3.1.1 Solver validation for single-lobed LOTTE airship hull

The tri-lobed geometries being simulated were based on combining three LOTTE airship hulls, while retaining the same volume as their single lobe counterpart. Thus, in order to ascertain validity of the aerodynamic inferences acquired for the tri-lobed models from current study, a comprehensive analysis is carried out in this section by comparing the numerical and experimental [Reference Funk, Lutz and Wagner57] results of SL_Lo between $\alpha = {0^ \circ }$ to ${20^ \circ }$ in Fig. 10. The sub-figures in Fig. 10(a) include the numerical results that were derived during an earlier phase of this study using SimScale® solver [Reference Tripathi, M, Pandey and Pant44]. As noted in the Fig. 10(a), both the solvers were able to capture the aerodynamic trends up-to $\alpha = {10^ \circ }$ . However, the OpenFOAM results showed an improvement of $ \approx 6{\rm{\% }}$ in the case of ${C_D}$ and $24{\rm{\% }}$ in the case of ${C_L}$ . Moreover, owing to the available computational resources, the simulations related to cloud based SimScale solver were only limited to $\alpha $ variations up-to ${10^ \circ }$ . The differences in these results were attributed to the usage of a fairly coarse mesh for the SimScale case leading to a ${y^ + }$ value of more than 30, and hence the boundary layer variations were not captured accurately, leading to significant disparity. The OpenFOAM results were within $1.6{\rm{\% }}$ for ${C_D}$ and $1.9{\rm{\% }}$ for ${C_L}$ compared to the experimental results.

Figure 10.

Solver validation through comparison of numerical and experimental aerodynamic coefficients for LOTTE bare-hull [Reference Funk, Lutz and Wagner57].

Figure 11 makes use of a drag polar plot to further validate the accuracy of employed numerical solver. The drag polar plot demonstrates the presence of similarity across the y-axis as well as a parabolic variation. These aerodynamic trends are at par with the expected trends as the low-aspect ratio and symmetrical geometry of SL_Lo, should lead to flow similarity across the positive and negative angles of attack whilst following a parabolic trend.

Figure 11.

Drag polar plot for LOTTE airship hull.

Qualitative validation with respect to the solver being able to capture the actual flow-physics can be gleaned from the axial pressure coefficient ( ${C_P}$ ) distribution along the lower and upper surface contours (Fig. 12(a)). As seen in Fig. 12(b), the ${C_P}$ distribution plot is similar for both the upper and lower surface plots with a negligible difference. Owing to the axi-symmetry nature of the hull as well as $\alpha $ being zero, this trend is at par with the expected trends. This further indicates the efficacy of current solver to capture ${C_P}$ distribution plots for the LOTTE airship. These investigations indicated the accuracy of employed solver and mesh configuration to capture aerodynamic trends related to LOTTE profile and its variants across the prescribed flow conditions that is, $\alpha \le {10^ \circ }$ and low subsonic flow. Hence, further investigations related to the tri-lobed geometries also made use of similar mesh and solver setup to carry out the required aerodynamic investigations.

Figure 12.

Solver validation through comparison of lower and surface plots of LOTTE single lobe hull at zero angle.

A preliminary step towards setting up the solver configuration implemented in this study involved numerical investigations related to the Zhiyuan-1 airship hull [Reference Liu, Fu, Zhu and Wang70]. Specifics related to this study have been elaborated in Appendix A.2. This study demonstrated increased corroboration between the experimental and numerical results especially ${C_L}$ values. Although some differences associated with the drag forces were observed at lower angles – linked with limitations in simulating flow transition – this study provided critical insights for setting up the final computational model. The increased accuracy of this solver in the case of SL_Lo simulations compared to SL_Zi results could be attributed to the fact that the experimental setup related to SL_Lo made use of a fully turbulent flow, enabling increased similarity of experimental and numerical flow conditions leading to increased accuracy.

3.3.1 Effect of forward skewness on the aerodynamics of tri-lobed hull

It can be seen in Fig. 14(a) that the overall ${C_L}$ undergoes a significant reduction (averaged at $57{\rm{\% }}$ ) for skewed fore tri-lobed hulls. In-fact, lift curve slope ( ${C_{L_{\alpha}}}$ ) of the skewed hull also reduces by an average of $69{\rm{\% }}$ leading to the reduced ${C_L}$ value. Moreover, this reduction is higher as the skewness angle increases (Fig. 15). It can also be noted that the overall trend for both the skewed variants is non-linear, whereas the plot pertaining to TL_Lo is more linear. This is indicative of increased effects pertaining to flow intermixing as well as wake vortices for the skewed fore airship hull STL_F10. Since flow intermixing would lead to a reduction in pressure differential across the upper and lower surfaces, this would explain their reduced ${C_L}$ values. In fact, this effect is maximum for STL_F10 as indicated by its higher deviation from the baseline model. Thus, as the stern portion of the side-lobes move upstream, hindrance to the flow escaping towards the upper surface as well as towards the sides is altered leading to flow-intermixing, and hence the reduced ${C_L}$ value. Further assertions regarding the change in flow intermixing pattern leading to this reduced ${C_L}$ will be established in a later section using the flow-field and ${C_P}$ distribution plots (Section 3.5.1).

Figure 14.

Comparison of aerodynamic coefficients pertaining to different skewed fore tri-lobed hull variants at different angles of attack.

Figure 15.

Comparison of lift-curve slope of skewed fore tr-lobed hull against the baseline tri-lobed hull.

Skewing the side-lobes in forward direction leads to a reduction in ${C_D}$ value at all angles for both the variants as seen in Fig. 14(b). The averaged reduction between baseline and STL_F05 is $6.23{\rm{\% }}$ , which enhances to $10.6{\rm{\% }}$ in the case of STL_F10. The presence of lower ${C_D}$ at $\alpha = 0^ \circ $ ( ${C_{{D_0}}}$ ) is indicative of lowered pressure drag at zero angle. Thus, upstream placement of side-lobes leads to reduced base drag value, and hence the lower ${C_D}$ at all angles. Explanation regarding the underlying reasons will be carried out later through ${C_P}$ distribution and flow-field plots. The trend associated with STL_F05 indicates a higher ${C_D}$ enhancement with $\alpha $ increment beyond $\alpha = {2^ \circ }$ , indicating the onset of increased pressure drag for the tri-lobed hull with lower skewness. The underlying reasons associated with increased intermixing related to this ${C_D}$ reduction trend will be further discussed in the flow-field visualisation plots.

