SHOCK TRAIN LENGTH MEASUREMENTS AND IMPROVED CORRELATIONS FOR RECTANGULAR
DUCTS
ISABE-2015-20231
Jonathan S. Geerts∗ , Kenneth H. Yu†
University of Maryland, College Park, MD 20740
Abstract
Widely accepted empirical relations for shock train lengths
in constant-area isolators are based on centerline pressure
measurements. They also do not account for the effect of
aspect ratio or boundary layer asymmetry in non-circular isolators. In this paper, new shock train length measurements are
reported for rectangular isolators with incoming Mach numbers
of up to 2.5 and duct aspect ratios of 3 and 6. Simultaneous
visualization of the shock trains from both the minor- and
major-axis perspectives revealed the boundary flow separation
initiating from the corners, at a location approximately one
duct height ahead of the ensuing shock trains in the core flow.
Incorporating the effects of aspect ratio on boundary layer
flow as well as corner flow separation on shock train length,
a modified relation for shock train length for rectangular
isolators is proposed. The results improved the agreement
with the present data, by reducing the root-mean-squared error of the correlation results by 50% in the case of aspect ratio 6.
compression inlet, a constant cross-sectional isolator, a combustor capable of supporting both supersonic and subsonic burning
with a supersonic outlet, and an expansion exhaust. Operating
across a wide flight envelope (Mach numbers, altitudes, dynamic
pressures etc.), the system must be adequately robust to avoid
Combustion-Inlet Interaction (CII), one of the prime catalysts
of engine unstart. The component in the system responsible for
mitigating the unstart risk is the isolator, located between the
lower pressure inlet and higher pressure combustor. The isolator, as the name implies, is meant to isolate the combustor and
inlet and avoid the occurrence of CII by containing a complex
flow structure known as a shock train.
The shock train is a vital component in the initial region of
the dual-mode scramjet operational envelope (i.e. the subsonic
combustion or ramjet-mode). Proper formation and position
control of the shock train allows the incoming flow to compress
to higher pressures prior to combustion, avoid the severe stagnation pressure losses associated with a single normal shock, and
shock down to subsonic bulk velocities for ramjet-mode combustion. Controlling the position of this shock train is equally
important, as uncontrolled oscillations and propagation can impart high pressure and thermal loads on the engine structure
and uncontrolled upstream propagation can severely affect the
inlet flow field, resulting in loss of mass flow and potential engine unstart. [1]
The shock train can be formed through a number of
processes. An upstream mixed-compression inlet causing multiple shock/boundary-layer interactions (SBLIs) can impose
severe adverse pressure gradient on the internal boundary
layer at each interaction in the isolator. More often, it is the
combustion of fuels creating large back pressures that call for
the presence of the shock system to increase the thermodynamic potential of the incoming flow. Large local pressure
surges downstream of the isolator caused by chemical energy
release due to combustion processes can cause boundary-layer
separation within both the combustor and isolator. The
resulting separation causes incoming flow to turn into itself
and generate a series of shock structures (Fig. 1) [2]. The
traditionally visualized shape (as viewed from a single planar
perspective) of the primary shock structure in the shock train
is predominantly depended on the incoming Mach number,
with the degree of normal shock bifurcation increasing with
Mach number [3]. A comprehensive review regarding isolator
performance, flow behavior, and terminology can be found in
the review paper by Matsuo et al. [4].
Nomenclature
AR
D
DH
H
M
P
W
Reθ
RMS E
S
St
Sc
T
U
δ
δ∗
θ
aspect ratio
isolator diameter
hydraulic diameter
isolator height (minor axis)
Mach number
pressure
isolator width (major axis)
momentum thick. Reynolds number
root mean square error
total (psuedo) shock train length
centerline (pseudo) shock train length
corner flow separation length
temperature
flow velocity
99% velocity thickness
displacement thickness
momentum thickness
Subscripts
i
w
0
1
2
3
∞
current static pressure tap
wall
stagnation flow parameter
lower/upper wall parameter
isolator inlet plane station & side wall parameter
isolator exit plane station
freestream flow parameter
Isolator shock train Fuel injection
Freestream
Introduction
Exhaust
Forebody
oblique shock
A. Background & Motivation
Dual mode scramjet engines are a prime candidate in the
search for a next-generation propulsion system capable of supporting hypersonic orbital access, cruise missile propulsion, and
accelerated global transportation. Sketched in a simplified twodimensional form in Fig. 1, the dual-mode cycle consists of a
Figure 1: Simplified 2D scramjet schematic (adapted from Ref.
[5]).
∗ Graduate Research Assistant, Department of Aerospace Engineering, [email protected]
† Associate Professor, Department of Aerospace Engineering
B. Isolator Flow Three-Dimensionality
To mitigate the risk of CII from occurring in a complex, con-
i
2
Compression Inlet
1
3
Isolator
e
4
Combustor
Exhaust Nozzle
2
fined three-dimensional flowfield, the shock train leading edge
structure must be accurately located. Prior to the implementation of such a shock train control scheme, the isolator must be
designed long enough to contain the shock train but not excessively long in order to avoid excess weight and supersonic drag
penalty. Empirical relations for shock train length are often
used to initialize the isolator design for certain target incoming
flow parameters. To more accurately predict the total length
of the shock train, as well as improve reliability of shock train
location control schemes, the shock train leading edge structure
in duct with rectangular cross-section and potentially complex
corner flows must be thoroughly understood.
