28.7.3. Euler equations

28.7.3.1. Thermodynamics and constitutive equations

todo Reference to Physics: Thermodynamics bbook

Thermodynamics

Principles of thermodynamics and differential of the internal energy

The differential of the internal energy as a function of \(s, \rho\) as independent thermodynamic variables, \(e(\rho, s)\) reads

(28.3)\[ de = T \, ds + \frac{p}{\rho^2} d \rho \ , \]

from the principles of thermodynamics, and the definition of the entropy \(s\) in classical thermodynamics. Following Gibbs’ mathematical foundation of classical thermodynamics, the partial derivatives of the internal energy can be related to temperature, pressure and density

(28.4)\[ T = \left( \frac{\partial e}{\partial s} \right)_\rho \quad , \quad \frac{p}{\rho^2} = \left( \frac{\partial e}{\partial \rho} \right)_s . \]

Speed of sound, \(\ a(\rho, s)\).

As it should be clear from the study of Euler equations as an hyperbolic system the speed of sound \(a\) can be defined as

(28.5)\[ a^2(\rho, s) = \left( \frac{\partial p}{\partial \rho} \right)_s \ . \]

As an example, the definition of the speed of sound appears in two of the eigenvalues of the system, as shown in (28.16) using the convective form (28.15) of the equations using \((\rho, \vec{u}, s)\) as primary variables.

Speed of sound, and change of independent thermodynamic variables

All the relations in this box mainly exploit rule of derivatives of composite functions and basic thermodynamic relations.

As an example, if the pressure is defined as a function of \((\rho, e)\), and the relation \(e(\rho,s)\) is known, with pressure can be written as \(p(\rho, e) = p(\rho, e(\rho,s))\). With the derivation of the composite function, an alternative expression of the speed of sound follows

(28.6)\[\begin{split}\begin{aligned} a^2 & = \left( \frac{\partial p}{\partial \rho} \right)_s = \\ & = \left( \frac{\partial p}{\partial \rho} \right)_e + \left( \frac{\partial e}{\partial \rho} \right)_s \left( \frac{\partial p}{\partial e} \right)_\rho = \\ & = \left( \frac{\partial p}{\partial \rho} \right)_e + \frac{p}{\rho^2} \left( \frac{\partial p}{\partial e} \right)_\rho \ . \end{aligned}\end{split}\]

Conservative variables of Euler equations are \((\rho, \vec{m}, E^t) = \left( \rho, \rho \vec{u}, \rho \left( e + \frac{|\vec{u}^2}{2} \right) \right)\). Pressure as a function of conservative variable reads

\[p = \Pi \left( \rho, \vec{m}, E^t \right) = \Pi \left( \rho, \rho \vec{u}, \rho e(\rho,s) + \rho \frac{|\vec{u}|^2}{2} \right)\]

Constitutive equations for an ideal Newtonian fluid

• Stress tensor \(\mathbb{T} = - p \mathbb{I}\)

• Stress vector \(\vec{t}_{\hat{n}} = \hat{n} \cdot \mathbb{T} = - p \hat{n}\)

• Heat conduction flux \(\vec{q} = \vec{0}\)

28.7.3.2. Integral equations

Conservative variables and flux are respectively

(28.7)\[\begin{split} \mathbf{u} = \begin{bmatrix} \rho \\ \vec{m} \\ E^t \end{bmatrix} \end{split}\]

(28.8)\[\begin{split} \mathbf{F} = \begin{bmatrix} \vec{m} \\ \frac{\vec{m} \otimes \vec{m}}{\rho} + p \mathbb{I} \\ H^t \frac{\vec{m}}{\rho} \end{bmatrix} \end{split}\]

