3.1. Validation and grid independence

The validation and verification of the present methodology was presented in detail by Kumar et al. (Reference Kumar, Roy and Ali2016a) and Kumar & Roy (Reference Kumar and Roy2016) for a number of flow problems including flow past oscillating cylinders for Newtonian fluids. Here, we present the validation cases for flow of Newtonian and Carreau fluids ($Cu = 1.0$, $n = 0.25$) over a stationary cylinder at Reynolds number $100$. This Reynolds number falls in the regime of two-dimensional (2-D) laminar vortex-shedding for both the fluids (Coelho & Pinho Reference Coelho and Pinho2003a). All subsequent calculations on an oscillating cylinder are performed at this Reynolds number.

3.1.1. Vortex-shedding frequency

Table 2 reports the respective Strouhal numbers for Newtonian and shear-thinning fluids obtained using four different grid sizes, and compares them with the available literature data. Based on this observation, a grid size of $D/50$ (near the cylinder) has been considered for all subsequent studies reported in this paper.

Table 2.

Strouhal numbers for flow past a stationary circular cylinder at $Re = 100$.

3.1.2. Viscosity and vorticity contours

The time-averaged viscosity contour (in Pa s) in a Carreau fluid is shown in figure 6 for the flow over the stationary cylinder. The distribution of the viscosity is similar to the behaviour reported by Bailoor et al. (Reference Bailoor, Seo and Mittal2019). It is observed that the low-viscosity region is concentrated around the upstream stagnation point and in the wake region. On the other hand, the low-shear far-wake and free stream locations have nearly constant viscosity ($\mu _{0}$). Low values of viscosity in the wake zone also result in a shorter formation length compared with that for the Newtonian fluid. Figure 7 shows an instantaneous distribution of vorticity. These observations are also similar to the results of Bailoor et al. (Reference Bailoor, Seo and Mittal2019).

Figure 6.

Time-averaged viscosity contour for a Carreau fluid over a fixed cylinder.

Figure 7.

Instantaneous vorticity contour for a Carreau fluid over a fixed cylinder.

3.2. Domain independence test

Three different domain sizes are considered with respective widths, $H = 12D$, $16D$ and $20D$. The flow features are apparently the same for all the domains for Newtonian and Carreau fluids. Average drag coefficient and r.m.s. of the lift coefficient are calculated for flow over the fixed cylinder and compared with the reported data in table 3. It can be noticed that for both Newtonian and Carreau fluids, mean and r.m.s. of the lift and drag coefficients for $H = 16D$ and $20D$ agree well with the available data. We consider the domain configuration with $H = 16D$ for performing all subsequent calculations. It should also be mentioned that force variations over the oscillating cylinder and the vortex patterns are also observed to be domain independent.

Table 3.

Lift and drag coefficients (average and r.m.s. values) for flow past a stationary circular cylinder at $Re = 100$ with different domain sizes.

3.3. Investigations on the lock-in phenomena

Lock-in phenomena for the forced vibration of a cylinder in the uniform flow of a Newtonian fluid have been investigated in detail by a number of researchers (Bishop & Hassan Reference Bishop and Hassan1964; Koopmann Reference Koopmann1967; Karnidakis & Triantafyllou Reference Karnidakis and Triantafyllou1989). There have been certain discrepancies in the lock-in criterion proposed by different authors. To address this issue, Kumar et al. (Reference Kumar, Navrose and Mittal2016b) proposed an inclusive criterion based on the peak frequencies of the power spectrum of lift force. Accordingly, two conditions are checked: $(a)$ the dominant frequency in the power spectrum of the lift coefficient is the same as the cylinder oscillation frequency and $(b)$ if any other peak frequencies are present in the power spectrum, they are integer multiples of the cylinder oscillation frequency. If both conditions are satisfied, the flow is categorized to be in the lock-in mode. If condition $(a)$ is satisfied but not $(b)$, the flow is said to be in the transition state. For all other cases, the system is considered to be in the non-lock-in state. We have attempted to identify the lock-in states for both Newtonian and Carreau fluids over a range of amplitudes and frequencies of the oscillation using this criterion. We also present the Lissajous phase to identify the lock-in phenomena with better clarity.

