Energy Dissipation Effect in the One-Dimensional Limit of the Energy Equation in Turbulent Compressible Flow

The transportation of natural gas through high pressure transmission pipelines has been modeled by numerically solving the conservation equations for mass, momentum, and energy for one-dimensional compressible viscous heat conducting flow. Since the one-dimensional version is a result of averages over the pipe cross-section and the flow is normally turbulent, the order of averaging in space and time is an issue; in particular, for the dissipation term. The Reynolds decomposition and time averaging should be performed first, followed by the contraction to the one-dimensional version by the cross-sectional averaging. The result is a correction factor, which is close to unity, on the usual expression of the dissipation term in the energy equation. This factor will, to some extent, affect the temperature distribution along the pipeline. For low Reynolds numbers (⁠$Re≃104$⁠) it reduces the dissipation by as much as 7%, irrespective of roughness. For high Reynolds numbers (Re ≥ 10⁷) and roughness in the high range of the micron decade, the dissipation is increased by 10%. If the pipeline is also thermally isolated such that the flow can be considered adiabatic, the effect of turbulent dissipation gains further importance.