Numerical solution of the Navier‐Stokes equation for flow past spheres: Part I. Viscous flow around spheres with and without radial mass efflux

Abstract

This study was undertaken to ascertain the accuracy of finite-difference solutions for flow around spherical particles in the intermediate Reynolds number range. Comparison of the results with experimental data on drag coefficients, frontal stagnation pressure, and wake geometry indicated good agreement. The approximate solutions, in which the Galerkin method and asymptotic analytical predictions were utilized, were evaluated by using the finite-difference solutions as a standard. These methods were used to calculate the effect of uniform and nonuniform mass efflux on the drag and flow characteristics around a sphere. Theoretical solutions indicated that nonuniform mass efflux can significantly reduce the drag on a submerged object. Ranges of applicability of the approximate methods were established.The laws of motion of a single spherical partical in an undisturbed fluid stream (that is, one in which there is no secondary motion) have been the sub'ect of many investigations (1, 2 ) . The macroscopic hy a! rodynnmical characteristics, exemplified by the drag coefficient, are wellestablished over a large Reynolds number range by numerous experimental studies ( 3 to 7). Obtained from these data are a number of quite accurate experimental correlations (8 to 10) over the range of interest in the present study (0 7 N R~ 7 500). More detailed investigations (11 to 14) reveal not only widely differing flow patterns in the various Reynolds number regimes, but also indicate that the region of present interest is very little understood. This lack of knowledge is best exemplified by conflicting evidence of the first appearance of a vortex ring (1 5 to 1 7 ) .We believe that the most reliable experimental investigation of the wake region is that of Taneda ( 1 7).An analytical solution of the complete Navier-Stokes equations is impossible at the present time owing to their nonlinearity. Although numerous approximate solutions have been obtained (18 to 21, 22 to 39), virtually the only exact solutions available are those of Stokes ( 4 0 ) (that is, N R e z I ) and the potential flow solution. The most successful approximate solutions have been obtained by use of the boundary-layer assumptions (22 to 3 2 ) . Although the literature contains an almost endless number of techniques for solving the boundary-layer equations, the most rigorous and accurate solution is that of Frossling (32). For the frontal stagnation point the theoretical treatment of Ilomann ( 4 2 ) probably yields the most accurate results.Although the lower Reynolds number limit of applicability of the boundary-layer solutions should be well above 200, they have been widely applied well below this value. Experimental justification of this extrapolation, for example, by the comparison of theoretical and experimental drag coefficients, has not been possible as the boundarylayer solutions are only applicable up to the flow separation point. At present there is no adequate mathematical description of the wake region. Thus, the boundary-layer solutions, although yield...

Dispersed Flow Film Boiling

“…Theoretical work by Hamielec [40] for low Reynolds number flows, indicated that droplet drag should decrease with increasing evaporation rates. However, experimental evidence presented by Yuen [41] suggests that evaporation has little effect on drag coefficients for mass transfer numbers, B, up to 3.…”

Numerical simulation of fuel droplet extinction due to forced convection

“…The following data were used: (1) Drag coefficients for solid spheres at Reynolds numbers (Re ∞ = dU ∞ /ν ∞ ) of 0.1, 1, 10, 100, and 300 were compared with numerical results in the literature [43][44][45][46][47] and with the correlations of Clift et al [48]. (2) The dimensions of the attached vortex on the downstream side of the sphere were compared with the experimental results of Taneda [49] reported in [48].…”

Simple viscous flows: From boundary layers to the renormalization group

“…There are many other numerical calculations of the drag coefficient for a sphere, including: Dennis (Dennis and Walker, 1971), Le Clair (Le Clair and Hamielec, 1970;Pruppacher and Le Clair, 1970), Hamielec (Hamielec et al, 1967), Rimon (Rimon and Cheng, 1969), Jenson (Jenson, 1959), and Kawaguti (Kawaguti, 1950 It should also be noted that there is a community of researchers interested in sedimentation and settling velocities who have studied the drag on a sphere. In a contribution to this literature, Brown reviews all of the authors discussed here, as he tabulates C D for R < 5000 (Brown and Lawler, 2003).…”