Technology & Engineering
Drag on a Sphere
Drag on a sphere refers to the resistance force that opposes the motion of a sphere through a fluid medium, such as air or water. The magnitude of the drag force depends on the size, shape, and speed of the sphere, as well as the properties of the fluid. Understanding drag on a sphere is important in designing efficient vehicles and structures.
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4 Key excerpts on "Drag on a Sphere"
eBook - PDF- J.H. Vincent(Author)
- 1995(Publication Date)
- Pergamon(Publisher)
The drag force acting on the particle may be derived from solutions of the Navier-Stokes equations for the air flow in the particle-fluid system in question. Mathematically this is achieved by determining the distributions of the local static pressure (normal) and viscous (tangential) forces over the surface of the particle. For very slow ('creeping') flow approaching at velocity 72 The motion of airborne particles Figure 4.1. Schematic to show the forces acting on a spherical particle moving in air. v and passing over a sphere of diameter d, integration of these forces yields the well-known Stokes' law for the overall drag force F D = - 37rd ~v (4.1) where the Reynolds' number for the particle dvpair Rep = (4.2) is very small (Rep < 1). Particles for which this applies are sometimes referred to as 'Stokesian'. In Equation (4.1), the minus sign indicates that the drag force is acting in the direction opposing the particle's motion. Strictly, this expression should be modified by three factors. The first derives from the fact that, in reality, the air surrounding the particle is not continuous but is made up of individual gas molecules which are in Figure 4.2. Diagram to illustrate the phenomenon of 'slip', showing the contrasting cases where (a) d >> mfp, and (b) d < mfp. 73 Aerosol sciencefor industrial hygienists random thermal motion. On the one hand, if the particle is large enough (i.e., much greater than the mean free path between gas molecules, mfp), it experiences the air as a continuum. This is because it cannot 'recognise' individual collisions with gas molecules. On the other hand, for particles which are so small that d is of the same order of magnitude or less than mfp, the nature of particle motion may be envisaged as 'slip' between successive collisions with the gas molecules. These contrasting scenarios are illustrated in Figure 4.2.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
For high velocities — or more precisely, at high Reynolds numbers — the overall drag of an object is characterized by a dimensionless number called the drag coefficient, and is calculated using the drag equation. Assuming a more-or-less constant drag coefficient, drag will vary as the square of velocity. Thus, the resultant power needed to overcome this drag will vary as the cube of velocity. The standard equation for drag is one half the coefficient of drag multiplied by the fluid mass density, the cross sectional area of the specified item, and the square of the velocity. Wind resistance is a layman's term used to describe drag. Its use is often vague, and is usually used in a relative sense ( e.g., a badminton shuttlecock has more wind resistance than a squash ball). Drag at high velocity Explanation of drag by NASA. ____________________ WORLD TECHNOLOGIES ____________________ The drag equation calculates the force experienced by an object moving through a fluid at relatively large velocity (i.e. high Reynolds number, R e > ~1000), also called quadratic drag . The equation is attributed to Lord Rayleigh, who originally used L 2 in place of A ( L being some length). The force on a moving object due to a fluid is: where is the force of drag, is the density of the fluid, is the speed of the object relative to the fluid, is the reference area, is the drag coefficient (a dimensionless parameter, e.g. 0.25 to 0.45 for a car) The reference area A is often defined as the area of the orthographic projection of the object — on a plane perpendicular to the direction of motion — e.g. for objects with a simple shape, such as a sphere, this is the cross sectional area. Sometimes different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given.
- Efstathios E. Michaelides, Martin Sommerfeld, Berend van Wachem(Authors)
- 2022(Publication Date)
- CRC Press(Publisher)
u = U far away from the sphere. The analytical expression of the velocity field may be used to determine the pressure field and the fluid stress tensor. Integration of the normal and shear stresses on the surface of the fluid sphere yields the total hydrodynamic force exerted by the fluid on the viscous sphere as follows(4.8)= π DF →DU →μ c.3 λ + 2λ + 1Since the viscous sphere is at rest, the fluid velocity, U, is the relative velocity of the sphere with respect to the fluid and, hence, Rer = DUρc /μc and U = u − v. For the more general case when the viscous spheres are not stationary, their scalar drag coefficient is defined as(4.9)C D==F →D1 2πa 2ρ c|u →−v →| 2.8 ( 3 λ + 2 )Re r( λ + 1 )At the limit λ→∞, Eqs. (4.8) and (4.9) yield the so-called “Stokes drag” for solid spheres(4.10)= 6 π a (F →Du →−v →)μ c= 3 π D (u →−v →)μ c⇒C D=.24Re rThe corresponding drag coefficient for a spherical inviscid sphere (e.g. a bubble) is CD = 16/Rer and the magnitude of the drag force on the inviscid sphere is FD = 2πD(u − v)μc . The latter is sometimes referred to as the form drag or pressure drag, while the difference of the two expressions, which is equal to πD(u − v)μc , is referred to as the friction drag. The form drag is due to the pressure distribution over the surface of the sphere, and the friction drag is the result of the viscous stress field at the surface of the sphere. While some publications make this distinction between the two parts of the drag force, it must be noted that the drag force is a single entity that arises from the hydrodynamic interactions between the fluid and the sphere and not two different forces.A solid or fluid sphere in a carrier fluid with uniform velocity U.FIGURE4.14.1.2 S
TEADYDRAGONSPHERICALATFINITEREYNOLDSNUMBERSAt finite Rer , the fluid has significant inertia that cannot be neglected, the fore-aft symmetry of the flow field around the sphere breaks down, and the stream function and the corresponding velocity field cannot be determined analytically. Experimental observations for both solid spheres and viscous spheres have proven that, even at low Rer , a wake is formed behind the sphere. This is a steady wake that becomes stronger as Rer
eBook - PDF- Chao Zhu, Liang-Shih Fan, Zhao Yu(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
4.2.1 Hydrodynamic Forces of a Pair of Spheres Describing the flow interactions between a particle of interest and its adjacent sur- rounding particles necessitates modification of the drag and other hydrodynamic forces. Correction factors are introduced to account for the deviation of the flow field from the isolated particle scenario. In general, this correction factor is a func- tion of the particle Reynolds number, the ratio of particle to fluid density, as well as the separation distance and orientation between the pair of particles. The most significant interaction between a pair of particles is represented in cases where a particle approaches another head-on or is located in its wake. Two iden- tical spheres approaching head-on can be modeled as a symmetric problem where the particles have trajectories normal to the plane of symmetry, as shown in Figure 4.2-1. For particles in the creeping flow regime and when the particle diameter is much smaller than the distance from the symmetric plan (i.e., d p << y), the drag coefficient of a sphere moving normal to the plane of symmetry can be estimated to the first order as (Brenner, 1962): C D = 24 Re p 1 + 1 2 d p y . (4.2-1) As two identical spheres approach head-on in a colinear fashion, the particles decelerate as the fluid pressure increases. Additional hydrodynamic forces, such as the added mass force and fluid pressure force, will influence the dynamic head-on approach and resulting collision. An approximated analysis of the added mass and the fluid pressure force can be obtained from the related case of a sphere moving normally toward a wall in an inviscid fluid.
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