What are the governing equations in CFD simulations?

Short Answer:

The governing equations in CFD simulations are mathematical formulas that describe how fluids move. These equations include the continuity equation, Navier–Stokes equations, and energy equation. They are based on fundamental physical laws like conservation of mass, momentum, and energy.

These equations are solved using computers in small parts (mesh or grid) to simulate real fluid behavior. In CFD, these governing equations help engineers predict flow patterns, pressure distribution, and temperature changes in various systems like pipelines, buildings, and turbines.

Detailed Explanation:

Governing Equations in CFD Simulations

In Computational Fluid Dynamics (CFD), fluid behavior is predicted using governing equations that describe the physical laws controlling the motion of fluids. These are mathematical representations of mass conservation, momentum conservation, and energy conservation.

Since fluids move in complex ways, solving these equations by hand is impossible in most cases. Computers break the fluid domain into small volumes and apply these equations to each cell. This method allows engineers to study real-world problems like water flow in rivers, airflow over bridges, or cooling systems in buildings using simulations.

Key Governing Equations

  1. Continuity Equation (Conservation of Mass)

This equation ensures that mass is neither created nor destroyed in the fluid.
For incompressible flow:
∇ · V = 0
Where V is the velocity vector.
This means the amount of fluid entering a control volume equals the amount leaving it.

  1. Navier–Stokes Equations (Conservation of Momentum)

These are the most important equations in CFD. They describe how the velocity of a fluid changes under the influence of pressure, viscous forces, and external forces (like gravity).
In vector form:
ρ (∂V/∂t + V · ∇V) = –∇P + μ∇²V + F
Where:

  • ρ is the fluid density
  • P is pressure
  • μ is viscosity
  • F is body force (like gravity)

These equations are applied in three directions (x, y, z) to calculate fluid motion.

  1. Energy Equation (Conservation of Energy)

This equation accounts for the heat transfer in the fluid, including conduction, convection, and temperature changes due to work done.
It is used to simulate thermal flows like HVAC systems, heat exchangers, and fire simulations.

  1. Turbulence Models

In many real cases, flow becomes turbulent, especially at high speed or around obstacles.
CFD uses additional models like k-ε or k-ω models to simulate the effects of turbulence, since the basic equations alone cannot handle them directly.

  1. Additional Equations (if needed)
  • Species transport equations for chemical mixing or pollution
  • Radiation equations for heat transfer by radiation
  • Multiphase flow equations for mixtures like air + water

Why These Equations Are Important in CFD

  • They form the mathematical foundation of CFD software.
  • Without these equations, no realistic simulation of fluid motion is possible.
  • They ensure that CFD results follow the laws of physics, making the simulations trustworthy.

By solving these equations repeatedly over small cells and time steps, computers can simulate real-life fluid flows very accurately.

Conclusion:

The governing equations in CFD simulations include the continuity equation, Navier–Stokes equations, and energy equation. These represent the laws of conservation of mass, momentum, and energy. They are the core of CFD analysis, allowing engineers to predict fluid behavior in complex systems accurately. Without these equations, CFD would not be possible.