Boundary layer thickness flat plate

Figure 1.25 shows the boundary layer that develops over a flat plate placed in, and aligned parallel to, the fluid having a uniform velocity upstream of the plate. Flow over the wall of a pipe or tube is similar but eventually the boundary layer reaches the centre-line. Although most of the change in the velocity component vx parallel to the wall takes place over a short distance from the wall, it does continue to rise and tends gradually to the value vx in the fluid distant from the wall (the free stream). Consequently, if a boundary layer thickness is to be defined it has to be done in some arbitrary but useful way. The normal definition of the boundary layer thickness is that it is the distance from the solid boundary to the location where vx has risen to 99 per cent of the free stream velocity v . The locus of such points is shown in Figure 1.25. It should be appreciated that this is a time averaged distance the thickness of the boundary layer fluctuates owing to the velocity fluctuations.

Momentum boundary layer calculations are useful to estimate the skin friction on a number of objects, such as on a ship hull, airplane fuselage and wings, a water surface, and a terrestrial surface. Once we know the boundary layer thickness, occurring where the velocity is 99% of the free-stream velocity, skin friction coefficient and the skin friction drag on the solid surface can be calculated. Estimate the laminar boundary layer thickness of a 1-m-long, thin flat plate moving through a calm atmosphere at 20 m/s. 

Laminar boundary layer theory assumes that a uniform flow (V = constant) approaches a flat plate. A laminar flow region develops near the plate where the thickness of the laminar boundary layer increases with thickness along the plate, as developed in Example 4.2. If we assign 5 to be the boundary layer thickness, or the distance from the plate where the velocity is equal to 0.99 times the velocity that approached the plate, and 5c to be the concentration boundary layer thickness, then we can see that both 5 and 5c are functions of distance, x, from the leading edge, as shown in Figure 8.11. 

If a fluid such as air flows over a flat plate placed with its surface parallel to the stream, particles in the vicinity of the surface are slowed down by viscous forces. Fluid particles adjacent to the surface stick to it and have zero velocity relative to the boundary. Other fluid particles are retarded as a result of sliding over the immobilised particles. The effects of viscous forces originating at the boundary extend for a certain distance (5, the boundary layer thickness). The effects of viscous forces originating at the boundary are not extensive and the velocity soon approaches free stream velocity. 

For example, the inviscid solution for flow over a flat plate is simply that the velocity is constant everywhere and equal to the velocity in the undisturbed flow ahead of the plate, say wi. In calculating the boundary layer on a flat plate, therefore, the outer boundary condition is that u must tend to u at large v. The terr large y is meant to imply outside the boundary layer , the boundary layer thickness. S, being by assumption small. 

It may be recalled that it was deduced from the similarity solution for flow over a flat plate that Six) = 5/jRex. The difference between the value of the coefficient in this equation, i.e., 5, and the value in Eq. (3.136), i.e., 4.64, has no real significance since, in deriving the similarity solution result, it was arbitrarily assumed that the boundary layer thickness was the distance from the wall at which u became equal to 0.99 m. 

Air at 300 K Ad 1 atm flows at a velocity of 2 m/s along a flat plate which has a length of 0.2 m. The plate is kept at a temperature of 330 K. Plot the variations of the velocity and thermal boundary layer thicknesses along the plate. 

Consider transition in the boundary layer flow over a flat plate. Using the expression for the thickness of a laminar boundary layer on a flat plate given in Chapter 3, find the value of the Reynolds number based on the boundary layer thickness at which transition begins. 

A vertical flat plate is maintained at a uniform surface temperature and is exposed to air at standard ambient pressure. At a distance of 10 cm from the leading edge of the plate the boundary layer thickness is 2 cm. Estimate the thickness of the boundary layer at a distance of 25 cm from the leading edge. Assume a laminar boundary layer flow. 

A low-speed wind tunnel is to be designed to study boundary layers up to Re., = 107 with air at 1 atm and 25°C. The maximum flow velocity which can be expected from an existing fan system is 30 m/s. How long must the flat-plate test-section be to produce the required Reynolds numbers What will the maximum boundary-layer thickness be under these conditions What would the maximum boundary-layer thicknesses be for flow velocities at 7 and 12 m/s ... 