Traditional analyses use the terms ‘normal’ and ‘oblique’
shock trains based on observations of the primary shock train
structure from a single planar perspective. To further reduce
the sense of flow three-dimensionality, conventional visualization techniques such as schlieren and shadowgraphy integrate
disturbances across the entire optical path length. Techniques
offering the capability to visualize the flowfield at a single ‘plane
’ along the optical axis, such as Focusing Schlieren [6, 7] and
Planar Mie- and Rayleigh-Scattering [8, 9], have been successfully used in compressible flow applications, yet they do not
offer a simultaneous global field of view in three-dimensions as
only one plane (perpendicular to the traditional optical axis) is
visualized at a time.
To address this shortcoming, a novel multiplane shadowgraph technique was employed by the author to visualize the
shock train structure simultaneously along the major and minor axis of rectangular isolator ducts with inflow Mach number
of 2.4 and aspect ratios 3 and 6. Detailed discussion of the
structures visualized and accompanying quantitative dynamic
pressure measurements along the duct minor axis can be found
in Ref. [10]. This work is referenced often throughout the discussions to follow. A result of the flow visualization technique is
shown in Fig. 2 for the aspect ratio 3 case and briefly discussed
to introduce the motivation behind thoroughly understanding
the shock train leading edge front. The two perspectives in Fig.
2 were acquired simultaneously using the same beam of light,
with the XY-plane (top) representing the traditional side-view
perspective of the duct vertical and longitudinal axis, and the
XZ-plane (bottom) representing the top-view perspective of the
duct lateral and longitudinal axis. Since it is a shadowgraph,
the second derivative of density gradient is visualized, characterized by the dark and light streaks accompanying each shock
structure (resembling the maximum and minimum bounds of
second derivative across the shock front).
The XY-plane shows what appears to be a primary oblique
shock train structure, with upper and lower boundary layer separation (1u, 1l) resulting in a right- and left-running oblique
shock (2u, 2l) intersection (3), which in turn spawns upper and
lower refracted shock features (4u, 4l). Interaction of these refracted features with the separated boundary layer reflect into a
coalesced normal structure (5). The re-acceleration mechanism
leading to the secondary normal shock is explained in Ref. [10].
The three-dimensionality of the flow features are recognized in
the XZ-plane image which represents the starboard half of the
channel width. Boundary layer separation points (1u, 1l) occur
well before the establishment of shock structure near the centerregion (5), as the upper and lower walls oblique shocks originate
at X=540mm and 560mm respectively, and the center feature
terminates at X=595mm. The opposing family oblique shocks
(2u, 2l) are seen traveling inboard, joining at the intersection
point (3) at [X,Z]=[580,35]mm. The coalesced refracted structure then makes a second transition (5) at [X,Z]=[595,20]mm.
The secondary normal shock is clearly visible (6), with thermodynamic differences between the flow behind the primary normal component (5, 5’, 5”) and the primary oblique component
(2, 3) causing the secondary normal shock to deflect (6’).
From the snapshot described in Fig. 2 and the detailed
flow analysis presented described in Ref. [10], it is clear that
the Mach 2.4 shock train leading edge front can no longer be
identified as a system of pure oblique shocks or highly bifur-
Centerline
Window
Figure 2: AR 3.0 multiplane shadowgraph. XZ-plane representing the starboard half of the isolator duct. Flow is left to right,
dimensions in mm. Detailed description of visualization setup
and results analysis can be found in Ref. [10]
cated normal shocks. It is instead a hybrid oblique/normal
shock front, one in which corner flow separation precedes
center-flow separation. Accurately predicting the full length
of the isolator shock train, including the upstream corner
flow separation, is critical in designing an isolator robust and
reliable enough to mitigate CII across a wide flight regime.
This in turns means that wall-mounted pressure sensors used
to derive empirical relations of shock train length must be
oriented along more than just the longitudinal centerline,
as the single axis orientation would not capture the preceding corner flow separation. The work presented below is
a look at introducing a modification to existing empirical
relations for shock train length that accounts for the state
of the corner boundary layer in a rectangular cross-section duct.
C. Previous Work
From early work on ramjet (diffuser) terminal shock position sensors for the YF-12 family of aircraft [11, 12] to more
recent development efforts of the Propulsion System Controller
(PSC) logic for the X-43A scramjet engine controller [13, 14],
pressure measurements inside the internally confined compressible inlet/isolator flowfield are used to keep the isolator from operating above a certain threshold (i.e. shock train leading edge
not exceeding a certain longitudinal position). This is achieved
by either controlling the inlet flowfield (as in the former), or the
downstream combustion dynamics (as in the latter).
Prior to the implementation of such techniques, estimates
of shock train lengths at and off design conditions allow the
designer to construct an isolator that is long enough to contain
the entire shock train (in addition to safety margin), but not
excessively long to avoid unnecessary weight and drag penalties. This becomes especially important as the cycle transitions
to scramjet-mode and the shock structure is not a predominant
feature inside the isolator since the entire flow path is supersonic. The isolator could then be designed as a constant area
combustor with injectors upstream of the isolator inlet plane
accompanied with a step for flame holding [15].
The classic empirical relation for shock train length in axisymmetric (circular) ducts, presented in Eq. 1 as introduced by
Waltrup & Billig in 1973 [16], with the exception of Mach and
pressure subscripts matching the station designations in Fig. 1,
is widely considered to be the fore-most empirical relation for
shock train length estimation.
