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327lec-drag - 530.327 Introduction to Fluid Mechanics Su...

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Unformatted text preview: 530.327 - Introduction to Fluid Mechanics - Su Boundary layers: Pressure gradients and separation. Reading: Text, § 9.6 and 9.7. You will remember from our earlier discussion of inviscid flows that steady, uniform, incompressible flow of inviscid fluid past a body resulted in no drag on the body, even though in the real world we know that putting a body in a flow gives us drag. We called this “D’Alembert’s Paradox”. At the time, we learned that the proper handling of drag forces in such a situation requires consideration of viscosity. In class we decomposed the drag force into surface friction, which has to do with viscous shear stresses, and pressure forces. It’s obvious, then, that the viscosity of a fluid leads to surface friction. What’s not so obvious is that the fluid viscosity also explains the pressure forces. The explanation for this lies in the concept of boundary layer separation that we talked about earlier in the context of the first lab. We’ll start by looking at the mechanisms by which a boundary layer either remains ‘attached’ to a surface, or separates. A boundary layer can remain ‘attached’ to a surface if the flow streamlines in the boundary layer have no difficulty following the surface contours. To understand what might cause the boundary layer to be unable to follow the surface contours, consider Fig. 1, depicting the boundary layer on a flat plate, where the mean flow is from left to right. In the leftmost cartoon, Figure 1: Representation of boundary layer separation. the flow is moving in the positive x-direction, pushed along by the negative pressure gradient, dp/dx < 0. The velocity gradient at the surface, du/dy ( y = 0), is positive. Now suppose we increase the pressure gradient (i.e. make it more positive). As the pressure gradient changes sign and becomes positive, it will want to push the flow in the negative x-direction. If the flow has some inertia, it will tend to push against this adverse pressure gradient. At some point, these effects will be in balance, and the situation in the middle cartoon in Fig. 1 will pertain, where du/dy at the surface is zero. As the pressure gradient is increased past this point, we get flow reversal near the surface (the cartoon on the right). When this happens, the left-to-right boundary layer flow can no longer follow the contour of the surface. Instead, the boundary layer is diverted upward to go around the region of reversed flow. This process is calledaround the region of reversed flow....
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327lec-drag - 530.327 Introduction to Fluid Mechanics Su...

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