3.3.1. Lock-in behaviour with frequency variation

Earlier studies suggest that the lock-in phenomenon is observed when the oscillation frequency is close to the Strouhal frequency. So, to compare the lock-in behaviour in Newtonian and Carreau fluids, we have varied the frequency ratio as $f^{*} = 0.6$, $0.8$, $1.0$ and $1.2$ while keeping the oscillation amplitude fixed to a moderate value ($A^{*}=0.4$). The power spectrum of the lift coefficient for these different cases is shown in figure 8. In all of these cases, we can observe a peak frequency at ${f}/{f_y}= 1.0$, which is the oscillation frequency of the cylinder. However, at a low-frequency ratio ($f^{*} = 0.6$), other peaks are observed at higher frequency which are not integer multiples of the oscillation frequency. At this low frequency and low-oscillation amplitude, the cylinder inertia effects are small and, therefore, the stationary vortex-shedding mode is dominant. The peak frequency is observed at ${f}/{f_y} = 0.167$, which is the same as ${f}/{f_{St}} = 1$ (as $f^{*}={f_y}/{f_{St}} = 0.6$ here). Consequently, this case corresponds to a non-locked-in case in both fluids. The moderate frequencies, $f^{*} = 0.8$ and $1.0$, correspond to locked-in cases with the only spectrum peak present at the oscillation frequency of the cylinder. At a further higher frequency, $f^*= 1.2$, smaller low-frequency peaks are observed near the main peak frequency (${f}/{f_y} = 1.0$), which result in a departure from the lock-in behaviour. These fluctuations are more dominant in the Carreau fluids compared with the Newtonian case. We further look into the Lissajous pattern (figure 9) to discuss this departure.

Figure 8.

Power spectrum of lift coefficient (PSD): (a,b) $A^{*} = 0.4$ and $f^{*} = 0.6$; (c,d) $A^{*} = 0.4$ and $f^{*} = 0.8$; (e,f) $A^{*} = 0.4$ and $f^{*} = 1.0$; (g,h) $A^{*} = 0.4$ and $f^{*} = 1.2$, for (a,c,e,g) Newtonian (black line) and (b,d,f,h) Carreau fluids (red line).

Figure 9.

Lissajous patterns – lift coefficient with the oscillation displacement of cylinder: (a) $A^{*} = 0.4$ and $f^{*} = 0.6$; (b) $A^{*} = 0.4$ and $f^{*} = 0.8$; (c) $A^{*} = 0.4$ and $f^{*} = 1.0$; (d) $A^{*} = 0.4$ and $f^{*} = 1.2$, for Newtonian (black line) and Carreau fluids (red line).

In the cases with $f^* = 0.8$ and $f^* = 1.0$, where a single dominant peak is observed at ${f}/{f_{St}} = 1.0$ and no peak is observed at a frequency lower than the oscillation frequency (figure 8c–f), the Lissajous patterns show a single curve indicating a constant phase difference between the cylinder displacement and lift force. The observations are similar for both Newtonian and Carreau fluids. These are the locked-in cases. At $f^* = 0.6$, both the fluids show higher frequency peaks in the lift spectrum and the Lissajous patterns show a non-repeating aperiodic behaviour and this is identified as a non-locked-in case for both the fluids. At higher oscillation frequency ($f^* = 1.2$), we observe small-amplitude noise near the main dominant peak at ${f}/{f_y} = 1.0$. The Lissajous plot for this case also reflects this departure from lock-in behaviour showing multiple lines instead of a single repeating curve. This non-lock-in behaviour is more pronounced in the Carreau fluid.

3.3.2. Lock-in behaviour with amplitude variation

At moderate-oscillation amplitude ($A^{*} = 0.4$), both the fluids exhibit lock-in behaviour when the oscillation frequency matches (or is close to) the Strouhal frequency. Now, to investigate the lock-in behaviour for different amplitude ratios, we keep the oscillation frequency fixed at $f^{*} = 1.0$ and increase the amplitude. In figure 10, the power spectrum of the lift coefficient is presented for the amplitude ratios of 0.6, 1.0 and 1.2 with $f^*= {f_y}/{f_{St}} = 1.0$. The low-amplitude ratio case ($A^* = 0.6$) presents a single dominant peak at ${f}/{f_y} = 1.0$ indicating locked-in flow. At a higher amplitude ratio, $A^* = 1.0$, the Carreau fluid shows another peak at ${f}/{f_y} = 3.0$, which is an integer multiple of the peak dominant frequency, so this case is still locked-in. However, at an even higher amplitude ratio ($A^* = 1.2$), the dominant peak in the Carreau fluid is still at ${f}/{f_y} = 3.0$. This clearly shows a deviation from the locked-in mode as per criterion (a) discussed above. For Newtonian fluids, although peaks are visible at higher frequencies, the dominant frequency is still at ${f}/{f_y} = 1.0$ satisfying the lock-in criteria. The Lissajous patterns in figure 11 also confirm these observations.

Figure 10.