S t (M22 − 1)Re1/4
P3
P3
Θ
= 50(
− 1) + 170(
− 1)2
D1/2 θ1/2
P2
P2
(1)
Through varying the flow parameters including inflow Mach
3
number (M), momentum thickness boundary layer (Reθ ), duct
diameter (D), and momentum thickness of the upstream boundary layer (θ), it was found that the shock train length for a given
pressure ratio (P3 /P2 ) varies directly with θ1/2 D1/2 and inversely
with (M 2 − 1)Re1/4
to form the quadratic expression in Eq. 1. It
θ
is worth noting that the term ‘shock train length’ as used by
Waltrup & Billig corresponds to the entire region of pressure
rise, which includes the visible shock structure region of the
shock train as well as the subsequent mixing region in which
pressure rise continues, albeit at a reduced rate as compared
to the visible shock structure region. This combined region is
referred to as the ‘psuedo shock region’ as summarized in the
review paper by Matsuo et al. [4], wherein the ‘shock train region’ represents the visible shock portion of the pseudo region,
and the ‘mixing region’ represents the subsequent region of continued pressure rise. This nomenclature choice is visualized in
Fig. 3 (adapted from Ref [4]) and subsequently used in this
work. Thus, pseudo shock train length derived from the classic
relations is referred to as S t , whereas the additional upstream
corner flow separation length discussed in Fig. 2 is labeled S c .
Together, the total shock train length, S, is calculated.
affects the total pseudo shock train length. To develop such
modification, experimental data obtained in a Mach 2.4 aspect
ratio 3 and 6 isolator was compared to both empirical relations
(Eq. 1-2), with a modification proposed to account for the effects of both the minor- and major-duct axis boundary layer
adding to the total shock train length purely derived from traditional centerline pressure measurements. Furthermore, the
effect of aspect ratio is included through the use of the hydraulic diameter as the physical duct variable, rather than the
minor-axis length scale (duct height) alone.
Length scales of upstream corner separation (with respect
to centerline separation) observed from the multiplane shadowgraph visualization efforts (Fig. 2) are used in lieu of outboard wall static pressure measurements, which are planned
for upcoming tests. Likewise, boundary layer pitot-probe surveys are performed along the centerline representing the minoraxis values of θ and Reθ used in the relations discussed. Outboard and side-wall boundary layer surveys are planned to provide experimentally derived values of duct major-axis boundary
layer parameters. A preliminary parametric study varying the
and major-axis boundary layer parameters is presented and discussed.
Technical Approach & Experimental Setup
A. Wind Tunnel Facility & Isolator Models
The experimental study was performed in a cold flow, atmospheric indraft wind tunnel (volumetric capacity of 53m3 )
whose test section is shown in Fig. 4a-b. The Method Of Characteristics was used to design a Mach 2.5 nozzle contour of
minimum length, producing a supersonic, sharp corner nozzle
design with a flow straightening throat region. This model thus
presents a type of direct-connect facility in that the acceleration
to supersonic conditions is achieved by a converging-diverging
nozzle, and not by the typical oblique shock compression ramp
inlet geometry found in flight-worthy vehicles and certain experimental ground test facilities. For reference purposes, the
positive longitudinal x-axis, vertical y-axis, and lateral z-axis
are designated in the facility schematic. The origin is located
at the cross-sectional center of the isolator inlet plane, marked
by the location of nozzle wall zero slope change.
Figure 3: Illustration of pseudo shock train nomenclature
(adapted from Ref. [4]). Flow is left to right.
First explicitly written (and referenced) by Sullins & McLafferty [15] through the work of Billig [17], a modification of the
original empirical relation for pseudo shock train lengths in axisymmetric ducts was put forth for rectangular ducts, and is
expressed in Eq. 2. Of similar form to the original (Eq. 1), the
rectangular modification included the minor axis duct height
(H), as well as a 1/5th power Reθ dependence. Experimental
data for shock train length in an aspect ratio 2.5 duct with inflow Mach numbers of 2 and 2.85 and a 1.3 and 2.5mm step (to
account for the aforementioned scramjet-mode flame holding
capability) were in good agreement with the modified empirical
relation set forth in Eq. 2.
S t (M22 − 1)Re1/5
P3
P3
Θ
= 50(
− 1) + 170(
− 1)2
H 1/2 θ1/2
P2
P2
(2)
The motivation behind proposing an additional modification for rectangular ducts lies in the fact that Eq. 2 does not
account for both the minor- and major-axis duct and boundary
layer parameters. Due to the complex three-dimensional corner
flow, it was observed that both entities play a significant role in
the location of the upstream corner separation, which in turn
Figure 4: Simulated AR 3.0 (a) and 6.0 (b) isolator test model,
dimensions normalized with duct height. Flow is left to right.
Two isolator models were tested, with aspect ratios (ARs)
of (width/height) 3.0 (Fig. 4a) and 6.0 (Fig. 4b), duct heights
of 50.8 and 25.4mm, length-to-height ratios (L/H) of 13.75 and
27.5 to allow proper shock train formation, and tunnel unstart
times of 33 and 56 seconds respectively. The AR 6.0 is a lower
wall half nozzle extension of the AR 3.0 configuration maintaining the appropriate Mach-Area relationship nozzle throat
height. Principle isolator duct dimensions are also shown for
each aspect ratio. The upper wall is equipped with 16, 0.5mm
diameter centerline static pressure ports spaced 38mm apart,
connected to a 16 channel piezoelectric static pressure module
(Scanivalve DSA3217) through the means of 1.6mm diameter
4
stainless steel tubulations and supporting Tygon tubing. These
channels were used as a rudimentary approach to track the
leading edge shock train location through monitoring boundary
layer separation pressure rise and to provide a measurement of
unstart time (time at which the shock train exits the duct and
settles inside the diverging portion of the nozzle), which is used
to subsequently normalize the time scale in each AR case.
