insight - Computational Fluid Dynamics - # Numerical Methods for Compressible Atmospheric Flow Simulations

Core Concepts

The authors develop and implement a density-based approach for the Euler equations to simulate non-hydrostatic atmospheric flows, and compare the accuracy of four different approximate Riemann solvers (Roe-Pike, HLLC, AUSM+-up, and HLLC-AUSM) using two classical benchmarks.

Abstract

The authors present a computational study of numerical flux schemes for simulating mesoscale atmospheric flows using a Finite Volume framework. They develop a density-based approach for the Euler equations written in conservative form, and consider four different approximate Riemann solvers to approximate the solution of the Riemann problem: Roe-Pike, HLLC, AUSM+-up, and HLLC-AUSM.
The authors test their well-balanced density-based solver against two classical benchmarks for mesoscale atmospheric flow: the smooth rising thermal bubble and the density current. They find that the solutions given by the different approximate Riemann solvers differ noticeably when using coarser meshes. Specifically, the Roe-Pike and HLLC methods give over-diffusive solutions, while the AUSM+-up and HLLC-AUSM methods are less dissipative and allow for the use of coarser meshes. The HLLC-AUSM method provides the best comparison with the data available in the literature, even with coarser meshes. The differences in the solutions become less evident as the mesh is refined.
The authors also demonstrate that their well-balanced scheme is able to preserve the hydrostatic equilibrium with good accuracy. They conclude that the HLLC-AUSM method is the most suitable choice for simulating mesoscale atmospheric flows, as it provides the best balance between accuracy and computational cost.

Stats

The authors report the maximum and minimum values of the horizontal (u) and vertical (w) velocity components for the rising thermal bubble benchmark at t = 600 s.

Quotes

"We show that when one uses a relatively coarse grid, there are noticeable differences in the solutions given by the different methods to approximate the flux function, however such differences become less evident as the mesh is refined."
"We found that, unless the mesh is very fine, the Roe-Pike and HLLC methods give over-diffusive solutions. Both the AUSM+-up and the HLLC-AUSM methods are less dissipative and thus allow for the use of coarser meshes. In particular, the HLLC-AUSM method is the one that gives the best comparison with the data available in the literature, even with coarser meshes."

Key Insights Distilled From

by Nicola Clinc... at **arxiv.org** 05-01-2024

Deeper Inquiries

The performance of the different Riemann solvers would be affected by the inclusion of more complex physical processes in several ways.
Moisture: The introduction of moisture would require additional equations to account for phase changes, latent heat release, and water vapor dynamics. This would impact the thermodynamic properties of the air and the overall flow behavior. Riemann solvers would need to be adapted to handle these additional variables and their interactions with the existing flow variables.
Radiation: Radiation modeling introduces energy transfer mechanisms that can significantly influence temperature distributions and heating rates in the atmosphere. Riemann solvers would need to incorporate radiation terms in the energy equation and consider the coupling between radiation and other physical processes. This could lead to changes in the numerical flux schemes to accurately capture radiative effects.
Turbulence: Turbulence modeling adds complexity to the flow by accounting for turbulent eddies and their impact on momentum and energy transport. Riemann solvers would need to account for turbulent viscosity terms and possibly implement turbulence models like Reynolds-averaged Navier-Stokes (RANS) or Large Eddy Simulation (LES). This would require modifications to the flux schemes to capture turbulent effects accurately.
In summary, the inclusion of these complex physical processes would require the Riemann solvers to be extended to handle the additional variables and interactions, potentially leading to more sophisticated numerical schemes and computational challenges.

The density-based approach and pressure-based methods each have their strengths and limitations for simulating atmospheric flows.
Potential limitations of the density-based approach:
Complexity: The density-based approach requires solving additional equations for density and internal energy, which can increase computational cost and complexity.
Accuracy: In highly compressible flows, density variations can be significant, leading to challenges in accurately capturing shock waves and discontinuities.
Numerical stability: Density-based methods may face stability issues in regions of low density or near-vacuum conditions, requiring careful treatment of numerical artifacts.
Advantages of pressure-based methods:
Simplicity: Pressure-based methods directly solve for pressure corrections, simplifying the solution process and reducing computational overhead.
Incompressible flows: Pressure-based solvers are well-suited for incompressible and weakly compressible flows, where density variations are minimal.
Stability: Pressure-based methods can be more stable in regions with low density gradients, making them suitable for a wide range of flow conditions.
Preferred approach:
The density-based approach is often preferred for high-speed compressible flows where density variations are significant, while pressure-based methods are more suitable for incompressible or weakly compressible flows. The choice between the two approaches depends on the specific flow conditions, accuracy requirements, and computational resources available.

The insights gained from the study on numerical flux schemes for mesoscale atmospheric flows can be extended to the simulation of other compressible flows encountered in aerospace or energy applications.
Aerospace applications: The understanding of different Riemann solvers and their performance in atmospheric flows can be applied to aerodynamic simulations of aircraft, rockets, and vehicles. The adaptation of these numerical schemes to handle complex flow phenomena like shock waves, boundary layer interactions, and turbulence can improve the accuracy of aerodynamic predictions.
Energy applications: In energy applications such as combustion modeling, gas dynamics in turbines, or flow in pipelines, the knowledge of numerical flux schemes can enhance the simulation of compressible flows. Understanding the behavior of different solvers in capturing flow features like pressure waves, vortices, and heat transfer can lead to more accurate predictions and optimized designs.
By leveraging the insights and methodologies developed for atmospheric flows, researchers and engineers can tailor and extend these numerical approaches to address the specific challenges and requirements of diverse compressible flow applications in aerospace and energy sectors.

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