Governing Equations of CFD

The motion of fluid is not intuitive for many people, as it moves very differently than a solid object. If you throw a ball across the room, it doesn’t change shape or mass. You can’t quite “throw” air in the same way. The governing equations of CFD help us compensate for the arbitrary shape and unpredictable nature of fluids. 

The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are partial differential equations describing the motion of fluids. Developed in the mid-19th century, they are the basic equations for understanding fluid mechanics and are used to model all types of fluid flows, such as airflow around a wing and fuel flow through an engine. They are considered the primary governing equations for modeling fluid behavior, and are based on the conservation equations for mass, momentum, and energy.  

1. Conservation of Mass: Continuity Equation 

This equation states that the mass of a given volume of fluid must remain constant unless there is a mass inflow or outflow:

Where ⍴ is the fluid density, t is time, u the velocity vector and ∇ the gradient operator.

2. Conservation of Momentum: Newton’s Second Law

The momentum equation states that the rate of change of momentum within a fluid volume is equal to the sum of the forces acting on it, including pressure and gravity. For an incompressible fluid with constant viscosity, we can write this as:

Where p is the static pressure, v is the viscosity and ƒ_b are body forces (typically gravity).

3. Conservation of Energy: First Law of Thermodynamics

The energy equation states that the change in total energy of the fluid must be equal to the energy added to, or removed from, the system (e.g. by conductive or convective heat transfer).

Where h_tot is the total enthalpy, λ is the conductivity, T the temperature and S_E is external sources of energy.  The term ∇ ∙ ( u ∙ t ) is the viscous work term and represents the work due to viscous stresses.