Boundary-layer flow axisymmetric

H. K. Calkin, Axisymmetric Free Convection Boundary-Layer Flow Past Slender Bodies. Int. J. Heat Mass Transfer (11) 1141-1153,1968. 

Cone in Supersonic Flow. The preceding solutions for a flat plate may be applied to a cone in supersonic flow through the Mangier transformation [39], which in its most general form relates the boundary layer flow over an arbitrary axisymmetric body to an equivalent flow over a two-dimensional body. This transformation is contained in Eq. 6.89, which results in transformed axisymmetric momentum and energy equations equivalent to the two-dimensional equations (Eqs. 6.95 and 6.97). Hence, solutions of these equations are applicable to either a two-dimensional or an axisymmetric flow, the differences being contained solely in the coordinate transformations. 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

The burner in the test facility, shown in Fig. 18.1, is an axisymmetric nozzle, which is concentrically placed into the circular suction collar. To achieve a top-hat velocity profile with laminar boundary layer at the nozzle exit, a fourth order polynomial with a large contraction ratio of 31.6 1 and an exit diameter of D = 10.16 mm is used in the design. The suction collar assembly is connected to a vacuum pump through a series of solenoid valves so that a counterflow, which is in the opposite direction of the fuel-air mixture flow, can be established... 

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.

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. 

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. 

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 ... 

Given the scaling arguments in the previous sections, the axisymmetric channel-flow boundary-layer equations can be summarized as... 

The difference in these two configurations is in the flow pattern developed over the wafer. In Figure 17 the flow will develop as a classical boundary layer which will be thinner at the front and thicker towards the trailing edge. To whatever extent the deposition is diffusion controlled, it will be influenced by boundary layer thickness, so deposition rates may be higher near the front of the wafer than in the rear. For Figure 18 the flow pattern will be quite different and whatever nonuniformities occur will at least be axisymmetric. 

Ry i wj, and Rdp/dx - (dp/dx) reduces the axisymmetric equations to equations for planar boundary layers in the new variables. The indicated transformation concerning the pressure gradient means that the axisymmetric flow for a given pressure gradient corresponds to a two-dimensional flow under a different pressure gradient. If the subscript oo identifies external-flow variables just outside the boundary layer, then from equations (15), (16), and (17) we find that... 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

Unfortunately, however, there are a large number of different types of flow conditions for which the boundary-layer form of the heat transfer correlation (9-255) is not applicable. This applies, basically, to any flow configuration in which the body is completely surrounded by a region of closed streamlines (or pathlines, if the flow is not 2D or axisymmetric). We will discuss high-Peclet-number heat transfer in such cases in Section L. Here, we consider... 

Thus, in a manner entirely equivalent to the two-dimensional analysis, we seek rescaled equations in the inner (boundary-layer) region very near to the sphere surface within which the tangential velocity goes from the potential-flow value (3/2)sin0 to 0 at the body surface. The only difference from the previous analysis is in the detailed form of the Navier-Stokes and continuity equations for axisymmetric geometries. When expressed in terms of spherical coordinates, these equations are... 

When the fluid approaches the sphere from above, the fluid initially contacts the sphere at 0 = 0 (i.e., the stagnation point) because polar angle 6 is defined relative to the positive z axis. This is convenient because the mass transfer boundary layer thickness Sc is a function of 6, and 5c = 0 at 0 = 0. In the laminar and creeping flow regimes, the two-dimensional fluid dynamics problem is axisymmetric (i.e., about the z axis) with... 

The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

Both the Stokes flow solutions and the large Reynolds number boundary layer solutions assume that the flow around the bubble is axisymmetric. This assumption fails for some situations at large Reynolds numbers. Saflinan [20] and Hartunian and Sears [21] performed experiments on bubbles at large Reynolds numbers. For bubbles in distilled water, they found that bubbles larger than a critical size tend to spiral or zigzag. Hartunian and Sears reported the critical value of the equivalent spherical diameter to be 1.7 mm. Later, Duineveld [15] found that the critical value was 1.9 mm. In tap water, the critical value is smaller. Hartunian and Sears reported the value to be 1.3 mm. Table 1 summarizes experimental results for the onset of oscillations. 

---