Journal of Applied Mechanics and Technical Physics, Vol. 54, No. 2, pp. 195–206, 2013.

Original Russian Text c© N.N. Fedorova, I.A. Fedorchenko, A.V. Fedorov.

MATHEMATICAL MODELING OF JET INTERACTION

WITH A HIGH-ENTHALPY FLOW IN AN EXPANDING CHANNEL

UDC 533.6.011.5, 51-74N. N. Fedorovaa, b, I. A. Fedorchenkoa, b, and A. V. Fedorova

Abstract: Results of modeling the interaction of a plane supersonic jet with a supersonic turbulent

high-enthalpy flow in a channel are reported. The problem is solved in a two-dimensional formulation

at external flow Mach numbers M∞ = 2.6 and 2.8 and at high values of the total temperature of

the flow T0 = 1800–2000 K. The mathematical model includes full averaged Navier–Stokes equations

supplemented with a two-equation turbulence model and an equation that describes the transporta-

tion of the injected substance. The computations are performed by using the ANSYS Fluent 12.1

software package. Verification of the computational technique is performed against available experi-

mental results on transverse injection of nitrogen and helium jets. The computed and experimental

results are demonstrated to agree well. For the examined problems, in addition to surface distribu-

tions of characteristics, fields of flow parameters are obtained, which allow one to reproduce specific

features that can be hardly captured in experiments. Parametric studies show that an increase in

the angle of inclination and the mass flow rate of the jet leads to an increase in the depth of jet

penetration into the flow, but more intense separated flows and shock waves are observed in this

case.

Keywords: supersonic flows, mixing, internal flows, separation, turbulence, shock waves.

DOI: 10.1134/S002189441302003X

1. FORMULATION OF THE PROBLEM

The problem of transverse injection of a jet into a supersonic flow is of interest for analyzing the processes

that occur in vectored-thrust rocket engines, thruster-controlled flying vehicles, etc. Problems of this kind have also

to be solved for the development of hypersonic flying vehicles and their propulsion systems. For efficient operation

of combustion chambers of hypersonic flying vehicles, where chemical reactions proceed at supersonic flow velocities,

it is necessary to ensure a high degree of fuel mixing with the air flow within a short time interval. The choice of

the injection system is extremely important. Normal injection ensures deep penetration of the fuel jet into the main

flow and good mixing, but it leads to an appreciable loss of the total pressure and can be the reason for channel

choking. Therefore, in solving the problem, one has to study the process of jet injection into a supersonic flow with

chemical reactions being ignored. It should be noted that mathematical modeling plays a very important role at

the stage of the transition from laboratory to full-scale experiments, where it is necessary to take into account the

effects induced by geometric and gas-dynamic scaling.

aKhristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch, Russian Academy of Sci-ences, Novosibirsk, 630090 Russia; irina@itam.nsc.ru. bNovosibirsk State University of Architecture and CivilEngineering (Sibstrin), Novosibirsk, 630008 Russia. Translated from Prikladnaya Mekhanika i TekhnicheskayaFizika, Vol. 54, No. 2, pp. 32–45, March–April, 2013. Original article submitted April 28, 2012; revision submittedMay 30, 2012.

0021-8944/13/5402-0195 c© 2013 by Pleiades Publishing, Ltd. 195

U1

h

4

5

678910

1

2

3

Fig. 1. Flow pattern constructed in [1] on the basis of the analysis of experimental data: (1) turbulentboundary layer; (2) separation shock; (3) sonic line; (4) Mach disk; (5) reattachment shock; (6) primaryvortex in the separation region behind the jet; (7) secondary vortex in the separation region behind thejet; (8) secondary vortex in the separation region ahead of the jet; (9) primary vortex in the separationregion ahead of the jet; (10) tertiary vortex in the separation region ahead of the jet.

The wave structure of the flow formed near a strongly underexpanded jet injected normal to the supersonic

external flow is shown in Fig. 1 [1]. The jet escaping from the orifice and passing through the boundary layer becomes

expanded and penetrates into the supersonic external flow. A typical wave structure with a strong transverse shock

wave (Mach disk) is formed in the jet [2]. As the jet, like a step, is an obstacle for the main flow, a shock wave

is formed ahead of the jet, resulting in boundary layer separation. Secondary and tertiary vortices can be formed

inside the primary separation vortex formed ahead of the jet. Further downstream from the jet, there is another

separation of the flow, which is closed by the reattachment shock wave. A secondary vortex can be also formed

in this region in addition to the primary vortex. The number and sizes of secondary vortices in separation regions

substantially depend on the external flow velocity, jet intensity, jet pressure ratio, and degree of turbulence of the

boundary layer.

