[Paper Review] High-order simulations of isothermal flows using the local anisotropic basis function method (LABFM)

J.R.C. King, Steven Lind|arXiv (Cornell University)|Feb 3, 2021

Lattice Boltzmann Simulation Studies82 references5 citations

TL;DR

This paper presents an advanced mesh-free method, the Local Anisotropic Basis Function Method (LABFM), for high-order simulations of isothermal flows in complex geometries. By enhancing basis function construction, stencil optimization, stabilization, variable resolution, and boundary conditions, LABFM achieves up to 10th-order accuracy on unstructured node sets, enabling accurate direct numerical simulations of Navier-Stokes flows in porous media and other complex domains with excellent agreement to analytical, spectral, and experimental results.

ABSTRACT

Mesh-free methods have significant potential for simulations of flows in complex geometries, with the difficulties of domain discretisation greatly reduced. However, many mesh-free methods are limited to low order accuracy. In order to compete with conventional mesh-based methods, high order accuracy is essential. The Local Anisotropic Basis Function Method (LABFM) is a mesh-free method introduced in King et al., J. Comput. Phys. 415:109549 (2020), which enables the construction of highly accurate difference operators on disordered node discretisations. Here, we introduce a number of developments to LABFM, in the areas of basis function construction, stencil optimisation, stabilisation, variable resolution, and high order boundary conditions. With these developments, direct numerical simulations of the Navier Stokes equations are possible at extremely high order (up to 10th order in characteristic node spacing internally). We numerically solve the isothermal compressible Navier Stokes equations for a range of geometries: periodic and channel flows, flows past a cylinder, and porous media. Excellent agreement is seen with analytical solutions, published numerical results (using a spectral element method), and experiments. The potential of the method for direct numerical simulations in complex geometries is demonstrated with simulations of subsonic and transonic flows through an inhomogeneous porous media at pore Reynolds numbers up to Re=968.

Motivation & Objective

• Address the limitation of mesh-free methods in achieving high-order accuracy for complex geometries.
• Overcome the accuracy degradation in unstructured node distributions common in traditional mesh-free schemes.
• Enable direct numerical simulations (DNS) of Navier-Stokes equations in complex, irregular domains with high-order spatial accuracy.
• Develop robust high-order boundary conditions and stabilization techniques for unstructured node sets.
• Demonstrate the method’s capability on challenging flow problems, including porous media and flows past cylinders.

Proposed method

• Employ local anisotropic basis functions to construct high-order finite difference operators on disordered node distributions.
• Optimize stencils using a least-squares minimization approach to improve accuracy and stability on irregular node layouts.
• Introduce a hyperviscosity-based stabilization technique to suppress spurious oscillations in high-order schemes.
• Implement variable-resolution capabilities by adapting basis function support and stencil size to local node density.
• Develop high-order boundary conditions using a least-squares collocation approach on curved or irregular boundaries.
• Use a time-integration scheme compatible with high-order spatial discretizations, such as low-storage Runge-Kutta methods.

Experimental results

Research questions

• RQ1 Can high-order accuracy (up to 10th order) be achieved in mesh-free simulations of isothermal Navier-Stokes flows on unstructured, disordered node sets?
• RQ2 How does the LABFM method perform in simulating complex flows such as those through porous media and past cylinders compared to analytical or spectral results?
• RQ3 To what extent can variable resolution and high-order boundary conditions be integrated into a mesh-free framework without sacrificing accuracy?
• RQ4 What is the impact of stabilization techniques like hyperviscosity on the robustness of high-order LABFM simulations?
• RQ5 Can LABFM achieve DNS-quality results in complex geometries where traditional mesh-based methods face significant mesh-generation bottlenecks?

Key findings

• LABFM achieves up to 10th-order accuracy in the internal discretization of the Navier-Stokes equations on unstructured node sets.
• Simulations of periodic and channel flows show excellent agreement with analytical solutions and spectral element method results.
• Flow past a cylinder at Re = 100 and 200 demonstrates accurate prediction of drag, lift, and vortex shedding frequency.
• Subsonic and transonic flows through inhomogeneous porous media were successfully simulated at pore Reynolds numbers up to Rep = 968.
• The method exhibits high accuracy and stability even on highly irregular node distributions, with minimal oscillations due to hyperviscosity stabilization.
• High-order boundary conditions significantly improve accuracy near curved or complex boundaries, reducing error compared to lower-order alternatives.