2. Experimental work

3. Vorticity

A series of vorticity plots are shown in this section for angles of attack of ${\it\alpha}=5^{\circ }$ and ${\it\alpha}=10^{\circ }$ . In the vorticity plots, a counterclockwise sense of rotation is defined as positive (red) and a clockwise sense of rotation is defined as negative (blue). Images associated with each spanwise plane have been located adjacent to one another for comparison. Both the $y$ - and $z$ -axes have been made non-dimensional through dividing by the tubercle wavelength. The lower limit of the $z$ -axis values is governed by the foil curvature as well as the angle of attack, and hence is non-zero in most cases. For example, at $x^{\prime }/c=0.4$ and ${\it\alpha}=5^{\circ }$ , the distance in the $z$ -direction between the origin and the foil surface can be calculated as $z_{surface}=10.7~\text{mm}$ , using the following equation:

Here, $t(x)$ is the local thickness of the foil, which is evaluated at the point of interest. A schematic diagram is shown in figure 4 to illustrate the methodology used to derive equation (3.1). The corresponding non-dimensional value is $z/{\it\lambda}=0.36$ , which is close to the minimum value of the $z$ -axis in figure 5(a). Differences are due to the discretisation error associated with the size of the PIV interrogation windows.

Figure 4.

Schematic diagram showing geometric quantities used to derive (3.1).

Figure 5.

View of the vorticity field from behind the foil where the counterclockwise vortex core (red/upper) is positive and the clockwise vortex core (blue/lower) is negative. Angle of attack ${\it\alpha}=5^{\circ }$ , $Re=2230$ , image planes are shown for $x^{\prime }/c=0.4$ , 0.6, 0.8 and 1 (a–d).

Figure 5(a–d) shows that between tubercle peaks, there exist adjacent regions of positive (upper section) and negative (lower section) vorticity for all image planes. These regions of vorticity, marked ‘P’ in figure 5(a,b), will henceforth be described as the primary vorticity. With streamwise distance, vortex spreading leads to a decrease in the peak primary vorticity and an increase in the area over which vorticity is distributed. As shown in figure 5(a,b), vortex spreading is greater in the $y$ -direction than the $z$ -direction, where it is inhibited due to adjacent vorticity of opposite sign.

This vorticity of opposite sign (or secondary vorticity) has been marked with an asterix and is evident in the image planes of $x^{\prime }/c=0.4$ and $x^{\prime }/c=0.6$ in figure 5(a,b). This vorticity appears to be annihilated by the primary vorticity of opposite sign adjacent to it, as evidenced by its absence for image planes further aft. The process of annihilation is further evidenced by the closely spaced contours of vorticity between the primary and secondary structures in figure 5(a,b).

The asymmetrical appearance of figure 5(d) is consistent with the occurrence of unsteady effects at $x^{\prime }/c=1$ , which will be discussed in § 8. The asymmetry can also be attributed to possible differences in the flow patterns for adjacent wavelengths of the tubercle pattern. Flow visualisation images have shown that differences in the flow patterns exist for adjacent wavelengths of the tubercle pattern, but that alternate wavelengths demonstrate similar characteristics (Custodio Reference Custodio2007).

Another important feature that is visible in figure 5(d) is the existence of wall vorticity which is adjacent to the primary vortices and has an opposite sense of rotation. This wall vorticity is evident in the streamwise vorticity plots published by Skillen et al. (Reference Skillen, Revell, Pinelli, Piomelli and Favier2014), both at similar chordwise locations and at chordwise locations closer to the leading edge. The absence of wall vorticity near the leading edge for the PIV measurements reported here is due to resolution limitations.

Figure 6 shows that increasing the foil angle of attack to ${\it\alpha}=10^{\circ }$ gives rise to an increase in the magnitude of the peak vorticity associated with the counter-rotating primary vortices. Consistent with the results at ${\it\alpha}=5^{\circ }$ , vortex spreading occurs in the streamwise direction at ${\it\alpha}=10^{\circ }$ , and the spreading of vorticity is restricted in the $z$ -direction at $x^{\prime }/c=0.2$ and 0.4 due to the proximity of secondary vorticity of opposite sign. Flow asymmetry is evident in the results at ${\it\alpha}=10^{\circ }$ for the image planes close to the trailing edge, as was observed for ${\it\alpha}=5^{\circ }$ , and is attributed to the same mechanism.

Figure 6.

View of vorticity field from behind the foil where the counterclockwise vortex core (red/upper) is positive and the clockwise vortex core (blue/lower) is negative. Angle of attack ${\it\alpha}=10^{\circ }$ , $Re=2230$ , image planes are shown for $x^{\prime }/c=0.2$ , 0.4, 0.6, 0.8 and 1 (a–e).

Figure 7.

