# Interaction of Shock Train with Cavity Shear Layer in a Scramjet Isolator

Vignesh Ram Petha Sethuraman  Department of Aerospace System Engineering,
Sejong University, Seoul, South Korea - 05006.  Yosheph Yang  Department of
Aerospace Engineering, Sejong University, Seoul, South Korea - 05006.  Jae
Gang Kim<EMAIL_ADDRESS>Department of Aerospace System Engineering,
Sejong University, Seoul, South Korea - 05006.

###### Abstract

The interaction between the self-excited shock train flow and the cavity shear
layer in a scramjet isolator is investigated numerically using detached-eddy
simulations (DES). The effect of changing the position of the shock train by
controlling the back pressure ratio and the effect of changing the cavity
front wall angle are analyzed using unsteady statistics and modal analysis.
The propagation mechanism of the pressure disturbance was investigated by
spatiotemporal cross-correlation coefficient analysis. In the present
numerical study, a constant isolator section with a cavity front wall
($\theta$ = 90o and 60o) was considered, followed by a diffuser section
simulated at Mach number 2.2 with three different back pressure ratios (pb/p∞
= 0.7, 5.0, and 6.0). The change in back pressure provides three different
conditions (i.e., no shock train, shock train ends before the leading edge of
the cavity, and shock train present above the cavity). To understand the
unsteady dynamics of the interaction of the shear layer with the shock train,
the spatiotemporal trajectory of the wall pressure and the centerline pressure
distribution, the spatiotemporal cross-correlation coefficient, and the modal
analysis by dynamic mode decomposition are obtained. The results show that the
low-frequency shock train oscillation dominates the self-sustained cavity
oscillation. The spatiotemporal cross-correlation between the wall surface and
the center of the cavity bottom wall indicates the propagation of local
disturbances originating from the separated boundary layer caused by the shock
and the recirculation zone in the corners of the cavity. Dynamic mode
decomposition analysis shows the shear layer at the leading edge of the cavity
and the downstream propagation of large eddies from the cavity. It also shows
the pairing of coherent structures between the shock train and the
recirculation zone of the cavity.

††preprint: AIP/123-QED

##  I INTRODUCTION:

The presence of shock waves in high-speed internal flow devices such as the
supersonic combustion ramjet (scramjet) is common and unavoidable. The
scramjet engine consists of four main parts, namely an inlet, an isolator, the
combustion chamber, and a nozzle. The compression process occurs through
various types of shock waves such as oblique, incident, and reflected shock
structures that occur at the inlet. When a scramjet is operated under
supercritical flow conditions, the normal shock usually occurs in the
isolator, which is also referred to as terminal shock. Interacting with the
incoming boundary layer, the terminal shock undergoes three shock patterns
(i.e., a curved shock wave, a lambda shock wave, and a shock train). In the
case of the curved shock wave and the lambda shock wave, the interaction
between the normal shock and the boundary layer is quite weak. When the
interaction becomes too strong, a single normal shock bifurcates into several
successive normal shock structures known as a "shock train". If the isolator
section is long enough, the static pressure continues to increase until the
frictional force dominates. This region is called the "mixing region." The
formation of the shock train in a constant area channel and its properties are
described in detail by Matsuo et al. [1].

Unlike normal shocks, the length of the shock train is usually on the order of
several duct heights. For this reason, the isolator section is designed to
accommodate the entire shock train system. The isolator provides sufficient
static pressure in the combustion system to allow an efficient combustion
process through the shock train system. At the same time, this combustion
process not only causes the downstream static pressure to increase but also
causes the entire shock train system to move upstream toward the intake area,
resulting in engine unstart condition [2, 3, 4, 5, 6] or inlet buzz condition
[7, 8, 9, 10]. The steady-state flow characteristics of the shock train depend
mainly on the pressure ratio (p2/p1), inlet freestream Mach number (M∞), the
aspect ratio of the duct (AR = L/H), and the blockage ratio ($\beta$) [11, 12,
13, 14]. These parameters determine the position of the shock train, its
length, the number of successive shocks, and its structures. In recent
decades, the steady-state properties of the shock train, such as its position,
length, and static pressure rise, have been estimated using various one-
dimensional models [15, 16, 17] and empirical relations [18, 19, 20, 21].
However, shock train flows exhibit unsteadiness due to the separated boundary
layer and downstream pressure perturbations [16]. Such oscillatory flows can
be classified as high-amplitude shock wave oscillations. In contrast, the low-
amplitude shock wave oscillations are related to the induced shock wave over a
compression ramp or the oblique incident shock wave [22].

Studies related to shock train unsteadiness have recently attracted increasing
attention from researchers. In the case of a uniform, unperturbed inlet flow,
the shock train oscillates in phase with the frequency of the downstream
pressure perturbation and behaves like a second-order non-homogeneous
differential equation [23]. Varying the incoming flow properties [24, 25, 26,
27, 28, 29, 30, 31] and the downstream back pressure [32, 33, 34, 35, 36] with
background waves results in different types of shock train oscillations that
have a greater impact on the performance of the scramjet. The low-frequency
oscillations of shock train initiate asymmetric flowfield and lead to an
intrinsic istability[37]. Various active and passive control methods are used
to control such low-frequency high-amplitude oscillations of shock train[38,
39, 40]. One of the active control methods, such as partial removal boundary
layer flow through the suction slot, not only helps to alleviate the shock
wave oscillations but also reduces the length of the shock train [41, 42, 43].
Moreover, the shock train oscillates at the same frequency but out of phase
with the flame oscillation [44]. A major challenge faced by a dual-mode
scramjet engine is the stabilization of combustion (i.e., auto-ignition and
fuel mixing), which can be achieved by using a cavity downstream of the fuel
injection. Although the cavity flame holder has the advantage of reducing the
total pressure loss, aerodynamic heating [45, 46, 47, 44] and sudden
transition between ram to scram mode[48]. The mechanism of combustion
stabilization and mode transistion phenomenon in high-speed flows is not fully
understood. Before combustion stabilization can be studied, the mechanism of
interaction between the pre-combustion shock train and the cavity shear layer
must be understood. Both the self-sustaining oscillation of the shock train
and the oscillation of the cavity flow are completely different mechanisms.
Self-sustained oscillations of the shock train are triggered by a small
perturbation of the downstream flow or the local boundary layer flow. On the
other hand, the interaction between the shear layer and acoustic disturbances
from the recirculation zone leads to self-sustained oscillatory flow in the
cavity [49, 50, 51, 52]. Extensive experimental and numerical studies on
cavity flow oscillation and its control techniques can be found in the open
literature [53, 54, 55]. Most of these studies assume that the flow is free of
shock interactions. However, in scramjet flow, the cavity is subjected to
various types of shock interactions, such as incident shocks, and bow shocks
that appeared due to fuel injection [56], resulting in a more complex
oscillation mechanism. Cheng et al [57] conducted experimental studies on the
interaction of the induced shock with the shear layer of the cavity and found
that the incident shock alters the oscillatory behavior by reducing the local
momentum thickness and enhancing the acoustic feedback mechanism. Recent
numerical studies on the interaction of wedge-induced oblique shocks with the
shear layer of the cavity show an increase in static wall pressure of about
25% compared to the case without interaction [58]. Studies on the interaction
between normal and oblique shock waves and the shear layer in cavities are
limited and mostly deal with the interaction of low-amplitude shocks with the
cavity shear layer. As mentioned earlier, shock train flows are high-amplitude
flows that can oscillate on the order of the duct height. Therefore, the
cavity flows experience a repeated supersonic-subsonic flow condition with
shock train oscillation. Therefore, in the present work, the interaction
between the shock train and a cavity shear layer in an isolator is studied
with the following objectives:

  1. 1.

