058:0160 Chapter 7 Professor Fred Stern Fall 2010 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 velocityˆUi. Boundary-layer thickness arbitrarily defined by y = %99(where, %99is the value of y at u = 0.99U). Streamlines outside %99 will deflect an amount*(the displacement thickness). Thus the streamlines move outward fromHy at 0x to * HYyat 1xx .058:0160 Chapter 7 Professor Fred Stern Fall 2010 2 Conservation of mass: CSVndA-=0=00HHUdy udy Assuming incompressible flow (constant density), this relation simplifies to YY YdyUuUYdyUuUudyUH00 0 Note: *HY, we get the definition of displacement thickness: dyUuY0*1 *( 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: 0*udyUdy Flowrate between *and of inviscid flow=actual flowrate, i.e., inviscid flow rate about displacement body = equivalent viscous flow rate about actual body 0*0001*dyUuudyUdyUdy 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. δ* Lam=/3 δ δ* Turb=/8 Inviscid flow about δ* body058:0160 Chapter 7 Professor Fred Stern Fall 2010 3 Conservation of x-momentum: 00HYxCSFD uV ndA U Udy u udy - dyuHUDDragY022= Fluid force on plate = - Plate force on CV (fluid) Again assuming constant density and using continuity: YdyUuH0 dxdyuUdyuUDxwYY00022/ dyUuUuUDY102 where, is the momentum thickness (a function of x only), an important measure of the drag. dxCxxxUDCxfD02122dxdxCdxdCUCDfwf2212 2fCdxd dxdUw2 Per unit span Special case 2D momentum integral equation for px = 0058:0160 Chapter 7 Professor Fred Stern Fall 2010 4 Simple velocity profile approximations: )//2(22yyUu u(0) = 0 no slip u(δ) = U matching with outer flow uy(δ)=0 Use velocity profile to get Cf() and () and then integrate momentum integral equation to get (Rex) δ* = δ/3 θ = 2δ/15 H= δ*/θ= 5/2 )(2Re/46.1Re/73.0/Re/83.1/;/ReRe/5.5/3015);15/2(222/1/2/22/12/12/1*2/122LCCxxUxxUdxUdxddxddxdUUCUfLDxxxxfw 10% error, cf. Blasius058:0160 Chapter 7 Professor Fred Stern Fall 2010 5 7.3. Boundary layer approximations, equations and comments 2D NS, =constant, neglect g )(1)(10yyxxyxtyyxxyxtyxvvypvvuvvuuxpvuuuuvu Introduce non-dimensional variables that includes scales such that all variables are of O(1): /ReRe//Re/20******ULUpppUvUuuLtUtLyyLxx u = v = 0 x y U, ,058:0160 Chapter 7 Professor Fred Stern Fall 2010 6 The NS equations become (drop *) 201Re111()Re Re Rexyt x y x xx yyt x y y xx yyuvuuu vu p u uvuv vv p v v For large Re (BL assumptions) the underlined terms drop out and the BL equations are obtained. Therefore, y-momentum equation reduces to 0.. ( ,)()yxtxpie p p x tpUUU 2D BL equations: yyxtyxtyxuUUUvuuuuvu)(;0 Note: (1) U(x,t), p(x,t) impressed on BL by the external flow. (2) 022x: i.e. longitudinal (or stream-wise) 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 external flow. A marching solution is possible. (4) Boundary conditions From Euler/Bernoulli equation for external flow058:0160 Chapter 7 Professor Fred Stern Fall 2010 7 No slip: 0,0,,0, txvtxu Initial condition: 0,, yxu known Inlet condition: tyxu ,,0given at 0x Matching with outer flow: ,, ,ux t Uxt (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 y xX0 inletSolution by marching matching No slip δ058:0160 Chapter 7 Professor Fred Stern Fall 2010 8 Although BL equations have been written in Cartesian Coordinates, they apply to curved surfaces provided δ << 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 2yupR Assume u is a linear function of y: uUy 2222() (0)3dp U ydy RUppR Or 2;3pUR therefore, we require δ << R058:0160 Chapter 7 Professor Fred Stern Fall 2010 9 (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 → δ*(x), τw(x), θ(x),etc. and predicts separation if any (c) If separation present then no further information → must use inviscid models, BL equation in inverse
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