3.1 Envelope shape generation

The current study is based on the standard profiles of six airships shown in Fig. 1, viz., ZHIYUAN-1, LOTTE, Ellipsoid, Garg, Wang and NPL. In this study, the Gertler Series 58 shape generator [Reference Gertler45, Reference Landweber and Gertler46] has been considered for airship envelope generation. A detailed description of the shape generator and its associated design factors can be found in the existing literature [Reference Alam and Pant47, Reference Manikandan, Shah, Priyan, Singh and Pant48]. To obtain the final envelope surface, the curve is revolved around the desired axis to produce the axisymmetric shape of the body. The multi-lobed shape can be considered as several conventional bodies as shown in Fig. 2.

Figure 1. Standard envelope profiles of airships.

Figure 2. Different airship configurations.

Figure 3 illustrates the fundamental method for determining the necessary length of a specific shape to achieve a desired envelope volume, ensuring consistent volume across all models utilised in the simulation. All the models used in this study are rigid. The effects of the airship’s flexible nature, which leads to fluid-structure interaction, are beyond the scope of this study. This study encompasses not only the airship hull but also the geometries of the fins and gondola for conventional models. For the bi- and tri-lobed configurations, simulations were conducted using the hull in conjunction with the fins.

Figure 3. Procedure for envelope sizing.

3.2 Airship models

To ensure the accuracy of the CFD simulations, a rigorous validation process was implemented in this study. This process involved a comparative analysis between the numerical results for the ZHIYUAN-1 airship model shown in Fig. 4 and the experimental wind tunnel data presented in a prior study [Reference Wang, Fu, Duan and Shan49]. Meticulous attention was given to replicating the 3D model of the ZHIYUAN-1 to accurately reflect the dimensions of the physical model used in the wind tunnel experiments. This dimensional precision was critical for establishing a reliable benchmark for the CFD simulations. The geometric details of the CAD model are listed in Table 1.

Figure 4. Scaled CAD model of the ZHIYUAN-1.

Table 1. Geometry specifications of the ZHIYUAN-1 model

Constructing a multi-lobed airship requires a detailed design and modeling process to optimise both aerodynamic performance and structural integrity. Using SOLIDWORKS ^® CAD software, the airship’s envelope is created by importing shape coordinates and employing the revolve feature to form the smooth, curved hull. The gondola, which houses essential equipment and serves as the control and propulsion centre, is modeled by sketching its dimensions and extruding the sketch along the mid-plane. The fins, crucial for stabilisation, are designed with precise sketches based on specified dimensions shown in Fig. 5. The sketches are lofted to replicate the aerodynamic profile of the fins accurately. The geometrical parameters of the fins are given in Table 2.

Figure 5. Reference sketch of an airship fin (Reproduced from (Reference Wang, Fu, Duan and Shan49)).

Table 2. Specifications of fin geometry [Reference Wang, Fu, Duan and Shan49]

In this study, a standardised methodology was applied to all airship profiles. Initially, 3D models of single-lobed configurations shown in Fig. 6 were generated, ensuring a uniform hull volume for each model. To enable a thorough aerodynamic comparison, bi-lobed and tri-lobed models were subsequently developed using the parametric design methods detailed in the study [Reference Manikandan, Shah, Priyan, Singh and Pant48]. By maintaining a constant volume across all configurations, the study focused on evaluating the aerodynamic performance variations attributable to different lobe configurations.

Figure 6. Single-lobed (conventional) airship models with fins.

To achieve the objective, fins were integrated into the airship designs. The NACA 0010 aerofoil profile shown in Fig. 7 was selected for these fins due to its well-documented aerodynamic properties. The aerodynamic characteristics of NACA 0010 are presented in Fig. 8 obtained from the dataset of aerofoils provided by Kanak et al. [Reference Agarwal, Vijaykrishnan, Mohanty and Murugaiah50]. Different fin configurations were employed for single-lobed and multi-lobed airships during the modeling process. The single-lobed models featured a cross-type fin configuration, whereas the multi-lobed variants utilised a more complex ‘X-configuration’ as shown in Fig. 2. In the case of both aircraft and airships, the sizing of tail fins is typically performed using a method known as the tail volume coefficient approach [Reference Carichner and Nicolai51], which is based on historical data. The tail volume coefficient is a function of the envelope volume. Since the envelope volume is identical for all models considered in the present study, a uniform tail fin size is applied across all the models.

Figure 7. Schematic of the NACA 0010 aerofoil.

