12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

MODAL PROPERTIES OF CROSS-FLOW INSTABILITY IN COMPRESSIBLE BOUNDARYLAYERS
Zhiyong Liu
Aerodynamics Science Center
China Academy of Aerospace Aerodynamics
No. 17 Yungang west road, Fengtai District, Beijing 100074, Chinaliuzhiyongtv@163.com

ABSTRACTThe present work is concerned with the stability
characteristics of compressible boundary layers with crossflow
on a flat plate. Based on a boundary-layer model on a flat plate,
Linear Stability Theory (LST) is used to investigate the modal
properties of cross-flow instability. The results reveal thatasymmetry and negativeness of frequency are fundamental
features for crossflow-affected instability, and two theorems
are proposed to reveal the intrinsic details. Particularly, theconjugate mode is introduced to make the negative-frequency
disturbance suitable for physical applications. In addition, theassociation between cross-flow mode and Mack's mode isclarified. It is found that the cross-flow instability is integrated
with Tollmien-Schlichting (T-S) mode or the first mode, and it
cannot be distinguished as a single mode for most cases.

dominance of different instabilities with increasing strength ofcrossflow in base flow.
Such understanding about primary instability sheds light on
the research of receptivity (Schrader, Brandt & Henningson
2009; Balakumar & King 2012; Xu et al. 2018), secondary
instability (Högberg & Henningson 1998; Malik et al. 1999;
Xu et al. 2019) and flow control (Wassermann & Kloker 2002;
Schuele, Corke & Matlis 2013; Corke et al. 2018) of crossflow instability-dominated boundary layers. Numericalsimulations (Pruett, Chang & Streett 2000; Dinzl & Candler2017) and experimental tests (White & Saric 2005; Craig &Saric 2016) have benefited from the theoretical studies as well.
Nevertheless, some common and fundamental issues aboutcross-flow instability still need to be figured out. Firstly, boththe cross-flow mode and Mack mode exist in compressible 3-Dboundary layers, whereas the association and difference havenot been elucidated. The boundary-layer stability analyses on ayawed cone (Li et al. 2016) and an elliptic cone (Paredes et al.
2016) indicated distinct features between cross-flow instability
and Mack's second-mode instability. However, the comparison
between the evolvements of slow mode in 2-D (Ma & Zhong2005) and 3-D (Xu et al. 2018) compressible boundary layers
implies some connection existing between cross-flow modeand Mack mode. Moreover, the relationship between travelingcross-flow mode and Mack's first mode has not been addressed,
despite that the frequency band of the two modes overlaps
largely. Similar situation exists as well for traveling cross-flowmode and T-S mode in low-speed boundary layers. Secondly,
it is not well understood that how do the stability
characteristics change when crossflow plays a role in a 3-D
boundary layer. Specifically, the changes of interest include
quantitative variations of growth rate, unstable region anddominance of different instabilities with increasing strength ofcrossflow in base flow. Perhaps the most striking phenomenon
caused by crossflow is the occurrence of zero- and negativefrequency cross-flow modes, in which the zero-frequency(stationary) one was proved to be crucial to secondary
instability and nonlinear interactions (Malik, Li & Chang 1994;
Malik et al. 1999; Pruett, Chang & Streett 2000; Craig & Saric
2016). The critical value, if it exists, of crossflow to induce
cross-flow mode with non-positive frequency has not been
unraveled yet. Thirdly, significance and characteristic ofnegative-frequency cross-flow mode have not been
demonstrated in detail. Cross-flow instability with negativefrequency region has been discovered for long (Mack 1984;
Högberg & Henningson 1998; Itoh 1996). Mack (1984)
pointed out that the opposite sign of frequency indicatedopposite propagation directions of disturbances, which wasconsistent with the statement of Itoh (1996). However, theinterpretation about negative-frequency mode is not in direct
accordance with the eigen solution of LST equations. Moredetails need to be disclosed to support a more satisfactory
explanation.

