Turbulent Boundary Layer Equations

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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, the governing equations can be simplified to:

 $\frac{\partial \bar{u}}{\partial x}+\frac{\partial \bar{v}}{\partial y}=0$ (1)
 $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial y}=-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial x}+\nu \left( \frac{\partial ^{2}\bar{u}}{\partial x^{2}}+\frac{\partial ^{2}\bar{u}}{\partial y^{2}} \right)-\left( \frac{\partial \overline{{u}'^{2}}}{\partial x}+\frac{\partial \overline{{v}'{u}'}}{\partial y} \right)$ (1)
 $\bar{u}\frac{\partial \bar{v}}{\partial x}+\bar{v}\frac{\partial \bar{v}}{\partial y}=-\frac{1}{\rho }\frac{\partial \bar{p}}{\partial y}+\nu \left( \frac{\partial ^{2}\bar{v}}{\partial x^{2}}+\frac{\partial ^{2}\bar{v}}{\partial y^{2}} \right)-\left( \frac{\partial \overline{{v}'^{2}}}{\partial x}+\frac{\partial \overline{{u}'{v}'}}{\partial y} \right)$ (1)
 $\bar{u}\frac{\partial \bar{T}}{\partial x}+\bar{v}\frac{\partial \bar{T}}{\partial y}=\alpha \left( \frac{\partial ^{2}\bar{T}}{\partial x^{2}}+\frac{\partial ^{2}\bar{T}}{\partial y^{2}} \right)-\left( \frac{\partial \overline{{u}'{T}'}}{\partial x}+\frac{\partial \overline{{v}'{T}'}}{\partial y} \right)$ (1)
 $\bar{u}\frac{\partial \bar{\omega }}{\partial x}+\bar{v}\frac{\partial \bar{\omega }}{\partial y}=D\left( \frac{\partial ^{2}\bar{\omega }}{\partial x^{2}}+\frac{\partial ^{2}\bar{\omega }}{\partial y^{2}} \right)-\left( \frac{\partial \overline{{u}'{\omega }'}}{\partial x}+\frac{\partial \overline{{v}'{\omega }'}}{\partial y} \right)$ (1)

To obtain the turbulent boundary layer governing equations, scale analysis can be performed to eqs. (4.388) – (4.391). While the treatments of the time-averaged quantities are similar to the cases of laminar flow, special attention must be paid to the time averaging of the products of the fluctuations. It can be shown through a scale analysis that the first terms in the last parentheses on the right hand side of eqs. (4.388) – (4.391) are negligible compared to the second terms in the parentheses (Oosthuizen and Naylor, 1999). Therefore, eqs. (4.388) – (4.391) can be simplified to:

 $\bar{u}\frac{\partial \bar{u}}{\partial x}+\bar{v}\frac{\partial \bar{u}}{\partial y}=-\frac{1}{\rho }\frac{d\bar{p}}{dx}+\nu \frac{\partial ^{2}\bar{u}}{\partial y^{2}}-\frac{\partial \overline{{v}'{u}'}}{\partial y}$ (1)
 $\bar{u}\frac{\partial \bar{T}}{\partial x}+\bar{v}\frac{\partial \bar{T}}{\partial y}=\alpha \frac{\partial ^{2}\bar{T}}{\partial y^{2}}-\frac{\partial \overline{{v}'{T}'}}{\partial y}$ (1)
 $\bar{u}\frac{\partial \bar{\omega }}{\partial x}+\bar{v}\frac{\partial \bar{\omega }}{\partial y}=D\frac{\partial ^{2}\bar{\omega }}{\partial y^{2}}-\frac{\partial \overline{{v}'{\omega }'}}{\partial y}$ (1)

where eq. (4.389) became $\partial \bar{p}/\partial y=0$ and the partial derivative of time-averaged pressure in eq. (4.388) became: $\partial \bar{p}/\partial x=d\bar{p}/dx$ , which has been reflected in eq. (4.392). When molecules or eddies in a turbulent flow cross a control surface, they will carry momentum with them. Thus, the shear stress in a turbulent flow can be caused by both molecular and eddy level activities. While the molecular level activity is the only mechanism of shear stress, transport of momentum by eddies can only be found in turbulent flow. The time-averaged shear stress tensor can be expressed as

 $\mathbf{\bar{\tau }}=\mathbf{\bar{\tau }}^{\text{m}}+\mathbf{\bar{\tau }}^{\text{t}}$ (1)

where $\mathbf{\bar{\tau }}^{\text{m}}$ is the contribution of the molecular motion, and $\mathbf{\bar{\tau }}^{\text{t}}$ is caused by eddy level activity – referred to as Reynolds stress. For two-dimensional flow, the shear stress can be expressed as

 $\bar{\tau }_{yx}=\mu \frac{\partial \bar{u}}{\partial y}-\overline{\rho {u}'{v}'}$ (1)

Similarly, the heat flux in the turbulent flow can also be caused by molecular level activity (conduction) and eddy level activity. The time-averaged heat and mass flux can be respectively expressed as

 Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): {\bar{q}}''_{y}={\bar{q}}''_{y}^{m}+{\bar{q}}''_{y}^{t}=-k\frac{\partial \bar{T}}{\partial y}+\rho c_{p}\overline{{v}'{T}'} (1)

The mass flux can be obtained by a similar way:

 $\bar{{\dot{m}}''}_{y}=\bar{{\dot{m}}''}_{y}^{m}+\bar{{\dot{m}}''}_{y}^{t}=-\rho D\frac{\partial \bar{\omega }_{1}}{\partial y}+\rho \overline{{v}'{\omega }'_{1}}$ (1)