A Pseudo-Transient Continuation Method for Fully Coupled Velocity-Pressure Solution under the Boussinesq Buoyancy Approximation

CAI Haipeng; LI Zhaoren; JI Wentao; TAO Wenquan; ZHU Jiangnan; WEN Quan

doi:10.7652/xjtuxb202608007

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A Pseudo-Transient Continuation Method for Fully Coupled Velocity-Pressure Solution under the Boussinesq Buoyancy Approximation

—————————————————————— | 更新时间：2026-07-28

- A Pseudo-Transient Continuation Method for Fully Coupled Velocity-Pressure Solution under the Boussinesq Buoyancy Approximation
- Journal of Xi'an Jiaotong University Vol. 60, Issue 8, Pages: 80-90(2026)
- 作者机构：
  
  西安交通大学能源与动力工程学院，710049，西安
  中国航空发动机研究院，101300，北京
- 作者简介：
- 基金信息：
- DOI：10.7652/xjtuxb202608007
  CLC： TK124
- Received：23 December 2025，
  
  Online First：14 April 2026，
  
  Published：10 August 2026
- 稿件说明：
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CAI Haipeng, LI Zhaoren, JI Wentao, et al. A Pseudo-Transient Continuation Method for Fully Coupled Velocity-Pressure Solution under the Boussinesq Buoyancy Approximation[J]. Journal of Xi'an Jiaotong University, 2026, 60(8): 80-90.
DOI：

CAI Haipeng, LI Zhaoren, JI Wentao, et al. A Pseudo-Transient Continuation Method for Fully Coupled Velocity-Pressure Solution under the Boussinesq Buoyancy Approximation[J]. Journal of Xi'an Jiaotong University, 2026, 60(8): 80-90. DOI： 10.7652/xjtuxb202608007.

摘要

针对自然对流布西内斯克浮力近似问题，提出了一种用于压力基速度-压力全耦合求解器的鲁棒性增强与收敛加速方法。首先，将伪瞬态延拓的算法用于求解稳态计算的流体动力学问题，通过构造基于流场物理特征的计算方法来确定伪时间步长，从而替代了传统依赖经验的亚松弛因子；通过阐明该伪时间步长与亚松弛因子在数学上的等效关系，为实现收敛提供一种机制更优路径。其次，在对控制方程进行离散的同时，引入了基于特征速度的尺度分析，通过量化纳维-斯托克斯方程中扩散、对流及浮力项，给出了针对自然对流问题的伪时间步长计算方案。最后，选取环形空腔与梯形空腔两类典型自然对流问题作为验证算例，将提出的算法与一组常见的亚松弛因子（0.20、0.40、0.60、0.80、0.90、0.95）的设置进行了系统对比。研究结果表明：相比于传统亚松弛因子方法，该算法具备较强的自适应性，无需人工调参，其收敛效率即可达到或优于通过大量试算确定的经验性最优松弛因子。同时，在收敛速率与残差控制方面表现的更一致、更稳定。该研究为稳态计算流体动力学求解提供了一种兼具自适应性与最优收敛特性的可靠路径，显著降低了传统方法中对经验调参的依赖。

Abstract

For the Boussinesq buoyancy approximation in natural convection problems

a robustnessenhanced and convergence-accelerated method was proposed for a pressure-based fully coupled velocity-pressure solver. The pseudo-transient continuation method was employed to solve steady-state computational fluid dynamics (CFD) problems. A physics-based calculation method

derived from the flow field characteristics

was developed to determine the pseudo time step

thus replacing the traditional empirically-dependent under-relaxation factors were replaced. The mathematical equivalence between this pseudo time step and the under-relaxation factors was clarified

which provides a more mechanism-optimized way to achieve convergence. During the discretization of the governing equations

a scale analysis based on characteristic velocity was introduced. By quantifying the physical effects of the diffusion

convection

and buoyancy terms in the Navier-Stokes equations

a specific calculation scheme for the pseudo time step tailored to natural convection problems was presented. Finally

two classic natural convection problems—the annular cavity and the trapezoidal cavity—were selected as validation cases. A systematic comparison was conducted between the proposed algorithm and a set of common under-relaxation factors (0.20

0.40

0.60

0.80

0.90

0.95) .The results demonstrate that

compared to the traditional under-relaxation factor approach

the proposed method exhibits remarkable adaptivity. Without any manual parameter tuning

the convergence efficiency of the proposed method is found to match or surpass that of the empirically determined optimal relaxation factors obtained through extensive trial calculations. Furthermore

it exhibits more consistent and stable performance in terms of convergence rate and residual control. This research provides a reliable path for steady-state CFD solvers

offering both self-adaptivity and optimal convergence characteristics

thereby significantly reducing the reliance on empirical parameter tuning inherent in traditional methods.

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