Boundary layer thickness potential flow

Solution to the nondimensional axisymmetric stagnation-flow problem is plotted in Fig. 6.3. Since the viscous boundary layer merges asymptotically into the inviscid potential flow, there is not a distinct edge of the boundary layer. By convention, the boundary-layer thickness is defined as the point at which the radial velocity comes to 99% of its potential-flow value. From Fig. 6.3 it is apparent that the boundary-layer thickness S is approximately z 2. In addition to the boundary-layer thickness, a displacement thickness can be defined. The displacement thickness is the distance that the potential-flow field appears to be displaced from the surface due to the viscous boundary layer. If there were no viscous boundary layer (i.e., the inviscid flow persisted right to the surface), then the axial velocity profile would have a constant slope du/dz = —2. As shown in Fig. 6.3, projecting the constant axial-velocity slope to the surface obtains an intercept of u = 0 at approximately z = 0.55. Since the inviscid flow would have to come to zero velocity at the surface, z = 0.55 is the distance that the potential flow is displaced due to the viscous boundary layer. Otherwise, the potential flow is unaltered by the boundary layer. 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the solid surface is called the boundary layer. The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundary layer thickness is conventionally taken to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simplified by scaling arguments. Schlichting (Boundary Layer Theory, 8th ed., McGraw-Hill, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

We have seen how heat transfer and thus dry deposition of SO2 is reduced on large surfaces, due to the buildup of boundary layer thickness (which reduces the local gradients). However, there are economically important structural objects composed of many elements of small dimension which show the opposite effect. These include fence wire and fittings, towers made of structural shapes (pipe, angle iron, etc.), flagpoles, columns and the like. Haynie (11) considered different damage functions for different structural elements such as these, but only from the standpoint of their effect on the potential flow in the atmospheric boundary layer. The influence of shape and size act in addition to these effects, and could also change the velocity coefficients developed by Haynie (11), which were for turbulent flow. Fence wire, for example, as shown below, is more likely to have a laminar boundary layer. 

In the case of a retarded bubble surface, the velocity v exceeds that for a free surface -s/Re times since the boundary layer thickness differs slightly and the velocity gradient along the boundary layer is times greater. Thus, at a retarded surface a more substantial F can be expected. Mileva (1990) compares F and the pressing force (Dukhin Rylov 1971) for the case of a potential flow and concludes that Fl is negligible. But just from such a comparison it becomes evident that for retarded bubbles the pressing force is substantially less than F. ... 

The thin region near the body surface, which is known as the boundary layer, lends itself to relatively simple analysis by the very fact of its thinness relative to the dimensions of the body. A fundamental assumption of the boundary layer approximation is that the fluid immediately adjacent to the body surface is at rest relative to the body, an assumption that appears to be valid except for very low-pressure gases, when the mean free path of the gas molecules is large relative to the body [6]. Thus the hydrodynamic or velocity boundary layer 8 may be defined as the region in which the fluid velocity changes from its free-stream, or potential flow, value to zero at the body surface (Fig. 1.3). In reality there is no precise thickness to a boundary layer defined in this manner, since the velocity asymptotically approaches the free-stream value. In practice we simply imply that the boundary layer thickness is the distance in which most of the velocity change takes place. 

In the potential flow regime, where i> = sin 6> and Re is much larger, but the flow remains laminar, the dimensionless mass transfer boundary layer thickness is... 

Obtain analytical expressions for the liquid-phase mass transfer boundary layer thickness for (a) creeping flow, and (b) potential flow around a gas bubble of radius R. In which case will the boundary layer thickness be larger at the same relative position along the surface of the bubble ... 

Answer In the potential flow regime, the mass transfer boundary layer thickness at the equatorial position of the bubble is calculated as follows via equation (11-155) ... 

Reynolds number regime, on the order of 10 (in comparison to large aircraft, which may have Re values in excesses of 10 ). At low Re, viscous effects become dominant. In this regime, the aerodynamic efficiency of microrotors is significantly smaller than is seen for conventional rotorcraft on the macroscale [1]. Other aerodynamic consequences of low Re flow are that the thermal boundary layer thickness is large compared to the airfoil chord size, and can affect the potential flow region, and that the pressure will not necessarily be constant throughout the boundary layer. 

The potential production of sulfide depends on the biofilm thickness. If the flow velocity in a pressure main is over 0.8-1 ms-1, the corresponding biofilm is rather thin, typically 100-300 pm. However, high velocities also reduce the thickness of the diffusional boundary layer and the resistance against transport of substrates and products across the biofilm/water interphase. Totally, a high flow velocity will normally reduce the potential for sulfide formation. Furthermore, the flow conditions affect the air-water exchange processes, e.g., the emission of hydrogen sulfide (cf. Chapter 4). 

It is clear what the outer boundary value is, but it is not yet clear where (i.e., the value of z) it should be applied. The thickness of the viscous boundary layer is not known a-priori, so it is not known how far away from the surface the viscous layer extends and where the flow becomes fully inviscid. However, it is known that in the inviscid potential-flow region... 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

An obvious question that may occur to the reader is why the very simple method of integrating the viscous dissipation function has not been used earlier for calculation of the force on a solid body. The answer is that the method provides no real advantage except for the motion of a shear-stress-free bubble because the easily attained inviscid or potential-flow solution does not generally yield a correct first approximation to the dissipation. For the bubble, Vu T=0(l) everywhere to leading order, including the viscous boundary layer where the deviation from the inviscid solution yields only a correction of 0(Re x 2). For bodies with no-slip boundaries, on the other hand, Vu T is still 0(1) outside the boundary layer, but inside the boundary layer Vu T = O(Re). When integrated over the boundary layer, which is G(Re k2) in radial thickness, this produces an ()( / Re) contribution to the total dissipation,... 

For a potential flow around the cylinder, the approximate solution of equations (19.26) gives S - = 0.0625. However, at Re 1, a viscous boundary layer of thickness <5 r /V is formed near the cylinder surface, which causes a stronger curving of streamlines near the surface than we would expect for a potential flow. As a result, trajectories are pushed away from the surface. This means that the number of droplets reaching the surface in a unit time decreases. We can ac-... 

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