The Interaction of Turbulent Spots With Low-Speed Streaks¹

Transitional and turbulent flows occur in most engineering situations, and being able to accurately predict them is essential for technological progress across critical applications such as gas turbine engines [1], wind turbine blades [2], scramjet engines [3,4], and for the design of safer and more efficient aircraft [5,6]. Turbulent spots are regions of near-wall turbulence surrounded by laminar flow [7] that often occur during the final stages of both classical laminar-turbulent transition observed in quiet environments, as well as bypass transition occurring subject to large freestream turbulence intensities [8].

Turbulent spots were first observed in the context of bypass transition [9]. Some significant early experiments were those of Wygnanski et al. [10] who studied the shape, entrainment, and spreading mechanisms of spots, and those of Cantwell et al. [11] which revealed the self-similar structure of the spot and estimated the celerity of the spot. These were followed by the experiments on the evolution of spots under favorable pressure gradients [12,13]. Singer and Joslin [14] performed direct numerical simulations (DNS) of spot generation from a hairpin vortex.

Subsequent direct numerical simulations include that of Strand and Goldstein [15] which revealed the effect of riblets on spot spreading, and Goldstein et al. [16] provided insight into the spreading mechanisms from a vorticity point of view and demonstrated arresting spreading through (nonphysical) damping fins.

Other recent studies explored the effect of initial conditions, Reynolds number, and freestream turbulence on spot growth [17], shed insight into the hairpin vortex regeneration mechanism [18] as well as on the dynamics of the “calmed region,” the interaction of a spot with a two-dimensional cavity [19], and the analysis of the early development of turbulent spots in terms of high and low-speed streaks [20]. Turbulent spots thus remain a topic of current research, especially considering their importance in bypassing transition even in hypersonic flow. Goparaju and Gaitonde [21] studied the structure of spots in hypersonic boundary layers: on studying the acoustic, entropic, and vortical modes, they found that the vortical mode resembles spots in incompressible flow, suggesting that fundamental investigation of turbulent spots in incompressible flow may have a general relevance.

Most of these studies, however, involve studying the evolution of isolated (artificially generated) turbulent spots, surrounded by an undisturbed laminar boundary layer. In contrast, turbulent spots in a transitioning boundary layer usually evolve in the presence of other disturbances such as low-speed streaks. Low-speed streaks can originate from freestream disturbances or surface roughness and are often unstable. Owing to their importance in bypassing transition, streak instability has been a topic of separate study. Direct numerical simulation studies of streaks caused by freestream turbulence include Vaughan and Zaki [22] and Hack and Zaki [23], who classify the dominant streak instabilities as having a varicose inner mode and a sinuous outer mode. Berger et al. [24] performed a linear stability analysis of a low-speed streak downstream of a discrete roughness element (DRE) and described the two instabilities as a dominant high-frequency instability of the top shear layer and a subdominant low-frequency instability of the side shear layer, which were both found in the bandpass filtered hot-wire data. From a vorticity point of view, the top shear layer instability involves the back-and-forth motion of the perturbation at the top of the low-speed streak causing clumping of the “heads” of the vortex lines that straddle the streak. This causes the tilting of the legs (that contain wall-normal vorticity, $ωy)$⁠, causing a generation of the fluctuating streamwise vorticity via the source term containing $ωy ∂u/∂y$⁠. On the other hand, the sinuous mode involves side-to-side meandering of the streak that generates streamwise vorticity by tilting the spanwise vortex lines at the edge of the streak via the source term containing $ωz ∂w/∂x$⁠. Suryanarayanan et al. [25] thus described the varicose and sinuous instabilities as the clumping mode and swaying mode, respectively.

Streak instability and turbulent spot dynamics are both topics of engineering interest, as they are both relevant to bypass transition prediction and control, which in turn play a significant role in many engineering scenarios including gas turbine engines [26]. Furthermore, turbulent spots are often generated via the breakdown of low-speed streaks. Therefore, understanding how spots interact with streaks and in general evolve in a streaky environment is of interest and is the topic of the current paper.

