Angular momentum & RTT r X m M net , X = d t CV d V d r X - PowerPoint PPT Presentation

Angular momentum & RTT ⃗ r X m ∧휌⃗ M net , X = ⃗ d t ∭ CV d V d  r X m ∧휌(⃗ n )⃗ + ∬ CS ⃗ V rel ⋅ ⃗ V d A The vector sum of moments on the fluid = the rate of change of angular momentum in the cv + the net flow of angular momentum through the cs 1|25

Figure CC-0 Olivier Cleynen 2|25

Figure CC-0 Olivier Cleynen 3|25

r X m ∧ 휌(⃗ n )⃗ If ⃗ V rel ⋅ ⃗ V is uniform across each inlet/outlet: ⃗ r X m ∧ 휌⃗ M net = ⃗ d t ∭ CV d V d  { } { } m |⃗ m |⃗ + ∑ ⃗ r X m ∧ | ̇ − ∑ ⃗ r X m ∧ | ̇ V V out in 4|25

Photo CC-by-sa by Commons User:Timhall 5|25

Photo CC-by by JJ Harrison 6|25

Drawing CC-by-sa U:Tosaka 7|25

Photo by Andy Wolfe, US Army (public domain) 8|25

Figure CC-by by Commons User:Tosaka 9|25

Photo by Alex R. Lloyd, U.S. Air Force (public domain) 10|25

∼ A worked-out example for the angular momentum equation ∼ 11|25

Falcon Heavy take-o ff on 2019-04-10 Photo by nasa (public domain) 12|25

Falcon Heavy take-o ff on 2019-04-10 Photo CC - 0 by SpaceX 13|25

crs-8 landing on 2016-04-09 Photo CC-0 by SpaceX 14|25

crs-8 landing on 2016-04-09 Photo CC-0 by SpaceX 15|25

Figure CC-0 Olivier Cleynen 16|25

Figure CC-0 Olivier Cleynen 17|25

Figure CC-0 Olivier Cleynen 18|25

Remove unsteady terms, split inlet/outlet ⃗ r X m ∧ 휌⃗ M net , X = ⃗ d d t ∭ CV V d  r X m ∧ 휌(⃗ n )⃗ ⃗ V rel ⋅ ⃗ + ∬ CS V d A r X m ∧ 휌(⃗ n )⃗ ⃗ V rel ⋅ ⃗ = + ∬ in V d A r X m ∧ 휌(⃗ n )⃗ + ∬ out ⃗ V rel ⋅ ⃗ V d A r X m ∧ 휌 V ⟂ in ⃗ = + ∬ in ⃗ V d A r X m ∧ 휌 V ⟂ out ⃗ + ∬ out ⃗ V d A 19|25

Hoop #1: sign of V ⟂ ⃗ r X m ∧ 휌 V ⟂ in ⃗ M net , X = + ∬ in ⃗ V d A r X m ∧ 휌 V ⟂ out ⃗ + ∬ out ⃗ V d A r X m ∧ 휌| V ⟂ in |⃗ = − ∬ in ⃗ V d A r X m ∧ 휌| V ⟂ out |⃗ + ∬ out ⃗ V d A 20|25

Uniform velocity at inlets & outlets ⃗ r X m ∧ 휌| V ⟂ in |⃗ M net , X = − ∬ in ⃗ V d A r X m ∧ 휌| V ⟂ out |⃗ + ∬ out ⃗ V d A r X-inlet ∧ 휌| V ⟂ in |⃗ = − ⃗ V in A in r X-outlet ∧ 휌| V ⟂ out |⃗ + ⃗ V out A out r X-inlet ∧ ⃗ = − 휌| V ⟂ in | A in ⃗ V in r X-outlet ∧ ⃗ + 휌| V ⟂ out | A out ⃗ V out r X-inlet ∧ ⃗ = − ̇ m in ⃗ V in r X-outlet ∧ ⃗ + ̇ m out ⃗ V out 21|25

Vector cross-product: don’t panic ⃗ r X-inlet ∧ ⃗ r X-outlet ∧ ⃗ M net , X = − ̇ m in ⃗ V in + m out ⃗ ̇ V out r X-inlet ∧ ⃗ r X-outlet ∧ ⃗ = − ̇ m in ⃗ V in ⟂ r + m out ⃗ ̇ V out ⟂ r 22|25

All right-hand-side vectors in same plane: we drop vectors Hoop #2: sign of V ⟂ r ⃗ r X-inlet ∧ ⃗ r X-outlet ∧ ⃗ M net , X = − ̇ m in ⃗ V in ⟂ r + m out ⃗ ̇ V out ⟂ r M net , X = − ̇ m in r X-inlet × ( V in ⟂ r ) + ̇ m out r X-outlet × ( V out ⟂ r ) ( V ⟂ r positive clockwise, negative anticlockwise) ̇ M net , X = 0 + m out r X-outlet × (+ | V out ⟂ r |) (length in z -direction) 23|25

Numbers! M net , X = 0 + m out r X-outlet × (+ | V out ⟂ r |) ̇ = 0 + 2 , 5 × 40 × (+ | 155 cos 20 |) = + 14 , 7 kN m ⎛ ⎞ ⃗ ⎜ 0 ⎟ M net , X = ⎜ ⎟ 0 + 14 , 7 ⋅ 10 3 ⎝ ⎠ Moment exerted on fl uid leaving the rocket (moment on rocket by fl uid is opposite) 24|25

Angular momentum & RTT ⃗ r X m ∧ 휌⃗ M net , X = ⃗ d t ∭ CV d V d  r X m ∧ 휌 ( ⃗ n ) ⃗ + ∬ CS ⃗ V rel ⋅ ⃗ V d A The vector sum of moments on the fl uid = the rate of change of angular momentum in the cv + the net fl ow of angular momentum through the cs 25|25