INTRODUCTION
Crossflow is commonly encountered in three-dimensional(3-D) boundary layers, such as swept wings, yawed cones andhigh-speed flight vehicles with generally 3-D configurations. It
is well accepted that the existence of inflection point often
indicates an inviscid instability, which is usually called crossflow instability. In most of the 3-D boundary layers, cross-flow
instability co-exists with streamwise instabilities, like
Tollmien-Schlichting (T-S) waves and Mack's modes. While incontrast with streamwise instabilities, cross-flow instability
exhibits unstable disturbances that are traveling as well as
stationary. Linear Stability Theory (LST) shows larger
amplification rates for the traveling modes than the stationary
ones. Concerning the crossflow-induced boundary-layertransition, however, stationary waves play an important role in
low-turbulence environments, while the traveling waves
dominate in high-turbulence environments (Saric, Reed &White 2003).
Increasing understanding of cross-flow instability has beenachieved in past decades in experiments and numericalsimulations. Nevertheless, some common and fundamental
issues about cross-flow instability still need to be figured out.
Firstly, both the cross-flow mode and Mack mode exist incompressible 3-D boundary layers, whereas the association anddifference have not been elucidated. The boundary-layer
stability analyses on a yawed cone (Li et al. 2016) and an
elliptic cone (Paredes et al. 2016) indicated distinct features
between cross-flow instability and Mack's second-mode
instability. However, the comparison between the evolvementsof slow mode in 2-D (Ma & Zhong 2005) and 3-D (Xu et al.
2018) compressible boundary layers implies some connection
existing between cross-flow mode and Mack mode. Secondly,
it is not well understood that how do the stability
characteristics change when crossflow plays a role in a 3-D
boundary layer. Specifically, the changes of interest include
quantitative variations of growth rate, unstable region and1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
In addition, in order to shed some light on the generalcharacteristics of cross-flow instability, analyses are madeunder different Mach numbers, Reynolds numbers andtemperature boundary conditions. However, systematic
knowledge of cross-flow instability can hardly be obtained
from case-by-case investigations aforementioned. Mack (1984)
presented a comprehensive study of the linear stability of 2-D
compressible boundary layers, based on the mean flow with anexact solution. While for 3-D compressible boundary layers,
the counterpart research has been substantially hindered due tolacking a simplified solution of laminar base flow. Therefore,
an appropriate boundary-layer solution needs to be obtained toinitiate the comprehensive stability analysis of cross-flow
instability.
Following the research paradigm of Mack (1984), we
developed a model problem for compressible boundary layerswith parametric crossflow to study the linear instabilities dueto the presence of typical crossflow. The main purpose ofpresent work is to illustrate some fundamentals of cross-flow
instability and to clarify how it associates with the instability
modes already known in 2-D boundary layers.

compressible boundary-layer equations. By introducing stream
function and Illingworth-Stewartson transformation, andassuming Falkner-Skan external potential flow in x direction,
we obtain an ordinary-differential-equation (ODE) system
under local-similarity assumption (Liu, 2021).

 Nf   ff  

 Ng   fg   0,
 N  2 
2 
     S  1  N  f      K  1 S  N  g   Pr  f     S  1  f 2    K  1 S  g 2    0.
Boundary conditions are written as  0 : f  f   g  0,    w or    0,

   : f   g    1.

y
x

u
w

inviscid streamline

Q
P


(1)

(2)
(3)

(4)

(5)
Based on the solution of the ODE system, the streamwise
and crosswise velocity components can be written as
L2 f  cos2   g sin 2 (6)u,
L2 cos2   sin 2 L  g  f  cos  sin (7)w
.
L2 cos2   sin 2 Such flow model is named Falkner-Skan-Cooke(FSC)
boundary layer. The crossflow distributions under different
flow parameters are displayed in Fig. 2. The influence of local
swept angle on crossflow in an incompressible boundary layer
is demonstrated in Fig. 2(a). The crossflow deviates from zeroand gets stronger with the increase of local swept angle until alargest maximum is reached on the profile. The largest
maximum of crossflow occurs precisely at Λ=45o. In addition,
the distributions of crossflow under two swept angles, which
are mutually complementary, are identical. However, slightly
different results appear in compressible boundary layers, where
the largest maximum occurs at around Λ=70o and no special
relations are observed for different local swept angles (Fig.
2(b)). Pressure-gradient parameter affects the crossflow both inmagnitude and direction, as shown in Fig. 2(c). Favorablepressure gradients produce positive crossflow components,
while adverse pressure gradients act otherwise. Meanwhile, forpositive values of m, it is observed that pressure gradient withlarger magnitude produces stronger crossflow but thinnerboundary layer. Lastly in Fig. 2(d), wall cooling is found to
weaken the crossflow as well as thin the boundary layer.
The crossflow exhibits a variety of profiles under different
combinations of flow parameters, which provides rich diversityof base flow. Corresponding variations of crossflow-induced
stability characteristics are expected to appear in the 3-Dboundary layers.
The crossflow in current model is controlled by pressuregradient parameter m and local swept angle Λ. On the one hand,when the local swept angle vanishes, the system is reduced tothe one describing a 2-D compressible boundary layer with
self-similarity (Stewartson 1964). On the other hand, when
Mach number approaches zero, the system is reduced to the
Falkner-Skan-Cooke flow describing an incompressible 3-Dboundary layer (Cooke 1950). Therefore, the well-known
stability characteristics in classical boundary layers can beextended to 3-D compressible conditions with the aid ofcurrent base-flow model.

