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Zanetti, G.; Siviero, M.; Cavazzini, G.; Santolin, A. Development and Implementation of Inverse Design Method. Encyclopedia. Available online: https://encyclopedia.pub/entry/46689 (accessed on 07 August 2024).

Zanetti G, Siviero M, Cavazzini G, Santolin A. Development and Implementation of Inverse Design Method. Encyclopedia. Available at: https://encyclopedia.pub/entry/46689. Accessed August 07, 2024.

Zanetti, Giacomo, Monica Siviero, Giovanna Cavazzini, Alberto Santolin. "Development and Implementation of Inverse Design Method" *Encyclopedia*, https://encyclopedia.pub/entry/46689 (accessed August 07, 2024).

Zanetti, G., Siviero, M., Cavazzini, G., & Santolin, A. (2023, July 12). Development and Implementation of Inverse Design Method. In *Encyclopedia*. https://encyclopedia.pub/entry/46689

Zanetti, Giacomo, et al. "Development and Implementation of Inverse Design Method." *Encyclopedia*. Web. 12 July, 2023.

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The increasingly stringent requirements in terms of flexibility and efficiency for hydraulic turbines pose new challenges for designers. The inverse three-dimensional design strategy has recently demonstrated its effectiveness in improving the hydraulic machines performance hence representing a valid alternative to the traditional design method.

3D inverse design method
blade loading
Francis turbines
pump turbines

Hydropower represents the largest and historically most developed renewable energy source and is expected to play a key role in the ongoing energy transition ^{[1]}. With a share between 16 and 17% of the entire world’s energy production ^{[2]} and 53% of the whole renewable electricity generation ^{[3]}, hydropower will support the energy transition by providing both low-cost renewable electricity and large-scale energy storage to support grid stability, which is threatened by increasing electricity generation from intermittent and unpredictable energy sources such as wind and solar.

Despite the variety of hydraulic turbines that can be fitted in modern hydropower plants, the Francis type represents the most spread configuration due to its wide operating range in terms of both water discharge and head. Moreover, reversible pump turbines (RPTs) today are generally preferred over other solutions in the new generation of storage hydropower plants ^{[4]} due to their compactness and cost efficiency. However, hydropower and pumped hydropower plants are required to operate over wider operating ranges and with frequent start and stops, forcing the machines to operate far from their design conditions. Taking these considerations into account, it is easy to understand the new challenges faced in the design of Francis turbines and pump turbines.

The hydraulic design of Francis runners is typically represented by an iterative approach requiring considerable engineering input. Indeed, despite the rise in modern optimization techniques, mainly based on computational fluid dynamics (CFD), know-how still plays a fundamental role in the design process. The traditional design methodologies adopted in industry relies on the definition of the runner blade geometry in terms of the blade angle distributions both in the streamwise and spanwise directions, so as to meet the performance requirements. The process continues by iterating the runner geometry until a satisfactory compromise between the design objectives is achieved, after which expensive experimental verifications are undertaken ^{[5]}. In this scenario, it is clear that even a modest improvement in the design methodology can result in greater competitiveness and faster project development.

In the inverse design method (IDM) for hydraulic machinery, a series of system parameters are set to achieve the desired system behavior, e.g., secondary flow suppression or runner efficiency. In contrast to the traditional design approach, where geometrical parameters are treated as design variables, the IDM focuses on hydrodynamic parameters, guaranteeing more direct control over the machine hydrodynamics. Instead of defining a blade angle distribution, the designer specifies the runner blade loading distribution, hence indirectly defining the pressure distribution on the blades. This relationship between blade loading and pressure distribution represents the strength of the IDM method, since it results in a more physically intuitive method for reaching the desired flow field rather than defining the blade angles. Once the loading distribution is defined, the angle distribution of the runner blade is evaluated in order to define the runner geometry. Consequently, the numerical effort necessary to reach a satisfactory result via CFD analyses may be drastically reduced.
## 2. Development and Implementation of Inverse Design Method

The turbomachinery field is dominated by viscous, turbulent, three-dimensional, unsteady flow. Recent developments, both in experimental and computational techniques, have provided a better understanding of the three-dimensional flow fields in many types of turbomachinery. However, using these results to optimize the blade geometry still represents a challenge for turbomachinery designers. In fact, the current design practice consists of incremental changes to existing geometries, the impact of which on the flow field is evaluated using analysis methods, without the possibility of controlling—even in an approximate way—the machine fluid dynamics in the design phase. It is hence difficult for a designer to incorporate their knowledge of fluid dynamics directly into the optimization process, since there are no clear correlations between the blade shape and the fluid flow field.