Power spectrum of lift coefficient (PSD): (a,b) $A^{*} = 0.6$ and $f^{*} = 1.0$; (c,d) $A^{*} = 1.0$ and $f^{*} = 1.0$; and (e,f) $A^{*} = 1.2$ and $f^{*} = 1.0$, for (a,c,e) Newtonian (black line) and (b,d,f) Carreau fluids (red line).

Figure 11.

Lissajous patterns – lift coefficient with the oscillation displacement of cylinder: (a) $A^{*} = 0.6$ and $f^{*} = 1.0$; (b) $A^{*} = 1.0$ and $f^{*} = 1.0$; and (c) $A^{*} = 1.2$ and $f^{*} = 1.0$, for Newtonian (black line) and Carreau fluids (red line).

We can infer that at $f^{*}=1.0$, the lock-in behaviour is similar for both Newtonian and Carreau fluids up to $A^*=1.0$, except for minor deviations. However, as the amplitude ratio increases to $A^* = 1.2$, Newtonian fluid shows lock-in behaviour but the Carreau fluid behaves differently.

3.3.3. Variation in drag coefficient

figure 12 shows variation in the drag forces during different oscillation cycles. We can observe that the locked-in cases in figure 12(b,c) show synchronized behaviour of the drag variation with a frequency twice that of the oscillation frequency for both Carreau and Newtonian fluids. However, the non-locked-in case (figure 12a) shows aperiodic variation of the drag coefficient. Figure 12(d) shows a variation in drag coefficient for $A^{*} = 1.2$ and $f^{*} = 1.0$ for both fluids. The Newtonian fluid shows periodic variation with a constant peak amplitude over different cycles. However, for the Carreau fluid, different peak values of $C_d$ are observed during the oscillation cycle as this is a non-locked-in case. It should be mentioned that the cycle-averaged drag coefficients are slightly smaller in the Carreau fluid compared with the Newtonian fluid (table 4).

Figure 12.

Variation in drag coefficient during cylinder oscillation: (a) $A^* = 0.4\ {\rm and}\ \,f^* = 0.6$; (b) $A^* = 0.4\ {\rm and}\ \,f^* = 1.0$; (c) $A^* = 1.0\ {\rm and}\ \,f^* = 1.0$; and (d) $A^* = 1.2\ {\rm and}\ \,f^* = 1.0$.

Table 4.

Average drag coefficient ($C_{d,avg}$).

It is well known that the lock-in behaviour results from the synchronization of the vortex shedding with the cylinder oscillation cycle (Barbi et al. Reference Barbi, Favier, Maresca and Telionis1986; Yokoi & Kamemoto Reference Yokoi and Kamemoto1994). In the following sections, we investigate the vortex dynamics for different oscillation cases and try to observe the differences in the behaviour of Newtonian and Carreau fluids, especially at the higher oscillation amplitudes.

3.4. Vortex dynamics for different oscillation cases

Figure 13 presents the lock-in and non-lock-in modes in a $f^*-A^*$ map for both Newtonian and Carreau ($Cu=1.0$, $n=0.25$) fluids along with their respective vortex advection patterns. We can observe that the low- or moderate-amplitude locked-in cases result in synchronized 2S vortices. In these cases, the vortex pattern is similar for both the fluids with some differences in peak vorticity and vortex wavelengths (discussed in later subsections). Furthermore, in the non-lock-in cases at extreme frequencies ($\,f^*=0.6$ and 1.2) at $A^*=0.4$, we can observe the formation of pair vortices (P) in both the fluids. However, for the high-amplitude oscillation case ($A^*=1.2, f^*=1.0$) both fluids show a P+S mode, though the flow structures are substantially different. Vortices generated in the Carreau-fluid case show stronger asymmetry compared with the Newtonian-fluid case. This case also shows different lock-in behaviour in different fluids. We seek to understand these differences in more detail in a later subsection.

Figure 13.

Lock-in behaviour and vortex modes for Carreau ($Cu=1.0$, $n=0.25$) and Newtonian fluids over different amplitude and frequency ratios (cylinder is in the mean position).

3.4.1. Vortex dynamics of 2S mode during lock-in oscillation

The 2S-mode vortex dynamics is qualitatively similar in the Newtonian and Carreau fluids. Here, the shedding of two single (2S) vortices is synchronized with the oscillation cycle. In figure 14, we show vortex contours at different phases of an oscillation cycle for a Newtonian fluid at $f^* = 1.2$, $A^* = 0.4$. It should be mentioned that similar patterns of vortex motion are observed for all locked-in 2S-mode cases for both Newtonian and Carreau fluids. In figure 14, two predominant vortical regions, the positive shear layer (PS layer) and the negative shear layer (NS layer) can be observed along with a minor arc base region vorticity (ABRV) on the cylinder surface. The vorticity dynamics after the separation point is largely governed by the interaction of the ABRV with the shear layers present at the upper and lower surfaces of the cylinder (Krishnan et al. Reference Krishnan, Agrawal, Sharma and Sheridan2016, Reference Krishnan, Agrawal, Sharma, Thompson and Sheridan2019). Figure 14 depicts the growth and decay of the positive and negative ABRVs for a complete oscillation cycle.

