a b
Figure 3. (a) Schematic representation of polymer-induced turbulent drag reduction (DR) mechanism [32] (b) Illustration of turbulence suppression mechanism Source: [13].
Between the Prandtl-Karman Law and the maximum drag reduction curve is a roughly linear polymeric regime characterised by the wall shear stress\(\left(\tau_{w}\right)\) and the increment in slope\(\left(\delta\right)\) [33]. This regime is represented by Eq. (9).
\(\frac{1}{\sqrt[f]{}}=\left(4.0+\delta\right)\text{Re}\sqrt[f]{}\ -0.4-\delta\left[\frac{\sqrt[2d]{}}{v_{s}}\left(\frac{\tau_{w}^{*}}{\rho}\right)^{\frac{1}{2}}\right]\ \)(9)
The centrifugal forces associated with flow in curved pipes results in secondary flows which appear in the form of vortices [34]. Centrifugal force causes faster-moving fluids in the middle of the pipe to move to the outer wall while fluids in the outer wall to move to the centre resulting in secondary flow [10]. These vortices flow behaviour results in flow fluctuations and higher pressure drop in curved pipes compared with that in straight pipes of equivalent length. The higher pressure drops observed in curved pipes prompted  Shah and Zhou [30] to propose a modified Virk’s envelop. They replaced the Prandtl-Karman Law (Newtonian line in Fig. 1) with a Newtonian friction factor correlation for coiled tubing given by  Srinivasan [32]. Findings have revealed that maximum drag reduction asymptote (MDRA) for curved pipes is lower than that of straight pipes and depends on the curvature ratio of the pipe.  Shah and Zhou [33] proposed an expression for MDRA for flow of drag-reducing polymers (DRPs) in curved pipes as a function of curvature ratio given by Eq. (10). Fig. 2 shows the MDRAs for coils of various curvatures, as determined by Eq. (10).
\(\frac{1}{\sqrt[f]{}}=AN_{\text{Re}}\sqrt[f]{}\ +B\) (10)
where;\(A=\left[c_{1}+c_{2}\left(\frac{a}{R}\right)^{0.5}\right]^{-1},\ \ c_{1}=0.053109965,\ c_{2}=0.29465004\)and
\(B=\left[c_{3}+c_{4}\left(\frac{a}{R}\right)^{0.5}\right]^{-1},\ \ c_{3}=0.0309447,\ c_{4}=0.245746\)
when \(\left(\frac{a}{R}=0\right),\ \ A=18.83\) and \(B=32.32\), and Eq. (10) reduces approximately to the Virk’s MDRA.
Based on the redefined MDRA for curved pipes, Shah and Zhou [33] described a new drag reduction envelop. This drag reduction envelop is bounded by three lines – the laminar flow line, the MDRA for curved pipe and the zero-drag reduction line given by the Srinivasan [32] correlation for Newtonian turbulent flow in curved pipes. The laminar flow correlation chosen for their work was that of Liu and Masliyah [34].
Studies have shown that phenomenological models for MDRA, developed for polymers, are not applicable to surfactants. An interesting characteristic of surfactants is their higher shear viscosity compared to polymer solutions. This makes surfactant solution more shear rate dependent and makes the definition of the Reynolds number all the more difficult [21]. Zakin et al. [35] showed that fanning friction factor curves of most surfactants in straight pipes lies below the Virk MDRA. They proposed an MDRA for surfactant solutions in straight pipes given by:
\(f=0.32N_{\text{Re}}^{-0.55}\) (11)
Their work did not account for how viscosity depends on the shear rate in surfactants. Aguilar et al. (2006) used surfactant with viscosity similar to that of the solvent and recorded friction factors slightly lower than those given by the Zakin MDRA. They proposed a new correlation for MDRA given by;
\(f=0.18N_{\text{Re}}^{-0.50}\)(12)
Surfactant solutions exhibit higher MDRA than polymers. Kamel and Shah [36] therefore extended Zakin et al. [35] MDRA for straight pipes to curved pipes and proposed a correlation for MDRA for surfactant in coiled pipes given by Eq. (13).
\(f=\left[-32200.42\left(\frac{a}{R}\right)^{3}+1830.62\left(\frac{a}{R}\right)^{2}+0.32\right]N_{Re^{\prime}}^{\left[7210.95\left(\frac{a}{R}\right)^{3}-316.97\left(\frac{a}{R}\right)-0.55\right]}\)(13)
They further suggested a modified maximum drag reduction envelop for surfactant in coiled pipes bounded by Liu and Masliyah [34]’s equation for laminar flow, the Srinivasan et al. [32] correlation and Eq. (13).

