A
Numerical
Investigation
of
Premixed
Combustion
M.
R.
Nalim
Nat
i
onal
Research
Cou
ncil.
in
Wave
Rotors
D.
E.
Paxson
NASA
Lewis
Research
Center.
Cleveland.
OH
44135
Wave rotor cycles that utilize premixed combustion processes within the passages
are examined numerically using a one-dimensional
CFD·hased simulation. Internal-
co
mbustion wave rotors are envisioned
for
use as pressure-gain combustors
in
gas
turbine engines. The
simulation methodology is described, including a presentation
of
the assumed governing equations
for
the flow and reaction
in
the channels, the
numerical integration method used, and the modeling
of
extemal components such
as recirculation ducts. A number
of
cycle simulations are then presented that illustrate
both turbu/ent-deflagration
and
detonation modes
of
combustion. Estimates
of
perfor-
mance
and
rotor wall tempera
II/
res
for
the various cycles are made,
and
the advan-
tages and disadvantages
of
each are discussed.
Introduction
The
wave rotor is a device that utilizes unsteady wave motion
to exchange energy by direct work action between fluids. which
may be chemically inert
or
reacting.
It
consists
of
a number
of
channels arranged about an axis; by rota
ti
on the ends
of
the
channels are periodically ported to high and low-pressure ducts,
which generate and utilize waves in the channels. Because the
number
of
channels is large, the flow
in
the ducts
is
practically
steady. and is directed to other steady flow components. An
important feature is that as gases
of
a wide temperature range
flow through the rotor, the mean channel-wall temperature is
lower than the highest gas temperature. Rotational speed is low
relative to turbomachines, and the geometry usually simpler,
allowing greater strength and l
ower
cost. F
or
deta
il
ed descrip-
tions
of
wave rotor principles and app
li
cations see Shreeve and
Mathur ( 1
985),
Nalim
(1994),
and Welch
eta
!.
( 1995).
A wave rotor ac
ti
ng as a
pr
essure exchanger can be used
(together with a conventional combustor) as a topping un
it
to
enhance the performance
of
a gas turbine engine. Welch ct al.
( 1
995)
have presented simulations based on validated codes,
which indicate a substantial pressure gain possible between the
compressor and t
he
turbine. Similar pressure
ga
in could also be
obtained using an intemal-combustion wave rotor. In this case,
combustion
oc<:urs
sequentially within the wave channels, each
channel being periodically charged and discharged as
it
rotates
past-properly-sized-and-timed inlet and outlet ports. Simplified
combustion and wave processes are illustrated
in
the wave rotor
sketch in Fig.
I.
By accomp
li
shing
com
bustion on the rotor,
the external combustor needed in a pressure-exchanger topping
cycle is eliminated.
So is the associated dueling, w
hi
ch
mi
ght
be long a
nd
unmanageably hot in some designs.
Combu
stion
Mod
es.
The
implications
of
internal combus-
tion and the feasible combustion modes are discussed in Nalim
( 1995). Rapid combustion is essential to minimize residence
time and rotor size.
Th
e charge may be partially
or
fully pre-
mi
xed, and ignited by compression or
ot
her means. The feasible
modes resemble combus
ti
on in various types
of
internal com-
bustion (I
C)
engines. For relatively low i
nl
et temperature (less
than about
800 K f
or
hydrocarbon fuel
),
premixing and suffi-
cient turbulence are necessary to permit a high deflagration
Contributed by the International Gas Turbine Institute and presented at the
41
st
International
Ga.~
Turbine and Aeroengine Congress and Exhibition. Birmingham.
United Kingdom. June
10
-
13.
1996. Manuscript received at ASME Headquancrs
February 1996. Paper No. 96-GT-1 16. Associate Technical Editor:
J.
N. Shinn.
668 I Vol. 119, JULY 1997
fl
anne
speed when ignited by a
··spark''
of
residual
or
injected
hot gas. With higher inlet temperatures. a detonation mode be-
comes likely in premixed gas.
or
a nonpremixed, turbulent ·"die-
sel''
combustion mode may be used. Only premixed combustion
is considered here.