(a)
Isolator Outlet
P15
Tunnel Start
Steady State
Complete Unstart
Shock Train
Arrival P16
Figure 7: Aspect Ratio 3.0 shock train & pseudo-Shock length
at P3 /P2 = 5.32. Marker size represents approximate measurement uncertainty.
Figure 5: Time averaged upper wall centerline static pressure
measurements for aspect ratio 3 duct.
(b)
Figure 6: Time averaged upper wall centerline static pressure
measurements for aspect ratio 6 duct.
A time history of time-averaged, static wall pressure
distribution is shown in Figs. 5-6 for the 3.0 and 6.0 case
respectively. Longitudinal pressure tap position from the
isolator inlet is given in mm and subsequently normalized by
duct height in brackets. Since this study aimed at analyzing
continuous shock train oscillation and propagation, shock train
dynamics were driven by the inherent, continuous backpressure rise of the facility and was not user controlled. Shock
arrival is characterized by the pressure rise associated with
boundary layer separation. Due to the lower mass flow rate,
the backpressure rise occurs more gradually in the aspect ratio
6.0 case, holding the shock train in the diffuser section for a
longer time than the 3.0 case. All subsequent data presented
utilize a non-dimensionalized time, τ, normalized by the time
it takes for the isolator to unstart.
B. Pseudo Shock Train Length Measurement
The centerline static wall pressure taps outlined in Figs.
4-6 were used to estimate the length of the pseudo shock region
(S t ) by monitoring the pressure rise due to boundary layer
separation associated with the shock train formation. Figure 7
shows the state of the longitudinal distribution of wall static
pressure behavior for a given timestep τ (and thus a given
P3 /P2 ). Each marker represents a static wall pressure tap,
with the markersize approximately equal to the uncertainty
of the 16ch Scanivalve pressure module. The solid dark line
represents the static pressure at each tap (Pi ) normalized
with respect to the stagnation pressure (P0 ). To find the
location of the initial pressure rise corresponding to leading
edge shock location, the current static pressure behavior was
compared to both the steady state (prior to shock train arrival
in duct) static wall pressure as shown by the pick dotted line
((Pi /P0 ) steady ) and the difference of static wall pressure between
tap (i) and the next upstream tap (i-1), as indicated by the
red dashed line (∆P = Pi − Pi−1 ). The visible shock train length
(whose trailing edge is marked with a vertical dashed blue
line in Fig. 7) as derived from the schlieren and shadowgraph
visualization efforts, was approximately 203mm, accounting for
37% of the pseudo shock length which is measured as 548mm.
Static pressure rise, as measured at the wall, accounts for 80%
of the pressure rise expected behind a normal shock position
at the pseudo shock leading edge.
C. Boundary Layer Pitot Survey
The three visualization stations, whose results and analyses
are presented in Ref. [10] and whose center axes are longitudinally positioned 150mm (inlet), 350mm (middle), and 600mm
(outlet) from the isolator inlet plane (labeled in Fig. 4a-b),
are accompanied by steady-flow lower wall vertical pitot-probe
surveys performed along the centerline. Vertical position of
the total pressure probe ranged from the lower wall to the half
duct height. This was performed to characterize the state of
the incoming flow field and flow parameters at the inlet station are used to represent the minor-axis flow variables used in
the empirical relations. The probe had a thickness of 0.4mm,
an internal diameter of 0.8mm, and a flattened tip with height
of 0.4mm, and was accompanied by a wall static pressure tap
located one probe outer diameter upstream of the tip.
Flow parameters are presented in Table 1 and resemble the
center-axis values of the parameters considered. Given the ratio
of total probe to static wall pressure, the Rayleigh supersonic
pitot-probe formula was used to solve for the Mach number.
Assuming Mach 2.5 isentropic expansion from room conditions
and a Prandtl number of 0.7 (resulting in a recovery factor of
0.89), the Walz equation (Ref. [18]) was used to calculate the
temperature variations throughout the boundary layer, allowing
the subsequent calculation of freestream velocity. Classic displacement and momentum boundary layer integrals were used
to calculate the displacement thickness δ∗ and the momentum
5
thickness θ [19].
AR3.0
M∞
U∞ (m/s)
T ∞ (K)
Pw (Pa)
δ(mm)
δ∗ (mm)
θ(mm)
AR6.0
Inlet
Middle
Oulet
Inlet
Middle
Oulet
2.44
560
131
7802
4.17
1.01
0.28
2.39
554
134
8045
6.74
2.11
0.64
2.34
549
137
8410
9.90
3.38
1.01
2.38
542
141
7412
3.89
0.55
0.17
2.31
535
144
7697
5.94
1.02
0.36
2.27
528
149
7903
7.71
1.94
0.65
Table 1: Lower wall experimental pitot survey parameters (subscripts: ∞ represents freestream value, w symbolizes wall value
Results
Given that the flow parameters of Mach number, θ, and Reθ
in the original empirical relations for shock train length (Eq.
