Continuous Adjoint Methods for Turbulent Flows,Applied to Shape and Topology Optimization: Industrial Applications

This article focuses on the formulation, validation and application of the continuous adjoint method for turbulent flows in aero/hydrodynamic optimization. Though discrete adjoint has been extensively used in the past to compute objective function gradients with respect to (w.r.t.) the design variables under turbulent flow conditions, the development of the continuous adjoint variant for these flows is not widespread in the literature, hindering, to an extend, the computation of exact sensitivity derivatives. The article initially presents a general formulation of the continuous adjoint method for incompressible flows, under the commonly used assumption of “frozen turbulence”. Then, the necessary addenda are presented in order to deal with the differentiation of both low- and high-Reynolds (with wall functions) number turbulence models; the latter requires the introduction of the so-called “adjoint wall functions”. An approach to dealing with distance variations is also presented. The developed methods are initially validated in \(2D\) cases and then applied to industrial shape and topology optimization problems, originating from the automotive and hydraulic turbomachinery industries.     

A zonal RANS-LES method to determine the flow over a high-lift configuration

High Reynolds number flows are particularly challenging problems for large-eddy simulations (LES) since small-scale structures in thin and often transitional boundary layers are to be resolved. The range of the turbulent scales is enormous, especially when high-lift configuration flows are considered. For this reason, the prediction of high Reynolds number flow over the entire airfoil using LES requires huge computer resources. To remedy this problem a zonal RANS-LES method for the flow over an airfoil in high-lift configuration at Re_c=1.0×10⁶ is presented. In a first step, a 2D RANS solution is sought, from which boundary conditions are formulated for an embedded LES domain, which comprises the flap and a sub-part of the main airfoil. The turbulent fluctuations in the boundary layers at the inflow region of the LES domain are generated by controlled forcing terms, which use the turbulent shear stress profiles obtained from the RANS solution. The comparison with an LES solution for the full domain and with experimental data shows likewise results for the velocity profiles and wall pressure distributions. The zonal RANS-LES method reduces the computational effort of a full domain LES by approx. 50%.     

Optimization of the Rigid Body Mechanism in a Wobble Plate Compressor

A complex mechanical system is optimized with respect to its performance. The mechanism is a compressor, which is modeled as a multibody system. The optimization is first performed on a simplified 2D model, where it is possible to find analytical sensitivities, and the results indicate that the mechanism can be optimized. Optimization is finally performed with numerical sensitivities, from a full 3D mechanism simulation with 20 bodies, and the results show that the desired change of performance is obtained. For the optimization procedure the SLP method (sequential linear programming) is used with good results, and although the paper deals with optimization of a specific mechanism, the procedure can be modified to treat also other mechanical systems.     

Optimization of resource allocation for underlay device-to-device communications in cellular networks

Underlay device-to-device (D2D) communication in cellular networks has been considered as a promising technique that can improve the spectral efficiency of cellular systems and meet the growing demand for wireless local services. In underlay D2D, it is of primary importance to manage the mutual interference between cellular links and D2D links through effective resource allocation. While most of previous works on D2D resource allocation are developed based on the knowledge of the channel state information (CSI) on the interference channels as well as the desired channels, it is hard to obtain full CSI in practice. Accordingly, we consider D2D resource allocation schemes based on distance between nodes. In particular, we formulate two optimization problems for D2D resource allocation using the outage probability computed based on the distance information as cost functions. One is a linear sum assignment problem (LSAP) and the other is a linear bottleneck assignment problem (LBAP). By applying the graph theory, we provide efficient algorithms for solving the optimization problems. Numerical results are provided to show the effectiveness of the proposed optimization as compared to previously proposed distance-based resource allocation algorithms.     

Numerical simulation of phase separation and a priori two-phase LES filtering

This work reports on the potential application of Large Eddy Simulation (LES) in the calculation of turbulent isothermal two-phase flows, in the case where the large scales of each phase are resolved and small interface structures can be smaller than the mesh size. In comparison with single phase flows, application of LES to two-phase flow problems should account for the complex interaction between the interface and the turbulent motion. The complete filtered two-phase flow equations are formulated to deal with turbulence at the interface. Explicit filtering of 3D direct numerical simulations of a phase separation problem has been employed to evaluate the order of magnitude of the specific subgrid contributions. Analyses of the numerical results have been conducted to derive conclusions on the relative importance of the different subgrid scale contributions. Modeling issues and turbulent energy transfer across the interface are discussed.     

