Under consideration for publication in J. Fluid Mech.
Numerical Investigation of the Role of
Free-stream Turbulence on Boundar y-Layer
Separation
By W. B A L Z E R A N D H. F. F A S E L
Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ
85721, USA
(Received ??? and in revised form ???)
The aerodynamic performance of lifting surfaces operating at low Reynolds number conditions is impaired by laminar separation. In most cases, flow transition to turbulence
occurs in the separated shear layer as a result of a series of strong hydrondynamic instability mechanisms. Numerical investigations have become an integral part in the effort to enhance our understanding of the intricate interactions between separation and
transition. Due to the development of advanced numerical methods and increase in the
performance of supercomputers with parallel architecture, it has become very feasible
for low Reynolds number application (O(105)) to carry out direct numerical simulations
(DNS) such that all relevant spatial and temporal scales are resolved without the use of
turbulence modeling. Because the employed high-order accurate DNS are characterized
by very low levels of background noise, they lend themselves to transition research where
the amplification of small disturbances, sometimes even growing from numerical roundoff, can be examined in great detail. When comparing results from DNS and experiment,
however, it is beneficial, if not necessary, to increase the background disturbance levels
in the DNS to levels that are typical for the experiment. For the current work, a numerical model that emulates a realistic free-stream turbulent environment was adapted and
implemented into an existing Navier-Stokes code based on vorticity-velocity formulation.
The role free-stream turbulence (FST) plays in the transition process was then investigated for a laminar separation bubble forming on a flat plate. F S T was shown to cause the
formation of the well known Klebanoff mode that is represented by streamwise-elongated
streaks inside the boundary layer. Increasing the F S T levels led to accelerated transition,
a reduction in bubble size, and better agreement with the experiments. Moreover, the
stage of linear disturbance growth due to the inviscid shear-layer instability was found
to not be “bypassed”.
1. Introduction
The transition of regular, “smooth” fluid flow (laminar flow) into a state of chaotic,
random fluid motion (turbulent flow) remains one of the most elusive subjects in physics.
Due to the vast parameter space that influences flow transition and the chaotic non-linear
behavior of turbulent flow, advancing the physical understanding presents a formidable
challenge. For low Reynolds number flow problems ( Re c < 106; based on the chord
length, c, of an airfoil, for example) with strong adverse pressure gradients (APGs) in
the streamwise direction, matters are complicated by the fact that transition is often
initiated inside a laminar separated shear layer located in close vicinity to a surface.
A special case exists for laminar separation when transition and the associated increase
1
2
W. Balzer and H. F. Fasel
in wall-normal momentum exchange cause the reattachment of the separated flow. The
resulting region of enclosed fluid between separation and reattachment locations is denoted as laminar separation bubble (LSB). For a given geometry and APG the extent
of the separated region and the transition region are typically larger at lower Reynolds
number conditions. For sufficiently low Reynolds numbers and/or low APGs transition
to turbulence might not occur and the flow reattaches in a laminar fashion. For a variety of low Reynolds number flows in engineering applications, the appearance of LSBs
can be the decisive factor for performance limitations of the device. Typical examples
are low-pressure turbine ( L P T ) stages, small unmanned aerial vehicles (UAVs), high-lift
multi-element airfoil configurations, and diffusers. While small regions of separated flow
on a lifting surface, for example, may only have little effect on the aerodynamic performance, larger separation bubbles can cause a severe reduction in lift and an increase in
drag. In the worst case, the flow does not reattach to the surface leading to a behavior
known as “stall”. Stall can occur gradually (“soft stall”) or abruptly (“hard stall”). The
latter is dangerous and can have negative consequences such as, for example, a sudden
complete loss of control.
The general structure of a time-averaged separation bubble is presented in figure 1.
Figure 1 is an adaptation of a widely-used schematic presented by Horton (1968). As
a result of the APG and the viscous stresses, fluid particles in the near-wall region of
the attached laminar boundary layer are slowed down until the velocity profile becomes
inflectional at the separation point, S. For steady separation, this point is typically
defined as the first downstream location for which the skin friction vanishes and subsequently assumes negative values. Just downstream of separation, and below the dividing
streamline, the fluid is almost stagnant. This corresponds to a region often referred to
as the “dead-air” region. Due to the small velocities in this area, the pressure remains
almost constant, which typically results in a pronounced pressure plateau in the wallpressure distribution. The separated laminar shear layer is characterized by inflectional
reverse-flow profiles which support the amplification of small disturbances in the early
stages of transition. Farther downstream the shear layer becomes wavy and rolls up
into spanwise vortices (often referred to as “rollers”), which detach from the shear layer
and are convected downstream (vortex shedding). These vortices considerably increase
the wall-normal momentum exchange by transporting low-momentum fluid away from
the wall and high-momentum free-stream fluid towards the wall. In the time-averaged
picture, however, the process of shear-layer roll-up and shedding of vortices cannot be
seen. Here it is indicated by rapid reattachment and the appearance of a reverse-flow
vortex which represents an average over multiple such events. The mean velocity profiles
in the region of the reverse-flow vortex exhibit the largest reverse-flow magnitudes in
the separation bubble. In addition, the disturbance amplitudes are large, which gives
rise to three-dimensional (3D) secondary instabilities and nonlinear wave interactions.
These are the late stages of transition. Although transition is to be considered a process
rather than a local event, it is common to define a “transition onset point”, T, as, for
example, the location of the minimum skin-friction coefficient. The downstream extent
of the transition region depends on multiple factors such as the Reynolds numbers, the
APG, and the disturbance spectrum. Shortly downstream of transition, the flow reattaches due to both entrainment of free-stream fluid and increased wall-normal mixing.
The mean reattachment point, R, can be defined as the last downstream location for
which the (time-mean) skin-friction vanishes and subsequently assumes positive values.
Note that such a definition only has a meaning for the time-averaged flow field since
the flow is highly unsteady and the instantaneous location of reattachment can fluctuate
significantly.
The Role of Free-stream Turbulence on Boundary-Layer Separation
"Dead−Air" Region
Free Stream
S
Attached Laminar
Boundary Layer
Dividing Streamline
Reverse−Flow Vortex
T
Separated Laminar
Shear Layer
Boundary−Layer Edge
R
Transition
Redeveloping Turbulent
(non−linear stages) Boundary Layer
Figure 1. Mean-flow structure of a laminar separation bubble on a flat plate. Plotted are
streamlines, selected streamwise velocity profiles, as well as the boundary-layer thickness.
Since flow separation and transition are at work interactively, they should not be considered as independent challenges. On the one hand, the separated shear layer is highly
unstable with respect to small disturbances which can accelerate transition. On the other
hand, earlier transition can reduce or even eliminate separation. For example, reducing
the size of a LSB also results in less displacement of the inviscid potential flow, which
in turn can change the APG and thus directly affect separation. Understanding of the
inherent connection between separation and transition is therefore imperative for the
design and optimization of lifting surfaces and the development of efficient flow control
strategies. Moreover, transition and separation are strongly influenced by external factors
such as free-stream turbulence (FST), surface roughness both discrete (e.g. insect contamination) and distributed (e.g. sandpaper roughness), noise, vibration (e.g. caused by
a motor), and others. Consequently, the physical mechanisms associated with separation
control will also be affected by these factors. Driven by the need for more accurate predictions of separation and transition phenomena in “real-world” applications, the question
of how relevant transition/separation mechanisms for separated flows are affected by FS T
(as well as surface roughness) has received renewed attention in the recent years.
1.1. Transition in Laminar Separation Bubbles
The technical relevance of LSBs has motivated a larger number of experimental, theoretical, and numerical studies of the stability and transition process for this flow phenomenon. Recent overviews of the subject are given by Rist (1998) and Boiko et al.
(2002). In general, the transition process in wall-bounded external shear flows can be
understood as a progression of stages. Morkovin et al. (1994) discriminate between 5
different paths to turbulence (A-E) depending on the level of external disturbance excitation (see figure 2). The first stage common to all paths is called receptivity (Morkovin
1969). Receptivity denotes the process through which controlled and uncontrolled external disturbances (FST, sound, wall blowing and suction, etc.) enter the boundary layer
and generate disturbances of certain amplitudes, frequencies, and wavenumbers. The resulting disturbance spectrum greatly influences the ensuing transition scenario. Figure
2 shows a schematic that summarizes possible transition paths/scenarios observed for
laminar separation bubbles. Paths D and E are excluded from the following discussion
since it is assumed that forcing at very large amplitudes will quickly lead to breakdown
of the flow prior to separation and likely suppresses the separation bubble completely.
Such a transition scenario is often categorized as “bypass” transition (Morkovin 1979)
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W. Balzer and H. F. Fasel
Forcing environmental disturbances
amplitude
t:absoluteinstability
x:convectiveinstability
Transitionprocess
Receptivity
Transient Growth
(algebraic growth)
A
B
Primary Instability
C
(exponential growth)
D
Secondary Mechanisms
E
Bypass Mechanism
(mode interactions)
Breakdown
(coherent structures, higher instabilities)
Turbulence
LSB size
no LSB
(depending on Re & APG)
Figure 2. Paths to turbulence for incompressible wall-bounded shear flows. Adapted from
Morkovin et al. (1994).
since linear stages have no influence on the transition process and are bypassed on the
route to turbulence.
Path A in figure 2 corresponds to the transition scenario observed when the level of
environmental disturbances is very low (see also Marxen 2005, for a detailed description).
In this case, transition is initiated by the exponential amplification (modal growth) of
so-called Tollmien-Schlichting (TS) waves. The inflectional velocity profiles in LSBs give
rise to an inviscid instability mechanism (KH instability) with large amplification rates.
Especially in a situation where the inflection point moves far away from the wall, amplification rates can be orders of magnitude larger than for attached boundary layers.
Rist (1998) found good agreement between the exponential disturbance growth obtained
from 3D DNS and predictions from linear stability theory (LST). He concludes that for
typical LSB flows, non-parallel effects have only a minor influence.
Diwan & Ramesh (2009) provide experimental evidence for the fact that the origin of
the inflectional instability lies upstream of the separation bubble where TS waves are
generated (receptivity) and amplified due to a viscous instability in the attached APGregion of the boundary layer. On the other hand, the self-sustained vortex shedding seen
in both experiments and DNS must not necessarily be caused by the spatial growth of
The Role of Free-stream Turbulence on Boundary-Layer Separation
incoming disturbances (convective instability). Inflectional velocity profiles with a strong
reverse flow can also become absolutely unstable (Huerre & Monkewitz 1990). Absolute
instability refers to the amplification of small disturbances in time rather than in space.
Several authors have investigated the theoretical onset of absolute instability for a variety
of analytic reverse-flow profiles (Gaster 1992; Hammond & Redekopp 1998; Alam &
Sandham 2000). As a crude estimate, for the Reynolds number range of Re δ1 = 500− 1000,
reverse flow percentages in excess of 15 − 20% are required to observe absolute instability.
