Combustion, Explosion, and Shock Waves, Vol. 38, No. 3, pp. 269 277, 2002
Boundary-Layer Structure with Hydrogen Combustion
with Different Injection Intensities
É. P. Volchkov,1 V. V. Terekhov,1 and V. I. Terekhov1 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 influence 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 flow were simulated using the k µ turbu-
lence model with Chien s modification for low Reynolds numbers. The diffusion model
(infinitely large burning rate) was used to describe the chemical reaction process, but
the difference in diffusion coefficients of different 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 significantly deforms
the profiles of density and viscosity of the gas mixture. As the injection velocity
increases, the flame 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
The interest in studying boundary layers with in-
diffusion flames [3 7]. Heat release decreases turbulent
jection of chemically reacting substances is primarily
fluctuations in the flame front and, hence, reduces the
caused by extensive practical applications. The pro-
heat- and mass-transfer coefficients. All this confirms
cess of combustion of most liquid and solid fuels is ac-
the complex mechanism of interrelated processes caused
companied by injection of an evaporating or decompos-
by injection of mass on the surface, combustion, heat
ing substance from the wall and its afterburning in the
transfer, and diffusion of substances in a multispecies
gaseous phase inside the boundary layer. Depending on
gas mixture.
the normal velocity on the surface, the boundary-layer
By the moment, a certain progress has been
structure changes, and the flame front is shifted from
achieved in mathematical simulation of turbulent com-
the surface toward the outer edge of the boundary layer
bustion in the boundary layer [8, 9], and various mod-
as the injection parameter increases.
els of chemical reactions have been analyzed [10]. As a
The mechanism of action of porous injection on tur-
whole, good agreement with experimental data has been
bulent heat and mass transfer and friction for nonreact-
reached, and the main special features of the behavior
ing flows has been extensively studied [1, 2]. In the
of boundary-layer characteristics with combustion have
case of combustion, the pattern of the process becomes
been confirmed, namely, a large delay in transition from
more complicated. In this case, a rather strong effect is
the laminar to the turbulent flow regime and reduction
exerted by heat release in the reaction zone, which sig-
of turbulent fluctuations in the case of combustion, as
nificantly changes the thermophysical properties and,
compared to a nonreacting flow with the same velocity
as a consequence, characteristics of turbulence. This is
and composition of the injected substance.
At the same time, there are many issues in this field
1
Kutateladze Institute of Thermal Physics,
that require more profound investigation. One of them
Siberian Division, Russian Academy of Sciences,
is the analysis of the influence of intensity of fuel injec-
Novosibirsk 630090; [e-mail].
0010-5082/02/3803-0269 $27.00 © 2002 Plenum Publishing Corporation 269
270 Volchkov, Terekhov, and Terekhov
with increasing boundary-layer thickness.
Different concentrations of hydrogen on the wall
corresponded to different values of the injection param-
eter. The calculations were performed for the maximum
possible range of concentrations: (CH )w = 0.02 0.8.
2
The lower boundary of the concentration interval was
limited by the size of the computational grid; the flame
front could not be closer than the first point from the
wall. For high concentrations and, hence, injection pa-
rameters, a stable solution could not be obtained. These
Fig. 1. Flow patters in the case of hydrogen injection
values of concentrations corresponded to a wide range of
and combustion in the boundary layer.
injection parameters: b1 = 2jw/Á0U0cfr <" 0.1 4, where
=
jw is the mass flow from the wall, Á0 is the density of
the main flow, and cfr is the skin-friction coefficient.
