2.1 Generation of envelope profile

Any geometric parameterization model should have several desirable characteristics. It should be well-behaved and produce smooth and realistic shapes. It should be mathematically efficient, numerically stable, fast, accurate and consistent. It should require less number of variables to represent a large domain to contain optimum aerodynamic shapes for varying design conditions and constraints. Finally, it should allow the user to define the specification of design parameters such as leading-edge radius, boat-tail angle, and profile closure.

2.1.1 Class shape transformation method

The Class Shape Transformation (CST) parametrisation technique was developed around 15 years ago by Kulfan [Reference Kulfan25], an aerodynamicist at Boeing. It incorporates a geometric class/shape function transformation technique. The shape of a specified geometry can be defined using Bernstein polynomials representation of the unit shape function [Reference Kulfan and Bussoletti26]. The CST model is an optimisation technique where a shape can be optimised for analysis using a mathematical model. The optimisation is carried out to obtain an optimum and smooth cross-sectional curve of the desired profile. This method is widely used for the shape generation of aerofoils.

In this method, the first step is to convert the x and y coordinates of the given profile into non-dimensional spatial coordinates, which are defined as $\psi = x/L$ and $\zeta = y/L$ . Then, the component shape functions are defined as:

(1) \begin{align}{S_i}(\psi ) = {K_i}{\psi _i}{(1 - \psi )^{n - 1}}\end{align}

${K_i}$ is computed using the following expression:

(2) \begin{align}{K_i} = \left( {\begin{array}{*{20}{c}}n\\i\end{array}} \right) = \frac{{n!}}{{i!(n - i)!}}\end{align}

The overall shape function for the upper surface of the given profile is defined as

(3) \begin{align}Su(\psi ) = \sum\limits_{i = 1}^n A{u_i}.{S_i}(\psi )\end{align}

where $A{u_i}$ are the shape coefficients that can be determined using the least-square fit to match a specified geometry. Similarly, the lower surface is defined by the equations

(4) \begin{align}Sl(\psi ) = \sum\limits_{i = 1}^n A{l_i}.{S_i}(\psi )\end{align}

The generatrix of the given profile can be defined by the following expression:

(5) \begin{align}\zeta = C_{{N_2}}^{{N_1}}(\psi ).Su(\psi ) + \psi .\Delta {\eta _{TE}}\end{align}

where $C_{{N_2}}^{{N_1}}(\psi ) = {\psi ^{{N_1}}}{(1 - \psi )^{{N_2}}}$ .

To capture the specified shape, one can vary the values of the shape coefficients in such a way that the error between the specified shape and the approximate one defined by the shape generator is minimised. The function for the minimisation can be written as:

(6) \begin{align}f(x) = \sum\limits_{i = 1}^n \left[ {y({x_i}) - {y_i}} \right]\end{align}

where $y({x_i})$ is the value of the y-coordinate derived from the mathematical equations that define the particular envelope profile and ${y_i}$ is the value of the y-coordinate generated using the shape generator.

The value of shape coefficients that are required to replicate some standard geometries of a single lobed airship envelope using this method is shown in Table 1.

Table 1. CST-shape coefficients

The different combinations of ${N_1}$ and ${N_2}$ define a variety of geometric shapes like an aerofoil, elliptic aerofoil, biconvex aerofoil, and Sears-Haack body [Reference Kulfan and Bussoletti26]. The value of ${N_1}$ and ${N_2}$ was fixed to be 0.5 for the airship envelope profile.

2.1.2 Gertler method

Gertler et al. [Reference Gertler27, Reference Landweber and Gertler28] formulated a technique to generate a 2-D curve that can be revolved ${360^ \circ }$ to develop streamlined bodies-of-revolution for the hull design of high-submerged-speed submarines. This method involves five geometrical parameters as shown in Fig. 1, namely, the position of the maximum section (m), the nose radius ( ${R_0}$ ), the tail radius ( ${R_1}$ ), prismatic coefficient ( ${C_p}$ ) (the ratio of the volume of body-of-revolution to the volume of the cylinder which encloses the body), and the fineness ratio (length over diameter ratio, $L/D$ ), which can be used to define the shape of streamlined bodies-of-revolution.

Figure 1. Design parameters.

The generatrix of the airship models are derived from a sixth-degree polynomial of the form:

(7) \begin{align}{y^2} = {a_1}x + {a_2}{x^2} + {a_3}{x^3} + {a_4}{x^4} + {a_5}{x^5} + {a_6}{x^6}\end{align}

where, the x is the non-dimensional abscissa and y is the non-dimensional ordinate. The non-dimensional offsets $X/L$ vs. $Y/D$ are the same for all the fineness ratios, once the other four geometrical parameters have been fixed.

