Inviscid flow

Although no real fluid is inviscid, in some instances the fluid may be treated as such, and it is worthwhile to present some of the equations which apply in these circumstances. For example, in the flat-plate problem discussed above, the flow at a sufficiently large distance from the plate will behave as a nonvis-cous flow system. The reason for this behavior is that the velocity gradients normal to the flow direction are very small, and hence the viscous-shear forces are small. 

If a balance of forces is made on an element of incompressible fluid and these forces are set equal to the change in momentum of the fluid element, the Bernoulli equation for flow along a streamline results  

The Bernoulli equation is sometimes considered an energy equation because the V2/2gc term represents kinetic energy and the pressure represents potential energy however, it must be remembered that these terms are derived on the 

When the fluid is compressible, an energy equation must be written which will take into account changes in internal thermal energy of the system and the corresponding changes in temperature. For a one-dimensional flow system this equation is the steady-flow energy equation for a control volume, 

Q = heat added to control volume Wk = net external work done in the process v = specific volume of fluid 

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Exact Solutions to the Navier-Stokes Equations. As was tme for the inviscid flow equations, exact solutions to the Navier-Stokes equations are limited to fairly simple configurations that aHow for considerable simplification both in the equation and in the boundary conditions. For the important situation of steady, fully developed, laminar, Newtonian flow in a circular tube, for example, the Navier-Stokes equations reduce to... 

The Bernoulli equation can be written for incompressible, inviscid flow along a streamhne, where no shaft work is done. 

The Spin Velocity Under the inviscid flow assumption, where all fluid that enters the cyclone does so with approximately the same amount of momentum, a free vortex may be predicted for the spin velocity distribution as... 

Brayshaw, M.D., 1990. Numerical model for the inviscid flow of a fluid in a hydrocyclone to demonstrate the effects of changes in the vorticity function of the flow field on particle classification. International Journal of Mineral Processing, 29, 51. 

In creeping flow with the inertia term neglected, the velocity distribution rapidly reaches a steady value after a distance of r0 inside a capillary tube. At this stage the velocity distribution showed the typical parabolic shape characteristic of a Poiseuille flow. In the case of inviscid flow where inertia is the predominant term, it takes typically (depending on the Reynolds number) a distance of 20 to 50 diameters for the flow to be fully developed (Fig. 34). With the short capillary section ( 4r0) in the present design, the velocity front remains essentially unperturbed and the velocity along the symmetry axis, i.e. vx (y = 0), is identical to v0. 

When a very large bubble of gas is allowed to rise in a large expanse of liquid it is found that the bubble becomes rather flattened, having a spherical upper surface and a fairly flat lower surface, as shown in Figure 7.11a. This is characteristic of the fact that the bubble s motion through the liquid is dominated by inertial forces. Inviscid flow theory shows that the rise velocity is given by the expression... 

SMD = 0.07lf v/ gM - 1 pVul y/3 in cm (cgs units) / Derived by empirically correcting a theoretical equation of inviscid flows for viscosity with fan spray data of wax Valid for 3[Pg.259]

Inviscid Flow and Potential Flow Past a Sphere... 

Whereas inviscid flow is a useful reference point for high Reynolds number flows, a different simplification known as the creeping flow approximation applies at very low Re. From Eq. (1-3), the terms on the right-hand side dominate as Re 0, so that the convective derivative may be neglected. In dimensional... 

Control volume method Finite element method Boundary element method and analytic element method Designed for conditions with fluxes across interfaces of small, well-mixed elements - primarily used in fluid transport Extrapolates parameters between nodes. Predominant in the analysis of solids, and sometimes used in groundwater flow. Functions with Laplace s equation, which describes highly viscous flow, such as in groundwater, and inviscid flow, which occurs far from boundaries. 

For a steady, inviscid flow along a streamline, show that the general vector form of the Navier-Stokes equations reduces to the familiar Bernoulli equation ... 

Discuss the conditions under which the viscous terms vanish, leading to the Euler equations for inviscid flow,... 

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. 

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

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... 

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. 

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. 

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. 

The first choice, where the ratio tends to zero, would lead to an inviscid flow as all diffusion terms be negligible. For our purpose this is an uninteresting alternative because, without viscous effects, it would be impossible to support the no-slip condition at the tube wall. The third alternative, where the ratio tends toward infinity would lead to a conclusion that there are no convective effects—again, an uninteresting alternative. Thus we are left with the alternative that the ratio is order one. After choosing the channel radius ro as the characteristic radial length scale, the axial length-scale zs scales as... 

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. 

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... 

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 ... 

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. 

The velocity component in the x-direction shown in Fig. 10.9 can, because the boundary layer is assumed to be thin, be taken as equal to the velocity at the surface, i.e., as equal to the velocity that would exist at the surface at the value of x considered in inviscid flow over the surface (see discussion in Section 10.3 above). The boundary layer form of the full energy equation for porous media flow is derived using the same procedure as used in dealing with pure fluid flows, this procedure having been discussed in Chapter 2. Attention will be restricted to two-dimensional flow. 

The inviscid flow solution gives for this region ... 

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