Advertisement

Numerical simulation of thermally developing turbulent flow through a cylindrical tube

  • Ali Belhocine
  • Oday Ibraheem Abdullah
Original Paper

Abstract

A numerical study was conducted using the finite difference technique to examine the mechanism of energy transfer as well as turbulence in the case of fully developed turbulent flow in a circular tube with constant wall temperature and heat flow conditions. The methodology to solve this thermal problem is based on the energy equation a fluid of constant properties in an axisymmetric and two-dimensional stationary flow. From the mathematical side, a numerical technique for solving the problem of fluid–structure interaction with a fully developed turbulent incompressible Newtonian flow is described. The global equations and the initial and boundary conditions acting on the problem are configured in dimensionless form in order to predict the characteristics of the turbulent fluid flow inside the tube. Using Thomas’ algorithm, a program in FORTRAN was developed to numerically solve the discretized form of the system of equations describing the problem. Finally, using this elaborate program, we were able to simulate the flow characteristics, for changing parameters such as Reynolds, Prandtl and Peclet numbers along the pipe to obtain the important thermal model. These are discussed in detail in this work. Comparison of the results to published data shows that results are a good match to the published quantities.

Keywords

Finite difference method Nusselt number Fully developed turbulent flow Reynolds number Pipe flow 

List of symbols

Aj

Coefficient in Eq. (36)

Bj

Coefficient in Eq. (36)

Cj

Coefficient in Eq. (36)

Cp

Specific heat at constant pressure (J kg−1 K−1)

C1, C2

k–ε model constants

D

Inner diameter (m)

Dj

Coefficient in Eq. (36)

E

Inner energy (J kg−1), dimensionless variable

f

Fanning friction factor

F

Function

k

Turbulent kinetic energy (J kg−1)

L

Tube length (m)

M

Tridiagonal matrix of dimensions (N × N)

NuD

Nusselt number

NuiD

Local Nusselt number

P

Mean pressure (Pa)

Pr

Prandtl number

Prt

Turbulent Prandtl number

qω

Heat transfer rate at the wall

r

Radial coordinate (m)

R

Dimensionless radial coordinate

ReD

Reynolds number

t

Time (s)

T

Temperature (K)

Tb

Bulk temperature (K)

Tc

Centerline temperature (K)

Ti

Initial/entrance temperature (K)

Tω

Wall temperature (K)

\( \bar{T} \)

Wall temperature (K)

uc

Centerline mean velocity (m s−1)

ui

Mean velocity component (m s−1)

\( \bar{u} \)

Mean velocity (m s−1)

\( \bar{u}_{m} \)

Average velocity (m s−1)

U

Dimensionless velocity

\( \bar{v} \)

Radial velocity component (m s−1)

xi

Cartesian coordinate (m)

y +

Dimensionless distance from cell center to the nearest wall

z

Axial coordinate (m)

Z

Dimensionless axial coordinate

Greek symbols

α

Thermal diffusivity (m2 s−1)

δij

Kronecker symbol 

ρ

Density of fluid (kg m−3)

θ

Dimensionless temperature

ε

Turbulent dissipation rate (m3 s−2)

єh

Eddy viscosity (kg m−1 s−1)

µ

Dynamic viscosity (kg m−1 s−1)

µt

Eddy viscosity (kg m−1 s−1)

Φ

Scalar quantities

τω

Wall-shear stress

λ

Thermal conductivity (W m−1 K−1)

Subscripts

i, j, k

Unit direction vector of Cartesian coordinates

local

Local value

out

Outlet

t

Turbulence

wall

Tube wall

Notes

Acknowledgements

The authors declare that they have no conflicts of interest in conducting work to any organisation or funding bodies. The authors would like to thank the reviewers for their valuable comments.

Compliance with ethical standards

Conflict of interest

The authors declares there is no conflict of interest.