The TS waves grow until they saturate at amplitude levels of order O(0.1 − 0.3 U∞). It
is known (from L S T analysis, for example) that 2D disturbances are stronger amplified
than 3D disturbances in separated shear layers. Therefore, they typically saturate first,
leading to the commonly observed periodic shedding of spanwise coherent (2D) vortices.
At this stage secondary mechanisms set in and break the remaining symmetries of the
disturbed flow.
In many cases, separation and the associated deflection of the boundary layer results
in local streamline curvature in the vicinity of boundary-layer separation. The curvature
of the streamlines is concave and might give rise to steady, longitudinal vortices that
are amplified due to a centrifugal instability (G¨ortler instability; G¨ortler (1941)). While
Spalart & Strelets (2000) did not find any evidence of G¨ortler vortices in their 3D DNS
of an unsteady LSB, Pauley (1994) presented spanwise velocity contours which suggested
the formation of such structures. For a LSB on a flat plate subjected to a strong favorableto-adverse pressure gradient, Marxen et al. (2009) investigated the evolution of controlled
steady linear 3D disturbances and found strong evidence that a G¨ortler-type instability
due to streamline curvature is indeed possible.
The results by Marxen et al. (2009) can also serve as an example for Path B of the
transition process since the G¨ortler instability was preceded by transient growth of steady
streamwise disturbances in the favorable pressure gradient (F PG) region of the flow. This
growth mechanism provides higher initial amplitudes for the G¨ortler eigenmode growth.
Transient growth is associated with the non-normality of the linearized Navier-Stokes operator (see Trefethen et al. 1993; Schmid & Henningson 2001, for example). Superposition
of individual non-orthogonal eigenmodes can lead to algebraic growth for some downstream distance before exponential decay due to viscosity becomes the overwhelming
factor. A physical explanation for transient growth is provided by the so-called “lift-up”
effect (Ellingsen & Palm (1975); Landahl (1975)). This effect describes an inviscid mechanism through which the wall-normal displacement of a fluid particle in a shear layer
causes a perturbation in the streamwise velocity since initially the fluid particle will retain its horizontal momentum. The optimal growth (i.e. the optimal superposition; see
Andersson et al. (1999); Luchini (2000)) is obtained for pairs of counter-rotating streamwise vortices, which very effectively lift up low-velocity fluid from the wall and push down
high-velocity fluid towards the wall. This mechanism results in strong streamwise streaks
which modulate the boundary layer in the spanwise direction. Despite the possibly large,
non-linear amplitudes of the resultant flow structures, the transient growth theory is a
linear theory.
Path C corresponds to a transition scenario initiated by transient growth in absence
of any modal instabilities (TS, G¨ortler). McAuliffe & Yaras (2010), for example, showed
that for a “short” LSB under elevated levels of F S T (T u = 1.45% at separation), the
nature of the instability mechanisms changes from amplification due to the KH instability
(path A) to amplification of streamwise streaks. These streaks extend into the region of
laminar separated flow and initiate breakdown via the formation of turbulent spots.
In nature, disturbances from different paths will likely coexist and interact with each
other. Wind-tunnel experiments carried out for a flat-plate boundary layer by Kosorygin
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6
W. Balzer and H. F. Fasel
& Polyakov (1990), for example, show that in the intermediate range of free-stream
fluctuations (0.1% < T u < 0.7%) both TS waves (path A) and streamwise boundarylayer streaks (path C) can grow simultaneously. Wu et al. (1999) and Wu & Durbin
(2001) employed 3D DNS to investigate the separated flow in a low-pressure turbine stage
under the influence of turbulent, periodically passing wakes. They studied the connection
bewteen classical linear growth mechanisms (TS, KH) and transient growth, concluding
that transient growth plays an important role in this context. From recent DNS results
(Fasel 2002; Liu et al. 2008a,b) emerges a picture of complex interactions between the
streamwise streaks and TS waves. While steady streaks may have a stabilizing effect on
TS waves, unsteady streaks can enhance secondary instabilities of TS waves and promote
transition.
1.2. The Effect of Free-Stream Turbulence
The free-stream turbulence intensity (FSTI) can be relatively low for free-flight applications (< 0.1%) and wind-tunnel experiments (≃ 0.7%, and also lower) (Kurian &
Fransson 2009), but can also be very high, as in turbomachinery flow. For the latter,
FSTIs as high as 20% have been reported (e.g. Hourmouziadis 2000). Lower F S T levels
in the order of 10% have been measured in a LPT environment (Sharma 1998). Whereas
wind turbulences is rather uniform, the turbulence in turbomachinery flow is often concentrated in the airfoil wakes periodically impinging on downstream blade rows.
Historically, F S T has made it very difficult to identify linear stages of the transition
process (path A). Experimental evidence of TS waves, for example, was not at hand
until Schubauer & Skramstad (1948) managed to reduce the level of turbulent fluctuations in the free stream to “unusually low values” (∼ 1%). The success of Schubauer’s
experiments could in part be attributed to suggestions by Dryden (1936) who carried out
experiments at a much more turbulent free stream. He reported slow irregular u-velocity
fluctuations of large amplitude in the laminar region of the boundary layer that were
induced by the turbulence in the free stream. In fact, transition in wind-tunnel experiments typically seems to be preceded by such a low-frequency streaky modulation of the
boundary layer. Today, these streaks are commonly referred to as “Klebanoff modes” (Kmode) or “Klebanoff distortions” after P. S. Klebanoff who investigated this phenomenon
and described them as a periodic thickening/thinning of the boundary layer (Klebanoff &
Tidstrom 1959; Klebanoff 1971). From other experimental studies investigating the effect
of FST, it has been established that in addition to its low frequency and high amplitude,
urms = O(0.1U∞), the Klebanoff mode is also characterized by a distinct spanwise
spac√
ing of O(2δ99 − 4δ99) and an algebraic streamwise growth proportional to
x (Kendall
1985, 1990; Westin et al. 1994). Streamwise growth of the K-mode is associated with the
lift-up effect and transient growth mechanism described earlier.
In general, a shear layer adjacent to a wall is surprisingly resistant to fluctuations impinging from the free stream. The general observation that eddies convected by the free
stream are quickly damped out inside the boundary layer due to inviscid shearing and
viscous dissipation has been coined the “shear sheltering” effect by Hunt et al. (1996).
Using solutions from the continuous spectrum of the Orr-Sommerfeld equation, Jacobs
& Durbin (1998) demonstrated that at finite Reynolds numbers the boundary layer acts
as a low-pass filter, admitting predominantly low-frequency modes to penetrate into the
boundary layer. Due to streamwise stretching of the ingested free-stream vorticity and
transient growth (Saric et al. 2002), low-frequency, high-amplitude streamwise streaks
arise in the boundary layer (K-mode). TS waves, on the other hand, have higher frequencies and smaller scales than the energy-bearing long-wavelength disturbances that
are typical of F S T (and sound). Therefore some kind of coupling or rescaling of the free-
The Role of Free-stream Turbulence on Boundary-Layer Separation
stream disturbances is necessary in order to allow for energy transfer to the TS modes.
It is known that such a coupling can only occur in flow regions where non-parallel effects
are important, such as near the leading edge, a wall hump, or flow around the separation
point (e.g. Kerschen 1991).
When a highly disturbed free stream introduces finite non-linear disturbances in the
laminar boundary layer, the first stages of the transition process (linear growth) may be
“bypassed”†. Jacobs (1999) employed 3D DNS to investigate transition in a zero pressure
gradient, flat-plate boundary layer with a FS T I of T u ∼ 3.5%. His results indicate that
bypass transition might be initiated when a boundary layer that is highly distorted by
low-speed streaks has become susceptible to higher-frequency disturbances which now
enter the boundary layer from the free stream and trigger secondary instabilities of the
low-speed Klebanoff distortions. The feasibility of such a bypass mechanism was later
experimentally confirmed by Hernon et al. (2007).
Due to their distinct spanwise modulation and typical wall-normal u-velocity distributions, Klebanoff modes are often misconceived as streamwise vortices (also introduced
as such by Klebanoff & Tidstrom 1959). To the contrary, the disturbances rather take
the form of elongated streaks in the laminar boundary layer with comparatively low
levels of streamwise vorticity (Fasel 2002; Durbin & Wu 2007). On the other hand, the
elongated streaks associated with the K-mode are, just like streamwise vortices, found
to be weaker on convex walls and can lead to unstable G¨ortler vortices on concave walls
(Herbert 1997). From a transient growth point-of-view, the K-modes are not optimal
disturbances. In addition, the question of how the spanwise scaling for the free-stream
induced K-modes compares to the optimal spacing found from transient growth theory remains unanswered. In both cases, however, this spanwise scaling seems to be an
intrinsic property of the boundary layer.
Naturally, the effect of F S T is equally important in the presence of streamwise pressure
gradients and the formation of a laminar separation bubble (Mayle 1991). Elevated levels
of F S T have been reported to reduce the length of the separation bubble (Gault 1955;
Volino 2002) or even prevent separation completely by transitioning the flow upstream
of the separation location (Dong & Cumpsty 1990; Zaki et al. 2010). However, early
transition is not necessarily beneficial. The significant increase in skin friction and heattransfer rate associated with turbulent flow can have a detrimental effect on performance
and hardware durability (Sohn & Reshotko 1991).
The existence of low-frequency, large-amplitude streaks in the boundary layer and in
the separated shear layer has been confirmed in the experiments of H¨aggmark (2000)
and Volino (2002). In the experiments by H¨aggmark (2000), LSBs on a flat plate were
investigated with low noise and disturbance level (T u = 0.02%), and with grid-generated
turbulence of 1.5% intensity. Smoke visualization images from his work are reproduced
in figure 3. H¨aggmark (2000) reports that there is no strong evidence of two-dimensional
waves in the separated shear layer at T u = 1.5%. In addition, he performed a spectral
analysis and observed that lower frequencies are dominant in the case of increased FST.
Volino’s (2002) experimental investigations focused on the effect of low and high F S T
levels on the suction side separation of a low-pressure turbine blade. For a low Reynolds
number Re = 25, 000 (based on the blade’s suction surface length and exit velocity), he
reported that even in the case of high levels of F S T (T u = 9%), breakdown to turbulence
† Lately, the term “continuous mode” transition is also used (Durbin & Wu 2007) arguing that
bypass transition is preceded by the amplification of continuous modes of the Orr-Sommerfeld
spectrum rather than discrete modes (TS waves).
7
8
W. Balzer and H. F. Fasel
Figure 3. Smoke visualizations of a laminar separation bubble with low noise and disturbance
level (left, T u = 0.02%) and elevated levels of F S T (right, T u = 1.5%). Reproduced from
H¨aggmark (2000).
occured in the separated shear layer above the wall and that the flow did not reattach
upstream of the trailing edge.
