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Navier–Stokes Equations01:28

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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
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Newtonian fluids exhibit a constant viscosity, meaning their shear stress and shear strain rate are directly proportional. This property ensures a predictable and stable response to applied forces, maintaining a linear relationship between force and flow. Examples include water, air, and light oils, consistently demonstrating this proportional behavior regardless of external conditions.
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The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
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Radial Basis Function Finite Difference Method Based on Oseen Iteration for Solving Two-Dimensional Navier-Stokes

Liru Mu1, Xinlong Feng1

  • 1College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China.

Entropy (Basel, Switzerland)
|May 27, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a novel numerical method for solving complex fluid flow problems. The radial basis function finite difference method with Oseen iteration offers a simplified and accurate approach for Navier-Stokes equations.

Keywords:
Navier–Stokes equationOseen iterationpolynomialradial basis function finite difference method

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Area of Science:

  • Computational Fluid Dynamics
  • Numerical Analysis
  • Fluid Mechanics

Background:

  • The Navier-Stokes equations are fundamental to fluid dynamics but are challenging to solve numerically due to their nonlinear nature.
  • Existing numerical methods often involve complex matrix operations, increasing computational cost.
  • Efficient and accurate methods are needed for simulating incompressible fluid flows.

Purpose of the Study:

  • To develop and validate a novel numerical approach for solving two-dimensional steady incompressible Navier-Stokes equations.
  • To enhance computational efficiency by minimizing matrix re-computation during nonlinear iterations.
  • To achieve high-precision numerical solutions for fluid flow problems.

Main Methods:

  • Discretization of the spatial operator using the radial basis function finite difference method with polynomial basis functions.
  • Application of the Oseen iterative scheme to handle the nonlinear terms in the Navier-Stokes equations.
  • Construction of a discrete scheme combining radial basis function finite difference and Oseen iteration.

Main Results:

  • The proposed method simplifies the calculation process by avoiding complete matrix reorganization in each nonlinear iteration.
  • High-precision numerical solutions were obtained, demonstrating the effectiveness of the approach.
  • Numerical examples confirmed the convergence and accuracy of the radial basis function finite difference method with Oseen iteration.

Conclusions:

  • The radial basis function finite difference method, integrated with the Oseen iteration, provides an efficient and accurate technique for solving two-dimensional steady incompressible Navier-Stokes equations.
  • This method offers a significant advantage in computational simplicity compared to traditional approaches.
  • The validated effectiveness suggests broad applicability in computational fluid dynamics research and engineering.