Despite the reduced ${C_D}$ value, aerodynamic efficiency for both the skewed fore tri-lobed hull variants is significantly lower compared to TL_Lo with the depletion being higher at $ \approx 39{\rm{\% }}$ for the STL_F10 variant. Thus, the effect of mitigated ${C_L}$ value is more appreciable compared to drag alleviation leading to reduced aerodynamic efficiency for the skewed fore tri-lobed hulls. The pitching moment curve plot shown in Fig. 14(d) is evaluated about a point $40{\rm{\% }}$ aft along the middle-lobe axis. From this plot, it can be ascertained that the overall ${C_M}$ tends to increase at a particular $\alpha $ as the skewness angle increases. This is further corroborated by the bar plot presented in Fig. 16, wherein the overall pitching moment slope increases for higher skewness. This increment ranges as $7.5{\rm{\% }}$ for STL_F05 to $29{\rm{\% }}$ for the STL_F10 variant compared to the baseline model. Thus, it can be stated that static instability of the tri-lobed hull increases for the skewed fore tri-lobed hull variants.

Figure 16.

Comparison of pitching moment curve slope for skewed fore tri-lobed hull against the baseline tri-lobed hull.

3.3.2 Effect of aft skewness on the aerodynamics of tri-lobed hull

The aerodynamic trends related to skewed aft tri-lobed hulls are significantly different compared to the skewed fore tri-lobed hulls as seen in Fig. 17. Contrary to the skewed fore tri-lobed hulls, skewed aft tri-lobed hulls display significantly higher ${C_L}$ values compared to TL_Lo. This enhancement increases with the increase in aft-displacement of the side-lobes. Thus, the increment which is valued at $15.35{\rm{\% }}$ for STL_A05, enhances to $45{\rm{\% }}$ in the case of STL_A10 (calculated between $\alpha = {2^ \circ }$ to ${10^ \circ }$ ). At $\alpha = 0^ \circ $ , STL_A05 presents near zero ${C_L}$ , which is lower than the baseline model, whereas STL_A10 hull presents a significantly higher ${C_{L_{0}}}$ . Analogous to the skewed fore counterparts, skewed aft tri-lobed hull variants display a non-linear lift increment, which is indicative of a significant deviation of aerodynamics between baseline tri-lobed hull and the skewed aft tri-lobed hulls. It can be envisaged by having a look at the contours in Fig. 5 that the overlapping contour of the side-lobes in the skewed aft tri-lobed hull induced flow convergence towards the upper surface of the middle-lobe near the stern portion, resulting in lowering of upper surface pressure, hence the increased pressure differential leading to higher ${C_L}$ . This shall be explained later in the flow-field results. The lift carry-over effect (discussed in Ref. (Reference Funk, Lutz and Wagner57)) for the skewed aft tri-lobed hull also contributes to the increased ${C_L}$ value. It was shown in Ref. (Reference Funk, Lutz and Wagner57) that the presence of fin members on the sides of airship hulls lead to increased lift attributed to interaction of the tailplane with hull flow. In the case of skewed aft tri-lobed hulls, the side-lobes act like tail-planes leading to increased lift-carry-over. The plots in Fig. 17(a) indicates the presence of higher lifting slope at lower angles. This is corroborated by comparing the bar plot of  ${C_{{L_\alpha }}}$ values for the two skewed tri-lobed hulls in Fig. 18(a). Thus, the overall lifting performance of skewed aft tri-lobed hulls is better compared to the baseline model, and hence presents a productive option towards using such a configuration for future HA applications. Further explanation regarding these trends for STL_A10 will be undertaken later in the flow-field visualisation section.

Figure 17.

Comparison of aerodynamic coefficients pertaining to different skewed aft tri-lobed hull variants at different angles of attack.

Figure 18.

Comparison of lift and moment coefficient slopes pertaining to different skewed aft tri-lobed hull variants with the baseline tri-lobed model.

Figure 17(b) presents a significant increase in the ${C_D}$ value pertaining to skewed aft tri-lobed hulls, with the increment being $6.4{\rm{\% }}$ between TL_Lo and STL_A05, which jumps to $23{\rm{\% }}$ between TL_Lo and STL_A10. Thus, increased ${C_L}$ presented above is generated at the expense of higher ${C_D}$ value for the skewed aft tri-lobed hull variants. The uniform difference at all angles between ${C_D}$ of all the variants, indicates the presence of higher pressure drag at low angles, which could be attributed to wakes vortices being stronger for these hulls or the increased intensity of the flow stagnation region, and hence the higher drag. Much more clarity will be obtained using the flow-field visualisation plots later. Despite the heightened ${C_D}$ value, the overall lift-to-drag ratio is higher for all the skewed aft tri-lobed hulls (Fig. 17(c)) with an average increment of $ \approx 17{\rm{\% }}$ compared to TL_Lo. However, as $\alpha $ increases, difference between the baseline and skewed tri-lobed hull reduces and converges at $\alpha = {10^ \circ }$ . This further emphasises the suitability of current research being limited upto $\alpha = {10^ \circ }$ .

Trends associated with ${C_M}$ in Fig. 17(d) displays a subtle variation across the three variants. However, given the fact that overall ${C_{{M_\alpha }}}$ (Fig. 18(d)) is reduced for the skewed aft tri-lobed hulls, it can be stated that the overall static stability of the hull increases for the skewed aft tri-lobed hulls. It is also noteworthy, that the presence of skewed aft side-lobes will enable placement of the control surfaces at a larger distance from the volumetric centre leading to further augmentation of the overall static stability. Hence, displacing the side-lobes in the aft-direction has a significant impact on the overall aerodynamics of tri-lobed hulls leading to increased lift as well as improved aerodynamic efficiency and amelioration of the inherent static instability. Thus, aft-skewness seems to be a more productive step towards enhancing the performance of tri-lobed hulls compared to its baseline model.

