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Development of Tollmien-Schlichting disturbances in the presence of laminar
separation bubbles on an unswept infinite wavy wing
Christian Thomas and Shahid Mughal
Department of Mathematics, South Kensington Campus,
Imperial College London, London, SW7 2AZ, UK∗
Richard Ashworth
Airbus Group Innovations, Bristol, BS99 7AR, UK
The effect of long wavelength sinusoidal surface waviness on the development of Tollmien-
Schlichting (TS) wave instabilities is investigated. The analysis is based on the compressible flow
that forms over an unswept infinite wavy wing with surface variations of variable amplitude, wave-
length and phase. Boundary layer profiles are extracted directly from the solutions of a Navier-Stokes
solver, which allows a thorough parametric analysis to be undertaken. Many wavy surface configu-
rations are examined that can be sufficient to establish localised pockets of separated flow. Linear
stability analysis is undertaken using parabolised stability equations (PSE) and linearised Navier-
Stokes (LNS) methods, and surface waviness is generally found to enhance unstable behaviour.
Results of the two schemes are compared and criteria for PSE to establish accurate solutions in
separated flows are determined, which are based on the number of TS waves per wavelength of the
surface deformation. Relationships are formulated, relating the stability variations and the surface
parameters, which are consistent with previous observations regarding the growth of TS waves on
a flat plate. Additionally, some long wavelength surface deformations are found to stabilise TS
disturbances.
∗c.thomas@imperial.ac.uk
2
I. INTRODUCTION
Accurate prediction of boundary layer transition on aerofoils is critical to the improvement of future wing design.
Surface deformations that have been established by environmental conditions (roughness or hail stone impacts [1])
or industrial effects (steps, gaps and waviness) can potentially cause significant variations in the onset of transition,
which may have a severe impact on the flight performance characteristics. Several studies have examined the effect of
short-scale surface deformations (as listed in table I), while for the current investigation we explore the effect of long
wavelength variations on an unswept infinite wing.
Wind tunnel experiments carried out by Fage [2] examined the effect of various surface deformations (bulges,
hollows, ridges) on transition to turbulence. Experiments were conducted on both a flat plate with a small favourable
pressure gradient and an aerofoil. Using his observations, Fage derived the following relationship
h2x2
trRe 3
∞
λ= 8.1×1013;√λl/xtr <0.09,(1)
to describe the minimum height hrequired to affect the onset of boundary layer transition. Here λand ldenote
the respective length and location of the deformation, xtr is the chord location for transition and Re∞represents
the freestream Reynolds number. Further experiments by the Carmichael group [3–5] included both compressible
and pressure gradient effects on a wavy wing body. Using the experimental flight data Carmichael formulated the
expression
h2c1/2Re3/2
∞
λ= 51300,(2)
that defines the critical size of the surface waviness for a two-dimensional (2D) flow.
Holmes and collaborators [6, 7] reported on manufacturing tolerances and flight experiments that included various
step, gap and surface wave imperfections. It was found that the Tollmien-Schlichting (TS) wave instability was
significantly destabilised in the region of the surface wave that generates an adverse pressure gradient. Additionally,
increasing the height of step deformations on a 2D flat plate significantly augments the growth of the TS wave and
brings about the premature onset of transition [8].
The effect of wall waviness on the stability of the incompressible Blasius boundary layer was considered by Lessen
and Gangwani [9], who found that the critical Reynolds number for transition decreased with the height of the wave.
Using solutions of the compressible boundary layer equations, Lekoudis et al. [10] computed the effect of shallow
surface waves on the meanflow and obtained good agreement with the earlier experimental observations. However, for
surface variations that were sufficiently large to establish separation, the boundary layer method for generating the
basic state breaks down. This failing of the boundary layer computations is a direct consequence of the streamwise
marching numerical procedure [11]; thus limiting boundary layer and stability analysis for some forms of surface
deformations. However, such difficulties can be overcome by the implementation of alternative methods, including the
interactive boundary layer (IBL) procedure that allows the boundary layer equations to be solved even for separated
flows [12].
Several theoretical investigations considered the effect of a hump on a 2D flat plate without a streamwise pressure
gradient [13–15]. In these particular studies the undisturbed basic state was established using the IBL procedure,
while linear stability analysis was performed using a parallel flow approximation where the streamwise variation of
the meanflow was ignored. A locally based eNmethod [16–19] was utilised to predict the onset of boundary layer
transition and the theoretical findings were qualitatively similar to the observations of Fage [2].
The IBL methods were further utilised by Wie and Malik [20] to compute a subsonic 2D base flow over a flat-plate
with wavy surface variations. The effect of waviness on the growth of the linear TS wave instability was carried out
using parabolised stability equation (PSE) methods [21], which were based on a non-parallel boundary layer that took
into account the streamwise variations of the undisturbed flow. Several surface wave configurations and freestream
conditions were considered and it was determined that wavy deformations could establish significant increases in the
amplification rate of the disturbance. Additionally, Wie and Malik were able to derive an expression relating the
variation of the N-factor amplification rate
∆N=NWavy −NNon-Wavy,
and the physical dimensions of the surface wave for the flow over a flat-plate with a favourable pressure gradient:
∆N= 0.07nh2Re∞
λ,(3)
3
Reference Study Type Geometry Deformation Type
Fage [2] Experimental Flat Plate/Aerofoil Bulges/Hollows/Ridges
Carmichael Group [3–5] Experimental Aerofoil Waviness
Holmes Group [6, 7] Experimental Aerofoil Steps/Gaps
Wang & Gaster [8] Experimental Flat Plate Steps
Lessen & Gangwani [9] Theoretical Flat Plate Waviness
Nayfeh et al [13] Theoretical Flat Plate Humps
Cebeci & Egan [14] Theoretical Flat Plate Humps
Masad & Iyer [15] Theoretical Flat Plate Humps
Wie & Malik [20] Theoretical Flat Plate Waviness
Park Group [23, 24] Theoretical Flat Plate Humps
Brehm et al [25] Theoretical Flat Plate Ridges/Roughness
Gaster [26] Experimental Flat Plate Ridges/Roughness
TABLE I. Type of surface deformations investigated by previous investigators.
where the parameter ndefines the number of waves before the onset of transition xtr (given for the non-deformed
model). Comparing their expression for the allowable measure of surface waviness with that formulated by Fage
and Carmichael [2, 4], Wie and Malik concluded that Fage’s criteria (1) was quite restrictive and allows significantly
smaller waviness than the relationship conceived by Carmichael (2). It was suggested that expression (3) could be
used to estimate the effect on stability and transition for a limited range of surface dimensions, provided that surface
waviness did not enhance receptivity or excite nonlinear interaction with centrifugal instabilities that may develop as
a result of the surface waviness [22].
PSE methods were also employed by Park and collaborators [23, 24] to investigate the effects of a hump on both
the linear and nonlinear development of the TS wave instability. Both the height and length scales of the hump were
again critical to the amplification rate of the disturbances. Nonlinear interaction was found to greatly amplify the
size of the perturbation and it was concluded that this may cause the premature breakdown of the laminar flow.
Thomas et al. [27] considered the effect of surface waviness on the crossflow instability that develops in a com-
pressible flow on an infinite swept wing body. Their study was based on the flow solutions of an industrial Reynolds-
Averaged Navier-Stokes (RANS) formulation called TAU [28], where laminar flow was established by specifying the
onset of transition; the RANS scheme solves the laminar form of the Navier-Stokes equations. Boundary layer solu-
tions in this limit were then extracted directly from the TAU output and formatted for a stability analysis based on
both PSE and linearized Navier-Stokes (LNS) methods [29, 30]. Similar extraction methods have been developed by
Malik and co-workers [31, 32] for investigating the stability of the flow on flat plates and full aircraft configurations.
Thomas and collaborators [27] validated their ‘RANS extracted boundary layer’ (REBL) solutions against results of
a compressible boundary layer formulation [33] that applies a streamwise marching strategy [11]. As boundary layer
profiles were drawn directly from the TAU solutions, a thorough stability investigation was undertaken for several
surface configurations, including those wavy dimensions that established separation. It was shown that, depending on
the location used to draw comparisons, surface waviness of variable height, wavelength and phase could marginally
stabilise and destabilise the crossflow instability. Furthermore, utilising the eNmethod for estimating transition,
Thomas et al. found that surface waviness could generate variations in the chord location where laminar-turbulent
transition was established. However, stability variations ∆Nwere small compared with the observations of Wie and
Malik [20] on TS disturbances. Additionally, PSE solutions were in excellent agreement with the corresponding LNS
computations, with only minor differences arising for those flow systems containing small pockets of separation.
Recently, Brehm et al [25] undertook a theoretical investigation of 2D distributed wall roughness effects on a flat
plate, with the aim of corroborating the experimental findings of Gaster [34]. In their study they found that roughness
modelled as 2D rectangular or sinusoidal elements could establish separated flow. Their numerical results confirmed
Gaster’s observations that there exists a critical roughness height, beyond which the amplification of the TS wave
instability increases with increasing height. The effect of sinusoidal roughness distributions on the growth of the TS
wave was also considered by Gaster [26], who used the Orr-Sommerfeld equation to obtain solutions to the basic
state and undertake a perturbation analysis. Separated flow was found to form between sufficiently large roughness
elements, while the amplification rate of the perturbations was enhanced by the roughness.
