2.1 Rigid $\text{180}^{\circ }$ curved pipe

The $180^{\circ }$ curved tube with inner diameter of $d=12.7~\text{mm}$ and curvature ratio of $\unicode[STIX]{x1D6FF}=d/D=1/7$ was machined out of an acrylic block and polished to provide clear optical access. The cross-sectional measurements for this particular study were made at $45^{\circ }$ and $90^{\circ }$ along the curve. Schematics of the test section, laser sheet locations and camera directions are shown in figure 1.

2.2 Inflow conditions

A $94d$ length of straight pipe was attached to the curve inlet to minimize upstream effects and disturbances at the curved test section ( $d=12.7~\text{mm}$ ). Although the flow upstream of the curved section is fully developed (axially invariant), the flow inside the curve is developing. The inflow waveforms were produced by a gear pump (Ismatec model BVP-Z). Once the desired input voltage to the pump is obtained using the PID (proportional–integral–derivative) controller (Najjari & Plesniak Reference Najjari and Plesniak2017), the pump and PIV system are controlled by a customized LabView program and a USB data acquisition device (NI USB-6229).

Figure 1.

Schematic of experimental set-up, including test section, laser sheet location and camera direction.

2.3 Working fluid

The working fluid was a composition of water, glycerin and sodium iodide. The viscosity of the solution was set by the glycerin/water ratio, and the concentration of sodium iodide was varied to match the refractive index (RI) of the test section (sodium iodide concentration linearly increases the RI of glycerin–water mixture). Another advantage of adding sodium iodide to the solution is that it can prevent mould formation in the solution over time. Owing to oxidization of sodium iodide, the solution colour gradually changes over time, first to light yellow and then to red. Mixing approximately 0.5 g of sodium thiosulfate in 1 litre of the solution will restore the colour and make the fluid transparent again. It is found that the fluid with RI of 1.487 (measured by PAL-RI, Atago) matched the RI of our acrylic test sections perfectly.

The dynamic viscosity of the fluid was measured using a Discovery HR-2 (TA Instruments) rheometer with cone ( $1^{\circ }:00^{\prime }:11^{\prime \prime }$ and 40 mm diameter) and flat-plate geometry with $24~\unicode[STIX]{x03BC}\text{m}$ gap under steady shear for three times, and the results were averaged. The dynamic viscosity $\unicode[STIX]{x1D707}$ was found to be $\unicode[STIX]{x1D707}=6\times 10^{-3}\pm 0.3\,\%~\text{Pa}~\text{s}$ . The density of the fluid was measured and was equal to $\unicode[STIX]{x1D70C}=1713\pm 1.1\,\%~\text{kg}~\text{m}^{-3}$ , producing a kinematic viscosity of $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}=3.5\times 10^{-6}~\text{m}^{2}~\text{s}^{-1}$ .

2.4 Particle image velocimetry (PIV)

Phase-locked instantaneous velocity fields were measured using PIV. At each phase, 100 image pairs were acquired, which was shown to be sufficient for convergence (Najjari & Plesniak Reference Najjari and Plesniak2016). The PIV parameters are tabulated in table 1. DaVis 8 (LaVision) was used for the data acquisition and image post-processing.

Table 1.

PIV recording parameters.

For calibrating the PIV system, a custom-designed dotted map (with dot diameter of 0.5 mm and spacing of 1 mm) was created and printed on a transparency sheet. The printed map was then secured on a plastic fibre, allowing us to control the location of the calibration target accurately inside the test section. After the calibration target was placed inside the test section, the test section was filled with RI-matched fluid.

In the image post-processing step, the averaged background image was first subtracted, geometrical masking was then applied to the images, followed by the vector field calculation. A multi-pass option with decreasing window size from 64 pixels to 16 pixels with 50 % overlap was used for the interrogation area. This resulted in ${\approx}0.10~\text{mm}$ vector spacing and ${\approx}0.20~\text{mm}$ spatial resolution. In the final vector post-processing step, the vector allowable range, cross-correlation peak ratio criteria, median filter and $3\times 3$ smoothing are applied to the vector field.

The instantaneous velocity fields provided by DaVis software were input to an in-house Fortran code, which was used for further post-processing. In-plane spatial derivatives were calculated within the code using the second-order central finite difference method.

4.1 Boundary conditions

The simulations require inlet, outlet and wall boundary conditions. The inlet velocity profiles are defined at the inlet and a pressure outlet was used for the outlet. At the wall, either no-slip or slip conditions are enforced. A parabolic inlet velocity profile was used for steady-state flow simulations; inlet velocity profiles calculated analytically using the method in appendix B were used for transient flow simulations. After expressions for the inlet velocity profiles were obtained analytically, a user-defined-function (UDF) was written to introduce the correct inlet velocity profile at each physical time step for transient simulations. For these transient flows, the physical time step was set to $\unicode[STIX]{x0394}t=0.0005~\text{s}$ to satisfy the Courant–Friedrichs–Lewy (CFL) condition and guarantee convergence. The constructed geometry had a $5d$ straight inlet pipe upstream of the curve and a $10d$ straight pipe downstream of the curve to allow flow to develop and eliminate any upstream effect of the pressure outlet boundary condition.

Figure 4 shows analytical velocity profiles obtained by the method provided in appendix B. These profiles are similar to a plug profile at the beginning of impulsive flows, and as time passes the profile converges to the parabolic shape inherent in Poiseuille flow.

Figure 4.

Evolution of velocity profiles during the transient moderate-step flow rate at the inlet boundary condition obtained from analytical solution (appendix B). At $t=1.5~\text{s}$ , the flow has already reached the steady-state condition and the velocity profile has converged to a parabolic Poiseuille solution.

4.2 Grid independence

To obtain mesh-independent results, five different meshes (M1–M5, table 3) were generated for mesh convergence studies and the same simulations were performed using each one. Table 3 summarizes the parameters of each mesh and figure 5 shows the various mesh cross-sections. Meshes M1 and M5 are the coarsest and the finest mesh, respectively. Vorticity magnitude was computed for different meshes and tracked at several points inside the pipe. Figure 6 shows the vorticity magnitude at points P1–P3 (shown in figure 5) for five meshes. Refining the mesh from M4 to M5 does not change these results and, therefore, we conclude that converged results are achieved on the M4 mesh. However, since the elements near the core of the pipe in M4 are rather non-uniform in size, with rapid transition from small elements to larger ones, mesh M6 was generated to contain finer elements than M4 near the pipe core but coarser than M5. Ultimately, mesh M6 contains 3 043 840 elements and was used for all simulations.

Figure 5.

