derived for the Keplerian region and for the aerodynamic region, where the aerodynamic forces are dominant. Lagrange multipliers and state variables obtained in closed form for both expansions were matched in the overlap domain. A method for estimating the order of magnitude of multipliers in various regions was discussed and will be useful in applying singular perturbation methods to a wider class of optimal control problems. For unbounded control, the lift variation can be classified into four different programs depending on the terminal altitude. Results were compared with the numerical solutions obtained by the method of steepest descent. For bounded control, there exist 12 different sequences of arcs which reduce to those obtained in an earlier study as the drag coefficient becomes independent of angle of attack.