In the following investigation, PSE and LNS methods are utilised to study the effect of long wavelength surface
waviness on the growth of the TS wave instability. This expands upon the earlier theoretical studies that considered
short-scale deformations on a flat plate (refer to table I). The undisturbed flow on an unswept infinite wing of
variable surface waviness is established using the TAU industrial flow solver, where the surface variations can establish
localised pockets of separation. The suitability of the PSE approach for investigating disturbances in separated flows
4
TAU Solver
CoBL REBL
LNS
PSE
Stability Analysis
Unswept Infinite Wing
Generate Flow
Boundary Layer Methods
FIG. 1. Flow chart diagram illustrating the methodology for the boundary layer and stability analysis. CoBL - Compressible
Boundary Layer; REBL - RANS Extracted Boundary Layer; PSE - Parabolised Stability Equations; LNS - Linearised Navier-
Stokes.
is a contentious issue. PSE methods are based on a streamwise marching procedure that do not account for the
upstream propagation associated with boundary layer separation. Furthermore, separated flow can establish absolute
instability, for which PSE methods are most definitely not applicable. Hammond and Redekopp [35] suggest that
absolutely unstable behaviour will develop in laminar separation bubbles if the peak value of the reverse flow reaches
about 30% of the freestream magnitude, while Adam and Sandham [36] indicate that more than 15% reverse flow is
required for the onset of absolute instability.
On the applicability of the PSE approach for studying convective disturbances in systems with reverse flow, Wie and
Malik suggest that the PSE method may be able to successfully ‘step-over’ a small separation bubble. This particular
conclusion was drawn, based on the minimum step-size restriction derived by Li and Malik [37]; ∆x > 1/|αr|for a
stable solution. Here αrand ∆xrepresent the real part of the streamwise wavenumber and the associated step-size.
Li and Malik also proposed a Parabolised Navier-Stokes type strategy [38], whereby the streamwise pressure gradient
term is suppressed, leading to a relaxation of the step-size limit. However, Andersson et al [39] introduce a stabilisation
procedure that allows the pressure gradient term to be retained in the PSE formulation allowing numerically stable
solutions to be obtained for considerably smaller step-sizes than that established by the Li-Malik step-size criteria.
Furthermore, there are some indications from within the available literature that the PSE approach may be used to
calculate disturbances in separated flow systems to an acceptable degree of accuracy. For instance, Gao et al [23]
compared solutions of their PSE analysis with that established via direct numerical simulations, and found that the
PSE method can work well in flow systems with a small amount of boundary layer separation. Additionally, Wie and
Malik [20] state (though do not publish) that they successfully applied the PSE method through separation bubbles.
The remainder of this paper is outlined as follows. In the subsequent section we describe the routines for generating
the basic state on a wavy wing, and the PSE and LNS methods used to undertake a linear stability analysis of TS
disturbances. Boundary layer solutions are presented in §III for both non-separated and separated flow systems, while
results of both the PSE and LNS methods are discussed in §IV. PSE and LNS results are compared and we examine
the criteria necessary for PSE to establish solutions to a reasonable degree of accuracy in separated flows. Finally, we
conclude our investigation with several comments pertaining to the observations of our study.
II. FORMULATION
Compressible flow characteristics and boundary layer disturbances that develop on a wavy wing are determined
using several numerical schemes. Accurate and robust methods are required to successfully capture the effect of surface
waviness on the evolution of TS wave instabilities. In this section we discuss and highlight the essential ingredients,
as depicted in figure 1, for generating solutions and achieving the objectives of this investigation.
5
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
-0.6
-0.4
-0.2
0
0.2
0.4
c= 1m
¯x∗= 0.55c
¯x∗
x∗
y∗
¯y∗
∗
Laminar Turbulent
FIG. 2. Cross-sectional view of the unswept infinite wing model.
A. Wing Model
1. Non-Deformed Geometry
Figure 2 depicts a cross-sectional view of the wing geometry, used in this investigation, in 2D Cartesian coordinates
¯
x∗={¯x∗,¯y∗}[40] (asterisks denote dimensional properties). As our interest is in flow variations along the ¯x∗-direction
and 2D TS wave instabilities, the wing model and the resulting flow dynamics are assumed to be independent of a
third spanwise axis. Additionally, the angle of incidence is set equal to zero. Although the wing geometry represents
a simple aerofoil, it is reasonable to assume that the stability analysis illustrated in this paper will also appear (in one
form or other) on other wing geometries with comparable surface features and freestream characteristics. The chord
length of the model c∗= 1m, whilst the maximum thickness is approximately 0.15m near the chord centre. The flow
is assumed to be compressible with a Mach number M∞and a freestream Reynolds number Re∞based on the chord
c∗. Finally, the freestream temperature T∞= 300K for all flow conditions considered.
2. Imposing Surface Waviness
The surface of the wing is deformed using sinusoidal wave variations of the form
s=H∗sin{2πx∗/λ∗−ϕ}on y∗= 0,(4)
where x∗represents the chordwise surface axis and y∗is the wall-normal direction (as depicted in figure 2). Here H∗
is the amplitude (or half height) and λ∗is the wavelength of the wavy surface, which are both scaled on the chord
length c∗. Note that in all future discussion H=H∗/c∗and λ=λ∗/c∗are used to define the surface variations. The
parameter ϕis a phase shift, which when set to zero establishes a surface wave originating at the attachment-line
¯x∗=x∗= 0.
In subsequent sections we limit our analysis to surface configurations λ∈[0.1 : 0.1 : 0.4], H∈[0 : 0.0001 : 0.0006]
and four equally spaced phase shifts ϕ∈[−π/2 : π/2 : π]. Smaller wavelengths were not considered as significantly
denser meshes would have been required to establish accurate flow solutions (see §II A 3), while the amplitudes H
are relatively large and correspond to deformations of order one tenth of a millimetre. This particular characteristic
was implemented to ensure variations in the flow dynamics were sufficient enough to establish localised pockets of
separation.
6
3. Generating Flow Solutions
The steady flow, about the wavy wing, was generated using the TAU industrial flow solver [28]. The TAU program
solves the RANS system of equations for the flow that develops about 2D and 3D geometries. For the current
investigation the method was made applicable to the analysis of laminar flows by imposing a fixed transition line (as
depicted in figure 2) at x∗= 0.55c∗. This was implemented to avoid convergence issues that could arise with the
formation of severe separation along the trailing edge of a laminar wing. Although pockets of separated flow were
established within the troughs of some of the wavy surface geometries considered herein, they were relatively small.
Thus, laminar flow was established for x∗≤0.55c∗by setting the eddy viscosity in the TAU formulation to zero; the
RANS scheme thus reduces to solving the laminar Navier-Stokes equations. Downstream of the transition location,
the full RANS formulation was solved and a fully turbulent boundary layer was allowed to develop. However, in the
subsequent sections, the analysis of TS disturbances was restricted to the laminar flow domain.
A sufficiently dense unstructured mesh was established (about the wavy wing) to ensure accurate flow features
were captured by the TAU formulation. After careful experimentation it was determined that at least 40 mesh points
were required within the boundary layer region, to ensure that the flow dynamics were fully captured. Beyond the
boundary layer, the mesh density was significantly reduced and quite sparse in the far field limit (which was set at
about 50 chord lengths c∗from the wing). Along the length of the wing, approximately 500 mesh points were required
to generate accurate flow solutions. Decreasing the minimum wavelength λbeyond that considered herein would have
required this latter mesh specification to be greatly increased, leading to larger computational requirements.
Given a set of freestream flow conditions {M∞,Re∞}the TAU solver simulates the dimensional steady flow ¯
Q∗
B=
{¯
U,∗
B¯
P∗
B,¯
T∗
B}in Cartesian coordinates ¯
x∗. The vector ¯
U∗
Bdenotes the undisturbed dimensional velocity field, while
¯
P∗
Band ¯
T∗
Brespectively represent the pressure and temperature variables. A converged steady-state solution was
then obtained by using the Spalart-Allmaras-Edwards turbulence model [41]. Flow solutions were obtained subject to
satisfying no-slip conditions on the wing surface and far-field boundary conditions given by the freestream specification.
An explicit Runge-Kutta iterative scheme was then utilised, where it was assumed that a steady basic state was
achieved once residuals were of the order 10−8and less.
B. Boundary Layer Methods
A non-dimensional steady basic state QB={UB, PB, TB}(x) (in non-dimensional surface coordinates x={x, y})
was required to conduct both a PSE and LNS investigation of TS wave instabilities. Boundary layer profiles were
obtained by either solving a set of boundary layer equations or by carefully extracting profiles directly from the TAU
solutions. The former method is based on the results of the infinitely swept boundary layer equations (CoBL [33]),
where a surface pressure distribution is required as an initial input taken directly from the TAU output. These
particular equations were discretised using a fully implicit second-order accurate three-point backward differencing
scheme along the chordwise axis, whilst a two point second-order accurate method was utilised in the wall-normal
direction.
A second boundary layer method, REBL [27], is based entirely on the output generated by the TAU flow solver.
