The drag coefficient of a sphere: An approximation using Shanks transform

a r t i c l e i n f o An accurate model for the drag coefficient (C D) of a falling sphere is presented in terms of a non-linear rational fractional transform of the series of Goldstein (Proc. Roy. Soc. London A, 123, 225-235, 1929) to Oseen's equation. The coefficients of the six polynomial terms are improved through a direct fit to the experimental data of Roos and Willmarth (AIAA J., 9:285-290, 1971). The model predicts C D up to Reynolds number 100,000 with a standard deviation of 0.04. Results are compared with eight different formulations of other authors. The case of a free falling (or rising) sphere with constant velocity in an unbounded environment is relevant to numerous practical problems. Many applications in chemical and metallurgical processes, in sediment transport and deposition in channels and pipes require the specification of the drag coefficient and the settling velocity of spherical bubbles, drops or particles. Unfortunately, the Navier–Stokes (N.–S.) equations do not exhibit exact solutions for flows around bodies of finite size, for any range of particle Reynolds number (R p). At some extreme values of R p , approximate analytical methods can be used to derive equations that yield useful approximate solutions. For a spherical particle, these are frequently limited to R p b 30 (Liao [1]). In view of the above remarks, expressions for the drag coefficient of a sphere (C D) with a large interval of application need to be obtained from empirical or numerical data through regression techniques. The various approximations quoted in literature vary somewhat in form (Cheng [2]), but normally are expressed in terms of single power series expansions on R p. Piecewise or asymptotically matched small segments (Almedeij [3]) are also frequently used. Here, an approach is proposed to determine an empirical expression for the drag coefficient of a sphere based on Shanks transform [4]. This is a non-linear transform that is very effective to accelerate the convergence of slowly converging series. Shanks very early recognized the advantages of working with rational fractions of higher order provided more than three terms of a power series are known. For a falling sphere, his proposed 6-term expression for the drag coefficient based on the first five terms of Goldstein's series (see Eq. (6) below) agrees well with experiments up to R p = 10. An increase on the number of terms retained by a …