The shape coefficients ${a_1}$ , ${a_2}$ , …, ${a_6}$ for each profile are determined when the values for the geometrical parameters are assigned. The constraints applicable for the airship envelope are: $y = 0$ , when $x = 1$ ; $y = 1/2$ when $x = m$ ; and $dy/dx = 0$ when $x = m$ . The final equations are obtained in terms of the shape coefficients by satisfying the constraints applicable to the shape of the airship envelope:

(8) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{a_1} + {a_2} + {a_3} + \cdots + {a_n} = 0}\\[4pt]{{a_1}m + {a_2}{m^2} + {a_3}{m^3} + \cdots + {a_n}{m^n} = \frac{1}{4}}\\[4pt]{{a_1} + 2{a_2}m + 3{a_3}{m^2} + \cdots + n{a_n}{m^{n - 1}} = 0}\end{array}} \right.\end{align}

The radius of curvature for any generatrix may be evaluated from:

(9) \begin{align}R = \pm \frac{1}{{{d^2}X/d{Y^2}}}{\left[ {1 + {{\left( {\frac{{dX}}{{dY}}} \right)}^2}} \right]^{3/2}}\end{align}

In the dimensionless form, R in Equation (9) can be written as:

(10) \begin{align}r = \pm \frac{1}{{{d^2}x/d{y^2}}}{\left[ {1 + \frac{{{L^2}}}{{{D^2}}}{{\left( {\frac{{dx}}{{dy}}} \right)}^2}} \right]^{3/2}}\end{align}

where, r is the non-dimensional radius, L is the length, and D is the diameter.

The value of $dx/dy$ can be obtained from the successive differentiation of Equation (7) with respect to y:

(11) \begin{align}2y = \left( {{a_1} + 2{a_2}x + \cdots + n{a_n}{x^{n - 1}}} \right)\frac{{dx}}{{dy}}\end{align}

and

(12) \begin{align}2 = \left( {{a_1} + 2{a_2}x + \cdots + n{a_n}{x^{n - 1}}} \right)\frac{{{d^2}x}}{{d{y^2}}} + \left( {2{a_2} + \cdots + n(n - 1){a_n}{x^{n - 2}}} \right){\left( {\frac{{dx}}{{dy}}} \right)^2}\end{align}

When $x = 0$ , $dx/dy = 0$ and hence, from Equation (12) we get:

(13) \begin{align}\frac{{{d^2}x}}{{d{y^2}}} = 2a_1^{ - 1}\quad \quad {\rm{if}}\,{a_1} \ne 0\end{align}

Substituting the value of ${d^2}x/d{y^2}$ in Equation (10), we get:

(14) \begin{align}{a_1} = 2{r_0}\end{align}

where ${r_0}$ is the radius of curvature at the nose. If ${a_1} = 0$ , the body will have a pointed nose ( ${r_0} = 0$ ). Hence, Equation (14) is valid for both cases, i.e. ${a_1} \ne 0$ or ${a_1} = 0$ .

Similarly, when $x = 1$ , $y = 0$ and from Equation (11) $dx/dy = 0$ , unless

(15) \begin{align}{a_1} + 2{a_2} + \cdots + n{a_n} = 0\end{align}

Hence, Equations (10) and (12) give

(16) \begin{align}{a_1} + 2{a_2} + \cdots + n{a_n} = - 2{r_1}\end{align}

where ${r_1}$ is the radius of curvature at the tail.

The positive sign is taken in Equation (14) and the negative in Equation (16) because ${r_0}$ and ${r_1}$ are the positive values. ${a_1}$ is the slope of the sectional-area curve at $x = 0$ , and hence, is positive. Similarly, ${a_1} + 2{a_2} + \cdots + n{a_n}$ is the slope of the sectional-area curve at $x = 1$ , and hence, is negative.

The volume of the envelope ( ${V_{env}}$ ) can be expressed as:

(17) \begin{align}{V_{env}} = \int_0^1 \pi {Y^2}dX = \pi {D^2}L\int_0^1 {y^2}dx\end{align}

Substituting for ${y^2}$ from Equation (7), we get

(18) \begin{align}\frac{1}{2}{a_1} + \frac{1}{3}{a_2} + \cdots + {\frac{{n + 1}}{a_n}} = \frac{1}{4}{C_p}\end{align}

The formulated linear equations (Equations (8), (14), (16), and (18)) can be represented in a matrix form as:

(19) \begin{align}AY = B\end{align}

(20) \begin{align}\left[ {\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}1 &1& 1 &1&1&1\\[3pt]1&0&0&0&0&0\\[3pt]1&2&3&4&5&6\\[3pt]m&{{m^2}}&{{m^3}}&{{m^4}}&{{m^5}}&{{m^6}}\\[3pt]1&{2m}&{3{m^2}}&{4{m^3}}&{5{m^4}}&{6{m^5}}\\[3pt]{\dfrac{1}{2}}&{\dfrac{1}{3}}&{\dfrac{1}{4}}&{\dfrac{1}{5}}&{\dfrac{1}{6}}&{\dfrac{1}{7}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{a_1}}\\[3pt]{{a_2}}\\[3pt]{{a_3}}\\[3pt]{{a_4}}\\[3pt]{{a_5}}\\[3pt]{{a_6}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\[3pt]{2{r_0}}\\[3pt]{ - 2{r_1}}\\[3pt]{\dfrac{1}{4}}\\[3pt]0\\[3pt]{\frac{1}{4}{C_p}}\end{array}} \right]\end{align}

By solving these six linear equations simultaneously, we can obtain the value for six shape coefficients ${a_1}$ , ${a_2}$ , ${a_3}$ , ${a_4}$ , ${a_5}$ , and ${a_6}$ for the given geometrical parameters.

