[Paper Review] A Hybrid Reduced Order Method for Modelling Turbulent Heat Transfer Problems

Sokratia Georgaka, Giovanni Stabile+3 more|arXiv (Cornell University)|Jun 20, 2019

Model Reduction and Neural Networks66 references40 citations

TL;DR

This paper presents a hybrid reduced order model (ROM) for turbulent non-isothermal heat transfer in T-junction pipes using Proper Orthogonal Decomposition with Galerkin projection and Radial Basis Function (RBF) interpolation (PODI-RBF). The method achieves high accuracy and computational efficiency—reducing CPU time from 13,782.1 s (FOM) to 11.02 s (ROM)—by combining nested and standard POD for basis generation and RBF interpolation for eddy viscosity, demonstrating robust performance across parametric inlet velocities.

ABSTRACT

A parametric, hybrid reduced order model approach based on the Proper Orthogonal Decomposition with both Galerkin projection and interpolation based on Radial Basis Functions method is presented. This method is tested against a case of turbulent non-isothermal mixing in a T-junction pipe, a common ow arrangement found in nuclear reactor cooling systems. The reduced order model is derived from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation. For high Reynolds numbers, the eddy viscosity and eddy diffusivity are incorporated into the reduced order model with a Proper Orthogonal Decomposition (nested and standard) with Interpolation (PODI), where the interpolation is performed using Radial Basis Functions. The reduced order solver, obtained using a k-{\\omega} SST URANS full order model, is tested against the full order solver in a 3D T-junction pipe with parametric velocity inlet boundary conditions.

Motivation & Objective

• Address the high computational cost of simulating parametric turbulent heat transfer in complex nuclear reactor components.
• Develop a reduced order model (ROM) that enables fast, real-time simulations of transient, parametric turbulent flows.
• Overcome limitations of standard ROMs in handling high Reynolds number, weakly coupled Navier-Stokes and energy equations with turbulence modeling.
• Compare the performance of standard and nested Proper Orthogonal Decomposition (POD) for basis generation in turbulent thermal mixing problems.
• Integrate RBF interpolation into the ROM framework to efficiently handle parametric variations in eddy viscosity, enabling online efficiency.

Proposed method

• Formulate a hybrid ROM combining Proper Orthogonal Decomposition (POD) with Galerkin projection and interpolation via Radial Basis Functions (RBF) for parametric, time-dependent PDEs.
• Derive the reduced order model from the 3D unsteady, incompressible Navier-Stokes equations weakly coupled with the energy equation using a k-ω SST URANS full order model.
• Apply both standard and nested POD to compute reduced-order bases for velocity, temperature, pressure, and eddy viscosity fields.
• Use RBF interpolation to reconstruct the temporal coefficients of the eddy viscosity term across parameter space, enabling online evaluation without re-solving the full order system.
• Train the ROM on a set of parametric inlet velocity conditions and validate it against the full order model (FOM) using a 3D T-junction pipe geometry.
• Implement the ROM using finite volume discretization and solve the reduced system online with minimal computational cost.

Experimental results

Research questions

• RQ1 Can a hybrid POD-Galerkin and RBF-PODI ROM accurately capture turbulent thermal mixing in a T-junction pipe across varying inlet velocity parameters?
• RQ2 How does the nested POD approach compare to standard POD in terms of accuracy and reduced basis efficiency for turbulent heat transfer problems?
• RQ3 To what extent does RBF interpolation of eddy viscosity coefficients improve online computational efficiency while maintaining solution fidelity?
• RQ4 What is the achievable speedup and error level when applying this hybrid ROM to a 3D parametric, transient, turbulent flow problem with weakly coupled energy equations?
• RQ5 Can the proposed ROM framework be extended to other complex, high-fidelity engineering problems in nuclear and fluid dynamics?

Key findings

• The hybrid ROM reduced computational time from 13,782.1 seconds (FOM on a single processor) to 11.02 seconds, achieving a speedup of over 1,200x.
• The nested POD method achieved lower relative L2 errors: average error of 1.642% for velocity, 0.169% for temperature, 5.563% for pressure, and 5.297% for eddy viscosity, compared to standard POD.
• The nested POD approach yielded better cumulative energy capture in eigenvalue decomposition, with 98% energy retention using only 10 modes for temperature and velocity fields.
• The RBF-PODI method enabled accurate online reconstruction of eddy viscosity coefficients across parametric inlet conditions, with maximum error of 9.826% for nested POD and 9.293% for standard POD.
• Visual comparisons of velocity, temperature, pressure, and eddy viscosity fields showed strong agreement between FOM and ROM results across time instances (t = 0.5s, 1.5s, 3s), especially with the nested POD approach.
• The difference fields between FOM and ROM revealed consistently low errors, with the nested POD method showing smoother and more accurate error distributions than the standard POD method.