1.1. Flow Structure and Semi-Empirical Models

The problem of transverse injection of a jet into a supersonic flow has been studied by many authors (see [3]

and the references therein). The main results of theoretical and experimental studies necessary for further analysis

are summarized below.

Some authors who studied jet injection into a supersonic flow made attempts to develop simplified analytical

models. For instance, a relation for the depth of jet penetration into the flow depending on the external flow

parameters and on the flow rate of the jet was derived on the basis of the experiments [4] with due allowance for

the analogy between the flow near the jet and the flow around a blunted body. Similarly, based on the model of

an effective body embedded into the flow, Avduevskii et al. [5] derived a semi-empirical relation between the jet

barrel size and the Mach numbers of the flow and the jet. They calculated the critical value of the jet pressure

ratio n below which the separation region length depends both on the jet diameter and on the jet penetration

depth. Based on experimental data, Eremin et al. [6] constructed a simplified model for calculating the jet and

external flow parameters. The review and classification of the basic analytical models developed before the 1970s

can be found in [7].

1.2. Parametric Investigations

of the Influence of Various Factors on the Flow Structure

The study of the influence of the governing parameters on the flow pattern is extremely important for practice.

In accordance with [8–10], the most important factor affecting the flow structure is the ratio of the external flow

momentum to the jet momentum. It was noted [11, 12] that the penetration depth is mainly determined by the

momentum ratio and is less dependent on the boundary layer thickness, Mach number, and Reynolds number. It

was also found [12] that the ratios of temperatures and molecular weights of the injected and main flows affect the

concentration of the injected gas in the separation region. It was shown [13] that the depth of jet penetration into

196

the flow increases with increasing Mach number of the main flow. A weak effect of the jet temperature on the flow

pattern was noted in [14], where the efficiency of surface cooling by the injected gas was studied.

The influence of the state of the supersonic incoming flow ahead of the injection zone for different intensities

of the separation shock was studied in [15, 16]. Different types of the jet flow were found, depending on the value of

this parameter. Borovoy and Ryzhkova [17] concluded that the main parameters affecting the flow structure are the

velocities and densities of the main and injected flows. These parameters are also important for the mixing process.

The effect of the injector geometry on the mixing characteristics were studied in [10]. It was experimentally

demonstrated that a change in the geometry leads to a change in the transfer angle and mainly affects the flow

near the injector. Large-scale structures were found on the interface between the jet and the main flow, which

apparently play a certain role in mixing of the injected gas near the injection point. Gruber et al. [18] measured

the velocity of transportation of large-scale structures in the region of transverse injection. The analysis performed

in [19] shows that the time-averaged characteristics of the process are overestimated as compared with the degree

of instantaneous mixing. The best mixing in the near wake was found to be reached in the central part of the jet.

1.3. Numerical Modeling

Methods of numerical simulation used for solving gas-dynamic problems make it possible to capture fine

details of the flow structure and to determine the factors that affect the flow characteristics. The first numerical

calculations of the jet in a supersonic flow, based on the Navier–Stokes equations, were performed in the 1970s.

Drummond [20] modeled the exhaustion of a hydrogen jet with turbulent phenomena being ignored. Grasso and

Magi [21] calculated the parameters of a turbulent flow with injection of helium and hydrogen jets by using a two-

equation k−ε turbulence model. Baev et al. [22] modeled hydrogen injection into a supersonic flow of an oxidizer

with allowance for chemical reactions.

Models based on both averaged Navier–Stokes equations and large eddy simulation (LES) approaches are

currently used for turbulence description [23, 24].

Despite a large number of numerical investigations, the influence of high stagnation parameters, temper-

ature factor, and flow evolution in the presence of the upper wall and channel expansion have not be studied in

sufficient detail. The present paper describes the calculated results for injection of helium and hydrogen jets into

a supersonic flow in an expanding channel under conditions typical for short-duration high-enthalpy wind tunnels.