Integral equations for a control volume \(\ V \ \) at rest

Integral equation for a control volume at rest \(V\) immediately follows from the three principles of classial mechanics: conservation of mass, 2-nd principle of Newtonian dynamics, 1-st principle of thermodynamics

\[\begin{split}\begin{aligned} \frac{d}{dt} \int_{V} \rho + \oint_{\partial V} \rho \vec{u} \cdot \hat{n} & = 0 \\ \frac{d}{dt} \int_{V} \rho \vec{u} + \oint_{\partial V} \rho \vec{u} \vec{u} \cdot \hat{n} & = - \oint_{\partial V} p \hat{n} \\ \frac{d}{dt} \int_{V} \rho e^t + \oint_{\partial V} \rho e^t \vec{u} \cdot \hat{n} & = - \oint_{\partial V} p \vec{u} \cdot \hat{n} \\ \end{aligned}\end{split}\]

with the need for an equation of state with the expression of pressure \(p\) as a function of the dynamical variables \(p = \Pi(\rho, \vec{u}, e^t)\). Using conservative variables \((\rho, \vec{m}, E^t) = (\rho, \rho \vec{u}, \rho e^t)\)

\[\begin{split}\begin{aligned} \frac{d}{dt} \int_{V} \rho + \oint_{\partial V} \vec{m} \cdot \hat{n} & = 0 \\ \frac{d}{dt} \int_{V} \vec{m} + \oint_{\partial V} \left( \frac{\vec{m} \otimes \vec{m}}{\rho} + p \mathbb{I} \right) \cdot \hat{n} & = 0 \\ \frac{d}{dt} \int_{V} E^t + \oint_{\partial V} H^t \frac{\vec{m}}{\rho} \cdot \hat{n} & = 0 \ . \end{aligned}\end{split}\]

having defined the total enthalpy per unit volume

\[H^t = \rho h^t = \rho \left( e^t + \frac{p}{\rho} \right) = \rho \left( e + \frac{p}{\rho} + \frac{|\vec{u}|^2}{2} \right) = \rho \left( h + \frac{|\vec{v}|^2}{2} \right) \ .\]

Integral equations for an arbitrary domain \(\ v_t \ \)

Integral equation for an arbitrary domain \(v_t\), with Reynolds’ transport theorem

\[\begin{split}\begin{aligned} \frac{d}{dt} \int_{v_t} \rho + \oint_{\partial v_t} \rho ( \vec{u} - \vec{v}_b ) \cdot \hat{n} & = \\ \frac{d}{dt} \int_{v_t} \rho \vec{u} + \oint_{\partial v_t} \rho \vec{u} ( \vec{u} - \vec{v}_b ) \cdot \hat{n} & = \\ \frac{d}{dt} \int_{v_t} \rho e^t + \oint_{\partial v_t} \rho e^t ( \vec{u} - \vec{v}_b ) \cdot \hat{n} & = \\ \end{aligned}\end{split}\]

28.7.3.2.1. Jump conditions

Jump conditions (28.1) for Euler equations read

(28.9)\[\begin{split} \vec{v}_b \cdot \hat{n} \left[ \left( \begin{matrix} \rho \\ \vec{m} \\ E^t \end{matrix} \right) \right] = \hat{n} \cdot \left[ \left( \begin{matrix} \vec{m} \\\frac{\vec{m} \vec{m}}{\rho} + p \\ H^t \end{matrix} \right) \right] \end{split}\]

28.7.3.3. Differential equations

In regions where the fields are smooth, from integral to differential equations using theorems of differential calculus

28.7.3.3.1. Conservative form

Conservative form of the differential equations immediately follows introducing the expressions (28.7) of the conservative variables and (28.8) of the flux in the general expression of the conservative form of hyperbolic problems (28.2)

(28.10)\[\begin{split} \partial_t \begin{bmatrix} \rho \\ \vec{m} \\ E^t \end{bmatrix} + \nabla \cdot \begin{bmatrix} \vec{m} \\ \frac{\vec{m}\otimes\vec{m}}{\rho} + p \mathbb{I} \\ H^t \frac{\vec{m}}{\rho} \end{bmatrix} = \mathbf{0} \ . \end{split}\]