The relative transverse velocity on the wall and the in-
tion through the porous surface on the boundary-layer
jection parameter for hydrogen concentrations on the
structure, flame-front position, heat and mass transfer,
wall used in calculations in the case of injection with-
and friction. Very little information on this problem
out combustion and with combustion for a Reynolds
can be found in the literature, and most experimental
number Rex = 107 are listed in Tables 1 and 2. For
and theoretical works have been performed with a fixed
comparison, Tables 1 and 2 give also data for an inte-
injection velocity.
gral Reynolds number Re"" = Á0U0´""/µ0 and injection
Results of a numerical analysis of a turbulent flow
parameters, which were constructed on the basis of fric-
with hydrogen injection into the boundary layer are de-
tion in a standard boundary layer (b = 2jw/Á0U0cfr,0)
scribed in the present paper. Taking into account that
and under conditions considered (b1). Here ´"" is the
the process examined depends on many parameters, we
momentum thickness, µ0 is the viscosity of the main
concentrated the main attention in this work on the
flow, and cfr,0 is the friction coefficient in a standard
analysis of dynamic parameters: density and velocity
boundary layer.
profiles, flame-front coordinate, and skin friction.
As it follows from Tables 1 and 2, the relative
Two series of calculations were performed: injec-
flow of hydrogen from the wall jw/Á0U0 is compara-
tion without combustion and with combustion under
tively small both with and without combustion. Based
identical conditions on the wall and in the core flow.
on the estimates made using formulas of [1], the injec-
This allowed us to perform direct comparisons and re-
tion parameter for a flow without combustion and with
veal the effect of combustion on the flow structure.
the maximum concentration of hydrogen on the wall
(CH )w = 0.8 reached approximately half of its critical
2
value at which boundary-layer displacement occurs.
1. FORMULATION OF THE PROBLEM
In the present work, we used the Navier Stokes
AND GOVERNING EQUATIONS
equations in the boundary-layer approximation. This
model includes the following equations:
The flow pattern is shown in Fig. 1. Pure hydro-
gen is injected through a flat porous plate exposed to
 the continuity equation
a gradientless air flow with a velocity U0 and tempera-
"ÁU "ÁV
ture T0. All calculations were performed for the same
+ = 0; (1)
value of the air-flow velocity (U0 = 20 m/sec). Being "x "y
injected through the wall, hydrogen reacts with oxygen
 the equation of motion
contained in air inside the boundary layer in the com-
bustion front, and reaction products diffuse toward the
"U "U " "U dp
boundary-layer edge and toward the wall. To eliminate
ÁU + ÁV = (µ + µt) - ; (2)
"x "y "y "y dx
the influence of the temperature factor, all calculations
were performed for identical temperatures of the core
 the equation of energy written in terms of total en-
flow and the wall (T0 = Tw = 300 K) unchanged in
thalpies
the lengthwise direction. The lengthwise concentration
of injected hydrogen on the surface was also assumed
"H "H " µ µt "H
ÁU + ÁV = + + Q , (3)
to be constant [(CH )w = const]. Accordingly, the flux
2
"x "y "y Pr Prt "y
of the substance on the wall along the plate decreased
Boundary-Layer Structure with Hydrogen Combustion 271
TABLE 1
(CH2 )w jw, kg/(m2· sec) jw/Á0U0 cfr/2 b1 Re"" b
0.02 8.44 · 10-4 3.52 · 10-5 1.14 · 10-3 0.0308 1.14 · 104 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 Rex = 107.
TABLE 2
(CH2 )w jw, kg/(m2· sec) jw/Á0U0 cfr/2 b1 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 Rex = 107.
where whose rate is infinitely high. This corresponds to the
nsp
"Cj so-called diffusion model of combustion, but the diffu-
Q = (Lej - 1)Hj ,
sion coefficients of different substances composing the
"y
j=1
mixture were different, and the Lewis numbers differed
T
from unity (Le = 1).

k
0
The equation of mass transfer (4) was solved for
H = cpdT + "H298 Ci;
i
hydrogen, oxygen, and water, and the concentration of
i=1
298
inert nitrogen was found under the assumption that the
 the equation for concentration of the ith substance
sum of all concentrations was equal to unity.
