# Numerical simulation of thermally developing turbulent flow through a cylindrical tube

## 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

*A*_{j}Coefficient in Eq. (36)

*B*_{j}Coefficient in Eq. (36)

*C*_{j}Coefficient in Eq. (36)

- C
_{p} Specific heat at constant pressure (J kg

^{−1}K^{−1})*C*_{1},*C*_{2}k–ε model constants

*D*Inner diameter (m)

*D*_{j}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)

*Nu*_{D}Nusselt number

*Nu*_{iD}Local Nusselt number

*P*Mean pressure (Pa)

*Pr*Prandtl number

*Pr*_{t}Turbulent Prandtl number

*q*_{ω}Heat transfer rate at the wall

*r*Radial coordinate (m)

*R*Dimensionless radial coordinate

*Re*_{D}Reynolds number

*t*Time (s)

*T*Temperature (K)

*T*_{b}Bulk temperature (K)

*T*_{c}Centerline temperature (K)

*T*_{i}Initial/entrance temperature (K)

*T*_{ω}Wall temperature (K)

- \( \bar{T} \)
Wall temperature (K)

*u*_{c}Centerline mean velocity (m s

^{−1})*u*_{i}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})*x*_{i}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 (m

^{2}s^{−1})*δ*_{ij}Kronecker symbol

*ρ*Density of fluid (kg m

^{−3})*θ*Dimensionless temperature

*ε*Turbulent dissipation rate (m

^{3}s^{−2})*є*_{h}Eddy viscosity (kg m

^{−1}s^{−1})*µ*Dynamic viscosity (kg m

^{−1}s^{−1})*µ*_{t}Eddy viscosity (kg m

^{−1}s^{−1})*Φ*Scalar quantities

*τ*_{ω}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.

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