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 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 99 1 Conservation of mass V dA 0 udy Udy Assuming incompressible flow constant density this relation simplifies to UH udy U u U dy UY u U dy Note Y H we get the definition of boundary layer displacement thickness Y H 0 CS 0 Y Y Y 0 0 0 Y 0 u 1 dy 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 dy U 0 0 0 0 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 Conservation of x momentum F D u udy U Udy Drag D U H u dy Again assuming constant density and introducing Y x H 0 0 2 Y 2 0 Y H 0 u dy U 2 Y x Drag D u U u dy w dx 0 0 Y u D u 1 dy 2 0 U U U i e is the momentum thickness a function of x only an important measure of the drag d d x 2D 2 1 C f w C f xCD 2 C C dx 1 D U 2 x x x 0 f 2 d C f dx 2 dx U 2 w U 2 dx d dx Assume that the velocity profiles had an approximately parabolic shape 2 y y2 u x y U 2 2 We get 15 w u y y 0 0 y x 2 U v dx U 0 at x 0 d 15 Integrate from 0 to x assuming 12 5 5 x Ux the leading edge 5 5 Re1x 2 7 1 4 The laminar Boundary Layer Equations 1 For two dimensional incompressible flow u v 0 Continuity x y x momentum y momentum 2u 2u u u u 1 p u v 2 2 t x y x y x 2v 2v v v v 1 p u v 2 2 t x y y y x 3 2 Boundary Layer Approximations v u Velocities u u Rate of change x y v v x y Ux Re x 1 2v 2v v v v 1 p u v 2 2 t x y y x y Reynolds number small small small very small small or p p x only using Bernoulli s equation applied to p 0 y So the outer inviscid flow p dp dU U x dx dx For x momentum 2u 2u x 2 y 2 Then we get Prandtl s two boundary layer equations for twodimensional incompressible flow u v 0 x y u u u dU 1 u v U t x y dx y u y u u v y Laminar flow Notes Turbulent flow 1 u x t and p x t imposed on BL by the external flow 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 2 2 4 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 x 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 0 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 v and y Re Define dimensionless variables all of which to be of order unity if Re is large 1 2 5 x x L y y Re L t p p0 U 2 u p 2u u u u v t x y x y 2 tU L u u U u v 0 x y v v U p Re p y 0 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 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 x 7 2 Laminar Boundary Layer 7 2 1 Similarity solutions 2D steady incompressible u x v y 0 uu x vu y UU x u yy BCs u x 0 v x 0 0 u x U x inlet condition For Similarity y u x y F U x g x expect g x related to x Or in term of stream function u v y g x And for similarity U x g x f Then u Uf etc BC f 0 f 0 0 f 1 Substitute into the boundary layer equations yields y x y 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 6 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 …
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