A two-scalar description of the thermochemistry of premixed reactants is applied to a turbulent stream impinging on a wall whose temperature is arbitrary. The two scalars are the mass fraction of fuel and the mixture enthalpy. The chemical kinetic characteristics are described in terms of a crossover temperature T* suggested by activation energy asymptotics and related to the mean rate of strain. The system of describing equations is shown to involve two parameters, one related to the turbulent intensity and a second to the reciprocal of the system Reynolds number. After developing the equations and applicable boundary conditions for arbitrary values of these parameters, their small magnitude in flows of applied interest is exploited in an asymptotic analysis, which leads to the identification of three flow regions and to a reduced set of equations for each region. The location of the flame in these flows depends on the relative chemical kinetic rate, that is, on T*. If the rate of strain is suitably low, the flame is located in the region most remote from the wall and the wall temperature cannot influence combustion. At increased rates of strain, that is, increased values of T*, combustion occurs in the second region but the wall temperature again has no influence on combustion provided the wall temperature T(0) is less than T*. When T(0) exceeds T*, combustion is confined to the region closest to the wall and these two temperatures and the extent of that combustion as measured by the fuel mass fraction at the wall are related. Thus, we find that the wall temperature influences combustion only if it exceeds the applicable kinetic temperature. Representative numerical results for each region are presented to illustrate these characteristics.