5 As described in Dr Sepris Note the conditions leading to a similar solution

5 as described in dr sepris note the conditions

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(5) As described in Dr. Sepri’s Note, the conditions leading to a similar solution are: (i) B.C need to be similar ( ρ v ) 0 restricted in form (ii) I.C is similar, that is, can’t accept an arbitrary f 0 ( η )
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Advanced Fluid Mechanics Chapter 5-21 (iii) External pressure gradient must comply with β = const. (iv) Density profile is similar. As m = -0.091, 0 = y y u = U 0 = η ηη f = 0, therefore, the separation occurs. We conclude that If m > 0 0 > dX dU e , dx dp e = ρ U e dX dU e <0 accelerating flow If m < 0 (but 1/2 < m ) dX dU e < 0, dx dp e > 0 decelerating flow In this course, the flow is taken as incompressible ; therefore, the flow is a accelerated as it past a wedge and decelerated as it past a corner. 0 > dx du e nozzle) (subsonic 0 < dx du e diffuser) (subsonic However, as the flow is compressible , it will be different, e.g. 1 M 2 M shock ) M (M 2 1 > expansion 1 M 2 M ) M (M 2 1 < 2 1 T T > 2 1 T T but < diffser) c (supersoni nozzle) c (supersoni Since M= RT V γ V=M RT γ It hard to tell whether V 1 >V 2 or V 1 <V 2 But normally V 1 <V 2
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Advanced Fluid Mechanics Chapter 5-22 (In x-y plane, no gravity force acting) 0 > m 0 < m . 0 u . u ) ( X ) ( V U 0 > dx dU e 0 < dx dU e seperation , 0 0 = = y y u e U 1 2 3 4 5 . . For curve 1-4, 2-4 & 3-4, the velocity profile at 5 will not be the same. (Cannot determine the dx dU e >0 or dx dU e <0 from the slope of the local surface w.r.t the free-stream direction. It normally further upstream as shown)
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Advanced Fluid Mechanics Chapter 5-23 Problem: Show that ( δ * / τ w ) d p /d x represents the ratio of pressure force to wall friction force in the fluid in a boundary layer. Show that it is constant for any of the Falkner-Skan wedge flows. (J. schetz. P. 92, prob. 4.6)
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Advanced Fluid Mechanics Chapter 5-24 5.4 Flow in the wake of Flat Plate at zero incidence Preface: the B.L. equation can be applied not only in the region near a solid wall, but also in a region where the influence of friction is dominating exists in the interior of a fluid. Such a case occurs when two layers of fluid with different velocities meet, such as: wake and jet . Consider the flow in the wake of a flat plate at zero incidences l y U U U δ 1 A A 1 B B x h surface control = 0 v U u ) , ( y x u Want to find out: (1) the velocity profile in the wake . Assume: d p /d x = 0 For the mass flow rate: ( Σ = 0) At AA 1 section = ρ h 0 U d y (entering) At BB 1 section = - ρ h 0 u d y (leaving) At AB section = 0 At A 1 B 1 section = - ρ h 0 ( U - u )d y (To keep Σ mass = 0) Actually along A 1 - B 1 , the u = U , the mass must be more out to satisfy continuity m & = - ρ B A v ( x , h )d x
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Advanced Fluid Mechanics Chapter 5-25 For the x -momentum floe rate: At AA 1 section = ρ h 0 U 2 d y (entering) At BB 1 section = - ρ h 0 u 2 d y (leaving) At AB section = 0 At A 1 B 1 section = AB m & U = U [ - ρ h 0 ( U - u )d y ] = - ρ h 0 U ( U - u )d y Drag on the upper surface = Σ Rate of change of x -momentum in A 1 -B 1 -B-A = ρ h 0 u ( U - u )d y (5.17) In order to calculate the velocity profile, let us first assume a velocity defect u 1 ( x, y ) as u 1 ( x, y ) = U - u (x, y) (5.18) and u 1 << U
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