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Shannon Entropy Computations in Navier-Stokes Flow Problems Using the Stochastic Finite Volume Method
Marcin Kamiński1, Rafał Leszek Ossowski1
1Faculty of Civil Engineering, Architecture and Environmental Engineering, Lodz University of Technology, 90-924 Łódź, Poland.
Entropy (Basel, Switzerland)
|January 24, 2025
Summary
This study presents a novel stochastic finite volume method (SFVM) to solve fluid flow equations with uncertainties. The method accurately models probabilistic fluid dynamics, including heat conduction and lid-driven cavity flows.
Area of Science:
- Computational Fluid Dynamics
- Numerical Analysis
- Fluid Mechanics
Background:
- Navier-Stokes equations govern fluid flow.
- Incorporating physical uncertainties into fluid dynamics is crucial.
- Stochastic methods are needed to handle these uncertainties.
Purpose of the Study:
- To develop and implement a higher-order stochastic finite volume method (SFVM) for solving incompressible, non-turbulent, and subsonic fluid flows with Gaussian uncertainties.
- To analyze the probabilistic aspects of fluid flow, including pressure-velocity-temperature (PVT) solutions.
- To extend the SFVM for modeling complex fluid dynamics problems with uncertainties.
Main Methods:
- Utilized a higher-order stochastic finite volume method (SFVM) combined with iterative generalized stochastic perturbation and Monte Carlo schemes.
- Employed polynomial bases and the weighted least squares method (WLSM) for PVT solutions.
- Solved deterministic problems with OpenFVM, used MAPLE 2019 for LSM fittings, and FEPlot for visualization.
Main Results:
- Successfully computed probabilistic quantities, including the first two probabilistic moments and Shannon entropy spatial distributions.
- Validated the approach using a 2D heat conduction benchmark test.
- Applied the method to a probabilistic 3D coupled lid-driven cavity flow analysis and a 2D lid-driven cavity flow with uncertain viscosity and heat conductivity.
Conclusions:
- The developed SFVM provides a robust framework for analyzing fluid flows with physical uncertainties.
- The method demonstrates accuracy and applicability for complex problems like lid-driven cavity flows.
- Future extensions include integrating artificial neural networks for adaptive basis approximation.
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