Chapter 7 Flow Past Immersed Bodies 7 1 Boundary Layer Theory 7 1 1 Introduction 1 Boundary layer flows External flows around bodies immersed in a fluid stream have viscous shear and no slip effects near the body surfaces and in its wake but be nearly inviscid far from the body 2 Features flow unconfined free to expand no matter how thick the viscous layers grow 3 Applications of BL theory aerodynamics airplanes rockets projectiles hydrodynamics ships submarines torpedos transportation automobiles trucks cycles wind engineering buildings bridges water towers and ocean engineering buoys breakwaters cables 7 1 2 Flat Plate Integral Analysis Displacement Thickness To gain much insight and quantitative information about boundary layers by making a broad brush momentum analysis of the flow of a viscous fluid at high Re past a flat plate Shear layer thickness as 99 streamlines outside this shear layer will deflect an amount the displacement thickness Thus the streamline move outward from y H at x 0 to y Y H at x x1 1 Conservation of mass Y V dA 0 0 CS H udy Udy 0 Assuming incompressible flow constant density this relation simplifies to UH udy U u U dy UY u U dy Y Y Y 0 0 0 Note Y H we get the definition of boundary layer displacement thickness Y u 1 dy 0 U is an important measure of effect of BL on external flow To see this more clearly considering an alternate derivation based on an equivalent discharge argument 0 Udy udy a Discharge between and of inviscid flow actual discharge u Udy Udy udy 1 0 0 0 0 U dy w o BL displacement effect actual discharge For 3D flow in addition to a it must also be explicitly argued that is a stream surface of the inviscid flow continued from outside of the BL 7 1 3 Momentum thickness 2 Conservation of x momentum Fx D u udy U Udy Y H 0 0 Y Drag D U 2 H u 2 dy 0 Y u H Again assuming constant density and introducing 0 U dy Drag D u U u dy w dx Y x 0 0 Y u D u 1 0 U U dy U 2 i e is the momentum thickness a function of x only an important measure of the drag 2D 2 1 CD C dx U 2 x x x 0 f x Cf w 1 U 2 2 d C f dx 2 Cf d xCD 2 d dx dx w U 2 d dx Assume that the velocity profiles had an approximately parabolic shape 3 2 y y2 u x y U 2 0 y x 2 u 2 U w We get y y 0 15 v d 15 dx U Integrate from 0 to x assuming 0 at x 0 the leading edge 12 5 5 5 5 1 2 x Re x Ux 7 1 4 The laminar Boundary Layer Equations 1 For two dimensional incompressible flow u v 0 Continuity x y 2u 2u 1 p u u u x momentum t u x v y x x 2 y 2 2v 2v 1 p v v v y momentum t u x v y y x 2 y 2 2 Boundary Layer Approximations v u Velocities u u v v Rate of change y x x y Ux 1 Reynolds number Re x 2v 2v 1 p v v v 2 2 u v y y x t x y small small small very small small 4 p 0 or p p x only using Bernoulli s equation applied So y to the outer inviscid flow p dp dU U x dx dx For x momentum 2u 2u 2 x 2 y Then we get Prandtl s two boundary layer equations for twodimensional incompressible flow u v 0 x y u dU 1 u u u v U t x y dx y uy u y u v Laminar flow Turbulent flow Notes 1 u x t and p x t imposed on BL by the external flow 2 2 2 The term x in the x momentum equation has been neglected i e Streamwise diffusion is neglected 3 Due to 2 the equations are parabolic in x Physically this means all downstream influences are lost other than that contained in the external flow A marching solution is possible 4 Boundary conditions 5 patching inlet Solution by marching y x X0 No slip No slip u x 0 t v x 0 t 0 Inlet condition u x0 y t given at same x0 Patching with outer flow u x t U x t Initial condition u x y 0 v x y 0 known 5 Practical use of Boundary Layer theory For a given body geometry A Inviscid theory gives p x t B Boundary layer theory gives x t w x t x t etc and predicts separation if any C If separation present no further information must use inviscid models or NS equations D If separation is absent integration of w x frictional resistance body and inviscid theory gives p x t and can go back to step B for more accurate BL calculation using viscous inviscid iteration 3 Dimensionless form of boundary layer equations Recall in 7 1 3 we have 12 5 5 x Ux 5 5 Re1x 2 The sizes of the various terms u and x unity 6 v and y Re Define dimensionless variables all of which to be of order unity if Re is large p p0 x y tU u v Re t Re p x y u v U 2 L L L U U u p 2u u v u u u v 0 2 t x y x x y y 1 2 p 0 y Detailed derivations by yourself recommended note some terms dropped due to large Reynolds number 4 Limitations of the boundary layer equations 1 The Reynolds number must be large Re x 1000 2 If outer flow is decelerating dU dx 0 dp dx 0 a point may be reached where wall shear stress approaches zero the separation point Beyond this point the boundary layer approximations are not accurate 7 2 Laminar Boundary Layer 7 2 1 Similarity solutions 2D steady incompressible ux vy 0 uu x vu y UU x u yy BCs u x 0 v x 0 0 u x U x inlet condition y u x y F g x related to x For Similarity U x g x expect 7 Or in term of stream function And for similarity U x g x f u y Uf Then u y v x y g x etc BC f 0 f 0 0 f 1 Substitute into the boundary layer equations yields y yx x yy UU x yyy x U x gf Ug x f U gx f y Uf yy Uf g yyy Uf g 2 xy U x f Uf g x g Assemble them together Uf U x f Uf gx U x gf Ug x f Ug x f Uf g g UU …
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