28.7.3.3.2. Convective forms

Convective form with conservative variables

Using Cartesian coordinates for a 2-dimensional domain,

\[\begin{split}\begin{cases} \partial_t \rho + \partial_x m_x + \partial_y m_y = 0 \\ \partial_t m_x + \partial_x \left( \frac{m_x^2}{\rho} + \Pi \right) + \partial_y \left( \frac{m_x m_y}{\rho} \right) = 0 \\ \partial_t m_y + \partial_x \left( \frac{m_x m_y}{\rho} \right) + \partial_y \left( \frac{m_y^2}{\rho} + \Pi \right) = 0 \\ \partial_t E^t + \partial_x \left( \left( E^t + \Pi \right) \frac{m_x}{\rho} \right) + \partial_y \left( \left( E^t + \Pi \right) \frac{m_y}{\rho} \right) = 0 \end{cases}\end{split}\]

\[\begin{split}\begin{cases} \partial_t \rho + \partial_x m_x + \partial_y m_y = 0 \\ \partial_t m_x - \frac{m_x^2}{\rho^2} \partial_x \rho + 2 \frac{m_x}{\rho} \partial_x m_x + \partial_x \Pi - \frac{m_x m_y}{\rho^2} \partial_y \rho + \frac{m_y}{\rho} \partial_y m_x + \frac{m_x}{\rho} \partial_y m_y = 0 \\ \partial_t m_x - \frac{m_x m_y}{\rho^2} \partial_x \rho + \frac{m_y}{\rho} \partial_x m_x + \frac{m_x}{\rho} \partial_x m_y - \frac{m_y^2}{\rho^2} \partial_y \rho + 2 \frac{m_y}{\rho} \partial_y m_y + \partial_y \Pi= 0 \\ \partial_t E^t - u h^t \partial_x \rho + h^t \partial_x m_x + u \partial_x E^t + u \partial_x \Pi - v h^t \partial_y \rho + h^t \partial_y m_y + v \partial_y E^t + v \partial_y \Pi = 0 \end{cases}\end{split}\]

and using the matrix formalism

(28.11)\[\begin{split}\begin{aligned} \partial_t \begin{bmatrix} \rho \\ m_x \\ m_y \\ E^t \end{bmatrix} & + \begin{bmatrix} 0 & 1 & 0 & 0 \\ -u^2 + \Pi_{\rho} & 2 u + \Pi_u & \Pi_v & \Pi_{E^t} \\ -uv & v & u & \cdot \\ (-h^t + \Pi_\rho ) u & h^t + u \Pi_{m_x} & u \Pi_{m_y} & u ( 1 + \Pi_{E^t} ) \\ \end{bmatrix} \partial_x \begin{bmatrix} \rho \\ m_x \\ m_y \\ E^t \end{bmatrix} + \\ & + \begin{bmatrix} 0 & 0 & 1 & 0 \\ -uv & v & u & \cdot \\ -v^2 + \Pi_{\rho} & \Pi_u & 2 v + \Pi_v & \Pi_{E^t} \\ ( -h^t + \Pi_\rho ) v & v \Pi_{m_x} & h^t + v \Pi_{m_y} & v ( 1 + \Pi_{E^t} ) \\ \end{bmatrix} \partial_y \begin{bmatrix} \rho \\ m_x \\ m_y \\ E^t \end{bmatrix} = \mathbf{0} \end{aligned}\end{split}\]

Convective form with primary variables \(\ (\rho, \vec{u}, e)\).

(28.12)\[\begin{split}\begin{cases} \partial_t \rho + \rho \nabla \cdot \vec{u} + \vec{u} \cdot \nabla \rho = 0 \\ \partial_t \vec{u} + \vec{u} \cdot \nabla \vec{u} + \frac{1}{\rho} \nabla p = \vec{0} \\ \partial_t e + \vec{u} \cdot \nabla e + \frac{1}{\rho} p \nabla \cdot \vec{u} = 0 \ . \end{cases}\end{split}\]

or using Cartesian coordinates and matrix formalism

(28.13)\[\begin{split}\begin{aligned} \partial_t \begin{bmatrix} \rho \\ u \\ v \\ e \end{bmatrix} + \begin{bmatrix} u & \rho & \cdot & \cdot \\ \frac{p_\rho|_e}{\rho} & u & \cdot & \frac{p_e|_\rho}{\rho} \\ \cdot & \cdot & u & \cdot \\ \cdot & \frac{p}{\rho} & \cdot & u \\ \end{bmatrix} \partial_x \begin{bmatrix} \rho \\ u \\ v \\ e \end{bmatrix} + \begin{bmatrix} v & \cdot & \rho & \cdot \\ \cdot & v & \cdot & \cdot \\ \frac{p_\rho|_e}{\rho} & \cdot & v & \frac{p_e|_\rho}{\rho} \\ \cdot & \cdot & \frac{p}{\rho} & v \\ \end{bmatrix} \partial_y \begin{bmatrix} \rho \\ u \\ v \\ e \end{bmatrix} = \mathbf{0} \end{aligned}\end{split}\]

Convective form with primary variables \(\ (\rho, \vec{u}, s)\).