As the name implies, the direct design approach involves the direct control of the geometric parameters defining the three-dimensional geometry of the blades. However, even a slight change in one parameter at one location can greatly affect the flow field upstream and/or downstream. This difficulty is further amplified by the large number of geometric variables defining the three-dimensional blade geometry, whose simultaneous impact on the flow field cannot be controlled during the design phase. Therefore, the direct design approach represents a time-consuming method mainly based on designer experience, due to the lack of a direct relationship between the geometric control parameters and the internal flow characteristics.

In contrast, the inverse design method (IDM) uses the fluid dynamic parameters of the blade load distribution as input data. Furthermore, the influence of other geometric design parameters such as the meridional shape and number of blades can be analyzed more independently for the same blade loading pattern.

The advantages of such an approach are evident. The IDM has several important features compared to conventional design methods ^{[6]}^{[7]}, such as being controlled by hydrodynamic parameters. Moreover, the IDM can handle 3D flow effects, allowing the designers to gain a good understanding of the secondary flow development. Thanks to such an approach, the costs associated with optimization processes can be reduced.

Although traditional and inverse design approaches generally share the same goals, such as increasing hydraulic efficiency, there are clear differences in the final blade geometries. The blade angle distribution of a conventionally designed impeller is generally quite uniform, whereas blades designed using the IDM generally exhibit greater complexity, with significant blade angle variations in both the flow and span directions. However, before entering into the theoretical aspects of the method, it is interesting to briefly present an account of its development and implementation in the fluid machine field.

Originally developed for the hydraulic design of hydrofoils, the first computational implementations of the IDM for hydraulic pumps are dated between the 1980s and 1990s. These first codes used IDMs in a general design method developed for two-dimensional incompressible potential flows.

However, the first appearance of IDMs for impeller design dates back to World War II, when the group of scientists led by Werner von Braun used a similar approach for the design of the V-1 and V-2 rockets ^{[8]}. Since the 1950s, several two-dimensional and later quasi-three-dimensional inverse design methods have been developed thanks to various researchers ^{[9]}^{[10]}. They were able to further improve this method and significantly increase the design quality using the approach established by Wu ^{[11]} in 1952 to calculate the flows combination on relative stream surfaces upstream or midway in blade rows.

The application of the IDM for turbomachinery design was first studied in 1984 by Hawthorne et al. ^{[12]} and Tan et al. ^{[13]}. These researchers initially based their studies on two-dimensional flow calculations assuming a non-viscous and incompressible flow. However, because of the strong simplifications introduced into the fluid flow evaluation, the contribution of these first applications in the blade design was limited: the blade geometry was determined imposing the flow conditions at the machine walls without considering the influence of the boundary layer or the blade thickness ^{[13]}.

Thanks to the contribution of several researchers, such as Borges ^{[14]}^{[15]} and Zangeneh et al. ^{[16]}^{[17]}^{[18]}, the IDM then evolved into a three-dimensional method based on the potential flow theory.

In particular, in 1990, Borges ^{[14]} proposed a design strategy based on setting the distribution of the mean swirl over the runner meridional section. This approach was applied in the design of a slow-speed radial turbine and validated in an experimental campaign ^{[15]}. The IDM approach allowed to increase the hydraulic efficiency by about 1.4% in comparison with the conventional design strategy. Starting from these promising results, in 1991, Borges ^{[19]} presented a through-flow (hub-to-shroud) inverse method for the design of a rotor of a mixed flow pump. The fluid was considered inviscid, incompressible, and irrotational at the inlet and the blade thickness was neglected. Borges was able to obtain reasonable pressure distributions on the blade surfaces with a small amount of CPU time.

In 1996, Zangeneh et al. ^{[16]} adopted a new implementation of the three-dimensional compressible inverse design method to study the design of a radial and mixed flow turbomachine. Their method considered as input the distribution of the circumferentially averaged swirl velocity 𝑟𝐶𝑢

on the meridional geometry of the impeller, while the corresponding blade geometry was then interactively calculated. Here, two different approaches were proposed for the solution of the flow field: In the first approach, named approximate, the variation in the density in the pitch was neglected, thus resulting in a simplified and time-saving algorithm. In the second approach, named exact, the velocities and densities in the entire three-dimensional flow field were evaluated using a fast Fourier transform over the tangential direction. These two approaches were applied on the case of a high-speed radial turbine (subsonic), resulting in two blade geometries with almost negligible differences. The flows resulting from the two geometries were computed using a three-dimensional inviscid Euler model. Both showed a good correlation between the estimated and resulting distributions of 𝑟𝐶𝑢
and a negligible fluid field variation, confirming the effectiveness of the method, which was applied in several application cases ranging from the field of compressible and non-compressible fluid turbomachines (operating machines and prime movers).

