3.3.1. Twist angle vs number of cells and type of dislocations

Computations have been carried out for the $sAR=5$ wing with varying twist, $\delta$. In all cases, the angle of attack of the root section, $\alpha _r$, is held at the same value while that at wing tip is $\alpha _r+\delta$. For the range of twist angles considered in the study, the local geometric angle of attack at the wing tip becomes too large for $\alpha _r > 14^\circ$. Therefore, we consider $\alpha _r=8^\circ$ and $14^\circ$, while the wing twist for each $\alpha$ is varied in the range $-6^\circ \leqslant \delta \leqslant 6^\circ$. We recall that vortex shedding for an EEW commences at $\alpha \sim 8^\circ$ and the flow at this $\alpha$ is associated with a single-cell vortex shedding. We also note from figure 4, for the untwisted wing, that the difference between the vortex shedding frequency of central and end cells ($St_C-St_E$) is maximum at $\alpha =14^\circ$. The geometric angle of attack varies linearly across the span between the root and wing tip. We explore the effect of twist on the number of cells, vortex shedding frequency and nature of dislocation. The $Q$ iso-surface of the instantaneous flow coloured with spanwise vorticity for certain twist angles, with $\alpha _r=14^\circ$, is shown in figure 8 along with the spanwise variation of $St$. For ready reference, the local $\alpha$ at different locations is marked on the plot showing $St$. Also shown in the plot is $St$ for the EEW at the corresponding local geometric angle of attack at that span location. We recall that $St$ for the EEW decreases with increase in $\alpha$ (see figure 4). Therefore, for the wing with twist, it is expected that $St$ at the wing tip should be larger than that at the root for $\delta <0^\circ$ and lower for $\delta >0^\circ$. Indeed, figure 8 shows that the variation of $St$ along the span, for various $\delta$, is consistent with this observation. The difference in the shedding frequency at the tip and root is correlated with the number of cells along the span – higher difference leads to a larger number of cells. Figure 10(a) shows the variation of number of cells with twist angle along with the Strouhal number for the vortex shedding frequency in each cell. A single cell forms for $\delta = -2^\circ$ wherein, unlike for the untwisted wing ($\delta =0^\circ$, $\alpha =14^\circ$, see figure 2c), $St$ is uniform along the span and the flow is devoid of dislocations (see figure 8c,d). Two cells are observed for moderate twist. With an increase in twist, the difference between $St$ for the end and central cells increases. As a result, the central cell splits into two cells – middle and central. Dislocations appear at the border of cells. They are of three types: reverse fork-type ($D_{rf}$) (Pandi & Mittal Reference Pandi and Mittal2023), mixed-type ($D_{rf-rfc}$) and fork-type ($D_f$) (Williamson Reference Williamson1989; Mittal et al. Reference Mittal, Pandi and Hore2021; Pandi & Mittal Reference Pandi and Mittal2023). The range of twist for which they are observed is marked in figure 10. The dislocations are generated at the same spanwise location for $\delta <-2^\circ$ and at varying locations for $\delta >-2^\circ$. Further, the frequency of occurrence of dislocations increases with an increase in the twist angle. The mixed-type and fork-type dislocations are described in more detail in §§ 3.3.2 and 3.3.3, respectively.

Figure 8.

Flow past a twisted $sAR=5$ wing at $Re=1000$ with $\alpha _r=14^\circ$: $Q(=0.25)$ iso-surface for an instantaneous flow coloured with spanwise vorticity ($\omega _z=\pm 2$) for $\delta =$ (a) $-6^\circ$, (c) $-2^\circ$, (e) $2^\circ$ and (g) $6^\circ$. The dislocations, central, middle and end cells are identified in these images. The non-dimensional vortex shedding frequency across the span is plotted in (b,d,f,h) for $\delta =-6^\circ$, $-2^\circ$, $2^\circ$ and $6^\circ$, respectively. The angle of attack along the span of the twisted wing is shown in the right axis of these images. Also marked in black and the cyan colour solid symbol are the vortex shedding frequencies within the central and end cell, respectively, of the untwisted wing at the corresponding local angle of attack. The $St$ value of EEW is also plotted.

