Measurement of Drag about a Body immersed in a fluid

Consider abody such as an aerofoil placed in a flow, which could be a in a wind tunnel. Far from the body the flow is uniform and inviscid. As the flow approaches the body many dramatic changes take place. The flow will start to depart from uniformity. But as the flow negotiates the body viscosity comes into play. Consequently, the velocity on the body surface is zero. The velocity catches up with the freestream speed as we move away from the body. In other words, a boundary layer develops. A boundary layer is not static. It grows as the flow moves downstream. When the flow leaves the body the centreline velocity is not zero anymore. It starts to build up slowly. This is the Wake region. If a velocity profile is measured across the wake by carrying out what is called a Wake Traverse, we see that it resembles that shown in Fig.3.28. The wake profile thus carries signatures of the viscous effect.

Figure 3.28: Measurement of Drag about an immersed body

If a force balance is conducted in a region surrounding the body/ aerofoil then a force imbalance is evident. This should be related to Drag.

Consider the body/aerofoil placed in a wind tunnel. Let us prescribe a control volume ABCD surrounding it. The left and right hand boundaries AB and CD are far from the body. As a result the flow is uniform ( at a speed $U_\infty$ ) on AB. At the right hand boundary CD is the wake with the velocity profile as sketched. We assume that the top and bottom boundaries of the control volume,AD and BC are far away from the body and the vertical component of velocity namely v is zero across them.

	$\displaystyle \int_{CS} \rho V.dA~=~0$
i.e.,
	$\displaystyle \int_{AB}\rho V.dA~+~\int_{BC}\rho V.dA~+~\int_{CD}\rho V.dA~+~\int_{DA}\rho V.dA=0$
since v component of velocity along BC and AD is zero, the equation reduces to
	$\displaystyle \int_{AB}\rho V.dA~+~\int_{CD}\rho V.dA~=~0$
	$\displaystyle \int_A^B u ~dy + \int_C^D u~ dy~=~0$
leading to	$\displaystyle \int_A^B u~dy ~=~\int_C^D u~dy$	(3.89)

$\displaystyle F_{sx}~+~F_{bx}~$	$\displaystyle =~\int_{AB}u \rho V.dA~+~u \int_{BC}\rho V.dA~+~u \int_{CD}\rho V.dA~+~u \int_{DA}\rho V.dA$	(3.90)

i.e., $\displaystyle F_{sx}~+~F_{bx}~$	$\displaystyle =~\int_{AB}u~ U_{\infty}dy~+~ \int_{BC}u~\rho~ v~ dx+~ \int_{CD}\rho~u ~u ~dy~+~ \int_{DA}u~\rho~v~dx$	(3.91)
since v = 0 on BC and AD, we have
$\displaystyle F_{sx}~+~F_{bx}~$	$\displaystyle =~\int_{AB}u~ U_{\infty}dy~+~ \int_{CD}\rho~u ~u ~dy~$	(3.92)

The body force F_bx on the control volume is zero. The surface forces are drag and that due to pressure. Since we have assumed that pressure is uniform, the latter is zero. Further length AB = length CD, allowing us to combine the integrals on the RHS. Thus we have,

$\displaystyle D~=~\int_C^D \rho~u~(u~-~U_\infty)~dy$

(3.93)

In effect the velocities below C and that above D will be uniform and equal to $U_\infty$ . Consequently the above equation could also be written as

$\displaystyle D~=~\int_\infty^\infty \rho~u~(u~-~U_\infty)~dy$

(3.94)

A flaw in the above analysis should be apparent to you. Look at Eqn.3.89. This cannot be true. The mass flow going through AB at a uniform velocity $U_\infty$ cannot be equal to that across CDwhere the velocities are smaller than $U_\infty$ . Some mass has to escape through AD and BC. In other words our assumption of v = 0 on AD and BC is faulty. The equation for drag that we have obtained is inaccurate as a consequence. A more acceptable estimate for drag can be obtained by considering the v component of velocity on AD and BC. The other method is to make these boundaries streamlines of flow. Then $AB \neq CD$ . This is left as an exercise.