3.0 Selected concepts in the incompressible flow dynamics

--- tags: Mathematical modeling and optimization title: 3.0 Selected concepts in the incompressible flow dynamics --- Chapter 3. Selected concepts in the incompressible flow dynamics === ### Introduction Fluid mechanical components are present in a wide range of engineering systems. However, the underlying principles can vary significantly and are often non-intuitive. Chapter 3 addresses the key fluid dynamics concepts that are widely applied in practice, specifically with the focus on incompressible flow regime. Whereas numerical methods are indispensable for the analysis of complex systems, applying analytical solutions for certain flow patterns can reduce modeling complexity while still capturing the essential transport phenomena. Section 3.1 begins with hydrostatics and hydraulic principles, revisiting foundational elements from fluid mechanics lecture. The discussion starts with Torricelli’s law and the presence of pressure difference along the height. Extending this concept with the body submerged in the fluid medium, buoyancy becomes the main phenomenon to be considered. Engineering applications related to buoyancy are rather standalone, mostly seen in the marine engineering regarding floating object stability. The dynamics can be highly complex due to diverse impacts and floating object goemetry. Comparing to that, Hydraulics is straight forward. The fluid applied in the device mainly is to transfer the pressure. Force is then created while the pressure is working on the surface. Section 3.2 obtains the fundamental hydrodynamics concepts. Started with laminar viscous flow, particularly in thin tubes, which is a scenario frequently encountered in medical applications or hydraulic devices with liquid supply. As flow velocity increases, the Bernoulli equation becomes relevant for describing momentum-dominated flow regimes. Its mathematical form lends itself to various transformations, making it applicable across a wide range of engineering systems. Bridging to the realistics, the neglected effects are descirbed by empirical formulations. Extending the spectrum, when viscous forces dominate, their influence becomes both significant and beneficial from a design perspective. Lubrication problems serve as a typical example, where Reynolds’ equation provides a practical modeling framework. The focus then shifts in Section 3.3 to aerodynamic phenomena, where the turbulent boundary layer is of central focus. Usually, aerodynamics refer to physical properties over immerse bodies. The resulting force and torque intergrated over the surface drives the objective to move and rotate. A common modeling strategy for aerodynamic impacts is to apply the coefficient to describe physical properties, e.g. lift, drag, pressure drop, etc. However, finding out these coefficients usually require experimental data and/or accurate CFD (computational fluid dynamics) simulation. On the other hand, revealing these coefficients helps us to manage specific technical disciplines or assess more complex systems. Essential topics discussed in this section include turbulent flow in pipes, for which the firction contribution can be characterized using tools such as the Moody diagram, external ballistics in calculating projectile trajectory and blade element momentum (BEM) theory to estimate turbine and propeller performance. Such engineering models offer a first-order assessment of system characteristics instead of expensive high fidelity CFD simulation. --- ### Content of the chapter #### 3.1 Disciplines in hydrostatics and hydrodynamics - Hydrostatic principle - Buoyancy and floating object stability - Hydraulic system #### 3.2 Disciplines in hydrostatics and hydrodynamics - Laminar pipe flow and Hagen-poiseuille equation - Streamline concept and Bernoulli equation - Lubrication film and Reynolds equation #### 3.3 Boundary layer phenomena and aerodynamic applications - Turbulence on wall bounded flows and Moody's diagram - External ballistics and the trajectory of flying objects - Airfoil profile, blade element theory towards propeller propulsion system --- #### Tutorials: At the end of chapter, we will applied the specific knowledge in the fluid mechanics to deal with some realistic engineering challenges: ### 1. Kaplan turbine and water power system _A low-head river power plant uses a Kaplan turbine to convert hydraulic energy into mechanical shaft power. The water dam creates a head of 12m, with the planed flow rate as 45 m^3/s._ <img src="https://d9-wret.s3.us-west-2.amazonaws.com/assets/palladium/production/s3fs-public/styles/full_width/public/thumbnails/image/wss-ws-hydro-typical-powerplant.gif?itok=nOlmoIUi" width="50%"> <img src="https://frontend-assets.simscale.com/media/2019/02/kaplan-300x211.jpg" width="45%"> _The turbine is synchronized with a power grid, which restricts the turbine in a rotational speed of 90 rpm. Technical details of the turbine states:_ Runner diameter: $D=6.0 m \Rightarrow R = 3\,\text{m}$ Hub ratio: $r_\text{hub}/R = 0.33 \Rightarrow r_\text{hub} = 1.0\,\text{m}$ Number of Blades: $B=5$ Rotational speed: $n=90 \text{rpm}$ Twist of turbine blade: from $12°$(hub) to $3°$(tip), linear Generic lift and drag coefficient of the hydrofoil given: $$ \begin{align} C_L =& \ 1.4\ \tanh(\frac{6\ \alpha}{1.4})\\ \\ C_D =& \ 0.01 + 0.025\ C_L(\alpha)^2 \end{align} $$ _To analyze the system, we want to know:_ __*1.How high is the potential hydraulich power from the net head?*__ __*2.How should the blades pitched to achieve the power of 0.4MW?*__ __*3.To characterize the turbine performance, how is the pitch-flow rate-power relation?*__ ### 2. Rocket ballistics [__BACK TO CONTENT__](https://hackmd.io/@SamuelChang/H1LvI_eYn)