Inviscid layers

Early stability analyses employed the classical Kelvin-Helmholtz (K-H) theory for two inviscid layers (Kordyban and Ranov [33], Kordyban [34], Wallis and Dobson [35]). However, in referring to gas-liquid flows, pyp C 1, and assuming that the interfacial disturbance velocity equals the (slower) liquid layer velocity, the liquid destabilizing contribution has been degenerated. This results in a rather simple Bernoulli-type transitional criteria, whereby the suction forces in the gas-... 

The general structure of stability Equation 18 remains unchanged when different quasi-steady models are applied for the various shear stresses terms. Moreover, even when the viscous effects are completely ignored, resorting to an inviscid K-H stability type of analysis, the structure of the resulting stability condition. Equation 18, is still maintained while Equation 19 for attains different expression. For instance, the long wave K-H stability analysis on two inviscid layers (rectangular channel) yields ... 

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 sohd 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 boundaiy layer thickness is conventionally t en 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 simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

Hiemenz (in 1911) first recognized that the relatively simple analysis for the inviscid flow approaching a stagnation plane could be extended to include a viscous boundary layer [429]. An essential feature of the Hiemenz analysis is that the inviscid flow is relatively unaffected by the viscous interactions near the surface. As far as the inviscid flow is concerned, the thin viscous boundary layer changes the apparent position of the surface. Other than that, the inviscid flow is essentially unperturbed. Thus knowledge of the inviscid-flow solution, which is quite simple, provides boundary conditions for the viscous boundary layer. The inviscid and viscous behavior can be knitted together in a way that reduces the Navier-Stokes equations to a system of ordinary differential equations. 

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... 

The stagnation-surface boundary values are u =0 and V = 0. At the outer edge of the boundary layer, the inviscid flow provides that V = 1. The extent of the domain (i.e., Zend) still needs to be determined, but it needs only to be done once and for all in the nondimensional setting. The nondimensional axial velocity gradient in the inviscid region is du/dz = -2. 

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. 

The axisymmetric Hiemenz solution assumes an inviscid outer flow field. The outer flow, which the inner viscous boundary layer sees, has a constant scaled radial velocity V = 1 and an outer axial velocity that decreases linearly with the distance from the stagnation surface. 

Fig. 6.4 Streamlines for two axisymmetric Hiemenz stagnation flow situations having different outer velocity gradients, one at a = 1 s 1 and the other at a = 5 s—. Both cases are for air flow at atmospheric pressure and T = 300 K. The streamlines are plotted to an axial height of 3 cm and a radius of 10 cm. However, the solution itself has infinite radial extent in both the axial and radial directions. In both cases the streamlines are separated by 2jt A l = 2.0 x 10-5 kg/s. The shape of the scaled radial velocities V = v/r is plotted on the right of the figures. The maximum value of the scaled radial velocity is Vmax = a/2. Even though streamlines show curvature everywhere, the viscous region is confined to the boundary layer defined by the region of V variation. Outside of this region the flow behaves as though it is inviscid.

The traditional view of stagnation flow was presented in the previous sections, that is, a semi-infinite inviscid potential flow interacting with a thin viscous boundary layer near the stagnation surface. In that case there is no physically based characteristic length scale. 

Figure 6.7 shows the axial and radial velocity profiles for several Reynolds numbers. Clearly, as the Reynolds number increases, the boundary layer, as evidenced by the V profile, moves closer and closer to the surface. Perhaps less clear is the fact that the upper regions of the flow behaves as if the fluid were inviscid. Whenever dV /dz is a constant, as it is in the upper areas of the gap, the only viscous term (i.e second-derivative term) in Eq. 6.81 vanishes since d2V/dx2 = 0. Therefore the remaining equations represent only inviscid flow. Recall that the vorticity anywhere within the gap region is... 

This behavior stems from the fact that there is an essentially inviscid region between the inlet manifold and viscous boundary layer near the surface. As the Reynolds number increases, the viscous layer becomes thinner. As the Reynolds number decreases below around 10, the viscous layer fills the entire gap. For sufficiently low Reynolds number, the fluid flow becomes negligible and the heat transfer is characterized by thermal conduction. In that limit, Nu = 1. 

Fig. 6.10 Nondimensional axial-velocity gradients and scaled radial velocities at the viscous boundary-layer edge as a function of Reynolds number in a finite-gap stagnation flow. The Prandtl number is Pr = 0.7 and the flow is isothermal in all cases. The outer edge of the boundary layer is defined in two ways. One is the z position of maximum V velocity and the other is the z. at which T — 0.01. As Re - oo, du/dz - —2 and V — 1, which are the values in the inviscid semi-infinite stagnation flow regions.

