058 0160 Professor Fred Stern Fall 2010 Chapter 7 1 Chapter 7 Boundary Layer Theory 7 1 Introduction Boundary layer flows External flows around streamlined bodies at high Re have viscous shear and no slip effects confined close to the body surfaces and its wake but are nearly inviscid far from the body Applications of BL theory aerodynamics airplanes rockets projectiles hydrodynamics ships submarines torpedoes transportation automobiles trucks cycles wind engineering buildings bridges water towers and ocean engineering buoys breakwaters cables 7 2 Flat Plate Momentum Integral Analysis Laminar approximate solution Consider flow of a viscous fluid at high Re past a flat plate i e flat plate fixed in a uniform stream of velocity Ui Boundary layer thickness arbitrarily defined by y 99 where 99 is the value of y at u 0 99U Streamlines outside 99 will deflect an amount the displacement thickness Thus the streamlines move outward from y H at x 0 to y Y H at x x1 058 0160 Professor Fred Stern Chapter 7 2 Fall 2010 Conservation of mass H H V ndA 0 0 Udy 0 udy CS Assuming incompressible flow constant density this relation simplifies to UH udy U u U dy UY u U dy Note Y Y Y 0 0 0 Y H we get the definition of displacement thickness u 0Y 1 dy U a function of x only is an important measure of effect of BL on external flow To see this more clearly consider an alternate derivation based on an equivalent discharge flow rate argument Lam 3 Turb 8 0 Udy udy Inviscid flow about body Flowrate between and of inviscid flow actual flowrate i e inviscid flow rate about displacement body equivalent viscous flow rate about actual body u Udy Udy udy 1 0 0 0 0 U dy w o BL displacement effect actual discharge For 3D flow in addition it must also be explicitly required that is a stream surface of the inviscid flow continued from outside of the BL 058 0160 Professor Fred Stern Chapter 7 3 Fall 2010 Conservation of x momentum Fx D CS H Y 0 0 uV ndA U Udy u udy Y Drag D U 2 H 0 u 2 dy Fluid force on plate Plate force on CV fluid Y u H Again assuming constant density and using continuity 0 U dy Y x 2 Y D U 0 u Udy u 2 dy 0 w dx 0 D 2 U u Y u 0 1 U U dy where is the momentum thickness a function of x only an important measure of the drag 2D 2 1 CD C dx Per unit span 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 Special case 2D momentum integral equation for px 0 058 0160 Professor Fred Stern Chapter 7 4 Fall 2010 Simple velocity profile approximations u U 2 y y 2 2 u 0 0 u U uy 0 no slip matching with outer flow Use velocity profile to get Cf and and then integrate momentum integral equation to get Rex 3 2 15 H 5 2 w 2 U 2 U d d Cf 2 2 2 15 2 dx dx 1 2 U 15 dx d U 30 dx 2 U x 5 5 Re1x 2 Re x Ux x 1 83 Re1x 2 x 0 73 Re1x 2 C D 1 46 Re1L 2 2C f L 10 error cf Blasius 058 0160 Professor Fred Stern Chapter 7 5 Fall 2010 7 3 Boundary layer approximations equations and comments U y x u v 0 2D NS constant neglect g ux vy 0 1 p u xx u yy x 1 p vt uv x vv y v xx v yy y ut uu x vu y Introduce non dimensional variables that includes scales such that all variables are of O 1 x x L y y Re L t tU L u u U v Re U p p0 p U 2 Re UL 058 0160 Professor Fred Stern Fall 2010 Chapter 7 6 The NS equations become drop ux vy 0 1 u xx u yy Re 1 1 1 vt uvx vv y p y 2 vxx v yy Re Re Re ut uu x vu y px For large Re BL assumptions the underlined terms drop out and the BL equations are obtained Therefore y momentum equation reduces to py 0 i e p p x t px U t UU x From Euler Bernoulli equation for external flow 2D BL equations u x v y 0 ut uu x vu y U t UU x u yy Note 1 2 3 4 U x t p x t impressed on BL by the external flow 2 0 i e longitudinal or stream wise diffusion is 2 x neglected Due to 2 the equations are parabolic in x Physically this means all downstream influences are lost other than that contained in external flow A marching solution is possible Boundary conditions 058 0160 Professor Fred Stern Chapter 7 7 Fall 2010 matching inlet Solution by marching y x X0 No slip No slip u x 0 t v x 0 t 0 Initial condition u x y 0 known Inlet condition u x0 y t given at x0 Matching with outer flow u x t U x t 5 When applying the boundary layer equations one must keep in mind the restrictions imposed on them due to the basic BL assumptions not applicable for thick BL or separated flows although they can be used to estimate occurrence of separation 6 Curvilinear coordinates 058 0160 Professor Fred Stern Chapter 7 8 Fall 2010 Although BL equations have been written in Cartesian Coordinates they apply to curved surfaces provided R and x y are curvilinear coordinates measured along and normal to the surface respectively In such a system we would find under the BL assumptions py u 2 R Assume u is a linear function of y u Uy dp U 2 y 2 dy R 2 p p 0 U 2 3R Or p U 2 3R therefore we require R 058 0160 Professor Fred Stern Chapter 7 9 Fall 2010 7 Practical use of the BL theory For a given body geometry a Inviscid theory gives p x integration gives L D 0 b BL theory gives x w x x etc and predicts separation if any c If separation present then no further information must use inviscid models BL equation in inverse mode or NS equation d If separation is absent integration of w x frictional resistance body inviscid theory gives p x can go back to 2 for more accurate BL calculation including viscous inviscid interaction 8 Separation and shear stress At the wall u v 0 u yy 1 px 1st derivative u gives w w u y w 0 separation 2nd derivative u depends on px w 058 0160 Professor Fred Stern Fall 2010 Chapter 7 10 Inflection point 7 4 Laminar Boundary …
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