Revision as of 14:03, 7 June 2012

Work in progress, come back later ! =)

Motivation

H-Allegro uses the compressible and reactive Navier-Stokes equations in order to solve two-phase combustion. It also takes into account a simplified chemistry of the reaction and the propagation of acoustic waves.

H-Allegro was developped by Eric Albin and Marianne Sjostrand.

Equations and Hypotheses

Equations

• ${\displaystyle {\dfrac {\partial {\rho }}{\partial {t}}}+{\dfrac {\partial {{\rho }U_{j}}}{\partial {x_{j}}}}=0}$

• ${\displaystyle {\dfrac {\partial {{\rho }U_{i}}}{\partial {t}}}+{\dfrac {\partial {\rho }U_{i}U_{j}}{\partial {x_{j}}}}+{\dfrac {\partial {P}}{\partial {x_{i}}}}={\dfrac {\partial {\tau _{ij}}}{\partial {x_{j}}}}+S_{i}}$

• ${\displaystyle {\dfrac {\partial {{\rho }E}}{\partial {t}}}+{\dfrac {\partial {(P+{\rho }E)}U_{j}}{\partial {x_{j}}}}={\dfrac {\partial {q_{j}}}{\partial {x_{j}}}}+{\dfrac {\partial {\tau _{ij}U_{i}}}{\partial {x_{j}}}}+S_{5}}$

• ${\displaystyle {\dfrac {\partial {{\rho }Y_{k}}}{\partial {t}}}+{\dfrac {\partial {{\rho }Y_{k}U_{j}}}{\partial {x_{j}}}}={\dfrac {\partial {q_{j}^{k}}}{\partial {x_{j}}}}+S_{k}}$

Hypotheses

The hypotheses to simplify the Navier-Stokes equations are :

• The total energy balance according to Poinsot-Veynante is described below, if ${\displaystyle E=e_{s}+{\dfrac {1}{2}}u_{i}^{2}}$ :

${\displaystyle {\rho }{\dfrac {DE}{Dt}}={\dfrac {\partial {{\rho }E}}{\partial {t}}}+{\dfrac {\partial {{\rho }u_{i}E}}{\partial {x_{i}}}}={\dot {\omega _{T}}}+{\dfrac {\partial }{\partial {x_{i}}}}({\lambda }{\dfrac {\partial {T}}{\partial {x_{i}}}})-{\dfrac {\partial }{\partial {x_{i}}}}({\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i})+{\dfrac {\partial {\sigma _{ij}}u_{i}}{\partial {x_{j}}}}+{\dot {Q}}+{\rho }{\sum }_{k-1}^{N}Y_{k}f_{k,i}(u_{i}+V_{k,i})}$

The term ${\displaystyle {\dfrac {\partial }{\partial {x_{i}}}}({\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i})}$ can be ignored with regard to ${\displaystyle {\dot {\omega _{T}}}}$, which is the heat release due to combustion.

This release can be defined by these equations : ${\displaystyle {\dot {\omega _{T}}}=-\sum _{k=1}^{N}\Delta h_{f,k}^{0}{\dot {\omega _{k}}}=\nu _{F}M_{F}{\dot {\omega }}Q_{m}}$, where ${\displaystyle Q_{m}}$ is the specific heat of reaction, ${\displaystyle \Delta h_{f,k}^{0}}$ is the standard enthalpy of formation and ${\displaystyle {\dot {\omega _{k}}}}$ is the reaction rate.

${\displaystyle {\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i}}$ is the power produced by volume forces ${\displaystyle f_{k}}$ on species k, considered as non-existent since there are no volume forces.

${\displaystyle {\dot {Q}}}$ is the heat source term which is null here since there is no electric spark, laser or radiative flux.

${\displaystyle Y_{k}}$ is the mass fraction of the species k.

The tensor ${\displaystyle \sigma _{ij}}$ is composed of the tensor of constraints and the tensor of pressure : ${\displaystyle \sigma _{ij}=\tau _{ij}-P\delta _{ij}}$

Thanks to these hypotheses, the total energy balance becomes :

${\displaystyle {\dfrac {\partial \rho E}{\partial t}}+{\dfrac {\partial (P+\rho E)U_{j}}{\partial x_{j}}}={\dfrac {\partial q_{j}}{\partial x_{j}}}+{\dfrac {\partial \tau _{ij}U_{i}}{\partial x_{j}}}+S_{5}}$.

With ${\displaystyle S_{5}}$ corresponding to the heat release due to combustion, ${\displaystyle {\dot {\omega _{T}}}}$.

• The term ${\displaystyle S_{i}^{*}}$ is not used. It usually defines other forces like electromagnetical forces or weight.

• ${\displaystyle S_{k}}$ is either a fuel or a combustive source term.

${\displaystyle S_{6}=\nu _{F}M_{F}{\dot {\omega }}}$ if it is a fuel source term or ${\displaystyle S_{7}=\nu _{O}M_{O}{\dot {\omega }}}$ if it is the oxydant one.

