External flows past objects encompass a variety of fluid mechanics phenomena. The character of the flow field depends on the shape of the object. Even the simplest shaped objects, like a sphere, produce rather complex flows. For a given shaped object, the flow pattern and related forces depend strongly on various parameters such as size, orientation, speed and fluid properties. By the concepts of dimensional analysis, the important dimensionless parameters are the Reynolds Number (ratio of inertia forces and viscous forces), the Mach Number (ratio of flow velocity and speed of sound) and sometimes the Froude Number (ratio of inertia forces and gravity forces). The study of flow around immersed bodies has a wide variety of engineering applications but in terms of heat and mass transfer, spheres are the most important. Thus most of this article deals with spheres.
There is a great similarity in the development of flow pattern at increasing Reynolds number between a sphere and a circular cylinder (or tube), except for the vortex street associated with the latter and other two-dimensional bodies, which is not formed for three-dimensional bodies, instead a vortex ring occurs, which for a sphere is formed at about Re_{D} = 24 (see Figure 1) and becomes unstable at about Re_{D} = 200 when it tends to move downstream of the body and is immediately replaced by a new vortex ring.
Figure 1. Observed lengths of the region of closed streamlines behind a sphere. From Taneda S. (1956).
No regular pattern of motion like the vortex street forms in the wake of a sphere (or of any three-dimensional body), although there is a general impression that vorticity is shed from the standing ring-vortex like a succession of distorted vortex loops not symmetrical around the central axis (see Vortex Shedding). This flow process does not give rise to vibrations of the sphere. At large Reynolds number, flow over the front half of a sphere may be divided into a thin Boundary Layer region where viscous forces are dominant, and an outer region, where the flow corresponds to that of an inviscid fluid.
Pressure decreases over the front half of the sphere from the stagnation point onwards, thus having a stabilizing effect on the boundary layer, which remains laminar up to about Re_{d} = 5·10^{5}. Beyond the minimum pressure point on the sphere surface, the boundary layer is subjected to an adverse pressure gradient and the fluid decelerates. Later, flow separation occurs. At low Re_{D}, the separation point is located at the rear stagnation point and with increasing Re_{D} moves forward and reaches φ ≈ 80° from the front stagnation point at Re_{D} ≈ 1000. Pressure drag begins to dominate and the Drag Coefficient becomes almost independent of the Reynolds number until about Re_{D} = 5·10^{5}, when the transition from Laminar to Turbulent Flow occurs before separation. As a result, the point of separation moves to the rear, making the wake smaller and abruptly reducing drag coefficient.
At very low Re_{D}, during the so-called creeping flow, inertia forces are assumed to be negligible; hence, the governing equations for the flow (Navier-Stokes equations and the continuity equation) are greatly simplified. Stokes succeeded in obtaining a solution for these equations and the drag force has been found to be
where R is the radius of the sphere, U_{∞} the freestream velocity and η the dynamic viscosity of the fluid. See also Stokes' Law for Solid Spheres and Spherical Bubbles. This relationship may be used up to Re_{D} = 0.5 with negligible error. This flow range is often called the Stokes flow. The drag coefficient is defined as:
where A is the projected area (equal to πR^{2}), which can be expressed as:
where Re_{D} is
Equation (1) is known as Stokes' law. Sometimes Eq. (3) is also referred to as Stokes' Law.
An extension of Eq. (3) has been formulated using a method of successive approximations to the governing equations (still at low Re_{D}). This formula reads
Equation (5) may be used up to Re_{D} ≈ 100. At higher values of Re_{D}, it is necessary to rely on empirical expressions based on experiments. Figure 2 provides a graph of C_{D} versus Re_{D} for a sphere.
When applied to particle mechanics, the following regimes are introduced:
Several other expressions for drag coefficient may be found in other literature.
For blunt bodies like a sphere, an increase in surface roughness may cause a decrease in drag. The transition to turbulent boundary layer flow occurs at a lower Reynolds number than for a smooth sphere. One effect is the wake region behind the sphere becomes considerably narrower and overall drag is reduced.
Similar to a circular cylinder, the inviscid flow field around a sphere can also be determined analytically. The velocity components are:
where θ is measured from the forward stagnation point, r is the radial coordinate, r_{o} is sphere radius and U_{∞} is the free-stream velocity. The maximum velocity occurs at θ = π/2 and is
Experimentally, however, the maximum value is about 1.3 U_{∞} and occurs upstream of θ = π/2.
Velocity distribution in the laminar boundary layer over the front part of a sphere can be calculated using a series expansion technique. It requires, however, an experimentally determined pressure distribution if accurate values of skin friction and boundary layer thickness are to be achieved.
REFERENCES
Batchelor, G. K. (1970) An Introduction to Fluid Dynamics. Cambridge University Press.
Munson, B. R., Young, D. F. and Okiishi, T. H. (1990) Fundamentals of Fluid Mechanics. J. Wiley & Sons.
Taneda S.. (1956) Rep. Res. Inst. Appl. Mech., Kyushu Univ, 4, 99.
White, F. M. (1994) Fluid Mechanics. 3rd Edn. McGraw-Hill.
References
- Batchelor, G. K. (1970) An Introduction to Fluid Dynamics. Cambridge University Press.
- Munson, B. R., Young, D. F. and Okiishi, T. H. (1990) Fundamentals of Fluid Mechanics. J. Wiley & Sons.
- Taneda S.. (1956) Rep. Res. Inst. Appl. Mech., Kyushu Univ, 4, 99.
- White, F. M. (1994) Fluid Mechanics. 3rd Edn. McGraw-Hill.