With the aim of understanding transition caused by multiple discrete roughness elements (such as an insect splatter on an aircraft wing leading edge), Suryanarayan et al. [25] performed direct numerical simulations of the destabilization of low-speed streaks generated by a subcritical roughness element by the wakes of taller discrete roughness elements. It was found that when a subcritical low-speed streak is near a supercritical streak that is naturally becoming unstable via the dominant clumping mode, it gets “infected” and also experiences the same varicose clumping instability. On the other hand, if the subcritical streak is infected by a turbulent wedge, it experiences the sinuous swaying instability. In this backdrop, we aim to understand the mechanism by which a low-speed streak is destabilized by a turbulent spot. While there are similarities in spreading mechanisms between turbulent spots and wedges [16], the spot is a moving disturbance whereas the wedge is pinned by a static roughness and is statistically stationary.

Figure 1 shows a DNS snapshot of a turbulent spot interacting with multiple low-speed streaks. As indicated in Fig. 1, there appears to be evidence that the streaks are destabilized by the spot. This simulation, however, raises several questions including the effect of the streak on the lateral spreading of the spot, the instability mechanisms by which the streak is destabilized, and the upstream (feedback) effects of spots. With the objective of understanding this complex evolution and answering the broad questions on the evolution of turbulent spots in a realistic transition scenario, this paper considers, in detail, the interaction of a turbulent spot, artificially generated in a zero-pressure gradient laminar (Blasius) boundary layer, with a low-speed streak downstream of a subcritical static, discrete roughness element. We first describe the computational setup in Sec. 2, before discussing the simulation results in Sec. 3. After a discussion of the overall flow features and time and ensemble-averaged mean and fluctuating velocity fields in Sec. 3.1, the results are analyzed using time traces of velocity perturbations and control volume evaluations of vorticity production terms in Sec. 3.2. This is followed by a discussion of simulations with nonphysical forcing and shielding strips to further probe the mechanisms and suggest strategies for mitigation of streak instability in the presence of turbulent spots.

The present simulations are performed using an in-house developed immersed boundary (IB) pseudo-spectral DNS code based on the channel flow algorithm of Kim et al. [27] and originally presented in Refs. [28] and [29]. It has been extensively used in roughness-induced transition studies and validated across experiments [30,31]. The streamwise and spanwise directions use Fourier modes whereas the wall-normal direction uses Chebyshev polynomials. The nonlinear terms are evaluated in real space. The time advancement is carried out using Crank–Nicholson for viscous terms and Adams-Bashforth for the nonlinear terms.

where $α$ and $β$ are gain parameters.

The computational setup is shown in Fig. 2. A buffer region is used to generate a quiet Blasius boundary layer with Reynolds number based on the displacement thickness, $Reδ *=U∞δ0*/ν=1460$ at the inflow. Turbulent spots are generated by “temporary” roughness elements with height $k=0.53δ0*$ called the spot generator and streaks are generated by low-amplitude static roughness elements with $k=0.43δ0*$⁠, called the streak generator. Both discrete roughness elements (DREs) are created using IB forces, and have the same planform—they are serrated rectangular prisms inclined at 45 deg with a long edge of approximately $5δ0*$ and a short edge of approximately $2δ0*$⁠. The maximum width measured along the spanwise direction is $4.6δ0*$⁠. The DRE geometry used in the present work is identical to the DRE geometry used in recent roughness-induced boundary layer transition studies [25,30,31]. This geometry was chosen in these previous studies to mimic environmental debris such as insect impacts on aircraft wing leading edges and to facilitate carefully matched experiments. It is chosen in the present work to enable a direct comparison with our previous studies, particularly those on wedge-streak interactions [25].

The streak generator is placed 25.7 $δ0*$ downstream and 13.71 $δ0*$ to the left of the spot generator. The temporary roughness elements are placed at $t=0$ and disappear at $tU∞/δ0*∼10$⁠. The grid size (in 3/2 de-aliased physical space) is 1152 × 128 × 384 and computational domain spans $246.7δ0*×8.22 δ0*×41.12 δ0*$ in streamwise (including the 41.12 $δ0*$ buffer zone), wall-normal and spanwise directions. In real space, the grid is uniformly spaced in the streamwise and spanwise directions but is nonuniform in the wall-normal direction with points clustered close to the wall. The $y+$ of the first grid point from the wall is less than 0.5 for fully turbulent flow. The roughness elements are well-resolved with 472 grid points on the planform and 17 grid points in the wall-normal direction (19 for the spot generator). The grid is shown alongside a sample instantaneous snapshot as an inset in Fig. 2; it can be observed that the grid adequately resolves the near-wall streaks in the turbulent flow. The streamwise extent of the spot development (i.e., the end of the domain from the end of the spot generator) is $186.31 δ0*$⁠.