Next, we present the $Q$ iso-surface of the instantaneous flow coloured with spanwise vorticity for various twist angles in figure 9 for $\alpha _r=8^\circ$. The vortices for the untwisted finite wing at $\alpha =8^\circ$ are weaker compared with that for the EEW and the wake consists of a single cell (figure 9c). The negative twist ($\delta <0^\circ$), where the local angle of attack at the tip ($\alpha _t$) is less than $8^\circ$, further weakens the vortex shedding. We note that the $Re=1000$ flow past an untwisted EEW is steady for $\alpha <7^\circ$. The vortex shedding is completely suppressed for $\delta <-2^\circ$ and the flow attains a steady state. In the case of positive twist ($\delta >0^\circ$), vortex shedding occurs in two cells for $\delta =2^\circ$ and three cells for $\delta >2^\circ$. The spanwise variation of $St$ is shown in figures 9(e) and 9(g) for $\delta =2^\circ$ and $6^\circ$. Also shown in the plot is $St$ for the EEW at the corresponding local geometric angle of attack at that span location. The number of cells formed in the positive twist regime is similar to the case of a twisted wing with $\alpha _r=14^\circ$. Another common feature between the two $\alpha _r$ is the fork-type dislocation for $\delta >0^\circ$. The variation of non-dimensional vortex shedding frequency within the various cells, with $\delta$, is presented in figure 10(b). The hatched region in this image represents the steady-state regime. The variation in the number of cells of vortex shedding with twist is also marked in the figure. It is interesting to compare the trend of the variation of $St$ in various cells with $\delta$ for $\alpha _r=8^\circ$ and $14^\circ$. In both cases, the central-cell frequency ($St_C$) assumes a value close to that for the untwisted wing, which in turn is close to the $St$ value for the EEW. The end-cell frequency, $St_E$, for $\alpha =14^\circ$ follows the variation observed for the untwisted wing at the corresponding span-averaged $\alpha$. However, a significant departure is seen for $\alpha =8^\circ$ when the twist angle is $\delta =6^\circ$. The value of $St_E$, in this case, is significantly lower than the corresponding $St_E$ for an untwisted wing. We suspect that this is because of the onset of the transition of the flow from a two- to three-cell shedding. We note that the number of vortex shedding cells increases with an increase in twist and as a result the end-cell frequency for the twisted wing undergoes a departure from the value for the untwisted wing where the number of cells changes. An observation related to this phenomenon can also be made for the $2^\circ$ twist for $\alpha _r=8^\circ$ (figure 9e), where the vortex shedding occurs in two cells despite the very small frequency difference between the end and central cells. Again, we suspect that this twist angle is close to the transition between one- and two-cell shedding.

Figure 9.

Flow past a twisted $sAR=5$ wing at $Re=1000$ with $\alpha _r=8^\circ$: $Q(=0.25)$ iso-surface for an instantaneous flow coloured with spanwise vorticity ($\omega _z=\pm 2$) for $\delta =$ (a) $-4^\circ$, (b) $-2^\circ$, (c) $0^\circ$, (d) $2^\circ$ and (f) $6^\circ$. The dislocations, central, middle and end cells are identified in these images. The non-dimensional vortex shedding frequency across the span is plotted in (e,g) for $\delta =2^\circ$ and $6^\circ$, respectively. The angle of attack along the span of the twisted wing is shown on the right axis of these images. Also marked in black and the cyan colour solid symbol are the vortex shedding frequencies within the central and end cell, respectively, of the untwisted wing at the corresponding local angle of attack. The value of $St$ of EEW is also plotted.

Figure 10.

Flow past a twisted $sAR=5$ wing at $Re=1000$: variation of non-dimensional vortex shedding frequency (solid symbol) in the central cell ($St_C$), middle cell ($St_M$) and end cell ($St_E$) with twist angle ($\delta$) for $\alpha _r=$ (a) $14^\circ$ and (b) $8^\circ$. The regimes are classified based on the type of dislocation. Here, $D_{rf}$, $D_{rf-rfc}$ and $D_f$ indicate reverse fork-type, mixed-type (a mix of reverse-fork- and connected reverse fork-type) and fork-type dislocations. The variation of the number of cells with twist angle is marked. The corresponding angle of attack at the tip ($\alpha _t$) for each $\delta$ is marked along the upper $x$-axis. Also plotted is the end-cell ($St_E$) frequency of the untwisted wing (hollow symbol) with respect to $\alpha _t$. The hatched region in (b) denotes the steady-state regime.

The spanwise instabilities in flow past EEW and a finite wing have been presented in earlier work (see e.g. Son et al. Reference Son, Gao, Gursul, Cantwell, Wang and Sherwin2022; Pandi & Mittal Reference Pandi and Mittal2023). Here, we discuss the effect of wing twist on these instabilities. Figures 2(c), 2(d) and 2(e) show the evolution of the hairpin vortices, associated with mode C instability, with $\alpha$ for the EEW. They become stronger with an increase in $\alpha$ and show increased disorder. The interaction of the wing-tip vortices, for the finite wing, with the vortex shedding results in the weakening of these hairpin vortices, as illustrated in figure 2. For $\alpha =14^\circ$, the finite wing is devoid of hairpin vortices near the root section. They form for $\alpha =16^\circ$ and beyond. The effect of twist on the hairpin vortices can be seen from figure 8. With the wing root oriented at $14^\circ$ local angle, a negative twist reduces the 3-D instabilities related to the undulation of the primary vortices and the hairpin vortices. Positive twist, on the other hand, results in an increase in 3-D instabilities. Figure 9 shows the effect of twist on the wing while the root chord is oriented at $8^\circ$ local angle of attack. Negative twist weakens vortex shedding and quenches three-dimensionality. Positive twist results in increased unsteadiness. Despite the vortices being oblique there are neither any spanwise undulations near the root section, nor any hairpin vortices. Increase in positive twist is accompanied by an increase in the number of cells of vortex shedding and the formation of linkages between vortices of opposite polarity near to the wing-tip region.

