4.1.1 Non-seam side

Figure 12(a) shows that on the non-seam side of the ball there is a white stripe caused by a flow separation. The flow separates at the front of this stripe and the recirculating flow, trapped beneath the separated shear layer, has low heat transfer so the surface of the ball remains cold. This result agrees with Scobie et al.'s sketch for the non-seam side shown in figure 11 and confirms that the separation angle on this side of the ball is approximately $80^{\circ }$. Figure 12(a–c) also shows that the position of the laminar separation on the non-seam side does not change between $Re=1.42\times 10^5$ and $Re=1.89\times 10^5$. This agrees with Achenbach's result for a smooth sphere, figure 10, which shows that the separation angle remains constant at $82^{\circ }$ for cases where $Re$ is below the critical value.

Figure 12. Seam and non-seam side IR images of a new, red Dukes cricket ball, ball I in figure 6, undergoing conventional swing at varying Reynolds number. Flow is left to right and red dashed lines illustrate eventual separation position.

Measurements using model balls instrumented with pressure tappings by Reference Scobie, Pickering, Almond and LockScobie et al. (2013), see figure 9, and Reference Deshpande, Shakya and MittalDeshpande et al. (2018), have also shown that the separation on the non-seam side occurs at approximately $80^{\circ }$. Figure 8(d) from Reference Jackson, Harberd, Lock and ScobieJackson et al. (2020) shows that the separation angle on the non-seam side at $Re=1.1\times 10^5$ is $95^{\circ }$. The authors explain that this is because the surface roughness is greater than a new ball and links this to Achenbach's roughened sphere results in figure 10. For the separation angle to increase to $95^{\circ }$ implies that transition has begun but this idea is not discussed. It is also noted that the model ball tested by Jackson et al. is a replica of the one used for the measurements in figure 9. The roughness of these balls is reported to be the same, however, Scobie et al.'s tests at $Re=0.20\times 10^5$ and $Re=1.47\times 10^5$ both show a laminar separation at approximately $80^{\circ }$. The discrepancy between these results and the PIV result in figure 8(d) is not explained.

4.1.2 Seam side

Existing publications discussing cricket ball swing often state that the boundary layer on the seam side is ‘tripped’ to turbulence by the seam. Indeed, Reference HorlockHorlock (1973), Reference Deshpande, Shakya and MittalDeshpande et al. (2018) and Reference SayersSayers (2001) test model cricket balls where the seam is represented by a wire trip. Reference Son, Choi, Jeon and ChoiSon et al. (2011) describe two ways in which a wire trip can delay boundary layer separation and reduce drag: first, the disturbance from the trip can cause transition along the separated shear layer resulting in a separation bubble and turbulent reattachment. Alternatively, disturbances from the trip can cause transition to turbulence before separation, with no separation bubble formed. Deshpande et al. are the only authors to suggest the presence of a laminar separation bubble on the seam side, while others state that the boundary layer is tripped with no mention of a separation bubble.

Another description of the flow on the seam side of the ball, provided by Reference Scobie, Shelley, Jackson, Hughes and LockScobie et al. (2020) and Reference Jackson, Harberd, Lock and ScobieJackson et al. (2020), links the seam side behaviour to the rough sphere in figure 10 (black line with square symbols) so that, for conventional swing, the seam side boundary layer separates further back on the ball than the smooth, non-seam side (red line). Achenbach's data for the rough sphere show a separation angle of $100+/-5^{\circ }$ in the range $Re=1-2\times 10^5$. However, Jackson's PIV measurement of a model cricket ball, figure 8(d), and Scobie et al.'s measurements using an instrumented model ball, figure 9, show a separation angle of $120+/-5^{\circ }$. This is the value used in the diagram in figure 11. The difference between the rough sphere and cricket ball experiments is not discussed and we conclude that Achenbach's rough sphere results are unsuitable for modelling the seam side of a cricket ball, since the separation angles differ by approximately $20^{\circ }$.

Figure 12(d–f) presents Whittle Laboratory IR images of the seam side of the ball for conventional swing at varying $Re$. These results show streamwise ‘streaks’ on the front half of the ball, which are caused by coherent structures shed by the seam. These appear dark in the image, where warm flow from the free stream is mixed down to the cool surface of the ball, implying that the seam acts more like a row of vortex generators than a wire trip. Just behind the midpoint of the ball there is a laminar separation bubble, indicated by the white stripe in the IR images. The separation bubble is closed with a turbulent boundary layer reattachment, shown by the dark region of high heat transfer, before the flow eventually separates towards the back of the ball.

The results in figure 12 show that streamwise flow structures are involved in the separation and reattachment of the boundary layer flow on the seam side. This type of behaviour is widely reported in the literature outside of cricket ball aerodynamics, and reviews are provided by Reference Gad-El-Hak and BushnellGad-El-Hak & Bushnell (1991) and Reference YangYang (2019). Much of this ‘boundary layer control’ research is concerned with improving the performance of aircraft wings, however, Reference Joubert and HoffmanJoubert & Hoffman (1962) show that vortex generators placed along a cylinder, $20^{\circ }$ from the stagnation point, are able to reduce drag by 50 % at $Re=1.3\times 10^5$ compared with a smooth cylinder. We hypothesise that the cylinder drag reduction measured by Joubert et al. is due to the same kind of flow physics seen in figure 12(d–f): the coherent streamwise structures shed by the seam/vortex generators interact with the separated shear layer causing transition, a turbulent boundary layer reattachment and eventual separation towards the rear of the ball/cylinder. This is different to the wire trip or rough sphere mechanisms where transverse perturbations cause the transition to turbulence.

