Adjoint shape optimization coupled with LES-adapted RANS of a U-bend duct for pressure loss reduction
Introduction
Product developments are nowadays assisted by computer software to speed-up the design process. Problems involving fluid dynamics can be simulated numerically, avoiding expensive experimental campaigns when many different design configurations need to be analyzed. This enables the use of computer aided optimization algorithms to search autonomously for design improvements. Usually the relationship between shape and its effect on performance is rather complex and beyond the capability of human designers to be accurately predicted, which is why design processes driven by human instincts are mainly characterised by a trial and error process. On the other hand, numerical shape optimization algorithms, especially when guided by gradient information, allow to obtain performance improvements in a few cycles adjusting the design. As such, optimization routines have gained sufficient maturity to be deployed in an industrial context in the design process [1], [2], reducing significantly the design time and the number of experimental tests needed.
The accuracy of the optimization result largely depends on the numerical model used to simulate the flow. Reynolds Averaged Navier-Stokes (RANS) simulations are commonly used both in academia and industry due to the acceptable numerical cost [3], [4], [5]. This allows to explore a wide range of different configurations within the context of a design activity [1]. The prediction capability of RANS is however limited: highly three dimensional flows with secondary motions, flows with strong pressure gradients on the boundary layer, or flows with strong separations for instance are not well predicted by the steady state RANS approach and may guide the optimization algorithm to an erroneous optimal solution. More accuracy is possible by means of a Large Eddy Simulation (LES) approach, where large unsteady turbulent structures are resolved. However, due to larger spatial discretization requirements and because the simulation is time-resolved, the computational cost increases significantly compared to RANS. The role LES has within the design process was therefore initially quasi-similar to an experimental validation, and mainly performed in a final stage. But as computational power keeps increasing, LES within an optimization framework is becoming possible. As an example, Marsden et al. applied a gradient-free optimization in conjunction with LES to reduce the noise generated by turbulent flow over a hydrofoil trailing edge [6]. Goit and Meyer used LES with the aim of increasing the total energy extraction in wind farms [7]. Collis and Chang considered LES for determining the optimal control of a turbulent flow for drag reduction problems [8].
Two main strategies exist when dealing with optimization problems. A first strategy requires only the evaluation of the objective function and does not need the computation of its gradient. Such optimization strategies are mainly non-deterministic and require in general a large amount of simulations to be performed to reach an optimum, while the degrees of freedom given to the design are rather limited. A second strategy is based on the calculation of the gradient of the objective function. This strategy gives to the optimizer a direct hint on how the design needs to be changed to reach a certain improvement. This leads to a more efficient algorithm, especially when the gradient is computed using the adjoint approach, with a cost nearly independent of the number of control variables. For this reason the application of gradient based optimization is very attractive within a LES context.
The adjoint method had been originally developed by Lions [9] in the context of control theory. The first application in the aerodynamics field goes back to Pironneau [10], who studied the energy dissipation due to a small hump on a body in a uniform steady flow with the aim to obtain the optimal condition for different drag minimization problems. Jameson introduced the technique in the aeronautical field [11] developing the adjoint counterpart of the Euler equations. The work of Jameson aimed to show how the design problem could be tackled through control theory, in which in particular the control is the shape of the boundary. The design problem considered was the search for an airfoil profile which gives a prescribed pressure distribution. The adjoint method continued to grow in the aeronautical industry framework bringing its use for: complete aircraft optimization [12], problems considering the compressible Navier-Stokes equations [13], aero-structural optimization [14], [15]. The examples illustrated above are related to the aeronautical field but applications can be found in many other different domains, such as turbomachinery [16], [17], automotive [18], [19], [20], energy [7], naval [21] and thermal exchange [22], [23]. In the context of shape optimization, the coupling with deformation techniques [24], [25] is necessary after the evaluation of the direction of improvement.
The application of gradient based optimization techniques using LES is however not widespread and it is so far mainly limited to control problems [7], [8] with short time-averaging windows. The reason is that the adjoint method becomes unstable and diverges due to the ”butterfly effect” [26], [27]. Due to the chaotic nature of the flow, small perturbations in flow quantities, whether they come from numerical round-off or from a perturbed mesh, will increase exponentially in time such that after a certain time the perturbed flow will be totally different from the unperturbed one. The objective function for an optimization is typically a time averaged flow quantity, and hence, for a limited averaging period, it may lead to a completely different objective function value for a slightly perturbed shape. This remains so for the linearized model and thus the sensitivities computed through the adjoint method become unreliable.
