In high-speed flows in the channels the
areas of interaction between the boundary layer and shock waves arise. The
interaction of an inclined shock wave (oblique shock) with the boundary layer
has been studied for several decades [1–6]. A strong pressure gradient caused
by the impact of a shock wave on the boundary layer can lead to the flow
separation. This phenomenon is accompanied by an increase in the dynamic load
on the surface, its high local heating, and a rise in the drag coefficient. In
addition, the interaction of the shock wave and the boundary layer can cause
unsteady formation of vortices [3, 4]. In rocket propulsion systems, these
effects affect the flow in the nozzles of supersonic engines [6]. In a number
of experimental studies of flows of this class [3, 4] and works on numerical
modeling [1, 2, 6, 7], it was found that in the separation zone the pressure
and the heat transfer coefficient increase sharply. Generally, numerical
simulation reflects the main features of the studied flows correctly, however
the calculated values in developed separation zones generated by
strong shock waves differ from experimental ones [5]. As a result, further
investigation of the interaction of shock waves with boundary layers is
required.

The usage of low-temperature plasma of
surface discharges of various types (plasma actuators) in plasma aerodynamics
is aimed to correct the flow regime, including the reduce of dynamic and
thermal loads on the surface, controlling the laminar-turbulent transition in
the boundary layer, the position of separation zones and shock waves [8- 10], and
the processes of combustion [11]. Pulsed distributed surface sliding discharge
of nanosecond duration (plasma sheet) can be used as an actuator to influence
the flow [9]. It consists of channels sliding along the dielectric surface and
forming a plasma layer comparable in thickness with the boundary layer of a
supersonic flow in a shock tube (~ 0.5 mm). Surface sliding discharge glow can
visualize a laminar and turbulent boundary layer [12]. The short duration of
nanosecond discharges (~ 100 ns) makes it possible to use the registration of
radiation to visualize the structure of unsteady supersonic plane [12] and
three-dimensional flows [13]. The radiation of glow-type discharges with a
duration of more than one millisecond is mainly used to visualize the structure
of stationary flows [10, 14].

The purpose of this work was an
experimental study of the spatial distribution of the glow of a pulsed surface
sliding discharge in inhomogeneous supersonic flows with an oblique shock wave
in the discharge chamber of the shock tube. It was important to study the
fundamental physical processes under the development of a nanosecond discharge
in an inhomogeneous medium, and to study the structure of the near-surface flow
in the area of a pulsed energy input to develop flow control methods in such conditions.
Numerical simulations of the flow fields were carried out to compare the
density field with the discharge glow structure. The possibility to visualize
the interaction region of the boundary layer with an oblique shock wave based
on radiation registration is discussed.

The experiments were carried out using a
shock tube with a low-pressure chamber with a length of 3 m, a rectangular
channel with an internal cross section of 24´48 mm^{2}, and a discharge
chamber of the same cross section (Fig. 1 a) [9, 12]. Supersonic air flows were
created behind plane shock waves with Mach numbers 2.8-4.2. The flow velocities
were 700-1150 m/s, Mach numbers of the flows were 1.30-1.60, the density was
0.10-0.14 kg/m^{3}. Under these conditions, the duration of a uniform
co-current flow behind the shock-wave front was 200–500 µs, and the length of
this flow behind the shock-wave front was about 30 cm [12]. In a shock tube the
thickness of the boundary layer increases from zero at the shock wave front in
the direction of the contact surface, and the boundary layer becomes turbulent
at a certain distance from the shock wave front [15]. The Reynolds number of
the flow was ~ 10^{5}, estimated from the size of the shock tube
channel. The thickness of the laminar boundary layer on the channel walls did
not exceed 1 mm [9, 12].

In the discharge chamber, on the lower and
upper walls, flat electrodes of surface sliding discharges 10 cm long with an
inter-electrode distance of 3 cm were located. The side walls of the discharge
chamber on a 17-cm-long area were quartz glasses with a transmission band of
200-2800 nm, which allowed recording discharge radiation and spectra. The registration
of the integral discharge glow was performed using photo cameras located at
different angles to the camera windows. The upper plasma sheet was initiated in
the experiments, its radiation was investigated in the boundary layer
interacting with an oblique shock (inclined shock wave) (Fig. 1b). The
discharge spectra were recorded by AvaSpec-2048FT spectrometer with wavelength
range 174-1100 nm. The fiber-optic cable (UV/VIZ, 100 μm diameter) was placed at an angle to the plane of the upper plasma
sheet.

a)

b)

Fig. 1. a) Scheme of the shock tube and
diagnostic equipment; b) the flow structure in the discharge chamber with an
obstacle on the bottom wall.

