≠ ω constant ≠ s 0 ≠ ∇ s
Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei ( 李雨 ) Equations in a non-inertial frame So far we have derived our equations in an inertial frame. In non-inertial frame, the continuity equation, the energy equation, and the equations of state remain unchanged. For the momentum equation, the acceleration term becomes: dt d dt d dt d dt d c I u r Ω r Ω Ω u Ω u u + × + × × + × + = ) ( 2 ↑ ↑ Coriolis acceleration centrifugal acceleration linear acceleration of the center of the non-inertial coordinates acceleration observed in the non-inertial frame angular acceleration of the non-inertial frame
Vorticity and Circulation (for understanding the movie “Vorticity” by Shapiro, from this to last page) Vorticity, , represents twice the spinning rate of an infinitesimal fluid element. Circulation, Γ , represent the strength of vorticity contained inside a closed path C ( t ), defined as = ∇ × u ω ( ) C t d Γ = ⋅ u r ( ) ( ) ( ) ( ) . ( ) . C t A t A t A t If is regular inside C(t), d d n singula dA If is inside C t , nd r A Γ = ⋅ = ∇ × ⋅ = ⋅ Γ ≠ ⋅ u u r u A u ω ω Circulation is helpful in understanding the force on body, and in experimental quantification.
Example: Free vortex (2D flow) 2 ( ) 0 ( ) Velocity field in polar coordinates: 0, . ( = constant) 2 We have: 0, , and 0 because is singular at 0. r C t A t B u u B r d u rd B ndA u r θ π θ θ π θ = = ∇ × = Γ = ⋅ = = Γ ≠ ⋅ = = u u r ω = ω Note: A fluid element may travel on a circular streamline while having zero vorticity. Vorticity is proportional to the angular velocity of a fluid element about its principal axes ( 自轉 ), not about an axis through some other reference point ( 公轉 ). Thus a fluid particle which is traveling on a circular streamline will have zero vorticity, provided that it does not revolve about its center of gravity as it moves.