The calculations are performed under the “cold wall” conditions realized in such facilities, which can affect the wave

structure. The influence of the ratio of momenta of the jet and the main flow J and the angle of jet injection α on

the mixing characteristics is also studied numerically.

2. MATHEMATICAL MODEL AND NUMERICAL METHOD

Full averaged Navier–Stokes equations in a two-dimensional formulation were used in this work for modeling

the problem of jet injection. The Navier–Stokes equations were closed with the k−ω SST turbulence model. As the

flow is a mixture of two different gases (air and injected gas), the equation of conservation of the mass fraction of

the injected gas with allowance for turbulent diffusion was used.

The calculations were performed with the ANSYS CFD (Fluent) 12.1 software package. An implicit scheme

was used for approximation in time. Spatial discretization of convective terms was performed by the high-order

Roe [25] and AUSM [26] schemes. A regular tetragonal computational grid refined toward the injection orifice and

channel surfaces was used. There were at least eight nodes on the width of the injection slot. As a developed

turbulent flow was studied, particular attention was paid to resolving the viscous laminar sublayer near the upper

and lower channel walls. For this purpose, we controlled the parameter y+ (law-of-the-wall variable) in the first

computational node near the surface; the value of this parameter in all computations did not exceed unity.

197

Table 1. Basic parameters of the jet and external flow [4]

Flow domainTest case 1 Test case 2

CN2P , kPa M T0, K CHe P , kPa M T0, K

External flow 0 6.8 2.61 988 0 6.8 2.61 988Jet 1 84.5, 158.7, 302.7 1 943, 935, 928 1 151.4, 286.6 1 942, 948

0.02

0

0.04

0.06

_0.05 0 0.05

0 M0.6 1.2 1.8 2.4 3.0 500 7875 15250 22625 30000

100 270 440 610 780 950

y, m

x, m

T, Ê

p, Pa

(a) (b)

(c)

0.02

0

0.04

0.06

_0.05 0 0.05

y, m

x, m

0.02

0

0.04

0.06

_0.05 0 0.05

y, m

x, m

5

6

72

3

5

6

72 5

8

114

6

2

7

2

1

Fig. 2. Mach number contours (a), pressure contours (b), and temperature contours (c) in thecase of injection of a nitrogen jet into an air flow at M∞ = 2.61: (1) supersonic flow; (2) Machdisk; (3) separation of the turbulent boundary layer; (4) recirculation zone; (5) separation shock;(6) jet-induced shock wave; (7) reattachment shock; (8) local pressure peak.

3. VERIFICATION OF THE MATHEMATICAL APPROACH

The mathematical model and algorithm were tested against the experimental data [4] on slotted injection of

nitrogen and helium jets from the plate surface into a supersonic air flow with Mach numbers M∞ = 2.61–4.54. At

the center of the plate whose width was 457 mm, there was a slot 0.27 mm wide. The computations were performed

for several experimentally studied cases under the conditions described in Table 1.

For the case of nitrogen injection with the static pressure in the jet Pjet = 84.5 kPa, the computation was

performed on a sequence of consecutively refined grids. Grid refinement was performed by using a function of

adaptation in terms of the pressure gradient of the ANSYS Fluent software package. The pressure distributions on

the wall obtained on four grids with different numbers of nodes (77 · 103, 104 · 103, 133 · 103, and 174 · 103) show

that the difference in results decreases from 3 to 1% as the number of nodes is increased.

Figure 2a shows the calculated Mach number contours. The nitrogen jet escaping from the slot forms a

supersonic flow region 1, which is typical for underexpanded supersonic jets [1, 2]. The velocity of the flow bounded

by the contact surfaces of the jet increases, and compression waves reflected from these surfaces generate internal

198

123

123

2

121

pw/p1pw/p1

x, m x, m

(a)

_0.15 _0.10 _0.05 0.05 0.10 0.10

0.5

1.0

1.5

2.0

2.5

00

(b)

_0.10 _0.05 0.05

1

2

3

00

Fig. 3. Calculated (curves) and experimental [4] (points) distributions of pressure on the plate pw fordifferent ratios of pressures in the external and injected flows: (a) nitrogen jet with p0j/p1 = 23.5 (1),44.5 (2), and 82.9 (3); (b) helium jet with p0j/p1 = 45.6 (1) and 86.5 (2).

shocks. After that, a strong transverse shock 2 (Mach disk) is formed, and the flow velocity behind this shock

becomes subsonic. The jet is an obstacle for the supersonic flow, which leads to the emergence of a shock wave and

to separation of the turbulent boundary layer 3. Above the recirculation zone 4, one can see the separation shock 5

and the jet-induced shock wave 6. The separation region behind the jet is closed by the reattachment shock 7.

