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I'm trying to work out the value of $dT/dt$ where $T$ is the total kinetic energy of the flow and $t$ is time, for a fluid motion with velocity $\underline{u}$, pressure $p$ and constant density $\rho$ inside a closed region with a fixed solid boundary, where $\underline{u}$ satisfies the Euler Equations; $$ u_t + (u \cdot \nabla )u = \frac{-1}{\rho} \nabla p $$ $$\nabla \cdot u = 0 $$

I've started the problem but am now stuck, I'm told I need to use two vector identities and the divergence theorem. Here are my partial workings,

$$T = \frac{1}{2}\rho\iiint_Vu^2dV$$ $$\implies \frac{dT}{dt} = \frac{d}{dt} (\frac{1}{2}\rho\iiint_Vu^2dV) $$ $$ = \frac{1}{2}\rho\iiint_V \frac{d}{dt}(u^2)dV $$ $$ = \frac{1}{2}\rho\iiint_V 2u \cdot u_t dV $$ $$ = \rho\iiint_V u \cdot u_t dV $$ Any pointers would be appreciated, thank you.

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You did alright, but you also need the other terms. Let me present the most usual derivation: multiplying the momentum conservation equation by $\mathbf{u}$, $$\mathbf{u}_t\cdot\mathbf{u} + \mathbf{u} \cdot \nabla \cdot (\mathbf{u} \mathbf{u}) = -\frac{1}{\rho} \mathbf{u} \cdot \nabla p.$$

Using $\nabla \cdot \mathbf{u}=0$ (from the continuity equation) and $T=\rho \mathbf{u}\cdot\mathbf{u}/2=\rho \| \mathbf{u} \|^2/2$, $$T_t + \nabla \cdot (\mathbf{u} T) = -\frac{1}{2} \nabla \cdot (\mathbf{u} p).$$ The identity $\nabla \cdot(\mathbf{u} p) = \mathbf{u} \cdot \nabla p + p \nabla \cdot \mathbf{u}$ is probably the one you mentioned.

Integrating this equation along the volume $V$, $$\int_V T_t\ dV =- \int_V \nabla \cdot (\mathbf{u} T) dV -\frac{1}{2} \int_V\nabla \cdot (\mathbf{u} p) dV,$$ and using the divergence theorem, $$\int_V T_t\ dV = - \oint_{\partial V} \mathbf{u} T \cdot \mathbf{n}\ dS -\frac{1}{2} \oint_{\partial V}\mathbf{u} p \cdot \mathbf{n} \ dS.$$ Since $V$ is a fixed volume, $$\frac{d}{dt} \int_V T\ dV = - \oint_{\partial V} \mathbf{u} T \cdot \mathbf{n}\ dS -\frac{1}{2} \oint_{\partial V}\mathbf{u} p \cdot \mathbf{n} \ dS,$$ which is the kinectic energy conservation equation in integral form.

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