a b

Figure 13. a. Particle paths projected on the cross-section of flow for m = 1 (full line) and m = 0 (dash line). N and N’ represent neutral points for m = 1 and m = 0 respectively. Source:[98], b. Paths of particles in the central plane for varying m. Source: [98].

No comprehensive work exists in open literature that investigates the effect of polymer and surfactant concentration on drag reduction in bends. Munekata et al. [95] investigated the flow of surfactant solution (CTAB) in 90^o square-cross-section bend where reduced drag reduction as well as delayed onset of drag reduction with increase in concentration was reported. Only two concentrations were tested and so a reasonable conclusion cannot be drawn. Their plots showed reduced critical Reynolds number (or flow velocity) with decrease in concentration. The effect of concentration of drag-reducing agents for flow in bends such as 45^o and 180^o remain unclear. Notwithstanding the limited literature in this area, DRA concentration is expected to impact DR efficiency since concentration can influence flow redevelopment and the overall ability of the DRA to supress turbulence [10].

Bend angles and curvatures is known to influence centrifugal forces in the bends and consequently affects fluid redistribution in and around the bend. While in coils, increased curvature delays the unset of turbulence and supresses turbulence, in the case of bends, increased curvature may result in increased flow fluctuations, even under laminar flow conditions. There are contracting reports from the limited studies that have been carried out to investigate the effect of curvature ratio hence, its effect remains unclear. For low Reynolds number flows, Jones and Davies [96] reported that the effect of curvature on drag reduction is negligible. Nonetheless, the observed drag is higher than that which occurred in straight pipes. Yokoyama and Tomita [97] studied the flow of polyethylene-oxide in a 360^o bend of varying curvature ratios. They recorded a decrease in drag reduction with increase in curvature ratio. The drag reduction recorded was predominant at high Reynolds numbers. Only three curvatures were tested and so the effect of curvatures on the effectiveness of drag-reducing agents in 360^o bends was inconclusive. In the knowledge of the Authors, the effect of curvature ratio for flow in 45^o, 90^o and 180^obends had not been reported in open literature. Another area of interest is in determining the effect of pipe diameter on drag reduction in elbows. This is because; the effect of flow separation is expected to reduce with increase in pipe diameter.

Application of micro-bubbles for DR in curved pipe flows have received little scholarly attention and the effect of micro-bubbles on pressure losses as well as the mechanism of micro-bubble DR remain unclear. It was highlighted earlier that, the action of micro-bubbles on turbulent flows is similar, in a number of ways, to that of polymers and surfactant DRAs. To this end, it is expected that micro-bubbles will result in significant DR in curved pipe flows. The application of micro-bubbles for drag reduction in helical coils was first carried out by Shatat et al. [75] and Shatat et al. [98] using hydrocyclone effect to generate micro-bubbles. Their investigation involved three helical coils of curvature ratios 0.025, 0.05 and 0.1. They reported that, though there was significant drag reduction in helical coils by injection of micro-bubbles, this drag reduction is less than that in straight pipes under similar conditions of flow. The reduced drag reduction in helical coils is linked to centrifugal forces (resulting in suppressed turbulence) associated with the flow. Though the theories for micro-bubble drag reduction in helical coils are in agreement with existing micro-bubble DR theories for other geometries, further research is needed to establish these theories. The effect of various parameters such as pipe geometry, micro-bubble fraction, flow rate and micro-bubble size remains unclear due to the limited research in this area. Based on the limited data available, only a brief outline of the effect of these parameters is presented in this review.

Fig. 12a gives an illustration of the effect of curvature on effectiveness of micro-bubble drag reduction in helical coils. It can be observed that increase in curvature resulted in decrease in drag reduction as well as a shift of both the onset of drag reduction and maximum drag reduction to higher values of Reynolds number. The figure also shows higher drag reduction in straight pipes compared to helical coils. Though there is limited data on the effect of curvature, two important hydrodynamics properties may play important roles: first, unlike flows of polymer and surfactant solutions in curved pipes, gravity/centrifugal forces may result in significant phase separation (micro-bubble and liquid phases) for the case of micro-bubble DR. If this occurs, the concentration distribution of micro-bubbles (particularly in the buffer region where it is most effective) becomes inhomogeneous and this is likely to reduce DR efficiency; second, the curvature effects in coils is expected to suppress turbulence and thus it should be expected that the percentage DR is affected by the degree of curvature. Further study is, however, required to fully investigate the effect of curvature ratio on drag reduction in curves.

The effect of micro-bubble fraction is illustrated in Fig. 12b. It can be seen that the effect of air fraction on the onset of drag reduction is insignificant. However, the air fraction has a profound effect on the percentage drag reduction and the range of Reynolds numbers over which drag reduction occurs. In general, the percentage drag reduction increased with increase in air fraction. Again, additional data is needed in order to understand the effect of micro-bubble fraction on drag reduction since very scanty reports are available.