To simulate turbulence, we used the k µ model
"Ci "Ci " µ µt "Ci
with Chien s modification for low Reynolds num-
ÁU + ÁV = + + ‡i. (4)
"x "y "y Sci Sct "y
bers [12]. The equations for the transfer of turbulent
Here U and V are the longitudinal and transverse com- kinetic energy k and its dissipation rate µ have the form
0
ponents of velocity, H is the total enthalpy, "H298 is the
"k "k " "k
ÁU + ÁV = (µ + µt)
enthalpy of formation of the substance, Ci is the mass
"x "y "y "y
concentration of components of the mixture, cp is the
2
"U
heat capacity at constant pressure, and ‡i is the rate of
+ µt - Áµ, (5)
formation of the ith substance; the parameters marked "y
by the subscript  t refer to turbulent values. Molec-
"µ "µ " µt "µ
Ü Ü Ü
ular viscosity, Prandtl number Pr, and Schmidt num-
ÁU + ÁV = µ +
"x "y "y 1.3 "y
ber Sci for the ith substance, which enter these equa-
tions, were calculated using semi-empirical formulas for
2
"U µ µ +
Ü Ü
+ c1f1µt - Ác2f2µ - 2µ e-0.5y , (6)
Ü
a multispecies mixture of gases [11], and the density was
"y k y2
calculated using the equation of state for an ideal gas.
where
The preliminary calculations showed that combus-
k
tion under the above conditions is mainly determined t
f1 = 1, f2 = 1 - 0.22e-(Re /6)2, µ = µ + 2½ ,
Ü
y2
by the mixing rate of the reagents, since the rate of
chemical reactions is rather high. Therefore, in all sub-
c1 = 1.35, c2 = 1.8,
sequent calculations, we assumed that there is only one
reaction
k2 yuÄ Á Ák2
µt = cµfµ , y+ = , Ret = ,
fuel + oxidizer products
µ µw µµ
Ü Ü
272 Volchkov, Terekhov, and Terekhov
+
fµ = 1 - e-0.0115y ,
"u
uÄ = Äw/Áw, Äw = µw .
"y w
The turbulent Prandtl Prt and Schmidt Sct 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:
 in the initial cross section (x = 0),
U = U0, V = V0, T = T0, k = k0, µ = µ0,
and the values of turbulent characteristics were set such
that the law of turbulence decay in the external flow
corresponded to most typical experiments;
Fig. 2. Skin friction versus the Reynolds number for dif-
ferent concentrations of hydrogen on the wall: the open
 at the boundary-layer edge (y = ´),
and filled points refer to injection with and without com-
bustion, respectively.
U = U0, T = T0, CO = 0.23, CN = 0.77,
2 2
and "Åš/"y = 0 for remaining variables;
 at the wall (y = 0),
case of hydrogen injection without combustion are also
U = 0, k = 0, µ = 0, CH = (CH )w, T = Tw, given.
2 2
We note the main features of the behavior of skin
ÁD "CH
2
(jw)H = . friction. As in boundary layers without combustion, an
2
1 - (CH )w "y w
2
increase in concentration on the wall or injection inten-
The advantage of the turbulence model used is that sity leads to a decrease in friction. However, in con-
it allows one to calculate the flow at low Reynolds num- trast to nonreacting boundary layers, combustion sig-
bers. Therefore, it was possible to perform calculations nificantly delays the laminar turbulent transition, and
including the laminar portion of the flow. In the present in the case of low injection intensities, it occurs at
work, we used the transformation of coordinates and the Rex = (3 5)·106, which is an order of magnitude greater
numerical method proposed by Denny and Landis [13]. than in the absence of combustion. With increasing
Before the main calculations, we made a series of injection intensity, as in the case of injection without
test calculations, where the workability of the numeri- combustion, the transition is shifted to the region of
cal algorithm and turbulence model was tested. Friction low Reynolds numbers.