Using the thermodynamic relation \(de = T ds + \frac{p}{\rho^2} d \rho\) and combining mass and internal energy equations, the differential equation \(D_t s = 0\) follows for the entropy, having introduced the material derivative \(D_t \_ = \partial_t \_ + \vec{u} \cdot \nabla \_\). The convective form of the equations becomes

(28.14)\[\begin{split}\begin{cases} \partial_t \rho + \rho \nabla \cdot \vec{u} + \vec{u} \cdot \nabla \rho = 0 \\ \partial_t \vec{u} + \vec{u} \cdot \nabla \vec{u} + \frac{1}{\rho} \nabla p = \vec{0} \\ \partial_t s + \vec{u} \cdot \nabla s = 0 \ . \end{cases}\end{split}\]

or using Cartesian coordinates and matrix formalism in a 2-dimensional domain

(28.15)\[\begin{split}\begin{aligned} \partial_t \begin{bmatrix} \rho \\ u \\ v \\ s \end{bmatrix} + \begin{bmatrix} u & \rho & \cdot & \cdot \\ \frac{p_\rho|_s}{\rho} & u & \cdot & \frac{p_s|_\rho}{\rho} \\ \cdot & \cdot & u & \cdot \\ \cdot & \cdot & \cdot & u \\ \end{bmatrix} \partial_x \begin{bmatrix} \rho \\ u \\ v \\ s \end{bmatrix} + \begin{bmatrix} v & \cdot & \rho & \cdot \\ \cdot & v & \cdot & \cdot \\ \frac{p_\rho|_s}{\rho} & \cdot & v & \frac{p_s|_\rho}{\rho} \\ \cdot & \cdot & \cdot & v \\ \end{bmatrix} \partial_y \begin{bmatrix} \rho \\ u \\ v \\ s \end{bmatrix} = \mathbf{0} \end{aligned}\end{split}\]

Using this expression of the equations, it’s pretty easy to find the eigenvalues of the system, evaluating the eigenvalues of the matrix

\[\begin{split} \mathbf{A}^{\mathbf{v}}_{\hat{n}} = n_x \mathbf{A}^\mathbf{v}_x + n_y \mathbf{A}^\mathbf{v}_y = \begin{bmatrix} u_n & \rho \mathbf{n}^T & \cdot \\ \frac{p_\rho|_s}{\rho} \mathbf{n} & u_n \mathbf{I}_2 & \frac{p_s|_\rho}{\rho} \mathbf{n} \\ \cdot & \cdot & u_n \end{bmatrix} \end{split}\]

as the zeros of the following equation,

\[0 = | \mathbf{A}_{\hat{n}}^{\mathbf{v}} - s \mathbf{I}_4| = ( u_n - s )^2 \left( ( u_n - s )^2 - \left.\frac{\partial p}{\partial \rho}\right|_s \right) = ( u_n - s )^2 \left( ( u_n - s )^2 - a^2 \right) \ ,\]

having used the definition of the spped of sound (28.5). Thus the eigenvalues of the system are

(28.16)\[\begin{split}\begin{aligned} s_{1,4} & = u_n \mp a \\ s_{2,3} & = u_n \ . \end{aligned}\end{split}\]

28.7.3.3.3. Spectral decomposition

It may be convenient to evaluate the spectrum using physical variables \(\mathbf{v} = (\rho, \vec{u}, e)\) and then use the rules for the spectral decomposition after a change of variables to get the spectrum in terms of the conservative variables.