To show how one might proceed to analyze a new problem to obtain an important functional relationship from the differential equations, consider the problem of determining the hydrodynamic-boundary-layer thickness for flow over a flat plate. This problem was solved in Chap. 5, but we now wish to make an order-of-magnitude analysis of the differential equations to obtain the functional form of the solution. The momentum equation... 

Let us first consider the simple flat plate with a liquid metal flowing across it. The Prandtl number for liquid metals is very low, of the order of 0.01. so that the thermal-boundary-layer thickness should be substantially larger than the hydrodynamic-boundary-layer-thickness. The situation results from the high values of thermal conductivity for liquid metals and is depicted in Fig. 6-15. Since the ratio of 8/8, is small, the velocity profile has a very blunt shape over most of the thermal boundary layer. As a first approximation, then, we might assume a slug-flow model for calculation of the heat transfer i.e., we take... 

For vertical surfaces, the Nusselt and Grashof numbers are formed with L, the height of the surface as the characteristic dimension. If the boundary-layer thickness is not large compared with the diameter of the cylinder, the heat transfer may be calculated with the same relations used for vertical plates. The general criterion is that a vertical cylinder may be treated as a vertical flat plate [13] when... 

Now consider the flat plate shown in Fig. 12-3. The plate surface is maintained at the constant temperature Tw, the free-stream temperature is 7U, and the thermal-boundary-layer thickness is designated by the conventional symbol 5,. To simplify the analysis, we consider low-speed incompressible flow so that the viscous-heating effects are negligible. The integral energy equation then becomes... 

Consider the following flow field over a flat plate that is excited simultaneously at the wall y = 0 and at the free stream y = Y) as shown in Fig.2.24, where Y is significantly larger than the boundary layer thickness. At the wall, a time-periodic blowing-suction device is placed at x = xq defined in a coordinate system fixed at the leading edge of the plate. The circular frequency of excitation of the wall device is lvq such that the transverse velocity oscillation at the wall is set up as. 

Instead of the numerical factor 4.0 in Equation 7.10, hydrodynamic theory predicts a factor near 6.0 for the effective boundary layer thickness adjacent to a flat plate (both numbers increase about 3% per 10°C Schlichting and Gersten, 2003). However, wind tunnel measurements under an appropriate turbulence intensity, as well as field measurements, indicate that 4.0 is more suitable for leaves. This divergence from theory relates to the relatively small size of leaves, their irregular shape, leaf curl, leaf flutter, and, most important, the high turbulence intensity under field conditions. Moreover, the dependency of 6bl on /° 5, which applies to large flat surfaces, does... 

A laminar boundary layer develops on the upwind side of a cylinder (Fig. 7-8). This layer is analogous to the laminar sublayer for flat plates (Fig. 7-6), and air movements in it can be described analytically. On the downwind side of the cylinder, the airflow becomes turbulent, can be opposite in direction to the wind, and in general is quite difficult to analyze. Nevertheless, an effective boundary layer thickness can be estimated for the whole cylinder (to avoid end effects, the cylinder is assumed to be infinitely long). For turbulence intensities appropriate to field conditions, in mm can be represented as follows for a cylinder ... 

Figure 7-9. Relationship between Reynolds number (Re) and Nusselt number (Nu) for flat leaves under field conditions or for metal plates shaped like leaves and placed in wind tunnels (turbulence intensity of about 0.4 in all cases). Air temperatures in the boundary layer were 20 to 25°C. The curve, which represents the best fit for the data, indicates that Nu = 0.97Re°5 (see text for interpretation in terms of boundary layer thickness).

Note that /i, is proportional to Re and thus to. v- - for laminar flow. Therefore, is infinite at the leading edge (jc = 0) and decreases by a factor of.r in the flow direction. The variation of the boundary layer thickness 5 and the friction and heat transfer coefficients along an isothermal flat plate are shown in Fig. 7-9. The local friction and heat transfer coefficients are higher in... 

