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Convective overshoot mixing is a critical ingredient of stellar structure models, but is treated in most cases by ad hoc extensions of the mixing-length theory for convection. Advanced theories which are both more physical and numerically treatable are needed. Convective flows in stellar interiors are highly turbulent. This poses a number of numerical challenges for the modelling of convection in stellar interiors. We include an effective turbulence model into a 1D stellar evolution code in order to treat non-local effects within the same theory. We use a turbulent convection model which relies on the solution of second order moment equations. We implement this into a state of the art 1D stellar evolution code. To overcome a deficit in the original form of the model, we take the dissipation due to buoyancy waves in the overshooting zone into account. We compute stellar models of intermediate mass main-sequence stars between 1.5 and 8 $M_\odot$. Overshoot mixing from the convective core and modified temperature gradients within and above it emerge naturally as a solution of the turbulent convection model equations. For a given set of model parameters the overshooting extent determined from the turbulent convection model is comparable to other overshooting descriptions, the free parameters of which had been adjusted to match observations. The relative size of the mixed cores decreases with decreasing stellar mass without additional adjustments. We find that the dissipation by buoyancy waves constitutes a necessary and relevant extension of the turbulent convection model in use.