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BOUNDARY LAYERRef: Frank M. White, Fluid Mechanics, McGraw-Hill, Inc, 4th edition1Boundary Layer

Ludwig Prandtl introduced concept of Boundary Layer in 1904 for the first time.

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For high Reynolds Number flow, two length scales1. Far from surface, viscous forces are unimportant and inertial forces dominate.2. Near the surface, viscous forces are comparable to inertial forces3Examples boundary layers (layers where there is velocity gradient) in Chemical Engineering

1. Development of fully developed laminar flow (a) and turbulent flow (b) in circular pipes

Higher flow induces higher du/dy close to the wall higher shear stress higher pump power in case of power of pump.

Momentum boundary layer of flow in pipes5

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Boundary layer in front part of the particle2. Development of boundary layer around spherical particles when Re is increasedSeparation point of boundary layer

Q hot Q coldThTi,wallTo,wallTcRegion I : Hot Liquid-Solid ConvectionNEWTONS LAW OF CCOLING

Region II : Conduction Across Copper WallFOURIERS LAW

Region III: Solid Cold Liquid ConvectionNEWTONS LAW OF COOLING

3. Thermal boundary layer in Heat Transfer

8Heat moves from hot fluid to a surface by convection through thermal BL, by conduction through the wall, and then from the surface to the cold fluid by convection through thermal BL

The ratio of diffusivity of momentum to the diffusivity of heat. Also thickness ratio of momentum-boundary-layer and thermal-boundary-layer is Prandtl numberFor convection-controlled, higher fluid velocity higher du/dy thinner momentum boundary layer lower heat transfer resistance higher convective heat transfer (h) larger heat convectionFor diffusion-controlled, when heat diffusion > momentum diffusion (k-controlled), thickness of the thermal BL > the momentum BL.

Pr =namcpkConvection- controlled, momentum BL dependentDiffusion-controlled, momentum or thermal BL controlled7.1. Reynolds-Number and Geometry EffectsThe technique of boundary-layer (BL) analysis can be used to compute viscous effects near solid walls and to patch (to join) these onto the outer inviscid motion. In Fig. 7.1 a uniform stream U moves parallel to a sharp flat plate of length L. If the Reynolds number UL/ is low (Fig. 7.1a), the viscous region is very broad and extends far ahead and to the sides of the plate due to retardation of the oncoming stream greatly by the plate.

Fig. 7.1 Comparison of flow past a sharp flat plate at low and highReynolds numbers: (a) laminar, low-Re flow; (b) high-Re flowAt a high-Reynolds-number flow (Fig. 7.1b) the viscous layers, either laminar or turbulent, are very thin, thinner even than the drawing shows.We define the boundary layer thickness as the locus of points where the velocity u parallel to the plate reaches 99 % of the external velocity U. As we shall see in Sec. 7.4, the accepted formulas for flat-plate flow are

where Rex = Ux/ is called the local Reynolds number of the flow along the plate surface. The turbulent-flow formula applies for Rex > approximately 106.

The blanks indicate that the formula is not applicable. In all cases these boundary layers are so thin that their displacement effect on the outer inviscid layer is negligible.Thus the pressure distribution along the plate can be computed from inviscid theory as if the boundary layer were not even there.

This external pressure field acts as a forcing function in the momentum equation along the surface.For slender bodies, such as plates and airfoils parallel to the oncoming stream, the assumption of negligible interaction between the boundary layer and the outer pressure is an excellent approximation because pressure along plate is constant.For a blunt-body (bluff body) flow, however, there is a pressure distribution over the surface body, so the pressure must be taken into account.

Figure 7.2 shows two sketches of flow past a two- or three-dimensional blunt body. In the idealized sketch (7.2a), there is a thin film of boundary layer about the body and a narrow sheet of viscous wake in the rear.In a actual flow (Fig. 7.2b), the boundary layer is thin on the front, or windward, side of the body, where the pressure decreases along the surface (favorable pressure gradient).

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Fig. 7.2 Illustration of the strong interaction between viscous and inviscid regions in the rear of blunt bodyflow: (a) idealized and definitely false picture of blunt-body flow (according to Bernoullis law); (b) actual picture of blunt body flow.But in the rear the boundary layer encounters increasing pressure (adverse pressure gradient) and breaks off, or separates, into a broad, pulsating wake.The mainstream is deflected by this wake, so that the external flow is quite different from the prediction from inviscid theory with the addition of a thin boundary layer

7.2. von Karmans Momentum-Integral EstimatesA boundary layer of unknown thickness grows along the sharp flat plate in Fig. 7.3. The no-slip wall condition retards the flow, making it into a rounded profile u(y), which merges into the external velocity U constant at a thickness y (x).By utilizing the control volume of Fig. 7.3, we found (without making any assumptions about laminar versus turbulent flow) that the drag force on the plate is given by the momentum integral across the exit plane due to the change of velocity from U to u(x,y)