Transformation of variables. The tranformation between physical variables \(\mathbf{v} = (\rho, \vec{u}, e)\) and the conservative variables \(\mathbf{u} = ( \rho, \vec{m}, E^t )\) reads

\[\begin{split}\begin{aligned} \mathbf{v}(\mathbf{u}) = \begin{bmatrix} \rho \\ \vec{u} \\ e \end{bmatrix} = \begin{bmatrix} \rho \\ \frac{\vec{m}}{\rho} \\ \frac{E^t}{\rho} - \frac{|\vec{m}|^2}{2 \rho^2} \end{bmatrix} \qquad , \qquad \mathbf{u}(\mathbf{v}) = \begin{bmatrix} \rho \\ \vec{m} \\ E^t \end{bmatrix} = \begin{bmatrix} \rho \\ \rho \vec{u} \\ \rho \left( e + \frac{|\vec{u}|^2}{2} \right) \end{bmatrix} \ . \end{aligned}\end{split}\]

Gradients of the transformations.

\[\begin{split} \mathbf{U}_{\mathbf{v}} = \begin{bmatrix} 1 & \cdot & \cdot & \cdot \\ u & \rho & \cdot & \cdot \\ v & \cdot & \rho & \cdot \\ e + \frac{|\mathbf{u}|^2}{2} & \rho u & \rho v & \rho \\ \end{bmatrix} = \begin{bmatrix} 1 & \cdot & \cdot \\ \mathbf{u} & \mathbf{I}_2 & \cdot \\ e^t & \rho \mathbf{u}^T & \rho \end{bmatrix} \end{split}\]

\[\begin{split} \mathbf{V}_{\mathbf{u}} = \mathbf{U}_{\mathbf{v}}^{-1} = \begin{bmatrix} 1 & \cdot & \cdot & \cdot \\ -\frac{u}{\rho} & \frac{1}{\rho} & \cdot & \cdot \\ -\frac{v}{\rho} & \cdot & \frac{1}{\rho} & \cdot \\ \frac{1}{\rho} \left( \frac{|\mathbf{u}|^2}{2} - e \right) & -\frac{ u}{\rho} & -\frac{ v}{\rho} & \frac{1}{\rho} \\ \end{bmatrix} = \begin{bmatrix} 1 & \cdot & \cdot \\ - \frac{\mathbf{u}}{\rho} & \frac{1}{\rho} \mathbf{I}_2 & \cdot \\ \frac{1}{\rho} \left( |\mathbf{u}|^2 - e^t \right) & - \frac{\mathbf{u}^T}{\rho} & \frac{1}{\rho} \end{bmatrix} \end{split}\]

Spectral decomposition of \(\mathbf{A}_{\hat{n}}^{\mathbf{v}}\), using \(\mathbf{v}=(\rho, \vec{u}, e)\). The normal convective matrix using physical variables \(\mathbf{v}\) reads

\[\begin{split}\mathbf{A}_{\hat{n}}^{\mathbf{v}} = \begin{bmatrix} u_n & \rho \mathbf{n}^T & \cdot \\ \frac{1}{\rho}\partial_\rho p|_e \mathbf{n} & u_n \mathbf{I} & \frac{1}{\rho}\partial_e p|_{\rho} \mathbf{n} \\ \cdot & \frac{p}{\rho} \mathbf{n}^T & u_n \end{bmatrix} \ ,\end{split}\]

Eigenvalues

In a 2-dimensional domain, the system of equations is 4-dimensional. As shown with the convective form (28.15) using \((\rho, \vec{u}, s)\) as primary variables, the eigenvalues of the system are

\[\left\{ u_n - a, u_n, u_n, u_n + a \right\} \ .\]

In a 3-dimensional domain, the system of equations is 5-dimensional, and its eigenvalues are

\[\left\{ u_n - a, u_n, u_n, u_n, u_n + a \right\} \ .\]

Right eigenvectors

In a 2-dimensional domain

\[\begin{split} \mathbf{R}^\mathbf{v} = \begin{bmatrix} \rho & \rho & 0 & \rho \\ -a n_x & 0 & a n_y & a n_x \\ -a n_y & 0 &-a n_x & a n_y \\ \frac{p}{\rho} & -\rho \frac{\partial_\rho p|_e}{\partial_e p|_{\rho}} & 0 & \frac{p}{\rho} \end{bmatrix} = \begin{bmatrix} \rho & \rho & 0 & \rho \\ -a \mathbf{n} & \mathbf{0} & a \mathbf{t} & a \mathbf{n} \\ \frac{p}{\rho} & -\rho \frac{\partial_\rho p|_e}{\partial_e p|_{\rho}} & 0 & \frac{p}{\rho} \end{bmatrix} \end{split}\]