1-2) are those at the inlet of the isolator, flow and boundary
layer parameters obtained at the inlet of the isolator aspect
ratios (Table 1) are used to solve the empirical relations for
shock train length. Error bars scale with the distance between
centerline wall static pressure taps (30mm). Root Mean
Square Error (RMSE) quantitatively evaluating the correlation
strength between the empirical curve to the experimental data
are presented for each relationship in the low (P3 /P2 < 4.25),
middle (4.25 < P3 /P2 < 5.25), and high (P3 /P2 > 5.25) pressure
ratio regimes, with results summarized in Table 3. Results
are listed in order of discussion below, accompanied by a case
number. Critical pressure ratios of 4.25 and 5.25 were chosen in
this study to highlight the behavior of the pseudo shock train
length. In the following discussion, Waltrup & Billig’s original
1973 relationship for axisymmetric ducts is referred to as Waltrup’s relation, Eq. 1, or simply the circular relation. Billig’s
later modification for rectangular ducts is referred to as Billig’s
relation, Eq. 2, or the original rectangular modification. Results are presented below and a new modification to the existing
Waltrup relation is proposed to account for additional shock
train lengths due to corner flow separation in rectangular ducts.
A. Original Empirical Relations
First, experimental measures of shock train length derived
from centerline pressure measurement was compared to the predictions of Eqs. 1-2 (cases I-II) in Fig. 8. Hydraulic diameter
(DH ) is substituted for duct diameter (D) in Eq. 1 given the
rectangular cross-section of the experimental apparatus. This
is also the first step in accounting for the aspect ratio of the
duct. For the aspect ratio 3 case, the strongest correlation
between empirical and experimental falls in the middle pressure ratio region for the original circular cross-sectional relation
by Waltrup. Differences between corner flow and center-body
flow are expected to be larger at larger pressure ratios. Confined supersonic flows in circular duct are expected to maintain
greater boundary layer symmetry around the duct perimeter
than their rectangular counterparts throughout the pressure ratio regime. For the middle pressure ratio regime, the uniformity
of the boundary layer can tend to follow the circular relation
(reduction in Case I RMSE of 70% compared to Case II), while
at higher pressure ratios the influence of the corner flow separation can cause the circular relation to under predict shock train
length as it does not account for the more severe boundary layer
asymmetry and corner flow effects (reduction in Case II RMSE
of 45% as compared to Case I).
Neither the work by Billig that initially discussed the rectangular modification (Ref [17]) nor the work by Sullins that
first presented the modified rectangular relation in mathematical form (Ref [15]) explicitly state that corner flow and threedimensional effects are taken into account in its formation. The
modification was intended to minimize departure from the original empirical relation for circular ducts which was based on an
extensive test matrix. The minor axis duct parameter H (duct
height), and the centerline values of θ and <θ maintained from
the original relation, run the risk of not fully capturing the
contribution of corner flow separation to isolator pseudo shock
train length in rectangular ducts.
Depending on aspect ratio, upstream corner flow separation
can impact measurements along the centerline since weak compression waves stemming from the separated corner boundary
layer can impart pressure gradients along the duct minor axis.
Due to the 1/5th power dependence of Reθ , the original modification in Eq. 2 predicts longer pseudo shock train lengths
for the same initial conditions of M, θ, Reθ and the same pressure ratios. Even if only centerline measurements are used, the
impact of corner flow separation can affect the centerline measurements, prompting the experiments to sense the leading edge
of the shock system earlier causing a longer overall length. The
characteristics of corner flow separation are thus emphasized in
the larger pressure ratio regime and the modified empirical relation by Billig provides a stronger correlation in Fig. 8 (RMSE
decreases by 45% in the higher pressure ratio regimes between
Case I & Case II).
The correlation for the aspect ratio 6 case is more consistent with the rectangular modification than the original circular
relationship across all pressure ratio regimes. Dynamic pressure measurements performed along the minor axis in Ref [10]
showed that corner flow effects were emphasized with an increase in aspect ratio. Pressure rise due to boundary layer
separation was recorded at the outboard station an increased
length of time prior to centerline measurement, as compared to
the aspect ratio 3 case. Thus, the rectangular modification that
could account, even if implicitly, for the effects of corner flow
separation provides a better fit for the increased aspect ratio.
Although centerline pressure measurements may sense
the upstream corner flow separation to some extent, more
accurate representation of the upstream separation length
scale is required to obtain a more representative value of total
pseudo-shock train length. Visualization efforts presented in
Ref. [10] are used to quantify this contribution.
Figure 8: Original empirical relations of Waltrup & Billig (Eq.
1 [16]) and Billig (Eq. 2 [17]) compared to centerline static wall
pressure derived pseudo shock length for aspect ratios 3 and 6.
Hydraulic duct diameter (DH ) is used in Eq. 1 in place of duct
diameter (D). Comparison referred to as Cases I & II in Table
3.
B. Corner Flow Separation
6
Visualization of the corner flow separation at the upstream
station (station 1 in Fig. 4) was used to derive an additional
‘corner flow separation’ length (S c ). Examples of corner flow
separation in both aspect ratios are shown in Figs. 9-10. It is
shown that corner flow separation occurs approximately 40mm
ahead of the center flow separation, meaning that shock train
length derived from centerline pressure measurement (S t ) would
be approximately 40mm less than the full shock train length S
(S = S t + S c ).
Sc = 188-146mm
Sc = 42mm = 0.82H
Sc = 42mm = 16.8AR
SC
ST
Figure 9: Upstream visualization of corner flow separation in
aspect ratio 3.0 isolator. Flow left to right, dimensions in mm.
[10]
SC
of the corner flow boundary layer to account for the upstream
separation component.
