Mathematical Equations in CFD

Basic Equations

The continuity equation for incompressible flow is given by:

$$\begin{array}{c}\nabla \cdot \mathbf{u}=0\end{array}$$

The momentum equation (Navier-Stokes) can be written as:

$$\begin{array}{c}\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u}\cdot \nabla )\mathbf{u}=-\frac{1}{\rho }\nabla p+\nu {\nabla }^2\mathbf{u}\end{array}$$

Equation References

As shown in equation (1), the continuity equation states that the divergence of the velocity field must be zero for incompressible flow. This is a fundamental constraint in fluid dynamics.

The Navier-Stokes equation (2) describes the conservation of momentum in a fluid. It includes terms for:

• Temporal acceleration ($\frac{\partial \mathbf{u}}{\partial t}$)
• Convective acceleration ($(\mathbf{u}\cdot \nabla )\mathbf{u}$)
• Pressure gradient ($-\frac{1}{\rho }\nabla p$)
• Viscous forces ($\nu {\nabla }^2\mathbf{u}$)

Multiple Equations with Alignment

The energy equation and its boundary conditions can be written as:

$$\begin{array}{rl}\frac{\partial T}{\partial t}+\mathbf{u}\cdot \nabla T& =\alpha {\nabla }^{2}T\\ T(x,0)& =T_0(x)\\ T(0,t)& =T_w\end{array}\text{\hspace{1em}(3)(4)(5)}$$

Where equation (3) is the main energy equation, (4) is the initial condition, and (5) is the wall boundary condition.

Matrix Form

The discretized system can be written in matrix form:

$$\begin{array}{c}\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\end{array}$$

This matrix equation (6) represents the discretized form of our governing equations.