Exercise 10: Potential Flow Example 1: Half body ... - KTH Mechanics

c) Calculate the shape of a free surface at atmospheric pressure p 0 .Find the difference in z between r = 0 and r → ∞{r = 0 : p 0 = −ρgz 0 + C 1⇒ z ∞ − z 0 = C 2 − C 1= ω2 a 2r → ∞ : p 0 = −ρgz ∞ + C 2 ρg gDetermine the shape of the free surface:{p0 = ρω2 r 22− ρgz + C 1 r < a z ∼ r 2 ⇒ z = ω2 r 22g+ C1−p0ρgp 0 = − ρω2 a 42r− ρgz + C 2 2 r > a z ∼ 1r⇒ z = − ω2 a 42 2gr 2Set z = 0 at r = 0. Then C 1 = p 0 and we further get+ C2−p0ρgz = ω2 r 22gr < aandz = − ω2 a 42gr 2 + C 2 − C 1= − ω2 a 4ρg 2gr 2 + ω2 a 2= ω2 a 2 ) (1 − a2g g 2r 2r > aSo we have⎧⎨z(r) =⎩ω 2 r 22gω 2 a 2gr < a( )1 − a22r 2r > a .z0.10.090.080.070.060.050.040.030.020.0100 0.5 1 1.5 2 2.5 3rFigure 2: The free surface of a Rankine vortex with ω = a = 1 and g = 9.82.Example 2Show that the inviscid vorticity equationreduces to the equationin the case of axisymmetric flowD¯ωDt= (¯ω · ∇)ū)D ω= 0Dt(rū = u r (r, z, t)ē r + u z (r, z, t)ē z .3