ST
Sc = 134-90mm
Sc = 44mm = 1.73H
Sc = 44mm = 7.3AR
Figure 10: Upstream visualization of corner flow separation in
aspect ratio 6.0 isolator. Flow left to right, dimensions in mm.
[10]
Full shock train length is plotted against the original
empirical relations in Fig. 11. With the additional corner
length separation added, Billig’s rectangular modification
(Case IV) shows a stronger correlation for the aspect ratio
3 case, with an RMSE reduction of 24% across the entire
pressure ratio regime compared to the modification without
S c added (Case II). The largest improvements lies in the
middle pressure area, where RMSE is reduced by 64%. The
aspect ratio 6 case does not respond to the addition of S c as
favorably, with correlation weakening and RMSE increasing
by over 200%. The original modified rectangular relation in
Eq. 2, relying solely on centerline flow parameters, under
predicts the experimental data across the entire pressure ratio
regime. This under prediction prompts the introduction of a
modification to the existing relations that can account for both
lower and higher aspect ratio behavior and associated flow
three-dimensionality. This modification needs to account for
the aspect ratio of the rectangular duct, as well as the state
Figure 11: Original empirical relations of Waltrup & Billig (Eq.
1 [16]) and Billig (Eq. 2 [17]) compared to total shock train
length (S t + S c ). Hydraulic duct diameter (DH ) is used in Eq. 1
in place of duct diameter (D). Comparison referred to as Cases
III & IV in Table 3.
C. Corner θ0 Modification
Waltrup’s original empirical relation was derived from circular cross-section isolator experimental data taken across a
wide range of conditions. Uniformity of the boundary layer
around the duct inner perimeter (relatively to the expected
non-uniformity in a rectangular duct) reduces the presence of
three-dimensional features such as those observed in Figs. 910. If Waltrup’s original relation is to be used for rectangular
isolator work, a modification factor accounting for the state of
the corner flow must be included. This is accomplished by the
inclusion of the minor- and major-axis momentum boundary
layer. The necessity of including the aspect ratio dependence
can be accomplished through the introduction of the hydraulic
diameter (DH ) calculated as the ratio of twice the duct perimeter over the sum of the minor and major axis. Finally, the
proven 1/4th power dependence of Reθ is maintained.
The proposed modification is presented in Eq. 3, solved for
S (total shock train length). The traditional centerline boundary layer θ is replaced by a ‘corner’ θ0 , calculated as the diagonal
between the minor-axis θ (θ1 ) and the major-axis θ (θ2 ). The
minor-axis θ1 remains the experimentally obtained θ used in
previous comparisons and listed in Table 1. The nomenclature
is visualized in the insert to Fig. 12 and calculated
by taking
q
the square root of the sum of squares [θ0 = θ12 + θ22 ]. The additional corner separation length (S c ) is added to the relation,
resulting in a non-zero shock train length for zero pressure ratio. The momentum thickness Reynolds number, Reθ0 is likewise
calculated with respect to θ0 .
"
# r 0
P3
P3
θ
DH
S = 50(
− 1) + 170(
− 1)2 (
)
+ S c (3)
P2
P2
DH (M22 − 1))Re1/4
θ0
Fig. 12 represents the case of a symmetrical
boundary
√
layer, where θ1 = θ2 and, since θ = θ1 , θ0 = 2θ (Case V in
Table 3). Using an empirical relation for shock train length
originally derived from data obtained in cylindrical ducts, a
7
fit is provided for rectangular ducts that takes into account
the upstream corner flow separation and the state of the corner boundary layer. The comparisons from Fig. 11 are shown
for reference, with the proposed modification plotted in thicker
lines. When comparing Case V to Case III, for the symmetrical
case of θ = θ1 = θ2 , improvements in RMSE performance are
shown across the entire pressure ratio regime and both aspect
ratios. Correlation in the aspect ratio 3 case is especially strong,
with the biggest improvement of RMSE occurring in the high
pressure ratio regime, where three-dimensional features are expected to be most prominent. A reduction in RMSE of 77%
is observed in the higher pressure region, with a total RMSE
improvement of 65%. Total RMSE improvements in the aspect
ratio 6 case is even higher, at 77%. As was previously seen,
three-dimensional effects due to corner flow separation are emphasized in the higher aspect ratio case, and this echos the observation made in Fig. 12 and Table 3 together with the need
for a modification to the original circular relation to account for
these effects.
be thicker than the minor axis boundary layer. The strength
of three-dimensional flow features at the higher pressure ratios
correspond to the increase in major axis momentum boundary
layer thickness.
Finally, a comparison between the original rectangular modification (Case IV) and the proposed modification with accompanying parametric study (Case V-XI) is required to comment
on the performance and applicability of the work discussed.
Comparing Case IV with the symmetric boundary layer Case
V, improvements in total RMSE of 11% and 52% are observed
for aspect ratio 3 and 6 respectively. Assuming a θ2 value 10%
larger or smaller than θ1 do not significantly alter the performance of the aspect ratio 3 case, but significant reductions in
performance are seen for a ± 25 and 50% change in θ2 .
Improvements for the aspect ratio 6.0 case are significant
as θ2 is reduced, with a total RMSE decrease of 65% comparing
the θ2 = .75θ1 case (Case X) with the original modification
(Case VI). The observed performance of the θ0 modification
further supplements the experimental observations made in
Ref. [10] in that the higher aspect ratio displays a more
prominent case of corner flow separation, adding to the total
pseudo-shock train length estimate.