Figure 14.

Vorticity dynamics of the 2S mode – snapshots of vortex contours over a complete oscillation cycle ($f^*=0.8, A^*=0.4$, Newtonian fluid). PABRV, positive signed arc base region vorticity; NABRV, negative signed arc base region vorticity; PS layer, positive shear layer; and NS layer, negative shear layer. The growth and decay of PABRV are shown in circles in panels (a–d,i), while the growth and decay of NABRV are shown in circles in panels (e–h). Vorticity scale is the same as that in figure 13.

At the beginning of the present cycle, PABRV is observed over the cylinder surface (wake side), which grows further in the subsequent phases and reaches a peak (figure 14a–c) and then decays (figure 14d). Through these steps, it feeds vorticity to the lower PS layer and the PS layer undergoes a stretching. Note that the growth and the decay of PABRV and stretching of the PS layer are also associated with the movement of the NS layer. The NS layer weakens by the growth of PABRV and it breaks down to form an S vortex during this phase. The mechanism of vortex shedding arising from activities of the base region vorticity is discussed in detail by Krishnan et al. (Reference Krishnan, Agrawal, Sharma, Thompson and Sheridan2019) for flow past a square cylinder. In the next half of the cycle, a NABRV is observed which results in the formation of the other S vortex. The mode of vortex shedding is similar in Carreau fluids.

The locations of the vortex cores are identified using the $Q$-criterion. Hunt, Wray & Moin (Reference Hunt, Wray and Moin1988) formulated the criterion for identification of vortices in an incompressible flow. Here, $Q = (\|\varOmega \|^2 - \|S\|^2)/2$ is a measure of the local rotation rate in excess of the strain rate. When $Q > 0$, the vortex is identified as a region where there is more rotation than straining. Figure 15 shows the location of the vortices as observed using the $Q$-criterion and maximum vorticity magnitude for all 2S-mode cases. The distance between two consecutive vortex cores is measured as half of the vortex wavelength, as marked in figure 15. This wavelength is nearly constant for a particular flow case over the passage of vortices. The Carreau fluid shows a 4–10 % shorter wavelength than the Newtonian fluid. This can be attributed to the fact that the shear-thinning nature of Carreau fluids reduce the viscosity near the vortices and therefore restricts their diffusion.

Figure 15.

Vorticity and $Q$-criterion contour for Newtonian and Carreau fluids at different amplitudes and frequencies. Vortex cores are shown with a maximum peak of the $Q$-criteria contour. Inner vortex separation lengths are also shown. (a) $A^{*} = 0.4, f^{*} = 0.8$; (b) $A^{*} = 0.4, f^{*} = 1.0$; (c) $A^{*} = 0.6, f^{*} = 1.0$; and (d) $A^{*} = 1.0, f^{*} = 1.0$.

3.4.2. Vortex dynamics for non-locked-in-mode formation of a pair of vortices

Figure 16 shows the formation and advection of vortices in the strongly non-locked-in case ($A^{*}=0.4, f^{*}=0.6$). The flow structures show consecutive shedding of the single (S) and pair (P) vortices but the vortex-shedding cycle does not synchronize with the cylinder oscillation cycle. The pattern of P and S vortices are similar for both the Newtonian and Carreau fluids. However, the inner vortex separation lengths are shorter for the Carreau fluid. It is important to observe that the ABRV does not show periodicity over a complete oscillation cycle. A combined ABRV (CABRV) is observed, where both positive and negative vorticities exist simultaneously at the cylinder arc base. Owing to the non-synchronization of the growth and decay of ABRV with cylinder oscillation, the variation of lift force does not synchronize with the cylinder oscillation cycle. Hence, this oscillation shows a non-lock-in mode. Formation of CABRV is also observed in the non-locked-in case with $A^*=0.4, f^*=1.2$, where 2P vortices are observed in both fluids. However, the smaller values of wake-region viscosity in Carreau fluids result in a shorter vortex formation length and introduces higher levels of fluctuations.

Figure 16.

