Boundary-Layer Structure with Hydrogen Combustion with Different Injection Intensities, CHEMIA I PIROTECHNIKA, ...
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Combustion, Explosion, and Shock Waves, Vol. 38, No. 3, pp. 269{277, 2002
Boundary-Layer Structure with Hydrogen Combustion
with Dierent Injection Intensities
E. P. Volchkov,
1
V. V. Terekhov,
1
and V. I. Terekhov
1
UDC 536.461:661.98
Translated from Fizika Goreniya i Vzryva, Vol. 38, No. 3, pp. 20{29, May{June, 2002.
Original article submitted November 15, 2001.
Results of numerical simulation of the inuence of intensity of hydrogen injection
through a porous surface in the case of hydrogen burning in the boundary layer are
presented. Turbulent characteristics of the ow were simulated using the k{" turbu-
lence model with Chien's modication for low Reynolds numbers. The diusion model
(innitely large burning rate) was used to describe the chemical reaction process, but
the dierence in diusion coecients of dierent substances was taken into account.
A comparison of injection with and without combustion shows that the presence of
a heat-release front delays the laminar{turbulent transition and signicantly deforms
the proles of density and viscosity of the gas mixture. As the injection velocity
increases, the ame front is shifted from the porous surface toward the outer edge of
the boundary layer. The contributions of injection itself and combustion to reduction
of skin friction are analyzed.
Key words: boundary layer, combustion, porous injection, heat and mass transfer,
friction.
INTRODUCTION
evidenced by available experimental data on the turbu-
lent structure and heat and mass transfer in near-wall
diusion ames [3{7]. Heat release decreases turbulent
uctuations in the ame front and, hence, reduces the
heat- and mass-transfer coecients. All this conrms
the complex mechanism of interrelated processes caused
by injection of mass on the surface, combustion, heat
transfer, and diusion of substances in a multispecies
gas mixture.
By the moment, a certain progress has been
achieved in mathematical simulation of turbulent com-
bustion in the boundary layer [8, 9], and various mod-
els of chemical reactions have been analyzed [10]. As a
whole, good agreement with experimental data has been
reached, and the main special features of the behavior
of boundary-layer characteristics with combustion have
been conrmed, namely, a large delay in transition from
the laminar to the turbulent ow regime and reduction
of turbulent uctuations in the case of combustion, as
compared to a nonreacting ow with the same velocity
and composition of the injected substance.
At the same time, there are many issues in this eld
that require more profound investigation. One of them
is the analysis of the inuence of intensity of fuel injec-
The interest in studying boundary layers with in-
jection of chemically reacting substances is primarily
caused by extensive practical applications. The pro-
cess of combustion of most liquid and solid fuels is ac-
companied by injection of an evaporating or decompos-
ing substance from the wall and its afterburning in the
gaseous phase inside the boundary layer. Depending on
the normal velocity on the surface, the boundary-layer
structure changes, and the ame front is shifted from
the surface toward the outer edge of the boundary layer
as the injection parameter increases.
The mechanism of action of porous injection on tur-
bulent heat and mass transfer and friction for nonreact-
ing ows has been extensively studied [1, 2]. In the
case of combustion, the pattern of the process becomes
more complicated. In this case, a rather strong eect is
exerted by heat release in the reaction zone, which sig-
nicantly changes the thermophysical properties and,
as a consequence, characteristics of turbulence. This is
1
Kutateladze Institute of Thermal Physics,
Siberian Division, Russian Academy of Sciences,
Novosibirsk 630090; vt@itp.nsc.ru.
0010-5082/02/3803-0269 $27.00
c
2002
Plenum Publishing Corporation
269
270
Volchkov, Terekhov, and Terekhov
Fig. 1. Flow patters in the case of hydrogen injection
and combustion in the boundary layer.
with increasing boundary-layer thickness.
Dierent concentrations of hydrogen on the wall
corresponded to dierent values of the injection param-
eter. The calculations were performed for the maximum
possible range of concentrations: (C
H
2
)
w
= 0:02{0:8.
The lower boundary of the concentration interval was
limited by the size of the computational grid; the ame
front could not be closer than the rst point from the
wall. For high concentrations and, hence, injection pa-
rameters, a stable solution could not be obtained. These
values of concentrations corresponded to a wide range of
injection parameters: b
1
= 2j
w
=
0
U
0
c
fr
= 0:1{4, where
j
w
is the mass ow from the wall,
0
is the density of
the main ow, and c
fr
is the skin-friction coecient.