Six different meshes shown at the cross-section of the pipe. Meshes M1 and M5 are the coarsest and the finest mesh, respectively. Mesh M6 is finer than M4 but coarser than M5, and this mesh is used for the simulations presented herein. The convergence of vorticity magnitude at points P1–P3 is shown in figure 6.

Table 3.

Details of meshes M1–M5 that were used for the mesh convergence study. Mesh M6, which is finer than M4 in the cross-section, was used to run the final simulations.

Figure 6.

Vorticity magnitude along the streamwise direction at points P1–P3 shown in figure 5 for meshes M1–M5.

Figure 7.

Comparison of in-plane velocity magnitude ( $V$ ) from experimental PIV results (a) and CFD results (b) at three time instants ( $t=0.105$ , 0.525 and 1.365 s) for mid-step flow, at the $90^{\circ }$ cross-section of the curve. The experimental and simulation results match well with the correlation coefficient (4.2) of $c=97.9\,\%$ , 98.4 % and 97.4 % for $t=0.105$ , 0.525 and 1.365 s, respectively. The distribution of differences (c) between experimental and numerical in-plane velocity magnitude $V_{exp}(i,j)-V_{cfd}(i,j)$ .

4.3 CFD validation

The in-plane velocity and vorticity results of our simulations are validated by the experimental results to benchmark the numerical simulations. To perform the comparisons, numerical data are transferred to a structured Cartesian mesh that exactly resembles the experimental PIV data mesh. The side-by-side comparisons of in-plane velocity maps are provided in figure 7, as well as the probability density function (p.d.f.) of error ( $e_{ij}=V_{exp}(i,j)-V_{cfd}(i,j)$ ), which is a point-by-point difference of the in-plane velocity magnitude of experimental and CFD results. Figure 7 shows the in-plane velocity magnitude at the $90^{\circ }$ cross-section for the transient mid-step flow rate at three phases. Excellent agreement between the PIV and CFD results was obtained. The spatial correlation between the numerical and experimental results is also calculated for both velocity and vorticity maps over time using 2-D Pearson’s correlation given in (4.2), where the overbar refers to spatial average:

(4.2) $$\begin{eqnarray}c=\frac{\displaystyle \mathop{\sum }_{i}\mathop{\sum }_{j}(V_{exp}(i,j)-\overline{V}_{exp})\times (V_{cfd}(i,j)-\overline{V}_{cfd})}{\sqrt{\displaystyle \mathop{\sum }_{i}\mathop{\sum }_{j}(V_{exp}(i,j)-\overline{V}_{exp})^{2}}\sqrt{\displaystyle \mathop{\sum }_{i}\mathop{\sum }_{j}(V_{cfd}(i,j)-\overline{V}_{cfd})^{2}}}.\end{eqnarray}$$

The calculated correlations for velocity are 97.9 %, 98.4 % and 97.4 % for $t=0.105$ , 0.525 and 1.365 s, respectively, further validating our results. Moreover, the p.d.f. of error shown in figure 7 demonstrates sufficiently low error.

Comparison of vorticity results at different instants in time for mid-step flow rate from PIV and CFD data is shown in figure 8, which again shows an excellent match. The correlations between experimental and numerical streamwise vorticity fields are $c=97\,\%$ , 98.1 % and 98 % for $t=0.105$ , 0.525 and 1.365 s, respectively, which show a sufficiently high correlation.

Figure 8.

Comparison of streamwise vorticity component from experimental PIV results (a) and CFD results (b) at three instants in time ( $t=0.105$ , 0.525 and 1.365 s) for mid-step flow, at the $90^{\circ }$ cross-section of the curve. Small magnitudes of streamwise vorticity ( $|\unicode[STIX]{x1D714}_{n}|<3.2~\text{s}^{-1}$ ) are shown by white colour to delineate positive and negative regions. The experimental and simulation results agree with each other, with a correlation coefficient (4.2) of $c=97\,\%$ , 98.1 % and 98 % for $t=0.105$ , 0.525 and 1.365 s, respectively.

5.1 Two-dimensional vortex ID

The $d_{2}$ method (discriminant of non-real eigenvalues of velocity gradient matrix (Vollmers Reference Vollmers2001)) defined in (5.2) is a Galilean-invariant method used to identify the vortical structures in the secondary flow using 2-D data:

(5.2) $$\begin{eqnarray}d_{2}=\left(\frac{\unicode[STIX]{x2202}u_{x_{local}}}{\unicode[STIX]{x2202}x_{local}}+\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}Z}\right)^{2}-4\left(\frac{\unicode[STIX]{x2202}u_{x_{local}}}{\unicode[STIX]{x2202}x_{local}}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}Z}-\frac{\unicode[STIX]{x2202}u_{x_{local}}}{\unicode[STIX]{x2202}Z}\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}x_{local}}\right).\end{eqnarray}$$

This method is based on the swirling strength criterion (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999) definition. A negative value of $d_{2}$ indicates a vortex region. Since $d_{2}$ can only be applied to 2-D data, if local coordinates were not aligned with the main coordinate axes (e.g.  $45^{\circ }$ cross-section), a directional derivative formulation was used to calculate $d_{2}$ .

5.2 Three-dimensional vortex ID

In order to identify vortices in 3-D data, several methods such as $Q$ -criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988), swirling strength criterion (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999) and $\unicode[STIX]{x1D706}_{2}$ criterion (Jeong & Hussain Reference Jeong and Hussain1995) are available. For a detailed review on different vortex ID methods, see Chakraborty, Balachandar & Adrian (Reference Chakraborty, Balachandar and Adrian2005). In this study we used $\unicode[STIX]{x1D706}_{2}$ and swirling strength criteria for identifying vortices in three dimensions.

6.1 Evolution of vorticity

Figures 10 and 11 show the vorticity components normal to each plane for four instants in time under no-slip and slip wall conditions, respectively, for moderate-step flow rate. Supplemental movies 1 and 2 (available online at https://doi.org/10.1017/jfm.2019.510) also show these vorticity plots under these wall conditions, respectively. In figure 10, the vorticity component in the $Z$ direction ( $\unicode[STIX]{x1D714}_{z}$ , shown in the axial slices) is strong and generated due to the no-slip wall, and is initially concentrated in the vicinity of the wall while the streamwise vorticity ( $\unicode[STIX]{x1D714}_{n}$ ) in the cross-sections is weak. As time passes, $\unicode[STIX]{x1D714}_{n}$ strengthens and $\unicode[STIX]{x1D714}_{z}$ is transported to the interior regions. During this transient flow, positive $\unicode[STIX]{x1D714}_{z}$ (at the inner wall) forms a branch away from the inner wall at $t=0.24~\text{s}$ around the $45^{\circ }$ cross-section for the no-slip condition (figures 10 b and 12). This branch grows in time and reaches the $90^{\circ }$ cross-section at $t=0.315~\text{s}$ (figures 10 c and 12). The branch of $\unicode[STIX]{x1D714}_{z}$ at the interior region coincides with the location of the Lyne-type vortex at the $90^{\circ }$ cross-section of the pipe shown by $\unicode[STIX]{x1D714}_{n}$ in figure 10(c,d). Later in this section, a vortex ID isosurface will confirm that this branch of $\unicode[STIX]{x1D714}_{z}$ is attributable to the Lyne-type vortex.

