Numerical Study of Nusselt Number in a Heated Pipe With the Use of Variable-Order Resolution

We present results for a numerical study of turbulent heat transfer in a pipe with a constant heat flux at the wall. Nusselt numbers are computed for Reynolds numbers between 5,000 and 15,000 over a wide range of Prandtl numbers and the results are compared to the Dittus-Boelter relation. The simulations are based on the spectral element method in which the velocity and pressure are represented by tensor-product polynomials of degree $N$ in each of $E$ elements. Typical values of $N$ are in the range 4 to 20 and, for this study, $E$ is between 4000 and 12000. We examine potential savings of using elevated resolution for the temperature field only, which is particularly interesting for the case $Pr > 1$. Specifically, for water flow, one has as Peclet to Reynolds number ratio of approximately six, which implies a need for elevated resolution of the temperature field. This study explores the relative convergence rates and costs when the polynomial order for the temperature field, $N_t$, is increased with respect to $N$. We identify the optimal ratio, $N_t/N$, as a function of Prandtl number as grid convergence is attained.