Combu
stion
Sim
ul
ation
Goal
s. This work is a step t
owar
d
simulation
of
internal-combustion wave rotors. Numerical mod-
eling
of
combustion is a cha
ll
enging research area. There is a
strong interaction between the energy release by chemical reac-
t
ion
and the dynamics of fluid motion. Localized energy release
creates steep gradients
in
temperature and other properties,
which drive transport
of
species. momentum, and energy. Fluid
turbulence. when
pr
esen
t,
may
in
teract with the reaction to the
extent
of
dominating its rate. A wide range
of
time scales a
nd
length scales are important for different phenomen
a.
In an internal-combustion wave rotor,
the
large-amplitude
nonsteady motion typical
of
wave rotors combines with inter-
mittem combustion. This imposes a heavy
comp
utational bur-
den. especially
fo
r multidimensional calculations. In the case
of
shock-induced reaction and detonation, the numerical problems
typica
ll
y associated with shock resolution and ensuring accurate
shock-speed are compo
un
ded by the chemical reaction. Because
chemical induction time is a sensitive function
of
temperature,
it is a challenge to obtain accurate chemistry near a shock wnen
there is locally poor accuracy for energy and temperature. In
the case
of
turbulence-enhanced deflagration. the
nu
merical rep-
resentation
of
turbulence. as we
ll
as the choice
of
a combustion
model
that appropriately combines chemical kinetic and turbu-
len
ce
effects, are both difficult issues. The computa
ti
on should
prop
er
ly resolve the flame thi
ck
ness and the mul
ti
ple time scales
and length scales (acoustic,
dif
fusion, and reaction). Ideally,
adaptive
griddjng is needed to efficiently compute the flame
ptup:1g1tion:
The
present work attempts only the simplest. one-dimen-
sional, representation of combustion that is compatible with an
existing wave-rotor design and simulation
code
(Paxson, 1995a;
Paxson and Wilson, I 995 ). This uniform-grid, one-dimensional
code has already allowed much progress in designing
pressure-
exchange cycles, with rapid turnaround
of
co
mputations on a
single-processor workstation.
It
was desirable to create a similar
tool for preliminary analysis and design
of
internal-combustion
wave cycles.
£t
is acknowledged that any one-dimensional repre-
sentation
of
combus
ti
on processes
wi
ll
necessarily be rather
crude, particularly when turbulence
do
minates. The intended
approach is to select the model parameters to achieve sim
ul
a
ti
on
Transactions
of
the
ASME
Stator End Plate
Shoelc
wave
genmted
by closure
of
out
fl
ow co
mpresses
incomina
ch&rae
Fig. 1 Internal
combustion
wave
rotor
Outflow
Manifold
of
a desired combustion rate, and then
est
imate the required
chemical and turbulence properties.
This paper is focused on the design of
pr
emixed-charge wave
cycles, on understanding the
fl
ow dynamics relating
th
e heat
release to the pressure waves, and on estimating overall
perfor-
mance and material temperatures. The wave rotor model, gov-
erning equations for tlow a
nd
combustion
in
the channels. and
the numerical method used a
re
described. A number
of
dctla-
gra
ti
on
a
nd
detonation mode cycle simulations are
th
en pre-
sente
d.
The predicted pressure gains and wa
ll
temperatures arc
compared and
th
e advantages and disadvantages
of
the
va
ri
ous
cycles are discussed.
Wave Rotor Model
Th
e present model is based on a
p
n:viuu~
wave rotor simula-
tion model for nonreacting gases (Paxson, 1992,
l995a).
rn
this model, one-dimensional
co
mputation using a high-resolu-
ti
on
CFD
technique is performed for a single cha
nn
el, neglect-
ing interactions between channels. Losses due
to
the finite pas-
sage-opening time, leakage to the casing through the end gaps,
heat transfer to the channel
wa
ll
s, a
nd
boundary layer viscous
losses, are a
ll
tr
eated by experime
nt
ally validated submodels
(Paxson and Wilson. 1995). In addition
to
the CFD treatment
of
the
fl
ow in
th
e channel. the cavities and the channel walls
are treated by lumped-parameter models, and the ducts are
mod-
eled
as
steady, constant-ar
ea
flows to obtain flow-homogeniza-
ti
on losses. The overall pressure gain is calculated using aver-
aged stagnation quantities
co
mputed
fr
om the absolute frame
of
reference. which takes
th
e rotational speed into account.
