## Boundary Layers
### 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]]
## Pipe Flow
Also known as Bournoulli, energy, or head equation. Variant of the first law
$\frac{P_{2} - P_{1}}{\rho g} + \alpha \frac{v_{2}^{2} - v_{1}^{2}}{2g} + z_{2} - z_{1} =h_{f} - h_{p}$
$\alpha=\left\{{\begin{array}{l l}{2}&{{\mathrm{ laminar}}}\\ {1.06}&{{\mathrm{turbulent}}}\\ {1}&{{\mathrm{uniform}}}\end{array}}\right.$
$h_{p} = \text{Pump Power} = \frac{\lvert \dot{W}_{p} \rvert}{\dot{m}_{p}g}$
| Term to Drop | Condition |
| ------------------------------- | --------------------------------- |
| $\frac{P}{\rho g}
lt;br> | Ambient / free surface / outlet |
| ${} \alpha \frac{v^{2}}{2g} {}$ | Slow flow / Cross-section is wide |
| ${} z_{2}-z_{1} {}$ | no height change |
| $h_{f}$ | No losses |
### Losses
$f \frac{L}{D}$ term is major losses, $\sum K_{i}$ term is minor losses
$
h_{f} = \left( f \frac{L}{D} + \sum K_{i} \right) \frac{V^{2}}{2g}
$
### Friction Factor
For laminar flow, we have
$
f = \frac{64}{\mathrm{Re}}
$
For turbulent flow, we can use the colebrook equation:
$\frac{1}{f^{1/2}}=\,-2.0\,\log\left(\frac{\epsilon/d}{3.7}+\frac{2.51}{\mathrm{Re}_{d}f^{1/2}}\right)$
or Haaland's Formula, an approximation to 2% accuracy
$f\!=\!\!\left[-1.8\log_{_{10}}\!\left(\frac{6.9}{\mathrm{Re}_{{D}}}\!+\!\left(\frac{\varepsilon\,/\,D}{3.7}\right)^{\!1.11}\right)\right]^{\!-2}$
### Moody Chart
![[Pasted image 20250303015200.png]]
### Pipe Flow Pressure Drop
1. Use known flow rate to calculate Reynolds Number
2. Identify whether flow is laminar or turbulent
3. Determine friction factor
4. Use $h_{f}$ to determine friction head loss
5. Sum minor loss coefficients
6. Determine total pressure drop using head equation
### Pipe Chains
1. Net flow into any junction must be zero (KCL)
2. Net head loss around any closed loop must be zero (KVL)
3. Head losses must satisfy Moody and minor losses (V=IR)
### Pipe Entrance Length
For Laminar Flow:
$
\frac{L_{e}}{d} \approx 0.06 \mathrm{Re}_{d}
$
For turbulent flow, we have
$
\frac{L_{e}}{d} \approx 4.4 \mathrm{Re}_{d}^{1/6}
$
![[Pasted image 20250302224518.png|500]]
### CDA references
![[Pasted image 20250310221213.png|525]] ![[Pasted image 20250310221727.png|525]]