Vorticity contours during $A^{*} = 0.4, f^{*} = 0.6$ oscillation cycle for Newtonian and Carreau fluids.

3.4.3. Vortex dynamics of (P+S) mode – $f^* = 1.0$, $A^* = 1.2$

Oscillations with $A^* = 1.2$, $f^* = 1.0$ show a P+S vortex mode in both fluids. Interestingly, this high-amplitude oscillation is the only case in our investigation in which the Newtonian fluid shows lock-in behaviour but the Carreau fluid behaves differently (figure 11c). Figure 17 shows that in the first half of the cycle, P vortices are advected and the later half sheds an S vortex. The vortex patterns in Newtonian and Carreau fluids are visibly different in this oscillation cycle. The Newtonian fluid shows a connected region of negative vorticity joining the S vortex with the negative vortex of the P pair (figure 17b). The S vortices in the Newtonian fluid are weaker and strained. A $Q$-criteria-based visualization also helps us to identify these vortices distinctively (figure 18). In the Carreau fluid, the vortices are discrete and follow a strong asymmetric pattern. The Carreau fluid vortices are stronger and they orient themselves quickly along the flow direction as they move downstream. The fast Fourier transformation (FFT) of the downstream location streamwise velocity is also presented (figure 19). In the Newtonian fluid, we can see that the peak frequency is twice that of the oscillation frequency, which indicates the advection of P and S vortices during one complete oscillation cycle. In the Carreau fluid, FFT shows multiple harmonics with a higher level of unsteadiness. The formation of S and P vortices as well as the breaking down of the negatively connected region of vorticity (one S and one of the P pair) are reflected by this signal. This activity further contributes to the fluctuations in the lift force (figure 11c) and sets in the no-lock-in behaviour.

Figure 17.

Vorticity contours during an $A^{*} = 1.2, f^{*} = 1.0$ oscillation cycle for Newtonian and Carreau fluids.

Figure 18.

$Q$-criterion map to locate the vortex cores in the $A^{*}=1.2, f^{*}=1.0$ case.

Figure 19.

FFT of streamwise velocity in the wake region for $A^{*}=1.2, f^{*}=1.0$ Newtonian (a) and Carreau (b) fluids.

The ABRV shows an interesting behaviour for this case (figure 17). We can see that both the positive and negative ABRV are aligned with the cylinder surface near the bottom (or top) half of the cylinder during its upward (or downward) motion. The formation of ABRV in this location on the cylinder surface arises from the higher acceleration of the cylinder in this oscillation cycle. Pressure contour and streamline maps show a low-pressure region at this location and a saddle point in its vicinity (figure 20). The low-pressure zone is wider in the Newtonian fluid while disconnected and stronger negative pressure regions are visible during the motion of the Carreau fluid. The pressure distribution reflects the differences in the vortex patterns in these two fluids. We will try to explain this behaviour through the vorticity transport equation in a later subsection.

Figure 20.

Pressure contour and streamlines for $A^{*}=1.2, f^{*}=1.0$ Newtonian (a) and Carreau (b) fluids.

3.5. Relation of arc base region vorticity and the vortex modes

Following the discussions in the previous section, we can attribute the features of the vortex motion in flow past an oscillating cylinder to two key factors: (i) the behaviour of ABRV, which is responsible for the formation of the S or P vortices; and (ii) the viscous diffusion, which governs the spreading of the vortices in the downstream region. The formation of the vortices is largely governed by the interaction of the ABRV with the streamwise shear layers. For the locked-in cases, the PABRV interacts with the negative shear layer and forms a negative S vortex during the first half of the cycle. In the next half, a positive S vortex is formed similarly. The non-locked-in cases show formation of P vortices. These vortices arise from the formation of CABRV. When these combined vortices show an overlapping pattern, i.e. the PABRV forms an envelop for the NABRV (which is attached to the cylinder surface), both the vortices simultaneously cut the PS and NS layers and form a P vortex. In the other case, where both the positive and negative ABRV are side by side aligned over the cylinder surface, a P vortex is formed during the first half of the cycle and an S vortex is formed in the next half. This behaviour, listed in table 5, is similar in both the Newtonian and Carreau fluids.

Table 5.

Effects of arc base region vorticity on vortex shedding.

