面向叶轮机气动形状精细设计的伴随方法及其应用研究
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摘要
精细设计能够进一步提升叶片气动性能和负荷能力。但三维流场分析结合优化算法寻优的叶轮机气动形状设计方法在设计参数多时存在计算量大的问题。隶属于梯度法的伴随方法因计算量与设计参数数目基本无关,近年来在叶轮机气动形状设计领域受到重视。本论文对基于伴随方法的多级叶轮机气动形状设计方法进行研究,以期缩短设计周期,并为三维叶片精细设计提供技术支撑。包括五部分内容,具体如下：
1.一般敏感性模型
比较了伴随方法与传统梯度法计算敏感性原理的异同。结果表明伴随方法的计算量相当于求解两倍的流动控制方程,具有较低的计算成本。进一步利用伴随方法经过代数运算推导了目标函数与设计变量的一般敏感性模型,获得伴随方程、伴随边界条件、敏感性的一般表达式。
2.叶轮机气动形状设计方法
给出了正／反问题目标函数；由罚函数约束压比和流量以保持伴随方法低计算成本优势；选取不存在叶片重构问题的Hicks-Henne峰值函数作为叶片形状扰动参数化方法。在不影响附面层预测合理性的情况下,选用计算量较少的薄层简化粘性体力、壁面函数经验关系式和流动滑移固壁边界等细化一般敏感性模型。进而推导了封闭伴随方程求解所需的伴随排间掺混界面方法,实现多级叶轮机敏感性分析。结合最速下降法寻优,获得了基于伴随方法的多级叶轮机气动形状设计一体化方法,可用于多级叶轮机叶片排气动匹配设计。
3.叶轮机气动形状设计软件
给定三维叶片形状扰动参数化方案,采用代数方法生成简单H型网格。利用时间推进法和有限体积法求解薄层简化N-S方程和伴随方程；根据特征速度传播方向给定边界条件；采用局部时间步长、多重网格和二阶／四阶光顺等方法加速数值求解收敛速度。借助复变函数方法,由流场和伴随场后处理计算敏感性值。结合最速下降法寻优,并由形状扰动参数化方法更新叶片几何。最终开发了多级叶轮机三维粘性气动形状设计软件,包含三个子求解器,分别求解薄层简化N-S方程、伴随方程和敏感性。
4.模型、方法及软件验证
首先,以NASA35号跨音压气机级考核流场求解程序。其次,以涡轮环形叶栅和1+1/2级轴流压气机两个算例,选取反问题目标函数,分别由伴随方法和有限差分方法计算敏感性值。最后,再次以涡轮环形叶栅和1+1/2级轴流压气机两个算例实施反问题优化设计。结果表明,流场求解程序能够捕捉关键流动特征,且主要气动性能参数与试验值符合良好；伴随方法和有限差分法的敏感性值误差不超过10%,满足工程精度需求；反问题中优化后的叶型与目标叶型基本重合,且二者压力分布一致,由此验证气动形状设计方法及软件的有效性。
5.叶轮机气动形状精细设计应用探索
在给定压比和流量下,叶轮机性能追求高气动效率。以进、出口质量平均熵增为目标函数,以压比和流量作为约束,初步尝试叶轮机三维叶片精细设计。算例包括NASA67号风扇转子叶片、NASA35号跨音压气机级、1+1/2级轴流压气机、5+1/2级跨音压气机、1+1对转涡轮前三排叶片。优化后绝热效率分别提高1.38、1.41、0.91、1.30和0.42个百分点,流量和压比大致在容许范围内。分析性能提升的机制,跨/超音流动中通过减小峰值马赫数、降低激波强度以及削弱流动分离等提高气动性能；亚音流动中通过改善攻角分布等减小流动损失。
最后,总结论文工作,并对以伴随方法为核心的叶轮机气动形状设计发展趋势进行展望。
Detail design can further improve blade aerodynamic loading. But turbomachin-ery aerodynamic shape design based on optimization algorithms and three-dimensional flow analysis is always too time-consuming. Manipulating hun-dreds of design variables to implement blade detail design with low computing cost is desirable for routine design. Differing with traditional gradient methods, the compu-ting cost of adjoint method is regardless of the number of design variables. In recent years, it has received considerable attentions in turbomachinery community. This pa-per presents investigations on multi-stage turbomachinery aerodynamic shape design using adjoint method, aiming to reduce design cycle and implement blade detail de-sign. The main contents with five parts are as follows:
1. A general sensitivity-analysis model
Compare the sensitivity-analysis principle of adjoint method and traditional gra-dient methods. Results show that the computing cost of adjoint method is low and roughly equivalent to solve two sets of flow equations. Further, derive a general sen-sitivity-analysis model of objective function to design variables based on adjoint method. The model includes adjoint equations, adjoint boundary conditions, and a general sensitivity expression.