3.3.3 Effect of upward and downward skewness of side-lobes on the aerodynamics of tri-lobed hull

This section makes use of aerodynamic coefficient bar plots (Fig. 19) for STL_U10 and STL_D10 plotted at $\alpha = {0^ \circ }$ , ${4^ \circ }$ , and ${8^ \circ }$ and compare them against TL_Lo. The ${C_L}$ bar plot in Fig. 19(a) presents a significantly reduced ${C_L}$ value for STL_U10 variant at $\alpha = {0^ \circ }$ and ${4^ \circ }$ . However, at $\alpha = {8^ \circ }$ , it displays a net ${C_L}$ increment of $4{\rm{\% }}$ compared to the baseline case. Understanding related to the negative lift associated with STL_U10 is provided in Appendix A.3 The skewed downward variant, STL_D10 displays higher ${C_L}$ value at all the angles with an averaged ${C_L}$ increment of $ \approx 88{\rm{\% }}$ . The overall ${C_D}$ value of both the skewed tri-lobes are significantly higher compared to the baseline model (averaged at $41.55{\rm{\% }}$ ). In fact, the overall drag penalty is highest for STL_U10 at all angles with an average increment of $43.5{\rm{\% }}$ compared to baseline model, whereas STL_D10 presented an averaged increment of $ \approx 39.6{\rm{\% }}$ . The aerodynamic efficiency of STL_D10 is better compared to both the variants at $\alpha = {0^ \circ }$ (Fig. 19(c)). However, at higher angles, the overall aerodynamic efficiency is higher for the baseline model. The upward and downward displacement of side-lobes lead to increased drag at all angles indicating the presence of heightened pressure drag, and hence leading to lowered aerodynamic efficiency. Thus, displacing the side-lobes in the normal direction seems to be deleterious for the overall aerodynamic efficiency. Hence, further analysis related to detailed aerodynamic comparison between the baseline tri-lobed hull and skewed tri-lobed hull will be carried out for the skewed fore and skewed aft tri-lobed hull variants only.

Figure 19.

Comparison of aerodynamic coefficients pertaining to skewed up and skewed down tri-lobed hulls at different angles of attack with baseline tri-lobed hulls.

3.4.1 Surface pressure coefficient plots: 3D analysis

Figures 20(a)–(i) and 21(a)–(c) compare the surface-wise ${C_P}$ plots of different variants at various angles to gain visual insight into the flow variations across a lifting body. Considering the heightened impact of external pressure distribution on the overall structural rigidity as well as shape retention of the tri-lobed hull, the aerodynamic assertions drawn hereinafter are based on the assumption of implementing appropriate load stiffeners like internal ribs and laminate tailoring to retain the overall shape. Further, discussions regarding this aspect are covered in a later section (Section 3.6.1).

Figure 20.

Front-view of the pressure coefficient distribution over the hull surfaces at different angles of attack.

Figure 21.

Back-view of the pressure coefficient distribution over hull surfaces at different angles of attack (legends are same as that of Fig. 20).

All the variants in Fig. 20 present stagnation region at the nose portion followed by pressure decrement as flow accelerates past the nose region. This is followed by pressure recovery beyond the point of maximum diameter, leading to the generation of adverse pressure gradient near the stern portion. Notably, a subtle ridged pattern appears near the nose in the 3-D surface ${C_P}$ renders, which could be attributed to the dependence of these contours on the overall surface meshing and contour interpolation. However, later poly-fitted 2-D ${C_P}$ curves (Section 3.4.2) exhibit the expected smooth pressure transition from the high stagnation region pressure value to a lower downstream value, consistent with the underlying 3-D distribution. These localised perturbations could be further reduced with targeted mesh refinement in future work. Given the close agreement between the 2-D and 3-D trends and the convergence of the global pressure behaviour, it can be concluded that these ridged features have a minimal impact on the overall ${C_P}$ trends.

As $\alpha $ increases, the stagnation region intensity on the upper surfaces reduce, and the flow is accelerated to a lower pressure. This leads to an increase in pressure difference across the upper and lower surfaces leading to increased ${C_L}$ . This lowering of ${C_P}$ value over the upper surface leads to higher adverse pressure gradient conditions near the stern portion, leading to an increase in ${C_D}$ value as $\alpha $ increases. Moreover, the plots also demonstrate the presence of flow corresponding to low ${C_P}$ (accelerated flow) near the intersecting curves pertaining to the tri-lobed variants, which is more intense as $\alpha $ increases. Thus, the tri-lobed ${C_P}$ distribution plots are associated with diverse regions of ${C_P}$ variations. It is the variation of intensity related to these regions across the three variants that would govern their inherent aerodynamic characteristics.

The stagnation region is similar for side-lobes across all variants. However, stagnation region corresponding to middle-lobes displays appreciable variation between the different variants. As the side-lobes move downstream, the intensity of this region increases. For instance, compare the Fig. 20(d) through 20(f). Due to this variation, the downstream pressure decrement is slower for STL_A10 compared to STL_F10. In the case of STL_A10, at $\alpha = {0^ \circ }$ and ${4^ \circ }$ presented in Fig. 20(c) and (f), respectively, a slower ${C_P}$ decrement, and hence a bigger stagnation region is presented. This would amount to increased ${C_D}$ value for skewed aft tri-lobed hulls as displayed earlier in Fig. 17(b). In the case of STL_F10, the stagnation region is bigger compared to the baseline model that is TL_Lo. However, the presence of funneling effect between the two side-lobes, leads to higher velocity between the lobes, and hence reduced ${C_P}$ value on the middle-lobe nose. This accelerated flow would lead to overall drag reduction, which would be smaller than TL_Lo. The relevant high velocity region dissipates as the side-lobes get shifted aft due to which the impact of this parameter is lower for STL_A10. Thus, the drag is mainly influenced by intensity of the stagnation region in the case of skewed aft tri-lobed hull variants, whereas the influence of flow acceleration between the lobes near the nose has a greater influence on the overall ${C_D}$ characteristics for STL_F10. These differences become more appreciable as $\alpha $ increases to $\alpha = {10^ \circ }$ (Fig. 20(g) through 20(i)).

