A Hybrid Adjoint Network Model for Thermoacoustic Optimization

2021 ◽  
Author(s):  
Felicitas Schaefer ◽  
Luca Magri ◽  
Wolfgang Polifke
Keyword(s):  
Network Model ◽  
Hybrid Approach ◽  
Adjoint System ◽  
Scattering Matrix ◽  
Adjoint Equations ◽  
Jump Conditions ◽  
Discrete Adjoint ◽  
Direct Scattering ◽  
Annular Combustor ◽  
Continuous Adjoint

Abstract A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

2021 ◽  
Author(s):  
Fellcitas Schäfer ◽  
Luca Magri ◽  
Wolfgang Polifke
Keyword(s):  
Network Model ◽  
Hybrid Approach ◽  
Adjoint System ◽  
Scattering Matrix ◽  
Adjoint Equations ◽  
Jump Conditions ◽  
Discrete Adjoint ◽  
Direct Scattering ◽  
Annular Combustor ◽  
Continuous Adjoint

Abstract A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved whilst maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

Adjoint-Based Optimization Method With Linearized SST Turbulence Model and a Frozen Gamma-Theta Transition Model Approach for Turbomachinery Design

2015 ◽  
Author(s):  
Pengfei Zhang ◽  
Juan Lu ◽  
Zhiduo Wang ◽  
Liming Song ◽  
Zhenping Feng
Keyword(s):  
Turbulence Model ◽  
Entropy Generation ◽  
Adjoint System ◽  
Optimization Method ◽  
Adjoint Equations ◽  
Transition Model ◽  
Transition Flow ◽  
Flow Optimization ◽  
Sst Turbulence Model ◽  
Continuous Adjoint

In this paper, based on the grid node coordinates variation and Jacobian Matrices, the turbulent continuous adjoint method with linearized turbulence model is studied and developed to fully account for the variation of turbulent eddy viscosity. The corresponding adjoint equations, boundary conditions and the final sensitivities are formulated with a general expression. To implement the adjoint optimization of the transition flow, a flow solver combining the transition model with the turbulence model is employed, and an adjoint optimization framework with linearized SST turbulence model and a frozen Gamma-Theta transition model is established. In order to choose an appropriate objective for the transition flow optimization, four objectives are studied, including the entropy generation, the total pressure loss coefficient, the field integral of turbulent kinetic energy, the area ratio of transition and turbulent regions to the suciton side. And finally the entropy generation is adopted as the objective. Then, the derivation of the adjoint system for the entropy generation optimization is presented. To demonstrate the validity of the adjoint system for transition flow, four shape optimizations for the bypass transitions and the separation-induced transition are implemented. A 2D isentropic case for bypass transitions is conducted to compares the performances of the fully turbulent adjoint system and the frozen Gamma-Theta transition adjoint system, while the other isothermal case is performed to take the aerodynamic and heat transfer issues into account together. The case of separation-induced transition is performed and also consistent well with its flow mechanism. The four optimization results illustrate the effectiveness of the adjoint system for the transition flow optimization, which can improves the performance of overall cascades and the transition region.

Inverse Problem for Isentropic Mach-Number on Blade Wall in Aerodynamic Shape Design of Turbomachinery Cascades by Using Adjoint Method

2011 ◽  
Author(s):  
Haitao Li ◽  
Liming Song ◽  
Pengfei Zhang ◽  
Zhenping Feng
Keyword(s):  
Inverse Problem ◽  
Mach Number ◽  
Objective Function ◽  
Adjoint System ◽  
Boundary Integral ◽  
Adjoint Equations ◽  
Grid Node ◽  
Time Step ◽  
Flow Quantity ◽  
Continuous Adjoint

This paper presents an approach of the continuous adjoint system deduction based on the variation in grid node coordinates, in which the variation in the gradient of flow quantity is converted into the gradient of the variation in flow quantity and the gradient of the variation in grid node coordinates, which avoids the coordinate system transforming in the traditional derivation process of adjoint system and make the adjoint system much more sententious. By introducing the Jacobian matrix of viscous flux to the gradient of flow variables, the adjoint system for turbomachinery aerodynamic design optimization governed by compressible Navier-Stokes equations is derived in details. Given the general expression of objective functions consisted of both boundary integral and field integral, the adjoint equations and their boundary conditions are derived, and the final expression of the objective function gradient including only boundary integrals is formulated to reduce the CPU cost. Then the adjoint system is numerically solved by using the finite volume method with an explicit 5-step Runge-Kutta scheme and Riemann approximate solution of Roe’s scheme combined with multi-grid technique and local time step to accelerate the convergence procedure. Finally, the application of the method is illustrated through a turbine cascade inverse problem with an objective function of isentropic Mach number distribution on the blade wall.

