Adjoint-based aerodynamic shape optimization including transition to turbulence effects

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Abstract

The inclusion of transition to turbulence effects in aerodynamic shape optimization makes it possible to use it as a tool for the design of airframes with laminar flow. Modified Reynolds-Averaged Navier–Stokes (RANS) models that consider transition to turbulence have gained traction in the computational fluid dynamics (CFD) community. These models enable the computation of transitional flows without the need for external modules. In this work, we use a smooth version of the amplification factor transport (AFT) model, called AFT-S, to perform gradient-based aerodynamic shape optimization (ASO) of airfoils in subsonic and transonic flow conditions. We investigate the benefits of including transition effects into the optimization process and assess the impact of losing laminar flow when early transition to turbulence occurs due to surface contamination. Our results indicate that our design optimization approach yields lower drag airfoils when transition effects are considered. For the transonic case, the optimizer trades between shock wave strength and laminar flow extension to minimize drag.

Introduction

The inclusion of transition to turbulence effects in the aerodynamic design process is currently a major research area in the aerospace industry. The design of laminar flow airframes relies on the ability to predict transition to turbulence. It is expected that the use of laminar flow technologies, combined with turbulence and separation control techniques, will lead to a 15% total drag reduction for typical jetliners at cruise conditions [1].

Nonetheless, the importance of including transition to turbulence effects in daily computational fluid dynamics (CFD) simulations extends beyond laminar flow technology. There is evidence that considerable regions of laminar flow are present over the wings of a prototypical jet airplane configuration not designed to achieve laminar flow [2], [3]. This was observed for different flight conditions, with mean aerodynamic chord-based Reynolds numbers from 3 to 20 million and Mach numbers from 0.5 to 0.74. Therefore, including transition effects in CFD studies increases agreement with experimental data [2], [4].

The inclusion of transition effects into CFD simulations also benefits the analysis of high-lift configurations. In high-lift devices, each element experiences reduced Reynolds numbers, so that regions of laminar and transitional flow appear [5], [6]. Therefore, it is necessary to include a transition prediction capability to any computational tool used in high-lift design [4]. Computational results that include transition prediction in high-lift analysis through the use of an eN method have demonstrated the importance of transitional effects in computational analysis of high-lift surfaces [7], [8]. Results in these papers indicated that an improved prediction capability, with good agreement with experimental data, is obtained when transition is included. More recently, numerical results showed that including transition to turbulence in the analysis of 3-D high-lift configurations improves correlation with experimental data when compared to fully-turbulent simulations [4].

Reynolds–Averaged Navier–Stokes (RANS) turbulence models, which are commonly used in engineering applications, are the result of a Favre time-averaging of the original Navier–Stokes equations. Modeling of transition to turbulence is performed through the inclusion of additional transport equations, generally supported by empirical correlations. Langtry and Menter [9], [10], [11], [12] proposed a transition model where two additional transport equations are used to estimate transition onset and region extent. Transition onset is triggered by the momentum thickness Reynolds number (Reθ) transport equation, and the intermittency (γ) transport equation is used to estimate the extent of the transition region. This model is coupled to the shear stress transport (SST) turbulence model [13].

A RANS transition model based on linear stability theory was proposed by Coder and Maughmer [14]. They proposed a transport equation for the approximate N-factor envelope, n˜, based on the work of Drela and Giles [15], with the Spalart–Allmaras [16] (SA) turbulence model as the base for the new two-equation transition model, called amplification factor transport (AFT). The AFT model was then augmented with an additional transport equation for the modified intermittency, γ˜, to improve its robustness when complex flows are considered [17]. The AFT model uses the notion of a local boundary layer shape factor, HL, which is then mapped to the real shape factor, H12. When compared with the original Langtry–Menter transition model, the AFT model has three transport equations, compared to four transport equations for the former. The AFT model was proposed as a tool to study external aerodynamic flows. However, the Langtry–Menter model was originally developed to investigate transitional flows in turbomachinery, where the turbulence levels are higher than those typically found in wind tunnels or in free flight.