The surface bound ${C_P}$ distribution pattern associated with the stern portion also displays appreciable differences across the three variants. As the side-lobes move aft, presence of a lower ${C_P}$ value on the middle-lobes for STL_A10 (Fig. 20(g)) compared to STL_F10 (Fig. 20(i) indicates the presence of lowered flow intermixing between the side-lobe flow and middle lobe flow for STL_A10. The presence of this lower ${C_P}$ would explain the presence of higher ${C_L}$ for skewed aft tri-lobed hull as indicated earlier in Fig. 17(a). The other two variants display uniform ${C_P}$ value distribution on the three lobes. This assertion is further qualified by observing the ${C_P}$ distribution contours in Fig. 20. This lowering of value at the stern region for STL_A10 (Fig. 21(c)) compared to STL_F10 (Fig. 21(a)) can be explained by the difference in relative placement of the side-lobes compared to the middle-lobe. Thus, due to the upstream placement of middle-lobe pertaining to STL_A10 (which is at a higher pressure compared to the side-lobes at a particular distance along the hull), higher ${C_P}$ flow from middle-lobe move towards the side-lobes. This would amount to lowered flow intermixing across the upper and lower surfaces of middle-lobe at the stern for STL_A10 leading to a lower ${C_P}$ value. This could also be related to the intersecting planes converging towards the middle-lobe for STL_A10, leading to lowered ${C_P}$ value. Au contraire, downstream location of the middle-lobe stern section for STL_F10, which is at a higher ${C_P}$ , induces increased flow intermixing over the middle lobe stern section leading to higher ${C_P}$ values, amplified by the presence of this region within the wake of side-lobes. Further corroboration related to this intermixing will be explained using the two-dimensional ${C_P}$ distribution plots in a later section. These plots emphasise that as the side-lobes move aft, the overall flow becomes more stagnant at the nose region, followed by increased differences associated with flow intermixing pattern in the stern region, leading to significant aerodynamic deviations between the three variants.

3.4.2 Pressure coefficient distribution at longitudinal vertical plane maximum diameter section of middle and side-lobes

The three-dimensional ${C_P}$ contours mentioned above provided a global overview of the overall flow behaviour associated with the three tri-lobed variants. In order to obtain a zoomed quantitative assessment of the associated leading edge suction region, pressure recovery in the middle region as well as stern region’s flow separation characteristics, further analysis was carried out using two-dimensional ${C_P}$ distribution plots. This section compares the ${C_P}$ distribution plots along the suction and pressure surfaces of the maximum-diameter section, located on the longitudinal vertical planes of side-lobe and middle-lobe, denoted as SP and MP, respectively in Fig. 22. The distribution line plots in Fig. 23 have been evaluated at SP and MP locations of the three variants for $\alpha = 0^{0}$ (Fig. 23(a) and 23(b)), ${4^ \circ }$ (Fig. 23(c) and 23(d)) and ${8^ \circ }$ (Fig. 23(e) and 23(f)).

Figure 22.

Planes used to plot the pressure distribution plot.

Figure 23.

Comparison of pressure coefficient distribution pertaining to the different skewed hull variants with baseline model plotted along central plane of middle-lobe and side-lobe hulls for different angles of attack.

In-line with the established flow-physics related to airship hull, ${C_P}$ starts reducing near the nose portion, as flow accelerates. Beyond $x/{L_h} = 0.11$ , the overall surface contour gradient reduces leading to a constant ${C_P}$ value. As flow moves beyond $x/{L_h} = 0.3$ , it tends to accelerate towards the maximum diameter region leading to further ${C_P}$ decrement up to $x/{L_h} = 0.5$ . However, beyond the maximum diameter length, the flow starts decelerating leading to ${C_P}$ increment. Interestingly, all the plots demonstrate constant ${C_P}$ or reduction of ${C_P}$ trends beyond $x/{L_h} = 0.83$ and minimal differences between the upper and lower surface ${C_P}$ values due to flow intermixing.

At $\alpha = {0^ \circ }$ , the value of ${C_P}$ at the nose on plane pertaining to MP is significantly lower for STL_F10 ( $ \approx 0.24$ ) compared to STL_A10 ( $ \approx 0.78$ ) as seen in Fig. 23(a). This further corroborates the presence of reduced relieving effect for STL_A10, leading to higher ${C_P}$ at the nose region similar to BM, and hence higher ${C_{{D_0}}}$ . The funneling effect within the side-lobes for STL_F10 leads to reduced ${C_P}$ value, and hence lower ${C_D}$ . Reduction of ${C_P}$ beyond $x/{L_h} = 0.3$ (as discussed above) is relatively steeper for STL_A10 and BM, which could be attributed to the converging intersection contour towards MP leading to increased flow velocity, and hence the lower ${C_P}$ value (also presented earlier in Fig. 21(c)). STL_F10 presents the lowest streamwise ${C_P}$ decrement owing to the diverging intersecting contours. This is also indicative of high pressure flow from the side-lobes moving towards the middle-lobe leading to curtailed ${C_P}$ decrement. This will be corroborated later through the usage of velocity contours. Beyond $x/{L_h} = 0.5$ , all variants demonstrate similar ${C_P}$ increment trends. Although this increment is steeper for STL_A10, the overall value is still lower compared to STL_F10. Thus, as the side-lobes move downstream, adverse pressure gradient over the upper surface attributed to the middle-lobe reduces. The higher ${C_P}$ value under the lower surface for stern portion of STL_A10 is indicative of reduced flow velocity leading to increased ${C_P}$ value retained across a longer length. This effect is more pronounced for STL_A10 because of reduced inter-mixing, which leads to increased ${C_L}$ value at $\alpha = {0^ \circ }$ (illustrated earlier in Fig. 17(a)). Thus, the unique geometry of STL_A10 leads to the stern portion of all lobes being mostly affected by their own flow structure. This leads to increased effect of the vortices on the overall lift leading to generation of higher lift at $\alpha = {0^ \circ }$ . The overall trends display minimum ${C_P}$ variation across the length for STL_F10 leading to lowest ${C_L}$ as well as lower ${C_D}$ . This can be understood by referring to Fig. 27(g) and Fig. 24. Due to the downstream location of middle-lobes, flow attached with the side-lobes at a particular lengthwise location has higher ${C_P}$ compared to middle-lobes. This leads to the side-lobes’ HP (high pressure) flow mixing with middle-lobes’ LP (low pressure) flow leading to an earlier increase in ${C_P}$ , increased intermixing, and hence the higher ${C_P}$ value over MP compared to other variants. Discrepancies associated with different regions of the three variants are significantly lower at SP (Fig. 23(b)), with all the variants displaying similar ${C_P}$ variation along the length. This could be attributed to the increased intermixing phenomena on the side-lobes, further establishing the relevance of middle-lobe on controlling the tri-lobed hull geometry’s aerodynamics.