and heat transfer were consecutively calculated for an Intense heat release in the boundary layer also af-
impermeable plate and for a porous surface with injec- fects the character of friction distribution along the
tion of identical and foreign gases. The results of all plate. The analysis shows that the exponent in the fric-
calculations were in good agreement with the data on tion law St <" Re-n in the laminar flow region varies
velocity profiles and friction laws for various injection within n = 0.35 0.5; in the turbulent regime, this ex-
parameters known from the literature [1, 2]. 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-
2. CALCULATION RESULTS
tion in the boundary layer with combustion are plotted
AND DISCUSSION
in Fig. 3. An increase in injection intensity shifts the
The friction coefficient versus the Reynolds number reaction front from the wall toward the boundary-layer
Rex in the case of hydrogen injection and combustion edge, and the value of temperature itself also increases
in the boundary layer is plotted in Fig. 2, which shows with increasing injection intensity. If the hydrogen con-
the data for different concentrations of hydrogen on the centration on the wall is rather high [(CH )w > 0.5], the
2
wall and, hence, for different intensities of injection. For temperature in the front stabilizes and approaches the
comparison, the calculation results for friction in the adiabatic temperature of hydrogen combustion in air.
Boundary-Layer Structure with Hydrogen Combustion 273
Fig. 3. Temperature profiles in the boundary layer
with hydrogen combustion.
The calculations show that, in the case of a fixed
injection parameter, the Reynolds number has a weak
effect on the value of the maximum temperature in the
combustion front; for (CH )w > 0.5, the latter remains
2
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 effect of combustion on den-
Fig. 4. Density distribution of the gas mixture in the
sity profiles can be noted both for low and high concen-
boundary layer without combustion (a) and with com-
trations of hydrogen on the wall. At low concentrations,
bustion of hydrogen (b) for Rex = 107 (the arrows indi-
intense heat release in the front significantly decreases
cate the position of the flame front).
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 acting boundary layer. The position of the combustion
the low-density region significantly increases, and this front, as is demonstrated in Fig. 3, changes depending
region occupies the major part of the boundary layer. on hydrogen concentration on the wall. More detailed
The presence of the heat-release front leads also to an data on the behavior of the flame-front coordinate along
increase in viscosity of the gas mixture, which, together the wetted surface for different injection parameters are
with the decrease in gas density, delays (in terms of the presented in Fig. 5. With increasing injection parame-
Reynolds number) the laminar turbulent transition in ter, the front coordinate is shifted away from the wall,
a reacting boundary layer. If the Reynolds number is and the displacement may be rather considerable. Ob-
determined on the basis of density and viscosity in the viously, depending on the position of the flame front
combustion front rather than on the basis of external (in the near-wall or external region), it exerts different
flow parameters, it approximately corresponds to the influence on the transfer characteristics, Therefore, cal-
Reynolds number of transition from a laminar to a tur- culation of the flame-front position is one of the impor-
bulent flow for the case of hydrogen injection without tant problems of predicting characteristics of boundary
combustion. layers with combustion.
One important parameter that affects the flow The Schwab Zel dovich diffusion model allows one
structure in a reacting boundary layer is the position of to determine analytically the relative velocity in the
the heat-release front over the height of the boundary front. In the case of fuel injection into an air stream, the
layer. The coordinate of the flame front was determined expression for the dimensionless velocity in the flame
by the position of the temperature maximum in a re- front is [14]
274 Volchkov, Terekhov, and Terekhov
Fig. 5. Position of the flame front along the plate for
Fig. 7. Evolution of velocity profiles in the bound-
different injection intensities.
ary layer with hydrogen combustion: the solid curves
refer to velocities in the laminar sublayer u+ = y+
and turbulent core u+ = 5.75 log y+ + 5.5 of the
classical turbulent boundary layer without injection
or combustion (the arrows indicate the flame-front
position).
determined for different substances. The diffusion Stan-
ton number was found by solving the diffusion equa-
tion (4) on the wall
"Ci
Std = - ÁDi Á0U0(Ci,0 - Ci,w),
i
"y w
where Di is the diffusion coefficient of the ith substance.
As it follows from Fig. 6, the use of dynamic perme-
ability yields good agreement with the diffusion model.
The diffusion model also yields good agreement in terms
of the limiting value Éf = 0.9721 for intense injection
Fig. 6. Effect of the injection parameter on the
(b1 "). If diffusion permeability is used, good agree-
flame-front coordinate.
ment with the Schwab Zel dovich model is observed in
the case of intense or moderate injection (b1 > 0.1),
whereas a large difference is observed for low-intensity
Uf ½(CO )0(1 + b1)
2
injection.