BASE FLOW
A model problem for compressible boundary layer isestablished on an infinite flat plate (Fig. 1). The Cartesian
coordinate system is set up as follows. The y direction points
normal to the plate. Pressure gradient is equipped in parallelwith the plate. The direction of pressure gradient is designatedas x direction in Cartesian coordinate system, and the pressure
gradient occupies the area over the plate. The z direction points
normal to the x-y plane. The incoming flow consists of two
velocity components in x and z directions, respectively.
Depending on favorable or adverse pressure gradient, theinviscid potential flow accelerates or decelerate in x direction
on the plate, while the z-direction velocity keeps unchanged.
The pressure gradient in misalignment with velocity direction
curves the inviscid streamlines over the flat plate. Similar flow
configuration was used to study cross-flow instabilities inincompressible (Wassermann & Kloker 2003) andcompressible (Tempelmann, Hanifi & Henningson 2012; Xu etal. 2018) conditions.

z2m
 f 2    , m  1 S

(ue,ref, 0, we)Figure 1. The Cartesian coordinate system on a semi-infinite
flat plate. The local swept angle Λ denotes the angle between
directions of velocity and pressure gradient in potential flow.
The subscripts '∞', 'e' and 'ref' represent incoming flow,boundary-layer edge and reference point, respectively.
For the infinite conditions in z direction, the variables ofthe steady laminar base flow are independent of z. Reference
point P is appointed at x=xref, where the local flow is concerned.
The base flow is governed by quasi-three-dimensional2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022(d ) 0

(a) 0

-0.01

-0.01

=10
o
=20
o
=30
o
=40
o
=45
o
=50
o
=60
o
=70
o
=80

-0.02

-0.03

-0.02

Tw=1
Tw=2
Tw=3
Tw=4
Tw=5
Tw=6
Tw=Tad
Tw=8

w

w

o

-0.03

-0.04
-0.04

0

2

4

6

8

0

y

(b)

10

15

20

y

0

Figure 2. Variations of cross-flow profile with (a) local swept
angle at Ma=0.001 (m=0.1), (b) local swept angle at Ma=6.0(Tw=Tad, m=0.1), (c) pressure-gradient parameter at Ma=6.0(Tw=Tad, Λ=30o) and (d) wall temperature at Ma=6.0 (m=0.1,
Λ=30o). Tw≈7.0 for an adiabatic wall in this case.

-0.02

-0.04

=10
o
=20
o
=30
o
=40
o
=50
o
=60
o
=70
o
=80

w

o

-0.06

-0.08

-0.1

5

0

5

10

15

LINEAR STABILITY THEORY
The modal properties of boundary-layer instabilities areanalyzed with LST, which is well known and introduced inquite a lot literatures, say Mack (1984). Two facts arepresented as theorems to elaborate the characteristics ofcrossflow-related instability. The proofs of the two theorems
are straightforward, thus not presented here.
The Conjugate-Solution Theorem is stated as below.
Given ,  ,  ,   as the solution of LST equation, the

20

y



(c ) 0.04



conjugate solution ,  ,   ,  exists.

 denotes the conjugate of α, and so do  and  . The
related solution is named conjugate solution, or conjugate
mode to each other. Considering all of the parameters ascomplex, it can be seen that the conjugate solution of, ,  ,  is actually the negative and conjugate combination.

0.02

w

0

m=-0.05
m=0.00m=0.05m=0.10m=0.15m=0.20m=0.25

-0.02

-0.04

-0.06
0

5

10

15

20

Therefore, when a mode with positive frequency is found,
there must exist a negative counterpart, and vice versa. Basedon such association, a negative-frequency disturbance, whichmight be deficient in physical significance, relates to a
positive-frequency disturbance with opposite wave-number
vector and equal growth rate (-αi). This provides alternativeinterpretation for the disturbance with negative frequency. By
revisiting Mack's assertion, it can be concluded that thenegative frequency indeed signifies propagation-direction
reversal of disturbance, which, however, corresponds to adifferent eigen solution. LST results about conjugate mode aredemonstrated in next section. It should also be noted that thefact about conjugate mode holds for linear stability in allparallel flows, as long as equation and boundary conditions arehomogeneous.
Compared with 2-D flows, three-dimensionality of baseflow in 3-D flows gives rise to deviation of stability
characteristics. Different from the three-dimensionality of
Mack's first mode in 2-D boundary layers, whose mostamplified disturbance is three-dimensional, crossflow in 3-Dbase flows results in three-dimensionality of stability
characteristics in two folds. Firstly, the cross-wise perturbation
velocity ŵ is non-zero for zero cross-wise wave number β.