At present, complex aircraft configurations are optimized considering high-fidelity, RANS-based CFD calculations. As an example, Lyu et al. [18] were able to perform a lift-constrained drag minimization of the Common Research Model (CRM) wing with a RANS turbulence model. The corresponding wing-body-tail configuration was also optimized [19]. An aerostructural optimization has also been performed by Kenway and Martins [20], and a boundary layer ingestion propulsion system was investigated using adjoint-based optimization [21]. When performing gradient-based aerodynamic shape optimization (ASO), adjoint methods are the most efficient, since their computational cost is mostly independent of the number of design variables.

Adjoint methods were first used within the optimal control community, with the works of Lions [22] and Bryson and Ho [23]. Adjoint methods were then used to solve structural optimization problems [24], [25]. The use of the adjoint method in fluid mechanics was first introduced by Pironneau [26], who derived the adjoints of the Stokes equations and of the Euler equations [27]. In 1988, Jameson [28] extended the method to inviscid compressible flows, making it suitable for transonic airfoil design. The adjoint method was then applied to the Navier–Stokes equations by Jameson et al. [29] and by Nielsen and Anderson [30].

The inclusion of laminar turbulent transition in ASO computations is desirable because it enables the exploitation of laminar flow for drag minimization. However, the literature results considering high-fidelity aerodynamic shape optimization with transition to turbulence are sparse [31]. Dodbele [32] proposed an optimization method to design axisymmetric bodies with high transition Reynolds numbers in subsonic, compressible flow. The proposed approach used the Granville transition criterion [33] to compute the gradients of the objective function and an eN method to compute the functional value at the end of each design iteration. The use of the Granville transition criterion to compute the gradients was motivated by its reduced computational cost when compared to the eN method. Green et al. [34] proposed an approach to design airfoils with a large laminar flow region while satisfying geometric and aerodynamic constraints. Their approach is based on the prescription of a pressure distribution that stabilizes flow instabilities, and stability analysis was used to predict the transition location. Kroo and Sturdza [35] used a design-oriented aerodynamic analysis to perform direct optimization of supersonic wings employing laminar flow. They coupled a boundary layer solver with an Euler solver and a numerical optimization tool, and designed a wing with a minimum total drag at Mach 1.6.

Amoignon et al. [36] used a boundary layer solver coupled to a transition tool based on the parabolized stability equations (PSE) to delay transition based on an Euler solver. In their framework, the pressure coefficient distribution from an Euler solver is used as a boundary condition for a boundary layer solver. The boundary layer solver then provides the PSE code the boundary layer velocity profiles needed for the stability computation. In the reverse direction, their adjoint PSE code feeds the adjoint boundary layer code, which finally provides inputs to the adjoint Euler solver. They delayed transition by minimizing a measure of the disturbance kinetic energy of a chosen disturbance, which is computed using the PSE.

Driver and Zingg [37] used MSES [38], which is an Euler code augmented with boundary layer corrections, to predict transition location using an eN approach. The predicted transition front was then used in a RANS solver that uses a Newton–Krylov discrete-adjoint optimization algorithm. Their algorithm was then used to design airfoils with maximum lift-to-drag ratio, endurance factor, and lift coefficient. In that work, the transition prediction module was loosely coupled to the RANS solver.

Lee and Jameson [39], [40] coupled a transition module based on two eN-database methods considering Tollmien–Schlichting (TS) waves and crossflow vortices with a flow solver to predict and prescribe transition locations automatically. This study showed that natural laminar flow optimized airfoil and wings not only presented improvements in the single design point, but also performed well in off-design conditions. Their flow solver used the Baldwin–Lomax turbulence model [41], an algebraic, 0-equation model. They used a computational approach that creates turbulent patches starting at the transition location specified by the transition module. Because of this, the blending between laminar and turbulent regions was not smooth. Additionally, the gradient computations used in their work did not consider the transition prediction itself.

Khayatzadeh and Nadarajah [42], [43] used the γ-Re˜θ model [9] to perform aerodynamic shape optimization of natural laminar flow airfoils. This model is also commonly referred to as the Langtry–Menter transition model. They applied their framework to the design of low Reynolds NLF airfoils that exhibit separation bubbles. They also showed that, for subsonic flows where viscous drag dominates, turbulent kinetic energy can be an alternative to total drag as an objective function.

