---
tags: Mathematical modeling and optimization
title: Boundary layer phenomena and aerodynamic applications
---

## 3.3 Boundary layer phenomena and aerodynamic applications

As the velocity increases, the flow filed becomes unstable and turbulence occurs. The appearance of turbulence indeed enhances the complexity. However, the conventional methodologies, e.g. Bernoulli equation, are still valid for large scale of engineering assignments.

Where turbulence really plays the central role, is for flows around the immerse bodies. In such cases, turbulent blundary layer is of central focus. Forces derived by integrating pressure and shear stress distribution dominate the system performance, e.g. airplane flying condition with the lift and drag force generated over wings. For such cases, the geometrical effects and optimization are the main challenges. This class of engineering discipline is also known as aerodynamics.

Since there is no analytical solution frasible for most of the turbulent flow types, the general strategies to deal with aerodynamic challenges are wind tunnel measurements and/or CFD simulations. The physical characteristics, are often represented in the energy form incoporated with corresponding coefficients, in the form of:
$$
\frac{\alpha}{\mathrm{A}}= c_\alpha\ \left(\frac{1}{2} \rho \ u^2 \right)
$$
where
- $\alpha$ is the physical property, which can be pressure, lift, drag etc.
- $\mathrm{A}$ is the reference characteristics, applied to normalize the formulation.
- $\rho$, $u$ are the medium density and flow velocity
- $c_\alpha$ is the dimensionless coefficient, which is conditionally not always a constant value and should be acquired by CFD simulations or wind tunnel experiments.

For instance, the pressure coefficient is defined to describe the pressure drop after flow passing an objective, where
$$
\Delta p - p_\infty = c_p \ \left(\frac{1}{2} \rho \ U^2 \right)
$$
and the $U$ refers to the incoming flow velocity as well as the $p_\infty$ to the environement pressure.

<!--
#### Reynolds number

Reynolds number is the dimensionless key paramter to distinguish turbulent state from laminar patterns. The definition of Reynolds number reads:
$$
\text{Re} = \frac{\rho\ U\ L}{\mu}
$$

Where:
- $\rho$ is the density of the fluid ,
- $U$ is the velocity of the fluid,
- $L$ is a characteristic length scale of the flow geometry, and
- $\mu$ is the dynamic viscosity of the fluid.

Physically speaking, the Reynolds number shows the ratio of inertia ($\rho U$) to viscous force ($\mu$). Once the inertia force dominates, every disturbance will be amplified and the flow becomes unstable. Whereas, the viscous effects can dampen down the perturbation, and the flow stays in a stable and smooth state.

The border of these two condition is described by critical Reynolds number $\text{Re}_c$. Where the turbulent state for $\text{Re} \gg \text{Re}_c$ and the laminar for $\text{Re} \ll \text{Re}_c$.

The definition of Reynolds number varies __from flow type to flow type__, as well as the determination of critical Reynolds number. It always depends on the flow type and the focus of the investigation. e.g.
- the $\text{Re}_c$ for __pipe flow__ reads $2000 \sim 2500$, defined as with diameter ($d$) to $\displaystyle{\text{Re} = \frac{\rho U d}{\mu}}$, but
- $\text{Re}_c \sim 5773$ for __plane channel flow__, where the half channel height is taken as the length scale in the Reynolds number definition.
-->

### 3.3.1 Turbulence on wall bounded flow types

Turbulent flow in the near wall region used to be a classical discipline in the turbulence research community. The mechanism behind is highly complex and serves as fundamental challenges in demystifying the physics and modeling in the CFD framework.