The REBL scheme extracts the dimensional TAU solutions and transforms the results from the dimensional Cartesian
coordinates ¯
x∗to the non-dimensional surface fitted coordinate system x={x, y}. The extraction procedure was
achieved by implementing several geometric transformations and boundary-layer properties. Paraview filters [42] were
used that perform several processes during the boundary-layer extraction process, while arithmetic operations were
performed using Python to the default 15 decimal place accuracy. Firstly, unit normals ¯
x∗
n={¯x∗
n,¯y∗
n}were generated
at all points along the wing surface. Dimensional flow profiles were then extracted along each normal and transformed
using the matrix operator
A=¯y∗
n¯x∗
n
−¯x∗
n¯y∗
n,
to give
x∗=A¯
x∗and U∗
B=A¯
U∗
B,
where the dimensional vector U∗
B={U∗
B, V ∗
B}represents the 2D undisturbed velocity field in surface fitted coordinates
x∗, while P∗
B=¯
P∗
Band T∗
B=¯
T∗
B. Non-dimensional quantities were then obtained by scaling flow properties on their
boundary layer edge values, based on the location where the U∗
B-velocity field attains 99% of its maximum. Thus, the
non-dimensional velocity field
UB=U∗
B
U∗
e
,
7
where the subscript edenotes the boundary layer edge characteristics. Hence, at each chord location UB→1 as
y→ ∞. Similar representations were formulated for the non-dimensional pressure PB=P∗
B/P ∗
eand temperature
TB=T∗
B/T ∗
e.
For the subsequent discussion of the PSE and LNS formulation, non-dimensional coordinates xwere obtained by
scaling on a locally defined boundary layer thickness δ∗that was measured about a fixed location:
x=x∗/δ∗.
However, in §III and IV, the base flow and stability calculations are presented in terms of a coordinate system scaled on
the chord length c∗;x=x∗/c∗. Note that the same character has been used to represent non-dimensional coordinates
based on δ∗and c∗, which has been implemented to avoid introducing further notation. Additionally, results are
presented in x=x∗/c∗coordinates as this provides a better means of illustrating the flow dynamics.
C. Stability Methods
The stability of the undisturbed flow QBwas undertaken by considering infinitesimally small time periodic pertur-
bations
q′(x, y, t) = q(x, y) exp{−iωt},
where q={u, v, p, T }and ωrepresents a local non-dimensional frequency given as
ω=2πf ∗δ∗
U∗
e
.
The parameter f∗is the dimensional frequency of the perturbation that is measured per unit hertz (Hz). For
presentation purposes we introduce a non-dimensional global frequency
f=2πf ∗ν∗
∞
U∗2
∞
,(5)
where U∗
∞and ν∗
∞respectively denote the freestream velocity and kinematic viscosity.
1. PSE Theory
Linear PSE perturbations qare decomposed as
q=˜
q(x, y)E(x) + c.c,(6)
where ˜
qrepresents a slowly varying in xamplitude function and Eis a wave function of the form
E(x) = exp ix
x0
α(ζ) dζ.(7)
The real and imaginary parts of αrespectively denote the wavenumber and growth rate of the TS disturbance, while
x0represents the critical location for the onset of the instability.
The linear PSE formula for the shape function ˜
qis represented as
L˜
q+M∂˜
q
∂x = 0,(8)
where Land Mare differential matrix operators in the wall-normal y-direction (a detailed description of the matrix
operators are given in Mughal [43]). Both Land Mare dependent on the curvature of the surface that are embodied
in the terms κand χthat respectively represent the local body curvature and
χ=1
1−κy .(9)
8
The system of equations (8) are then closed by the integral condition
∞
0˜
q†·
∂˜
q
∂x dy∞
0
˜
q†·˜
qdy= 0,(10)
where †denotes the complex conjugate form. Equation (8) is then solved using a marching procedure where the
wavenumber αis determined at each x-position using the iterative scheme
αk+1 =αk+i∞
0˜
q†·
∂˜
q
∂x dy∞
0
˜
q†·˜
qdy. (11)
The imaginary part of αdenotes the growth of the TS wave instability and is used to perform an N-factor calculation
[17, 44] that is given by the expression
N=−x
x0
αi(ζ) dζ. (12)
For the subsequent stability analysis the maximum absolute value of the u-velocity perturbation field, |u|max, is used
to draw direct comparisons between solutions of the PSE and LNS formulations.
The system of PSE equations (8) are solved subject to the no-slip conditions at the wall
u=v=T= 0 on y= 0,(13a)
while the Dirichlet conditions are imposed in the freestream
u=v=T→0 as y→ ∞.(13b)
The pressure pis also assumed to satisfy the Dirichlet condition in the freestream.
Chordwise step-sizes ∆xwere implemented, where the Li-Malik [37] stability restriction criteria ∆x > 1/|αr|was
imposed to obtain numerically stable solutions. It should be noted that PSE analysis was also successfully undertaken
for smaller ∆xthan that imposed by the Li-Malik criterion. However, it was found that the best PSE-LNS comparisons
arose for calculations based on ∆xthat satisfied this criterion. As the size of the wavenumber αris dependant on
the frequency of the TS wave, special care was required to ensure that the step-size limitation was satisfied for those
PSE results presented. Additionally, to improve the numerical robustness of the method, the effect of the chordwise
pressure gradient ∂p/∂x was suppressed.
2. LNS Scheme
The LNS formulation for a compressible flow is presented within the appendices. In the far-field limit, Dirichlet
conditions are imposed, where perturbations are assumed to have decayed to a negligible magnitude. On y= 0, the
no-slip condition is generally enforced, where u=v=T= 0. Perturbations are then excited by a small periodic
forcing, where the no-slip condition is defined as
u′=−h(x, t)UB,y (x, 0), v′=∂h(x, t)
∂t and T′=−h(x, t)TB ,y(x, 0) on y= 0,(14a)
⇒u=−h(x)UB,y (x, 0), v =−iωh(x) and T=−h(x)TB,y (x, 0) on y= 0,(14b)
and the function h(x) represents a normalised Gaussian distribution of the form
h(x) = 10−6exp{−0.5([x−xf]/σ)2}/√2πσ2,(15)
where σ= 10 and xfprescribe the variance and centre of the wall forcing, respectively.
The LNS system of equations were discretised and solved in the manner described by Mughal and Ashworth [30].
High order finite difference methods were implemented along the x-axis, while a pseudo-spectral approach was utilised
in the wall normal y-direction. Along the x-direction, up to 8000 points were used over the chord domain considered
and 81 points was deemed sufficient in yto accurately resolve solutions. LNS solutions were then computed by
decomposing the discretised formulation as a large lower-upper block factorisation matrix.
9
0 0.1 0.2 0.3 0.4 0.5
0
0.2
0.4
0.6
0.8 (a)
0 0.1 0.2 0.3 0.4 0.5
0
0.2
0.4
0.6
0.8
1(b)
FIG. 3. Flow characteristics obtained using REBL for the surface configurations {λ, H}={0,0}(solid line), {0.1,0.0001}
(dashed), {0.2,0.0002}(chain) and {0.4,0.0004}(dotted). (a) Surface pressure coefficient Cp; (b) Skin friction coefficient
CfRe1/2
∞.
0.2
0.2
0.2
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
0.99
0.99
0.99
0.99
0.99
x
0 0.1 0.2 0.3 0.4 0.5
y
10-3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FIG. 4. Non-dimensional base flow UBobtained using REBL over a non-deformed wing in the {x, y}-plane.
III. BASE FLOW
A. Effect of Shallow Surface Waviness
Figure 3 illustrates the pressure Cpand skin friction CfRe1/2
∞coefficients based on the boundary layer solutions
generated by the REBL extraction method. Three wavy surface configurations are considered, where the freestream
conditions are defined as {M∞,Re∞}={0.7,5×106}and the phase shift ϕ= 0. Dashed lines depict solutions
established over a surface with {λ, H}={0.1,0.0001}, while chain and dotted lines respectively represent the surface
variations {λ, H}={0.2,0.0002}and {0.4,0.0004}. The solid lines display the results corresponding to that estab-
lished on the non-deformed wing. Solutions are plotted against the x-direction and the effect of the sinusoidal wavy
wall is shown to be mirrored in the two illustrated flow components. Furthermore, waviness can cause significant
variations from the results obtained for the non-deformed wing. In particular the skin friction coefficient displays
relatively large variations, with respective increases and decreases in CfRe1/2
∞corresponding to the flow passing over
the crests and troughs of the wavy surface.
The non-dimensional UB-velocity field for the non-deformed wing is plotted in figure 4 as a contour map in the
10
FIG. 5. Non-dimensional base flow UBobtained using REBL in the {x, y}-plane. (a) {λ, H }={0.2,0.0002}; (b) {0.4,0.0004}.
FIG. 6. (a) Illustration of the base flow UBobtained using REBL in the {x, y}-plane, over a wavy surface with {λ, H}=
{0.1,0.0003}. (b) Flow separation depicted within the troughs of the surface.