Magnitudes of the maximum positive and negative vorticity peaks for ${\it\alpha}=5^{\circ }$ and ${\it\alpha}=10^{\circ },Re=2230$ .

Figure 7 shows the magnitudes of the positive and negative primary vortex peaks corresponding to the contour plots shown in figures 5 and 6. The magnitudes of both vorticity peaks decrease with downstream distance for the image planes nearer to the leading edge ( $x^{\prime }/c=0.2$ – $0.6$ ) at ${\it\alpha}=5^{\circ }$ and ${\it\alpha}=10^{\circ }$ . In accordance with Morton (Reference Morton1984), the presence of strong pressure gradients near a foil leading edge results in the production of vorticity, giving rise to large values of the vorticity field in this region. This vorticity is, then, transported through viscous diffusion as well as the vortex stretching and tilting phenomena into the flow field. With the presence of large velocity gradients, in particular near the leading edge, the mechanism for the decay of peak vorticity is activated, as reflected in figure 7. Axial compression of streamwise vortices caused by the presence of an adverse pressure gradient on the suction surface of the foil leads to an increase in the cross-sectional area of the vortices, as shown in figures 5 and 6, and a corresponding reduction in the peak vorticity. The reduction in peak vorticity is also influenced by annihilation of the secondary vorticity, observed in figures 5(a,b) and 6(a,b), as well as diffusion and annihilation between the streamwise vortices themselves. Towards the trailing edge, the magnitude of the peak vorticity increases as a result of the vortex tilting phenomenon, as will be discussed in § 7.

4. Circulation of streamwise vortices

The integration paths for calculating the counterclockwise and clockwise circulation are shown superimposed on the vorticity contour plots for ${\it\alpha}=5^{\circ }$ in figure 8(a–d). The vorticity plots are shown with the same resolution as in figure 5, where the increment between contour lines is 0.1  $\text{s}^{-1}$ . The corresponding regions enclosed by the integration paths are of equal size and the left-hand border of each plot aligns with the foil surface. This ensures that all significant vorticity contributions are captured within the integral as $x^{\prime }/c$ changes. Extending the right-hand border of these regions has negligible impact on the results. The regions of integration for an angle of attack of ${\it\alpha}=10^{\circ }$ are not shown for the sake of brevity.

Figure 8.

Path of integration and enclosed region for ${\it\alpha}=5^{\circ }$ , $Re=2230$ , image planes $x^{\prime }/c=0.4$ , 0.6, 0.8 and 1 (a–d). Red/upper, integration path for calculating the circulation of the counterclockwise vortex; blue/lower, integration path for calculating the circulation of the clockwise vortex. The vorticity contours are plotted with the same resolution as figure 5, where the separation between contour lines is 0.1  $\text{s}^{-1}$ .

Figure 9.

Circulation variation, ${\it\Gamma}$ , with angle of attack and chordwise position.

Results from the integration for ${\it\alpha}=5^{\circ }$ and ${\it\alpha}=10^{\circ }$ are depicted in figure 9, where the magnitude of the circulation is shown for both the positive and negative primary vortex cores. The circulation of the clockwise and anticlockwise primary vortices increases both with increasing angle of attack and with streamwise distance from the leading edge. These results were also observed when the region of integration encompassed the primary vorticity only, indicating that the secondary vorticity and wall vorticity have negligible impact on the magnitude of the circulation. To understand the reason behind the increase in circulation, it is first necessary to analyse the flow pattern in detail, and this will be done in § 7. Subsequently, the streamwise increase in circulation will be explained with reference to the identified flow mechanism.

Figure 9 shows that the circulation of the negative vortex is slightly higher than that of the positive vortex for the image planes at $x^{\prime }/c=0.8$ and $x^{\prime }/c=1$ at ${\it\alpha}=10^{\circ }$ . This is related to the asymmetry of the primary vortices which is apparent in figures 6(d) and 6(e).

A streamwise increase in circulation of the primary vortices has also been observed for delta wings prior to the point at which vortex breakdown occurs (Visser & Nelson Reference Visser and Nelson1993; Johari & Moreira Reference Johari and Moreira1998). The chord Reynolds numbers investigated were $Re=1.9\times 10^{\text{5}}$ and $Re=2.5\times 10^{\text{5}}$ , and the circulation increase was approximately linear. An approximately linear increase in circulation is also evident in figure 9.