A small dielectric obstacle in the form of
a rectangular parallelepiped measuring 48.0´6.2´1.9 mm^{3} (Fig. 1b) was placed on the bottom wall of the
discharge chamber at a distance of about 1.5 cm from the leading edge of the
lower electrode. The long part of the obstacle was perpendicular to the
discharge camera glasses. After diffraction of a plane shock wave on an
obstacle for ~ 200 μs, supersonic flow was
established by a co-current flow. The quasi-stationary flow field in the
channel of the discharge chamber contained an oblique shock wave that
interacted with the boundary layer on the upper wall of the discharge chamber
(Fig. 1b). The diagrams of two types of interaction that are realized in the
interaction of an oblique shock wave with a boundary layer are shown in Fig. 2.
In the case of a laminar boundary layer, the interaction with the boundary
layer can be continual, but contains a region of low density (Fig. 2 a); in the
other case, an interaction occurs with the formation of separation zone (Fig. 2
b).

A pulsed surface sliding discharge was
initiated on the upper wall of the discharge chamber in the time range of
70–1200 μs after the shock wave passed the
obstacle. The discharge triggering was synchronized with the passage of the
shock wave front from the signals of the piezoelectric pressure sensors in the
shock tube channel. The initiation of the discharge was carried out at a given
time after the shock wave passed through the test area in the discharge
chamber, including the stage of quasistationary supersonic flow around the
obstacle.

a) b)

Fig. 2. Diagrams of the interaction of an
oblique shock wave with a boundary layer: a) without separation; b) with the
separation zone.

** **

A pulsed surface sliding discharge was developed
in a thin gas layer near the surface of the dielectric when a pulsed voltage of
25 kV was applied to the discharge electrodes, providing a significant specific
energy input. The discharge current was recorded by a low-inductive shunt of a
special design. The discharge current was ~ 1 kA, the duration did not exceed
500 ns (Fig. 3). The main energy input to the gas occurs within 120-150 ns of
the first half period of the discharge current, i.e. almost instantaneously as
compared with the characteristic gas-dynamic flow times in the boundary layer
[9, 12]. The electric field applied to the discharge gap was E = 8.33 kV/cm,
and the reduced electric field *E**/N* under the experimental conditions reached
3·10^{-19} V·m^{2}, where *N* is the concentration of
molecules). Under such conditions, the processes of excitation of electronic
levels of nitrogen molecules N_{2} and oxygen O_{2}, ionization
and dissociation of molecules [8, 16] actively occur in discharges in the air.
This leads to the presence of significant ultraviolet part in the emission
spectra due to the emission of molecular nitrogen and atomic lines (Fig. 4).

The ionization rate, on which the electron
concentration in the discharge plasma and conductivity depend, is determined by
the local reduced electric field *E**/N*. Therefore, the density distribution
in the medium affects the structure of the discharge radiation. The photographs
of the discharge in still air show that the discharge consists of parallel
channels of close intensity that completely fill up the discharge gap with a
length of 10 cm (Fig. 5 a, b). In an inhomogeneous flow with an oblique shock
wave, the discharge was developed as a single intense channel (Fig. 5c), the
direction of development of the channel is perpendicular to the flow velocity.
The geometry of the discharge current depended on the density distribution in
the discharge region.

Fig. 3. Current waveforms in still air at a
pressure of 5 and 60 torr (density of 0.008 and 0.10 kg/m3).

The analysis of the discharge spectra
showed that the main part of the discharge radiation consists of the bands of
the second positive nitrogen system (C^{3}Ï_{u} →B^{3}Ï_{g} transition) in the wavelength range
280-500 nm (Fig. 4). In the spectra there are intense lines of atoms of oxygen,
nitrogen, hydrogen, formed during the dissociation of molecules. It should be
noted that the radiation of the visible part of the spectrum (400–750 nm) is
recorded in the photo images (Fig. 3), the second positive nitrogen system makes
a significant contribution. The radiation lifetime of the emitting level of
C3Pu of molecular nitrogen is about 40 ns. During the passage of the discharge
current and radiation from this level, no significant displacement of the
structural elements of the high-speed flow occur.

Fig. 4. The emission spectra of the
discharge: 1– in a supersonic flow with a Mach number of 1.38; 2 – in still
air; density is 0.10 kg/m3.

Fig. 5. Photo images of a sliding discharge
glow in the still air at a pressure of 33 (a) and 70 Torr (b) (density,
respectively, 0.06 and 0.12 kg / m3) and in a supersonic airflow with an
oblique shock wave (c) at a density of 0.12 kg / m3 and flow Mach number 1.40.

