Under consideration for publication in J. Fluid Mech. 1

Numerical Investigation of the Role ofFree-stream Turbulence on Boundary-Layer

Separation

By W. BALZER AND H. F. FASEL

Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ85721, USA

(Received ??? and in revised form ???)

The aerodynamic performance of lifting surfaces operating at low Reynolds number con-ditions is impaired by laminar separation. In most cases, flow transition to turbulenceoccurs in the separated shear layer as a result of a series of strong hydrondynamic in-stability mechanisms. Numerical investigations have become an integral part in the ef-fort to enhance our understanding of the intricate interactions between separation andtransition. Due to the development of advanced numerical methods and increase in theperformance of supercomputers with parallel architecture, it has become very feasiblefor 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 ofturbulence modeling. Because the employed high-order accurate DNS are characterizedby very low levels of background noise, they lend themselves to transition research wherethe amplification of small disturbances, sometimes even growing from numerical round-off, 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 levelsin the DNS to levels that are typical for the experiment. For the current work, a numer-ical model that emulates a realistic free-stream turbulent environment was adapted andimplemented into an existing Navier-Stokes code based on vorticity-velocity formulation.The role free-stream turbulence (FST) plays in the transition process was then investi-gated for a laminar separation bubble forming on a flat plate. FST was shown to cause theformation of the well known Klebanoff mode that is represented by streamwise-elongatedstreaks inside the boundary layer. Increasing the FST levels led to accelerated transition,a reduction in bubble size, and better agreement with the experiments. Moreover, thestage of linear disturbance growth due to the inviscid shear-layer instability was foundto 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-linearbehavior of turbulent flow, advancing the physical understanding presents a formidablechallenge. For low Reynolds number flow problems (Rec < 106; based on the chordlength, c, of an airfoil, for example) with strong adverse pressure gradients (APGs) inthe streamwise direction, matters are complicated by the fact that transition is ofteninitiated 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

2 W. Balzer and H. F. Fasel

in wall-normal momentum exchange cause the reattachment of the separated flow. Theresulting region of enclosed fluid between separation and reattachment locations is de-noted as laminar separation bubble (LSB). For a given geometry and APG the extentof the separated region and the transition region are typically larger at lower Reynoldsnumber conditions. For sufficiently low Reynolds numbers and/or low APGs transitionto turbulence might not occur and the flow reattaches in a laminar fashion. For a vari-ety of low Reynolds number flows in engineering applications, the appearance of LSBscan be the decisive factor for performance limitations of the device. Typical examplesare low-pressure turbine (LPT) stages, small unmanned aerial vehicles (UAVs), high-liftmulti-element airfoil configurations, and diffusers. While small regions of separated flowon a lifting surface, for example, may only have little effect on the aerodynamic perfor-mance, larger separation bubbles can cause a severe reduction in lift and an increase indrag. In the worst case, the flow does not reattach to the surface leading to a behaviorknown as “stall”. Stall can occur gradually (“soft stall”) or abruptly (“hard stall”). Thelatter is dangerous and can have negative consequences such as, for example, a suddencomplete 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). Asa result of the APG and the viscous stresses, fluid particles in the near-wall region ofthe attached laminar boundary layer are slowed down until the velocity profile becomesinflectional at the separation point, S. For steady separation, this point is typicallydefined as the first downstream location for which the skin friction vanishes and subse-quently assumes negative values. Just downstream of separation, and below the dividingstreamline, the fluid is almost stagnant. This corresponds to a region often referred toas the “dead-air” region. Due to the small velocities in this area, the pressure remainsalmost constant, which typically results in a pronounced pressure plateau in the wall-pressure distribution. The separated laminar shear layer is characterized by inflectionalreverse-flow profiles which support the amplification of small disturbances in the earlystages of transition. Farther downstream the shear layer becomes wavy and rolls upinto spanwise vortices (often referred to as “rollers”), which detach from the shear layerand are convected downstream (vortex shedding). These vortices considerably increasethe wall-normal momentum exchange by transporting low-momentum fluid away fromthe wall and high-momentum free-stream fluid towards the wall. In the time-averagedpicture, however, the process of shear-layer roll-up and shedding of vortices cannot beseen. Here it is indicated by rapid reattachment and the appearance of a reverse-flowvortex which represents an average over multiple such events. The mean velocity profilesin the region of the reverse-flow vortex exhibit the largest reverse-flow magnitudes inthe separation bubble. In addition, the disturbance amplitudes are large, which givesrise to three-dimensional (3D) secondary instabilities and nonlinear wave interactions.These are the late stages of transition. Although transition is to be considered a processrather than a local event, it is common to define a “transition onset point”, T, as, forexample, the location of the minimum skin-friction coefficient. The downstream extentof the transition region depends on multiple factors such as the Reynolds numbers, theAPG, and the disturbance spectrum. Shortly downstream of transition, the flow reat-taches 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 forwhich 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 sincethe flow is highly unsteady and the instantaneous location of reattachment can fluctuatesignificantly.

The Role of Free-stream Turbulence on Boundary-Layer Separation 3

S RT

Boundary LayerRedeveloping Turbulent

Shear LayerSeparated Laminar

Boundary LayerAttached Laminar

(non−linear stages)Transition

Dividing Streamline"Dead−Air" RegionFree Stream Reverse−Flow Vortex Boundary−Layer Edge

Figure 1. Mean-flow structure of a laminar separation bubble on a flat plate. Plotted arestreamlines, selected streamwise velocity profiles, as well as the boundary-layer thickness.

Since flow separation and transition are at work interactively, they should not be con-sidered as independent challenges. On the one hand, the separated shear layer is highlyunstable with respect to small disturbances which can accelerate transition. On the otherhand, earlier transition can reduce or even eliminate separation. For example, reducingthe size of a LSB also results in less displacement of the inviscid potential flow, whichin turn can change the APG and thus directly affect separation. Understanding of theinherent connection between separation and transition is therefore imperative for thedesign and optimization of lifting surfaces and the development of efficient flow controlstrategies. Moreover, transition and separation are strongly influenced by external factorssuch as free-stream turbulence (FST), surface roughness both discrete (e.g. insect con-tamination) and distributed (e.g. sandpaper roughness), noise, vibration (e.g. caused bya motor), and others. Consequently, the physical mechanisms associated with separationcontrol will also be affected by these factors. Driven by the need for more accurate predic-tions of separation and transition phenomena in “real-world” applications, the questionof how relevant transition/separation mechanisms for separated flows are affected by FST(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, theo-retical, and numerical studies of the stability and transition process for this flow phe-nomenon. 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 beunderstood as a progression of stages. Morkovin et al. (1994) discriminate between 5different paths to turbulence (A-E) depending on the level of external disturbance exci-tation (see figure 2). The first stage common to all paths is called receptivity (Morkovin1969). Receptivity denotes the process through which controlled and uncontrolled exter-nal disturbances (FST, sound, wall blowing and suction, etc.) enter the boundary layerand generate disturbances of certain amplitudes, frequencies, and wavenumbers. The re-sulting disturbance spectrum greatly influences the ensuing transition scenario. Figure2 shows a schematic that summarizes possible transition paths/scenarios observed forlaminar separation bubbles. Paths D and E are excluded from the following discussionsince it is assumed that forcing at very large amplitudes will quickly lead to breakdownof the flow prior to separation and likely suppresses the separation bubble completely.Such a transition scenario is often categorized as “bypass” transition (Morkovin 1979)

4 W. Balzer and H. F. Fasel

BA

E

x: c

onve

ctiv

e in

stab

ility

t:

abs

olut

e in

stab

ility

Tra

nsiti

on p

roce

ss

Secondary Mechanisms Bypass Mechanism

Breakdown

Receptivity

Transient Growth

Primary Instability

Forcing environmental disturbances

(mode interactions)

(exponential growth)

(algebraic growth)

(coherent structures, higher instabilities)

C D

(depending on Re & APG)

Turbulence

amplitude

no LSBLSB size

Figure 2. Paths to turbulence for incompressible wall-bounded shear flows. Adapted fromMorkovin et al. (1994).

since linear stages have no influence on the transition process and are bypassed on theroute to turbulence.

Path A in figure 2 corresponds to the transition scenario observed when the level ofenvironmental disturbances is very low (see also Marxen 2005, for a detailed description).In this case, transition is initiated by the exponential amplification (modal growth) ofso-called Tollmien-Schlichting (TS) waves. The inflectional velocity profiles in LSBs giverise 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, am-plification rates can be orders of magnitude larger than for attached boundary layers.Rist (1998) found good agreement between the exponential disturbance growth obtainedfrom 3D DNS and predictions from linear stability theory (LST). He concludes that fortypical LSB flows, non-parallel effects have only a minor influence.

Diwan & Ramesh (2009) provide experimental evidence for the fact that the origin ofthe inflectional instability lies upstream of the separation bubble where TS waves aregenerated (receptivity) and amplified due to a viscous instability in the attached APG-region of the boundary layer. On the other hand, the self-sustained vortex shedding seenin both experiments and DNS must not necessarily be caused by the spatial growth of

The Role of Free-stream Turbulence on Boundary-Layer Separation 5

incoming disturbances (convective instability). Inflectional velocity profiles with a strongreverse flow can also become absolutely unstable (Huerre & Monkewitz 1990). Absoluteinstability 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 varietyof analytic reverse-flow profiles (Gaster 1992; Hammond & Redekopp 1998; Alam &Sandham 2000). As a crude estimate, for the Reynolds number range of Reδ1

= 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∞). Itis known (from LST analysis, for example) that 2D disturbances are stronger amplifiedthan 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 thedisturbed flow.

In many cases, separation and the associated deflection of the boundary layer resultsin local streamline curvature in the vicinity of boundary-layer separation. The curvatureof the streamlines is concave and might give rise to steady, longitudinal vortices thatare amplified due to a centrifugal instability (Gortler instability; Gortler (1941)). WhileSpalart & Strelets (2000) did not find any evidence of Gortler vortices in their 3D DNSof an unsteady LSB, Pauley (1994) presented spanwise velocity contours which suggestedthe formation of such structures. For a LSB on a flat plate subjected to a strong favorable-to-adverse pressure gradient, Marxen et al. (2009) investigated the evolution of controlledsteady linear 3D disturbances and found strong evidence that a Gortler-type instabilitydue to streamline curvature is indeed possible.