Flow visualisation and pressure tapping measurements by Reference Deshpande, Shakya and MittalDeshpande et al. (2018) also indicate the presence of a laminar separation bubble on the seam side. However, these results appear to have been overlooked by other researchers since they are recorded using an aluminium model ball, where the switch from conventional to reverse swing does not occur until approximately $Re=4\times 10^5$, an unrealistically high value for cricket. Scobie et al.'s measurements in figure 9, show that at $Re=1.47\times 10^5$ (labelled CS – 67 mph) there is a flattening of the pressure profile between $80^{\circ }$ and $90^{\circ }$ on the seam side. This is indicative of a laminar separation bubble, however, this feature is not discussed in Scobie et al.'s paper (Reference Scobie, Pickering, Almond and LockScobie et al. 2013) or the same group's subsequent work (Reference Jackson, Harberd, Lock and ScobieJackson et al. 2020; Reference Scobie, Shelley, Jackson, Hughes and LockScobie et al. 2020).

Overall, the description of seam side flow for conventional swing, summarised in figure 11, where the seam is said to trip the boundary layer, should be updated. The IR images in figure 12, supported by previous pressure tapping measurements (Reference Scobie, Pickering, Almond and LockScobie et al. 2013; Reference Deshpande, Shakya and MittalDeshpande et al. 2018), show that coherent streamwise structures are shed from the seam and that these cause a turbulent reattachment behind a laminar separation bubble.

4.1.3 Flow asymmetry

Combining the descriptions for the non-seam and seam sides leads to an updated explanation for conventional swing. Figure 13 shows a top down sketch of the flow past a cricket ball. The laminar separation on the non-seam side was hypothesised in the earliest papers on cricket ball swing in the 1950s and has been shown to occur at approximately $80^{\circ }$ when $Re$ is below the critical value for that side of the ball.

Figure 13. Top down sketch of flow past a cricket ball for conventional swing updated using Whittle Laboratory results.

On the seam side, established explanations that the boundary layer transitions to turbulence because the seam acts like a wire trip, or that the seam side behaves like a rough sphere, should be revised. Figure 13 shows coherent streamwise structures, shed from the seam, which close a separation bubble and result in a turbulent boundary layer separation at approximately $120^{\circ }$.

The asymmetry in the boundary layer separation results in a sideways force which causes the ball to swing in the direction that the seam is pointing. While the non-seam side flow is below a critical $Re$, the separation points on both sides of a new ball do not change greatly for $Re=1.42\times 10^5$ to $Re=1.89\times 10^5$, as shown in figure 12. This explains why $C_{S}$, shown in figure 6 for new balls, does not vary significantly, e.g. for a new red Dukes ball, tested at the Whittle Laboratory, there is less than 5 % variation in $C_{S}$ between $Re=1.42\times 10^5$ and $Re=1.92\times 10^5$. Figure 7 shows that used balls can also exhibit conventional swing with $C_{S}$ between 0.2 and 0.35, however, the switch to reverse swing occurs at lower $Re$ and is less sudden than for new balls. This is because a roughened non-seam side transitions more gradually and at lower $Re$, as shown for rough spheres by Achenbach in figure 10.

4.2.1 Non-seam side

On the non-seam side, figure 15 shows that the laminar boundary layer separates at $95^{\circ }$, forms a laminar separation bubble, reattaches as a turbulent boundary layer and then separates to form a wake at $135^{\circ }$. This explanation is based on several measurements reported in the literature: figure 9 shows that at $Re=1.95\times 10^5$ (89 mph – RS), a constant pressure is recorded between $-90^{\circ }$ and $-100^{\circ }$, indicating the presence of a separation bubble; figure 14(a) shows a laminar separation bubble on the non-seam side using oil flow visualisation; the red region on the non-seam side in figure 14(b) indicates a separation bubble and reattachment; and in figure 14(c) PIV measurements show a laminar separation, reattachment and eventual turbulent separation at $130^{\circ }$. The Whittle Laboratory IR image of the non-seam side of the new ball in figure 16(a) matches the flow mechanism sketched in figure 15: the white stripe followed by a dark region indicates a laminar separation bubble and turbulent reattachment. These results agree qualitatively with Achenbach's measurements (Reference AchenbachAchenbach 1974) for smooth spheres above a critical $Re$ of $2\unicode{x2013}3\times 10^5$, shown in figure 10, where the flow becomes turbulent in the shear layer of the laminar separation, causing a turbulent boundary layer to reattach then finally separate at $120^{\circ }$.

Figure 16(c) shows an IR image of the non-seam side of a ball artificially roughened to represent an old ball. Roughness is generated with a sand blaster and is measured using a profilometer to be $k/D = 55\times 10^{-5}$ for the cases shown in figure 16(c,e,f), where $k$ is the peak-to-peak value of the surface profile. Further details of the generation and measurement of surface roughness are given in Appendix A. Visualisation for this condition has not been presented in the literature before, although force measurements for old or artificially roughened balls, are reproduced in figure 7. In figure 16(c), a region of laminar flow, which appears light grey because the heat transfer coefficient is low, is observed at the front of the ball. The surface roughness then causes the boundary layer to become turbulent, there is no laminar separation bubble and the turbulent separation is moved forwards compared with the new ball case shown in figure 16(a). This behaviour is consistent with Achenbach's results in figure 10, where the separation angle for the rough spheres are 10–$20^{\circ }$ further forwards than for the smooth sphere for cases where $Re$ is above the critical value.

Scobie et al.'s sketch of the non-seam side flow in figure 15 captures the case when the ball is new or its shine has been maintained. However, a second drawing is required to describe the non-seam side flow as the ball becomes older and more worn and the separation point moves forwards.