The spot simulations are initialized with a Blasius solution over the entire domain. For simulations that involve interaction of spots with streaks, a simulation with only the streak generator is first run until it reaches a steady solution, and then the spot is introduced.

The code and setup used in this work have been extensively validated for roughness-induced boundary layer transition across a series of recent studies carried out at the same inflow Reynolds numbers and performed at the same grid resolution. It was found [30] that the statistics on transition location and turbulent wedge spread rates hardly varied when the grid resolution was doubled. Detailed agreement was observed on the flow field near roughness elements between the DNS and water tunnel dye visualization [30]. Favorable agreement existed between DNS and experiments carried out at the Klebanoff-Saric Wind Tunnel at Texas A&M on the effectiveness of transition mitigation strategies [31]. The grid resolution used in the present study is comparable to that used by Strand and Goldstein [15] who used the same code and demonstrated grid convergence for turbulent spots.

In the case of statistically stationary turbulent flows such as a turbulent boundary layer, or a turbulent wedge, long-time averages provide reliable statistics. As a temporally evolving flow with no symmetries (when there is a low-speed streak on one side), ensemble averaging is required. Due to the expense of direct numerical simulations, the number of realizations is usually small, for example, Strand and Goldstein [15] considered a four-member ensemble. In the present study, eight similar but slightly different spots are created by varying the time over which the temporary roughness exists by $±2%$⁠. The difference is small enough to not alter the size, overall shape, or virtual origin of the resulting turbulent spot, but is adequate to create a distinct “realization” with different small-scale turbulence, as seen in Fig. 3.

The present ensemble size ensures that statistics that represent the overall flow features such as time-averaged mean and RMS velocity are nearly converged. For example, the peak $urms′$ within the streak at $x/δ0*=57.4$ from the spot generator only changes slightly from 8.2% to 8.3% when the ensemble size is increased from three to eight. Vorticity fluxes computed at the edge of the spot, on the other hand, have a larger variability across realizations based on the exact width of the spot achieved in that realization, and should not be thus considered to provide a precise quantitative estimate, but nevertheless provide useful insights on the nature of the underlying mechanisms.

The left panel of Fig. 4 shows the evolution of the spot through the undisturbed laminar boundary layer. The spot attains the characteristic arrowhead shape within the first quarter of the domain, continues to grow as it moves downstream, and eventually fills up the spanwise extent of the domain at $tU∞/δ0*=246.71$ (Fig. 4(a3)). It is important to note that due to the periodicity in the spanwise direction, the results should perhaps be interpreted as an evolution of a periodic (spanwise) array of spots that eventually merge and form a turbulent boundary layer. Regardless, this study will focus on the evolution and interaction of the spot with the streak primarily before $tU∞/δ0* ≲200$⁠, or those interactions occurring within the first half of the streamwise extent of the domain, and thus the effect of the finite span of the simulations is not expected to affect the present analysis. As the spot appears to achieve a mature arrowhead shape at the time it interacts with the streak, the shape of the spot generator is unlikely to affect the results. While the detailed shape of the streak generator is unlikely to affect the insights drawn in this study, the effect of the flatness of the streak generator will be explored in future work.

The right panel of Fig. 4 shows the interaction of the spot with the streak once the second static DRE has been emplaced. At early times (Fig. 4(b1)) the streak is not significantly influenced by the presence of the spot, but as the spot grows laterally, its associated velocity perturbations begin to influence the streak (Fig. 4(b2)). It can be observed that at this stage the streak experiences side-to-side sinuous perturbation (inferred from the u-isosurface over the streak in Fig. 4(b2), around the location where the streak is closest to the spot) as well as a hint of shorter wavelength (higher frequency) perturbations (seen in the $ωx$ contours). The spot does not appear to be greatly affected by the streak—i.e., there does not appear to be a significant enhancement in its spanwise spread due to the streak destabilization. At longer times (Fig. 4(b3)), once the spot has grown into and through the streak it can be observed to continue destabilizing the streak upstream, and these instabilities appear to be of the shorter-wavelength kind. As a result, while the spot does not appear to spread faster, its influence at a given streamwise location may last longer in the presence of streaks.