3.3.2. Mixed-type ($D_{rf-rfc}$) dislocation

A dislocation forms at the boundary of the adjacent cells with varying vortex shedding frequency. The dislocations are periodically generated at the same spanwise location for $\delta =-4^\circ$. Spanwise and temporal variation of the $v$-component of velocity in the near wake is shown in figure 11(a). Two instances of the formation of dislocations, due to the appearance of an extra vortex in the end cell at $t = 70.79$ and $110.132$, are identified in the image. They are marked as $D_1$ and $D_2$. The non-dimensional frequency of appearance of the dislocations estimated from the average time period is $0.0254$. The beat frequency in the time signal near the boundary of the cells can also be used to estimate the frequency of dislocation (Williamson Reference Williamson1989; Behara & Mittal Reference Behara and Mittal2010; Mittal et al. Reference Mittal, Pandi and Hore2021; Pandi & Mittal Reference Pandi and Mittal2023). The frequency so estimated from the time history of the cross-flow component of velocity is $0.0264$. We note that the two estimates are in good agreement. To further explore the dislocation $D_1$, we consider the $Q$ iso-surface of the flow at two time instants shortly after $D_1$ is formed. The advection of the spanwise vortices along with the related dislocation can be seen in figures 11(b) and 11(c). Here, $D_1$ is of reverse fork-type ($D_{rf}$) at $t = 72.164$ (figure 11b). It connects two vortices from the end cell to a single vortex of the same polarity in the central cell. As $D_1$ convects downstream in the wake, a linkage forms between vortices of opposite polarity, resulting in a ring-type vortex structure (see figure 11(c) corresponding to $t = 74.664$). This, in addition to the already existing reverse fork-type structure, leads to the mixed-type dislocation. To the best of our knowledge, this type of dislocation is being reported for the first time.

Figure 11.

Flow past a $sAR=5$ wing with $\delta =-4^\circ$ and $\alpha _r=14^\circ$ at $Re=1000$: (a) spatio-temporal variation of $v$ obtained using a probe placed at $x/c=1.25$, $y/c=-0.068$. The $Q(=0.25)$ iso-surface for instantaneous flow coloured with the spanwise component of vorticity ($\omega _z=\pm 2$) at various time instants is presented in (b,c). The reverse fork-type ($D_{rf}$) dislocation in (b) convects and transforms to a connected reverse fork-type ($D_{rfc}$) dislocation with an additional ring-like vortex structure highlighted in (c).

3.3.3. Fork-type ($D_f$) dislocation

Fork-type dislocations ($D_f$) are observed for wings with $\delta \geqslant 0^\circ$. In all cases the dislocations form at varying spanwise locations at different time instants. However, once formed, they convect downstream along the same spanwise location at which they are created. We consider a representative case of $\delta = 4^\circ$. The spanwise and time variation of $v$ in the near wake, presented in figure 12(a), reveals a three-cell structure of the flow. The dislocations, that form at the boundaries of the cells, are identified. The ones that are created at the boundary of the end and middle cells are referred to by the superscript ‘1’, i.e. $D^1$ while those that form at the junction of the middle and central cells are identified by the superscript ‘2’, i.e. $D^2$. The subscript, in each of the two types, refers to the chronological sequence in which the dislocations are created. Dislocations $D^1_{1-5}$ are formed at various spanwise locations between $z/c = 1.92$ and $2.68$. The time interval between the occurrence of successive dislocations also varies. The average dislocation frequency is estimated to be $0.098$. The dislocations $D^2$, that demarcate middle and central cells, also behave in a similar manner. For example, $D^2_1$ and $D^2_2$ form at different spanwise locations at $z/c = 0.42$ and $1.16$, respectively. The average frequency of occurrence of dislocation $D^2$ is estimated to be $0.044$. The $Q$ iso-surface coloured with spanwise vorticity for the instantaneous flow is shown in figures 12(b) and 12(c) at $t=62.812$ and $73.396$, respectively. These images show that both, dislocations $D^1$ and $D^2$ are of fork-type.

Figure 12.

Flow past a $sAR=5$ wing with $\delta =4^\circ$ and $\alpha _r=14^\circ$ at $Re=1000$: (a) spatio-temporal variation of $v$ obtained using a probe placed at $x/c=1.25$, $y/c=-0.068$. The $Q(=0.25)$ iso-surface for instantaneous flow coloured with the spanwise component of vorticity ($\omega _z=\pm 2$) at various time instants is presented in (b,c). Also marked are the dislocation $D_1^{1}$ in (b) $D_2^{1}$ and $D_1^{2}$ in (c).

