Measurement of Drag about a Body immersed in a fluidConsider 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. 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
subsubsectionAnalysis We make the following assumptions.
Continuity Equation. Since the flow is steady, we have,
Momentum equation On applying the momentum equation to the control volume we have
The body force Fbx 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,
In effect the velocities below C and that above D will be uniform
and equal to
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 (c) Aerospace, Mechanical & Mechatronic Engg. 2005 University of Sydney |