Figure 10.

Vorticity component normal to each plane for moderate-step flow rate with no-slip wall condition at (a) $t=0.015~\text{s}$ , (b) $t=0.24~\text{s}$ , (c) $t=0.315~\text{s}$ and (d)  $t=0.9~\text{s}$ . Four cross-sectional slices are shown at $30^{\circ }$ , $45^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ locations, and the axial slice location is at $Z=-2.5~\text{mm}$ , indicated by the black line in the cross-sectional views. Vorticity component along the $Z$ direction (shown in axial slice) forms a branch at $t\approx 0.24~\text{s}$ between $30^{\circ }$ and $60^{\circ }$ near the inner wall. This vorticity region is connected to the Lyne-type vortex region later in time (see movie 1 for more phases).

Figure 11.

Vorticity component normal to each plane for moderate-step flow rate with slip wall condition at (a) $t=0.48~\text{s}$ , (b) $t=0.66~\text{s}$ , (c) $t=0.9~\text{s}$ and (d)  $t=1.2~\text{s}$ . Four cross-sectional slices are shown at $30^{\circ }$ , $45^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ locations, and the axial slice location is at $Z=-2.5~\text{mm}$ , indicated by the black line in the cross-sectional views. Vorticity component along the $Z$ direction (shown in axial slice) which is introduced by inlet velocity profile propagates through the curved pipe and forms the Lyne-type vortex region later in time (see movie 2).

The slip wall model is investigated in order to determine whether the Lyne-type vortex is associated with the vorticity that is generated at the inner wall of the curved pipe or if it can form even without wall vorticity. For the slip wall model, figure 11, similar behaviour of vorticity is observed, with some differences. Firstly, due to the elimination of the wall contribution in the generation of vorticity, the overall vorticity level is lower than that under the no-slip wall condition. Additionally, since the vorticity is not produced at the wall, it is either convected into the curved section (i.e. vorticity introduced by the inlet velocity profile) or generated in the curved section. Figure 11(a–d) and movie 2 show that positive and negative $\unicode[STIX]{x1D714}_{z}$ propagate into the curve from the inlet of the curve as time passes, and this vorticity region is not connected to the wall near the inlet. Therefore, this vorticity is initially introduced in the flow by the physical velocity profile introduced at the inlet boundary and then transported to the curved section. Another major difference between the slip and no-slip wall results is the latency in the formation of Lyne-type vortex in the slip model. For instance, the Lyne-type vortex first appeared at the $90^{\circ }$ cross-section at $t\approx 0.315~\text{s}$ for the no-slip wall, and at $t\approx 0.78~\text{s}$ for the slip wall model. Therefore, although the vorticity generated at the wall increases the strength of Lyne-type vortex and accelerates its formation, the Lyne-type vortex can be generated without the wall vorticity present albeit with lower strength and slower in time.

Figure 12.

Vorticity $Z$ component ( $\unicode[STIX]{x1D714}_{z}$ ) for the moderate-step flow rate with no-slip wall condition at (a) $t=0.015~\text{s}$ , (b) $t=0.21~\text{s}$ , (c) $t=0.24~\text{s}$ and (d)  $t=0.315~\text{s}$ at the axial slice of $Z=-2.5~\text{mm}$ . Points A, B, C and D are placed at the location of the $\unicode[STIX]{x1D714}_{z}$ branch at $t\gtrsim 0.21~\text{s}$ , which corresponds to the location of the Lyne-type vortex. The balance of vorticity transport equation terms are shown at these four points in figure 16 to support our hypothesis of vorticity convection.

Spatiotemporal evolution of $\unicode[STIX]{x1D714}_{z}$ in the slip wall model shows that positive $\unicode[STIX]{x1D714}_{z}$ appears at the inner wall region near the entrance to the curve (it is convected from the inlet boundary). When this shear layer enters the curved section, it moves away from the inner wall due to wall curvature, and when the flow rate is high enough, this shear layer peels off because it cannot follow the wall curvature throughout the entire curve. This shear layer is then convected away from the wall to the interior of the flow. In this region $\unicode[STIX]{x1D714}_{z}$ contributes to the formation of the Lyne-type vortex. Similar flow physics occur in the no-slip wall model (figure 10), and because new vorticity is generated along the inner wall in this case, $\unicode[STIX]{x1D714}_{z}$ vorticity contours appear to branch off from the inner wall. Another mechanism that contributes to the branching of $\unicode[STIX]{x1D714}_{z}$ and transporting it to the interior region is the secondary flow induced by the Dean vortices which convects the vorticity from the wall region to the interior.

At the $90^{\circ }$ cross-section in figure 11(b), and $60^{\circ }$ cross-section in figure 11(c,d), the streamwise vorticity (which is part of the deformed-Dean vortex) is pushed away from the centreline of the pipe as the Lyne-type vortex forms. The Lyne-type vortex needs space in the cross-section of the pipe downstream of $\unicode[STIX]{x1D719}=30^{\circ }$ , which results in a decrease of the area of the deformed-Dean vortex. As the area of the deformed-Dean vortex is reduced, its circulation also declines over time when the Lyne-type vortex is forming (see figure 24).

6.1.1 Vorticity transport equation results, moderate-step flow

Figure 12 shows $\unicode[STIX]{x1D714}_{z}$ for moderate-step flow with a no-slip condition at four instants in time, where the branching from positive $\unicode[STIX]{x1D714}_{z}$ at the inner wall proximity is clear. This branch contributes to the formation of the Lyne-type vortex. Four points are placed at specific locations within this vorticity region, and the distributions of vorticity transport equation terms for these four points are shown in figure 16.

