Combustion, Explosion, and Shock Waves, Vol. 32, No. 4, 1996
NUMERICAL MODELING
OF SUPERSONIC-FLOW
IN A TWO-DIMENSIONAL
OF WALL TANGENTIAL
DECELERATION
DUCT IN COMBUSTION
HYDROGEN
JETS
UDC 536.46
O. M. Kolesnikov
Results are presented of a numerical modeling of the ignition and combustion of underexpanded
turbulent hydrogen jets injected into supersonic air flow (M = 2.63) along the walls of a twodimensional duct. Calculations were performed by numerical integration of reduced NavierStokes equations using the method of global iterations. The kinetic mechanism of hydrogen
combustion in air involved 13 reactions. In the calculations the duct height was varied. In a
fairly narrow duct, the static pressure increased with flow deceleration to subsonic velocities
due to ignition and combustion. The influence of combustion on the pressure distribution in the
transverse direction is ambiguous. Initially, combustion increases the pressure nonuniformity
(a new oblique shock wave occurs), while, downstream, the pressure profile is flattened out due
to the appearance of a subsonic layer near the flame front.
A two-dimensional duct with supersonic flow at the inlet and with two slot-injected underexpanded
hydrogen jets was studied. The following problem was posed. What is the effect of jet ignition on the. flow?
Can the appearance of wide high-temperature subsonic layers and their interaction with shocks cause either
flow deceleration to subsonic velocities or even the formation of a pseudoshock wave? The duct height may
be the most important parameter in the problem. The limiting cases are obvious. These are combustion of a
slot-injected hydrogen jet in an unconfined supersonic flow [1] and flow choking in a narrow duct in the course
of injection and ignition of underexpanded hydrogen jets.
Numerical modeling is performed by solution of parabolized (PNS) or reduced (RNS) Navier-Stokes
equations which are derived from a full system of equations by rejecting insignificant viscous terms with
derivatives along the flow direction. The use of different terms for the same system of equations is now generally
accepted (the English abbreviation PNS means Parabo].ized Navier-Stokes and RNS Reduced Navier-Stokes
equations). The first term is used for parabolic problems and the second for elliptic problems with, e.g.,
a strong viscous-inviscid interaction [2]. In a two-dimensional approach the system of dimensionless PNS
equations has the form
opu
op.
Opu2 opus_
+0V =°'
Opuv
Opv 2
O~ + 0~--
+ 0y
0
1
0x
Op +
Oy 3 R e 0 y
o(o ) ,
I~1
Oy '
Zhukovskii Central Aerohydrodynamic Institute, Zhukovskii 140160. Translated from Fizika Goreniya
i Vzryva, Vol. 32, No. 4, pp. 47-54, July-August, 1996. Original article submitted May 31, 1995; revision
submitted January 29, 1996.
0010-5082/96/3204-0399 $15.00 ~) 1997 PlenumPublishingCorporation
399
OpuT
OpvT
Op.
( 7 o o - 1)ML f
Sp'~
Re \JOy
%
m1 0 ( O T )
qoRe Oy %.2 -~y
#2 (~i
OI~OT
+~
cpi --~y) c3y
1
c, Re ~ hi#i,
z
OpuYi
Oz
OpvYi
Oy -
I
0 (#20~'~
Re 0 v
1 Wi,
+
where ~tl = # + p a r e and #2 = (1/Pr)# + (1/Prt)peRe; Pr and Prt are the molecular and turbulent Prandtl
numbers equal to 0.74 and 1, respectively; I~ is the concentration of the ith component; 7~ is the ratio of
/ *
the specific heats of the incoming flow; hi = /c~dT + hoi; and hoi is the specific heat of formation. The
velocity components u and v, the density p, the temperature T, the molecular-viscosity coefficient #, and the
specific heat % are referred to the corresponding values of the incoming flow, and the pressure p to p~u 2. The
turbulent-viscosity coefficient e is referred to Luc¢ (L being the characteristic dimension), and the formation
rate of the ith component l~i is made dimensionless by dividing it by the c o m p l e x / ~ / L 2.
