in the description of locations in a flow field where the flow speed becomes

In the description of locations in a flow field where

This preview shows page 118 - 121 out of 164 pages.

in the description of locations in a flow field where the flow speed becomes zero, as noted above. Such locations are called stagnation points ; we have depicted such a point in Fig. 4.5 along with its associated streamline, termed the stagnation streamline . We note that there is another stagnation point (and streamline) not shown in the figure on the opposite side of the sphere. In this case, velocity is zero at all points along this streamline due to inviscid flow. (The reader should consider what would happen along this streamline in a viscous flow.)
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4.2. BERNOULLI’S EQUATION 113 4.2.2 Example applications of Bernoulli’s equation In this section we will present two rather standard examples of employing Bernoulli’s equation to solve practical problems. In the first we will analyze a device for measuring air speed of aircraft, and in the second we consider a simple system for transferring liquids between two containers without using a pump. Such systems often involve what is termed a syphon. Analysis of a Pitot Tube In this subsection we present an example problem associated with the analysis of a pitot tube via Bernoulli’s equation. Pitot tubes are simple devices for measuring flow speed. EXAMPLE 4.5 In Fig. 4.6 we present a sketch of a pitot tube. This consists of two concentric stagnation streamline location #1 stagnation point pressure measurement location #2, port for static measurement stagnation pressure U Figure 4.6: Sketch of pitot tube. cylinders, the inner one of which is open to oncoming air that stagnates in the cylinder. Thus, the stagnation pressure can be measured at the end of this inner cylinder. The outer cylinder is closed to oncoming air but has several holes in its surface that permit measurement of static pressure from the flow passing these holes. We will assume density, ρ , is known since if temperature is available (say, measured with a thermocouple), ρ can be calculated from an equation of state. It is desired to find the flow speed U . We first write Bernoulli’s equation between the locations 1 and 2 indicated in the figure. For gases in which hydrostatic effects are usually negligible this takes the form of Eq. (4.14); i.e. , p 1 + ρ 2 U 2 1 = p 2 + ρ 2 U 2 2 . Now, in the present case we can assume that the pressure at location 1 is essentially the stagnation pressure. This is on the stagnation streamline, and it is close to the entrance to the pitot tube. (If this tube is small in diameter, the flow will be completely stagnant all the way to the tube entrance.) Furthermore, at any location where the pressure is the stagnation pressure the speed must be zero, by definition. Thus, in the above formula we can consider p 1 to be known; it is measured, say, with a pressure transducer, and U 1 = 0. Next, we observe that the streamline(s) passing the pitot tube outer cylinder where the static pressure is measured have started at locations where the speed is U , the desired unknown value. So, in the right-hand side of the above equation we take p 2 to be (measured) static pressure, and U 2 = U . This results in p 1 = p 2 + ρ 2 U 2 ,
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114 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS
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