Figure 13 shows the negative convection term (to represent its value when moved to the right-hand side of (6.1)) for the same flow conditions as above. If we treat $\unicode[STIX]{x1D714}_{z}$ as a scalar, then we can describe its transport by the convection term. The convection term is zero at the wall because of the no-slip condition and peaks near the wall. The $\unicode[STIX]{x1D714}_{z}$ component of vorticity is being transported away from the wall first by viscous diffusion and afterwards by convection in the interior region. Therefore, convection peaks slightly away from the wall at $t=0.015~\text{s}$ . Figure 13 shows a branch forming in convection between points B and C at $t=0.21~\text{s}$ , while at this time the plot in figure 12 does not show the formation of the $\unicode[STIX]{x1D714}_{z}$ branch.

Figure 13.

Negative convection of $\unicode[STIX]{x1D714}_{z}$ in (6.1) for the moderate-step flow rate with no-slip wall condition at (a) $t=0.015~\text{s}$ , (b) $t=0.21~\text{s}$ , (c) $t=0.24~\text{s}$ and (d)  $t=0.315~\text{s}$ at the axial slice of $Z=-2.5~\text{mm}$ .

It is important to note that the diffusion of vorticity is occurring as a consequence of the diffusion of momentum via molecular interactions, which changes the velocity gradient (Morton Reference Morton1984):

(6.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}}{\unicode[STIX]{x2202}t}+(\unicode[STIX]{x1D66A}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\unicode[STIX]{x1D714}_{z}=(\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\unicode[STIX]{x1D735})w+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D714}_{z}.\end{eqnarray}$$

The tilting and stretching term ( $(\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\unicode[STIX]{x1D735})w$ ) in the vorticity transport equation is responsible for intensification/dilution of vorticity by the act of stretching/compression and rotation of the vorticity vector, which results in an increase in one component and corresponding decrease in (an)other component(s). Figure 14 shows this term for four instants in time at $Z=-2.5~\text{mm}$ . At this axial slice the tilting/stretching term is positive but has lower values than the convection term in the location of Lyne-type vortex during its formation phases (points A–D).

Figure 14.

Tilting and stretching term in (6.1) for the moderate-step flow rate with no-slip wall condition at (a) $t=0.015~\text{s}$ , (b) $t=0.21~\text{s}$ , (c) $t=0.24~\text{s}$ and (d) $t=0.315~\text{s}$ at the axial slice of $Z=-2.5~\text{mm}$ .

The diffusion term ( $\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D714}_{z}$ ) in (6.1) is shown in figure 15 for the same flow conditions as in figure 12. It is clear that at $t=0.015~\text{s}$ the vorticity is concentrated in the wall region, and hence the diffusion is at its maximum at the wall. Figure 15 insets show zoomed views of the diffusion term where a negative narrow region of diffusion exists. Since $\unicode[STIX]{x1D714}_{z}$ is positive at the inner wall, the diffusion term is negative in this region, indicating that the vorticity generated because of the wall is being transported to the interior regions by the act of viscous diffusion of momentum. On the outer wall, $\unicode[STIX]{x1D714}_{z}$ is negative, and since the diffusion always reduces the extrema magnitude over time, the diffusion term is positive at this region, which counteracts the growth of $\unicode[STIX]{x1D714}_{z}$ by reducing the magnitude of $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}/\unicode[STIX]{x2202}t$ . The contributions of vorticity transport terms to the changes in the vorticity field should be interpreted with respect to the sign of vorticity components locally.

Figure 15.

Diffusion of $\unicode[STIX]{x1D714}_{z}$ in (6.1) for the moderate-step flow rate with no-slip wall condition at (a) $t=0.015~\text{s}$ , (b) $t=0.21~\text{s}$ , (c) $t=0.24~\text{s}$ and (d) $t=0.315~\text{s}$ at the axial slice of $Z=-2.5~\text{mm}$ . Insets show that the diffusion is high in close vicinity to the pipe wall at $t=0.015~\text{s}$ which transports $\unicode[STIX]{x1D714}_{z}$ into the interior region from the wall.

In order to evaluate the contribution of each term in the vorticity transport equation to the variations of $\unicode[STIX]{x1D714}_{z}$ in the Lyne-type vortex location, figure 16 illustrates the balance of each term over time at the four points (A–D) shown in figures 12–15. Figure 16 shows that at those four points along the branch of $\unicode[STIX]{x1D714}_{z}$ , $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}/\unicode[STIX]{x2202}t$ is highly correlated with the convection term. Therefore, $\unicode[STIX]{x1D714}_{z}$ is being primarily transported by convection to the Lyne-type vortex location during its formation period. A similar analysis was performed (not shown here) for axial slices at $Z=0$ and $Z=-1~\text{mm}$ , and similar results were obtained. Our analysis revealed that at axial slices closer to the centreline $Z=0~\text{m}$ , the effect of stretching to increase $\unicode[STIX]{x1D714}_{z}$ in the Lyne-type vortex location is greater but convection remains the dominant term in transporting $\unicode[STIX]{x1D714}_{z}$ during the formation of the Lyne-type vortex.

The same analysis was performed for the slip wall model (with viscous inlet profile that generates vorticity only at the inlet), and similar results were obtained (not shown here), which indicated that $\unicode[STIX]{x1D714}_{z}$ vorticity in the Lyne-type vortex region was transported primarily by convection from the inlet to that location and then strengthened by the stretching term. The major difference between the slip and no-slip models is that for the slip wall model the high concentration of $\unicode[STIX]{x1D714}_{z}$ vorticity is near the wall (not on the wall), and hence the diffusion term in the slip model is not as strongly negative at the inner wall.

Figure 16.

Balance of vorticity transport equation terms for the four points shown in figure 12. The convection term (solid green line) contributes most to $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}_{z}/\unicode[STIX]{x2202}t$ .

6.2 Vortex identification

The analysis above revealed how the $\unicode[STIX]{x1D714}_{z}$ vorticity component is transported to the Lyne-type vortex in the curved pipe. In this section the morphology of various vortices is illustrated by different vortex ID methods, i.e.  $\unicode[STIX]{x1D706}_{2}$ , $\unicode[STIX]{x1D706}_{ci}$ and $d_{2}$ .

Figures 17 and 18 and supplementary movies 3 and 4 show the isosurface of $\unicode[STIX]{x1D706}_{2}$ at four instants in time for the moderate-step flow rate with no-slip and slip wall conditions, respectively. The isosurface is coloured by streamwise vorticity ( $\unicode[STIX]{x1D714}_{n}$ ) and the formation phases of the Lyne-type vortex are illustrated. At $90^{\circ }$ , $\unicode[STIX]{x1D714}_{n}$ is shown along with shaded regions that depict vortex cores detected by $\unicode[STIX]{x1D706}_{2}$ . The chosen isosurface and cross-sectional view show that the Lyne-type vortex displays weak $\unicode[STIX]{x1D714}_{n}$ but strong $\unicode[STIX]{x1D714}_{z}$ vorticity. In the location where $\unicode[STIX]{x1D714}_{n}$ is negligible (shown in grey/white colour) vorticity is mostly aligned in the $Z$ direction, and hence the white region on $\unicode[STIX]{x1D706}_{2}$ isosurface corresponds to high magnitude of $\unicode[STIX]{x1D714}_{z}$ .