The turbulent-viscosity coefficient is calculated using the Sebesi-Smith algebraic model. The kinetic
mechanism of hydrogen combustion in air consists of 13 reactions involving nine components (02, H2, H20,
O, H, OH, HO2, H202, and N2):
1)H + 02 = O H + H ,
2) O + H2 = O H + H,
3) OH + H2 = H20 + H,
4) 20H = H20 + O,
5) H2 + M = 2H + M,
9) H2 + O2 = 2OH,
6) H20 + M = H + OH + M, 10) H + 02 + M = HO2 + M,
7)OH+M=O+H+M,
11) 2OH + M = H202 + M,
8) 02 + M = 20 + M,
12) HO2 + H2 = H202 + H,
13) HO2 + H20 = H202 + OH,
where M is any of these. Nitrogen was considered inert.
The averaged rates of chemical reactions in turbulent flow are calculated with allowance for the
intermittence effects using the simphfied model of [3]. This model was shown [4] to give better agreement
between calculated and experimental data compared with a quasi-laminar approximation.
The main advantage of PNS equations over full Navier-Stokes equations is that at supersonic flow
velocities the former refer to a mixed hyperbolic-parabolic type and, hence, can be sotved by highly effective
methods. For subsonic velocities, both the PNS system and full Navier-Stokes equations are elliptic. The
mechanism of upstream transfer of perturbations also arises in problems of viscous-inviscid interaction because
of the presence of the term 01O/Ox in the equation of the longitudinal momentum component. Nevertheless,
using an appropriate regularization procedure (see [5]), one can solve the original system of equations by
marching methods, provided that the subsonic sublayer is comparatively thin and the pressure gradients are
not large. Otherwise, it is necessary to discard the formulation of the Cauchy problem and use the method
of global iterations [2, 6]. In the present paper, as in [6], the solution is found by repeated calculation of the
entire region of integration from the inlet to outlet section using the pressure distribution along each subsonic
layer obtained in the previous marching. The term Op/Oz is approximated as
Op
Oz -
~
Pi -- Pi-1 + (1 -- ~) Pi+l -- Pi
Ax
Az
where i is the number of grid points in the x direction, ~ = 7M2/(t + (7 - 1) M 2) for M ~< 1, and ~ = t for
M > 1. It is seen that, owing to the second term, the finite-difference statement is elliptic. In the marching
solution of the PNS system, the second term is omitted [5]. In the method of global iterations, io/+1 is taken
from the previous iteration, and this allows the same marching algorithm to be used. The iteration process is
400
P/~
o.
2.0
b
~Pcl
1.0
!
0
t
100
200
t
t
1
300
400
500
!
600 x / h
Fig. 1. Diagram of flow (a) and pressure distributions on the wall and in the
symmetry plane without combustion (b): H = 15 cm; h = 0.5 cm; n = 6.
p/p,
a
3
p
b
f~
, - \ r-~j'"
r,
7
1,,.,'\
Ar
'I,.
.....- ............
1
0
--~
Pw
i
100
non combustion
=
200
J
300 x / h
4
800
t
900
t
1000
1 laO0
x/h
Fig. 2. Pressure distribution on the wall and in the symmetry plane in hydrogen
combustion: H = 15 cm; h = 0.5 cm; n = 6.
terminated when the difference between pressure distributions in several successive iterations becomes small.
The former calculations of hydrogen jet combustion show that the method of [6] does not work in its
original form in which the pressure distribution along a single layer (along the wall) is taken as Pi+l. Ignition
and combustion give rise to new subsonic layers. As a result, the mechanism of viscous-inviscid interaction
becomes more complicated. Retention of the entire pressure domain in memory and use of these data to
approximate ap/Ox made this method efficient.
K strong viscous-inviscid interactions are accompanied by the formation of separated zones, the methods
based on the flow-marching procedure involve new difficulties. In the recirculation zone, the characteristics of
convection are directed counterflow and are suppressed by the classical FLARE approximation [7] in which
convective terms within small separated zones can be ignored.
The solution of the initial nonlinear equations in terms of partial derivatives was based on
quasilinearization using the Newton-Raphson method followed by implicit numerical integration. A finitedifference scheme of a second-order approximation was employed in the transverse direction, and that of a
first order was used in the longitudinal direction. The difference equations of conservation of mass, momentum,
and energy were solved simultaneously by vector sweep. Only nine equations of conservation of concentrations
were detached, and each of these was solved by scalar sweep.