It should be noted that the calculated flow pattern agrees well with that shown in Fig. 1. Figure 2b shows the

static pressure contours. The maximum deceleration of the flow is observed in the vicinity of the frontal edge of

the jet, where a local pressure peak 8 is formed owing to meeting of two oppositely directed flows. The maximum

temperature T = 950 K is reached near the frontal separation point (Fig. 2c). Figure 3 shows the experimental and

calculated pressure distributions on the plate surface at M∞ = 2.61 for nitrogen and helium jets. It is seen that the

pressure values and the separation region lengths are in good agreement both ahead of the jet and behind it. As

the jet injection pressure increases, the jet–flow interaction intensity and the separation region length increase.

4. FORMULATION OF THE PROBLEM AND BOUNDARY CONDITIONS

Two test cases were considered. In test case 1, a hydrogen jet is injected into a channel with a cavity. In

test case 2, a helium jet is injected into a channel with a backward-facing step. The corresponding computational

domains are shown in Fig. 4. It is seen that the cavity has a gently sloping rear wall, and the channel behind the

cavity slightly expands. The profiles of all parameters of the turbulent flow with the Mach numbers M∞ = 2.5 (test

case 1) and M∞ = 2.8 (test case 2) with allowance for the presence of the boundary layers on the lower and upper

surfaces were prescribed in the input section. The no-slip conditions for velocity and the temperature Tw = 300 K

were set on the walls. The basic parameters of the jet and external flow for test cases 1 and 2 are listed in Table 2.

In test case 1, we studied the influence of the jet pressure on the resultant flow structure under prescribed

initial and boundary conditions. The computations were performed for three values of the jet pressure corresponding

to three values of the ratio of the jet momentum to the external flow momentum J = ρju2j/(ρ1u

21) (ρ is the

density and u is the flow velocity; the subscripts j and 1 refer to the jet parameters and external flow parameters,

respectively). In test case 2, we varied the jet injection angle: α = 30, 60, and 90◦.

199

(b)

(a)

x, m

y, m

x, m

4

5

0.05

_0.10 _0.05_0.15 0 0.05 0.10 0.15 0.20

0.10

y, m

0

0.05

0.10

0_0.05 0 0.05 0.10 0.15 0.20 0.25

1

2

3

4

51

2

3

Fig. 4. Computational domains for test case 1 (a) and test case 2 (b): (1) input boundary; (2) locationof the slot through which the jet is injected; (3) upper wall; (4) lower wall; (5) output boundary.

Table 2. Basic parameters of the jet and external flow

Flow domain

Test case 1 Test case 2

CH2P ,

MPaM J T0,

KCHe

P ,MPa

M α, degT0,K

External flow 0 0.185 2.5 — 1800 0 0.11 2.8 — 2000

Jet 1 2.6, 3.8, 5.5 1.0 2.35, 3.50, 5.00 300 1 1.72 1.4 30, 60, 90 293

5. CALCULATION OF HYDROGEN INJECTION INTO A CHANNEL

WITH A CAVITY (TEST CASE 1)

Figure 5 shows the Mach number contours obtained in flow calculations for different values of J . In the

vicinity of the injection point, there arises a shock 1, which impinges onto the upper wall, resulting in boundary

layer separation. At J = 2.6 and 3.5, there is a reattachment shock 2 behind the jet, which interacts further

downstream with the shock reflected from the upper wall 5. A compression wave 3 is formed near the rear wall of

the cavity owing to interaction of the mixing layer with the inclined surface. Consecutive reflections of shock waves

are observed in the downstream direction. At J = 5, the separation shocks emanating from the upper and lower

surfaces interact in an irregular manner, forming the normal shock 4. As the mass flow rate of the jet is increased,