To numerically simulate the shock train flows in the proximity of wall mounted
cavity in an isolator.

  2. 2.

To compare the transient characteristics of shock train–shear layer
interaction with different back pressure ratios and cavity front wall angle.

  3. 3.

To investigate the power spectrum along the flow field and identification of
common features between the different cases.

  4. 4.

To analyze the mechanism of local disturbance propagation using the two-point
cross-correlation method.

  5. 5.

To identify the dominant spatiotemporal modes using the dynamic mode
decomposition (DMD) method.

The outcome of the present work can help to understand the effects of the
interaction of the shock train with the cavity shear flow and its behavior.
The paper is organized as follows. In Sec.II, the problem is briefly
introduced with the flow field theory ( Sec.II.1), geometrical details
(Sec.II.2), and flow configuration (Sec.II.3). The numerical procedure, grid
independence study, and reliability of the unsteady simulation are described
in Sec.III. The steady and unsteady statistics, the spectral analysis of the
shock train shear layer interactions, the mechanism of the disturbance
propagation, and the modal analysis are described in Sec.IV. In the last
section (Sec.V), the major outcomes of the present study are narrated.

##  II PROBLEM STATEMENT

To compare the dynamics of the shock train and shear layer interaction, a
problem statement has been constructed using the numerical study of the
scramjet isolator along with a cavity (see Fig.1). A terminal shock in the
scramjet isolator interacts with the boundary layer and forms a shock train
(at M∞ $>$ 1.5) [1]. The increase in back pressure leads to an upstream
movement of the shock train. The current study focuses on the shock train
present upstream of the cavity (similar to the ramjet mode) and the shock
train present above the cavity location (similar to the scramjet mode). Two
different variations of the angle of the cavity front wall are used ($\theta$
= 90o and 60o), as shown in Fig.2. At pb/p∞ = 0.7, the cavity experiences
complete supersonic flow with no shock train in the isolator section. At pb/p∞
= 5.0 and 6.0, the shock train is located near and upstream of the cavity,
respectively. The present cases were simulated as cold flow without combustion
or fuel injection. A brief description of the interaction between the shock
train and the cavity shear layer is discussed in Sec.II.1.

Figure 1:  A simple schematic of shock train - cavity shear layer interaction.
(1) Shock train region (2) mixing region (3) cavity recirculation region (4)
incoming flow boundary layer thickness (5) shock train leading shock
separation region (6) diffuser section flow separation region (7) shear layer
due to separated boundary layer of shock train region (8) cavity shear layer

###  II.1 Shock train - shear flow interaction

Figure.1 shows a simple schematic of shock train flows in an isolator with a
cavity. After interacting with the incoming boundary layer, a normal shock
undergoes a bifurcation process at a freestream Mach number greater than 1.5.
The location of the shock train is controlled by the ratio of upstream to
downstream pressure (pb/p∞) and duct blockage ratio ($\beta$). Each successive
shock structure includes a short compression and re-acceleration region. As
described earlier in Sec.I, the shock train is subject to self-excited
oscillation due to local pressure perturbations. At the same time, the
supersonic flow across the cavity results in a self-sustained oscillation due
to the interaction between the shear layer and acoustic disturbances from the
cavity front wall. The combination of this low-frequency and high-frequency
oscillatory flow may correlate, or one of the two flows may dominate the flow
field.

###  II.2 Geometrical details

The geometrical details of the isolator section with cavity and downstream
diffuser are taken from the experimental work of Micka et al.[44] and Fotia et
al.[45]. The isolator section is a constant area duct with a height (H) of
25.4 mm and a length (Liso) of 402.5 mm. The cavity section is an open type
with a length (L) to depth (D) ratio [L/D] of 4 and a depth (D) of 12.7 mm.
The downstream portion of the cavity is a single-sided diffuser with a lower
wall divergence angle of 4o and a divergence section length (Ldiv) of 349.0
mm. The origin of the present geometry is considered at the leading edge of
the cavity. The geometrical details are shown in Fig.2. The measurements are
taken at various locations. The locations where the wall pressure data were
measured are labeled as ‘S’ with a point number (1$\sim$22). In addition, the
streamwise rakes ranging from -20.0 $<$ [x/D] $<$ 15.5 for [y/D] = 0.01 and
1.0 are extracted to understand the influence of the interaction between the
shock train and the shear layer.

Figure 2:  Schematic of the present numerical domain along with the
geometrical details, boundary conditions, and pressure probing stations
($S_{1}\sim S_{22}$). (not to scale)

###  II.3 Flow configuration

The present numerical simulations are performed at a freestream Mach number
(M∞) of 2.2. The boundary layer is allowed to grow from the inlet region. The
value of boundary layer thickness ($\delta$/D) is 0.24 at (x/D) = -20.0 from
the bottom wall obtained after the steady-state solution is reached. The
boundary layer thickness value is taken as the height at which the velocity
reaches 98% of the freestream velocity (u∞= 544 m/s). The back pressure ratio
[pb/p∞ = 0.7 (no shock train), 5.0, and 6.0] is changed to obtain different
flow conditions in the isolator. Other important input parameters can be found
in Table.1.

