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Drag Coefficient

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Drag coefficient is a dimensionless factor of proportionality between overall hydrodynamic force vector on a body in a liquid or gas flow and the product of reference area S of the body (commonly at midship section) and velocity head q

where , and vs are the velocity vectors of the fluid and the body, is the relative velocity of the body, ρ the liquid (gas) density, S the midship section area of the body, and Cd the drag coefficient.

This relation follows from similarity theory and is extensively used in engineering for simplified calculation of the force acting on a body or a particle in liquid or gas in which it moves.

In practice, drag coefficient is calculated in most cases using empirical relations generalizing experimental data. The most widely studied case is the sphere. Figure 1 graphs the dependence of drag coefficient for a sphere and a cylinder in crossflow on the Reynolds Number Re = ρuD/η, where D is the sphere (cylinder) diameter, η the viscosity of liquid, and . The drag coefficient decreases drastically from extremely high values at small Re numbers, to unity and lower at Re > 103. For Re < 0.2, Stokes has derived a theoretical formula for drag coefficient for a sphere:

Here, a purely viscous nonseparating flow occurs. The drag is governed by a high molecular friction of the liquid, the effect of which extends far upstream. With increasing Re number, inertial forces begin to predominate over viscosity forces and a laminar boundary layer is originated. Now, viscous forces are manifested only in this fairly thin layer. Flow beyond the boundary layer is virtually not affected by viscosity. Flow separation in the stern (point S in Figure 1) also occurs. As Re grows, the area of separation increases and attains the highest values at Re ~ 103; the drag coefficient in this case no longer diminishes and even slightly increases, remaining close to 0.4 for the range 2 × 103 < Re < 2 × 105.

Drag coefficient for cylinders (1) and spheres (2) as a function of Reynolds number (Re).

Figure 1. Drag coefficient for cylinders (1) and spheres (2) as a function of Reynolds number (Re).

In the 0.2 < Re < 2 × 103 range, an approximation formula for calculating a drag coefficient for a sphere is:

(1)

If Re continues to increase, the situation arises (at Re ~ 2 × 105) when the laminar boundary layer becomes partially turbulent in the nonseparating flow region of the sphere. The velocity profile in the turbulent boundary layer is fuller and better resists a positive pressure gradient. The area of separation is sharply displaced toward the stern, thereby drastically decreasing the drag coefficient. A self-similarity regime sets in and with further enhancement of the Re number, drag coefficient remains unchanged.

At high gas velocitiy, the drag coefficient also depends on the Mach number Ma = u/a, where a is the velocity of acoustic waves in the gas. At Ma < 1, a formula approximating a vast body of experimental data is:

where Cd0 is calculated using Eq. (1), has gained acceptance.

Drag coefficient is strongly affected by a body's shape. It is taken into account via the sphericity coefficient, which is the ratio of the sphere surface area of the same volume as the body relative to the body's surface area. For a tetrahedron, this is 0.67; for a cube, 0.806; and for octahedron, 0.85. Introduction of the sphericity coefficient in reality means the changing from an irregular shape of the body to some equivalent spherical shape, with sphere diameter taken as a reference dimension for determining the Re number and the midship section area.

Bodies of irregular shape, e.g., those of great length or twisted ones, move by tortuous trajectories and rotate, substantially changing the drag coefficient.

Where there are two spheres in close proximity, the drag coefficient is increased as shown in Figure 2.

Figure 2. 

Following from:

Immersed Bodies, Flow Around and Drag

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D in A-Z Index
Immeresed Bodies, Flow Around and Drag in Fundamentals
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