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%99δ, streamlines outside this shear layer will deflect an amount(the displacement thickness). Thus the streamline move outward from at *δHy =0=x to at . *δδ+=== HYy1xx = 1Conservation of mass: ∫∫CSρV•dA=0= ∫∫−HYUdyudy00ρρAssuming incompressible flow (constant density), this relation simplifies to () (∫∫ ∫−+=−+==YY YdyUuUYdyUuUudyUH00 0) Note: Y, we get the definition of boundary layer displacement thickness: *δ+= HdyUuY∫∞→⎟⎠⎞⎜⎝⎛−=0*1δ 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: *δδ δ* ∫∫=δδδ0*udyUdy (a) Discharge between and *δδof inviscid flow=actual discharge. ∫∫∫∫⎟⎠⎞⎜⎝⎛−=⇒=−δδδδδ0*0001*dyUuudyUdyUdy 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. *δ 7.1.3 Momentum thickness 2Conservation of x-momentum: () (∫∫∑−=−=HYxUdyUudyuDF00ρρ) dyuHUDDragY∫−==022ρρ Again assuming constant density and introducing: ∫=YdyUuH0 ()dxdyuUuDDragxwY∫∫=−==00τρ dyUuUuUDY⎟⎠⎞⎜⎝⎛−==∫→∞102θρ i.e. θ is the momentum thickness (a function of x only), an important measure of the drag. dxCxxxUDCxfD∫===02122θρ()dxdxCdxdCUCDfwfθρτ2212==⇒= 2fCdxd=θ dxdUwθρτ2= Assume that the velocity profiles had an approximately parabolic shape 3()⎟⎟⎠⎞⎜⎜⎝⎛−≈222,δδyyUyxu ()xyδ≤≤0 We get δθ152≈, δµµτUyuyw20≈∂∂== dxUvd 15≈δδ Integrate from 0 to x, assuming 0=δat 0=x, the leading edge: 2/121Re5.55.5xUxx=⎟⎠⎞⎜⎝⎛≈νδ 7.1.4. The laminar-Boundary Layer Equations 1. For two-dimensional incompressible flow: Continuity: 0=∂∂+∂∂yvxu x-momentum: ⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂−=∂∂+∂∂+∂∂22221yuxuxpyuvxuutuνρ y-momentum: ⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂−=∂∂+∂∂+∂∂22221yvxvypyvvxvutvνρ 2. Boundary-Layer Approximations Velocities: uv<< Rate of change: yuxu∂∂<<∂∂ yvxv∂∂<<∂∂ Reynolds number: 1Re >>=νUxx ⎟⎟⎠⎞⎜⎜⎝⎛∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂−=∂∂+∂∂+∂∂22221yvxvypyvvxvutvννρ small small small very small small 4So, 0=∂∂yp or only, using Bernoulli’s equation applied to the outer inviscid flow: ()xpp = dxdUUdxdpxpρ−==∂∂ For x-momentum, 2222yuxu∂∂<<∂∂ Then we get Prandtl’s two boundary layer equations for two-dimensional incompressible flow: 0=∂∂+∂∂yvxu ydxdUUyuvxuutu∂∂+≈∂∂+∂∂+∂∂τρ1 ⎪⎩⎪⎨⎧=∂∂−∂∂yuvuyuµρµτ'' Turbulent flow Laminar flow Notes: (1). and (txu ,)()txp , imposed on BL by the external flow. (2). The term 22x∂∂in the x-momentum equation has been neglected, i.e. Streamwise diffusion is neglected. (3). Due to (2), the equations are parabolic in . Physically this means x all downstream influences are lost other than that contained in the external flow. A marching solution is possible. (4). Boundary conditions: 5No slip: ()()0,0,,0,== txvtxu No slip y xX0 δ patching Solution by marching inlet Inlet condition: given at same (tyxu ,,0)0x Patching with outer flow: ()()txUtxu ,,,=∞ Initial condition: , ()0,, yxu()0,, yxv known. (5). Practical use of Boundary Layer theory For a given body geometry A. Inviscid theory gives()txp , B. Boundary layer theory givesÆ()tx,*δ, ()txw,τ, (tx,)θ, etc. and predicts separation if any. C. If separation present—no further informationÆmust use inviscid models or NS equations. D. If separation is absent, integration of ()xwτÆfrictional resistance; body+and inviscid theory givesÆ and can *δ(txp ,) go back to step B for more accurate BL calculation using viscous-inviscid iteration. 3. Dimensionless form of boundary layer equations Recall in 7.1.3, we have 2/121Re5.55.5xUxx=⎟⎠⎞⎜⎝⎛≈νδ The sizes of the various terms: and u)(unityxϑ= 6and v)(Re21−=ϑy Define dimensionless variables, all of which to be of order unity if Re is large Lxx =* Re*Lyy = LtUt =* Uuu =* Re*Uvv = 20*Upppρ−= 0****=∂∂+∂∂yvxu 2**2**********yuxpyuvxuutu∂∂+∂∂−=∂∂+∂∂+∂∂ **0yp∂∂−= 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, . 1000Re >x(2). If outer flow is decelerating (,0<dxdU 0>dxdp ), a point may be reached where wall shear stress approaches zero, the separation point. Beyond this point, the boundary-layer approximations are not accurate. 7.2. Laminar Boundary Layer 7.2.1. Similarity solutions (2D, steady, incompressible) 0=+yxvu yyxyxuUUvuuuν+=+ BCs: ()()00,0, == xvxu ()(xUxu)=∞, + inlet condition For Similarity ()()()⎟⎟⎠⎞⎜⎜⎝⎛=xgyFxUyxu , expect ()xgrelated to ()xδ 7Or in term of stream function ψ: yuψ=, xvψ−= And for similarity ()()()ηψfxgxU=,
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