Zangeneh et al. successfully applied a derivative of the mean swirl distribution to suppress secondary flows in a mixing pump impeller ^{[20]} and a compressor diffuser ^{[21]}. In addition, Goto et al. ^{[22]} investigated a different approach to redesign pump diffuser vanes with the aim of suppressing flow separation. Based on this last model, Zangeneh et al. ^{[23]} attempted to design a centrifugal compressor with diffuser blades.

In 1998, Demeulenaere et al. ^{[24]} developed an IDM that incorporated the specified pressure distribution instead of an average vortex distribution to design compressor and turbine blades using the Euler model for three-dimensional inviscid flows. De Vito et al. ^{[25]} combined the method in ^{[24]} with a direct two-dimensional Navier–Stokes method in an iterative scheme to redesign a turbine blade.

Dang et al. developed a particular IDM by using the Euler model for two-dimensional cascades ^{[26]} and then for fully three-dimensional geometries ^{[27]} by employing a specified pressure distribution and thickness distribution.

The IDM was then extended for other different blade types by several researchers such as Jiang et al. ^{[28]}. Damle et al. ^{[29]} used the same approach to increase the efficiency of a first stage rotor of a centrifugal compressor. Then, Daneshkhah et al. ^{[30]} developed a two-dimensional inverse Navier–Stokes method using a given pressure and thickness distribution to redesign a subsonic turbine and a transonic compressor. In addition, Wang et al. ^{[31]} presented a 3D inverse method based on the Navier–Stokes equations using a prescribed pressure distribution for three-dimensional cascades.

The benefits derived by the direct imposition of the blade load and/or of the static pressure distribution have been repeatedly proven over the years.

As an example, in the optimization proposed by Tiow et al. ^{[32]} for the high-speed NASA rotos 67, the blade load distribution was found to be strictly related to both the location and intensity of the shock, hence demonstrating its effectiveness as a design parameter. Other interesting applications can be found for several specific applications such as pumps ^{[33]}^{[34]}^{[35]}, turbines, pump turbines, compressors ^{[21]}^{[23]}^{[36]}, diffusers ^{[22]}^{[37]}^{[38]}, and inducers ^{[39]}^{[40]}.

Despite the fact that in such examples the flow evaluation was based on potential flow theory without viscosity, the fluid viscous effect can be introduced by considering further parameters related to the viscosity of the fluid, such as the clogging distribution, the vorticity, or the entropy gradient ^{[41]}.

However, it is important to emphasize how the IDM is based on different approaches to the flow field calculation. Specifically, the IDM can be classified into four different categories: the “inviscid” category, as described in the references by Pàscoa et al. ^{[42]} and also by Zangeneh et al. ^{[43]}; the “viscous” category ^{[30]}^{[44]}^{[45]}; the “compressible” category, analyzed in particular by Zangeneh et al. ^{[16]}; and the “incompressible” category, studied in the initial phase by researchers such as Hawthorne et al., Tan et al., and Borges ^{[12]}^{[13]}^{[14]}^{[15]}^{[19]}.

Despite the intense evolution witnessed here from the first two-dimensional implementations with incompressible flows to the most recent three-dimensional applications with viscous and/or compressible flows, the concept behind the IDM remains the same. Furthermore, all applications currently do not take into account the unsteady behavior of the flow and thus do not directly address the important aspect of the instability of turbomachinery.

Finally, in 2020, Leguizamón and Avellan ^{[5]} proposed an open-source implementation of the IDM method specifically developed for Francis hydraulic turbines, representing the first non-proprietary software available to the public.

Nowadays, the notable development of computational fluid dynamics (CFD) has made it possible to evaluate the fluid dynamics in turbomachinery in ever greater detail at an ever lower computational cost. In this scenario, the combination of the IDM and CFD would represent a powerful tool for turbomachinery development and optimization. However, since a fully three-dimensional turbulence flow calculation approach cannot be easily directly introduced into the IDM, the trial-and-error design process of turbomachinery still represents the only one viable solution.

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