#### Universal wall law
Theoretical investigations in the last century found out the general mean velocity profile for turbulent flow in the wall region as:
$$
\begin{align}
\frac{\bar{u}}{u_\tau}&= \frac{y\ u _\tau}{\nu}\qquad \qquad &&\text{for} \qquad \frac{y\ u _\tau}{\nu} < 5 \qquad {\text{viscous sub-layer}}\\
\\
\frac{\bar{u}}{u_\tau}&=\frac{1}{\kappa}\ \ln (\frac{y\ u _\tau}{\nu}) + C^+ \qquad &&\text{for} \qquad \frac{y\ u _\tau}{\nu} > 30 \qquad {\text{log-law region}}
\end{align}
$$
<img style="float: right;" src="https://live.staticflickr.com/65535/55046524976_c88b253fd3_b.jpg">

where
- $\bar{u}$ is the mean velocity,
- $\tau_w$ is the wall shear stress, defined as $\displaystyle \tau_w = \mu \frac{\partial \bar{u}}{\partial y}$
- $u_\tau$ is the friction velocity or shear velocity, defined as $\displaystyle u_\tau = \sqrt{\frac{\tau_w}{\rho}}$
- $κ$ is the Von Kármán constant ($\sim0.41$),
- $C^+$ is a constant ($\sim 5.5$)

$yu _\tau / \nu$ is also known as $y^+$. It is the dimensionless length representative to indicate the affacting zone. This term is commonly used in the CFD framework to illustrate the normal-to-wall distance of the first cell, and check if it is suitable for the applied turbulence model and wall function.

<br/>

Reformulate the mean velocity profile in the log-law region, the constant $C^+$ can be involved into the log sign. Thus,
$$
\bar{u} = \frac{u_\tau}{\kappa}\ \ln(\frac{y}{y_0})\qquad \text{with}\qquad y_0 = \frac{\nu}{u_\tau \ \left(\exp(C^+)\right)^\kappa}
$$

The $y_0$ representation opens a window to describe the mean velocity profile under rough surface. The concept of [roughness length](https://en.wikipedia.org/wiki/Log_wind_profile) is also widely used in the wind engineering symbolizing the effects of the terain.

#### Moody's diagram

Following the idea of Darcy-Weisbach equation in [the last section](https://hackmd.io/5S8TxljrRaKLuvOXV7RYLw?view#Frictional-contribution), the pressure drop in relation of Darcy friction factor is formulated to
$$
{\displaystyle \Delta p=f_{D}{\frac {\rho u^{2}}{2}}{\frac {L}{D}},}
$$
which can be further applied into turbulent pipe flow. This is proposed by [Colebrook et al.](https://en.wikipedia.org/wiki/Darcy_friction_factor_formulae#Colebrook%E2%80%93White_equation), and summarized into the [Moody's diagram](https://en.wikipedia.org/wiki/Moody_chart).
<br/>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Moody_EN.svg/1280px-Moody_EN.svg.png">

The chart simply states the frictional effects in a pipe with the dependency of Reynolds number $(\text{Re})$ and relative roughness $(\epsilon)$.

The border of laminar to turbulent flow is represented in Reynolds number $({Re}_p)$ for around 2000-4000. The Reynolds number for pipe flow is defined as
$$
Re_p = \frac{\rho \ D\ U_{\text{bulk}}}{\mu}
$$
where
- $\rho$ is the fluid density
- $\mu$ is the dynamic viscosity, which can be temperature dependent. and
- $U_{\text{bulk}}$ is the averaged velocity in the pipe.

Moody's chart is widely used in the pipe relevant applications. By utilizing the formula, one can roughly estimate the pressure drop along the pipe and serve this as baseline information for pump and pipe system implementation.

__<ins>Scenario: Dynamics in pipe system__

Non-steady effects in the pipe system can sometimes be very critical. The dynamics can be created from the pump, valve or the mechanical metering system, where pulsation is found in the downstream. Pulsation may cause severe issues like vibrations, noise, fatigue, component damage (valves, pumps), and even pipe rupture, leading to inefficiency and safety hazards. [More information for pipe vibration issue.](https://www.dnv.com/services/piping-vibrations-110735/)

Especially, the pressure drop due to boundary layer (and roughness) effects can be underestimated from the steady state estimation. This can lead to design issues or unexpected critical fluid state, e.g. cavitation.

