Governing Equations for Internal Turbulent Flow

Velocity profiles and friction factor in turbulent flow in a circular tube.

The generalized governing equations for three-dimensional turbulent flow have been presented in fundamentals of turbulence. For two-dimensional steady-state turbulent flow inside a cylindrical coordinate system (see figure on the right), the governing equations are[1]:

 $\frac{\partial \bar{u}}{\partial x}+\frac{1}{r}\frac{\partial (r\bar{v})}{\partial r}=0$ (1)
 $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial r}=-\frac{1}{\rho }\frac{d\bar{p}}{dx}+\frac{1}{r}\frac{\partial }{\partial r}\left[ r(\nu +{{\varepsilon }_{M}})\frac{\partial \bar{u}}{\partial r} \right]$ (2)
 $\bar{u}\frac{\partial \bar{T}}{\partial x}+\bar{v}\frac{\partial \bar{T}}{\partial r}=\frac{1}{r}\frac{\partial }{\partial r}\left[ r(\alpha +{{\varepsilon }_{H}})\frac{\partial \bar{T}}{\partial r} \right]$ (3)

which can be obtained by using the boundary layer theory. The second order derivatives of $\bar{u}\text{ and }\bar{T}$in the x-direction have been dropped based on the similar arguments for laminar flow in a duct. The time-averaged pressure is not a function of r, but is a function of x only. It can also be observed from eqs. (2) and (3) that the both momentum and energy diffusions are governed by molecular and eddy diffusions. Similar to the cases of external turbulent boundary layers, the momentum and thermal eddy diffusivities, ${{\varepsilon }_{M}}\text{ and }{{\varepsilon }_{H}}$, are defined as:

 $-\overline{\rho {u}'{v}'}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial r}$ (4)
 $-\rho {{c}_{p}}\overline{{v}'{T}'}=\rho {{c}_{p}}{{\varepsilon }_{H}}\frac{\partial \bar{T}}{\partial r}$ (5)

where u', v' and T' are the fluctuations of axial velocity, radial velocity, and temperature, respectively. Appropriate turbulent models in either algebraic or differential equation forms must be employed to obtain the eddy diffusivities.

References

1. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.