Figure 17.

Isosurface of $\unicode[STIX]{x1D706}_{2}=-90$ coloured by streamwise vorticity ( $\unicode[STIX]{x1D714}_{n}$ ) component for the moderate-step flow rate with no-slip wall condition at four instants in time. At $90^{\circ }$ , $\unicode[STIX]{x1D714}_{n}$ is shown and the shaded areas correspond to $\unicode[STIX]{x1D706}_{2}\leqslant -90$ , which identifies the vortex core regions. This figure shows the formation phase of the Lyne-type vortex. Movie 3 shows more phases and evolution in time.

According to figures 17 and 18 (and movies 3 and 4), the deformed-Dean vortex forms first prior to the Lyne-type vortex. The Lyne-type vortex forms at around $45^{\circ }$ at $t=0.024~\text{s}$ and then develops downstream for the no-slip wall condition. As discussed before, this process occurs later in time for the slip wall model at around the $60^{\circ }$ location. In these figures the isosurface of $\unicode[STIX]{x1D706}_{2}$ at the Lyne-type location during its formation is mostly white coloured, suggesting the vorticity axis in Lyne type at its origin is aligned with the $Z$ direction. This will be discussed later in further detail.

Figure 18.

Isosurface of $\unicode[STIX]{x1D706}_{2}=-40$ coloured by streamwise vorticity ( $\unicode[STIX]{x1D714}_{n}$ ) component for the moderate-step flow rate with slip wall condition at four instants in time. At $90^{\circ }$ , $\unicode[STIX]{x1D714}_{n}$ is shown and the shaded areas correspond to $\unicode[STIX]{x1D706}_{2}\leqslant -40$ , identifying the vortex core region.

Application of almost any Eulerian-based vortex ID method involves some degree of subjectivity, because a threshold on the criterion must be defined. In the $\unicode[STIX]{x1D706}_{2}$ plots (figures 17 and 18) we have chosen the negative isosurface value such that it identifies the regions of interest. If all $\unicode[STIX]{x1D706}_{2}<0$ contours are plotted, all of these identified vortices will be connected together. In order to study this difficulty in more detail, various vortex ID methods are used, and for better illustrations, the identified vortices are mostly shown at the $90^{\circ }$ cross-section for steady-state conditions of moderate-step, mid-step and high-step flow rates. Figure 19 shows the identified vortices at $Re=640$ , 898 and 1322 via $\unicode[STIX]{x1D706}_{2}$ , $\unicode[STIX]{x1D706}_{ci}$ and $d_{2}$ . The thresholds were chosen objectively based on the definition of each method. The $\unicode[STIX]{x1D706}_{2}$ and $\unicode[STIX]{x1D706}_{ci}$ results, which are based on the 3-D data, show a connected region for all vortices, but $d_{2}$ results, based on 2-D cross-sectional in-plane data, show separated vortices. This suggests that those connecting regions that are not identified by the $d_{2}$ method have axes of rotation that lie in the cross-sectional plane (i.e. the plane of rotation is normal to the cross-sectional plane); therefore, it cannot be identified by the 2-D $d_{2}$ method. However, regions identified by the $d_{2}$ cannot be inferred to rotate solely in the cross-sectional planes, and might only have a weak rotational component in those planes.

Figure 19 also shows that the regions identified by $\unicode[STIX]{x1D706}_{2}$ and $\unicode[STIX]{x1D706}_{ci}$ are similar to each other, where $\unicode[STIX]{x1D706}_{2}$ provides a smaller area with more distinct vortices. Now the question arises: What is the axis of rotation for each vortex? Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999) and Chakraborty et al. (Reference Chakraborty, Balachandar and Adrian2005) suggested that if the velocity gradient tensor has a complex conjugate eigenpair of ( $\unicode[STIX]{x1D706}_{cr}\pm \text{i}\unicode[STIX]{x1D706}_{ci}$ , $\unicode[STIX]{x1D742}_{cr}\pm \text{i}\unicode[STIX]{x1D742}_{ci}$ ), where the $\unicode[STIX]{x1D742}_{cr}\pm \text{i}\unicode[STIX]{x1D742}_{ci}$ are eigenvectors associated with $\unicode[STIX]{x1D706}_{cr}\pm \text{i}\unicode[STIX]{x1D706}_{ci}$ , the plane spanned by the ( $\unicode[STIX]{x1D742}_{cr},\unicode[STIX]{x1D742}_{ci}$ ) is the plane of rotation. Therefore, the vector $\unicode[STIX]{x1D73B}_{n}=\unicode[STIX]{x1D742}_{cr}\times \unicode[STIX]{x1D742}_{ci}/(|\unicode[STIX]{x1D742}_{cr}||\unicode[STIX]{x1D742}_{ci}|)$ is parallel to the axis of rotation.

Figure 19.

Vortex regions at the $90^{\circ }$ cross-section for three steady-state flow rates identified by (a) $\unicode[STIX]{x1D706}_{2}<0$ , (b) $\unicode[STIX]{x1D706}_{ci}>0$ and (c) $d_{2}<0$ . The $\unicode[STIX]{x1D706}_{2}$ and $\unicode[STIX]{x1D706}_{ci}$ values based on 3-D data show a connected region for identified vortices, and the $d_{2}$ method based on 2-D data shows separated identified vortices. Here 2-D streamlines are superimposed on the $d_{2}$ contour for illustration.

Since the $\unicode[STIX]{x1D706}_{2}$ and $\unicode[STIX]{x1D706}_{ci}$ methods yield only scalar values and do not provide any information about the axis of rotation of each vortex, we identify the orientation of vortex axes by $\unicode[STIX]{x1D73B}_{n}$ . Figure 20 shows $\unicode[STIX]{x1D73B}_{n}$ vectors coloured by $\unicode[STIX]{x1D714}_{n}$ and contour plots of $\unicode[STIX]{x1D706}_{ci}>0$ at $90^{\circ }$ for three $Re$ . This figure shows that the axis of the identified vortex located near the centreline of the pipe ( $Z=0~\text{m}$ ) is mostly parallel to the $Z$ direction with insignificant streamwise vorticity (white vertical lines at the centre of the cross-section). Figure 20 also reveals that the Lyne-type vortex axis is mostly aligned with the $Z$ direction with a weak component in the streamwise direction. As $Re$ increases, the round-shaped head of the deformed-Dean vortex splits from its elongated part and the split-Dean vortex axis becomes more aligned with the streamwise direction.