The density distribution in the medium
affects the development of nanosecond discharges. The inhomogeneity of the
density leads to the inhomogeneity of the reduced electric field *E/N* and, consequently, the heterogeneity
of the local conductivity of the medium (the concentration of electrons).
Increased conductivity in areas of low density can lead to a change in the
geometry of the current of a pulsed surface sliding discharge [17]. The
excitation of electron levels of molecules by electrons leads to an increased
intensity of radiation from low-density regions of the flow, hence visualizing
them [12, 13, 17]. The spatial distribution of radiation provides information
about the instantaneous density distribution in the region of the discharge radiation
(in terms of gas dynamics). So, the glow of a pulsed surface sliding discharge
in a boundary layer of a uniform supersonic flow clearly visualizes its
structure and the region of a laminar – turbulent transition at a certain
distance from the shock wave front [13].

The spatial distribution of a sliding
discharge radiation in an inhomogeneous supersonic flow in the channel of a
shock tube was structurally different at different stages of the formation of a
flow with an oblique shock wave. In the case when at the moment of initiation
of the discharge the front of a plane shock wave traveling through the channel
was inside the discharge region or went a short distance beyond it, the
discharge current flowed principally ahead of the wave front. The main
radiation of the discharge was concentrated in this region. In fig. 6a, the
discharge glow is shown in the form of a U-shaped channel near the shock wave
front. At the same time, a weak discharge glow was observed at a distance of
5–6 cm behind the shock wave front, which visualized a zone of low density in
the region of interaction of an oblique shock wave with a laminar boundary
layer (Fig. 6a). At later stages of the development of the flow, oblique shock
wave interacted with the turbulent boundary layer, forming a separation region
with a lower density. The discharge glow was concentrated in a single intense
channel with a width of less than 10 mm (Fig. 6b-d). The left boundary of the
channel corresponds to a straight line of intersection of the inclined shock
wave with the boundary layer, as shown in the photo image obtained by the
filter (Fig. 6c). When initiating a discharge after the end of a uniform
co-current flow with the Mach number of the flow decreasing, the discharge
channel moves upstream, in order to be located closer to the obstacle (Fig.
6g).

An analysis of experimental photo images
showed that the structure of the glow of the discharge plasma depends on the
flow parameters and is determined by the density distribution in the
near-surface flow. The position and geometry of the glow region in the boundary
layer of the flow were determined on the basis of digital processing of
registered photo images from both sides of the discharge chamber. The distance
Dk from the beginning of the obstacle on the lower wall of the channel (see
Fig. 1 b) to the discharge glow region on the upper wall of the discharge
chamber and the length of this region along the flow direction at different
stages of the flow (Fig. 7) were measured. These values were measured
by scanning images along the OX axis using a Python language program. Fig. 1 b highlights
the boundaries of the regions of the glow in three-color ranges.

Fig. 6. Photo images of a sliding discharge
glow in a flow with an oblique shock wave at discharge initiation times 76 (a),
93 (b), 140 (c), 240 (d), 455 μs (e) (from the
moment of the shock wave diffraction on an obstacle). The density is 0.13
kg/m3, the Mach number of flow is 1.48, the flow is directed from left to
right. The image (c) was registered through an optical filter that transmits
radiation with a wavelength of 405 nm.

Fig. 7. The coordinate of the region of
maximum glow of the discharge channel as a function of time T for flows with
Mach numbers 1.36-1.42 (1), 1.45-1.47 (2); the calculated position of the
density minimum (3) and the length of the region of low density (4) for the
flow Mach number 1.46. The density is 0.120.01
kg/m3. (T = 0 corresponds to the moment when the shock wave contacts the
obstacle.)

Dependence between the distance
Dk and the time T for two ranges of flow Mach numbers is shown in fig. 7 (T is
the time from the beginning of the diffraction of a plane shock wave on an
obstacle until the moment of the discharge). The obtained values
of Dk are within (27–38) ±2 mm under the experimental conditions, being in the region of
interaction of an oblique shock wave with a boundary layer. The position of the
discharge glow area varies slightly with time in the range of 150-400 μs from the beginning of the diffraction of the shock wave. The
structure of the discharge glow changes with the increasing distance from the
shock front due to a change in the type of interaction of an oblique shock wave
with the boundary layer during the transition from the laminar to the turbulent
regime. The discharge channel radiation spectrum in this case is characterized
by an increased intensity of the continuum and atomic lines of the visible
range (Fig. 4), indicating an increased concentration of electrons.