Details for \( \ d\)-dimensional domain

Given the eigenvalues of the system in a \(d\)-dimensional (\(d = 2, 3\)), the right eigenvectors of the matrix \(\mathbf{A}^{\mathbf{v}}_{\hat{n}}\),

\[\begin{split} \mathbf{A}^{\mathbf{v}}_{\hat{n}} = n_i \mathbf{A}^\mathbf{v}_i = \begin{bmatrix} u_n & \rho \mathbf{n}^T & \cdot \\ \frac{p_\rho|_s}{\rho} \mathbf{n} & u_n \mathbf{I}_d & \frac{p_s|_\rho}{\rho} \mathbf{n} \\ \cdot & \frac{p}{\rho}\mathbf{n}^T & u_n \end{bmatrix} \end{split}\]

are evaluated. For \(s_{1,d+2} = u_n \mp a\), the singular system

\[\begin{split}\mathbf{0} = \begin{bmatrix} \pm a & \rho \mathbf{n}^T & \cdot \\ \frac{p_\rho|_e}{\rho} \mathbf{n} & \pm a \mathbf{I}_3 & \frac{p_e|_\rho}{\rho} \mathbf{n} \\ \cdot & \frac{p}{\rho}\mathbf{n}^T & \pm a \end{bmatrix} \begin{bmatrix} \tilde{\rho} \\ \tilde{\mathbf{u}} \\ \tilde{e} \end{bmatrix} \ , \end{split}\]

has the non-trivial solution - it should be easy to prove it recalling the expression (28.6) of the speed of sound

\[\begin{split} \begin{bmatrix} \tilde{\rho}_{1,d+2} \\ \tilde{\vec{u}}_{1,d+2} \\ \tilde{e}_{1,d+2} \end{bmatrix} = \begin{bmatrix} \rho \\ \mp a \mathbf{n} \\ \frac{p}{\rho} \end{bmatrix} \ . \end{split}\]

For \(s_{2:d+1} = u_n\), the solution of singular system

\[\begin{split}\mathbf{0} = \begin{bmatrix} \cdot & \rho \mathbf{n}^T & \cdot \\ \frac{p_\rho|_e}{\rho} \mathbf{n} & \mathbf{0}_d & \frac{p_e|_\rho}{\rho} \mathbf{n} \\ \cdot & \frac{p}{\rho}\mathbf{n}^T & \cdot \end{bmatrix} \begin{bmatrix} \tilde{\rho} \\ \tilde{\mathbf{u}} \\ \tilde{e} \end{bmatrix} \ , \end{split}\]

is a linear combination of the basis of the kernel of the matrix. It shouldn’t be hard to prove that the kernel is spanned by the right eigenvectors

\[\begin{split}\mathbf{r}_2 = \begin{bmatrix} \rho \\ \mathbf{0} \\ - \rho \frac{p_\rho|_e}{p_e|_\rho} \end{bmatrix} \ ,\end{split}\]

\[\begin{split}\mathbf{r}_{2+i} = \begin{bmatrix} \cdot \\ a \mathbf{t}_i \\ \cdot \end{bmatrix} \quad , \quad i = 1:d \ ,\end{split}\]

with \(\mathbf{t}_{i}\) \(d-1\) independent “tangent vectors”, orthogonal w.r.t. the unit “normal vector” \(\mathbf{n}\).

Left eigenvectors

In a 2-dimensional domain

\[\begin{split} \mathbf{L}^\mathbf{v} = \begin{bmatrix} \frac{P_\rho}{2 \rho a^2} & - \frac{1}{2a} \mathbf{n}^T & \frac{P_e}{2 \rho a^2} \\ \frac{P}{\rho^2} \frac{P_e}{\rho a^2} & \mathbf{0}^T & - \frac{P_e}{\rho a^2} \\ 0 & \frac{\mathbf{t}^T}{a} & 0 \\ \frac{P_\rho}{2 \rho a^2} & + \frac{1}{2a} \mathbf{n}^T & \frac{P_e}{2 \rho a^2} \\ \end{bmatrix} \end{split}\]

Spectral decomposition of \(\mathbf{A}_{\hat{n}}^{\mathbf{u}}\), using \(\mathbf{u}=(\rho, \vec{m}, E^t)\).