3.6. Generation of vorticity during oscillation cycles and enstrophy

Vortex patterns near the cylinder surface are independent of the rheology of the fluid (Newtonian or Carreau). However, these patterns, as well as the formation of ABRVs, are strongly related with the cylinder motion. Therefore, it will be useful to find a relation between the cylinder motion and the generation of vorticity on the cylinder surface. In a 2-D flow, the vorticity flux from a curved surface (where $n$ is the surface normal coordinate) can be defined as

(3.1)\begin{equation} \sigma_z = \frac{\partial \omega_z}{\partial n}. \end{equation}

Here, $\sigma _{z}$ is the measure of the amount of vorticity that is entering the flow from the solid walls. Now, using the momentum equations, the vorticity gradient can be related to the pressure gradient on the surface of a stationary object (where $\boldsymbol {V} = 0$) and the vorticity flux can be obtained as

(3.2)\begin{equation} \frac{\partial p}{\partial s} = \mu \frac{\partial \omega_z}{\partial n} = \mu \sigma_z, \end{equation}

where $s$ is the coordinate along the surface tangent. Further, for an accelerating surface, the local inertia force augments the vorticity flux as

(3.3)\begin{equation} \mu \sigma_z = \frac{\partial p}{\partial s} + \rho a_s, \end{equation}

where $a_s$ is the tangential acceleration of the surface.

Proof of the above relation may be found in the paper by Panton (Reference Panton2013). Therefore, the influx of vorticity from the cylinder wall is a function of the surface pressure gradient and the tangential component of the cylinder acceleration. Coelho & Pinho (Reference Coelho and Pinho2004) showed that for flow over fixed cylinders, the pressure gradient in a shear-thinning flow is moderately higher than that in a Newtonian flow. Also, the non-zero values of the pressure gradient are mostly restricted within the minimum pressure point and base pressure region on the wake side of the cylinder. However, as the viscosity of the Carreau fluid reduces near the solid wall, owing to higher strain rates, the vorticity flux increases as per (3.2) and (3.3) respectively for the fixed and oscillating cylinder. Consequently, non-Newtonian shear-thinning fluids show higher vorticity values in the vicinity of the cylinder (observed by Chhabra et al. (Reference Chhabra, Soares and Ferreira2004) and Bharti et al. (Reference Bharti, Chhabra and Eswaran2007) for fixed cylinders) compared with Newtonian fluids. For moving cylinders, this difference is further enhanced when the cylinder acceleration is high ((3.3)). So, the vorticity formation patterns remain similar for both the fluids but the magnitude of vorticity is higher in a Carreau fluid.

Enstrophy for this 2-D flow is obtained as

(3.4)\begin{equation} E = \int_{A} \omega_{z}^{2} \,\textrm{d}A. \end{equation}

Enstrophy is proportional to the total rotational energy of the vortices in the flow. The higher influx of vorticity ($\sigma _z$), in the case of Carreau fluids, results in higher levels of enstrophy of the flow. Figure 21 presents the enstrophy variation during the oscillation cycles for both Newtonian and Carreau fluids. Apart from the strongly non-locked-in case in figure 21(a) ($f^* = 0.6$, $A^* = 0.4$), enstrophy shows a periodic behaviour for both Newtonian and Carreau fluids. However, its magnitude is higher for Carreau fluids owing to the lower value of viscosity. It is also important to mention here that we have obtained a 20–30% reduction of the average viscosity in the Carreau fluid compared with the Newtonian fluid over all the oscillation cases. At higher oscillation frequency and amplitude, the enstrophy increases for both Newtonian and Carreau fluids as the vorticity flux increases owing to the cylinder acceleration ((3.3)). During a sinusoidal cylinder oscillation cycle, the magnitude of the acceleration attains a maximum value twice and results in the two peaks in the enstrophy plots. A full cycle of growth and decay of both the positive and negative ABRVs are observed during a complete oscillation cycle for all these cases except for the low-frequency low-amplitude non-locked-in case. In this case, the vorticity flux is mostly governed by the pressure gradient (owing to lower levels of cylinder acceleration) and thus is not forced to follow the oscillation cycle.

Figure 21.

Enstrophy fluctuation during cylinder oscillation cycles for Newtonian (black line) and Carreau fluids (red line): (a) $A^{*} = 0.4, f^{*} = 0.6$; (b) $A^{*} = 0.4, f^{*} = 1.0$; (c) $A^{*} = 1.0, f^{*} = 1.0$; and (d) $A^{*} = 1.2, f^{*} = 1.0$.

3.7. Vorticity transport rate analysis

Figure 22 shows the snapshots of viscosity contours of the Carreau fluid at the mean location of the cylinder during its course of upward motion for the different oscillation cases. It can be seen that the viscosity levels are lower in the vicinity of the cylinder, especially at the wake zone. The primary shear layers and the arc base region vortices show almost a 50 % reduced viscosity. Lower levels of viscosity are also found near the vortex cores.

Figure 22.

