---
title: Governing equations
summary: Gyre solves the compressible Euler and Navier–Stokes equations in conservation form, non-dimensionalised by the free stream, and uses their entropy structure as the organising principle of the discretization.
section: theory
order: 1
level: 2
keywords: [Navier-Stokes, Euler, entropy variables, non-dimensionalisation]
---
<!-- gyre docs: expanded Markdown of theory/governing-equations.md (generated tables, verification numbers and chart data are written out). Base level of this page: L2 Practice; passages at another level are bracketed by 'begin level N' / 'end level N' comments and start with [LN Name]. Built from revision ceb1f4b0be. -->

Gyre models a calorically perfect gas in $d = 2$ or $3$ space dimensions. The unknowns are the
conserved variables

$$
\mathbf U = \begin{pmatrix} \rho \\ \rho \mathbf u \\ \rho E \end{pmatrix}, \qquad
p = (\gamma - 1)\left(\rho E - \tfrac12 \rho\,|\mathbf u|^2\right), \qquad
T = \frac{p}{\rho R},
$$

and the governing equations are a system of conservation laws with a source $\mathbf s$ that is
zero except in manufactured-solution studies:

$$
\partial_t \mathbf U + \nabla\cdot\mathbf F(\mathbf U) - \nabla\cdot\mathbf F_v(\mathbf U, \nabla \mathbf W) = \mathbf s .
$$

For turbulent flow the same system is read as the Reynolds-averaged equations: the state
gains one transported variable, $\rho\tilde\nu$, and the viscosity and conductivity gain an
eddy contribution. That closure has its own page,
[Turbulence closure](turbulence-model.md); everything below describes the laminar system it
extends.

<!-- begin level 2 (Practice) -->
**[L2 Practice] Fluxes**

The inviscid flux in Cartesian direction $b$ and the viscous flux are

$$
\mathbf F_b = \begin{pmatrix} \rho u_b \\ \rho \mathbf u\, u_b + p\,\mathbf e_b \\ (\rho E + p)\, u_b \end{pmatrix},
\qquad
\mathbf F_{v,b} = \begin{pmatrix} 0 \\ \boldsymbol\tau_{\cdot b} \\ \mathbf u\cdot\boldsymbol\tau_{\cdot b} + \kappa\, \partial_b T \end{pmatrix},
$$

with the Newtonian stress under Stokes' hypothesis and Fourier conduction,

$$
\boldsymbol\tau = \mu\left(\nabla\mathbf u + \nabla\mathbf u^{\mathsf T} - \tfrac23 (\nabla\cdot\mathbf u)\,\mathbf I\right),
\qquad \kappa = \frac{\mu\, c_p}{Pr}, \quad c_p = \frac{\gamma R}{\gamma - 1}.
$$

The viscous flux depends on the gradient of only $d+1$ quantities, the *gradient variables*
$\mathbf W = (\mathbf u, T)$. The discretization lifts and differentiates exactly these.

<!-- end level 2 -->

<!-- begin level 2 (Practice) -->
**[L2 Practice] Non-dimensionalisation**

Reference density, velocity and temperature are the free-stream values, and the reference length
is one mesh unit (the chord of the built-in aerofoils, the diameter-independent unit of the
cylinder mesh). Consequently

$$
\rho_\infty = 1,\quad |\mathbf u_\infty| = 1,\quad T_\infty = 1,\quad
p_\infty = \frac{1}{\gamma M_\infty^2},\quad R = \frac{1}{\gamma M_\infty^2},\quad \mu = \frac{1}{Re}.
$$

Force coefficients are therefore $C = 2F / S_\text{ref}$, and the pressure coefficient is
$c_p = 2\,(p - p_\infty)$. The Mach number, the angle of attack $\alpha$ (which sets the direction
of $\mathbf u_\infty$ and of the lift/drag axes), the Reynolds number, $\gamma$ and $Pr$ are entries
of one flat parameter dictionary `th`; because it is a JAX pytree, *each entry is a differentiable
input* without any further code.

<!-- end level 2 -->

<!-- begin level 4 (Theory) -->
**[L4 Theory] Entropy structure**

The discretization is organised around the mathematical entropy

$$
\eta(\mathbf U) = -\frac{\rho s}{\gamma - 1}, \qquad s = \ln p - \gamma \ln \rho ,
$$

which is strictly convex for $\rho, p > 0$. Its gradient defines the *entropy variables*

$$
\mathbf w = \frac{\partial \eta}{\partial \mathbf U}
 = \left( \frac{\gamma - s}{\gamma - 1} - \frac{\rho |\mathbf u|^2}{2p},\; \frac{\rho \mathbf u}{p},\; -\frac{\rho}{p} \right)^{\mathsf T}.
$$

Contracting the Euler equations with $\mathbf w$ gives a scalar conservation law for smooth
solutions, $\partial_t \eta + \nabla\cdot(\eta\,\mathbf u) = 0$, because $\mathbf w^{\mathsf T}
\partial_{\mathbf U}\mathbf F_b = \partial_{\mathbf U}(\eta u_b)$. The *entropy potential*
$\psi_b = \mathbf w^{\mathsf T}\mathbf F_b - \eta u_b = \rho u_b$ is what the entropy-conservation
condition for [two-point fluxes](two-point-fluxes.md) is written in. With viscosity and heat
conduction the entropy can only decay: $\tfrac{\mathrm d}{\mathrm dt}\int \eta \le 0$ up to boundary
terms. A scheme that reproduces this inequality discretely cannot blow up through the
mechanism that destabilises under-resolved high-order methods (aliasing-driven entropy
production of the wrong sign). Which parts of this Gyre reproduces discretely, and under
which conditions, is stated with the [interface fluxes](two-point-fluxes.md#interface-fluxes):
the inviscid volume terms always, the interface terms provably only with `es_roe` or `llf`, the viscous
terms not at all. The unit test `test_entropy_variables_are_gradient_of_entropy`
checks the formula above against `jax.grad` of $\eta$.

<!-- end level 4 -->

<!-- begin level 3 (Mechanics) -->
**[L3 Mechanics] What `th` contains**

| key | meaning | default |
|---|---|---|
| `gamma` | ratio of specific heats | 1.4 |
| `mach` | free-stream Mach number | 0.5 |
| `alpha` | angle of attack in radians (API accepts `alpha_deg`) | 0 |
| `re` | Reynolds number per unit length; $10^{30}$ when inviscid | — |
| `pr` | Prandtl number | 0.72 |
| `p_back`, `pt_in`, `tt_in`, `t_wall` | boundary data relative to free-stream values | case dependent |
| `suth` | Sutherland constant $S/T_\infty$; absent means constant viscosity | absent |

<!-- end level 3 -->