Right eigenvectors

In a 2-dimensional domain

\[\begin{split} \mathbf{R}^\mathbf{u} = \begin{bmatrix} \rho & \rho & 0 & \rho \\ \vec{m} - \rho a \hat{n} & \vec{m} & \rho a \hat{t} & \vec{m} + \rho a \hat{n} \\ H^t - m_n a & \frac{|\vec{m}|^2}{\rho} - \rho \frac{\Pi_{\rho}}{\Pi_{E^t}} & a m_t & H^t + m_n a \end{bmatrix} \end{split}\]

Left eigenvectors

In a 2-dimensional domain

\[\begin{split} \mathbf{L}^\mathbf{u} = \frac{1}{2 \rho a^2} \begin{bmatrix} \Pi_\rho + a u_n & - a n_x + \Pi_{m_x} & - a n_y + \Pi_{m_y} & \Pi_{E^t} \\ 2 a^2 - 2 \Pi_{\rho} & - 2 \Pi_{m_x} & 2 - \Pi_{m_y} & - 2 \Pi_{E^t} \\ - 2 a u_t & 2 a t_x & 2 a t_y & 0 \\ \Pi_\rho - a u_n & a n_x + \Pi_{m_x} & a n_y + \Pi_{m_y} & \Pi_{E^t} \\ \end{bmatrix} \end{split}\]

Some algebra

Right eigenvectors w.r.t. conservative variables are

\[\begin{split}\begin{aligned} \mathbf{R}^\mathbf{u} = \mathbf{U}_{\mathbf{v}} \mathbf{R}^{\mathbf{v}} & = \begin{bmatrix} \rho & \rho & 0 & \rho \\ \rho u - \rho a n_x & \rho u & \rho a n_y & \rho u + \rho a n_x \\ \rho u - \rho a n_y & \rho v &-\rho a n_x & \rho u + \rho a n_y \\ \rho e^t - \rho u_n a + p & \rho e^t - \rho^2 \frac{P_\rho}{P_e} & \rho a u_t & \rho e^t + \rho u_n a + p \end{bmatrix} = \\ & = \begin{bmatrix} \rho & \rho & 0 & \rho \\ \rho u - \rho a n_x & \rho u & \rho a n_y & \rho u + \rho a n_x \\ \rho u - \rho a n_y & \rho v &-\rho a n_x & \rho u + \rho a n_y \\ \rho h^t - \rho u_n a & \rho e^t - \rho^2 \frac{P_\rho}{P_e} & \rho a u_t & \rho h^t + \rho u_n a \end{bmatrix} \ , \end{aligned}\end{split}\]

with \(h^t = e^t + \frac{|\mathbf{u}|^2}{2}\), and

\[\rho e^t - \rho^2 \frac{P_\rho}{P_e} = \rho e^t + P - P - \rho^2 \frac{P_\rho}{P_e} = \rho h^t - \frac{\rho^2}{P_e} \left( \frac{P}{\rho^2} P_e + P_\rho \right) = \rho h^t - \frac{\rho^2}{P_e} a^2 \ .\]

Or using the derivatives of \(\Pi(\rho, \mathbf{m}, E^t)\),

\[\begin{split}\begin{aligned} E^t - \rho^2 \frac{P_\rho}{P_e} & = E^t - \rho^2 \frac{\partial_\rho \Pi - \partial_{E^t} \Pi (-e^t + |\mathbf{u}|^2)}{\rho \partial_{E^t} \Pi } = \\ & = E^t - \rho \frac{\partial_\rho \Pi}{\partial_{E^t} \Pi} - E^t + \rho |\mathbf{u}|^2 = \\ & = \rho |\mathbf{u}|^2 - \rho \frac{\partial_\rho \Pi}{\partial_{E^t} \Pi} \ . \end{aligned}\end{split}\]

