Stagnation flows axisymmetric

Deriving the axisymmetric stagnation-flow equations begins with the steady-state three-dimensional Navier-Stokes equations (Eqs. 3.58, 3.60, and 3.60), but considering flow only in the z-r plane. In general, there may be a circumferential velocity component ui, but there cannot be variations of any variable in the circumferential direction 0. The derivation depends on two principal conjectures. First, the velocity field is presumed to be described in terms of a streamfunction that has the separable form... 

The axisymmetric inviscid stagnation flow is described in terms of a stream function having the form... 

Fig. 6.3 Nondimensional axial and radial velocity profiles for the axisymmetric stagnation flow in the semi-infinite half plane above a solid surface. The flow is approaching the surface axially (i.e., u < 0) and flowing radially outward (i.e., V > 0). The temperature profile, which is the result of solving the thermal-energy equation, is discussed in Section 6.3.6.

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. 

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.

Fig. 6. 5 A stencil that illustrates the finite-difference discretization of the semi-infinite-domain axisymmetric stagnation flow problem.

For a two-dimensional axisymmetric problem, all other vorticity components vanish exactly. For the stagnation-flow problem, it has been established that du/dr = 0 thus cog = dv/dz =rdV/dz. It is apparent that 2 = cog/r is a function of z alone. Thus, like the radial velocity, the scaled vorticity also exhibits a radially independent boundary layer. 

For the steady, constant-viscosity, axisymmetric stagnation flow, assuming no body forces, the vorticity equation emerges as a scalar equation for the circumferential vorticity field,... 

Fig. 6.9 Correlation of the Nusselt number with the Reynolds number for the axisymmetric stagnation flow in a finite gap. The Prandtl number is Pr = 0.7.

In the foregoing sections the discussion of axisymmetric stagnation flows has concerned four subcases of the same general problem—semi-infinite or finite domains and rotation or no rotation. The intent of this section is to focus attention on the fact that with suitable choices of length and velocity scales these problems can be collapsed to a common representation. Generally speaking, the length scale is called L and the velocity scale is called U. Thus nondimensional variables are defined as... 

Fig. 6.18 Comparison of general flow patterns for planar and axisymmetric, finite-gap, stagnation flow.

The discussion in this chapter has been dominated by axisymmetric flow. However, there is analogous behavior for planar stagnation flow in two-dimensional cartesian coordinates. In fact Hiemenz s original work was for planar stagnation flow in a semi-infinite region. The planar flow illustrated in Fig. 6.18 is for a finite domain. 

Develop a numerical solution to describe the axisymmetric, semi-infinite, isothermal, stagnation flow of air with strain rates in the range 1/s < a < 1000/s. Solve the problem in physical variables (i.e., not nondimensional) using constant properties evaluated at T = 300 K. 

In an ideal stagnation flow, a certain amount of the flow that enters through the inlet manifold can leave without entering the thermal or mass-transfer boundary layers above the surface. For an axisymmetric, finite-gap, flow, determine how the bypass fraction depends on the separation distance and the inlet velocity. 

Derive the nondimensional thermal-energy equation for an axisymmetric, semi-infinite stagnation flow of a constant-property incompressible fluid. 

Based on solutions of the semi-infinite, axisymmetric, stagnation flow problem for strain rates in the range Is-1 < a < 100s 1, evaluate and plot axial profiles of the scaled vorticity ft = cog/r. Explain why the vorticity cog vanishes on the centerline. 

Following the general approach discussed in Section 6.2 for axisymmetric flows, derive the general equations for planar stagnation flow. The planar equations are summarized, but not derived, in Section 6.9. Discuss the differences and similarities between the two stagnation flows. 

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

V Scaled radial velocity in axisymmetric stagnation flow 1/s... 

In retrospect, the effect of the change of variables has been to deform the velocity field from axisymmetric stagnation flow over a sphere to linear shear flow along a flat plate. The main advantage of the new coordinates is that the coefficients of the derivatives in (32) are independent of X, and consequently Duhamel s theorem can be applied. Thus, the following procedure can be used (1) solve equation (32) subject to a uniform surface concentration (2) extend this solution to one valid for an arbitrary, nonuniform surface concentration by applying Duhamel s theorem (3) select the surface concentration which satisfies (33a). 

Steady extensional deformations can be created by impinging two liquid streams, creating a stagnation flow. Figure 7.7.1 illustrates both axisymmetric and planar stagnation flow. These flows are not homogeneous. A material element near the central part of the flow... 

Fiber spinning (Figure 7.5.2) approximates one end of the axisymmetric stagnation flow. The tubeless siphon is a little closer. But neither has a stagnation point. Ideally we want to confine a fluid to flow within the stream surfaces indicated in Figure 7.7.1. For planar stagnation these surfaces are defined (Winter et al., 1979) by the relation... 

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

If exceeds a critical value of order unity, reverse flow occurs near the forward stagnation point for oblate and prolate axisymmetric bodies (M8, M9), leading to formation of an upstream separation bubble. ... 

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. 13.12 Opposed-flow diffusion flame between parallel, axisymmetric, burner faces that are fabricated as honeycomb monoliths. As illustrated, the flame is positioned on the oxidizer side of the stagnation plane. However, depending on the flow conditions as well as the fuel and oxidizer composition, the flame may form on the fuel side of the stagnation plane.

Vertical CVD Reactors. Models of vertical reactors fall into two broad groups. In the first group, the flow field is assumed to be described by the one-dimensional similarity solution to one of the classical axisymmetric flows rotating-disk flow, impinging-jet flow, or stagnation point flow (222). A detailed chemical mechanism is included in the model. In the second category, the finite dimension of the susceptor and the presence of the reactor walls are included in a detailed treatment of axisymmetric flow phenomena, including inertia- and buoyancy-driven recirculations, whereas the chemical mechanism is simplified to a few surface and gas-phase reactions. 

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