16100lectre26_cg

16100lectre26_cg - Effect of Turbulent Fluctuations on Mean...

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Effect of Turbulent Fluctuations on Mean Flow: Reynolds-Averaging In a turbulent flow, we can define the mean, steady flow as: 0 1 (, ,) l im (, ,,) T T uxyz uxyztd t T →∞ = This allows us to split the flow properties into a mean and a fluctuating part: mean turbulent part fluctuating part (,,,) (,,) uxyzt uxyz u xyzt vxyzt vxyz v xyzt wxyzt wxyz w xyzt pxyzt pxyz p xyzt =+ ±²³ ² ´± ² ³ ² ´ Note: the mean of u is zero: {} 00 11 lim lim TT uu u u u dt u dt →∞ →∞ −= ∫∫ 0 1 lim T T u udt u u T →∞ ±²³²´ N 0 uuu = 0 u =⇐ mean of fluctuations is zero. Now, we will develop equations which govern the mean flow and try to develop some insight into how the fluctuations alter the mean flow equations. Let’s start with incompressible flow and look at the x momentum: x momentum: 222 1 u p u u u uvw txyz x x y z ν ρ ∂∂∂ +++ = + + + ∂∂
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16100lectre26_cg - Effect of Turbulent Fluctuations on Mean...

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