The results by Marxen et al. (2009) can also serve as an example for Path B of thetransition process since the Gortler instability was preceded by transient growth of steadystreamwise disturbances in the favorable pressure gradient (FPG) region of the flow. Thisgrowth mechanism provides higher initial amplitudes for the Gortler eigenmode growth.Transient growth is associated with the non-normality of the linearized Navier-Stokes op-erator (see Trefethen et al. 1993; Schmid & Henningson 2001, for example). Superpositionof individual non-orthogonal eigenmodes can lead to algebraic growth for some down-stream distance before exponential decay due to viscosity becomes the overwhelmingfactor. 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 mech-anism through which the wall-normal displacement of a fluid particle in a shear layercauses a perturbation in the streamwise velocity since initially the fluid particle will re-tain its horizontal momentum. The optimal growth (i.e. the optimal superposition; seeAndersson et al. (1999); Luchini (2000)) is obtained for pairs of counter-rotating stream-wise vortices, which very effectively lift up low-velocity fluid from the wall and push downhigh-velocity fluid towards the wall. This mechanism results in strong streamwise streakswhich 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 alinear theory.

Path C corresponds to a transition scenario initiated by transient growth in absenceof any modal instabilities (TS, Gortler). McAuliffe & Yaras (2010), for example, showedthat for a “short” LSB under elevated levels of FST (Tu = 1.45% at separation), thenature 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 oflaminar separated flow and initiate breakdown via the formation of turbulent spots.

In nature, disturbances from different paths will likely coexist and interact with eachother. Wind-tunnel experiments carried out for a flat-plate boundary layer by Kosorygin

6 W. Balzer and H. F. Fasel

& Polyakov (1990), for example, show that in the intermediate range of free-streamfluctuations (0.1% < Tu < 0.7%) both TS waves (path A) and streamwise boundary-layer 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 stageunder the influence of turbulent, periodically passing wakes. They studied the connectionbewteen classical linear growth mechanisms (TS, KH) and transient growth, concludingthat 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 thestreamwise streaks and TS waves. While steady streaks may have a stabilizing effect onTS waves, unsteady streaks can enhance secondary instabilities of TS waves and promotetransition.

1.2. The Effect of Free-Stream Turbulence

The free-stream turbulence intensity (FSTI) can be relatively low for free-flight appli-cations (< 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 FST levelsin the order of 10% have been measured in a LPT environment (Sharma 1998). Whereaswind turbulences is rather uniform, the turbulence in turbomachinery flow is often con-centrated in the airfoil wakes periodically impinging on downstream blade rows.

Historically, FST has made it very difficult to identify linear stages of the transitionprocess (path A). Experimental evidence of TS waves, for example, was not at handuntil Schubauer & Skramstad (1948) managed to reduce the level of turbulent fluctua-tions in the free stream to “unusually low values” (∼ 1%). The success of Schubauer’sexperiments could in part be attributed to suggestions by Dryden (1936) who carried outexperiments at a much more turbulent free stream. He reported slow irregular u-velocityfluctuations of large amplitude in the laminar region of the boundary layer that wereinduced by the turbulence in the free stream. In fact, transition in wind-tunnel experi-ments typically seems to be preceded by such a low-frequency streaky modulation of theboundary layer. Today, these streaks are commonly referred to as “Klebanoff modes” (K-mode) or “Klebanoff distortions” after P. S. Klebanoff who investigated this phenomenonand described them as a periodic thickening/thinning of the boundary layer (Klebanoff &Tidstrom 1959; Klebanoff 1971). From other experimental studies investigating the effectof 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 thelift-up effect and transient growth mechanism described earlier.

In general, a shear layer adjacent to a wall is surprisingly resistant to fluctuations im-pinging from the free stream. The general observation that eddies convected by the freestream are quickly damped out inside the boundary layer due to inviscid shearing andviscous 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 actsas a low-pass filter, admitting predominantly low-frequency modes to penetrate into theboundary layer. Due to streamwise stretching of the ingested free-stream vorticity andtransient growth (Saric et al. 2002), low-frequency, high-amplitude streamwise streaksarise in the boundary layer (K-mode). TS waves, on the other hand, have higher fre-quencies and smaller scales than the energy-bearing long-wavelength disturbances thatare typical of FST (and sound). Therefore some kind of coupling or rescaling of the free-

The Role of Free-stream Turbulence on Boundary-Layer Separation 7

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 effectsare important, such as near the leading edge, a wall hump, or flow around the separationpoint (e.g. Kerschen 1991).

When a highly disturbed free stream introduces finite non-linear disturbances in thelaminar 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 pressuregradient, flat-plate boundary layer with a FSTI of Tu ∼ 3.5%. His results indicate thatbypass transition might be initiated when a boundary layer that is highly distorted bylow-speed streaks has become susceptible to higher-frequency disturbances which nowenter the boundary layer from the free stream and trigger secondary instabilities of thelow-speed Klebanoff distortions. The feasibility of such a bypass mechanism was laterexperimentally confirmed by Hernon et al. (2007).

Due to their distinct spanwise modulation and typical wall-normal u-velocity distri-butions, Klebanoff modes are often misconceived as streamwise vortices (also introducedas such by Klebanoff & Tidstrom 1959). To the contrary, the disturbances rather takethe form of elongated streaks in the laminar boundary layer with comparatively lowlevels of streamwise vorticity (Fasel 2002; Durbin & Wu 2007). On the other hand, theelongated streaks associated with the K-mode are, just like streamwise vortices, foundto be weaker on convex walls and can lead to unstable Gortler vortices on concave walls(Herbert 1997). From a transient growth point-of-view, the K-modes are not optimaldisturbances. In addition, the question of how the spanwise scaling for the free-streaminduced K-modes compares to the optimal spacing found from transient growth the-ory remains unanswered. In both cases, however, this spanwise scaling seems to be anintrinsic property of the boundary layer.

Naturally, the effect of FST is equally important in the presence of streamwise pressuregradients and the formation of a laminar separation bubble (Mayle 1991). Elevated levelsof FST have been reported to reduce the length of the separation bubble (Gault 1955;Volino 2002) or even prevent separation completely by transitioning the flow upstreamof the separation location (Dong & Cumpsty 1990; Zaki et al. 2010). However, earlytransition is not necessarily beneficial. The significant increase in skin friction and heat-transfer rate associated with turbulent flow can have a detrimental effect on performanceand hardware durability (Sohn & Reshotko 1991).

The existence of low-frequency, large-amplitude streaks in the boundary layer and inthe separated shear layer has been confirmed in the experiments of Haggmark (2000)and Volino (2002). In the experiments by Haggmark (2000), LSBs on a flat plate wereinvestigated with low noise and disturbance level (Tu = 0.02%), and with grid-generatedturbulence of 1.5% intensity. Smoke visualization images from his work are reproducedin figure 3. Haggmark (2000) reports that there is no strong evidence of two-dimensionalwaves in the separated shear layer at Tu = 1.5%. In addition, he performed a spectralanalysis 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 FSTlevels on the suction side separation of a low-pressure turbine blade. For a low Reynoldsnumber Re = 25, 000 (based on the blade’s suction surface length and exit velocity), hereported that even in the case of high levels of FST (Tu = 9%), breakdown to turbulence

† Lately, the term “continuous mode” transition is also used (Durbin & Wu 2007) arguing thatbypass transition is preceded by the amplification of continuous modes of the Orr-Sommerfeldspectrum rather than discrete modes (TS waves).

8 W. Balzer and H. F. Fasel

Figure 3. Smoke visualizations of a laminar separation bubble with low noise and disturbancelevel (left, Tu = 0.02%) and elevated levels of FST (right, Tu = 1.5%). Reproduced fromHaggmark (2000).

occured in the separated shear layer above the wall and that the flow did not reattachupstream of the trailing edge.

2. Governing equations

For the present investigation, the three-dimensional, incompressible, time-dependentNavier-Stokes (NS) equations are solved using direct numerical simulations.

∇ · v = 0 (2.1a)

∂v

∂t+ (v · ∇)v = −∇p +

1

Re∇2v. (2.1b)

In these equations, all quantities are non-dimensionalized by a global reference length,L∞, and a reference velocity, U∞,

x = L∞x∗, t =t∗U∞

L∞

, v =v∗

U∞

, p =p∗ − p∗

∞

ρU2∞

, (2.2)

where dimensional quantities are denoted by an asterisk and p∗∞

is the pressure at someconvenient reference point in the fluid. The Reynolds number in equation 2.1a is definedas Re = L∞U∞/ν where ν is the kinematic viscosity.

Depending on the application and geometry, numerous numerical methods to solve theincompressible 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 nonambiguousboundary conditions for pressure have led to alternative formulations in which pressureas a dependent variable is eliminated. One such alternative, employed in the presentinvestigation, is the vorticity-velocity formulation, which is obtained by taking the curl ofthe momentum equation (2.1a) and taking into consideration the fact that both velocityand vorticity vectors are solenoidal. With the definition of vorticity as the negative curlof the velocity vector, ω = −∇ × v, the set of governing equation in vorticity-velocityformulation can be derived,

∂ω

∂t+ (v · ∇)ω = (ω · ∇)v +

1

Re∇2

ω (2.3a)

∇2v = ∇× ω. (2.3b)

The advantages of the vorticity-velocity methods have been reviewed and discussed byFasel (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 CartesianNS solver first developed by Meitz (1996). Improvements to the solver, in particular theparallel algorithm, were contributed by Postl (2005) and Balzer (2011). Details of thenumerical solution procedure other than discussed in this section can be found in thesereferences. The code hase been thoroughly validated and employed successfully for numer-ous investigations of transitional and turbulent shear flows, in particular boundary-layertransition (Meitz & Fasel 2000; Fasel 2002), laminar and turbulent separation bubbles(Fasel & Postl 2006; Postl & Fasel 2006), as well as separation control using vortexgenerator 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 spatialderivatives in the streamwise (∂i/∂xi) and in the wall-normal (∂i/∂yi) direction arediscretized using fourth-order accurate compact differences (Lele 1992). An exception arethe streamwise first derivatives of the non-linear terms, which are discretized using fourth-order accurate split compact differences resulting in improved stability characteristicscompared to standard compact differences. Most computational time is expended forobtaining the solution of the wall-normal velocity component, which is governed by aPoisson equation. A solution is obtained by a direct method using standard compactdifferences and Fourier sine transforms.

