Boundary Layer Theory

Boundary–layer thickness relation $$ \boxed{\;\delta \sim \sqrt{\frac{\nu\,L}{U_\infty}}\qquad\Longleftrightarrow\qquad \frac{\delta}{L}\sim \frac{1}{\sqrt{\mathrm{Re}_L}},\ \ \mathrm{Re}_L=\dfrac{U_\infty L}{\nu}\;} $$ for a steady, incompressible, laminar boundary layer with external speed $U_\infty$ over a body of length scale $L$

Advective term $u\,\dfrac{du}{dx}$	Viscous term $\nu\,\dfrac{d^2u}{dy^2}$	Balance
Order: $u=\mathcal O(U_\infty),\ \dfrac{du}{dx}=\mathcal O(U_\infty/L)$ $\Rightarrow u\,\dfrac{du}{dx}=\mathcal O\!\left(\tfrac{U_\infty^2}{L}\right)$	Order: $\dfrac{d^2u}{dy^2}=\mathcal O(U_\infty/\delta^2)$ $\Rightarrow \nu\,\dfrac{d^2u}{dy^2}=\mathcal O\!\left(\tfrac{\nu U_\infty}{\delta^2}\right)$	$\mathcal O\!\left(\tfrac{U_\infty^2}{L}\right)\sim \mathcal O\!\left(\tfrac{\nu U_\infty}{\delta^2}\right)$
Discretize: $\displaystyle u\,\dfrac{du}{dx} \approx \underbrace{u(x,y)}_{\mathcal O(U_\infty)} \ \underbrace{\tfrac{u(x+\Delta x,y)-u(x,y)}{\Delta x}}_{\mathcal O(U_\infty/L)},\ \Delta x\sim L$ $\underbrace{U_\infty}_{u(x,y)}\, \frac{U_\infty}{L}\, \Big(\tfrac{\cancel{\Delta x}}{\cancel{\Delta x}}\Big) = \frac{U_\infty^2}{L}$	Discretize: $\displaystyle \nu\,\dfrac{d^2u}{dy^2} \approx \nu\,\underbrace{\tfrac{u(x,y+\Delta y)-2u(x,y)+u(x,y-\Delta y)}{(\Delta y)^2}}_{\mathcal O(U_\infty/\delta^2)},\ \Delta y\sim \delta$ $\nu\,\frac{U_\infty}{\delta^2}\, \Big(\tfrac{\cancel{(\Delta y)^2}}{\cancel{(\Delta y)^2}}\Big) = \frac{\nu\,U_\infty}{\delta^2}$	$\tfrac{U_\infty\cdot U_\infty}{L}\ \sim\ \tfrac{\nu\cdot U_\infty}{\delta^2}$
$\tfrac{U_\infty\cdot U_\infty}{L}$	$\tfrac{\nu\cdot U_\infty}{\delta^2}$	$\tfrac{\cancel{U_\infty}\,U_\infty}{L}\sim \tfrac{\nu\,\cancel{U_\infty}}{\delta^2} $ $\Rightarrow\ \tfrac{U_\infty}{L}\sim \tfrac{\nu}{\delta^2}$ $\Rightarrow\delta^2\sim \tfrac{\nu L}{U_\infty}$ $\Rightarrow\ \boxed{\delta\sim \sqrt{\tfrac{\nu L}{U_\infty}}}$ $ \tfrac{\delta}{L}\sim \mathrm{Re}_L^{-1/2}$

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Advective term \(u\,\dfrac{du}{dx}\)	Viscous term \(\nu\,\dfrac{d^2u}{dy^2}\)	Balance
Order: \(u=\mathcal O(U_\infty),\ \dfrac{du}{dx}=\mathcal O(U_\infty/L)\) \(\Rightarrow u\,\dfrac{du}{dx}=\mathcal O\!\left(\tfrac{U_\infty^2}{L}\right)\)	Order: \(\dfrac{d^2u}{dy^2}=\mathcal O(U_\infty/\delta^2)\) \(\Rightarrow \nu\,\dfrac{d^2u}{dy^2}=\mathcal O\!\left(\tfrac{\nu U_\infty}{\delta^2}\right)\)	\(\mathcal O\!\left(\tfrac{U_\infty^2}{L}\right)\sim \mathcal O\!\left(\tfrac{\nu U_\infty}{\delta^2}\right)\)
Discretize: \(\displaystyle u\,\dfrac{du}{dx} \approx \underbrace{u(x,y)}_{\mathcal O(U_\infty)} \ \underbrace{\tfrac{u(x+\Delta x,y)-u(x,y)}{\Delta x}}_{\mathcal O(U_\infty/L)},\ \Delta x\sim L\) \(\underbrace{U_\infty}_{u(x,y)}\, \frac{U_\infty}{L}\, \Big(\tfrac{\cancel{\Delta x}}{\cancel{\Delta x}}\Big) = \frac{U_\infty^2}{L}\)	Discretize: \(\displaystyle \nu\,\dfrac{d^2u}{dy^2} \approx \nu\,\underbrace{\tfrac{u(x,y+\Delta y)-2u(x,y)+u(x,y-\Delta y)}{(\Delta y)^2}}_{\mathcal O(U_\infty/\delta^2)},\ \Delta y\sim \delta\) \(\nu\,\frac{U_\infty}{\delta^2}\, \Big(\tfrac{\cancel{(\Delta y)^2}}{\cancel{(\Delta y)^2}}\Big) = \frac{\nu\,U_\infty}{\delta^2}\)	\(\tfrac{U_\infty\cdot U_\infty}{L}\ \sim\ \tfrac{\nu\cdot U_\infty}{\delta^2}\)
\(\tfrac{U_\infty\cdot U_\infty}{L}\)	\(\tfrac{\nu\cdot U_\infty}{\delta^2}\)	\(\tfrac{\cancel{U_\infty}\,U_\infty}{L}\sim \tfrac{\nu\,\cancel{U_\infty}}{\delta^2} \) \(\Rightarrow\ \tfrac{U_\infty}{L}\sim \tfrac{\nu}{\delta^2}\) \(\Rightarrow\delta^2\sim \tfrac{\nu L}{U_\infty}\) \(\Rightarrow\ \boxed{\delta\sim \sqrt{\tfrac{\nu L}{U_\infty}}}\) \( \tfrac{\delta}{L}\sim \mathrm{Re}_L^{-1/2}\)