5. Computational fluid dynamics model

ANSYS (2014) was employed to model the flow over the full-span tubercled and baseline NACA 0021 foils at $Re=2230$ for the two attack angles described in the experiments ( ${\it\alpha}=5^{\circ }$ and ${\it\alpha}=10^{\circ }$ ). To this end, the transitional shear stress transport (SST) model ${\it\gamma}$ – $Re_{{\it\theta}}$ (Langtry & Menter Reference Langtry and Menter2009), was used for solving the steady Reynolds-averaged Navier–Stokes equations. Here, ${\it\gamma}$ is defined as the intermittency, which determines the percentage of time that the flow is turbulent, and $Re_{{\it\theta}}$ is the momentum thickness Reynolds number. The ${\it\gamma}$ – $Re_{{\it\theta}}$ model was successful in capturing the pressure distribution and flow characteristics, including the vorticity field around a wavy foil, in our previous study (Rostamzadeh et al. Reference Rostamzadeh, Kelso, Dally and Hansen2012).

Figure 10.

Boundary-layer plots for CFD results at an angle of attack of ${\it\alpha}=10^{\circ }$ for the foil cross-section corresponding to the tubercle peak, for (a)  $x^{\prime }/c=0.0924$ , (b)  $x^{\prime }/c=0.216$ , (c)  $x^{\prime }/c=0.3329$ and (d)  $x^{\prime }/c=0.4604$ .

The boundary-layer profile for the CFD results at ${\it\alpha}=10^{\circ }$ is shown in figure 10 for a series of spanwise planes at the peak cross-section. The parabolic flat-plate approximation to the laminar boundary-layer profile and the turbulent flat-plate approximation, $u/U_{\infty }=(y/{\it\delta})^{1/7}$ , are also shown for comparison. It can be seen that the boundary layer in the CFD results most closely resembles a laminar profile. This was also found to be the case at the trough and mid cross-sections. This supports the choice of the transitional SST model, since a laminar boundary layer would be expected at the low Reynolds number investigated in this study, and a transitional model allows specification of the free-stream turbulence intensity. Initially, a laminar model was implemented, but it was found that this model did not resolve the streamwise vortices correctly, leading to discrepancies between the computational and experimental results.

Figure 11.

(a) The computational domain with the coordinate system positioned at the leading edge. (b,c) The grid system in the vicinity of the tubercled foil (b) and unmodified NACA 0021 foil (c).

To discretise the transport equations, ANSYS CFX 14.5 uses an element-based finite volume method (ANSYS 2014). High-resolution algorithms were implemented for the advection and turbulence numerical schemes. A C-grid topology with hexahedral elements was adopted as the computational domain, with far-field boundaries located at 12 chord lengths from the trailing edges of the foils, as shown in figure 11. The angle of attack was changed by rotating the model and maintaining constant values for the free-stream velocity components. The surfaces of the foils were assigned the no-slip boundary condition, and periodic conditions were applied to the side planes.

Figure 12.

Force monitor plots for (a) the lift coefficient, $C_{L}$ , and (b) the drag coefficient, $C_{D}$ , for ${\it\alpha}=10^{\circ }$ .

As flow separation was expected to occur, the boundary layer had to be resolved with sufficient detail, hence the grid was refined near the wall boundaries such that $y^{+}<1$ . To assess convergence, the behaviour of the force monitor points was considered (see figure 12) along with a residual target equal to $10^{-7}$ for the momentum and SST transport equations, and $1.15\times 10^{-6}$ for the intermittency transport equation. The force monitor points are the lift coefficient, $C_{L}$ , and the drag coefficient, $C_{D}$ .

The geometric and the grid characteristics associated with the foils are provided in table 1. Although the mesh sensitivity analysis showed no difference in the behaviour of the forces with a lower-density grid test case (total number of hexahedral elements equal to 2859 012) for the tubercled foil, the adopted grid for simulation was capable of showing finer details of the vorticity field near the foil surface, which was of particular interest in this study. The flow mechanism described in this work was consistently detected across all of the test cases, including one with a much higher number of elements (10 748 400).

Table 1.

Grid and geometric details of the modelled foils.

Figure 13.

Vorticity at $Re=2230$ , image plane $x^{\prime }/c=0.4$ , for (a) ${\it\alpha}=5^{\circ }$ (CFD), (b) ${\it\alpha}=5^{\circ }$ (experiment), (c) ${\it\alpha}=10^{\circ }$ (CFD), (d) ${\it\alpha}=10^{\circ }$ (experiment).

6. Pressure forces

This section presents the chordwise pressure distributions corresponding to a tubercle peak, a trough and midway between. The data were obtained from the CFD simulation described in § 5. Analysis of the surface pressure distributions allows identification of the point at which the adverse pressure gradient initiates for each cross-section, which is relevant to the vortex stretching mechanism. The chordwise pressure distributions show the existence of a spanwise variation in pressure, and this is explored further through plotting surface contours of the spanwise pressure gradient. The spanwise pressure gradient is shown to result in the generation of streamwise vorticity at the surface. Analysis of the surface pressure data also enables investigation into the spanwise variation in lift.

Figure 17.

