Inviscid flow approximation

Clearly, in the limit Re y> 1, the leading-order approximation for the solution to this problem is identical to the inviscid flow problem for a solid sphere. Although the no-slip boundary condition has been replaced in the present problem with the zero-shear-stress condition, (10-197), this has no influence on the leading-order inviscid flow approximation because the potential-flow solution can, in any case, only satisfy the kinematic condition u n = 0 at r = 1. Hence the first approximation in the outer part of the domain where the bubble radius is an appropriate characteristic length scale is precisely the same as for the noslip sphere, namely, (10-155) and (10-156). However, this solution does not satisfy the zero-shear-stress condition (10-197) at the bubble surface, and thus it is clear that the inviscid flow equations do not provide a uniformly valid approximation to the Navier-Stokes... 

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... 

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... 

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. 

Parallel Flow Approximation and Inviscid Instability Theorems... 

Auton [7], Thomas et al [152] and Auton et al [8] determined a lift force due to inviscid flow around a sphere. In an Eulerian model formulation this lift force parameterization is usually approximated for dilute suspensions, giving ... 

Flow around single cylinders is the elementary model for (he fibrous filter and is the geometry of interest for deposition on pipes, wires, and other such objects in an air flow (Chapter 3). The flow patterns at low and high Reynolds numbers differ significantly, and thi.s affects impaction efficiencies. For Re > 100. the velocity distribution outside the velocity boundary layer can be approximated by inviscid flow theory. This approximates the velocity distribution best over the front end of the cylinder which controls the impaction efficiency. The components of the velocity in the direction of the mainstream flow, x, and normal to the main flow, y, are... 

But now we can obtain a first approximation to the integral term in (10-256) by using the inviscid flow solution. In particular, we have shown that... 

The amazing feature of (10-260) is that it is obtained entirely from the inviscid flow solution - the boundary-layer analysis does play an important role in demonstrating that the volume integral of 4>, based on the inviscid velocity field, will provide a valid first approximation to the total viscous dissipation but does not enter directly. 

Neither the r9 component of the rate-of-strain tensor nor the simple velocity gradient dvg/dr vanishes at the gas-liquid interface. This is expected for inviscid flow because viscous stress is not considered, even in the presence of a signiflcant velocity gradient. Once again, the leading term in the polynomial expansion for vg, given by (11-126), is used to approximate the tangential velocity component for flow of an incompressible fluid adjacent to a zero-shear interface ... 

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.. 

A common approximation in many flow field computations at high fluid velocities is to consider that inertial forces dominate the flow and to neglect viscous forces (inviscid approximation). Since solvent viscosity is a variable in some of the experiments discussed here, the above approximation may be not be valid throughout and viscous forces are explicitly considered in the flow equations. Results of computations showed, nevertheless, that even with viscous solvents such as bis-(2-ethyl-hexyl)-phtalate with qi = 65 mPa s, viscous forces do not affect the flow field unless tbe fluid velocity drops below a few m s"1 at the orifice. This limit is generally more than one order of magnitude lower than the actual range used in the present investigations. 

As the Reynolds number increases, dV/dz is approximately a constant over a greater region. In the limit of infinite Reynolds number, the entire gap flow is inviscid. These solutions are represented as dashed lines in Fig. 6.7. When the viscous term vanishes, the... 

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,... 

The both considered limit situations can be encountered in numerous problems of convective heat transfer they are schematically shown in Figure 3.1. One can see that in the case Pr — 0, which approximately takes place for liquid metals (e.g., mercury), one can neglect the dynamic boundary layer in the calculation of the temperature boundary layer and replace the velocity profile v(x, y) by the velocity v<, (x) of the inviscid outer flow. As Pr-)- oo, which corresponds to the case of strongly viscous fluids (e.g., glycerin), the temperature boundary layer is very thin and lies inside the dynamic boundary layer, where the velocity increases linearly with the distance from the plate surface. 

Boundary Layer Concept. The transfer of heat between a solid body and a liquid or gas flow is a problem whose consideration involves the science of fluid motion. On the physical motion of the fluid there is superimposed a flow of heat, and the two fields interact. In order to determine the temperature distribution and then the heat transfer coefficient (Eq. 1.14) it is necessary to combine the equations of motion with the energy conservation equation. However, a complete solution for the flow of a viscous fluid about a body poses considerable mathematical difficulty for all but the most simple flow geometries. A great practical breakthrough was made when Prandtl discovered that for most applications the influence of viscosity is confined to an extremely thin region very close to the body and that the remainder of the flow field could to a good approximation be treated as inviscid, i.e., could be calculated by the method of potential flow theory. 

An alternative explanation, which is consistent with the inviscid approximation, was given in [14]. It was hypothesized that ambient phase inertia causes the periphery of the deformed drop to be deflected in the direction of the flow, thereby forming a sheet. Following this, the sheet breaks into ligaments and then individual fragments. This mechanism seems to be confirmed by recent numerical simulations [15]. 

The governing equations for the flow are obtained by one-dimensional approximations of conservation laws for mass, momentum, and energy. Often the additional assumptions of inviscid, adiabatic flow are invoked, and as a consequence, the flow is regarded as isentropic throughout. From the conservation laws, one may derive an important relation between the cross-sectional area, the velocity, and the local Mach number ... 

It is obvious that the system of governing equations in the multi-tempierature approach is considerably simpler than the corresponding system in the state-to-state approach, since it contains much fewer equations. In the zero-order approximation of the Chapman-Enskog method, the system of governing equations takes the form typical for inviscid non-conductive flows. In this case equations (85), (86) read ... 

Tuck established a mathematical expression for squat with a slender body theory. The slender body theory assumes that the beam, draft, and water depth are very small relative to ship length. This theory uses potential flow where the continuity equation becomes Laplace s equation. The flow is taken to be inviscid and incompressible and is steady and irrotational. In restricted water, the problem is divided into the inner and the outer problems, following a technique of matched asymptotic expansions to construct an approximate solution. The inner problem deals with flow very close to the ship. The potential is only a function of y and 2 in the Cartesian coordinate system. In the cross-flow sections, the potential function... 

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