4.2.2 Seam side

Figure 15 shows a turbulent boundary layer separation on the seam side at $110^{\circ }$ and this is also labelled on the pressure measurements shown in figure 9 for the reverse swing case. As with the conventional swing results, Reference Scobie, Pickering, Almond and LockScobie et al. (2013) do not discuss the flattening of the pressure profile observed between $80^{\circ }$ and $90^{\circ }$ which suggests the presence of a laminar separation bubble.

The Whittle Laboratory IR image in figure 16(d) shows that the seam side behaviour of a new ball at $Re=1.95\times 10^5$ is similar to that in the conventional swing regime at $Re=1.89\times 10^5$, shown in figure 12, with streaks shed from the seam, a separation bubble, turbulent reattachment and eventual separation. This is consistent with the flattened pressure profile between $80^{\circ }$ and $90^{\circ }$ seen in figure 9 for reverse swing.

Figure 16(e,f) shows IR images of the seam side of the ball for cases where artificial roughening has been applied. Here, the roughness causes the boundary layer to be turbulent towards the front of the ball, there is no separation bubble and turbulent reattachment and the separation point is moved forward compared with the new ball case shown in figure 16(d). This result is similar to the flow visualisation on the seam side shown in figure 14(a,b). The coherent structures shed from the seam are not as clear in figure 16(e,f) as the other seam side cases. This is because the rough surface causes a turbulent boundary layer to develop and the heat transfer coefficient is increased. This reduces the contrast in heat transfer caused by the coherent flow structures so the streaks are not as noticeable in the IR image. The geometry of the seam is not changed by the roughening process used here so flattening or damaging the seam is not responsible for the streaks being less visible and the changes in measured force coefficient.

Scobie et al.'s seam side sketch of the flow in figure 15 should be updated to include a laminar separation bubble for cases where the ball is new or its shine has been maintained. A second sketch is required to describe the seam side flow as the surface becomes older and more worn and the separation point moves forwards.

4.2.3 Flow Asymmetry

Measurements of swing force coefficient for old balls, collated in figure 7, show that reverse swing magnitude varies more than conventional swing magnitude. This is due to the different conditions shown in figure 16 which ‘bracket’ the boundary layer behaviours that can occur on each side of the ball. To illustrate this, top down sketches of the flow are presented in figure 17 for the three cases recorded in figure 16.

Figure 17. Top down sketches of flow past a cricket ball for reverse swing updated using Whittle Laboratory results. (a) New ball with shiny seam and non-seam sides; (b) doctored ball with rough seam side and shiny non-seam side; (c) used ball with rough seam and non-seam sides.

For a new ball which reverse swings, the separation on both sides is towards the back; the seam side separation is slightly in front of the non-seam side, resulting in a small $C_{S}$ of $-$0.08. With the doctored ball, the separation point on the seam side is brought forward by roughness, increasing the asymmetry and resulting in $C_{S}=-0.31$. For the old ball, the separation on the non-seam side has also moved forwards, meaning that the asymmetry is reduced so that $C_{S}$ is only $-$0.09. This updated model for reverse swing, shown in figure 17, allows previous literature results, including real and model balls, to be reassessed and grouped.

Different brands of new ball tested by Reference Scobie, Shelley, Jackson, Hughes and LockScobie et al. (2020) and Reference Deshpande, Shakya and MittalDeshpande et al. (2018), balls A–E in figure 6, have shiny seam and non-seam sides and $C_{S}$ between 0 and $-$0.1 above a critical $Re$. The sketch in figure 17(a) captures these balls’ behaviour.

The model ball tested by Reference Scobie, Pickering, Almond and LockScobie et al. (2013) and Reference Jackson, Harberd, Lock and ScobieJackson et al. (2020) has a laminar separation bubble on the non-seam side, as shown with pressure measurements in figure 9 and PIV in figure 14. No PIV visualisation of the seam side is provided, however, there is evidence in figure 9 of a laminar separation bubble, as discussed above. The model ball's roughness is said to represent 25 overs of wear and $C_{S}$ is measured to be between $-$0.12 and $-$0.18. Similarly, the 8–10 and 25 over old balls, labelled A and B in figure 7, have $C_{S}$ between $-$0.1 and $-$0.13 for $Re>1.8\times 10^5$. These balls are likely to sit somewhere between the sketches in figure 17(a) and figure 17(b), where the laminar separation bubble on the non-seam side is present but the separation point on the seam side is moved forwards, compared with a new ball, due to roughness. For these cases, reverse swing magnitude could be increased by roughening the seam side more to bring the separation point further forward.

Reference Deshpande, Shakya and MittalDeshpande et al. (2018) also tested a seam side roughened ball, ball H in figure 7, which has $C_{S}$ around $-$0.2 for $Re>1.95\times 10^5$. This test is similar to the doctored ball shown in figure 16(b) so will have behaviour represented by the sketch in figure 17(b). Old balls provided by a professional cricket team and tested by Reference Scobie, Pickering, Almond and LockScobie et al. (2013), balls E–G in figure 7, have swing force coefficients between $-$0.2 and $-$0.3 and are likely to sit with this group as well. Reference Tadrist, Sampara, Ashraf and AndrianneTadrist et al. (2020) measured swing force coefficients between $-$0.22 and $-$0.34 at $Re=2.4\times 10^5$ for cricket balls at $0^{\circ }$ seam angle with one side roughened and the other left as new, balls J and K in figure 7. These tests are also best represented by the sketch in figure 17(b).

Old balls tested by Reference Scobie, Shelley, Jackson, Hughes and LockScobie et al. (2020), have swing force coefficients between 0 and $-$0.1, for example ball D in figure 7. Photographs of these balls show they are worn on both sides, with prominent gaps in the quarter seam, and they are therefore likely to behave as shown in figure 17(c). Reference Deshpande, Shakya and MittalDeshpande et al. (2018) tested a ball where both sides were artificially roughened, resulting in swing force coefficients between 0 and 0.1 for $Re>2\times 10^5$; this is ball I in figure 7. The behaviour of this ball is best captured by figure 17(c), but where the non-seam side is rough enough to move its separation point in front of the seam side, causing a small, positive swing force.