Figure 8. Aerodynamic characteristics of the NACA 0010 at Re = $2.58 \times {10^6}$ .

To establish a uniform reference for performance evaluation, a standardised methodology for calculating key aerodynamic parameters was employed. The reference length for each airship was computed using the formula ${V^{1/3}}$ , and the reference area was determined using the formula ${V^{2/3}}$ , where V represents the volume of the hull containing the lifting gas. Importantly, the value of $V$ was held constant across all airship models, facilitating direct comparison of aerodynamic performance metrics such as lift and drag coefficients. This meticulous approach resulted in a robust and well-defined framework for comparative analysis.

4.1 Governing equations

The present study utilised $ANSYS$ Fluent 2023 and $OpenFOAM$ ^® for numerical simulations of airship aerodynamics, focusing on a three-dimensional, steady-state, incompressible flow. The use of OpenFOAM flow solver in the present study is motivated by two main reasons: first, to validate the results obtained from ANSYS Fluent, and second, to overcome computational limitations related to the number of cells/elements available during the meshing of the computational domain under the academic license of ANSYS Fluent (Fluent can handle up to 512 k cells). Mathematically, any three-dimensional flow can be described by a set of five non-linear differential equations, which are intricately coupled and highly complex to solve [Reference Gupta, Tripathi and Pant25, Reference Tripathi, Misra and Sucheendran52]. These equations are known as the Navier-Stokes equations, and they are divided into one mass continuity equation (Equation 1), three momentum conservation equations (Equation 2), and one energy conservation equation (Equation 3).

Mass Continuity:

(1) \begin{align}\frac{{\partial \rho }}{{\partial t}} + \frac{\partial }{{\partial {x_j}}}\left[ {\rho {u_j}} \right] = 0\end{align}

Equations of Motion:

(2) \begin{align}\frac{\partial }{{\partial t}}\left( {\rho {u_i}} \right) + \frac{\partial }{{\partial {x_j}}}\left[ {\rho {u_i}{u_j} + p{\delta _{ij}} - {\tau _{ij}}} \right] = 0,{\rm{\;\;\;\;}}i = 1,2,3\end{align}

Energy Conservation:

(3) \begin{align}\frac{\partial }{{\partial t}}\left( {\rho {e_0}} \right) + \frac{\partial }{{\partial {x_j}}}\left[ {\rho {u_j}{e_0} + {u_j}p + {q_j} - {u_i}{\tau _{ij}}} \right] = 0\end{align}

where $t$ represents the time variable. The variables ${x_i}$ and ${u_i}$ refer to the ${i^{th}}$ components of the spatial and velocity vectors, respectively. The symbols $\rho $ , $p$ , and $\tau $ stand for density, pressure and viscous shear stress, respectively. The total energy is represented by ${e_0}$ , and the heat flux is denoted by $q$ .

The steady-state values for flow variables such as velocity and pressure in the present study were determined using the RANS equations. These equations replace the original flow variables in the Navier-Stokes equations with the sum of their time-averaged values and fluctuating components.

4.2 Flow conditions

The present study utilised flow conditions consistent with the experimental testing conducted by Wang et al. [Reference Wang, Fu, Duan and Shan49] on the airship model with ZHIYUAN-1 profile. This approach ensures alignment between computational and experimental results, thereby facilitating the validation of the CFD analysis. By maintaining uniform flow conditions across all airship analyses, the study enables fair comparisons between different designs. The flow conditions are summarised in Table 3.

Table 3. Flow conditions for CFD simulations

4.3 Computational domain

A domain sensitivity study in CFD ensures simulation accuracy by analysing how domain size and configuration affect results shown in Fig. 9. By varying upstream/lateral length ( $b$ ) and downstream length ( $l$ ) relative to a reference length ( $L$ ), optimal sizes were identified. Results showed that $b = 6L$ and $l = 12L$ minimised error to 2.08%, balancing accuracy and computational resources.

Figure 9. Sensitivity analysis of the computational domain.

The cylindrical computational domain shown in Fig. 10 selected for simulating airflow around an airship offers several advantages, including efficient representation of the flow field and minimised computational complexity. Extending boundaries $6L$ upstream, $12L$ downstream and $6L$ laterally ensures a sufficiently large region to capture wake and boundary layer interactions. To enhance mesh resolution near the airship, a body of influence (BOI) with dimensions $1.5L \times 0.6L \times 0.6L$ was used. This localised refinement allows for finer meshing in critical areas, accurately capturing flow details like boundary layer separation and wake formation while maintaining computational efficiency in less critical regions. Leveraging the axisymmetric nature of the airship, the mesh was generated for only half the body, reducing computation time while still capturing essential flow characteristics.

