# Inviscid & Incompressible flow

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1 < 3.1. Introduction and Road Map > Basic aspects of inviscid, incompressible flow Bernoulli s Equation Laplaces s Equation Some Elementary flows Some simple applications 1.Venturi 2. Low-speed wind tunnel 3. Pitot tube General philosophy and use in solving problems Synthesis of complex flows from a superposition of elementary flows Uniform flow Sources and sinks Doublet Vortex flow Flow over semiinfinite and ovalshaped bodies Nonlifiting flow over a cylinder Lifiting flow over a cylinder Kutta- Joukowski theorem General nonlifting bodies; source panel numerical technique Aerodynamics 2015 fall - 1 -

2 < 3.2. Bernoulli s Equation > From the momentum equation For an inviscid and steady flow without no body force : x-dir. Momentum equation Aerodynamics 2015 fall - 2 -

3 < 3.2. Bernoulli s Equation > Multiply dx From the equation of streamline Aerodynamics 2015 fall - 3 -

4 < 3.2. Bernoulli s Equation > : x-dir. Similarly, : y-dir. : z-dir. Aerodynamics 2015 fall - 4 -

5 < 3.2. Bernoulli s Equation > along a streamline If incompressible, along a streamline If irrotational, everywhere Aerodynamics 2015 fall - 5 -

6 < 3.3. The Venturi and Low-Speed Wind Tunnel > Venturi tube Assume) 1. Quasi-one-dimensional flow (the properties are uniform across the x-section) 2. Inviscid flow 3. Steady flow Continuity equation steady Aerodynamics 2015 fall - 6 -

7 < 3.3. The Venturi and Low-Speed Wind Tunnel > Venturi tube r 1 r 2 at the wall Aerodynamics 2015 fall - 7 -

8 < 3.3. The Venturi and Low-Speed Wind Tunnel > Venturi tube Incompressible Use two equations 1. Continuity eq. 2. Bernoulli s eq. Aerodynamics 2015 fall - 8 -

9 < 3.3. The Venturi and Low-Speed Wind Tunnel > Low-speed wind tunnel How to measure the pressure difference? Manometer Aerodynamics 2015 fall - 9 -

10 < 3.4. Pitot Tube > Low-speed wind tunnel p 1 p 0 V 1 Dynamic pressure Static pressure Total pressure Aerodynamics 2015 fall

11 < 3.5. Pressure Coefficient > Low-speed wind tunnel : Over all the range of speed For incompressible flow : works for M<0.3 (low subsonic) * freestream * stagnation point Aerodynamics 2015 fall

12 < 3.6. Condition on Velocity for Incompressible Flow > Continuity equation Incompressible Aerodynamics 2015 fall

13 < 3.7. Laplace s Equation > Laplace s equation * In 2D, * In irrotational flow, Note) 1. For irrotational, incompressible flow, and are both solutions of Laplace equation. 2. Since Laplace equation is linear, the solution can be superimposed, so that any complex flow is expressed by adding elementary. Aerodynamics 2015 fall

14 < 3.7. Laplace s Equation > i) Infinity B.C. ii) wall B.C. Flow tangency condition slope of the streamline Aerodynamics 2015 fall

15 < 3.9. Uniform Flow > =const. =const. can be set to zero zero Aerodynamics 2015 fall - 1 -

16 < 3.9. Uniform Flow > Curve C h l =const. =const. Evaluate G in a uniform flow Aerodynamics 2015 fall - 2 -

17 < Source/Sink Flow > Mass flow Volume flow rate = : Source strength Volume flow rate per unit length Aerodynamics 2015 fall - 3 -

18 < Source/Sink Flow > For sink flow, set L to -L Aerodynamics 2015 fall - 4 -

19 < A Uniform Flow with a Source and Sink> Uniform flow + Souce Aerodynamics 2015 fall - 5 -

20 < A Uniform Flow with a Source and Sink> Uniform flow + Souce At the stagnation point Streamline going through the stagnation point So, the body surface can be replaced by a streamline Aerodynamics 2015 fall - 6 -

21 < A Uniform Flow with a Source and Sink> Uniform flow + Souce + Sink Rankine oval Aerodynamics 2015 fall - 7 -

22 < Doublet Flow > Uniform flow + Souce + Sink Aerodynamics 2015 fall - 8 -

23 < Nonlifting Flow over a Circular Cylinder > Aerodynamics 2015 fall - 9 -

24 < Nonlifting Flow over a Circular Cylinder > Let At the stagnation point, & Stagnation streamline : Aerodynamics 2015 fall

25 < Nonlifting Flow over a Circular Cylinder > Now, check the pressure distribution at the surface of the cylinder. R No Lift No Drag * In the real situation, no lift is acceptable. But no drag makes non-sense. d Alembert Paradox in 18c what happens in real life? the role of viscosity makes drag Aerodynamics 2015 fall

26 < Vortex Flow > By definition, (NOTE) 1. Vortex flow is irrotational except its origin 2. Circulation is positive-clockwise Aerodynamics 2015 fall - 1 -

27 < Vortex Flow > What happens at r=0? For 2D flow G is the same for all the circulation streamlines As Aerodynamics 2015 fall - 2 -

28 < Lifting Flow over a Cylinder > Uniform flow + doublet + vortex Aerodynamics 2015 fall - 3 -

29 < Lifting Flow over a Cylinder > Stagnation point Aerodynamics 2015 fall - 4 -

30 < Lifting Flow over a Cylinder > At surface, The drag on a cylinder is zero, regardless of whether or not having circulation in inviscid, irrotational and incompressible flow. Aerodynamics 2015 fall - 5 -

31 < Lifting Flow over a Cylinder > Sectional Lift, Kutta-Joukowski Theorem Aerodynamics 2015 fall - 6 -

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