{x, y}-plane. Here we have used the coordinate scaling x=x∗/c∗to present flow solutions, and since the chord
length c∗= 1m, the coordinate axis can represent both dimensional and non-dimensional planes. Thus, the boundary
layer is approximately 1mm thick. The undisturbed flow development over two wavy surfaces is depicted in figure 5,
where the sinusoidal wall deformations are given as {λ, H }={0.2,0.0002}and {0.4,0.0004}, respectively. The y-axis
has now been transformed to include the wavy surface variations, to help visualise behaviour of the flow as it passes
over the crests and troughs of the surface. The reader is reminded that the vertical scale is very much smaller than
that along the horizontal and if the flow was drawn using an equal axis the surface variation would be very difficult
to distinguish. Nevertheless, the wavy wall has a relatively large impact on the flow development. In particular, the
boundary layer thickness is found to vary significantly as the flow develops downstream, decreasing about the surface
crests and increasing significantly near the troughs of the wavy wall. This behaviour is to be expected, as favourable
and adverse pressure gradients are established about the respective crests and troughs of the surface.
B. Separated Flows
Negative valued skin friction was not established for those wavy configurations drawn in figure 3(b). However, it
was found that this eventually materialises for sufficiently large amplitudes H; finite regions of reversed flow form
within the troughs of the wavy surface. Figure 6 depicts contours of the steady UB-velocity field that develops over
the wavy surface with {λ, H}={0.1,0.0003}. The non-dimensional velocity field is plotted for both an extensive
spatial range in (a) and for a concentrated region that highlights flow reversal in (b). Stationary bounded regions of
11
10-5 10-4 10-3 10-2
Re1/2
∞(HM∞)2n
0
0.05
0.1
0.15
0.2
−min(UB)
FIG. 7. Minimum value of the base flow UBagainst the function Re1/2
∞(HM∞)2n.
separation form within the troughs of the wavy surface, progressively increasing in magnitude and volume as the flow
convects downstream. However, the maximum magnitude of the reverse flow depicted in figure 6 is less than 10%
of the freestream amplitude, which is less than the suggested 15 −30% requirement to trigger absolute instability
[35, 36]. Hence, we should only expect convective disturbances to develop over this particular wavy wing.
For λ= 0.1 and {M∞,Re∞}={0.7,5×106}, amplitudes H≥0.0002 were sufficient to establish reverse flow, while
for λ≥0.2, amplitudes H > 0.0006 were necessary. Greater precision in the value of Hneeded to generate separation
was not feasible, as we were only able to consider a finite number of surface deformations due to limited access to
the TAU flow solver. Nevertheless, a crude relationship between the magnitude of the separation bubbles and the
flow specifications can be deduced. Figure 7 displays the minimum value of the base flow UBthat forms within the
troughs of the wavy surface against the expression Re1/2
∞(HM∞)2n. A logarithmic-linear scaling has been utilised
to map the computations, while ndenotes the number of waves between the attachment-line and the location that
UBis a minimum. For instance, in figure 6 reverse flow is strongest about the centre of the troughs, located about
x= 0.275, 0.375 and 0.475; hence, n= 2.75, 3.75 and 4.75, respectively. Given the logarithmic-linear relationship
in figure 7 we find that reverse flow first appears for Re1/2
∞(HM∞)2nof the order 10−4. As the subsequent study
concerns only convectively growing TS disturbances, flow systems with −min(UB)>0.15 were discounted from the
following stability analysis, as there was a possibility that they could establish absolute instability [35, 36].
The skin friction coefficient and pressure gradient PB,x =P∗
B,x/(ρ∗
BU∗2
B) associated with the REBL generated flow
illustrated in figure 6 are plotted in figure 8 using dashed curves, while the dotted curves represent the corresponding
results computed using the CoBL boundary layer method. The form of the surface waviness is also included in figure
8(c) to help illustrate flow characteristics. Over the chord range 0 ≤x≤0.23 the two sets of results are almost
identical. However, the solution from the CoBL scheme ends abruptly about x= 0.23, which is indicated by a cross
marker in figure 8(a, b). Boundary layer methods (that include CoBL) are based on a chordwise marching procedure
[11, 33] that fail for particularly strong adverse pressure gradients. For the case considered here, CoBL breaks down
when PB,x ≈1.3 and before the onset of boundary layer separation; the skin friction coefficient is positive about
the chord location that CoBL exhibits non-convergence. Thus, separated flow is not necessarily required for the
boundary layer method to fail. Nevertheless, using the REBL procedure (based on extracting the basic state directly
from the TAU flow solutions), we are able to construct complete flow profiles up to the end of the chord domain
considered. Relatively strong variations in both CfRe1/2
∞and PB,x are captured by the REBL procedure. The skin
friction coefficient and pressure gradient respectively decrease and increase in size along the downward slopes of the
surface waviness, while the opposite behaviour is found along the upward slopes. Strong adverse pressure gradients
(positive PB,x) form within the troughs of the surface and negative valued skin friction is found to develop about the
three chord locations that correspond to regions of separated flow depicted in figure 6(b).
12
0 0.1 0.2 0.3 0.4 0.5
x
s
(c)
0 0.1 0.2 0.3 0.4 0.5
-3
-2
-1
0
1
2
PB,x
(b)
0 0.1 0.2 0.3 0.4 0.5
-0.5
0
0.5
1
1.5
CfRe1/2
∞
(a)
FIG. 8. Flow characteristics for the surface configuration {λ, H}={0.1,0.0003}obtained using REBL (dashed) and CoBL
(dotted). Cross-markers indicate the location that the CoBL method breaks down. (a) Skin friction coefficient CfRe1/2
∞; (b)
Pressure gradient PB,x; (c) Surface variation s.
IV. RESULTS
A. REBL versus CoBL Stability Calculations
Stability analysis was undertaken for various surface configurations and non-dimensional frequencies f∈[1 :
150] ×10−6(that corresponds to dimensional frequencies f∗∈[1 : 30]kHz). Unless stated otherwise, the freestream
flow conditions were unchanged from that specified earlier; {M∞,Re ∞}={0.7,5×106}. Figure 9 compares the growth
rates αiand N-factor amplification rates of TS disturbances generated on three wavy surfaces for f= 34 ×10−6.
The form of the surface waviness (4) has been included in the illustration to help draw conclusions, while the choice
of frequency was made based on the strongest growing TS wave at x≡xref = 0.55 for the non-deformed wing
(solid lines). Note that xr ef represents a chord reference location that is used to compare stability calculations.
For all three cases considered the flow remains attached at all chord locations x≤xref . Thus, the PSE method
can be utilised as there is no upstream propagation and we would only expect convective disturbances to develop
[35, 36]. The dashed lines illustrate stability results based on the REBL generated boundary layers, while dotted
lines depict the corresponding solutions for base flows established by CoBL. Results of the two sets of calculations
are indistinguishable, at least up to the chord location (highlighted by a cross marker) that both methods were
successfully able to compute solutions. However, at the cross markers, the CoBL method for generating the base
flow breaks down, as the numerical model fails due to the appearance of a relatively large adverse pressure gradient
(which is not necessarily sufficient to establish separation). Thus, downstream of the cross markers, PSE analysis
could only be applied to the boundary layer solution established by REBL. For the remainder of this investigation we
only consider linear stability of boundary layers generated by REBL.
Figure 9 also illustrates the strong influence of the surface waviness on the growth of the disturbance. Both αiand
13
0.2 0.4
αi
-0.01
0
0.01
0.02 (a)
0.2 0.4
-0.01
0
0.01
0.02 (b)
0.2 0.4
-0.01
0
0.01
0.02 (c)
0.2 0.4
N
0
2
4
6
0.2 0.4
0
2
4
6
0.2 0.4
0
2
4
6
x
0.2 0.4
s
x
0.2 0.4
x
0.2 0.4
FIG. 9. Stability comparisons for base flows generated by REBL (dashed) and CoBL (dotted), for f= 34 ×10−6. Surface
configurations are given as (a) {λ, H}={0.1,0.0001}; (b) {0.2,0.0002}; (c) {0.4,0.0004}. Cross-markers indicate the location
that the CoBL method breaks down, and the solid curves represent the solutions associated with the non-deformed wing.
Nfluctuate in magnitude as the disturbance propagates downstream. The size of αiincreases along the downward
slopes of the wavy surface, where strong adverse pressure gradients arise. Growth rates then attain local maxima
about the troughs of the wave. Along the upward slopes of the wavy surface, favourable pressure gradients are
generated that cause αito decrease. Local minimum growth rates are then found about the crests of the wave. The
fluctuating growth rate then causes dips and rises in the N-factor, with the respective local maxima and minima
located immediately downstream of the surface troughs and crests.
B. LNS versus PSE Analysis for Separated Flow Systems
Figure 10 depicts the development of LNS generated u-velocity perturbation fields on five wavy wings. Tollmien-
Schlichting disturbances were excited using a normalised Gaussian roughness centred about xf= 0.15, for f=
34 ×10−6. The wavelength λ= 0.1 in all cases, while the amplitude Hincreases in size from (a) through to (e).
The wall-normal y-direction has again been deformed to include the sinusoidal surface variations, and the u-velocity
fields have been normalised on their respective maximum absolute values |u|max determined at each x-location. In the
latter three subplots, regions of separation have been highlighted using solid black contours. For small H, disturbances
display behaviour consistent with the expected TS wave evolution; the magnitude of the perturbation is largest within
the boundary layer. However, as Hincreases, the disturbance forms two equally strong peaks located about the surface
troughs. This particular observation is best illustrated in figure 10(e) where the disturbance splits into two separate
components as it emerges from the crests of the surface. The lower structure forms within the surface troughs, whilst
the upper component develops directly above. As the disturbance approaches the end of each trough, the two parts
of the TS wave re-coalesce. Similar behaviour was observed by Wie and Malik [20] on a wavy flat-plate, who stated
that the disturbance was a mixed TS-Rayleigh type instability.