The program based on this algorithm was tested repeatedly, in particular, by comparison with the
results of the well-known experimental study of hydrogen combustion in supersonic air flow [8]. Fair agreement
between concentration and temperature profiles and the pressure distributions along the wall [4] was obtained.
We now consider the main results of numerical modeling of flow in two-dimensional ducts of various
heights (2H = 12-30 cm) with the following air-flow parameters at the inlet: M= = 2.63; Ta = 1010 K;
pa = 1.74- 105 Pa. Hydrogen was injected from two slots of height h = 0.5 cm located on opposite walls
(Fig. 1) with the following inlet parameters: Mj = 1; Tj = 670 K; pj/pa = 2 or 6. The thickness of the wall
401
p/p~
a
M
2
b
2.5
1.5 ~ x
= 9 cm
2
]
.
481
.
.
.
.
.
,
--
1
,r-
0.5 ~
o
577
o.'2
0:4
-"
0.5
0:6
o:8
,/0ym
0
0:2
d6
0:4
o:s
1/oy/H
Fig. 3. Profiles of velocity (a) and Mach number (b) in duct cross sections:
H = 15 cm and n = 6.
8
M
2.5
t
Mcl
6
2
4
1.5
j,,v
1
2
0
!
I
I
I
t
100
200
300
400
500
0.5
x / h 600
Fig. 4. Distributions of pressure and Mach number along the flow:
Ta = 890 K; H = 15 cm; h = 0.5 cm; n = 2.
boundary layer was assumed to be 1 cm up to the point of hydrogen injection.
To determine the heat-release effect on gas dynamics, calculations without chemical reactions were
performed. Figure lb shows the static-pressure distribution along both the wall pw(x) and the line of symmetry
pcz(x). The periodic structure of the distribution is caused by the shock generated by the underexpanded jet.
Jet expansion near the nozzle exit leads to a rapid increase in the Mach number, so that the flow in the
duct is supersonic everywhere except for a very narrow subsonic wMl part of the boundary layer. Thus, the
marching solution of the PNS system suits this case well. Global iterations do not lead to a marked correction
of the results. The process of mixing between the hydrogen jet and the air flow is rather inert, i.e., hydrogen
is absent even at a distance of 300 cm in the central part of the duct that occupies one-third of the area.
The distance at which ignition occurs depends significantly on the jet off-design parameters (n = pj/pa):
it is 28 and 12 cm for n = 2 and 6, respectively. In the combustion calculations, the onset of ignition can be
determined by the profiles of water-vapor concentration and by the appearance of a new peak in the pressure
distribution along the duct wall by comparison of the results with and without combustion (Fig. 2).
The cause of the strong effect of the injection pressure on the ignition delay is as follows. A large
pressure drop increases the difference between the flow and jet velocities and, hence, the turbulent-viscosity
coefficient. As a result, a hydrogen-air mixture suitable for ignition is formed more rapidly. The data on the
effect of the value of n on the ignition delay were obtained when the static temperature of the external flow
was sufficient for ignition. When it is lower (the total temperature is assumed to be higher than the ignition
temperature), the effect of the parameter n is reversed, as is shown in [1].
The strongest shock was caused by hydrogen ignition. The region in which this shock interacts with the
mixing layer after reflection from the plane of symmetry is a serious barrier to the solution of PNS equations
by marching methods. A rise in the mixing-layer temperature due to ignition leads to a decrease in the Mach
402
8
8
6
10
4
2
I
0
2oo
4~
6~o
I
~/h 8oo
Fig. 5. Pressure distribution along the wall in the symmetry plane at various duct
heights: M~ = 2.63; n = 6; h = 0.5 cm.
number. The interaction between this hot layer and the shock causes a further decrease in the Mach number
until a comparatively wide subsonic region forms. Since the solution accuracy and the stability of the marching
algorithm decrease, it is necessary to use the method of global iterations.
Combustion initially increases the pressure nonuniforrnity (a new shock appears), but, downstream,
weakens all oblique shocks, so that after several reflections from the wall the shocks vanish altogether. The
pressure in the transverse direction becomes more uniform. This is clearly seen in Fig. 2, which shows the
pressure distribution along the wall and the line of symmetry in the initial (x = 0-150 cm) and final (z =
400-600 cm) regions. Such a strong weakening of shocks is not observed without combustion (Fig. lb).