2. Governing equations
For the present investigation, the three-dimensional, incompressible, time-dependent
Navier-Stokes (NS) equations are solved using direct numerical simulations.
∇· v = 0
(2.1a)
∂v
1 2
(2.1b)
+ (v · ∇) v = −∇p +
∇ v.
Re
∂t
In these equations, all quantities are non-dimensionalized by a global reference length,
L∞, and a reference velocity, U∞,
p∗ − p∗∞ ,
p=
,
,
(2.2)
2
ρU∞
L
U∞∗
∞
∗U
t
v
∞
x = L∞x∗ ,
t =denoted by anv asterisk
=
where dimensional
quantities are
and p is the pressure at some
∗∞
convenient reference point in the fluid. The Reynolds number in equation 2.1a is defined
as Re = L∞U∞/ν where ν is the kinematic viscosity.
Depending on the application and geometry, numerous numerical methods to solve the
incompressible NS equations have been developed and employed over the past decades.
Although most of these methods are based on primitive-variable approaches (solving for v
and p), none has emerged as an universal approach. Difficulties for defining nonambiguous
boundary conditions for pressure have led to alternative formulations in which pressure
as a dependent variable is eliminated. One such alternative, employed in the present
investigation, is the vorticity-velocity formulation, which is obtained by taking the curl of
the momentum equation (2.1a) and taking into consideration the fact that both velocity
and vorticity vectors are solenoidal. With the definition of vorticity as the negative curl
of the velocity vector, ω = −∇ × v , the set of governing equation in vorticity-velocity
formulation can be derived,
∂ω + (v ) = (
·∇ ω
ω · ∇) v +
∂t
2
∇ v = ∇ × ω.
1 2
∇ω
Re
(2.3a)
(2.3b)
The advantages of the vorticity-velocity methods have been reviewed and discussed by
Fasel (1980), Speziale (1987) and Gatski (1991).
The Role of Free-stream Turbulence on Boundary-Layer Separation
9
3. Numerical Method
Numerical solutions to the governing equations are obtained by employing a Cartesian
NS solver first developed by Meitz (1996). Improvements to the solver, in particular the
parallel algorithm, were contributed by Postl (2005) and Balzer (2011). Details of the
numerical solution procedure other than discussed in this section can be found in these
references. The code hase been thoroughly validated and employed successfully for numerous investigations of transitional and turbulent shear flows, in particular boundary-layer
transition (Meitz & Fasel 2000; Fasel 2002), laminar and turbulent separation bubbles
(Fasel & Postl 2006; Postl & Fasel 2006), as well as separation control using vortex
generator jets (Postl et al. 2011).
The numerical solution of the governing equations is advanced in time using an explicit,
fourth-order accurate Runge-Kutta method (see Ferziger 1998, for example). The spatial
derivatives in the streamwise (∂i/∂x i) and in the wall-normal (∂i/∂y i) direction are
discretized using fourth-order accurate compact differences (Lele 1992). An exception are
the streamwise first derivatives of the non-linear terms, which are discretized using fourthorder accurate split compact differences resulting in improved stability characteristics
compared to standard compact differences. Most computational time is expended for
obtaining the solution of the wall-normal velocity component, which is governed by a
Poisson equation. A solution is obtained by a direct method using standard compact
differences and Fourier sine transforms.
In the spanwise direction, z, the flow is assumed to be periodic. Each variable is
represented by a total of 2K + 1 Fourier modes: the 2D spanwise average (zeroth Fourier
mode), K symmetric Fourier cosine and K antisymmetric Fourier sine modes.
k=0
φ (x, y, z, t) = Φ
(x, y, t) +
K
k
2K
k
Φ (x, y, t) cos γ z +
k=1
k
k
Φ (x, y, t) sin γ z (3.1)
k=K+1
Fast Fourier Transforms (F F Ts) are employed to convert each variable from spectral
space to physical space and back. The Fourier representation of the spanwise direction
reduces the governing equations to a set of two-dimensional equations for each Fourier
mode. The set of 2-D problems is solved in parallel using 2K + 1 processors on a modern
supercomputing platform. For the calculation of the nonlinear terms, the flow field has
to be transformed from spectral to physical space (and back) before each sub step of
the time integration. This requires redistributing of the entire three-dimensional arrays
among the processors, which requires extensive inter-processor communication realized
by using the message passing interface (MPI). To avoid aliasing errors, the nonlinear
terms in physical space are computed on ≈ 3K spanwise collocation points. Note that
only the non-linear terms are calculated in physical space. All other computational work
(differentiation, time integration, imposition of boundary conditions etc.) takes place in
spectral space.
3.1. Boundary conditions
The spectral approach in the spanwise direction implies the following periodic boundary
conditions at the spanwise boundaries, z = ±λ z /2, for all variables and their derivatives:
(3.2a)
φ (x, y, −λ z /2, t) = φ (x, y, λ z /2, t)
n
∂ φ
∂zn
(x, y, − λ z /2, t) =
n
∂z
∂n φ
(x, y, λ z /2, t) ;
n = 1, 2, . . .
(3.2b)
At the inflow boundary, x = x 0, and initially in the entire computation domain, the
10
W. Balzer and H. F. Fasel
velocity and vorticity components of a 2D, steady, laminar base flow are prescribed. This
initial flow field is obtained by solving for the similarity solution of Blasius boundarylayer equations. In addition, the use of fourth-order accurate compact finite differences
for the spatial discretization requires to provide the streamwise derivatives for some of
the flow variables at the inflow.
At the inflow boundary, all velocity and vorticity components are prescribed as Dirichlet conditions. For maintaining the fourth-order accuracy of the code, the streamwise
derivatives are prescribed as well. To prevent reflections from the outflow boundary, turbulent fluctuations are damped out in a buffer domain using the approach proposed by
Meitz & Fasel (2000). At the upper boundary, a Neumann condition is prescribed for
the wall-normal velocity, ∂v/∂y = f (x), and irrotational flow is assumed, ω = 0. Since
the equations for the streamwise (u) and spanwise (w) velocity components reduce to
ordinary differential equations in the streamwise direction, no free stream boundary conditions are required for these variables. At the wall, no-slip and no-penetration conditions
are imposed.
For simulating realistic, free-stream turbulent inflow conditions, an unsteady disturbance field is imposed onto the steady, undisturbed base flow profile:
k
k
U ,V ,W
k
T
T
′k
(x 0, y, t) =
Ωkx, Ωky, Ωkz
T
(x 0, y, t) =
(x 0, y, t)
′k T
′k
u ,v ,w
(x 0, y, t)
T
[UB, VB, 0] (x 0, y) + u′ k , v′ k , w′ k
T
[0, 0, Ωz,B] (xT0, y) + ωx′ k , ωy ′ k , ωz ′ k
k
k
k =0
∀k = 0 ,
(3.3a)
T
(x 0, y, t)
k
ωx′ , ωy ′ , ωz ′ (x 0, y, t)
k =0
∀k =(3.3b)
0.
The disturbances fields are designed to model isotropic free-stream turbulence described
in the next section.
4. A numerical model of free-stream turbulence
For the generation of realistic free-stream disturbance fields, the method of Jacobs
(1999) was adopted and implemented in the present Navier-Stokes code. In this approach,
F S T is modelled by introducing random velocity and vorticity disturbances which are
designed to satisfy continuity, and represent isotropic turbulence with a specified energy
spectrum. The method is based on an expansion of the disturbance velocity, v ′ , in terms
of its Fourier coefficients,
ik· x
′
v (x, t) =
v (k , t) e
,
(4.1)
k ˆ
where k is the wavenumber vector with length k =
kx2 + ky2 + k2. Turbulent fluctuations
z
are only introduced at the inflow boundary plane (x 0, y, z) of the computational domain.
The inflow boundary condition is a superposition of a steady, undisturbed base flow VB
(e.g a Blasius boundary-layer profile) and the unsteady disturbance velocity:
′
V(x 0, y, z, t) = VB(y) + v (x 0, y, z, t) .
(4.2)
In the same manner, the inflow boundary condition for the vorticity field becomes
′
Ω(x 0, y, z, t) = Ω B(y) + ω (x 0, y, z, t) ,
where ω′ is determined from the disturbance velocity field as ω′ = −∇ × v ′ .
(4.3)
The Role of Free-stream Turbulence on Boundary-Layer Separation
11
The dispersion relation kx = β/U ∞ can be used to express the streamwise wavenumber
in terms of the frequency. Consequently, kxx is replaced by −βt at the inflow boundary,
i k yy i k zz − i β t
′
v (x 0, y, z, t) =
A(|k|) e
β
kz
e
e
,
(4.4)
ky
where A(|k|) represent the coefficients/weights for the eigenfunctions that need to be
determined. The implementation of the spanwise Fourier modes, ei k zz , is straightforward
since the numerical model assumes periodicity of the flow field in z-direction. However,
the expansion in wall-normal direction requires a modification due to the presence of the
wall. Jacobs (1999) suggested to use local solutions from the continuous spectrum of the
Orr-Sommerfeld operator, ˜,v and the Squire operator, ω, ˜to write the expansion in terms
of the wall-normal eigenfunctions Φ, i.e.
i k zz −iωt
′
v (x 0, y, z, t) =
A(|k|) Φ(y; β, kz , ky ) e
β
kz
e
.
(4.5)
ky
The solutions to the homogeneous OS equation and the homogeneous SQ equation are
T
Φ O S = [Φu , Φv , Φw ]O S =
Φ SQ = [Φu , Φv , Φw ]TSQ =
iα ∂v˜ iγ ∂˜v
, ˜,
v 2
k 2 ∂y
k ∂y
iγ
T
iα
ω˜y , 0, − 2 ω˜y
k
k
2
T
and
(4.6a)
,
(4.6b)
respectively. These solutions can be superposed to find a new set of eigenfunctions,
Φ=
1
EΦ
iθ
iθ
e 1 cos(ϕ) Φ O S + e 2 sin(ϕ) Φ SQ ,
(4.7)
where, for the purpose of the numerical model, θ1, θ2 and ϕ are uniformly distributed
random numbers in the range [−π, π] which provide arbitrary weights and phase shifts
for the eigenfunctions. The normalization is such that the energy of each disturbance
mode is unity. This leads to an expression for EΦ:
EΦ =
1
ym a x
y max
0
∗
∗
1
(Φu Φu + Φv Φ∗v+ Φw Φw ) dy .
2
(4.8)
Since Φ carries the velocity vector information, the Fourier amplitude can be written
as a scalar, A(|k|) in equation 4.5. As will be shown, A (|k|) can be used to distribute
turbulent kinetic energy among the disturbance modes. A common choice for the analytic
form of the energy spectrum is the von K´arm´an spectrum,
(k L) 4
2
E(k) ∼ T u L
17
C [1 + (k L) 2] 6
.