25

y

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
Secondly, the linear growths of disturbances with opposite
wave angles are not identical with respect to the streamwise
direction. The somewhat straightforward results are supportedby the fact that asymmetry of eigen solution arises due to theappearance of crossflow. This fact can be inferred from theSymmetry Theorem below.
In a 2-D base flow (w=0), given ,  ,  ,  ˆ , uˆ, vˆ, wˆ , Tˆ as



the



solution

LST





equation, a symmetric solutionˆ
exists. Especially if   0 ,,  ,   ,  ˆ , uˆ, vˆ,  wˆ , T

offrequency regions. This agrees with the result of Ma & Zhong(2003). Mack's second mode is observed to be more unstable
than the first mode for 2-D disturbances, with and without theeffect of crossflow. However, crossflow seems to stabilize bothmodes and shift the unstable frequency band in comparison
with the 2-D boundary layers. Phase velocities are plotted forMode S as well as Mode F in Fig. 4(b). Apparent deviation isobserved for 2-D and 3-D base flows. While the local swept
angle seems to make little effect on the phase velocity in 3-Dbase flows.



(a)

wˆ  0 .

0.002

As a result, the stability characteristics, such as growth rate,
wave angle and phase velocity, are symmetrically distributedwith respect to β. This also implies that the disturbances
pertinent to the symmetry solutions propagate with symmetricshape and identical growth rate in symmetric directions withrespect to the streamwise direction. When crossflow plays arole in base flow, the proof of the theorem is ruined andconsequently, the symmetry of the solution is broken. LST
results about the asymmetry are demonstrated repeatedly inwhat follows.

Mode S

-i

0.001

0

-0.001RESULTS
Based on the base flow introduced above, spatial theory ofviscous stability analysis is investigated. The boundary layer at
Mae=4.5 is firstly considered to investigate the effects ofcrossflow on Mack's first and second modes, which arecommonly encountered in 2-D conditions. The flow condition
is chosen to be the one in wind tunnel (Kendall 1975), i.e.,
Te*=65.15K and unit Reynolds number Ree*=7.2×106m-1. The
superscript '*' denotes the dimensional quantity and an
adiabatic wall is assumed. This condition is also used by Ma &Zhong (2003) for boundary-layer stability analysis. The
subscript 'e' will be dropped in what follows.

-0.002
0

(b)

0.02

0.04

r

0.06

0.08

0.1

m=0.0,=0o
m=0.1,=0o
m=0.1,=5o
m=0.1,=10o
m=0.1,=15o
m=0.1,=30o
m=0.1,=45o
m=0.1,=60o
m=0.1,=75o

0.004

-i

0.003

Basic characteristics of linear stability
0.002

Based on the knowledge of eigenmode in 2-D
compressible boundary layers, global method (Malik 1990) is
firstly used to identify the unstable mode from 2-D to 3-D
conditions. Fig. 3 shows the spectra of eigen-values obtainedwith QZ algorithm. A discrete mode with positive growth rateis discovered for a 2-D boundary layer and highlighted by acircle in Fig. 3(a). The unstable mode is the well-known
Mack's first mode, which is also the slow mode (Mode S)
according to the terminology proposed by Fedorov & Tumin(2011). The result resembles the one obtained by Ma & Zhong(2003) except the fast mode (Mode F), which lies among thecontinuous spectra on the left of Fig. 3(a) and it is notdiscernible unless the frequency is moderately large. Eigenvalue spectra are also obtained for 3-D boundary layers withcrossflow by imposing pressure gradient and increasing local
swept angle. Mode S is still found to be the only unstable modein each case and its trajectory is illustrated in Fig. 3(b). Startingfrom the 2-D condition without and with pressure gradient, themode travels away due to the appearance of crossflow. Basedon such observations, Mode S is identified to be the target
eigenmode for crossflow induced instability analysis. An
iterative technique will be used to solve the eigen-value
problem for higher efficiency, and the result obtained by global
method can serve as a first guess.
The growth rate and phase velocity are calculated for 2-D
mode (Fig. 4). Mack's first and second modes, both of whichcorrespond to Mode S, locate obviously at two distinct

0.001

0.048

0.05

0.052

r

Figure 3. Eigen-value spectra for boundary layers with
Ma=4.5, Re=2000, ω=0.04 and β=0.1. (a) Discrete andcontinuous spectra for a 2-D base flow. (b) Trajectory of Mode
S for 3-D base flows.
(a)om=0.0,=0
m=0.1,=15o
m=0.1,=45oo
m=0.1,=75

0.004

Second mode

-i

0.003

0.002

0.001

0

4

First mode

0

0.05

0.1

0.15



0.2

0.25

0.3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
m=0.0,=0o,Mode S
m=0.0,=0o,Mode F
m=0.1,=15o,Mode S
m=0.1,=15o,Mode F
m=0.1,=45o,Mode S
om=0.1,=45 ,Mode Fom=0.1,=75 ,Mode S
om=0.1,=75 ,Mode F

(b)
1.3

1.2

cph

1.1

1

0.9

0.8

0.7

0.05

0.1

0.15

0.2



0.25

0.3

0.35

Figure 5. Growth-rate contours of the unstable modes inboundary layers (Ma=4.5, Re=2000, m=0.1) for (a) Λ=0o and(b) Λ=60o.