Pressure distribution for the unmodified (unmod) and tubercled foil at ${\it\alpha}=10^{\circ }$ , where ‘peak’, ‘trough’ and ‘mid’ correspond to the cross-sections at the tubercle peak, trough and midway between. Here, $C_{p}$ is the pressure coefficient.

The chordwise variation in pressure for both the unmodified and tubercled foils at ${\it\alpha}=10^{\circ }$ is shown in figure 17. The most negative values of the peak pressure developed over the suction surface of the tubercled foil are associated with the trough cross-section. The lowest minimum suction peak occurs at the peak cross-section, and the pressure values for the middle cross-section lie somewhere between. This is consistent with results obtained at a higher Reynolds number on a foil with the same profile shape (Hansen Reference Hansen2012). It must be noted that the slight overprediction in the pressure coefficient at the stagnation points is due to numerical error and was not resolved by refining the CFD grids or extending the domain boundaries up to 40 chord lengths. The impact of this error is negligible on the underlying flow mechanism reported in this work.

The location of the suction peak along the trough cross-section is further aft than for the other cross-sections and the unmodified foil. Conversely, the suction peak along the peak cross-section is slightly closer to the leading edge. This variation is attributed to differing positions of maximum thickness as well as changes in chord length with spanwise location. These parameters define the position at which the adverse pressure gradient initiates. At this point, compression of streamwise vortices is initiated, resulting in a larger area over which vorticity is distributed and a reduction in peak vorticity, as discussed in § 3.

Figure 18.

Spanwise pressure gradient on the surface of the foil at ${\it\alpha}=10^{\circ },Re=2230$ , where the highest values occur at the foil leading edge.

The difference in magnitude and location of the suction peaks for the mid, peak and trough cross-sections gives rise to a spanwise pressure gradient which initiates at the leading edge and persists up to approximately $x^{\prime }/c=0.2$ . The contour plot shown in figure 18 indicates the magnitude of the spanwise pressure gradient in the region of the leading edge. It is also evident that a spanwise pressure gradient exists near the trailing edge; however, its magnitude is significantly lower.

The presence of a spanwise pressure gradient results in direct production of streamwise vorticity, as can be seen by the spanwise momentum equation at the wall expressed in (6.1) (Gad-el-Hak 2006), where the coordinate axes are consistent with figure 1 in § 2,

The first and third terms on the left-hand side of the equation are equal to zero due to the absence of flow injected from the wall, $v_{w}$ , and constant viscosity, ${\it\mu}$ . Therefore, the spanwise pressure gradient, $\partial p/\partial y$ , can be seen to affect the curvature of the velocity profile shown on the right-hand side of the equation, which is the same as the flux of streamwise vorticity from the wall (Gad-el-Hak 2006).

Figure 19.

Variation in the lift coefficient, $C_{L}$ , with the spanwise location.

Figure 20.

(a) Shear stress lines, (b) shear stress contours and (c) vortex lines, ${\it\alpha}=5^{\circ },Re=2230$ .

Table 2.

Lift and drag forces at ${\it\alpha}=10^{\circ }$ calculated in the CFD simulation.

Results from the CFD simulation shown in table 2 indicate that the lift coefficient achieved by the foil with tubercles is higher than that achieved by the unmodified foil. In addition, the efficiency of the tubercled foil, $L/D$ , is marginally better than that of the unmodified foil, despite the slight increase in the associated drag. The increased lift for the tubercled foil is attributed to the greater flow attachment that is maintained behind the tubercle peaks. Integration of the pressure at the peak, trough and midway between (mid) cross-sections confirms that the highest proportion of lift is generated at the peak cross-section, as shown in figure 19. The spanwise variation in the circulation and hence lift coefficient is also highlighted in figure 19. The increased drag for the foil with tubercles is related to the presence of streamwise vortices and the associated increase in surface shear stress.

7. Analysis of flow characteristics

Regions of separation and attachment of the flow on the foil can be identified from the surface shear stress lines and contours shown in figures 20(a,b) and 21(a,b). These images were obtained from the CFD simulation described in § 5. Behind the tubercle trough, separation occurs much earlier than behind the peak, which is consistent with previous results (Johari et al. Reference Johari, Henoch, Custodio and Levshin2007; van Nierop et al. Reference van Nierop, Alben and Brenner2008). The surface shear stress lines show that a large separation zone forms ahead of a pair of foci (F), into which the flow spirals, to be lifted from the surface. The upstream extent of the separation zone is defined by a saddle point (S).

The foci mark the locations at which concentrated vortices separate from the surface of the foil, which occurs at $x^{\prime }/c\sim 0.8$ and $x^{\prime }/c\sim 0.6$ for ${\it\alpha}=5^{\circ }$ and ${\it\alpha}=10^{\circ }$ respectively. Downstream of the foci, aligned with the tubercle peaks, lie a pair of bifurcation lines (BL) which define the spanwise extent of the separation zone. In the following discussion we will describe the downstream evolution of both surface vorticity and vorticity in the separated shear layer. This analysis provides insight into the mechanisms responsible for the presence of streamwise vorticity in the numerical and experimental results and how these may explain the experimental observations of streamwise vorticity at various spanwise planes above the foil.