Figure 24.

Pressure gradient in lateral direction leading to flow intermixing on the middle-lobe for STL_F10.

As $\alpha $ increases to ${4^ \circ }$ , differences between the lower and upper surfaces increase leading to ${C_L}$ enhancement for all the variants. Upper surface ${C_P}$ plots at MP (Fig. 23(c)) for STL_A10 and BM demonstrates significant lowering over the upper surface between nose and $x/{L_h} = 0.5$ section, while STL_F10 exhibits a significantly lower variation of ${C_P}$ till $x/{L_h} = 0.5$ corresponding to a net variation of $82{\rm{\% }}$ . The lowest ${C_P}$ value at nose (due to flow acceleration) followed by this gradual ${C_P}$ variation upto $x/{L_h} = 0.5$ for STL_F10 is again indicative of lowered ${C_D}$ value. Differences between the upper and lower surface flow upto $x/{L_h} = 0.5$ (which governs the overall ${C_L}$ value) is found to be significantly higher for STL_A10. Beyond $x/{L_h} = 0.5$ , plots pertaining to STL_F10 presents the earliest increase in upper surface ${C_P}$ value, which indicates the onset of earlier adverse pressure gradient conditions. This shall be corroborated later in the velocity flow-field Section 3.5.2. In the case of BM and STL_A10 upper surface plots, ${C_P}$ increment is deferred and is steeper. Lower surface plots pertaining to STL_A10 presents higher ${C_P}$ value near the stern portion, again indicative of the presence of lower velocity. This could further indicate the presence of adverse pressure conditions beneath the lower surface also. Owing to the reduced lobe-wise intermixing pertaining to middle-lobe for STL_A10, the flow behind the maximum diameter suffers enhanced adverse pressure gradient conditions leading to this phenomena. This is indicative of increased wake vortex impact for aft skewed airship hull variants. In the case of STL_F10 and BM, this difference is significantly higher and also opposite. Thus, the plots demonstrate lower ${C_P}$ under their lower surfaces, and higher ${C_P}$ over their upper surfaces. Thus, the overall flow contour of lower surface region beyond the maximum diameter is more favourable for these variants, leading to ${C_P}$ reduction under the lower surface. This amounts to reduced ${C_L}$ values for these variants. In-fact, it is the combination of increased positive lift on the nose region and reduced negative lift on the stern portion that leads to significant lift enhancement for STL_A10 at $\alpha = {4^ \circ }$ presented earlier in Fig. 17(a). Analogous to MP, SP plots in Fig. 23(d) again indicate the presence of increased pressure difference between the upper and lower surface plots. However, these differences are significantly lowered at this plane. Moreover, due to increased flow intermixing, differences between the respective upper and lower surface plots across the three variants are significantly lower at this plane. This intermixing is more intense near the stern portion leading to convergence of all the plots with a subtly reduced value for lower surface plots pertaining to STL_F10 and BM, further leading to reduced ${C_L}$ values compared to STL_A10.

As $\alpha $ increases to ${8^ \circ }$ (Fig. 23(e) and 23(f)), the same trends related to increased differences between the upper surface and lower surface plots at MP and SP, followed by increased differences between the variants at MP is repeated. Plots pertaining to MP re-establishes the presence of reduced flow intermixing at the stern portion for STL_A10. In fact, differences between the upper surface and lower surface plots amounting to negative lift is significantly higher at this angle for STL_F10 and BM. Interestingly, STL_F10 displays a significantly lower ${C_P}$ value at the nose for its lower surface. In fact, its value increases across the entire length. Due to this appreciably higher adverse pressure gradient, the upper surface flow separates at the stern region, which is indicated by the presence of constant positive ${C_P}$ value over the upper surface-stern portion.

Thus, skewing the side-lobes in aft direction leads to increased upper surface and lower surface ${C_P}$ differences as well as deviations related to flow intermixing. These plots can also be used as reference plots to implement flow control techniques to control the pressure variation leading to improved aerodynamic performance like implementation of passive devices behind the maximum diameter region to mitigate adverse pressure condition.

3.5.1 Pressure coefficient flow-field in lateral vertical plane

This section makes use of ${C_P}$ contour plots obtained by slicing the computational domain along two lateral vertical planes depicted as red planes in Fig. 25. These lateral planes were placed at ${L_h}/4$ , and $3{L_h}/4$ locations from the nose of side-lobe hulls. Figures 26 and 27 compare the ${C_P}$ flow-field contour plots for STL_F10, TL_Lo and STL_F10 hulls at $\alpha = {0^ \circ }$ , ${4^ \circ }$ , and ${8^ \circ }$ corresponding to the aforementioned lateral planes. Before proceeding towards the detailed explanation of this analysis, it is essential to review an important concept related to wall bounded incompressible flow called the three-dimensional relieving effect [Reference Anderson71] (hereinafter referred to as ‘relieving effect’), mentioned earlier in Section 3.2. Flow approaching a streamlined three-dimensional body like a conventional single-lobe hull is able to ‘relieve’ some of its initial stagnation pressure by moving in the side as well as suction and pressure surface directions, leading to reduced velocity increment across the body. On the other hand, blunt bodies like a tri-lobed hull, amounts to restricted relieving effect leading to increased flow stagnation, increased velocity adjacent to the upper surfaces, and a higher ${C_D}$ value. As mentioned earlier by the authors in Ref.(Reference Tripathi, M, Pandey and Pant44), this reduced relieving effect for tri-lobed hulls was the biggest reason for its increased drag penalties. Thus, the impact of shifting side-lobe members on this effect, needs diligent consideration while comparing the lateral ${C_P}$ plots pertaining to skewed airship hulls.

Figure 26.

Pressure coefficient distribution at 25% ${L_h}$ hull length location for skewed fore, skewed aft and baseline tri-lobed hulls at different angles of attack.