${\displaystyle M_{F}}$ and ${\displaystyle M_{O}}$ are respectively the molar mass of fuel and oxidant.

• ${\displaystyle {\dot {\omega }}}$ can be determined by the Arrhenius law :

${\displaystyle {\dot {\omega }}=B_{1}(\phi )\rho Y_{F}Y_{O}e^{\dfrac {\beta }{\alpha }}e^{\dfrac {-T_{a}}{T}},B_{1}(\phi )=cst}$

• The heat flows due to mass fraction gradients and the species molecular distribution due to temperature gradients are neglicted (cf : Soret and Dufour).

• The termal conductivity, ${\displaystyle \lambda }$ is given by :

${\displaystyle \lambda ={\dfrac {\mu C_{p}}{P_{r}}}}$

Where ${\displaystyle P_{r}}$ is the Prandtl number :

${\displaystyle P_{r}={\dfrac {\nu }{D_{th}}}}$

This number compares the momemtum distribution with the heat distribution.

• The Lewis number for the species k, ${\displaystyle Le_{k}}$, is :

${\displaystyle Le_{k}={\dfrac {\lambda }{\rho C_{P}D_{k}}}}$ ${\displaystyle ={\dfrac {D_{th}}{D_{k}}}}$

Where ${\displaystyle D_{k}}$ is the species diffusion.

We will consider the Lewis number egals to 1 in order to simplify the physics of the pre-mixture flame.

• The Schmidt number, ${\displaystyle S_{ck}}$, is :

${\displaystyle S_{ck}=P_{r}Le_{k}}$

• From the relation of these last three numbers, we can deduct that :

${\displaystyle D_{th}=D_{k}}$ and ${\displaystyle D_{k}={\dfrac {D}{\rho }}}$

${\displaystyle P_{r}={\dfrac {\mu }{D}}}$

Therefore, the diffusion coefficient can be defined as :

${\displaystyle D={\dfrac {\mu }{S_{ck}}}}$

• We use the Law of Fick which is a simplified diffusion law :

${\displaystyle {\vec {\phi }}=-D.{\vec {\nabla }}C}$

${\displaystyle {\vec {\phi }}}$ is the flux of particles density.

C is the particles density.

Thus the diffusion coefficients of the various species are characterized by the number of Lewis.

• We consider that the gas is a perfect, viscous, reagent and diatomic gas. Consequently as the gas is viscous, the compressible effects are not dominating. What involves that the bulk viscosity is useless.

• The gas is reagent thus the mixture of various species is not isothermal, they must be individually followed. It implies that the calorific capacities depend on the temperature and on the composition.

• For a diatomic gas, the calorific capacities can be defined as follows :

${\displaystyle C_{P}={\dfrac {7}{2}}r}$

${\displaystyle C_{V}={\dfrac {5}{2}}r}$

• The gas respects the law of perfect gases :

${\displaystyle P=\rho rT}$

• The combustion is an irreversible transformation whose the creation of entropy is compensated with an entropy given up by the system to the outside, because of a thermal transfer, thus we can consider that the transformation is isentropic. Consequently, the law of Laplace for the thermodynamics is applicable :

${\displaystyle {\dfrac {C_{P}}{C_{V}}}=\gamma =1.4}$

• According to the power law, the dynamic viscosity depends only on the temperature :

${\displaystyle \mu =\mu _{0}\left({\dfrac {T}{T_{0}}}\right)^{\alpha _{s}}}$

${\displaystyle \alpha _{s}=0.76}$

• The tensor of the constraints by respecting the hypothesis of a Newtonian fluid is :

${\displaystyle \tau _{ij}=\mu \left({\dfrac {\partial U_{i}}{\partial x_{j}}}+{\dfrac {\partial U_{j}}{\partial x_{i}}}\right)-{\dfrac {2}{3}}\mu {\dfrac {\partial U_{k}}{\partial x_{k}}}\delta _{ij}}$

The Kronecker symbol, ${\displaystyle \delta _{ij}}$, egals 1 if i=j, 0 else.

• The acoustic Reynolds number is defined such as :

${\displaystyle R_{c}={\dfrac {cL}{\nu }}}$

c is the speed of sound : 340 m.${\displaystyle s^{-1}}$, L is the characteristic length and ${\displaystyle \nu }$ is the kinematic viscosity of air : 1.45e-5 m².${\displaystyle s^{-1}}$.

Models

• Mixing : constant Schmidt number

• Chemistry : one step Arrhenius' law

• Viscosity : power law

Numerics

• Spatial : 6th-order finite difference scheme

• Temporal : 3rd order explicit time integration (Runge-Kutta3)

• Chemical :
  • one step Arrhenius' law
  • partially premixed gas

• Boundary conditions : the 3D-NSCBC processing, applied in a referential attached to the local streamlines.

Data Structures

Software Engineering

Gallery

Performances