The relative transverse velocity on the wall and the in-
jection parameter for hydrogen concentrations on the
wall used in calculations in the case of injection with-
out combustion and with combustion for a Reynolds
number Re
x
= 10
7
are listed in Tables 1 and 2. For
comparison, Tables 1 and 2 give also data for an inte-
gral Reynolds number Re
=
0
U
0
=
0
and injection
parameters, which were constructed on the basis of fric-
tion in a standard boundary layer (b = 2j
w
=
0
U
0
c
fr;0
)
and under conditions considered (b
1
). Here
is the
momentum thickness,
0
is the viscosity of the main
ow, and c
fr;0
is the friction coecient in a standard
boundary layer.
As it follows from Tables 1 and 2, the relative
ow of hydrogen from the wall j
w
=
0
U
0
is compara-
tively small both with and without combustion. Based
on the estimates made using formulas of [1], the injec-
tion parameter for a ow without combustion and with
the maximum concentration of hydrogen on the wall
(C
H
2
)
w
= 0:8 reached approximately half of its critical
value at which boundary-layer displacement occurs.
In the present work, we used the Navier{Stokes
equations in the boundary-layer approximation. This
model includes the following equations:
| the continuity equation
tion through the porous surface on the boundary-layer
structure, ame-front position, heat and mass transfer,
and friction. Very little information on this problem
can be found in the literature, and most experimental
and theoretical works have been performed with a xed
injection velocity.
Results of a numerical analysis of a turbulent ow
with hydrogen injection into the boundary layer are de-
scribed in the present paper. Taking into account that
the process examined depends on many parameters, we
concentrated the main attention in this work on the
analysis of dynamic parameters: density and velocity
proles, ame-front coordinate, and skin friction.
Two series of calculations were performed: injec-
tion without combustion and with combustion under
identical conditions on the wall and in the core ow.
This allowed us to perform direct comparisons and re-
veal the eect of combustion on the ow structure.
1. FORMULATION OF THE PROBLEM
AND GOVERNING EQUATIONS
The ow pattern is shown in Fig. 1. Pure hydro-
gen is injected through a at porous plate exposed to
a gradientless air ow with a velocity U
0
and tempera-
ture T
0
. All calculations were performed for the same
value of the air-ow velocity (U
0
= 20 m/sec). Being
injected through the wall, hydrogen reacts with oxygen
contained in air inside the boundary layer in the com-
bustion front, and reaction products diuse toward the
boundary-layer edge and toward the wall. To eliminate
the inuence of the temperature factor, all calculations
were performed for identical temperatures of the core
ow and the wall (T
0
= T
w
= 300 K) unchanged in
the lengthwise direction. The lengthwise concentration
of injected hydrogen on the surface was also assumed
to be constant [(C
H
2
)
w
= const]. Accordingly, the ux
of the substance on the wall along the plate decreased
@U
@x
+
@V
= 0;
(1)
@y
| the equation of motion
U
@U
@x
+ V
@U
@y
=
@
h
( +
t
)
@U
@y
i
dp
@y
dx
;
(2)
| the equation of energy written in terms of total en-
thalpies
@x
+ V
@H
@y
=
@
h
Pr
+
t
Pr
t
@H
@y
+ Q
i
; (3)
@y
U
@H
Boundary-Layer Structure with Hydrogen Combustion
271
TABLE 1
(C
H
2
)
w
j
w
, kg/(m
2
sec) j
w
=
0
U
0
c
fr
=2 b
1
Re
b
0.02
8:44 10
4
3:52 10
5
1:14 10
3
0:0308 1:14 10
4
2:85 10
2
0.1
2:92 10
3
1:21 10
4
7:64 10
4
0.159
8573
0.091
0.5
6:30 10
3
2:62 10
4
2:30 10
4
1.14
6865
0.186
0.8
8:11 10
3
3:38 10
4
9:02 10
5
3.75
6997
0.241
Note. Injection without combustion for Re
x
= 10
7
.
TABLE 2
(C
H
2
)
w
j
w
, kg/(m
2
sec) j
w
=
0
U
0
c
fr
=2 b
1
Re
b
0.02
6:91 10
4
2:88 10
5
3:05 10
4
0.0944 2193 1:87 10
3
0.1
1:38 10
3
5:75 10
4
2:57 10
4
0.223
1950 2:98 10
2
0.5
3:47 10
3
1:44 10
4
1:16 10
4
1.246
1870 7:39 10
2
0.8
5:49 10
3
2:28 10
4
5:67 10
5
4.04
2366
0.124
Note. Injection with combustion for Re
x
= 10
7
.
where
ns
X
whose rate is innitely high. This corresponds to the
so-called diusion model of combustion, but the diu-
sion coecients of dierent substances composing the
mixture were dierent, and the Lewis numbers diered
from unity (Le 6= 1).
