Fully-developed Turbulent Pipe Flow Using a Zero-Equation Model

Aim of this study is to evaluate a zero-equation turbulence model. A fully-developed turbulent pipe flow was simulated. Uncertainty was approximated through grid-independence and model validation. Results for mean axial velocity, u and Reynolds stress had maximum error of 5%, while results for the friction factor had negligible error. The mean axial velocity was shown to increase and extend farther in the outer layer with increasing Reynolds number, up to 10. There was no effect of Reynolds number on u below wall distance, Y, of 100. Similar to the friction velocity, peak of the Reynolds stress was shown to increase and extend farther in the outer layer with increasing Reynolds number. There was no effect of Reynolds number on Reynolds stress below wall distance of 20. The new turbulence model is equally applicable to developing and external flows using the same constant. For wallbounded flows, the constant is a function of wall roughness.


INTRODUCTION
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 (1975).
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 pipe flow have been performed recently, including Eggels et al. (1993), Loulou et al. (1997) and Wu and Moin (2008).The latter has carried out DNS on a turbulent pipe flow at Reynolds number of 44,000, which is the largest among the three studies.Mean velocity, Reynolds stresses and turbulent intensities are presented and discussed, along with visualization of flow structure.Good agreement was attained with the Princeton Superpipe data on mean flow statistics and Lawn (1971) data on turbulence intensities.Large Eddy Simulation (LES) is another tool that somewhat bridges between DNS and Reynolds-averaged Navier-Stokes (RANS) methods.In LES, large turbulent structures in the flow field are resolved, while the effect of Sub-Grid Scales (SGS) are modeled.LES investigation, for example, has been carried out by Rudman and Blackburn (1999) using LES on a turbulent pipe flow at Reynolds number of 38,000.Mean velocity and Reynolds stresses are presented and discussed, along with visualization of flow structure.Results were reported to compare favorably with measurements.
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, Hinze (1975), for relating the Reynolds stresses to the average flow filed.In turn, the eddy viscosity is determined by using any of a variety of techniques, including the zero-equation, one-equation and two-equation models, most notably the k- model.While such models vary in complexity, they share several shortcomings, including isotropy of the eddy viscosity and the lack of generality in wall treatment.Such shortcomings lead to poor results in separated flows and other non-equilibrium turbulent boundary layers, Yamamoto et al. (2008).
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 and costly due to the need for solving six additional transport equations along with many unknown terms.For more on the subject of turbulence modeling, the reader is referred to, for example, Launder and Spalding (1972).
In this study, the accuracy of a zero-equation turbulence model is assessed.Unlike typical eddyviscosity models, the proposed model does not require the solution of additional transport equations and requires one constant which is strictly a function of wall roughness.Moreover, the model does not require a wall function because the momentum equation is integrated throughout the flow field.The new model is equally applicable to external flows.For simplicity, steady, axisymmetric and fully-developed pipe flow is considered.
Theory: 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.μ t = CRe t μ is the eddy viscosity (Alammar, 2008).C is a non-dimensional function of the wall roughness.For a smooth wall, it is a constant.For isotropic roughness, it is a different constant.
and d is the distance to the nearest wall.C, therefore, takes the value of that location at the wall.When Eq. ( 2) is normalized, the shear stresses result in the following: The second term in the brackets is a nondimensional number attributed to turbulence.Clearly, this term dominates at high Reynolds numbers.In absence of walls, one plausible length scale would be the mean free path.This would give rise to secondorder effects that would be negligible in presence of walls.
Using cylindrical coordinates, Fig. 1 and assuming steady, axisymmetric and fully-developed pipe flow, we have (after integration once with respect to r): The constant of integration vanishes due to symmetry condition at the center point.The eddy viscosity is given by: where c = 0.016, d is the distance from the wall and the pressure gradient is constant.

Numerical procedure and uncertainty analysis:
There are mainly two sources of uncertainty in Computational Fluid Dynamics (CFD), namely modeling and numerical, Stern et al. (1999).Modeling uncertainty can be approximated through theoretical or experimental validation while numerical uncertainty can be approximated through grid independence.
Numerical uncertainty has two main sources, namely truncation and round-off errors.Higher order schemes have less truncation error.In explicit schemes, roundoff error increases with increasing iterations and is reduced by increasing significant digits (machine precision).Equation (4) was solved using the Euler secondorder algorithm with the no-slip boundary condition.The fully-developed mean axial velocity is depicted in Fig. 2 for Reynolds number of 44,000, along with DNS results of Wu and Moin (2008).The discrepancy is <±5%.Such discrepancy could be attributed, in part, to transitional effects in the buffer layer.At the center of the pipe, the discrepancy could be attributed to the enforcement of symmetry condition.The Reynolds stress is shown in Fig. 3 for Reynolds number of 44,000.Again, good agreement is attained between the current simulation and the DNS results, with the discrepancy restricted within the buffer layer and is <±5%.
A grid-independence test is depicted in Fig. 4 for three different, non-uniform cell sizes, namely 500, 1000 and 2000.The error in u + is mostly in the laminar sub-layer and is shown to be negligible with 2000 cells for Reynolds number of 10 6 .Hence, we can conclude that the overall uncertainty in the current numerical results is ±5%.

RESULTS AND DISCUSSION
The fully-developed mean axial velocity is depicted in Fig. 5 for various Reynolds numbers up to 10 6 .The mean velocity is shown to increase and extend farther in the outer layer with increasing Reynolds number.This is in agreement with published measurements, e.g., Laufer (1954).There is no  Moody (1944).The agreement is excellent.The Reynolds stress is depicted in Fig. 6 for various Reynolds numbers up to 10 6 .Similar to the mean velocity, peak of the Reynolds stress is shown to increase and extend farther in the outer layer with increasing Reynolds number.There is no effect of Reynolds number on the profiles below wall distance of 20.This is different from the case of u + where the change was negligible below wall distance of 100.

CONCLUSION
Using a zero-equation turbulence model, fullydeveloped turbulent pipe flow was simulated.Results for the mean axial velocity and Reynolds stress had maximum error of 5%, while results for the friction factor had negligible error.Mean axial velocity was shown to increase and extend farther in the outer layer with increasing Reynolds number.There was no effect of Reynolds number on u + below wall distance of 100.Similar to the mean axial velocity, peak of the Reynolds stress was shown to increase and extend farther in the outer layer with increasing Reynolds number.There was no effect of Reynolds number on the profiles below wall distance of 20.The new turbulence model is equally applicable to developing and external flows using the same constant for smooth walls.For wallbounded flows, the constant is a function of wall roughness.This study was conducted in the year 2012 at the college of Engineering, King Saud University, Riyadh main campus.

Fig. 1 :
Fig. 1: Schematic of the pipe with fully-developed flow

Fig. 5 :
Fig. 5: Mean axial velocity profiles and friction factor for various Reynolds numbers C = A non-dimensional function of wall roughness D = Pipe diameter, m d = Normal distance from the wall, m f = Friction factor = τ w/ 0.5ρU 2 Re = Reynolds number = UρD/μ Re t = Non-dimensional parameter = velocity component, m/s u = Mean axial velocity, m/s u + = Normalized mean axial velocity = u/U * x i = Cartesian coordinate, m y + = Non-dimensional wall distance = rU * ρ/μ r = Radial distance, m μ = Fluid dynamic viscosity, Pa s ρ = Fluid density, kg/m 3 τ w = Wall shear stress, Pa