Some General Points about Drag and Lift

Drag on a SphereAs noted in Chapter 7 of your textbook, the common dimensional parameters often used to correlate the

drag force acting on an object, FD, are given by the density, ρ; the relative velocity of the object to the fluid,V ; the fluid viscosity, µ; the size of the object, given by a reference length `:

FD = f (V, ρ, µ, `)

When the Buckingham Pi theorem is applied, the result is:

Cd ≡FD

12ρV

2`2= f∗

(ρV `

µ

)where Re = ρV `/µ is the familiar Reynolds number, and a factor of 1/2 is introduced to make the group1/2ρV 2 equal to the dynamic pressure of the flow. The quantity `2 has the dimensions of an area, which forblunt (non-streamlined) objects, is taken to be the projected frontal area of the object. When the object isa sphere, the relevant length scale is taken to be the diameter, ` = D, and the Drag coefficient is redefinedslightly to use the projected frontal area of the sphere:

Cd ≡FD

12ρV

2 π4D

2= f∗

(ρV D

µ

)

Figure 1: Drag coefficient for a smooth sphere, showing three different empirical correlations relative to experi-mental data.

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The drag coefficient measured for a smooth sphere is shown as a function of the Reynolds number inFigure 1. As shown, there is large variation in the value of Cd as one moves from very small towards verylarge Re. Over the range from Re > 1000 to Re < 250, 000, Cd is approximately constant with a valuenear 0.5. Above the value Re > 250, 000, the drag coefficient drops suddenly by over a factor of 5, which isofter referred to as the ”drag crisis”. This results from a change in the boundary layer behavior and alteredflow separation as explained in Chapter 9 of your text. On the low end, Re < 1000, viscous effects startto become important and likewise change the flow. For very small Re (less than 0.1), the viscous effectsdominate, and one can find an analytical solution which predicts the drag should vary as Cd = 24/Re.

Using a constant drag coefficient is a common approximation, but caution should be exercised if therange of Reynolds numbers is outside the range noted. Numerous curve fits to the above data exist, all withvarying degrees of complexity and accuracy. A simple one that is accurate to within approximately 10% isgiven by White (FM White, 1991, Viscous Fluid Mechanics, McGraw Hill, New York):

Cd =24

Re+

1

1 +√Re

+ 0.4

A more involved expression given by Cheng (NS Cheng, 2008, Powder Technology,189(3), pp. 395–398.) hasan average error of less than 2.5%:

Cd =24

Re(1 + 0.27Re)

0.43+ 0.47

[1− exp

(−0.04Re0.38

)]Drag vector components

By definition, the drag is the net hydrodynamic force component in the direction of the oncoming flow asseen by object. Figure 2 shows the path of an object xp(t) = (xp(t), yp(t)) that is moving through a velocity

field Vf = (uf , vf ). Note that here vectors are denoted in bold face so that, for example, V = ~V . At thelocation of the object at any fixed time, such as the point indicated by a red dot in Figure 2a, there is alocal velocity of the fluid, Vf , as well as the velocity of the object, given by Vp = dxp/dt. These velocitiesare illustrated Figure 2b, where they are depicted as viewed by a fixed, non-moving observer.

Figure 2: (a) An object whose path is marked by the red path line is moving within the velocity field given bythe blue streamlines. The location of the object at a given time t is marked by the red dot on the curved line.b) Absolute reference frame showing fluid velocity vectors (blue) and particle velocity (red) from a stationaryobserver. c) Relative observer reference frame, showing the velocity of the fluid relative to an observer movingwith the particle (Vrel, green) and the resulting drag force (FD, orange)

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Now consider what the local fluid velocity appears to be from the point of view of an observer movingwith the object. As shown in Figure 2c, this velocity is given by

Vf −dxpdt

.

To convince yourself that this is true, note that if the object were moving at the same velocity as the localfluid, then the relative velocity is zero.

The drag on an object depends on the velocity of the fluid surrounding it as measured by an observeron the object. In particular, the direction of the drag force is in the direction of the local, relative fluidvelocity. For example, if you feel a wind blowing on you, the drag force points in the direction that the windis blowing. In some circumstances the total force of the fluid on a body is not confined to the direction ofthe drag, in other words, there is a force component pointing perpendicular to the drag direction. This forceis referred to as the lift force and its presence has a lot to do with the shape of the body itself. For example,if the body is symmetric to the wind direction, its unlikely to have a lift force. But if the object is in theshape of a wing, it may have a very significant lift force.

As noted in the previous section, for all except the slowest moving objects, it is an empirical fact thatthe magnitude of the drag FD satisfies the generic relationship

FD = Cd1

2ρAU2

rel

where Cd is an empirically determined drag coefficient, A is either the planform or projected area of thebody in the relative flow direction (it depends on the shape considered) and

Vrel ≡∣∣∣∣Vf −

dxpdt

∣∣∣∣ =√

(uf − dxp/dt)2 + (vf − dyp/dt)2 =√u2rel + v2rel

is the magnitude of the relative flow velocity, as shown in Figure 3.

Figure 3: Relative velocity component.

The two formulas above give the magnitude of the force, but we need the force as a vector pointing inthe direction of the flow. To accomplish this, we generalize the drag law to the form:

FD = Cd1

2ρA

∣∣∣∣Vf −dxpdt

∣∣∣∣2 τ̂

where

τ̂ =Vf − dxp

dt∣∣∣Vf − dxp

dt

∣∣∣3

is a unit vector pointing in the direction of the relative fluid velocity. This can also be expressed in terms ofits unit cosine angles (see Figure 3) by:

τ̂ = cos (θ) i + sin (θ) j =uf − xp

dt√(uf − dxp

dt )2 + (vf − dypdt )2

i +vf − yp

dt√(uf − dxp

dt )2 + (vf − dypdt )2

j

Substituting for τ̂ it is found that the drag is:

FD = Cd1

2ρA

(Vf −

dxpdt

) ∣∣∣∣Vf −dxpdt

∣∣∣∣ .In 2D this gives the components of the drag force in the x and y direction that can be written explicitly as

FD,x = Cd1

2ρA

(uf −

dxpdt

)√(uf −

dxpdt

)2 + (vf −dypdt

)2

and

FD,y = Cd1

2ρA

(vf −

dypdt

)√(uf −

dxpdt

)2 + (vf −dypdt

)2

These relations show that if the object is moving horizontally (taken as the x direction) in a horizontal wind,then FD,y = 0. Moreover, if the object is moving slower than the wind and uf > 0 then FD,x is in the +xdirection. Conversely, the drag is in the −x direction if the object moves faster than the wind.

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