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 Integral Analysis Laminar approximate solution 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 Boundary layer thickness arbitrarily defined by y where is the value of y at u 0 99U Streamlines outside 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 99 99 Conservation of mass 1 99 V dA 0 H udy H Udy 0 0 CS 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 displacement thickness Y Y Y 0 0 0 u 0Y 1 U dy 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 argument Laminar Turbulent 0 Udy udy Inviscid flow about body a Discharge between and of inviscid flow actual discharge inviscid flow rate about displacement body equivalent viscous flow rate about actual body 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 required that is a stream surface of the inviscid flow continued from outside of the BL Conservation of x momentum H Y Fx D u V d A 0 U Udy 0 u udy CS Y Drag D U 2 H 0 u 2 dy Force on plate Force on CV Again assuming constant density and introducing 2 Y H 0 u dy U Y Y x D U 2 0 u Udy u 2 dy 0 w dx 0 D U 2 Y u 0 1 U u dy U where 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 CD C f dx 1 dx span dx U 2 Per unit U 2 x x x0 2 d C f dx 2 w U 2 d dx Special case 2D momentum integral equation for px 0 Simple velocity profile approximations u U 2 y y 2 2 u 0 0 no slip u U matching with outer flow uy 0 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 U 0 7 3 Boundary layer approximations equations and comments y x 10 error cf Blasius 3 Tw u v 0 NS 2D u x v y 0 1 p ut uu x vu y g x T T0 u xx u yy x 1 p vt uvx vv y g y T T0 vxx v yy y Cv Tt uTx vTy k Txx Tyy 2u x2 2v 2y u y vx 2 Introduce non dimensional variables that includes scales such that all variables are of O 1 x x L y T T0 T T0 y Re t tU L T L Tw T0 T v u u U v U Re UL Ec p po Re p U 2 U2 U2 Fi Cv T gi L The NS equations become drop u x v y 0 1 ut uu x vu y p x TT Fx u xx u yy Re 1 1 1 vt uv x vv y p y TT Fy v v yy 2 xx Re Re Re 1 1 Tt uTx vTy Txx Tyy Ec u 2y O 1 Re Re Pr Pr 4 For large Re BL assumptions the underlined terms drop out and the BL equations are obtained Also buoyancy terms free convection can be neglected except at very low velocities or over very large scales e g atmospheric BL Therefore y momentum equation reduces to p y 0 i e p p x t p x U t UU x From Euler equation for external flow 2D BL equations u x v y 0 ut uu x vu y U t UU x g x T T0 u yy Cv Tt uTx vTy kTyy u 2y Note 1 2 3 4 U x t 2 x 2 0 p x t impressed on BL by the external flow i e longitudinal or stream wise diffusion is 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 matching inlet Solution by marching y x X0 No slip No slip u x 0 t v x 0 t 0 T x 0 t Tw x t or qw x t kTy Initial condition u x y 0 T x y 0 known 5 w Inlet condition u x0 y t T x0 y 0 given at same x u x t U x t 0 Matching with outer flow T x t T x t e 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 Although BL equations have been written in Cartesian Coordinates they apply to curved surfaces provided R and x y are curvilinear coordinates measured along the normal to the surface respectively In such a system we would find under the BL assumptions py u 2 R p p 0 U 2 R or 6 p U 7 8 2 R therefore we require R 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 of body and inviscid theory gives p x and can go back to 2 for more accurate BL calculation including viscous inviscid interaction Separation and shear stress 1 At the wall u v 0 u yy px 1st derivative gives w w u y w w 0 separation 2nd derivative depends on px 7 Inflection point 7 4 Laminar Boundary Layer Similarity …
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