Éf = = 1 - . (7)
U0 b1[1 + ½(CO )0]
2 Note that relation (7) is valid both for the lami-
nar and turbulent flow regimes. Numerical calculations
Here ½ is the stoichiometric ratio and (CO )0 is the mass
2
also confirm that this equation is universal and demon-
concentration of oxygen in the core flow.
strate rather weak influence of the Reynolds number
In the case of diffusion combustion of hydrogen in
and, hence, the laminar or turbulent flow regime on
air, we have (CO )0 = 0.23; if the reaction has the sim-
2
the value of Éf, though according to the data of Fig. 5,
plest form 2H2 + O2 2H2O and ½ = 1/8, Eq. (7)
the distance from the wall at which the flame front is
becomes
formed depends substantially on the longitudinal co-
1 + b1
Éf = 1 - 0.0279 . (8) ordinate and, correspondingly, on the Reynolds num-
b1
ber Rex. This gives grounds to use Eqs. (7) and (8) to
The values of Éf calculated by Eq. (8) are plot- evaluate the velocity in the flame front and also [by
ted in Fig. 6; the same figure shows the results of nu- virtue of similarity of the process of heat and mass
merical calculations. The determining parameter in transfer and friction, which was accepted in deriving
numerical calculations was the dynamic permeability Eq. (7)] the total enthalpies and concentrations of chem-
b1 = 2jw/Á0U0cfr or its diffuse analog b1d = jw/Á0U0Std ical elements.
Boundary-Layer Structure with Hydrogen Combustion 275
Fig. 8. Velocity profiles in the boundary layer with injection and combustion of hydrogen for Rex = 107 and
(CH2)w = 0.02 (a), 0.1 (b), 0.5 (c), and 0.8 (d); the filled and open points refer to injection without and with
combustion, respectively (the arrows indicate the flame-front position).
The presence of the flame front in the boundary logarithmic coordinates is demonstrated in Fig. 8, where
layer leads to principal changes in velocity profiles. This the numerical results in the presence and absence of
can be deduced from the data of Fig. 7, which shows the combustion are compared directly, the remaining pa-
evolution of velocity profiles along the plate in the case rameters being identical. For low concentrations of hy-
of combustion of the injected hydrogen. The velocity drogen on the wall (see Fig. 8a, weak injection), the
profiles are plotted here in universal semi-logarithmic velocity profile without combustion is close to the stan-
coordinates where u+ = u/uÄ , uÄ = Äw/Áw, and dard distribution in the laminar sublayer and in the
y+ = ÁwyuÄ /µw. For comparison, Fig. 7 also shows internal logarithmic region. In the case of combustion,
the velocity profiles in the laminar sublayer u+ = y+ the profile is cardinally different. As in the case of low
and in the turbulent core u+ = 5.75 log y+ + 5.5 of the Reynolds numbers (see Fig. 7), there is no logarithmic
classical turbulent boundary layer without injection and region, which indicates the dominant influence of heat
combustion. release on the flow structure.
It is seen from Fig. 7 that the velocity profile for In the case of moderate injection (see Fig. 8b
Rex d" 106 has no traditional internal zone with a loga- and c), the velocity profiles in universal coordinates re-
rithmic section. This indicates that the flow regime is main almost unchanged in the absence of combustion.
close to laminar, which is caused by intense heat release This result is of interest in itself, since it is known [1, 2]
in the boundary layer. With increasing Reynolds num- that injection involves strong deformations of the veloc-
ber, the profiles deviate from the laminar distribution, ity profile. A possible explanation can be the mutual
which testifies to flow turbulization. Similar trends have influence of injection and inhomogeneity of the composi-
been already noted in Fig. 2. tion. The presence of the flame front gradually shifts the
The influence of injection intensity on the dis- profiles closer to each other, and the difference becomes
tribution of velocities in the boundary layer in semi- minimum in the case of intense injection (see Fig. 8d),
276 Volchkov, Terekhov, and Terekhov
nificant. The effect is amplified by the joint action of
these factors. At the same time, in the case of intense
injection, the calculation results for reacting and non-
reacting flows become closer, and the contribution of
combustion decreases, since the flame front is displaced
from the wall toward the boundary-layer edge. In addi-
tion, the effects of significant reduction of friction in the
case of hydrogen combustion are manifested already for
comparatively low-intensity injection of fuel, which is a
rather favorable factor from the practical point of view.
