40.流體力學(xué)教程-introduction to aerospace engineering dte4321lecture

1、AE 2350Lecture Notes #6April 19, 1999We have studied.Meaning of pressible FlowHow and why speed of the flow affects compressibilityStreamlines and stream tubesContinuity EquationTopics to be studiedConservation of MomentumEulers equationConservation of EnergyBernoullis equation (even though Euler th

2、ought of it first.)Practical Applications of Bernoullis EquationRead Chapter 6 in the text!ContinuityStation 1Density r1Velocity V1Area A1Station 2Density r2Velocity V2Area A2Mass Flow Rate In = Mass Flow Rate Outr1 V1 A1 = r2 V2 A2 Momentum EquationBased on Newtons Second Law:Rate of Change of Mome

3、ntum of a particle= Forces acting on itConsider an infinitesimally small slice of stream tube in space.Rate of change of momentum of the fluid particles within thisstream tube must be due to forces acting on it.Momentum Equation (Contd.)Density rvelocity VArea =ADensity r+drvelocity V+dVArea =A+dAMa

4、ss Flow Rate in = Mass Flow rate outr VA = (r+dr)(V+dV)(A+dA)Momentum rate in=Mass flow rate times velocity= rV2AMomentum Rate out=Mass flow rate times velocity= r VA (V+dV)Momentum Equation (Contd.)Density rvelocity VArea =ADensity r+drvelocity V+dVArea =A+dAMomentum rate in=Mass flow rate times ve

5、locity= rV2AMomentum Rate out=Mass flow rate times velocity= r VA (V+dV)Rate of change of momentum within this element = Momentum rate out - Momentum rate in = r VA (V+dV) - rV2A = r VA dVMomentum Equation (Contd.)Density rvelocity VArea =ADensity r+drvelocity V+dVArea =A+dARate of change of momentu

6、m as fluid particlesflow through this element= r VA dVBy Newtons law, this momentum change must be caused byforces acting on this stream tube. Forces acting on the Stream tube Pressuretimesarea=pA(p+dp)(A+dA)Horizontal Force = Pressure times area of the ring=(p+dp/2)dAArea of this ring = dANet force

7、 = pA + (p+dp/2)dA-(p+dp)(A+dA)=- Adp - dp dA/2-AdpProduct of two small numbersMomentum EquationFrom the previous slides,Rate of change of momentum when fluid particles flowthrough the stream tube = rAVdVForces acting on the stream tube = -AdpWe have neglected all other forces - viscous, gravity, el

8、ectricaland magnetic forces.Equating the two factors, we get: rVdV+dp=0This equation is called the Eulers EquationBernoullis EquationEuler equation: rVdV + dp = 0For pressible flows, this equation may be integrated:Kinetic Energy + Pressure Energy = ConstantBernoullisEquationApplications of Bernoullis EquationSee examples 6.1 through 6.4 in the text.We will do more worked out examples on Wednesday, April 21.Important Applications include:Pitot TubeVenturi MeterFlow over airfoilsPitot tubes are used on aircraft as a speedometer. The Venturi MeterIt is used to mea