Our updated model allows the reverse swing magnitude to be linked to a range of ball conditions. However, the quantified relationship between surface roughness, $Re$ and separation angle on both sides of the ball, has not been investigated systematically and this represents an area for future work.

5.1.1 Spin mounted wind tunnel tests

Rotating a cricket ball in a wind tunnel requires a mounting point to be located on the axis of rotation in the centre of the non-seam and/or seam side. This interferes with the boundary layer aerodynamics in an important region of the ball and limits the useful measurements which can be taken. Reference Sayers and HillSayers & Hill (1999) performed this type of experiment with a rotating mount passing through both sides of the ball. Their results show that at 30 m s$^{-1}$ and $0^{\circ }$ seam angle, lift force, which acts perpendicular to the swing and drag forces, increases from less than 0.1 N at 0 Hz rotation rate, to between 0.3 and 0.45 N at 8.3–16.7 Hz, with these forces only ‘weakly dependent on the speed of rotation’. These results are explained by the Magnus effect which causes balls in tennis, baseball, football and golf to curve in the air, as described in Reference Mehta and PallisMehta & Pallis (2001), Reference Alam, Chowdhury, Moria and FussAlam et al. (2010a,  Reference Alam, Ho, Smith, Subic, Chowdhury and Kumar2012), Reference Bearman and HarveyBearman & Harvey (1976) and Reference MehtaMehta (1985). Reference MehtaMehta (2014) discusses how slow bowlers, as well as unusual side-arm bowlers, are able to use the Magnus effect to make cricket balls deviate laterally, but these phenomena are not the focus of this review. Further tests by Sayers and Hill, on drag and swing force, are limited by the parasitic force and disruption to the boundary layers caused by the mounting system for rotation. Reference Jackson, Harberd, Lock and ScobieJackson et al. (2020) test a rotating, two-times scale model cricket ball which is mounted on one side. No force measurements are recorded and instead a PIV system is used to visualise the flow past the ball in two planes: from below, so that the non-seam side flow can be observed while the ball rotates, and from the side, so the flow in a plane aligned with the seam can be investigated in relation to the Magnus effect. A non-dimensional spin rate based on the ratio of tangential velocity at the top and bottom of the ball to the ball speed is defined: $\alpha _{Jackson} = 0.5 \omega D / U$, using the nomenclature given in figure 3.

Jackson et al.'s results, reproduced in figure 18, show that the laminar separation, turbulent reattachment and turbulent separation on the non-seam side are unaffected by spin rates of $\alpha _{Jackson}=0\unicode{x2013}0.12$ with $Re=1.69\times 10^5$ and seam angle $15^{\circ }$. This result refutes a hypothesis by Reference MehtaMehta (2014) that rotation will stop the laminar separation bubble from forming. The second set of Jackson et al.'s tests, less relevant for this review, provide flow visualisation for the Magnus effect and support the results of Sayers and Hill which show how backspin generates lift.

Figure 18. The PIV time-averaged velocity magnitude and streamlines on the non-seam side at $Re=1.69\times 10^5$ and the seam angle $15^{\circ }$ for varying spin rates. Separation and reattachment angles shown to nearest $5^{\circ }$ (Reference Jackson, Harberd, Lock and ScobieJackson et al. 2020) with (a) $\alpha _{Jackson}=0.0$; (b) $\alpha _{Jackson}=0.06$; (c) $\alpha _{Jackson}=0.12$. Yellow contours indicate high velocity, dark blue contours indicate zero velocity.

5.1.2 Projection wind tunnel tests

In 1982, two pieces of work were published which overcame some of the challenges of testing rotating cricket balls in a wind tunnel. Reference BartonBarton (1982) and Reference BentleyBentley (1982) both performed experiments in which a cricket ball was rolled down an inclined track before dropping through the open jet of a wind tunnel. The ball spin rate was set by the distance rolled and the seam angle adjusted by moving the track angle relative to the direction of the wind tunnel jet. The forces acting on the ball were calculated by measuring the position that the ball lands relative to a datum and using trajectory equations. Both studies report results for a range of variables: Barton tested 17 different balls at 8 speeds up to 30 m s$^{-1}$ (67 mph, $Re = 1.45\times 10^5$), at seam angles of $0^{\circ }$, $15^{\circ }$ and $30^{\circ }$ and at spin rates of 2.1, 4.9 and 9.3 Hz, while Bentley et al. tested 23 balls in total, of which 5 were tested at 7 speeds up to 40 m s$^{-1}$ (89 mph, $Re = 1.93\times 10^5$), at seam angles of $0^{\circ }$, $10^{\circ }$, $20^{\circ }$ and $30^{\circ }$ and at spin rates of 0, 2.6, 5, 9.1, 11.4 and 14.2 Hz. Barton also qualitatively investigated the effect of seam wobble and Bentley et al. conducted experiments with a turbulence grid, discussed in the next section on atmospheric conditions.

Barton and Bentley et al.'s projection tests are inherently less repeatable than stationary wind tunnel experiments and a spread of results is reported. Barton notes that some of the scatter is due to the method of projection: with the wind tunnel turned off, the balls land inside a strip 0.043 m wide, 43 % of the maximum deflection recorded when the wind tunnel was operating. On the other hand, Bentley et al. show that 4 pairs of balls ‘of the same quality’ have force coefficients which vary only 6 % on average at 30 m s$^{-1}$ with seam angle $20^{\circ }$ and spin rate 5 Hz. Both studies acknowledge the uncertainty in their results, however, through repeated tests and averaging, trends are observed and conclusions drawn.