3.5.1. The case of a $sAR=5$ untwisted wing at $\alpha =14^\circ$ for various Re

Figure 14 shows the variation of time-averaged force coefficients and their fluctuations with $Re$. The lift coefficient (figure 14a) is maximum at $Re=100$. It decreases with increase in $Re$ up to $Re=375$ and it increases thereafter. A similar trend in the variation of lift coefficient with $Re$ was reported by Srinath & Mittal (Reference Srinath and Mittal2009) for a NACA 0012 airfoil at $\alpha =4^\circ$. The drag coefficient decreases with an increase in $Re$ (please refer to figure 14b). The fluctuation in the force coefficients increases with an increase in $Re$. Vortex shedding has a more significant impact on the lift than the drag. We explore the variation of the lift coefficient with $Re$. The time-averaged surface pressure distribution on the airfoil at the wing root is presented in figure 15(a) for various $Re$. Also shown is the separation point estimated from the location where skin friction vanishes on the airfoil surface. The separation point moves upstream towards the leading edge of the airfoil with an increase in $Re$. An interesting feature at low $Re$ is the increased pressure ($\overline {C_p}>1$) on the lower surface of the airfoil resulting in enhanced lift. Issa (Reference Issa1995) showed that the stagnation pressure is higher in viscous flow ($C_p > 1$). This was demonstrated by Oudheusden (Reference Oudheusden1996) for a cylinder and a sphere at low $Re$. The pressure on the lower surface decreases with an increase in $Re$. The vortex shedding increases the suction on the aft region of the airfoil for $Re\ge 375$. The spanwise variation of the local sectional lift coefficient is presented in figure 15(b) for various $Re$. At low $Re$, the sectional local lift coefficient is maximum at the root and it decreases monotonically towards the wing tip. This is in line with Prandtl's LLT (Prandtl Reference Prandtl1918, Reference Prandtl1921; Bertin Reference Bertin2002; Anderson Reference Anderson2017). With an increase in $Re$, the variation becomes non-monotonic with a local peak near the tip due to the streamwise vortices (Pandi & Mittal Reference Pandi and Mittal2023).

Figure 14.

Flow past an untwisted $sAR=5$ wing at $\alpha =14^\circ$: variation of time-averaged (a) $\overline {C_L}$, (b) $\overline {C_D}$ and (c) $C_{Lrms}$ and $C_{Drms}$ with $Re$.

Figure 15.

Flow past an untwisted $sAR=5$ wing at $\alpha =14^\circ$: time-averaged (a) pressure distribution at the wing root ($z/c=0$) and (b) spanwise variation of sectional local lift coefficient for various $Re$. The solid circles in (a) denote the point of separation identified from the location on the airfoil surface where the skin friction is zero.

3.5.2. The case of a $sAR=5$ wing at various $\alpha$ and $\delta$

Figure 16 shows the variation of time-averaged aerodynamic force coefficients for the end-to-end and $sAR = 5$ wing with angle of attack for $Re = 1000$. The data for the twisted wing, with wing root $\alpha _r=8^\circ$ and $14^\circ$, are plotted with respect to the span-averaged geometric angle of attack defined as $\langle \alpha \rangle =\alpha _r+\delta /2$. Owing to the downwash generated by the wing-tip vortices, the finite wing experiences a reduction in effective angle of attack at each spanwise section. Therefore, the lift coefficient of the finite untwisted wing is lower than that for the EEW at each $\alpha$. The slope of the $C_L-\alpha$ curve at low $Re$ is significantly lower than the value predicted by the thin airfoil theory ($=2{\rm \pi}$) (Bertin Reference Bertin2002; Anderson Reference Anderson2017). At $Re=1000$, its value is $0.78{\rm \pi}$. Maji et al. (Reference Maji, Pandi and Mittal2024) presented the pressure distribution on a NACA 0012 airfoil obtained from direct numerical simulation for $100 \leqslant Re \leqslant 6000$ and compared it with the prediction from inviscid flow using the second-order vortex panel method. The flow at low $Re$ is associated with decreased suction over the upper surface of the airfoil which in turn lowers its lift significantly as compared with that at high $Re$ or the inviscid flow. The wing-tip vortex weakens the vortex shedding near the wing tip thereby decreasing the overall unsteadiness of the flow over the wing (Taira & Colonius Reference Taira and Colonius2009; Zhang et al. Reference Zhang, Hayostek, Amitay, He, Theofilis and Taira2020b; Pandi & Mittal Reference Pandi and Mittal2023). This is evident from figure 19 that shows a significantly lower r.m.s. value of the force coefficients at each $\alpha$ for the finite wing. Pandi & Mittal (Reference Pandi and Mittal2023) showed that the contribution of pressure to the overall drag is significant, and that it reduces with decrease in flow unsteadiness. As a result, despite the induced drag generated due to tilting of the local lift, the drag coefficient for the finite wing is lower than that for EEW at each $\alpha$ (see figure 16b). Wing twist, in the form of a wash-out, is often utilized in high $Re$ flows to avoid stall of the flow at the wing tip. It was shown by Maji et al. (Reference Maji, Pandi and Mittal2024), by compiling data from earlier studies, that unlike at high $Re$, the flow past a wing at low $Re$ does not experience an abrupt stall with increase in angle of attack. Therefore, at low $Re$, the wing twist can be utilized to enhance the performance of the aerial vehicle.

Figure 16.