_Following the [scenario in the last section](https://hackmd.io/5S8TxljrRaKLuvOXV7RYLw#Scenario-Pipe-with-diameter-adaption)with sinusoid pulsation from the upstream flow._

_The pulsation is desribed in the velocity form, as_
$$
\begin{align}
u_{\text{pulse}} = &\ A_\text{mplitude} \sin(\omega \ t)\qquad \text{, with}\\
\\
\omega = & \ 2\pi \ f
\end{align}
$$

_How is the corresponding pressure loss at the 11m downstream position?_
<img src="https://live.staticflickr.com/65535/54846906237_b8b4b90fc9_b.jpg">

[<ins>__Sample Implementation__](https://cocalc.com/share/public_paths/2f38b8fb7a25fc6c0aa8f0e1eb1f2c908e8ee62e)

Steady Δp: 1.852 bar
Quasi-steady Δp: 1.945 bar (+5.0 %)
Dynamic Δp: 1.944 bar (+5.0 %)

<img src= "https://live.staticflickr.com/65535/55051460613_bf72772c66_c.jpg">

### 3.3.2 Utilization of physical coefficients

In the field of vehicle design, drag reduction is on the central concern. The drag coefficient $(C_D)$ is defined as

$$
C_D = \frac{F_D}{\frac{1}{2}A \rho u² },
$$
where
- $A$ is the affacting area [m²]
- $F_D$ is the drag force [Nt]

Since the drag force always work in the opposite direction to the trajectory. The vector representation is formulated to

$$
\vec{F}_D = - C_D \ \left(\frac{1}{2}A \rho |\vec{u}|^2 \right) \frac{\vec{u}}{|\vec{u}|}
$$

<br/>

A commonly read table of the development of vehicle shape and their corresponding drag coefficient acquired on the wind tunnel shows:

[<ins>_Source link_](https://media.licdn.com/dms/image/v2/D5622AQGdgXwaxDJt6g/feedshare-shrink_800/feedshare-shrink_800/0/1700856051264?e=1770249600&v=beta&t=JoSzOmSqp7px-3iD93ymwTE8UDEDxLXt1h7NQBmkUxc)

By applying this value, automotive engineers can simulate the driving condition and optimize the comfort.

#### External ballistics
External ballistics is a discipline based on the drag effects. It studies a projectile's flight path after it leaves the barrel, focusing on how gravity, air resistance, wind, and the projectile's own spin and shape affect its trajectory from the muzzle to the target, determining accuracy and impact.
<img src="https://cdn.britannica.com/30/178630-050-D12B8390/bullet-rifle-trajectory-effects-gravity-forces-path.jpg">

One crucial factor is the shape of projectile. As it is listed below, different shapes of object suffer different level of drag force.
<img src="https://ozaner.github.io/assets/images/physics/drag_coefficients.png" width="50%">

The mechanism behind is straight forwards. During the flight, the force acting on the projectile are mainly the gravitational and drag force. Gravitation always points in the vertical direction, and the drag force states in the opposite direction of the flight, as long as the shape stays axis symmetric to the flight direction.

The trajectory can be approached using the numerical methods. In each iteration, the velocity and its direction shall be calculated under the affecting forces (drag and gravitation) and integrated to the displacement.

<img src="https://live.staticflickr.com/65535/55053496980_7863d37da2_b.jpg" width="90%">

[__A comparison__](https://cocalc.com/share/public_paths/7cebf62148f50f6b148b0ff342c27dfefa25085f) of projectile shape is shown below with 10mm diameter and 10g weight, starting on 300 m/s speed with 2.5° lifting angle. The projectile type with smaller drag coeffient can fly longer and sustain higher velocity under the same initial condition. The geometry has significant impact.
<img src="https://live.staticflickr.com/65535/55052247182_b4e8831a47_b.jpg">

#### Magnus Effects
Lateral effects in the barrel ballistics do not draw lots of attension, since the impacts are relatively small in comparison with drag. However, for samll flying speed projectiles, the spanwise impact can be significant.

Classical scenarios are the curved balls in the baseball and football games, where the playing spin the ball while pitching/kicking to achieve lateral displacement. Due to the spin, streamlines along the sides of the objective are not equally long, causing the speed difference and creating the pressure difference. As a result, a force in generated normal to the flow direction, which is known the the [Magnus effects](https://en.wikipedia.org/wiki/Magnus_effect).