(4.9)
A turbulent length scale, L, is used to normalize k. The free-stream turbulence intensity,
defined as
1
3
(u2) ∞ + (v2) ∞ + (w2) ∞ =
1
(q /2) ,
(4.10)
3
2
2
u′ i, u′ i =′
2
is used to normalize the spectrum. The quantity q /23= 12 ui, u′ i is
the turbulent kinetic
energy (often also denoted by k). For isotropic turbulence, the free-stream turbulent
Tu =
intensities are similar, i.e. (u2) ∞ ≈ (v2) ∞ ≈ (w2) ∞; therefore, T u =
(u2) ∞. The
12
W. Balzer and H. F. Fasel
E(k)
ky
-5/3
~k
k
2
Emax(u )
~k
4
k
kx
k1 k 1.157
max
L11
k
k2
k
kz
Figure 4. Left: Typical von K´arm´an energy spectrum (—). The symbols ( ) mark the range
of wavenumbers realized in a numerical simulation. Right: schematic of the energy shell model.
Energy is distributed among a limited number of disturbance modes located on equidistant,
spherical wavenumber shells.
normalization constant C is defined by integration of the spectrum
√π Γ 1
4
x
2 ∞
3
C =
.
17
1 7 dx =
2 6
4Γ 6
3 0 (1 + x )
(4.11)
The turbulent length scale, L, is related to the turbulent integral length scale L11 by
L=
55C
L11 ,
9π
(4.12)
where the integral length scale is defined by the longitudinal two-point correlation (mixing
length theory; see Tennekes & Lumley 1972, p. 44, for a detailed derivation):
L11 =
∞
0
u(x)u(x + r)
dr .
u2
(4.13)
This leads to the following expression for the energy spectrum,
E(k) = (u2) ∞L11
1.196(k L 11) 4
17
[0.558 + (k L 11) 2] 6
.
(4.14)
Figure 4 (left ) shows the schematic of a typical von K´arm´an energy spectrum. For large
4
scales (small k) the spectrum is asymptotically proportional to k . For small scales (large
k) it approaches Kolmogorovs’s (−5/3)-law. It can be deduced that the choice of L11
controls the size (wavenumber) of the most energetic structures (km a x ≈ 1.157L11) while
T u controls the amplitude levels of these disturbances.
The turbulent kinetic energy is obtained by integrating over the entire 3D energy
spectrum,
2
q
=
2
∞
0
E(k)dk = lim
∆k→ 0
E(|k|)∆k .
(4.15)
K
Since in a numerical simulation only part of the spectrum can be captured due to con-
The Role of Free-stream Turbulence on Boundary-Layer Separation
13
straints in domain size and resolution, the turbulent kinetic energy is approximated,
2
q
≈
2
E(|k|)∆k .
(4.16)
k
The 3D energy distribution can be envisioned as a sphere with concentric shells of
radius k (see figure 4, right ) and continuously distributed energy E(k). For the generation
of inflow disturbance fields a discrete number, N sh , of equidistant wavenumber shells
(k1, k1 +∆k, . . . , k2) was chosen on which energy is distributed discretely among a limited
number, N p, of disturbance modes. Modes on the same shell are given by identical total
wavenumbers k =
β2 + ky2 + k2 and the turbulent kinetic energy
2
z
q
1
=
2
2
2
k
uˆ iuˆ
∗
i
N p A(|k|) .
=2 k
(4.17)
Equations 4.15 and 4.17 lead to a formula for A(|k|),
A(|k|) =
2E(k)∆k
.
(4.18)
Np
It can be seen from figure 4 (right ) that the disturbance modes on each wavenumber
shell are distributed along discrete and equidistant lines of spanwise wavenumbers, kz .
It was found to be most efficient when kz was selected according to the discretization
of the z-direction used in the Navier-Stokes code. In doing so, computational time was
saved by avoiding additional FFT calls and inter-processor communication during each
timestep. Within limits, the values for kx and ky are chosen randomly along lines of
constant kz . Since F S T is characterized by very low frequencies (see Westin et al. 1994,
for example), it was decided to set the lower bound of possible streamwise wavenumbers
to kx,min = 0, i.e. the lowest frequency which is introduced at the inflow boundary
of the computational domain is β = kxU∞=0 (steady disturbance). The upper bound
of the range of streamwise wavenumbers depends on the streamwise discretization. If
one wavelength of a disturbance travelling with the speed of the free stream should be
represented by at least five grid points in x-direction, then kx,max = π/( 2 ∆ x). For the
wall-normal direction, ky = 0 must be avoided since it does not correspond to any physical
+
eigenvalue. The upper limit, ky, m a x = 2π n/y m a x where n ∈ Z , is chosen with respect to
the domain height, ym a x , and the grid resolution in y. Continuous modes corresponding
to wall-normal wave numbers ky > ky, m a x are not resolved in the free stream (especially
on strongly stretched grids) and are therefore not considered.
Upon choosing the parameters L11 and T u, the inflow velocity and vorticity fields can
be completely determined. Figure 5 presents an elevated-surface plot with amplitudes
and gray scale based on the instantaneous spanwise disturbance velocity, w′ . This figure
emphasizes that the current method only introduces disturbances in the free stream, but
leaves the inside of the boundary layer almost undisturbed.
Note that the disturbance field plotted in figure 5 also vanishes near the free-stream
boundary. It was found that high disturbance levels close to ym a x can lead to numerical
instabilities. The reason for these numerical difficulties is the lacking of appropriate freestream boundary conditions. While for a single disturbance mode it appears straightforward to obtain an x-dependent free-stream condition from the theoretical eigenbehavior,
such an approach is highly complicated if many modes, possibly with the same spanwise
wavenumber and/or frequency are present. In this case a decomposition of the entire flow
field into continuous OS and SQ modes would be required at each timestep to apply the
14
W. Balzer and H. F. Fasel
′
Figure 5. Instantaneous elevated-surface plot of the spanwise disturbance velocity, w , used for
introducing F S T in the presented simulations. The edge of the boundary layer of the undisturbed
flow is indicated by the boundary layer thickness, δ99.
correct boundary condition to each disturbance mode. Such a decomposition is computationally expensive and non-trivial since the continuous modes are not orthogonal. As
an alternative approach, a smooth ramping function, R(y), was multiplied to each eigenT
function (˜, ω) before computing the eigenfunctions Φ O S and Φ SQ which are used for
the expansion
v ˜ of the disturbance field. Note that the use of a ramping function does not
prevent the disturbance field from being divergence-free, since the continuity equation is
applied after R(y) is imposed onto the eigenfunctions.
For the present study a total of 1200 continuous modes (N sh = 40 equidistant wavenumber shells with N p = 30 disturbance modes on each shell) were used for generation of the
inflow disturbance field. Figure 6 presents the theoretical (equation 4.14) and discrete
energy spectrum E(k) for all presented cases with T u = 0. Energy values for the discrete
spectrum are larger than the theoretical values, which results from the condition that
both curves should have the same turbulent kinetic energy (equations 4.15 and 4.16), i.e.
2
q
=
2
∞
0
E(k)dk ≈
E(|k|)∆k .
k
The smallest and largest spanwise wavenumbers which are resolved by the DNS (γ1,DNS
and γ64,DNS, respectively) are indicated by vertical dotted lines in figure 6. The maximum
of the energy spectrum appears shifted towards low values of spanwise wavenumbers, i.e.
the eddies carrying the most energy will be represented predominantly by small spanwise
wavenumbers in the DNS. In fact, for the first wavenumber shell the total wavenumber
is smaller than γ1,DNS, which means that all 30 modes on this shell are two-dimensional
disturbances in the DNS, i.e. γ0,DNS = 0.
Note that the choice of wavenumber triplets is the same for all cases with T u = 0. In
other words, for all simulations the free-stream disturbance distribution is the same while
the amplitudes/energy increases with increasing levels of free-stream turbulence. The
The Role of Free-stream Turbulence on Boundary-Layer Separation
-3
10
1, DNS
64, DNS
-4
10
-5
10
-6
10
-7
E(k)
10
-8
10
-9
10
-10
10 1
10
100
1000
k
Figure 6. Spectral kinetic energy spectrum. The lines show the theoretical curve (see equation
4.14) and symbols correspond to the wavenumbers resolved in the numerical simulation. ( )
case FST-0.05 ; ( ) case FST-0.5 ; ( ) case FST-2.5.
lower and upper bounds for the wall-normal wavenumber were chosen as ky, m i n = π/2
and ky, m a x = 24π to avoid continuous mode solutions with either very large wavelengths
compared to the computational domain (i.e. the wavelength of the oscillation is significantly larger than ym a x ) or very short wavelengths that could not be resolved on the
computational grid. Resolution limitations also exist for the streamwise and spanwise
direction.
5. Simulation Setup
The present computational setup for the DNS of a flat-plate LSB was motivated by
the wind-tunnel experiments of Gaster (1966). The computational domain and grid for
these numerical simulations can be seen in figure 7. In the experiments, an inverted airfoil
was mounted above a flat plate to accelerate and decelerate the flow between the airfoil
and the flat plate. Because circulation control was applied on the airfoil, flow separation
and reattachment occurred on the flat plate. This is illustrated in figure 7 by plotting
instantaneous gray-scale contours of the spanwise-averaged ωz -vorticity component. Figure 7 also shows the location of a disturbance slot upstream of separation that was used
for active flow control simulations (results are omitted from this manuscript; see Balzer
(2011) for details).
Also shown in figure 7 is the computational grid. In the streamwise direction the
baseline grid had 2001 equidistantly distributed points in the range 5.0 ≤ x ≤ 23.0. The
height of the domain was ym a x = 2.0 (≈ 19δ99, 0) and exponential grid stretching was
employed in the wall-normal direction using 256 grid points. At the solid wall, y = 0, noslip and no-penetration boundary conditions were applied. The spanwise domain width
was chosen as Lz = 2.0 (≈ 19δ99, 0) and was resolved using K = 64 Fourier cosine and
K = 64 Fourier sine modes. The number of collocation points was K = 200. The total
number of grid points for these simulations was approximately 102 million.
One case was computed with a finer grid (207 million points) in order to demonstrate
grid convergence. The refined grid had 2161 and 320 grid points in the streamwise and
15
16
W. Balzer and H. F. Fasel
Figure 7. Simulation setup motivated by the experiments of Gaster (1966). For the computational grid, every 10th grid line is shown in both streamwise and wall-normal direction. Also
shown are instantaneous gray-scale contours of the spanwise-averaged ωz -vorticity component.
wall-normal directions, respectively. For the spectral discretization a total of 2K +1 = 193
Fourier modes was employed using K = 300 collocation points. Results obtained with the
fine grid will be presented and compared to the results obtained with the regular grid.
By variation of the tunnel speed, Gaster (1966) generated a series of short and long
separation bubbles. The present investigation focuses on a long bubble (Gaster’s case VI,
series I). The velocity scale was chosen according to the tunnel speed in the experiment,
U∞ = 6.64 [m/s], and the reference length scale was L∞ = 1 [in] = 0.0254 [m]. This
length scale was chosen because the experimental data was reported in inches. It is,
however, not a scale with physical meaning. Marxen (2005) addressed the question of
relevant scales for LSBs developing on a flat-plate beneath a displacement body (e.g.
