Calculating Heat Flux numerically using Fourier's Law for a 2D Domain

Question

I have computed in a 2d mesh the velocity and temperature, denoted by the matrices U and T respectively. Say, for a square domain, how would I go about computing the individual heat flux components at each of the boundary points? I am currently using the FDM scheme $$ q = -k\nabla T $$ $$ q = -k \frac{T_{i+1,j}-T_{i,j}}{\Delta x} $$ but it seems to be yielding incorrect values.

Answer 1

Figured it out - revise average temperature on the basis of velocity - $T_m$ must also factor in $U_m$