Fluid_Kinematics_II_2022_Notes

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Class Objectives Fluid Kinematics II: Acceleration of Fluid Motions • Acceleration • Material Derivative • Acceleration under Streamline Coordinates

Fluid Kinematics II Fluid motion involves position , velocity , of the fluid acceleration (Text Chapter 4.1, pp 204 & 206)

(1) Physical Intuition for Acceleration Field  ) , ( t x a u u x    garden hose faucet t t x u   ) , ( nozzle Temporal (Local) derivative Spatial (Advective) derivative Total acceleration Example 4-2 Text

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( , , , ) Du Dv Dw Dt Dt a x y z t D i t j k    Math Components of Acceleration Field u u x    v v u v v w x y z          t w    DV Dt  Du Dt  Dv Dt  Dw Dt  v w z u u y       ) , , , ( t z y x a x ) , , , ( t z y x a y ) , , , ( t z y x a z u t    v t    w w u v w w x y z         

A velocity field is given by the following x y-plane: 𝑉 = (0.5 − 2? + 3?) Ƹ ? + (−2 + ? + 2?) Ƹ ? Calculate the acceleration at point (x, y) = (-1, 2). x u u u v u t x y a 0.5 2 3 u x y 2 2 v x y 14 y v v u v v t x y a ( ) (0.5 2 3 ) ( 2 2 ( ) ) 1 2 x y x y 2 ( ) (0 ( ) .5 2 3 3 ) ( 2 2 ) x y x y 6.5 Example 4-3 Text       , 0.5 0.8 1.5 0.8 V u v x i y j        

𝑉 = 6(1 + 0.4? 2 )(1 − 0.5𝑡) x u v u u u t x y a 2 2 6(1 0.4 ) 6(1 0.4 )(1 0.5 ) ( (1 0 ( ) .5 ) 0 6 0.8 ) .5 x x t x t Given the velocity Ask for the acceleration in the nozzle. Recall the acceleration can be obtained by “Material Derivative of velocity” 2 2 ( 6(1 0.4 ) (1 0.5 ) 6 0.8 ) [ 5 ] 0. t x x

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(2) Material Derivative The rate at which that variable ( velocity u , concentration C or temperature T ) changes for a particle following flow motions Example: A temperature change in the car x u = 10 mph • Temperature T is warming as 1 o F/hr • Temperature T spatial gradient is 0.04 o F/mile Madison Florida, Disney D T Dt  What is the rate of temperature change in the car? 0 1 0.4 10(0.04) 1.4( ) 1 / F hr      D Dt  t   u v w x y z          t T   T u x    u = 120 mph 0 (0.04) 1 4. 1 8 1 5.8( / 20 ) F hr     

(3) Streamline coordinates ˆ s ˆ n D t V D R x y z Radius of curvature Examp A steady water flows through a 180 o pipe le: What are a s and a n for the circular motions? R=20 ft V=10 ft/s 0 10 10 ) (        s s V V a s s s ) sec / ( 5 20 10 ) ( 2 2 2 ft R V a s n    ˆ s D t V s D  a  ˆ s s D V Dt  ˆ s t V D s D  ˆ n a n  ˆ s a s  ( + ) ˆ s s s V V V s t s      2 ( + ˆ ) n s V V n t R    (Text – pp 204 & 206)

2 0.4 (0.4s ) t V e   ( + ) s s s s s a V V V t      2 ( + ) n s n V V t a R    Determine a s and a n for the flow in the curved pipes. 2 0.4 2 0.4 0.4 0.4 (-0.4) 0.4 (0.8 ) t t t s e s e s e        2 3 0.4 0.4 0.16 0.32 t t s s e e      2 0.4 2 2 (0.4 ) t s e R   R = 5 m • Along a streamline • Normal to a streamline 1 (sec) t  2 - 0.00755 m/s  @ 1, 6 t s   2 - 0.01863 m/s  @ 1, 6, 5 t s R   

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