The final equation for the airship envelope shape can be written as:

(21) \begin{align}y(x) = D\sqrt {{a_1}\left( {\frac{x}{L}} \right) + {a_2}{{\left( {\frac{x}{L}} \right)}^2} + {a_3}{{\left( {\frac{x}{L}} \right)}^3} + {a_4}{{\left( {\frac{x}{L}} \right)}^4} + {a_5}{{\left( {\frac{x}{L}} \right)}^5} + {a_6}{{\left( {\frac{x}{L}} \right)}^6}}\end{align}

To obtain the final envelope surface, the curve generated from Equation (21) 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. A detailed description of the standard envelope profiles represented in Table 2 is given in the 6.

Table 2. Gertler shape coefficients

The shape coefficients for standard envelope profiles obtained through Gertler’s methods are presented in Table 2. Table 3 shows the list of design parameters obtained from the Gertler-shape generator for standard envelope profiles.

Table 3. Gertler design parameters

Figure 2 shows that the standard envelope profiles can be generated using CST and Gertler’s method. The CST method is more flexible than Gertler’s method because the latter method will fail to follow the target curve with a sharp trailing edge as shown in Fig. 2(d).

Figure 2. Standard envelope profile generated using CST and Gertler method.

2.2 Estimation of multi-lobed envelope volume

There are many methods that can be found in the existing literature to estimate the volume of the envelope of conventional and unconventional airship geometries. The generatrix for the envelope profile can be generated using the CST method as discussed in Section 2.1.1 or Gertler’s method (refer to Section 2.1.2). From the curve defining the profile of the envelope, the surface area and volume can be estimated using the classical approach. The volume of the envelope ( ${V_{env}}$ ) with axisymmetric body of revolution is given as:

(22) \begin{align}{V_{env}} = \pi \int_0^1 {y^2}dx\end{align}

where y represents the y-coordinates of the given profile generated using shape generators.

The surface area of the envelope is calculated as:

(23) \begin{align}{S_{env}} = 2\pi \int_0^1 y{\left[ {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right]^{1/2}}dx\end{align}

The volume of an airship with the axisymmetric body of revolution can also be estimated through Gertler’s method using the following expression:

(24) \begin{align}{V_{env}} = \frac{{\pi L{D^2}{C_p}}}{4}\end{align}

where L is the length of the envelope, D is the diameter of the envelope, and ${C_p}$ is the prismatic coefficient.

Table 4 shows the difference in the value of envelope volume computed using the classical and Gertler’s method for the NPL envelope profile. The classical method is the most convenient method to estimate the volume and surface area for a given generatrix of any profile. Estimation of the envelope volume for a multi-lobed airship involved deduction of the volume of intersection, which is not a straightforward task. To find the volume of a multi-lobed envelope, the intersection volume between the lobes must be calculated. By subtracting the intersection volume from the volume of the lobes, the total volume of the multi-lobed envelope can be calculated. Similarly, the estimation of the wetted surface area of a multi-lobed envelope is a non-trivial task. It is a difficult procedure to calculate the wetted surface area from the surface area of the lobes shown in Equation (23). To compute the wetted surface area, the elemental approach is one of the easiest approaches in which the envelope will be divided into elements. The area of each element will be calculated from its x, y and z coordinates.

Table 4. Comparion between classical method and Gertler’s method

The present study compares two different methods to estimate the intersection volume between the lobes. The first method is based on the Monte Carlo method. The latter is based on an analytical approach developed by the authors. In the Monte Carlo method, a random number of points are generated into a domain, viz., a box placed at the intersection of two lobes. All the random points must be checked whether they belong to the intersection or not. From the volume of the defined box, the number of points falling into the intersection, and the total number of generated points, the approximate volume of the intersection between the lobes can be estimated. The volume of the envelope computed by both methods is in good agreement with each other; however, the Monte Carlo method was far more computationally expensive. A comparison between the two methods was carried out for 100 different geometries of tri-lobed envelopes, whose geometry parameters shown in Fig. 3 and 4 were varied.

Figure 3. Multi-lobed envelope geometry (front view).

Figure 4. Multi-lobed envelope geometry (top view).

For the implementation of the Monto Carlo method and analytical method, the ellipsoid was chosen as a base profile for each lobe. The geometry function is based on the ellipsoid equation (ellipsoid rotated about the z-axis by angle $\theta $ ):

(25) \begin{align}\frac{{{{\left[ {(x - {x_0})\cos (\theta ) + (y - {y_0})\sin (\theta )} \right]}^2}}}{{{a^2}}} + \frac{{{{\left[ {(x - {x_0})\sin (\theta ) + (y - {y_0})\cos (\theta )} \right]}^2}}}{{{b^2}}} + \frac{{\left[ {z - {z_0}} \right]}}{{{c^2}}} = 1\end{align}

In a simple form, it can be defined as

(26) \begin{align}{x_e} + {y_e} + {z_e} = 1\end{align}

The function shown in Equation (25) is to find the randomly generated points belonging to the ellipsoid (it does if Equation (26) is $ \lt = 1$ ).

In the analytical method, the airship envelope is divided into the number of segments in the longitudinal direction as shown in Fig. 5. Thereafter, the segments which fall inside the intersection volume are taken into consideration. The area of each segment is calculated using geometrical consideration shown in Fig. 6. Then the average area is calculated from the area of each segment in the intersection region. So, multiplying the average area by the length of the intersection defined by X will provide a nearly accurate volume of the intersection between the lobes.

Figure 5. Volume of intersection between the lobes.

Figure 6. Area of intersection between lobes.

