Figure 10 Influence of Turbulence on velocity history [5]
Reynolds equations
In the Navier – Stokes equations, the turbulence motion had been neglected and only the mean viscous stresses and the apparent turbulent stresses had been taken in the considerations. In this way, the laminar and turbulent fluid flows can be treated in a common frame work of the Navier – Stokes equations. Thus, if the turbulences stresses included in the equation of motion, then the resulted equation called the Reynolds equation.
From fluid mechanics textbooks; The normal stresses [6];
\(\sigma_{\text{xx}}=-P+2\mu\left(\frac{\partial u}{\partial x}\right)-\overset{\overline{}}{\rho{\overset{\acute{}}{u}}^{2}\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.3\right)\)
\(\sigma_{\text{yy}}=-P+2\mu\left(\frac{\partial v}{\partial y}\right)-\overset{\overline{}}{\rho{\overset{\acute{}}{v}}^{2}\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.4\right)\)
\(\sigma_{\text{zz}}=-P+2\mu\left(\frac{\partial u}{\partial x}\right)-\overset{\overline{}}{\rho{\overset{\acute{}}{w}}^{2}\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.5\right)\)
The shearing stresses [6];
\(\tau_{\text{xy}}=\tau_{\text{yx}}=\mu\left(\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}\right)-\overset{\overline{}}{\rho\overset{\acute{}}{u}\overset{\acute{}}{v}\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.6\right)\)
\(\tau_{\text{xz}}=\tau_{\text{zx}}=\mu\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)-\overset{\overline{}}{\rho\overset{\acute{}}{u}\overset{\acute{}}{w}\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.7\right)\)
\(\tau_{\text{yz}}=\tau_{\text{zy}}=\mu\left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)-\overset{\overline{}}{\rho\overset{\acute{}}{w}\overset{\acute{}}{v}\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.8\right)\)
If we substitute the above formulas eq. (4.3) – (4.8) in the Navier – Stokes equation, the Reynolds equations will be as indicated below;
\(\left(\frac{D\overset{\overline{}}{u}}{\text{Dt}}\right)=-\frac{1}{\rho}\frac{\partial\overset{\overline{}}{P}}{\partial x}+\nu\nabla^{2}\overset{\overline{}}{u}-\left[\frac{\partial}{\partial x}\left(\overset{\overline{}}{{\overset{\acute{}}{u}}^{2}}\right)+\frac{\partial}{\partial y}\left(\overset{\overline{}}{\overset{\acute{}}{u}\overset{\acute{}}{v}}\right)+\frac{\partial}{\partial z}\left(\overset{\overline{}}{\overset{\acute{}}{u}\overset{\acute{}}{w}}\right)\right]\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.9\right)\)
\(\left(\frac{D\overset{\overline{}}{v}}{\text{Dt}}\right)=-\frac{1}{\rho}\frac{\partial\overset{\overline{}}{P}}{\partial y}+\nu\nabla^{2}\overset{\overline{}}{v}-\left[\frac{\partial}{\partial x}\left(\overset{\overline{}}{\overset{\acute{}}{v}\overset{\acute{}}{u}}\right)+\frac{\partial}{\partial y}\left(\overset{\overline{}}{{\overset{\acute{}}{v}}^{2}}\right)+\frac{\partial}{\partial z}\left(\overset{\overline{}}{\overset{\acute{}}{v}\overset{\acute{}}{w}}\right)\right]\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.10\right)\)
\(\left(\frac{D\overset{\overline{}}{w}}{\text{Dt}}\right)=-\frac{1}{\rho}\frac{\partial\overset{\overline{}}{P}}{\partial z}+\nu\nabla^{2}\overset{\overline{}}{w}-\left[\frac{\partial}{\partial x}\left(\overset{\overline{}}{\overset{\acute{}}{w}\overset{\acute{}}{u}}\right)+\frac{\partial}{\partial y}\left(\overset{\overline{}}{\overset{\acute{}}{w}\overset{\acute{}}{v}}\right)+\frac{\partial}{\partial z}\left(\overset{\overline{}}{{\overset{\acute{}}{w}}^{2}}\right)\right]\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(4.11\right)\)
In the present work, Navier – Stokes equation will be used instead of Reynolds equations for the boundary layer analysis as it will be demonstrated in the next section.
  1. Estimation of boundary layer characteristics
  2. Displacement thickness\(\mathbf{\delta}^{\mathbf{*}}\)
It is the distance (y) by which the external free stream is effectively displaced to formulation of boundary layer.