The strong decay of oblique shocks in combustion is likely to be due to subsonic layers that arise in the
vicinity of the flame front. The interaction of these layers with the adjacent supersonic layers crossed by the
shocks attenuates the latter.
The air flow is not decelerated to subsonic velocities in a wide duct (H = 15 cm), although the
calculations are extended to the far downstream region. When the homogeneous mixture accumulated in
the mixing layer during induction burns up, the heat release decreases rapidly. The diffusion combustion
maintains only a very small rise in pressure. As is seen in Fig. 2b, the relative pressure P/Pa at a length of
~,,0.2 m increases from 5.7 to 6.7. At the inlet (x = 577 cm), the flow at the center of the duct is decelerated to
M = 1.1. Its temperature increases to 2000 K, and the flame front is located between the wall and the plane
of symmetry. Figure 3 shows the changes in the velocity and Mach-number profiles along the duct length.
As was mentioned above, the ignition of the homogeneous mixture in the mixing layer gives rise to
a shock. It is assumed that if the volume of this mixture increases for some reason, the intensity of the
corresponding shock also increases. This, in turn, can have a substantial effect on the flow characteristics
as a whole. Calculations were performed for a lowered static temperature of the air flow (Ta = 890 K).
Ignition was assumed to occur downstream, so that a large volume of combustible mixture had time to
accumulate in the mixing layer before the onset of ignition. Indeed, as follows from the pressure and Machnumber distributions along the wall and symmetry plane (Fig. 4), ignition occurs well past the place of
hydrogen delivery (x m t50 cm) and causes the appearance of a much stronger shock. As a result, the pressure
fluctuations in the flow increase sharply. The flow character, however, does not change fundamentally. The
supersonic core is preserved, and after the region of uniformity the pressure almost reaches the previous level.
The effect of the duct height is more important. As is seen in Fig. 5, which shows the pressure
distributions in ducts with H = 10, 8, and 6 cm, a decrease in the duct height leads to an increase in
the pressure fluctuations in the initial region and to a stronger flow deceleration in the diffusion region. In
ducts with H = 8 and 6 cm, the flow has time to decelerate to subsonic velocities at the distance considered
(400 cm). From this point, as follows from Fig. 6, which shows the pressure and Mach-number distributions
along the symmetry plane, the rise in pressure slows down. The pressure stabilizes at a level that is 8 times
higher than the initial level and is close to the static pressure behind the normal shock calculated from the
403
lJ~;
M
2.5
2
6
1,5
1
2
0
I
I
I
200
400
600
.....
x/h
0.5
80O
Fig. 6. Pressure and Mach-number distributions along the wall and
symmetry plane: H = 8 cm; h = 0.5 cm; Ma = 2.63; n = 6.
0.8
b
M
35OO
T,K
0.8
3O0O
0.6
2500
0.4
2000
0.2
1500
1,0
a
Y
0.6
0.4
H2 0
0.2
0
" ~ .
" 0;2 " o . ' 4
0--E :: --0;B: -
;'0 y / h
o
0'2
0;.
0:s
o;s y / ~
I000
Fig. 7. Profiles of concentration (a), Mach number and temperature (b) at the duct outlet
(x/h = 765): H = 8 cm; h = 0.5 cm; Ma = 2.63; n = 6.
initial air-flow parameters. Nevertheless, it is incorrect to discuss the appearance of a pseudoshock wave here.
Although the deceleration length reduces with decreasing duct height, it still does not reach the limiting value
that can be expected (about 10 duct heights) from the data on the pseudoshock length obtained from the
known experiments on duct throttling.
Figure 7 shows concentration, temperature, and Mach-number profiles at the outlet section for H =
8 cm. In this case, about half the incoming hydrogen burns up. The friction coefficient is positive, i.e.,
deceleration proceeds without formation of separated zones.
The main conclusion from these calculations is as follows. In a fairly narrow duct the combustion of
wall hydrogen jets via the mechanism of viscous-inviscid interaction causes a supersonic flow to decelerate to
subsonic velocities. Combustion leads to pressure uniformity in the transverse direction, although, initially, it
acts in the opposite direction.
This work was supported by the Russian Foundation for Fundamental Research.
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1.
2.
3.
404
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405