Viscosity contours for different oscillation cases in a Carreau fluid: (a) $A^{*} = 0.4, f^{*} = 0.6$; (b) $A^{*} = 0.4, f^{*} = 1.0$; (c) $A^{*} = 1.0, f^{*} = 1.0$; and (d) $A^{*} = 1.2, f^{*} = 1.0$.

While the smaller levels of viscosity in a Carreau fluid are responsible for higher vorticity generation near the cylinder wall and smaller inner separation length at the wake zone (figure 15), it should be noted that the patterns of vortex formation and advection of vortices do not show much difference from the Newtonian flow behaviour in most of the cases considered. (The supplementary movies 1–4 and 6–9 available at https://doi.org/10.1017/jfm.2021.485 are related to figures 22 and 23, showing the variations in viscosity and vorticity during the cylinder oscillation cycles.)

Figure 23.

Contours of ${\textrm {D}\omega _z}/{\textrm {D}t}$ for different oscillation cases – (a,c,e,g) Newtonian, (b,d,f,h) Carreau fluids: (a,b) $A^{*} = 0.4, f^{*} = 0.6$; (c,d) $A^{*} = 0.4, f^{*} = 1.0$; (e,f) $A^{*} = 1.0, f^{*} = 1.0$; and (g,h) $A^{*} = 1.2, f^{*} = 1.0$.

To understand the advection pattern and breaking of the vortices, we look into the vorticity transport equation ((3.5)):

(3.5)\begin{equation} \frac{\textrm{D}\omega_z}{\textrm{D}t} = \boldsymbol{\nabla} \boldsymbol{\cdot} (\eta (|\boldsymbol{\dot{\gamma}}|)\boldsymbol{\nabla} \omega_z) + \boldsymbol{\nabla} \boldsymbol{\cdot} f,\end{equation}

where $\eta (|\boldsymbol {\dot {\gamma }}|$ is the non-Newtonian kinematic viscosity and $f$ represents the non-Newtonian viscosity gradient term and is given as

(3.6) \begin{equation} f = \begin{bmatrix} \dfrac{\partial \eta|\boldsymbol{\dot{\gamma}}|}{\partial x} \left(\dfrac{\partial u}{\partial y} +\dfrac{\partial v}{\partial x}\right) -2\dfrac{\partial \eta|\boldsymbol{\dot{\gamma}}|}{\partial y}\dfrac{\partial u}{\partial x}\\ \dfrac{\partial \eta|\boldsymbol{\dot{\gamma}}|}{\partial y} \left(\dfrac{\partial u}{\partial y} +\dfrac{\partial v}{\partial x}\right) -2\dfrac{\partial \eta|\boldsymbol{\dot{\gamma}}|}{\partial x}\dfrac{\partial v}{\partial y} \end{bmatrix} . \end{equation}

As the flow is 2-D, only the $z$-component of vorticity and the $x$, $y$ components of velocity and co-ordinate are considered. This is to note that for the Newtonian fluid, kinematic viscosity is constant, $\eta$, and the vorticity transport equation is simply written as

(3.7)\begin{equation} \frac{\textrm{D}\omega_z}{\textrm{D}t} = \eta \nabla^2 \omega_z \end{equation}

In (3.5) and (3.7), the vorticity transport rate (VTR), ${\textrm {D}\omega _z}/{\textrm {D}t}$, is balanced by the net viscous diffusion of vorticity and it represents the rate of net growth and advection of the vorticity. For the Carreau fluids, we can express the right-hand side of (3.5) as a combination of viscosity multiplied by the Laplacian of vorticity (as in (3.7)) and another extra term owing to the gradients of non-Newtonian viscosity denoted as NN terms:

(3.8) \begin{equation} \frac{\textrm{D}\omega_z}{\textrm{D}z} = \underset{{term\ I}}{\underbrace{\eta ( |\boldsymbol{\dot{\gamma}}|) \varDelta^2\omega_z}} + \underset{{term\ II}}{\underbrace{{NN\ terms}}}. \end{equation}

In figure 23, we present the contours of VTR (${\textrm {D}\omega _z}/{\textrm {D}t}$) for Newtonian and Carreau fluids, respectively as obtained as the right-hand side of (3.7) and (3.8). It can be seen that the primary shear layers and ABRV show peaks of VTR for both the fluids owing to the high-vorticity gradients at these regions. However, the magnitudes of these VTR peaks (both positive and negative values) are higher in a Carreau fluid. In these cases, the high values of VTRs sharply reduce to smaller levels. Also, for high-frequency and high-amplitude cases (figure 23(b,d,f,h)) in a Carreau fluid, high-VTR zones are less spread in the streamwise direction. The case of $A^* = 1.2$, $f^* = 1.0$ is of particular interest here. At this oscillation cycle, both the fluids show P+S modes though the Newtonian fluid shows a connection between one of the P vortices and the S vortex but the Carreau fluid shows distinct strong vortices (figure 17). In figure 23(g,h), a higher rate of positive viscous diffusion of vorticity (or VTR) is observed between the P and S vortices in the Carreau fluid case. Owing to this high diffusion rate, the negative vorticity region between the P and S vortex (as shown in figure 19) diffuses quickly and distinct vortices are advected downstream.