Figure 6.13 illustrates the streamline patterns and velocity profiles for two rotation rates. The outer flow for the rotating disk is seen to be quite different from the semi-infinite stagnation-flow situation. In the rotating-disk case, the inviscid flow outside the viscous boundary layer has only uniform axial velocity. In the stagnation flow, the axial velocity varies linearly with the distance from the stagnation surface z and the scaled radial velocity v/r is a constant (cf. Fig. 6.6). The rotating-disk solutions reveal that as the rotation rate increases, the axial velocity increases in the outer flow and the boundary-layer thickness decreases as fi1/2 and f2-1/2, respectively.

The radial velocity profile is linear and the circumferential velocity is zero outside the viscous boundary layer, which indicates that the vorticity is constant in that region. Thus, for substantial ranges of the flow and rotation Reynolds numbers, the flow is inviscid, but rotational, outside the viscous boundary layer. For sufficiently low flow, the boundary-layer can grow to fill the gap, eliminating any region of inviscid flow. 

The discussion of stagnation flow usually considers flow that impinges on a solid surface. In genera], however, the surface itself is not needed for the stagnation-flow similarity to be valid. The opposed-flow situation illustrated in Fig. 6.19 is one in which the viscous boundary layer is in the interior of the domain, bounded by regions of inviscid flow on the top and the bottom. 

Figure 6.22 illustrates the solution to this problem for several Reynolds numbers. The boundary layer forms near the inlet boundary, owing to the axial no-slip condition. The inner portions of the flow (i.e., near the centerline) tend to behave as an inviscid fluid, as evidenced by the linear v profile. As expected, the boundary layer thins as the Reynolds number increases. 

Consider the relationship of the finite-gap flow in the region outside the boundary layer with the semi-infinite flow in the inviscid region. Evaluate and plot profiles of alternative measures of the strain rate as... 

While our primary interest in this text is internal flow, there are certain similarities with the classic aerodynamics-motivated external flows. Broadly speaking, the stagnation flows discussed in Chapter 6 are classified as boundary layers where the outer flow that establishes the stagnation flow has a principal flow direction that is normal to the solid surface. Outside the boundary layer, there is typically an outer region in which viscous effects are negligible. Even in confined flows (e.g., a stagnation-flow chemical-vapor-deposition reactor), it is the existence of an inviscid outer region that is responsible for some of the relatively simple correlations of diffusive behavior in the boundary layer, like heat and mass transfer to the deposition surface. 

The conditions on velocity at the outer edge" of the boundary layer arc a little more difficult to define because there is really an interaction between the boundary layer flow and the outer inviscid flow, i.e., because of the reduction in velocity near the surface, the outer flow is somewhat different from that in truly inviscid flow over the surface. However, as discussed above, in many cases, this effect can be ignored because the boundary layer remains thin and in such cases the inviscid flow over the surface considered is calculated and the value that this solution gives for the velocity on the surface at any position is used as the boundary condition at the outer edge" of the boundary layer, i.e., if ui(jc) is the surface velocity distribution given by the solution for inviscid flow over the surface then the boundary coofhtion on the boundary layer solution is... 

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. 

There are some cases where this approach fails. One such case is that in which significant regions of separated flow exist. In this case, although the boundary layer equations are adequate to describe the flow upstream of the separation point, the presence of the separated region alters the effective body shape for the outer inviscid flow and the velocity outside the boundary layer will be different from that given by the inviscid flow solution over the solid surface involved. For example, consider flow over a circular cylinder as shown in Fig. 2.16. Potential theory gives the velocity, ui, on the surface of the cylinder as ... 

Such flow can be treated adequately using the boundary layer assumptions but the freestream velocity gradients exist purely because of the boundary layer growth and the boundary layer and inviscid core flows must be simultaneously considered. There are, nevertheless, many important practical problems in which such interactions can be ignored. 

In passing, it should also be noted that since the flow outside the boundary is assumed to be inviscid and since vi is of a lower order of magnitude than u at the outer edge of the boundary layer, the Bernoulli equation gives ... 

In order to illustrate how these integral equations are derived, attention will be given to two-dimensional, constant fluid property flow. First, consider conservation of momentum. It is assumed that the flow consists of a boundary layer and an outer inviscid flow and that, because the boundary layer is thin, the pressure is constant across the boundary layer. The boundary layer is assumed to have a distinct edge in the present analysis. This is shown in Fig. 2.20. 

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