The maximum amplitude of the u-velocity field associated with disturbances plotted in figure 10 are illustrated in
figure 11. Dashed lines represent the solutions established using the LNS formulation, while the dotted lines depict the
corresponding PSE generated computations. Remarkably, the PSE and LNS results are (to the accuracy of the grid
scale used) identical over the chord range shown for all wavy configurations considered. This result is quite surprising
as the latter two flow systems contain rather large separation bubbles, and PSE methods are unable to resolve the
effects of upstream propagating structures. This would suggest that the perturbations are primarily convective and
that the regions of reverse flow do not engineer upstream disturbance development (at least not large enough to
14
FIG. 10. Disturbance development u/|u|max, in the {x, y}-plane for f= 34 ×10−6and λ= 0.1. (a) H= 0.0; (b) 0.0001; (c)
0.0002; (d) 0.0003; (e) 0.0004. Solid black curves highlight the local regions of separated flow.
establish significant variations between the PSE and LNS solutions). If upstream propagating perturbations and
absolutely unstable behaviour were excited by the regions of separation, we might expect large differences between
the two sets of calculations.
In order to achieve the excellent comparison between the two sets of solutions, the step size ∆xused to undertake the
PSE analysis was carefully chosen to give the best comparison with that established via LNS. Initially we considered
several step sizes and drew comparisons to determine the solution that gave the best accuracy. For the range of step
sizes considered, the PSE method was able to successfully step through the separation bubbles, achieving numerically
stable solutions for a range of ∆x. PSE calculations were then scaled about x= 0.3 to match that established by
the LNS model, and numerical differences between the two set of results were measured about x= 0.5 using the
expression
ϵ=
|u|max,LNS − |u|max,PSE
|u|max,LNS
×100%.
Differences ϵ, for those disturbances considered in figures 10-11, are plotted in figure 12 against |αr|∆x. Numerically
stable PSE calculations were obtained for smaller step-sizes than the Li-Malik critical value |αr|∆x= 1. However,
15
0.2 0.3 0.4 0.5
10-10
10-5
|u|max
(a)
0.2 0.3 0.4 0.5
10-10
10-5
(b)
0.2 0.3 0.4 0.5
x
10-10
10-5
|u|max
(c)
0.2 0.3 0.4 0.5
x
10-10
10-5
(d)
FIG. 11. Maximum absolute value of the u-velocity perturbation as a function of xfor f= 34 ×10−6and λ= 0.1. Solutions
generated for LNS (dashed lines) and PSE (dotted) methods. (a) H= 0.0001; (b) 0.0002; (c) 0.0003; (d) 0.0004.
0 0.5 1 1.5 2
|αr|∆x
0
10
20
30
40
50
ǫ(%)
H= 0.0001
H= 0.0002
H= 0.0003
H= 0.0004
FIG. 12. Differences ϵagainst the Li and Malik stability criterion |αr|∆xfor those disturbances considered in figures 10-11.
the smallest differences were obtained for |αr|∆x > 1. For the smallest amplitude, H= 0.0001, differences ϵwere less
than 1%, while ϵ < 3% was obtained for the largest amplitude, H= 0.0004. Some differences between the two sets
of calculations should be expected, as the PSE method utilises approximations that neglect the higher order terms.
Hence, for the frequency wave investigated above, the PSE formulation can be used to compute the magnitude of the
disturbance to a reasonable degree of accuracy.
In addition to the above observations, figure 11 reveals several interesting disturbance characteristics. Firstly, the
magnitude of the perturbations established for H= 0.0003 and 0.0004 are approximately the same near the end of the
chord domain shown. Although this behaviour is initially surprising, we can attribute this particular observation to
the large regions of separation that form within the troughs of the wavy surface; separation bubbles form a secondary
wall that affects the evolution of the perturbation, establishing smaller amplification rates than that which might
arise if separation did not occur. Secondly, it would appear that (at least for the frequency f= 34 ×10−6) regions
of separated flow do not excite upstream propagating disturbances or absolutely unstable behaviour. This particular
16
0.2 0.3 0.4 0.5
10-10
10-8
10-6
|u|max
(a1)
0.2 0.3 0.4 0.5
10-10
10-8
10-6 (a2)
0.2 0.3 0.4 0.5
x
10-10
10-8
10-6
|u|max
(a3)
0.2 0.3 0.4 0.5
x
10-10
10-8
10-6 (a4)
0.2 0.3 0.4 0.5
x
10-10
10-5
(b4)
0.2 0.3 0.4 0.5
x
10-10
10-5
|u|max
(b3)
0.2 0.3 0.4 0.5
10-10
10-5
|u|max
(b1)
0.2 0.3 0.4 0.5
10-10
10-5
(b2)
FIG. 13. Maximum absolute value of the u-velocity perturbation as a function of x. Solutions generated for (a) {λ, H }=
{0.1,0.0001}; (b) {0.1,0.0004}; for LNS (dashed lines) and PSE methods (dotted). (1) f= 31 ×10−6; (2) 42 ×10−6; (3)
52 ×10−6; (4) 63 ×10−6.
conclusion is drawn based on the fact that the PSE solutions are almost identical to that given by LNS, even though
the former numerical scheme does not account for upstream propagating disturbances.
Further comparisons are drawn between the PSE and LNS solutions in figure 13. The magnitudes of four distur-
bances generated on wavy wings with the amplitudes H= 0.0001 and 0.0004 are depicted in figures 13(a) and 13(b),
respectively. The line types are the same as that presented for figure 11, while the frequency f×106= 31,42,52 and
63. Note that the step size implemented for the PSE analysis was based on that which satisfied the Li-Malik stability
criterion and gave the smallest difference ϵ. The LNS and PSE solutions are (to accuracy of the illustration) identical
17
0 0.5 1 1.5 2 2.5
|αr|∆x
0
50
100
ǫ(%)
(b)
0 0.5 1 1.5 2 2.5
|αr|∆x
0
50
100
ǫ(%)
(a)
10-1 100101
H1/2Λ2
10-1
100
101
102
ǫ(%)
(c)
FIG. 14. (a, b) Differences ϵagainst the Li and Malik stability criterion |αr|∆xfor H= 0.0001 and 0.0004. Frequencies
f= 31 ×10−6(solid line), 42 ×10−6(dashed), 52 ×10−6(chain) and 63 ×10−6(dotted). (c) Differences ϵbased on the
optimum step-size conditions, as a function of H1/2Λ2for Re∞×10−6= 2.5, 5 and 7.5, and those frequencies considered in
(a, b). Λ = λ/λTS is the number of TS waves per wavelength λ. Cross and circle markers respectively represent non- and
separated flow systems.
for the H= 0.0001 configuration. The associated differences ϵare plotted against |αr|∆xin figure 14(a), and are less
than 5% for all four frequencies.
For the larger surface variation H= 0.0004, the two sets of solutions are again in excellent agreement for the
smallest of the frequencies considered (f= 31 ×10−6). However, as the TS frequency increases, the amplitudes of the
PSE and LNS solutions diverge and the PSE method would appear to under-predict the growth of the disturbance.
Furthermore, the magnitudes of ϵassociated with the larger frequencies (depicted in 14(b)) are found to approach
100%. Varying the step size ∆xwas found to have little effect on the accuracy of the PSE calculations, with the
method failing to establish converged solutions for step sizes outside of the limits considered in figure 14(b). Though
not shown here, we included the effect of the higher-order PSE terms (often neglected in stability analysis [43]) and
the chordwise pressure gradient that is usually suppressed; however their inclusion in the PSE analysis was not found
to improve the accuracy of the computations.
The evolution of the four TS perturbations established for H= 0.0004 are depicted in figure 15. The vertical
y-axis has again been deformed to include the surface variation and disturbances have been normalised using the local
maximum amplitude. The structure of each perturbation is qualitatively similar to that described earlier; disturbances
split into two equally strong components about the troughs of the surface and re-coalesce at the crests. Additionally,
upstream propagating structures do not appear to be generated for any of the frequencies considered; disturbances
develop only downstream and are convective. However, the wavelength λTS associated with each TS disturbance is
clearly distinct and is found to decrease in size with increasing frequency. For f= 31 ×10−6, depicted in figure
15(a), the wavelength of the disturbance is approximately 1/50th of the chord length c∗(or 2cm), with about five TS
waves established per surface wavelength λ= 0.1. Meanwhile, for the larger frequency f= 63 ×10−6(figure 15(d)),
λTS is approximately 1cm and ten waves develop per wavelength λ= 0.1. Thus, larger ϵ-differences coincide with a
decreasing disturbance wavelength.
