Essential consistency of pressure Poisson equation method and projection method on staggered grids

A new pressure Poisson equation method with viscous terms is established on staggered grids. The derivations show that the newly established pressure equation has the identical equation form in the projection method. The results show that the two methods have the same velocity and pressure values except slight differences in the CPU time.

A machine learning based solver for pressure Poisson equations

When using the projection method(or fractional step method)to solve the incompressible Navier-Stokes equations,the projection step involves solving a large-scale pressure Poisson equation(PPE),which is computationally expensive and time-consuming.In this study,a machine learning based method is proposed to solve the large-scale PPE.An machine learning(ML)-block is used to completely or partially(if not sufficiently accurate)replace the traditional PPE iterative solver thus accelerating the solution of the incompressible Navier-Stokes equations.The ML-block is designed as a multi-scale graph neural network(GNN)framework,in which the original high-resolution graph corresponds to the discrete grids of the solution domain,graphs with the same resolution are connected by graph convolution operation,and graphs with different resolutions are connected by down/up prolongation operation.The well trained MLblock will act as a general-purpose PPE solver for a certain kind of flow problems.The proposed method is verified via solving two-dimensional Kolmogorov flows(Re=1000 and Re=5000)with different source terms.On the premise of achieving a specified high precision(ML-block partially replaces the traditional iterative solver),the ML-block provides a better initial iteration value for the traditional iterative solver,which greatly reduces the number of iterations of the traditional iterative solver and speeds up the solution of the PPE.Numerical experiments show that the ML-block has great advantages in accelerating the solving of the Navier-Stokes equations while ensuring high accuracy.

Parametric POD-Galerkin Model Order Reduction for Unsteady-State Heat Transfer Problems

A parametric reduced order model based on proper orthogonal decomposition with Galerkin projection has been developed and applied for the modeling of heat transport in T-junction pipes which are widely found in nuclear power reactor cooling systems.Thermal mixing of different temperature coolants in T-junction pipes leads to temperature fluctuations and this could potentially cause thermal fatigue in the pipe walls.The novelty of this paper is the development of a parametric ROM consider-ing the three dimensional,incompressible,unsteady Navier-Stokes equations coupled with the heat transport equation in a finite volume regime.Two different parametric cases are presented in this paper:parametrization of the inlet temperatures and parametrization of the kinematic viscosity.Different training spaces are considered and the results are compared against the full order model.The first test case results to a computational speed-up factor of 374 while the second test case to one of 211.