The aim of this study is to evaluate a three-equation turbulence model applied to pipe flow. Uncertainty is approximated by comparing with published direct numerical simulation results for fully-developed average pipe flow. The model is based on the Reynolds averaged Navier-Stokes equations. Boussinesq hypothesis is invoked for determining the Reynolds stresses. Three local length scales are solved, based on which the eddy viscosity is calculated. There are two parameters in the model; one accounts for surface roughness and the other is possibly attributed to the fluid. Error in the mean axial velocity and Reynolds stress is found to be negligible.
The problem of turbulence dates back to the days of Claude-Louis Navier and George Gabriel Stokes, as well as others in the early nineteenth century. Searching for its solution, it was a source of great despair for many notably great scientists, including Werner Heisenberg, Horace Lamb, and many others. The complete description of turbulence remains one of the unsolved problems in modern physics. A great deal of early work on turbulence can be found, for example, in Hinze [
Recently, direct numerical simulation (DNS) has emerged as an indispensible tool to tackle turbulence directly, albeit at relatively low Reynolds numbers. Several DNS studies on turbulent flow have been performed recently, including Eggels et al. [
While DNS and LES are fairly accurate for modeling turbulent flows, they remain limited to relatively low-range Reynolds numbers. This drawback explains the wide-spread of turbulence modeling in industrial applications where the use of DNS techniques remains formidable. Turbulence modeling includes eddy viscosity models which utilize the Boussinesq hypothesis [
A second-order turbulence model, which also falls under RANS methods, is the Reynolds stress model. While the model relaxes the isotropic assumption, it remains more complicated with many unknown terms. For more on the subject of turbulence modeling, the reader is referred to, for example, Launder and Spalding [
In this paper, the accuracy of a three-equation turbulence model is assessed. Using the model, average turbulent flow through a pipe is simulated for Reynolds number of 44,000. Uncertainty is approximated by comparing with DNS results of Wu and Moin [
Starting with the incompressible Navier-Stokes equations in Cartesian index notation, and with Reynolds decomposition, averaging, and following Boussinesq hypothesis, we have
For simplicity, the normal stresses (except for the thermodynamic pressure) and body forces are neglected.
Hence, three local length scales are solved for, based on which the eddy viscosity is calculated. There are three equations for the turbulence length scales with their sources being the average strain rate, along with the molecular viscosity. C1 is a length parameter perhaps attributed to the fluid. C2 is another length parameter attributed to wall roughness.
The axisymmetric form of equations (1)-(3) were solved with a finite-volume solver using Gauss-Seidel iterative method, in conjunction with second-order schemes. 20,000 structured cells were used with y+ down to 0.4,
Along with DNS results of Wu and Moin [
In this paper, the accuracy of a three-equation turbulence model was assessed. Using the turbulence model, average turbulent flow through a pipe was simulated for Reynolds number of 44,000. Model results for mean axial velocity and Reynolds stress were compared with DNS results. The agreement was excellent. While the model was tested on incompressible axisymmetric flow, testing of the model is needed on more complex flows.