Figure 4. Distributions of (a) growth rate and (b) phasevelocity against frequency for 2-D modes (Ma=4.5, Re=2000,
β=0).

(a)

3-D stability characteristics are more concerned inboundary layers with crossflow. Based on the results of 2-D
mode, eigen-value problems are solved for non-zero crosswise
wave number, i.e., β≠0. The growth rate of Mode S is
contoured in Fig. 5 at Λ=60o, around which the maximum ofcrossflow in base flow reaches the largest value. The result of a
2-D boundary layer (m=0.1, Λ=0o) is also presented forcomparison. Two separate unstable regions with local maxima
are observed and identified to be Mack's first mode and second
mode, respectively. Symmetry in growth-rate distribution andnegativeness in frequency are two marked features. For the 2-D
base flow, the growth-rate distributions of Mack's first andsecond modes are strictly symmetric with respect to thehorizontal line β=0, as shown in Fig. 5(a). The unstable regionsof the first mode for positive and negative β coalesce. While
the two regions are separate at lower Reynolds numbers withstill symmetric distribution of growth rate. The result is ingood agreement with the prediction of the Symmetry Theorem
given above. Furthermore, the disturbance of crosswise
velocity in 2-D base flow is zero for the first and second mode
(Fig. 6), as predicted by the Symmetry Theorem.

0.08

2-D disturbance, 2-D base flow
3-D disturbance, 2-D base flow
2-D disturbance, 3-D base flow
3-D disturbance, 3-D base flow

|w'|

0.06

0.04

0.02

0
0

10

20

30

40

50

y
(b)
2-D disturbance, 2-D base flow
2-D disturbance, 3-D base flow
3-D disturbance, 3-D base flow

|w'|

0.04

0.02

0
0

10

20

30

40

y

Figure 6. Distribution of disturbing crosswise velocity for 2-D
and 3-D most amplified disturbance of the (a) first mode and(b) second mode. Λ=0o for 2-D base flow and Λ=60o for 3-Dbase flow (Ma=4.5, Re=2000, m=0.1).
In contrast, when crossflow plays a role in base flow, thesymmetry is broken for both first and second modes.
Meanwhile, the crosswise disturbing velocity becomes nonzero for 3-D disturbances in 2-D base flows, and for the 2-D
and 3-D disturbances in 3-D base flows (Fig. 6). Under theeffect of negative-value crossflow produced by favorablepressure gradient, the unstable regions move towards the512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
direction in which β gets larger (Fig. 5(b)), i.e., opposite tocrossflow direction. The asymmetry leads directly to the resultthat 3-D disturbance becomes the most amplified for thesecond mode, different from that in a 2-D boundary layer.
More striking changes are observed for the first mode, whose
unstable region is found to become almost one-sided. Firstly,
the maximum growth rate exceeds that of the second mode,
while the reverse is true in a 2-D case as demonstrated in Fig.
5(a). This implies that crossflow produces greater effect on thefirst mode than on the second mode. Secondly, the unstablefrequency band of the first mode extends to comprise the zeroand negative frequencies, which is unique in 3-D boundary
layers. The zero-frequency unstable mode is the commonly
encountered stationary cross-flow mode, and the more
amplified disturbance with positive frequency is the travelingcross-flow mode. However, it is noted that the so-called crossflow mode is no other than the slow mode, or Mack's firstmode, depending on the terminology chosen. Crossflow in baseflow does not essentially create a new unstable mode.
Therefore, it seems more appropriate to name a stationary or
traveling first mode rather than a cross-flow mode. This fact is
supported by the numerical findings in a 3-D high-speedboundary layer presented by Xu et al. (2018). They obtained
the disturbing crosswise velocity of Mode S, resembling thecurrent result as shown in Fig. 6(a). It was illustrated that
unstable traveling cross-flow wave can be evolved from ModeS upstream.
It might be argued that cross-flow mode is naturally
qualified to name the unstable mode particularly induced bycrossflow. Nevertheless, conventional usages of mode disagreewith this argument. On the one hand, based on the terminology
of Fedorov and Tumin (2011), the cross-flow mode is thesolution related to Mode S. Thus, it cannot stand as a mode
coequal to Mode S or Mode F. On the other hand, based on theterminology of Mack (1984), the cross-flow mode belongs to
the deviated Mack's first mode. No distinguishable features areobserved pertaining uniquely to cross-flow mode, if it exists.
Thus, it cannot be a coequal mode as the first or second mode
as well. Therefore, cross-flow mode is less appropriate tosignify the crossflow-induced unstable mode. This viewpoint
will gain more support from stability analysis in the following
subsections.