The section volume between the lobes is expressed as:

(27) \begin{align}{V_{sec}} = {\rm{Average\ area}} \times {\rm{Length\ of\ intersection}} = \left( {\frac{{sum({A_1}\,{:}\,{A_n}}}{n}} \right) \times X\end{align}

where ${A_1},{A_2},{A_3}, \ldots ,{A_n}$ is the cross-sectional area of segments in the intersection region and n represents the number of segments.

The total volume of the multi-lobed envelope can be written as:

(28) \begin{align}{V_{env}} = \sum\limits_{i = 1}^N {V_{lobe}} - {N_I} \times {V_{sec}}\end{align}

where N represents the number of lobes and ${N_I}$ represents the number of intersection regions. For double- and tri-lobed envelopes, the number of intersection regions equals one and two, respectively.

To determine the area of segments in the intersection region, from the cosine rule, we get:

(29) \begin{align}\alpha = 2(\mathop {\cos }\nolimits^{ - 1} (({R^2} + {h^2} - {d^2})/(2Rh)))\end{align}

(30) \begin{align}\beta = 2(\mathop {\cos }\nolimits^{ - 1} (({d^2} + {h^2} - {R^2})/(2dh)))\end{align}

In general, the area of a segment between two circles with a radius ‘r’ is defined as

(31) \begin{align}{A_{segment}} = {A_{sector}} - {A_{triangle}} = \left( {\frac{\theta }{2}r} \right) - \left( {\frac{1}{2}{r^2}\sin \theta } \right)\end{align}

Therefore, the area of segment ${S_1}$ and ${S_2}$ shown in Fig. 6 can be expressed as:

(32) \begin{align}{S_1} = \frac{1}{2}{d^2}\left( {\beta - \sin (\beta )} \right)\end{align}

(33) \begin{align}{S_2} = \frac{1}{2}{R^2}\left( {\alpha - \sin (\alpha )} \right)\end{align}

From the given data of envelope volume (fixed to be ${V_{env}} = 10,000{m^3}$ ), the x and y data points that represent the envelope profile and geometry dimensions are generated by back-and-forth calculations for four standard profiles, viz., Ellipsoid, NPL, Wang, and Zhiyuan-1 to compare the difference in the volume estimation between the proposed analytical method and CAD software. Table 5 represents the difference in the estimation of the volume using the analytical method. The x, y and other dimension data obtained from the analytical method were used to create the 3D geometry in CAD software.

Table 5. Comparion of envelope volume ( ${{\rm{m}}^3}$ ) between analytical method and CAD

For an unconventional airship configuration, it is difficult to determine the exact planform area and wetted area of the envelope. However, it can be approximated by an equivalent shape of an ellipsoid as shown in Fig. 7.

Figure 7. Equivalent ellipsoid (reproduced from Ref. (Reference Carichner and Nicolai29)).

The approximate planform area and wetted area of the envelope can be estimated analytically using the method proposed by Carichner and Nicolai [Reference Carichner and Nicolai29]. The wetted area of the multi-lobed envelope is estimated using the Equations (34)–(37).

(34) \begin{align}{S_{area}} = \pi \left( {({L^p}W_{eq}^p + {L^p}h{t^p} + W_{eq}^ph{t^p})} \right)\end{align}

where $p = 1.6075$

(35) \begin{align}{S_{wetted}} = \left( {{P_{lobes}}/{P_{ellipse}}} \right){S_{area}}\end{align}

where

(36) \begin{align}{P_{ellipse}} = \pi \left[ {3(a + b) - \sqrt {10ab + 3({a^2} + {b^2})} } \right]\end{align}

(37) \begin{align}{P_{lobes}} = {N_{lobes}}\left( {2\pi R} \right) - 4\left( {{L_{arc}}} \right)\end{align}

where R is the radius of a lobe and ${L_{arc}}$ is the arc length of a segment (red-dashed lines) between the lobes shown in Fig. 8.

Figure 8. Schematic of arc length of a segment.

From the length (L), equivalent width ( ${W_{eq}}$ ), and height (ht) of the envelope, the wetted surface area ( ${S_{wetted}}$ ) of the tri-lobed envelope is calculated by multiplying the surface area of the equivalent ellipsoid ( ${S_{area}}$ ) assuming the shape of a scalene ellipsoid by the ratio of the perimeter of lobes to the perimeter of the ellipse ( ${P_{lobes}}/{P_{ellipse}}$ ) shown in Fig. 9.

Figure 9. Cross-section of the tri-lobed envelope (reproduced from Ref. (Reference Carichner and Nicolai29)).

2.3 Design parameters

The design variables consist of parameters whose values are varied to describe the envelope geometry, size and layout of fins and solar panel layout. The number of design variables depends on the configuration of the airship, viz., conventional and unconventional. The conventional design represents an airship with the axisymmetric body of revolution as shown in Fig. 10(a). The unconventional design represents an airship with a non-axisymmetric body of revolution as shown in Fig. 10(b) and (c). To parameterise the geometry of a conventional airship including envelope, fins and solar panels, 14 (using the Gertler Series 58 Shape Generator for envelope geometry explained in the Section 2.1.2) or 17 (using the Class Shape Transformation (CST) method explained in the Section 2.1.1 for envelope geometry) design variables are required. the first 5 of which are related to the geometry of the envelope, and the next 4 are related to the length of the airship and the layout of the solar array. The last five variables are related to the fin design. In the case of an unconventional airship, viz., multi-lobed dynastat, 18 design variables are required.

