Fluid Mechanics Applications/B-34: Sudden contraction

Introduction
The sudden contraction in the area/diameter of a ﬂuid jet after it emerges from a circular aperture in a pressurized reservoir is called as vena contracta. Coefficient of contraction is the ratio of the cross sectional area of the jet at the vena contracta to the area of the orifice. The typical value may be taken as .64

Checking the vena contracta
Assumption P1A1v1 be the Pressure, Area and velocity of fluid in the tank,

P2A2v2 be the Pressure, Area and velocity of fluid in the orifice,

P3A3v3 be the Pressure, Area and velocity of fluid at the vena contracta,

Vena contracta plays a very important role in the minor losses in pipes. The diameter of the vena contracta is nearly equal to .64 times the diameter of the original hole. Assume fluid is incompressible

using continuity equation v1A1 =v2A2                                                        (1) and             v1  << v2                                                         (2) Ignoring energy losses due to viscosity,Bernoullis' holds for points along streamlines

P1 +1/2 d v12 = P2 +1/2dv22           (3)

where d is the(constant) density of fluid and P is the pressure using equation (2) and (3) v22 =2( P1 - P2 )/d                                                  (4) Considering momentum in the system, the mass flux =dvA so, momentum flux = dv2A The net flux bounded by area A1 and A2  dp/dt =   d(v22A2-v12A1)=dv22A2                                        (5) F ≈ P1A1 - [P1(A1 - A2)] = (P1 - P2)A2                        (6)

equating to equation (5)

v22= (P1-P2)/d           (7)

This is the contradiction with equation (4) based on conservation of energy. Then according to Torricelli this contradiction is resolved in nature by a contraction of fluid to area A3 after it passes through A2 The momentum flux is actually dp/dt =  d(v32A3-v12A1) ≈ dv32A3 ≈ 2P1A3           (8) According to Bernoulli's equation in the limit that P3 << P2 The force that causes this change is now F ≈ P1A1 - [P1(A1 - A2) + P3A3] = (P1A2 - P3A3) ≈ P1A2                          (9) we estimate vena contracta to be A3= A2/2

  http://www.physics.princeton.edu/~mcdonald/examples/vena_contracta.pdf