2. Turbomachinery aerodynamic shape design approach
The objective functions of direct and inverse problems are specified. The pres-sure-ratio and mass-flow-rate constraints are introduced as penalty functions to re-main the low computing cost of adjoint method.3D blade perturbation parameteriza-tion based on Hicks-Henne hump functions is used to fit blade profile perturbation without blade geometry reconstruction. The thin shear-layer viscous body force and wall function with flow slip boundary condition can predict the wall boundary layer properly with low computing cost, and are adopted to detail the general sensitivity model. Further, an adjoint mixing-plane treatment is proposed to close the solution of adjoint equations, and thus sensitivity analysis is achieved in multi-stage tur- bomachinery conveniently. Integrated with the simple steepest method, a frame of an adjoint-based multi-stage turbomachinery aerodynamic shape design approach is con-structed and can be used for stage-matching design.
3. Turbomachinery aerodynamic shape design software
Scheme of3D blade perturbation parameterization is given and simple H-type grid is generated using algebraic method. The numerical simulation of thin shear-layer N-S equations and adjoint equations is based on time-marching method and finite volume method. The way to implement boundary conditions depends on prorogation directions of characteristic information. Local time step method, multi-grid method, and blended second and fourth order numerical dissipation method are used to accel-erate the solving process. Sensitivity can be calculated as a post-process of flow field and adjoint field using complex method. And quantitative modifications of blade pro-files are searched by the simple steepest method and blade perturbation parameteriza-tion. Thus, turbomachinery aerodynamic shape design software is developed, includ-ing three main sub-solvers:flow solver, adjoint solver and sensitivity calculator.
4. Validation of the model, approach and software
Firstly, check flow-field prediction of NASA35transonic compressor stage us-ing flow solver against test data. Results show that it seizes key flow features and main aerodynamic parameters are agreed well. Secondly, calculate the value of sensi-tivity using adjoint method and finite difference method with the objective function of inverse problem for turbine annular cascade and1+1/2stage axial compressor. The error of the two method don't exceed10%. Thirdly, again implement inverse optimi-zation design of the two cases to show the capability of the approach and software. The optimized blade profiles and pressure distributions agree well with the target, and thus the approach and software are validated.
5. Applications to turbomachinery aerodynamic shape detail design
To increase aerodynamic efficiency and keep mass flow rate and stagnation pressure ratio unchanged, turbomachinery aerodynamic shape detail design aiming to minimize inlet and outlet mass-averaged entropy production rate and subjecting mass flow rate and stagnation pressure ratio constraints is investigated from lessons of NASA67rotor, NASA35transonic compressor stage,1+1/2stage transonic com-pressor,5+1/2stage transonic compressor and the first three rows of1+1stage coun-ter-rotating turbine. The optimised blades increases their adiabatic efficiency by1.38%、1.41%、0.91%、1.30%and0.42%points respectively while the two constraints are satisfied. The performance improvements are realized by minimizing peak Mach number, shock intensity and flow separation for transonic/supersonic flows, and fine tuning incidence angle distribution to reduce flow loss for subsonic flows.
At last, this paper concludes the overall work and posts some prospects of tur-bomachinery aerodynamic shape design using adjoint method.
1.一般敏感性模型
比较了伴随方法与传统梯度法计算敏感性原理的异同。结果表明伴随方法的计算量相当于求解两倍的流动控制方程,具有较低的计算成本。进一步利用伴随方法经过代数运算推导了目标函数与设计变量的一般敏感性模型,获得伴随方程、伴随边界条件、敏感性的一般表达式。
2.叶轮机气动形状设计方法
给出了正／反问题目标函数；由罚函数约束压比和流量以保持伴随方法低计算成本优势；选取不存在叶片重构问题的Hicks-Henne峰值函数作为叶片形状扰动参数化方法。在不影响附面层预测合理性的情况下,选用计算量较少的薄层简化粘性体力、壁面函数经验关系式和流动滑移固壁边界等细化一般敏感性模型。进而推导了封闭伴随方程求解所需的伴随排间掺混界面方法,实现多级叶轮机敏感性分析。结合最速下降法寻优,获得了基于伴随方法的多级叶轮机气动形状设计一体化方法,可用于多级叶轮机叶片排气动匹配设计。
3.叶轮机气动形状设计软件
给定三维叶片形状扰动参数化方案,采用代数方法生成简单H型网格。利用时间推进法和有限体积法求解薄层简化N-S方程和伴随方程；根据特征速度传播方向给定边界条件；采用局部时间步长、多重网格和二阶／四阶光顺等方法加速数值求解收敛速度。借助复变函数方法,由流场和伴随场后处理计算敏感性值。结合最速下降法寻优,并由形状扰动参数化方法更新叶片几何。最终开发了多级叶轮机三维粘性气动形状设计软件,包含三个子求解器,分别求解薄层简化N-S方程、伴随方程和敏感性。
4.模型、方法及软件验证
首先,以NASA35号跨音压气机级考核流场求解程序。其次,以涡轮环形叶栅和1+1/2级轴流压气机两个算例,选取反问题目标函数,分别由伴随方法和有限差分方法计算敏感性值。最后,再次以涡轮环形叶栅和1+1/2级轴流压气机两个算例实施反问题优化设计。结果表明,流场求解程序能够捕捉关键流动特征,且主要气动性能参数与试验值符合良好；伴随方法和有限差分法的敏感性值误差不超过10%,满足工程精度需求；反问题中优化后的叶型与目标叶型基本重合,且二者压力分布一致,由此验证气动形状设计方法及软件的有效性。
5.叶轮机气动形状精细设计应用探索
在给定压比和流量下,叶轮机性能追求高气动效率。以进、出口质量平均熵增为目标函数,以压比和流量作为约束,初步尝试叶轮机三维叶片精细设计。算例包括NASA67号风扇转子叶片、NASA35号跨音压气机级、1+1/2级轴流压气机、5+1/2级跨音压气机、1+1对转涡轮前三排叶片。优化后绝热效率分别提高1.38、1.41、0.91、1.30和0.42个百分点,流量和压比大致在容许范围内。分析性能提升的机制,跨/超音流动中通过减小峰值马赫数、降低激波强度以及削弱流动分离等提高气动性能；亚音流动中通过改善攻角分布等减小流动损失。
最后,总结论文工作,并对以伴随方法为核心的叶轮机气动形状设计发展趋势进行展望。