Figure 10. Airship geometry.

2.4 Multi-lobed envelope geometry

Apart from the shape variables that define the profile of a lobe of the tri-lobed envelope, the geometry of the tri-lobed envelope is defined by four design variables such as length of the envelope, (L), the fraction of distance of outer lobes position with respect to central lobe in the longitudinal direction, i.e. along the length of the body, I, the fraction of distance of outer lobes position with respect to central lobe in the lateral direction, (f), and the fraction of distance of outer lobes position with respect to central lobe in the vertical direction, (g). The design variables that define the envelope geometry and layout of the solar array over the surface of the tri-lobed envelope are geometrically explained in Fig. 11.

Figure 11. Trilobed envelope geometry.

2.5 Solar array model

A model based on an elemental approach has been developed for the generation of the solar array over the surface of the envelope and estimation of the area of the solar array for the multi-lobed airship. The starting and ending point for the array are decided from the length of the airship. The surface of the envelope between the starting and ending point of the array is divided into [(m-1) $ \times $ (n-1)] rectangular grids as shown in Fig. 12. For any element of the grid, length ( $\vec dl$ ) and width ( $\vec db$ ) vectors can be expressed in terms of position vector ( $\vec r$ ) as:

(38) \begin{align}\vec d{l_{ij}} = {\vec r_{i,j + 1}} - {\vec r_{i,j}}\end{align}

(39) \begin{align}\vec d{b_{ij}} = {\vec r_{i + 1,j}} - {\vec r_{i,j}}\end{align}

Figure 12. Schematic of solar array grid on the envelope.

For a conventional airship with the axisymmetric body of revolution, the solar array model is very simple without any geometrical complications. But for the multi-lobed configurations, there are several geometrical complications because of the orientation of the lobes with respect to one another. The solar array grid has to be generated over the surface of the envelope to estimate the energy available based on the orientation of each element of the generated grid with respect to the Sun vector. In the multi-lobed configuration, the lobes are merged together based on the value of e, f and g. Hence, it is not simple to generate the grid over the surface. The orientation of the lobes and merging points between the lobes at each section of the envelope have to be calculated using trigonometric relations as shown in Fig. 13.

Figure 13. Geometry of the solar array.

The entire array model can be parameterised using three design parameters, viz., the starting point of the array ( ${X_s}$ ), ending point of the array ( ${X_f}$ ) and intended angle of the array ( ${\theta _{array}}$ ). Figure 14 shows the solar array generation over the conventional and multi-lobed configurations.

Figure 14. Solar array over conventional and multi-lobed configurations.

2.6 Vehicle sizing and performance model

For a given set of design parameters and operating conditions, a design solution is obtained using an iterative design loop as shown in Fig. 15. The first stage in the design process involves the selection of the appropriate envelope profile and sizing of the envelope in order to produce the required lift. In this proposed methodology, the shape of the envelope profile and envelope geometry can be varied according to the mission requirements and constraints.

Figure 15. Schematic of vehicle design.

Many existing studies have limited the design space to a fixed envelope profile and geometry for the initial sizing. To select the ideal shape of an airship, several factors need to be considered including the size and mass of the airship, the materials it is made from, the lift (static and dynamic) and drag forces acting on the airship and the desired performance characteristics. One of the best approaches in selecting the ideal shape would be to use CFD simulations to model the flow of air around the airship and optimise the shape to minimise the drag and maximise the aerodynamic efficiency ( ${C_L}/{C_D}$ ). The simplest approach might be to use empirical data and mathematical models to predict the lift and drag forces acting on the airship and optimise the shape based on these predictions. There are many factors that can affect the aerodynamic performance of an airship, including the size and shape of the envelope, the size and layout of the tail surfaces, the selection of the propellers and the position and orientation of the engines. To obtain the ideal shape, all of these factors must be considered during the optimisation to meet the desired performance goals. It is important to note that the ideal shape for an airship will depend on the specific design constraints and performance goals of the airship.

Wind tunnel testing is one of the standard procedures to optimise the design of an airship body and is also a good means for accurately measuring the aerodynamic characteristics and moment. The larger size and high Reynolds number at low speed enhance the complexity of measuring the aerodynamic forces and moments using a wind tunnel [Reference Meng, Li, Zhang, Lv and Liu30].

Payload weight and power requirement are the two key parameters that drive the sizing process of any aircraft. For an airship, the maximum range and maximum cruise altitude define the required envelope volume and total mass. The choice of range decides the fuel requirement for conventional fuel-powered airships and the size of the energy management system for electric-powered airships. Maximum cruise altitude has a significant impact on the selection of envelope material and the choice of internal structures to withstand the required pressure difference that needs to be maintained at that high altitude.

The total mass of the airship can be derived by estimating the mass of each sub-system of the vehicle. There are a set of empirical relations to estimate the mass of sub-systems of airships presented by Carichner and Nicolai based on existing airship models in [Reference Carichner and Nicolai29]. The major mass contribution comes from the hull of the airship. The hull mass includes envelope mass, tail mass and ballonet mass. The envelope mass is calculated from the estimation of surface area, maximum stress requirement and envelope material density. The selection of envelope material plays a significant role in mass estimation. Ballonet mass estimation depends on the pressure differential required to maintain the required buoyancy and operating altitude. The tail mass is calculated from the material properties, internal frames and surface area of the fins.

