User talk:Ferrandm

Hypotheses

The hypotheses to simplify the Navier-Stokes equations are :

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

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

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

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

${\rho }{\sum }_{k=1}^{N}h_{s,k}Y_{k}V_{k,i}$ is the power produced by volume forces $f_{k}$ on species k.

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

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

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

Thanks to these hypotheses, the total energy balance becomes :

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

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

• The term $S_{i}^{*}$ is not used. Thanks to it we can define other forces like electromagnetical force or weight.

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

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

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

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

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

• 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, $\lambda$ is given by :

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

Where $P_{r}$ is the Prandtl number :

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

This number compares the momemtum distribution with the heat distribution.

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

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

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

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

• The Schmidt number, $S_{ck}$, is :

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

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

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

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

Therefore, the diffusion coefficient can be defined as :

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

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

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

${\vec {\phi }}$ 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 :

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

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

• The gas respects the law of perfect gases :

$P=\rho rT$

• 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 :

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

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

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

$\alpha _{s}=0.76$

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

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

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

• The acoustic Reynolds number is defined such as :

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

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

Closure equations

• $\rho E=\rho C_{V}T+{\dfrac {1}{2}}\rho U_{i}^{2}$

• $P=\rho rT$

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

     $\alpha _{s}=0.76$

• $q_{j}=\lambda {\dfrac {\partial T}{\partial x_{j}}}$

• $q_{j}^{k}=D{\dfrac {\partial Y_{k}}{\partial x_{j}}}$

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

Time discretization

A third order Runge-Kutta method is used. This one allows to improve the precision of time integration and the stability.

The equations used in H-Allegro are defined below :

${\dfrac {Q_{1}^{1}-Q_{1}}{\Delta t/3}}+{\dfrac {8}{5}}RHS(Q_{1})=0$

$Q^{2}=Q^{1}-{\dfrac {\Delta t}{4}}RHS(Q_{1})$

${\dfrac {Q_{1}^{2}-Q^{2}}{\Delta t/2}}+{\dfrac {5}{6}}RHS(Q_{1}^{1})$

${\dfrac {Q^{3}-Q^{2}}{\Delta t}}+{\dfrac {3}{4}}RHS(Q_{1}^{2})$

$Q^{i}$ is the matrix of gross conservative data at the ith step and at the time t.

$Q_{1}^{i}$ is the same matrix as before at the time $t+\Delta t$.

$Q^{i}$ and $Q_{1}^{i}$ are Np x Nvar matrix, with Np, number of points and Nvar, number of variables.

Parameters

Makefile

The makefile allows to compile routines and subroutines.

In order to compile, the command "make" will be typped in the terminal.

The default target of this file is "all : :".

The configuration of compilation can be modified :

• FC indicates what is the used language. For instance : if the language is fortran90, it would be written FC = mpif90.
• CCFLAGS and FFLAGS represent the options of compilation. If the used language is C, then the options would be indicated for the variable CCFLAGS, otherwise if the language is fortran90, then the options would be indicated for the variable FFLAGS.

For instance, FFLAGS = -O3 -ipo -p means that the used language is fortran90.

-p is an option of compilation. It is used in order to have gprof commande.

Since various computer can be used, the compilation have to be conditional. That is to say that several configurations of compilation are available in function of the computer.

The common files of the executable for calculation and the common files of the executable for postprocessing have to be defined.

For example : all : : Mezo3D df6_pospro

COMMON_OBJS += communs.o

COMMON_OBJS += lectus.o

Then Allegro files are compiled and then an executable is created.

Mezo3D : $(ALLEGRO_OBJS)

$(QUIET_LINK)$(FC) $(FFLAGS) -o $@ $(filter %.o,$ ˆ)

The same thing is done for the postprocessing files :

df6_pospro : $(POSTPRO_OBJS)

$(QUIET_LINK)$(FC) $(FFLAGS) -o $@ $(filter %.o,$ˆ)

An option named clean is available. This one allows to erase files with extensions .o, .mod, .d. The executable files, the solution files and the gmon.out files are also deleted.

It is possible to change this option if necessary .

Job

The job file is used if the calculation of the problem is made on a computer which is not local.

The extension of this file is .ll because it is a LoadLeveler file.

A job has to be created like this :

• Working name : locating more easily the calculations.

@ job_name = $NAME-$dir 

• Standard exit file :

@ output = \$(job_name).o\$(jobid)

• Error exit file :

@ error = \$(job_name).e\$(jobid)

• Working type :

@ job_type = MPICH

• Total number of wanted processors :

@ total_tasks = $NP

• Maximal elapsed time for the job :

@ wall_clock_limit = 30 :00 :00

• Maximal memory by core : if an error occurs because of the number of processors, this limit can be raised.

@ data_limit = 550mb

• Initial directory to send : this directory contains all the data for the calculation.

@ cri_initialdir = $PWD/$dir/IN

• Final directory for the results :

@ cri_finaldir = $PWD/$dir/OUT.