2D wing section). He pointed out that there exist two, independent length scales (and
consequently two relevant Reynolds numbers). The first scale is associated with the
distance of the displacement body from the leading edge of the flat plate. In a numerical
simulation, the integration domain does typically not include the leading edge. Rather
at some location upstream of the displacement body/pressure gradient, an inflow profile
is prescribed that matches important boundary-layer parameters (δ99, δ1, etc.) of the
approach flow. In other words, the distance of the displacement body from the leading
edge determines the local boundary-layer properties. In the present numerical setup,
for example, a Blasius similarity solution is prescribed at the inflow, x 0 = 5.0, with
δ1 ≈ 0.036. The corresponding local Reynolds number based on displacement thickness
is Re δ1 = 405. From Gaster’s (1966) experiments, only some values at the separation
location are known. Here, the boundary-layer edge velocity and Reynolds number based
on momentum thickness were ue,S = 1.30 and Re δ2 , S = 218. In the DNS, these values
are ue,S = 1.32 and Re δ2 , S = 189, respectively, which compare favorably to the values
measured in the experiments. The second relevant length scale is associated with the
streamwise dimension of the displacement body. The Reynolds number based on this
length scale can be related to the pressure gradient. The chord length of the wing section
in the experiments was C x = 5.5[in] and the global Reynolds number based on C x was
Re ≈ 62, 000.
For the numerical simulation, the displacement body is replaced with a free-stream
The Role of Free-stream Turbulence on Boundary-Layer Separation
17
0
-0.25
cp
-0.5
-0.75
-1
5
10
15
20
x
Figure 8. Time- and spanwise-averaged wall-pressure coefficient: (——) 3D DNS; (− − −)
inviscid solution; ( ) experiment separated; (+) experiment tripped to turbulence.
boundary condition that imposes a favorable-to-adverse pressure gradient on the mean
flow similar to the experiments. The streamwise pressure gradient was enforced by prescribing wall-normal blowing and suction at the free-stream boundary, ym a x . The freestream v-velocity distribution was found iteratively. A comparison of the inviscid and
viscous wall-pressure coefficient is presented in figure 8. The differences in the inviscid
pressure coefficient can be explained by displacement effects of the (tripped) attached
turbulent boundary layer in the experiments. For the separated flow, the accelerated
flow region and onset of separation (pressure plateau) agree very well between DNS and
experiment.
Despite the good agreement for the upstream part of the bubble, the DNS data indicates later reattachment compared to case V I of the experiments. In fact, it proved
very diffcult with the current iterative procedure to find a free-stream boundary condition which would yield a shorter bubble without sacrificing the reasonable agreement in
the upstream, laminar portion of the flow. From experience it is known that a better
comparison can be achieved when some background noise is introduced in the DNS†.
Therefore, instead of further trying to adjust the free-stream boundary condition, it was
decided to apply the numerical F S T model for the present case. Employing the F S T
model was preferred over traditional methods based on introducing selective or random
disturbances inside the boundary layer, since it is less artificial and is rather based on
relevant physical mechanisms.
Time- and spanwise-averaged streamwise velocity profiles as well as streamlines are
presented in figure 9. Also plotted is the distribution of the boundary-layer thickness. For
x < 9.5, δ99 decreases, which indicates the strong acceleration of the flow upstream of the
separation location, x S = 10.4. In the region of APG, the flow separates and reattaches
as a turbulent flow in the mean at x R = 15.54. The strong growth of the boundarylayer thickness and the full velocity profiles suggest that the flow indeed transitions to
turbulence. In his experiments, Gaster (1966) reported levels of turbulent free-stream
† Note that the choice of a particular inflow profile (Blasius, Falkner-Skan) can also have
a significant impact on the developing separated flow. The influence of the inflow profile was,
however, not investigated in more detail.
18
W. Balzer and H. F. Fasel
u
1.0
0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1
0.5
y
99
0.0
5
6
7
8
9
10
11
12
13
14
15
16
xS
17
18
19
20
21
xR
1.0
x
y
0.5
0.0
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
x
Figure 9. Time- and spanwise-averaged results of the uncontrolled separation bubble. Top:
streamwise velocity profiles. The dashed line indicates the boundary-layer thickness, δ99. Bottom:
in y-direction is shown and that the y-direction is scaled by a factor of 3.
`˛
˛
´
streamlines plotted as equidistant contours of ln ˛104 · ψ˛ + 1 . Note that only half of the domain
fluctuations in the order of T u = 0.05%. For the current flow, six different cases were
considered:
(1)case FST-0 : “quiet” case; no fluctuations in the free stream; T u = 0.
(2)case FST-0.05 : low F S T level; T u = 0.05%.
(3)case FST-0.5 : medium F S T level; T u = 0.5%.
(4)case FST-2.5 : high F S T level; T u = 2.5%.
(5)case FST-2.5b: T u = 2.5%; DNS are computed on the fine grid.
(6)case ZPG-2.5 : T u = 2.5%; DNS are computed without pressure gradient.
The generation of the inflow disturbance field for cases with T u = 0 was presented in
section 4. Comparisons between cases FST-2.5 and FST-2.5b will demonstrate grid convergence. Case ZPG-2.5 serves as a reference case to isolate effects that are associated
with the strong favorable-to-adverse streamwise pressure gradient. While the F S T intensity is varied, the turbulent integral length scale remains constant and was chosen
as L11 = 5δ1, 0. This is in the order of the values typically used in related investigations of FST-induced transition for attached boundary layers (see Brandt et al. 2004,
for example). It is, however, well known that a change in integral length scale can have
a significant impact on the transition location and transition mechanisms (Jon´aˇs et al.
2000; Brandt et al. 2004; Ovchinnikov et al. 2008). In experiments it is, in fact, rather
difficult to vary the turbulence intensity while maintaining the same turbulent characteristic length scales or vice versa (e.g Kurian & Fransson 2009). Since the effect of varying
turbulent length scales was neglected during the present numerical campaign, the results
by no means claim to offer a complete description of the problem.
19
0.5
0
y
1.0
The Role of Free-stream Turbulence on Boundary-Layer Separation
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
13.5
14.0
14.5
15.0
15.5
16.0
13.5
14.0
14.5
15.0
15.5
16.0
13.5
14.0
14.5
15.0
15.5
16.0
0.5
0
y
1.0
x
10.0
10.5
11.0
11.5
12.0
12.5
13.0
0.5
0
y
1.0
x
10.0
10.5
11.0
11.5
12.0
12.5
13.0
0.5
0
y
1.0
x
10.0
10.5
11.0
11.5
12.0
12.5
13.0
x
Figure 10. Contours of −10 ≤ ωz˘ ≤ 10, where ωz˘ = ln | ωz |, for DNS with various levels
of free-stream turbulent intensity T u. From top to bottom: case FST-0 ; case FST-0.05 ; case
FST-0.5 ; case FST-2.5. Dashed contour lines correspond to ωz ˘< 1.
6. Instantaneous Flow
For the purpose of illustration, a modified spanwise vorticity, ωz ,˘ is defined as
ω
˘ z = ln |ωz | .
(6.1)
Instantaneous contours of ω˘z -vorticity are presented in figure 10. The use of a logarithmic
scale demonstrates that upstream of transition, two different flow regions can be identified
for all cases: (1) an outer region of turbulent fluctuations (except in case FST-0 ), and
(2) an inner region of the laminar, separated boundary layer. The observation that the
vortical free-stream disturbances appear to be unable to enter the shear layer can be
explained by the so-called “shear sheltering” effect (Hunt et al. 1996), which describes
the effect that free-stream disturbances are rapidly damped out inside a shear layer due
to inviscid shearing and viscous dissipation.
The onset of transition is moved upstream when the level of F S T increases. This, in
turn, leads to a reduction of the separated flow region. Note that even for the highest
20
W. Balzer and H. F. Fasel
FS TI (case FST-2.5 ), the flow separates from the surface as a laminar boundary layer
and transition occurs in the separated shear layer above the surface. While the shear layer
still exhibits traces of the roll-up and the formation of spanwise-oriented vortices, the
reattaching turbulent boundary layer does not appear organized by dominant coherent
structures.
Free-stream turbulence induces low-frequency u-velocity distortions inside the laminar
boundary layer - the so-called Klebanoff modes/distortions. Figure 11 presents instantaneous gray-scale contours of the streamwise velocity component plotted in an x/z-plane
inside the boundary layer, at y = δ1(x), and near the edge of the boundary layer, at
y = δ99(x). Values for δ1 and δ99 were obtained from the time- and spanwise-averaged
flow fields (see also figure 15). The u-velocity contours inside the boundary layer indicate
a very sharply defined separation between laminar and turbulent flow regions demonstrating that transition occurs very rapidly over a short streamwise extent. Inside the
laminar separated flow region (x < 12) the flows with non-zero F S T are characterized
by streamwise elongated streaks associated with the K-mode. These structures are most
prominent for case FST-2.5 where they appear with a high amplitude and a distinct
spanwise scaling. For the case of very low FST, the Klebanoff distortions have much
smaller amplitudes and can only be visualized by looking at the disturbance velocity, u′ ,
which in this case was calculated by subtracting the spanwise mean from the streamwise
velocity, i.e. u′ = u − U 0. Gray-scale contours of u′ -velocity are presented in figure 12
for cases FST-0.05 and FST-2.5. Here the local acceleration (u′ > 0) and deceleration
(u′ < 0) that is associated with the K-mode can also be seen for the low-level F S T case.
Nevertheless, in this case the K-mode is weak and does not seem to be connected to the
turbulent flow region whereas in the case of high FST, the streaks seem to be directly
connected to the first-occurring turbulent flow structures in the region 12.0 < x < 13.0.
Since breakdown of the streaks and/or the appearance of turbulent spots was not observed in any of the present cases, it can be speculated that even for the case with the
highest F S T intensity, bypass transition can be ruled out as a viable path to turbulence.
From the flow visualizations in figure 10 and the u-velocity contours at y = δ1 presented in figure 11 it appears as if the separated shear layer loses its predominantly twodimensional character and experiences the onset of flow transition seemingly without the
shedding of spanwise vortices. However, the u-velocity contours plotted at y = δ99(x) in
figure 11 give room for speculation that the reattaching flow might still be organized by
such coherent flow structures. For the sake of argument, it is assumed that near the edge
of the boundary-layer the small-scale turbulence has died out such that the imprint of the
large-scale motion can be “picked up”. Whereas the gray-scale contours show a strong
2D spanwise coherence for cases FST-0 (x > 14) and FST-0.05 (x > 13), a staggered or
“checkerboard” pattern arises for cases FST-0.5 and FST-2.5 (x > 12.5). This observation suggests that for the cases with higher levels of F S T slightly oblique waves are more
dominant than 2D waves. This is often confirmed by carrying out a proper orthogonal
decomposition (POD) of the unsteady flow field, which was, however, not attempted for
the present cases.
