For simple turbulent shear flows, Boussinesq in 1877 proposed the idea of eddy viscosity (also called turbulent viscosity)

A simple turbulent shear flow is one in which one single shear-rate component dominates; examples include a two-dimensional turbulent mixing layer, a two-dimensional or axisymmetric jet, an axisymmetric wake, and a two-dimensional boundary layer over a wall. In these cases the Reynolds shear stress $-\rho\,\overline{u_1 u_2}$ plays the main role in the mean flow. By analogy with Newton's law of viscosity for a Newtonian fluid, Boussinesq proposed the turbulent viscosity hypothesis $$ -\rho\,\overline{u_1 u_2} \;=\; \mu_t\,\frac{\partial U_1}{\partial x_2} $$ where $\mu_t$ is the turbulent viscosity coefficient or eddy viscosity coefficient

For more complex turbulent flows, by analogy with the Newtonian law one further assumes that the Reynolds-stress tensor is related to the mean rate-of-strain tensor by $$ \overline{u_i u_j} =\frac{2}{3}\,k\,\delta_{ij} -\nu_t\!\left(\frac{\partial U_i}{\partial x_j} +\frac{\partial U_j}{\partial x_i}\right) $$ where $k$ is the turbulent kinetic energy per unit mass, $\delta_{ij}$ is the Kronecker delta, and $\nu_t=\mu_t/\rho$ is the kinematic eddy viscosity