The value of the total mass of the airship and the aerodynamic characteristics data are then passed into the Breguet Range Equation to obtain the revised gross weight. The process is iterated until a solution is found.

In this study, we have focused on the application of open-source software currently available to the public and based on the existing literature. The complete design and optimisation framework involves multiple disciplines that can be developed using publicly available open-source software. Figure 16 shows the interface between the software to achieve an optimal design using the parametric design approach.

Figure 16. Design framework based on open-source software(s).

With the given design parameters and specified mission requirements, Fig. 16 gives an idea of how a model undergoes the mentioned processes such as CAD modeling, meshing and analysis to generate an optimal design output.

By using the initial design parameters, the first step in the design process is to create a CAD model. CAD modeling is an important aspect of parametric design as it allows designers to make detailed, accurate representations of the model’s structure and components. For this FreeCAD or SALOME can be used for modeling as they are open-source and completely free to use. After finalising the CAD model, it undergoes the meshing process.

Meshing is the process of dividing a continuous physical space into discrete elements, such as points, lines or surfaces. In parametric design, meshing is often used to represent a design model as a collection of interconnected elements. This allows designers to analyse and optimise the design using computational techniques, such as FEA or CFD.

To save time, SALOME can do both modeling and meshing and later can be exported for analysis. Alternatively, the CAD model from FreeCAD can be imported into other open-source software, e.g. Gmsh and MeshLab for the meshing.

The analysis and simulation methods can be classified into CFD, FEM and thermal analysis. Computational fluid dynamics (CFD) is a numerical method used to simulate and analyse the flow of fluids, such as gases and liquids, through and around objects. It involves solving mathematical equations that describe the behaviour of fluids to predict the flow patterns, pressures and temperatures in a given system. To achieve this, software like OpenFOAM or SU2 can be used for studying the flow properties around the body, improving its aerodynamics of it, and getting an optimised shape reducing the drag and improving the lift.

The FEM is a numerical method used to solve problems involving the behaviour of structures and materials under load. It involves dividing the structure or material into smaller, simpler shapes called ‘finite elements’ and solving equations that describe the behaviour of each element. FEM is often used in conjunction with CFD to analyse the thermal behaviour of structures and materials, as well as their structural integrity. Software such as CalculiX is used for static structural analysis.

Thermal analysis is a subset of CFD that focuses on predicting and analysing the temperature distribution in a system. It is often used to design and optimise heat transfer systems, predict the performance of cooling systems, and identify areas of high-temperature gradients or thermal stress. This type of analysis can be simulated in the CFD tool or in Scilab/Python to get heat transfer results by solving the simultaneous energy equations.

For LTA systems such as airships, Added Mass (AM) is one of the important parameters during accelerated flight. AM is the mass of air that an airship displaces as it moves through the air. This consideration is very important in the design and performance of the body as it affects performance, stability and manoeuverability. To compute the AM of an airship, the shape and dimensions of the airship must be known, as well as the density of the air through which the airship will be moving. To calculate the AM, the meshed model is imported into Scilab/Python platform.

The initial and operating cost of an airship is also an important consideration, apart from the design and sizing. This depends on several factors such as size, material, the technology used and the operating environment of the system. Scilab/Python can be used for doing such tasks.

Finally, after sorting out all the workflow processes, the model and the supporting data are inserted into the MDOA (model, design, optimise, and analyse) framework, which uses a set of algorithms to produce an optimal design that fits in every aspect. It is a systematic approach to designing a product. This could be done using Scilab/Python by creating a machine learning model, which processes the given data to produce an optimised product.

4.1 Tail sizing

At the point of initial sizing, the airship has been sized (estimation of the envelope volume) based on its takeoff gross weight (TOGW). A configuration in terms of the number of lobes ( ${N_{lobes}}$ ) and fineness ratio ( $l/d$ ) has also been assumed. For the selected configuration, aerodynamic analysis and components weight estimation need to be performed to estimate the drag and fuel/solar array required to accomplish the mission, the empty weight, and the $c.g.$ location. Tail sizing is one of the important phases in the configuration design of airships since it typically accounts for $10 - 14\% $ of the empty weight and 20% of the airship ${C_{D0}}$ . The initial sizing of the tail fins is carried out using a procedure based on historical data called the tail volume coefficient approach. In the initial design process, the location of $c.g.$ and its movement with the change in weight of the airship is unknown. But it is predicted that the $c.g.$ will be close to the centre of buoyancy ( $c.b.$ ). From the historical data, the $c.b.$ is typically located at 45% of the airship length from the nose, and the moment arm varies between $36 - 43\% $ of the length of the airship. [Reference Carichner and Nicolai29] Figure 20 describes the tail sizing procedure given in Ref. (Reference Sadraey34).

Figure 20. Tail design procedure (adapted from Ref. (Reference Sadraey34)).

The horizontal tail ( ${C_{HT}}$ ) and vertical tail ( ${C_{VT}}$ ) coefficients are defined as follows:

(40) \begin{align}{C_{HT}} = \frac{{{l_{HT}}{S_{HT}}}}{{{l_b}V_{env}^{2/3}}}\end{align}

(41) \begin{align}{C_{VT}} = \frac{{{l_{VT}}{S_{VT}}}}{{{l_b}V_{env}^{2/3}}}\end{align}

where, ${l_{HT}}$ and ${l_{VT}}$ are the distance between the $c.g.$ and the quarter chord of the tail fin mean aerodynamic centre (m.a.c.), ${S_{HT}}$ and ${S_{VT}}$ are the projected planform area of the horizontal and vertical fins, respectively, and ${l_b}$ is the length of the airship.

