Chapter 7 Flow Past Immersed Bodies7.1. 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%99, streamlines outside this shear layer will deflect an amount*(the displacement thickness). Thus the streamline move outward fromHy at 0x to * HYyat 1xx .1Conservation of mass:CSρV-dA=0=HYUdyudy00Assuming incompressible flow (constant density), this relationsimplifies to Y Y YdyUuUYdyUuUudyUH0 0 0Note: *HY, we get the definition of boundary layer displacement thickness: dyUuY0*1 *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:0*udyUdy (a) Discharge between *and of inviscid flow=actual discharge.0*0001*dyUuudyUdyUdy w/o BL - displacement effect=actual dischargeFor 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.7.1.3 Momentum thickness Conservation of x-momentum: HYxUdyUudyuDF00dyuHUDDragY022 Again assuming constant density and introducing: YdyUuH0δ*δ2 dxdyuUuDDragxwY00dyUuUuUDY102i.e. is the momentum thickness (a function of x only), an important measure of the drag.dxCxxxUDCxfD02122 dxdxCdxdCUCDfwf2212 2fCdxd dxdUw2 Assume that the velocity profiles had an approximately parabolic shape 222,yyUyxu xy0We get 152, Uyuyw20dxUvd 15Integrate from 0 to x, assuming 0at 0x, the leading edge:2/121Re5.55.5xUxx7.1.4. The laminar-Boundary Layer Equations 1. For two-dimensional incompressible flow: Continuity: 0yvxu x-momentum: 22221yuxuxpyuvxuutu y-momentum: 22221yvxvypyvvxvutv32. Boundary-Layer Approximations Velocities: uv Rate of change: yuxu yvxv Reynolds number: 1Re Uxx 22221yvxvypyvvxvutv small small small very small small So, 0yp or xpp only, using Bernoulli’s equation applied to the outer inviscid flow: dxdUUdxdpxp For x-momentum, 2222yuxuThen we get Prandtl’s two boundary layer equations for two-dimensional incompressible flow:0yvxuydxdUUyuvxuutu1 yuvuyu''Notes: (1). txu , and txp , imposed on BL by the external flow.(2). The term 22xin the x-momentum equation has been neglected, i.e. Streamwise 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 the external flow. A marching solution is possible.(4). Boundary conditions:4Laminar flowTurbulent flowNo slip: 0,0,,0, txvtxu Inlet condition: tyxu ,,0 given at same0x 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 gives tx,*, txw,, tx,, etc. and predicts separation if any. C. If separation present—no further informationmust use inviscid models or NS equations. D. If separation is absent, integration of xwfrictional resistance; body+*and inviscid theory gives txp , and can go back to step B for more accurate BL calculation using viscous-inviscid iteration.3. Dimensionless form of boundary layer equationsRecall in 7.1.3, we have 2/121Re5.55.5xUxxThe sizes of the various terms: u and )(unityx v and )(Re21yDefine dimensionless variables, all of which to be of order unity if Re is large5yxX0inletSolution by marchingpatchingNo slipδLxx * Re*Lyy LtUt * Uuu * Re*Uvv 20*Uppp0****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 (,0dxdU0dxdp), 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
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