Figure 19 is processed from projection test data published in Bentley et al. for two cricket balls released at $20^{\circ }$ seam angle. Bentley et al. plot results with force normalised by ball weight against tunnel velocity in m/s and spin rate in Hz. These data are reprocessed in figure 19, assuming standard atmospheric conditions, to show contours of $C_{S}$ against $Re$ and reduced frequency. Reduced frequency for a spinning cricket ball is defined as the ratio of time taken for the flow to pass the ball to the time for the ball to complete a rotation. At a spin rate of 15 Hz and speed 30 m s$^{-1}$, $RF = \omega D / U = 0.035$, i.e. there are 28 flow through times for each revolution of the ball.

Figure 19. Swing force coefficient, $C_{S}$, against Reynolds number, $Re$, and reduced frequency reprocessed in non-dimensional form from data presented in Reference BentleyBentley (1982) for two balls at seam angle $20^{\circ }$ with (a) ball 8 and (b) ball 11.

The values of $C_{S}$ for ball 11, shown in figure 19(b), are positive across the range tested because $Re$ is not high enough to cause transition on the non-seam side of the ball. However, for ball 8, figure 19(a), the change from conventional swing to reverse swing occurs over a Reynolds number range of approximately $0.3\times 10^5$. These results indicate a more gradual switch from conventional to reverse swing than the sharp change seen in stationary wind tunnel tests plotted in figure 6 (the term ‘reverse swing’ was not in use in the early 1980s; however, Bentley et al. note the negative force coefficients measured at high Reynolds number and include a private communication from Imran Kahn, a well-known Pakistani cricketer, who confirms that he had observed the effect in practice).

Both Bentley et al. and Barton discuss which type of cricket ball is most likely to swing and they agree that a smooth non-seam side is required to maintain a laminar boundary layer for conventional swing. For example, Barton highlights a new ball with a ‘gaped’ quarter seam which has an average $C_{S} < 0.2$ and Bentley et al. find that a two-piece ball, constructed without quarter seams, has the highest overall swing magnitude when $C_{S}$ is averaged over a range of speeds, angles and spin rates. Both studies agree that the quarter seam, which Bentley et al. measure to be typically 0.1 mm wide and 0.3 mm deep, as well as other features such as scuff marks and logos, produce an ‘effective roughness’ when the ball is spinning. This causes transition on the non-seam side which stops conventional swing. Barton and Bentley et al. also test balls with different types of primary seam and while Barton suggests that balls with a ‘prominent’ seam swing more, Bentley et al. state that seam ‘stitching type’ is not as important as the smooth surface on the non-seam side.

The results reproduced in figure 20 are from the average of five different balls at seam angles of $10^{\circ }$ and $20^{\circ }$. Again, the data are reprocessed to show $C_{S}$ against $Re$ and reduced frequency. It is observed that swing is maximised at a seam angle of $20^{\circ }$, $Re = 1.22\times 10^5$ (25 m s$^{-1}$, 56 mph) and reduced frequency 0.033 (spin rate of 11.4 Hz). In a similar analysis, Barton recommends a ‘moderate’ spin rate of 5–8 Hz and ‘slight’ seam angle of $10^{\circ }$–$15^{\circ }$ to maximise swing. Hypotheses to explain variation in swing with seam angle are discussed in § 4.3, however, the effect of spin rate on swing has not been measured experimentally in any other work in the literature. Both papers note that spin helps to make the ball more stable and qualitatively observe that balls with zero or low spin rate (<5 Hz) wobbled more as they fell through the wind tunnel jet. Barton shows that increasing ball wobble reduces measured swing force and Bentley et al. state ‘It is almost always the difficulty in maintaining stability that is responsible for a new ball not swinging’. Both studies therefore suggest that increasing spin rate up to a point improves ball stability and this increases swing.

Figure 20. Swing force coefficient, $C_{S}$, averaged from 5 balls against Reynolds number, $Re$, and reduced frequency reprocessed in non-dimensional form from data presented in Reference BentleyBentley (1982) for two seam angles of (a) $10^{\circ }$ and (b) $20^{\circ }$.

To explain the optimum spin rate, Bentley et al. state that ‘too much spin is of course detrimental since the effective roughness on the ball's surface is increased’ and a similar argument is made by Barton, although in this discussion it is also hypothesised that ‘high rates of spin would almost inevitably be accompanied by wobbles and these too appear to diminish swing’. The idea that ‘effective roughness’ is increased by spin rate is not investigated further in these studies or anywhere else in the literature and it is not clear what physical mechanism is involved. Figure 20 also shows that the average swing coefficient is higher at seam angles of $20^{\circ }$. This is consistent with the idea that seam wobble reduces swing if it causes the seam angle to vary above and below $0^{\circ }$ during a delivery, effectively switching the seam and non-seam side. For a given amount of seam wobble, increasing seam angle reduces this effect as the mean angle is further from $0^{\circ }$.

The maximum reduced frequency in the tests is 0.063 and this drops below 0.04 for $Re > 1.22\times 10^5$ (25 m s$^{-1}$, 56 mph). This implies that the problem is quasi-steady, i.e. the flow through time is more than an order of magnitude quicker than the rotation time so that spin rate should not be an important factor. Separating the effects of mechanical stability and aerodynamics using the projection tests was difficult for Barton and Bentley et al. and it is also not clear if the trajectory equations used account for varying lift on the ball, caused by the Magnus effect changing with spin rate. Further work is required to confirm whether there is an optimum spin rate for cricket ball swing, and if there is, what mechanisms are responsible.