Figure 21.

(a) Shear stress lines, (b) shear stress contours and (c) vortex lines, ${\it\alpha}=10^{\circ },Re=2230$ .

The evolution of the vorticity vectors over the foil surface is shown by the vortex lines in figures 20(c) and 21(c). In these figures, vortex lines are coloured according to the sign of the streamwise vorticity, where red (darker) and green (lighter) are positive and negative respectively, in the $x^{\prime }$ -direction. For a conventional unswept foil without tubercles, vortex lines are initially aligned in the $y$ -direction, parallel to the leading edge, with positive vorticity, consistent with the mean shear. However, in the highly three-dimensional flow around the tubercles, the vortex lines are initially wavy, leading to streamwise vorticity components on the sides (A) of the tubercles themselves. A short distance downstream, the curvature becomes reversed (B) ahead of the separation zone, leading to small streamwise vorticity components of opposite sign to the streamwise vorticity on the tubercles.

Downstream of the saddle the vortex lines can be seen to bend sharply downstream (C), merging into a single line (M), representing the edge of the canopy-like separated vortex sheet that defines the separation zone. The edge of the canopy is also indicated in the associated surface shear stress contours. The downstream orientation of the vortex lines near the foil surface (C) leads to additional streamwise vorticity of the same sign as the primary vortices formed on the sides of the tubercles. In both figures 20 and 21 the vortex lines are seen to be turned over the entire length of the separation zone, supporting the observation that the circulation of the streamwise vortices formed adjacent to the separation zone increases with downstream distance, as shown in § 4.

In figures 20(c) and 21(c) the vortex lines in the reversed flow region, which are of opposite sign to those of the upstream boundary layer, are also seen to contribute to the generation of streamwise vorticity components (D) upstream of the foci. However, downstream of the foci, the vortex line curvature is reversed (E), leading to vorticity of a sign opposite to the primary vortices. The presence of strong streamwise vorticity of opposite sign beneath the primary vortices is confirmed by the experimental and numerical results for the ${\it\alpha}=10^{\circ }$ case in figure 14. Finally, the vortex lines near the trailing edge (F) at ${\it\alpha}=10^{\circ }$ in figure 21 are orientated such that the streamwise vorticity is of the same sign as observed at (C).

Figure 22.

Magnitude of the vortex turning terms for ${\it\alpha}=10^{\circ },Re=2230$ , in the vicinity of (a) the leading edge and (b) the trailing edge.

The turning of the surface vortex lines from a spanwise-dominant orientation towards the streamwise direction is caused by the deflection or skewing of the mean flow, which is described by the term ${\it\omega}_{y}(\partial u/\partial y)+{\it\omega}_{z}(\partial u/\partial z)$ in the vorticity transport equation (Perkins Reference Perkins1970). The magnitude of the vortex turning terms is shown to be largest in the region of the leading edge in figure 22(a). Near the trailing edge, after $x^{\prime }/c=0.6$ , the sign of the vortex turning terms changes, as shown in figure 22(b), which is consistent with the reversal of the vortex line curvature that occurs in this region. Direct generation of streamwise vorticity also occurs in both of these regions, as was discussed in § 6.

Figure 23.

(a) Three-dimensional vortex line interpretation, (b) vortex lines from CFD simulation and (c) vortex lines with normalised green (lighter) and red (darker) arrows indicating spanwise and streamwise orientation of vorticity respectively.

A three-dimensional vortex line interpretation is shown in figure 23(a) to provide a more detailed visualisation of the proposed flow mechanism. Along the asymptotic vortex line contour, the vortex lines lift from the foil surface, forming a canopy of vorticity above the separation zone. At the edge of the canopy-shaped vortex sheet, the forward flow and reverse flow vortex lines contribute to streamwise vorticity of the same sign, referred to as ‘primary vorticity’ in this study. This process occurs continuously along the sides of the canopy, consistent with the growth of the circulation of the primary vortices with streamwise distance (see figure 16).

A pair of foci is revealed by the surface streamlines in figure 23(a), and the primary vortices separate from the surface at these points. The vortex lines corresponding to the CFD results are shown in figure 23(b), and these results support the vortex line interpretation and show the canopy of vorticity that forms in the separation region aft of the tubercle troughs. The sign of vorticity on the vortex line ring is indicated in figure 23(c), and it is emphasised that this is different from the boundary-layer vorticity shown in figure 23(a).

Figure 24.

