A new OpenFOAM solver is presented, for the simulation of plasma cutting torches. The mathematical model that is introduced is based on the compressible Navier-Stokes equations coupled via source terms to the electric current conservation equation. Due to the conservative and hyperbolic nature of the model, a Godunov-type scheme is used for the first time in the context of plasma cutting simulation. The numerical method consists of a second-order Total Variation Diminishing (TVD) integration with flux Harten-Lax-van Leer-Contact (HLLC) Riemann solver for the flow conservation equations, coupled with a Laplace solver for the current conservation equation. An efficient formulation for the equation of state, accurately taking into account the plasma properties, is also presented. The solver is validated through a set of canonical test cases (shock tubes and 2D Riemann problems) and it is used to simulate a three-dimensional plasma cutting torch. Good agreement is found with the literature, with an improvement in the ability to deal with the shocks occurring during plasma cutting.