12.8 Combustion
401
that the density differences can be dealt with by means of the Boussinesq
approximation discussed earlier. In other systems (engine cylinders are the
most common example) there are large changes in both pressure and density.
It is possible to do direct numerical simulation of turbulent combusting
flows but only for very simple flows. It is important to note that the speed of
travel of a flame relative to the gas is rarely greater than 1 m/s (explosions
or detonations are an exception). This speed is much lower than the speed
of sound in the gas; usually, the fluid velocities are also well below the sound
speed. Then the Mach number is much less than unity and we have the
strange situation of a flow with large temperature and density changes that
is essentially incompressible.
It is possible to compute combusting flows by solving the compressible
equations of motion. This has been done (see Poinsot et al., 1991). The
problem is that, as we have pointed out earlier, most methods designed for
compressible flows become very inefficient when applied to low speed flows,
raising the cost of a simulation. For this reason, these simulations are very
expensive; this is especially so when the chemistry is simple. However, when
more realistic (and therefore more complex) chemistry is included, the range
of time scales associated with the chemical reaction is almost always very
large and this dictates that small time steps be used. In other words, the
equations are very stiff. In this case, the penalty for using compressible flow
methods may be largely eliminated.
An alternative approach is to introduce a low Mach number approximation (McMurtry et al., 1986). One starts with the equations describing a
compressible flow and assumes that all of the quantities to be computed may
be expressed as power series in the Mach number. This is a non- singular
perturbation theory so no special care is required. The results are, however,
somewhat surprising. To lowest (zeroth) order, the momentum equations reduce to the statement that the pressure p(O) is constant everywhere. This
is the thermodynamic pressure and the density and temperature of the gas
are related by its equation of state. The continuity equation has the compressible (variable density) form, which is no surprise. At the next order, the
momentum equations in their usual form are recovered but they contain only
the gradient of the first-order pressure, p('), which is essentially the dynamic
head found in the incompressible equations. These equations resemble the
incompressible Navier-Stokes equations and can be solved by methods that
have been given in this book.
In the theory of combustion, two idealized cases are distinguished. In the
first, the reactants are completely mixed before any reaction takes place and
we have the case of premixed flames. Internal combustion engines are close
to this limit. In premixed combustion, the reaction zone or flame propagates
relative to the fluid at the laminar flame speed. In the other case, the reactants
mix and react at the same time and one speaks of non-premixed combustion.
These two cases are quite different and are treated separately. Of course,
401
that the density differences can be dealt with by means of the Boussinesq
approximation discussed earlier. In other systems (engine cylinders are the
most common example) there are large changes in both pressure and density.
It is possible to do direct numerical simulation of turbulent combusting
flows but only for very simple flows. It is important to note that the speed of
travel of a flame relative to the gas is rarely greater than 1 m/s (explosions
or detonations are an exception). This speed is much lower than the speed
of sound in the gas; usually, the fluid velocities are also well below the sound
speed. Then the Mach number is much less than unity and we have the
strange situation of a flow with large temperature and density changes that
is essentially incompressible.
It is possible to compute combusting flows by solving the compressible
equations of motion. This has been done (see Poinsot et al., 1991). The
problem is that, as we have pointed out earlier, most methods designed for
compressible flows become very inefficient when applied to low speed flows,
raising the cost of a simulation. For this reason, these simulations are very
expensive; this is especially so when the chemistry is simple. However, when
more realistic (and therefore more complex) chemistry is included, the range
of time scales associated with the chemical reaction is almost always very
large and this dictates that small time steps be used. In other words, the
equations are very stiff. In this case, the penalty for using compressible flow
methods may be largely eliminated.
An alternative approach is to introduce a low Mach number approximation (McMurtry et al., 1986). One starts with the equations describing a
compressible flow and assumes that all of the quantities to be computed may
be expressed as power series in the Mach number. This is a non- singular
perturbation theory so no special care is required. The results are, however,
somewhat surprising. To lowest (zeroth) order, the momentum equations reduce to the statement that the pressure p(O) is constant everywhere. This
is the thermodynamic pressure and the density and temperature of the gas
are related by its equation of state. The continuity equation has the compressible (variable density) form, which is no surprise. At the next order, the
momentum equations in their usual form are recovered but they contain only
the gradient of the first-order pressure, p('), which is essentially the dynamic
head found in the incompressible equations. These equations resemble the
incompressible Navier-Stokes equations and can be solved by methods that
have been given in this book.
In the theory of combustion, two idealized cases are distinguished. In the
first, the reactants are completely mixed before any reaction takes place and
we have the case of premixed flames. Internal combustion engines are close
to this limit. In premixed combustion, the reaction zone or flame propagates
relative to the fluid at the laminar flame speed. In the other case, the reactants
mix and react at the same time and one speaks of non-premixed combustion.
These two cases are quite different and are treated separately. Of course,