In the spanwise direction, z, the flow is assumed to be periodic. Each variable isrepresented by a total of 2K +1 Fourier modes: the 2D spanwise average (zeroth Fouriermode), K symmetric Fourier cosine and K antisymmetric Fourier sine modes.

φ (x, y, z, t) = Φk=0 (x, y, t)+

K∑

k=1

Φk (x, y, t) cos(

γkz)

+

2K∑

k=K+1

Φk (x, y, t) sin(

γkz)

(3.1)

Fast Fourier Transforms (FFTs) are employed to convert each variable from spectralspace to physical space and back. The Fourier representation of the spanwise directionreduces the governing equations to a set of two-dimensional equations for each Fouriermode. The set of 2-D problems is solved in parallel using 2K +1 processors on a modernsupercomputing platform. For the calculation of the nonlinear terms, the flow field hasto be transformed from spectral to physical space (and back) before each sub step ofthe time integration. This requires redistributing of the entire three-dimensional arraysamong the processors, which requires extensive inter-processor communication realizedby using the message passing interface (MPI). To avoid aliasing errors, the nonlinearterms in physical space are computed on ≈ 3K spanwise collocation points. Note thatonly the non-linear terms are calculated in physical space. All other computational work(differentiation, time integration, imposition of boundary conditions etc.) takes place inspectral space.

3.1. Boundary conditions

The spectral approach in the spanwise direction implies the following periodic boundaryconditions at the spanwise boundaries, z = ±λz/2, for all variables and their derivatives:

φ (x, y,−λz/2, t) = φ (x, y, λz/2, t) (3.2a)

∂nφ

∂zn(x, y,−λz/2, t) =

∂nφ

∂zn(x, y, λz/2, t) ; n = 1, 2, . . . (3.2b)

At the inflow boundary, x = x0, 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. Thisinitial flow field is obtained by solving for the similarity solution of Blasius boundary-layer equations. In addition, the use of fourth-order accurate compact finite differencesfor the spatial discretization requires to provide the streamwise derivatives for some ofthe flow variables at the inflow.

At the inflow boundary, all velocity and vorticity components are prescribed as Dirich-let conditions. For maintaining the fourth-order accuracy of the code, the streamwisederivatives are prescribed as well. To prevent reflections from the outflow boundary, tur-bulent fluctuations are damped out in a buffer domain using the approach proposed byMeitz & Fasel (2000). At the upper boundary, a Neumann condition is prescribed forthe wall-normal velocity, ∂v/∂y = f(x), and irrotational flow is assumed, ω = 0. Sincethe equations for the streamwise (u) and spanwise (w) velocity components reduce toordinary differential equations in the streamwise direction, no free stream boundary con-ditions are required for these variables. At the wall, no-slip and no-penetration conditionsare imposed.

For simulating realistic, free-stream turbulent inflow conditions, an unsteady distur-bance field is imposed onto the steady, undisturbed base flow profile:

[

Uk, V k, W k]T

(x0, y, t) =

[UB, VB, 0]T (x0, y) +[

u′k, v′k, w′k]T

(x0, y, t) k = 0[

u′k, v′k, w′k]T

(x0, y, t) ∀k 6= 0 ,

(3.3a)

[

Ωkx, Ωk

y, Ωkz

]T(x0, y, t) =

[0, 0, Ωz,B]T

(x0, y) +[

ω′kx , ω′k

y , ω′kz

]T(x0, y, t) k = 0

[

ω′kx , ω′k

y , ω′kz

]T(x0, y, t) ∀k 6= 0 .

(3.3b)

The disturbances fields are designed to model isotropic free-stream turbulence describedin 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,FST is modelled by introducing random velocity and vorticity disturbances which aredesigned to satisfy continuity, and represent isotropic turbulence with a specified energyspectrum. The method is based on an expansion of the disturbance velocity, v′, in termsof its Fourier coefficients,

v′(x, t) =∑

k

v(k, t) eik·x , (4.1)

where k is the wavenumber vector with length k =√

k2x + k2

y + k2z . Turbulent fluctuations

are only introduced at the inflow boundary plane (x0, 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(x0, y, z, t) = VB(y) + v′(x0, y, z, t) . (4.2)

In the same manner, the inflow boundary condition for the vorticity field becomes

Ω(x0, y, z, t) = ΩB(y) + ω′(x0, y, z, t) , (4.3)

where ω′ is determined from the disturbance velocity field as ω

′ = −∇× v′.

The Role of Free-stream Turbulence on Boundary-Layer Separation 11

The dispersion relation kx = β/U∞ can be used to express the streamwise wavenumberin terms of the frequency. Consequently, kxx is replaced by −βt at the inflow boundary,

v′(x0, y, z, t) =∑

β

∑

kz

∑

ky

A(|k|) eikyyeikzze−iβt , (4.4)

where A(|k|) represent the coefficients/weights for the eigenfunctions that need to bedetermined. The implementation of the spanwise Fourier modes, eikzz , is straightforwardsince 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 thewall. Jacobs (1999) suggested to use local solutions from the continuous spectrum of theOrr-Sommerfeld operator, v, and the Squire operator, ω, to write the expansion in termsof the wall-normal eigenfunctions Φ, i.e.

v′(x0, y, z, t) =∑

β

∑

kz

∑

ky

A(|k|)Φ(y; β, kz , ky) eikzze−iωt . (4.5)

The solutions to the homogeneous OS equation and the homogeneous SQ equation are

ΦOS = [Φu, Φv, Φw]TOS =

[

iα

k2

∂v

∂y, v,

iγ

k2

∂v

∂y

]T

and (4.6a)

ΦSQ = [Φu, Φv, Φw]TSQ =

[

iγ

k2ωy, 0,− iα

k2ωy

]T

, (4.6b)

respectively. These solutions can be superposed to find a new set of eigenfunctions,

Φ =1

EΦ

[

eiθ1cos(ϕ)ΦOS + eiθ2sin(ϕ)ΦSQ

]

, (4.7)

where, for the purpose of the numerical model, θ1, θ2 and ϕ are uniformly distributedrandom numbers in the range [−π, π] which provide arbitrary weights and phase shiftsfor the eigenfunctions. The normalization is such that the energy of each disturbancemode is unity. This leads to an expression for EΦ:

EΦ =1

ymax

∫ ymax

0

1

2(ΦuΦ∗

u + ΦvΦ∗

v + ΦwΦ∗

w) dy . (4.8)

Since Φ carries the velocity vector information, the Fourier amplitude can be writtenas a scalar, A(|k|) in equation 4.5. As will be shown, A(|k|) can be used to distributeturbulent kinetic energy among the disturbance modes. A common choice for the analyticform of the energy spectrum is the von Karman spectrum,

E(k) ∼ Tu2 L(k L)4

C [1 + (k L)2]17

6

. (4.9)

A turbulent length scale, L, is used to normalize k. The free-stream turbulence intensity,defined as

Tu =

√

1

3

(

(u2)∞ + (v2)∞ + (w2)∞

)

=

√

1

3〈u′

i, u′

i〉 =

√

2

3(q2/2) , (4.10)

is used to normalize the spectrum. The quantity q2/2 = 1

2〈u′

i, u′

i〉 is the turbulent kineticenergy (often also denoted by k). For isotropic turbulence, the free-stream turbulent

intensities are similar, i.e. (u2)∞ ≈ (v2)∞ ≈ (w2)∞; therefore, Tu =

√

(u2)∞. The

12 W. Balzer and H. F. Fasel

E(k)

k

~k-5/3

k1 k2∆k

~k4

kmax≈1.157

L11

Emax≈(u2)∞

k

k z

k x

k y

k∆

Figure 4. Left: Typical von Karman energy spectrum (—). The symbols (E) mark the rangeof 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

C =2

3

∫

∞

0

x4

(1 + x2)17

6

dx =

√π Γ

(

1

3

)

4 Γ(

17

6

) . (4.11)

The turbulent length scale, L, is related to the turbulent integral length scale L11 by

L =55C

9πL11 , (4.12)

where the integral length scale is defined by the longitudinal two-point correlation (mixinglength theory; see Tennekes & Lumley 1972, p. 44, for a detailed derivation):

L11 =

∫

∞

0

u(x)u(x + r)

u2dr . (4.13)

This leads to the following expression for the energy spectrum,

E(k) = (u2)∞L11

1.196(kL11)4

[0.558 + (kL11)2]17

6

. (4.14)

Figure 4 (left) shows the schematic of a typical von Karman energy spectrum. For largescales (small k) the spectrum is asymptotically proportional to k4. For small scales (largek) 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 (kmax ≈ 1.157L11) whileTu controls the amplitude levels of these disturbances.

The turbulent kinetic energy is obtained by integrating over the entire 3D energyspectrum,

q2

2=

∫

∞

0

E(k)dk = lim∆k→0

∑

k

E(|k|)∆k . (4.15)

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,

q2

2≈

∑

k

E(|k|)∆k . (4.16)

The 3D energy distribution can be envisioned as a sphere with concentric shells ofradius k (see figure 4, right) and continuously distributed energy E(k). For the generationof inflow disturbance fields a discrete number, Nsh, of equidistant wavenumber shells(k1, k1+∆k, . . . , k2) was chosen on which energy is distributed discretely among a limitednumber, Np, of disturbance modes. Modes on the same shell are given by identical total

wavenumbers k =√

β2 + k2y + k2

z and the turbulent kinetic energy

q2

2=

1

2

∑

k

uiu∗

i =Np

2

∑

k

A(|k|)2 . (4.17)

Equations 4.15 and 4.17 lead to a formula for A(|k|),

A(|k|) =

√

2E(k)∆k

Np. (4.18)

It can be seen from figure 4 (right) that the disturbance modes on each wavenumbershell 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 discretizationof the z-direction used in the Navier-Stokes code. In doing so, computational time wassaved by avoiding additional FFT calls and inter-processor communication during eachtimestep. Within limits, the values for kx and ky are chosen randomly along lines ofconstant kz. Since FST is characterized by very low frequencies (see Westin et al. 1994,for example), it was decided to set the lower bound of possible streamwise wavenumbersto kx,min = 0, i.e. the lowest frequency which is introduced at the inflow boundaryof the computational domain is β = kxU∞=0 (steady disturbance). The upper boundof the range of streamwise wavenumbers depends on the streamwise discretization. Ifone wavelength of a disturbance travelling with the speed of the free stream should berepresented by at least five grid points in x-direction, then kx,max = π/(2∆x). For thewall-normal direction, ky = 0 must be avoided since it does not correspond to any physicaleigenvalue. The upper limit, ky,max = 2π n/ymax where n ∈ Z

+, is chosen with respect tothe domain height, ymax, and the grid resolution in y. Continuous modes correspondingto wall-normal wave numbers ky > ky,max are not resolved in the free stream (especiallyon strongly stretched grids) and are therefore not considered.

