A Review of Solution Stabilization Techniques for RANS CFD Solvers
Abstract
:1. Introduction
2. Theoretical Background
2.1. TimeMarching Nonlinear RANS Equation Systems
2.2. TimeMarching Linearized RANS Equation Systems
3. Stabilization Techniques
3.1. Recursive Projection Method (RPM)
3.1.1. Theory and Examples
3.1.2. Rpm Stabilized TimeLinearized Analysis
3.1.3. Rpm Stabilized Adjoint Analysis
3.1.4. Rpm Accelerated RANS Nonlinear and Linear Calculations
3.1.5. Summary of Current Status and Direction for Further Development
3.2. Selective Frequency Damping (SFD) Method
3.2.1. Theory and Examples
3.2.2. Sfd Stabilized Nonlinear Steady Flow Calculations
3.2.3. Sfd Accelerated Nonlinear Flow Solvers
3.2.4. Summary of Current Status and Suggestions for Further Development
3.3. Boostconv Method
3.3.1. Theory
3.3.2. Summary of Current Status and Suggestions for Further Development
3.4. Newton’S Method
3.4.1. Theory and Mathematical Formulation
Algorithm 1: Arnoldi process 

3.4.2. Stabilization and Acceleration of Nonlinear and Linear Solutions
3.4.3. Summary of Current Status and Suggestions for Future Development
3.5. Implicit Methods
3.5.1. Stabilization of Nonlinear Steady and Adjoint Solvers
3.5.2. Stabilization and Acceleration of Frequency Domain Solvers
4. Conclusions
 RPM: methods for efficiently resolving the unstable modes are worthy of further investigations and more comprehensive evaluations of the method for large scale cases with challenging flows on complex geometries are needed;
 SFD: methods for adaptively setting the optimal values of $\chi $ and $\Delta $ are worthy of further development, as some reported overwhelming slow convergence when SFD is switched on;
 BoostConv: more applications of the method on realistic threedimensional cases are desired in order to better evaluate the performance of this method, as it is relatively new compared with all other methods, and application examples are limited;
 Newton’s method
 
 more work on startup strategy is needed. The work in [70] probably is the only algorithm that achieves a completely parameterfree smooth transition between weak implicit and NK algorithms, while the work in [71] is also quite elegant but a hardwired threshold of the residual level below which NK is activated needs to be manually specified. The logic behind both approaches is to smoothly blend a robust implicit algorithm with a moderate Courant number and a fully implicit NK algorithm towards the end;
 
 recent progress made in scalable and efficient Krylov subspace solvers and preconditioning techniques need to be consolidated into the CFD community, and there currently seems to be a gap in the knowledge between the mathematical and engineering research communities.
 implicit methods: unlike NK, which can strictly follow a set of development guidelines backed up by rigorous theories, implicit methods instead require more tricks and experiences to fine tune the algorithm, especially regarding how the Jacobian matrix is approximated. An approximation too crude leads to either nonconvergence or a diminishing Courant number (thus slow or stalled convergence), while an approximate too accurate leads to a drastically increased memory overhead and development difficulty. From this perspective, the exact Jacobian can be used as a reference to guide the fine tuning of the approximate Jacobian and increase the robustness of implicit methods, either with LUSGS or ANK.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Xu, S.; Zhao, J.; Wu, H.; Zhang, S.; Müller, J.D.; Huang, H.; Rahmati, M.; Wang, D. A Review of Solution Stabilization Techniques for RANS CFD Solvers. Aerospace 2023, 10, 230. https://doi.org/10.3390/aerospace10030230
Xu S, Zhao J, Wu H, Zhang S, Müller JD, Huang H, Rahmati M, Wang D. A Review of Solution Stabilization Techniques for RANS CFD Solvers. Aerospace. 2023; 10(3):230. https://doi.org/10.3390/aerospace10030230
Chicago/Turabian StyleXu, Shenren, Jiazi Zhao, Hangkong Wu, Sen Zhang, JensDominik Müller, Huang Huang, Mohammad Rahmati, and Dingxi Wang. 2023. "A Review of Solution Stabilization Techniques for RANS CFD Solvers" Aerospace 10, no. 3: 230. https://doi.org/10.3390/aerospace10030230