the size of the separation regions on the upper and lower walls increases owing to enhancement of the interaction

intensity. At J = 3.5, the shock wave reflected from the upper wall 5 impinges onto the mixing layer and merges

with the compression wave 3. The thus-enhanced shock induces separation of the boundary layer from the upper

wall 6. At the same time, no separation on the upper wall occurs at J = 5, because the irregular reflection of

separation shocks ahead of the jet alters the flow pattern; as a result, the reflected shock 5 impinges onto the mixing

layer above the cavity at a more upstream point, near the leading edge of the cavity. In the case of interaction with

the mixing layer having a lower density, the shock 5 is reflected as an expansion wave. Therefore, the shock wave 3

has a moderate intensity and no separation occurs on the upper wall.

At J = 5, the length of the separation region 1 ahead of the jet is 32.8 mm, which is 2.5 times greater

than in the case with J = 2.6 (Fig. 6a). At moderate values of the mass flow rate (J = 2.6 and 3.5), the point of

200

(b)

(a)

y, m

y, m

0.02

0.04

0.08

0.06

0

0.04

0.02

0.08

0.06

0

(c)

x, m

y, m

_0.05 0 0.05 0.150.10

0.15

0.20 0.25

0.04

0.02

0.08

0.06

0

x, m_0.05 0 0.100.05 0.20 0.25

0.150.100.05 0.20 0.25 x, m_0.05 0

0 0.6 1.2 1.8 2.4 3.0 M

4

1

1

15

5

5

7

6

3

3

2

2 3

Fig. 5. Mach number contours for J = 2.6 (a), 3.5 (b), and 5.0 (c); (1) separation shock; (2) reat-tachment shock; (3) compression wave; (4) normal shock; (5) shock reflected from the upper wall;(6) separation of the boundary layer from the upper wall; (7) compression wave.

reattachment of the boundary layer behind the jet is located ahead of the leading edge of the cavity (see Figs. 6a

and 6b). The analysis of the flow in the vicinity of the jet at J = 5 (see Fig. 6c) shows that the jet penetrates into

the channel to a large depth and the second barrel of the underexpanded supersonic jet is formed.

It should be noted that the vortex structures inside the cavity are essentially different at J = 2.6 and 3.5.

At J = 2.6, there is one large vortex inside the cavity. At J = 3.5, a complex system of vortices is formed in the

cavity because of the secondary separation from the lower wall of the cavity (see Fig. 6b). The emergence of the

secondary separation leads to a change in the shape of the boundary of the mixing layer above the cavity and, as a

consequence, to the formation of a weak compression wave 7 (see Fig. 5b). At J = 5, the separation region behind

the jet 2 merges with the recirculation zone inside the cavity 3, forming a vast separation region (see Fig. 6c). The

shock reflected from the upper wall is incident into this region, which leads to “bulging” of the mixing layer and to

the emergence of a corner point 4. Figure 7 shows the isolines of the mass fraction of hydrogen in the channel at

J = 2.6 and 5.0. It is seen that a large mass flow rate of the jet leads to an increase in the depth of jet penetration

into the external flow and the thickness of the mixing layer of the gases.

It follows from the discussion that an increase in the mass flow rate of the jet leads to significant changes in

the wave pattern in the channel and in the vortex structure inside the cavity. With increasing J , the depth of jet

penetration into the external flow increases. We found a critical value of J at which the transition from regular to

Mach reflection of shock waves from the upper and lower walls occurs.

201

(b)

(a)

y, m

y, m

0.01

0.02

0.04

0.03

0

0.02

0.01

0.04

0.03

0

(c)

x, m

y, m

_0.02_0.04_0.06 0 0.02 0.04

0.04

0.06 0.08 0.10

0.02

0.01

0.04

0.03

0

x, m_0.02_0.04 0 0.02 0.06 0.08 0.10

0.04 x, m_0.02_0.04 0 0.02 0.06 0.08 0.10

3

21

21

1

4

2

Fig. 6. Streamlines for J = 2.6 (a), J = 3.5 (b), and J = 5.0 (c): (1) separation region ahead ofthe jet; (2) separation region behind the jet; (3) recirculation region inside the cavity; (4) cornerpoint; the arrow indicates the point of jet injection.