To quantify these qualitative observations, we first perform time-averages from $t=0 to 185.03 δ0*/U∞.$ As the spot moves and grows in all spatial directions with time, this kind of averaging must be carefully interpreted—more like a long exposure photograph than traditional turbulent statistics. However, this viewpoint provides a connection with turbulent wedge evolution and provides a measure of the spanwise spread of the spot as it evolves downstream. The time-averages are then used to compute fluctuations and then are ensemble averaged. To prevent any slight differences in the spot initiation across the different realizations contributing to the fluctuations, $urms′$ is first computed for each realization separately based on the time-average for that case. The $urms′$ fields computed for each case are then ensemble averaged. This process is then repeated for the cases with the low-speed streak, and the results are compared in Fig. 5. Streamwise velocity fluctuations are used in the present analysis as they are a potentially reliable transition onset marker [32].

The spot does not just spread into the streak, but destabilizes the streak at a distance, as evidenced by the nonzero $urms′$ occurring within the streak before the $urms′$ of the spot reaches this location. Examining the $urms′$ contours in detail, it can be seen that the fluctuations primarily appear on the side shear layers, i.e., regions of large $∂u/∂z$⁠. This is consistent with the qualitative observations in Fig. 4 that the streak is destabilized, at least initially, via the long wavelength sinuous instability which is associated with the side shear layers. The mechanism of the initial destabilization of the streak by a turbulent spot therefore appears to be similar to that caused by the turbulent wedge [25].

From the contours of the time averaged wall shear stress (Fig. 5), it can be seen that the spot appears to spread linearly with no significant change in spread rate due to the presence of the streak. Using a threshold value of 1.5 times the laminar wall shear stress to identify the spot edge, the spot was observed to spread linearly with a half-angle of 10 degrees between $x≈30δ0*$ to $90δ0*$ with or without the streak. While larger ensemble sizes and a bigger flow development length are required to ascertain the spread rate more accurately, the currently measured value is within the experimentally quoted range [7]. Beyond $x ≳100δ0*$⁠, the spot merges with the destabilized streak.

Figure 6(b) shows the time evolution of the velocity fluctuations at two probes on the low-speed streak. The locations of the probes are shown in Fig. 6(a). The timescales associated with the two instability modes are calculated from the linear stability theory and experimental results [24] for the low-speed streak in the wake of a discrete roughness element of similar shape and are indicated in Fig. 6(b) in a novel way: The lower frequency “swaying” mode is indicated by the spacing of the pink lines while the higher frequency “clumping” mode is indicated by the vertical green lines. It can be seen that the spot primarily forces the low-frequency mode. The streak's response contains both timescales but is dominated by the lower frequency swaying mode that is forced by the spot and greatly amplified compared to the isolated spot case.

These results, taken along with the contours of $urms′$ in Fig. 5, demonstrate that the spot initially destabilizes the streak by triggering the low-frequency subdominant sinuous swaying mode associated with the side shear layers, similar to a destabilization of a streak by a turbulent wedge [25].

Streamwise vortices play an important role in transition and streak instability [30]. In particular, both the clumping and swaying instability of the streak generate fluctuating streamwise vorticity that amplifies downstream, ultimately causing breakdown of the streak. While the streak itself is in part generated via lift-up created by a streamwise vortex in the near wake of the roughness element, this steady vortex is nearly dissipated at the location where the spot begins to perturb the streak. Thus, any streamwise vorticity observed in the vicinity of the streak during the streak-spot interaction is either due to the instability of the streak or is the streamwise vorticity of the spot itself as it spreads into the streak. To distinguish between the two possibilities, we study the evolution of $ωx2$ integrated over a small control volume (shown by the dashed lines in Fig. 6(a)) in a spatial location around where the streak would be present in both the isolated spot and the spot with streak cases. This is shown in the top panel of Fig. 6(c), where it can be seen that the streamwise vorticity in this volume persists for much longer when the streak is present confirming the qualitative observations of Fig. 4.