<img src="https://www.tec-science.com/wp-content/uploads/2021/04/en-gases-liquids-fluid-mechanics-magnus-effect-cylinder-superposition.jpg" width="60%">

Mathematical representation of Magnus effect is formulated to
$$
\vec{F}_M \propto \vec{\omega} \times \vec{u}.
$$
On the other hand, it can also be modeled as a manner of lift.
$$
F_L = C_D\ \left(\frac{1}{2} \rho A u^2\right )
$$
Also, it is found that $C_D$ is proportional the the spin parameter $\large \frac{\omega R}{u}$, so the force can be reformulated in
$$
\vec{F}_M = \kappa \left( \frac{1}{2} \rho A R |\vec{u}||\vec{\omega}|\right)\ \frac{\vec{\omega} \times \vec{u}}{|\vec{\omega} \times \vec{u}|}
$$
$\kappa$ is the proportion and used to take for smooth/rough sphere between $0.3$ to $0.55$.

#### <ins>__Scenario: Curved free kick__</ins>

Imagine a scenario of a free kick, the player is aiming of 15° launching angle with 5° of sideway to avoid the front defender. With an initial velocity of 30 m/s and spin of 5 rps, _how is the ball trajectory towards to goal?_

<img src="https://cdn.shopify.com/s/files/1/0961/3230/files/FreeKick_large.png?14523026631212072114">

[<ins> __Sample solution__](https://cocalc.com/share/public_paths/d7f12d0cca72725e1a4a8cd1649f73e98e17ec75)
<img src="https://live.staticflickr.com/65535/55056171609_72eeb83758_b.jpg">

### 3.3.3 Aerodynamics of airfoil profile

Lift and drag are the most important phyical porperties in the aerodynamic applications. Results out of lift-drag combination are force and/or torque, which brings up the aircrafts into the air or drives the turbine to generate power.

Airfoils are streamlined shapes designed to interact with airflow in a way that profitable in generating lift. They are characterized by their unique cross-sectional profiles, typically featuring a curved upper surface and a flatter lower surface.

<img src="https://d3i71xaburhd42.cloudfront.net/6b0b0f182296f2fbae6478fd9bf8c14ccc770eb4/23-Figure1.8-1.png" width=70%>

<br/>
<br/>
<br/>

Speaking of the shape of airfoil, __leading edge__ is the front edge of the airfoil, where the airflow first encounters. Since it's the region where the flow patterns start, it has a high impact on [stall](https://en.wikipedia.org/wiki/Stall_(fluid_dynamics)) behavior. The rear end is called the __trailing edge__, where around the upper and lower flow patterns shall reunion. The tailing edge shape affects where the reunion point occurs, correspondingly to the airfoil drag characteristics.

__Suction Side__ and __Pressure Side__ are the upper and lower surface connected from leading and trailing edge. As the airflow passes under the suction side, it accelerates and creates lower pressure according to Bernoulli's principle. The opposite pheomenon happens on the pressure side, where higher pressure region is created. The lift is the resulting pressure integrated over the airfoil shape.

Apart from the geometrical concept, some phyiscal characterisitcs accompanying the airfoil applications should also be mentioned:

<img src="https://mwi-inc.com/wp-content/uploads/2023/05/airfol-purpose-blog.jpg" width=70%>

<br/>

- __Angle of Attack (AoA)__: The angle of attack is the angle between the chord line of an airfoil and the direction of the oncoming airflow. It is a reference parameter that determines the aerodynamic characteristics of the airfoil, mainly lift and drag. Increasing the angle of attack typically increases lift until a critical angle is reached, beyond which the airflow separates from the airfoil, leading to a stall.
- __Lift__ and __drag__ are the aerodynamic force that acts _perpendicular and parallel to the direction of the oncoming airflow_ and is generated by the airfoil's shape and angle of attack.
- The definition of lift and drag are:
$$
\begin{align}
\vec{F_L}=\underbrace{\oint P \ (\vec{n} \cdot \vec{e_l})\ \vec{e_l}\ dA}_{\text{pressure driven}} + \underbrace{\oint \tau \ (\vec{t} \cdot \vec{e_l})\ \vec{e_l}\ dA}_{\text{viscous force driven}} = C_l \ (\frac{1}{2}\rho U^2) \ \vec{e_l} \ c\ L\\
\\
\vec{F_D}=\underbrace{\oint P \ (\vec{n} \cdot \vec{e_d})\ \vec{e_d}\ dA}_{\text{pressure driven}} + \underbrace{\oint \tau \ (\vec{t} \cdot \vec{e_d})\ \vec{e_d}\ dA}_{\text{viscous force driven}} = C_d \ (\frac{1}{2}\rho U^2)\ \vec{e_d}\ c\ L
\end{align}
$$
where
- $\vec{n}$, $\vec{t}$ are the unit vectors perpendicular and parallel to the surface of airfoil
- $\vec{e_l}$, $\vec{e_d}$ are the unit vectors perpendicular and parallel to the airflow direction
- $P$ and $\tau$ are the pressure and shear stress on the airfoil surface
- $\rho$, $U$ are the airflow density and velocity
- $C_l$ and $C_d$ are the lift and drag coefficient.
- $c$ and $L$ are the chord length and the spanwise length

