The space-borne missions have provided a wealth of highly accurate data. However, our inability to properly model the upper-most region of solar-like stars prevents us from making the best of these observations. This problem is called surface effect and a key ingredient to solve it is turbulent pressure for the computation of both the equilibrium models and the oscillations. While 3D hydrodynamic simulations help to include properly the turbulent pressure in the equilibrium models, the way this surface effect is included in the computation of stellar oscillations is still subject to uncertainties. We aim at determining how to properly include the effect of turbulent pressure and its Lagrangian perturbation in the adiabatic computation of the oscillations. We also discuss the validity of the gas-gamma model (GGM) and reduced gamma model (RGM) approximations, which have been used to compute adiabatic oscillations of equilibrium models including turbulent pressure. We use a patched model of the Sun with an inner part constructed by a 1D stellar evolution code (CESTAM) and an outer part by the 3D hydrodynamical code (CO$^5$BOLD). Then, the adiabatic oscillations are computed using the ADIPLS code for the GGM and RGM and with the MAD code imposing the adiabatic condition on an existing time-dependent convection (TDC) formalism. We show that the computation of the oscillations using the TDC formalism in the adiabatic limit improves significantly the agreement with the observed frequencies compared to the GGM and RGM. Of the components of the turbulent pressure perturbation, the perturbation of the density and advection term is found to contribute most to the frequency shift. We propose a formalism to evaluate the frequency shift due to the inclusion of the term with the turbulent pressure perturbation in the variational principle in order to extrapolate our result to other stars.
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