At $\alpha = {0^ \circ }$ in Fig. 26(a)–(c), flow-field for all the variants indicate the presence of similar flow across the upper surface and lower surfaces accounting for net zero ${C_L}$ value. However, the plots also demonstrate the presence of lower ${C_P}$ values over and below the middle-lobe. This lowering is maximum for STL_A10 (Fig. 26(c)) indicating reduced relieving effect. The initially side relieved flow from middle-lobes gets immediately blocked by the side-lobes downstream. This can be further corroborated by the presence of increased high-pressure area corresponding to surface pressure plots of STL_A10 as shown earlier in Fig. 20(c). This reduced relieving effect for STL_A10 leads to increased base drag, and hence the increase in ${C_{{D_0}}}$ (Fig. 17(b)). In the case of STL_F10, $C_{P}$ value pertaining to the middle lobes is smallest (Fig. 26(a)). Aft presence of middle-lobe leads to improved ‘relieving effect’ for the side-lobes followed by acceleration of flow within the lobes as flow approaches the middle-lobe. Consequently, the flow velocity is higher over and below the middle-lobes for STL_F10 (Fig. 26(a)). These variations could explain the overall ${C_D}$ trends with lower value being presented for STL_F10, and higher for STL_A10, in comparison to TL_Lo (Table 7).

In line with the surface bound ${C_P}$ plots discussed earlier, at higher angles ( $\alpha = {4^ \circ }$ and ${8^ \circ }$ ), the three variants display increased suction pressure, accounting for an increase in the ${C_L}$ value. Moreover, ${C_P}$ contours along the side-lobes are similar for the three variants, whereas the middle-lobe presents appreciably higher variations. Owing to increased funneling effect between the side-lobes followed by induced intermixing across the middle-lobe for STL_F10 as discussed earlier, Fig. 26(d) and 26(g) display significantly lower ${C_P}$ value over its middle-lobe. Despite this lower ${C_P}$ value compared to TL_Lo, increased intermixing near the nose portion, as well as a narrower ‘low ${C_P}$ ’ region contribute to lowering of the overall ${C_L}$ pertaining to STL_F10 as seen earlier in Fig. 14(a) and mentioned in Table 7. The reduction in ${C_P}$ value over the middle lobe is smaller for TL_Lo as seen in Fig. 26(e) and 26(h). Upper surface of the STL_A10 middle-lobe presents maximum pressure reduction (Figs 27(f) and 26(i)) across a wider area leading to highest ${C_L}$ increment with $\alpha $ . There is also presence of a small ‘low ${C_P}$ ’ region under the lower surface for STL_A10 at $\alpha = {4^ \circ }$ and ${8^ \circ }$ , further emphasising the presence of weakened relieving effect for STL_A10. However, at $\alpha = {8^ \circ }$ , this reduction of lower ${C_P}$ under the lower surface for STL_A10 is significantly curtailed. This analysis concludes some very important aerodynamic considerations related to side-lobe skewness for tri-lobed configurations. Improving the three-dimensional relieving effect by skewing the side-lobes in front direction to reduce ${C_D}$ value comes at the expense of mitigating the overall ${C_L}$ . Moreover, skewing the side-lobes in aft direction amounts to heightened differential between the upper surface and lower surface ${C_P}$ values at the cost of increased base drag value. This study establishes the relevance of optimised selection of side-lobe skewness from an aerodynamic stand-point, and importance of middle-lobes on the overall aerodynamic characteristics of a tri-lobed airship hull. It is envisaged that this study could prove crucial in improving the current state-of-the-art by making use of variable middle-lobe dimensions compared to the side-lobes.

As flow approaches the stern section at $75{\rm{\% }}{L_h}$ , the ${C_P}$ plots in Fig. 27 further indicate some interesting trends. At $\alpha = {0^ \circ }$ , flow contours associated with TL_Lo (Fig. 27c) and STL_A10 (Fig. 27(b)) display the presence of uniform contour across the entire surface indicative of the fact that flow across the upper surfaces and lower surfaces for all lobes have been intermixed. Owing to the increased middle-lobe area for STL_F10 (Fig. 27(a)), it presents lowest value for flow linked with the middle-lobe (Fig. 27(a)), whereas STL_A10 presents higher ${C_P}$ value close to the middle-lobes. The lower ${C_P}$ value for STL_F10 is indicative of favourable pressure gradient for STL_F10 leading to lower base drag compared to the other two variants. Contrarily, the increased ${C_P}$ value for STL_A10 especially beneath the lower surface is indicative of adverse pressure conditions, leading to increased ${C_D}$ . This would further indicate the presence of increased flow separation near the stern portion for STL_A10, which will be corroborated later in velocity flow-field section (Section 3.5.2). At such low angles, flow associated with the stern portion has a significant impact on the overall drag characteristics, indicated by trends in these plots.

Figure 27.

Pressure coefficient distribution at 75% ${L_h}$ hull length location for skewed fore, skewed aft and baseline tri-lobed hulls at different angles of attack (legends are same as that of Fig. 26).

As angle increases, STL_F10 in Fig. 27(d) and 27(g) presents significantly diverse contour variations compared to the other two variants. It presents lower $C_{P}$ value under the middle-lobes. This lowering of $C_{P}$ would explain the overall decrement of $C_{L}$ pertaining to skewed fore variants, indicated earlier in Fig. 14(a). On the other hand, TL_Lo (Fig. 27(h)) and STL_A10 (Fig. 27(i)) indicate the presence of lowered ${C_P}$ value along the side-lobes, indicative of increased flow intermixing on the side-lobes. The middle lobe of STL_A10 displays increased ${C_P}$ value. This increased ${C_P}$ value would amount to increased adverse pressure gradient condition, leading to higher flow separation, and hence the higher ${C_D}$ . In the case of STL_F10, middle-lobe leads to lowering of ${C_P}$ value, due to which flow-field close to the side-lobes, which is at higher ${C_P}$ , gets pushed towards the middle member. This trend is significant for flow under the middle-lobe because of mitigated flow separation associated with lower surfaces. In the case of STL_A10 and TL_Lo, lateral flow from side-lobe to middle-lobe or vice-versa is minimal. However, the presence of lower ${C_P}$ along the side-lobes is presented, indicative of flow intermixing across sides of the side-lobes. This would explain the increased variation of ${C_L}$ plots pertaining to STL_A10 discussed in Table 7.

3.5.2 Velocity contour distribution

In this section, SP and MP linked streamwise velocity component ( ${U_x}$ ) contours have been compared at $\alpha = {4^ \circ }$ (Fig. 28) and $\alpha = 8^o$ (Fig. 29) for STL_F10, BM (also referred as TL_Lo) and STL_A10.