Table 1:  Numerical input parameters.  Parameters | Values
---|---
Isolator length (Liso, mm) | 402.5
Isolator height (H, mm) | 25.4
Cavity length (Lcav, mm) | 50.8
Cavity depth (D, mm) | 12.7
Boundary layer thickness ($\delta$/D) | 0.24
Inlet total pressure (P01, kPa) | 303
Outlet static pressure (pb, kPa) | 20, 142, 170
Inlet total temperature (T01, K) | 300
Mach number (M∞, at x/D = -31.7) | 2.2
Inlet Reynolds number (Re x 10-7, /m) | 2.0

##  III NUMERICAL METHODOLOGY

The interaction of the shock train with the cavity shear layer is analyzed
numerically using a commercial fluid dynamics package of ANSYS Fluent®. The
unsteady statistics of the shock train flow field and the cavity shear layer
flow are accurately estimated using higher-order numerical techniques by
direct numerical simulation (DNS) and large eddy simulation (LES). LES
requires a rigorous grid configuration near the wall to resolve the boundary
layer, which is computationally expensive. Alternatively, a hybrid Reynold’s
Averaged Navier-Stokes (RANS)/LES-based method called Detached Eddy Simulation
(DES), can resolve turbulent features in the largely separated flow with low
computational cost and greater flexibility in grid generation [59]. DES
provides a RANS-based solution within the boundary layer region and switches
to LES in the separated boundary layer flows [60, 61]. Researchers performing
simulations of cavity flow using a hybrid RANS/LES method were extensively
reviewed by Lawson et al. [62]. They concluded that DES simulation provides a
better comparison with experimental results. Therefore, in the present study,
a similar numerical methodology is applied to determine the unsteady dynamics
of the shock train and the cavity shear layer interaction in a scramjet
isolator. Moreover, the shock train flows are mainly affected by the
streamwise component or the longitudinal component along the symmetry plane
[42]. Recent numerical studies on two-dimensional cavities have provided
sufficient insight into the shear layer flows in the cavity compared to the
experimental results [55, 58, 63]. Therefore, the present numerical study
focused on the two-dimensional flow field, while the effects in the span
(i.e., lateral instability, flow separation in the corner, and sidewall
effects) were not considered.

###  III.1 Numerical procedure and boundary conditions

A two-dimensional planar domain is computed along with the cavity, as
described in Sec.II.2. The initial time-averaged solution is obtained with a
RANS-based simulation, and the near-wall turbulence is resolved by the
k-$\omega$ SST model. The spatial and temporal discretization is performed
using a second-order upwind scheme and a fully implicit density-based method.
The convective fluxes across each grid cell are evaluated using the Advection
Upstream Splitting Method (AUSM) together with the Total Variation Diminishing
(TVD) slope limiter, which uses the minmod function to limit the overshoots
and undershoots at the cell faces [64]. After a steady-state solution was
obtained, the solver proceeded to the transient simulation. To improve
accuracy, the transient formulation is chosen as the second-order implicit
scheme. The additional advantage of solving the time integration terms with
the implicit scheme is that it is unconditionally stable with respect to the
time step size. Since the unsteady simulation is initialized from a steady-
state solution, a saturation time is required to develop a transient flow in
the computational domain and to obtain self-oscillating shock train flows in a
constant area duct [43]. A detailed discussion of the saturation time can be
found in Sec.III.3.

The working fluid is assumed to be perfect air obeying the ideal gas law. The
viscosity of the fluid is a function of temperature, which is determined using
Sutherland’s law with the three-coefficient method. The transient simulation
is performed using the SST k-$\omega$ based DES model with the standard
constants given in the solver. The model equations are as follows

$\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial
x_{i}}=\frac{\partial}{\partial
x_{j}}\Bigg{[}\Bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}}\Bigg{)}\frac{\partial
k}{\partial x_{j}}\Bigg{]}+G_{k}-\Delta_{k}$ (1)

$\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho\omega
u_{i})}{\partial x_{i}}=\frac{\partial}{\partial
x_{j}}\Bigg{[}\Bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\omega}}\Bigg{)}\frac{\partial\omega}{\partial
x_{j}}\Bigg{]}+G_{\omega}-\Delta_{\omega}+D_{\omega}$ (2)

where Gk represents the generation of turbulence kinetic energy due to mean
velocity gradients. The Gω accounts for the production of $\omega$.
${\Delta_{k}}$ and ${\Delta_{\omega}}$ represent the dissipation of k and
$\omega$ due to turbulence, respectively. ${\sigma_{k}}$ and
${\sigma_{\omega}}$ are the turbulent Prandtl numbers for k and $\omega$,
respectively. For DES simulation, the dissipation of kinetic energy
(${\Delta_{k}}$) is modified. Such that

$\Delta_{k}=\rho\beta^{*}k\omega F_{DES}$ (3)

where FDES is the maximum of ratio of turbulent length scale (Lt) to the local
grid spacing ($\Delta x$, $\Delta y$) and 1.0. The turbulent length scale is
defined from the RANS model and is given as

$L_{t}=\frac{\sqrt{k}}{\beta^{*}\omega}$ (4)

Together with the above model equation, it is necessary to introduce the
solution with realistic turbulence parameters. The turbulence intensity at the
inlet and the length scale are given as 2% and 0.0254 m (i.e., duct height, H)
respectively. To achieve smooth convergence and stability, the CFL number and
under relaxation factor are set to a lower value and linearly increased with
iteration for steady-state simulation using a full multi-grid (FMG)
initialization method. The residuals of the conservation equation and
turbulence equation were brought down below 10-6 for steady-state simulations.
After reaching the steady-state solution, a transient simulation is performed.
The time step size of 1.0e-6 s was chosen based on the physical convection
time of an air molecule moving along the duct. The physical time
($\Delta_{pt}$) can be expressed as the ratio between the length of the
isolator (Liso) and the freestream velocity (u∞), which is about 7.54e-4 s
(based on Liso = 0.4025 m and u∞ = 544.0 m/s). The chosen time step size is
much smaller than the physical time to ensure the prediction of unsteady
dynamics. The inlet region of the isolator is specified as the pressure inlet,
for which the stagnation properties P01 and T01 are specified. The outlet
boundary is considered as a non-reflecting pressure outlet boundary condition
to specify the static pressure or back pressure (pb). The isolator and cavity
walls are considered as adiabatic walls with no-slip boundary conditions.

###  III.2 Grid independence study

Figure 3:  Comparison of wall static pressure (p/P∞) along the cavity wall surface (x/H) with the experimental pressure distribution of Gruber et al.[46] for the following freestream values: P0 = 690 kPa, T0 = 300 K, M∞ = 3.0. Inside figure: The streamline inside the cavity shows two large counter-rotating vortices in comparison with Gruber et al.[46] Table 2:  Grid counts Grid | Domain | Cavity | First cell | Cell growth | Toal no. of
---|---|---|---|---|---
| Nx$\times$Ny | Nx$\times$Ny | distance ($\Delta$/D) | ratio | cells
coarse | 400$\times$150 | 200$\times$100 | 4.0 x 10-4 | 1.35 | 8.0 x 104
medium | 600$\times$175 | 250$\times$140 | 2.4 x 10-4 | 1.25 | 1.5 x 105
fine | 800$\times$200 | 300$\times$160 | 8.0 x 10-5 | 1.20 | 2.0 x 105
present | 1500$\times$200 | 300$\times$150 | 8.0 x 10-5 | 1.20 | 3.5 x 105

For the purpose of a grid independence study, a two-dimensional planar open
cavity is adopted from Gruber et al [59]. The cavity depth [D] is 8.9 mm and
the length-to-depth ratio (Lcav/D) is 3.0. The grid is created using Pointwise
commercial grid generation software and the near wall region is clustered to
resolve the boundary layer flow. The accuracy of the numerical solution
depends strongly on the distribution of grid cells in the fluid domain. The
maximum equiangular skewness of the grid cells is kept below 0.3. To ensure
that the present numerical methodology is independent of the grid resolution,
three different grid levels were generated and tested. The details of the grid
number can be found in Table.2. The total number of cells in the coarse,
medium and fine grids are 8.0E4, 1.5E5, and 2.0E5, respectively. The cavity
wall static pressure is plotted along the surface length and compared with the
experimental pressure distribution (see Fig.3). The static pressure
distribution between the three grid levels does not show any difference, and
the numerical solution shows clear grid independence with medium to fine grid
levels. The deviation of the pressure distribution in comparison with
experimental values is less than 5% and the differences are mainly due to the
three-dimensionality or the uncertainties in the experiments [58]. Therefore,
a grid similar to the fine grid level is sufficient to capture the flow field
and is adopted for the present numerical cases, and the grid spatial counts
along the x and y directions are given in Table.2. In order to capture the
shock train flow and cavity shear layer, extreme care was taken in the grid
generation. For the unsteady simulation, a time step size of ($\Delta t/T$ =
1e-3) is chosen and simulated for the above cavity case. The static wall
pressure at [x/D] = 1.5 (corresponding to [s/D] = 2.5) is monitored to obtain
the dominant frequency of the cavity oscillation. The frequency spectra of the
self-excited oscillation can be estimated using the semi-empirical formula of
Rossiter [65] and Heller et al [51]. It has been modified to account for the
effect of a higher Mach number on the oscillation frequency. Fig.4 shows the
dimensionless power spectrum ($fG_{xx}$(p)/(p∞)2) compared with the modified
Rossiter formula defined by equation (Eq.5).

Figure 4:  Power spectrum obtained from wall static pressure (p/p∞) at [x/D] =
1.5 from fine grid. n is the mode number(1$\sim$5). red dashed line represents
the modes obtained from the modified Rossiter’s formula (Eq.5).

$\frac{fL_{cav}}{u_{\infty}}=\frac{n-\alpha}{M_{\infty}(1+[(\gamma-1)/2]M_{\infty}^{2})^{-1/2}+1/k}$
(5)

where f is the frequency in Hz, Lcav is the length of the cavity, n is the
mode number for the cavity oscillation from the peak frequencies, ’k’ is an
empirical constant defined by the convection velocity versus the free stream
velocity, and $\alpha$ is the phase delay. The constants k and $\alpha$ for
the validation case are 0.57 and 0.25 respectively adopted from Rossiter et
al.[65]. The values of the calculated peak frequency and that calculated from
Rossiter’s formula are tabulated in Table.3, which shows that the difference
is less than 5% for the first five modes.

Table 3:  Comparison of cavity oscillation frequency with Rossiter’s empirical formula (Eq.5) mode number | Present CFD mode | Rossiter’s mode
---|---|---
(n) | fLcav/u∞ | fLcav/u∞
1 | 0.2172 | 0.2114
2 | 0.5874 | 0.4933
3 | 0.7567 | 0.7753
4 | 1.0412 | 1.0572
5 | 1.3180 | 1.3391

###  III.3 Unsteady numerical error estimation

Figure 5:  Probed wall static pressure [(p-pmean)/p∞] measurements along the
bottom wall surface (y/H= 1.0) at S5 and S6 (i.e., x/D = 18.89 and 20.5) for
the case (pb/p∞=6.0). The flow development time is required during a numerical
run switching from steady to unsteady simulation. Self-excited oscillatory
flow is observed after the time [t/T] $\geq$ 20\. Data sampling for unsteady
statistics is carried out after [t/T] $\geq$ 30\. The maximum to minimum
pressure extent shows the shock train oscillatory motion.

Large-scale unsteady simulation is bound to accumulation of error due to
numerical discretization and can be controlled by reducing the length scale to
an infinitely small value. Smirnov et al.[66] proposed a comprehensive
methodology to sort out the grid size effect on numerical precision and error
estimation. The poor grid resolution and time step size could lead to the
accumulation of numerical error within the unsteady simulation. The detailed
method for the estimation of numerical reliability can be found in Refs. [66].
This method is well suited for the application of scramjet simulations and is
adopted by various researchers recently [38, 67, 42]. Hence in the present
work, the author evaluates the total relative error of integration by assuming
maximum allowable error limits of 5%. The reliability (Rs) is the ratio of the
maximum allowable number of time steps (nmax) to the number of simulated time
steps (nsim). From the present grid resolution, the maximum allowable number
of time steps (nmax) is 1 $\times$ 1011. To resolve the low-frequency
oscillation of the shock train, a physical time of up to 50ms is simulated.
Thus, the reliability (Rs) of the present cases is 1 $\times$ 106, which
clearly shows that the present simulation has high reliability on unsteady
simulation. The unsteady wall static pressure measurement from the present
numerical simulation probed at two different points (S5 and S6) is plotted and
shown in Fig.5. In the present numerical calculation, a stable unsteady shock
oscillatory flow is obtained for time, [t/T] $\geq$ 20\. Therefore, to resolve
the unsteady dynamics, the data obtained about 50 ms after the saturation time
([t/T] $\geq$ 30) are processed.

###  III.4 Post Processing

Post-processing is performed using MATLAB R2021b®. The Fast Fourier Transform
(FFT), two-point cross-correlation, and Dynamic Mode Decomposition (DMD)
analysis are performed to extract the transient dynamics. From the two-
dimensional DES simulation, the set of data for each node at every time step
is exported in ASCII format. Due to the large size of the data, the exported
scattered data is converted into uniform gridded data with little or no loss
of spatial information. The spatial data are extracted only for a specific
region that includes the shock train and the cavity (-20 $\leq$ [x/D] $\leq$
15 and -1 $\leq$ [y/D] $\leq$ 2) and interpolated to a coarse grid of points
of equal distance (1001 $\times$ 101) that is used for modal analysis. On the
other hand, a set of scattered point data (S1 $\sim$ S22) and two probing
rakes ([y/D] = 0.01, 1.0) are extracted and used for FFT and cross-correlation
analysis.

##  IV RESULTS AND DISCUSSIONS

Numerical analysis was carried out to study the interaction between the shock
train and the cavity shear layer in a scramjet isolator. Cases were considered
with three different pressure ratios from upstream to downstream, controlled
by changing the outlet pressure or the back pressure (see Table.4 ), and two
different cavity geometries (see Fig.2). The cases can be distinguished into
the case without shock train, shock train ends before the cavity leading edge
and shock train near the cavity location. Even with a fixed pressure ratio,
the shock train experiences a self-excited oscillation and interacts with the
cavity shear layer flow. These flow features are discussed in the following
sections.

Table 4:  Case details Cases | Back Pressure | (pb)/(p∞) | Cavity front wall
---|---|---|---
| (pb), kPa |  | angle [$\theta$], deg
case 1 | 20 | 0.7 | 90
case 1a | 142 | 5.0 | 90
case 1b | 170 | 6.0 | 90
case 2 | 20 | 0.7 | 60
case 2a | 142 | 5.0 | 60
case 2b | 170 | 6.0 | 60

###  IV.1 Steady and Unsteady statistics

Figure 6:  Time-averaged static pressure measurements from RANS simulation are
plotted along (a) bottom wall [y/D] = 0, (b) centerline [y/D] = 1.0. The solid
line represents 90o

The time-averaged static pressure along the centerline (y/D = 1.0) and the
static pressure distribution at the bottom wall extracted from the steady-
state RANS k-$\omega$ SST simulation for different cases are compared and
shown in Fig.6. The static pressure distribution for the cavity front wall
angle of 90o is shown as a solid line and the case of 60o is shown as a dotted
line. The wall static pressure comparison among various cases is shown in
Fig.6 where the x-axis is defined as the surface length, and the origin (s/D =
0) refers to the cavity leading edge point. The pressure comparison graph of
cases (1a & 1b) shows that the location of the shock train is slightly lower
than the case (2a & 2b). This is because the angle of the cavity front wall of
60o makes the exit area smaller than the 90o case. So, for the given back
pressure (pb), the shock train leading shock moves to a forward location to
match the given back pressure at the outlet. From Fig.6, it can be seen that
the shock train in cases (1a & 2a) ends before the cavity leading edge. This
corresponds to a supersonic inlet operating at ramjet mode, where the
combustion process occurs at subsonic velocity. In cases (1b & 2b), the shock
train located above the cavity shows strong interaction with the cavity shear
layer and observed distorted shock peaks. The back pressure ratio (pb/p∞ =0.7)
shows supersonic flow around the cavity in a confined flow field without shock
interactions. The centerline pressure distribution shows a weak oblique shock
originating from the front wall of the cavity.

Figure 7:  Instantaneous flow field showing (a) Mach number [M] and (b)
normalized vorticity [$\omega D/u_{\infty}$] contour. (i$\sim$vi) represents
case 1,1a,1b,2,2a,2b respectively and details shown in Table.4. Various
distinct flow features marked are (1) shear layer induced weak shock wave, (2)
cavity primary recirculation, (3) cavity secondary recirculation (4) separated
boundary layer due to shock train leading shock wave, (5) extent of shock
train and mixing layer (pseudo shock wave), (6) reflected oblique shock wave,
(7) smeared shock train due to shear layer interaction, (8) large scale
vortical structures from shear layer, (9) small scale corner vortices, (10)
interaction of counter-rotating vortices inside the cavity, (11) distorted
large scale vortical structures travels downstream (12) small scale vortical
structures in cavity recirculation region, (13) vortical structures from the
downstream separated boundary layer

Later, with the steady convergence results from the RANS simulation, DES
combined with the k-$\omega$ SST formulation is switched on. The delayed
detached eddy simulation (DDES) is chosen as the shielding function [60].
After switching to the unsteady simulation, a saturation time of [t/T] = 30 is
run to achieve a stable unsteady solution, as previously described in
Sec.II.3. In order to understand the different flow features along the
isolator with cavity flow, the instantaneous Mach contour and vorticity
contour are shown in Fig.7 along with the quiver representing the velocity
vector. Fig.7 and 7 (i, ii, iii) show the Mach contour [M] and normalized
vorticity [$\omega D/u_{\infty}$] for the case where the cavity front wall has
an angle of 90o, and (iv, v, vi) for the case of 60o. The main flow features
such as the shear-induced shock wave, the shock train, and the recirculation
region in the cavity can be clearly seen. The leading shock structure of the
shock train is more oblique rather than normal along the core region of the
flow. This is evident from the centerline pressure plot (see Fig. 6), in which
the pressure rise due to the first shock is less than that of the second shock
wave. The formation of a normal shock train takes place when the freestream
Mach number (M) $\geq$ 1.5, an oblique shock train (M $\geq$ 2.5), and at Mach
number (M) between 1.7 and 2.2 represents a transition from a normal shock
wave to an ’X’ type shock train [68]. The transition from normal to oblique
shock train may also depend on the boundary layer blockage ratio ($\beta$).
For the present case, the boundary layer blockage ratio is calculated from the
following equation (Eq.6),

$\beta=1-\Bigg{[}1-\frac{2\delta^{*}}{H}\Bigg{]}^{2}$ (6)

where $\delta^{*}$ is the thickness of the boundary layer displacement
measured at [x/D] = -20, and H is the height of the duct, giving the blockage
ratio $\beta$ = 0.15. Controlling the blockage ratio by partially removing the
boundary layer can form a normal shock train [43]. It is also observed that
the shock train near the cavity location undergoes smeared successive shock
structure with the interaction of the cavity shear layer (see Fig.7v). This is
due to the longitudinal driving force of the cavity shear layer. The Mach and
vorticity contour also reveal the presence of a large recirculation region in
the cavity with multiple counter-rotating vortices. Distinct distortions of
the large-scale vortical structures are visible and marked as 11 in Fig.7v.

###  IV.2 Shock train and shear layer oscillatory characteristics

Figure 8:  Non-dimensional static pressure [p/p∞] trace over time [t/T] along
the bottom wall surface length [s/D]. [(a)-(c)] & [(d)-(f)] shows cavity with
front wall angle is 90o and 60o respectively. I) isolator section, II) cavity
rear wall, III) cavity bottom wall, IV) cavity front wall and V) diffuser
section. Observed flow features are 1) left running wave starts from front
wall corner and ends at rear wall corner point, 2) reflected wave from the
rear wall corner point, 3) & 4) flow converging point of right running and
left running wave at rear wall and front wall corner point respectively 5)
disturbance propagating downstream in the diffuser section 6) trace of shock
train oscillation 7) disturbance propagation towards isolator section 8) trace
of separated flow in the diffuser section 9) upstream and downstream location
of shock train, 10) maximum extent of shock train from wall pressure, 11)
coalescence of right and left running wave, 12) & 13) pressure rise due to
shear layer impingement on front wall surface, 14) upstream propagation of
disturbance from separated flow in the diffuser section.

The transient features extracted from the two-dimensional DES solution for a
period of 30 $\leq$ [t/T] $\leq$ 80 (T = 1ms), correspond to a physical time
of 50ms. The extracted data from the centerline rake and several probing
stations (S1 $\sim$ S22) are arranged as a three-dimensional matrix for post-
processing, as mentioned in Sec.III.4. The space-time contour (i.e., x-t) of
the static pressure distribution on the wall can be plotted to visualize the
shock train and oscillatory motion in the cavity. This can also help to
understand the mechanism of propagation of the pressure disturbance. The
oscillation of the shock train is caused by local pressure fluctuations in the
boundary layer flow. Any small disturbance that occurs in the downstream
region propagates upstream through the subsonic boundary layer flow. These
pressure fluctuations from upstream to downstream cause the self-excited shock
train oscillatory motion. Similarly, a separated shear layer flow with large-
scale vortical structures propagates downstream from the leading edge of the
cavity and collides with the trailing edge of the cavity. This results in an
acoustic disturbance that travels in the upstream direction.

Figure 9:  Non-dimensional static pressure [p/p∞] trace over time [t/T] along
the duct centerline [i.e., y/D = 1.0]. [(a)-(c)] & [(d)-(f)] shows cavity with
front wall angle is 90o and 60o. I) isolator section, II) cavity, III)
diffuser section. Observed flow features are 1) traces of shear-induced weak
shock wave, 2) successive shock from shock train, 3) traces of oblique shock
wave from the front wall surface 4) shock train cavity shear layer interaction
5) disturbance propagating downstream in the diffuser section (right running
wave) 6) disturbance propagating upstream in the isolator section (left
running wave) 7) coalescence of right and left running wave.

Figure.8 shows the static pressure distribution along the bottom wall traced
over time [t/T], where T = 1ms. The x-axis is taken as the wall surface
coordinate system (s/D) and the origin is set at the cavity leading edge
point. The space-time contour shows the propagation of the pressure
disturbance through the slope lines labeled (1,2,5,7) in Fig.8. The slope not
only shows the disturbance propagation direction but also indicates the global
propagation velocity. Here the global represents either for the total length
of the duct or the entire flow field domain. Along the flow field, there may
be a phase change in the disturbance propagation direction which can be seen
in the subsequent section from the two-point cross-correlation method. The
comparison between the different cases clearly shows the oscillatory motion of
the cavity and the shock train. For the case without a shock train (see
Fig.8(a) & (d)), only the time period of [t/T] = 2 is shown to indicate
various flow features in detail. This also conveys the frequency of self-
sustained oscillation is larger in the cavity. Figure.8(b) and (e) show the
sinusoidal motion of the shock train referred to as self-excited oscillation.
In the case where the front wall of the cavity has an angle of 60o (case 2b),
a large amplitude of the shock train oscillation is observed. It can also be
seen that the effects of the shear flow on the shock train are smaller in
cases 1a and 2a (i.e., pb/p∞ = 6.0) than in cases 1b and 2b (i.e., pb/p∞ =
5.0). At the same time, a strong influence of the diffuser section is observed
in cases 1b and 2b due to the separated boundary layer flow. The static
pressure along the centerline (y/D = 1.0 and -20 $\leq$ x/D $\leq$ 15) is
probed and a space-time contour (x-t) similar to the wall static pressure data
is constructed and shown in Fig.9. Figure.9(b,c,e,f) clearly shows successive
shocks oscillating almost in phase with each other. Due to the interaction
with a shear layer, the successive shocks tend to smear out and form a strong
compression shock at the trailing edge of the cavity. This is also confirmed
by Mach [M] contour and the vortex contour [$\omega$ D/u∞] (see Fig.7 and 7.
Moreover, the shock train oscillatory fashion is erratic in the case of the
back pressure ratio [pb/p∞ = 5.0].

Figure 10:  Non-dimensional power spectrum [fGxx(p)/p${}_{\infty}^{2}$] contour from centerline static pressure distribution (p/p∞). The dominant frequency from Rossiter’s empirical relation is shown with the dashed red line. Table 5:  Comparison of cavity oscillation frequency (case 1) with Rossiter’s empirical formula (Eq.5) mode number | Present CFD mode | Rossiter’s mode
---|---|---
(n) | fLcav/u∞ | fLcav/u∞
2 | 0.4780 | 0.4735
3 | 0.7536 | 0.7441

To analyze the oscillatory and dominant flow mechanism in detail, a spectral
analysis using the extracted static pressure data was performed at different
locations in the computational domain. Performing Fast Fourier Transformation
(FFT) of the pressure signal using equation (Eq.7) yields the power spectral
density (PSD) [43].

$G_{xx}(p)=\lim\limits_{\Gamma\to\infty}\frac{1}{\Gamma}|F_{\Gamma}(p(t))|^{2}$
(7)

where Gxx(p) is the power spectrum (unit: pa2/Hz), the total duration of data
acquisition is given by $\Gamma$, and |FΓ(p(t))|2 is the fast Fourier
transform of the given pressure signal. The power spectra for case 1 from the
static pressure data (p/p∞) data monitored at the center of the bottom wall of
the cavity (S17, at [x/D] = 2.0 and [y/D] = -1.0) are shown in Fig.10 along
with the Rossiter’s mode (from Eq.5). The second and third modes are dominant
and are relatively close to the empirical Rossiter frequency shown in Table.5
and other modes (1,4,5,6) appear to be split into several low-amplitude
spectra. With the spatiotemporal power spectrum, it becomes easier to
visualize the peak frequency range between the different cases. The two-
dimensional power spectrum from the available static pressure distribution in
the centerline is shown in Fig.11. The x-axis is the dimensionless surface
length (s/D), the y-axis is the dimensionless frequency (fD/u∞) (i.e., the
Strouhal number), and the contour values correspond to the frequency
multiplied power spectrum [Gxx(p)] non-dimensionalized by the square of
freestream static pressure (p∞ = 28 kPa). The peak frequencies are marked and
indicated with a dashed line. Fig.11(a and d) show the PSD without shock train
for the 90o and 60o cavity front wall angle cases respectively. The peak
frequency (fD/u∞) without shock train falls in the range of 0.122 $\sim$ 0.183
for case 1 and 0.036 $\sim$ 0.247 for case 2. The corresponding dimensional
frequency [f] is $>$ 5kHz, which falls under the category of high-frequency
oscillations. These high-frequency oscillations are prone to combustion
instabilities and buzz flow in the scramjet engine. It appeared that the peak
frequency was oriented towards the trailing edge of the cavity, and a
reduction in peak frequency [fD/u∞ $\sim$ 0.036] was observed for the cavity
front wall angle of 60o (as marked in Fig.11(d)). For the cases with a shock
train, the peak frequencies are shown near the shock train locations. The high
oscillatory frequency of the cavity flow seems to be naturally controlled by
the low-frequency oscillation of the shock train. A reduction of more than 50%
in cavity oscillation is observed. This in turn provides a longer time for the
fuel-oxidizer mixing and flame stabilization process. When the shock train is
present near the cavity, a strong interaction takes place between the cavity
shear flow and shock train, which is shown earlier in the x-t contour. The
power spectrum for cases 1b and 2b shows several peak frequency components due
to the shock train and shear layer interaction. Several patches of frequency
components in the diffuser region (see Fig.11(f)) clearly indicate that the
downstream separated flow oscillates in phase with the shock train or vice-
versa.

Figure 11:  Non-dimensional power spectrum [fGxx(p)/p${}_{\infty}^{2}$]
contour from centerline static pressure distribution (p/p∞). The dominant
frequency is marked with the dashed line showing

###  IV.3 Disturbance propagation mechanism

Figure 12:  a) Two-point cross-correlation coefficient between ij, jj and jk
points b) schematic of disturbance propagation direction

The acoustic disturbance in the scramjet isolator can propagate either
upstream or downstream at a finite velocity, which can initiate a local
oscillatory flow. Such disturbances can only propagate through the subsonic
boundary layer flows[43]. Although the x-t contour shows the convection
direction of the disturbance (1,2,5,7), as marked in Fig.8, it is possible to
estimate the propagation velocity of the pressure disturbance from the slope.
But a local disturbance that also causes the self-excited oscillatory motion
cannot be predicted. To understand the propagation mechanism of such a
disturbance, the extracted static wall pressure data can be processed for a
two-point cross-correlation (Cij) given by equation (Eq.8),

Figure 13:  Spatiotemporal cross-correlation coefficient contour analyzed from
bottom wall static pressure distribution (p/p∞). The maximum cross-correlation
coefficient at each spatial node is shown as a white dotted line.

$C_{ij}(\tau)=\frac{\lim\limits_{\zeta\to\infty}\frac{1}{\zeta}\int_{0}^{\zeta}p_{i}^{{}^{\prime}}(t)p_{j}^{{}^{\prime}}(t+\tau)dt}{\left(\lim\limits_{\zeta\to\infty}\frac{1}{\zeta}\int_{0}^{\zeta}p_{i}^{{}^{\prime}2}(t)dt\right)^{1/2}\left(\lim\limits_{\zeta\to\infty}\frac{1}{\zeta}\int_{0}^{\zeta}p_{j}^{{}^{\prime}2}(t)dt\right)^{1/2}}$
(8)

where ’i’ is the location cross-correlated with the reference location (’j’),
$\tau$ is the time delay between (i-j), $\zeta$ is the total duration of data
sampling time, pi and pj are the corresponding pressure-time histories at i
and j, respectively. In the present computational analysis, the cross-
correlation coefficients (Cij) are calculated with respect to the reference
point (j) and considered at the mid-location of the cavity bottom wall (i.e.,
[s/D] = 3.0). Figure.12 shows an example of the cross-correlation between the
leading-edge point and the trailing edge point with respect to the mid-point
of the cavity bottom wall. The delay time between two sampling points directly
indicates the propagation direction. In the present analysis, a positive time
delay means that the disturbance propagates upstream, and a negative time
delay means that the disturbance propagates downstream. For the same point,
the correlation of a signal is an autocorrelation and shows a time delay of
zero. Plotting two-dimensional cross-correlation coefficients at each point
along the bottom wall can clearly see the phase shift of the local disturbance
convection direction. Fig.13 shows the spatiotemporal distribution of the
cross-correlation coefficient with respect to the cavity bottom wall (at [s/D]
= 3.0). The y-axis shows the time lag of each spatial coordinate along the
bottom wall with the reference point. The white dotted line indicates the time
lag [$\tau$/T] of the peak cross-correlation coefficient [Cij]. As mentioned
earlier, the positive value of the time lag [$\tau$/T] indicates that the
disturbance propagates in the upstream direction and a negative time lag
[$\tau$/T] represents the disturbance propagates downstream. The black dashed
vertical line (at [s/D] =3.0) indicates the reference location from which the
cross-correlation coefficient is calculated. It can be seen that the
propagation of the disturbance is not a linear smooth change either upstream
or downstream. Fig.13 clearly shows a local phase shift in the propagation
direction of the disturbance. This is due to the interaction of counter-
rotating vortices in the separated boundary layer, recirculating vortices in
the cavity approaching the corner point, and the impingement of the shear
layer on the front wall or trailing edge point. At these points, a small-scale
vortex formed, triggering a local self-sustaining oscillation. The present
cases also show a small-scale disturbance marked as 5 in Fig.13. This can be
solved using higher-order numerical techniques such as direct numerical
simulation (DNS). The averaged propagation of the disturbance is indicated by
a black dashed arrow line, which may be the slope line of the x-t contour. A
sudden rise and dip in the time delay are due to the separation and
reattachment of the boundary layer. Fig.13(a,b,d,e) shows the propagation of
the disturbance in both directions with respect to the reference point. In the
presence of a shock near the cavity (cases 1b and 2b), the disturbance
propagates only in the upstream direction. This is due to the strong influence
of the separated flow in the diffuser region.

###  IV.4 Modal analysis

Figure 14:  Comparison of kinetic energy spectra [$\sigma_{j}$] and cumulative
energy fraction for different cases with respect to POD modes. ( for cases
details refer Table.4

Proper orthogonal decomposition (POD) is a technique for determining the
orthonormal basis from a set of input data. Further, DMD analysis can provide
sufficient information about dominant spatiotemporal modes between the shock
train and shear layer interaction based on their frequency spectra [69]. In
the present analysis, the spatial POD and DMD algorithms of Tu .et.al [70] are
used to identify the coherent structures based on their energy and frequency
content. For the analysis, a set of velocity fields extracted from the fluid
domain at each time step is reconstructed with an equidistance matrix of 1001
$\times$ 101 for a physical time of 50 ms. Later it is stacked in as a data
matrix of 1000 snapshots with $\Delta$t = 5e-5 s. The eigenvalues obtained
from the POD analysis are directly proportional to the total kinetic energy
(TKE) of each POD mode. Then the eigenvalues are sorted in the order of
highest to lowest to ensure that the most important modes consist of large-
scale flow structures. The calculated kinetic energy associated with each mode
number (n) is plotted in Fig.14 for 1000 modes. It can be seen that the total
kinetic energy is retained within the first 10 modes itself. For cases (1, 2,
and 2b) at least five main modes are required to associate more than 50% of
the total kinetic energy (Fig.14). For case 2a, the flow field within the
first five modes can be convective because it contains more than 90% of the
TKE. At the same time, more than 100 modes are required to capture 90% of the
TKE for case 1.

Figure 15:  Normalized temporal spectra from DMD analysis (fΘD/u∞ vs
$||\alpha_{\Theta}||$) for different cases (in the order of 12,1a,2a,1b,2b
refer Table.4). Discrete peak frequencies marked as I, II, III, IV are taken
for DMD modes visualization

Fig.15 shows normalized temporal spectra from DMD analysis (fΘD/u∞ vs
$||\alpha_{\Theta}||$) for different cases (in the order 1,2,1a,2a,1b,2b). In
the cases without the shock train, the cavity flow experiences high-frequency
oscillation, and in the other cases with the shock train shows the DMD
spectrum falls in the frequency range [fΘD/u∞] of 0.05. In the 90o cavity
case, the dominant frequency corresponding to n = 1,2,3,4 is observed and is
close to the Rossiter mode, as shown in the power spectral density plot
(Fig.10). In the case of the 60o cavity, the peak frequencies (II and IV)
marked in Fig.15(b) is clearly visible in the spectral contour (fD/u∞ $\sim$
0.036 and 0.247). The mode number (n) corresponding to 1, 5, and 6 is
dominant. The discrete peak for frequency [fΘD/u∞] $<$ 0.05 is due to the
oblique shock wave formed at the front wall, and higher frequencies are due to
the shear layer interaction. For cases (1a, 2a, 1b, and 2b), the most dominant
frequency was observed near $\sim$ 0.0018. The corresponding frequency is in
the limit of low frequencies caused by shock train oscillation (f $\sim$ 200
Hz) [20, 71, 42]. In the case of the 90o cavity (case 1a), a peak frequency
was observed near $\sim$ 0.028 (marked as III) in Fig.15(c), which also occurs
in all other cases (marked as IV) with shock train flows.

