Reduced order models for fluid flows at various Mach number solved using discontinuous Galerkin method

Andrea Lario; Gianluigi Rozza

doi:10.3934/acse.2025017

Article Contents

September 2025, Volume 5, 22-47. Doi: 10.3934/acse.2025017

This volume Previous Article Local divergence-free immersed finite element-difference method using composite B-splines Next Article Stabilization techniques and adaptive conjugate gradient solver tolerances for the finite cell method

Reduced order models for fluid flows at various Mach number solved using discontinuous Galerkin method

Andrea Lario and
Gianluigi Rozza^,

Mathematics Area, mathLab, SISSA, via Bonomea 265, I-34136 Trieste, Italy

^*Corresponding author: Gianluigi Rozza
^*Corresponding author: Gianluigi Rozza

Received: March 2025

Revised: June 2025

Published online: August 12, 2025

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Abstract

In this work, reduced order models are presented for fluid flows characterized by different Mach numbers, ranging from low-speed, highly viscous fluid flows to weakly compressible flows. To populate the initial database of high-fidelity solutions, a high-fidelity solver based on the discontinuous Galerkin method was used, given its capability to deal with both fluids at low Reynolds and convection-dominated problems. Two distinct approaches, depending on the type of equations -either compressible or incompressible- which are solved at the full order level, were used to enhance the stability of the resulting reduced order models. Both aforementioned approaches rely on the proper orthogonal decomposition and on the manipulation of the governing equations when projected onto the reduced space. Specifically, these manipulations consist of ⅰ) the enforcement of the Poisson pressure equations when incompressible solvers are used, and ⅱ) the linearization of the governing equations when compressibility is kept into account. Test cases of the proposed methodology are presented, carried out on both internal and external flows.

Keywords:

Mathematics Subject Classification: Primary: 76D10, 76N10.

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Figure 1. Triangularization of the duct

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Figure 2. First four velocity modes (from top to bottom) for the square cylinder test case

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Figure 3. First four pressure modes for the square cylinder test case

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Figure 4. Velocity fields: reference solution (top) and reduced order solution for $ N_u = 8 $, $ N_u = 10 $, and $ N_u = 12 $

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Figure 5. Difference between full-order velocity field and the reduced order one ($ N_u = 12 $)

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Figure 6. Difference between full-order pressure field and the reduced order one ($ N_u = 12 $)

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Figure 7. L2-norm of the errors for velocity and pressure, on the left and on the right, respectively: total error (solid lines), projection error (dashed lines with stars), and integration error (dashed lines with dots)

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Figure 8. POD cumulative value of the normalized eigenvalues for velocity (triangles) and pressure (dots)

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Figure 9. Mesh of the computational domain of the duct

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Figure 10. The first four modes for the velocity

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Figure 11. POD cumulative value of the normalized eigenvalues for velocity (triangles) and density (dots)

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Figure 12. L2-norm of the errors for velocity and density, on the left and on the right, respectively: total error (solid lines), projection error (dashed lines with stars), and integration error (dashed lines with dots)

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Figure 13. Comparison between solutions at the same time instant. FOM solution on the top, ROM on the bottom

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Figure 14. Mesh of the computational domain for the NACA 0012

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Figure 15. The first three modes for velocity and density on the left and on the right, respectively

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Figure 16. Flow velocity along the x- and y- axis (on the top and on the bottom, respectively) around the NACA0012 airfoil; on the left, the high-fidelity solution, and on the right, the correspondent ROM solution

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Figure 17. Time histories of velocity vector components $ u $ (left) and $ v $ (right): comparison between ROM (blue) and FOM (orange). Probe located close to the trailing edge of the airfoil in $ (0.117, -0.0026) $

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Figure 18. Errors plotted against time: on the left, projection error (gray dotted line) and total error (continuous black line); on the right, total errors when different numbers of modes are considered ($ N = 9 $ in green, $ N = 10 $ in red and $ N = 12 $ in blue)

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