Governing
Eq
uati
ons
for
Channel
Fl
ow
and
Premix
ed
Co
mbu
stion.
Th
e present model assumes a calorically perfect
gas, i.e., with a constant specific heat ratio (
'Y).
Th
e composition
of
the charge at any time
an
d l
oca
tion is described solely
by
a
reac
ti
on
progress variable
(z),
which c
ha
nges from I
(pu
re
react
ant)
to 0 (produ
ct)
as combustion occurs, similar to Colella
ct a
l.
( 1986). Th
us
there wi
ll
be
one
additional equation to
be
solv
ed
, besides the
Eu
ler equations of
th
e nonreacting model.
A simple representation
of
turbulence is included in the form
of
an eddy diffusivity.
The model nume
ri
cally integrates
th
e equations
of
motion
in
a single passage as it revolves past the ports and
wa
ll
s
th
at
comprise the ends
of
the wave rotor and establish the boundary
conditions for
th
e governi
ng
equations in the passage. Ports arc
specified by their circumferential location relative to
so
me
fi
xed
point
on
the
wave rotor casing, and by a representative pressure.
temperature, and reactant fraction. With each time step the
pas-
sage advances an an
gu
lar distan
ce
specified by
th
e angular ve-
locity. If the flow is into the passage,
th
e pressure and tempera-
tu
re are sp
ec
ified as stagnation values. If
th
e flow is o
ut
of the
Journal
of
Engineering
for
Gas
Turbines
and
Power
passage. only
th
e port pressure is required. and it is specified
as a static value. Determination
of
the direction
of
the port
fl
ow~
at each time step is discussed in Paxson ( 1992).
The governing equ'l
ti
ons
writt
en
in nondimension
aJ
form arc:
ow
DF
(w)
-=
+-=-=-
=
S(w)
Dr
fJx
- -
( 1 )
where
vecto
r
s~ and
E.
have the respective perfect-gas forms:
p
pu
w = p pu2
+ -
2
+
pzqo
y(
'Y -
I)
F =
pz
pu
!!.
+ pu 2
'}'
u(
p + pu
2
+ pzqo)
(y-
1) 2
pu-;.
(2)
(3)
T
he
nondimensionalization
of
pressure
(p),
de
nsity (
p),
and
velocity
(u)
has been
obt
ained using a referen
ce
state
p*,
p*.
and the corresponding sound speed a*. The distance has been
scaled by the passage length,
L.
Th
e t
im
e has been scaled using
the nominal wave
trans
it
time,
Ua*.
The h
eat
of reac
ti
on of
the reactant gas,
q
0
•
is
ass
umed to
be
a constant. An alte
rn
ative
fo
rmulation is possible. in which the heat
of
reaction is treated
like
an external heat source, and the che
mi
cal energy terna is
not use
d.
Although this simplifies the algebra and coding, and
th
e computations were check
ed
to
be equivalent,
th
e given for-
mulation is more consiste
nt
with
th
e usc
of
conserva
ti
on vari-
ables, and
ca
n be extended to treat multiple species and real
chemistry with variable
q
0
.
The source vector,
S(
w.
x)
includes
co
ntributions from the
chemical reac
ti
on
rate-:-
tu
rbulent eddy diffusion, and viscous
forces and heat transfer at the walls.
A leakage term is also
added for the end gas. Without leakage, the source term is
0
e,
a~
u
----
+ aotd pu l
O.?S
R
e*
fJx
2
-
t,
8
2
( u
2
T zqo)
~(~.x)=
--
- +
+-
Re*fJx2 2
(-y-
I)Pr
, Sc,
(4)
T
he
forms of
th
e wall
so
urce terms for viscous stress
anJ
heat transfer and their
coe
ffi
cients a
2
and a
3
are ba
sed
on semi-
em
pirical correlations.
The
expressions and definitions for the
combustion rate and eddy diffusion terms are discussed below.
Combusrion Rate. In general,
th
e
ra
te of combus
ti
on at a
given location in the flow will depend on the local composition,
temperatur
e,
pressure. and turbulence properties.