Figure 20.

Contours of $\unicode[STIX]{x1D706}_{ci}>0$ at the $90^{\circ }$ cross-section of the pipe for steady flows at (a) $Re=640$ , (b) $Re=898$ and (c) $Re=1322$ . The $\unicode[STIX]{x1D73B}_{n}$ vectors are parallel to the axis of rotation and are coloured by the streamwise vorticity (red lines anticlockwise and blue lines clockwise rotation). Three views shows the orientation of $\unicode[STIX]{x1D73B}_{n}$ vectors in three dimensions. DD, SD and LT refer to the deformed-Dean, split-Dean and Lyne-type vortices, respectively.

6.3 Lyne-type vortex

In order to determine the orientation of the Lyne-type vortex at its origin (the location in which it was formed initially), identified vortices are shown at multiple cross-sections in figure 21 for steady-state flow at the high-step flow rate. At $90^{\circ }$ , $\unicode[STIX]{x1D706}_{2}<0$ identifies three cores (dark blue) at the location of the Lyne-type vortex (shown by 1, 2 and 3) weakly connected, with a low magnitude of $\unicode[STIX]{x1D706}_{2}$ (light blue) in between. However, these three separate regions are strongly connected upstream at the $55^{\circ }$ cross-section and their axes of rotation are parallel and mostly aligned with the $Z$ direction. Therefore, the identified regions 1, 2 and 3 at the $90^{\circ }$ cross-section are part of one vortex upstream, rotating about an axis in the $Z$ direction. Figure 21 also shows that the spatial evolution of deformed-Dean vortex pairs through the curve and illustrates how the round-shaped head of the deformed-Dean vortex splits from its elongated part downstream of the $55^{\circ }$ cross-section and forms the split-Dean vortex system.

Figure 21.

Cross-sectional views of $\unicode[STIX]{x1D706}_{2}<0$ for steady-state flow at $Re=1322$ . The $\unicode[STIX]{x1D73B}_{n}$ vectors are parallel to the axis of rotation and are coloured by the streamwise vorticity (red lines anticlockwise and blue lines clockwise rotation). Front and side views of cross-sections (a) and a 3-D view (b) of the curve are illustrated, demonstrating the evolution of the LT vortex and how its orientation changes and gains an axial component downstream of $55^{\circ }$ . DD, SD and LT refer to the deformed-Dean, split-Dean and Lyne-type vortices, respectively.

Looking at the $55^{\circ }$ cross-section in figure 21 may suggest that vortices (both deformed-Dean vortices and the Lyne-type vortex) that have configurations similar to two antiparallel vortices and one orthogonal vortex are also part of one structure. However, the high-step transient flow reveals more information about the interaction of these vortices at the $55^{\circ }$ cross-section.

Figure 22 and movie 5 show $\unicode[STIX]{x1D706}_{2}<0$ at $55^{\circ }$ at eight phases during the high-step transient flow. At the early phases, two deformed-Dean vortices are detected. At $t=0.135~\text{s}$ the Lyne-type vortex starts to form near the centreline of the pipe and at $t=0.15~\text{s}$ the Lyne-type vortex is already formed at this cross-section. The Lyne-type vortex continues to grow in time while moving towards the outer wall until $t=0.645~\text{s}$ , where $\unicode[STIX]{x1D706}_{2}$ shows a connected region of Lyne-type and deformed-Dean vortices. After this phase, the $\unicode[STIX]{x1D706}_{2}$ plot is similar to the $55^{\circ }$ cross-section in figure 21. Therefore, figure 22 suggests that the Lyne-type vortex is different from the classical deformed-Dean vortices because the Lyne-type vortex formed later in time and was distinct from the deformed-Dean vortices seen at the beginning. However, the deformed-Dean vortices still play a role in the formation of the Lyne-type vortex.

Figure 22.

Contours of $\unicode[STIX]{x1D706}_{2}<0$ at $55^{\circ }$ at eight phases during the high-step transient flow. This figure shows that only the deformed-Dean (DD) vortices existed at the beginning of the flow. Later in time at $t=0.135~\text{s}$ , the Lyne-type (LT) vortex is forming. Towards the end at steady-state condition, $\unicode[STIX]{x1D706}_{2}$ shows a connected region of three vortices (in dark blue).

After analysing and determining the orientation of the Lyne-type vortex, it is necessary to discuss its uncommon shape based on 3-D vortex ID results. Figure 23 shows a schematic of a vortex on the left that is similar to the well-known hairpin vortex. The colour denotes the streamwise vorticity, where red and blue represent positive and negative vorticity, respectively. If the plane of investigation cuts through the head of this vortex (section A–A) the cross-sectional view shows two counter-rotating vortices and zero streamwise vorticity near the centreline (white region between section A–A). Since the head of this vortex rotates about the $Z$ direction, the cross-sectional view does not detect any vorticity at the centreline, so a 2-D vortex ID method will detect only two separate regions. However, if a 3-D vortex ID method is used, it will detect one connected region. If the head of this vortex stretches so that it connects the area between its legs (similar to the rightmost vortex in figure 23), vorticity in any cross-sectional view is similar to the cross-section A–A. This is the model of the Lyne-type vortex whose primary axis of rotation is in the $Z$ direction (figure 21, $55^{\circ }$ cross-section) and if we cut downstream of the head, three connected cores can be detected (regions 1, 2 and 3 in figure 21, $90^{\circ }$ cross-section). In figure 23, the isosurfaces of the Lyne-type vortex are shown at two values ( $\unicode[STIX]{x1D706}_{2}=-220$ and $\unicode[STIX]{x1D706}_{2}=-140$ ) to illustrate the correspondence between the schematic and the actual vortex.

