Boundary layer (BL) structure and development for parallel flow
over an infinite flat plate

• $u(y) = 0$ at the surface (no-slip condition)
• $u(y) = U$ far away from surface (free stream condition)

Convenient to define Reynolds number as $\text{Re}_x = x U / \nu$

• Laminar BL: $\text{Re}_x < 5 \times 10^5$:
• Turbulent BL: $\text{Re}_x > 5 \times 10^5$:

Critical Reynolds number varies with surface roughness
and upstream turbulence

Flat plate exposed to flow with free stream velocity, $U$

• Boundary layer thickness $\delta$ defined as:

$\delta = y~~~~~$ where $~~~~~u(y) = 0.99U$

Note: The 99% criterion is kind of arbitrary

Flat plate exposed to flow with free stream velocity, $U$

• Boundary layer displacement thickness $\delta^*$ defined as:

$\displaystyle \delta^* = \int_0^{\infty} \left(1-\dfrac{u}{U}\right) \text{d}y$

• Think as: Distance the wall should be moved outward to maintain the same volume flow rate for an inviscid flow
• Boundary layer momentum thickness $\theta$ defined as:

$\displaystyle \theta = \int_0^{\infty} \dfrac{u}{U} \left(1-\dfrac{u}{U}\right) \text{d}y$

• Think as: Distance the wall should be moved outward to maintain the same momentum flow rate for an inviscid flow

\(~~~~~~~~~~~~~\displaystyle \delta^*bU = \int_0^{\infty} \left( U-u\right) b \text{d}y \)

\(~~~~~~~~~~~~~~~~~~\displaystyle \delta^* = \int_0^{\infty} \left(1-\dfrac{u}{U}\right) \text{d}y \)

Divided into three regions away from the wall

1. Viscous sublayer: Laminar part, viscous effects
2. Buffer layer: Transitional part, both viscous and inertial effects
3. Log-law layer: Turbulent part, inertial effects dominant

Based on dimensional analysis, we obtain two dimensionless groups:

• Dimensionless velocity: $u^+ = \overline{u}/u^*$
• Dimensionless position: $y^+ = y u^*/\nu$

• Time-averaged velocity, $\overline{u}$
• Friction velocity, $u^*= \sqrt{\dfrac{\tau_w}{\rho}}~~~~~~~~~~$where $\tau_w = \mu \dfrac{\text{d}\overline{u}}{\text{d}y}|_{y=0}$
• Distance from wall, $y$

Velocity profiles in log-log plot

• Viscous sublayer: Linear profile, $u^+ = y^+~~$ (Eq. 1)
• Valid for $y^+ < 5$
• Log-law layer: Logarithmic profile, $u^+ = \dfrac{1}{2.5} \ln y^+ + 5.0~~$ (Eq. 2)
• Valid for $y^+ > 30$

Boundary layer thickness $\delta$ (Blasius solution)

\( \delta = \dfrac{5x}{\sqrt{\text{Re}_x}} \)

Boundary layer displacement thickness $\delta^*$ (Blasius solution)

\( \delta^* = \dfrac{1.721x}{\sqrt{\text{Re}_x}} \)

Boundary layer momentum thickness $\theta$ (Blasius solution)

\( \theta = \dfrac{0.664x}{\sqrt{\text{Re}_x}} \)

Note: Boundary layer thickness decreases with
increasing $\text{Re}_x$:
\( \delta \to 0 \) as \( \text{Re}_x \to \infty \)

Looking for an analytical solution..

$x$-direction:

\( \rho \) \( \bigg( \) \( u \dfrac{\partial u}{\partial x} \) \( + \) \( v \dfrac{\partial u}{\partial y} \) \( \bigg) = -\) \( \dfrac{\partial p}{\partial x} \) \( + \) \( \mu \) \( \bigg( \) \( \dfrac{\partial^2 u}{\partial x^2} \) \( + \) \( \dfrac{\partial^2 u}{\partial y^2} \) \( \bigg) \)

$y$-direction:

\( \rho \) \( \bigg( \) \( u \dfrac{\partial v}{\partial x} \) \( + \) \( v \dfrac{\partial v}{\partial y} \) \( \bigg) = -\) \( \dfrac{\partial p}{\partial y} \) \( + \) \( \mu \) \( \bigg( \) \( \dfrac{\partial^2 v}{\partial x^2} \) \( + \) \( \dfrac{\partial^2 v}{\partial y^2} \) \( + \) \( \bigg) \)

Continuity equation:

\(\dfrac{\partial u}{\partial x} \) \( + \) \(\dfrac{\partial v}{\partial y}\) \( = 0\)

Boundary conditions:

\(u=v=0 \) at \(y=0\)
\(u=U \) at \(y \rightarrow \infty\)