Barton and Bentley et al.'s projection tests, which include spin, are a more realistic representation of deliveries bowled in cricket matches than wind tunnel tests with cricket balls held in a fixed mount. This can be seen in the results in figures 19 and 20, which show continuous distributions of $C_{S}$, more like the ball tracking data in figure 2, than the results from fixed tests in figure 6. We hypothesise that this is due to two effects: first, for each spinning projection test, rotation angle, $\alpha$, and seam angle, $\theta$, vary continuously so the flow past the ball can switch between conventional and reverse swing states and the swing force is averaged out to an intermediate value. Second, from test to test (or in a match, delivery to delivery) the seam angle and wobble can vary, along with the condition of the ball, resulting in a distribution of $C_{S}$ for a given $Re$ which when averaged give the smooth distributions seen in figure 2. Barton and Bentley et al.'s work illustrates this behaviour but further work is required to develop a statistical model which can quantify the effect of realistic boundary conditions including ball condition, seam angle and seam wobble. The optimum spin rate, observed by Barton and Bentley et al., is not understood and future work could repeat the projection tests using motorised, computer-controlled mechanisms for spinning and dropping the balls, and modern high-speed cameras for tracking their trajectory.

5.1.3 Computational fluid dynamics simulations

Computational fluid dynamics can be used to simulate cricket ball swing where backspin is modelled using a rotating interface and unsteady calculation. Reference Latchman and PooransinghLatchman & Pooransingh (2015) simulate a cricket ball with a simplified geometry in which the primary seam is represented as a 20 mm wide concentric rim projecting 1 mm from the ball surface. Quantified details of the mesh, e.g. cell count and surface $y+$ are not provided. The solver used to simulate the flow is an incompressible, Reynolds-averaged Navier–Stokes (RANS) scheme with $\kappa - \epsilon$ turbulence model. The ball was modelled at velocities from 22 to 45 m s$^{-1}$, at seam angles of $10^{\circ }$, $20^{\circ }$ and $30^{\circ }$, and with rotation rates from 0 to 14 Hz. At $20^{\circ }$, increasing rotation rate from 0 to 2 Hz increases $C_{S}$ by 11 % but for the cases between 2 and 14 Hz there is less than 2 % variation. Without rotation, $C_{S}$ of 0.11, 0.21 and 0.39 are reported for $10^{\circ }$, $20^{\circ }$ and $30^{\circ }$, respectively. This result does not agree with the experimental results reviewed elsewhere in this paper and no explanation is given for the increase in $C_{S}$ with seam angle. There is also less than 1 % change in $C_{S}$ across the full range of velocities simulated for each seam angle and overall these physically unrealistic results lead the authors to conclude ‘that the $\kappa -\epsilon$ turbulence model may not be suitable for simulating the fluid flow around a cricket ball or that the model constants need to be redefined for this application’, where $\kappa$ is turbulent kinetic energy and $\epsilon$ is the turbulent kinetic energy dissipation rate.

A similar study using RANS with a $\kappa - \epsilon$ turbulence model is reported by Reference Kalburgi, Rathi, Narayan, Keni, Chethan and ZuberKalburgi et al. (2020). This work shows the boundary layers separating further forward on the non-seam side than the seam side with a seam angle of $20^{\circ }$ at velocities of 25, 29 and 36.5 m s$^{-1}$. Similar to Latchman and Pooransingh's work, however, the switch from conventional to reverse swing associated with transition on the non-seam side is not captured. Kalburgi et al. do not quantify swing force from their calculations and do not model rotation.

Reference Pahinkar and SrinivasanPahinkar & Srinivasan (2010) report CFD simulations of reverse swing using a simplified geometry, similar to that of Reference Latchman and PooransinghLatchman & Pooransingh (2015). Steady and unsteady RANS solutions are obtained using a Spalart–Allmaras turbulence model near the wall surface with large eddy simulation in the free-stream flow. The results show that for the velocities simulated, from 24 to 42 m s$^{-1}$, the force is negative and surface pressure plots show that this is because the flow separates further forwards on the seam side than the non-seam side. The pressure plots do not show any evidence of laminar separation and turbulent reattachment on either side of the ball and negative $C_{S}$ at all velocities indicates that this study is unable to capture transition on the non-seam side of the ball. Including ball rotation in the simulation increases the swing force magnitude but this effect is not explained.

Reference Tom, Ruishikesh, Jose and KumarTom et al. (2013) simulate cricket balls with smooth and rough surfaces where the rough surfaces have $k/D=100\times 10^{-5}$. There is no explanation of how these rough surfaces are implemented and details of the overall numerical scheme and mesh are not provided. Validation of CFD cases for conventional and reverse swing against ‘real life’ trajectory data is shown to be within 0.01 m throughout, although the source of the validation data is not provided. For conventional swing the ball is displaced $s=0.29$ m over a delivery length of $l=12.53$ m and using (3.1) the average swing force coefficient is calculated to be $C_{S} = 0.237$. The paper goes on to study the behaviour of a new ball, where both sides are smooth and the rotation rate is 6.4 Hz, and an old ball, where one side is smooth and one rough, and the rotation rate is 9.5 Hz. $C_{S}$ and $C_D$ are reported for cases at various seam angles and speeds and for the new ball a maximum magnitude of $C_{S}=0.05$ is reported. It is not clear how these results are made consistent with the trajectories reported in the validation section. Further details of the flow, such as the boundary layer states and separation points, and an explanation of the CFD set-up, are not provided so it is not possible to assess how well the simulations match reality.

