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Flowsquare 4.0: Theory and Computation
Yuki Minamoto (http://flowsquare.com)
12th December 2013
Flowsquare is a two-dimensional Computational Fluid Dynamics (CFD) software for unsteady, non-
reactive, reactive and subsonic/supersonic flows. The aim of this software is to provide a handy CFD
environment so that more people can get to know what CFD is like and simulate flows for their
educational and/or a

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Flowsquare 4.0: Theory and Computation
Yuki Minamoto (http://flowsquare.com)
12
th
December 2013
Flowsquare
is a two-dimensional Computational Fluid Dynamics (CFD) software for unsteady, non-reactive, reactive and subsonic/supersonic flows. The aim of this software is to provide a handy CFD environment so that more people can get to know what CFD is like and simulate flows for their educational and/or academic interests. This documentation includes theoretical and numerical aspects of the software, which are basis of the parameters in
grid.txt
. Although flow simulations can be carried out without understanding them, the users are recommended to read this documentation.
1. Non-reactive flows
First, numerical method for non-reactive flow simulation is explained.
1.1. Governing equations
In Flowsquare, the incompressible governing equations are solved. They consist of the continuity equation (mass conservation, and also Einstein notation is used):
,0
ii
u xt
(1) and the equation for momentum:
,
jiij j jii j
g x x puu xut
(2) where
i
u
is the velocity component in
i
-direction (m/s),
is the mixture density (kg/m
3
),
p
is the pressure (Pa),
i
g
is the external force in
i
-direction due to buoyancy (m/s
2
). The viscous term
ij
is written as:
,32
i j jiijk k ij
xu xu xu
(3) where
is the dynamic viscosity (kg/(m s)) and
ij
is the Kronecker delta. Note that the equation is solved assuming a constant density, although Eqs. (1) and (2) are written in conservative form. Equation (2) is solved (integrated in time) in two steps. In the first step, the equation is solved for
*
j
u
without the pressure term as:
,
*
jiij jii j
g xuu xut
(4) In the second step (time integration), the pressure term is included so that the mass conservation is taken into account in the second time integration:
.
*
j j
x put
(5) By calculating divergence of Eq. (5), the
Poisson’s equation for the correction pressure
*
p
is as:
,1.
*22
t t u x x p
d j ji
(6) where
t
is the time step and
d
, a user defined parameter, is typically unity. Once the corrected pressure field in obtained by solving Eq. (6), the corrected velocity in the next time step is computed from Eq. (5) as:
Flowsquare 4.0 Users’ Guide (2013)
2
/
10
.
**
t x puu
j j j
(7) The corrected velocity field satisfying the continuity equation (Eq. 1) is obtained. Note that in this numerical method, the continuity equation is not explicitly solved.
2. Reactive flows
In the simulation of reactive flows, Eqs. (1)
—
(7) are solved to obtain a velocity field. This means the flow field is solved based on the low-Mach number assumption (the flow has
to be “slow” compared to the speed of sound
). Also, changes of species and internal energy per unit mass need to be considered by solving the transport equations, due to the chemical reactions and heat release. The transport equations for mass fraction species
i
,
i
Y
and temperature
T
maybe written under appropriate assumptions as:
,
i jii ji j ji
xY D xY u xt Y
(8)
,
pT j p j j j
c xT c xT u xt T
(9) where the energy conservation is written in terms of temperature. The molecular diffusivity of species
i
, thermal conductivity and specific heat capacity at constant pressure are respectively denoted by
i
D
,
and
p
c
. Species and heat production rate per unit volume are denoted by
i
and
T
. For the simplest reaction system, where the fuel
F
and oxidiser
O
react to produce products
P
as:
,
P O F
(10) the total number of equations for Eqs. (8) and (9) is four. In Flowsquare, in order to reduce the number of equations to be solved, a progress variable and a mixture fraction are introduced. The progress variable is used for reactive flows with premixed mixtures, while the mixture fraction is used for reactive systems with non-premixed mixtures. These two modes of reactions and the reduced governing equation are explained in the next subsections.