The equation of mass transfer (4) was solved for
hydrogen, oxygen, and water, and the concentration of
inert nitrogen was found under the assumption that the
sum of all concentrations was equal to unity.
To simulate turbulence, we used the k{" model
with Chien's modication for low Reynolds num-
bers [12]. The equations for the transfer of turbulent
kinetic energy k and its dissipation rate " have the form
U
@k
Q =
(Le
j
1)H
j
@C
j
@y
;
j=1
k
X
h
Z
i
H =
c
p
dT + H
298
i
C
i
;
i=1
298
| the equation for concentration of the ith substance
@x
+ V
@C
i
@y
=
@
h
Sc
i
+
t
@C
i
@y
i
+ w
i
: (4)
@y
Sc
t
Here U and V are the longitudinal and transverse com-
ponents of velocity, H is the total enthalpy, H
298
is the
enthalpy of formation of the substance, C
i
is the mass
concentration of components of the mixture, c
p
is the
heat capacity at constant pressure, and w
i
is the rate of
formation of the ith substance; the parameters marked
by the subscript \t" refer to turbulent values. Molec-
ular viscosity, Prandtl number Pr, and Schmidt num-
ber Sc
i
for the ith substance, which enter these equa-
tions, were calculated using semi-empirical formulas for
a multispecies mixture of gases [11], and the density was
calculated using the equation of state for an ideal gas.
The preliminary calculations showed that combus-
tion under the above conditions is mainly determined
by the mixing rate of the reagents, since the rate of
chemical reactions is rather high. Therefore, in all sub-
sequent calculations, we assumed that there is only one
reaction
@x
+ V
@k
@y
=
@
h
( +
t
)
@k
@y
i
@y
+
t
@U
@y
2
"; (5)
U
@"
@x
+ V
@"
@y
=
@
h
+
t
1:3
@"
@y
i
@y
h
@U
@y
2
i
k
2
"
y
2
e
0:5y
+
; (6)
+
c
1
f
1
t
c
2
f
2
"
where
f
1
= 1; f
2
= 1 0:22e
(Re
t
=6)
2
; " = " + 2
k
y
2
;
c
1
= 1:35; c
2
= 1:8;
t
= c
f
k
2
"
; y
+
=
yu
w
; Re
t
=
k
2
"
;
fuel + oxidizer ! products
U
@C
i
"
272
Volchkov, Terekhov, and Terekhov
f
= 1 e
0:0115y
+
;
w
:
The turbulent Prandtl Pr
t
and Schmidt Sc
t
numbers
were assumed to be constant across the boundary layer
and equal to 0.9; the subscript \w" refers to quantities
on the wall.
The following boundary conditions were used for
this problem:
u
=
p
w
=
w
;
w
=
w
@u
@y
| in the initial cross section (x = 0),
U = U
0
; V = V
0
; T = T
0
; k = k
0
; " = "
0
;
and the values of turbulent characteristics were set such
that the law of turbulence decay in the external ow
corresponded to most typical experiments;
| at the boundary-layer edge (y = ),
Fig. 2. Skin friction versus the Reynolds number for dif-
ferent concentrations of hydrogen on the wall: the open
and lled points refer to injection with and without com-
bustion, respectively.
U = U
0
; T = T
0
; C
O
2
= 0:23; C
N
2
= 0:77;
and @=@y = 0 for remaining variables;
| at the wall (y = 0),
case of hydrogen injection without combustion are also
given.
We note the main features of the behavior of skin
friction. As in boundary layers without combustion, an
increase in concentration on the wall or injection inten-
sity leads to a decrease in friction. However, in con-
trast to nonreacting boundary layers, combustion sig-
nicantly delays the laminar{turbulent transition, and
in the case of low injection intensities, it occurs at
Re
x
= (3{5)10
6
, which is an order of magnitude greater
than in the absence of combustion. With increasing
injection intensity, as in the case of injection without
combustion, the transition is shifted to the region of
low Reynolds numbers.