Upon choosing the parameters L11 and Tu, the inflow velocity and vorticity fields canbe completely determined. Figure 5 presents an elevated-surface plot with amplitudesand gray scale based on the instantaneous spanwise disturbance velocity, w′. This figureemphasizes that the current method only introduces disturbances in the free stream, butleaves the inside of the boundary layer almost undisturbed.

Note that the disturbance field plotted in figure 5 also vanishes near the free-streamboundary. It was found that high disturbance levels close to ymax can lead to numericalinstabilities. The reason for these numerical difficulties is the lacking of appropriate free-stream boundary conditions. While for a single disturbance mode it appears straightfor-ward 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 spanwisewavenumber and/or frequency are present. In this case a decomposition of the entire flowfield 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 forintroducing FST in the presented simulations. The edge of the boundary layer of the undisturbedflow is indicated by the boundary layer thickness, δ99.

correct boundary condition to each disturbance mode. Such a decomposition is compu-tationally expensive and non-trivial since the continuous modes are not orthogonal. Asan alternative approach, a smooth ramping function, R(y), was multiplied to each eigen-function (v, ω)T before computing the eigenfunctions ΦOS and ΦSQ which are used forthe expansion of the disturbance field. Note that the use of a ramping function does notprevent the disturbance field from being divergence-free, since the continuity equation isapplied after R(y) is imposed onto the eigenfunctions.

For the present study a total of 1200 continuous modes (Nsh = 40 equidistant wavenum-ber shells with Np = 30 disturbance modes on each shell) were used for generation of theinflow disturbance field. Figure 6 presents the theoretical (equation 4.14) and discreteenergy spectrum E(k) for all presented cases with Tu 6= 0. Energy values for the discretespectrum are larger than the theoretical values, which results from the condition thatboth curves should have the same turbulent kinetic energy (equations 4.15 and 4.16), i.e.

q2

2=

∫

∞

0

E(k)dk ≈∑

k

E(|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 maximumof 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 spanwisewavenumbers in the DNS. In fact, for the first wavenumber shell the total wavenumberis smaller than γ1,DNS , which means that all 30 modes on this shell are two-dimensionaldisturbances in the DNS, i.e. γ0,DNS = 0.

Note that the choice of wavenumber triplets is the same for all cases with Tu 6= 0. Inother words, for all simulations the free-stream disturbance distribution is the same whilethe amplitudes/energy increases with increasing levels of free-stream turbulence. The

The Role of Free-stream Turbulence on Boundary-Layer Separation 15

1 10 100 1000k

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

E(k

)

γ1, DNS

γ64, DNS

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. (E)

case FST-0.05 ; () case FST-0.5 ; (A) case FST-2.5.

lower and upper bounds for the wall-normal wavenumber were chosen as ky,min = π/2and ky,max = 24π to avoid continuous mode solutions with either very large wavelengthscompared to the computational domain (i.e. the wavelength of the oscillation is signif-icantly larger than ymax) or very short wavelengths that could not be resolved on thecomputational grid. Resolution limitations also exist for the streamwise and spanwisedirection.

5. Simulation Setup

The present computational setup for the DNS of a flat-plate LSB was motivated bythe wind-tunnel experiments of Gaster (1966). The computational domain and grid forthese numerical simulations can be seen in figure 7. In the experiments, an inverted airfoilwas mounted above a flat plate to accelerate and decelerate the flow between the airfoiland the flat plate. Because circulation control was applied on the airfoil, flow separationand reattachment occurred on the flat plate. This is illustrated in figure 7 by plottinginstantaneous gray-scale contours of the spanwise-averaged ωz-vorticity component. Fig-ure 7 also shows the location of a disturbance slot upstream of separation that was usedfor 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 thebaseline grid had 2001 equidistantly distributed points in the range 5.0 ≤ x ≤ 23.0. Theheight of the domain was ymax = 2.0 (≈ 19δ99,0) and exponential grid stretching wasemployed in the wall-normal direction using 256 grid points. At the solid wall, y = 0, no-slip and no-penetration boundary conditions were applied. The spanwise domain widthwas chosen as Lz = 2.0 (≈ 19δ99,0) and was resolved using K = 64 Fourier cosine andK = 64 Fourier sine modes. The number of collocation points was K = 200. The totalnumber of grid points for these simulations was approximately 102 million.

One case was computed with a finer grid (207 million points) in order to demonstrategrid convergence. The refined grid had 2161 and 320 grid points in the streamwise and

16 W. Balzer and H. F. Fasel

Figure 7. Simulation setup motivated by the experiments of Gaster (1966). For the computa-tional grid, every 10th grid line is shown in both streamwise and wall-normal direction. Alsoshown 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 = 193Fourier modes was employed using K = 300 collocation points. Results obtained with thefine 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 longseparation 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]. Thislength 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 ofrelevant 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 (andconsequently two relevant Reynolds numbers). The first scale is associated with thedistance of the displacement body from the leading edge of the flat plate. In a numericalsimulation, the integration domain does typically not include the leading edge. Ratherat some location upstream of the displacement body/pressure gradient, an inflow profileis prescribed that matches important boundary-layer parameters (δ99, δ1, etc.) of theapproach flow. In other words, the distance of the displacement body from the leadingedge determines the local boundary-layer properties. In the present numerical setup,for example, a Blasius similarity solution is prescribed at the inflow, x0 = 5.0, withδ1 ≈ 0.036. The corresponding local Reynolds number based on displacement thicknessis Reδ1

= 405. From Gaster’s (1966) experiments, only some values at the separationlocation are known. Here, the boundary-layer edge velocity and Reynolds number basedon momentum thickness were ue,S = 1.30 and Reδ2,S = 218. In the DNS, these valuesare ue,S = 1.32 and Reδ2,S = 189, respectively, which compare favorably to the valuesmeasured in the experiments. The second relevant length scale is associated with thestreamwise dimension of the displacement body. The Reynolds number based on thislength scale can be related to the pressure gradient. The chord length of the wing sectionin the experiments was Cx = 5.5[in] and the global Reynolds number based on Cx wasRe ≈ 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

5 10 15 20x

-1

-0.75

-0.5

-0.25

0

c p

Figure 8. Time- and spanwise-averaged wall-pressure coefficient: (——) 3D DNS; (−−−)

inviscid solution; (u) experiment separated; (+) experiment tripped to turbulence.

boundary condition that imposes a favorable-to-adverse pressure gradient on the meanflow similar to the experiments. The streamwise pressure gradient was enforced by pre-scribing wall-normal blowing and suction at the free-stream boundary, ymax. The free-stream v-velocity distribution was found iteratively. A comparison of the inviscid andviscous wall-pressure coefficient is presented in figure 8. The differences in the inviscidpressure coefficient can be explained by displacement effects of the (tripped) attachedturbulent boundary layer in the experiments. For the separated flow, the acceleratedflow region and onset of separation (pressure plateau) agree very well between DNS andexperiment.

Despite the good agreement for the upstream part of the bubble, the DNS data in-dicates later reattachment compared to case VI of the experiments. In fact, it provedvery diffcult with the current iterative procedure to find a free-stream boundary condi-tion which would yield a shorter bubble without sacrificing the reasonable agreement inthe upstream, laminar portion of the flow. From experience it is known that a bettercomparison 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 wasdecided to apply the numerical FST model for the present case. Employing the FSTmodel was preferred over traditional methods based on introducing selective or randomdisturbances inside the boundary layer, since it is less artificial and is rather based onrelevant physical mechanisms.

Time- and spanwise-averaged streamwise velocity profiles as well as streamlines arepresented in figure 9. Also plotted is the distribution of the boundary-layer thickness. Forx < 9.5, δ99 decreases, which indicates the strong acceleration of the flow upstream of theseparation location, xS = 10.4. In the region of APG, the flow separates and reattachesas a turbulent flow in the mean at xR = 15.54. The strong growth of the boundary-layer thickness and the full velocity profiles suggest that the flow indeed transitions toturbulence. 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 havea 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

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

x

0.0

0.5

1.0

y

u

δ99

xS xR

0 0.5 1 0 0.5 1 0 0.5 10 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 10 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1

x5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

y

0.0

0.5

1.0

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:streamlines plotted as equidistant contours of ln

`˛

˛104 · ψ˛

˛ + 1´

. Note that only half of the domainin y-direction is shown and that the y-direction is scaled by a factor of 3.

fluctuations in the order of Tu = 0.05%. For the current flow, six different cases wereconsidered:

(1)case FST-0 : “quiet” case; no fluctuations in the free stream; Tu = 0.(2)case FST-0.05 : low FST level; Tu = 0.05%.(3)case FST-0.5 : medium FST level; Tu = 0.5%.(4)case FST-2.5 : high FST level; Tu = 2.5%.(5)case FST-2.5b: Tu = 2.5%; DNS are computed on the fine grid.(6)case ZPG-2.5 : Tu = 2.5%; DNS are computed without pressure gradient.

The generation of the inflow disturbance field for cases with Tu 6= 0 was presented insection 4. Comparisons between cases FST-2.5 and FST-2.5b will demonstrate grid con-vergence. Case ZPG-2.5 serves as a reference case to isolate effects that are associatedwith the strong favorable-to-adverse streamwise pressure gradient. While the FST in-tensity is varied, the turbulent integral length scale remains constant and was chosenas L11 = 5δ1,0. This is in the order of the values typically used in related investiga-tions 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 havea significant impact on the transition location and transition mechanisms (Jonas et al.

