A new discretization scheme of a Cartesian grid method for flow with heat transfer is proposed. The energy transport equation is discretized directly even in the boundary cells involving either the Dirichlet (isothermal) or the Neumann (iso-heat-flux/adiabatic) boundary conditions in order to ensure the energy conservation in those cells. The basic idea of this discretization is the same as the discretization scheme which is previously proposed by the present authors for boundary forcing in incompressible flow simulations. Moreover, the temperature gradients in both the normal and tangential directions at boundaries are required in the present method for representing the Neumann boundary condition on the Cartesian grids which do not necessarily coincide with the body geometries. The tangential components of the temperature gradients at boundaries are calculated by the extrapolations from the surrounding temperature field. Accuracy evaluations are conducted in a convective heat transfer problem in a flow between concentric cylindrical walls under the several different types of thermal boundary conditions applied at the inner and outer walls. It is confirmed that the present method significantly improves the accuracy orders for the temperature as well as the error magnitudes under both types of thermal boundary conditions. In particular, because the temperature gradients are correctly considered at the boundary cells where the Neumann boundary conditions are enforced, the same level of accuracy order is also maintained even in cases of non-uniformly distributed temperature at those boundaries.
