# Governing Equations for Internal Turbulent Flow

(Difference between revisions)
 Revision as of 02:47, 8 July 2010 (view source) (Created page with 'The generalized governing equations for three-dimensional turbulent flow have been presented in Chapter 2 (see Section 2.5). For two-dimensional steady-state turbulent flow insid…')← Older edit Current revision as of 09:15, 27 July 2010 (view source) (4 intermediate revisions not shown) Line 1: Line 1: - The generalized governing equations for three-dimensional turbulent flow have been presented in Chapter 2 (see Section 2.5). For two-dimensional steady-state turbulent flow inside a cylindrical coordinate system (see Fig. 5.2), the governing equations are: + [[Image:Fig5.2.png|thumb|300 px|alt=Velocity profiles and friction factor in turbulent flow in a circular tube. | 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 areFaghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 14: Line 16: $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial r}=-\frac{1}{\rho }\frac{d\bar{p}}{dx}+\frac{1}{r}\frac{\partial }{\partial r}\left[ r(\nu +{{\varepsilon }_{M}})\frac{\partial \bar{u}}{\partial r} \right]$ $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial r}=-\frac{1}{\rho }\frac{d\bar{p}}{dx}+\frac{1}{r}\frac{\partial }{\partial r}\left[ r(\nu +{{\varepsilon }_{M}})\frac{\partial \bar{u}}{\partial r} \right]$ - |{{EquationRef|(1)}} + |{{EquationRef|(2)}} |} |} Line 22: Line 24: $\bar{u}\frac{\partial \bar{T}}{\partial x}+\bar{v}\frac{\partial \bar{T}}{\partial r}=\frac{1}{r}\frac{\partial }{\partial r}\left[ r(\alpha +{{\varepsilon }_{H}})\frac{\partial \bar{T}}{\partial r} \right]$ $\bar{u}\frac{\partial \bar{T}}{\partial x}+\bar{v}\frac{\partial \bar{T}}{\partial r}=\frac{1}{r}\frac{\partial }{\partial r}\left[ r(\alpha +{{\varepsilon }_{H}})\frac{\partial \bar{T}}{\partial r} \right]$ - |{{EquationRef|(1)}} + |{{EquationRef|(3)}} |} |} - which can be obtained by using the boundary layer theory similar to Chapter 4. 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. (5.267) and (5.268) 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: + 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: {| class="wikitable" border="0" {| class="wikitable" border="0" Line 31: Line 33: $-\overline{\rho {u}'{v}'}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial r}$ $-\overline{\rho {u}'{v}'}=\rho {{\varepsilon }_{M}}\frac{\partial \bar{u}}{\partial r}$ - |{{EquationRef|(1)}} + |{{EquationRef|(4)}} |} |} Line 39: Line 41: $-\rho {{c}_{p}}\overline{{v}'{T}'}=\rho {{c}_{p}}{{\varepsilon }_{H}}\frac{\partial \bar{T}}{\partial r}$ $-\rho {{c}_{p}}\overline{{v}'{T}'}=\rho {{c}_{p}}{{\varepsilon }_{H}}\frac{\partial \bar{T}}{\partial r}$ - |{{EquationRef|(1)}} + |{{EquationRef|(5)}} |} |} where ${u}'\text{, }{v}'\text{ and }{T}'$ 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. where ${u}'\text{, }{v}'\text{ and }{T}'$ 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== + {{Reflist}}

## Current revision as of 09:15, 27 July 2010

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.