2000; Brandt et al. 2004; Ovchinnikov et al. 2008). In experiments it is, in fact, ratherdifficult to vary the turbulence intensity while maintaining the same turbulent character-istic length scales or vice versa (e.g Kurian & Fransson 2009). Since the effect of varyingturbulent length scales was neglected during the present numerical campaign, the resultsby no means claim to offer a complete description of the problem.

The Role of Free-stream Turbulence on Boundary-Layer Separation 19

x10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0

y0

0.5

1.0

x10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0

y0

0.5

1.0

x10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0

y0

0.5

1.0

x10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0

y0

0.5

1.0

Figure 10. Contours of −10 ≤ ωz ≤ 10, where ωz = ln |ωz|, for DNS with various levelsof free-stream turbulent intensity Tu. From top to bottom: case FST-0 ; case FST-0.05 ; caseFST-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

ωz = ln |ωz| . (6.1)

Instantaneous contours of ωz-vorticity are presented in figure 10. The use of a logarithmicscale demonstrates that upstream of transition, two different flow regions can be identifiedfor 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 thevortical free-stream disturbances appear to be unable to enter the shear layer can beexplained by the so-called “shear sheltering” effect (Hunt et al. 1996), which describesthe effect that free-stream disturbances are rapidly damped out inside a shear layer dueto inviscid shearing and viscous dissipation.

The onset of transition is moved upstream when the level of FST increases. This, inturn, leads to a reduction of the separated flow region. Note that even for the highest

20 W. Balzer and H. F. Fasel

FSTI (case FST-2.5 ), the flow separates from the surface as a laminar boundary layerand transition occurs in the separated shear layer above the surface. While the shear layerstill exhibits traces of the roll-up and the formation of spanwise-oriented vortices, thereattaching turbulent boundary layer does not appear organized by dominant coherentstructures.

Free-stream turbulence induces low-frequency u-velocity distortions inside the laminarboundary layer - the so-called Klebanoff modes/distortions. Figure 11 presents instanta-neous gray-scale contours of the streamwise velocity component plotted in an x/z-planeinside the boundary layer, at y = δ1(x), and near the edge of the boundary layer, aty = δ99(x). Values for δ1 and δ99 were obtained from the time- and spanwise-averagedflow fields (see also figure 15). The u-velocity contours inside the boundary layer indicatea very sharply defined separation between laminar and turbulent flow regions demon-strating that transition occurs very rapidly over a short streamwise extent. Inside thelaminar separated flow region (x < 12) the flows with non-zero FST are characterizedby streamwise elongated streaks associated with the K-mode. These structures are mostprominent for case FST-2.5 where they appear with a high amplitude and a distinctspanwise scaling. For the case of very low FST, the Klebanoff distortions have muchsmaller 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 streamwisevelocity, i.e. u′ = u − U0. Gray-scale contours of u′-velocity are presented in figure 12for 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 FST case.Nevertheless, in this case the K-mode is weak and does not seem to be connected to theturbulent flow region whereas in the case of high FST, the streaks seem to be directlyconnected 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 ob-served in any of the present cases, it can be speculated that even for the case with thehighest FST 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 pre-sented in figure 11 it appears as if the separated shear layer loses its predominantly two-dimensional character and experiences the onset of flow transition seemingly without theshedding of spanwise vortices. However, the u-velocity contours plotted at y = δ99(x) infigure 11 give room for speculation that the reattaching flow might still be organized bysuch coherent flow structures. For the sake of argument, it is assumed that near the edgeof the boundary-layer the small-scale turbulence has died out such that the imprint of thelarge-scale motion can be “picked up”. Whereas the gray-scale contours show a strong2D 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 observa-tion suggests that for the cases with higher levels of FST slightly oblique waves are moredominant than 2D waves. This is often confirmed by carrying out a proper orthogonaldecomposition (POD) of the unsteady flow field, which was, however, not attempted forthe present cases.

Figure 13 presents an alternative visualization of the streamwise streaks. Here, in-stantaneous u′-velocity contours are plotted in three z/y-planes for cases FST-0.05 andFST-2.5. The first plane is located in the accelerated flow region at x = 8. The secondplane is located closely upstream of separation at x = 10, and the third plane is locatedinside the separated shear-layer region shortly upstream of transition at x = 12. Thethick lines in these plots represent an estimate of the local boundary layer thickness. Forthe case with low FST, no significant modulation of the boundary-layer thickness can beseen 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 21

x9 10 11 12 13 14 15 16

z1

0-1

u0.0 0.2 0.4 0.6 0.8 1.0

x9 10 11 12 13 14 15 16

z1

0-1

u0.9 1.0 1.1 1.2 1.3 1.4

x9 10 11 12 13 14 15 16

z1

0-1

x9 10 11 12 13 14 15 16

z1

0-1

x9 10 11 12 13 14 15 16

z1

0-1

x9 10 11 12 13 14 15 16

z1

0-1

x9 10 11 12 13 14 15 16

z1

0-1

x9 10 11 12 13 14 15 16

z1

0-1

x9 10 11 12 13 14 15 16

z1

0-1

x9 10 11 12 13 14 15 16

z1

0-1

Figure 11. Instantaneous gray-scale contours of the streamwise velocity component, u, plottedin a x/z-plane inside the boundary layer (y = δ1(x), left) and at the edge of the boundarylayer (y = δ99(x), right). From top to bottom: cases FST-0, FST-0.05, FST-0.5, FST-2.5, andFST-2.5b.

x8 9 10 11 12 13 14 15

z1

0-1

x8 9 10 11 12 13 14 15

z1

0-1

xS xR xS xR

u’-0.1 -0.05 0.0 0.05 0.1

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,u′ = u− U0, in an x/z-plane at distance y = δ1(x) away from the wall.

22 W. Balzer and H. F. Fasel

z0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y0.

00.

10.

20.

30.

4

z0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y0.

00.

10.

20.

30.

4

z0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y0.

00.

10.

20.

30.

4

z0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y0.

00.

10.

20.

30.

4

z0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y0.

00.

10.

20.

30.

4

z0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

y0.

00.

10.

20.

30.

4

−0.02 < u’ < 0.02

−0.10 < u’ < 0.10

−0.10 < u’ < 0.10

−0.25 < u’ < 0.25

x=12

x=10

x=8

x

−0.002 < u’ < 0.002

−0.002 < u’ < 0.002

Figure 13. View from upstream. Instantaneous contours of streamwise disturbance velocity,u′ = u− U0, plotted in z/y-planes at x = 8, 10, 12 (from top to bottom). Dashed contour linesindicate negative u′-values. The thick line represents an estimate of the local boundary-layerthickness. Left: case FST-0.05 ; right: case FST-2.5.

very pronounced spanwise-periodic thickening and thinning of the boundary layer. Themost regular pattern of the K-mode is observed at x = 10 where the spanwise streakspacing, measured as the distance between two u′-velocity minima or maxima, is ap-proximately 0.25 < λz < 0.35. Based on the local boundary-layer thickness, the spacingis in the order of 2.5δ99 − 3.5δ99. A more accurate value will be determined in section8. Note that the spacing appears to be independent of the intensity of the free-streamfluctuations.

7. Time- and Spanwise-Averaged Results

The instantaneous flow visualizations presented in the previous section indicate a re-duction of the separation region for increased levels of FST. Time- and spanwise-averagedresults of the different flow scenarios are compared in this section to quantify this ob-servation. Streamlines are presented in figure 14. The mean downstream locations forseparation and reattachment are summarized in table 1 and are also included in figure14. For the case with the lowest FST level the separation length is already significantly re-duces by 30.5%. Increasing the FST 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 locationremains almost constant, although a general trend can be observed that separation isslightly delayed for increased FSTIs.

In addition to the reduction in length, the height of the separation bubble also de-creases. This can be seen from the streamline images but also from the boundary-layerand displacement thickness presented in figure 15. Prior to separation, the curves are al-most identical. Only for the case with 2.5% FSTI the laminar boundary layer is slightly

The Role of Free-stream Turbulence on Boundary-Layer Separation 23

xS xR

y0.

00.

5

xS xR

y0.

00.

5

xS xR

y0.

00.

5

x10.0 10.5

xS

11.0 11.5 12.0 12.5

xR

13.0 13.5 14.0 14.5 15.0 15.5

y0.

00.

5

Figure 14. Time- and spanwise-averaged streamlines for the uncontrolled flow with variousfree-stream turbulence intensities. Plotted are equidistant contours of ln

`˛

˛104 · ψ˛

˛ + 1´

. From topto bottom: cases FST-0, FST-0.05, FST-0.5, and FST-2.5. Downstream locations of separation,xS, and reattachment, xR, 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, xS, and reattachment, xR.

thicker than for the other cases. Overall, it seems that at the chosen FST levels theboundary-layer fluctuations associated with the K-mode have only a local effect but donot 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 for0.05% < Tu < 0.5%. Recall that the FSTI in the experiments was Tu ≈ 0.05%. Thenumerical FST model, therefore, proved itself to be very useful for obtaining a morerealistic separation bubble size for this case. Note, however, that the pressure level forthe reattaching turbulent boundary layer increases from case FST-0 to FST-2.5. Thiseffect is most likely a consequence of the numerical setup and in particular the free-stream boundary condition. At ymax, the same v-velocity distribution is prescribed for allcases. Therefore, all flows experience the same inviscid pressure gradient. However, this

24 W. Balzer and H. F. Fasel

6 8 10 12 14 16 18x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

δ 99 6 8 10x

0.08

0.09

0.1

0.11

δ 99

6 8 10 12 14 16 18x

0

0.1

0.2

0.3

0.4

δ 1 6 8 10x

0.02

0.03

δ 1

Figure 15. Time- and spanwise-averaged boundary-layer thickness (left) and displacement

thickness (right) for various levels of FST. (——) case FST-0 ; (E) case FST-0.05 ; () case

FST-0.5 ; (A) case FST-2.5.