Left eigenvectors w.r.t. conservative variables are

\[\begin{split}\begin{aligned} \mathbf{L}^\mathbf{u} = \mathbf{L}^{\mathbf{v}} \mathbf{V}_{\mathbf{u}} & = \begin{bmatrix} \frac{P_\rho}{2 \rho a^2} + \frac{u_n}{2 \rho a} + \frac{1}{2 \rho a^2} \frac{P_e}{\rho} (|\mathbf{u}|^2 - e^t) & - \frac{1}{2 \rho a} n_x - \frac{u}{2 \rho a^2}\frac{P_e}{\rho} & - \frac{1}{2 \rho a} n_y - \frac{v}{2 \rho a^2}\frac{P_e}{\rho} & \frac{P_e}{2 \rho^2 a^2} \\ \frac{P}{\rho^2}\frac{P_e}{\rho a^2} - \frac{P_e}{\rho a^2} \frac{|\mathbf{u}|^2 - e^t}{\rho} & \frac{P_e}{\rho a^2} \frac{u}{\rho} & \frac{P_e}{\rho a^2} \frac{v}{\rho} & - \frac{P_e}{\rho^2 a^2} \\ - \frac{u_t}{\rho a} & \frac{t_x}{\rho a} & \frac{t_y}{\rho a} & 0 \\ \frac{P_\rho}{2 \rho a^2} - \frac{u_n}{2 \rho a} + \frac{1}{2 \rho a^2} \frac{P_e}{\rho} (|\mathbf{u}|^2 - e^t) & \frac{1}{2 \rho a} n_x - \frac{u}{2 \rho a^2}\frac{P_e}{\rho} & \frac{1}{2 \rho a} n_y - \frac{v}{2 \rho a^2}\frac{P_e}{\rho} & \frac{P_e}{2 \rho^2 a^2} \\ \end{bmatrix} = \\ & = \frac{1}{2 \rho a^2} \begin{bmatrix} \Pi_\rho + a u_n & - a n_x + \Pi_{m_x} & - a n_y + \Pi_{m_y} & \Pi_{E^t} \\ 2 a^2 - 2 \Pi_{\rho} & - 2 \Pi_{m_x} & 2 - \Pi_{m_y} & - 2 \Pi_{E^t} \\ - 2 a u_t & 2 a t_x & 2 a t_y & 0 \\ \Pi_\rho - a u_n & a n_x + \Pi_{m_x} & a n_y + \Pi_{m_y} & \Pi_{E^t} \\ \end{bmatrix} \end{aligned}\end{split}\]

being

\[\begin{split}\begin{aligned} \Pi_\rho & = P_\rho + \frac{ P_e }{\rho} \left( - e^t + |\mathbf{u}|^2 \right) \\ \Pi_{\mathbf{m}} & = -\frac{\mathbf{u}}{\rho} P_e \\ \Pi_{E^t} & = \frac{1}{\rho} P_e \end{aligned}\end{split}\]

\[\begin{split}\begin{aligned} 2 \frac{P}{\rho^2} P_e - 2 P_e \frac{|\mathbf{u}|^2 - e^t}{\rho} & = 2 \frac{P}{\rho^2} P_e + 2 P_\rho - 2 \Pi_\rho = \\ & = 2 a^2 - 2 \Pi_\rho \end{aligned}\end{split}\]

and the speed of sound

\[a^2(\rho,s) = \partial_\rho P|_s\]

with \(P(\rho, e(\rho,s))\)

\[\begin{split}\begin{aligned} a^2 & = \partial_\rho P|_e + \partial_\rho e|_s \, \partial_e P|_\rho = \\ & = \partial_\rho P|_e + \frac{P}{\rho^2}\partial_e \, P|_\rho \ . \end{aligned}\end{split}\]

• Eigenvalues, right and left eigenvectors

• Characteristic lines/surfaces, and compatibility conditions

28.7.3.4. Riemann problem

• Useful in numerical schemes in finite volume methods, using Godunov flux

28.7.3.4.1. Linearization - Roe intermediate state

• Local linearization of the problem, to reduce the computational cost of solving non-linear Riemann problems at all the interfaces in a grid in FVM

28.7.3.5. Boundary conditions

• characteristic-based

• wall

• …