Th
e mecha-
nisms
of
combustion
are
quite differe
nt
for turbulent deflagra-
tion and for detonation
of
premixed charges. and different forms
of
the rate equation arc expect
ed
. In each case we represent
combustion by a finite-rat
e,
single-step reactio
n.
JULY 1997, Vol. 119 I 669
For the calculation
of
shock-ignited reaction and detonation.
the rate (R)
is assumed to be proportional to the reactant frac-
tion, and to have an Arrhenius-type dependence on temperature.
The
rate coefficient is based on available single-st
ep
reaction
kinet
ic
models.
Us
ually. a large activation
ene
rgy is assumed,
a
nd
ignition temperature kinetics are used. i.e., the rate coeffi-
cient is zero below a threshold (ignition) temperature ( T
0
),
and
is a constant (
K
0
)
above it. This mode
is
activated by settina
c.=OinEq.(4).
.,
For the calculation
of
turbulent deflagration. the turbulence
model described in the next subsection
is
ac
tivated. Here also,
ignition-temperature kinetics are used, but t
he
rate is assumed
to be proportional to both the reactant and the product fractions,
i.e ..
R
cc
z(
I
...:
z)
by setting c
1
= I in the sou
rce
vector. based
on the suggestion
of
Magnussen and H
je
rtager ( 1976). This
implies that the reaction can occur only at a propagating name
surface. The rate coefficient
mu
st be assigned phenomenologi-
cally. based on an estimate for the reaction timescale, which
may
be
influenced by both chemistry and turbulence.
No special model is used for the ignition process to initiate
a deflagration.
Tn
the cases
co
nsidered here,
ini
ti
ation takes
place by recirculation
of
hot combustion gas from leading chan-
nels, and by residual hot gas in the channel. Cavity leakage
was also observed to
in
itiate a flame in some simulations not
discussed here. The one-dimensional treatment does not capture
the penetration and vortex mixing effect
of
a
jet
of
hot gas
injected through an orifice smaller than the channel width.
Turbulence Model.
Th
e effect
of
turbulence is approxi-
mated by the usc of an eddy diffusiv
it
y, which results in
diff
u-
sive fluxes
of
z,
energy, and momentum, proportional to their
respecti vc streamwise gradients. In
Eq. (
4)
, Re *
is
a Reynolds
number based on the reference state
p*, a*, and
L:
€,
is
the
eddy viscosity scaled by the molecular viscosity.
The
formula-
tion pennits the use
of
different diffusivities for mass, momen-
tum, and heat, by specifying the turbulent Prandtl number Pr,
and turbulent Schmidt num
ber
Sc,.
Cl
ear
ly, such a model has little predictive value, because the
role
of
turbulence in
fl
ame propagation is much more complex
than simply eddy diffusion. Within the constraints
of
a
one
-
dimensional calculation, however. there is not much scope for
worthwhile sophistication.
It
is
comforting that t
he
name propa-
gation rates calculated showed an appropriate sensitivity to the
values for the model parameters. as discussed later. A more
detailed model and multidimensional computations arc needed
to examine the real physics
of
turb
ul
ent flame
pr
opagation.
Sy
stem
Mo
del a
nd
Wall Te
mp
erature
Calc
u
latio
n.
The
system layout for wave rotor simulation is shown schematically
in Fig.
2.
Boundary conditions for the end
or
port regions
of
the channel now are
ge
nerally supplied as stagnation states.
These are either provided directly by the user, as in the case
of
the port leading from the upstream compressor. or calculated
by lumped-capacitance models
of
the rotor hou
si
ng space a
nd
the recirculation passages. The space between the rotor and the
housing, to and from which leakage occurs, is lumped as a
single cavity. The pressure
dif
ferences between the cavity and
the channels, together with the specified gap between rotor and
endwalls, govern the leakage now via a sour
ce
t
erm
in the first
and last computational cells
of
each channel. T
he
recirculation
ducts arc also lumped together as
if
they were a single cavity.
