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Supernova explosions, active galactic nuclei jets, galaxy--galaxy interactions and cluster mergers can drive turbulence in the circumgalactic medium (CGM) and in the intracluster medium (ICM). However, the exact nature of turbulence forced by these sources and its impact on the different statistical properties of the CGM/ICM and their global thermodynamics is still unclear. To investigate the effects of different types of forcing, we conduct high resolution ($1008^3$ resolution elements) idealised hydrodynamic simulations with purely solenoidal (divergence-free) forcing, purely compressive (curl-free) forcing, and natural mixture forcing (equal fractions of the two components). The simulations also include radiative cooling. We study the impact of the three different forcing modes (sol, comp, mix) on the morphology of the gas, its temperature and density distributions, sources and sinks of enstrophy, i.e., solenoidal motions, as well as the kinematics of hot ($\sim10^7~\mathrm{K}$) X-ray emitting and cold ($\sim10^4~\mathrm{K}$) H$α$ emitting gas. We find that compressive forcing leads to stronger variations in density and temperature of the gas as compared to solenoidal forcing. The cold phase gas forms large-scale filamentary structures for compressive forcing and misty, small-scale clouds for solenoidal forcing. The cold phase gas has stronger large-scale velocities for compressive forcing. The natural mixture forcing shows kinematics and gas distributions intermediate between the two extremes, the cold-phase gas occurs as both large-scale filaments and small-scale misty clouds.