[유체] Chap 6. Energy in Potential Flow

Chap 6. Energy in Potential Flow

6.1 Equation of Energy & Power

• Consider Euler equation $\times \rho \vec u$ : Power ($dE\over dt$)

\[\text{x : }\left(\frac{du}{dt}=f_x-\frac{1}{\rho}\frac{\partial P}{\partial x}\right)\times \rho u\] \[\text{y}=\left(\frac{dv}{dt}=f_y-\frac{1}{\rho}\frac{\partial P}{\partial y}\right)\times \rho v\] \[\text{z : }\left(\frac{dw}{dt}=f_z-\frac{1}{\rho}\frac{\partial P}{\partial z}\right)\times \rho w\]

Force times $\rho u$ :

\[\begin{aligned}f_x\rho u+f_y\rho v+f_z\rho w &=\frac{\partial f}{\partial x}\rho u +\frac{\partial f}{\partial y}\rho v +\frac{\partial f}{\partial z}\rho w \\ &=\rho(\vec{u}\cdot\nabla)\vec f \\ &=\rho{d\vec f \over dt} - \rho\cancel{\partial \vec f \over \partial t} = \rho{d\vec f \over dt}\end{aligned}\]

Summing all axis :

\[\frac{1}{2}\rho\left\{\frac{d}{dt}(u^2+v^2+w^2)\right\}=\rho\frac{df}{dt}-(\vec{u}\cdot\nabla)P\]

where $\vec{f}=-g{z}$

\[\iiint_{C.V}\frac{d}{dt}\left\{\frac{\rho}{2}(u^2+v^2+w^2)+\rho gz\right\}d\forall=-\iiint_{M.V.}(\vec{u}\cdot\nabla)p\,d\forall = -\iint_{\partial M.V.}p\vec{u}\cdot\vec{n}\,ds\]

Physical Meaning : Time derivative of Kinetic + Potential energy can be expressed with pressure and velocity of its surface

Second divergence theorem

• First divergence theorem

\[\iiint_{C.V.}\nabla\cdot\vec{H}\,dV=\iint_{C.S.}\vec{H}\cdot\vec{n}\,dS\]

• Second divergence theorem

\[\iiint_{C.V.}\left[(\vec{G}\cdot\nabla)f+f(\nabla\cdot\vec{G})\right]dV=\iint_{C.S.}f\vec{G}\cdot\vec{n}\,dS\]

if $\vec G = \vec u, f=p$, then

\[\iiint_{C.V.}\left[(\vec{u}\cdot\nabla)P+\cancel{P(\nabla\cdot\vec{u})}\right]dV=\iint_{C.S.}P\vec{u}\cdot\vec{n}\,dS\] \[\therefore\iiint_{C.V.}(\vec{u}\cdot\nabla)P\,dV=\iint_{C.S.}P\vec{u}\cdot\vec{n}\,dS\]

6.2 Kinetic Energy

By 1st divergence theroem,

\[\iiint_{C.V.}\nabla\cdot\vec{G}\,dV=\iint_{C.S.}\vec{G}\cdot\vec{n}\,ds\]

then, let

\[\vec{G}=\left(\phi\frac{\partial\phi}{\partial x},\phi\frac{\partial\phi}{\partial y},\phi\frac{\partial\phi}{\partial z}\right)=\phi\nabla\phi\] \[\begin{aligned}\nabla\cdot\vec{G}&=\left(\frac{\partial\phi}{\partial x}\right)^2+\cancel{\phi\frac{\partial^2\phi}{\partial x^2}}+\left(\frac{\partial\phi}{\partial y}\right)^2+\cancel{\phi\frac{\partial^2\phi}{\partial y^2}}+\left(\frac{\partial\phi}{\partial z}\right)^2+\cancel{\phi\frac{\partial^2\phi}{\partial z^2} }\\&=u^2+v^2+w^2+\cancel{\nabla^2\phi}\end{aligned}\] \[\begin{aligned}\therefore\iiint_{C.V.}(u^2+v^2+w^2)\,dV&=\iint_{C.S.}\phi(\nabla\phi\cdot\vec{n})\,ds \\ &= \iint_{C.S.}\phi\frac{\partial\phi}{\partial n}\,ds\end{aligned}\] \[\frac{1}{2}\rho\iiint_{C.V.}(u^2+v^2+w^2)\,dV=\frac{1}{2}\rho\iint_{C.S.}\phi\frac{\partial\phi}{\partial n}\,ds\]

Kinetic Energy can be expressed with velocity potential at surface and normal velocity

$\nabla\phi\cdot\vec{n}=u_n$

Let $U = 1.0$ (unit velocity)

\[K.E.=\frac{1}{2}M_aU^2=\frac{1}{2}M_a=\frac{1}{2}\rho\iint_{C.S.}\phi\frac{\partial\phi}{\partial n}\,ds\] \[\therefore M_a=\rho\iint_{C.S.}\varphi\frac{\partial\varphi}{\partial n}\,ds\]

where $\varphi$ : Velocity potential at unit velocity

⇒ Added mass $m_a$ : function of shape