Intense heat release in the boundary layer also af-
fects the character of friction distribution along the
plate. The analysis shows that the exponent in the fric-
tion law St Re
n
in the laminar ow region varies
within n = 0:35{0:5; in the turbulent regime, this ex-
ponent varies within n = 0{0:16 depending on injection
intensity, lower parameters of injection corresponding
to lower values of n.
The calculation results for temperature distribu-
tion in the boundary layer with combustion are plotted
in Fig. 3. An increase in injection intensity shifts the
reaction front from the wall toward the boundary-layer
edge, and the value of temperature itself also increases
with increasing injection intensity. If the hydrogen con-
centration on the wall is rather high [(C
H
2
)
w
> 0:5], the
temperature in the front stabilizes and approaches the
adiabatic temperature of hydrogen combustion in air.
U = 0; k = 0; " = 0; C
H
2
= (C
H
2
)
w
; T = T
w
;
w
:
The advantage of the turbulence model used is that
it allows one to calculate the ow at low Reynolds num-
bers. Therefore, it was possible to perform calculations
including the laminar portion of the ow. In the present
work, we used the transformation of coordinates and the
numerical method proposed by Denny and Landis [13].
Before the main calculations, we made a series of
test calculations, where the workability of the numeri-
cal algorithm and turbulence model was tested. Friction
and heat transfer were consecutively calculated for an
impermeable plate and for a porous surface with injec-
tion of identical and foreign gases. The results of all
calculations were in good agreement with the data on
velocity proles and friction laws for various injection
parameters known from the literature [1, 2].
(j
w
)
H
2
=
D
1 (C
H
2
)
w
@C
H
2
@y
2. CALCULATION RESULTS
AND DISCUSSION
The friction coecient versus the Reynolds number
Re
x
in the case of hydrogen injection and combustion
in the boundary layer is plotted in Fig. 2, which shows
the data for dierent concentrations of hydrogen on the
wall and, hence, for dierent intensities of injection. For
comparison, the calculation results for friction in the
Boundary-Layer Structure with Hydrogen Combustion
273
Fig. 3. Temperature proles in the boundary layer
with hydrogen combustion.
The calculations show that, in the case of a xed
injection parameter, the Reynolds number has a weak
eect on the value of the maximum temperature in the
combustion front; for (C
H
2
)
w
> 0:5, the latter remains
almost unchanged over the length.
The presence of a heat-release front in the bound-
ary layer with combustion is primarily manifested in
the density distribution of the gas mixture. These dif-
ferences can be analyzed on the basis of Fig. 4, which
shows the calculation results for injection without and
with combustion. A strong eect of combustion on den-
sity proles can be noted both for low and high concen-
trations of hydrogen on the wall. At low concentrations,
intense heat release in the front signicantly decreases
the density in the near-wall region. As the hydrogen-
injection intensity increases, the density on the wall de-
creases, but in the case of combustion, the thickness of
the low-density region signicantly increases, and this
region occupies the major part of the boundary layer.
The presence of the heat-release front leads also to an
increase in viscosity of the gas mixture, which, together
with the decrease in gas density, delays (in terms of the
Reynolds number) the laminar{turbulent transition in
a reacting boundary layer. If the Reynolds number is
determined on the basis of density and viscosity in the
combustion front rather than on the basis of external
ow parameters, it approximately corresponds to the
Reynolds number of transition from a laminar to a tur-
bulent ow for the case of hydrogen injection without
combustion.
One important parameter that aects the ow
structure in a reacting boundary layer is the position of
the heat-release front over the height of the boundary
layer. The coordinate of the ame front was determined
by the position of the temperature maximum in a re-
Fig. 4. Density distribution of the gas mixture in the
boundary layer without combustion (a) and with com-
bustion of hydrogen (b) for Re
x
= 10
7
(the arrows indi-
cate the position of the ame front).
acting boundary layer. The position of the combustion
front, as is demonstrated in Fig. 3, changes depending
on hydrogen concentration on the wall. More detailed
data on the behavior of the ame-front coordinate along
the wetted surface for dierent injection parameters are
presented in Fig. 5. With increasing injection parame-
ter, the front coordinate is shifted away from the wall,
and the displacement may be rather considerable. Ob-
viously, depending on the position of the ame front
(in the near-wall or external region), it exerts dierent
inuence on the transfer characteristics, Therefore, cal-
culation of the ame-front position is one of the impor-
tant problems of predicting characteristics of boundary
layers with combustion.
