Ch08_2_TF - MAE130B Ch 08 (2) Turbulent Flow Changzheng...

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MAE130B – Ch 08 (2) Turbulent Flow Changzheng Huang
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Ch 08 (2) Changzheng Huang 2 Fully developed turbulent flow 1. Turbulent shear stress 2. Turbulent velocity profile 3. Example
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Ch 08 (2) Changzheng Huang 3 1. Turbulent shear stress Q A x A u t laminar transitional turbulent ( ) ,, ux y z = Vi ( ) ( ) ( ) , , , yzt v x yzt w x yzt = ++ j k Laminar flow: momentum transfer at molecular level, microscopic, well-defined streaklines. Turbulent flow: momentum transfer through eddies, macroscopic, random flow fluctuations. Re VD ρ µ = D 0 2100 4000 Laminar Transitional Turbulent
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Ch 08 (2) Changzheng Huang 4 () 2 p t ρµ ⎡⎤ +• = + ⎢⎥ ⎣⎦ V VV V 1. Turbulent shear stress ( )( ) ,, , , , (, ,,) ux yzt v x yzt w x yzt pp x y z t =++ = Vi j k If a pipe flow is turbulent flow, the flow field is unsteady, i.e., Recall that for fully developed pipe flow, the inertia terms (local acceleration and convective acceleration) must be equal to zero. 0 0 But for turbulent pipe flow, these terms are not always zero because of flow unsteadiness. So how can we have fully developed turbulent flow in pipes?
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Ch 08 (2) Changzheng Huang 5 1. Turbulent shear stress () ( ) ,, , , ux yzt u x y zu x =+ The answer is that when we deal with turbulent flow, we usually do time averaging for all flow quantities. Then we will have fully developed pipe flow as the time averaged inertia terms are equal to zero. So how do we do time averaging? Take u(x,y,z,t) as an example, u t 0 t 0 tT + u u instantaneous time velocity velocity averaged fluctuation velocity 0 0 1 , t uu x y z t d t T + = Here T is considerably longer than the longest fluctuations, but considerably shorter than any unsteadiness of the average velocity.
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Ch 08 (2) Changzheng Huang 6 1. Turbulent shear stress or uuu u u u = += Now that We have, () 00 0 0 11 1 1 0 tT tt t u u u dt udt udt u uT TT T T ++ + = =−= = ∫∫ 0 0 22 1 0 t uu u d T T + =− > 0 0 1 0 t uv u u v v dT T + ′′
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Ch 08 (2) Changzheng Huang 7 1. Turbulent shear stress Time average the Navier-Stokes equations, 22 uu v p u u
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This note was uploaded on 03/10/2009 for the course MAE 130B taught by Professor Huang during the Spring '09 term at UC Irvine.

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Ch08_2_TF - MAE130B Ch 08 (2) Turbulent Flow Changzheng...

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