1.0 Introduction

Locations of petroleum wells are sometimes several kilometres from processing plants. There is therefore need to transport single and multi-phase fluids through pipes, including pipe fittings such as bends, to processing plants and for separation [1]–[3]. A large percentage of the energy cost in petroleum production and transport results from pressure losses. Most of the pressure losses in pipeline flows are associated with the production of turbulence eddies resulting in non-axial components of flow. Unlike laminar flows where pumping power is directed at providing axial unidirectional fluid flow, turbulent flows are characterised by both axial and radial flows. The implication of this is loss of pumping power or increased drag. A common view is that any process mechanism that results in flow laminarization would also result in drag reduction [4].
Drag reduction (DR) is a process of reducing pressure losses associated with flows [5]. Additives, called drag-reducing agents (DRAs), are often used for drag reduction. After the pioneering work credited to Tom [5], several studies have examined the effect of DRAs on liquid flows through straight pipes and channels of various orientations [7], [8]. A few others investigated this effect in curved pipes [9]–[11]. Other methods of drag reduction involving pipe modifications such as riblets, dimples, wavering walls and amenable surfaces are also common [12]–[14].
Drag reduction (DR) as originally defined by [15] is given by Eq. (1).
\(\text{DR}\left(\%\right)=\frac{\left(\frac{\text{dp}}{\text{dl}}\right)_{s}-\left(\frac{\text{dp}}{\text{dl}}\right)_{\text{DRA}}}{\left(\frac{\text{dp}}{\text{dl}}\right)_{s}}\times 100\%\)(1)
where \(\left(\frac{\text{dp}}{\text{dl}}\right)_{s}\) and\(\left(\frac{\text{dp}}{\text{dl}}\right)_{\text{DRA}}\) are frictional pressure gradients for solvents and DRA solution respectively, under the same flow conditions. Where the viscosity and density of solvent and polymer solution are almost the same, Eq. (2) gives an equivalent measure of drag reduction.
\(\text{DR}\left(\%\right)=\frac{f_{s}-f_{\text{DRA}}}{f_{s}}\times 100\%\)(2)
where \(f_{s}\) is the fanning friction factor before the addition of DRA. fDRA is the fanning friction factor after the addition of DRA.
\(f=\frac{\tau_{w}}{\frac{1}{2}\rho U^{2}}\) (3)
The wall shear stress \(\tau_{w}\) is given by
\(\tau_{w}=\frac{dp}{4l}\) (4)
where d is the internal diameter of pipe and \(P\) is the frictional pressure drop over the pipe length l .
Eqs. (1) and (2) are referred to as pressure drop drag reduction and friction factor drag reduction [16].
A measure of drag reduction, in curved and straight pipes, called turbulence reduction drag (TRD) given by Eq. (5) is sometimes used [17].
\(\text{TRD}\left(\%\right)=\frac{f_{T}-f_{T\_DRA}}{f_{T}-f_{L}}\times 100\%\)(5)
where T and L denote turbulent and laminar flow of the solvent respectively.
The definition given by Eq. (5) enables comparison of only the degree of turbulence suppression in curved and straight pipes. In general, the difference between Eq. (2) and (5), for straight pipes is small. However, the respective difference is large in the case of flow in curved pipes due to suppressed turbulence and secondary flow effects [18]. It should be stated that at the same Reynolds number of flow, the degree of turbulence in straight pipes is higher than that in curved pipes [19].
Drag-reducing agents also influence turbulent heat transfer [20]–[22]. In certain applications, the effect of DRAs on heat transfer reduction (HTR) outweighs its effect on drag reduction [23]. Besides heat transfer and drag reduction, DRAs affects flow structure, phase-distribution and flow regime transitions [24]–[27].
Till date, most of the drag reduction studies have focussed on flows through vertical, horizontal, inclined and undulated pipes. Application of DRAs for flows in curved pipes has received little attention. Moreover, the flow of single and multiphase fluids through curved pipes is a common occurrence in the petroleum and chemical industries. Such a flow is associated with large pressure drop and pressure fluctuations among other effects [10]. It is important to gain insight into drag reduction in curved pipes to improve the economics of pipeline design and operation.  Fsadni [27] provided a brief review of pressure drop reduction studies for flow in helical coils. Besides this review, the Authors are not aware of any other reviews pertaining to drag reduction in curved pipes. Hence this work is devoted to the review of existing research on single and two-phase drag reduction for flows through curved pipes.