The Schwab{Zel'dovich diusion model allows one
to determine analytically the relative velocity in the
front. In the case of fuel injection into an air stream, the
expression for the dimensionless velocity in the ame
front is [14]
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Combustion, Explosion, and Shock Waves, Vol. 38, No. 3, pp. 269{277, 2002
Boundary-Layer Structure with Hydrogen Combustion
with Dierent Injection Intensities
E. P. Volchkov,
1
V. V. Terekhov,
1
and V. I. Terekhov
1
UDC 536.461:661.98
Translated from Fizika Goreniya i Vzryva, Vol. 38, No. 3, pp. 20{29, May{June, 2002.
Original article submitted November 15, 2001.
Results of numerical simulation of the inuence of intensity of hydrogen injection
through a porous surface in the case of hydrogen burning in the boundary layer are
presented. Turbulent characteristics of the ow were simulated using the k{" turbu-
lence model with Chien's modication for low Reynolds numbers. The diusion model
(innitely large burning rate) was used to describe the chemical reaction process, but
the dierence in diusion coecients of dierent substances was taken into account.
A comparison of injection with and without combustion shows that the presence of
a heat-release front delays the laminar{turbulent transition and signicantly deforms
the proles of density and viscosity of the gas mixture. As the injection velocity
increases, the ame front is shifted from the porous surface toward the outer edge of
the boundary layer. The contributions of injection itself and combustion to reduction
of skin friction are analyzed.
Key words: boundary layer, combustion, porous injection, heat and mass transfer,
friction.
INTRODUCTION
evidenced by available experimental data on the turbu-
lent structure and heat and mass transfer in near-wall
diusion ames [3{7]. Heat release decreases turbulent
uctuations in the ame front and, hence, reduces the
heat- and mass-transfer coecients. All this conrms
the complex mechanism of interrelated processes caused
by injection of mass on the surface, combustion, heat
transfer, and diusion of substances in a multispecies
gas mixture.
By the moment, a certain progress has been
achieved in mathematical simulation of turbulent com-
bustion in the boundary layer [8, 9], and various mod-
els of chemical reactions have been analyzed [10]. As a
whole, good agreement with experimental data has been
reached, and the main special features of the behavior
of boundary-layer characteristics with combustion have
been conrmed, namely, a large delay in transition from
the laminar to the turbulent ow regime and reduction
of turbulent uctuations in the case of combustion, as
compared to a nonreacting ow with the same velocity
and composition of the injected substance.
At the same time, there are many issues in this eld
that require more profound investigation. One of them
is the analysis of the inuence of intensity of fuel injec-
The interest in studying boundary layers with in-
jection of chemically reacting substances is primarily
caused by extensive practical applications. The pro-
cess of combustion of most liquid and solid fuels is ac-
companied by injection of an evaporating or decompos-
ing substance from the wall and its afterburning in the
gaseous phase inside the boundary layer. Depending on
the normal velocity on the surface, the boundary-layer
structure changes, and the ame front is shifted from
the surface toward the outer edge of the boundary layer
as the injection parameter increases.
The mechanism of action of porous injection on tur-
bulent heat and mass transfer and friction for nonreact-
ing ows has been extensively studied [1, 2]. In the
case of combustion, the pattern of the process becomes
more complicated. In this case, a rather strong eect is
exerted by heat release in the reaction zone, which sig-
nicantly changes the thermophysical properties and,
as a consequence, characteristics of turbulence. This is
1
Kutateladze Institute of Thermal Physics,
Siberian Division, Russian Academy of Sciences,
Novosibirsk 630090; vt@itp.nsc.ru.
0010-5082/02/3803-0269 $27.00
c
2002
Plenum Publishing Corporation
269
270
Volchkov, Terekhov, and Terekhov
Fig. 1. Flow patters in the case of hydrogen injection
and combustion in the boundary layer.
with increasing boundary-layer thickness.
Dierent concentrations of hydrogen on the wall
corresponded to dierent values of the injection param-
eter. The calculations were performed for the maximum
possible range of concentrations: (C
H
2
)
w
= 0:02{0:8.
The lower boundary of the concentration interval was
limited by the size of the computational grid; the ame
front could not be closer than the rst point from the
wall. For high concentrations and, hence, injection pa-
rameters, a stable solution could not be obtained. These
values of concentrations corresponded to a wide range of
injection parameters: b
1
= 2j
w
=
0
U
0
c
fr
= 0:1{4, where
j
w
is the mass ow from the wall,
0
is the density of
the main ow, and c
fr
is the skin-friction coecient.