The use of LES within a gradient based optimization approach hence remains an active field of research. Many solutions have been proposed, of which the Least Square Shadowing Technique [28], [29] is the most promising. This technique makes use of the shadowing lemma [30], [31], which states that for each numerical trajectory in a chaotic dynamic system, a true trajectory can be found which is very close to the former one with slightly different initial conditions, even though the numerical trajectory would diverge exponentially from the true one if the same initial condition would be used. Applying this technique to LES, it implies that, when the grid is perturbed, a flow solution can be found near to the original one if a slightly different initial flow field would be used. As a consequence, the derivative of the objective function calculated is meaningful and can be used to drive the solution to its optimum. The Least Square Shadowing technique requires thus to find the initial flow field such that it shadows the original flow field time-evolution. This is an optimization problem with a very large set of degrees of freedom, which would significantly increase the overall computational cost. Some methods to reduce the cost of the optimization of the initial field have been proposed [32], [33], [34]. Nonetheless, to date the Least Square Shadowing Technique can only be applied to very small numerical domains, far below the current interest.
For this reason, another approach is chosen in the present work. Two observations lay at the basis of the proposed method: 1) RANS models fail to predict the time-averaged LES flow field mainly because of an inapt turbulence model, and 2) the gradients can in many cases be approximatively obtained using a frozen turbulence approach, which considers the turbulent viscosity invariant under small shape perturbations [35], [36]. The novel approach presented here includes an initial step where a suitable RANS model is sought to fit the time-averaged flow field of the LES simulation. Hereto, the turbulent viscosity of each cell in the RANS simulation is adapted to yield a similar flow field as the LES result. This step requires an optimization and is similar to the work reported in Hayek et al. [37], in which a discrete adjoint formulation is presented. Once the RANS model is established, the adjoint method is used to compute the surface sensitivities under the assumption of constant turbulent viscosity. An optimizer allows then to update the shape, after which the full procedure is repeated. A new LES calculation is thus performed, followed by a RANS simulation adapting the turbulent viscosity, the computation of the adjoint field and of the surface sensitivities and, finally, the shape modification. The procedure is repeated until no significant improvement is further obtained. The proposed strategy allows to combine the reliability of a LES evaluation to a low computational cost optimization routine through the link with a RANS approach.
The method is applied to the optimization of a U-bend, being a prototype test case for which RANS is proven to be insufficiently reliable. The U-bend test case is described in Verstraete et al. and Coletti et al. [38], [39] and optimized to reduce the pressure drop which arises from the significant flow turning [38]. A large reduction in pressure drop was achieved by a classical RANS-based optimization algorithm through shape modifications that suppressed the flow separation in the bend. However, a recirculation bubble, not detected by the RANS evaluation, is still present inside the design as revealed by the experimental validation through Particle Image Velocimetry (PIV) [39]. On the other hand, simulations using LES could replicate with high accuracy the experimental findings [40]. To further improve the U-bend design, a LES based approach is thus inevitable.
The paper is structured as follows: first the U-bend test case will be described in more detail. A following section focuses on the LES-based optimization approach used. The subsequent section then applies the approach to the U-bend and finally some conclusions are drawn.
Section snippets
The U-bend test case
Gas turbines are cooled by internal flows with air which is usually bled from the compressor, leading to a loss in thermodynamic efficiency. The design of the internal cooling system can be improved by reducing the internal losses and thus reducing the amount of work needed from the compressor. The U-bends which connect the different coolant passages are significant contributors to the pressure losses inside the system [41]. As a consequence, they have been the subject of many optimization
Towards LES-based optimization
The aim of the present work is to apply a LES-based optimization starting from the already optimized geometry of Verstraete et al. [38]. As the separation bubble has already been significantly reduced, only small changes are expected and therefore local gradient based optimization methods are more favourable. Despite the small design variations expected, many degrees of freedom need to be given to the shape to allow local adaptation for avoiding separation, which additionally favours gradient
Optimization results of the LES-based optimization
The previously outlined algorithms define an optimization strategy within a LES context. The methodology is applied to the U-bend test case optimized in Verstraete et al. [38] and aims to reach further improvements by a more accurate evaluation of the flow field.
Conclusions
The use of the RANS approach to evaluate the flow field characteristics in an optimization framework could lead to inaccurate results. A comparison between the mean velocity field obtained with RANS and LES and the experimental data available in literature shows that the RANS approach fails to correctly evaluate the flow features inside a U-bend geometry. A validation study in an optimized geometry highlights the presence of a recirculation region inside the bend both in the experiments and in
CRediT authorship contribution statement
G. Alessi: Conceptualization, Software, Validation, Formal analysis, Writing – original draft. T. Verstraete: Methodology, Validation, Resources, Writing – review & editing, Supervision. L. Koloszar: Software, Validation, Writing – review & editing, Supervision. B. Blocken: Validation, Writing – review & editing, Supervision. J.P.A.J. van Beeck: Validation, Writing – review & editing, Supervision.
Declaration of Competing Interest
None.
Acknowledgement
This work was supported by the Fonds Wetenschappelijk Onderzoek Vlaanderen - FWO (SB Fellowship 1S 581 16N).
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