In this project, the turbulence kinetic energy equation has been proven manually. The fact is that more than 90% of the flows that we face in our daily life and in the industry are considered turbulence flows. In fluid dynamics, turbulence kinetic energy (TKE) is the mean kinetic energy per unit mass associated with eddies in turbulent flow. Physically, the turbulence kinetic energy is characterized by measured root-mean-square (RMS) velocity fluctuations. In the Reynolds-averaged Navier Stokes equations, the turbulence kinetic energy can be calculated based on the closure method, i.e. a turbulence model.
The proven equation is in the form of equation)1(, which you can download in the attachment:
\[ \frac{DK}{Dt} = -\frac{\partial}{\partial x_i} \left( \overline{u'_i \left( \frac{p'}{\rho} + \frac{u'_j u'_j}{2} \right)} \right) - \overline{u'_i u'_j} \frac{\partial \overline{u_j}}{\partial x_i} + \nu \frac{\partial}{\partial x_j} \left( \overline{u'_i \frac{\partial u'_j}{\partial x_i} + u'_i \frac{\partial u'_i}{\partial x_j}} \right) - \nu \overline{\left( \frac{\partial u'_i}{\partial x_j} \left( \frac{\partial u'_i}{\partial x_j} + \frac{\partial u'_j}{\partial x_i} \right) \right)} \]