A stagnation pressure loss proportional to the square
of
the
mass
fl
ow is imposed on t
he
fl
ow
going through the recirculation
loop. The downstr
eam
turbine could also
be
modeled as a cavity
and valve; however, in this study, the exhaust port pressure was
held constant based on previous calculations for topping cycles
(Paxson, 1995b). In this paper. the term lumped-cap.tcitance
implies that the kinetic
ene
rgy
of
the flows in the components
is n
eg
lected. Thus, they may
be
model
ed
using only mass and
energy conservation equations.
670
I Vol. 119, JULY 1997
p in
T
in
z
(I)
from
in
Compressor
(inlet)
Hot Gas Recirculation
Loop
Passages
Center Cavity
Wave Rotor
F
ig
. 2 Sys
tem
layout
(4)
To
Twbine
(exhaust)
Dri
ve
Motor
p
4
The
stagnation boundary conditions supplied by the user or
component models are used by each channel in the
CF
D code
to determine the state
of
the so-called image cell at the next
instant
of
time.
The
code is capable
of
assessing whether a
given condition wi
ll
lead to inflow or outflow in a given channel.
This allows robust operation
of
the simulation even in so
me
off-design conditions where a portion of the flow in a given
port may be into the rotor and a portion
out
of
the rotor. For
outflow conditions, o
nl
y the boundary pressure is used, and it
is treated as a static value.
For
inflow conditions some account-
ing is made
if
the flow in the duct is not ali
gned
with the passage
(i.e., shaft work into the system). For both i
nfl
ow and outflow
conditions accounting is made at the boundaries for the effects
on the flow
of
those channels, which are only partia
ll
y opened
to a port: so-called
fi
nite opening effects.
Th
e ducts leading to
and from the ports are assumed loss free
(isent
r
opic);
however,
a constant area
mixing calculation is used in outflow ports to
account for losses
due
to non-uniformities in the flow.
A lumped capacitance method is also used to track the
wa
ll
temperatures. as described by Paxson ( 1995a), except that t
he
channel side wa
ll
s also contribute to the
he
at transfer, and are
assumed to be at the s
am
e temperature as the
up
per and lower
wa
ll
s. Longitudinal
co
nduction in the rotor is not a
ll
owed in
order to obtain the worst case wa
ll
t
em
pe
ratures. Thus, each
slice
of
the rotor that is in contact with a channel computational
cell is treated as a separate Jump.
The
computed steady-state
wa
ll
temperature may be thought
of
as
a time-averaged gas
temperature,
but
weighted for heat transfer.
Numerical M
et
hod
Equation (
I)
is integrated numerica
ll
y as follows:
~
+
I
- •
(j"
L"
)
/::it
n A
~I
-
~~
- _i+l/2 -
1-112
-;;::- + S1
ul
uX
-
where the numerical
fl
ux estimate is
I
A,R
<X
- 2 ::t:
l+l/2
(5)
+
~~
( (A)/+1
~~+I
+
[A)/~/)
(6)
Tr
ansac
tion
s of
the
ASME
and
the
numerical sour
ce
term is
s• = !
(3S
" - s·
I)
_ , 2 _ , _ ,
(7)
The term
c£.R
""
in Eq.
(5)
refers
to
the flux-limited dissipation
ba
se
d
on
the approximate Ri
ema
nn
so
lver
of
Roe ( 1
986)
for
Eq. ( I ) without a sour
ce
vector.
Th
e matrix
[A]
is the Jaco
bi
an
of
the flux vect
or
F.
Th
e superscript n and subscript i are indexes
for
the
discrete iemporal and spatial steps, respectively.
Th
is
sc
heme
has the advantage
of
being formally second-or
de
r accu-
rate in time and spa
ce
when the flow is smooth yet maintaining
the
high
resolution
of
Roe's
method in the vicinity
of
shock
wave
s.
Furthermore, as the
so
ur
ce
st
r
eng
th approaches zero. the
sc
heme becom
es
monotonic, which is physically correct.
Th
e
re
are additional requirements on the numerical scheme
to preserve the physical meaning of
z,
which
are
not inherent
in the governing equations:
z
sho
uld remain in the range from
0 to
l.
and the combustion sour
ce
term f
or
the rate
of
change
of
z in Eq. (
4)
should be negative
or
zero. These
ar
e usually
satisfied by the usc
of
physically meaningful initial and bound-
ary
con
ditions,
we
ll
-behaved
so
ur
ce
term discretization, and a
stable numerical
sc
heme with monotonic
so
ur
ce
-free behavior.
