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