D. Summary of Results
Table 3 summarizes the RMSE analysis for the cases discussed above and can be referenced throughout the discussion
above. Cases I & II correspond to the original Waltrup & Bilig
relation for circular isolators (Eq. 1) and Billig’s rectangular
modification (Eq. 2) respectively, as compared to the experimental measure of pseudo shock-train length derived from centerline measurement. Cases III & IV represent the aforementioned empirical relations but compared to the experimental
measure of total shock train length, including the upstream corner separation. Case V corresponds to the proposed modification of Waltrup & Billig’s original empirical relation to include
the state of corner boundary layer (Eq. 3) for θ = θ1 = θ2 . Finally, Cases VI-VIII correspond to Eq. 3 for θ2 = 1.1, 1.25, and
1.5θ1 , and Cases IX-XI correspond to Eq. 3 for θ2 = 0.9, 0.75,
and 0.5θ1 respectively.
AR3.0
Figure 12: Original empirical relations of Waltrup & Billig (Eq.
1 [16]) with corner θ0 modification.
For a facility with a two-dimensional nozzle, a non-uniform
boundary layer can be expected. Nozzle bounded upper and
lower wall boundary layers (duct minor axis boundary layers)
are expected to differ from the sidewall boundary layer (duct
major axis boundary layer). Likewise, the differences in wall
material for the upper and lower walls (shop ground Aluminum
6061) and sidewalls (BK 7 glass) would result in different momentum deficit profiles due to the different viscous dissipation
effects.
Although differences in boundary layer profiles would be
minimum at the isolator inlet, where θ0 would be considered,
cases where θ2 6= θ1 must be considered. In lieu of detailed side
wall boundary layer pitot survey planned for follow-up work,
a few cases varying the value of θ2 are presented in Table 3 in
terms of their RMSE values. As before, θ1 = θ and corresponds
to the experimentally determined values shown in Table 1.
θ2 values for 10, 25, and 50% above and below θ1 are considered. It is shown that for θ2 < θ1 (Cases IX-XI), the empirical relation offers more agreeable fits to the experimental data in the
middle pressure ratio regime than for the θ2 > θ1 cases (Cases
VI-VIII). However, the higher pressure ratio regimes are characterized by stronger correlations for the θ2 > θ1 cases. Due to the
two-dimensional nozzle bounding the minor axis (δ1 ) boundary
layer, it is expected that the major axis (δ2 ) boundary layer will
AR6.0
Case
Low
Middle
High
Low
Middle
High
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
31.6
22.2
70.1
37.2
19.0
18.3
18.1
20.2
20.1
22.4
26.7
18.7
61.2
50.4
22.4
26.8
32.1
40.2
54.0
21.9
15.7
12.4
76.8
42.7
114.0
34.1
38.1
34.8
32.7
37.2
42.4
50.0
60.0
9.8
5.9
48.6
35.2
7.1
9.1
12.2
17.7
5.3
3.8
5.8
20.4
8.8
59.0
34.4
10.7
14.2
19.8
29.7
8.0
7.1
12.6
33.7
30.2
66.0
34.5
32.5
36.6
43.7
56.7
29.2
26.4
27.4
Table 2: Root Mean Square Error Analysis results for aspect
ratio 3 and 6. ‘Low’ pressure ratio represents P3 /P2 < 4.25,
‘Middle’ pressure ratio represents 4.25 < P3 /P2 < 5.25, and ‘
High’ represents P3 /P2 > 5.25
Discussion & Conclusions
The ability to predict shock train length for specific incoming flow parameters is critical in designing an isolator that is
capable of containing the shock train present in ramjet-mode
tasked with supporting the required pre-combustion pressure
rise and avoiding CII events. Classic empirical relations for
shock train length presented by Waltrup & Billig (Ref [16]) for
8
AR3.0
AR6.0
Case
Low
Middle
High
Low
Middle
High
I
II
III
IV
V
31.6
22.2
70.1
37.2
19.0
18.7
61.2
50.4
22.4
26.8
76.8
42.7
114.0
34.1
38.1
9.8
5.9
48.6
35.2
7.1
20.4
8.8
59.0
34.4
10.7
33.7
30.2
66.0
34.5
32.5
Table 3: Root Mean Square Error Analysis results for aspect
ratio 3 and 6. ‘Low’ pressure ratio represents P3 /P2 < 4.25,
‘Middle’ pressure ratio represents 4.25 < P3 /P2 < 5.25, and ‘
High’ represents P3 /P2 > 5.25
circular cross-section isolators and a later modified for rectangular cross-sections by Billig ( [17]) were compared to experimental data inside a Mach 2.4 isolator duct of aspect ratio 3 and
6. Centerline static pressure measurements were first used to
calculate the traditional, centerline-derived pseudo shock train
length, maintaining the original nomenclature of S t . Visualization of upstream corner flow separation length (S c ) presented in
Ref. [10] were then used to calculate a total pseudo shock train
length (S). Finally, a modification to Waltrup’s original relation
for circular cross-section isolators is presented to account for the
state of the corner flow boundary layer. This modification includes both minor- and major- axis contributions, as boundary
layer non-uniformity along the inner duct perimeter is expected
to play a role in the effect of upstream corner flow separation
on total pseudo-shock length. Additionally, the dependence on
aspect ratio is introduced through the hydraulic diameter.