When the maximum of crossflow exceeds a threshold value(around 0.012 under such condition), stationary first mode
begins to be unstable. At even larger cross-flow strength, thestationary first mode gets more unstable with broader
bandwidth of crosswise wave number, and negative-frequency
disturbances become unstable as well, as shown in Fig. 5(b).
According to the Conjugate-Solution Theorem presented above,a conjugate solution can be easily obtained based on a known
solution by turning the parameters into their negative andconjugate counterparts. Fig. 7 shows the growth-rate contoursof Mode S as well as the conjugate Mode S. The contours arerotationally symmetric, while the two modes are related todifferent eigen solutions. Thus, the growth-rate contours cannot
connect with each other continuously. The plane in Fig. 7 isdivided into quadrant I, II, III and IV. The unstable wave ofMode S with negative frequency are surrounded with dashedlines in quadrant II, and the counterpart surrounded with solidlines in quadrant IV bears positive frequency. Therefore, when
the negative-frequency mode needs to be considered, a more
physical choice is to consider its conjugate mode with positivefrequency and opposite wave numbers. The puzzle of negativefrequency mode is thus solved by resorting to the conjugate
mode. In addition, as the usually-encountered unstable wave ofMode S in quadrant I has positive frequency, its conjugate
mode in quadrant III is far less concerned. Exceptions may be
encountered in boundary layers with adverse pressure gradient.
In such cases, the negative-frequency unstable region is
stretched from quadrant IV to quadrant III, thus the counterpart
in quadrant I is worth consideration. It is also observed that,
despite the rotational symmetry of conjugate modes, symmetry
with respect to crosswise wave number does not exist. This
indicates that the distributions of growth rate and unstablecrosswise-wavenumber band are not symmetric for a given
disturbance, unless its frequency is zero. The current results
combined with the Conjugate-Solution Theorem, provides analternative approach to understand and treat negativefrequency mode. The understanding is in accordance with thatof Mack (1984) and Itoh (1996). However, evidence has long
been absent until it is supplemented theoretically in current
work.
It should be noted that conjugate mode is significant in 3-D
boundary layers regardless of Mach numbers. Borodulin et al.
(2019) measured experimentally the unsteady cross-flow
instability in incompressible boundary layers on a swept airfoil.
The crosswise-wavenumber spectra reveal two unstable modes
for a given positive frequency. The mode with negative
crosswise wave number is named Mode 1 therein, and theother Mode 2. The Mode 1 has smaller growth rate andnarrower unstable crosswise-wavenumber bandwidth. Besides,
the two modes exhibit symmetry only for quasi-stationary
disturbances. Based on such observations, it is believed that theMode 2 is the cross-flow mode, as called in that paper, and theMode 1 is the conjugate mode.
The boundaries of growth-rate contours are extracted todenote unstable regions and plotted in Fig. 8(a) for various
Reynolds numbers. The unstable region enlarges along with
the increase of Reynolds number, and the stationary
disturbance emerges at a Reynolds number of around 400. At
moderately small Reynolds numbers, the unstable region of thefirst mode is completely one-sided. The maximum growth rates
and corresponding wave angles are calculated for the first andsecond modes. The 2-D and 3-D results are plotted forcomparison, as shown in Fig. 8(b) and Fig. 8(c). The growth
rates of both modes are found to increase with Reynolds
number in a 3-D boundary layer, which indicates a stabilizing
effect of viscosity as in a 2-D case. Thus, the crossflow seems

Figure 7. Growth-rate contour of conjugate modes (Mode S,
Ma=4.5, Re=2000, m=0.1, Λ=60o).
With the increase of cross-flow strength in base flow, theunstable region of the first mode deviates from that shown in
Fig. 5(a). For weak crossflow, stationary mode stays stable.

612th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
not to alter the viscous effect. While the crossflow does change
the specific instability characteristics, like growth rate andwave angle. Apparent discrepancies in maximum growth rate
are observed for both modes (Fig. 8(b)). The first-mode growthrate in a 3-D base flow is unanimously larger than that in a 2-D
base flow. The reverse is true for the second mode. The
contrary effects mean that the first mode is destabilized, while
the second mode is stabilized by crossflow. Consequently, thefirst mode becomes more unstable than the second mode in a
broader Reynolds-number range. In addition, the mostamplified disturbances for both modes are found to propagate
in more tilted directions under the effect of crossflow. The
wave angle of the second-mode disturbance with maximum
growth rate varies between 4o and 5o, compared with the exact
0o in a 2-D case. This result quantitatively reveals theasymmetry of the second mode as discussed previously.

