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Documentation of the main branch

These pages follow Mallard's main branch, which is ahead of release 0.8.0: some features they describe come with the next release.

Theory manual

The equations Mallard solves and the methods it solves them with, as implemented on main: every model and scheme of the solver, with the formulas, constants and defaults of the source code it cites. The user guide explains how to use them; Numerical methods collects measured stability limits, accuracy studies and known limitations; the sources are in References.

Chapters

  1. Governing equations: the compressible Navier-Stokes equations, the single-gas constitutive laws and the axisymmetric form.
  2. Thermodynamics and transport of mixtures: thermally perfect mixtures, mixture-averaged transport, thermal diffusion and double flux.
  3. Chemistry and radiation: finite-rate kinetics, the RODAS integrator, Strang and SIMPLER splitting, optically thin radiation.
  4. Large-eddy simulation: the filtered equations, SGS models, the dynamic procedure, thickened flames, the partially stirred reactor and synthetic inflow turbulence.
  5. Finite-volume discretization and reconstruction: face quadrature, curved boundaries, MUSCL, TENO-E and the gradients of the viscous terms.
  6. Numerical fluxes: the Riemann solvers, the hybrid central-upwind flux, species fluxes and viscous fluxes.
  7. Time integration: Runge-Kutta methods, the time step and the order of operations within a step.
  8. Boundary conditions and sources: exterior states, characteristic boundaries, body forces and sponges.

Notation

  • \(\rho\), \(\mathbf{u}\), \(p\), \(T\): density, velocity, pressure, temperature
  • \(e\), \(E\), \(h\), \(H\): specific internal energy, total energy \(e + \tfrac12\lvert\mathbf{u}\rvert^2\), enthalpy \(e + p/\rho\), total enthalpy \(E + p/\rho\)
  • \(a\), \(\gamma\), \(M\): sound speed, ratio of specific heats, Mach number
  • \(R\), \(R_u\): specific and universal gas constants
  • \(Y_k\), \(X_k\), \(C_k\), \(W_k\), \(\overline{W}\): mass and mole fractions, concentration \(\rho Y_k/W_k\), molar mass of species \(k\) and of the mixture
  • \(\dot\omega_k\), \(\dot\omega_T\): mass production rate of species \(k\), heat release rate
  • \(\mu\), \(\lambda\), \(D_k\), \(D_k^T\): viscosity, thermal conductivity, mixture-averaged and thermal diffusion coefficients
  • \(\boldsymbol{\tau}\), \(\mathbf{q}\), \(\mathbf{j}_k\): viscous stress, heat flux, diffusive mass flux of species \(k\)
  • \(\mathbf{U}\), \(\mathbf{W}\): conservative variables, primitive variables \([\rho, \mathbf{u}, p]\)
  • \(\mathbf{F}\), \(\mathbf{F}^v\), \(\mathbf{S}\), \(\hat{\mathbf{F}}\): convective and viscous fluxes, sources, numerical flux
  • \(\Omega_i\), \(\lvert\Omega_i\rvert\), \(\mathbf{x}_i\), \(\overline{\mathbf{U}}_i\): cell \(i\), its volume, centroid and cell average
  • \(f\), \(A_f\), \(\mathbf{n}_f\): face, its area, its unit normal from its first (left, L) cell to its second (right, R)
  • \(\mathbf{x}_{f,q}\), \(w_{f,q}\): face quadrature points and weights
  • \(\Delta t\), \(\mathrm{CFL}\): time step, Courant number
  • \(\Delta\), \(\mu_t\), \(\nu_t\), \(Pr_t\), \(Sc_t\): LES filter width, eddy viscosity (dynamic and kinematic), turbulent Prandtl and Schmidt numbers
  • \(F\), \(\mathcal{E}\), \(\Omega\): thickening factor, efficiency and flame sensor of the thickened flame model

Vectors and tensors are bold; \(\delta_{ij}\) is the Kronecker delta and \(S_{ij} = \tfrac12(\partial_j u_i + \partial_i u_j)\) the strain rate. Units are SI. Chapter 4 alone distinguishes filtered quantities (\(\overline{\phi}\), and Favre-filtered \(\widetilde{\phi}\)); elsewhere filter symbols are dropped. Equations are numbered within each chapter.