For the Carreau fluid, the viscosity is a strong function of strain rate, and hence, the second term in (3.8) is related to the local strain rates. Therefore, the viscous diffusion for a Carreau fluid is augmented owing to the higher levels of local straining although viscosity can be of reduced magnitude at these regions. To quantify the relative contribution of both the components of viscous diffusion (term I and term II from (3.8)), we propose a correlation function:

(3.9) \begin{equation} \varOmega_c = \frac{4\times {term\ I} \times {term\ II}}{({term\ I}+ {term\ II})^2}. \end{equation}

If both the terms are equal, then this correlation, $\varOmega _c$, reaches a maximum value of unity. If both the terms are of opposite sign, then it shows a negative value. A high-negative value of $\varOmega _c$ implies that even when the individual terms are high, they are of opposite sign and hence they cancel each other, and the net viscous diffusion (or VTR) is small. Now, we compute $\varOmega _c$ for different oscillation cases of the Carreau fluid and present the instantaneous contours in figure 24. It should be mentioned that when both the terms are very small (${<}0.05$), the correlation value is clipped to zero for smoothness. Figure 24 identifies the zones with higher (${>}0.5$) and lower ($<= -1$) values of $\varOmega _c$. The side-by-side appearance of these two zones explains the high value of VTR (owing to same-signed values of both the terms) and its sharp drop (owing to opposite-signed comparable values of the terms). The negative values of $\varOmega _c$ obtained near the vortex cores (figure 24b,d) results in reduction of the viscous diffusion and the vortices retain their strength during advection downstream. In contrast, the decay of Newtonian vortices is much faster owing to a higher viscous diffusion. Also, the negative $\varOmega _c$ region is restricted mainly in the close vicinity of the vortex cores in the $A^* = 1.2$, $f^* = 1.0$ case. Therefore, the diffusion between the vortices is higher (as observed in figure 24d) and results in the formation asymmetric P+S vortices in Carreau fluids.

Figure 24.

Contours of $\varOmega _{c}$ for different oscillation cases in a Carreau fluid: (a) $A^{*} = 0.4, f^{*} = 0.6$; (b) $A^{*} = 0.4, f^{*} = 1.0$; (c) $A^{*} = 1.0, f^{*} = 1.0$; and (d) $A^{*} = 1.2, f^{*} = 1.0$.

3.8. Effect of Carreau fluid properties

Here, we present investigations on the effect of Carreau number ($Cu$) and power index ($n$) on the flow behaviour. The lock-in behaviour and vortex patterns are found to be similar over a range of Carreau fluid properties: $Cu = 1, 10$ and $n = 0.25, 0.5$. However, we can notice quantitative differences in terms of inner vortex separation lengths as shown in figure 25 for $A^* = 0.4$, $f^* = 1.0$. We can observe that the distance between the vortices reduce with smaller $n$ and higher $Cu$ values. However, the effect of $n$ is smaller at higher $Cu$. It should be mentioned that Pantokratoras (Reference Pantokratoras2016) observed that flow properties do not show much variation with respect to power index at higher $Cu$ for $Re > 40$. Nevertheless, the viscosity contours show strong sensitivity to $Cu$ and the wake region shows smaller viscosity levels at $Cu = 10$ (figure 26a). It can be inferred that the smaller viscosity in the wake region results in higher levels of vorticity near the vortex cores for $Cu = 10$ (figure 25). However, we can further observe that the vorticity transport rate is of same order for both Carreau numbers (figure 26b) and as a result of this, the vortex patterns are also similar in these two cases. We further inform that this similarity is observed in all oscillation cases for these different sets of $Cu$ and $n$ values (the supplementary movies 2, 5, 7 and 10 in the supplementary material are related to figure 26).

Figure 25.

Vorticity contours for different $Cu$ and $n$ values with $A^{*} = 0.4, f^{*} = 1.0$.

Figure 26.

(a,b) Viscosity contours and (c,d) VTR for different $Cu$ with $n = 0.25$ and $A^{*} = 0.4, f^{*} = 1.0$.