Discrete Adjoint Solution of Unsteady Turbulent Flow in Compressor

2015 ◽  
Author(s):  
Can Ma ◽  
Xinrong Su ◽  
Xin Yuan
Keyword(s):  
Unsteady Flow ◽  
Stokes Equations ◽  
Flow Simulation ◽  
Adjoint Equations ◽  
Navier Stokes ◽  
Aerodynamic Optimization ◽  
Flow Equations ◽  
Discrete Adjoint ◽  
Continuous Adjoint ◽  
Unsteady Flow Simulation

Unsteady blade row interactions play an important role in the performance of the compressor stages. However, due to the large cost of the unsteady flow simulation, most aerodynamic optimizations of the compressor are based on the steady flow simulation. This paper adopts the time spectral method to reduce the cost of the unsteady flow simulation and a discrete adjoint solver based on the unsteady flow solver has been developed. The unsteady flow equations and the adjoint equations are solved using an in-house code. The in-house code is based on the finite volume method and solves the URANS (Unsteady Reynolds Averaged Navier-Stokes) equations on the multi-block structured mesh. For spatial discretization the 3rd order WENO (Weighted Essentially Nonoscillatory) upwind scheme is used for reconstruction and the convective flux is computed with Roe’s approximate Riemann solver. The widely used one-equation Spalart-Allmaras turbulence model is adopted for the flow simulation. For the adjoint solution, the constant-eddy viscosity assumption is adopted and only the main flow adjoint equations are solved. The adjoint equations are formed in a discrete manner, which leads to more accurate discrete objective function sensitivity than the continuous adjoint method. The present work serves as an essential part of the system for efficient unsteady aerodynamic optimization of turbomachinery.

S2 Surface Quasi-3D Aerodynamic Design Using the Continuous Adjoint Method for Multi-Stage Turbine

2014 ◽  
Author(s):  
Lei Chen ◽  
Jiang Chen
Keyword(s):  
Objective Function ◽  
Adjoint Method ◽  
Adjoint Equations ◽  
Design System ◽  
Shape Design ◽  
Aerodynamic Design ◽  
Aerodynamic Shape ◽  
Multi Stage ◽  
Gradient Based ◽  
Continuous Adjoint

The adjoint method eliminates the dependence of the gradient of the objective function with respect to design variables on the flow field making the obtainment of the gradient both accurate and fast. For this reason, the adjoint method has become the focus of attention in recent years. This paper develops a continuous adjoint formulation for through-flow aerodynamic shape design in a multi-stage gas turbine environment based on a S2 surface quasi-3D problem governed by the Euler equations with source terms. Given the general expression of the objective function calculated via a boundary integral, the adjoint equations and their boundary conditions are derived in detail by introducing adjoint variable vectors. As a result, the final expression of the objective function gradient only includes the terms pertinent to those physical shape variations that are calculated by metric variations. The adjoint system is solved numerically by a finite-difference method with explicit Euler time-marching scheme and a Jameson spatial scheme which employs first and third order dissipative flux. Integrating the blade stagger angles and passage perturbation parameterization with the simple steepest decent method, a gradient-based aerodynamic shape design system is constructed. Finally, the application of the adjoint method is validated through a 5-stage turbine blade and passage optimization with an objective function of entropy generation. The result demonstrates that the gradient-based system can be used for turbine aerodynamic design.

scholarly journals Derivation of the Adjoint Drift Flux Equations for Multiphase Flow

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 31 ◽  
Author(s):  
Shenan Grossberg ◽  
Daniel S. Jarman ◽  
Gavin R. Tabor
Keyword(s):  
Multiphase Flow ◽  
Objective Function ◽  
Settling Velocity ◽  
Flow Problem ◽  
Adjoint System ◽  
Drift Flux ◽  
Adjoint Approach ◽  
Continuous Adjoint ◽  
First Time ◽  
Mathematical Derivation

The continuous adjoint approach is a technique for calculating the sensitivity of a flow to changes in input parameters, most commonly changes of geometry. Here we present for the first time the mathematical derivation of the adjoint system for multiphase flow modeled by the commonly used drift flux equations, together with the adjoint boundary conditions necessary to solve a generic multiphase flow problem. The objective function is defined for such a system, and specific examples derived for commonly used settling velocity formulations such as the Takacs and Dahl models. We also discuss the use of these equations for a complete optimisation process.

