## Overview ### Laminar BL: ($\mathrm{Re}<5\times 10^{5}$) Blasius solution $\delta_{BL}=f(x,\mathrm{Re}_{x})$ ### Turbulent BL: ($\mathrm{Re}>5\times 10^{5}$) Law of the wall: $u^{\dagger} = \frac{u}{u^{\star} }$, $y^{\dagger} = \frac{yu^{\star}}{\nu}$, and $u^{\star} = \sqrt{ \frac{\tau_{w}}{\rho} }$ **Viscous sublayer** $u^{\dagger} = y^{\dagger}=\delta_{\text{sublayer}}$ **Overlap layer** $u^{\dagger} = \frac{1}{\kappa} \ln y^{\dagger} + B$ $\kappa \approx 0.41$, $B \approx 5.0$ **Outer Turbulent Layer** $\frac{1}{7}$ power law $\delta_{x,t}=f(\mathrm{Re}_{x},x)$ ## Flat Plate Theory Transition from laminar to turbulent flow happens at the critical reynold's number: $@R \mathrm{Re}_{\text{cr}} = \frac{\rho U_{\infty}x_\text{cr}}{\mu}=10^{6} $ ![[Pasted image 20250309160639.png|700]] ![[Pasted image 20250310103150.png]] ### Momentum Integral Derived from momentum and mass conservation for incompressible fluid flow. $ \begin{align} \sum F_{x} = -D &= \rho \int _{0}^{h} U_{0}(-U_{0})b \, dy + \rho \int_{0}^{\delta} u^{2}b \, dy \\ &= \rho b U_{0} * \underbrace{ (U_{0}h) }_{ \int_{0}^{\delta} u \, dy } - \rho b\int_{0}^{\delta} u^{2} \, dy \\ &= \rho b \int_{0}^{\delta} u(U_{0}-u) \, dy \end{align} $ ### Momentum Thickness Measure of total plate drag, defined by $ D = \rho bU^{2}\theta $ Then the momentum integral gives us $ \theta = \int_{0}^{\delta} \frac{u}{U}\left( 1-\frac{u}{U} \right) \, dy $ Since $D(x) = b \int_{0}^{x} \tau_{w} \, dx$, we have $\tau_{w}=\frac{1}{b} \frac{dD}{dx}$, and thus $ \tau_{w} = \rho U^{2} \frac{d\theta}{dx} $ ### Skin Friction Analogous to friction factor in ducts $ c_{f} = \frac{2\tau_{w}}{\rho U^{2}} = 2 \frac{d\theta}{dx} $ ### Boundary Layer Under Karman's parabolic velocity profile assumption for laminar flow, we have $ u(x,y) \approx U\left( \frac{2y}{\delta} - \frac{y^{2}}{\delta^{2}} \right) $ For turbulent flows, we assume a logarithmic overlap layer which can be solved to obtain an estimate within 10% of the exact Blasius solution shown below: $ \frac{\delta}{x} \approx \begin{cases} \frac{5.0}{\sqrt{ \mathrm{Re}_{x} }} & & \text{Laminar} \\ \\ \frac{0.16}{\mathrm{Re}_{x}^{1/7}} & & \text{Turbulent} \end{cases} $ ### Displacement Thickness Measure of outer streamline deflection $ \delta = h + \delta^{\star} $ $ \delta^{\star} = \int_{0}^{\delta} \left( 1-\frac{u}{U} \right) \, dy $ ![[Pasted image 20250310095343.png|395]] ## Turbulent Flow Define time mean $\bar{u} = \frac{1}{T}\int_{0}^{T} u \, dt$ and fluctuation $u'=u - \bar{u}$ Then Navier-Stokes simiplifies to $ \rho \frac{d\bar{u}}{dt} \approx -\frac{ \partial \bar{p} }{ \partial x } + \rho g_{x} + \frac{ \partial \tau }{ \partial y } $ which gives us $ \begin{align} \tau &= \mu \frac{ \partial \bar{u} }{ \partial y } -\rho \overline{u'v'} \\ &=\tau_{\text{lam}} + \tau_{\text{turb}} \end{align} $ ![[Pasted image 20250310113100.png|500]] 1. Wall layer: Viscous shear dominates, $\tau_{lam}\gg \tau_{turb}$ 2. Overlap Layer: Both matter 3. Outer layer: Turbulent shear dominates, $\tau_{\text{turb}} \gg \tau_{\text{lam}}$ ### Friction Velocity Dimensional analysis shows that $u$ is independent of shear layer thickness, and thus the quantity *friction velocity*, $u^{\star}$, is coined which has units of velocity but isn't actually a flow velocity. $ u^{\star} = \sqrt{ \frac{\tau_{w}}{\rho} } $ $ \begin{align} u^{\dagger} = \frac{u}{u^{\star} } & & & y^{\dagger} = \frac{yu^{\star}}{\nu} \end{align} $ ### Wall Layer Good for $y^{\dagger}<5$ $ u^{\dagger} = y^{\dagger}=\delta_{\text{sublayer}} $ $ \frac{ \partial v }{ \partial y } = \tau_{w} $ This usually covers less than 2% of the profile and is negligible. ### Logarithmic Overlap Layer (Law of the wall) $ u^{\dagger} = \frac{1}{\kappa} \ln y^{\dagger} + B $ $\kappa \approx 0.41$ $B \approx 5.0$ ![[Pasted image 20250310111827.png|500]] ### Skin-Friction Law Derived from log-overlap layer law and skin friction coefficient $ \sqrt{ \frac{2}{c_{f}} } \approx 2.44 \ln\left( \mathrm{Re_{\delta}}\sqrt{ \frac{c_{f}}{2} } \right) + 5 $ But much easier to use a power-law approximation $ c_{f} \approx 0.02 \mathrm{Re_{\delta} ^{-1/6}} $ ### Pressure Gradients ![[Pasted image 20250310220248.png|525]]