Level set topology optimization of cooling channels using the Darcy flow model

Structural and Multidisciplinary Optimization - The level set topology optimization method for 2D and 3D cooling channels, considering convective heat transfer for high Reynolds number flows, is...     

A numerical strategy to combine high-order schemes, complex geometry and parallel computing for high resolution DNS of fractal generated turbulence

Impressive advances in parallel platform architectures over the past decade have made Direct Numerical Simulation (DNS) a powerful tool which can provide full access to the spatial structure of turbulent flows with complex geometries. An innovative approach which combines high-order schemes and a dual domain decomposition method is presented in this paper and is applied to DNS of multiscale-generated turbulent flows by a fractal grid. These DNS illustrate the applicability of our approach to the simulation of complex turbulent flows and provide results which are compared with recent laboratory experiments thus providing new insights for the interpretation of the experimental measurements.     

Domain decomposition for near-wall turbulent flows

Modeling near-wall high-Reynolds-number turbulent flows is a time-consuming problem. A domain decomposition approach is developed to overcome the problem. The original computational domain is split into a near-wall (inner) subdomain and an outer subdomain. The developed approach is applied to a model 2D equation simulating major peculiarities of near-wall high-Reynolds-number flows. On the base of the Calderon–Ryaben’kii potential theory it is possible to consider the near-wall (inner) problem independently on the outer problem. The influence of the inner problem can exactly be represented by a pseudo-differential equation formulated on the intermediate boundary. In a 1D case, it leads to the wall functions represented by Robin boundary conditions, which can be determined either analytically or numerically. It is important that the wall functions (or boundary conditions) are mesh independent and can be realized in a separate routine. Thus, the original problem can only be solved in the outer domain with some specific nonlocal boundary conditions called nonlocal wall functions. The technique can be extended to 3D problems straightforward.     

Numerical analysis of the acoustic field of reacting flows via acoustic perturbation equations

The acoustic radiation of a turbulent non-premixed flame using a hybrid method is numerically simulated. The two-step method consists of an incompressible large-eddy simulation (LES) and acoustic perturbation equations (APE), which are reformulated to account for reacting flow effects (APE-RF). In reacting flows, hybrid methods to compute acoustic radiation have some advantages compared to the direct simulation of the acoustic field using a compressible LES. Considering the different characteristic length scales optimized schemes can be applied to each subproblem, i.e., to the hydrodynamic and the acoustic problem. This is of interest since the fluid mechanics is governed by the combustion process and to compute the highly intricate chemistry tailor-made schemes with reasonable computational costs combustion models can be implemented to resolve this phenomenon by, for instance, preprocessed databases for the chemical reactions, like in the steady flamelet approach.The APE-RF system possesses several source terms on the right-hand side (RHS), which are thoroughly discussed to their relation to various sound mechanisms. The acoustic sources describe the impact of unsteady heat release, non-isomolar combustion, species diffusion, heat diffusion, viscous effects, non-uniform mean flow and non-constant combustion pressure effects, and the influence of acceleration of density inhomogeneities. Moreover, an additional source term within the APE-RF pressure-density relation can be identified to describe the local acoustic wave amplification due to acoustic-flame interaction. It is evidenced that the well-known Rayleigh criterion can be directly given by this source.The unsteady heat release is shown to occur in the total time derivative of the density that is directly provided from an LES solution. By analyzing via the two-step method the acoustic field of an open turbulent non-premixed flame being generated just by the total temporal derivative of the density, and by comparing the numerical data with experimental findings the total substantial derivative is shown to describe for a wide frequency range the essential sound propagation caused by reacting flows. Nevertheless, it is also discussed that to simulate all the details over the complete frequency range additional source mechanisms occurring on the RHS of the APE-RF system are to be considered in the investigation.     

Shape design optimization of an expansion step in a channel with moving boundaries for a viscous flow

A shape design optimization problem for viscous flows has been investigated in the present study. An analytical shape design sensitivity expression has been derived for a general integral functional by using the adjoint variable method and the material derivative concept of optimization. A channel flow problem with a backward facing step and adversely moving boundary wall is taken as an example. The shape profile of the expansion step, represented by a fourth-degree polynomial, is optimized in order to minimize the total viscous dissipation in the flow field. Numerical discretizations of the primary (flow) and adjoint problems are achieved by using the Galerkin FEM method. A balancing upwinding technique is also used in the equations. Numerical results are provided in various graphical forms at relatively low Reynolds numbers. It is concluded that the proposed general method of solution for shape design optimization problems is applicable to physical systems described by nonlinear equations.