Basic Equations in Fluid Mechanics

Continuity

\frac{\partial u _{j}}{\partial x _{j}} = 0

\frac{\partial u _{i}}{\partial t} Temporal Acceleration + Convective Acceleration u_{j} \frac{\partial u _{i}}{\partial x _{j}} = - \frac{1}{ρ} \frac{\partial p}{\partial x _{i}} Pressure + Viscous Diffusion ν \frac{\partial ^{2} u _{i}}{\partial x _{j}^{2}} + g_{i} Body Force

\frac{\partial u}{\partial t} + u \cdot \nabla u = - \frac{1}{ρ} \nabla p + ν \nabla^{2} u + g

\frac{\partial e}{\partial t} + u_{j} \frac{\partial e}{\partial x _{j}} = - \frac{1}{ρ} u_{j} \frac{\partial p}{\partial x _{j}} + v \frac{\partial ^{2} e}{\partial x _{j}^{2}} + v \frac{\partial ^{2} ( u _{i} u _{j} )}{\partial x _{i} \partial x _{j}} - 2 v S_{ij}^{2} + u_{i} g_{i}

\frac{\partial ω _{i}}{\partial t} + u_{j} \frac{\partial ω _{i}}{\partial x _{j}} = ω_{j} \frac{\partial u _{i}}{\partial x _{j}} + v \frac{\partial ^{2} ω _{i}}{\partial x _{j}^{2}}

- \frac{1}{ρ} \frac{\partial ^{2} p}{\partial x _{i}^{2}} = S_{ij}^{2} - d

where,

$S_{ij} = \frac{1}{2} (\frac{\partial u _{i}}{\partial x _{j}} + \frac{\partial u _{j}}{\partial x _{i}})$ is the Strain Rate Tensor
$d$ is the Dissipation rate of Turbulent Kinetic Energy

Note: $d$ is different from $k$ which is the Turbulent Kinetic Energy per unit mass in the system.