Figure 13 presents an alternative visualization of the streamwise streaks. Here, instantaneous u′ -velocity contours are plotted in three z/y-planes for cases FST-0.05 and
FST-2.5. The first plane is located in the accelerated flow region at x = 8. The second
plane is located closely upstream of separation at x = 10, and the third plane is located
inside the separated shear-layer region shortly upstream of transition at x = 12. The
thick lines in these plots represent an estimate of the local boundary layer thickness. For
the case with low FST, no significant modulation of the boundary-layer thickness can be
seen while for the case with high FST, the high-amplitude streamwise streaks lead to a
The Role of Free-stream Turbulence on Boundary-Layer Separation
u
0.4
u
0.6
0.8
1.0
0.9
10
11
12
13
14
15
16
9
1.3
1.4
10
11
12
13
14
15
16
13
14
15
16
13
14
15
16
13
14
15
16
13
14
15
16
1
x
-1
10
11
12
13
14
15
16
9
10
11
12
-1
1
x
9
10
11
12
13
14
15
16
9
10
11
12
x
-1
9
10
11
12
13
14
15
16
9
10
11
12
x
-1
9
10
11
12
13
14
15
16
9
10
11
12
x
1
x
1
0
0
z
-1
1
x
1
0
0
z
-1
1
x
1
0
z
-1
1
x
0
z
1.2
0
z
0
z
-1
1
x
9
z
1.1
0
z
0
z
9
z
1.0
-1
0.2
-1
0.0
21
Figure 11. Instantaneous gray-scale contours of the streamwise velocity component, u, plotted
in a x/z-plane inside the boundary layer (y = δ1(x ), left) and at the edge of the boundary
layer (y = δ99(x ), right). From top to bottom: cases FST-0, FST-0.05, FST-0.5, FST-2.5, and
FST-2.5b.
u’
-0.1
0.0
-1
xR
10
11
12
x
13
14
0
z
9
15
8
1
8
1
0.1
xS
0
z
0.05
xR
-1
xS
-0.05
9
10
11
12
13
14
15
x
Figure 12. Visualization of the Klebanoff mode for cases FST-0.05 (left) and FST-2.5 (right).
Plotted are instantaneous gray-scale contours of the streamwise disturbance velocity component,
′
0
u = u − U , in an x/z-plane at distance y = δ1(x ) away from the wall.
22
W. Balzer and H. F. Fasel
0.4
−0.10 < u’ < 0.10
0.4
−0.002 < u’ < 0.002
0.3
0.3
0.4
0.6
0.7
0.8
0.9
1.0
0.0
0.0
z
0.5
0.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
0.9
1.0
0.6
0.7
0.8
0.9
1.0
z
−0.10 < u’ < 0.10
0.3
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.0
z
0.1
0.2
0.3
0.4
0.5
z
−0.25 < u’ < 0.25
0.4
−0.02 < u’ < 0.02
0.4
0.0
0.4
0.2
y
0.2
0.3
x=12
0.3
y
y
0.3
0.1
0.1
0.2
0.1
0.0
0.2
0.1
0.2
0.3
0.4
0.5
z
0.6
0.7
0.8
0.9
1.0
0.2
y
0.2
0.3
x=10
0.1
0.0
0.1
0.4
−0.002 < u’ < 0.002
0.4
0.0
0.5
0.2
y
0.2
0.1
0.1
0.1
0.0
0.0
x
0.1
y
0.2
0.3
x=8
0.1
0.2
0.3
0.4
0.5
z
0.0
0.0
Figure 13. View from upstream. Instantaneous contours of streamwise disturbance velocity,
′
0
u = u − U , plotted in z/y-planes at x = 8, 10, 12 (from top to bottom). Dashed contour lines
indicate negative u′ -values. The thick line represents an estimate of the local boundary-layer
thickness. Left: case FST-0.05 ; right: case FST-2.5.
very pronounced spanwise-periodic thickening and thinning of the boundary layer. The
most regular pattern of the K-mode is observed at x = 10 where the spanwise streak
spacing, measured as the distance between two u′ -velocity minima or maxima, is approximately 0.25 < λ z < 0.35. Based on the local boundary-layer thickness, the spacing
is in the order of 2.5δ99 − 3.5δ99. A more accurate value will be determined in section
8. Note that the spacing appears to be independent of the intensity of the free-stream
fluctuations.
7. Time- and Spanwise-Averaged Results
The instantaneous flow visualizations presented in the previous section indicate a reduction of the separation region for increased levels of FST. Time- and spanwise-averaged
results of the different flow scenarios are compared in this section to quantify this observation. Streamlines are presented in figure 14. The mean downstream locations for
separation and reattachment are summarized in table 1 and are also included in figure
14. For the case with the lowest F S T level the separation length is already significantly reduces by 30.5%. Increasing the F S T level leads to a further reduction: 43.0% and 52.7%
for cases FST-0.5 and FST-2.5, respectively. Note that the mean separation location
remains almost constant, although a general trend can be observed that separation is
slightly delayed for increased FSTIs.
In addition to the reduction in length, the height of the separation bubble also decreases. This can be seen from the streamline images but also from the boundary-layer
and displacement thickness presented in figure 15. Prior to separation, the curves are almost identical. Only for the case with 2.5% FS T I the laminar boundary layer is slightly
23
0.5
xR
0.0
y
The Role of Free-stream Turbulence on Boundary-Layer Separation
xS
0.5
xR
0.0
y
xS
0.5
xR
0.5
0.0
y
0.0
y
xS
xS
10.0 10.5
xR
11.0 11.5 12.0 12.5
13.0 13.5 14.0 14.5 15.0 15.5
x
Figure 14. Time- and spanwise-averaged streamlines for the uncontrolled
`˛4 flow ˛with´ various
free-stream turbulence intensities. Plotted are equidistant contours of ln ˛10 · ψ˛ + 1 . From top
to bottom: cases FST-0, FST-0.05, FST-0.5, and FST-2.5. Downstream locations of separation,
x S, and reattachment, x R, are indicated as well.
FST-0 FST-0.05 FST-0.5 FST-2.5
xS
10.40
10.45
10.50
10.56
xR
15.54
14.02
13.43
12.99
Table 1. Time-and spanwise-averaged locations of separation, x S, and reattachment, x R.
thicker than for the other cases. Overall, it seems that at the chosen F S T levels the
boundary-layer fluctuations associated with the K-mode have only a local effect but do
not significantly alter the time and spanwise mean of the flow field prior to separation.
Distributions of wall-pressure and skin-friction coefficients are plotted in figure 16.
The agreement with the experimental results for the wall-pressure coefficient is best for
0.05% < T u < 0.5%. Recall that the FS T I in the experiments was T u ≈ 0.05%. The
numerical F S T model, therefore, proved itself to be very useful for obtaining a more
realistic separation bubble size for this case. Note, however, that the pressure level for
the reattaching turbulent boundary layer increases from case FST-0 to FST-2.5. This
effect is most likely a consequence of the numerical setup and in particular the freestream boundary condition. At ym a x , the same v-velocity distribution is prescribed for all
cases. Therefore, all flows experience the same inviscid pressure gradient. However, this
24
W. Balzer and H. F. Fasel
0.8
0.4
0.6
0.1
0.09
0.5
0.02
0.08
8
x
10
0.2
6
8
x
10
6
8
10
1

0.4
6
0.03
0.3
1
0.11

0.7
0.3
0.1
0.2
0.1
0
6
8
10
12
14
16
0
18
x
12
14
16
18
x
0.25
0.02
0
0.015
-0.25
0.01
-0.5
0.005
Cf
Cp
Figure 15. Time- and spanwise-averaged boundary-layer thickness (left) and displacement
thickness (right) for various levels of FS T. (——) case FST-0 ; ( ) case FST-0.05 ; ( ) case
FST-0.5 ; ( ) case FST-2.5.
-0.75
0
-1
-0.005
-1.25
6
8
10
12
14
x
16
18
-0.01
6
8
10
12
14
16
18
x
Figure 16. Time- and spanwise-averaged wall-pr
essure coefficient (left) and skin-friction coefficient (right) for various levels of FS T. ( ) experiment; (—) case FST-0 ; (
( ) case FST-0.5 ; ( ) case FST-2.5.
) case FST-0.05 ;
approach does not take into account that the separation bubble, and the displacement
effect on the outer flow field, are drastically reduced when F S T and/or flow control
are applied (viscous effects). The wall-normal suction at the free-stream boundary that
enforces the APG is therefore likely overpronounced in these cases and causes the slightly
higher pressure values.
8. Turbulence Statistics
The downstream development of selected spanwise-averaged Reynolds stresses as well
as the turbulent kinetic energy, q/2, is depicted in figure 17. Plotted are values for the
wall-normal maxima inside the boundary layer. Note that especially for high levels of
FST, a global disturbance maximum could be located in the free stream rather than near
the wall. With increasing F S T the disturbance level inside the boundary layer increases
as well. Although the fluctuations in the free stream are nearly isotropic, inside the
boundary layer v′ , v′ is approximately one order of magnitude smaller than u′ , u′ and
w′ , w′ . As indicated by the location of x R in figure 17, reattachment of the flow occurs
shortly after the disturbance components have reached their maximum amplitudes. The
distance between the maximum and x R reduces as the F S T increases. This could be
explained by the fact that for larger F S T levels the separation bubble is shallower and
The Role of Free-stream Turbulence on Boundary-Layer Separation
25
v’,v’max
q/2max
w’,w’max
u’,u’max
that transition occurs closer to the surface such that the turbulent mixing leads to a
more rapid and “effective” reattachment of the bubble.
For x < 12, the u′ , u′ Reynolds-stress component exhibits algebraic streamwise
growth which is most likely the consequence of the lift-up mechanism (Ellingsen & Palm
1975; Landahl 1975) and transient growth. Evidence for such a conclusion is provided by
the fact that only u′ , u′ shows this distinct streamwise growth while amplification of the
other Reynolds-stresses remains rather modest†. The comparison between case ZPG-2.5
(zero pressure gradient) and FST-2.5 shows that in the region of F PG, x < 9, transient
growth is alleviated. However, in the region of APG and separated mean flow, transient
growth appears significantly stronger than for the ZPG case.
