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Open AccessArticle

Shape Transformation Approaches for Fluid Dynamic Optimization

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Peter Marvin Müller

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Georgios Bletsos

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Thomas Rung

Aerospace 2023, 10(6), 519; https://doi.org/10.3390/aerospace10060519 - 31 May 2023

Viewed by 198

Abstract

The contribution is devoted to combined shape- and mesh-update strategies for parameter-free (CAD-free) shape optimization methods. Three different strategies to translate the shape sensitivities computed by adjoint shape optimization procedures into simultaneous updates of both the shape and the discretized domain are employed [...] Read more.

The contribution is devoted to combined shape- and mesh-update strategies for parameter-free (CAD-free) shape optimization methods. Three different strategies to translate the shape sensitivities computed by adjoint shape optimization procedures into simultaneous updates of both the shape and the discretized domain are employed in combination with a mesh-morphing strategy. Considered methods involve a linear Steklov–Poincaré (Hilbert space) approach, a recently suggested highly non-linear p-Laplace (Banach space) method, and a hybrid variant which updates the shape in Hilbert space. The methods are scrutinized for optimizing the power loss of a two-dimensional bent duct flow using an unstructured, locally refined grid that initially displays favorable grid properties. Optimization results are compared with respect to the optimization convergence, the computational effort, and the preservation of the mesh quality during the optimization sequence. Results indicate that all methods reach, approximately, the same converged optimal solution, which reduces the objective function by about 18% for this classical benchmark example. However, as regards the preservation of the mesh quality, more advanced Banach space methods are advantageous in comparison to Hilbert space methods even when the shape update is performed in Hilbert space to save costs. In specific, while the computational cost of the Banach space method and the hybrid method is about 3.5 and 2.5 times the cost of the pure Hilbert space method, respectively, the grid quality metrics are 2 times and 1.7 times improved for the Banach space and hybrid method, respectively. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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Open AccessArticle

Aerodynamic Optimization Design of Supersonic Wing Based on Discrete Adjoint

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Aerospace 2023, 10(5), 420; https://doi.org/10.3390/aerospace10050420 - 29 Apr 2023

Viewed by 695

Abstract

Reducing fuel consumption and improving the economy by effectively reducing cruising drag is the main objective of the aerodynamic design of supersonic civil aircraft. In this paper, the aerodynamic optimization design system based on the Reynolds-Averaged Navier–Stokes (RANS) equation and discrete adjoint theory [...] Read more.

Reducing fuel consumption and improving the economy by effectively reducing cruising drag is the main objective of the aerodynamic design of supersonic civil aircraft. In this paper, the aerodynamic optimization design system based on the Reynolds-Averaged Navier–Stokes (RANS) equation and discrete adjoint theory is applied to supersonic wing design. Based on this system, a single-point optimization design study of aerodynamic drag reduction in cruise conditions was carried out for two typical supersonic wing layouts, subsonic leading edge and supersonic leading edge, and the drag reduction reached 3.78% and 4.53%, respectively. The aerodynamic design characteristics of different types of supersonic wings were explored from the perspectives of wing load, twist angle distribution, pressure distribution, airfoil shape characteristics, and flow field characteristics. The optimization results show that the drag reduction of the subsonic leading edge configuration is dominated by the induced drag, while the optimizer mainly focuses on reducing the shock wave drag for the supersonic leading edge configuration. By comparing the sensitivity analysis of lift and drag coefficients to airfoil deformation with the optimization results, the optimized dominant directions of two types of supersonic wings are qualitatively analyzed. The derivatives obtained from discrete adjoint equations are useful to elaborate the design tendency and the reason for the trade-off generation of supersonic wings under specific layouts and engineering constraints, which provides a reference for the design of supersonic wings in the future. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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Open AccessArticle

Explaining the Lack of Mesh Convergence of Inviscid Adjoint Solutions near Solid Walls for Subcritical Flows

by

Carlos Lozano

and

Jorge Ponsin

Aerospace 2023, 10(5), 392; https://doi.org/10.3390/aerospace10050392 - 24 Apr 2023

Viewed by 398

Abstract

Numerical solutions to the adjoint Euler equations have been found to diverge with mesh refinement near walls for a variety of flow conditions and geometry configurations. The issue is reviewed, and an explanation is provided by comparing a numerical incompressible adjoint solution with [...] Read more.