Study and Verification of Continuous Adjoint System for Aerodynamic Optimization in Turbomachinery Cascades

2013 ◽  
Author(s):  
Pengfei Zhang ◽  
Haitao Li ◽  
Zhenping Feng
Keyword(s):  
Adjoint Method ◽  
Adjoint System ◽  
Inviscid Flow ◽  
Laminar Flows ◽  
Aerodynamic Optimization ◽  
Adjoint Systems ◽  
Inverse Designs ◽  
Turbine Design ◽  
2D And 3D ◽  
Continuous Adjoint

This paper is a further study of the authors’ previous work on the continuous adjoint method based on the variation in grid node coordinates and Jacobi Matrices of the flow fluxes. This method simplifies the derivation and expression of the adjoint system, and reduces the computation cost. In this paper, the differences between the presented and the traditional methods are analyzed in details by comparing the derivation processes and the adjoint systems. In order to demonstrate the reliability and accuracy of the adjoint system deduced by the authors, the presented method is applied to different optimal problems, which include two inverse designs and two shape optimizations in both 2D and 3D cascades. The inverse designs are implemented by giving the isentropic Mach number distributions along the blade wall for 2D inviscid flow and 3D laminar flow. The shape optimizations are implemented with the objective function of the entropy generation in flow passage for 2D and 3D laminar flows. In the 3D optimal case, this method is validated by supersonic turbine design case with and without mass flow rate constraint. The numerical results testify the accuracy of this adjoint method, which includes only the boundary integrals, and furthermore, the universality and portability of this adjoint system for inverse designs and shape optimizations are demonstrated.

A continuous adjoint-based aeroacoustic shape optimization for multi-mode duct acoustics

2017 ◽  
Vol 232 (21) ◽  
pp. 3897-3914 ◽  
Author(s):  
Sheng Qiu
Keyword(s):  
Optimization Method ◽  
Adjoint Equations ◽  
Duct Acoustics ◽  
Design Variables ◽  
Adjoint Variables ◽  
Duct Geometry ◽  
The Cost ◽  
Continuous Adjoint ◽  
Multi Mode ◽  
Inlet Duct

A multi-mode adjoint-based optimization method is proposed for the noise reduction optimization in multi-mode duct acoustics problems. The objective is to minimize the amplitude of sound from an inlet duct on the wall and integral line while maintaining the aerodynamic performance. The complete detailed derivation of the adjoint equations and their corresponding adjoint boundary conditions are presented firstly based on the multi-mode linear Euler equations. With the solved adjoint variables, the final expression of the cost function gradient with respect to the design variables is formulated. The sensitivity derivative computed by the continuous adjoint method is validated by comparing with that obtained using finite difference method. Up to 50 design variables are involved in the adjoint optimization to ensurely provide an adequate design space. And a quasi-Newton Broyden–Fletcher–Goldfarb–Shanno algorithm is utilized to determine an improved intake duct geometry based on the objective function gradient provided by the adjoint solution. Finally, two multi-mode optimization of a typical inlet duct confirms the flexibility of the multi-mode adjoint-based framework and the efficiency of the multi-mode adjoint-based technique.

Concurrent Blade Aerodynamic-Aero-elastic Design Optimization Using Adjoint Method

2010 ◽  
Vol 133 (1) ◽  
Author(s):  
L. He ◽  
D. X. Wang
Keyword(s):  
Unsteady Flow ◽  
Solution Method ◽  
Stokes Equations ◽  
Adjoint System ◽  
Elastic Stability ◽  
Adjoint Equations ◽  
Navier Stokes ◽  
Simple Extension ◽  
Mean Flow ◽  
Elastic Design

Increasing aerothermal and aero-elastic performance requirements and constraints are closely linked in modern blading designs. There is thus a need for more concurrent interaction between the disciplines at earlier stages of a design process. Presented in this paper are the development, validation, and demonstration of the adjoint approach to concurrent blading aerodynamic and aero-elastic design optimizations. A nonlinear harmonic phase solution method is adopted to solve the unsteady Reynolds-averaged Navier–Stokes equations. The flow field response in terms of both the mean aerothermal performance and aero-elastic stability to a geometrical perturbation can be obtained by three “steadylike” flow solutions at three distinctive temporal phases. This unsteady flow solution method is computationally very efficient and provides a convenient and consistent base for formulating the corresponding adjoint equations. The adjoint system for the unsteady flow solver is solved effectively by a relatively simple extension of the method and techniques previously developed for a steady flow adjoint solver. As a result, the sensitivities of both the steady (time-mean) flow loss and the aerodynamic damping/forcing to detailed blade geometry changes can be very efficiently obtained by solving equivalently three steadylike adjoint equations. Several case studies are presented to illustrate the validity and effectiveness of this new concurrent approach.

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