Given the above observations we attempt to determine a relationship between ϵ, the surface configuration and the TS
18
FIG. 15. Disturbance development u/|u|max, in the {x, y }-plane for {λ, H}={0.1,0.0004}. (a) f= 31 ×10−6; (b) 42 ×10−6;
(c) 52 ×10−6; (d) 63 ×10−6. Solid black curves highlight the local regions of separated flow.
wavelength. Differences ϵare computed for all amplitudes H∈[0 : 0.0001 : 0.0004], frequencies f×106= 31,42,52,63
and Reynolds numbers Re∞×10−6= 2.5,5 and 7.5. The step sizes ∆xused in the PSE calculations are again based
on those solutions that give the best comparison with LNS. The resulting ϵ-calculations are plotted against H1/2Λ2
in figure 14(c) using a log-log scaling. Cross and circle markers respectively represent non- and separated flow
systems, while Λ = λ/λTS is the number of TS waves per wavelength λ. Surprisingly, ϵis found to be approximately
proportional (on the log-log scaling) to the function H1/2Λ2and is independent of the Reynolds number. For small
H(non-separated flows) and sufficiently small Λ, ϵ < 10%. However, for separated flows (larger H) and large Λ, the
ϵ-differences increase greatly. Hence, our PSE-LNS analysis suggests that the accuracy of the PSE method is both
dependant on the size of the surface variation (that establishes reverse flow), and the wavelength of the TS wave. If
λTS is sufficiently large (about 2cm for the model considered herein), then PSE can be utilised to give very accurate
solutions, including for those flow systems with large separation.
Reverse flow was not established for wavelengths λ≥0.2 and amplitudes H≤0.0006, which was the upper
limit of our investigation. Nevertheless, we might expect that for parameter settings that establish separation, similar
differences between the LNS and PSE calculations will be observed that are again dependant on the flow specifications
and the wavelength of the TS wave instability.
19
0.2 0.4
0.04
0.06
0.08
0.1
αr
(a)
0.2 0.4
0.04
0.06
0.08
0.1 (b)
0.2 0.4
0.04
0.06
0.08
0.1 (c)
0.2 0.4
x
0.2 0.4
x
0.2 0.4
x
s
FIG. 16. The disturbance wavenumber αrfor frequencies f= 31 ×10−6(solid thick line) 42 ×10−6(dashed) 52 ×10−6
(chain) 63 ×10−6(dotted). (a) {λ, H }={0.1,0.0001}; (b) {0.2,0.0002}; (c) {0.4,0.0004}. Thinner line types depict the results
established on the non-deformed wing.
C. PSE Analysis for Non-Separated Flow Systems
For the remainder of this investigation, we only consider the development of disturbances in non-separated boundary
layers; i.e. surface variations with a wavelength λ≥0.2 or {λ, H }={0.1,0.0001}. Additionally, the step-size ∆x
used to undertake the following PSE analysis was carefully selected to satisfy the Li-Malik stability criterion.
1. Variations in the Disturbance Wavenumber
Figure 16 illustrates the evolutionary paths of the disturbance wavenumber αr(thicker line types) that develops
on three wavy surfaces. Calculations are given for four frequencies f, while the form of the surface waviness (4) has
again been included to draw conclusions. The corresponding solutions on the non-deformed wing are drawn using
thinner line types. Once again surface waviness establishes oscillatory behaviour in the disturbance characteristics,
as the wavenumber increases and decreases as it develops downstream. Matching the oscillations in αrwith the form
of the surface variation, αris found to grow towards a peak within the troughs of the surface wave and decreases to
a local minimum about the crests. Furthermore, the surface waviness would appear to establish greater variations in
αrfor larger frequencies. This particular observation is best illustrated in figure 16(a), by comparing the oscillatory
variations found for f= 31 ×10−6(solid curve) with that established for f= 63 ×10−6(dotted). The latter
result clearly depicts stronger fluctuations (compared to the non-deformed wing) that are spike-like in appearance.
Additionally, as the wavelength λincreases, the variations in αrare significantly damped.
2. Non-Parallel and Curvature Effects
Non-parallel and surface curvature effects are examined in figure 17. Four wavy wall configurations are considered,
where the frequency f= 34 ×10−6. Solid line types represent calculations based on the parallel flow approximation,
while dashed and dotted lines respectively illustrate the non-parallel flow results without and with curvature effects.
Surface curvature effects are included within the stability calculations by defining the parameters κand χin (9) in
terms of the wavy wing geometry. Similarly, they are removed from the two formulations by setting κ= 0 and χ= 1.
20
x
0.2 0.3 0.4 0.5
N
0
1
2
3
4
5
6(a)
Parallel
Non-Parallel without curvature
Non-Parallel with curvature
x
0.2 0.3 0.4 0.5
N
0
1
2
3
4
5
6(b)
x
0.2 0.3 0.4 0.5
N
0
1
2
3
4
5
6(c)
x
0.2 0.3 0.4 0.5
N
0
1
2
3
4
5
6(d)
FIG. 17. Stability N-factor calculations for perturbations with frequency f= 34 ×10−6. Parallel flow results (solid line),
non-parallel without curvature effects (dashed) and non-parallel with curvature effects (dotted). (a) {λ, H }={0,0}; (b)
{0.1,0.0001}; (c) {0.2,0.0002}; (d) {0.4,0.0004}.
Non-parallel flow effects are found to be very small, while curvature effects would appear to have no effect on the
relative sizes of the N-factor.
3. Neutral Stability Curves
Neutral stability curves are drawn in figure 18 in the {x, f }-plane for ϕ= 0, where the unstable parameter space
is enclosed by the curves. Three wavelengths λand varying amplitudes Hare considered, while results for the
non-deformed wing are drawn using a solid curve. The illustrations highlight several interesting characteristics that
are directly related to the form of the surface waviness. Firstly, the wavy walls establish a number of bounded
regions of instability. Secondly, as the amplitude Hincreases, the bounded regions of instability grow in size and
unstable disturbances are obtained at larger frequencies. At a fixed frequency the flow develops downstream along
the x-direction and passes through alternating stable and unstable sections of the stability parameter space. Thus,
waviness forms finite regions of instability. In figure 18(a), surface waviness forms two bounded neutral stability
curves that depict significantly different behaviour to the solution drawn for the non-deformed wing. However, as
the wavelength increases, the effect of waviness would appear to diminish (as depicted in figure 18(c)); the solutions
appear qualitatively similar to that established on the non-wavy wing. The chordwise oscillating pressure gradient
that is engineered by the sinusoidal surface variation is the primary cause of these finitely bounded neutral stability
curves. Waviness establishes both adverse and favourable pressure gradients that are sufficiently strong to destabilise
and re-stabilise TS disturbances.
21
0.1 0.2 0.3 0.4 0.5
0
50
100
150
f106
(a)
H= 0
H= 0.0001
H= 0.0002
H= 0.0003
H= 0.0004
0.1 0.2 0.3 0.4 0.5
0
50
100
150
f106
(b)
H= 0
H= 0.0001
H= 0.0002
H= 0.0004
H= 0.0006
0.1 0.2 0.3 0.4 0.5
x
0
50
100
150
f106
(c)
H= 0
H= 0.0001
H= 0.0002
H= 0.0004
H= 0.0006
FIG. 18. Neutral stability curves for (a) λ= 0.2; (b) 0.3; (c) 0.4 and variable H.
4. Effect of Amplitude and Wavelength
Figure 19 illustrates the effect of variable amplitude Hand wavelength λon the TS disturbance. The surface
configurations considered in the three subplots are given as λ∈[0.2:0.1 : 0.4], H∈[0 : 0.0001 : 0.0006] and ϕ= 0.
The N-factor amplification rates are established by constructing envelopes of the strongest growing disturbances for
all frequencies f∈[1 : 150] ×10−6. Effects of surface waviness are again mirrored in the stability calculations, as
increases and decreases in the growth coincide with the respective regions of an adverse (near surface troughs) and
favourable pressure gradient (crests). The onset of the TS wave instability can be forced to appear at a smaller
x-position (as shown in figure 19(b) and λ= 0.3) or it can be delayed to locations downstream of that found on the
non-wavy wing (as depicted in figure 19(a) and λ= 0.2). Furthermore, an unstable TS wave can be re-stabilised over
some sections of the chord, before becoming unstable again further downstream. This particular observation is best
illustrated in figure 19(b) about 0.3< x < 0.4.
Although waviness can suppress the initial onset of the TS wave instability, it is generally found that once the
disturbance emerges, the amplification rate of the TS wave is enhanced. This is particularly true for λ≤0.3 and as
the amplitude Hincreases. For instance, the surface configuration {λ, H }={0.2,0.0003}generates a disturbance
(represented by a dotted line in figure 19(a)) with an amplification factor N= 11 at xref , while the non-wavy wing
model establishes N≈4.5. Thus, waviness can considerably destabilise the TS disturbance. Furthermore, it was
22
0.1 0.2 0.3 0.4 0.5
0
5
10
N
(a)
H= 0
H= 0.0001
H= 0.0002
H= 0.0003
H= 0.0004
0.1 0.2 0.3 0.4 0.5
0
5
10
N
(b)
H= 0
H= 0.0001
H= 0.0002
H= 0.0003
H= 0.0004
H= 0.0005
H= 0.0006
0.1 0.2 0.3 0.4 0.5
x
0
5
10
N
(c)
H= 0
H= 0.0001
H= 0.0002
H= 0.0003
H= 0.0004
H= 0.0005
H= 0.0006
FIG. 19. Stability N-factor calculations for surface waviness configurations (a) λ= 0.2; (b) λ= 0.3; (c) λ= 0.4 and variable
H.
determined that the ∆Nvariations at a fixed chord position are proportional to the square of the amplitude H, which
is consistent with the conclusions of Wie and Malik [20]. Additionally, larger positive variations ∆Nare found for
smaller wavelengths λ. Hence, sinusoidal surface waviness generally increases the growth of the TS wave instability,
which may in turn trigger the premature onset of transition to turbulence.