A si
mple
first
-o
rd
er
stiffness
sc
heme was created for the com-
bustion
so
ur
ce
t
erm,
by dividing
it
by a factor ( I + K
0
6.t),
to
ensure positivity
of
z.
tn practice, this was found unnecessary
because
the ti
me
ste
p for stabj.lity
of
the Riemann solution was
always
much smaller than the reaction time scale.
Th
ere are
situations. unrelated to the source term,
in
which numerical
integration
of
the Ri
ema
nn
problem results in
sl
ightly out-of-
range
z.
eve
n with a monotonic
sc
h
eme
( Larrouturou, 1991);
h
oweve
r, in the simulations to be
pr
esented, they did not arise.
Since
the time
co
nstants associated with transients in the
wall temperatures and cavity
pr
openies
arc much lar
ge
r than a
complete
wave
cycle. these quantities are treated as constants
for
each
wa
ve
cycle, and then updated using simple Eul
er
inte-
gration (
Pa
xson. 1995a).
The
actual rotor thermal in
enia
and
cavity vol
um
es
do not a
ff
ect a steady-state solution. Hence, the
smallest valu
es
that a
ll
ow stable computation are used for rapid
co
nvergence to a periodic, zero-net-
fl
ux
so
lution, and steady-
state wall, cavity, and
duct
properties.
Te
st Cases
and
Gri
d Iitdependence. A number
of
reac-
ti
on-wave
test problems were solved to ensure that the numeri-
ca
l
sc
heme was stable and
pr
o
du
ce
d meaningful solutions.
These included the d
eve
lopment
of
detonations in various
frames of reference. For a direct te
st
of
accuracy in
comp
uting
detonation speed, the boundary and initial conditions were
se
t
up to
matc
h a steady Chapman- J
oug
uet
(C-J)
detonation in
a perfect gas with
'Y
= 1.2,
and
fixed heat release, q
0
= 30,
where the
ref
erence state is that
of
the unburned gas. After a
brief transient, due to the fa
ct
that the pre
sc
ribed initial
ste
p
profile neglects the thickness
of
the reaction zone. the detonation
becomes steady with the propagation speed corr
ec
tly matching
the prescribed inflow sp
ee
d.
The
classical Zeldovich
-vo
n Neu-
mann- Doring
(ZN
D) structure appears, with the computed von
Neu
m
ann
pr
essure spike within 3 percent of the theoretical
value of
30.9
(with
6.x
= 0.005, K
0
= 30.0, T
0
=
2.0T
1
).
While the detonation speed was independent
of
reaction rate
parameters and grid spacing,
the pressure spike was underpre-
dicted for coarser grids
or
lower·T
0
.
Computations of overdriven
and underdriven detonations were al
so
qualita
ti
vely correct.
It
is not
equa
ll
y straightforward to
dir
ec
tl
y compare turbulent
flame computations with any
data
or
with r
ea
listic models, as the
one-dim
ensional model rate
par
amete
rs have limited physical
meani
ng.
In
a sense, however, the inver
se
of K
0
can
be
related
to the dominant reaction time scale, which is expected to
be
the turbulen
ce
time scale.
It
has been found experimentally
(H
eywood)
that highly turbu
le
nt premixed flames, such as th
ose
found in IC engines,
ha
ve flame speed
(s,)
com
parable to the
turbulen
ce
intensity, u
',
which should imply
Journal
of
Engineering
for
Gas
Turb
i
ne
s
and
Power
S
1
~
II'
OC
~
(
8)
VR
e*
In computations
of
flame propagation in a closed tube. this
relationship was seen to
be
r
oug
hl
y preserved with a proportion-
ality constant in the
range
of
1.
0 to
1.
2, for valu
es
of f., from
250 to 1000, and K
0
from 8 to
40,
all other variabl
es
being
fix
ed at typical values ( Re * = 8.3 X I 0
6
, Pr, =
Sc
, = 1.0, q
0
=
4.0
, To = 1.5,
'Y
=
1.33)
.