The studies reviewed in this subsection show that CFD can be used to investigate the effect of spin rate on swing. However, predicting cricket ball swing with CFD has proved challenging and to date, only simplified geometries have been simulated with models used to represent surface roughness. The sub- and super-critical flow regimes on the non-seam side of the ball also have to be captured accurately using the same turbulence modelling scheme so that the switch from conventional to reverse swing can be predicted. Only Reference Tom, Ruishikesh, Jose and KumarTom et al.'s (Reference Tom, Ruishikesh, Jose and Kumar2013) results demonstrate this behaviour but unfortunately their paper does not provide information about CFD set-up or detailed analysis of the flow. In conclusion, a comprehensive study of cricket ball swing using CFD has yet to be published and no agreed or established approach has been presented. This represents a significant area for future work.

5.2.1 Macroscopic conditions

These are atmospheric conditions which change slowly during play and which affect a whole cricket ground uniformly. ‘Slowly’ means that properties can be considered constant through periods of play of the order an hour long. Properties which affect the cricket ball on this time scale are atmospheric temperature, pressure and humidity. Cricketers often talk about ‘cloud cover’ but this is not included in this discussion as it is difficult to quantify and does not directly affect the air flow around the ball. However, the relationship between cloud cover and other variables will be considered where relevant.

The Reynolds number, $Re$, defined in (2.2), is a function of air density and dynamic viscosity, quantities which are affected by the macroscopic properties listed above. The density and viscosity for humid air can be calculated from pressure, temperature and humidity using formulae from Reference TsilingirisTsilingiris (2008). For example, an increase in air temperature from $25\,^{\circ }$C to $35\,^{\circ }$C, with fixed atmospheric pressure of 101.2 kPa, reduces $Re$ for a ball delivered at 40 m s$^{-1}$ (89 mph) from $1.85\times 10^5$ to $1.74\times 10^5$, a decrease of 6 %. For a match played at an altitude of 1500 m, as is the case at Centurion in South Africa, the atmospheric pressure is reduced to 85 kPa. Here, at a fixed temperature of $25\,^{\circ }$C, a 40 m s$^{-1}$ delivery has $Re = 1.58\times 10^5$, a decrease of 15 % compared with the case at standard sea level conditions. Finally, for the case with $25\,^{\circ }$C and atmospheric pressure of 101.2 kPa, increasing relative humidity from 0 % to 90 % increases $Re$ from $1.85\times 10^5$ to $1.86\times 10^5$, an increase of 0.6 %. These results show that changes in temperature and pressure should be taken into account when the bowling speed is converted to $Re$, however, the effect of humidity can be neglected. Similarly, for a given value of force coefficient, the force varies linearly with density so humidity has little effect while temperature and pressure should be accounted for. This explains why the ball flies further in hot, high altitude venues since reduced air density reduces the drag force on the ball.

The impact of humidity beyond its effect on $Re$ has been considered in a number of studies. Reference BinnieBinnie (1976) concluded that a condensation shock could upset a laminar boundary layer if the relative humidity is ‘nearly 100 %’. However, this mechanism would reduce the likelihood of conventional swing in humid conditions, as the critical $Re$ for the non-seam side boundary layer would be reduced. Reference BentleyBentley (1982) and Reference James, MacDonald and HartJames, MacDonald & Hart (2012) investigate the idea that the seam swells as it absorbs moisture from the air but their tests showed no measurable change in geometry due to humidity. Furthermore, changes to seam geometry are unlikely to affect the non-seam side flow which is the dominant factor in determining whether conventional swing can occur. Reference Moore and NeedesMoore & Needes (1973) measured no difference in swing force, at fixed $Re$, on days with different specific humidity levels between 20 % and 50 %, and Reference Sherwin and SprostonSherwin & Sproston (1982) found no effect on the magnitude of swing with humidity varying between 61 % and 100 %. More recent tests by Reference Scobie, Shelley, Jackson, Hughes and LockScobie et al. (2020) tested a new ball at 31 %, 51 % and 73 % relative humidity and showed that there was ‘no significant effect’ on $C_{S}$ or $Re$ at which the ball switches from conventional to reverse swing.

Finally, Reference SteadmanSteadman (1979) studied differences in perceived temperature due to changes in humidity and wind speed. An increase in wind speed of 4 m s$^{-1}$ was found to have a similar effect on perceived temperature as a drop of 20 %–30 % in humidity. This effect shows that days which are felt to be humid are also likely to have low wind speeds. The effect of wind on cricket ball swing is a transient condition and is considered in the next subsection.

5.2.2 Transient conditions

Transient conditions. Atmospheric conditions which vary from one delivery to the next, or during each delivery, are considered to be ‘transient’. They are quantified with variables associated with the air flow across the pitch: wind speed, wind direction and atmospheric turbulence, and need to be measured locally and time accurately.

The effect of wind speed on a cricket ball trajectory is included in the trajectory equations developed by Reference BakerBaker (2010). These show that, for a ball travelling 18 m released at 33.5 m s$^{-1}$ (75 mph, $Re=1.64\times 10^5$), the inclusion of a 4 m s$^{-1}$ perpendicular cross-wind in the model changes the lateral displacement by approximately 0.18 m. This is 19 % of the total lateral displacement for a swing force coefficient of 0.3 in such a cross-wind. Baker's analysis assumes that the drag and swing force coefficients are defined as a function of $Re$ only. However, a cross-wind also changes the incidence of the oncoming air to the ball. This has the same effect as changing seam angle and with a ball speed of 33.5 m s$^{-1}$ and a perpendicular cross-wind of 4 m s$^{-1}$ the angle change is $6.8^{\circ }$. As discussed in the previous section, $C_{S}$ is sensitive to seam angle and a case where the wind changes the effective seam angle from positive to negative would change the direction and magnitude of the swing force. This effect is likely to be more important than that captured by Baker's existing model and should be included in future cricket ball trajectory modelling.

