Inviscid flow regions

The strain-rate parameter a is a constant in the inviscid flow region... 

Both analyses lead to the same heat-transfer coefficient, as long as the Reynolds number is large enough to support an inviscid flow region in the finite-gap configuration. 

The development of viscous and inviscid regious of flow as a result of iu-serting h flat plate parallel into a fluid stream of uniform velocity is shown in Fig. 6 7. The fluid slicks to the plate on both sides because of the no slip con dition, and the thin boundary layer in which the viscous effects are significant near the plate surface is the viscous flow region. The regiou of flow on both sides away from the plate and unaffected by the presence of the plate is the inviscid flow region. 

As in the previous section we denote the tangential velocity component within the boundary layer as n to distinguish it from the tangential component ue in the outer, inviscid flow region. The boundary-layer solution is also required to match the outer inviscid solution, namely,... 

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. 

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... 

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.

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. 

As long as there is a region of inviscid flow in the region above the viscous boundary... 

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. 

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. 

Here CO is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irro-tational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects may be neglected. Such flows without viscous effects are called inviscid flows. 

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 ... 

The inviscid flow solution gives for this region ... 

The flow of an originally uniform fluid stream over a fl.K plate, and the regions of viscous flow (next to the plate on both sides) and inviscid flow (away from the plate). 

A number of authors, however, have represented forest or urban canopy layers by porous regions of distributed force (or drag) [206, 213, 217, 318, 320, 576, 662], The advantage of such an approach is that it permits inclusion of a canopy sublayer without the use of excessive and costly grid resolution. Yamada [662] and Shaw and Schumann [576] introduced the approach in order to add vegetation to meso-scale models of complex terrain. Jeram et al. [320] used the concept in 2d calculations for inviscid flow and constant eddy diffusivity flow estimates of the up and downwind penetration of flow within simple urban areas. 

By analyzing the motion of a small panicle in the region near the stagnation point, it can be shown that for an inviscid How, theory predicts that impaction does not occur until a critical Stokes number is reached. For an inviscid flow, the first term in an expansion of the velocity along the streamline in the plane of symmetry which leads to the stagnation point is... 

This completes the solution to 0(Re l/2). It should be noted that the first two terms in (10-246) are, in fact, nothing but the first two terms in the inviscid solution, evaluated in the inner region, namely, (10-213). Thus, to 0(Re x/2), we see that the solution in the complete domain consists of the inviscid solution (10-155) and (10-156), with an 0(Re l/2) viscous correction in the inner boundary-layer region to satisfy the zero-shear-stress boundary condition at the bubble surface. Because the viscous correction in the inner region is only C)( Re l/2), the governing equation for it is linear. Hence, unlike the no-slip boundary layers considered earlier in this chapter, it is possible to obtain an analytic solution for the leading-order departure from the inviscid flow solution. 

The governing inviscid flow equations of continuity, energy, and momentum (in conservation form) in gas region are Euler Equations (9)-(l 1) ... 

The identity between condition (41) and condition (38) for stable dynamic wave indicates that the region of well-posedness coincides with that of stable dynamic waves, c > 0. The region of c < 0, corresponds to unstable waves and their evolution, as formulated by the initial value set of equations, is ill-posed. As the stability condition for inviscid flows (obtained with = 0) is equivalent to that of stable dynamic waves, the well-posedness condition (with = 0, = 1) is... 

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. 

Outside the jet and away from the boundaries of the workbench the flow will behave as if it is inviscid and hence potential flow is appropriate. Further, in the central region of the workbench we expect the airflow to be approximately two-dimensional, which has been confirmed by the above experimental investigations. In practice it is expected that the worker will be releasing contaminant in this region and hence the assumption of two-dimensional flow" appears to be sound. Under these assumptions the nondimensional stream function F satisfies Laplace s equation, i.e.. 

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