AA311.Lecture4 - AA 311 Lecture 4: Basic Aerodynamics,...

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Unformatted text preview: AA 311 Lecture 4: Basic Aerodynamics, Measurement of Airspeed Reading: [1] Chapters 4.1-4.3 and 4.11. In this lecture we review a couple of fundamental physical principles, and apply them to derive some fundamental equations used in aerodynamics. These principles are • Conservation of mass. • Newton’s second law (force equals rate of change of momentum). • Conservation of energy. Consider Figure 1 showing a fluid element in a flow, that has velocity V . The path of this fluid element traces out a trajectory, which is called a streamline . Fluid velocity is always tangent to a streamline, and therefore streamlines cannot cross . A collection of streamlines passing through a simple closed contour is called a stream tube (Figure 2). By the definition of streamlines, no fluid can pass through the walls of the stream tube, only through the faces. (a) From [1]. (b) From [1]. Figure 1: Streamlines, and path of a fluid particle. Figure 2: Stream tube. Continuity equation The continuity equation is based on a very simple physical principle: mass can neither be destroyed or created . Hence, if a given amount of mass enters a fixed volume, then it either accumulates there and increases the density in the region, or has to exit somehow. Consider an imaginary circle drawn perpendicular to the flow direction as shown in Figure 2 ( A 1 area region), and follow the 1 streamlines of the flow some distance downstream. Further downstream consider another circle ( A 2 area region), drawn, again, perpendicular to the flow. These two circular faces, and the streamlines connecting them define a stream tube. The cross-sectional area of these circles may change in general. The mass flow rate into the stream tube has to equal the mass flow rate out of the stream tube. Since no flow can occur through the walls of the tube, all the mass transport must take place through the circular faces. At point 1 the cross-sectional area of the stream tube is A 1 , the flow velocity is V 1 , and density is ρ 1 , and at point 2 the cross-sectional area of the stream tube is A 2 , the flow velocity is V 2 , and density is ρ 2 . In a small infinitesimal time dt , the total mass of fluid that enters at point 1 is the density, ρ 1 , times the volume that is swept out in time dt . Since the volume of the cylinder is ( A 1 V 1 dt ), we can write dm 1 = ρ 1 ( A 1 V 1 dt ) , where dm 1 is the amount of mass that enters the tube at point 1. Hence the mass flow rate is dm 1 dt = ˙ m 1 = ρ 1 A 1 V 1 . Similarly, dm 2 dt = ˙ m 2 = ρ 2 A 2 V 2 . By conservation of mass, we have ρ 1 A 1 V 1 = ρ 2 A 2 V 2 . (1) The above relation is extremely useful in solving many fluid problems. When dealing with flow in pipes and ducts, then the stream tubes are naturally defined by the constraints of pipe walls. But stream tubes do not have to be bounded by a solid wall. For example, the shaded region above the airfoil shown in Figure 3 is a stream tube....
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AA311.Lecture4 - AA 311 Lecture 4: Basic Aerodynamics,...

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