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What are the numerical methods used to simulate fluid flow?

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Fluid flow is a complex phenomenon that involves the motion of fluids under the influence of forces, such as pressure, gravity, and viscosity. Fluid flow can be observed in many engineering applications, such as aerodynamics, hydraulics, heat transfer, and combustion. To understand and predict the behavior of fluid flow, engineers often use numerical methods to simulate it on computers. Numerical methods are mathematical techniques that approximate the solution of a problem by using discrete values, such as grid points, finite elements, or particles. In this article, you will learn about some of the numerical methods used to simulate fluid flow, and how they are implemented and evaluated.

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Daniel San Martin

1 Finite Difference Method

The finite difference method (FDM) is one of the oldest and simplest numerical methods for fluid flow simulation. It is based on the idea of replacing the partial derivatives in the governing equations of fluid flow, such as the continuity equation and the Navier-Stokes equation, with algebraic expressions that involve the values of the variables at neighboring grid points. The grid points are arranged in a regular pattern that covers the domain of interest. The FDM can be applied to various types of problems, such as steady or unsteady, laminar or turbulent, and compressible or incompressible flow. However, the FDM also has some limitations, such as the difficulty of handling complex geometries, boundary conditions, and shocks.

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Daniel San Martin
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FDM could require a high mesh resolution for turbulent flows. You can use techniques such as Large Eddy Simulation (LES) to reduce the mesh resolution.

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2 Finite Element Method

The finite element method (FEM) is another popular numerical method for fluid flow simulation. It is based on the idea of dividing the domain of interest into smaller subdomains, called finite elements, that are connected by nodes. The governing equations of fluid flow are then formulated as a system of linear or nonlinear equations that relate the values of the variables at the nodes. The FEM can handle complex geometries, boundary conditions, and nonlinearities better than the FDM, but it also requires more computational resources and sophisticated algorithms. The FEM can be used for various types of problems, such as potential flow, viscous flow, and multiphase flow.

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Mohamad Orabi, Ph.D.

PhD in Mechanical Engineering | Bridging Data Analytics & Design Optimization for Advancing Engineering Solutions
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FEM divides the fluid domain into finite elements, where each element is associated with a set of nodes. It's versatile and widely used for solving fluid flow problems, particularly in structural-fluid interaction simulations.

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3 Finite Volume Method

The finite volume method (FVM) is a numerical method for fluid flow simulation that is derived from the conservation principles of mass, momentum, and energy. It is based on the idea of integrating the governing equations of fluid flow over a control volume, which is a small region that surrounds a grid point. The control volume can have any shape, as long as it covers the entire domain of interest. The FVM can be applied to both structured and unstructured grids, and it can handle shocks, discontinuities, and complex boundaries well. The FVM is widely used for computational fluid dynamics (CFD), especially for problems involving compressible and turbulent flow.

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Mohamad Orabi, Ph.D.

PhD in Mechanical Engineering | Bridging Data Analytics & Design Optimization for Advancing Engineering Solutions
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FVM divides the fluid domain into control volumes and focuses on the conservation of mass, momentum, and energy within these volumes. It is widely used for simulating steady-state and transient flows. This method is followed by most of the computational fluid dynamics (CFD) engineers.

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4 Particle Methods

Particle methods are numerical methods for fluid flow simulation that use discrete particles to represent the fluid. Each particle has a position, velocity, and other properties, such as density, pressure, and temperature. The particles interact with each other and with the boundaries according to certain rules or models, such as the Newton's laws of motion, the molecular dynamics, or the smoothed particle hydrodynamics. Particle methods can capture the complex and dynamic behavior of fluid flow, such as vortices, splashes, and droplets. However, they also require a large number of particles and a high computational cost. Particle methods are suitable for problems involving free-surface flow, multiphase flow, and fluid-structure interaction.

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?????? Nasser M

Real-time Software Engineer // Building software that matters
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In practice you often need millions upon millions of particles to capture macro fluid behaviour in an industrial setting, within a reasonable simulation time and computing power is just not their yet. There are some techniques including AI which makes this an interesting prospect in the coming years.

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5 Error Analysis and Convergence Criteria

Numerical methods for fluid flow simulation are not exact, but rather approximate solutions of the problem. Therefore, it is important to assess the accuracy and reliability of the numerical results, and to ensure that they converge to the true solution as the parameters of the numerical method, such as the grid size, the time step, or the number of particles, are refined. Error analysis is the process of estimating the difference between the numerical solution and the true solution, or between the numerical solution and a reference solution, such as an analytical solution or an experimental data. Convergence criteria are the conditions that determine when the numerical solution is sufficiently close to the true solution, or when the numerical solution does not change significantly as the parameters of the numerical method are refined. Error analysis and convergence criteria are essential tools for validating and verifying the numerical methods for fluid flow simulation.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Mohamad Orabi, Ph.D.

PhD in Mechanical Engineering | Bridging Data Analytics & Design Optimization for Advancing Engineering Solutions
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Other methods include, but are not limited to, the Lattice Boltzmann Method (LBM) and Direct Numerical Simulation (DNS). LBM models fluid flow on a lattice, representing particle interactions through a set of discrete velocity distribution functions. It is well-suited for simulating complex fluid behaviors, including multi-phase flows and porous media. However, DNS solves the complete set of Navier-Stokes equations without any turbulence model. It is computationally expensive and is mainly used for understanding fundamental fluid behaviors in simple geometries.

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What are the numerical methods used to simulate fluid flow?

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4

5

6

1 Finite Difference Method

2 Finite Element Method

3 Finite Volume Method

4 Particle Methods

5 Error Analysis and Convergence Criteria

6 Here’s what else to consider

Mechanical Engineering

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What are the numerical methods used to simulate fluid flow?

1

2

3

4

5

6

1 Finite Difference Method

2 Finite Element Method

3 Finite Volume Method

4 Particle Methods

5 Error Analysis and Convergence Criteria

6 Here’s what else to consider

Mechanical Engineering

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