6 8 10 12 14 16 18x

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

Cp

6 8 10 12 14 16 18x

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Cf

Figure 16. Time- and spanwise-averaged wall-pressure coefficient (left) and skin-friction coef-

ficient (right) for various levels of FST. (u) experiment; (—) case FST-0 ; (E) case FST-0.05 ;

() case FST-0.5 ; (A) case FST-2.5.

approach does not take into account that the separation bubble, and the displacementeffect on the outer flow field, are drastically reduced when FST and/or flow controlare applied (viscous effects). The wall-normal suction at the free-stream boundary thatenforces the APG is therefore likely overpronounced in these cases and causes the slightlyhigher pressure values.

8. Turbulence Statistics

The downstream development of selected spanwise-averaged Reynolds stresses as wellas the turbulent kinetic energy, q/2, is depicted in figure 17. Plotted are values for thewall-normal maxima inside the boundary layer. Note that especially for high levels ofFST, a global disturbance maximum could be located in the free stream rather than nearthe wall. With increasing FST the disturbance level inside the boundary layer increasesas well. Although the fluctuations in the free stream are nearly isotropic, inside theboundary layer 〈v′, v′〉 is approximately one order of magnitude smaller than 〈u′, u′〉 and〈w′, w′〉. As indicated by the location of xR in figure 17, reattachment of the flow occursshortly after the disturbance components have reached their maximum amplitudes. Thedistance between the maximum and xR reduces as the FST increases. This could beexplained by the fact that for larger FST levels the separation bubble is shallower and

The Role of Free-stream Turbulence on Boundary-Layer Separation 25

that transition occurs closer to the surface such that the turbulent mixing leads to amore rapid and “effective” reattachment of the bubble.

For x < 12, the 〈u′, u′〉 Reynolds-stress component exhibits algebraic streamwisegrowth which is most likely the consequence of the lift-up mechanism (Ellingsen & Palm1975; Landahl 1975) and transient growth. Evidence for such a conclusion is provided bythe fact that only 〈u′, u′〉 shows this distinct streamwise growth while amplification of theother 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 FPG, x < 9, transientgrowth is alleviated. However, in the region of APG and separated mean flow, transientgrowth appears significantly stronger than for the ZPG case.

While the growth rates are almost identical for all cases in the FPG region of the flowfield, they are seen to increase in the APG region as the FST level decreases. One ex-planation for this behavior could be that the separation bubble grows as FST decreases,which, in turn, might give rise to higher amplification rates. Such a conclusion is cor-roborated by similar observations for the inviscid KH shear-layer instability, i.e. a longerbubble typically gives rise to larger KH growth rates. It is also noted that the onset of v-velocity and w-velocity disturbance growth commences farther upstream when the levelof FST decreases. Another interesting observation is that the strong streamwise growthof 〈v′, v′〉 for case FST-0.05 in the range 9.5 < x < 12 has no counterpart in the othercases. The growth rates in this flow region are initially identical for the u-velocity and thev-velocity disturbance. It is not clear at this stage how to interpret these results. On theone hand, it can be speculated that a second instability mechanism (possibly a Gortler-type instability) might exist near the separation point for case FST-0.05. On the otherhand, the similarity between the case without any FST (case FST-0 ) and case FST-0.05

indicates that the behavior might also be caused by the disturbance equilibrium of thenumerical scheme. The disturbance development seen in figure 17 also hints to the factthat transient growth alone does not explain transition to turbulence for the currentcases. For x > 12, the algebraic growth is overtaken by a strong exponential growthequally observed for all Reynolds-stress components. In section 10, it will be shown thatthis 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 presentedin figure 18. The disturbance profiles are normalized by their respective maxima insidethe boundary/shear layer and the y-direction is normalized by the displacement thicknessin these graphs. For the ZPG case (figure 18d) the profiles are nearly self-similar overa great portion of the boundary layer with a maximum at y/δ1 ≈ 1.3. Moreover, theprofiles compare very well to the urms-distribution obtained from optimal disturbancegrowth theory (Luchini 2000, e.g.). It is, in fact, a common observation that the shapeof the disturbance profile tends to assume the shape of the optimal perturbation even ifthe initial perturbation is far from optimal. Recall that in optimal perturbation theorythe u-velocity streaks are a response to counter-rotating vortices inside the boundarylayer, whereas in the DNS the streak formation is triggered by the fluctuations in thefree stream. This also explains the discrepancy of the curves for y → ymax. The profilesare not self-similar (with respect to the scaling y/δ1) for the cases with strong favorable-to-adverse pressure gradient (see figure 18a − c). However, the similarity of the profileshapes in the attached flow region suggests that the disturbances are caused by thesame 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 disturbancecomponents that are aligned with the mean flow. Modal growth, on the other hand, is typicallycharacterized by very similar growth rates for all velocity components.

26 W. Balzer and H. F. Fasel

(a) (b)

6 8 10 12 14 16x

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

⟨u’,u

’⟩ max

xSxR

6 8 10 12 14 16x

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

⟨v’,v

’⟩m

ax

xSxR

(c) (d)

6 8 10 12 14 16x

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

⟨w’,w

’⟩m

ax

xSxR

6 8 10 12 14 16x

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

q/2 m

axxS

xR

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 ; (E) case FST-0.05 ; () case FST-0.5 ; (A) case FST-2.5 ; (q)case FST-2.5b; (+) case ZPG-2.5. Downstream locations of separation, xS, and reattachment,xR, are also indicated.

scale in these cases, it shows that as the flow separates the maximum of the disturbancealigns with the location of δ1. Note that this alignment is already completed at x =10 in the case of low FST (case FST-0.05 ) while for the other cases it occurs fartherdownstream. Since the displacement thickness is commensurable with the location of thewall-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 stronglycorrelated in the spanwise direction. The spanwise spacing of the Klebanoff distortionscan 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, r), (8.1)

where the vector separation r is aligned with the spanwise direction. Figure 19 showscorrelations obtained for two y-locations: (1) inside, and (2) outside the boundary layer,

The Role of Free-stream Turbulence on Boundary-Layer Separation 27

(a) (b) (c) (d)

0 0.5 1

urms

/urms,max

0 0.5 1

urms

/urms,max

0

1

2

3

4

5

y/δ 1

0 0.5 1

urms

/urms,max

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 (E); (d) case ZPG-2.5. Results fromoptimal disturbance growth theory (Luchini 2000) are also included (+).

both plotted at x = 10. The inner y-location corresponds to the wall-normal maximumof the urms-velocity profiles and the outer location corresponds to y = 1.5δ99. Fromflow visualizations (e.g. Kendall 1985) it is known that the spanwise streak spacing isapproximately twice the distance to the first minimum in the correlation function. Thefirst 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 obtainedfrom the u′-velocity contours presented in figure 13. Note that the fluctuations in the freestream are uncorrelated. Results are also presented for the fine grid and the case of ZPGin 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 caseZPG-2.5. Thus, the spanwise spacing is almost identical for cases ZPG-2.5 and FST-

2.5. This in turn implies that the selection of the spanwise scale is independent of thepressure gradient. Furthermore, the values obtained for the spacing agree well with thedistinct 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 widthand boundary-layer thickness is Lz/δ99 ≈ 21. The K-mode spacing of 3.4δ99 thereforesuggests, 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, and19 shows that, in a statistical sense, the numerical solutions are grid independent.

28 W. Balzer and H. F. Fasel

0 2 4 6 8 10

∆z/δ99

-0.5

-0.25

0

0.25

0.5

0.75

1R

uu

0 1 2 3 4

∆z/δ99

-0.5

-0.25

0

0.25

0.5

0.75

1

Ruu

Figure 19. Spanwise two-point correlation function, Ruu, at x = 10. Symbols correspond toRuu-values inside the boundary layer (max. of urms), and lines correspond to Ruu-values outside

the boundary layer (y = 1.5δ99). (E, −−−) case FST-0.05 ; (, ——) case FST-0.5 ; (A, −·−·−)

case FST-2.5 ; (q) 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 thewall-normal disturbance velocity component, v′ = v−v, were analyzed to detect dominantfrequencies in the flow field. These time signals were taken inside the separated shearlayer 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/x-diagrams. White-colored contours correspond to regions of vanishing disturbance velocitywhich, most often, indicate a change of sign of the v′-velocity. Compared to the case withTu = 0, similar dynamic behavior is observed for the case of low FST (Tu = 0.05%).Here, the separated shear layer is relatively calm for x < 11.5 followed by the distincttime-periodic pattern associated with vortex shedding. Cases with elevated FST levelsshow an irregular flow pattern upstream of x = 11.5. However, farther downstream thesame time-periodic behavior as for the other cases becomes the prominent feature.

To determine the frequency spectrum associated with these velocity time signals, twospectral methods were employed. First, a discrete Fourier transform (DFT) of the signalswas performed, i.e.

v′β =1

N

N−1∑

n=0

v′n eiβtn , Ev′ =1

2v′β v′⋆β , (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 wasemployed. The maximum entropy method (MEM) has its origin in information theorywhere it is used for signal processing of data with high levels of broadband noise. MEMestimates 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 asthe 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 DFT andMEM (with M = 40) for time signals at selected downstream locations. For the flowwith Tu = 0 a distinct peak at the fundamental frequency f = 0.92 develops in thestreamwise direction. The dimensional frequency is scaled by the reference values for free-

The Role of Free-stream Turbulence on Boundary-Layer Separation 29

(a) (b)

x9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0

t0.

05.

010

.015

.020

.025

.0

x9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0

t0.

05.

010

.015

.020

.025

.0

x9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5

t0.

05.

010

.015

.020

.025

.0ln(|v’|)

-12 -10 -8 -6 -4 -2 0

x9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0

t0.

05.

010

.015

.020

.025

.0

(c) (d)

Figure 20. Spatio-temporal development of shear-layer disturbances. Plotted are logarithmicgray-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 frequencyare based on, for example, the separation bubble length, lb. However, as Postl (2005)pointed out, lb is not a meaningful length scale for the uncontrolled DNS cases sincethe low background disturbance levels in the DNS typically result in significantly largerbubbles than in the experiments, where FST and sound disturbances are omnipresent.

As the turbulence in the free stream increases, a dominant frequency, although lesspronounced, can still be discerned from these spectra. The value of the frequency in-creases 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 FST levels, and, in turn,smaller bubbles promote smaller disturbance wavelengths and higher frequencies. Sucha conclusion can be verified by a comparison of LST results for the different mean flows.