2.1 Drag-reducing Agents (DRAs)

Drag-reducing agents include additives such as polymers, surfactant, fibres and micro-bubbles. The use of polymer as a drag-reducing agent is most common because only small concentrations is needed to produce drag reduction [13], [39], [40]. Drag-reducing agents can either be soluble or insoluble resulting in homogeneous and heterogeneous fluids mixtures respectively [41]. The benefits of DRAs include reduced operation cost and ease in application [42]. Its application in oil and gas ranges from petroleum product transport to enhance oil recovery [43].

2.1.1 Polymer DRAs

Synthetic and natural polymers are classes of polymer DRAs. Examples of synthetic polymers include; polyethylene oxide (PEO), polyisobutylene (PIB), polyacrylamide (PAM), partially hydrolysed polyacrylamide (HPAM) etc. Synthetic polymers generally produce high percentage drag reduction. They are, however, mostly non-biodegradable thereby posing environmental challenges. Natural polymers include; carboxymethylcellulose (CMC), guar gum (GG), xanthan gum (XG), tragacanth, karaya, locust bean, chitosan and okra [14]. Natural polymers are biodegradable thus making them environmentally friendly [41]. However, this biodegradability reduces their shelf life thus reduces their effectiveness for long-distance transport. Grafting the artificial polymers into the rigid structures of natural polymers have been suggested as a means of controlling biodegradation [14], [44]. Recent advances in polymer technology have seen the rise in high performance biodegradable polymers. Some of the recent synthesis have been centred around improved cross-linking of polymer chains [45], [46]. A common characteristic of DRAs is the increase in efficiency with increase in molecular weight of polymer. A drawback of polymers DRAs is their susceptibility to both chemical and mechanical degradation. High molecular weight (Mwt > 106) polymers are the most commonly employed DRAs possibly because of their unique rheological properties which makes them effective and economical [14]. Various theories exist seeking to explain the mechanism of polymer drag reduction. These theories includes those based on shear thinning, viscoelasticity, vortex stretching, molecular stretching, flow anisotropy and turbulence suppression [16], [32].
A number of researchers have tried to explain the mechanism of polymer DR by molecular stretching of polymer molecules. In this model, the shear-hardening characteristic of drag-reducing polymers (DRPs) is assumed to increase resistance to extensional flow, thereby inhibiting turbulent burst at the near wall region. The Lumley [44] model, which is based polymeric chain extension, suggest that DR involves increased elongational viscosity. This results in increased thickness of the viscous sub-layer which dampens and suppresses small eddies and turbulent fluctuations. The overall effect is higher turbulence dissipation, reduction of both velocity gradient and shear stress near the wall and consequently reduction of drag. It has also been suggested that stretching of polymer molecules results in the storage of elastic energy (see Fig. 3a) emanating from flow very close to the wall [48]. Thus if there is sufficient relaxation time, the elastic energy is transported to the buffer layer and dissipated there by the vortex motion resulting in DR ([49].
A number of proposed DRP drag reduction mechanisms are based on polymer’s spring like behaviour. A bead-spring model was used by Armstrong and Jhon [47] to describe the mechanism of DR. The polymer molecule is assumed to be a chain of identical beads linked by an arbitrary spring potential. Here the effect of the stochastic velocity field on the polymer molecule is associated with arenormalisation of the connector potential and the dumb-bell probability density is derived for the arbitrary connector potential. At certain degree of turbulence, the second moment of the probability density becomes infinite. The renormalisation of the connection potential between the beads reduces the connection force, thus making the beads extend (or polymer molecules expand). A mechanism analogues to the dumb-bell model wherein stretched polymer molecule are simplified as springs with masses at their ends was also proposed by [49]. The theory assumes that there is a balance between centrifugal stretching force and centripetal restoring force acting on rotating polymer chains. The rotational flow kinetic energy is converted to polymer elastic energy and subsequently becomes damped by the surrounding viscous fluids when the polymer relaxes.
