A nivel de estudio de datos

Pressure drag is caused by the difference in pressure at the front of the object compared to the rear of the object and is strongly correlated to the object’s shape. The pressure drag is proportional to the frontal area of the object, so is therefore large for blunt bodies, small for streamlined bodies, and zero for thin, flat plates parallel to the flow.

Base drag is part of the pressure drag and occurs at the base of projectile shaped objects. Base drag largely depends on the the length of the forebody, its surface conditions, and the ratio of base-to-body diameter [Hoerner, 1965, p3-19]. For bodies with a base diameter smaller than that of the forebody, the drag may be considered similar to that of a parallel-sided shape. For three-dimensional bodies the base drag coefficient is a function of the forebody drag coefficient, as shown in Figure 3.1.

Figure 3.1: Base Drag Coefficient as a Function of Forebody Drag Coefficient [Hoerner, 1965]

Skin friction drag is caused by shear stress on the wall of the object as the air molecules collide with the surface of the object. Friction drag depends on the orientation of the object as well as the surface roughness,

and the friction drag is nearly zero for flat surfaces normal to the flow and maximum for flat surfaces parallel to the flow. Skin friction drag is also dependent on the viscosity of the fluid, and increases with increasing viscosity. For laminar flow over flat plates the skin friction coefficient can be calculated using Equation 3.16, where Re is the Reynolds number andxis the distance from the leading edge at which the Reynolds number

is taken. (Cf)laminar= 0.664 √ Rex (3.16) For the transition-to-turbulence region the skin friction drag, DSF, for a flat plate of length 1m can be

determined using Equation 3.17:

DSF =qAs % 0.074 Re15 L −1740_Re L & (3.17)

where q is the dynamic pressure,q= 1 2ρV

2_{, andA}

sis the surface area. Equations 3.16 and 3.17 can be used

to calculate the skin friction drag for an athlete, where As= 0.2025H0.725m0.425 [Pendergast et al., 2006].

At low speeds or high viscosity (low Reynolds number) viscous stress is the predominant parameter deter- mining the drag of a body so skin friction drag will contribute to a higher proportion of overall drag, as can be seen in Figure 3.2 [Hoerner, 1965, p2-1].

Figure 3.2: Skin Friction Coefficient; (a) in viscous flow, (b) with laminar, (c) with turbulent boundary layer flow, (d) cylinder in axial flow [Hoerner, 1965, p2-1]

In general, for a flat plate parallel to the flow drag is due almost entirely to skin friction drag; for a flat plate normal to the flow drag is due entirely to pressure drag; and for a cylinder normal to the flow drag is due to a combination of skin friction and pressure drag, although pressure drag dominates.

The total drag coefficient,Cd, is a dimensionless quantity that is used to quantify the total drag of an object,

comprising of pressure drag and skin friction drag, and is calculated byCd= 1 D

2ρAV2, where D is the drag (N), ρis the air density (kg/m3_{), A is the frontal area (m}2_{) and V is the velocity (m/s). The total drag coefficient}

cylinder (although closer to the drag coefficient of a flat plate normal to the flow). The drag coefficient is also related to the Reynolds number, as shown in Figure 3.3, where the minimum drag coefficient for both the long, flat plate parallel to the flow and circular cylinder in the spanwise orientation occurs of orderRe= 105_.

(a) Flat Plate [Roymech, 2011]

(b) Circular Cylinder in the Spanwise Orientation [Propeller- Guard, 2011]

Figure 3.3: Drag Coefficient as a Function of Reynolds Number

For spanwise cylinders of finite length, the relationship between drag coefficient, Cd, and diameter to span

ratio, d

l, can be seen in Figure 3.4 [Hoerner, 1965, p3-16].

Figure 3.4: Drag Coefficient of Rectangular Plates and Circular Cylinders as a Function of Height (or Diameter) to Span Ratio [Hoerner, 1965, p2-1]

For parallel sided shapes, such as those shown in Figure 3.5, the total drag consists of frictional drag on the surface of the body and base drag [Hoerner, 1965, p3-12]. Figure 3.5 shows the relationship between drag coefficient, Cd, and fineness ratio, _dl, for cylindrical bodies in axial flow. It is clear that as the ratio _dl

increases, the total drag comprises of a greater proportion of friction drag than base drag.

Figure 3.5: Drag Coefficients of Cylindrical Bodies in Axial Flow as a Function of Fineness Ratio

The Reynolds number of the parts of the body of a cyclist can be calculated usingRe=U D_ν , where U is the

cycling speed (m/s), D is the characteristic linear dimension (m), and ν is the kinematic viscosity (m2_s−1_).

The calculated Reynolds numbers for each body part at cycling speeds between 40kph and 70kph are shown in Table 3.7. A comparison between the calculated Reynolds numbers for the body parts of a cyclist (Table 3.7) and the drag coefficient as a function of Reynolds number for flat plates and cylinders (Figure 3.3) shows that all body parts of a cyclist at cycling speeds between 40kph and 70kph lie within the drag crisis region. Using Table 3.7 and Figures 3.3, 3.4 and 3.5, a drag coefficient of Cd = 0.0012 would be expected for the

torso, where as for the upper arms, thighs and calves a drag coefficient ofCd= 0.6−1.2would be expected,

depending on the surface roughness, diameter and cycling speed.

Re (x105₎

Body Part Shape Length (m) Diameter (m) l/d d/l 40kph 50kph 60kph 70kph

Head Sphere NA 0.28 NA NA 2.05 2.57 3.08 3.59

Torso Flat plate 0.8 NA NA NA 5.86 7.33 8.8 10.26

Forearm Streamwise cylinder 0.37 0.076 4.9 0.2 2.71 3.39 4.07 4.75

Upper Arm Spanwise cylinder 0.25 0.09 2.8 0.4 0.66 0.82 0.99 1.15

Thigh Spanwise cylinder 0.4 0.16 2.5 0.4 1.17 1.47 1.76 2.05

Calf Spanwise cylinder 0.4 0.1 4.0 0.3 0.73 0.92 1.1 1.28

Table 3.7: Calculated Reynolds numbers for body parts of a cyclist