Figure 28.

Comparison of velocity distribution plots at middle-lobe and side-lobe vertical planes pertaining to different tri-lobed hull variants for angle-of-attack = ${4^ \circ }$ .

Figure 29.

Comparison of velocity distribution plots at middle-lobe and side-lobe vertical planes pertaining to different tri-lobed hull variants for angle-of-attack = ${8^ \circ }$ .

In-line with the aforementioned trends, flow-field contours related to MP exhibits higher deviations within the different variants especially near the stern. Stagnation bubble upstream of the nose is seen to be the smallest for STL_F10 (Fig. 28(a)), followed by higher velocity across a wider region over the upper surface. This is followed by increased lowering of ${U_x}$ downstream of the maximum diameter region. Higher ${U_x}$ over upper surface leads to a steeper pressure difference between the flow at maximum diameter region and the wake region, which leads to increased adverse pressure gradient, and hence the higher flow separation behind the maximum diameter (indicated by A in Fig. 28(a)). Furthermore, tendency of the high-pressure flow from side-lobes to approach towards the middle-lobes, as seen earlier in Fig. 23(a), further contributes to lowering of the velocity over the upper surfaces of STL_F10. This reduced $U_{x}$ velocity over the upper surface is further indicative of the higher $C_{P}$ values on the upper surface (indicated earlier in Fig. 23(c) and 23(e)) leading to reduced $C_{L}$ value (Fig. 14(a)). Despite the presence of reduced flow velocity region beyond the maximum diameter portion, the increased velocity between the lobes at the nose (indicated by reduced value) leads to significant increment in the relieving effect, and hence the reduced $C_{D}$ value. For BM (Fig. 28(c)), the stagnation region is bigger, followed by reduced velocity increment compared to STL_F10. Though STL_A10 (Fig. 28(e)) presents subdued flow separation at the nose portion, a flow separation region is presented below the lower surface near the stern (B in Fig. 28(e)). The increased adverse pressure condition beneath the lower surfaces presented for STL_A10 (mentioned in Table 7) leads to flow separation beyond the maximum diameter region. Due to the lack of intermixing, this low velocity flow is not energised leading to subdued increment of  $C_P$ value (refer Fig. 23(c) and 23(f) for $C_{P}$ distribution) and the visible flow separation region. The presence of this increased flow separation region leads to the higher $C_D$ for STL_A10 presented earlier in Fig. 17(b).

In the case of SP contours, STL_F10 (Fig. 28(b)) and STL_A10 (Fig. 28(f)) exhibits lower nose stagnation region compared to BM (Fig. 28(d)). The stern region shows increased flow intermixing leading to lowered velocity fluctuations for all variants. However, contrary to trends presented at MP, stern region velocity fluctuations at SP increase from STL_F10 (C: Fig. 28(b)) to STL_A10 (D: Fig. 28(f)). In fact, due to the increased intermixing along the sides at this plane, STL_A10 presents no visible flow separation feature beneath the lower surface. The increased flow separation beneath the lower surface at MP followed by increased flow separation near the stern at SP for STL_A10 leads to the higher ${C_D}$ value.

As $\alpha $ increases to ${8^ \circ }$ (Fig. 29), stagnation region intensity increases for all the variants leading to the increase in $C_{D}$ value. The region corresponding to increased flow separation near the stern for STL_F10 (see E in Fig. 29(a)) is appreciably significant at this angle. Similar to the trends described earlier, STL_A10 presents increased flow separation attached to the lower surface (F in Fig. 29(e)) near the stern region, amounting to increased $C_{D}$ value.

Flow associated with SP again demonstrates reduced velocity fluctuations indicative of increased flow intermixing between the upper and lower surface flow. Fluctuations downstream of the stern still poses subtle differences across the three variants. Velocity fluctuations for STL_F10 are lowest followed by increased fluctuations for BM and STL_A10 (compare regions G, H, and I in Fig. 29(b), 29(d) and 29(f), respectively). Owing to the downstream location of side-lobes for STL_A10, interactions between the wake regions associated with side-lobes and middle lobes are reduced. Hence, the flow intermixing is significantly reduced. In the case of BM, the lateral motion is restricted, leading to reduced intermixing, and hence the higher streamwise fluctuations. Flow entertainment across the stern region from lower to upper surface is more significant for STL_F10. This would lead to reduced differences between upper and lower surface pressures leading to reduced ${C_L}$ .

It is notable that the velocity fluctuations indicated tendency of the flow to separate for upper surfaces in the case of STL_F10, whereas the same is presented under the lower surface for STL_A10. These vortices would generate a subtle longitudinal force in the x-axis direction (Fig. 30). Owing to the reduced aspect ratio of these hulls, generation of these vortices at the stern region leads to increased impact on the overall aerodynamics [Reference Gabriel and Mueller72, Reference Tripathi, Sucheendran and Misra73]. Notably, the relative placement of these vortices (emphasised in Fig. 30), leads to a force acting above the moment centre in the case of STL_F10 leading to increased pitch instability, whereas it acts below the moment centre in the case of STL_A10 leading to restoring moment. This leads to a net increase in the pitching moment stability for STL_A10 compared to BM presented earlier in Fig. 14(d), whereas the pitching moment instability is further degraded for STL_F10.

Figure 30.

Stern portion non-linear lift components.

Thus, notable comprehensions regarding the underlying flow-physics for the different tri-lobed variants were accomplished through this analysis. These results further corroborated the presence of increased flow intermixing on the middle-lobes for STL_F10, followed by increased flow separation beneath the lower surface for STL_A10 leading to variations with respect to the respective aerodynamic trends. These assertions can be utilised by future airship designers to reduce the flow separation associated with middle-lobe through implementation of flow control techniques at appropriate locations.