Detail design can further improve blade aerodynamic loading. But turbomachin-ery aerodynamic shape design based on optimization algorithms and three-dimensional flow analysis is always too time-consuming. Manipulating hun-dreds of design variables to implement blade detail design with low computing cost is desirable for routine design. Differing with traditional gradient methods, the compu-ting cost of adjoint method is regardless of the number of design variables. In recent years, it has received considerable attentions in turbomachinery community. This pa-per presents investigations on multi-stage turbomachinery aerodynamic shape design using adjoint method, aiming to reduce design cycle and implement blade detail de-sign. The main contents with five parts are as follows:
1. A general sensitivity-analysis model
Compare the sensitivity-analysis principle of adjoint method and traditional gra-dient methods. Results show that the computing cost of adjoint method is low and roughly equivalent to solve two sets of flow equations. Further, derive a general sen-sitivity-analysis model of objective function to design variables based on adjoint method. The model includes adjoint equations, adjoint boundary conditions, and a general sensitivity expression.
2. Turbomachinery aerodynamic shape design approach
The objective functions of direct and inverse problems are specified. The pres-sure-ratio and mass-flow-rate constraints are introduced as penalty functions to re-main the low computing cost of adjoint method.3D blade perturbation parameteriza-tion based on Hicks-Henne hump functions is used to fit blade profile perturbation without blade geometry reconstruction. The thin shear-layer viscous body force and wall function with flow slip boundary condition can predict the wall boundary layer properly with low computing cost, and are adopted to detail the general sensitivity model. Further, an adjoint mixing-plane treatment is proposed to close the solution of adjoint equations, and thus sensitivity analysis is achieved in multi-stage tur- bomachinery conveniently. Integrated with the simple steepest method, a frame of an adjoint-based multi-stage turbomachinery aerodynamic shape design approach is con-structed and can be used for stage-matching design.
3. Turbomachinery aerodynamic shape design software
Scheme of3D blade perturbation parameterization is given and simple H-type grid is generated using algebraic method. The numerical simulation of thin shear-layer N-S equations and adjoint equations is based on time-marching method and finite volume method. The way to implement boundary conditions depends on prorogation directions of characteristic information. Local time step method, multi-grid method, and blended second and fourth order numerical dissipation method are used to accel-erate the solving process. Sensitivity can be calculated as a post-process of flow field and adjoint field using complex method. And quantitative modifications of blade pro-files are searched by the simple steepest method and blade perturbation parameteriza-tion. Thus, turbomachinery aerodynamic shape design software is developed, includ-ing three main sub-solvers:flow solver, adjoint solver and sensitivity calculator.
4. Validation of the model, approach and software
Firstly, check flow-field prediction of NASA35transonic compressor stage us-ing flow solver against test data. Results show that it seizes key flow features and main aerodynamic parameters are agreed well. Secondly, calculate the value of sensi-tivity using adjoint method and finite difference method with the objective function of inverse problem for turbine annular cascade and1+1/2stage axial compressor. The error of the two method don't exceed10%. Thirdly, again implement inverse optimi-zation design of the two cases to show the capability of the approach and software. The optimized blade profiles and pressure distributions agree well with the target, and thus the approach and software are validated.
5. Applications to turbomachinery aerodynamic shape detail design
To increase aerodynamic efficiency and keep mass flow rate and stagnation pressure ratio unchanged, turbomachinery aerodynamic shape detail design aiming to minimize inlet and outlet mass-averaged entropy production rate and subjecting mass flow rate and stagnation pressure ratio constraints is investigated from lessons of NASA67rotor, NASA35transonic compressor stage,1+1/2stage transonic com-pressor,5+1/2stage transonic compressor and the first three rows of1+1stage coun-ter-rotating turbine. The optimised blades increases their adiabatic efficiency by1.38%、1.41%、0.91%、1.30%and0.42%points respectively while the two constraints are satisfied. The performance improvements are realized by minimizing peak Mach number, shock intensity and flow separation for transonic/supersonic flows, and fine tuning incidence angle distribution to reduce flow loss for subsonic flows.
At last, this paper concludes the overall work and posts some prospects of tur-bomachinery aerodynamic shape design using adjoint method.
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