We consider the direct numerical simulation (DNS) of a homogeneously turbulent flow in combination with a premixed flame. The combustion takes place in the flamelet regime which means that combustion occurs in a very thin layer, called the flame front. The position of the flame front is modelled by means of the $G$-equation, in which the flame front is represented by an isosurface $G_0$ of a scalar field $G({\bm x},t)$. The flow is described by the Navier–Stokes equations in the low-Mach-number limit, which allows for the inclusion of gas expansion due to the temperature increase by the combustion. The advantage of the low-Mach-number approximation is that efficient numerical methods, used for incompressible flows, can be applied to solve the discretized equations.

The calculations are carried out in a box with homogeneous isotropic turbulence. In addition, a uniform mean velocity is imposed with a inflow boundary condition at $x\,{=}\,0$. The inflow velocity is adjusted such that the mean position of the flame is stabilized at a fixed position. This allows us to use time averaging to obtain accurate statistics, which are very difficult to obtain when the flame is allowed to propagate. In the $y$- and $z$-direction, periodic boundary conditions are applied.

The numerical code has been checked with a well-known theoretical result, the so-called Darrieus–Landau instability of a thin flame front. The results show a good agreement between the computed growth rate and the theoretical value when the thickness of the flame front is much smaller than the wavelength of the disturbance. When this condition is not met, the growth rate becomes lower than the theory in agreement with the restriction under which the theory is valid.

For the computations in homogeneous turbulence, the results show an increase in the turbulent flame speed with increasing turbulent intensity at the position of the flame front. This is in good agreement with experimental data and theory. The turbulent flame speed shows also an increase as a function of the heat release parameter. This is because disturbances on the flame front, induced by the turbulence, are enhanced by the Darrieus–Landau instability.

The budgets of the turbulent kinetic energy and the enstrophy show that the expansion of the gas across the flame front suppresses the turbulence. At higher expansion rates, turbulence in the direction of the mean velocity increases and as a result turbulence becomes strongly anisotropic. The increase is due to two processes. The first is the influence of the Darrieus–Landau instability already mentioned. The second is the baroclinic production of vorticity owing to the flame front density and pressure gradients not being aligned.