Fig. 7.3 Growth of a boundary layer on a flat plate.Drag force = V x mass rate = ((U u) x b (width) x u x y). This represents how much the momentum is lost due to friction where b is the plate width into the paper and the integration is carried out along a vertical plane at a constant x (von Krmn, 1921).Imagine that the flow remains at uniform velocity U, but the surface of the plate is moved upwards to consider momentum flux reduction = momentum flux loss boundary layer actually does. Then, the drag force

Momentum thickness is defined as the loss of momentum per unit width b divided by U2 up to x concerned.

By comparing this with Eq. (7.4) Krmn arrived at what is now called the momentum integral relation for flat-plate boundary-layer flow (w )

It is valid for either laminar or turbulent flat-plate flow.

(von Krmn, 1921).

Vertical variablehorisontal variable

Variables in momentum integral relation (7.5)Local, vertical- variableLocal, horisontal variableEq. 7.9 is the desired thickness estimate. It is all approximate, of course, part of Krmns momentum-integral theory [7], but it is startlingly accurate, being only 10 percent higher than the known exact solution for laminar flat-plate flow, which we gave as Eq. (7.1a).By combining Eqs. (7.9) and (7.7) we also obtain a shear-stress estimate along the plate

Again this estimate, in spite of the crudeness of the profile assumption (7.6) is only 10 % > the known exact laminar-plate-flow solution cf = 0.664/Rex1/2, treated in Sec. 7.4. The dimensionless quantity cf, called the skin-friction coefficient, is analogous to the friction factor f in ducts.A boundary layer can be judged as thin if, say, the ratio /x < about 0.1. /x = 0.1 = 5.0/Rex1/2 or Rex > 2500. For Rex < 2500 we can estimate that boundary-layer theory fails because the thick layer has a significant effect on the outer inviscid flow (thickness creates pressure distribution across the boundary layer).The upper limit on Rex for laminar flow is about 3 x 106, where measurements on a smooth flat plate [8] show that the flow undergoes transition to a turbulent boundary layer. From 3 x 106 upward the turbulent Reynolds number may be arbitrarily large, and a practical limit at present is 5 x 109 for oil supertankers.Displacement ThicknessAnother interesting effect of a boundary layer is its small but finite displacement of the outer streamlines. As shown in Fig. 7.4, outer streamlines must deflect outward a distance *(x) to satisfy conservation of mass between the inlet and outlet as a result of fluid entrainment from fluid flow to boundary layer.

The quantity * is called the displacement thickness of the boundary layer. To relate it to u(y), cancel and b from Eq. (7.11), evaluate the left integral, and add and subtract U from the right integrand:

Bernoulli eq. appliesNavier Stokes eq. appliesFig. 7.4 Displacement effect of a boundary layer. Fluid entrainment occurs from free fluid flow to the boundary layer, so mass rate at 0 = mass rate at 1Hypothetical layer attributed to fluid entrainment

1Fluid entrainmentDisplacement thickness is defined as the loss of mass from main flow per unit width divided by U2 due to the presence of the growing boundary layer. Imagine that the flow remains at uniform velocity U, but the surface of the plate (wall) is moved upwards (displaced) * to consider mass flux reduction = mass flux loss of the main flow the BL generates.

Local, vertical variableIntroducing von Karmans profile approximation (7.6) into (7.12), we obtain by integration the approximate result

These estimates are only 6% away from the exact solutions for laminar flat-plate flow given in Sec. 7.4: * = 0.344 = 1.721x/Rex1/2. Since * 0dp/dx = 0dU/dx = 0dp/dx > 0dU/dx < 0dp/dx > 0dU/dx < 0As if all curves were pushed upwardEXAMPLE 7.3A sharp flat plate with L =1 m and b = 3 m is immersed parallel to a stream of velocity 2 m/s.Find the drag on one side of the plate, and at the trailing edge find the thicknesses , *, and for (a) air, =1.23 kg/m3 and =1.46x10-5 m2/s, and (b) water, =1000 kg/m3 and =1.02 x 10-6 m2/s.Part a.

Part b.

The drag is 215 x more for water in spite of the higher Reynolds number and lower drag coefficient because water is 57 x more viscous and 813 x denser than air. From Eq. (7.26), in laminar flow, it should have (57)1/2(813)1/2 = 7.53(28.5) = 215 x more drag.

The water layer is 3.8 times thinner than the air layer, which reflects the square root of the 14.3ratio of air to water kinematic viscosity. (14.3 = 3.8)