Flow past end-to-end and $sAR=5$ wings at $Re=1000$: variation of time-averaged (a) $\overline {C_L}$, (b) $\overline {C_D}$ and with angle of attack. The data for the wing with twist are plotted with respect to the span-averaged angle of attack ($\langle \alpha \rangle =\alpha _r+\delta /2$). The twist angle ($\delta$) is marked on the upper $x$-axis for $\alpha _r=8^\circ$ and $14^\circ$.

We explore the effect of twist on the performance of a finite wing. Figure 16 shows that, despite the difference in flow structure, surprisingly, the overall force coefficients for the twisted and untwisted wings are comparable. Therefore, their aerodynamic efficiencies (ratio of lift to drag) are very similar as well. The trend in the force coefficients of the twisted wing is very similar for $\alpha _r=8^\circ$ and $14^\circ$. Therefore, first, we consider a twisted wing with $\alpha _r=14^\circ$ at low to moderate twist angles ($-4^\circ \leqslant \delta \leqslant 4^\circ$).

3.5.3 The case of $sAR=5$ wing with $\alpha_r=14^{\rm o}$ and low to moderate twist $(-4^{\rm o} \leqslant \delta \leqslant 4^{\rm o})$

Table 3 lists various parameters for the twisted and untwisted wing for two representative angles of attack ($=12^\circ$ and $16^\circ$). In addition to the force and moment coefficients, also listed in the table are the spanwise and chordwise location of the resultant force acting on the wing, estimated from the distribution of the local force coefficients for the time-averaged flow. The untwisted wing at each of these two values of $\alpha$ is compared with the respective twisted wing placed at the same span-averaged angle of attack $\langle \alpha \rangle$. The geometric angles of attack at the root and tip sections for the twisted wing with $\langle \alpha \rangle = 12^\circ$ are $\alpha _r=14^\circ$ and $\alpha _t=10^\circ$ corresponding to a twist of $\delta =-4^\circ$. These values for $\langle \alpha \rangle =16^\circ$ are $\alpha _r=14^\circ$, $\alpha _t=18^\circ$ and $\delta =4^\circ$. We note from the table that the time-averaged force as well as pitch ($C_m$) and roll ($C_r$) moment coefficients for the half-span of the wing, with and without twist, are comparable. The effect of twist appears to average out over the span. The rolling moment is specifically important from the point of view of structural design of the wing to sustain the bending moment. A wing with negative twist has marginally higher lift and lower moment coefficients.

Table 3.

Flow past $sAR=5$ wing at $Re = 1000$: time-averaged force coefficients, their r.m.s. and aerodynamic efficiency for untwisted and twisted wings. Also listed are the pitch ($C_m$), roll ($C_r$) and yaw ($C_y$) moment coefficients at mid-chord and at plane of symmetry ($z/c=0$). Here, $X_F/c$ and $Z_F/c$ denote the chordwise and spanwise location of the resultant force acting on the wing.

How does this compare with the prediction from the LLT for a finite wing with twist at low $Re$? Table 4 lists the various parameters for the two sets of wings predicted by LLT. Unlike in the classical LLT, here, we use the lift curve slope for the corresponding airfoil section from the viscous flow computations for the EEW at the same $Re$. It is $0.78 {\rm \pi}$, estimated from the best linear fit to the lift variation for $0^\circ \leqslant \alpha \leqslant 20^\circ$. This is explained in more detail in our earlier work (Pandi & Mittal Reference Pandi and Mittal2023). While the lift coefficients for the two sets of wings are comparable (see table 4), the rolling moment coefficient is larger for the wing with positive twist and smaller for negative twist. A wing with larger local $\alpha$ near the wing tip generates larger rolling moment compared with an untwisted wing. Also listed in table 4 is the induced drag coefficient. It can be observed that wash-in increases the induced drag coefficient compared with untwisted wing while wash-out has negligible effect. In contrast, the viscous flow computations show that the wing twist has negligible effect on the drag coefficient at $Re=1000$ (see table 3). The lift and rolling moment coefficients predicted using LLT are relatively larger than the values at these low $Re$.

Table 4.

Predictions using LLT for $sAR=5$ wing at $Re = 1000$: induced drag ($C_{Di}$) and lift coefficient for untwisted and twisted wing. The roll moment coefficient ($C_r$) and the spanwise location ($Z_F$) of the resultant force acting on the wing are also listed.

