Effective adjoint approaches for computational fluid dynamics

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.

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

Review of design optimization methods for turbomachinery aerodynamics

The adjoint method for blade design optimization will be described in this two-part paper. The main objective is to develop the capability of carrying out aerodynamic blading shape design optimization in a multistage turbomachinery environment. To this end, an adjoint mixing-plane treatment has been proposed. In the first part, the numerical elements pertinent to the present approach will be described. Attention is paid to the exactly opposite propagation of the adjoint characteristics against the physical flow characteristics, providing a simple and consistent guidance in the adjoint method development and applications. The adjoint mixing-plane treatment is formulated to have the two fundamental features of its counterpart in the physical flow domain: conservation and nonreflectiveness across the interface. The adjoint solver is verified by comparing gradient results with a direct finite difference method and through a 2D inverse design. The adjoint mixing-plane treatment is verified by comparing gradient results against those by the finite difference method for a 2D compressor stage. The redesign of the 2D compressor stage further demonstrates the validity of the adjoint mixing-plane treatment and the benefit of using it in a multi-bladerow environment. DOI: 10.1115/1.3072498

Adjoint Aerodynamic Design Optimization for Blades in Multistage Turbomachines—Part I: Methodology and Verification

The adjoint method is an efficient approach for computing derivatives that allow gradient-based optimization to handle systems parameterized with a large number of design variables. Despite this ad...

DAFoam: An Open-Source Adjoint Framework for Multidisciplinary Design Optimization with OpenFOAM

This is the second part of a two-part paper. First, the design-optimization system basedon the adjoint gradient solution approach as described in Part I is introduced. Severaltest cases are studied for further validation and demonstration of the methodology andimplementation. The base-line adjoint method as applied to realistic 3D conﬁgurations isdemonstrated in the redesign of the NASA rotor 67 at a near-choke condition, leading toa 1.77% efﬁciency gain. The proposed adjoint mixing plane is applied to the redesign ofa transonic compressor stage (DLR compressor stage) and an IGV-rotor-stator conﬁgu-ration of a Siemens industrial compressor at a single-operating point, both producingmeasurably positive efﬁciency gains. An examination on the choice of the operating massﬂow condition as the basis for the performance optimization, however, highlights thelimitation of the single-point approach for practical applications. For the three-rowcompressor conﬁguration, a near peak-efﬁciency point based redesign leads to a mea-surable reduction in the choke mass ﬂow, while a near-choke point based redesign leadsto a signiﬁcant performance drop in other ﬂow conditions. Subsequently, a parallelmultipoint approach is implemented. The results show that a two-point design optimiza-tion can produce a consistently better performance over a whole range of mass ﬂowconditions compared with the original design. In the ﬁnal case, the effectiveness of thepresent method and system is demonstrated by a redesign applied to a seven-row indus-trial compressor at the design point, leading to a remarkable 2.4% efﬁciency gain.

Adjoint Aerodynamic Design Optimization for Blades in Multistage Turbomachines—Part II: Validation and Application

Solution Stabilization and Convergence Acceleration for the Harmonic Balance Equation System

This paper presents efficient alternative numerical methods for an implicit solution of the harmonic balance equation system for analysing temporal periodic unsteady flows. The proposed method employs approximate factorisation to decouple the common residual term and the time spectral source term of a harmonic balance equation system when it is discretised implicitly. With this approximate factorisation, the complexity of implicit solution of the discrete system is greatly reduced. The common residual term can be dealt with using a lower-upper symmetric-Gauss-Seidel (LU-SGS) method and the time spectral source term is integrated using a Jacobi iteration (JI) or one step Gauss-Seidel (GS) iteration, leading to the LU-SGS/JI method or LU-SGS/GS method. The NASA stage 35 compressor and the 1.5 stage Aachen turbine were used to demonstrate the effectiveness of the proposed methods in stabilising solution and its advantages in comparison with the existing lower-upper symmetric-Gauss-Seidel/block Jacobi...

Implicit solution of harmonic balance equation system using the LU-SGS method and one-step Jacobi/Gauss-Seidel iteration

Presented in this paper is an improved method for dealing with a mixing plane that exists between computational domains of two adjacent blade rows of a multiple-blade-row turbomachine. The method makes use of the semidiscrete flow equation updating scheme to convert flux differences across an interrow interface to conservative flow variable incrementals then to characteristic variable perturbations. Therefore, the proposed method bears more physics and is much more robust than any known method of its kind. As a result, reverse flow can be accommodated by this method naturally without any special treatment. Two existing methods are also included to provide a clear illustration of the differences and advantages of the new method. Two numerical test cases using a transonic compressor stage are presented to investigate the robustness of the new method and its influence on solution convergence, accuracy, and time cost in comparison with the two existing methods. [DOI: 10.1115/1.4026170]

An Improved Mixing-Plane Method for Analyzing Steady Flow Through Multiple-Blade-Row Turbomachines

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