The tail volume coefficient is a function of the tail surface area ( ${S_{HT}}$ ), and the tail surface area is readily determined using the Equations (40) and (41). From the historical data of existing airships, Carichner and Nicolai [Reference Carichner and Nicolai29] estimated the horizontal tail ( ${C_{HT}}$ ) and vertical tail ( ${C_{VT}}$ ) coefficients as a function of airship volume. From the curve fitting to the historical data, the horizontal and vertical tail volume coefficient as a function of envelope volume is derived as:

(42) \begin{align}{C_{HT}} = - 0.0051x + 0.0717\end{align}

(43) \begin{align}{C_{VT}} = - 0.0049x + 0.0641\end{align}

where, ${\rm{x}} = 10^6 $ /envelope volume (in $f{t^3}$ ).

For the present study, the reference envelope volume was taken from the solved example of a hybrid airship given in Chapter 12 of Ref. (Reference Carichner and Nicolai29). The horizontal and vertical tail volume coefficients for the reference envelope volume are listed in Table 8. From the given airship volume, three different tri-lobed envelope geometry were developed.

Table 8. Tail volume coefficient

4.2 Aft tail configuration

The selection of tail configuration is the first step in the sizing of the tail as shown in Fig. 20. An aft tail has several configurations that are designed to satisfy the design requirements. Each configuration has unique advantages and disadvantages.

Typically, a hybrid airship has four tail surfaces or two pairs of tails. The X-tail configuration shown in Fig. 21(c) has a dihedral angle of ${45^ \circ }$ , with the advantage of each tail surface contributing to both lateral-directional and longitudinal stability. One more advantage of the X-tail is the reduction in the overall height of the airship. The two important objectives that need to be achieved at the end of the design process of an airship are low weight and low drag. The wetted area of an airship significantly affects the drag and weight.

Figure 21. Tail configurations for multi-lobed airship.

Y-tail configuration is the best tail configuration for conventional airships from the aerodynamic point of view [Reference Carichner and Nicolai29, Reference Yanxiang, Yanchu, Jianghua and Xiangqiang35], because it has a lower zero-lift drag coefficient and high lift curve slope than other configurations shown in Fig. 22(b). Next to the Y-tail, the X-tail configuration has the lowest zero-lift drag coefficient and high lift curve slope as shown in Fig. 22(a). The suitable tail configuration for the multi-lobed hybrid airship configuration has to be explored by performing the aerodynamics analysis of an airship with different tail configurations.

Figure 22. Drag and lift curve characteristics for different tail configuration.

The second step is to determine the horizontal and vertical tail surface area. The tail surface area can be calculated from:

(44) \begin{align}{S_{HT}} = \frac{{{C_{HT}}{l_b}V_{env}^{2/3}}}{{{l_{HT}}}}\end{align}

(45) \begin{align}{S_{VT}} = \frac{{{C_{VT}}{l_b}V_{env}^{2/3}}}{{{l_{VT}}}}\end{align}

For the X-tail configuration, the surface area ( ${S_A}$ ) of each fin ( ${{\rm{A}}_1}$ , ${{\rm{A}}_2}$ , ${{\rm{A}}_3}$ , and ${{\rm{A}}_4}$ ) can be obtained from:

(46) \begin{align}{S_A} = \sqrt {{{\left( {\frac{{{S_{HT}}}}{4}} \right)}^2} + {{\left( {\frac{{{S_{VT}}}}{4}} \right)}^2}} \end{align}

Similarly, for the Y-tail configuration, the surface area of a fin and the inclination angle are calculated as:

(47) \begin{align}{S_A} = \frac{{ - \frac{2}{3}{S_{VT}} + \sqrt {{{\left( {\frac{2}{3}{S_{VT}}} \right)}^2} + \frac{4}{3}\left( {S_{VT}^2 + S_{HT}^2} \right)} }}{2}\end{align}

(48) \begin{align}\theta = \arccos \left( {\frac{{{S_{HT}}}}{{2{S_A}}}} \right)\end{align}

The mean aerodynamic chord shown in Fig. 24 for the tail can be expressed as:

(49) \begin{align}MAC = \left( {\frac{2}{3}} \right)\frac{{{C_R}\left( {1 + \lambda + {\lambda ^2}} \right)}}{{1 + \lambda }}\end{align}

where, ${C_R}$ is the root chord of the fin and $\lambda $ , the tail taper ratio.

Figure 23. Tail geometry and surface area.

Figure 24. Tail geometry.

Taper ratio is a design parameter of the horizontal and vertical tails similar to the wing of an aircraft which has a significant impact on the overall stability and control, aerodynamic performance, tail aerodynamic efficiency and weight as well as the centre of gravity location. [Reference Sadraey34] The tail taper ratio is defined as the ratio of the tail tip chord ( ${C_T}$ ) to the root chord ( ${C_R}$ ). In this study, the tail taper ratio of 0.6 is used. Figure 25 shows the standardised schematic view of fin geometry.

(50) \begin{align}\lambda = \frac{{{C_T}}}{{{C_R}}}\end{align}

Figure 25. Schematic view of a Fin.