Figure 17(a) shows the spanwise variation of the local lift coefficient for the time-averaged flow of untwisted and twisted wings for various angles of attack. The data for the twisted wing are ascribed to its span-averaged angle of attack. We make two observations from the figure. The first is non-monotonicity of the variation. It was shown by Pandi & Mittal (Reference Pandi and Mittal2023), for the untwisted wing, that such a variation is due to streamwise vortices at various spanwise locations in the time-averaged flow. The second observation is about the local lift coefficient at the wing root. We recall that the angle of attack at the wing root, for the twisted wing, is $14^\circ$ for all $\delta$. Yet, as seen in figure 17(a), the local lift coefficient at the wing root varies significantly with change in $\delta$. It increases with increase in $\delta$. How, then, does the twist affect the overall lift coefficient for the wing? The local $\alpha$ increases outboard across the span for $\delta >0^\circ$ and decreases for $\delta <0^\circ$. Therefore, as expected, for $\delta < 0^\circ$, the local lift coefficient is higher near the wing root and lower near the wing tip compared with that for an untwisted wing oriented at the corresponding $\langle \alpha \rangle$. The trend is opposite for $\delta >0^\circ$. The spanwise distribution of local drag coefficient shows a similar trend (figure 17b). Interestingly, the increase and decrease of the local force coefficients for the twisted wing over the span, compared with the untwisted one, nearly cancel each other. As a result, the overall lift coefficients of the twisted and untwisted wings are almost equal (see figure 16). The local sectional geometric angle of attack for the twisted wing (for $\delta =-4^\circ$) is marked along the upper $x$-axis in figures 17(a) and 17(b). It is $14^\circ$, $12^\circ$ and $10^\circ$ at the root, mid-span and tip of the wing, respectively. We note that the sectional coefficients at these spanwise sections are different than those for the untwisted wing. For example, $C_l$ at the wing root is lower than that of the untwisted wing and, marginally higher at the mid-span and wing tip. An opposite trend is seen for $\delta =4^\circ$ wing (not shown here). This demonstrates that the wing loading for the twisted wing cannot be directly inferred from the data for the untwisted wing. The spanwise location of the resultant force coefficients and their values, estimated from the local distribution of lift and drag coefficients, are marked via solid and hollow square symbols for the untwisted and twisted wing, respectively. The spanwise variations of local roll ($C_r$) and yaw ($C_y$) moment coefficients for the time-averaged flow of untwisted and twisted wings for various angles of attack are shown in figure 18. The moment is estimated about the mid-chord location and at plane of symmetry ($z/c=0$). Similar to the sectional local force coefficients, the contribution from the wing-tip region is larger for the twisted wing for $\delta > 0^\circ$ compared with that for an untwisted wing. The overall moment coefficients are marginally lower for $\delta < 0^\circ$ compared with the untwisted wing.

Figure 17.

Flow past a $sAR=5$ wing at $Re=1000$: spanwise variation of time-averaged sectional force coefficient (a) $\overline {c_l}$, (b) $\overline {c_d}$ at various $\alpha$. The data for the twisted wing are plotted in broken line and are ascribed to the span-averaged angle of attack ($\langle \alpha \rangle = 14^\circ +\delta /2$). Also listed in the legend is the angle of attack at the tip of the twisted wing ($\alpha _t$). The local angle of attack at root, for the twisted wing, is $\alpha _r=14^\circ$. Data for the untwisted wing are shown in solid line. The solid and hollow square symbols mark the value as well as the spanwise location of the resultant force coefficient for untwisted and twisted wings, respectively. The local sectional geometric angle of attack across the span of $\delta =-4^\circ$ wing is marked along the upper $x$-axis in (a,b).

Figure 18.

Flow past a $sAR=5$ wing at $Re=1000$: spanwise variation of time-averaged sectional moment coefficient (a) $\overline {c_r}$, (b) $\overline {c_y}$ at various $\alpha$. The data for the twisted wing are plotted in broken line and is ascribed to the span-averaged angle of attack ($\langle \alpha \rangle = 14^\circ +\delta /2$). Also listed in the legend is the angle of attack at the tip of the twisted wing ($\alpha _t$). The local angle of attack at root, for the twisted wing, is $\alpha _r=14^\circ$. Data for the untwisted wing are shown in solid line.

Figure 19 illustrates that, compared with an untwisted wing, unsteadiness is lower for a wing with positive twist ($\delta >0^\circ$). To explore this further, we estimate the net unsteadiness in the wake via volume integration. Figure 20(a) shows the iso-surface of $\overline {v'v'} (=0.3,0.4,0.5)$ for an EEW at $\alpha = 20^\circ$. The span length is $5.0c$. The volume considered for determining the net unsteadiness is marked in the image. The extent of the volume is $5c$, $4c$ and $5c$ along the streamwise, cross-flow and spanwise direction, respectively. Here, $o'$ denotes the origin of the volume and it coincides with the trailing edge of the airfoil at $z/c=0$ for an EEW and at mid-span for an untwisted and twisted finite wing. The net unsteadiness $\widetilde {(v'v')_V}$ is estimated using the following expression:

(3.1)\begin{equation} {\widetilde{(v'v')}_V}=\frac{\displaystyle\int\int\int \overline{v'v'} \,{\rm d}V}{\displaystyle\int\int\int {\rm d}V}. \end{equation}