(a) Three-dimensional vortex lines coloured by streamwise vorticity in the vicinity of $x^{\prime }/c=0.6$ and $x^{\prime }/c=0.8,{\it\alpha}=10^{\circ }$ . (a) Three-dimensional view, (b) side view and (c) top view.

The orientation of the vortex line ring is shown to vary with streamwise distance in figure 24, which depicts vortex lines coloured by streamwise vorticity magnitude for ${\it\alpha}=10^{\circ }$ . The black line shows a streamline section, corresponding to the same material point moving with the flow from approximately $x^{\prime }/c=0.6$ to $x^{\prime }/c=0.8$ . The end points of the streamline section are connected to vortex lines that form closed loops. Figure 24(a–c) indicates that the streamwise vorticity associated with the same material point increases with chordwise distance, which is consistent with the results presented in figure 15. The side and top views shown in figures 24(b) and 24(c) respectively indicate that the vortex line rings turn in such a way that they assume stronger streamwise components near the tops in the downstream direction. The turning of the vortex lines can be explained with reference to the streamwise vorticity transport equation expressed in Perkins (Reference Perkins1970). Here, the tilting of vorticity into the streamwise direction is accounted for by the product of the spanwise vorticity with the spanwise variation in the streamwise velocity, ${\it\omega}_{y}(\partial u/\partial y)$ , and the product of the normal vorticity with the normal variation in the streamwise velocity, ${\it\omega}_{z}(\partial u/\partial z)$ . This is the same mechanism that is responsible for turning of the surface vortex lines from a spanwise-dominant orientation towards the streamwise direction in the surface flow.

Figure 25.

Three-dimensional vortex lines coloured by streamwise vorticity in the vicinity of $x^{\prime }/c=0.07$ and $x^{\prime }/c=0.17,{\it\alpha}=10^{\circ }$ . (a) Top view and (b) side view.

The deflection or skewing of the mean flow is also evident in figure 25(a,b), which depicts a streamline section, indicated by a black line, corresponding to the same material point as it travels between vortex lines shown at approximately $x^{\prime }/c=0.07$ to 0.17. It is evident that for this material point, the magnitude of the vorticity reduces with chordwise distance, while the orientation of the tangent also changes, as shown in figure 25(b). This drop in streamwise vorticity is due to vortex compression as well as tilting. It should be noted that in the side view the transparency of the foil has been adjusted to view the streamline more clearly.

Figure 26.

(a) Isosurfaces of spanwise vorticity in the reverse flow region and below the foil at ${\it\alpha}=5^{\circ }$ , (b,c) isosurfaces of streamwise vorticity in green (lighter) and red (darker) depicting opposite signs of vorticity values.

The deformation and separation of the forward and reversed flow boundary-layer vortex sheets are shown in figure 26(a–c) for the numerical data. Figure 26(a) illustrates the canopy-shaped vortex sheet that forms over the separation zone. Figures 26(b) and 26(c) depict the formation of streamwise vorticity on the surface of the tubercled foil at ${\it\alpha}=5^{\circ }$ , with the two signs of vorticity shown from different viewing angles. The deformation of the individual vortex lines near the surface agrees with the isosurface contours shown in this figure.

Figure 27.

Three-dimensional streamlines at ${\it\alpha}=5^{\circ }$ : (a) formation of streamwise vortices and their trajectory downstream, (b) end view of streamwise vortex formation.

A plot of the three-dimensional streamlines is shown in figure 27(a). The origins of the streamwise vortices from the separation zone can be seen in this image. The reverse flow in the separation region is clearly evident. Figure 27(b) shows the end view of the streamwise vortex formation, emphasising the lift-off of the vortices from the surface of the foil.

Figure 28.

Owl face of the first kind, after Perry & Hornung (Reference Perry and Hornung1984).

The topology shown in figure 28 resembles an ‘owl face of the first kind’, first defined by Perry & Hornung (Reference Perry and Hornung1984). The central vertical plane of the owl-face pattern corresponds to the tubercle trough in the present study. It can be seen that the primary vortices create upwash along the vertical plane, which confirms that the sense of rotation of the streamwise vortices is consistent with the current results. The structure is similar to that of stall cells on wings (Perry & Hornung Reference Perry and Hornung1984; Manolesos & Voutsinas Reference Manolesos and Voutsinas2014), which are classified as an owl face of the second kind (Perry & Hornung Reference Perry and Hornung1984), although the present structure is somewhat less complex.

Figure 29.

Flow visualisation for the $A8{\it\lambda}30$ tubercle configuration at an angle of attack ${\it\alpha}=10^{\circ }$ and $Re=4370$ . The patterns are frames from a video sequence which are spaced by 3 s. The images show the development of streamwise vortices behind the tubercle peaks, and it is shown that asymmetry develops over time. Flow is from left to right.