3.6.1 Comparison of load distribution aspects for different tri-lobed hull variants

It can be concluded from the aforementioned analysis that tri-lobed hull increases the available lifting surface area while reducing frontal drag area compared to three isolated hulls. The external aerodynamic pressure distribution, shaped by Bernoulli effects, stagnation zones and wake regions, affect the overall lift, drag and pitching moment characteristics. However, the assertions drawn earlier were based on assumption of a rigid airship hull surface. For a semi-rigid surface, any internal pressure imbalance would deform the hull surface, thereby altering external pressure distribution and degrading aerodynamic performance. Thus, 3D and 2D external pressure coefficient ( ${C_P}$ ) distribution plots discussed in Sections 3.4.1 and 3.4.2, respectively, would have a significant influence on the overall structural load distribution, shape-retention strategies, and aerodynamic stability of the tri-lobed hull. To enhance the engineering applicability of this study, the current section provides a brief qualitative overview of how variations in ${C_P}$ for different skewed tri-lobed configurations can affect the selection of the final hull geometry from a structural integrity standpoint.

The authors emphasise that the following discussion is qualitative in nature; a comprehensive finite element analysis (FEA) study is recommended for future work to quantitatively validate these observations.

Load Distribution and Shape Retention: The internal pressure of the lifting gas primarily governs the structural rigidity of an airship hull. However, the complex geometry of the tri-lobed configuration, combined with the anisotropic external ${C_P}$ distribution, makes these hulls sensitive to both static and cyclic loading effects. The three lobes share loads through a common internal framework or membrane, which dictates how pressure variations influence load transfer, membrane compression, local bending, buckling and cyclic fatigue characteristics. The following discussion briefly summarises the structural implications for each skewed configuration.

Aft-Skewed Tri-lobed Hulls

1. 1. The deferred pressure reduction at the nose (as observed for STL_A10) would result in higher local compressive loads, increased susceptibility to nose buckling, with the overall cyclic fatigue exposure remaining on the lower side.

2. 2. Subtle pressure fluctuations across the nose indicates reduced bending near the junctions, which can be managed through load distribution techniques like diaphragms.

3. 3. The reduced streamwise pressure variation across the mid-lobe surfaces leads to reduced bending stresses, thereby improving fatigue life.

4. 4. Overall, aft-skewed tri-lobed hulls exhibit a more rigid structural behaviour, with emphasis on localised reinforcement around the nose and junctions rather than global stiffening.

Fore-Skewed Tri-lobed Hulls

1. 1. The dynamic low-pressure region at the nose would amount to increased cyclic stress amplitudes, leading to increased fatigue loading on the nose section.

2. 2. The pronounced ${C_P}$ discontinuity at the lobe junctions would cause increased bending, and hence degrading the long-term shape retention and structural uniformity characteristics of the lobes.

3. 3. The steeper streamwise pressure gradients would cause higher longitudinal bending and buckling cycles, further amplifying fatigue susceptibility.

4. 4. Thus, load regulation in this configuration is more complex, requiring engineered nose-skin design, optimised junction load paths and regular monitoring of structural integrity to ensure long-term durability.

In summary, side-lobe skewness linked with differing ${C_P}$ distribution trends will strongly influence the load distribution strategies of these hulls with regards to fatigue behaviour, and shape-retention characteristics. Aft-skewed variants provide improved rigidity characteristics with localised stress concerns at the nose and junctions, whereas fore-skewed variants exhibit higher flexibility and are amenable to cyclic fatigue and bending-deformations. These conclusions provide a preliminary way forward to optimising internal structural layout and reinforcement design for skewed tri-lobed airship configurations that would enable prolonged shape retention and improved fatigue characteristics. Given the varied sensitivity of different skewed airship variants towards these structural parameters, the authors would recommend carrying out a dedicated FEM analysis in the future. Such an approach would provide enhanced understanding regarding the structural attributes of skewed tri-lobed airships, enabling efficient retention of the optimised aerodynamic shape.

3.6.2 Comparison of stability characteristics

A comparison of the pitching moment slope ( ${C_{{M_\alpha }}}$ ) for the variants are presented in Fig. 32. A posteriori, airships are linked with a positive ${C_{{M_\alpha }}}$ , indicative of an inherently unstable lifting body. However, skewing the side-lobes in the aft direction leads to curtailed pitching instability. In fact, there is an average decrement of $ \approx 8.2{\rm{\% }}$ for ${C_{M\alpha }}$ between STL_F10 to STL_A10. This decrement is averaged at $ \approx 19{\rm{\% }}$ for the skewed aft variants compared to TL_Lo. Thus, as the side-lobes move aft, the tri-lobed hulls become more stable. Increased ${C_P}$ value at the nose and reduced stream-wise pressure difference for the skewed-aft hulls are associated with increased restoring moment as well as reduced bending moment fluctuations, respectively. These along with the presence of lower surface lift indicated earlier lead to the higher longitudinal stability for skewed-aft tri-lobed hulls. This would lead to reduced weight penalties associated with the fin control surfaces for STL_A10. Additionally the presence of fin surface on the skewed aft side-lobes would lead to increased moment arm for the control surfaces. This will further amount to reduction of the fin surface area required to provide the necessary stabilising moment. Thus, skewing the airships in aft-direction presents additional benefits with respect to longitudinal stability of the airships. However, given the reduced pitch stability (Fig. 32) and reduced $C_L/C_D$ enhancement at higher angles (Fig. 31(c)) between STL_A10 compared to STL_A05, along with the presence of reduced $C_{L_{\alpha}}$ (Fig. 31(a)) performance of STL_A10, indicates a limiting value on aft displacement of the side-lobes to achieve aerodynamic performance enhancement.

Figure 32.

Comparison of pitching moment coefficient slope for all skewed hulls.

It can be envisaged that employing a design optimisation scheme like the one discussed in Ref. (Reference Kumar, Tripathi, Pant, Tripathi and Kiran Babu74) to discern the most optimum skewness of side-lobes along with side-lobe hull dimensions would lead to improved aerodynamic efficiency whilst retaining higher ${C_{{L_\alpha }}}$ as well as mitigated ${C_D}$ .

Another important aspect to be considered while selecting the final tri-lobed hull geometry would be to analyse the impact of these variants on the overall yaw stability. The increased nose stagnation pressure for STL_A10 would lead to improved weathervaning effect, whereas smoother longitudinal pressure variation would lead to reduced yaw-inducing moments. As a result, the overall yaw-stability would seem to increase for skewed aft variants. Notably, these assertions are only based on subjective analysis of the ${C_P}$ distribution plots discussed earlier. Detailed analysis regarding this aspect can be carried out in the future through the analysis of skewed tri-lobed hulls subjected to side-wind load conditions.