Similar to flow of polymer and surfactant solutions in curved pipes, where DR is reported predominantly in the turbulent flow regime, the limited reports on the application of micro-bubbles in curved pipe DR also report it in the turbulent flow regime. Since the degree of turbulence increases with increase in flow rate, it is expected that flow rate will affect the efficiency of micro-bubble DR. It can be seen from Figs. 12a and 12b that drag reduction occurs above a critical Reynolds number and increases with Reynolds number until a maximum drag reduction is achieved. Further increase in Reynolds number decreases the drag reduction. At very high Reynolds numbers, there is increased centrifugal forces [103] resulting in lower shear stress near the inner wall and higher shear stress in the region close to the outer wall. The implication of this is the uneven distribution of air bubbles and thus reduced drag reduction.

In the knowledge of the Authors, no published research is available that investigates the effect of micro-bubble size on DR in curved pipes. There is therefore need for more research to enhance understanding of any possible effect of micro-bubble size on DR [28]. In the application of micro-bubbles as drag-reducing agents for straight channel flow, conflicting reports exist on its effect on DR. It suffices to say, however, that bubble behaviour is size dependent, thus DR is expected to be influenced by micro-bubble size. In general, small sized bubbles will be better retained in the liquid under the action of centrifugal forces. Hence, it would be expected that the smaller the size of the bubbles the more effective it’ll be as a DRA.

A number of early researchers chose to present their results in terms of flow rates rather than drag. The limited studies in this area have focussed on the application of polymer DRAs in curved pipe flows. There appears to be an agreement among the limited reports that addition of DRPs results in increased flow rate particularly at low and moderate Dean numbers [63], [100].

Barnes and Walters [60] reported that, for fully developed turbulent flows in curved pipes, there is decrease in flow rate after adding polymer. It was suggested that the suppression of turbulence may have an adverse effect on the flow rate at high Reynolds numbers. Given that a number of recent studies have reported DR in the turbulent flow regime, it is possible that the polymer used in that study has degraded at the turbulent flow conditions studied. They also reported an increase in flow rate with increase in polymer concentration and a negligible influence of pipe curvature on the effectiveness of the DRPs in the laminar and transition flow regimes. Though further research is required to understand the effect of fluid characteristics and pipe geometry on the flow rate of DRAs, the limited research available suggest that flow rates would increase in the region of DR.

The secondary flow observed for the flow of Newtonian fluids in bends and curves results from centrifugal forces associated with such flow. The secondary flow of spiral form superimposes on the axial primary flow and there is also reduction in flow rate as a result of higher dissipation resulting from secondary flow compared to primary flow. The maximum axial velocity in curves and bends is shifted to the outer side of the curve. As Dean number increases the secondary flow become more confined to a thin area near the pipe wall [19]. At higher Reynolds number additional pairs of vortices appear and multiple solutions exist [104].

It has been suggested that drag-reducing agents would have an effect on secondary flows [10]. At high flow rates, the secondary flow field can be categorised into two regions. These are the shear free mid-region and the offside boundary layer region [105]. The non-Newtonian characteristic of fluid changes the thickness of the shedding layer. For pseudo-plastic fluids the shedding layer becomes thicker, whereas for dilatant fluid flow it is thinner than that of Newtonian fluids. This thickening or thinning effect may, to a small or large extent, alter the secondary flow. Fig. 13a shows the paths of fluid particles projected on the cross section of the pipe. The extremes of m=1 and m=0represents viscoelastic and Newtonian viscous liquids respectively. It is seen from the figure that, the effect of elasticity (measured roughly by m ) on the projected streamlines is small. However, the neutral point for the viscoelastic liquid is slightly nearer to the outer edge of the pipe compared to that for the Newtonian liquid.

The elasticity of the liquid has a profound effect on the pitch of the spirals in which the liquid particles move along the central plane (Fig. 13b). Fig. 13b shows that a decrease in m leads to a major increase in the curvature of the streamlines in the central plane. The main effect of elasticity on the flow of viscoelastic liquids through a curved pipe is to decrease the curvature of the streamlines in the central plane and to increase the fluid flux through the pipe [63], [98].

For a third-order fluid (see Coleman and Noll [103]), Jones [80] presented correlation (Eq. 15) for the streamline function, which describes the secondary flow in the cross-section of curved pipes.

\(\Psi=\frac{2L\alpha_{1}}{\text{ρa}}\left[\left(\frac{1}{144}+\frac{\alpha_{3}^{{}^{\prime}}}{48}\right)r_{1}-\left(\frac{1}{64}+\frac{\alpha_{3}^{{}^{\prime}}}{24}\right)r_{1}^{3}+\left(\frac{1}{96}+\frac{\alpha_{3}^{{}^{\prime}}}{48}\right)r_{1}^{5}-\frac{r_{1}^{7}}{576}\right]\cos\cos\ \alpha\ \)(15)

Eq. 15 indicates that, for third-order fluids, the non-Newtonian effect on secondary flow streamline could be associated mainly to the elastic behaviour (\(\alpha_{3}\)) of the fluid.