The
flame spee
ds
computed arc
also
se
nsitive to changes in q
0
and T
0
•
Fo
r this same range
of
variables, grid-independent solutions were achieved for val u
es
of
6.x
less than 0.005, i.e.,
200
cells in a channel length.
Coar
se
r
grids resulted in exaggerated, grid-dependent flame speed.
Deflagration Mode Wave
Cy
cl
es
Several possible
wave
cycles using turbulent deflagration
were simulated. Because it
is
likely that low-pressure-ratio en-
gines will use this
mode
(Na
lim, 1995).
the
simulations
as
-
sumed
a design similar to a throughflow pressure-exchange
wave rotor optimized for a
small eng
in
e with an
up
stream com-
pre
ss
or
pr
essure ratio
of
approximately 8
(Paxso
n, 1995b
).
The
major
de
sign parameters for such a reference
wave
rotor.
li
sted
in Table
l, are retained
in
nondimensional form except as noted.
The
ref
erence state
fo
r nondimensionalizati
on
of va
ri
ables
is
the stagnation state
of
the inlet to the
wave
rotor. In all the
simulations, Re
* = 8.3 x I 0
6
,
Pr
, =
Sc
, = 1.0,
'Y
= 1.353.
The
simulations are pr
esented
as space- time contour diagrams
of
gas density and reactant fracti
on,
over a full rotor revolution,
with positive time in
the
upward vertical direction. The port
timings are indicated by
t!he
br
ea
ks in the side borders represent-
ing the end plates.
It
is
noted that the gas
dyna
mi
cs
of
a pressur
e-
gain wave r
otor
a
ll
ows,
in each cycle, only
pania
l discharge
of
the combustion gas to
the
higher-pre
ss
ure exhaust port, while
fresh charge
enrerc:
from the l
owe
r-pressure inlet pon.
Fast-Burn
Reve
r
se
-F low Cycle. In this mode, the wave
rotor is
de
signed for o
ppo
sed pairs
of
reverse-flow cycles, with
the one inflow
and one outflow port at each end
of
the rotor,
as illustrated in the
co
mputed wave diagram
of
Fig. 3. With
exact
sy
mm
etry of the
port
placement and the resulting gas
dynamics, there will be a resident layer
of
gas, which moves
from side
to
side but
does
not leave the channel.
Each wave cycle is required to
be
completed in about the
same half-revolution t
ime
as the corresponding
pr
essure-ex-
change cycle. This requires very fast combustion, and almost
instant ignition. The
cycle
is designed to provide
hot
gas r
ec
ir
cu-
l
at
ion from leading cha
nnel
s, via a transfer passage, to create
a torch
jet
into the premixed charge.
It
is assumed that the h
ot
residual ga
s.
heated by
combus
tion and repeated compression.
also initiates a flame
in
the c
ha
rge.
Th
e
iUu
strated simulation
was obtained by setting
K
0
= 28.0, T
0
= 1.5, and f., = I 000.0.
Th
e inl
et
mixture
is
uniform, and q
0
is 3.42 to
pr
ovide an overa
ll
temperature ratio
of
2.2. Ba
se
d on Eq.
(8),
and examination
of
the simulation, the corresponding flame speed is estimated to
be
25
mJ
s for the candidate engine operating at standard ambient
Table 1 Reference wave r
oto
r dimensions and design performance
Mean
Rotor Radius
Rotor
Length
R
oto
r
Pa
ssage Height
Rotational
Speed
Cycles/Revolution
Nu
mber
of
Passages
Mass
Aow R
ate
P
.J
P
1
(F
ig.
2)
T
.rr
l
8.15cm.
15.24 cm
2.
18cm.
16800
rpm
2
52
::
2.3
kgls
1.23
2.21
(3.2
in
.)
(6.0 in.)
(0.86 in.)
(5.0 lbm/s)
JU
LY
1997, Vol. 119 I 671
c.ircumferential position
Fig. 3 F
ast-b
um
rev
erse-
flow
cycle
temperature. This is at the high end
of
common IC engine
experience, where
10 m/s might be more typical.
A
fa
st
-bum throughflow cycle, which completes combustion
of
the charge in each half-revolution, would be very simil
ar
to
this r
eve
rse-flow cycle in flame pattern and performance.