The variation of ${\widetilde {(v'v')}_V}$ with $\alpha$ is presented in figure 20(b) for untwisted and twisted finite wing. The data of the twisted wing are ascribed to its span-averaged angle of attack ($\langle \alpha \rangle$). The corresponding twist angle for wings with $\alpha _r=8^\circ$ and $14^\circ$ is highlighted along the upper $x$-axis. The data for the EEW are also plotted. As expected, it has significantly larger unsteadiness compared with a finite wing. We now look at the effect of twist on the unsteadiness. The finite wing with positive twist has lower net unsteadiness compared with that for an untwisted wing. For negative twist and with $\alpha _r=8^\circ$, the span-averaged angle of attack $\langle \alpha \rangle$ is less than $8^\circ$. The unsteadiness in the flow vanishes for $\delta <0^\circ$ because the span-averaged angle of attack falls below the critical angle of attack (${\sim }7^\circ$) for the onset of vortex shedding for an EEW. We now investigate the spanwise variation of local unsteadiness for various cases and the same is presented in figure 20(c). This is obtained via area integration of $\overline {v'v'}$ at each spanwise section of the wing. We refer to this area average at a spanwise section as ${\widetilde {(v'v')}_S}$. For all cases of finite wing, with and without twist, the unsteadiness is very low near the wing tip owing to the weakening of vortex shedding by the wing-tip vortex. Compared with the untwisted wing, the unsteadiness near the root of the wing is lower for $\delta >0^\circ$ and higher for $\delta <0^\circ$. For example, ${\widetilde {(v'v')}_S}$ at wing root for the untwisted wing at $\alpha =16^\circ$ is $0.0687$ while it is lower ($=0.0587$) for the twisted wing at $\langle \alpha =16^\circ$ and $\delta =4^\circ$. The reduced $\overline {v'v'}$ in the vicinity of the wing-root results in lower overall unsteadiness for a finite wing with positive twist. In contrast, the negative twist does not have any significant effect on the net unsteadiness. Given that the time-averaged force and moment coefficients for the twisted and untwisted wing, for a low to moderate twist, are comparable, a positive twist (wash-in) is preferable as it results in marginally lower net unsteadiness.

Figure 19.

Flow past end-to-end and $sAR=5$ wings at $Re=1000$: variation of (a) $C_{Lrms}$, (b) $C_{Drms}$ with angle of attack. The data for the wing with twist are plotted with respect to the span-averaged angle of attack ($\langle \alpha \rangle = \alpha _r+\delta /2$). The twist angle ($\delta$) is marked on the upper $x$-axis for $\alpha _r=8^\circ$ and $14^\circ$.

Figure 20.

Flow past wing at $Re=1000$: (a) iso-surface of $\overline {v'v'}(=0.3,0.4,0.5)$ for EEW at $\alpha =20^\circ$. Also, highlighted is the region considered for estimating the net unsteadiness. Here, $o'$ is the origin and it coincides with the trailing edge of the airfoil at $z/c=0$ for an EEW and at mid-span for an untwisted and a twisted wing. (b) Variation of ${\overline {(v'v')}_V}$ with $\alpha$ for untwisted wing and $\langle \alpha \rangle$ for twisted wing. The twist angle is marked along the upper $x$-axis for wings with $\alpha _r=8^\circ$ and $14^\circ$, respectively. The spanwise variation of local sectional ${\widetilde {(v'v')}_S}$ is presented in (c) for untwisted and twisted wings with $\alpha _r=14^\circ$. The data for the twisted wing are plotted in broken line and is ascribed to the span-averaged angle of attack ($\langle \alpha \rangle = 14^\circ +\delta /2$). The local geometric angle of attack at the tip ($\alpha _t$) is also highlighted.

3.5.4 The case of $sAR=5$ wing with $\langle\alpha \rangle =11^{\circ}$ and high twist $(\delta=\pm 6^{\circ})$

We explore the effect of high twist on the aerodynamic performance of a wing placed at a span-averaged angle of attack of $\langle \alpha \rangle =11^\circ$. For the wash-out configuration ($\delta =-6^\circ$), we consider $\alpha _r=14^\circ$ and $\alpha _t=8^\circ$. For the wash-in configuration ($\delta =6^\circ$), the angles at the root and wing tip are $\alpha _r=8^\circ$ and $\alpha _t=14^\circ$. Figure 16 shows that the time-averaged lift and drag coefficients for the wing with $\delta =-6^\circ$ are very similar to those for the untwisted wing. The coefficients are lower for the wing with $\delta =6^\circ$. The values are listed in table 5. Lift is lower by 11 %, drag by 8 % and rolling moment at the root by 7 % approximately. The surface pressure superimposed with the skin-friction lines for the time-averaged flow are presented in figure 21 for both the configurations of the twist along with the untwisted wing at $\alpha =11^\circ$. Despite the proximity of the span-averaged force coefficients between $\delta =0^\circ$ and $-6^\circ$, the flows in all the three cases are significantly different. The flow separation at the root is significantly delayed for $\delta =-6^\circ$ compared with the untwisted wing while it is preponed for $\delta =6^\circ$. Compared with the twisted wing, the wing with $\delta =-6^\circ$ experiences more suction at the wing root while the suction is lower for the wing with positive twist. An opposite trend is seen near the wing tip.

Table 5.

Flow past a $sAR=5$ wing at $Re = 1000$: time-averaged force coefficients, their r.m.s. and aerodynamic efficiency for untwisted and twisted wings. Also listed are the pitch ($C_m$), roll ($C_r$) and yaw ($C_y$) moment coefficients at mid-chord and at plane of symmetry ($z/c=0$). Here, $X_F/c$ and $Z_F/c$ denote the chordwise and spanwise location of the resultant force acting on the wing.