Stability calculations of the longer wavelength λ= 0.4 (plotted in figure 19(c)) are found to behave very differently
to that given for the smaller wavelengths; this particular surface configuration significantly dampens the disturbance
growth. About xref the wavy surface {λ, H}={0.4,0.0006}(that is represented by a solid line with square symbols)
establishes N≈2.5, which is almost half of that given for the non-deformed wing. Hence, this particular surface
variation is stabilising and may be a mechanism for delaying the onset of boundary layer transition. This observation
was unanticipated as previous studies (Wie and Malik [20] amongst others) generally found that waviness destabilises
the TS disturbances.
5. Effect of Phase
The effect of a variable phase shift ϕis considered in figure 20, where the N-factor envelopes are plotted for
λ∈[0.2:0.1:0.4] and H= 0.0003. From the three illustrations it is immediately obvious that the phase of the
23
0.1 0.2 0.3 0.4 0.5
0
2
4
6
8
N
(a)
0.1 0.2 0.3 0.4 0.5
0
2
4
6
8
N
(b)
0.1 0.2 0.3 0.4 0.5
x
0
2
4
6
8
N
(c)
FIG. 20. Stability N-factor calculations for surface waviness configurations (a) {λ, H}={0.2,0.0003}; (b) {0.3,0.0003}; (c)
{0.4,0.0003}, where the phase shift ϕ= 0 (dashed), π/2 (chain), π(dotted), −π/2 (solid with cross symbols). The solid line
represents the solution on the non-deformed wing model.
surface waviness can have a significant impact on the evolution of the disturbance, as varying ϕcan cause the onset
and size of the instability to vary greatly. For λ= 0.2 unstable behaviour is first excited for a phase shift ϕ=π/2,
while ϕ= 0 generates the greatest delay in the onset of the instability. However, this does not necessarily imply that
these surface configurations will respectively engineer the strongest or weakest growing disturbances. For instance,
about xref , the TS wave generated on the wing with surface dimensions {λ, H, ϕ}={0.2,0.0003, π/2}(dotted line)
has a smaller N-factor than that established for the other phase shifts considered, even though disturbances appear
first for this particular configuration.
Figure 21 depicts the stability variations ∆Nagainst the phase shift ϕ. Computations are measured about the
reference location xref where the symbols illustrate the actual computations, with spline fitting used to plot a best-
curve-fit. The phase of the surface waviness is found to establish significant variations in the TS wave growth
rate. Although waviness generally establishes large positive N-factor variations, some configurations are stabilising;
{λ, H, ϕ}={0.3,0.0003, π},{0.4,0.0003,0}and {0.4,0.0003, π /2}. Thus as suggested above, it may be possible to
engineer a beneficial form of waviness that can be used to reduce instability and suppress the onset of transition.
24
-3 -2 -1 0 1 2 3
0
5
10
∆N
(a)
-3 -2 -1 0 1 2 3
0
2
4
6
8
∆N
(b)
-3 -2 -1 0 1 2 3
φ
-2
0
2
4
6
∆N
(c)
FIG. 21. Stability variations ∆Nas a function of the phase shift ϕmeasured about xref = 0.55. (a) λ= 0.2; (b) λ= 0.3;
(c) λ= 0.4. The marker symbols ×,⃝,□,♢,∗represent solutions of the respective amplitudes H= 0.1,0.2,0.3,0.4 and 0.5.
Dashed lines are spline fitted curves, linking the solutions of the same H.
6. Freestream Effects
Table II tabulates the strongest growing frequencies and associated N-factor about xref for variable Re ∞and M∞.
In figure 22(a, c) N-factor envelopes are plotted for both the non-wavy wing model (thin line types) and surface
dimensions {λ, H, ϕ}={0.2,0.0002,0}(thick), where the flow conditions are as specified in the caption. The size
of Nincreases with the Reynolds number, for both the non-wavy and wavy surface. However, in figure 22(c) the
strongest growing disturbance is found for M∞= 0.25 (dashed lines), with smaller N-factors found for lower and
higher valued Mach numbers.
Stability variations ∆Nare plotted against Re∞and M∞in figure 22(b, d). The ∆Ncomputations are again
based on the N-factor calculations established about xref . Cross symbols represent actual results with spline fitting
used to draw a best-curve-fit. For wavelengths λ≤0.3 surface waviness is destabilising for all values of Re∞and
M∞considered, while surfaces with a wavelength λ= 0.4 again engineer a decrease in the N-factor. In figure 22(b)
the magnitude of ∆Nis found to increase proportionally with Re∞, which is again consistent with the conclusions
drawn by Wie and Malik [20]. Additionally, the absolute size of ∆Nincreases with the Mach number, particularly
for M∞>0.5.
25
Re∞×10−6M∞f∗(kHz) f×106N
0.5 0.7 3 155 0.8
1 0.7 3.5 90 1.4
2.5 0.7 5 52 2.7
5 0.7 6.5 34 4.5
7.5 0.7 7 24 5.5
10 0.7 7 18 6.6
5 0.1 1 36 4.3
5 0.25 2.5 36 7.6
5 0.5 4.5 33 6.4
TABLE II. Strongest growing disturbance for variable freestream conditions about xref = 0.55 on the non-deformed wing.
0.25 0.3 0.35 0.4 0.45 0.5 0.55
x
0
5
N
(a)
0.25 0.3 0.35 0.4 0.45 0.5 0.55
x
0
5
N
(c)
0246810
Re∞×10−6
-2
0
2
4
6
∆N
(b)
0.1 0.2 0.3 0.4 0.5 0.6 0.7
M∞
-2
0
2
4
∆N
(d)
FIG. 22. (a) Stability N-factor calculations for Re∞×10−6= 0.5 (solid lines), 2.5 (dashed) and 7.5 (chain) with M∞= 0.7.
The fainter line types represent the solutions on the non-wavy surface and the thicker line types correspond to {λ, H, ϕ}=
{0.2,0.0002,0}. (b) Stability variation ∆Nagainst Re∞for {λ, H, ϕ}={0.2,0.0002,0}(solid), {0.3,0.0003,0}(dashed) and
{0.4,0.0004,0}(chain). (c) Stability N-factor calculations for M∞= 0.1 (solid lines), 0.25 (dashed) and 0.5 (chain) and
Re∞×10−6= 5. Line types are the same as (a). (d) Stability variation ∆Nagainst M∞. Line types are the same as (b).
26
0 2 4 6 8 10 12
nH2Re∞/λ
0
2
4
6
8
10
∆N
FIG. 23. Comparison between the variation ∆N(measured about xref ) and nH2Re∞/λ, for f= 34 ×10−6(f∗= 6.5kHz) and
M∞= 0.7.
7. Comparison with Wie &Malik
Given the above stability variations, we attempt to correlate results into an expression similar to that developed
by Wie and Malik [20] for the flow over a wavy flat plate. The Wie and Malik expression (3) relates to a fixed
freestream Mach number M∞= 0.7. Additionally, their analysis was applied to a fixed frequency that corresponded
to the strongest growing disturbance on the non-deformed plate. Thus, to facilitate a comparable relationship, we
only compare calculations established for M∞= 0.7 and f= 34 ×10−6(f∗= 6.5kHz). Furthermore, due to the
form of the waviness implemented in this study (imposed along the length of the wing with an origin located at
the leading-edge), we must make some assumptions regarding how to collate comparable surface configurations and
stability results.
PSE calculations for f= 34 ×10−6measured about xref = 0.55, confirm that ∆Nis proportional to both Re∞and
the square of the amplitude H(or h= 2Hused in Wie and Malik). However, relating other surface characteristics
(wavelength λand number of waves nfrom the leading-edge) was more problematic. As surface waviness was
implemented with a variable phase shift, the form of the wave at xref could differ quite significantly. Thus, to
successfully relate ∆Nfor variable surface dimensions it was necessary to compare only those wavy configurations
with a comparable surface structure at xref . Circle symbols in figure 23 mark the stability variations ∆Nagainst
the function nH2Re∞/λ for those wavy surfaces with a trough centred near xref . For instance, the surface wave
configuration {λ, H, ϕ}={0.2, H, 0}forms a trough centred at xref , where the number of waves nover the chord
range 0 ≤x≤xref is given as n= 100xref /λ ≡2.75. For those surface configurations that fall into this category, the
corresponding values of ∆Nare approximately located about the dashed line given as
∆N= 0.92nH2Re∞/λ. (16)
Thus, if a trough is centred about the reference location xref , the ∆Nvariations (for a fixed frequency) on the wavy
wing are in reasonably good agreement with the analysis of Wie and Malik. However, after repeating the above
procedure for surface configurations with a different wavy structure about xref , this particular relationship did not
hold and the stability variations were scattered throughout the (nH2Re∞/λ, ∆N)-parameter space.
