Euler Equations
The integral form
Integral equations are equations with unkown functions and integral quantities. They are equivalent forms of differential equations but are much more natural to show in the context of fluid mechanics.
Definition 1. The Euler equations in integral form, are:
\[\int_{D} [U(x,t_2)-U(x,t_1)] dx = -\int_{t_1}^{t_2} \left(\int_{\partial D} F(x,t)\cdot ndS\right) dt\]where
\[U= \begin{bmatrix} \rho \\ \rho u \\ \rho v \\ \rho w \\ \rho e \end{bmatrix} \quad F = \begin{bmatrix} \rho u & \rho v & \rho w\\ \rho u^2 + p & \rho uv & \rho uw \\ \rho vu & \rho v^2 + p & \rho vw \\ \rho wu & \rho wv & \rho w^2 + p \\ (\rho e+p)u &( \rho e+p)v & (\rho e+p) w \end{bmatrix}\]are the vector of conserved quantities and flux tensor respectively.
- \(u,v,w : \mathbb{R}^3 \to \mathbb{R}\) is the velocity field
- \(\rho : \mathbb{R}^3 \to \mathbb{R}\) is the density
- \(p: \mathbb{R}^3 \to \mathbb{R}\) is the pressure(force per unit area applied normal to any surface at the point)
- \(e: \mathbb{R}^3 \to \mathbb{R}\) is the energy
- \(D\subset \mathbb{R}^3\) is the spatial domain
The Euler equation is the mathematical statement of conservation over three fundamental quantities:
- mass
- momentum
- energy
Conservation of mass
Definition 2. Change in total mass in \(D\) in time interval \([t_1, t_2]\) \(=\) net mass passying through boundaries of \(D\) in time interval \([t_1, t_2]\)
Mathematically, that is:
\[\int_{D} [\rho(x,t_2)-\rho(x,t_1)] dx = -\int_{t_1}^{t_2} \left\{\int_{\partial D} \rho \begin{bmatrix} u \\ v \\ w \end{bmatrix} \cdot n dS \right\}dt\]which gets the first row of the flux tensor.
Conservation of Momentum
Definition 3. Change in total momentum in \(D\) in time interval \([t_1, t_2]\) \(=\) net momentum flow through boundaries of \(D\) in time interval \([t_1, t_2]\) \(+\) net momentum change due to pressure on boundaries of \(D\)
Mathematically, consider momentum in the \(x\)-coordinate:
\[\int_D [\rho(x,t_2)u-\rho(x,t_1)u]dx = - \int_{t_1}^{t_2} \left\{ \int_{\partial D} \left( \rho u \begin{bmatrix} u \\ v \\ w \\ \end{bmatrix} + \begin{bmatrix} p \\ 0 \\ 0 \end{bmatrix} \right) \cdot ndS \right\} dt\]The flux tensor is somewhat hard to understand, but it reduces to a much more interpretable form when considering the one-dimensional case.
Example 1. If we consider the 1 dimensional case, we have:
\[\int_a^b [U(x,t_2)-U(x,t_1)] dx = -\int_{t_1}^{t_2} F(U(b,t))-F(U(a,t)) dt\]where
\[U= \begin{bmatrix} \rho \\ \rho u \\ \rho e \end{bmatrix} \quad F = \begin{bmatrix} \rho u \\ \rho u^2 + p \\ (\rho e+p)u \\ \end{bmatrix}\]The conservation form
To write the integral form into differential form, we need a key tool:
The divergence theorem for tensor
Theorem 1. Let \(F\) be a second-order cartesian tensor of shape \(m\times n\)
\[\int_{D} \nabla \cdot F dx = \int_{\partial D} F \cdot n dS\]where \(\nabla \cdot F\) takes the divergence of each row(each row is a vector field \(F_j:\mathbb{R}^n\to \mathbb{R}^n\)):
\[\nabla \cdot \begin{bmatrix} F_1 \\ \vdots \\ F_m \end{bmatrix} = \nabla \cdot \begin{bmatrix} \nabla\cdot F_1 \\ \vdots \\ \nabla \cdot F_m \end{bmatrix}\]Applying this to the R.H.S of the integral form:
\[\begin{align} -\int_{t_1}^{t_2} \left(\int_{\partial D} F(U(x,t))\cdot ndS\right) dt &= -\int_{t_1}^{t_2} \int_D \nabla \cdot F dx dt \end{align}\]Also the L.H.S can be rewritten by fundamental theorem of calculus:
\[\int_{D} [U(x,t_2)-U(x,t_1)] dx = \int_D \int_{t_1}^{t_2} U_t dt dx\]By Fubini’s theorem, we get:
\[\int_D \int_{t_1}^{t_2} (U_t - \nabla \cdot F dt)dx= 0\]Since \(D\) is arbitrary we get:
Definition 4.
\[U_t - \nabla_x \cdot F = 0\]