Fluid Flow in Coiled Tubing: CFD Simulation

The fluid flow in coiled tubing is featured by the secondary flow that is caused by the centrifugal forces in the curved geometry. The secondary flows of the spiral form are superimposed on the axial primary flow.

Dean^(1,2)was the first to theoretically study the fluid flow problem in coiled pipe. Using a successive approximation approach, Dean obtained an analytical solution which is essentially a perturbation solution over the Poiseuille flow of straight pipe. He also introduced a dynamic similarity parameter, Equation (Available in full paper), where "a" and "R" are radii of the pipe and curvature. W₀ is the maximum axial velocity in the cross-section and νis kinematic viscosity. This parameter (K) was later called the Dean number. There have been several versions of Dean number, but the most common one is defined as: NDe = NRe(a/R)^1/2.

Since Dean's classical work, numerous studies on coiled pipe flow have been reported in the literature. These various studies can be categorized according to the factors considered in each study. These factors can be geometrical effects (curvature ratio, helicity, and torsion), Reynolds number, Dean number, and flow regime (laminar or turbulent) as well as fluid properties (Newtonian and non-Newtonian). If the helicity can be neglected and the curvature ratio is small, the governing equations can be simplified and reduced to equations about stream-function and the axial velocity. Dean obtained a series solution which is perturbed over the Poiseuille flow. Numerous investigations followed the Dean's approach for analytical analyses - McConalogue and Srivastava⁽³⁾, Larrain and Bonilla⁽⁴⁾, van Dyke⁽⁵⁾, Wang⁽⁶⁾, Dennis and Ng ⁽⁷⁾, Kao ⁽⁸⁾, Germano ^(9,10), and Tuttle ⁽¹¹⁾. These theoretical studies can be called series solutions or perturbation solutions. Among these Wang ⁽⁶⁾, Germano ^(9,10), Kao ⁽⁸⁾, and Tuttle ⁽¹¹⁾ considered the helicity or torsion effect. The perturbative parameters can be Dean number and/or curvature ratio. The series expansions can be power series or Fourier series. The limitation with the series expansion approach is that it becomes extremely laborious to extend it to higher orders.

One alternative to the theoretical approach is numerical method in which the governing equations for flow in curved pipe are discretized and solved numerically in the specified domain. These studies include Truesdell and Adler⁽¹²⁾, Austin and Seader ⁽¹³⁾, Greenspan ⁽¹⁴⁾, Patankar et al. ⁽¹⁵⁾, Collins and Dennis ⁽¹⁶⁾, Dennis ⁽¹⁷⁾, Soh and Berger ⁽¹⁸⁾, and Liu and Masliyah ⁽¹⁹⁾.