It should be noted that similar results on the in-
fluence of heat release on reduction of friction in the
boundary layer were obtained by Larin and Levin [15].
In comparing the calculation results with the data
on hydrogen combustion [5, 16], which have a quali-
tatively similar form, we faced problems described in
detail in [17]. The reason is that these experiments
Fig. 9. Relative function of friction versus injection in-
were performed for very high injection parameters, and
tensity: the filled and open points refer to injection
without and with combustion, respectively.
it was not possible to obtain stable solutions for these
regime parameters (jw/Á0U0 H" 0.01). For conditions
of the present calculations (see Table 2), the relative
but the velocity profile without combustion starts to
deviate noticeably from the classical logarithmic distri- transverse velocity of hydrogen on the wall was almost
two orders lower. In [17], similar calculations could be
bution.
performed only using the mixing-length model. Ap-
Deformations of density and velocity profiles
parently, the problem of hydrogen combustion with in-
caused by heat release inside the boundary layer lead to
significant changes in integral characteristics: displace- tense injection can be solved in an elliptical formula-
´
tion. Therefore, a comparison with available experi-
ment thickness ´" = (1 - ÁÉ)dy, momentum thickness
Ü
mental data has not yet been performed.
0
´
´"" = ÁÉ(1 - É)dy, and shape factor H = ´"/´"". An
Ü
0
analysis shows that, in the case of low-intensity injec-
CONCLUSIONS
tion without combustion, the value of H is close to the
standard value for the boundary layer both in the lam-
Based on numerical studies of the boundary layer
inar (H H" 2.7) and turbulent (H H" 1.28) flow regimes.
with injection and combustion of hydrogen with differ-
In the region of moderate and high-intensity injection
ent injection intensities, it was found that the pres-
of hydrogen without combustion, the value of the shape
ence of the heat-release zone significantly delays the
factor increases by an order of magnitude as compared
laminar turbulent transition. A change in injection in-
to an impermeable plate. An even more considerable
tensity leads to a displacement of the flame front over
effect is observed during combustion as the value of H
the boundary-layer height, which considerably deforms
increases (H > 100). In this case, a complex behav-
the density and velocity profiles of the gas mixture. The
ior of the shape factor depending on both the Reynolds
latter, in turn, is the reason for complicated laws of the
number and injection intensity is observed.
behavior of integral parameters of the boundary layer.
All the above-noted properties of the flow struc-
Contributions to the overall reduction of friction are
ture in the boundary layer with combustion are man-
demonstrated, which are caused by injection and com-
ifested in the behavior of the relative friction coeffi-
bustion in the boundary layer in the presence of a flame
cient. Its dependence on injection intensity is shown
front. Hydrogen combustion with even moderate injec-
in Fig. 9. Here cfr/cfr,0 is the relative function of fric-
tion velocities (hundredths of a percent of the main flow
tion (cfr,0 = 0.0576Re-0.2 is the friction on an imper-
x
velocity) significantly decreases the friction drag. The
meable isothermal plate in a developed turbulent flow).
effects of friction reduction are particularly strong in
Using the data of Fig. 9, one can analyze the contri-
the region of low-intensity injection parameters.
bution of injection and combustion to reduction of the
This work was supported by the Russian Founda-
friction coefficient. The individual action of each factor
tion for Fundamental Research (Grant Nos. 01-02-06509
leads to a decrease in friction, and this decrease is sig-
and 99-02-17171).
Boundary-Layer Structure with Hydrogen Combustion 277
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