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3-2Hi-Re-bl.tex
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\section{Viscous Flow at High Reynolds Numbers}
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Let us first give a heuristic estimates of boundary layer in steady flows.

Consider a particle near the wall to be influenced by viscosity.  After traveling a distance $x$ from the edge, it has been under viscous influence for a time of $t = x/U$.  Let $U$ be large.  For finite $x$, $t$ is small so that vorticity is spread sideways to the width $\lp \nu t \rp^{1/2} \sim \lp \nu x/U \rp^{1/2}$.  Let us define this width to be the boundary layer, which has   thickness $ \delta  = O\lp \nu x/U \rp^{1/2}$.

Alternatively we start from Navier-Stokes equations :
\be \f{\p u}{\p x} + \f{\p v}{\p y} = 0 \ee\be  u \, \f{\p u}{\p x} + v \, \f{\p u}{\p y} = - \f{1}{\rho} \, \f{\p p}{\p x} + \nu \, \lp \f{\p^2 u}{\p x^2} + \f{\p^2 u}{\p y^2} \rp \ee
\be  u \, \f{\p v}{\p x} + v \, \f{\p v}{\p y} = - \f{1}{\rho} \, \f{\p p}{\p y} + \nu \, \lp \f{\p^2 v}{\p x^2} + \f{\p^2 u}{\p y^2} \rp \ee
When viscosity is important $y = O(\delta), x =O(L)$, convective inertia is comparable to    viscous stresses.

 From continuity
\[ \f{u}{L} \sim \f{v}{\delta} \]
From $x-$momentum
\begin{eqnarray*}
u \, u_x & \sim & \nu \, u_{yy} \\
\f{U^2}{L} & \sim & \nu \, \f{U}{\delta^2} \end{eqnarray*} 
Therefore,
\be \delta   \sim  \lp \nu L/U \rp^{1/2} \ee 
and \be \f{\delta}{L}  \sim   \lp \f{\nu}{UL} \rp^{1/2} =  {Re}^{1/2}  . \ee
Shear stress on the awall :
\[  \f{\tau_0}{\rho} = \nu \left. \f{\p u}{\p y} \right|_0 = \nu \, \f{U}{\delta}  \sim \, \nu U \, \sqrt{\f{\overline{U}}{\nu L}}\]
Hence the drag coefficient is, 
\[ C_D = \f{\tau_0}{\f{1}{2} \rho U^2} = 2 \, \sqrt{\f{\overline{\nu}}{Ux}} = \f{2}{\mbox{Re} } . \]
For water $\nu = 10^{-5} \, \mbox{ft}^2/\mbox{sec}$.
Let $U = 1 \; \mbox{ft/sec} \quad L = 1 \; \mbox{ft}$, then $\mbox{Re} = 10^5$.  Hence,
\[ O\lp \f{\delta}{L} \rp \, \propto \, \f{1}{\sqrt{\mbox{Re}}} \, \sim \, \f{1}{3} \, 10^{-2} \qquad (\delta \, \sim \, 0.003 \; \mbox{ft}) \]
and
\[  {C_D \, \sim \, 0.003}. \]

Experiments for flat plates (Schlichting, p. 133) show that: $C_D \, \sim \, 0.002$, but experiments for a circular cylinder show that $C_D \, \approx \, 0(1)$ because flow is separated for  most $\mbox{Re}$ .

\subsection{Systematic Boundary-layer Approximation}

Let $u = O(U), \; x = O(L), \; y = 0(\delta)$.  From continuity, $v = O(U\delta /L)$.
  Let $u \rightarrow Uu, \; v \rightarrow \, \f{U\delta}{L} \, v, \; x \rightarrow \, Lx, \; y \rightarrow \, \delta y$
\be \f{U}{L} \, \lp u_x + v_y \rp = 0 . \label{Eq:19.1} \ee
\be \f{U^2}{L} \, \lp uu_x + vu_y \rp = - \f{P}{\rho L} \, \f{\p p}{\p x} + \f{\nu U}{L^2} \, u_{xx} + \f{\nu U}{\delta^2} \, u_{yy} . \label{Eq:19.2} \ee
\be \f{\delta}{L} \, \f{U^2}{L} \, \lp uv_x + vv_y \rp = - \f{P}{\rho \delta} \, p_y + \f{\nu U}{L^2} \, \f{\delta}{L} \, v_{xx} + \f{\nu U}{\delta^2} \, \f{\delta}{L} \, v_{yy} . \label{Eq:19.3} \ee
From Eqn. (\ref{Eq:19.1})
\be u_x + v_y = 0 . \label{Eq:19.4} \ee
From Eqn. (\ref{Eq:19.2})
\be uu_x + vu_y = - \f{P}{\rho U^2} \, p_x + \f{1}{\mbox{Re}} \, \lp u_{xx} + \f{L^2}{\delta^2} \, u_{yy} \rp . \label{Eq:19.5} \ee
From Eqn. (\ref{Eq:19.3})
\be uv_x + vv_y = - \f{PL^2}{\rho \delta^2 U^2} \, p_y + \f{1}{\mbox{Re}} \, \lp v_{xx} + \f{L^2}{\delta^2} \, v_{yy} \rp . \label{Eq:19.6} \ee
\noindent To keep the dominant viscous stress term in Eqn. (\ref{Eq:19.5}), we must have
\be \lp \f{\delta}{L} \rp^2 = \f{1}{\mbox{Re}} \qquad \mbox{or} \qquad \f{\delta}{L} = \mbox{Re}^{-1/2} . \label{Eq:19.7} \ee
From Eqn. (\ref{Eq:19.6})
\be p_y = O \lp \f{\delta^2}{L^2} \rp \label{Eq:19.8} \ee
and from Eqn. (\ref{Eq:19.5}) 
\be uu_x + vu_y = - \f{P}{\rho U^2} \, p_x + u_{yy} . \label{Eq:19.9} \ee
In physical variables, we have to leading order
\be u_x + v_y = 0 \label{Eq:19.10} \ee
\be uu_x + vu_y = - \f{1}{\rho} \, p_x + \nu u_{yy} \label{Eq:19.11} \ee
The pressure is constant across the boundary layer and must be the same as the pressure just outside.  In the inviscid outer flow
\be UU_x + VU_y = - \f{1}{\rho} \, p_x . \label{Eq:19.13} \ee
Since $V = 0$ on the wall,  $p_x = -\rho UU_x$.  Hence, inside the boundary layer:
\be uu_x + vu_y = UU_x + \nu u_{yy} . \label{Eq:19.14} \ee
 
This is the classical boundary layer approximation for high Re flows, due to Prandtl (1905). 











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