Figure 10. Computational domain used for the simulation.

Adiabatic no-slip conditions were applied to the airship surfaces, ensuring airflow adherence and no heat transfer. Pressure far-field boundary conditions were used for the inlet, outlet, and far-field regions, simulating unobstructed airflow and consistency with atmospheric conditions. This approach, with its tailored computational domain, mesh refinement and boundary conditions, ensures accurate simulation of aerodynamic interactions around the airship while optimising computational efficiency, enabling effective analysis and design optimisation.

4.4 Mesh generation

The simulated flow includes low subsonic speeds (Mach number $ \lt $ 0.2) and high volumetric Reynolds number ( $R{e_v}$ ) characteristics. These conditions cause boundary layer variations and flow separation, leading to eddies and vortices that significantly impact the overall aerodynamics. Therefore, designing an appropriate mesh to accurately capture these complex flow variations was the most challenging and crucial step in the current numerical investigations. A poly-hexcore mesh was generated using $ANSYS$ Fluent meshing for its efficiency and superior gradient approximation, which is crucial for airship aerodynamics. This mesh topology reduces the overall mesh count compared to tetrahedral or polyhedral meshes, enhancing computational efficiency and resource management. The poly-hexcore mesh minimises numerical diffusion, preserving flow feature sharpness and simulation fidelity. Refinement was focused near airship walls and fins to accurately capture curvature and thin trailing edges without excessive computational costs. To capture boundary layer flow variations, 34 inflation layers were projected with a growth rate of 1.2 and an initial layer height of 1.33e-5 m leading to a total inflation layer thickness of 2.93e-2 m. The final mesh had around 2.4 million elements, a maximum skewness of 0.54, minimum orthogonal quality over 0.1, and an average Y+ value under 1. Figure 11 shows the fine mesh for the analysis setup displaying the resolution of the curved surfaces, boundary layer growth, and the finer mesh near the airship.

Figure 11. Schematic of the discretised flow domain.

4.5 Solver setup

The SIMPLE algorithm was applied for pressure-velocity coupling in the simulation, ensuring stable and reliable solutions for fluid dynamics problems. The $k$ - $\varepsilon $ standard turbulence model was selected for its balance between computational efficiency and accuracy in capturing flow characteristics, validated through a comparative study with Spalart-Allmaras (SA) and SST $k$ - $\omega $ models. The rationale for choosing the $k$ - $\varepsilon $ turbulence model is explained in Section 5.2. Convergence was achieved when normalised residuals were reduced by five orders of magnitude, ensuring solution stability. This framework provides a rigorous basis for analysing airship aerodynamics.

5.1 Grid selection study

To reduce the computational cost, grid selection studies are essential in airship aerodynamics. Airships operate across various flow regimes, requiring robust simulations for accurate performance predictions. To assess grid convergence, this analysis compares three levels of grid refinement for the ZHIYUAN-1 shaped hull with fins and gondola configuration. The first grid, being coarse, consisted of 0.9 million elements. The second, a finer grid, had 2.4 million elements. The third and finest grid included 4 million elements. Comparing the drag coefficient values of the ZHIYUAN-1 airship at a 0° angle-of-attack showed that fine and very fine meshes provided better results shown in Fig. 12, indicating solution convergence. A fine mesh with 2.4 million elements was employed for all simulations of the conventional models, resulting in a 6.8% error at a zero-degree angle-of-attack for the ZHIYUAN-1 model when compared to experimental data. This mesh strikes a balance between computational cost and accuracy, thereby validating the CFD analysis and ensuring reliable results for airship performance evaluation. The simulations were conducted on a computer with a 12^th-generation Intel Core i7 processor (14 cores) and 16 GB of RAM.

Figure 12. Grid selection test.

5.2 Comparison of turbulence models

In this study three turbulence models were compared, namely Spalart-Allmaras [Reference Spalart and Allmaras53, Reference Dacles-Mariani, Zilliac, Chow and Bradshaw54], SST $k$ - $\omega $ [Reference Menter55, Reference Menter56] and $k$ - $\varepsilon $ [Reference Jones and Launder57, Reference Launder and Sharma58] to ensure accurate and reliable CFD simulations for airship aerodynamics. The aerodynamic data from these models were compared with experimental results from wind tunnel testing on the ZHIYUAN-1 airship to identify the most suitable model for predicting flow behaviour and aerodynamic characteristics. Figure 13 illustrates the performance of each turbulence model in capturing key aerodynamic parameters, such as lift and drag coefficients, across various angles of attack. The standard $k$ - $\varepsilon $ turbulence model outdid the others in accurately predicting flow behaviour and aerodynamic characteristics.

