The k-L turbulence model, where k is the turbulent kinetic energy and L represents the turbulent eddy scale length, is a two-equation turbulence model that has been proposed to simulate turbulence induced by Rayleigh-Taylor (RT) and Richtmyer Meshkov (RM) instabilities, which play an important role in the implosions of inertial confinement fusion (ICF) capsule targets. There are three free parameters in the k-L model, and in this paper, I calibrate them independently by comparing with RT and RM data from the linear electric motor (LEM) experiments together with classical Kelvin-Helmoholtz (KH) data. To perform this calibration, I numerically solved the equations of one-dimensional (1D) Lagrangian hydrodynamics, in a manner similar to that of contemporary ICF codes, together with the k-L turbulence model. With the three free parameters determined, I show that the k-L model is successful in describing both shear-driven and buoyancy-driven instabilities, capturing the experimentally observed separation between bubbles and spikes at high Atwood number for the RT case, as well as the temporal mix width recorded in RM shock tube experiments.