[유체] Chap 5. Basics of Ideal Flow

Chap 5. Basics of Ideal Flow

5.1 Ideal Fluid

• Invicid (No viscosity)
• Incompressible($\rho$ : constant)

5.2 Irrotationallity

\[\omega_z = \frac{1}{2}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)\] \[\vec{\omega}=\frac{1}{2}\nabla\times\vec{u}= 0\]

• Irrrotational flow : No rotation

5.3 Velocity Potential

• $\nabla\phi=\vec{u}$ ($\phi$ : Velocity potential, scalar)
• flows low → high
• Irrotationality (automatically satisfied)

\[\begin{aligned}\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}&= \frac{\partial}{\partial x}\frac{\partial \phi}{\partial y} - \frac{\partial}{\partial y}\frac{\partial \phi}{\partial x} \\&=0\end{aligned}\] \[\begin{aligned}\nabla\times\vec{u}=0\quad\Rightarrow\quad&\nabla\times\nabla\phi=0 \\ \nabla \times \nabla ( \cdot ) &= 0\end{aligned}\]

Ideal Flow(이상 유동 = potential flow)

1. Ideal fluid (invicid, incompressible)
2. Irrotationality

5.4 Laplace Equation

• Starting from continuity equation

\[\begin{aligned}\nabla\cdot\vec{u}&=0 \\\nabla\cdot\nabla\phi&=0 \\\nabla^2\phi&=0\end{aligned}\]

• Governing equation of ideal flow
• Solution of Laplace equation satisfies superposition property.

5.5 Bernoulli Equation

\[\frac{\partial\vec{u}}{\partial t}+(\vec{u}\cdot\nabla)\vec{u}= -\frac{1}{\rho}\nabla P+\vec{f}+\cancel{\nu\nabla^2\vec{u}}=0\]

• Zeroed out for ideal flow (invicid)
• $\nabla\vec F=\vec f, \vec f = -gz$

\[\text{x-axis : }\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\]

By irrotationality, $\frac{\partial v}{\partial x}=\frac{\partial u}{\partial y}$

\[\begin{aligned}&= \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial x}+w\frac{\partial w}{\partial x} \\&= \frac{\partial u}{\partial t}+\frac{1}{2}\left[\frac{\partial u^2}{\partial x}+\frac{\partial v^2}{\partial x}+\frac{\partial w^2}{\partial x}\right] \\ &=-\frac{1}{\rho}\frac{\partial P}{\partial x}+\frac{\partial F}{\partial x}\end{aligned}\]

Ettracting $\partial \over \partial x$ from every term,

\[\frac{\partial}{\partial x}\left\{\frac{\partial\phi}{\partial t}+\frac{1}{2}\left[\left(\frac{\partial\phi}{\partial x}\right)^2+\left(\frac{\partial\phi}{\partial y}\right)^2+\left(\frac{\partial\phi}{\partial z}\right)^2\right]+\frac{P}{\rho}-F\right\}=0\] \[P=-\rho\left(\frac{\partial\phi}{\partial t}+\frac{1}{2}|\nabla\phi|^2+gz\right)+\mathrm{const}\]

• Pressure being automatically calculated when $\phi$ is determined.