Now that the orientation and morphology of the Lyne-type vortex have been determined, we can compare it to the vortex system originally reported by Lyne (Reference Lyne1971). To study the classical Lyne vortex, oscillatory flow with $\unicode[STIX]{x1D6FC}=17$ was simulated and the results are presented in appendix A. There are major differences between the Lyne-type vortex in steady, transient or pulsatile flows and the original classical Lyne vortex pair in oscillatory flow. The main difference is that the rotation axis of the Lyne vortex is aligned with the streamwise direction, whereas the Lyne-type vortex axis is oriented mostly in the $Z$ direction. The Lyne vortex forms slowly in the inviscid region of the flow, while the Lyne-type vortex has significant vorticity magnitude that was delivered via vorticity convection and vortex stretching. Another major difference is revealed by the $\unicode[STIX]{x1D706}_{2}$ method, which cannot detect the Lyne vortex but can detect the Lyne-type vortex, suggesting that there is a fundamental difference between these two vortices. Therefore, the Lyne-type vortex reported in previous studies (e.g. Boiron et al. Reference Boiron, Deplano and Pelissier2007; Glenn et al. Reference Glenn, Bulusu, Shu and Plesniak2012; van Wyk et al. Reference van Wyk, Prahl Wittberg, Bulusu, Fuchs and Plesniak2015; Najjari et al. Reference Najjari, Hinke, Bulusu and Plesniak2016) is not the same as the Lyne vortex in oscillatory flow despite their visual similarities in 2-D cross-sectional slices. Fully 3-D data coupled with vortex ID methods were required to elucidate this structure in pulsatile flows.

Figure 23.

Schematic of a vortex (similar to the hairpin vortex) in which the vortex head grows and its legs approach one another to the point where the vortex head stretches and fills the area between the legs. The colour denotes the streamwise vorticity and section A–A is cutting through the vortex head. At the centre, the rotation of the head is purely in the $X$ – $Y$ plane (along the $Z$ direction). This schematic represents the Lyne-type vortex. The simulated Lyne-type vortex, for $Re=640$ illustrated by $\unicode[STIX]{x1D706}_{2}$ isosurfaces of $\unicode[STIX]{x1D706}_{2}=-220$ and $\unicode[STIX]{x1D706}_{2}=-140$ and coloured by $\unicode[STIX]{x1D714}_{n}$ , demonstrates its morphology.

6.4 Vortex circulation

Figure 24 plots the circulation ( $\unicode[STIX]{x1D6E4}$ ) of each vortex, calculated by (5.1) associated with the rotation in the secondary flow direction regardless of the actual vortex orientation. This helps us to investigate the formation time of vortices and is not necessarily correlated to the total vortex strength, because the main orientation of the Lyne-type vortex, for example, is not in the streamwise direction and $d_{2}$ , which identifies the vortex region for $\unicode[STIX]{x1D6E4}$ , only detects part of this vortex. Firstly, since the Lyne-type vortex axis is weakly aligned in the streamwise direction, the $d_{2}$ method (based on 2-D cross-sectional data) cannot detect the Lyne-type vortex when the flow rate is low. Secondly, circulation of the Lyne-type vortex in the cross-section is lower than that of the deformed-Dean vortex.

Figure 24.

Circulation of each vortex at $32^{\circ }$ and $90^{\circ }$ calculated in the cross-sections (in upper half of the cross-section, $Z>0$ ) with no-slip wall condition, for (a) moderate-step flow, (b) mid-step flow and (c) high-step flow rate. Each plot shows the formation time of various vortices obtained by methods explained in § 5. The area of the vortex was defined by the $d_{2}$ method and $\unicode[STIX]{x1D714}_{n}$ was used in the summation in (5.1). Deformed-Dean, split-Dean and Lyne-type vortices are designated by DD, SD and LT, respectively. After the SD was split from the DD vortex, the grey (coloured) dashed-dotted lines represent the summation of SD and DD circulations.

Figure 24 shows that vortices form earlier in time by increasing $Re$ from moderate-step to high-step flow rate. The Lyne-type vortex at $90^{\circ }$ forms at around $t=0.3~\text{s}$ and $t=0.15~\text{s}$ for moderate-step and high-step flows, respectively. Similarly, the deformed-Dean vortex forms quickly for higher flow rates and, by increasing the step flow rate, $\unicode[STIX]{x2202}\unicode[STIX]{x1D6E4}/\unicode[STIX]{x2202}t$ also increases. The phase of peak circulation of the deformed-Dean vortex also occurs earlier in time; for the moderate-step flow rate the peak in deformed-Dean circulation is at $t\simeq 2.5~\text{s}$ , whereas the peak for high-step flow rate occurs at $t\simeq 1.5~\text{s}$ , which is related to the formation of the Lyne-type vortex. It was discussed that during formation of the Lyne-type vortex, the deformed-Dean vortex was pushed away from the centre of the pipe by the Lyne-type vortex, causing a reduction in the area occupied by the deformed-Dean vortex. This is the reason why the circulation of the latter has a peak/plateau immediately before the appearance of the Lyne-type vortex in the cross-sections. However, this trend is absent in the $32^{\circ }$ cross-section and the circulation monotonically increases over time. It was also discussed that the Lyne-type vortex forms downstream of $30^{\circ }$ ; therefore, at the $32^{\circ }$ location (in figure 24) the Lyne-type vortex has not formed in the cross-section and cannot affect the circulation of the deformed-Dean vortex. In figure 24, for mid-step and high-step flow rates when the split-Dean vortex forms after splitting, the aggregate circulation of deformed-Dean and split-Dean follows the trend of deformed-Dean circulation before splitting. So, two vortices maintain their relative rotation after splitting in steady flow. The results also show that splitting occurs sooner for high-step flow rate when compared to mid-step flow rate. Several mechanisms responsible for vortex splitting and formation of the split-Dean vortex from deformed-Dean were discussed in Cox (Reference Cox2017) and are not the focus of the current study.

Figure 25.

Cylindrical coordinate orientation. The centrifugal and radial pressure gradient forces are shown schematically in the cross-section.

6.5 Analysis and further discussion

In this section the flow dynamics of both moderate-step flow and pure oscillatory flow (where a classical Lyne vortex pair forms) are investigated. Appendix A fully describes the velocity, vorticity and vortex ID results under pure oscillatory flow where the classical Lyne vortex is revisited and analysed with the new vortex ID method. It is known that the centrifugal force plays an important role in generating Dean vortices, and the balance of centrifugal and radial pressure gradient forces has been investigated in several studies, e.g. Boiron et al. (Reference Boiron, Deplano and Pelissier2007) and Alastruey et al. (Reference Alastruey, Siggers, Peiffer, Doorly and Sherwin2012). While the current numerical results were obtained by solving the governing equations in Cartesian coordinates, other researchers have used toroidal coordinates to simulate flow in a curved pipe. However, the centrifugal force does not appear in the momentum equation in Cartesian coordinates, and the pure centrifugal force term only appears on the centreline ( $z=0$ ) in toroidal coordinates (Boiron et al. Reference Boiron, Deplano and Pelissier2007). To circumvent this issue, cylindrical coordinates are used to provide the centrifugal force at any point in the cross-sections without being limited only to the symmetry plane. Figure 25 shows the coordinate system used for our momentum balance and (6.2) and (6.3) describe the relation between Cartesian and cylindrical coordinates for $0^{\circ }<\unicode[STIX]{x1D719}<180^{\circ }$ :