Figure 13. Comparison of different turbulence models.

The SST $k$ - $\omega $ model, while estimating lift coefficients adequately, consistently reported higher drag values compared to the other models and experimental data. This discrepancy is attributed to premature estimating of the transition point from laminar to turbulent flow, affecting overall drag estimation. In contrast, the $k$ - $\varepsilon $ model demonstrated superior performance in predicting both lift and drag coefficients, closely aligning with experimental data. Its consistent performance led to its selection for further analysis. The comparison underscores the importance of choosing the right turbulence model for accurate CFD simulations of airship aerodynamics. The $k$ - $\varepsilon $ model’s selection ensures confidence in the predictive capabilities of the simulations, paving the way for further analysis and optimisation of airship designs.

5.3 Validation with experimental data

Validation with experimental data is a crucial step in ensuring the accuracy and reliability of CFD simulations for airship aerodynamics. This study compares numerical solutions with experimental data obtained from controlled wind tunnel tests [Reference Wang, Fu, Duan and Shan49] to evaluate the congruence between simulated and real-world aerodynamic behaviours. Figure 14 illustrates a detailed comparison between the numerical predictions and experimental data, demonstrating a high degree of correlation. A thorough analysis revealed that the maximum deviation between the numerical and experimental results was 6.8% at a zero-degree angle-of-attack and an average error of 3%, indicating a good agreement between the simulated and experimental aerodynamic characteristics of the airship. The numerical results from CFD, presented in Fig. 14, were obtained using the $k$ - $\varepsilon $ turbulence model.

Figure 14. Validation of CFD results for ZHIYUAN-1 against experimental data from (Reference Wang, Fu, Duan and Shan49).

5.4 Numerical results

The analysis focuses on key aerodynamic coefficients, particularly lift and drag, crucial for airship performance and efficiency. Lift combines buoyancy, influenced by the density difference between lifting gas and air, and aerodynamic lift, affected by the airship’s shape. Drag, opposing the airship’s motion, impacts thrust requirements, efficiency and fuel consumption. Minimising drag is essential for better endurance. The analysis compares various airship configurations by examining these coefficients, aiding in design optimisation for specific operational needs and conditions. This serves as a foundational step in enhancing airship performance and efficiency.

5.4.1 Aerodynamic performance of conventional and multi-lobed airships

The geometric data has been calculated for the specified envelope volume of 0.2935 m ${{\rm{\;}}^3}$ , and the results are presented in Table 4.

Table 4. Geometric dimensions of conventional configurations

Figure 15 illustrates the changes in aerodynamic parameters for conventional airships as the angle-of-attack (AoA) varies from −10° to 10°. These angles were chosen to encompass all possible flight scenarios for the airship.

Figure 15. Aerodynamic characteristics of conventional airships.

After validating the solver’s ability to capture aerodynamic coefficients for the conventional (single-lobed) ZHIYUAN-1 model with fins and gondola using experimental data [Reference Wang, Fu, Duan and Shan49], additional verification was performed on other airship models, including fins and gondola. Comparative analyses using Fluent and OpenFOAM software under identical flow conditions showed that the results from both software were in good agreement shown in Fig. 16. This confirmed the reliability and accuracy of the CFD simulations. The average error percentages between the data obtained from the two different solvers are shown in Table 5. $OpenFOAM$ ^® is an open-source CFD software developed by OpenCFD Ltd. since 2004. It offers flexible solvers for various engineering applications, including compressibility, turbulence, thermal, chemical reactions, solid mechanics and electromagnetic fluctuations. OpenFOAM includes tools for pre-processing (blockMesh, snappyHexMesh), solving, and post-processing (ParaView). It supports parallel processing, reducing time and memory constraints, and allows custom solver design.

Figure 16. Comparison of simulation results between Fluent and OpenFOAM.