Figure 21.

Flow past a $sAR=5$ wing: pressure distribution and skin-friction lines on the upper surface of the wing for $(\delta, \alpha _r, \alpha _t) = (a)$ ($0^\circ, 11^\circ, 11^\circ$), (b) ($-6^\circ, 14^\circ, 8^\circ$) and (c) ($6^\circ, 8^\circ, 14^\circ$). The wing root and wing tip are marked in (b).

The spanwise variation of the time-averaged local lift coefficient ($\overline {c_l}$) for the wing with various twists is presented in figure 22(a). Also shown, for reference, is the distribution for the untwisted wing at various $\alpha$. The square symbols represent the spanwise location of the resultant lift coefficient ($\overline {C_L}$) and its value. For $\delta =-6^\circ$ and $\alpha _r=14^\circ$ the increase in lift near the wing root, compared with the untwisted wing at $\alpha =11^\circ$, is accompanied by a loss near the wing tip. As a result, the span-averaged lift of the untwisted wing and the wing with $\delta =-6^\circ$ is very similar. The wing with $\delta =6^\circ$ and $\alpha _r=8^\circ$, however, generates a lower span-averaged lift. Interestingly, its lift distribution is similar to the untwisted wing near the wing tip but is significantly lower at the wing root. The lower lift is attributed to weaker vortex shedding at the wing root at $\alpha _r=8^\circ$ compared with the untwisted wing at $\alpha =11^\circ$. Stronger spanwise vortices are associated with higher suction resulting in increased lift. Figure 22(a) brings out another interesting point that the load distribution at each span location on a twisted wing is not the same as that on an untwisted wing at the corresponding $\alpha$. For example, even at the wing root, $\overline {c_l}$ on the twisted wing is different from that for the untwisted wing at the corresponding local $\alpha$.

Figure 22.

Flow past a $sAR=5$ wing at $Re=1000$: spanwise variation of (a) time-averaged sectional force coefficient $\overline {c_l}$. The data for the twisted wing are ascribed to the span-averaged angle of attack ($\langle \alpha \rangle =\alpha _r+\delta /2$). Also listed in the legend is the angle of attack at the root ($\alpha _r$) and tip of the twisted wing ($\alpha _t$). The solid and hollow symbols mark the value as well as the spanwise location of the resultant force coefficient for the untwisted and twisted wing, respectively. The spanwise variation of local sectional ${\widetilde {(v'v')}_S}$ is presented in (b) for untwisted and twisted wings. The data for the twisted wing are plotted in broken line.

The skin-friction lines shown in figure 21 bring out significant difference in the flow separation for the various twist angles. We explore the difference in the unsteadiness in the various cases by studying the spanwise variation of ${\widetilde {(v'v')}_S}$ shown in figure 22(b). We recall that the unsteadiness in the flow for an untwisted wing is very low at $\alpha = 8^\circ$ and increases with an increase in $\alpha$. For the wing with $\alpha _r=8^\circ$ and $\delta =6^\circ$ ($\alpha _t = 14^\circ, \langle \alpha \rangle =11^\circ$) the unsteadiness near the wing root is higher compared with that of the untwisted wing at $\alpha =8^\circ$. However, it is significantly smaller compared with that of the untwisted wing at $\alpha =11^\circ$, for most of the span. Interestingly, the unsteadiness for the twisted wing is larger in the wing-tip region compared with the untwisted wing at $\alpha =14^\circ$. For the wing with $\alpha _r=14^\circ$ and $\delta =-6^\circ$ ($\alpha _t = 8^\circ, \langle \alpha \rangle =11^\circ$) the unsteadiness at the root is lower than for the untwisted wing at $\alpha = 14^\circ$. It is much higher at the wing tip compared with unsteadiness in that region for the untwisted wing at $\alpha =8^\circ$. The net unsteadiness is the smallest for the wing with positive twist (see figure 20b). The difference between the unsteadiness for untwisted and twisted wings for the same local angle of attack is related to the cellular nature of shedding. The attributes of the vortex shedding for the wings with $\delta =0^\circ$, $\delta =-6^\circ$ and $\delta =6^\circ$ are listed in table 5. Despite the very significant difference in the vortex shedding structure and the vortex dislocations, the time-averaged force coefficients are not too different for two of the three configurations of the wing. Overall, the force coefficients for wing with negative twist are comparable to those for the untwisted wing. The wing with positive twist has lower force coefficients and r.m.s. The rolling moment for the twisted wing, both $\delta =-6^\circ$ and $\delta =6^\circ$, is lower than that for the untwisted wing. Despite the resultant lift acting further outboard, the wing with positive twist experiences lower rolling moment compared with the $\delta =-6^\circ$ configuration due to reduced lift. A large twist, whether wash-in or wash-out, results in lower net unsteadiness and rolling moment and is, therefore, preferred over an untwisted wing. A positive twist has lower net unsteadiness, but also a lower time-averaged lift compared with a wing with negative twist.