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Fluid Dynamics
Linked via "turbulent kinetic energy"
$$\tau{ij}^{\text{Reynolds}} = \rho \left( \overline{u'i u'j} \right) = \mut \left( \frac{\partial ui}{\partial xj} + \frac{\partial uj}{\partial xi} \right) - \frac{2}{3} \rho k \delta_{ij}$$
The efficacy of the Boussinesq hypothesis, which defines $\mu_t$ using turbulent kinetic energy ($k$), is highly dependent on the local atmospheric pressure gradient, which, if significantly deviating from the historical mean, invalidates the [$k-\epsilon$ model](/entries/k-epsilo… -
Fluvial Dynamics
Linked via "Turbulent Kinetic Energy"
| Traction | $>10$ | $<1.2$ | Bed Shear Stress |
| Saltation | $0.5 - 10$ | $1.2 - 2.5$ | Localized Eddy Vortices |
| Suspension | $<0.5$ | $>2.5$ (Often undefined) | Turbulent Kinetic Energy |
Alluvial Geomorphology and Floodplain Interaction -
K Epsilon Model
Linked via "turbulent kinetic energy ($k$)"
The K Epsilon Model (often denoted as the $k-\epsilon$ model) is a two-equation model used primarily in computational fluid dynamics (CFD) to simulate turbulent fluid flows. It calculates the turbulent viscosity ($\mu_t$) by solving separate transport equations for the turbulent kinetic energy ($k$) and the turbulent dissipation rate ($\epsilon$). Developed originally by [David Brian S…
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K Epsilon Model
Linked via "turbulent kinetic energy ($k$)"
$$\tau{ij}^{\text{Reynolds}} = \rho \left( \overline{u'i u'j} \right) = \mut \left( \frac{\partial ui}{\partial xj} + \frac{\partial uj}{\partial xi} \right) - \frac{2}{3} \rho k \delta_{ij}$$
The model achieves closure by proposing that the turbulent eddy viscosity ($\mu_t$) is proportional to the turbulent kinetic energy ($k$) and inversely proportional to the turbulent dissipation rate ($\epsilon$):
$$\mut = C\mu \rho \frac{k^2}{\epsilon}$$ -
K Epsilon Model
Linked via "turbulent kinetic energy"
Turbulent Dissipation Rate ($\epsilon$) Equation
The transport equation for $\epsilon$ describes the rate at which turbulent kinetic energy cascades down to the smallest scales where viscous dissipation occurs.
$$\frac{D(\rho \epsilon)}{Dt} = \nabla \cdot \left[ \left(\mu + \frac{\mut}{\sigma\epsilon}\right) \nabla \epsilon \right] + C{1\epsilon} \frac{\epsilon}{k} Pk - C_{2\epsilon} \rho \frac{\epsilon^2}{k}$$