At the same time, an increase in the intensity of the shock waves and the transition to the irregular reflection

type lead to the emergence of a zone with a high static pressure (Fig. 8), resulting in higher total pressure losses.

The calculations show that a further increase in J leads to an increase in the sizes of the straightline segment of

the shock wave and separation regions, finally resulting in channel choking.

6. CALCULATION OF HELIUM INJECTION INTO THE CHANNEL

WITH A BACKWARD-FACING STEP (TEST CASE 2)

We studied the influence of the helium jet injection angle on the flow in a channel with a backward-facing

step. The mass flow rate of the jet is fixed (J = 3). Figure 9 shows the Mach number contours for α = 30, 60,

and 90◦. As the jet injection angle α is increased, the intensity of the jet-induced shock wave 3 increases, leading

to an increase in the size of the separation region 1 in the vicinity of the jet. At α = 60◦, the shock wave 3

is more intense, and its incidence onto the upper wall of the channel leads to boundary layer separation 2 (see

Fig. 9b). A separation shock 6 arises on the upper wall ahead of the separation region. The shock waves 3 and 6

202

(b)

(a)

x, m

y, m

x, m

_0.05 0

0 1.00 CH20.750.500.25

0.05 0.10 0.15 0.20 0.25

y, m

0.02

0.08

0.06

0.04

0_0.05 0 0.05 0.10 0.15 0.20 0.25

0.02

0.08

0.06

0.04

0

Fig. 7. Isolines of the mass fraction of hydrogen for J = 2.6 (a) and 5.0 (b).

(b)

(a)

x, m

y, m

x, m

_0.05 0 0.05 0.10 0.15 0.20 0.25

y, m

p, Pa

0.02

0.08

0.06

0.04

0_0.05 0 0.05 0.10 0.15 0.20 0.25

0.02

0.08

0.06

0.04

0

105 1.725.106

Fig. 8. Pressure isolines for J = 2.6 (a) and 5.0 (b).

interact, generating a zone with an elevated pressure at the channel center. Behind the separation region 2, there

is a reattachment shock 4, which interacts with the shock wave 5 closing the recirculation zone behind the step. As

the jet injection angle is increased to α = 60◦, the size of the separation region behind the jet increases. As in test

case 1, this region merges with the main recirculation zone behind the backward-facing step. The separation shock

6 incident onto this region leads to its “bulging” and increases its length.

203

(b)

(a)

x, m

y, m

x, m

_0.05 0 0.05 0.10 0.15 0.20

y, m

0.02

0.06

0.04

0

_0.05 0 0.05

0 0.5 1.0 1.5 2.0 2.5 3.0 M

0 0.825 1.650 2.475 M

0 0.825 1.650 2.475 M

0.10 0.15 0.20

0.02

0.06

0.04

0

(c)

x, m

y, m

_0.05 0 0.05 0.10 0.15 0.20

0.02

0.06

0.04

0

3

3

3

4

4

4

2

2

2

1

1

1

5

5

5

6

6

Fig. 9. Mach number contours and streamlines for different angles of helium injection for α =30◦ (a), 60◦ (b), and 90◦ (c): (1) separation region; (2) region of boundary layer separation;(3) jet-induced shock wave; (4) reattachment shock; (5) shock wave closing the recirculation zonebehind the step; (6) separation shock.

A further increase in the jet injection angle to α = 90◦ at a fixed flow rate produces an effect similar to an

increase in the mass flow rate described in Section 5. The second barrel is formed inside the jet, and the helium jet

penetrates into the air flow to a greater depth. The sizes of the separation regions 1 and 2 increase, resulting in a

more upstream position of the point of intersection of the separation shocks 3 and 6, as compared with the cases

of smaller injection angles. As a result, the point of arrival of the shock wave 6 on the lower wall is shifted to the

jet injection region rather than to the expanding part of the channel, as it happens at α = 60◦, and no “bulging”

of the recirculation zone behind the step occurs.