The comparison between the DFT and MEM results shows that MEM can be a valuabletool to determine whether preferred frequencies exist in a highly disturbed flow. For caseFST-2.5 at x = 12.5, for example, the time signal is extremely noisy, but MEM is stillable to detect the shedding frequency. A comparison with the (−5/3)-decay law showsthat at x = 12.5, and especially at x = 13.0, the inertial subrange is well developed whichindicates fully turbulent flow.

30 W. Balzer and H. F. Fasel

0.01 0.1 1 10 100f

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Ev’

0.01 0.1 1 10 100f

10-12

10-10

10-8

10-6

10-4

10-2

100

102

ME

M P

ower

x=13.0

x=13.5

x=14.0 ~f-5/3

f=0.92 f=0.92

0.01 0.1 1 10 100f

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Ev’

0.01 0.1 1 10 100f

10-12

10-10

10-8

10-6

10-4

10-2

100

102

ME

M P

ower

x=12.0

x=12.5

x=13.0 ~f-5/3

0.01 0.1 1 10 100f

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Ev’

0.01 0.1 1 10 100f

10-12

10-10

10-8

10-6

10-4

10-2

100

102

ME

M P

ower

x=12.0

x=12.5

x=13.0

~f-5/3

0.01 0.1 1 10 100f

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Ev’

0.01 0.1 1 10 100f

10-12

10-10

10-8

10-6

10-4

10-2

100

102

ME

M P

ower

x=12.0

x=12.5

x=13.0

~f-5/3

Figure 21. Spectral energy obtained using discrete Fourier Transform (DFT, left) and maxi-mum entropy method (MEM, right) of order M = 40. From top to bottom: case FST-0 ; caseFST-0.05 ; case FST-0.5 ; case FST-2.5.

10. Inviscid Shear-Layer Instability Mechanism

Independent of the FST level, a dominant frequency was found for all cases in theprevious section. Moreover, the frequency remained close to the natural shedding fre-quency 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 31

6 8 10 12 14 16x

10-6

10-5

10-4

10-3

10-2

10-1

100

Au1,

0

xSxR

6 8 10 12 14 16x

10-6

10-5

10-4

10-3

10-2

10-1

100

Au1,

1

xSxR

Figure 22. Downstream development of the streamwise disturbance velocity. Plotted are wal-l-normal maxima of the Fourier amplitude, A1,0

u (left), and A1,1u (right). (——) case FST-0 ;

(−−−) LST of case FST-0 ; (E) case FST-0.05 ; () case FST-0.5 ; (A) case FST-2.5. Down-stream locations of separation, xS, and reattachment, xR, are indicated as well.

spectrum of separation bubbles can typically be associated with the inviscid shear-layerinstability. To confirm this conjecture, DNS and LST results are compared for 2D and3D disturbance waves with the fundamental frequency, in particular Fourier modes (1, 0)and (1, 1). The downstream development of the maximum u′-velocity Fourier amplitudeof these modes, i.e. A1,0

u and A1,1u , is presented in figure 22. Note that A1,0

u and A1,1u

represent the maximum inside the boundary/shear layer. As the level of FST increases,the 2D and 3D disturbances in the attached laminar boundary layer exhibit higher am-plitude levels without experiencing significant downstream growth. This is contrary tothe behavior of the Klebanoff distortions, which were shown to have a distinct growth inthe region of attached flow (see figure 17a). However, farther downstream, inside the sep-arated shear layer, the disturbances grow exponentially until they saturate all at aboutthe same amplitude level (all cases with Tu 6= 0). The LST results included in figure 22for case FST-0 show that the exponential growth of the 2D and 3D modes agrees wellwith linear theory. Note that the LST analysis of all other cases yields similar growthcurves, which are omitted in figure 22. For cases FST-0.05, FST-0.5 and FST-2.5, theonset of exponential growth is at x ≈ 11.8. This is upstream of the location for whichdisturbances in the unforced case FST-0 exhibit exponential growth, x > 12.7. Althoughexponential growth is observed farther upstream for the cases with non-zero FST, it stilloccurs significantly downstream of the separation location and the location for whichLST predicts the onset of exponential growth, x ≈ 10 (not shown in figure 22). Thisdemonstrates that the vortical, turbulent fluctuations in the free stream are not veryeffective in forcing TS waves with the “proper” frequency. Rather, as shown in section8, 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 andremains 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 structuresin the reattaching turbulent flow, which was previously confirmed for similar flows usingproper 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 fartherdownstream. Note, however, that for the Fourier transform all flows were sampled withthe same frequency, f = 0.92. The results might change in favor of the 2D mode whenthe sampling is carried out with the slightly higher frequencies that were detected forcases FST-0.05, FST-0.5 and FST-2.5 in section 9.

32 W. Balzer and H. F. Fasel

(a) (b) (c) (d)

0 0.5 1

uA/A

u, U

B

0

1

2

3

y/δ 1

0 0.5 1

uA/A

u, U

B

0

1

2

3

0 0.5 1

uA/A

u, U

B

0

1

2

3

0 0.5 1

uA/A

u, U

B

0

1

2

3

Figure 23. Wall-normal distribution of u′-velocity Fourier amplitudes obtained from DNS(——) and LST (− − −) for the 2D mode (1, 0). Profiles were extracted at the downstreamlocations where A1,0

u = 8 ·10−3 and scaled by their respective maximum, A1,0u . 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 formodes (1, 0) and (1, 1). These profiles were extracted from the region of exponentialgrowth for which the maxima of the disturbance amplitudes have reached A1,0

u = A1,1u =

8 · 10−3. The shape of the amplitude distributions agrees reasonably well with the u′-velocity eigenfunctions obtained from LST. Notable differences in the wall-normal lo-cation of the disturbance peaks exist for case FST-2.5. Nevertheless, the reasonableagreement between LST and DNS for the region of exponential growth confirms that inall cases both 2D and weakly oblique 3D disturbances grow due to the linear, inviscidshear-layer instability mechanism. Therefore, it can be concluded that even for the high-est level of FST considered in the present work, the primary linear stage of the transitionprocess is not bypassed. Consequently, unsteady flow control techniques which are aimedat exploiting the shear-layer instability might still be effective means to reduce/eliminateseparation in these cases.

11. Conclusions

A numerical model for realistic free-stream turbulence(FST) was introduced in an in-compressible Navier-Stokes solver in order to investigate the effect of FST on separation,in particular the fundamental hydrodynamic instabilities that are known to govern thetransition process for laminar separation bubbles. This was done by prescribing physicallymeaningful free-stream turbulent velocity and vorticity fields at the inflow boundary. Thepresent work also represents the first attempt to introduce such a model into a solverbased on the vorticity-velocity formulation.

For the simulations of a separation bubble subject to various levels of FST, a flat-plate model was considered. Elevated FST levels led to the generation of streamwiseboundary-layer streaks (Klebanoff modes). The streaks were characterized by a very low

The Role of Free-stream Turbulence on Boundary-Layer Separation 33

(a) (b) (c) (d)

0 0.5 1

uA/A

u

0

1

2

3

y/δ 1

0 0.5 1

uA/A

u

0

1

2

3

0 0.5 1

uA/A

u

0

1

2

3

0 0.5 1

uA/A

u

0

1

2

3

Figure 24. Wall-normal distribution of u′-velocity Fourier amplitudes obtained from DNS(——) and LST (− − −) for the 3D mode (1, 1). Profiles were extracted at the downstreamlocations where A1,1

u = 8 · 10−3 and scaled by their respective maximum A1,1u . (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 FST levels caused higher K-mode amplitudes inside the boundary layer.In the approach flow, but especially inside the separated flow region, strong algebraicgrowth of the streaks was observed. At the same time, the presence of the inviscid shear-layer instability was detected for all cases. Thus, for the separation bubble, transitionto turbulence was a consequence of both the primary shear-layer instability and theenhanced three-dimensional disturbance level, in particular the streamwise streaks causedby the FST. Even very small FST levels caused a significant reduction of the size of theseparation bubble, which demonstrated that free-stream turbulence can have a significanteffect on LSB flows.

Further parameter studies including, for example, the effect of the free-stream turbu-lent length scales are required to build upon the gained experience and understanding.Nevertheless, the presented results are an important cornerstone in the ongoing effortto emulate physically ever more realistic and challenging flow environments by usingnumerical simulations.

This work was supported by the Air Force Office of Scientific Research (AFOSR) undergrant 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 NationalCenter for Computational Sciences (NCCS).

REFERENCES

Alam, M. & Sandham, N. D. 2000 Direct numerical simulation of ‘short’ laminar separationbubbles with turbulent reattachment. J. Fluid Mech. 403, 223–250.

Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypasstransition in boundary layers. Phys. Fluids 11 (1), 134–150.

34 W. Balzer and H. F. Fasel

Balzer, W. 2011 Numerical investigation of the role of free-stream turbulence on boundary-layer separation and separation control. PhD thesis, University of Arizona.

Boiko, A. V., Grek, G. R., Dogval, A. V. & Kozlov, V. V. 2002 The Origin of Turbulencein near-wall Flows, 1st edn. Springer, Berlin, Heidelberg, New York.

Brandt, L., Schlatter, P. & Henningson, D. S. 2004 Transition in boundary layers subjectto free-stream turbulence. J. Fluid Mech. 517, 167–198.

Diwan, S. S. & Ramesh, O. N. 2009 On the origin of the inflectional instability of a laminarseparation bubble. J. Fluid Mech. 629, 263–298.

Dong, Y. & Cumpsty, N. A. 1990 Compressor blade boundary layer: Part2 - measurementswith incident wakes. ASME J. Turbomachinery 112, 231–241.

Dryden, H. L. 1936 Air flow in the boundary layer near a plate. Tech. Rep. 562. NACA.Durbin, P. & Wu, X. 2007 Transition beneath vortical disturbances. Ann. Rev. Fluid Mech.

39, 107–128.Ellingsen, T. & Palm, E. 1975 Stability of linear flows. Phys. Fluids 18 (4), 487–488.Fasel, H. F. 1980 Recent developments in the numerical solution of the navier-stokes equa-

tions and hydrodynamic stability problems. In Computational fluid dynamics., pp. 167–279.Hemisphere Publishing Corp.

