integral_thicknesses

integral_thicknesses - Integral Thicknesses 1 Definitions...

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Unformatted text preview: Integral Thicknesses 1 Definitions The details of the velocity profile u(y) at any x location are rarely significant in engineering applications. The most significant quantities are integral thicknesses which describe the mass flux, momentum flux, and kinetic energy flux in the shear layer: 6* = / <1 — pu >dy displacement thickness peue 0 = / (1 — 3) pu dy momentum thickness as peue u2 pu _ . . 6* = / 1 — —2- dy kinetic energy thickness “5 peue 2 Integral Thickness Interpretation These thicknesses appear when comparing the mass, momentum, and kinetic energy flows in a shear layer and a corresponding potential flow. 2.1 Mass flow comparison Figure 1 shows the mass flux passing between the vertical extent y = 0 . . .ye for inviscid and viscous flows with the same edge velocity. Figure 1: Comparison of inviscid and viscous mass flows - . ye - m1 : /dm 2 0 pudy : m1 = peueye m1 _ peue6* . . ye ye mv = /dm = 0 pudy = peueye — /0 (pane—pawl; The viscous mass flow is decreased by an amount equal to the mass defect peu66*. 2.2 Momentum flow comparison Figure 2 shows the momentum flux carried by the mass flow passing between 3; = 0 . . .ye of the inviscid case. The viscous case capture height is increased by 6* so that the comparison is done at the same mass flows. In each case, the momentum flow can be considered to be the force acting on a barrier which arrests the flow velocity to zero. momentum extractor (barrier) Figure 2: Comparison of inviscid and viscous momentum flows, at the same mass flow ye F1 = fudr'n = pu2dy = F1 = peuiye 0 I ye+6‘ 2 2 yea—Hr 2 FV = fudm = /0 pa dy = peueye — f0 (ue—wudy = F: — peuee The viscous momentum flow is decreased by an amount equal to the momentum defect peugfi. 2.3 Kinetic energy flow comparison Figure 3 shows the kinetic energy flux carried by the mass flow passing between 3/ = 0 . . .ye of the inviscid case. The viscous case capture height is again increased by 6* so that the comparison is done at the same mass flows. In each case, the kinetic energy flow can be considered to be the power generated on an array of perfect windmills which reversibly bring the flow velocity to zero. 1 , 961 1 P1 = /§u2dm = /O §W3dy = P1 = Epeugye _ 1 2 . “+6.1 3 1 3 glad—6‘1 2 2 1 3 * PV — dm 2 /0 Ep'u dy — ipeueye —/O 5 (ue—u )pudy = P1 — ipeuefl The viscous kinetic energy flow is decreased by an amount equal to the kinetic energy defect 1 3 >1< Epeuefi . K.E. extractor (windmill array) mom Figure 3: Comparison of inviscid and viscous kinetic energy flows, at the same mass flow ...
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integral_thicknesses - Integral Thicknesses 1 Definitions...

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