The purpose of the numerical simulation was
to determine the shock-wave structure of the flow in the channel with an
obstacle under experimental conditions and subsequent comparison with the
results of the study of the surface sliding discharge in non-uniform supersonic
flows.

Mathematical modeling of gas flow in the
channel was carried out on the basis of two-dimensional Navier-Stokes equations,
which describe the unsteady flow of a viscous compressible gas with appropriate
initial and boundary conditions and relations of thermal and caloric equations
of state [18]. A perfect gas (air) model was used with a constant adiabatic
index (γ = 1.4) and a Prandtl number Pr = 0.72.
The dependence of the viscosity coefficient on temperature was described by the
Sutherland formula [18]. Numerical calculations of flows behind plane shock
waves were carried out in a channel with a height of 24 mm and a length of 216
mm under experimental conditions. In the numerical simulation a plane shock
wave diffracted on a 2´6 mm obstacle located on the lower wall of the channel. Fig. 8 shows
the successive stages of the interaction of a plane shock wave with an
obstacle. After diffraction of a shock wave for 210-230 μs, a quasi-stationary flow around the obstacle with a supersonic
flow with an inclined shock wave is established in the channel. Boundary layers
are formed on the upper and lower walls, disturbed by the interaction with
oblique shock waves. On the calculated flow fields, the dynamic color scale is
used thus the maximum value of the given value corresponds to the red color,
the minimum corresponds to the blue.

In the experiments the discharge was
initiated on the upper wall of the channel in the region corresponding to the
dimensionless coordinates of the computational domain from 1.9 to 6.0
(dimensions are related to the 24 mm channel height). In this area the oblique
shock from the obstacle interacts with the boundary layer on the upper wall of
the channel (Fig. 8). This interaction leads to the formation of a region of
low density or flow separation region (Fig. 9), located at a distance of 38-48
mm from the leading edge of the obstacle at steady state. The length of the
low-density region (along the flow direction) was 5-15 mm, depending on the
time after the diffraction of the initial shock wave on the obstacle. The region
of interaction of an oblique shock wave with a boundary layer on the upper wall
of the channel is shown on an enlarged scale in Fig. 9 b. While conducting the
calculations the thickness of the undisturbed boundary layer did not exceed 0.2
mm, and the separation zone was no more than 1.5 mm thick. The minimum density
in this area was 38-62% of the incident flow density.

* *

Fig. 8. The field of the local Mach number in
establishing the flow around an obstacle with a flow behind the shock wave with
a Mach number of 3.45: the time from the moment the shock wave contacts the
obstacle 30, 80, 180, 310 μs (from top to bottom).

a)

b)

Fig. 9. The calculated density field at the quasi-stationary
stage of the flow (a), an enlarged fragment of the region of interaction of an
oblique shock wave with the boundary layer on the upper wall of the channel
(b). The Mach number of the initial shock wave is 3.45, the time from the
moment the shock wave contacts the obstacle is 310 μs.

The dependence of the calculated position
of a low-density region on time is shown in Fig. 7. The length and position of
this region depends on time: first, it forms as an extended zone within 22-43
mm from an obstacle, then shifts downstream in the process of flow
establishing, and is located at a distance of 36-42 mm from an obstacle at the steady-state
flow (150-350 μs from the diffraction moment). According
to the fig. 7 the experimental dependence of the location of the discharge glow
region on time clearly correlates with the dependence of the position of the low-density
region on time in the calculated data. Thus, a comparison of the experimental
data with the calculated indicates that the glow geometry of a pulsed surface
sliding discharge coincides with the geometry of a low-density zone such as its
location and length. This allows to visualize the structure of the boundary
layer area under study based on the registration of discharge radiation
developing inside the boundary layer. The nanosecond discharge duration, which
is much shorter than the characteristic gas-dynamic times, makes it possible to
investigate non-stationary flows with shock waves.

The effect of inhomogeneous supersonic flow
in a channel behind shock waves with Mach numbers 2.8–4.2 on the spatial
structure of radiation of a surface sliding discharge of nanosecond duration is
studied experimentally. An analysis of the discharge glow in the region of the
interaction of an oblique shock wave with the boundary layer, including the
discharge in the form of an intense channel in the zone of separation of the
boundary layer is carried out.

It is shown that the glow of a pulsed
surface sliding discharge allows visualizing the region of interaction of an
oblique shock wave with a boundary layer at different stages of an unsteady
flow. Registration of discharge radiation at different angles can provide
information about the three-dimensional structure of the boundary layer zone.
The advantages of the visualization method on the base of the registration of
radiation from the discharge plasma are that the plasma locates directly into
the boundary layer, and the short discharge duration, which makes it possible
to investigate non-stationary flows.