2.1 Premixed reacting flows
A chemical reaction is called premixed reaction when a fuel and oxidiser are fully mixed before the reaction take place. For example, premixed reaction (combustion) can be observed in traditional gas stove and heater burners. One of the important features of this reaction mode is that a reaction front (a flame in combustion) can freely propagate toward unburnt mixture (see Fig. 1a), or mixture can ignite at any locations if there is enough activation energy. This is not the case in non-premixed systems as explained in the next subsection. For premixed mixture, reaction can be expressed by using reaction progress variables. A progress variable, often denoted by
c
, is a normalised scalar which shows the extent of reaction progress in premixed reactive systems. Although there are several definitions for progress variables, following definitions are frequently used:
uO F O F Y
Y Y c
O F
,//
1
/
,
b P P Y
Y Y c
P
,
, and
ubuT
T T T T c
, (11) where the subscripts,
u
and
b
denote unburnt and burnt mixtures, and
F
,
O
and
Flowsquare 4.0 Users’ Guide (2013)
3
/
10
P
denote fuel, oxidiser and product, respectively. As clearly seen from Eq. (11), a progress variable is 0 in the unburnt mixture and 1 in the fully burnt mixture. Figure 1b shows a typical progress variable variation for one-dimensional premixed flame. Flame front (reaction front) is often defined using a iso-contour of
6.0~3.0
c
. Lewis number
Le
is defined as the ratio of thermal diffusivity to mass diffusivity. If unity Lewis number, constant pressure, adiabatic conditions and
T Y Y
cccc
P O F
/
are assumed, Eqs (8) and (9) are reduced to one transport equation of progress variable as:
,
c j j j j
xc D xcu xt c
(12) where
c
is the reaction rate of
c
, and
D
is the diffusivity of the progress variable (under the above assumptions, it can be either thermal diffusivity or mass diffusivity of species). Therefore, once a progress variable field is given, temperature and species mass fraction field can be obtained using Eq. (11). The reaction rate
c
is modelled in the software as follows:
cnac
cT T T k
)1(exp
, (13) where
k
,
a
T
and
n
are mixture specific and constant for the single step reaction explained in Eq. (10). In Eq. (13),
c
is the mixture fraction between a fuel-oxidiser mixture and air (inert gas). Although the mixture fraction is explained in the next subsection, this may be used to represent surrounding air in premixed combustion. In
1
c
regions, the local mixture consists of only the fuel-oxidiser mixture, while only the air exists in
0
c
. By multiplying
c
to the reaction rate as in Eq. (13), partially premixed reaction such as
Figure 1:
Schematic illustrations of (a) a premixed flame and (b) one-dimensional variation of progress variable.
Figure 2:
Schematic illustrations of (a) a non-premixed flame, (b) one-dimensional variations of mixture fraction, and (c) temperature and mass fraction variations in a mixture fraction space.
premixed combustion in surrounding air (pure-air stream) can be considered. When no pure-air stream is set in the computational domain,
c
is automatically set as unity. The scalar transport equation, Eq. (12) may be solved for non-reactive cases without source
Flowsquare 4.0 Users’ Guide (2013)
4
/
10
term
c
when local scalar boundary condition is used (see Sec. 4.5.4).
2.2 Non-premixed reacting flows
In contrast to premixed reacting flows, non-premixed reactive systems do not require premixing of fuel and oxidiser before reactions. A schematic illustration of a typical non-premixed flame is shown in Fig. 2a. There are fuel and oxidiser streams and reaction takes place only at the location where the fuel and oxidiser meet. Using the given chemical reaction model in Eq. (10), the mass production rate of fuel, oxidiser, product and heat release are related as:
Q s s
T P O F
1
, (14) where
Q
is the heating value of fuel and
s
is the mass of oxidiser required per unit mass of fuel, defined as
F O
W W s
/
. Here,
i
W
is the molar mass of species
i
. Under the unity Lewis number assumption, the conserved scalar
is defined from any pair of variables as:
O F O F
Y sY
,
, (15)
sY Y
P F P F
1
,
, (16)
sY Y
P O P O
1
,
, (17)
QT cY
p F T F
,
, (18)
QT scY
pOO F
,
. (19)
is called conserved scalar because no source term appears in the transport equations of these conserved scalars (such transport equation can be obtained from Eqs. 8 and 9); for above
O F
,
, the transport equations of
F
sY
and
O
Y
are written as:
,
F j F j F j j F
s x sY D x sY u xt sY
(20)
.
O jO jO j jO
xY D xY u xt Y
(21) By combining Eqs. (20) and (21) using the relation in Eq. (14), it is clear that the source terms are cancelled out and following is obtained:
jO F jO F j jO F
x D xu xt
,,,
(22) Once a conserved scalar is defined, a normalised conserved scalar
is obtained. This normalised conserved scalar is generally called mixture fraction, and written as:
OX FU OX
, (23) where the subscripts
FU
and
OX
denote fuel and oxidiser streams, respectively. Figure 2b shows a one-dimensional mixture fraction variation for a non-premixed flame shown in Fig. 2a. With the above definition of mixture fraction,
1
corresponds to the fuel stream and
0
corresponds to the oxidiser stream. As shown in Fig. 2b, the flame location in non-premixed systems is often assumed as the location where
st
. Here,
st
is the stoichiometric mixture fraction, at which there is the exact amount of oxidiser to convert unit quantity of fuel into product. The stoichiometric mixture fraction is written as:

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