The thickness of the velocity boundary layer is normally defined as the distance from the solid body to the fluid layer at which the flow velocity reaches 99% of the free stream velocity, as illustrated in Figure 2.17. For a flat-plate body emerging in an incompressible and laminar fluid, the boundary layer thickness is given by... 

In general, (10-89) requires an initial profile atx = 0 (corresponding to the most upstream point in the boundary layer). Evidently, if a similarity solution does exist, the boundary condition (10-97) must represent this initial condition, as well as the boundary condition for Y oo. This implies that g(0) = 0 that is, the boundary-layer thickness must go to zero as x 0. This can occur only for a body that has a pointed leading edge (such as the flat plate). 

Problem 10-4. Boundary Layer on a Flat Plate in an Accelerating Flow, Uoo = Ax. Consider flow past a flat plate in the throat of a 2D channel as depicted in the figure. If the free-stream velocity is given by = ax, where x is the distance from the leading edge and X is a constant, show that the flow in the boundary layer on the plate is governed by an ODE. Solve for the streamfunction numerically. How does the boundary-layer thickness grow with x How does the shear stress vary with x ... 

We also recall that the boundary layer thickness for the flat plate viscous layer is... 

In addition to the boundary-layer thickness 5, two other thicknesses occur frequently in the boundary-layer literature the displacement thickness S and the momentum thickness 6. To see the meaning of the displaicement thickness, consider the streamlines for the laminar boundary layer on a flat plate, as sketched in Fig. 11.5. 

By assuming that the velocity was a function of y yj xv)X Blasius was able to solve Prandtl s equations for the steady-flow laminar boundary layer on a flat plate. He found that the laminar boundary-layer thickness is proportional to the square root of the length down the plate. 

For boundary layers on curved surfaces, the pressure will change with distance. This greatly complicates the solution of the boundary-layer equations compared with that on a flat plate (in which dPIdx was zero), and so very few exact solutions are known for such boundary layers. Some estimate of the behavior of such boundary layers is given by several methods. To illustrate, we apply them to the laminar boundary layer on a flat plate, where we can compare the results with Blasius exact solution. These methods begin by assuming a velocity profile of the form V tV where S is the boundary-layer thickness. 

A polymer solution (density 1022 kg/m ) flows over the surface of a flat plate at a free stream velocity of 2.25 m/s. Estimate the laminar boundary layer thickness and surface shear stress at a point 300 mm downstream from tlie leading edge of the plate. Determine the total drag force on the plate from the leading edge to this point. What is the effect of doubling the free stream velocity ... 

The velocity distribution for flow over a large, smooth, flat plate is shown schematically in Figure 7.17. The boundary layer thickness S is conventionally (arbitrarily) defined as the distance from the surface at which the velocity is 0 99 of the bulk velocity. 

Many additional studies have been conducted with the boundary layer model by taking into account the variation of physical properties with composition (or temperature) and by relaxing the assumption that Vy = 0 at y = 0 when mass transfer is occurring. Under conditions of high mass transfer rates one finds that mass transfer to the plate decreases the thickness of the mass transfer boundary layer while a mass flux away from the wall increases the boundary layer thickness The analogous problem of uniform flux at the plate has also been solved. Skelland describes a number of additional mass transfer boundary layer problems such as developing hydrodynamic and mass transfin- profiles in the entrance region of parallel flat plates and round tubes. 

In fluid flow, one may consider a flat plate over which the fluid is flowing (Figure 8.1). Then fluid particles making contact with the surface are presumed to have zero velocity. These particles further act to retard the motion of particles in the adjacent fluid layers until, at some distance, y = 8 from the solid surface, the fluid velocity is no longer influenced by the surface. The quantity 8 is the boundary layer thickness. This retardation of fluid motion is associated with shear stresses, r, acting in planes parallel to the fluid velocity (see Figure 8.1). Also, as one moves away from the solid surface in the y-direction, the x-component of the fluid velocity, , increases until it approaches the free stream value u. This is the velocity boundary layer and is expected to develop whenever there is fluid flow over a surface. 

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