The work was supported by RFBR grant
19-08-00661.

**1.
****Zheltovodov A. A. **Properties of two- and three-dimensional separation flows at
supersonic velocities // Fluid Dynamics. 1979. V. 14. No 3. P. 357-364.

**2.
****Zubin M. A., Ostapenko N. A.** Structure of flow in the separation region resulting from interaction
of a normal shock wave with a boundary layer in a corner // Fluid Dynamics. 1979.
V. 14. No 3. P. 365-371.

**3.
****Kubota H., Stollery J.** An experimental study of the interaction between a glancing shockwave
and a turbulent boundary layer // J. Fluid Mech. 1982. 116-431-58.

**4.
****Dupont P., Haddad C., Debiève
J.-F.** Space and time organization in a
shock-induced separated boundary layer. J. Fluid Mech. 2006. 559. P.
255-277.

**5.
****Borovoy V.Ya., Mosharov V.N, Noev A.Yu., Radchenko
V.N.** Laminar-turbulent flow over wedges mounted on
sharp and blunt plates // Fluid Dynamics. 2009. V. 44. No 3. P. 382-396.

**6.
****Hadjadj A., Perrot Y., Verma S.** Numerical study of shock/boundary layer interaction in supersonic
overexpanded nozzles // Aerosp. Sci. Technol. 2015. V.
42. P.
158-168.

**7.
****Bosnyakov**** ****S****.****M****., ****Babulin**** ****A****.****A****., ****Vlasenko**** ****V****.****V****., ****Engulatova**** ****M****.****F****., ****Matyash**** ****S****.****V****, ****Mikhaylov**** ****S****.****V****. ** On the accuracy of numerical simulation of the boundary layer
separation on a finite-width wedge. // Mathematical Models and Computer Simulations. 2016. V. 8. No 3. P.
238–248.

**8.
****Bayoda D., Benard N., and Moreau E.** Nanosecond pulsed sliding dielectric barrier discharge plasma
actuator for airflow control: Electrical, optical, and mechanical
characteristics // J. Appl.
Phys. 2015. V. 118. 063301.

**9.
****Mursenkova I.V., Znamenskaya I.A. and
Lutsky A.E. **Influence of shock waves from plasma
actuators on transonic and supersonic airflow // J. Phys. D: Appl. Phys. 2018.
V. 51. N 5. 105201.

**10.
**** Leger L., Sellam M., Barbosa E. and
Depussay E.** Visualization by discharge illumination
technique and modification by plasma actuator of rarefied Mach 2 airflow around
a cylinder // Meas. Sci. Technol. 2013. V. 24. N 6. 065401

**11.
**** Ju Y. and Sun W.** Plasma assisted combustion: Dynamics and chemistry // Progress in Energy and Combustion
Science. 2015. V. 48. P.
21-83.

**12.
**** Znamenskaya I.A., Latfullin D.F. and
Mursenkova I.V.** Laminar-Turbulent Transition in a
Supersonic Boundary Layer during Initiation of a Pulsed Surface Discharge. // Technical Physics Letters. 2008. V. 34. No. 8. P.
668–670.

**13.
**** Znamenskaya I., Mursenkova I., Kuli-Zade
T., Kolycheva A.** Vizualization of 3D non-stationary
flow in shock tube using nanosecond volume discharge // Shock Waves. 2009. P. 533-538.

**14.
**** Nishio M., Sezaki S., Nakamura H.** Visualization of flow structure around a hypersonic re-entry
capsule using the electrical discharge method // Journal of Visualization. 2004.
V. 7. N 2. P. 151-158.

**15.
**** Bazhenova T.V., Gvozdeva L.G.** Nestacionarnye vzaimodejstviya udarnyh voln. M., 1977. 274 p. [In Russian]

**16.
**** Raizer Yu. P. Gas Discharge Physics **(Springer, Berlin, 1991)

**17.
**** Mursenkova I.V., Sazonov A.S., and Liao Yu.
**The Effect of Pulsed Sliding Surface Discharges on
Supersonic Airflow past a Thin Wedge in Shock Tube. // Technical Physics
Letters, 2018 V. 44. No. 2. Ð. 157–159.

18.
** Glushko G.S., Ivanov I.E., Kryukov I.A. **Computational method for turbulent supersonic flows // Mathematical Models and Computer Simulations. 2010. V. 2. No 4. P.
407–422.