The distributions of pressure and skin friction coefficient on the lower surface of the channel are shown in

Fig. 10. As was noted above, the size of the separation region ahead of the jet, which can be identified on the

basis of the region of negative values of cf , increases as the injection angle is increased, which is manifested as the

presence of a constant pressure region for different angles of jet injection. At α = 90◦, the maximum size of the

separation region ahead of the jet is reached, whereas the size of the recirculation zone behind the step is smaller

than in the case with α = 60◦. At the same time, at α = 90◦, the reattachment shock 4 behind the separation

region near the upper wall is more intense and induces boundary layer separation from the lower wall approximately

204

123

cfpw/p1

x, m x, m

(a)

_0.1 0.20.1 0.2

0.5

1.0

1.5

2.0

2.5

3.0

00

(b)

_0.2 _0.1 0.1

0

0.002

_0.0020

Fig. 10. Distributions of pressure (a) and skin friction coefficient (b) on the lower wall of thechannel for different angles of jet injection: α = 30 (1), 60 (2), and 90◦ (3).

at x = 0.1 m (see Fig. 9c). A comparison of the base pressure levels behind the step shows that the smallest value

pb/p1 = 0.2 is reached at α = 30◦ (see Fig. 10a). At greater angles α, the base pressure increases because of the

incidence of shock waves reflected from the upper wall onto the recirculation zone behind the step.

Thus, as the angle of jet injection is increased, the jet penetration depth increases, a higher degree of mixing

of the flows at the channel exit is observed, and the mixing layer of the gases becomes thicker. At the same time,

at greater angles of jet injection, more intense shock waves are formed in the channel and, as a consequence, the

total pressure losses increase.

7. CONCLUSIONS

Injection of hydrogen and helium jets into variable-section channels of different geometries under condi-

tions of a high-enthalpy supersonic flow was numerically simulated on the basis of two-dimensional full averaged

Navier–Stokes equations.

Verification of the model and numerical method was performed against experimental results on slotted

injection of nitrogen and helium jets into a supersonic air flow. It was demonstrated that the model reproduces

the wave structure of the flow. The experimental and calculated pressure distributions on the plate surface are

in good agreement.

The flow structures obtained in test cases with injection of hydrogen and helium jets into variable-section

channels were analyzed in detail. An increase in the mass flow rate of the jet leads, on the one hand, to deeper

penetration of the jet into the flow and improvement of mixing of the main and injected gases and, on the other

hand, to greater total pressure losses. Similar effects were observed in the case of increasing the jet injection angle

at a fixed flow rate. An increase in the momentum of the jet or the injection angle alters the wave pattern of the

flow and the vortex structure in the expanding part of the channel.

Two types of the flow with regular and irregular regimes of reflection of separation shocks arising ahead of

the jet were identified. For the external flow Mach number M∞ = 2.5 and a fixed angle of jet injection α = 90◦,the critical ratio of momenta J at which the interaction regime changes from regular to Mach reflection is in the

interval J = 3.5–5.0.

The results of the present study can be used to explain the mechanisms of mixing of gas jets with the external

supersonic flow and to obtain an idea about the details of the complex wave pattern in the channel.

This work was supported by the Russian Foundation for Basic Research (Grant Nos. 12-07-00571-a

and 12-08-00955-a).

205

REFERENCES

1. J. A. Schetz, P. F. Hawkins, and H. Lehman, “Structure of Highly Underexpanded Transverse Jets in a Super-sonic Stream,” AIAA J., No. 5, 882–884 (1967).

2. V. G. Dulov and G. A. Luk’yanov, Gas Dynamics of Exhaustion Processes (Nauka, Novosibirsk, 1984) [inRussian].

3. V. Viti, R. Neel, and J. A. Schetz, “Detailed Flow Physics of the Supersonic Jet Interaction Flow Field,” Phys.Fluids 21, 046101 (2009).

4. E. E. Zukoski and F. W. Spaid, “Secondary Injection of Gases into a Supersonic Flow,” AIAA J. 2 (10),1689–1696 (1964).

5. V. S. Avduevskii, K. I. Medvedev, and M. N. Polyanskii, “Interaction of a Supersonic Flow with a TransverseJet Injected through a Circular Orifice in a Flat Plate,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3,193–197 (1970).

6. V. V. Eremin, Yu. M. Lipnitskii, A. N. Pokrovskii, et al., “Investigation of Interaction of a Plane TransverseGas Jet with a Supersonic Flow,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 4, 103–110 (1975).