The relative transverse velocity on the wall and the in-
jection parameter for hydrogen concentrations on the
wall used in calculations in the case of injection with-
out combustion and with combustion for a Reynolds
number Re
x
= 10
7
are listed in Tables 1 and 2. For
comparison, Tables 1 and 2 give also data for an inte-
gral Reynolds number Re
=
0
U
0
=
0
and injection
parameters, which were constructed on the basis of fric-
tion in a standard boundary layer (b = 2j
w
=
0
U
0
c
fr;0
)
and under conditions considered (b
1
). Here
is the
momentum thickness,
0
is the viscosity of the main
ow, and c
fr;0
is the friction coecient in a standard
boundary layer.
As it follows from Tables 1 and 2, the relative
ow of hydrogen from the wall j
w
=
0
U
0
is compara-
tively small both with and without combustion. Based
on the estimates made using formulas of [1], the injec-
tion parameter for a ow without combustion and with
the maximum concentration of hydrogen on the wall
(C
H
2
)
w
= 0:8 reached approximately half of its critical
value at which boundary-layer displacement occurs.
In the present work, we used the Navier{Stokes
equations in the boundary-layer approximation. This
model includes the following equations:
| the continuity equation
tion through the porous surface on the boundary-layer
structure, ame-front position, heat and mass transfer,
and friction. Very little information on this problem
can be found in the literature, and most experimental
and theoretical works have been performed with a xed
injection velocity.
Results of a numerical analysis of a turbulent ow
with hydrogen injection into the boundary layer are de-
scribed in the present paper. Taking into account that
the process examined depends on many parameters, we
concentrated the main attention in this work on the
analysis of dynamic parameters: density and velocity
proles, ame-front coordinate, and skin friction.
Two series of calculations were performed: injec-
tion without combustion and with combustion under
identical conditions on the wall and in the core ow.
This allowed us to perform direct comparisons and re-
veal the eect of combustion on the ow structure.
1. FORMULATION OF THE PROBLEM
AND GOVERNING EQUATIONS
The ow pattern is shown in Fig. 1. Pure hydro-
gen is injected through a at porous plate exposed to
a gradientless air ow with a velocity U
0
and tempera-
ture T
0
. All calculations were performed for the same
value of the air-ow velocity (U
0
= 20 m/sec). Being
injected through the wall, hydrogen reacts with oxygen
contained in air inside the boundary layer in the com-
bustion front, and reaction products diuse toward the
boundary-layer edge and toward the wall. To eliminate
the inuence of the temperature factor, all calculations
were performed for identical temperatures of the core
ow and the wall (T
0
= T
w
= 300 K) unchanged in
the lengthwise direction. The lengthwise concentration
of injected hydrogen on the surface was also assumed
to be constant [(C
H
2
)
w
= const]. Accordingly, the ux
of the substance on the wall along the plate decreased
@U
@x
+
@V
= 0;
(1)
@y
| the equation of motion
U
@U
@x
+ V
@U
@y
=
@
h
( +
t
)
@U
@y
i
dp
@y
dx
;
(2)
| the equation of energy written in terms of total en-
thalpies
@x
+ V
@H
@y
=
@
h
Pr
+
t
Pr
t
@H
@y
+ Q
i
; (3)
@y
U
@H
Boundary-Layer Structure with Hydrogen Combustion
271
TABLE 1
(C
H
2
)
w
j
w
, kg/(m
2
sec) j
w
=
0
U
0
c
fr
=2 b
1
Re
b
0.02
8:44 10
4
3:52 10
5
1:14 10
3
0:0308 1:14 10
4
2:85 10
2
0.1
2:92 10
3
1:21 10
4
7:64 10
4
0.159
8573
0.091
0.5
6:30 10
3
2:62 10
4
2:30 10
4
1.14
6865
0.186
0.8
8:11 10
3
3:38 10
4
9:02 10
5
3.75
6997
0.241
Note. Injection without combustion for Re
x
= 10
7
.
TABLE 2
(C
H
2
)
w
j
w
, kg/(m
2
sec) j
w
=
0
U
0
c
fr
=2 b
1
Re
b
0.02
6:91 10
4
2:88 10
5
3:05 10
4
0.0944 2193 1:87 10
3
0.1
1:38 10
3
5:75 10
4
2:57 10
4
0.223
1950 2:98 10
2
0.5
3:47 10
3
1:44 10
4
1:16 10
4
1.246
1870 7:39 10
2
0.8
5:49 10
3
2:28 10
4
5:67 10
5
4.04
2366
0.124
Note. Injection with combustion for Re
x
= 10
7
.
where
ns
X
whose rate is innitely high. This corresponds to the
so-called diusion model of combustion, but the diu-
sion coecients of dierent substances composing the
mixture were dierent, and the Lewis numbers diered
from unity (Le 6= 1).
