The introduction of the "frozen viscosity assumption" can simplify the derivation of the adjoint equation and the differentiation of flow solver subroutines, but can also lead to computational errors of sensitivities and sometimes even solution instability. To investigate the effect of laminar viscosity and eddy viscosity on computational accuracy of sensitivities, this paper presents a study on three different frozen viscosity approaches:the Frozen Laminar Viscosity approach(FLV), Frozen Eddy Viscosity approach(FEV) and Frozen Laminar and Eddy Viscosity approach(FLEV). First, the adjoint equations corresponding to full turbulence and three different frozen viscosity approaches are derived based upon the nonlinear flow equations and the objective function in an algebraic form. Then, we introduce how to use the algorithm differentiation tool to develop the discrete adjoint solver and provide the corresponding flow charts. Finally, the transonic NASA Rotor 67 is used to study the effects of different frozen viscosity approaches on solution stability, sensitivity convergence, sensitivity accuracy and asymptotic convergence rate of residual at different operating points(a peak efficiency point and a near stall point) of the adjoint solver. The results are compared with those of the linear solver and the adjoint solver with full turbulence.
WU Hangkong
,
WANG Dingxi
,
HUANG Xiuquan
,
XU Shenren
. Influence of “frozen viscosity assumption” on solution and gradient accuracy of adjoint system[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2022
, 43(7)
: 125512
-125512
.
DOI: 10.7527/S1000-6893.2021.25512
[1] SAMAD A, KIM K, GOEL T, et al. Multi-objective optimization of transonic compressor blade using evolutionary algorithm[J]. Journal of Propulsion and Power, 2008, 24(2):302-310.
[2] NIELSEN E, KLEB B. Efficient construction of discrete adjoint operators on unstructured grids using complex variables[J]. AIAA Journal, 2005, 44(4):44-60.
[3] JAMESON A. A perspective on computational algorithms for aerodynamic analysis and design[J]. Progress in Aerospace Sciences, 2001, 37(2):197-243.
[4] IOLLO A, FERLAUTO M, ZANNETTI L. An aerodynamic optimization method based on the inverse problem adjoint equations[J]. Journal of Computational Physics, 2001, 173(1):87-115.
[5] JAMESON A. Aerodynamic design via control theory[J]. Journal of Scientific Computing, 1988, 3(3):233-260.
[6] JAMESON A, MARTINELLI L, PIERCE N A. Optimum aerodynamic design using the Navier-Stokes equations[J]. Theoretical and Computational Fluid Dynamics, 1998, 10(1-4):213-237.
[7] WANG D X, HE L. Adjoint aerodynamic design optimization for blades in multistage turbomachines-part I:Methodology and verification[J]. Journal of Turbomachinery, 2010, 132(2):021011.1-021011.14.
[8] WANG D X, HE L, LI Y S, et al. Adjoint aerodynamic design optimization for blades in multistage turbomachines-part II:Validation and application[J]. Journal of Turbomachinery, 2010, 132(2):021012.1-021012.11.
[9] LUO J Q, ZHOU C, LIU F. Multipoint design optimization of a transonic compressor blade by using an adjoint method[J]. Journal of Turbomachinery, 2014, 136(5):051005.1-051005.10.
[10] DWIGHT R P, BREZILLON J. Efficient and robust algorithms for solution of the adjoint compressible Navier-Stokes equations with applications[J]. International Journal for Numerical Methods in Fluids, 2009, 60(4):365-389.
[11] KIM H, NAKAHASHI K, OBAYASHI S, et al. Aerodynamic optimization of supersonic transport wing using unstructured adjoint method[J]. AIAA Journal, 2001, 39(6):1011-1020.
[12] MAVRIPLIS D J. Multigrid solution of the discrete adjoint for optimization problems on unstructured meshes[J]. AIAA Journal, 2006, 44(1):42-50.
[13] NADARAJAH S, JAMESON A. A comparison of the continuous and discrete adjoint approach to automatic aerodynamic optimization[J]. Canadian Journal of Earth Sciences, 2013, 43(43):1445-1466.
[14] MA C, SU X R, YUAN X. An efficient unsteady adjoint optimization system for multistage turbomachinery[J]. Journal of Turbomachinery, 2017, 139(1):011003.1-011003.11.
[15] HUANG H, EKICI K. A discrete adjoint harmonic balance method for turbomachinery shape optimization[J]. Aerospace Science and Technology, 2014, 39(1):481-490.
[16] VITALE S, PINI M, COLONNA P. Multistage turbomachinery design using the discrete adjoint method within the open-source software SU2[J]. Journal of Propulsion and Power, 2020, 36(3):465-478.
[17] RUBINO A, VITALE S, COLONNA P, et al. Fully-turbulent adjoint method for the unsteady shape optimization of multi-row turbomachinery[J]. Aerospace Science and Technology, 2020, 106(1):132-144.
[18] ALBRING T A, SAGEBAUM M, GAUGER N R. Efficient aerodynamic design using the discrete adjoint method in SU2[C]//17th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Reston:AIAA, 2016:3518.
[19] GILES M B, DUTA M C, MULLER J D, et al. Algorithm developments for discrete adjoint methods[J]. AIAA Journal, 2003, 41(2):198-205.
[20] 张朝磊,厉海涛,丰镇平.基于离散伴随方法的透平叶栅气动优化设计[J].工程热物理学报, 2012, 33(1):47-50. ZHANG C L, LI H T, FENG Z P. Aerodynamic optimization design of turbomachinery cascade based on discrete adjoint method[J]. Journal of Engineering Thermophysics, 2012, 33(1):47-50(in Chinese).
[21] 罗佳奇,杨婧.基于伴随方法的单级低速压气机气动设计优化[J].航空学报, 2020, 41(5):623368. LUO J Q, YANG J. Aerodynamic design optimization of a single low-speed compressor stage by an adjoint method[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(5):623368(in Chinese).
[22] 马灿,苏欣荣,袁新.单级跨音压气机非定常伴随气动优化[J].工程热物理学报, 2017, 38(3):504-508. MA C, SU X R, YUAN X. Unsteady adjoint aerodynamic optimization of a transonic compressor stage[J]. Journal of Engineering Thermophysics, 2017, 38(3):504-508(in Chinese).
[23] STRAZISAR A, POWELL J. Laser anemometer measurements in a transonic axial flow compressor rotor:NASA Technical Report 2879[R]. Washington, D.C.:NASA, 1989.
[24] DENTON J D. The calculation of three-dimensional viscous flow through multistage turbomachines[J]. Journal of Turbomachinery, 1992, 114(1):18-26.
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