7. F. S. Billing, R. S. Orth, and M. Lasky, “A Unified Analysis of Gaseous Jet Penetration,” AIAA J. 9 (6),1048–1058 (1971).

8. N. E. Hawk and J. L. Amick, “Two-Dimensional Secondary Jet Interaction with a Supersonic Stream,” AIAA J.5 (4), 655–660 (1967).

9. A. I. Glagolev, A. I. Zubkov, and Yu. A. Panov, “Interaction of a Gas Jet Escaping from an orifice in a FlatPlate with a Supersonic Flow,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 99–103 (1968).

10. M. R. Gruber, A. S. Nejad, T. H. Chen, and J. C. Dutton, “Mixing and Penetration Studies of Sonic Jets in aMach 2 Freestream,” AIAA Paper No. 1994-709 (1994).

11. J. R. Sterrett, J. B. Burber, D. W. Alston, and D. J. Romeo, “Experimental Investigation of Secondary Jets fromTwo-Dimensional Nozzles with Various Exit Mach Numbers for Hypersonic Control Application,” Tech. NoteNASA No. D-3795, Washington (1967).

12. W. J. Thayer and R. C. Corlett, “Gas Dynamic and Transport Phenomena in the Two-Dimensional Jet Inter-action Flowfield,” AIAA J. 10 (4), 488–493 (1972).

13. R. Portz and C. Segal, “Penetration of Gaseous Jets in Supersonic Flows,” AIAA J. 44 (10), 2426–2429 (2006).14. R. J. Goldstein, G. Shavit, and T. S. Chen, “Film-Cooling Effectiveness with Injection through a Porous

Section,” J. Heat Transfer 87, 353–361 (1965).15. V. I. Penzin, “Experimental Study of Transverse Injection into a Supersonic Flow in a Channel,” Uch. Zap.

TsAGI 4 (6), 112–118 (1973).16. V. Ya. Borovoy and M. V. Ryzhkova, “Gas Flow and Heat Transfer on a Cone near a Transverse Jet with a

Laminar State of the Boundary Layer,” Uch. Zap. TsAGI 5 (4), 48–58 (1974).17. A. Ya. Nadyrshin and Z. G. Shaihutdinov, “Mixing of a Supersonic Flow with a Transverse Jet Injected through

a Circular Orifice in a Flat Plate,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1, 14–18 (1975).18. M. R. Gruber, A. S. Nejad, T. H. Chen, and J. C. Dutton, “Large Structure Convection Velocity Measurements

in Compressible Transverse Injection Flowfields,” Exp. Fluids 22, 397–407 (1997).19. W. M. Van Lerberghe, J. G. Santiago, J. C. Dutton, and R. P. Lucht, “Mixing of a Sonic Transverse Jet Injected

into a Supersonic Flow,” AIAA J. 38 (3), 470–479 (2000).20. J. P. Drummond, “Numerical Solution for Perpendicual Sonic Hydrogen Injection into a Ducted Supersonic

Airstream,” AIAA J. 17 (5), 531–533 (1979).21. F. Grasso and V. Magi, “Simulation of Transverse Gas Injection in Turbulent Supersonic Air Flows,” AIAA J.

33 (1), 56–62 (1995).22. V. K. Baev, V. I. Golovichev, and P. K. Tret’yakov, “Combustion in a Supersonic Flow,” Fiz. Goreniya Vzryva

23 (5), 5–15 (1987) [Combust., Explos., Shock Waves 23 (5), 511–520 (1987)].23. Z. A. Rana, B. Thornber, and D. Drikakis, “Transverse Jet Injection into a Supersonic Turbulent Cross-Flow,”

Phys. Fluids 23, 046103 (2011).24. A. O. Beketaeva and A. Zh. Naimanova, “Numerical Study of Spatial Supersonic Flow of a Perfect Gas with

Transverse Injection of Jets,” Prikl. Mekh. Tekh. Fiz. 52 (6), 58–68 (2011) [Appl. Mech. Tech. Phys. 52 (6),896–904 (2011)].

25. R. L. Roe, “Characteristic Based Schemes for the Euler Equations,” Annual Rev. Fluid Mech. 18, 337–365(1986).

26. M. S. Liou, “A Sequel to AUSM: AUSM+,” J. Comput. Phys. 129, 364–382 (1996).

206