Numerical solutions to the adjoint Euler equations have been found to diverge with mesh refinement near walls for a variety of flow conditions and geometry configurations. The issue is reviewed, and an explanation is provided by comparing a numerical incompressible adjoint solution with an analytic adjoint solution, showing that the anomaly observed in numerical computations is caused by a divergence of the analytic solution at the wall. The singularity causing this divergence is of the same type as the well-known singularity along the incoming stagnation streamline, and both originate at the adjoint singularity at the trailing edge. The argument is extended to cover the fully compressible case, in subcritical flow conditions, by presenting an analytic solution that follows the same structure as the incompressible one. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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Open AccessArticle

Adjoint-Based Aerodynamic Design Optimization and Drag Reduction Analysis of a Military Transport Aircraft Afterbody

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Aerospace 2023, 10(4), 331; https://doi.org/10.3390/aerospace10040331 - 27 Mar 2023

Cited by 1 | Viewed by 725

Abstract

Based on the adjoint method, the afterbody of a military transport aircraft was optimized and designed to meet engineering constraints under real flight conditions. Guidance for the key design parameters of the afterbody of the military transport aircraft is given. The vortex dynamics [...] Read more.

Based on the adjoint method, the afterbody of a military transport aircraft was optimized and designed to meet engineering constraints under real flight conditions. Guidance for the key design parameters of the afterbody of the military transport aircraft is given. The vortex dynamics and boundary layer extraction methods were used to analyze the optimization results of military transport aircraft. It was found that, upstream of the vortex shedding point, the circumferential accumulation process of the vorticity is weakened. The position of the vortex shedding and the appearance of the saddle line are delayed by reducing the circumferential inverse pressure gradient and the intensity of the crossflow. The afterbody vortex system of the optimized configuration is further away from the surface. Meanwhile, the distance between the counter-rotated vortex decreases, and the upwashing speed of the vortex core is smaller. Therefore, vortex-induced drag is reduced. Finally, compared with the initial configuration, the optimized configuration has a relative drag reduction of 23.2%. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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Open AccessArticle

Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method

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Aerospace 2023, 10(3), 280; https://doi.org/10.3390/aerospace10030280 - 11 Mar 2023

Viewed by 688

Abstract

Precise and inexpensive uncertainty quantification (UQ) is crucial for robust optimization of compressor blades and to control manufacturing tolerances. This study looks into the suitability of MC−adj−nonlinear, a nonlinear adjoint-based approach, to precisely and rapidly assess the performance discrepancies of a transonic compressor [...] Read more.

Precise and inexpensive uncertainty quantification (UQ) is crucial for robust optimization of compressor blades and to control manufacturing tolerances. This study looks into the suitability of MC−adj−nonlinear, a nonlinear adjoint-based approach, to precisely and rapidly assess the performance discrepancies of a transonic compressor blade section, arising from geometric alterations, and building upon previous research. In order to assess the practicality and illustrate the benefits of the adjoint-based nonlinear approach, its proficiency and precision are gauged against two other methodologies, the adjoint-based linear approach (MC−adj−linear) and the high-fidelity nonlinear Computational Fluid Dynamics (MC−CFD) method. The MC−adj−nonlinear methodology exhibits impressive generalization capabilities. The MC−adj−nonlinear method offers a great balance between precision and time efficiency, since it is more precise than the MC−adj−linear method in both design and near-stall conditions, yet requires approximately a thirtieth of the time of the MC−CFD method. Finally, the MC−adj−nonlinear method was utilized to conduct fast UQ analyses of the section at four distinct speeds to quantify the performance uncertainty for the compressor map. It is found that aerodynamic performance is more sensitive to geometric deviations at high speeds than at low speeds. The impact of the geometric deviations is generally detrimental to the mean efficiency. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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Open AccessArticle

Shock Equations and Jump Conditions for the 2D Adjoint Euler Equations

by

Carlos Lozano

and

Jorge Ponsin

Aerospace 2023, 10(3), 267; https://doi.org/10.3390/aerospace10030267 - 10 Mar 2023

Viewed by 512

Abstract

This paper considers the formulation of the adjoint problem in two dimensions when there are shocks in the flow solution. For typical cost functions, the adjoint variables are continuous at shocks, wherein they have to obey an internal boundary condition, but their derivatives [...] Read more.