Fasel, H. F. 2002 Numerical investigation of the interaction of the Klebanoff-mode with aTollmien-Schlichting wave. J. Fluid Mech. 450, 1–33.

Fasel, H. F. & Postl, D. 2006 Interaction of separation and transition in boundary layers:Direct numerical simulations. In Laminar-Turbulent Transition, Proceedings of the IUTAMSymposium, pp. 71–88. Bangalore, India: Springer.

Ferziger, J. H. 1998 Numerical Methods for Engineering Application, 2nd edn. John Wiley &Sons.

Gaster, M. 1966 The structure and behaviour of laminar separation bubbles. In AGARD CP4 , pp. 813–854.

Gaster, M. 1992 Stability of velocity profiles with reverse flow. In Street, Hussaini, Kumar(Eds.): Instability, Transition and turbulence, pp. 212–215. Springer.

Gatski, T. B. 1991 Review of incompressible fluid flow computations using the vortici-velocityformulation. Applied Numer. Meth. 7, 227–239.

Gault, D. 1955 An experimental investigation of regions of separated laminar flow. NACA TN3505.

Ghil, M., Allen, M. R., Dettinger, M. D., Ide, K., Kondrashov, D., Mann, M. E.,Robertson, A. W., Saunders, A., Tian, Y., Varadi, F. & Yiou, P. 2002 Advancedspectral methods for climatic time series. Rev. Geophys. 40(1).

Gortler, H. 1941 Instabilitat laminarer Grenzschichten an konkaven Wanden gegenuber gewis-sen dreidimensionalen Storungen. ZAMM 21 (4), 250–252.

Haggmark, C. 2000 Investigations of disturbances developing in a laminar separation bubbleflow. PhD thesis, KTH, Department of Mechanics, Stockholm.

Hammond, D. A. & Redekopp, L. G. 1998 Local and global instability properties of separationbubbles. Eur. J. Mech. - B/Fluids 17 (2), 145–164.

Herbert, T. 1997 Parabolized stability equations. Ann. Rev. Fluid Mech. 29, 245–283.Hernon, D., Walsh, E. J. & McEligot, D. M. 2007 Experimental investigation into the

routes to bypass transition and the shear-sheltering phenomenon. J. Fluid Mech. 591,461–479.

Horton, H. P. 1968 Laminar separation in two and three-dimensional incompressible flow.PhD thesis, University of London.

Hourmouziadis, J. 2000 Das DFG Verbundvorhaben Instationare Stromung in Turbomaschi-nen. Tech. Rep. JT2000-030. DGLR, Deutscher Luft-und Raumfahrtkongress, Leipzig, Ger-many, September 18-21.

Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developingflows. Ann. Rev. Fluid Mech. 22, 473–537.

Hunt, J. C. R., Durbin, P. A., Kevlahan, N. K.-R. & Fernando, H. J. S. 1996 Non-localeffects of shear in turbulent flows. In Sixth European Turbulence Conference, Lausanne,Switzerland .

Jacobs, R. G. 1999 Bypass transition phenomena studied by computer simulation. PhD thesis,Stanford University.

The Role of Free-stream Turbulence on Boundary-Layer Separation 35

Jacobs, R. G. & Durbin, P. A. 1998 Shear sheltering and the continuous spectrum of theOrr-Sommerfeld equation. Phys. Fluids 10 (8), 2006–2011.

Jonas, P., Mazur, O. & Uruba, V. 2000 On the receptivity of the by-pass transition to thelength scale of the outer stream turbulence. Eur. J. Mech. - B/Fluids 19 (5), 707–722.

Kendall, J. M. 1985 Experimental study of disturbances produced in a pre-transitional laminarboundary layer by weak freestream turbulence. AIAA Paper 1985-1695.

Kendall, J. M. 1990 Boundary layer receptivity to freestream turbulence. AIAA Paper 1990-1504.

Kerschen, E. J. 1991 Linear and nonlinear receptivity to vortical free-stream disturbances. InASME FED, vol. 114, Boundary Layer Stability and Transition to Turbulence.

Klebanoff, P. 1971 Effect of freestream turbulence on the laminar boundary layer. Bull. Am.Phys. Soc. 10, No. 11.

Klebanoff, P. S. & Tidstrom, K. D. 1959 Evolution of amplified waves leading to transitionin a boundary layer with zero pressure gradient. NASA TN D 195.

Kosorygin, V. S. & Polyakov, N. P. 1990 Laminar boundary layers in turbulent flows.In Laminar-Turbulent Transition, Proceedings of the IUTAM Symposium, pp. 573–578.Toulouse, France: Springer.

Kurian, T. & Fransson, J. H. M. 2009 Grid-generated turbulence revisited. Fluid DynamicsResearch 41 (2), 021403.

Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28, 735–756.Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comp.

Phys. 103, 16–42.Liu, Y., Zaki, T. A. & Durbin, P. A. 2008a Boundary layer transition by interaction of

discrete and continuous modes. J. Fluid Mech. 604, 199–233.Liu, Y., Zaki, T. A. & Durbin, P. A. 2008b Floquet analysis of secondary instability of

boundary layer distorted by Klebanoff streaks and Tollmien-Schlichting waves. Phys. Fluids20 (12), 124102.

Luchini, P. 2000 Reynolds number independent instability of the boundary layer over a flatsurface, part2: Optimal perturbations. J. Fluid Mech. 404, 289–309.

Marxen, O. 2005 Numerical studies of physical effects related to the controlled transitionprocess in laminar separation bubbles. PhD thesis, Universitat Stuttgart.

Marxen, O., Lang, M., Rist, U., Levin, O. & Henningson, D. S. 2009 Mechanisms for spa-tial steady three-dimensional disturbance growth in a non-parallel and separating boundarylayer. J. Fluid Mech. 634, 165–189.

Mayle, R. E. 1991 The role of laminar-turbulent transition in gas turbine engines. ASME J.Turbomachinery 113, 509–537.

McAuliffe, B. R. & Yaras, M. I. 2010 Transition mechanisms in separation bubbles underlow- and elevated-freestream turbulence. ASME J. Turbomachinery 132, 011004–011010.

Meitz, H. 1996 Numerical investigation of suction in a transitional flat-plate boundary layer.PhD thesis, University of Arizona.

Meitz, H. & Fasel, H. F. 2000 A compact-difference scheme for the Navier-Stokes equationsin vorticity-velocity formulation. J. Comp. Phys. 157, 371–403.

Morkovin, M. V. 1969 The many faces of transition. In Viscous Drag Reduction (ed. C. S.Wells), pp. 1–31. New York: Plenum Press.

Morkovin, M. V. 1979 On the question of instabilities upstream of cylindrical bodies. Tech.Rep. CR-3231. NASA.

Morkovin, M. V., Reshotko, E. & Herbert, T. 1994 Transition in open flow systmes - areassessment. Bull. Am. Phys. Soc 39, 1882.

Ovchinnikov, V., Choudhari, M. M. & Piomelli, U. 2008 Numerical simulations ofboundary-layer bypass transition due to high-amplitude free-stream turbulence. J. FluidMech. 613, 135–169.

Pauley, L. L. 1994 Response of two-dimensional separation to three-dimensional disturbances.ASMS J. Fluids Eng. 116, 433–438.

Postl, D. 2005 Numerical investigation of laminar separation control using vortex generatorjets. PhD thesis, University of Arizona.

Postl, D., Balzer, W. & Fasel, H. F. 2011 Control of laminar separation using pulsed vortexgenerator jets: Direct numerical simulations. J. Fluid Mech. 676, 81–109.

36 W. Balzer and H. F. Fasel

Postl, D. & Fasel, H. F. 2006 Direct numerical simulation of turbulent flow separation froma wall-mounted hump. AIAA J. 44 (2), 263–272.

Rist, U. 1998 Zur Instabilitat und Transition in laminaren Abloseblasen. Habilitation, Univer-sitat Stuttgart, Aachen: Shaker Verlag, 1999 (Berichte aus der Luft- und Raumfahrttech-nik).

Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestreamdisturbances. Ann. Rev. Fluid Mech. 34, 291–319.

Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Schubauer, G. B. & Skramstad, H. K. 1948 Laminar boundary layer oscillations and tran-

sition on a flat plate. Tech. Rep. 909. NASA.Sharma, O. 1998 Impact of Reynolds number on LP turbine performance. In Proceedings of the

1997 Minnowbrook II Workshop on Boundary Layer Transition in Turbomachines. NASACP-1998-206958: 65-69.

Sohn, K. & Reshotko, E. 1991 Experimental study of boundary layer transition with elevatedfreestream turbulence on a heated flat plate. Tech. Rep.. NASA, CR-187068.

Spalart, P. R. & Strelets, M. K. 2000 Mechanisms of transition and heat transfer in aseparation bubble. J. Fluid Mech. 403, 329–349.

Speziale, C. G. 1987 On the advantages of the vorticity velocity formulation of the equationsof fluid-dynamics. J. Comp. Phys. 73, 476–480.

Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence. MIT Press.Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic

stability without eigenvalues. Science 261 (5121), 578–584.Volino, R. J. 2002 Separated flow transition under simulated low-pressure turbine airfoil condi-

tions part1: Mean flow and turbulence statistics. ASME J. Turbomachinery. 124, 645–655.Westin, K. J. A., Boiko, A. V., Klingmann, B., B. G., Kozlov, V., V. & Alfredsson,

P. H. 1994 Experiments in a boundary layer subjected to free-stream turbulence. part 1.boundary layer structure and receptivity. J. Fluid Mech. 281, 193–218.

Wu, X. & Durbin, P. A. 2001 Evidence of longitudinal vortices evolved from distorted wakesin a turbine passage. J. Fluid Mech. 446, 199–228.

Wu, X., Jacobs, R., Hunt, J. & Durbin, P. A. 1999 Simulation of boundary layer transitioninduced by periodically passing wakes. J. Fluid Mech. 398, 109–153.

Zaki, T. A., Wissink, J. G., W., R. & Durbin, P. A. 2010 Direct numerical simulation oftransition in a compressor cascade: the influence of free-stream turbulence. J. Fluid Mech.665, 57–98.