The principles of free-stream turbulence are presented by Reference TaylorTaylor (1922,  Reference Taylor1935,  Reference Taylor1938) as a statistical model to describe fluctuations in a flow where the time accurate flow field is split into its mean and fluctuating components as shown in (5.1). The turbulence intensity is then defined in (5.2) as the ratio of the root mean square of the fluctuation to the mean value

(5.1)$$\begin{gather} U(t) = \bar{U} + U', \end{gather}$$

(5.2)$$\begin{gather}TI = \frac{(U')_{RMS}}{\bar{U}}. \end{gather}$$

Papers on cricket ball swing by Reference BakerBaker (2010), Reference James, MacDonald and HartJames et al. (2012) and Reference Scobie, Shelley, Jackson, Hughes and LockScobie et al. (2020) hypothesise that free-stream turbulence is generated from turbulent convection from a hot cricket pitch. This is linked to the belief that swing is less likely to occur on clear, sunny days. Scobie et al. cite work by Reference IbbetsonIbbetson (1978) where meteorological measurements show velocity fluctuations up to 0.75 m s$^{-1}$ which equate to 4 % turbulence intensity for a ball delivered at 38 m s$^{-1}$ (85 mph). The discussion in the paper by Scobie et al. attributes the fluctuations to ‘convective micro-turbulence’, although, in the paper itself, Ibbetson says this is due to thermal convection and spatial variations of the wind velocity, without separating these effects out.

Reference WangWang (1982) investigated buoyancy flows generated by a hot plate in a cross-flow which is analogous to a hot cricket pitch with wind blowing across it. Turbulent fluctuations are produced over the heated metal plate when free convection, driven by a temperature difference between the plate and the air, dominates the effect of flow across the plate. This effect is characterised by the ratio of the Grashof number, $Gr_x$, to the Reynolds number of the flow across the plate, $Re_x$, and Wang's experiments show that convection driven turbulence occurs when

(5.3)\begin{equation} \frac{Gr_{x}}{(Re_{x})^{5/3}}>55.3. \end{equation}

For example, with a wind speed of 5 m s$^{-1}$, the condition in (5.3) is met when ground temperature is $50\,^{\circ }$C greater than the air temperature. To test whether this condition is likely to occur in a cricket match, measurements of air and ground temperature and wind speed were taken at two professional cricket venues in England in the summer of 2020. Details of this test are included in Appendix A. Figure 21 shows that the critical value for the convection criterion from Wang's analysis was only exceeded once during the four days tested, with a maximum value of 56 and an overall mean of 14.9. The temperature difference between the ground and air is therefore not expected to drive free-stream turbulence on the ‘unremarkable’ English summer days that the data were collected on. Further work is required to establish whether ground to air temperature differences in hot countries, such as India or Australia, are large enough to drive free-stream turbulence when the wind speed is low.

Figure 21. Plots of (a) ground to air temperature difference, (b) wind speed and (c) convection criterion (Reference WangWang 1982) measured on four separate days at two professional cricket venues in England.

Figure 22 shows wind tunnel measurements reported by Reference Scobie, Shelley, Jackson, Hughes and LockScobie et al. (2020) in which a new ball, labelled (A), is tested with a turbulence grid which generates free-stream turbulence intensity of approximately 5 %. $C_{S}$ recorded is between $-$0.10 and $-$0.05 so conventional swing does not occur for $Re>1.1\times 10^5$. The same ball, tested with a turbulence intensity of approximately 1 % when no turbulence grid is used, is labelled (B) in figure 6 and has $C_{S}$ between 0.28 and 0.30 up to $Re=2.1\times 10^5$. The difference in behaviour with and without the turbulence grid is because the increased turbulence reduces the critical $Re$ for the non-seam side transition. This results in a laminar separation bubble and reattachment on the non-seam side at reduced $Re$ and limits the flow asymmetry so that small negative $C_{S}$ are recorded. Similar results were recorded for $Re<1.3\times 10^5$ by Reference BentleyBentley (1982) using the projection test approach and turbulence grids which gave turbulence intensities of 1.6 % and 3.1 %. These data are non-dimensionalised and reproduced in figure 22, labelled (B) and (C).

Figure 22. Swing force coefficient, $C_{S}$, plotted against Reynolds number, $Re$, for wind tunnel experiments using grids to increase free-stream turbulence intensity reported by Reference Scobie, Shelley, Jackson, Hughes and LockScobie et al. (2020): A; Reference BentleyBentley (1982): B,C; and from Whittle Laboratory tests: D. The IR images of the non-seam side are presented in figure 23 for the three cases highlighted with solid, diamond-shaped markers.

To validate Scobie et al.'s results, a turbulence grid which generates 2.2 % turbulence intensity was added to the Whittle Laboratory test section and force and IR measurements recorded. The data, labelled (D) in figure 22, show $C_{S}$ between $-$0.15 and $-$0.09 for $Re=1.44\unicode{x2013}1.93 \times 10^5$ and the same qualitative trend as Scobie et al.'s measurements. The IR images in figure 23 confirm that the non-seam side is above the critical value at $Re=1.38\times 10^5$ and has a laminar separation bubble and reattachment. Overall, free-stream turbulence prevents conventional swing for $Re>1.2\times 10^5$ and the flow past the ball is best represented by the sketch in figure 17(a).

Figure 23. Non-seam side IR image of a new cricket ball with grid-generated turbulent intensity of 2.2 % for varying $Re$. Flow is left to right and red dashed lines illustrate eventual separation position.

Studies using turbulence grids show that free-stream turbulence intensity could be an important factor in determining whether a ball will swing. However, there is no published work on the range of turbulence intensities and length scales a cricket ball is likely to encounter, no study of how this turbulence is generated, and no systematic investigation of how turbulence intensity and length scale affect the non-seam side critical $Re$. All of these topics should be the subject of future work.