Figure 29 presents hydrogen bubble visualisation of the flow above the tubercled foil, showing clearly the rollup of vortices in a trough for the $A8{\it\lambda}30$ tubercle case. The appearance of the streaklines and the variations between the images demonstrate that the flow pattern varies with time. Asymmetry in the flow can be seen to increase with time in this image sequence; however, in later image frames (not shown here) the vortices appear to be symmetrical again. Therefore, the mean pattern consists of two counter-rotating vortices between the tubercle peaks, in agreement with the observations and interpretations made elsewhere in this paper. On the other hand, there is a certain degree of time dependence in the flow pattern, as has been alluded to previously, and this will be discussed further in § 8.

8. Time dependence of flow patterns

The vorticity and circulation results presented in §§ 3 and 4 were derived from velocity-vector fields which represented the average of 2085 image pairs. For a laser pulse rate of 10 Hz, the total acquisition time considered was approximately 209 s. The reason for averaging such a large number of samples was to reduce the uncertainty associated with imaging in a cross-flow plane. Based on a comparison between the time-dependent and averaged results, it was found that the averaged flow field data reflect the general flow behaviour. Quantification of the degree of unsteadiness was achieved through use of a peak detection algorithm that located the position of peak vorticity for each primary vortex in each instantaneous image. It was found that the mean position of these peaks corresponds well to the position of peak vorticity when the velocity fields are temporally averaged. The standard deviation is within reasonable limits and the movement of the vortices is best described as a fluctuation about the mean, as will be shown in § 8.1. Results are presented in table 3, and the units are consistent with other figures in this paper, whereby the $y$ - and $z$ -coordinates have been normalised by the tubercle wavelength.

Table 3.

Mean position and standard deviation of peak vorticity. The position of peak vorticity for temporally averaged experimental data is given in parentheses. All values are normalised by the tubercle wavelength of 30 mm.

Despite the general agreement between the mean and instantaneous results, some aspects of the flow patterns are influenced by unsteady effects, and these will be explored in this section.

9. Discussion and conclusions

The combined experimental and numerical studies have identified the salient features of the flow structure produced by a NACA 0021 foil with tubercles at a Reynolds number of 2230. The presence of the tubercles leads to an increased flow velocity along the trough, and a larger adverse pressure gradient. This in turn causes boundary-layer separation to occur in the trough at a low angle of attack. These findings are consistent with previous research, where the larger adverse pressure gradient in the trough region was attributed to the relatively short chord length at this spanwise location (van Nierop et al. Reference van Nierop, Alben and Brenner2008). Hence, the same pressure difference needs to be overcome over a shorter distance behind a tubercle trough relative to a tubercle peak (van Nierop et al. Reference van Nierop, Alben and Brenner2008). Previous studies have also identified the presence of a spanwise pressure gradient (Skillen et al. Reference Skillen, Revell, Pinelli, Piomelli and Favier2014), and the results presented here indicate that this is most significant in the region of the leading edge, where the minimum suction peak occurs. It has been shown that the spanwise pressure gradients, set up by the tubercles, interact with the vorticity distribution, leading to a complex flow pattern. The numerical simulation of the flow is in strong agreement with the experimental observations and allows close examination of the flow structure. The resulting three-dimensional pattern bears a strong similarity to the owl face of the first kind, first described by Perry & Hornung (Reference Perry and Hornung1984).

Analysis of the flow field data obtained from both PIV experiments and CFD modelling reveals regions of vorticity of alternate sign in the troughs between tubercles for all spanwise planes investigated. The presence of these streamwise vortices has been shown in previous work (Custodio Reference Custodio2007; Pedro & Kobayashi Reference Pedro and Kobayashi2008; Stanway Reference Stanway2008; Skillen et al. Reference Skillen, Revell, Pinelli, Piomelli and Favier2014); however, the spanwise location of vorticity was not clear in these results and the vorticity was not quantified as it has been in this study. The current work also shows for the first time that the counter-rotating streamwise vortices lift away from the surface at a certain chordwise location, corresponding to the large foci in the surface streamline (shear stress) pattern, the locations of which are governed by the angle of attack of the foil. At this point, the flow field becomes highly unsteady and the distribution of vorticity in the region of the primary vortices varies with time, a feature that has not been previously identified.

Considering the averaged experimental results, the magnitude of the peak primary vorticity has been shown to decrease in the streamwise direction, while the region of vorticity expands in area. This effect can be attributed to axial compression of the streamwise vortices caused by the adverse pressure gradient on the suction surface of the foil, vortex tilting and viscous diffusion of vorticity. The results also indicate that the circulation of the primary pair of streamwise vortices increases with streamwise distance, achieving a maximum value at the trailing edge. The increase in circulation occurs due to continued influx of boundary-layer vorticity which occurs along the sides of the shear-layer canopy. An increasing vortex circulation in the streamwise direction has also been observed for delta wings (Visser & Nelson Reference Visser and Nelson1993; Johari & Moreira Reference Johari and Moreira1998), but this has not been shown for a tubercled wing prior to this study. This supports the theory that tubercles can be modelled as delta wings with a rounded leading edge (Weber et al. Reference Weber, Howle, Murray and Miklosovic2011).