(c )



0.3

0.2

Re=400Re=800
Re=1200
Re=1600
Re=2000

0.2

o

15

0
2

10

3

Re

10

4

Inconsistency can be observed about the effect ofcrossflow on second mode by comparing Fig. 8 with Fig. 5.
Under the effect of crossflow, the second-mode maximum
growth rate in a 3-D boundary layer is higher in Fig. 5, while
lower in Fig. 8, than that in a 2-D boundary layer. This isbecause the 2-D base flow is subject to a favorable pressure
gradient in Fig. 5, compared with the absence of pressure
gradient for the 2-D base flow in Fig. 8. Accompanied by
imposing an additional local swept angle Λ to create crossflow,
the streamwise pressure gradient parameter decreases tom·cosΛ in Fig. 5. As pointed out by Zurigat, Neyfeh & Masad(1992), a streamwise favorable pressure gradient has astabilizing effect on the second mode in a 2-D compressible
boundary layer. According to this fact, decreasing favorablepressure gradient in streamwise direction suggests theincreasingly unstable second mode, as shown in Fig. 5. In
contrast, increasing streamwise pressure gradient is imposed to
produce crossflow in Fig. 8, thus an decrease in second-mode
growth rate is observed.

0.3

(b)
0.006

30

Figure 8. Characteristics of unstable modes at various
Reynolds numbers (Ma=4.5, Λ=30o, Tw=Tad). m=0.1 for 3-Dbase flows and m=0 for 2-D base flows. (a) Unstable ω-βregion; (b) Maximum growth rate; (c) Wave angle.

-0.1



45

( )

(o)
10

0

0.1

max, 1st mode, 3D
max, 2nd mode, 3D
max, 1st mode, 2D
max, 2nd mode, 2D

0Second mode

0

60

15

0.1

-0.1

60

30

0.5

First mode

75

45

(a)

0.4

75

-i,max, 1st mode, 3D-i,max, 2nd mode, 3D-i,max, 1st mode, 2D-i,max, 2nd mode, 2D

-i

0.004

Effects of Mach number
0.002

0
102

103

Re

In 2-D boundary layers, the behaviors of the first and second
modes are well-known at varying Mach numbers. The firstmode, which reduces to T-S wave in subsonic flows (Fedorov
2011), exists in whole range of Mach number. In comparison,
the second mode appears when Ma>2.2 and becomes dominant
in hypersonic flows. However, limited understanding has beenrecognized about the crossflow-affected instabilities in 3-D
compressible boundary layers. In this section, we investigate
the Mach number effect on the stability in boundary layerswith crossflow.
The incompressible boundary layer is firstly considered by
setting Mach number to be 0.001. As mentioned above, thebase flow in current model reduces to FSC flow when Mach
number approaches zero. At such low Mach numbers, Mack's
second mode is not expected. In contrast, the T-S wave, which
is more often termed as T-S mode or the first mode in thispaper, is the main concern. In order to reveal the increasing
effect of crossflow, a 2-D base flow is preliminarily considered
and crossflow is added gradually. Fig. 9 shows the growth-rate
contours for different local swept angles, corresponding to thecross-flow maximum of 0, 0.013 and 0.034 respectively.
Starting with the use of global method, the unstable modal

104

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
solution is found to be related with Mode S for all three cases.
In a 2-D base flow, the growth rate of T-S mode issymmetrically distributed (Fig. 9(a)), and the most amplified
wave is 2-D as reported by Mack (1984). Apart from the T-S
mode, a new unstable region arises with the increase ofcrossflow, as shown in Fig. 9(b). Again, the asymmetry ofgrowth-rate distribution for T-S mode is observed due tocrossflow. Despite the seeming disconnection of the two
unstable regions, they belong to a common solution, thus they
are actually connected by the stable region in between. The
newly emerging unstable region bears lower frequency, higher
growth rate and larger crosswise wave number than the T-S
mode region. By comparing with Mack's (1984) result about
FSC boundary layers, it can be identified that this region
corresponds to cross-flow instability. At even stronger
crossflow, the two unstable regions coalesce, accompanying
with the decrease on the number of peaks from two to one (Fig.
9(c)). The cross-flow instability and the T-S instability cannot
be distinguished in this case.
The appearance of cross-flow instability at Ma=0.001 differs
strikingly from the crossflow-affected instability at Ma=4.5 asshown in the preceding subsection. In the highly supersonicboundary layers with crossflow, Mack's first mode deviates toexhibit some characteristics of cross-flow instability, despitethat the cross-flow instability can hardly be distinguished.
While in the incompressible boundary layers, crossflow
produces an additional cross-flow instability competing with TS instability. The cross-flow instability occupies a separate
unstable region at a moderate strength of crossflow. A local
maximum of growth rate always exists for cross-flow
instability, corresponding to the most amplified traveling crossflow wave. Although the T-S mode becomes fairly moreunstable with the increase of crossflow, the intensity of crossflow instability grows even more sharply. Finally, the unstableregion of cross-flow instability enlarges and swallows that of
T-S instability.

