atl_eqn_euler_module Module

Description of the Euler equation system for inviscid compressible flows.

The flow field is described in terms of the conservative variables density, momentum and energy. The Euler equations require the definition of the following parameters:

• Isentropic expansion coefficient isen_coef
• Gas constant r
• Heat capacity at constant volume cv
• Numerical flux numflux
• Linearization limit linear_limit
• Linearization indicator linearization_indicator
• Wether to ensure positivity modally ensure_positivity
• Porosity for modeling geometries as porous materials porosity
• Viscous permeability for the porous material viscous_permeability
• Thermal permeability for the porous material thermal_permeability
• Definition of the material (to model geometries) material

There are three different implementations for the numerical flux available:

• 'hll' Harten/Lax/van-Leer flux
• 'godunov' Solve the nonlinear Riemann problem
• 'lax_friedrich' Lax/Friedrichs flux, not recommended and may lead to instabilities, especially at boundaries

It is possible to adaptively linearize the flux in elements according to its state. This is activated by setting the linear_limit to some value larger than 0. The default is to use no adaptive linearization (linear_limit=0). There are three different indicators available that may be used to decide the linearization of elements. linearization_indicator may be:

• 'error' A modal estimation for the terms neglected by linearization. This involves all variables and is the default.
• 'energy' Use the deviation of the energy to decide the linearization.
• 'density' Use the deviation of the density to decide the linearization.

The deviations are considered in relation to the cell mean value. Thus, setting linear_limit = 0.01 and linearization_indicator = 'density' would only compute the linearized flux in elements where the maximal deviation of the state is guaranteed to be less or equal to one percent of the mean density in that cell.

For the material the following variables need to be defined:

• characteristic masking function, has to 1 where a geometry is to be found and 0 elsewhere.
• relax_velocity the velocity of the object
• relax_temperature the temperature of the object

If the characteristic is 0 everywhere, the other settings are irrelevant. Each material component is of tem_variable_type. However, these can be also simply defined as space-time functions, which can just be constants. See the tem_variable_module and tem_spacetime_fun_module for more details.

If ensure_positivity = true is set, the state will be scaled such that the deviation of the state is guaranteed to be smaller than the mean for density and energy such that these are positive. While this is a computationally cheap measure, you should be aware that it is not sufficient to avoid unphysical states, as the pressure may still be negative.

The following equation names are implementing the Euler equations and require the settings described above:

• euler (3D)
• euler_2d (2D)
• euler_1d (1D)
• loclineuler Locally linearized (3D)
• loclineuler_1d Locally linearized (1D)

The locally linearized variants use fluxes linearized to the cell means within the elements, while between elements the nonlinear flux is computed.

An example configuration for Euler equations is shown below:

```lua gamma = 1.4 -- isentropic expansion coefficient R = 1.0 -- gas constant equation = { name = 'euler', isen_coef = gamma, r = R, cv = R/(gamma-1), numflux = 'hll',

linear_limit      = 0.001,
linearization_indicator = 'error',

ensure_positivity = false, -- Ensure that density and energy remain
                           -- positive by only considering higher modes up
                           -- to the point where positivity is guaranteed.

porosity          = 1.0,   -- Porosity to use in material modelling for
                           -- wall representation in elements.
viscous_permeability = 1.0e-6, -- Viscous permeability for the porous
                               -- medium to represent wall geometries in
                               -- elements.
thermal_permeability = 1.0e-3, -- Thermal permeability for the material to
                               -- represent walls.
material = {
  -- Description of the material distribution to define obstacles inside
  -- the domain.
  characteristic = 0.0, -- Masking function (may be a variable) that
                        -- describes where Material is to be found
                        -- (Chi(x,y,z)), should be 1 inside material and 0
                        -- everywhere else.
  relax_velocity = {0, 0, 0}, -- Velocity of the obstacle.
  relax_temperature = 0.0     -- Temperature of the obstacle.
}

}