A common view is that interaction of polymer with turbulence (resulting in flow laminarization ) is the main reason for its efficiency as a drag-reducing agent [42]. The complex rheological properties of DRPs such as viscosity and elasticity play important role in the process [14]. The non-axial component of turbulent flows results in wasteful turbulent eddy dissipation and the implication of this is increased drag [16]. The ability of DRP to induce flow laminarizationtranslates to reduction of wasteful energy dissipation and consequently DR. In effect, the action of DRPs in flow laminarization is to reduce radial velocity fluctuations and Reynolds stresses [32], [41], [42].
The anisotropic behaviour of DRP solutions, where shear rate, structure and viscosity of the solution are directionally dependent, have been used to explain polymer DR. Here the effect of DRPs is to alter the turbulence structure and reduce drag [51]. Models based on the finite elastic non-linear extensibility-Peterlin (FENE-P) have also been used to explain the mechanism of polymer DR. Here pre-averaging approximation is applied to a suspension of non-interacting finitely extensive non-linear elastic dumb-bells, thus accounting for the finite extensibility of the molecule [52]. The FENE-P model has been used by Li et al. [50] as viscoelastic polymer conformation tensor equation.
A few numerical simulation studies have been carried out to shed more light on DR mechanism. In the Brownian dynamic simulation studies of Terrapon et al. [51] it was demonstrated that polymers experience significant straining around the vortices resulting in molecular stretching. As polymer molecules stretches around the vortices, by upward and downward fluid motion, there is extraction of energy from the near-wall vortices. Numerical studies has also been carried out to describe the systematic storage and release of energy to the flow by polymer [55], [56]. Energy storage occurs at the near-wall vortices, while the release of energy occurs at the very-near-wall region. Numerical studies was also used to show that polymer mixing acts as a relaxation mechanism for DR [57]. Direct numerical simulation was used to investigate the roles of shear stress/shear rate anisotropy and elasticity on DR [58]. The hypothesis is that, when polymer stretches, the viscous anisotropic effect produces change in turbulent structures and change in entropy which in turn results in DR. To shed more light on the mechanism of DR and explain certain observed behaviours, various studies have been carried out using laser Doppler velocimetry (LDV) and particle image velocimetry (PIV) [41], [59]–[62].
Overall, it appears that more than one of the suggested mechanisms is involved in DR. Notwithstanding the mechanism(s), polymers do stretch in the flow thereby absorbing the energy in the streak. This inhibits turbulent burst formation (Fig. 3b) in the buffer region and results in turbulence suppression.
The above reports details efforts to explain the DR mechanism via investigations of flows in straight pipes. Similar to straight pipes, DR by polymer solutions in curved pipes and channels have been linked with the dampening of turbulent intensities [63]. A few suggested mechanisms for polymer drag reduction in the laminar flow regime of curved flow exist. The general understanding is that for DRAs to be effective in the laminar flow regime of curved pipe flows, there must be an interaction between the DRAs and secondary flow stream lines. A few early studies investigated the effect of DRAs on secondary flows but the conclusions are inconsistent and mostly speculative [64]–[66]. Frictional losses as well as secondary flow losses contributes to pressure losses in hydrodynamically developed flows in coils. In the case of undeveloped flows in and after bends, additional form-drag exist due to flow redistribution. The effect of DRAs on each of these competing forces is a subject of investigation by the authors using a dedicated flow loop at the University of California Berkeley.