The equation of mass transfer (4) was solved for
hydrogen, oxygen, and water, and the concentration of
inert nitrogen was found under the assumption that the
sum of all concentrations was equal to unity.
To simulate turbulence, we used the k{" model
with Chien's modication for low Reynolds num-
bers [12]. The equations for the transfer of turbulent
kinetic energy k and its dissipation rate " have the form
U
@k
Q =
(Le
j
1)H
j
@C
j
@y
;
j=1
k
X
h
Z
i
H =
c
p
dT + H
298
i
C
i
;
i=1
298
| the equation for concentration of the ith substance
@x
+ V
@C
i
@y
=
@
h
Sc
i
+
t
@C
i
@y
i
+ w
i
: (4)
@y
Sc
t
Here U and V are the longitudinal and transverse com-
ponents of velocity, H is the total enthalpy, H
298
is the
enthalpy of formation of the substance, C
i
is the mass
concentration of components of the mixture, c
p
is the
heat capacity at constant pressure, and w
i
is the rate of
formation of the ith substance; the parameters marked
by the subscript \t" refer to turbulent values. Molec-
ular viscosity, Prandtl number Pr, and Schmidt num-
ber Sc
i
for the ith substance, which enter these equa-
tions, were calculated using semi-empirical formulas for
a multispecies mixture of gases [11], and the density was
calculated using the equation of state for an ideal gas.
The preliminary calculations showed that combus-
tion under the above conditions is mainly determined
by the mixing rate of the reagents, since the rate of
chemical reactions is rather high. Therefore, in all sub-
sequent calculations, we assumed that there is only one
reaction
@x
+ V
@k
@y
=
@
h
( +
t
)
@k
@y
i
@y
+
t
@U
@y
2
"; (5)
U
@"
@x
+ V
@"
@y
=
@
h
+
t
1:3
@"
@y
i
@y
h
@U
@y
2
i
k
2
"
y
2
e
0:5y
+
; (6)
+
c
1
f
1
t
c
2
f
2
"
where
f
1
= 1; f
2
= 1 0:22e
(Re
t
=6)
2
; " = " + 2
k
y
2
;
c
1
= 1:35; c
2
= 1:8;
t
= c
f
k
2
"
; y
+
=
yu
w
; Re
t
=
k
2
"
;
fuel + oxidizer ! products
U
@C
i
"
272
Volchkov, Terekhov, and Terekhov
f
= 1 e
0:0115y
+
;
w
:
The turbulent Prandtl Pr
t
and Schmidt Sc
t
numbers
were assumed to be constant across the boundary layer
and equal to 0.9; the subscript \w" refers to quantities
on the wall.
The following boundary conditions were used for
this problem:
u
=
p
w
=
w
;
w
=
w
@u
@y
| in the initial cross section (x = 0),
U = U
0
; V = V
0
; T = T
0
; k = k
0
; " = "
0
;
and the values of turbulent characteristics were set such
that the law of turbulence decay in the external ow
corresponded to most typical experiments;
| at the boundary-layer edge (y = ),
Fig. 2. Skin friction versus the Reynolds number for dif-
ferent concentrations of hydrogen on the wall: the open
and lled points refer to injection with and without com-
bustion, respectively.
U = U
0
; T = T
0
; C
O
2
= 0:23; C
N
2
= 0:77;
and @=@y = 0 for remaining variables;
| at the wall (y = 0),
case of hydrogen injection without combustion are also
given.
We note the main features of the behavior of skin
friction. As in boundary layers without combustion, an
increase in concentration on the wall or injection inten-
sity leads to a decrease in friction. However, in con-
trast to nonreacting boundary layers, combustion sig-
nicantly delays the laminar{turbulent transition, and
in the case of low injection intensities, it occurs at
Re
x
= (3{5)10
6
, which is an order of magnitude greater
than in the absence of combustion. With increasing
injection intensity, as in the case of injection without
combustion, the transition is shifted to the region of
low Reynolds numbers.
Intense heat release in the boundary layer also af-
fects the character of friction distribution along the
plate. The analysis shows that the exponent in the fric-
tion law St Re
n
in the laminar ow region varies
within n = 0:35{0:5; in the turbulent regime, this ex-
ponent varies within n = 0{0:16 depending on injection
intensity, lower parameters of injection corresponding
to lower values of n.