Table 5. Comparison between Ansys Fluent and OpenFOAM

The comparison between simulation data and experimental results for the ZHIYUAN-1 model, including fins and gondola, as presented in Fig. 14, indicates that the domain size, mesh settings and boundary conditions employed in ANSYS Fluent are well-suited for models featuring fins and gondola. However, these settings were found to be inadequate for accurately capturing the flow behaviour over the bare hull. As shown in Fig. 17, there is a significant discrepancy between the results obtained from Fluent and OpenFOAM. The OpenFOAM simulations for the bare hull exhibit a strong agreement with experimental data, with an error of approximately 7.7%, significantly outperforming Fluent, which shows much larger deviations. Consequently, OpenFOAM was chosen to predict the aerodynamic characteristics of bare hull models for all profiles. The simulations were performed using a mesh setup shown in Fig. 18 consisting of 6.1 million hexahedral cells. SnappyHexMesh was selected for mesh generation due to its reliable method for defining final layer thickness and its robust control over the meshing process. The SimpleFoam solver, coupled with the k-epsilon turbulence model, was employed for all simulations. Standard Dirichlet and Neumann boundary conditions, appropriate for steady incompressible flow, were applied across all cases. The solver operates using the SIMPLE algorithm. To ensure stability and convergence, the solver was relaxed with a pressure relaxation factor of 0.3 and a relaxation factor of 0.7 for both momentum and turbulence variables. This relaxation strategy helps to mitigate residuals, preventing divergence and enhancing the stability of the solution. All simulations were iterated until the pressure, momentum and turbulence residuals were reduced to a threshold of 1e-5. The initial conditions for the simulation setup in OpenFOAM are identical to those used in Fluent (refer to Table 3).

Figure 17. Variation of drag coefficient with angle-of-attack for the bare hull of the ZHIYUAN-1 model.

Figure 18. Schematic of the discretised computational domain used in OpenFOAM.

The impact of the gondola and fins was also analysed by comparing three configurations: hull alone, hull with fins and hull with both fins and a gondola. The results presented in Fig. 19 indicated that excluding the gondola reduced the drag coefficient by $3 - 17$ %, and excluding the fins resulted in a $47 - 53$ % reduction in drag coefficient.

5.4.2 Aerodynamic comparison of multi-lobed profiles

The study also compared bi-lobed and tri-lobed configurations presented in Fig. 20 for the specified envelope volume. All simulations for the bi-lobed and tri-lobed models were conducted using OpenFOAM. The geometric dimensions were adjusted to maintain a constant volume across all configurations. For an envelope volume of (V = 0.2935 ${m^3}$ ), the geometric values are provided in Table 6, where $f$ denotes the lateral distance between the two lobes. The value of $f$ is assumed to be 0.3 times the diameter of a lobe ( $D$ ), represented as $f = 0.3 \times D$ . The data shown in Figs. 21a and b suggest that the fins are the main contributors to the increased drag, which increased drag coefficient by $28 - 57$ % for bi-lobed models and $22 - 54$ % for tri-lobed models.

Table 6. Geometric dimensions of the multi-lobed configuration

Figure 19. Impact of gondola and fins on the drag characteristics of conventional airships.

Figure 20. Multi-lobed configurations.

Figure 21. Comparison of aerodynamic characteristics for multi-lobed airship configurations.

Both the bi-lobed and tri-lobed models exhibit similar aerodynamic efficiencies shown in Fig. 22 across the different profiles. The main difference lies in the slight variations in lift-to-drag ratios at different angles of attack, with the tri-lobed model showing slightly better performance at lower angles.

Figure 22. Aerodynamic efficiency of multi-lobed airship configurations.

5.4.3 Aerodynamic comparison of conventional and multi-lobed airships

The drag polar plots presented in Fig. 23 for the conventional, bi-lobed and tri-lobed configurations indicate that the tri-lobed model exhibits a superior lift-to-drag ratio ( $L/{D_{max}}$ ) compared to both the single-lobed and bi-lobed models.

Figure 23. Drag polar plot.

A collective drag polar plot, presented in Fig. 24, is generated for both bi-lobed and tri-lobed configurations across various shape profiles to assess their aerodynamic performance. This analysis is essential for identifying the shape profile that maximises the lift-to-drag ratio within these configurations. Among the evaluated shape profiles, the tri-lobed model featuring the Garg profile consistently demonstrate superior lift-to-drag ratios over the tested range of angles of attack.

Figure 24. Drag polar plot for bare hulls.

Figure 25 compares the aerodynamic characteristics of single-lobed airships (conventional) with multi-lobed airships. An increase of $47 - 69$ % in drag coefficient is observed for a bi-lobed configuration compared to a conventional configuration. An even higher increase of $5 - 6$ % in drag coefficient is observed for a tri-lobed configuration compared to a bi-lobed one.

Figure 25. Comparison between conventional and multi-lobed airships.