5.1. exec/hydro – incompressible fluctuating hydrodynamics

Purpose

Solves the incompressible fluctuating Navier–Stokes (or, in the overdamped limit, Stokes) equations on a staggered grid. It is the simplest FHDeX application and the best one to start with.

Libraries

src_hydro, src_gmres, src_rng, src_analysis, src_common; src_multispec is also compiled in.

Algorithm

Each time step is a predictor–corrector scheme. Viscous terms are treated semi-implicitly (Crank–Nicolson) and advection is treated explicitly with a trapezoidal correction. Both stages solve a Stokes saddle-point system for velocity and pressure with preconditioned GMRES from src_gmres. algorithm_type = 0 selects the inertial scheme; other values select the overdamped (Stokes) limit, in which the deterministic case needs only a single solve per step. Thermal fluctuations enter through the stochastic momentum flux (StochMomFlux), scaled by variance_coef_mom.

Key parameters

Parameter	Meaning
prob_type	Initial condition: 0 = fluid at rest, 1 = vortex, 2 and 3 = Kelvin–Helmholtz (sine / smooth); see exec/hydro/Init.cpp.
visc_coef, visc_type	Viscosity and form of the viscous operator (1 = \(\nabla\cdot\beta\nabla\), 2 = symmetric, 3 = symmetric with bulk viscosity; negative values allow spatially varying coefficients).
variance_coef_mom, initial_variance_mom	Amplitude of the stochastic momentum flux and of the initial velocity fluctuations (0 = deterministic).
k_B, T_init	Boltzmann constant and temperature, which set the noise strength.
bc_vel_lo/hi	Velocity boundary conditions per direction: -1 periodic, 1 slip, 2 no-slip.
struct_fact_int, n_steps_skip	Accumulate structure factors every struct_fact_int steps after skipping n_steps_skip steps of equilibration.
gmres_*, stag_mg_*, mg_*	Linear solver controls (gmres).

Example inputs

inputs_regression_vortex_2d/3d (deterministic vortex), inputs_regression_equil_2d/3d and inputs_incomp_equil_fluct_* (equilibrium fluctuations), inputs_shearwall_* (shear flow), inputs_turbtest_* (forced turbulence).

Convergence tests

run_convergence_tests.sh runs coarse, medium and fine versions of inputs_dtrmnistic_convergence_*d, and compute_convergence_rate.sh compares them with the AMReX tool DiffSameDomainRefinedStag. The procedure is described in exec/hydro/README.CONVERGENCE.