- Based on the definition above, __Polar diagram__ of an airfoil plots the lift coefficient ($C_l$ ) and drag coefficient ($C_d$ ) against the angle of attack or each other to illustrate the aerodynamic characteristics. Typically for lift coefficient is from negative to positive values, and for drag a bowl shaped evolution.

<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Aerodynamic_data_of_Aerofoil_BET.svg/1280px-Aerodynamic_data_of_Aerofoil_BET.svg.png" width=90%>

- Stall Region: The stall region of the polar diagram represents the range of angles of attack where the airfoil experiences flow separation and a significant decrease in lift. This is characterized by a sudden increase in drag and a reduction in lift coefficient. The point where the stall occurs is known as the stall angle of attack.
- Acquisition of lift-drag polar can be either via wind tunnel experiments or numerically determinated. Methods such as CFD are widely used incorporated with specific turbulence model. Special methodologies such as [Xfoil](https://en.wikipedia.org/wiki/XFOIL) can also deliver decent representation, both in invicid and vicouse conditions.

The airfoil aerodynamics is mainly triggered by the turbulent boundary layer phenomena. The shear stress, which affecting the lift and drag, is a turbulence characteristics. This also means the polar diagram varies in different Reynolds number. Handling of airfoil polar data should be very carefully.

<!--
<img src="https://aerospaceengineeringblog.com/wp-content/uploads/2016/06/boundarylayeraerofoil.jpg">
-->

### Applications of airfoil aerodynamics

<img src="https://airplanegroundschools.com/Flight-Aerodynamics/Figure-3-47-flight-control-surfaces.JPG" width=100%>
<br/>
<br/>

The everyday seen airfoil applications are the flying aircrafts. The wings are responsible to generate lift consistently. By controlling the angle of attack (i.e. tilding the balde parts), pilot is capable to steer the lift of the aircraft for up- and downwards movement, as well as to stablize the dynamics.

__Propeller characteristics and Blade Element Theory([BEM](https://en.wikipedia.org/wiki/Blade_element_theory))__:

Lift and drag characteristics can be also utilized in the rotational systems, i.e. propeller or turbine blades, depending on the working principles.

Extending the airfoil profile to a propeller blade, which can be considered as a summation of airfoil profiles of small spanwise sections. The thrust the propeller generates depends on to shape of the propeller, i.e. each applied airfoil profiles.

<img src="https://cla.aero/wp-content/uploads/2023/04/ATRprop1-2-2-1024x576.jpg" width = "40%"> <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/65/Propeller_blade_BET.svg/400px-Propeller_blade_BET.svg.png" width = "59%">

The trust and torque are the aerodynamic perperties when the propeller operates. Differ to the aircraft wing application, the inflow velocity for airfoil also has to consider rotational speed, where $\vec{W}_{\text{ref}}=\vec{U}_{\text{in}} + \vec{\omega}\times \vec{r}$. The angle $\phi$ refers to the resulting flow direction based on $\vec{U}_{\text{in}}$ and $\vec{\omega}\times \vec{r}$

<img src="https://seanny1986.files.wordpress.com/2017/06/bem-diag.jpg?w=840" width = 60%>