Airships employ aerofoils operating at a relatively low Re range between ${10^4}$ to ${10^6}$ . In the present study, NACA 0018 aerofoil shown in Fig. 26 is used.

Figure 26. Schematic of NACA 0018 aerofoil.

Tail layout and the hull-fin interaction have a significant impact on the dynamics (stability and control) of the airship [Reference Yanxiang, Yanchu, Jianghua and Xiangqiang35]. Tail sizing is not a direct method and has to be refined with the change and progress in the design process. The parameters involved in the sizing of the tail surface shown in Fig. 27 and to estimate its effect on the aerodynamics of a full vehicle are the number of tail pairs ( ${N_{fins}}$ ), exposed tail area ( ${S_T}$ ), total tail area ( ${S_T}{)_{total}}$ , tail moment arm from $c.b.$ to tail ( ${l_T}$ ), tail span ( ${b_T}$ ), tail aspect ratio ( $A{R_T}$ ), tail sweep angle at a maximum thickness ( $\Delta $ ), tail lift curve slope ( ${C_{L\alpha }}{)_{tail}}$ , tail dihedral angle ( $\Gamma $ ), reference volume (Vol) and length of the body ( ${{\rm{l}}_b}$ ).

Figure 27. Tail surface.

In this study, the tail moment arm ( ${l_T}$ ) is set to be 40% of the airship length based on the observation of the typical moment arm of existing airships.

4.3 Effect of fins on airship drag

This section carries out a preliminary evaluation of the effects of fin sizing on the overall aerodynamics of a tri-lobed geometry through a series of numerical investigations carried out at subsonic and high Reynolds number flow conditions. The study compares the drag coefficients at zero degrees angle-of-attack for three tri-lobed airship models pertaining to different hull profiles, namely Ellipsoid, LOTTE and NPL tri-lobed models. Figure 28 presents the three-dimensional view of the three tri-lobed hull variants for both bare hull and hull with fin configurations. The three models have been designed using the algorithm mentioned earlier while keeping the same volume across the different variants. Flow similarity across the three variants has been achieved using the same Re under the same Mach number (M) conditions that are $M \lt 0.3$ . Values related to the other parameters associated with the simulated flow conditions are given in Table 9. Fin sizing was accomplished by keeping the volumetric tail coefficient constant at 0.0662 for the horizontal tail and 0.0589 for the vertical tail. Fins for all the models were placed at a distance of 40% from the centre of buoyancy. The latter is located at a distance of 45% location on the central axis of the hull.

Figure 28. Tri-lobed geometry with fins.

Table 9. Flow Conditions

4.4 Mesh setup

Mesh was generated using the snappyHexMesh tool of OpenFOAM. Mesh density was varied according to the flow gradient associated with the geometry. Thus, the mesh was dense closer to the walls, followed by a higher density near the rear portion of the hull shown in Fig. 29. Furthermore, to efficiently snap the mesh and capture flow variations pertaining to the curve corresponding to the intersection of the two lobes, the mesh was refined in the region close to this intersection using a spherical refinement volume shown in Fig. 30. The inflation layer shown in Fig. 31 was generated close to the walls to capture flow variations related to the boundary layer of the simulated geometry. This layer was adjusted such that the ${{\rm{y}}^ + }$ was less than 5. Based on these refinements the cell for this particular study ranged between 18 million to 35 million cells with a higher mesh count being used for the hull with fins.

Figure 29. Variable mesh density.

Figure 30. Mesh refinement at rear portion.

Figure 31. Inflation layer close to the model.

During the simulations, the inlet, outlet and far-field boundaries were assigned the free-stream boundary conditions, whereas the wall was assigned the no-slip boundary condition. After generating the mesh and defining the boundary conditions, steady-state solutions pertaining to this study were acquired using the simpleFOAM solver while making use of the two-equation SST k- $\omega $ turbulence model for carrying out Reynolds-Averaged Navier-Stokes evaluations. Further details related to the solver setup can be acquired from a previous study carried out using a similar solver setup in [Reference Gupta, Tripathi and Pant36]. The solver was deemed to have converged upon acquiring a stable, normalised residual of five orders of magnitude.

Preliminary conclusions regarding the aerodynamic effect of fin placement on the scaled-down models of the tri-lobed models of different hull profiles were drawn out by comparing the drag coefficients for these variants at zero degrees angle-of-attack.

4.5 Results and discussions

For the given envelope volume of ${V_{env}} = 26,476{m^3}$ , the geometry data are calculated and the values are given in Table 10. The variable f represents the distance between the lobes in the lateral direction.

The values of ${C_{DV}}$ obtained from CFD simulations for bare hull models and models with fins are shown in Fig. 32.

Table 10. Geometry data of five different models

Figure 32. Comparison of volumetric drag coefficient.

The outcomes of the CFD simulation for bare hull models and models with fins are given in Table 11.

Table 11. Volumetric drag coefficient

To investigate the effect of fins on the aerodynamics of multi-lobed airships, numerical simulations were carried out for three different models of the tri-lobed airship with and without fins designed for the same envelope volume. It is interesting to note that there is a difference of $20 - 30\% $ in the volumetric drag coefficient between the bare hull and hull with fins. This study on fin sizing serves as a baseline for the initial sizing of multi-lobed hybrid airships for various applications.