2.1.2 Surfactant DRAs

Surfactants are surface-active chemical agents of relatively low molecular weight which alters the surface tension of the liquid in which it dissolves [67]. They assume various structures in solution such as spherical micelles, rod-like micelles, crystals, emulsions and vesicles depending on the concentration, temperature, salinity etc. [38]. The classes of surfactants are ionic (examples; anionic, cationic and zwitterionic) and non-ionic surfactants. When compared to polymer they have higher resistance to mechanical degradation [68] and are thermodynamically stable [23]. This is due the their ability to self-repair after degradation [21], [69]. The efficiency of surfactants in reducing drag depends on its concentration, temperature, geometry of flow channel, size of micelles and bond strength. Some early investors [70], [71] linked the mechanism of drag reduction in surfactants to the viscoelastic rheology of the solution. However, drag reduction has since been observed in non-viscoelastic surfactants [72]. The ability of surfactants to act as drag reducers is associated with the formation of thread-like micelles. These micelles changes the structure of turbulent flow at the near wall region [34], [68]. It has been suggested that surfactants drag reduction is achieved when micelles, under shear stress, line up in the direction of flow and build a huge network structure (the so-called shear-induced state) [42], [73]. This leads to a damping of radial turbulence and subsequently reducing pressure loss. Fig. 4a shows surfactant molecules and micelles structures while Fig. 4b show the transmission electron microscope (TEM) image of surfactant micelles. Different surfactants show different response or characteristics under the influence of shear. For example, the viscosity of Habon G decreased under prolonged shearing or mixing while that of the mixture Ethoquad T 13/ sodium salicylate (NaSal) increased after prolonged shearing in a rotational viscometer. The effective velocity range for which various surfactants produce drag reduction depends on the concentration and age of the surfactants [74]. The effectiveness of surfactants as drag-reducing agents is negatively influenced by disturbances in the flow, though sensitivity of surfactants to disturbances differs. This is important in bend-flow applications where there are high disturbances resulting from the bend. As reported by Gasljevic and Matthys [9], additional drag results from the flow of surfactant solutions in the region of high flow disturbance after the bend.
Cationic surfactants are by far the most commonly used drag-reducing surfactants DRS . Cationic surfactants combined with suitable counter-ions are effective drag reducers [75]. The applicability of anionic surfactants in aqueous or hydrocarbon solutions depends on their molecular weight. In general, low-molecular weight surfactants are used as drag-reducing agents. Very low-molecular weight (< 10 carbon atoms in chain) anionic surfactants are too soluble to have substantial surface effect and thus results in small drag reduction [42]. The surface-active portion of zwitterionic surfactants carry opposing charges on it as well as a subgroup derived from imidazoline. Zwitterionic surfactants are more environmentally friendly than the cationic ones. However at the recommended (low) concentration, they are very sensitive to upstream disturbances (as is common in bends) in the flow which may impede their drag-reducing capabilities [74]. Non-ionic surfactants are known to be chemically, mechanically and thermally stable in comparison with ionic surfactants. In addition non-ionic surfactants do not precipitate in the presence of calcium ions [41]. Non-ionic surfactants are only applicable over a limited range of temperature and concentrations and may be susceptible to chemical degradation [14]. Glycolic acid ethoxylate, Arquad 16–50 Cetyltrimethylammonium chloride (CTAC), Ethoquad O12, Soya-N(CH3)3Cl and Sodium oleate are some examples of commonly used surfactants [14]. Van der Plas [73] recently defined some essential characteristics required by viscoelastic surfactants for them to be effective DRAs in petroleum applications. It is safe to assume that insight into micelle formation and rheological properties of surfactants are essential to understanding the mechanism for drag reduction of surfactant solutions. Due to the high shear stresses observed in curved pipes, surfactants are more suitable as drag-reducing agents for flow through bends than polymers [53].

2.1.3 Micro-bubbles DRAs

The application of air in micro-bubbles drag reduction is environmentally friendly and cheaper compared to polymers and surfactants [77], [78]. Micro-bubbles have diameters less than ten-microns and exhibit behaviours different from those of larger size bubbles. These differences are seen in their chemical and physical characteristics such as the tendency to remain suspended in the liquid phase over longer periods of time [78]. The first work published on the application of micro-bubbles as drag-reducing agents was by [79]. The mechanism of drag reduction by micro-bubbles is not yet well understood. Similar to other drag reduction techniques, the purpose of micro-bubble injection is to alter the structure of the boundary layer. It had been suggested that micro-bubbles reduce drag by altering both laminar and turbulent boundary-layer characteristics [79]. It has been reported that, injecting air bubbles results in an increase in kinematic viscosity and decrease in the turbulent Reynolds number in the buffer layer [80]. This results in thickening of the viscous sub-layer and decrease in the velocity gradient at the wall. Hassan and Ortiz-Villafuerte [74] used particle image velocimetry (PIV) to study the effect of injecting low void fraction micro-bubbles into the boundary layer of a channel flow. Some of their results showed some similarities with drag reduction behaviour by polymers or surfactants as well as reports of some earlier investigations [80], [81]. These similarities include thickening of the buffer layer as well as upward shift of the log-law region. They stated that the micro-bubble layer formed at the top of the channel was not responsible for the drag reduction recorded. This micro-bubble layer served to reduce the slip between the micro-bubbles and the liquid. The major contribution to drag reduction is the accumulation of micro-bubbles in a critical zone within the buffer layer. The interaction of micro-bubbles with turbulence in the buffer layer is responsible for the observed DR. in general, injection of micro bubbles reduces turbulent energies with the shear in the boundary layer remaining unchanged [82]. There appears to be some agreement on the mechanism of micro-bubble drag reduction especially as it relates to thickening of the viscous sub-layer and turbulence suppression.