Drag Coefficient

Drag force is non-dimensionalised as

$$C_D~=~{{ DRAG} \over {1/2 \rho_\infty U_\infty ^2 A}}$$ (6.6)

where C_D is defined as Drag Coefficient. $U_\infty$ is the free stream speed, $\rho_\infty$ is the free stream density, A is the area. What area to use depends upon the application. In case of a cylinder it is the projected area normal to flow. For a flow past a thin flat plate, it will be the area of plate exposed to flow.

Figure 6.9: Flow past a Circular Cylinder at various Reynolds Numbers

The relative importance of the two kinds of drag is very apparent in case of flow over a circular cylinder or a sphere. The flow depends strongly upon Reynolds number as is clear from Fig.6.9. When the Reynolds numbers are small (1 and below)the flow behaves like a potential flow. There is no separation. The drag is all due to skin friction. As the Reynolds number is increased this drag decreases. At Reynolds numbers around 2 - 30, there is a separation of boundary layer, but the wake is of a limited length. The eddies formed seem fixed behind the cylinder. For Reynolds numbers close to 40 -70, there is a periodic oscillation of the wake. For higher Reynolds numbers the eddies break off from the cylinder. As the Reynolds number is increased, the eddies are continuously shed from the cylinder and washed downstream. Two rows of vortices are formed called the Vortex Street. Now the pressure drag contributes to almost 90% of the total drag. The value of C_D reaches a minimum of around 0.9 at a Reynolds number of around 2000. Increasing the Reynolds numbers further results in large angular velocities and a degeneration of vortices into turbulence.

Figure 6.10: Flow past a Circular Cylinder at various Reynolds Numbers, continued

In the Reynolds number range 10⁴ to 10⁵ one sees a laminar boundary layer to the left of the vertical centreline of the cylinder (Fig. 6.10). The flow separates at point S, which makes an angle of about 80⁰ with the centre of the cylinder. A wide wake is seen downstream. The pressure in the separated regions is almost constant. The observed C_P distribution is shown in Fig.4.31A. The net pressure difference p_A - p_B contributes to pressure drag. A dramatic change takes place when the Reynolds number is around 2x10⁵ when the boundary layer becomes turbulent before separation. Now the separation is postponed since a turbulent boundary layer is able to sustain for a longer time than a laminar flow. The point of separation S now is found at 130⁰ as shown in Fig.6.10. Notice that the wake has now narrowed. The C_P distribution indicates that the pressure in the wake is now higher than that for the laminar case (Fig.4.31A.). The consequence is that C_D is now reduced to about 0.3.

This reduction in drag around the cylinder is exploited in golf. The purpose of providing dimples on golf ball is to trip turbulence in order to decrease drag. Bowlers in cricket, especially the ones that bowl swings would like to have one side of the ball shining than the other. The idea is to keep flow on one side of the ball laminar and the other one turbulent. The ball is to swing from the laminar to the turbulent side.

It is clear that decrease in pressure drag can be achieved by delaying or stopping separation of flow. One of the strategies developed is to streamline the body. An aerofoil surface is an excellent example while the birds and fish are natural examples of this.

Figures 6.11 gives the numerical values of C_D for some of the familiar two-dimensional shapes. It is clear that C_D depends upon the orientation of the object to the flow. C_D values for some of the three-dimensional objects are given in Fig. 6.12.

Figure 6.11: C_Dvalues for familiar two-dimensional objects.

Figure 6.12: C_D Values for familiar three-dimensional objects

 

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney