JOURNAL OF PROPULSION AND POWER
Vol. 37, No. 4, July–August 2021
Computational Analysis of Unstart in Variable-Geometry Inlet
Jonathan P. Reardon∗
Lockheed Martin Aeronautics Company, Palmdale, California 93599
and
Joseph A. Schetz† and Kevin Todd Lowe‡
Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0203
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
https://doi.org/10.2514/1.B38214
Computational fluid dynamics was used to study the flow through a scaled, mixed-compression, high-speed inlet with a
rotating cowl at Mach 4.0 conditions. First, steady Reynolds-averaged Navier–Stokes computations were undertaken
with a range of popular turbulence models including the Spalart–Allmaras model, the realizable k–ε model, the cubic k–ε
model, and the Menter shear stress transport (SST) model to assess the impact on the inlet operating state. It was found
that two models, the Spalart–Allmaras model and the Menter SST model, predicted the inlet to become unstarted at a
cowl angle where the experimental data indicated the inlet remained started. Next, steady-state flow structures were
studied at three discrete cowl positions, identifying highly three-dimensional flow features including regions of separated
flow and spanwise gradients that became stronger as the cowl opened. Finally, the study culminated with the
development of the new transient model, which allowed for time-accurate investigation into the unstart, restart, and
hysteresis. The evolution of the separation bubbles was shown to be a major factor in the hysteresis, causing the inlet to
restart at an angle different from where it unstarted. The utility of unsteady Reynolds-averaged Navier–Stokes
computations to capture the complex time-dependent details of such flows was demonstrated.
τw
ω
ωc
Nomenclature
a
cf
H
Hm
Hth
k
L
Lc
Lr
M
P
T
u
u
u
x
y
y
z
α
β
δ
δ
ε
θ
θc
μ
ρ
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
speed of sound, ft/s
skin friction coefficient
shape factor
inlet model height, in.
inlet throat height, in.
turbulent kinetic energy, ft2 ∕s2
inlet isolator length, in.
inlet cowl length, in.
inlet ramp length, in.
Mach number
pressure, psf
temperature, R
velocity, ft/s
friction velocity, ft/s
nondimensional wall velocity
axial coordinate, ft
normal coordinate, ft
nondimensional wall distance
spanwise coordinate, ft
inlet ramp angle, deg
cowl convergence angle, deg
boundary-layer thickness, in.
displacement thickness, in.
turbulent dissipation rate, ft2 ∕s3
momentum thickness, in.
cowl angle, deg
dynamic viscosity, slug∕ft ⋅ s
density, slug∕ft3
=
=
=
wall shear stress, psf
specific dissipation rate, s−1
cowl rotation rate, rad/s
Subscripts
t
w
1
=
=
=
stagnation quantity
evaluated at wall
freestream condition
I.
T
Introduction
O REACH hypersonic velocities, the development of ramjet and
scramjet engines has been pursued for many years [1,2]. However, a single, fixed ramjet/scramjet geometry is still not suitable for
the large range of Mach numbers that are traversed by proposed
hypersonic vehicles [1]. One solution to this problem is the introduction of variable geometry that allows for tailoring of the engine
flowpath to the local flight conditions, thus improving efficiency.
Associated with these concepts is an array of technical challenges. At
the forefront of these challenges is the inlet unstart phenomenon. An
inlet is considered started when the shock system is contained within
the inlet, and the inlet becomes unstarted if the shock train is ejected
out of the inlet [3]. This can occur for many reasons including
increases in backpressure or, for a variable geometry inlet, too great
a change in the contraction ratio. The unstart process is a very violent
and rapid transient process that significantly reduces the engine
performance and can be detrimental to the vehicle by causing combustion to be interrupted [4]. Therefore, understanding the unstart
process and having the ability to predict the transient flowfield during
the unstart is of great importance to ramjet and scramjet development.
Computational fluid dynamics (CFD) has played a growing role in
the design and analysis process of aerospace vehicles [5,6]. Accordingly, there has been significant work on using CFD to analyze ramjet
and scramjet engines and components. CFD is vital for the analysis of
high-speed inlets because of the detailed flowfield calculations that
are possible, even in complex geometries. It also allows for study at
conditions that may not be achievable in ground test facilities [7] and
for data acquisition that may be difficult to obtain with experimental
measurements.
Commonly, to analyze a variable geometry inlet with CFD, a
quasi-steady approach is used. This approach uses a series of steadystate solutions at various geometric configurations that the inlet may
take and is based on the fact that the local flow time scale is often
Received 14 August 2020; revision received 24 November 2020; accepted
for publication 30 November 2020; published online Open Access 20 January
2021. Copyright © 2021 by Lockheed Martin Corporation. Published by
the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3876 to
initiate your request. See also AIAA Rights and Permissions www.aiaa.org/
randp.
*Aeronautical Engineer Senior, Advanced Development Programs.
Member AIAA.
†
Holder of the Fred D. Durham Chair, Department of Aerospace and Ocean
Engineering. Lifetime Fellow AIAA.
‡
Associate Professor, Department of Aerospace and Ocean Engineering.
Associate Fellow AIAA.
564
565
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
REARDON, SCHETZ, AND LOWE
much smaller than the time scale of the motion of the variable
geometry component. However, this approach does not provide a
means for capturing the actual transients during the unstart and restart
nor the hysteresis as the inlet progresses from one state to the next.
Now, with advances in high-performance computing resources,
time-accurate solutions to the unsteady Navier–Stokes equations are
becoming more common. For example, Neaves et al. [8] analyzed the
same inlet considered in the current Paper focusing on the transient
response of the inlet due to unstart that was brought about by backpressure through the closing of a throttling device. Additionally,
steady-state and transient CFD calculations in two and three dimensions were conducted by Hoeger [9] to characterize the effect of
backpressure on the shock train development and unstart in a separate
rectangular inlet/isolator configuration. Riley et al. [10,11] and
Yentsch and Gaitonde [12,13] studied inlet unstart and mode transition of dual-mode ramjets with fixed geometry using transient CFD
solutions.
Despite the advances made in the previously mentioned works which
focused on fixed geometry inlets, the transient analysis of variable
geometry inlets using CFD has received less attention. One example
is the work of Grainger et al. [14], in which a two-dimensional, transient
CFD analysis with overset meshes was used to analyze the starting
characteristics of an experimentally tested inlet with a sliding door
geometry. Also, Xianhong et al. [15] used a similar modeling approach
to investigate the mode transition of a turbine-based combined cycle
engine.
The current Paper seeks to extend the use of CFD in analyzing
variable geometry inlets to include the transient response of an inlet to
a rotating cowl. The experimental rectangular inlet from Ref. [16] that
includes a rotating cowl was chosen as a model problem and analyzed
using quasi-steady and transient computational methods. The first
goal of the current Paper was to develop a high-fidelity computational
model that allowed for a time-accurate prediction of inlet performance in the presence of variable geometry. The second goal was to
then use this model to gain insights into the inherently transient
processes of unstart, restart, and hysteresis. The approach seeks to
incorporate unsteady, Reynolds-averaged, Navier–Stokes solutions
coupled with relative motion through the use of an overset mesh
approach.
Section II will briefly describe the inlet to be studied, and Sec. III
will describe the computational analysis approach to be taken. Next,
the analysis approach will be validated and verified in Sec. IV using
established techniques for scientific computing. Following this, the
main results will be presented in Sec. V. To start, quasi-steady
analysis results are shown and compared with experimental data
from Ref. [16]. These results were obtained to assess the impact of
an array of turbulence models on the computed inlet performance, to
study the flowfield structure in the inlet at three discrete cowl angles,
and to act as baselines for comparisons made with the subsequent
transient results. Then, the transient model is used to study the timedependent aspects of the inlet unstart and restart processes as well as
the associated hysteresis due to the motion of a rotating cowl. These
results allow for time-accurate interrogations of the flowfield as the
flow adjusts to the change in position of the rotating cowl. Finally,
conclusions are drawn in Sec. VI.
II.
Overview of Experimental Case Studied
In the current work, the inlet configuration studied experimentally
in Ref. [16] is analyzed using CFD to assess its performance in
capturing the flowfield in a variable geometry inlet. Reference [16]
is an extensive parametric study of a combination of inlet cowl and
isolator arrangements that consisted of 250 configurations [16].
The geometries tested were at 2% scale of a generic hypersonic
vehicle. The inlet was planar and consisted of a compression ramp, a
rotating cowl, side walls, an isolator section, and a diffuser section as
seen in Fig. 1. The width of the inlet was 2.0 in., and the throat height
was 0.4 in., which led to an aspect ratio of 5.0 in the throat/isolator
section. The inlet ramp was 9.77 in. long with an 11 deg angle from
the horizontal to provide compression. There were three different
cowls tested: the long cowl was 4.4 in. in length (Lc ∕H th 11.00),
the medium cowl was 3.9 in. in length (Lc ∕Hth 9.75), and the short
cowl was 2.5 in. in length (Lc ∕Hth 6.25). All configurations
consisted of mixed external and internal compression. The cowls
were actuated so that they could be rotated about a hinge, and the cowl
angle from the horizontal was defined as θc . Another angle defining
the cowl rotational position was the cowl convergence angle β,
defined as the angle between the cowl and the inlet ramp. Because
the inlet ramp angle was fixed at 11 deg, the geometric relationship
between the cowl angle (from the horizontal) and the cowl convergence angle was β 11 deg −θc . The cowls could be rotated
between angles of θc 0 deg (horizontal) to 11 deg (parallel to
ramp). Following the throat and isolator section was a 20 deg diffuser
ramp that led into a constant area rectangular duct. At the end of the
Fig. 1 Schematic of the rectangular scramjet inlet used in the current paper from Ref. [16].
566
REARDON, SCHETZ, AND LOWE
duct was a throttling device that could be used to backpressure
the inlet.
In addition to different cowl configurations, the study also tested
various isolator designs and lengths. First, a simple constant area
isolator was used. Next, an isolator with a constant area section
followed by a 6 deg expansion was used. The isolator lengths were
1.08 in. (L∕H th 2.7), 1.88 in. (L∕H th 4.7), 3.48 in. (L∕H th 8.7), and 6.68 in. (L∕H th 16.7). Each cowl and isolator was also
tested with and without rearward-facing steps to be representative of
fuel injector features. The steps were not included in the current
computations, so they will not be addressed further here.
The inlet was instrumented with 110 wall static pressure taps on the
inlet ramp, side walls, cowl, and isolator. The effect of the boundarylayer thickness was obtained by testing with and without at a 12 in.
foreplate positioned ahead of the inlet ramp. The experimental work
was conducted in the NASA Langley Research Center Mach 4 Blowdown Facility. The evaluation of the inlet was conducted at nominal test
conditions of Pt 200 2 psia and M 4.03 0.02.
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
III.
Fig. 3 Computational mesh of the inlet of Ref. [16] used for the quasisteady model.
Analysis Approach
To conduct the analysis of the inlet flow features in the variable
geometry inlet case, a computational study was performed using the
commercial CFD code, CFD++, provided by Metacomp Technologies. The code is capable of solving incompressible and compressible
flows with inviscid, laminar, or turbulent options. It also has additional capabilities to include chemical reactions. It can compute
solutions on both structured and unstructured meshes and can accommodate overset meshes, which will be used in the transient work.
A. Computational Domain and Mesh for Quasi-Steady Model
For the quasi-steady analysis, computational meshes were created
for three distinct cowl angles, θc 8, 1.2, and 0 deg (β 3; 9.8; 11 deg). The computational domain and boundary conditions
for the 8 deg model are shown in Fig. 2. All computations were
conducted with the long cowl (Lc ∕H th 11.0), with no inlet foreplate (thin boundary layer), and with the short isolator length
(L∕H th 2.7). The long cowl was chosen because of the longer
shock train it developed before the throat, and the short isolator length
was chosen to minimize the size of the computational mesh needed.
The computational domain included the full three-dimensional
geometry of the inlet; symmetry was not imposed at the inlet centerline because when the inlet becomes unstarted the flowfield is asymmetric due to large separated flow regions.
The inflow boundary is placed ahead of the compression ramp so
as to avoid a singularity where the inflow meets the viscous wall.
Before the compression ramp and along the sides of the ramp are
boundaries used to guide the flow into the inlet such that no spillage
occurs. The inlet side walls begin at x∕H th 11.8. The leading edges
of the side walls are accurately modeled to the 0.005 in. radius of the
Fig. 2 Computational domain of the inlet of Ref. [16] developed for the
quasi-steady model.
test geometry. The cowl lower surface is fully modeled; however, the
upper surface was truncated. This was because the current area of
focus was the flow through the inlet. The cowl extends the full width
of the inlet and joins with the side walls so that no flow passes
between the cowl and side walls.
The grids used were created in Gridgen. The mesh was fully
structured with clustering at the leading edges and no-slip walls to
resolve the large gradients due to the shocks and the growth of the
boundary layer. Details of the grid are shown in Fig. 3. The first cell
height above the no-slip walls was set to achieve a y < 1 along the
ramp
an iterative approach (y ≡ ρw yu ∕μw , where u ≡
pusing
τw ∕ρ is the friction velocity [17]). This iterative approach included
computing a steady-state flow solution at the desired flow condition,
assessing the y value, reducing the first cell height, and repeating the
process until the y was less than 1 on the inlet ramp. The cell spacing
grew from this first cell height using a hyperbolic tangent distribution
function. The 8 deg cowl grid was created using 13 structured blocks
with a total of 8,878,023 cells. The 1.2 and 0 deg cowl grids were
created using 12 structured blocks with a total of 8,560,223 cells.
Additional details about the grids are given in Ref. [18].
B. Computational Domain and Mesh for Transient Model
The computational domain for the transient model was developed
in a manner similar to what was done for the quasi-steady model. The
main difference between the computational domains for the two
models was how the cowl was incorporated. For the transient model,
the full cowl was modeled, both the upper and lower surfaces. In
addition, surrounding the cowl was a flowthrough boundary to allow
for communication between the two grids when using the overset
Fig. 4 Computational domain of the inlet of Ref. [16] developed for the
transient model.
REARDON, SCHETZ, AND LOWE
Fig. 5 Computational mesh of the inlet of Ref. [16] used for the transient
model.
567
computing the viscous effects on the fences and modeling the gap
between the fence and inlet side walls.
All walls of the inlet and the cowl were set to viscous, adiabatic
wall boundary conditions. The use of an adiabatic wall was justified
in the fact that to obtain the velocity and Mach number profiles the
experimental data were processed assuming an adiabatic wall. This
will be further discussed in a later section. The outflow plane at the
end of the constant area duct and at the end of the domain at the upper
surface of the cowl were set to supersonic outflow boundary conditions. Note that, by applying a supersonic outflow at the end of the
constant area duct, the effect of backpressure was neglected. This was
done to isolate the effects of the cowl on the unstart process. This is
consistent with the experimental data of Ref. [16] that was chosen for
comparisons which was obtained when the throttling device was fully
open. For the transient model with the overset cowl, the flowthrough
boundary surrounding the cowl was assigned a patched and overset
boundary condition to allow for interpolation of the flow variables
between the overset cowl and background body-side grids.
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
D. Turbulence Models
mesh approach. See Fig. 4. In this model, a small gap of 0.001 in. on
either side of the cowl was included to allow for the cowl to move
independently of the inlet ramp and side walls. This gap is consistent
with the experimental model dimensions provided in Ref. [16].
Fully structured grids were created in Gridgen for both the inlet
body and cowl. To allow for relative motion of the cowl, an overset
mesh approach was used. The cowl and body-side portions of the
inlet were meshed separately and then combined in CFD++ where the
cutting and blanking process was conducted to remove cells from
the interiors of the solid bodies. Additional details on the cutting and
blanking process in CFD++ can be found in Ref. [19]. The two grids
yielded a total of 12,418,317 cells: 10,534,944 cells for the 12-block
body-side grid and 1,883,373 cells for the 37-block cowl grid.
See Fig. 5.
C. Numerical Setup and Boundary Conditions
To conduct the quasi-steady simulations, the equations for viscous,
compressible flow, the Navier–Stokes equations, were solved using a
steady-state approach at different cowl angles. Local time-stepping
was used in conjunction with a multigrid convergence accelerator.
Temporal smoothing was used so that after each iteration the flow
solution was updated by 75% of the computed solution at the next
step. Spatial discretization was blended from first order initially to
allow for the solution to develop to second order. The Courant–
Friedrichs–Lewy number was also ramped to facilitate the development of the flowfield from the freestream initial conditions. Finally,
the fluxes were evaluated using the Harten-Lax-van-Leer-Contact
(HLLC) method, and a continuous Total Variation Diminishing
(TVD) limiter was used. For brevity, the reader is directed to Ref. [18]
for the formulation of the Navier–Stokes equations used in this Paper.
For the transient model, time-accurate solutions were obtained
using an implicit, dual-time-stepping approach. At each time step,
a specified number of subiterations was conducted to converge the
flow solution before moving to the next time step. The baseline time
step was chosen to be 10 μs to balance computational time while
accurately capturing the transient physics to be studied. The impact of
time step on the computed solutions will be shown later.
The inflow boundaries were set to characteristics-based inflow/
outflow boundary conditions where the freestream temperature,
pressure, and velocity were specified. The sides of the computational
domain ahead of the inlet side walls were modeled as inviscid wall
boundary conditions to simulate the effect of the fences that were
used in the experimental setup and that are visible in Fig. 1. As noted
in Ref. [16], the purpose of the fences was to contain the ramp shock
and to prevent spanwise flow spillage from occurring. Gaps between
the fences and the inlet side walls were used to remove the fence
boundary layer so that it would not enter the inlet [16]. The inviscid
wall boundary conditions in the computational setup were used to
simulate this effect without the additional computational expense of
Because of the large range in time and spatial scales of turbulent
flow, a Reynolds-averaged Navier–Stokes (RANS) approach was
taken. However, no single turbulence model will be appropriate for
every flow [20]. Therefore, one of the major goals of this work was to
assess various RANS turbulence models and compare the computed
results with the experimental data. It was desired to conduct this study
using the quasi-steady model due to the increased computational cost
of the transient model.
A range of turbulence models was used to provide closure for the
turbulent correlation terms. The one-equation model considered was
the Spalart–Allmaras model, while three two-equation models were
considered: the Menter SST model, the realizable k–ε model, and the
cubic k–ε model. These models were chosen because of their widespread use in the CFD community and their applicability for the
current problem of study. The Spalart–Allmaras, Menter SST, and
realizable k–ε models all follow the Boussinesq approximation,
which equates the turbulent Reynolds stresses to the mean flow strain
rate linearly through an eddy viscosity coefficient. The cubic k–ε
model adds in additional higher-order, nonlinear terms to the Boussinesq approximation to account for anisotropy in the normal Reynolds stresses and streamline curvature effects [11]. Because of the
number of models used, the exact formulations as implemented in
CFD++ are not reproduced here but can be found in Refs. [18,19],
which are readily accessible to the reader online.
E. Flow Conditions
All computations were conducted at the nominal wind tunnel
condition: P 182.26 psf, T 124.76R, u 2206.61 ft∕s, and
M 4.03. Using the length of the inlet ramp leading edge to the
throat as a characteristic length (9.77 in.) and the freestream velocity,
a characteristic flow through time of the inlet can be calculated as
0.37 ms. This time scale can be used to relate the flow residence time
to the scale of inlet motion applied in the transient model. This will be
presented in a later section.
The computational domain was initialized to these conditions to
begin the solution. In all computations, air was modeled as a real gas
allowing for temperature -dependent specific heats and viscosity.
Turbulence variables were initialized to represent wind tunnel conditions by assuming a freestream turbulence level of 0.03 and a
turbulent to laminar viscosity ratio of 50, except for the Spalart–
Allmaras model, which used a value of 5 [19]. This was due to the fact
that the Spalart–Allmaras model does not include a destruction term
in the model [21]. The turbulence intensity value chosen is consistent
with the range investigated in Ref. [22], which cites an upper value of
about 10% to be representative of wind tunnel flow with screens that
are used to generate turbulence. It should be noted that the exact value
for these variables is still a matter of active research as these quantities, particularly the turbulent to laminar viscosity, are difficult to
measure [22]. Finally, for the transient results, steady-state solutions
568
REARDON, SCHETZ, AND LOWE
were first obtained and used as initial conditions for the beginning of
the time-accurate computations.
IV.
Verification and Validation
Because of the new analysis approach used for the transient model,
established verification and validation methods were applied to the
current problem. Errors and uncertainties due to spatial discretization
error, temporal discretization error, and iterative error were assessed
and are presented in the following sections. The spatial discretization
error was conducted using the transient model (overset cowl grid) in
the steady-state limit. The temporal error was conducted for the
transient model on a single grid with the time-dependent cowl rotational motion applied. Finally, the iterative error assessment was
conducted on the transient problem using a single grid and single
time step with varying levels of subiterations. This approach was
taken so that the spatial, temporal, and iterative errors could be
assessed individually.
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
A. Spatial Discretization Error
The traditional method of assessing spatial discretization error is to
conduct a grid convergence study. In this Paper, the grid convergence
study was conducted using the transient model (overset cowl model)
in the steady-state limit. The effect of temporal error is presented in
the following section. A series of three grids was created, coarse,
medium (baseline), and fine, such that each refined grid essentially
doubled the number of cells in all three coordinate directions. The
number of cells in each grid along with the effective refinement ratio,
defined as the cube root of the ratio of the number of cells of the
current grid to previous grid, rij N i ∕N j 1∕3 , is shown in Table 1.
These three grids were used to compute the steady-state flow
solution with the cowl fixed at an angle of 8 deg below the horizontal.
To assess grid convergence, this study initially used the mass flow
rate as the parameter of study as shown in Table 2. With these three
solutions, the observed order of accuracy could be computed. The
refinement factors were not exactly 2.0 due to the complex topology
of the structured grids. The observed order of accuracy p was
computed from the relationship adapted from Refs. [6,23] for nonuniform refinement factors,
f3 − f 2
f − f1
rp12 2p
(1)
p
r23 − 1
r12 − 1
where f1 , f2 , and f3 correspond to the mass flow rate on the fine,
medium, and coarse grids, respectively. The observed order of accuracy was found to be p 0.996. Although a second-order accurate
spatial scheme was used, the order of accuracy was essentially first
order. This was found to be consistent with the work of Ref. [24],
which found that the order of accuracy in an isolator flow solution
was first order despite the use of a second-order scheme. The cause of
this was stated to be due to the use of the TVD flux limiter [24].
Table 1
Grid characteristics used to assess
spatial discretization error
Grid
No. of cells Refinement ratio
Coarse (No. 3)
1,450,113
——
Medium (No. 2) 12,418,317
r23 2.05
Fine (No. 1)
142,978,764
r12 2.25
Table 2
Mass flow rate through the inlet for each grid
Grid
Coarse (No. 3)
Medium (No. 2)
Fine (No. 1)
Exact (Richardson)
Mass flow rate, lbm∕s
RDE, %
0.939609
0.945980
0.949358
0.952077
−1.31
−0.64
−0.29
——
Table 3
Spatial GCI results
Grid
Coarse, GCI23
Fine, GCI12
rp12 GCI12
GCI, %
1.93
0.86
1.94
The exact mass flow rate could then be calculated using the
Richardson extrapolation [6],
fexact f1 f 1 − f2
rp − 1
(2)
The exact mass flow rate was found to be 0.952077 lbm∕s. The
results are also displayed as a percent difference from the exact
solution by creating the relative discretization error (RDE),
RDEi fi − fexact
100
fexact
(3)
The medium grid, which was the baseline grid, showed less than 1%
difference from the exact value (see Table 2).
Another indicator of grid convergence is the grid convergence
index (GCI) [23]. To compute the GCI, the relative errors between
the solution on successive grids i and j, εij fj − fi ∕fi , were was
computed. These values were then used to compute the GCI using
GCIij FSjεij j∕rp − 1 . The GCI includes a factor of safety FS
chosen here as 3 because the observed order of accuracy did not
match the formal order of accuracy [23]. The computed GCI are
presented in Table 3. Roache [23] shows that the satisfaction of the
condition GCI23 rp12 GCI12 indicates that the grids have reached
the asymptotic range. The computed value rp12 GCI12 1.94, which
agreed well with GCI23 1.93, indicating the grids had reached the
asymptotic limit. Additional discretization results showing the discretization error based on the surface pressures throughout the inlet
can be found in Ref. [18].
B. Temporal Discretization Error
A time-step convergence study was conducted in a manner similar
to the grid convergence study for the transient model. For this study, a
transient computation with the cowl motion was conducted. The cowl
was initially positioned at 8 deg below the horizontal and was opened
at a constant rate of 10 rad∕s. When the cowl reached the horizontal
position, it was held there for 30 ms, after which it was closed until it
reached the original 8 deg position again.
Solutions were conducted using a coarse (100 μs), baseline
(10 μs), and fine (1 μs) time step on the baseline grid to compute
the observed order of accuracy in time. The baseline time step was
chosen to balance the desire to capture all of the pertinent flow
physics with the challenge associated with increased computational
time due to fine time steps. Using the convective velocity of pressure
waves in the flow, u a, and an average cell length in the grid, a time
step of 1.22 μs was computed. However, because of the significant
computational time associated with this time step, 10 μs was chosen
as the baseline. Justification of this decision is given later in this
section.
The time-averaged mass flow rate through the inlet from 0 to 65 ms
was used to compute the observed order of accuracy; see Table 4.
Because the temporal refinement was uniform, the observed order of
accuracy could be computed from a more simplified equation [6,23],
p
ln f3 − f2 ∕f2 − f1 ln r
(4)
The observed order of accuracy was found to be 1.35. Again, despite
the fact that the dual-time-stepping approach was a second-order
accurate scheme, singularities in the computations like shock waves
569
REARDON, SCHETZ, AND LOWE
Table 4
Time-averaged mass flow rate through
the inlet for each time step
Time step
Coarse (Δt 100 μs)
Medium (Δt 10 μs)
Fine (Δt 1 μs)
Exact (Richardson)
Time-averaged mass flow rate, lbm∕s RDE, %
0.881567
11.668
0.793594
0.525
0.789638
0.024
0.789451
——
Table 5
Temporal GCI results
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
Time step
Coarse, GCI23
Fine, GCI12
rp12 GCI12
GCI, %
1.55
0.07
1.57
resulted in the observed order of accuracy being closer to first-order
accurate.
The exact value from the Richardson extrapolation was again
computed and used to calculate the relative discretization error for
the three different time steps (see Table 4). The baseline time step of
10 μs shows only 0.5% difference in the time-averaged mass flow
rate. The GCI was also calculated for the temporal error estimate as
seen in Table 5. The solutions were again found to be within the
asymptotic range validating the Richardson extrapolation with
rp12 GCI12 1.57 compared to GCI23 1.55.
Additional results in Ref. [18] assessed temporal discretization
based on the instantaneous time histories of the solutions with the
different time steps. It was found that the coarsest time step predicted
a delayed unstart as compared to the other time steps and the unstart
process was smeared over a much longer time. However, the point of
unstart compared well between the solution computed with the baseline (10 μs) time step and the fine (1 μs) time steps. Also, the fine
time step showed large oscillations after the inlet became unstarted
that were not predicted by the baseline time step. This was determined to be driven by motion of the separation bubble and shock that
developed when the inlet became unstarted. The motion of the shock
wave occurs on the time scale related to the convective velocity of
pressure waves that was mentioned previously. As stated earlier in
this section, using this velocity to compute the flow time scale on the
current grid yielded a time step of 1.22 μs. Thus, the baseline time
step of 10 μs was larger than this characteristic time and resulted in a
time-averaged solution during this period. However, despite these
differences, the point of unstart agreed well between the baseline and
fine time steps. Therefore, it was concluded that if the exact flow
response while the inlet was unstarted was not needed the baseline
time step could be used to accurately capture the unstart points. This
is important because the fine time step (1 μs) incurred a significant
computational cost, making it essentially unfeasible for engineering
analyses at this time. Therefore, the baseline time step of 10 μs was
chosen in this Paper. For additional details, the reader is referred
to Ref. [18].
C. Iterative Error
All of the transient computations were conducted using an
implicit, dual-time-stepping approach where a specified number of
subiterations at each time step was used before the solution was
advanced to the next physical time step. This approach is preferred
as it remains stable at larger time steps than are required for a purely
explicit approach [19]. However, this led to an additional source of
error, namely, the convergence during the subiterations. To assess the
iterative error in the transient model, the same transient computation
with cowl motion described in the previous section was recomputed
with a varying number of subiterations. The current work used 20
subiterations as a baseline. The uncertainty due to iterative error
during the subiterations was assessed by computing solutions at the
baseline 10 μs time step with 10, 20, and 30 subiterations. For all
Table 6
Cowl angle and peak mass flow rate computed
with different subiterations
No. of
Cowl angle θc at peak Time at peak mass
Peak mass
subiterations mass flow rate, deg
flow rate, s
flow rate, lbm∕s
10
−0.0042
0.01411
1.9072
20
−0.0042
0.01405
1.9038
30
−0.0042
0.01404
1.8993
cases, the subiteration convergence criterion was set to one order of
magnitude. Generally, for 20 subiterations, convergence was seen to
be about half an order of magnitude or slightly more. As the number
of subiterations was increased, the subiteration convergence level
was seen to increase by approximately a quarter of an order of
magnitude. Table 6 shows the computed peak mass flow rates, the
time of the peak mass flow rates, and the cowl angle at which the peak
mass flow rates occurred. The agreement is quite good, showing that
the baseline number of subiterations was adequate.
V.
Results
The first results presented are quasi-steady solutions at the three
discrete cowl angles, θc 8, 1.2, and 0 deg (β 3, 9.8, and 11 deg).
The results are to be used first to assess the performance and effect of
an array of turbulence models on the computed performance, second
to study the flowfield structure in the inlet at three discrete cowl
angles, and third to be used as baselines for comparisons with the
subsequent transient results. Following the quasi-steady results, the
time-accurate results of the transient model are presented to show
the time-dependent response of the inlet to the cowl rotational motion.
A. Results from Quasi-Steady Model
1. Turbulence Model Assessment
It is a well-known fact that the choice of turbulence model in a
RANS computation can have a significant impact on the computed
flowfield, especially in high-speed applications where shock/
boundary-layer interactions and the induced flow separations are
present [24–26]. A major goal of the current work was to assess the
performance of several of the popular turbulence models for use in
computing the flowfield through a high-speed inlet. The results were
used to select a turbulence model for the time-dependent computations presented later.
To begin, computations were conducted on the inlet with the cowl
removed. This was done to compare with the experimental work in
Ref. [16], which included local boundary-layer profile measurements
on the inlet ramp in the absence of the cowl. The experimental profiles
were obtained at 6.81 in. aft of the leading edge of the inlet ramp using a
pitot pressure probe. Because of the finite probe geometry, the measurements began at 0.016 in. off the ramp surface. The measurements
indicated that the boundary layer extended to about 25% of the throat
height for the case where no foreplate was present [16].
The computational model used for this study can be seen in
Ref. [18], and boundary-layer profiles were extracted at the same
location as the measurements. A comparison of the computed and
experimental boundary-layer profiles is shown in Fig. 6. It can be
seen that the computed solutions agree reasonably well with the
experimental data shown in the figure by the solid markers. The
profiles were also plotted nondimensionally in law-of-the-wall scaling, shown in Fig. 7. An effective velocity ueff computed using the
van Driest transformation, was applied to the data so that they could
be plotted with the incompressible log law [17]. The vertical axis is
the nondimensional wall velocity u ≡ ueff ∕u, and the horizonal
axis is nondimensional wall height y . It can be observed that the
cubic k–ε model most closely matches the log law.
A quantitative comparison with the experimental data is shown in
Table 7 with various boundary-layer thicknesses and the skin friction
coefficient. The computed boundary-layer thickness was taken to be
the point at which the velocity reached 99% of the value behind the
oblique shock induced by the ramp, consistent with the experimental
570
REARDON, SCHETZ, AND LOWE
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
Fig. 6 Mach number profile computed with a range of turbulence
models and compared to experimental data from Ref. [16].
Fig. 7 Law-of-the-wall plot of the computed boundary-layer profiles
using different turbulence models.
correlation to the current results yields a shape factor of 5.57, very
close to the value obtained by each turbulent model used.
In addition, the computed skin fiction values are shown to be lower
than reported from the experiment. A possible explanation for this
could lie in the assumed boundary condition at the wall. Specifically,
the walls were modeled as adiabatic. However, if the walls in the
experiment were below the recovery temperature, this would lead to a
larger skin fiction coefficient than reported by the computations.
Although there is a discrepancy in the momentum thickness, the
computed boundary-layer profiles were deemed validated from the
other comparisons.
It was also desired to assess the influence of the turbulence model
on the inlet performance. Computations were conducted at two cowl
angles, θc 8 and 1.2 deg (β 3 and 9.8 deg). The third cowl angle
of θc 0 deg was not considered because the experimental data
indicated the inlet to be unstarted at that condition. Comparisons of
the centerline wall pressures, normalized by the freestream pressure,
are shown in Fig. 8 as a function of axial length down the inlet for the
two cowl angles considered. For the first cowl angle, θc 8 deg
(Fig. 8a), the computations generally agree with the measured data.
The axial locations of the pressure rise seem to be captured well for all
turbulence models, indicating the inviscid flowfield and shock structure are accurately computed. However, the magnitudes of the pressure peaks at the shock/boundary-layer interaction are generally
underpredicted, and the expansion following the interaction is generally overpredicted by the computations, indicating a discrepancy in
how the viscous/inviscid interactions are modeled.
At the second cowl angle, θc 1.2 deg (Fig. 8b), much more
noticeable differences between the turbulence models can be seen.
The shock/boundary-layer interaction is much stronger because the
cowl, which is now nearly horizontal, causes the flow to turn more
aggressively. The two k–ε models follow the experimental data,
which indicates the inlet to be in the started mode. However, the
Menter SST and Spalart–Allmaras models do not follow the experimental data as they predict the inlet to become unstarted. Therefore,
it is seen that the choice of turbulence model can have a dramatic
impact on inlet performance predictions, especially in the presence of
strong shock/boundary-layer interactions. Based on these results, the
cubic k–ε model was chosen as the baseline for the remainder of this
work and for the follow-up transient work. In addition, this model has
been used previously in similar applications, Refs. [12,13,28] being
examples that cite the model’s nonlinearity to be important in scramjet flowpaths.
2. Quasi-Steady Results at 8 Deg Cowl Angle
Table 7
Boundary-layer characteristics for flow over the inlet ramp
Data source
Experiment (Ref. [16])
Cubic k–ε model
Realizable k–ε model
Spalart–Allmaras model
Menter SST model
δ, in.
0.1
0.1055
0.1124
0.1133
0.1041
δ ; in.
0.036
0.03219
0.03466
0.03507
0.03332
θ; in.
0.0185
0.0058
0.0062
0.0063
0.0059
cf
0.00207
0.00124
0.00131
0.00132
0.00126
H
1.946
5.550
5.590
5.567
5.647
work. The experimental value for the skin friction coefficient was not
obtained from direct measurement but rather from an empirical
correlation due to inadequate near-wall resolution of the boundarylayer measurements [16].
The boundary-layer thickness and the displacement thickness agree
well with the experimental data for all turbulence models. In this
configuration, the Menter SST model matches the boundary-layer
thickness closest with 4.0% difference, and the Spalart–Allmaras
model matches the displacement thickness closest with 2.6% difference. However, the momentum thicknesses show large discrepancies
with the experimental data, which causes large differences in the shape
factor, H δ ∕θ. Despite the discrepancy, it is interesting to note that
the computed shape factors are consistent with additional computational findings. Reference [27] developed a correlation for H as
a function of Mach number and wall temperature. Applying that
After downselecting the cubic k–ε model, a more in-depth investigation into the flowfields at each cowl angle was conducted using
the quasi-steady modeling approach. Figure 9 shows the computed
Mach number contours at the inlet centerline plane. External compression is achieved through the ramp shock, which reduces the Mach
number from the freestream value of 4.03–3.23. At the 8 deg position,
the cowl initiates a shock train through a series of reflected oblique
shock waves that creates the inlet internal compression. The shock
train consists of four waves before entering the throat/isolator section.
In addition to the shock train, the side walls of the inlet also create
shocks that lead to further compression of the flow as it enters the
inlet. Along the inlet ramp, no separation is seen to occur at this cowl
angle due to the relatively weak shock/boundary-layer interactions.
However, at the end of the throat/isolator section, a small separation
region of subsonic flow is seen to exist. The features described
previously are consistent with previous computations presented
in Ref. [8].
Figure 10 depicts the inlet surfaces with contours of the logarithm
of the pressure ratio [log 10P∕P1 , along with simulated surface oil
flow lines. The inlet left side wall and cowl have been removed from
the figure to make the inlet ramp visible. In the isolator section, the
convergence of the oil flow lines indicates the separation regions. The
three-dimensional nature of the flow is also visible with regions of
spanwise flow developed first at the leading edges of the inlet side
walls where the side wall shocks establish. A second region of
spanwise flow is developed at the point where the cowl shock
REARDON, SCHETZ, AND LOWE
571
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
Fig. 8 Comparison of the normalized centerline wall pressures computed with various turbulence models and the experimental data of Ref. [16] for the
cowl positioned at a) 8 and b) 1.2 deg below the horizontal.
Fig. 9 Computed centerline Mach number contours for the cowl positioned at 8 deg below the horizontal.
Fig. 11 Computed centerline Mach number contours for the cowl
positioned at 1.2 deg below the horizontal.
3. Quasi-Steady Results at 1.2 Deg Cowl Angle
Fig. 10 Simulated oil flow lines and logarithm of the pressure ratio
contours along the inlet ramp and right side wall for the cowl positioned
at 8 deg below the horizontal.
impinges on the inlet ramp, creating a shock/boundary-layer interaction. This flow pattern develops a nonuniform spanwise pressure
distribution in the inlet. It will be shown in the discussions that follow
that these regions of spanwise flow will develop into separation
bubbles and corner flows which are believed to influence the unstart
process.
At the cowl angle of θc 1.2 deg, the flow through the inlet
maintains many of the same features as when the cowl is at 8 deg; see
the centerline Mach number contours in Fig. 11. However, in the
1.2 deg position, the ramp external compression shock is only slightly
above the cowl. This leads to shock/shock interaction between the
ramp shock and the upper cowl shock, which can be classified as
the type VI interaction based on Edney’s classification [29–31]. The
lower side of the cowl shock still sets up a shock train of oblique
waves as it did at the 8 deg position; however, now due to the steep
angles, only two waves exist before the throat/isolator section. Also,
it is seen that the second wave in the shock train, which is reflected off
the underside of the cowl, is essentially cancelled by the expansion
wave at the inlet ramp shoulder.
Now that the cowl is at a steeper angle to the ramp, a stronger cowl
shock is developed leading to a much stronger shock/boundary-layer
interaction. This imposes a larger adverse pressure gradient on the
boundary layer and subsequently results in a small separation region.
Separated flow can also be seen inside the isolator like in the 8 deg
position, but now the separation is closer to the inlet shoulder where
the second shock train wave impinges. Also, unlike when the cowl
was set to 8 deg, regions of separation have developed along the side
walls of the inlet which are clearly visible in Fig. 12 with the
logarithm of the pressure ratio on the surfaces and simulated oil flow
lines.
A highly three-dimensional flow is developed as corner separations
cause blockage, forcing more flow into the centerline of the inlet. This
is seen in Fig. 12, in which, at the cowl shock interaction region, fluid is
drawn into the large separation bubble from the side walls. The
separations at the side walls actually extend forward of the cowl shock
impingement interaction as disturbances from the adverse pressure
gradient propagate upstream in the subsonic portion of the boundary
572
REARDON, SCHETZ, AND LOWE
external compression wave due to the ramp still exists; however, the
internal oblique shock train has been forced upstream and out of the
inlet. Ahead of the inlet exists a large separation bubble and separation shock that diverts the flow around the inlet, leading to significant
spillage and thus a reduction in mass flow rate. Because of the
massively separated flow, the flowfield is inherently unsteady.
B. Results from Transient Model
Before conducting the time-accurate computations, the transient
model in the steady-state limit was first compared to the quasi-steady
model at the three cowl angles previously analyzed. The result of this
comparison can be found in Ref. [18] and indicated that the transient
model, with the overset cowl, was consistent with the quasi-steady
model in this limit as expected.
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
1. Transient Inlet Unstart and Restart
Fig. 12 Simulated oil flow lines and logarithm of the pressure ratio
contours along the inlet ramp and right side wall for the cowl positioned
at 1.2 deg below the horizontal.
layer. The results strongly suggest that the development of these
separation regions, in both the centerline and corners, eventually lead
to inlet unstart as they create a blockage to the flow entering the inlet,
increasing the contraction ratio above what the inlet can withstand.
4. Quasi-Steady Results at 0 Deg Cowl Angle
The last cowl angle analyzed was the horizontal position, θc 0 deg. In this configuration, the inlet was predicted to be unstarted by
the computations, which agrees with the experimental data. When the
inlet is unstarted, the flowfield changes drastically; see Fig. 13. The
Fig. 13 Computed centerline Mach number contours for the cowl at the
horizontal position.
For the time-accurate computations, the cowl motion followed
closely what was used in the experiment to determine the unstart
and restart angles. In the experiment, the cowl was initially parallel to
the ramp and opened slowly until the inlet unstarted, after which the
inlet was closed until it restarted [18]. In the current transient computations, the cowl was initially set to an angle of θc 8 deg, and a
converged steady-state solution was obtained to be used as the initial
conditions for the transient computations. Then, starting at 0 s, the
inlet cowl was rotated at a rate of 10 rad∕s from the 8 deg position to
the horizontal position (θc 0 deg), which was reached at 13.97 ms
(see Fig. 14a). After the cowl reached the horizontal, it was held there
for 30 ms so that the unsteady flow could reach a stable oscillatory
pattern. Then, beginning at 43.97 ms, the inlet cowl was rotated back
to the 8 deg position at the same rate at which it was opened. It should
be noted that, because the exact rotation rate used in the experimental
study was not stated, several rotation rates were assessed. As stated in
Ref. [18], these were 1, 5, and 10 rad∕s. The main features to be
described in the following sections were not significantly different at
these other rates: therefore, only the 10 rad∕s results will be presented here. The reader is referred to Ref. [18] for additional details.
During the rotation, the inlet was initially started, but it became
unstarted as the contraction ratio became too great, and the internal
shock train was ejected from the inlet. The point of unstart was near
the horizontal cowl position and was determined as the point of peak
mass flow rate, which occurred at 14.05 s. As the cowl closed, the
inlet restarted, and the standing shock outside of the inlet was
swallowed.
The transient response of the mass flow rate through the inlet
during the cowl rotation process is shown in Fig. 14b. As the inlet
cowl is initially opened and the capture area increases, the mass flow
rate also increases until it reaches a maximum. At this point, there is a
significant drop in the mass flow rate as the inlet becomes unstarted,
and significant spillage occurs due to the development of a separation
bubble which effectively blocks flow from entering the inlet. As the
Fig. 14 Time-history of a) cowl motion and b) mass flow rate during computation.
REARDON, SCHETZ, AND LOWE
unstarted flowfield is set up, the mass flow rate oscillates at a
significantly lower value than when the inlet is started. This is the
reason unstarted operation should be avoided in ramjet and scramjet
operation. As the cowl closes, the unstarted flowfield persists. Eventually, the mass flow rate begins to recover, and the inlet restarts such
that the started mass flow rate is achieved. The inlet was found to
unstart at a cowl angle of θc 0 deg and to restart at an angle of
8 deg (β 11 and 3 deg, respectively). This seems to agree well with
the data reported in Ref. [16], in which inlet unstart and restart angles
are reported to be θc 0.8–1.0 and 7.9–8.4 deg (β 10–10.2 and
2.6–3.1 deg), respectively. It should be noted that in the current Paper
the point of peak mass flow rate immediately before the steep drop in
mass flow rate was deemed the unstart point, and the point at which
the mass flow rate recovered to the steady-state started value was
deemed to be the restart point. The following sections will give a
more in-depth discussion of the phases of inlet operation including
unstart, restart, and the hysteresis between the two.
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
2. Details of Inlet Unstart
The cowl begins to rotate open at time t 0 s, and the mass flow
rate initially increases as the capture area of the inlet increases. At
t 13.97 ms, the inlet cowl has reached the horizontal position, and
the mass flow rate continues to increase slightly, even though the
573
cowl has stopped opening. Then, at t 14.05 ms, the mass flow rate
peaks and drops off suddenly. This is the point of inlet unstart where
the separated flow regions at the points of shock/boundary-layer
interaction and along the side walls have reduced the flowthrough
area of the inlet, thus choking it.
A time series of centerline flow cross-sectional images showing
the unstart process is given in Fig. 15. At this point in time, the cowl
has already reached the horizontal position and is being held fixed.
The contours shown are reported at increments of 0.5 ms to highlight
the rapid unstart process. The first frame shown is at 14.0 ms, and the
cowl is already in the horizontal position. In the next frame at 14.5 ms,
the development of a separation region at the inlet shoulder is visible.
This separation bubble begins to move forward as the inlet unstarts
and grows in size, creating an increasing flow constriction. As this
happens, the shock train inside the inlet is disrupted, as seen in the
15.0 to 16.0 ms frames. At 16.0 ms, the separation bubble has moved
forward enough to cause the separation shock to interact with the
cowl shock creating a lambda shock pattern that spans the entire
height of the inlet. After 16.5 ms, all of the shock waves have been
pushed upstream and out of the inlet, and a large, unsteady separation
region is set up. This creates a separation shock followed by significant spillage around the inlet causing the dramatic reduction in mass
flow rate previously mentioned.
Fig. 15 Centerline Mach number contours during inlet unstart.
574
REARDON, SCHETZ, AND LOWE
The evolution of the separated flow regions is shown in Fig. 16.
The inlet ramp, isolator section, and right side wall are shown with
contours of the logarithm of the pressure ratio, and simulated oil flow
lines are shown to indicate regions of separated flow. The cowl and
upper surfaces are removed for clarity. The time series begins at
12.0 ms, which is when the inlet cowl is positioned at 1.12 deg below
the horizontal. In the simulated oil flow patterns, significant spanwise
flow can be seen at the main shock/boundary-layer interaction point.
Also, the shoulder separation bubble draws in the flow from the
corner regions between the inlet ramp and side walls, leading to
additional spanwise flow. These features are identified in the 12.0
to 13.0 ms frames. As the inlet unstart progresses, vortices on either
side of the shoulder separation bubble develop and grow as seen in the
13.5 and 14.0 ms frames. As they continue to grow, the vortices begin
to draw in fluid from the centerline of the inlet. The vortices propagate
forward and merge with the main shock/boundary-layer interaction
region and ultimately leave the inlet as the inlet becomes fully
unstarted. After 16.0 ms, the inlet has become unstarted and the large
unsteady separation pattern is visible.
At this point, it should be mentioned that the three-dimensional
development of these vortices and separation regions is central to
the unstart process. This was illustrated in Ref. [18], in which
two-dimensional models were compared with the current threedimensional results. It was found that without the three-dimensional
growth of these separation regions, particularly at the inlet shoulder,
the two-dimensional models predicted the inlet to remain started for
all cowl angles. This conclusion is consistent with important experimental observations such as those reported in Ref. [32], indicating
that two-dimensionality, or even symmetry, does not occur in real,
internal shock-containing flows. Therefore, capturing the threedimensionality of the problem is key in computational modeling of
such flows.
3. Details of Inlet Restart
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
The inlet reaches the horizontal position at 13.97 ms, after which
the inlet unstarts. The cowl is then held in the horizontal position until
43.97 ms when it begins to rotate closed. The flowfield during the
cowl closing is shown in Fig. 17. The series begins at 53 ms, at which
point the cowl is at 5.17 deg below the horizontal and closing. As the
cowl is closed, the separation bubble ahead of the inlet is reduced in
size due to the development of a favorable pressure gradient as the
flow accelerates through the inlet. The inlet remains unstarted even as
it passes through the angle at which it unstarted. The cowl reaches its
initial 8 deg position at 57.94 ms. At this point, the ramp shock has
reached the shock-on-lip condition with the cowl, and all spillage is
stopped. Therefore, the separation bubble is drawn back into the inlet,
and the inlet regains the started operating mode.
4. Hysteresis
Fig. 16 Simulated oil flow lines on the inlet ramp and right side wall
during the unstart process highlighting the motion of the separation
bubbles.
As mentioned previously, the inlet progresses past the angle of
unstart before being able to restart. This indicates that, in addition to
the transient nature of the unstart and restart processes, the inlet
operation is also affected by hysteresis. This is manifested in that
progressing from the started operation to the unstarted operation does
not follow the same path as progressing from the unstarted operation
to the started operation. Figure 18 shows the mass flow rate as a
function of the cowl angle during the cowl rotation loop highlighting
this hysteresis. The loop begins on the right-hand side of the figure at
the 8 deg cowl positions. The path proceeds along the upper branch as
the cowl angle is reduced (i.e., when the cowl is opening), and the
mass flow rate is seen to increase until it unstarts at the horizontal
cowl position. Then, after the unstart, the loop continues along the
lower branch as the cowl closes. The mass flow rate is nearly constant
until the cowl has reached an angle of about 4 deg below the
horizontal. When the loop reaches the 8 deg point again, the mass
flow rate suddenly increases back to the initial started value. It is
interesting to note that in Fig. 18 and in Fig. 14b shown previously the
mass flow rate undershoots after unstarting and then overshoots after
restarting before reaching steady values. This phenomenon requires
additional investigation but is possibly driven by the regions of
separated flow that require a time delay before they are fully formed
and positioned.
Along with the unsteady hysteresis loop, three individual data
points can also be seen in Fig. 18. These are the quasi-steady computational results at the three discrete cowl angles reported previously. First, note that the results from the two types of analyses agree
well at the discrete angles. That is, the current transient results are
consistent with the previous quasi-steady results. However, it is
important to note that the quasi-steady analysis method does not
have the capability of computing the dynamic response of the inlet as
it progresses through the restart process. Clearly, the physics of
the hysteresis requires a transient approach so that the solution is
marched forward in time from the unstarted solution. Hence, this new
application of a time-accurate model allows for capturing important
details that are inherently transient in nature.
575
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
REARDON, SCHETZ, AND LOWE
Fig. 17 Centerline Mach number contours during inlet restart.
VI.
Fig. 18 Mass flow rate through the inlet during cowl rotation, plotted as
a function of cowl angle.
Conclusions
The main goals of the current Paper were to develop computational
models to compute the flow through a high-speed inlet with a rotation
cowl in the quasi-steady limit as well as to use time-accurate computations to gain insight into the transient processes that cannot be
captured in the quasi-steady limit. For the quasi-steady model, understanding the sensitivity of the solutions to different turbulence models
was a prime objective. It was found that the turbulence model can
have a large effect on the computed solution because of different
boundary-layer thicknesses. This was especially noticeable near the
inlet unstart cowl angle. In particular, Spalart–Allmaras and Menter
SST models predicted the inlet to unstart before the cubic and
realizable k–ε models and the experimental data. In addition, the
detailed flow structure through the inlet was assessed at three discrete
cowl angles. This series of steady-state computations showed the
development of three-dimensional flow features including separation
bubbles and spanwise flows as the cowl was opened and positioned
closer to the horizontal. It was found that the quasi-steady CFD was
able to correctly predict the started or unstarted operating mode for
the different cowl angles considered. The knowledge gained from the
576
REARDON, SCHETZ, AND LOWE
quasi-steady work was used in the development of a transient computation of the inlet with the rotating cowl modeled.
The second part of this Paper focused on the development of timeaccurate computations to analyze the time-dependent flowfield in the
inlet. State-of-the-art applied CFD methods including unsteady solutions to the Reynolds-averaged Navier–Stokes equations and overset mesh motion were used to do this. The model was used to explore
the dynamics of the flowfield during rotational motion of the cowl,
including unsteady response of the inlet to unstart and restart processes, as well as the hysteresis that exists between the two states.
The hysteresis was seen to manifest itself as a difference in the angle
at which the inlet unstarted and that at which it restarted. Timedependent evolution of the separation bubbles identified in the quasisteady model was observed and shown to play the primary role in
creating hysteresis in the system. Thus, this Paper has shown the
importance of the extension of time-accurate computations to the
study of variable-geometry inlets.
Downloaded by Uur Akolu on July 17, 2021 | http://arc.aiaa.org | DOI: 10.2514/1.B38214
References
[1] Fry, R. S., “A Century of Ramjet Propulsion Technology Evolution,”
Journal of Propulsion and Power, Vol. 20, No. 1, 2004, pp. 27–58.
https://doi.org/10.2514/1.9178
[2] Curran, E. T., “Scramjet Engines: The First Forty Years,” Journal of
Propulsion and Power, Vol. 17, No. 6, 2001, pp. 1138–1148.
https://doi.org/10.2514/2.5875
[3] Heiser, W. H., and Pratt, D. T., Hypersonic Airbreathing Propulsion,
AIAA Education Series, AIAA, Washington, D.C., 1994, pp. 242–251.
https://doi.org/10.2514/4.470356
[4] Zhang, Q. F., Tan, H. J., Chen, H., Yuan, Y. Q., and Zhang, Y. C.,
“Unstart Process of a Rectangular Hypersonic Inlet at Different Mach
Numbers,” AIAA Journal, Vol. 54, No. 12, 2016, pp. 3681–3691.
https://doi.org/10.2514/1.J055005
[5] Pletcher, R. H., Tannehill, J. C., and Anderson, D. A., Computational
Fluid Mechanics and Heat Transfer, 3rd ed., CRC Press, Boca Raton,
FL, 2013, pp. 3–12.
[6] Roy, C. J., “Review of Code and Solution Verification Procedures for
Computational Simulation,” Journal of Computational Physics,
Vol. 205, No. 1, 2005, pp. 131–156.
https://doi.org/10.1016/j.jcp.2004.10.036
[7] Reinartz, B. U., Herrmann, C. D., Ballmann, J., and Koschel, W. W.,
“Aerodynamic Performance Analysis of a Hypersonic Inlet Using
Computation and Experiment,” Journal of Propulsion and Power,
Vol. 19, No. 5, 2003, pp. 868–875.
https://doi.org/10.2514/2.6177
[8] Neaves, M. D., McRae, D. S., and Edwards, J. D., “High-Speed Inlet
Unstart Calculations Using an Implicit Solution Adaptive Mesh Algorithm,” AIAA Paper 2001-0825, 2001.
https://doi.org/10.2514/6.2001-825
[9] Hoeger, T. C., “CFD Transient Simulation of an Isolator Shock Train in a
Scramjet Engine,” Ph.D. Dissertation, U.S. Air Force Inst. of Technology, Wright–Patterson AFB, OH, 2012.
[10] Riley, L. P., Hagenmaier, M. A., Donbar, J. M., and Gaitonde, D. V., “A
Computational Investigation of Unstart in a Dual-Mode Scramjet,”
AIAA Paper 2016-1901, Jan. 2016.
https://doi.org/10.2514/6.2016-1901
[11] Riley, L. P., Gaitonde, D. V., Hagenmaier, M. A., and Donbar, J. M.,
“Isolator Dynamics During Unstart of a Dual Mode Scramjet,” Journal
of Propulsion and Power, Vol. 34, No. 6, 2018, pp. 1409–1427.
https://doi.org/10.2514/1.B36888
[12] Yentsch, R. J., and Gaitonde, D. V., “Unsteady Three-Dimensional
Mode Transition Phenomena in a Scramjet Flowpath,” Journal of
Propulsion and Power, Vol. 31, No. 1, 2015, pp. 104–122.
https://doi.org/10.2514/1.B35205
[13] Yentsch, R. J., and Gaitonde, D. V., “Unsteady Three-Dimensional
Phenomena in Mode-Transition Simulations of the HIFiRE-2 Scramjet
Flowpath,” AIAA Paper 2013-3753, July 2013.
https://doi.org/10.2514/6.2013-3753
[14] Grainger, A. L., Brieschenk, S., Boyce, R. R., Malpress, R., and
Buttsworth, D., “Investigation into the Flow Physics of Hypersonic
Variable Geometry Inlet Starting,” AIAA Paper 2014-3230, June 2014.
https://doi.org/10.2514/6.2014-3230
[15] Xianhong, X., Yuan, L., and Zhansen, Q., “Aerodynamic Design and
Numerical Simulation of Over-Under Turbine-Based Combined-Cycle
(TBCC) Inlet Mode Transition,” Procedia Engineering, Vol. 99,
Jan. 2015, pp. 129–136.
https://doi.org/10.1016/j.proeng.2014.12.516
[16] Emami, S., Trexler, C. A., Auslender, A. H., and Weidner, J. P., “Experimental Investigation of Inlet-Combustor Isolators for a Dual-Mode
Scramjet at a Mach Number of 4,” NASA TP-3502, May 1995.
[17] Schetz, J. A., Boundary Layer Analysis, 2nd ed., AIAA Education
Series, AIAA, Reston, VA, 2011, pp. 240–251.
https://doi.org/10.2514/4.868245
[18] Reardon, J. P., “Computational Analysis of Transient Unstart/Restart
Characteristics in a Variable Geometry, High-Speed Inlet,” Ph.D. Dissertation, Aerospace and Ocean Engineering Dept., Virginia Polytechnic Inst. and State Univ., Blacksburg, VA, 2019, https://vtechworks.lib.
vt.edu/handle/10919/95883
[19] METACOMP, CFD++ User Manual, Ver. 18.1, METACOMP, https://
mix.metacomptech.com/ [accessed 10 Jan. 2021].
[20] Wilcox, D. C., Turbulence Modeling for CFD, DCW Industries,
La Canada, CA, 1993, Chaps. 6, 8.
[21] Spalart, P., and Allmaras, S., “A One-Equation Turbulence Model for
Aerodynamic Flows,” AIAA Paper 1992-439, 1992.
https://doi.org/10.2514/6.1992-439
[22] Roy, C. J., and Blottner, F. G., “Methodology for Turbulence Model
Validation: Application to Hypersonic Flows,” Journal of Spacecraft
and Rockets, Vol. 40, No. 3, 2003, pp. 313–325.
https://doi.org/10.2514/2.3966
[23] Roache, P. J., Verification and Validation in Computational Science and
Engineering, Hermosa, Albuquerque, NM, 1998, Chaps. 2, 4, 5, 6.
[24] Di Stefano, M. A., Hosder, S., and Baurle, R. A., “Effect of Turbulence
Model Uncertainty on Scramjet Isolator Flowfield Analysis,” Journal of
Propulsion and Power, Vol. 36, No. 1, 2020, pp. 109–122.
https://doi.org/10.2514/1.B37597
[25] Yentsch, R. J., Gaitonde, D. V., and Kimmel, R., “Performance of
Turbulence Modeling in Simulation of the HIFiRE-1 Flight Test,”
Journal of Spacecraft and Rockets, Vol. 51, No. 1, 2014, pp. 117–127.
https://doi.org/10.2514/1.A32535
[26] Roy, C. J., and Blottner, F. G., “Review and Assessment of Turbulence
Models for Hypersonic Flows,” Progress in Aerospace Sciences,
Vol. 42, Nos. 7–8, 2006, pp. 469–530.
https://doi.org/10.1016/j.paerosci.2006.12.002
[27] Shahab, M. F., Lehnasch, G., Gatski, T. B., and Comte, P., “Statistical
Characteristics of an Isothermal, Supersonic Developing Boundary
Layer Flow from DNS Data,” Flow, Turbulence and Combustion,
Vol. 86, Nos. 3–4, pp. 369–397.
https://doi.org/10.1007/s10494-011-9329-0
[28] Yentsch, R. J., and Gaitonde, D. V., “Numerical Investigation of the
HIFiRE-2 Scramjet Flowpath,” AIAA Paper 2013-0119, Jan. 2013.
https://doi.org/10.2514/6.2013-119
[29] Edney, B. E., “Anomalous Heat Transfer and Pressure Distributions on
Blunt Bodies at Hypersonic Speeds in the Presence of an Impinging
Shock,” Aeronautical Research Inst. of Sweden (FFA), Flygtekniska
Forsoksanstalten, Rept. FFA-115, Stockholm, 1968.
[30] Edney, B. E., “Effects of Shock Impingement on the Heat Transfer
Around Blunt Bodies,” AIAA Journal, Vol. 6, No. 1, 1968, pp. 15–21.
https://doi.org/10.2514/3.4435
[31] Lind, C. A., “Effect of Geometry on the Unsteady Type IV Shock
Interaction,” Journal of Aircraft, Vol. 34, No. 1, 1997, pp. 64–71.
https://doi.org/10.2514/2.2136
[32] Babinsky, H., Oorebeek, J., and Cottingham, T., “Corner Effects in
Reflecting Oblique Shock-Wave/Boundary-Layer Interactions,” AIAA
Paper 2013-0859, 2013.
https://doi.org/10.2514/6.2013-859
V. Raman
Associate Editor