《流体动力学》（英文版）integral thicknesses

Integral Thicknesses 1 Definitions The details of the velocity profile u(y) at any a location are rarely significant in engineering applications. The most significant quantities are integral thicknesses which describe the mass Aux, momentum flux, and kinetic energy flux in the shear layer displacement thickness Pete dy momentum thickness Pe 0·= kinetic energy thickne 2 Integral Thickness Interpretation These thicknesses appear when comparing the mass, momentum, and kinetic energy fows in a shear layer and a corresponding potential flow 2.1 Mass flow comparison Figure 1 shows the mass fux passing between the vertical extent y=0... Je for inviscid and viscous fows with the same velocity. ye ye Figure 1: Comparison of inviscid and viscous mass flows dy= rir pewee pu dy u dy I- peu The viscous mass fow is decreased by an amount equal to the mass defect pete 8

2.2 Momentum flow comparison Figure 2 shows the momentum flux carried by the mass fow passing between y=0... ye of the inviscid case. The viscous case capture height is increased by 8'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) y y ye F Figure 2: Comparison of inviscid and viscous momentum flows, at the same mass flow dn pu2 dy= F= peuzye ye+6° Fy=udn "p2d=n2- (ue -u)oudy= Fr-Peu2e The viscous momentum flow is decreased by an amount equal to the momentum defect Peu20 2.3 Kinetic energy flow comparison Figure 3 shows the kinetic energy flux carried by the mass flow passing between y=0.ye of the inviscid case. The viscous case capture height is again increased by 8 so that the comparison is done at the same mass Rows. 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. P adrn pudy= P= peudye e+6*1 udn= v+61e (2-2)mdy=B-2 The viscous kinetic energy flow is decreased by an amount equal to the kinetic energy defect dEuce

K.E. extractor (windmill array) y De ye mom Figure 3: Comparison of inviscid and viscous kinetic energy flows, at the same mass flow