Figure 9. Growth-rate contours of the unstable modes inincompressible boundary layers for (a) Λ=0o, (b) Λ=10o and (c)Λ=30o (m=0.1, Re=2000).
The growth-rate distributions are contoured for unstablemodes in compressible boundary layers at various Mach
numbers in Fig. 10. The result in each case is obtained byincreasing gradually the strength of crossflow from zero. Thus,
the presented instabilities can be traced and identified based on
2-D results. Under such moderate strength of crossflow, theintensities are comparable for different instabilities, if there
exists more than one instability in the case. Cross-flow modes
are shown for cases with Mach numbers less than 1.5. Whilefor larger Mach numbers, the cross-flow instability cannot be
distinguished to be a mode, as discussed previously. The crossflow mode and T-S mode get entangled at Ma=1.0 (Fig. 10(b)),
and the mode entanglement is also observed for the first modeand second mode at Ma=6.0 (Fig. 10(f)).
The name of cross-flow mode is usually encountered inincompressible and compressible boundary layers. However,current results reveal that the cross-flow instability cannot berecognized as an individual mode in most cases. The
phenomenal similarity underpins the proposition of naming thecross-flow instability as a mode at low speeds. However, isshould be borne in mind that the cross-flow instability is
integrated in the T-S mode or the first mode. Particularly inhighly supersonic boundary layers, the usage of cross-flowmode is ambiguous.

8

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

Figure 10. Growth-rate contours of the unstable modes incompressible boundary layers (Re=2000, m=0.1, Tw=Tad) with(a) Ma=0.5, Λ=10o; (b) Ma=1.0, Λ=10o; (c) Ma=1.5, Λ=15o; (d)Ma=3.0, Λ=15o; (e) Ma=4.5, Λ=15o and (f ) Ma=6.0, Λ=15o.
CONCLUDING REMARKSThe stability characteristics of 3-D boundary layers subject
to parametric crossflow have been investigated. A base-flow
model on a flat plate is firstly designed to create boundary
layer with crossflow. Considering that the crossflow is
intrinsically resulted from crosswise pressure gradient,potentially fewest parameters, i.e., pressure-gradient parameterand local swept angle, are chosen to parameterize crossflow inthe model. The model base flow is governed by an ODE
system, which permits local similarity for the boundary layer.

912th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
Using the proposed model, we obtained a family of 3-D
boundary layer, which can be reduced to the incompressible
FSC flow and 2-D compressible self-similar boundary-layer
flow. Based on such base flows, we extended the stability
analysis from in 2-D cases to in 3-D cases at various Mach
numbers.
Asymmetry and negativeness of frequency are found to befundamental features for cross-flow instability in 3-D boundary
layers. Two theorems, i.e., Symmetry Theorem and ConjugateSolution Theorem are primarily proposed to reveal the intrinsic
details. Both theorems have been proved theoretically andnumerically. The symmetric distributions of stability
characteristics in 2-D flows are demonstrated by Symmetry
Theorem and LST results. The occurrence of instability
asymmetry in a 3-D flow is believed to be attributed tocrossflow, whose existence ruins the symmetry of the base flow.
The occurrence of negative-frequency disturbance is revealed
repeatedly in the results of stability analysis. According to
Conjugate-Solution Theorem, a conjugate solution withpositive frequency exists for a negative-frequency solution.
Therefore, although the negative-frequency disturbance per se
is notorious for lack of physical meaning, its conjugate
counterpart is suggested to be resorted to for physicalapplications. Meanwhile, the Conjugate-Solution Theorem
further indicates opposite propagating directions for thenegative- and corresponding positive-frequency disturbances,which is accordant to what has been suppositionally recognized.
Based on a large number of calculations, it is revealed thatthe occurrence of crossflow does not essentially create a new
discrete solution. All of the LST solutions obtained in thispaper are found to be related to Mode S, according to the modeterminology proposed by Fedorov and Tumin (2011). However,the crossflow does give rise to different behaviors of Mode S
from those in 2-D boundary layers. The differences are in two
folds. Firstly, the cross-flow instability, which is characterized
as containing zero and negative frequency band, emerges. And
it is found to be integrated in T-S mode at low Mach numbers
or the first mode at high Mach numbers. The cross-flow
instability can hardly be identified as an individual mode formost cases, even based on a loose definition of mode like
Mack's first and second mode. But exceptions are found toexist for moderate crossflow in favorably pressure-gradientboundary layers with Mach number less than 1.5. Secondly, thecrossflow seems to destabilize the T-S/first and second modes
in general, by expanding unstable regions and increasing
growth rates. Meanwhile, the unstable modes become more
oblique towards crosswise or the opposite direction, dependingon favorable or adverse pressure gradients. In comparison, thefirst mode is much more greatly affected by crossflow than thesecond mode.

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