While the growth rates are almost identical for all cases in the F P G region of the flow
field, they are seen to increase in the A PG region as the F S T level decreases. One explanation for this behavior could be that the separation bubble grows as F S T decreases,
which, in turn, might give rise to higher amplification rates. Such a conclusion is corroborated by similar observations for the inviscid KH shear-layer instability, i.e. a longer
bubble typically gives rise to larger KH growth rates. It is also noted that the onset of vvelocity and w-velocity disturbance growth commences farther upstream when the level
of F S T decreases. Another interesting observation is that the strong streamwise growth
of v′ , v′ for case FST-0.05 in the range 9.5 < x < 12 has no counterpart in the other
cases. The growth rates in this flow region are initially identical for the u-velocity and the
v-velocity disturbance. It is not clear at this stage how to interpret these results. On the
one hand, it can be speculated that a second instability mechanism (possibly a G¨ortlertype instability) might exist near the separation point for case FST-0.05. On the other
hand, the similarity between the case without any F S T (case FST-0 ) and case FST-0.05
indicates that the behavior might also be caused by the disturbance equilibrium of the
numerical scheme. The disturbance development seen in figure 17 also hints to the fact
that transient growth alone does not explain transition to turbulence for the current
cases. For x > 12, the algebraic growth is overtaken by a strong exponential growth
equally observed for all Reynolds-stress components. In section 10, it will be shown that
this growth is, in fact, associated with the KH shear-layer instability.
/
Wall-normal urms-profiles (urms = u′ , u′ 1 2) extracted at x = 8, 10, 12 are presented
in figure 18. The disturbance profiles are normalized by their respective maxima inside
the boundary/shear layer and the y-direction is normalized by the displacement thickness
in these graphs. For the ZPG case (figure 18d) the profiles are nearly self-similar over
a great portion of the boundary layer with a maximum at y/δ1 ≈ 1.3. Moreover, the
profiles compare very well to the urms-distribution obtained from optimal disturbance
growth theory (Luchini 2000, e.g.). It is, in fact, a common observation that the shape
of the disturbance profile tends to assume the shape of the optimal perturbation even if
the initial perturbation is far from optimal. Recall that in optimal perturbation theory
the u-velocity streaks are a response to counter-rotating vortices inside the boundary
layer, whereas in the DNS the streak formation is triggered by the fluctuations in the
free stream. This also explains the discrepancy of the curves for y → ym a x . The profiles
are not self-similar (with respect to the scaling y/δ1) for the cases with strong favorableto-adverse pressure gradient (see figure 18a − c). However, the similarity of the profile
shapes in the attached flow region suggests that the disturbances are caused by the
same mechanism (lift-up effect). Although the displacement thickness is not a meaningful
† Marxen et al. (2009) point out that transient growth acts strongest on those disturbance
components that are aligned with the mean flow. Modal growth, on the other hand, is typically
characterized by very similar growth rates for all velocity components.
26
W. Balzer and H. F. Fasel
(a)
10
(b)
xR
xS
0
10
-1
xR
xS
0
-1
10
10
-2
-2
10
10
-3
-3
10
10
-4
-4
10
10
10
-6
10
-7
10
v’,v’max
-5
u’,u’max
-5
10
-6
10
-7
10
-8
-8
10
10
6
8
10
12
14
16
6
x
(c)
10
10
14
16
xR
xS
0
10
-1
12
x
(d)
xR
xS
0
8
-1
10
10
-2
-2
10
10
-3
-3
10
10
-4
-4
10
w’,w’max
10
10
-6
10
-7
10
-5
q/2max
-5
10
-6
10
-7
10
-8
-8
10
10
6
8
10
12
14
16
6
8
10
12
14
16
x
x
Figure 17. Downstream development of selected spanwise-averaged Reynolds stresses. (a)
u′ , u′ ; (b) v′ , v′ ; (c) w′ , w′ ; (d) q/2. Plotted are the maximum values inside the boundary
layer, y < δ99. (—) case FST-0 ; ( ) case FST-0.05 ; ( ) case FST-0.5 ; ( ) case FST-2.5 ; ( )
case FST-2.5b; ( +) case ZPG-2.5. Downstream locations of separation, x S, and reattachment,
x R, are also indicated.
scale in these cases, it shows that as the flow separates the maximum of the disturbance
aligns with the location of δ1. Note that this alignment is already completed at x =
10 in the case of low F S T (case FST-0.05 ) while for the other cases it occurs farther
downstream. Since the displacement thickness is commensurable with the location of the
wall-normal maximum of shear, this shows that the streamwise streaks get “pushed up”
as the boundary layer separates and grow within the shear layer.
From hot-wire measurements it is known that the streamwise streaks are strongly
correlated in the spanwise direction. The spanwise spacing of the Klebanoff distortions
can therefore be assessed by evaluating the spanwise two-point correlation function, Ruu ,
of the streamwise velocity:
Ruu (x, r) =
u( x)u( x, r)
u2(x)
u2(x,
,
(8.1)
r)
where the vector separation r is aligned with the spanwise direction. Figure 19 shows
correlations obtained for two y-locations: (1) inside, and (2) outside the boundary layer,
The Role of Free-stream Turbulence on Boundary-Layer Separation
(a)
(b)
(c)
27
(d)
5
4
y/1
3
2
1
0
0
0.5
urms/urms,max
1 0
0.5
urms/urms,max
1 0
0.5
urms/urms,max
1 0
0.5
1
urms/urms,max
Figure 18. Wall-normal urms-profiles at x = 8 (− − −), x = 10 (− − −), and x = 12 (− · − · −).
The profiles are normalized by the local maximum inside the boundary layer. (a) case FST-0.05 ;
(b) case FST-0.5 ; (c) cases FST-2.5 (lines) and FST-2.5b ( ); (d) case ZPG-2.5. Results from
optimal disturbance growth theory (Luchini 2000) are also included (+).
both plotted at x = 10. The inner y-location corresponds to the wall-normal maximum
of the urms-velocity profiles and the outer location corresponds to y = 1.5δ99. From
flow visualizations (e.g. Kendall 1985) it is known that the spanwise streak spacing is
approximately twice the distance to the first minimum in the correlation function. The
first minimum is located at ∆z/δ99 = 1.7 for all cases with non-zero pressure gradient.
Therefore the spacing is approximately 3.4δ99, which agrees with the estimate obtained
from the u′ -velocity contours presented in figure 13. Note that the fluctuations in the free
stream are uncorrelated. Results are also presented for the fine grid and the case of ZPG
in figure 19 ( right ). For case ZPG-2.5 the streak spacing is in the order of 2δ99. Note,
however, that the boundary layer is approximately 1.6 times thicker at x = 10 in case
ZPG-2.5. Thus, the spanwise spacing is almost identical for cases ZPG-2.5 and FST2.5. This in turn implies that the selection of the spanwise scale is independent of the
pressure gradient. Furthermore, the values obtained for the spacing agree well with the
distinct spanwise scaling of O(2δ99 − 4δ99) reported for ZPG flat-plate boundary layers
(Kendall 1985, 1990; Westin et al. 1994). At x = 10 the local ratio of domain width
and boundary-layer thickness is Lz /δ99 ≈ 21. The K-mode spacing of 3.4δ99 therefore
suggests, that approximately 6 − 7 streaks are resolved in the computational domain.
This result agrees with the flow visualizations in figures 11, 12, and 13.
The excellent agreement between cases FST-2.5 and FST-2.5b in figures 17, 18, and
19 shows that, in a statistical sense, the numerical solutions are grid independent.
W. Balzer and H. F. Fasel
1
1
0.75
0.75
0.5
0.5
0.25
0.25
Ruu
Ruu
28
0
0
-0.25
-0.25
-0.5
-0.5
0
2
4
6
8
10
0
1
z/99
2
3
4
z/99
Figure 19. Spanwise two-point correlation function, Ruu, at x = 10. Symbols correspond to
Ruu-values inside the boundary layer (max. of urms), and lines correspond to Ruu-values outside
the boundary layer (y = 1.5δ99). ( , −−−) case FST-0.05 ; ( , ——) case FST-0.5 ; ( , −·−·−)
case FST-2.5 ; ( ) case FST-2.5b; ( +) case ZPG-2.5b.
9. Spectral Analysis
In order to assess the dynamic behavior of the separation bubble, time signals of the
wall-normal disturbance velocity component, v′ = v−¯, vwere analyzed to detect dominant
frequencies in the flow field. These time signals were taken inside the separated shear
layer at the center of the domain, z = 0, and at a distance y = δ1 away from the wall.
Gray-scale contours of the recorded time signals are presented in figure 20 in form of t/xdiagrams. White-colored contours correspond to regions of vanishing disturbance velocity
which, most often, indicate a change of sign of the v′ -velocity. Compared to the case with
T u = 0, similar dynamic behavior is observed for the case of low F S T (T u = 0.05%).
Here, the separated shear layer is relatively calm for x < 11.5 followed by the distinct
time-periodic pattern associated with vortex shedding. Cases with elevated F S T levels
show an irregular flow pattern upstream of x = 11.5. However, farther downstream the
same time-periodic behavior as for the other cases becomes the prominent feature.
To determine the frequency spectrum associated with these velocity time signals, two
spectral methods were employed. First, a discrete Fourier transform (DF T) of the signals
was performed, i.e.
vβ ′ =
1N −1
N
n =0
vn ei β t n ,
Ev ′ =
1
2
vβ′ vβ′⋆ ,
(9.1)
where N is the number of samples in the time signal, β = 2πf is the circular frequency,
and Ev ′ is the spectral energy. Second, a so-called maximum entropy spectral analysis was
employed. The maximum entropy method (MEM) has its origin in information theory
where it is used for signal processing of data with high levels of broadband noise. MEM
estimates the spectral density of a discrete time signal by considering the most random,
or least predictable, process that has the same M + 1 autocorrelation coefficients as
the given time series, where M is called the order of the method (Ghil et al. 2002).
The disturbance spectra plotted in figure 21 were generated by performing a D F T and
MEM (with M = 40) for time signals at selected downstream locations. For the flow
with T u = 0 a distinct peak at the fundamental frequency f = 0.92 develops in the
streamwise direction. The dimensional frequency is scaled by the reference values for free-
The Role of Free-stream Turbulence on Boundary-Layer Separation
29
ln(|v’|)
-12
-8
-6
-4
-2
0
(b)
9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5
0.0
0.0
5.0
5.0
10.0
10.0
t
t
15.0
15.0
20.0
20.0
25.0
25.0
(a)
-10
9.5
10.0
10.5
11.0
x
12.5
13.0
12.0
12.5
13.0
10.0
10.5
11.0
11.5
x
12.0
12.5
13.0
0.0
5.0
10.0
10.0
t
t
15.0
15.0
20.0
20.0
25.0
25.0
(d)
5.0
0.0
12.0
x
(c)
9.5
11.5
9.5
10.0
10.5
11.0
11.5
x
Figure 20. Spatio-temporal development of shear-layer disturbances. Plotted are logarithmic
′
gray-scale contours of the wall-normal disturbance velocity component, v , at y = δ1 and z = 0.