References

  1. 1.
    Cagin, S., Fischer, X., Delacourt, E., Bourabaa, N., Morin, C., Coutellier, D., Carre, B., Loume, S.: A new reduced model of scavenging to optimize cylinder design. Simul T. Soc. Mod. Sim. 92(6), 1–14 (2016)Google Scholar
  2. 2.
    Cagin, S., Bourabaa, N., Delacourt, E., Morin, C., Fischer, X., Coutellier, D., Carré, B., Loumé, S.: Scavenging process analysis in a 2-stroke engine by CFD approach for a parametric 0D model development. J. Appl. Fluid. Mech. 9(1), 69–80 (2016)Google Scholar
  3. 3.
    Choi, H., Moin, P.: Effects of the computational time step on numerical solutions of turbulent flow. J. Comp. Phys. 113, 1–4 (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    García, M., Duque, J., Pierre, B., Figueroa, P.: Computational steering of CFD simulations using a grid computing environment. Int. J. Interact. Des. Manuf. 9(3), 235–245 (2015)CrossRefGoogle Scholar
  5. 5.
    Cucinotta, F., Nigrelli, V., Sfravara, F.: Numerical prediction of ventilated planing flat plates for the design of air cavity ships. Int. J. Interact. Des. Manuf. 12, 1–12 (2017)Google Scholar
  6. 6.
    Cherrad, N., Benchabane, A.: Interactive process to control the evaporating temperature of refrigerant for solar adsorption cooling machine with new correlation. Int. J. Interact. Des. Manuf. 12, 1–7 (2017)Google Scholar
  7. 7.
    Bahar, Y.N., Landrieu, J., Pére, C., Nicolle, C.: CAD data workflow toward the thermal simulation and visualization in virtual reality. Int. J. Interact. Des. Manuf. 8(4), 283–292 (2014)CrossRefGoogle Scholar
  8. 8.
    Amaya, A.F.D., Torres, A.G.D., Maya, D.A.A.: First and second thermodynamic law analyses applied to spark ignition engines modelling and emissions prediction. Int. J. Interact. Des. Manuf. 10(4), 401–415 (2016)CrossRefGoogle Scholar
  9. 9.
    Zhang, W., Cheng, C., Du, X., Chen, X.: Experiment and simulation of milling temperature field on hardened steel die with sinusoidal surface. Int. J. Interact. Des. Manuf. 12, 1–9 (2017)Google Scholar
  10. 10.
    Afzal, N.: Wake layer in a thermal turbulent boundary layer with pressure gradient. Heat Mass Transf. 35(4), 281–288 (1999)CrossRefGoogle Scholar
  11. 11.
    Gbadebo, S.A., Said, S.A.M., Habib, M.A.: Average Nusselt number correlation in the thermal entrance region of steady and pulsating turbulent pipe flows. Heat Mass Transf. 35(5), 377–381 (1999)CrossRefGoogle Scholar
  12. 12.
    Yan, B.H., Gu, H.Y., Yu, L.: Numerical research of turbulent heat transfer in rectangular channels in ocean environment. Heat Mass Transf. 47(7), 821–831 (2011)CrossRefGoogle Scholar
  13. 13.
    Yan, B.H., Yu, Y.Q., Gu, H.Y., Yang, Y.H., Yu, L.: Simulation of turbulent flow and heat transfer in channels between rod bundles. Heat Mass Transf. 47(3), 343–349 (2011)CrossRefGoogle Scholar
  14. 14.
    Weigand, B., Wrona, F.: The extended Graetz problem with piecewise constant wall heat flux for laminar and turbulent flows inside concentric annuli. Heat Mass Transf. 39(4), 313–320 (2003)CrossRefGoogle Scholar
  15. 15.
    Habib, M.A., Attya, A.M., Said, S.A.M., Eid, A.I., Aly, A.Z.: Heat transfer characteristics and Nusselt number correlation of turbulent pulsating pipe air flows. Heat Mass Transf. 40(3–4), 307–318 (2004)CrossRefGoogle Scholar
  16. 16.
    Hui, G., Liejin, G.: Numerical investigation of developing turbulent flow in a helical square duct with large curvature. J. Therm. Sci. 10(1), 1–6 (2001)CrossRefGoogle Scholar
  17. 17.
    Roy, G., Vo-Ngoc, D., Bravine, V.: A numerical analysis of turbulent compressible radial channel flow with particular reference to pneumatic controllers. J. Therm. Sci. 13(1), 24–29 (2004)CrossRefGoogle Scholar
  18. 18.
    Eiamsa-ard, S., Changcharoen, W.: Analysis of turbulent heat transfer and fluid flow in channels with various ribbed internal surfaces. J. Therm. Sci. 20(3), 260–267 (2011)CrossRefGoogle Scholar
  19. 19.
    Taler, D.: Simple power-type heat transfer correlations for turbulent pipe flow in tubes. J. Therm. Sci. 26(4), 339–348 (2017)CrossRefGoogle Scholar
  20. 20.
    Tian, R., Dai, X., Wang, D., Shi, L.: Study of variable turbulent Prandtl number model for heat transfer to supercritical fluids in vertical tubes. J. Therm. Sci. 27(3), 213–222 (2018)CrossRefGoogle Scholar
  21. 21.
    Taler, D.: A new heat transfer correlation for transition and turbulent fluid flow in tubes. Int. J. Therm. Sci. 108(2016), 108–122 (2016)CrossRefGoogle Scholar
  22. 22.
    Belhocine, A., Wan Omar, W.Z.: Numerical study of heat convective mass transfer in a fully developed laminar flow with constant wall temperature. Case Stud. Therm. Eng. 6, 116–127 (2015)CrossRefGoogle Scholar
  23. 23.
    Belhocine, A.: Numerical study of heat transfer in fully developed laminar flow inside a circular tube. Int. J. Adv. Manuf. Technol. 85(9), 2681–2692 (2016)CrossRefGoogle Scholar
  24. 24.
    Belhocine, A., Wan Omar, W.Z.: Exact Graetz problem solution by using hypergeometric function. Int. J. Heat Technol. 35(2), 347–353 (2017)CrossRefGoogle Scholar
  25. 25.
    Belhocine, A., Abdullah, O.I.: Similarity and numerical analysis of the generalized Levèque problem to predict the thermal boundary layer. Int. J. Interact. Des. Manuf. 12(3), 235–245 (2018)CrossRefGoogle Scholar
  26. 26.
    Bryant, D.B., Sparrow, E.M., Gorman, J.M.: Turbulent pipe flow in the presence of centerline velocity overshoot and wall-shear undershoot. Int. J. Therm. Sci. 125, 218–230 (2018)CrossRefGoogle Scholar
  27. 27.
    Wilcox, D.C.: Turbulence Modeling for CFD, 2nd edn. DCW Industries, La Canada (1998)Google Scholar
  28. 28.
    Kays, W.M., Crawford, M.E.: Convective Heat and Mass Transfer, 3rd edn. McGraw-Hill, New York (1993)Google Scholar

Copyright information

© Springer-Verlag France SAS, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of Sciences and Technology of OranOranAlgeria
  2. 2.System Technologies and Mechanical Design MethodologyHamburg University of TechnologyHamburgGermany

Personalised recommendations