In the propeller application, thrust is the inquiring force which drive the aircraft forwards. Refering to the lift and drag characteristics of each applied airfoil profile, the thrust $(T)$ and torque $(Q)$ can be formulated in:
$$
\begin{align}
\displaystyle T = \int_{r_{\text{hub}}}^{r_{\text{tip}}} dT = & \int_{r_{\text{hub}}}^{r_{\text{tip}}} \left(L(r) \cos\phi - D(r)\sin\phi \right) dr\\
\\
= & \int_{r_{\text{hub}}}^{r_{\text{tip}}} \ \frac{1}{2} \rho \ (V_{rel}(r))^2​\ c(r) \ \left(\underbrace{ C_l(r, \alpha) \cos\phi}_{\text{lift driven}} \ - \underbrace{C_d(r, \alpha) \sin\phi}_{\text{drag driven}} \right)\ dr \\
\\
\displaystyle Q = \int_{r_{\text{hub}}}^{r_{\text{tip}}} dq = & \int_{r_{\text{hub}}}^{r_{\text{tip}}} \left(L(r) \sin\phi + D(r)\cos\phi \right) \ r\ dr\\
\\
= & \int_{r_{\text{hub}}}^{r_{\text{tip}}} \ \frac{1}{2} \rho \ (V_{rel}(r))^2​\ c(r) \ r \left(\underbrace{ C_l(r, \alpha) \cos\phi}_{\text{lift driven}} \ - \underbrace{C_d(r, \alpha) \sin\phi}_{\text{drag driven}} \right)\ dr
\end{align}
$$

where
- $L(r)$ and $D(r)$ are the lift and drag force on the corresponding radius.
- $C_l(r, \alpha)$ and $C_d(r, \alpha)$ as the lift and drag coefficients on corresponding the radius and the angle of attack, serving as look-up-table diagram.

Thinking on the movement the propeller engine generates also causes relative velocity, the $\vec{U}_{in}$ should both obtain the incoming wind speed and the thrust generated moving speed, as well as $\vec{\omega}$ for both rotating speed and the aerodynamics induced torque. Though, these (incoming wind speed and rotational torque) are comparatively small and negligible in high speed aircraft applications.

#### <ins> __Scenario: Scenario: Aircrafts Propeller Design Analysis__ </ins>

A P-47 Thunderbolt aircraft equipped with a Hamilton standard four blade propeller engine.

<img src="https://upload.wikimedia.org/wikipedia/commons/0/07/P47_Thunderbolt_-_Chino_2014_%28cropped%29.jpg" width = 56.5%> <img src="https://www.aircorpsaviation.com/wp-content/uploads/image007-10.jpg" width = "40%">

Taking the radius of hub and propeller are 0.15m and 1.8m,
Assumed:
- The chord length varied linearly along the blade's span. Near the root, the chord was approximately 0.45 meters (18 inches), tapering to about 0.15 meters (6 inches) near the tip.
- Blade without [twisted featured](https://howthingsfly.si.edu/ask-an-explainer/why-there-twist-propeller).
- The blade utilized NACA 16-series airfoils. The lift and drag coefficient polar assumed as
$$
\begin{align}
C_l(\alpha) &= 0.1 + 0.08 \ \alpha \qquad \text{for}\quad -12° < \alpha <12°\\
C_d(\alpha) &= 0.02 + 0.05 \ \alpha\\
\end{align}
$$

_For flying speed in the range from 0 to 300 m/s, rotational speed of 0 to 3000 rpm, how is the performance (thrust and torque) of the propeller?_

For untwisted blades, since the radial velocity increase along with radius $r$, the angle of attack decreases towards the tip. As shown below.

<img src="https://live.staticflickr.com/65535/55066279489_24be6faf2b_b.jpg">

The high AoA (in this case over 12°) is unfovored since it is in the stall region where there is no lift generated. So the blade design usually optimized to a twisted shape where the leading edge is formed against the incoming flow direction.

<img src="https://live.staticflickr.com/65535/55066303739_5fbc4e1a58_b.jpg">

[<ins>__Sample implementation__</ins>](https://cocalc.com/share/public_paths/98ddb90fc5c356331bd6e59301f224cd74343bfd)
Based on this stand point, we can involve the drag coefficient of the entire aircraft and lift coefficient of the wing to evaluate the flying dynamics and further, to optimize the blade shape for specific flying condition.