This paper considers the formulation of the adjoint problem in two dimensions when there are shocks in the flow solution. For typical cost functions, the adjoint variables are continuous at shocks, wherein they have to obey an internal boundary condition, but their derivatives may be discontinuous. The derivation of the adjoint shock equations is reviewed and detailed predictions for the behavior of the gradients of the adjoint variables at shocks are obtained as jump conditions for the normal adjoint gradients in terms of the tangent gradients. Several numerical computations on a very fine mesh are used to illustrate the behavior of numerical adjoint solutions at shocks. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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Open AccessArticle

Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver

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Aerospace 2023, 10(2), 106; https://doi.org/10.3390/aerospace10020106 - 21 Jan 2023

Cited by 1 | Viewed by 730

Abstract

This paper proposes an approach that combines manual differentiation (MD) and automatic differentiation (AD) to develop an efficient and accurate multi-row discrete adjoint solver. In this approach, the structures of adjoint codes generated using an AD tool are first analyzed. Then, the AD-generated [...] Read more.

This paper proposes an approach that combines manual differentiation (MD) and automatic differentiation (AD) to develop an efficient and accurate multi-row discrete adjoint solver. In this approach, the structures of adjoint codes generated using an AD tool are first analyzed. Then, the AD-generated codes are manually adjusted to reduce memory and CPU time consumption. This manual adjustment is performed by replacing the automatically generated low-efficient differentiated codes with manually developed ones. To demonstrate the effectiveness of the proposed approach, the single-stage transonic compressor–NASA Stage 35 and the 1.5-stage Aachen turbine–are used. The solution information exchange at a rotor-stator/stator-rotor interface is achieved by a conservative, non-reflective, and robust discrete adjoint mixing plane method. The results show that the discrete adjoint solver developed by hybrid automatic and manual differentiation is more economical in computational cost than that developed purely by an AD tool and has higher sensitivity accuracy than the adjoint solver with the constant eddy viscosity (CEV) assumption. Moreover, the multi-row turbomachinery design optimizations can be efficiently performed by the discrete adjoint solver developed by the hybrid automatic and manual differentiation. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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Review

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Open AccessReview

A Review of Solution Stabilization Techniques for RANS CFD Solvers

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Aerospace 2023, 10(3), 230; https://doi.org/10.3390/aerospace10030230 - 26 Feb 2023

Viewed by 888

Abstract

Nonlinear, time-linearized and adjoint Reynolds-averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) solvers are widely used to assess and improve the aerodynamic and aeroelastic performance of aircrafts and turbomachines. While RANS CFD solver technologies are relatively mature for applications at design conditions where the [...] Read more.

Nonlinear, time-linearized and adjoint Reynolds-averaged Navier-Stokes (RANS) computational fluid dynamics (CFD) solvers are widely used to assess and improve the aerodynamic and aeroelastic performance of aircrafts and turbomachines. While RANS CFD solver technologies are relatively mature for applications at design conditions where the flow is benign, their use in off-design conditions, featuring flow instabilities, such as separations and shock wave/boundary layer interactions, still faces many challenges, with tight residual convergence being a major difficulty. To cope with this, several solver stabilization techniques have been proposed. However, a systematic and comparative study of these techniques has not been reported, to some extent hindering the wide deployment of these methods for industrial applications. In this paper, we critically review the existing methods for solver convergence stabilization, with the main purpose of explaining the rationale behind the algorithms and providing a systematic view of the seemingly different methods. Specifically, mathematical formulations and implementation details of these methods, example applications, and the pros and cons of the methods are discussed in detail, along with suggestions for further improvements. This review is expected to give CFD method developers an overview of the various solution stabilization methods and application engineers an idea how to choose a suitable method for their respective applications. Full article

(This article belongs to the Special Issue Adjoint Method for Aerodynamic Design and Other Applications in CFD)

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