058:0160 Chapter 7 Professor Fred Stern Fall 2009 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 = %99δ(where, %99δis the value of y at u = 0.99U). Streamlines outside %99δ will deflect an amount*δ(the displacement thickness). Thus the streamlines move outward fromHy = at 0=x to *δδ+=== HYyat 1xx =.058:0160 Chapter 7 Professor Fred Stern Fall 2009 2 Conservation of mass: CSVndAρ•∫=0=00HHUdy udyδρρ∗+−+∫∫ Assuming incompressible flow (constant density), this relation simplifies to () ()∫∫ ∫−+=−+==YY YdyUuUYdyUuUudyUH00 0 Note: *δ+=HY, we get the definition of displacement thickness: dyUuY∫⎟⎠⎞⎜⎝⎛−=0*1δ *δis 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: ∫∫=δδδ0*udyUdy Flowrate between *δand δof inviscid flow=actual flowrate, i.e., inviscid flow rate about displacement body = equivalent viscous flow rate about actual body ∫∫∫∫⎟⎠⎞⎜⎝⎛−=⇒=−δδδδδ0*0001*dyUuudyUdyUdy w/o BL - displacement effect=actual discharge For 3D flow, in addition it must also be explicitly required that *δis a stream surface of the inviscid flow continued from outside of the BL. δ* Lam=δ/3 δ δ* Turb=δ/8 Inviscid flow about δ* body058:0160 Chapter 7 Professor Fred Stern Fall 2009 3 Conservation of x-momentum: () ()00HYxCSFD uV ndA U Udy u udyρρρ=− = • =− +∑∫∫∫ dyuHUDDragY∫−==022ρρ= Fluid force on plate = - Plate force on CV (fluid) Again assuming constant density and using continuity: ∫=YdyUuH0 dxdyuUdyuUDxwYY∫∫∫=−=00022/τρ dyUuUuUDY⎟⎠⎞⎜⎝⎛−==∫102θρ where, θ is the momentum thickness (a function of x only), an important measure of the drag. dxCxxxUDCxfD∫===02122θρ()dxdxCdxdCUCDfwfθρτ2212==⇒= 2fCdxd=θ dxdUwθρτ2= Per unit span Special case 2D momentum integral equation for px = 0058:0160 Chapter 7 Professor Fred Stern Fall 2009 4 Simple velocity profile approximations: )//2(22δδyyUu −= u(0) = 0 no slip u(δ) = U matching with outer flow uy(δ)=0 Use velocity profile to get Cf(δ) and θ(δ) and then integrate momentum integral equation to get δ(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========∴===⇒=θδϑδρμδρμδδδθρδμδμτ 10% error, cf. Blasius058:0160 Chapter 7 Professor Fred Stern Fall 2009 5 7.3. Boundary layer approximations, equations and comments 2D NS, ρ=constant, neglect g )(1)(10yyxxyxtyyxxyxtyxvvypvvuvvuuxpvuuuuvu++∂∂−=++++∂∂−=++=+ϑρϑρ Introduce non-dimensional variables that includes scales such that all variables are of O(1): ϑρϑ/ReRe//Re/20******ULUpppUvUuuLtUtLyyLxx=−====== u = v = 0 x y U, ρ,μ058:0160 Chapter 7 Professor Fred Stern Fall 2009 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 tpUUUρ==⇒=− + 2D BL equations: yyxtyxtyxuUUUvuuuuvuϑ++=++=+)(;0 Note: (1) U(x,t), p(x,t) impressed on BL by the external flow. (2) 022=∂∂x: 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 2009 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 2009 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 2yupRρ= Assume u is a linear function of y: uUyδ= 2222() (0)3dp U ydy RUppRρδρδδ=−∝ Or 2;3pURδρΔ∝ therefore, we require δ << R058:0160 Chapter 7 Professor Fred Stern Fall 2009 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 mode, or NS equation. (d) If separation is absent, integration of τw(x) → frictional resistance body + δ* , 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 → 1yyxupμ= 1st derivative u gives τw
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