It has also been suggested that an analogy existed between the counter-rotating secondary flow vortex superimposed on the primary flow in curved pipes and the vortex pair at the near wall region of turbulent shear flow in straight pipes [65]. Since drag reduction is a phenomenon of the near wall region where the flow is primarily a shear flow, it is suggested that any mechanism that results in this phenomenon would also affect the secondary flow in curved pipes, at least in the laminar flow regime. It should be stated here that this assumption relies on the notion that secondary flow, like turbulent flow, is dissipative. Though a few other studies [87], [107], [108] made brief mention of the effects of DRAs on secondary flows, there is insufficient data from which concrete conclusions can be drawn.

There are very few studies on the effect of DRAs on secondary flows in bends and though the limited reports agree that such effects exist, there is no clarity on whether DRAs suppresses or enhances secondary flows. In the study carried out by Jones and Davies [96] using very dilute polyacrylamide and Kezan solution, the onset of non-Newtonian effects was around Dean number of 300. This is the region where secondary flow with a Newtonian fluid is sufficiently strong enough to cause appreciable deviation from Poiseuille flow . Munekata et al. [95] in their study of viscoelastic fluid flow in square-section elbow bends suggested that centrifugal effects are suppressed by viscoelastic effect of the fluid flow. They reported that secondary flow for Newtonian fluids increases gradually downstream while for viscoelastic fluids it decreases slightly resulting in DR. Their result was not corroborated by any other research findings, and further study is therefore required.

Studies show that, flow transition in curved pipes occurs at much higher Reynolds number than in straight pipes. There is also delayed onset of turbulence with increase in curvature. Taylor [106] in one of the early researches in this area showed that streamline motion persisted to Reynolds number of about 6000 in curved pipe of\(\frac{\text{\ a}}{R}=\frac{1}{18}\). The mechanism by which turbulence is produced in curved flow varies with the location in the curves [110]. Turbulence near the inner wall results from gradual superposition of higher order frequencies on the fundamental frequency. On the other hand, turbulence, near the outer wall, results from high frequency bursts near the outer wall. The sinusoidal oscillations near the inner wall always precedes the turbulent bursts [19]. The transition region for flow of Newtonian fluids in straight pipes is associated with violent flashes which is not the case in curved pipes. Also, the pressure fluctuation for fully developed turbulent flow in curved pipes is relatively damped.

The transition from laminar to turbulent flow regime in curved pipes is gradual and sometimes difficult to identify. This transition is even more gradual in the case of non-Newtonian drag-reducing fluid flow in curved pipes [33], [88], [111]. A delayed and gradual transition from laminar to turbulent regime occurs for flow of DRAs through curved pipes [23]. Two factors could be responsible for this: turbulence suppression in curved flow geometry, and effect of drag-reducing agent on flow transition. Effect of DRAs on flow transition in curves and bends depends on the curvature of the bend and concentration of drag-reducing agent. Fig. 5 shows that the critical Reynolds number decreases with curvature and increases with concentration of surfactant. Transition to turbulent flow occurred when the wall shear of the DRA exceeded the critical wall shear stress under strong mechanical load at high Reynolds numbers.

The critical Reynolds numbers also depend on the temperature especially in the turbulent regime. In separate experiments conducted by Inaba et al. [70] and Aly et al. [22] using surfactants in the temperature range of 5 – 20^oC, it was observed that critical modified Reynolds number \(N_{\text{Re}_{\text{crit}}}^{{}^{\prime}}\) increases with increase in temperature. This is associated with the critical wall shear stress at the wall which increases with temperature.

Several theoretical and empirical models are available for predicting friction factor of non-Newtonian fluids through curved pipes. In majority of the correlations, friction factors are simple functions of the Dean number and curvature ratio, \(\frac{a}{R}\), of the pipe. In general, at low Dean number the friction factor can be defined as a sole function of Dean number, \(N_{\text{Dn}}\). At higher Dean numbers, the frictional characteristic of flow not only depend on \(N_{\text{Dn}}\), but also on \(\frac{a}{R}\). Most of these correlations appear in the form of ratios of friction factors in curved pipes to that in straight pipes at the same conditions. Some researchers [86], [112], [113] presented friction factor correlation for drag-reducing fluids in both straight and curved pipes in terms of the Deborah number \(N_{\text{De}}\) defined as:

\(N_{\text{De}}=\frac{\text{characteristic\ fluid\ time}}{\text{characteristic\ flow\ time}}\)(16)

Table 1 presents a summary of friction factor correlation for non-Newtonian fluids in curved pipes.