Recent research results [44], [45], [46] have used the Langtry–Menter model to perform finite-difference gradient-based aerodynamic optimization, which represents a costly approach. Rashad and Zingg [31] used a 2-D RANS solver coupled to a simplified eN method or to the Arnal–Habiballah–Delcourt criterion [47] in a discrete adjoint capability to perform airfoil natural laminar flow optimization. A simplified eN method was used to perform adjoint-based airfoil aerodynamic shape optimization with transition effects in a recent work [48].

In this work, we use a smooth variant of the original AFT model, referred to as AFT-S, to perform aerodynamic shape optimization. We propose a RANS-based, self-contained aerodynamic shape optimization framework that is able to consider transition to turbulence effects. This framework is unique in that it does not resort to external modules to include transition in the CFD computations, uses gradient-based optimization in which the transition transport equations are part of the adjoint formulation, and considers the flow stability-based AFT-S model. We perform aerodynamic shape optimization of airfoils with flight conditions ranging from subsonic to transonic regimes. These simulations include fully-turbulent and transitional cases. We demonstrate the benefits of including transition in the optimization process by comparing the results. We investigate the effects of early transition caused by flow contamination by running the natural laminar flow airfoils in fully-turbulent mode and show that airfoils optimized while considering transition exhibit similar performance to airfoils optimized without considering transition when the flow conditions make it impossible to achieve significant laminar flow.

The rest of the paper is organized as follows. In Sec. 2, we introduce the AFT model. We present our aerodynamic shape optimization framework in Sec. 3 Aerodynamic shape optimization results for transitional and turbulent flows are shown in Sec. 4. We conclude this paper with final remarks in Sec. 5.

Section snippets

Amplification Factor Transport Transition model

The Amplification Factor Transport (AFT) model was first proposed by Coder and Maughmer [14]. The original model was modified so that no isentropic flow assumption was needed and that Galilean invariance was achieved [49]. In a next version, the transport equation for the modified intermittency, γ˜=ln(γ), was introduced to improve robustness when complex flows are considered [50]. A modified intermittency function was used so that the model could be implemented with different solver

Aerodynamic shape optimization framework

Our aerodynamic shape optimization (ASO) framework uses ADflow as the flow solver. ADflow is an open source, in-house developed CFD solver [61]. ADflow has options to solve Euler, laminar Navier–Stokes, and RANS equations in steady, unsteady, and time-spectral modes, with multiblock structured and overset meshes. The inviscid fluxes are discretized by using three different numerical schemes: the scalar Jameson–Schmidt–Turkel [62] (JST) artificial dissipation scheme, a matrix dissipation scheme

Aerodynamic shape optimization with transition to turbulence effects

Our objective is to reduce drag rather than delaying transition, even though these are correlated objectives in subsonic flow. For subsonic flows with minor separation regions, drag is dominated by skin friction effects. Therefore, transition delay leads to drag reduction. In transonic cases, the wave drag is relevant and should be considered in the optimization process [45]. This suggests a compromise between the extent of the laminar region and shock wave strength, since extended laminar

Conclusions

Reynolds-Averaged Navier–Stokes (RANS) models, which are currently used to account for turbulence effects in the majority of CFD simulations across the industry, assume a fully-turbulent flow field. By doing so, any information on the laminar and transitional flows that precede the fully-turbulent state is lost in the modeling process. This drawback results from the Favre averaging procedure used to derive the RANS equations, which filters out all flow stability modes that, once amplified, may

Declaration of Competing Interest

The authors certify that no conflict of interest is present in the present manuscript.

Acknowledgements

The authors gratefully acknowledge the support provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, Brazil, under Research Grant No. 205552/2014-5. The first author thanks Dr. Ping He, Dr. Charles Mader, and Dr. Eirikur Jonsson for the insightful discussions on adjoint methods and optimization. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by The National Science Foundation grant number ACI-1548562.

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