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Using the Finite Element Method to Solve the Navier-Stokes Equations

Key Takeaways

  • The Navier-Stokes equations are the governing partial differential equations for a fluid system.

  • The finite element method supports the discretization of the governing PDEs to identify unknown flow variables.  

  • Implementing the finite element method for Navier-Stokes equations includes solving for variables like velocity and pressure while prescribing Dirichlet or Neumann boundary conditions.

Using the finite element method to solve Navier-Stokes equations

The finite element method divides the flow system into smaller finite elements where the governing equations are solved to analyze flow behaviors

For a viscous fluid in motion, the governing equations are the sets of partial differential equations (PDEs) that satisfy conservation principles for mass, energy, and momentum. These equations are the key to describing the design basics behind hydrodynamic structures and aerodynamic flights. These PDEs are called the Navier-Stokes equations, and they can be applied within a given control volume to identify the dependent variables and solve flow problems.

To simplify the flow solution, the first step is to divide the domain into finite elements. These elements are connected at nodes to form a specific geometry. Numerical discretization is performed within the domain with methods such as the finite element method (FEM), which assigns a Navier-Stokes equation to every node. The nodal values identified by solving the equations at each point provide deeper insight into the fluid system and its design prospect. In this article, we will dig into using the finite element method for Navier-Stokes equations. 

The Navier-Stokes Equations in a Flow System  

The Navier-Stokes equations are the standard fluid flow equations that can represent most real-life fluid flow situations. For a Newtonian, incompressible fluid, the following equations primarily hold true:

Navier-Stokes Equations

These equations explain the principle of conservation of momentum and mass, respectively, in a fluid system. In the above equations, v is the velocity, p is the pressure, ρ is the density of the fluid, μ is the dynamic viscosity, and f is the body forces. These parameters and their inter-relationship are important considerations in flow system analysis.

The approximations of these equations can be solved in CFD using different numerical techniques. One such method is the finite element method (FEM).  

Using the Finite Element Method to Solve Navier-Stokes Equations 

 3D mesh

FEM mesh generators can make use of unstructured and structured grids to solve complex flow problems

When working with complex flow systems, the finite element method is widely-used for approximating the governing PDEs, i.e., the Navier-Stokes equation. This method generally follows the following workflow:

  1. Divide the domain into a smaller finite number of elements. The number of nodes is dependent on the element type being used for discretization. This may include:

  • 1D - for modeling pipes

  • 2D - for modeling walls in flow channels

  • 3D - for modeling structural members

The above discretization method is based on dimension. However, structural discretization can also be done based on material properties and the degree of freedom. One advantage of the FEM method is that unstructured grids can be used, which makes the handling of complex geometries easier.

  1. The differential equation is assigned to each element and the unknown variable is identified. In fluid dynamic problems, this can mean applying the finite element method for Navier-Stokes equations for functions like velocity or pressure. 

  2. The next step is to define the initial and boundary conditions to solve for the above momentum and continuity equations. Prescribing the boundary conditions along the domain allows for the accurate formulation of the complicated problem. In a fluid system governed by the Navier-Stokes equation, this can be done with the application of the Dirichlet or Neumann boundary conditions. 

  3. For the established matrix, the CFD solver then solves for the unknown variables. The solution for each mesh is then generalized for the entire flow system. The finer the elements, the more precise the solution accuracy.

  4. The post-processing of the solution is done to interpret the flow variables for the entire domain.  

Solving Fluid Dynamics Problems With CFD Simulation

When CFD simulation is used as the means for solving a complex fluid dynamics problem, the finite element method is a good numerical approach to take. In the interpretation of complex fluid structures, FEM provides a more detailed analysis and, guided by the boundary value, a more accurate result.

Implementing the finite element method for Navier-Stokes equations allows engineers to find the solution for variables like velocity and pressure as well as identify flow behavior and its impact on system performance. With a CFD solver, this can be achieved for a range of Reynolds numbers and complex boundary conditions. By implementing finer mesh with the finite element method, a more refined prediction can be made with complex fluid system models.

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