(6.2) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\begin{array}{@{}c@{}}X=r\cos \unicode[STIX]{x1D719},\\ Y=r\sin \unicode[STIX]{x1D719},\\ r=\sqrt{X^{2}+Y^{2}},\end{array}\right\} & \displaystyle\end{eqnarray}$$

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\begin{array}{@{}c@{}}V_{r}=u\cos \unicode[STIX]{x1D719}+v\sin \unicode[STIX]{x1D719},\\ V_{\unicode[STIX]{x1D719}}=V_{n}=-u\sin \unicode[STIX]{x1D719}+v\cos \unicode[STIX]{x1D719}.\end{array}\right\} & \displaystyle\end{eqnarray}$$

In cylindrical coordinates the $r$ -momentum equation provides the centrifugal and radial pressure gradient (RPG) explicitly. Since the simulations were performed in Cartesian coordinates, velocity and pressure gradients are available with respect to $X$ , $Y$ and $Z$ . Therefore, the $r$ -momentum equation is transformed and written based on derivatives in Cartesian coordinates. To obtain the proper transformed equation shown in (6.4), the chain rule, directional derivatives and the equations cast in cylindrical coordinates were used:

(6.4) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}V_{r}}{\unicode[STIX]{x2202}t} & = & \displaystyle \frac{1}{r}\left(u^{2}\sin ^{2}\unicode[STIX]{x1D719}+v\left(v\cos ^{2}\unicode[STIX]{x1D719}+X\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}Y}+Y\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}Y}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\left.u\left(-v\sin 2\unicode[STIX]{x1D719}+X\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}X}+Y\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}X}\right)\right)+w\frac{\unicode[STIX]{x2202}V_{r}}{\unicode[STIX]{x2202}Z}+\frac{V_{n}^{2}}{r}-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}r}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D708}\left(\left(\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}X^{2}}+\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}Y^{2}}+\frac{\unicode[STIX]{x2202}^{2}u}{\unicode[STIX]{x2202}Z^{2}}\right)\cos \unicode[STIX]{x1D719}+\left(\frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}X^{2}}+\frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}Y^{2}}+\frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}Z^{2}}\right)\sin \unicode[STIX]{x1D719}\right).\end{eqnarray}$$

The first two terms on the right-hand side in (6.4) indicate the convective force in the $r$ direction, while the second ( $V_{n}^{2}/r$ ) and third ( $-(1/\unicode[STIX]{x1D70C})\,\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}r$ ) terms are centrifugal and radial pressure gradient forces, respectively. The last term corresponds to the viscous force in the $r$ direction. The radial pressure gradient can be written based on derivatives with respect to $X$ and $Y$ in (6.5), and $\unicode[STIX]{x2202}V_{r}/\unicode[STIX]{x2202}Z=\cos \unicode[STIX]{x1D719}\,\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}Z+\sin \unicode[STIX]{x1D719}\,\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}Z$ :

(6.5) $$\begin{eqnarray}-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}r}=-\frac{1}{\unicode[STIX]{x1D70C}}\left(\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}X}\frac{\unicode[STIX]{x2202}X}{\unicode[STIX]{x2202}r}+\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}Y}\frac{\unicode[STIX]{x2202}Y}{\unicode[STIX]{x2202}r}\right).\end{eqnarray}$$

Equation (6.4) is applicable at any point in the flow and the force balance at the $90^{\circ }$ cross-section can be obtained using (6.6), where $r$ and $V_{r}$ can be substituted by $Y$ and $v$ , respectively:

(6.6) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}t}=-\left(\frac{u^{2}}{Y}+u\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}X}+v\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}Y}+w\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}Z}\right)+\frac{u^{2}}{Y}-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}Y}+\unicode[STIX]{x1D708}\left(\frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}X^{2}}+\frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}Y^{2}}+\frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}Z^{2}}\right).\end{eqnarray}$$

Figure 26.

Momentum balance in $r$ direction on $Z=0$ and $Z=2.5~\text{mm}$ lines at $\unicode[STIX]{x1D719}=90^{\circ }$ for moderate-step flow rate. Balance of centrifugal force and radial pressure gradient is shown in the top half of the cross-section and streamwise vorticity is shown in the bottom half. The horizontal dashed line shows $Z=2.5~\text{mm}$ .

Figure 26 shows the force balance in $r$ direction at the $90^{\circ }$ cross-section of the pipe on two horizontal lines ( $Z=0$ and 2.5 mm). The cross-sectional view shows the balance of centrifugal and radial pressure gradient terms. At $t=0.015~\text{s}$ , the flow is accelerating and the velocity profile has not developed yet and is similar to a plug flow profile at $90^{\circ }$ . This causes the centrifugal force profile to be almost flat in the core region and balanced by RPG. During the transient flow at $t=0.24~\text{s}$ , the Lyne-type vortex forms, and similar to what Boiron et al. (Reference Boiron, Deplano and Pelissier2007) observed, the RPG dominates the centrifugal term at the location of the Lyne-type vortex. During the transient flow, the convective term is significant, and near the inner wall region its magnitude is comparable to centrifugal and RPG terms. At this time, near the inner wall (at $x_{local}\approx -4~\text{mm}$ and $Z=2.5~\text{mm}$ ) where the deformed-Dean vortex is located, the RPG is small and even becomes positive (exerting force towards the outer wall). The small, positive values of RPG ( $-(1/\unicode[STIX]{x1D70C})\,\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}r$ ) in this region are attributable to the local pressure minimum in the core of the deformed-Dean vortex.

Figure 27.

Momentum balance in $r$ direction on $Z=0$ and $Z=2.5~\text{mm}$ lines at $\unicode[STIX]{x1D719}=90^{\circ }$ for oscillatory Lyne flow. Balance of centrifugal force and radial pressure gradient is shown in the top half of the cross-section and streamwise vorticity is shown in the bottom half. The horizontal dashed line shows $Z=2.5~\text{mm}$ .

The force balance in $r$ direction for the oscillatory flow (classical Lyne problem) is shown in figure 27, which illustrates that the centrifugal and pressure gradient forces are well balanced by each other. Insets in figures 26 and 27 show that the viscous force is near zero for both oscillatory Lyne and moderate-step flow except at the near-wall regions. Also, the convective term is near zero for the oscillatory Lyne case as well, but non-zero for moderate-step flow where Lyne-type vortex forms. This convective term may have a role in formation of the Lyne-type vortex.