The numerical study provides novel insights into the flow structure and evolution of vorticity in the flow cases under investigation. The computed sectional velocity and vorticity fields show a strong correspondence to the experimental results. An analysis of the surface vortex lines reveals a number of regions where local turning of the vorticity leads to the formation of streamwise vorticity in the boundary layer, beginning on the sides of the tubercles and occurring at various locations all the way to the trailing edge. The results also demonstrate a source of reverse-sign secondary vorticity beneath the primary vortices downstream of the foci, which was shown to occur near the wall in both the experimental and numerical results. These observations are corroborated by the isosurfaces of streamwise and spanwise vorticity presented in figure 26. The turning of the boundary-layer vorticity from a spanwise-dominant orientation towards the streamwise direction suggests the development of Prandtl’s secondary flow of the first type. This is a quasi-inviscid mechanism which is activated by the presence of skewness in the flow, arising from spanwise pressure gradients. This mechanism has been observed to operate for a foil with tubercles in the transitional flow regime (Rostamzadeh et al. Reference Rostamzadeh, Hansen, Kelso and Dally2014) and is also observed to be at play in the laminar flow regime investigated in this study. The local turning of vorticity was also observed by Favier et al. (Reference Favier, Pinelli and Piomelli2012); however, these researchers referred to vorticity that was initially normal to the surface, rather than being aligned in the spanwise direction. The generation of the vorticity is not attributed to a Kelvin Helmholtz instability which was referred to by Favier et al. (Reference Favier, Pinelli and Piomelli2012).

In general, there is strong correspondence between the experimental and numerically derived magnitudes and trends of the peak vorticity and circulation of the primary vortices. The main differences lie towards the trailing edge of the foil in the ${\it\alpha}=10^{\circ }$ case. In the numerical results, the slight decrease in the circulation between $x^{\prime }/c=0.8$ and 1 is attributed to a source of negative vorticity in the reverse flow, as shown in figure 26(b,c). In the experimental results, no sudden drop is observed, but the overall magnitude of primary vortex circulation is generally lower than in the corresponding CFD results. While some of these differences may be due to inadequacies in the transitional SST closure model used in the numerical study, the effects of unsteadiness are also likely to be a significant cause. In particular, the time-variant nature of the flow pattern demonstrated in § 8 indicates the high degree of unsteadiness near the trailing edge for both ${\it\alpha}=5^{\circ }$ and ${\it\alpha}=10^{\circ }$ . Unsteadiness of this magnitude, greater in the ${\it\alpha}=10^{\circ }$ case, will result in axial and lateral oscillation of the flow pattern, leading to a higher rate of generation, diffusion and annihilation of the vortices in the flow, consistent with the structure shown in figure 30. From a measurement perspective, the unsteadiness will lead to attenuation and spreading of the time-averaged vorticity across the measurement plane as well as in the streamwise direction, obscuring the instantaneous flow structure. In fact, due to the strong influence of unsteadiness in this flow, it is likely that small errors in the computed flow structure due to the choice of closure model, such as the axial location of the flow separation, will not affect the validity of the CFD results as an interpretive tool.

The aerodynamic performance of a NACA 0021 foil with $A8{\it\lambda}30$ tubercles has been shown to be slightly better in comparison to an unmodified foil under the conditions of this study. Improved performance has also been observed in previous studies under different flow conditions (Watts & Fish Reference Watts and Fish2001; Miklosovic et al. Reference Miklosovic, Murray, Howle and Fish2004). Integration of the pressure distributions at various chordwise locations has revealed that the largest amount of lift is generated at a cross-section corresponding to a tubercle peak. This is consistent with the observation that flow separation occurs further aft at this location. A delayed separation at the peak cross-section has also been observed in previous studies (Johari et al. Reference Johari, Henoch, Custodio and Levshin2007; van Nierop et al. Reference van Nierop, Alben and Brenner2008; Hansen et al. Reference Hansen, Kelso and Dally2011).

In spite of the strong correspondence between the experimentally and numerically determined flow features and trends, there are still a number of unresolved features, such as the regions of vorticity marked with an asterisk in figures 5 and 6. Further investigation into the time-dependent characteristics of the flow field surrounding a foil with leading edge tubercles is suggested for future work. In particular, it would be useful to study the three-dimensional evolution of the streamwise vortices in the unsteady flow to gain more insight into the formation and annihilation processes which were touched upon briefly in this paper.