Slow-Burn
Throu
ghflow Single Cycle. This design is in-
tended for relatively sl
ow
-burning mixtures and conditions.
It
has only one cycle
per
revolution, with the inlet and exhaust
ports on opposite ends. In the simulation illustrated
in
Fi
g.
4,
the inlet charge is stratified so that the middle one-fifth
of
the
air has no fuel
(z
=
0).
I
gn
ition is
si
milar to the last case. Here,
the flames have about thrice the time to complete combustion.
and a fifth less distance to travel. Flame temperatures are higher.
with
q
0
= 4.275, to retain the overall temperature ratio
of
2.2.
The
simulation shown used K
0
= 6.0, T
0
=
1.5
, and c, = 500.0,
consistent with a flame speed about one-third that
of
the fast-
bum simulation, based on
Eq.
(8).
Combustion is completed
before discharge and, at this level
of
diffusivity, some tempera-
ture stratification persists in the exhaust.
The mass flow rate in this wave rotor
wi
ll
be
half that
of
the
reference design. or
co
nv
ersely, a given flow rate will dictate
double the rotor size. This may be a crippling penalty, and a
solution to the problem is presented
in
the next case.
circumferential position
Fig. 4
Slow
-burn throughflow
single
cycle
672
I
Vol.
119
, JULY 1997
ci
rc
umferential position
Fig. 5 Slow-burn throughflow dual
cycle
Slow-Burn Throughflow Dual Cycle.
If
combustion is
very slow, combustion
of
a charge introduced in one cycle may
not
be completed before the next discharge process. In a two-
port
pressure-gain cycle, optimized for an overall temperature
ratio
of
2.2, o
nl
y about
60
percent of the
gas
in the channel is
discharged in each cycle. Therefore, a dual throughflow cycle
may be envisioned,
in
which the fresh charge introduced at one
end
of
the channel is burned over a period
of
two cycles, such
that all the gas discharged
at
th
e other
end
ha
s completely
burned before
fi
nal expansion. The throughput mass flow rate
of
the reference design is now recovered.
Th
e corresponding
simulation, shown in Fig.
5,
used K
0
= I 0.0, T
0
= 1.5, and
E,
= 500.0. Equation
(8)
implies flame speed about two-fifths that
of
the fast-bum case, which is roughly consistent with the rela-
tive combustion durations
of
the two cases,
if
it is noted that
combustion is slowed during the low-pressure period. In this
case, recirculation loops are provided at both ends to ensure
quick ignition and complete combustion.
Detonation Mode Wave Cycles
The
de
tonation mode is
li
kely
to
be used for high-pressure-
ratio engines with inlet temperature close to the autoignition
temperature for the
fu
el used. For the purpose of this paper,
however, most
of
the design parameters
of
the reference small-
engine wave-rotor (Table
I)
are retained in nondimensional
form. We also keep Re*
= 8.3 X
10
6
,
Pr, = Sc, =
1.
0,
'Y
=
1.353. The throughput mass flow rate is approximately doubled
relative to the reference design by doubling the number
of
cycles
per revolution
(to
four) to take advantage
of
the rapidity
of
detonative combustion. The rotor speed is adjusted slightly to
match the strong combustion-driven waves. The contour dia-
grams
in
th
is section cover only half a revolution, and tempera-
ture is shown instead
of
density because detonation involves
unly a
s.Jig
ht 'change (increase) in
de
nsity.
Throughflow
Cycle.
Th
e throughflow cycle presented in
Fig. 6 has a stratified inlet charge, with no fuel in the first one-
fifth
of
the port duration
to
avoid the possibility
of
flashback
or
prematu
re
ignition
of
the nearly detonable charge. This buf-
fers the fuel from the residual hot gas in
the
channel.
To
com-
pensate and maintain the overall temperature ratio
at
about 2.2,
the
sim
ulation uses q
0
= 3.785. The reaction rate used is K
0
=
I 0.0, with T
0
=
1.2
and E, = I 00.0. At this low level
of
diffusiv-
ity, the temperature stratification due
to· the buffer layer persists
in
the exhaust.
The detonation is initiated by coalescing compression waves
generated by closi
ng
the exhaust port while there is still signifi-
Transactions
of
the ASME