The calculation results for temperature distribu-
tion in the boundary layer with combustion are plotted
in Fig. 3. An increase in injection intensity shifts the
reaction front from the wall toward the boundary-layer
edge, and the value of temperature itself also increases
with increasing injection intensity. If the hydrogen con-
centration on the wall is rather high [(C
H
2
)
w
> 0:5], the
temperature in the front stabilizes and approaches the
adiabatic temperature of hydrogen combustion in air.
U = 0; k = 0; " = 0; C
H
2
= (C
H
2
)
w
; T = T
w
;
w
:
The advantage of the turbulence model used is that
it allows one to calculate the ow at low Reynolds num-
bers. Therefore, it was possible to perform calculations
including the laminar portion of the ow. In the present
work, we used the transformation of coordinates and the
numerical method proposed by Denny and Landis [13].
Before the main calculations, we made a series of
test calculations, where the workability of the numeri-
cal algorithm and turbulence model was tested. Friction
and heat transfer were consecutively calculated for an
impermeable plate and for a porous surface with injec-
tion of identical and foreign gases. The results of all
calculations were in good agreement with the data on
velocity proles and friction laws for various injection
parameters known from the literature [1, 2].
(j
w
)
H
2
=
D
1 (C
H
2
)
w
@C
H
2
@y
2. CALCULATION RESULTS
AND DISCUSSION
The friction coecient versus the Reynolds number
Re
x
in the case of hydrogen injection and combustion
in the boundary layer is plotted in Fig. 2, which shows
the data for dierent concentrations of hydrogen on the
wall and, hence, for dierent intensities of injection. For
comparison, the calculation results for friction in the
Boundary-Layer Structure with Hydrogen Combustion
273
Fig. 3. Temperature proles in the boundary layer
with hydrogen combustion.
The calculations show that, in the case of a xed
injection parameter, the Reynolds number has a weak
eect on the value of the maximum temperature in the
combustion front; for (C
H
2
)
w
> 0:5, the latter remains
almost unchanged over the length.
The presence of a heat-release front in the bound-
ary layer with combustion is primarily manifested in
the density distribution of the gas mixture. These dif-
ferences can be analyzed on the basis of Fig. 4, which
shows the calculation results for injection without and
with combustion. A strong eect of combustion on den-
sity proles can be noted both for low and high concen-
trations of hydrogen on the wall. At low concentrations,
intense heat release in the front signicantly decreases
the density in the near-wall region. As the hydrogen-
injection intensity increases, the density on the wall de-
creases, but in the case of combustion, the thickness of
the low-density region signicantly increases, and this
region occupies the major part of the boundary layer.
The presence of the heat-release front leads also to an
increase in viscosity of the gas mixture, which, together
with the decrease in gas density, delays (in terms of the
Reynolds number) the laminar{turbulent transition in
a reacting boundary layer. If the Reynolds number is
determined on the basis of density and viscosity in the
combustion front rather than on the basis of external
ow parameters, it approximately corresponds to the
Reynolds number of transition from a laminar to a tur-
bulent ow for the case of hydrogen injection without
combustion.
One important parameter that aects the ow
structure in a reacting boundary layer is the position of
the heat-release front over the height of the boundary
layer. The coordinate of the ame front was determined
by the position of the temperature maximum in a re-
Fig. 4. Density distribution of the gas mixture in the
boundary layer without combustion (a) and with com-
bustion of hydrogen (b) for Re
x
= 10
7
(the arrows indi-
cate the position of the ame front).
acting boundary layer. The position of the combustion
front, as is demonstrated in Fig. 3, changes depending
on hydrogen concentration on the wall. More detailed
data on the behavior of the ame-front coordinate along
the wetted surface for dierent injection parameters are
presented in Fig. 5. With increasing injection parame-
ter, the front coordinate is shifted away from the wall,
and the displacement may be rather considerable. Ob-
viously, depending on the position of the ame front
(in the near-wall or external region), it exerts dierent
inuence on the transfer characteristics, Therefore, cal-
culation of the ame-front position is one of the impor-
tant problems of predicting characteristics of boundary
layers with combustion.
The Schwab{Zel'dovich diusion model allows one
to determine analytically the relative velocity in the
front. In the case of fuel injection into an air stream, the
expression for the dimensionless velocity in the ame
front is [14]
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