8. First Disturbance to give N= 4.5
Figure 24 depicts the chord location x≡xN=4.5against the corresponding disturbance frequency fthat first
establishes an amplification factor N= 4.5. For instance, on the non-wavy wing, the strongest growing frequency
f= 34 ×10−6gives N= 4.5 at the reference location xref ≡xN=4.5. However, for wavy surfaces, the frequency
responsible for the strongest growing TS wave was found to vary significantly. The corresponding results are marked
27
30 35 40 45 50 55
f106
0.3
0.35
0.4
0.45
0.5
0.55
xN=4.5
Clean
λ= 0.1
λ= 0.2
λ= 0.3
λ= 0.4
FIG. 24. The chord location xN=4.5plotted against the frequency fthat first establishes N= 4.5, for variable surface wave
configurations. The freestream conditions {M∞,Re∞}={0.7,5×106}.
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
-0.5
0
0.5
PB,x
(b)
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
s
(a)
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
x
0
2
4
N
(c)
FIG. 25. Flow characteristics for the surface configuration {λ, H, ϕ}={0.4,0.0006,0}(solid lines) and the non-deformed wing
(dashed). (a) Form of the surface deformation s; (b) Pressure gradient PB,x; (c) N-factor analysis.
in figure 23 for variable surface configurations and {Re∞, M∞}={5×106,0.7}. The chord location xN=4.5is found
to decrease with increasing f.
28
V. CONCLUSIONS
The effect of long wavelength surface variations on an unswept infinite wing have been investigated. Compressible
boundary layers were established using solutions of an industrial flow solver [28], while the stability and development
of TS disturbances were studied using PSE and LNS methods. Wavy deformations were generally found to enhance
the growth of the TS instability. This was especially true for smaller wavelength and larger amplitude surface
variations. Varying the phase of the sinusoidal surface waviness was also found to engineer large differences in the
TS disturbance development. Furthermore, the stability variations on some wavy surface configurations, measured
about a fixed location and for a constant frequency, were found to behave in a manner consistent with the analysis
and relationships formed by Wie and Malik [20] on a wavy flat plate. Thus, their simplified flat-plate model provides
a very good prediction for the TS wave variations on a wavy wing.
Although waviness was generally found to destabilise TS disturbances, some longer wavelength deformations estab-
lished a stabilising effect that may be used to delay the onset of transition beyond that specified on the non-deformed
wing. Those surface variations that damped the growth of the TS wave were found to create a stronger favorable
pressure gradient about sections of the wing that had previously been relatively weak or adverse. The stabilising
effect is demonstrated in figure 25 that depicts the form of the surface variation sand the associated pressure gradient
PB,x and N-factor envelope that develop on the wavy wing {λ, H, ϕ}={0.4,0.0006,0}. Along the upward slopes of
the wavy surface, PB,x is found to decrease in size and attains smaller values than that depicted on the non-deformed
wing. The strong favourable pressure gradient then suppresses the growth of the disturbance and the N-factor is
significantly reduced. Hence, this particular surface configuration establishes a smaller amplification rate over the
chord range considered. The stabilising effect illustrated for this particular configuration (and some others) was quite
surprising as waviness has previously been thought to only destabilise the TS wave instability. However, we should
acknowledge that many of the earlier studies were concerned with the flow development on flat-plate geometries and
that the application of surface waviness to an already curved wing body may explain why a stabilising effect can be
achieved in some instances.
Boundary layers were extracted directly from the solutions of a full Navier-Stokes solver, allowing us to investigate
the evolution of disturbances in separated boundary layers. The extent of the separation bubbles that could form
within the troughs of the wavy surface was found to increase with the amplitude H; for surface wavelengths λ= 0.1,
H≥0.002 was sufficient to establish separated flow. Although the PSE model does not account for the upstream
propagation associated with reverse flow, we were still able to successfully apply the PSE method in most cases.
Furthermore, for flows with small separation bubbles, PSE calculations were shown to be in excellent agreement with
the LNS computations provided the wavelength of the TS disturbance was sufficiently long. Differences between the
PSE and LNS modelling were shown to be proportional to H1/2Λ2on a log-log mapping, where Λ is the ratio of the
surface wavelength λto the TS wavelength λTS. Thus, the accuracy of the PSE model was found to be dependant on
the wavelength of the TS wave; if Λ is relatively small, then the PSE method could accurately compute the disturbance
development in separated boundary layers. However, for Λ ≳5 (shorter TS wavelengths λTS), the PSE method was
found to greatly under predict the amplification rate of the TS disturbance.
ACKNOWLEDGMENTS
This work has been supported by the EPSRC funded LFC-UK project: Development of Under-Pinning Technology
for Laminar Flow Control, grant EP/I037946/1 and by the TSB funded ALFET project 113022. We thank the referees
for many helpful suggestions concerning the presentation and improvement of our results.
Appendix A: LNS Formulation
The LNS continuity, momentum and energy equations in a 2D compressible flow are given as
χux+χρB,x
ρB
u+ρB,y
ρB−κχv+vy+χUBpx+VBpy
PB−(χUBTx+VBTy)
TB−
iω −χ(UB,x −κVB)−VB,y +(χUBTB ,x +VBTB,y )
TBp
PB
+
iω −χ(UB,x −κVB)−VB,y −χUBρB ,x +VBρB,y
ρB
+(χUBTB,x +VBTB,y )
TBT
TB
= 0,(A1a)
29
uyy +rχ2uxx
Re +sχvxy
Re +ρB
µBPBχUB(κVB−UB,x)−VBUB,y p−χ
µB
px−ρB
µBχ(UB,x −κVB)−iω+
κ2χ2
Re −κχµB,y
µBRe u−ρBVB
µB
+κχ
Re −µB,y
µBRe uy+ρB
µBκχUB−UB,y −rκχ2µB ,x
µBRe v+mχµB,x
µBRe vy−
ρBUB
µB−rχµB,x
µBRe χux−eκχ
Re −µB,y
µBRe χvx+fT
µBRe mVB,y +r(χUB,x −κχVB)χTx+
UB,y +χ(VB,x +κUB)Ty+fT T
µBRe χTB,x(rχ[UB,x −κVB] + mVB,y) + TB,y (χ[κUB+VB,x ] + UB,y )−
ρB
µBTBκχUBVB−χUBUB,x−VBUB,y +fT
µBRe UB,yy −κχ(κχUB+UB,y )−eκχ2VB ,x+sχVB,xy +rχ2UB ,xxT= 0,
(A1b)
rvyy +χ2vxx
Re +sχuxy
Re −ρB
µBPBχUB(κUB+VB,x) + VBVB,y p−py
µB−ρB
µBVB,y −iω+
rκ2χ2
Re +mκχµB,y
µBRe v−ρBVB
µB
+rκχ
Re −rµB,y
µBRe vy−ρB
µB2κχUB+χVB,x−κχ2µB,x
µBRe u+χµB,x
µBRe uy−
ρBUB
µB−χµB,x
µBRe χvx+eκχ
Re +mµB,y
µBRe χux+fT
µBRe rVB,y +mχ(UB,x −κVB)Ty+
χ(κUB+VB,x) + UB,y χTx+fT T
µBRe TB,y(mχ[UB,x −κVB] + rVB ,y ) + χTB,x (χ[κUB+VB,x ] + UB,y )+
ρB
µBTBχUB(κUB+VB,x)+ VBVB ,y +fT
µBRe r(VB ,yy −κχ[κχVB+VB,y])+ eκχ2UB ,x +sχUB,xy +χ2VB,xx T= 0,
(A1c)
Tyy +χ2Txx
Re +Γ
µBχUBpx+VBpy−iωΓ
µB
+σρB
µBPBχUBTB,x +VBTB,y p+
χ(fTTB,x +µB,x )
µBRe −σρBUB
µBχTx+fTTB,y +µB,y
µBRe −σρBVB
µB−κχ
Re Ty+
χ(ΓPB,x −σρBTB ,x)
µB
+2Γ
Re κχ(χ[κUB+VB,x] + UB ,y )u+
ΓPB,y −σρBTB ,y
µB
+2Γ
Re κχ(rχ[κVB−UB,x]−mVB,y )v+
2Γ
Re χ(κUB+VB,x) + UB ,y uy+2Γ
Re mχ(UB,x −κVB) + rVB,y vy+
2Γ
Re rχ(UB,x −κVB) + mVB,y χux+2Γ
Re χ(κUB+VB,x) + UB ,y χvx+
σρB
µBχUBTB,x +VBTB,y
TB
+iω+fT T
µBRe T2
B,y +χ2T2
B,x+χ2TB,xx +TB,y y −κχTB,y +
Γκ2χ2U2
B+rV 2
B,y +rχ2(UB ,x−κVB)2+2mχVB ,y (UB,x −κVB)+2κχUB(UB,y +χVB ,x)+(UB ,y +χVB,x )2fT
µBRe T= 0,
(A1d)
where Re is the Reynolds number based on the boundary layer thickness δ∗. Subscripts xand ydenote the derivatives
along the respective chordwise and wall-normal directions. The parameters Γ = (γ−1)M2
∞σ,e=r+ 1, r=s+ 1,
s=m+ 1 and m=−2/3. Here µB=f(T) is the dependence of the dynamic viscosity on the temperature and
30
fT=dµB/dT ,fT T =d2µB/dT 2. Further, σrepresents the Prandtl number and γis the ratio of the specific heats.
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