Figure 16:  Normalized spatial DMD modes [$\Theta$] corresponding to the
velocity field. Marked labels are given in the description.

Figure.16 shows the DMD modes in the flow field from discrete DMD peaks
(labeled as I, II, III, IV in Fig.15). The corresponding frequency value is
noted, and the contour values are scaled between -1.0 and 1.0, along with the
quiver plot of the velocity field. For cases 1 and 2, mode (I) shows a strong
vortical structure (marked as 1) inside the cavity. These vortical structures
are very similar to the secondary vortices inside the cavity (see Fig.7). The
most dominant mode seems to emanate from the secondary vortices. In case 1,
modes II and III show the coherent structures (marked as 2) originating from
the shear-induced weak shocks. In the fourth mode (IV), the coherent
structures (marked as 2) convect downstream to the front wall (marked as 4) of
the cavity and form a new structure (marked as 3) near the rear wall of the
cavity. Similarly, in case 2, the dominant structure appears to convect
downstream (Fig.16(b)). An oblique and reflected shock structure is also
clearly visible in all four modes presented here (I $\sim$ IV), and in mode
(IV) the shear layer interacts with the weak compression waves downstream of
the cavity. In cases 1a and 2a, the discrete dominant modes are due to the
shock train flow field. The fourth mode (IV) shows a weak coherent structure
in the diffuser section. In the case of a 90o cavity, the modes (I and II)
appear to be opposite in nature, while the coherent structure of the shock
train in the modes (II and III) is exactly the same. The structure of the
shock train (marked as 7) appears to be the same except for its length in case
2a (Fig.16(d)). When the shock train is present close to the cavity, the
interaction with the shear layer is predominant. Case 1b shows that the
coherent structure from the shock train strongly correlates with the cavity
flow field (marked as 9) and the pairing of coherent structures between the
shock train and cavity recirculation flow is observed. These coherent
structures moved downstream with subsequent dominant modes. In the case of the
60o cavity, the coherent structures are smeared out due to the large amplitude
of shock train oscillation. In both case 1b and case 2b, large vortical
structures can be seen and convecting towards the upstream direction. These
structures strongly induce the oscillatory motion of the shock train.

###  IV.5 Limitations and Future Scope

In the present study, a two-dimensional flow field in a scramjet isolator with
a cavity was investigated. The findings are related to the interaction between
the shock train- cavity shear layer, where the flow is mainly dominated by
longitudinal oscillatory motions. In an actual flow field, the effects of
sidewalls and flow separation from corners can cause lateral oscillations. The
present results may change relatively in the dominant spectral values due to
the inclusion of sidewalls, fuel injection, combustion, and flame propagation.
Therefore, it is necessary to extend such studies with high-fidelity
calculations along with parametric studies that include different cavity
sizes, free stream Mach number, and fuel injection. Such studies will be
conducted to characterize the flow field and left as a future research scope.

##  V CONCLUSIONS

The interaction of the shock train with the cavity shear layer in a scramjet
isolator is analyzed numerically using a commercial flow solver. To resolve
the unsteady dynamics of the oscillating flow field, the hybrid RANS/LES-based
methodology is applied. The effect of the position of the shock train and the
angle of the cavity front wall is investigated. The position of the shock
train in the isolator is achieved by changing the back pressure ratio (pb/p∞ =
5.0 and 6.0). The summary of the main results is presented below.

  1. 1.

The time-averaged static pressure distribution along the centerline (y/D =
1.0) shows that the maximum pressure increase due to the first shock is less
than that of the second shock. The structure of the first shock appears to be
oblique because the freestream Mach number range falls in the category of
transition between normal and oblique shock train. The transition from normal
to oblique shock train may occur due to the duct blockage ratio.

  2. 2.

In the space-time contour, the propagation of the pressure perturbations can
be clearly seen. The pressure fluctuations caused by the cavity propagate in
both upstream and downstream directions. The successive shock structure and
its oscillatory motion are distinctly visible from the spatiotemporal pressure
distribution along the centerline. The strong interaction between the shock
train and the cavity shear layer leads to an irregular oscillation at a
backpressure ratio (pb/p∞) of 5.0. At a backpressure ratio of 6.0, however,
the oscillation of the shock train is sinusoidal.

  3. 3.

The power spectrum of the cavity flow without shock train case (pb/p∞ = 0.7)
shows a high-frequency oscillation. The peak frequencies are compared with
Rossiter’s mode for the angle ($\theta$) of the cavity front wall of 90o. It
is possible to identify the first five modes, with modes (n = 2 and 3) being
dominant and correlating well with Rossiter’s mode. The spatiotemporal power
spectrum clearly shows that the low-frequency high-amplitude shock train flow
is dominant. The frequency of cavity flow oscillation is reduced by more than
50%. The interaction between the shock train and cavity flow tend to reduces
the cavity oscillation frequency which can help in fuel-oxidizer mixing and
flame stabilization.

  4. 4.

Through the spatiotemporal cross-correlation coefficient, the time delay
between the pressure disturbance shows a local phase shift in the disturbance
propagation direction. These local disturbances initiate fluid flow
instabilities.

  5. 5.

Modal analysis is carried out by using proper orthogonal decomposition and
dynamic mode decomposition (DMD) methods. The cumulative energy fraction shows
that more than 100 modes are required to capture 90% of the dominant flow
structures for case 1. In cases (1a and 2a), where the shock train occurs well
upstream of the cavity, only 10 modes are required. In cases (1b and 2b), 25
modes are required to capture 90% of the dominant flow field structures. The
cases with large modes requirement are due to the flow consisting of various
ranges of turbulent length scales.

  6. 6.

Using dynamic mode decomposition (DMD), the spectra of several discrete
dominant modes are analyzed. DMD analysis of the case with the back pressure
ratio of 0.7, shows the secondary vortices inside the cavity are dominant.
While the back pressure ratio of 5.0 and 6.0, the peak dominant mode is due to
the shock train flow field. A strong similarity and pairing of coherent
structures exist between the shock train and the cavity flow for the back
pressure ratio of 5.0.

## ACKNOWLEDGMENTS

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korean government (MSIT) (No. NRF-2020R1F1A1075716). The
first author would like to thank Dr.S.K.Karthick and Jintu K James for
providing insight on computations and post-processing techniques.

## DATA AVAILABILITY STATEMENT

The data that supports the findings of this study are available from the
corresponding author upon reasonable request.

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