Root Mean Square Error (RMSE) analysis for three regions
of pressure ratio magnitude was used to quantify the correlation strength between experimental data and empirical relation
and used as a preliminary performance parameter. Comparing the pure centerline derived shock train length experimental data to the original relations (Fig. 8), the original circular
cross-section relation provided a relatively strong correlation
for the middle pressure ratio regime. The increased effects of
flow three-dimensionality and corner flow separation at higher
pressure ratios however, cause the correlation to shift in favor of
the original rectangular modification. The original modification
presented by Billig (Eq. 2 predicts longer shock trains for the
same inflow parameters due to the 1/5th power dependence of
the Reθ term. The effects of three-dimensional flow features and
the presence of upstream corner flow separation causing longer
isolator shock trains are expected to be negligible in the more
uniform boundary layer of the circular duct, with a 1/4th power
dependence of the Reθ term
Adding the experimentally derived corner shock train length
magnitude, S c , to the centerline derived calculations of pseudo
shock train length (S t ) provides a value of total pseudo shock
train length (S). As expected, the rectangular modification proposed by Billig provides a much stronger correlation than its
original circular counterpart when accounting for the upstream
separation length. Comparison between the modification with
and without S c (Case II and IV respectively) was used as a
benchmark in evaluating the applicability of the modification
for different aspect ratios. While the empirical relation still over
predicted pseudo shock train length, agreement in the aspect ratio 3 case was improved by 24% across the entire pressure ratio
regime. The aspect ratio 6 case did not exhibit similar improvements, as the empirical relation under predicted pseudo shock
train length and total RMSE increased by over 200% compared
to the case where no additional S c was added.
To expand on the Waltrup’s original empirical relation for
circular ducts, an additional modification for rectangular crosssection ducts is introduced to include the effects of both aspect ratio magnitude and corner flow composition by including
duct minor- and major-axis boundary layer parameters (Eq.
3). Maintaining the same 1/4th power dependence, the contri-
butions of both the minor- and major-axis are considered in
the forms of θ1 and θ2 , contributing to a θ0 value that is substituted for the original centerline θ. Inclusion of the θ0 model
increased both θ0 and Reθ0 , as compared to the traditional centerline θ and Reθ . The effects of aspect ratio is included in the
form of the hydraulic diameter. Substituting the minor-axis
duct height (H) with the larger hydraulic diameter (DH ) results
in longer
shock train lengths.
Combining these contributions,
"
#
q
1
θ0
the DH ( DH ) 2
term on the right hand side of Eq. 3
1/4
(M −1))Re 0
θ
contributes to a longer pseudo shock train length estimate for
both aspect ratios than the original rectangular modification
using H, θ, and θ0 .
Under the symmetric boundary layer assumption, where
θ1 = θ2 , correlation between the experimental data for total
shock train length and the proposed modification is stronger
than the original rectangular modification put forth in Eq. 2,
with reduction in RMSE of 11% and 52% respectively in the
aspect ratio 3 and 6 case. The previously observed emphasis on
three-dimensional corner flow separation effects in the aspect
ratio 6 case from Ref. [10] is further presented in the present
analysis, as large improvements in correlation strength are obtained by including the corner flow boundary layer parameter.
Follow up work is to include detailed side-wall boundary
layer pitot surveys to ascertain the differences between minorand major-axis momentum boundary layer thickness, as only
experimentally derived boundary layer parameters on the lower
wall centerline (minor axis) were used in the present student. To
explore the capabilities of the corner θ0 modification presented,
a brief parametric study varying the value of θ2 with respect to
θ1 is considered. In the aspect ratio 3 case, only small deviations
from the symmetric θ0 performance are observed by increasing
or decreasing θ2 by 10% of θ1 , while the correlation weakens as
greater changes to θ2 are introduced.
The sensitivity of the higher aspect ratio 6 case to the contributions made by the corner boundary layer is observed in the
significant improvements in correlation strength brought about
by the symmetric θ0 assumption. Further improvements are obtained when θ2 < θ1 with the correlation RMSE decreasing by
65% as compared to the original rectangular modification when
θ2 = 0.75θ1 . Due to the 2D nozzle profile bounding the minoraxis boundary layer, the major-axis boundary layer is expected
to be fuller (δ2 > δ1 ), yet differences in minor- and major-axis
wall composition (shop ground 6061 Aluminum and BK-7 glass
respectively) can cause the minor axis to experience a larger
momentum thickness (θ) as θ represents a local measure of the
momentum deficit in the boundary layer as compared to the inviscid fluid stream. The momentum thickness integral is more
sensitive to the shape of the velocity profile than its displacement counterpart, and thus potentially larger drag forces and
viscous effects on the minor-axis wall can increase the momentum thickness of the minor-axis boundary layer. This hypothesis will be examined with the planned side-wall boundary layer
pitot probe survey.
It has been shown in both Ref. [10] and the present work
that rectangular aspect ratio isolator design and performance
analysis can no longer be based on observations and measurements made along a single axis. Three-dimensional corner flow
must be taken into account to improve both shock train length
and leading edge location estimation. Improved understanding
of the shock train/boundary-layer interaction along the major
and minor axis and in the corner regions will contribute greatly
to the development of more robust isolator design relations and
shock train position control schemes.
Acknowledgements
The study was conducted as part of the UMD-AEDC Hypersonic Center of Testing Excellence project, sponsored by
AFOSR, TRMC, and AEDC. The authors acknowledge the
financial support from the Air Force Office of Scientific Research under grant FA9550-10-1-0535, managed by Dr. Michael
9
Kendra. The authors also acknowledge the valuable technical
support received from Dr. Allen Winkelmann of the University
of Maryland and Mike Smith and Dr. Eric Marineau of the
AEDC Hypervelocity Wind Tunnel 9.
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