(a) case FST-0 ; (b) case FST-0.05 ; (c) case FST-0.5 ; and (d) case FST-2.5.
stream speed and length scale, f = f ∗ L∞/U ∞. Other common scalings for the frequency
are based on, for example, the separation bubble length, l b. However, as Postl (2005)
pointed out, l b is not a meaningful length scale for the uncontrolled DNS cases since
the low background disturbance levels in the DNS typically result in significantly larger
bubbles than in the experiments, where F S T and sound disturbances are omnipresent.
As the turbulence in the free stream increases, a dominant frequency, although less
pronounced, can still be discerned from these spectra. The value of the frequency increases slightly to f ≈ 1.1 for case FST-2.5. An increase in frequency can be expected,
since the size of the separation bubble is reduced for higher F S T levels, and, in turn,
smaller bubbles promote smaller disturbance wavelengths and higher frequencies. Such
a conclusion can be verified by a comparison of L S T results for the different mean flows.
The comparison between the D F T and MEM results shows that MEM can be a valuable
tool to determine whether preferred frequencies exist in a highly disturbed flow. For case
FST-2.5 at x = 12.5, for example, the time signal is extremely noisy, but MEM is still
able to detect the shedding frequency. A comparison with the (−5/3)-decay law shows
that at x = 12.5, and especially at x = 13.0, the inertial subrange is well developed which
indicates fully turbulent flow.
30
W. Balzer and H. F. Fasel
f=0.92
-2
10
0
-4
10
10
-6
MEMPower
10
Ev’
-8
10
-10
10
-12
10
-4
x=13.5
-6
x=13.0
10
10
-5/3
~f
-8
10
-10
1
10
100
10 0.01
f
0.1
1
10
100
f
2
10
10
0
-4
10
10
-6
MEMPower
10
-8
Ev’
x=14.0
10
-12
0.1
-2
10
-10
10
-12
10
-5/3
-2
x=13.0
-4
x=12.5
-6
x=12.0
10
10
10
~f
-8
10
-10
10
-14
10 0.01
-12
0.1
1
10
100
10 0.01
f
-2
0.1
1
10
100
f
2
10
10
0
-4
10
10
x=13.0
~f
-2
-6
MEMPower
10
-8
Ev’
-2
10
-14
10 0.01
10
-10
10
-12
10
10
-5/3
x=12.5
-4
10
x=12.0
-6
10
-8
10
-10
10
-14
10 0.01
-12
0.1
1
10
100
10 0.01
f
-2
0.1
1
10
100
f
2
10
10
0
-4
10
10
-6
MEMPower
10
-8
Ev’
f=0.92
2
10
10
-10
10
-12
10
x=13.0
-2
x=12.5
-4
x=12.0
10
~f
-5/3
10
-6
10
-8
10
-10
10
-14
10 0.01
-12
0.1
1
10
100
10 0.01
0.1
f
1
10
100
f
Figure 21. Spectral energy obtained using discrete Fourier Transform ( D F T, left) and maximum entropy method (MEM, right) of order M = 40. From top to bottom: case FST-0 ; case
FST-0.05 ; case FST-0.5 ; case FST-2.5.
10. Inviscid Shear-Layer Instability Mechanism
Independent of the F S T level, a dominant frequency was found for all cases in the
previous section. Moreover, the frequency remained close to the natural shedding frequency of the unforced case FST-0, i.e. f = 0.92. A dominant peak in the frequency
The Role of Free-stream Turbulence on Boundary-Layer Separation
xR
xS
0
10
10
-1
-1
10
10
-2
-2
10
10
1,1
10
Au
1,0
-3
Au
31
xR
xS
0
-4
10
-5
-3
10
-4
10
-5
10
10
-6
-6
10
10
6
8
10
12
x
14
16
6
8
10
12
14
16
x
Figure 22. Downstream development of the streamwise
disturbance
velocity. Plotted are wal,
,
l-normal maxima of the Fourier amplitude, A u 10 (left), and A u 11 (right). (——) case FST-0 ;
(− − −) L S T of case FST-0 ; ( ) case FST-0.05 ; ( ) case FST-0.5 ; ( ) case FST-2.5 . Downstream locations of separation, x S, and reattachment, x R, are indicated as well.
spectrum of separation bubbles can typically be associated with the inviscid shear-layer
instability. To confirm this conjecture, DNS and L S T results are compared for 2D and
3D disturbance waves with the fundamental frequency, in particular Fourier modes (1, 0)
and (1, 1). The downstream development of the maximum u′ -velocity Fourier amplitude
,
,
,
,
of these modes, i.e. A u10 and A u11, is presented in figure 22. Note that A u 01 and A u 11
represent the maximum inside the boundary/shear layer. As the level of F S T increases,
the 2D and 3D disturbances in the attached laminar boundary layer exhibit higher amplitude levels without experiencing significant downstream growth. This is contrary to
the behavior of the Klebanoff distortions, which were shown to have a distinct growth in
the region of attached flow (see figure 17a). However, farther downstream, inside the separated shear layer, the disturbances grow exponentially until they saturate all at about
the same amplitude level (all cases with T u = 0). The L S T results included in figure 22
for case FST-0 show that the exponential growth of the 2D and 3D modes agrees well
with linear theory. Note that the L S T analysis of all other cases yields similar growth
curves, which are omitted in figure 22. For cases FST-0.05, FST-0.5 and FST-2.5, the
onset of exponential growth is at x ≈ 11.8. This is upstream of the location for which
disturbances in the unforced case FST-0 exhibit exponential growth, x > 12.7. Although
exponential growth is observed farther upstream for the cases with non-zero FST, it still
occurs significantly downstream of the separation location and the location for which
L S T predicts the onset of exponential growth, x ≈ 10 (not shown in figure 22). This
demonstrates that the vortical, turbulent fluctuations in the free stream are not very
effective in forcing TS waves with the “proper” frequency. Rather, as shown in section
8, they are responsible for the formation of streamwise streaks.
It is interesting to observe, that for case FST-0 the 2D mode, (1, 0), saturates and
remains at amplitude levels of one order of magnitude larger than the 3D mode, (1, 1).
This can be seen as an indication for the existence of predominantly 2D flow structures
in the reattaching turbulent flow, which was previously confirmed for similar flows using
proper orthogonal decomposition techniques (add reference). In all other cases, however,
the amplitude levels for the 2D and 3D modes are very similar at reattachment and farther
downstream. Note, however, that for the Fourier transform all flows were sampled with
the same frequency, f = 0.92. The results might change in favor of the 2D mode when
the sampling is carried out with the slightly higher frequencies that were detected for
cases FST-0.05, FST-0.5 and FST-2.5 in section 9.
32
W. Balzer and H. F. Fasel
(a)
(b)
(c)
(d)
3
3
3
2
2
2
2
1
1
1
1
y/1
3
0 0
0.5
1
uA/Au, UB
0 0
0.5
1
uA/Au, UB
0 0
0.5
1
uA/Au, UB
0 0
0.5
1
uA/Au, UB
′
Figure 23. Wall-normal distribution of u -velocity Fourier amplitudes obtained from DNS
(——) and L S T (−, − −) for the 2D mode (1, 0). Profiles were extracted , at the downstream
0
−3
0
locations where A u1 = 8 · 10 and scaled by their respective maximum, A u 1. Also shown is the
local mean flow profile, UB (− · − · −). (a) case FST-0 ; (b) case FST-0.05 ; (c) case FST-0.5 ;
and (d) case FST-2.5.
Figures 23 and 24 show wall-normal u′ -velocity Fourier amplitude distributions for
modes (1, 0) and (1, 1). These profiles were extracted from the region of exponential
,
,
growth for which the maxima of the disturbance amplitudes have reached A u 01= A u 11=
−3
8 · 10 . The shape of the amplitude distributions agrees reasonably well with the u′ velocity eigenfunctions obtained from LST. Notable differences in the wall-normal location of the disturbance peaks exist for case FST-2.5. Nevertheless, the reasonable
agreement between L S T and DNS for the region of exponential growth confirms that in
all cases both 2D and weakly oblique 3D disturbances grow due to the linear, inviscid
shear-layer instability mechanism. Therefore, it can be concluded that even for the highest level of F S T considered in the present work, the primary linear stage of the transition
process is not bypassed. Consequently, unsteady flow control techniques which are aimed
at exploiting the shear-layer instability might still be effective means to reduce/eliminate
separation in these cases.
11. Conclusions
A numerical model for realistic free-stream turbulence(FST) was introduced in an incompressible Navier-Stokes solver in order to investigate the effect of F S T on separation,
in particular the fundamental hydrodynamic instabilities that are known to govern the
transition process for laminar separation bubbles. This was done by prescribing physically
meaningful free-stream turbulent velocity and vorticity fields at the inflow boundary. The
present work also represents the first attempt to introduce such a model into a solver
based on the vorticity-velocity formulation.
For the simulations of a separation bubble subject to various levels of FST, a flatplate model was considered. Elevated F S T levels led to the generation of streamwise
boundary-layer streaks (Klebanoff modes). The streaks were characterized by a very low
The Role of Free-stream Turbulence on Boundary-Layer Separation
(a)
(b)
(c)
33
(d)
3
3
3
2
2
2
2
1
1
1
1
y/1
3
0 0
0.5
uA/Au
1
0 0
0.5
1
0 0
uA/Au
0.5
uA/Au
1
0 0
0.5
1
uA/Au
Figure 24. Wall-normal distribution of u′ -velocity Fourier amplitudes obtained from DNS
(——) and L S T (− − −) for the 3D mode (1, 1). Profiles were extracted at the downstream
,1
,1
−3
locations where A u1 = 8 · 10 and scaled by their respective maximum A u 1. (a) case FST-0 ;
(b) case FST-0.05 ; (c) case FST-0.5 ; and (d) case FST-2.5.
frequency, a long streamwise wavelength, and a spanwise spacing in the order of 3δ99 −
4δ99. Increasing F S T levels caused higher K-mode amplitudes inside the boundary layer.
In the approach flow, but especially inside the separated flow region, strong algebraic
growth of the streaks was observed. At the same time, the presence of the inviscid shearlayer instability was detected for all cases. Thus, for the separation bubble, transition
to turbulence was a consequence of both the primary shear-layer instability and the
enhanced three-dimensional disturbance level, in particular the streamwise streaks caused
by the FST. Even very small F S T levels caused a significant reduction of the size of the
separation bubble, which demonstrated that free-stream turbulence can have a significant
effect on LSB flows.
Further parameter studies including, for example, the effect of the free-stream turbulent length scales are required to build upon the gained experience and understanding.
Nevertheless, the presented results are an important cornerstone in the ongoing effort
to emulate physically ever more realistic and challenging flow environments by using
numerical simulations.
This work was supported by the Air Force Office of Scientific Research (AFOSR) under
grant number FA9550-09-1-0214. Douglas R. Smith served as the program manager.
Computer time was provided by the US Army Space and Missile Defense Command
(SMDC), the Arctic Region Supercomputing Center (ARSC), as well as the National
Center for Computational Sciences (NCCS).
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