Figure 6(c) compares the evolution of the production terms of $ωx2$⁠. It can be observed that for the case with the low-speed streak, the integrated $ωx2$ tends to increase earlier, consistent with the observation that the spot is destabilizing the streak before reaching it. The integrated $ωx2$ also persists longer compared to the case without the streak suggesting that the streak continues to remain unstable for 50 to $100 δ0*/U∞$ after the spot has passed the control volume. The peak value of the integrated $ωx2$ is slightly larger for the case with the streak, but this is within the uncertainty of the ensemble averaging (this specific statistic is very sensitive to the precise location of the edge of the spot causing considerable variation from one realization to another).

The source term $ωxωz∂w/∂x$ is clearly dominant at early and late times for the case with the low-speed streak, suggesting that it is the term associated with the streak destabilization. The production term $ωxωz∂w/∂x$ can be interpreted as the generation of streamwise vorticity by the tilting of vortex lines due to the variation of the spanwise velocity with streamwise distance, consistent with the swaying mechanism.

The interaction between the spot and the streak may be expected to depend on the distance between them. To shed light on this dependence, we repeat the spot-streak interaction simulation with a “fence”—a z-normal plane on which spanwise velocity is enforced to be 0 using the immersed boundary forces. (No forces are applied in the streamwise or wall-normal directions, so the fence acts like a slip surface.) The fence is placed approximately midway between the spot and streak generators. The fence does not have a significant effect on the streak itself as evidenced (Fig. 7(Right)) by the mean velocity profile immediately downstream of the streak generator being nearly identical to the case without the fence. The fence, however, greatly reduces “communication” between the spot and the streak and suppresses the spot evolution to the left of the plane. Note that the streak can still be perturbed from its left by the right edge of spot (i.e., the spot in the next spanwise period) due to the periodic boundary conditions in z. However, the distance of this periodic interaction is over four times the distance between the left edge of the spot in the computational domain and the streak. The results of this simulation are shown in Fig. 7. It can be seen at $tU∞/δ0*=185$⁠, the streamwise vorticity perturbations on the streak (Fig. 7(Left)) are hardly noticeable in the same colorbar as the base simulation. The analysis of the velocity perturbations shows that the streak still undergoes the same swaying instability, but the amplitude of the perturbations is significantly weaker. For example, the maximum $urms′$ within the streak in the third cut plane (⁠$x′/δ0*=57.4$⁠) is 8.3% in the case without the fence and is 4.1% in the case where the fence is present. These numbers must be cautiously interpreted as there is an additional effect of the displacement of fluid in the spot traveling at a different speed compared to the surrounding laminar flow in a finite spanwise domain.

Regardless, this simulation suggests that reducing the spreading rate of spots, such as by using riblets [15] may have the additional benefit of reducing their interaction with neighboring streaks.

A second mitigation strategy involves reducing the streak amplitude before it encounters the spot. If the streaks are generated by surface roughness, the use of shielding strips [31] has been demonstrated to be effective in reducing the streak amplitude. Shielding strips are low amplitude flat strips or distributed roughness that work by disrupting the steady streamwise vortices generated by the roughness elements. Since the streaks are primarily generated by the lift-up of the spanwise vorticity by the DRE-caused streamwise vortex, the resulting streak amplitude is also reduced. While the target application of shielding strips is the mitigation of transition induced by large roughness elements, we explore another circumstance in which they may be useful. Figure 8 shows the disruptive effect of a finite span shielding strip (of height 0.119 $δ0*)$ on the steady streamwise vortex generated by the static roughness (streak generator) and thus the strip's mitigating effect on the streak amplitude. The interaction of the turbulent spot with this shielded streak is shown in Fig. 9. The perturbation amplitudes are slightly reduced compared to the unshielded streak (e.g., maximum $urms′$ within the streak in the third cut plane is reduced from 8.3% for the unshielded case to 7.5% for the shielded case). While these differences are modest, they suggest pathways of mitigation of the spot-streak interactions: Either keep the spot away from the streak, weaken the streak, or damp the laterally spreading disturbances caused by the spot.

The interaction between a turbulent spot and a (subcritical) low-speed streak is studied in detail using direct numerical simulations. For the configurations and parameters considered, the spot is observed to destabilize the streak at a distance, before eventually spreading into it. The spot is observed to primarily force the subdominant low-frequency sinuous swaying instability of the streak during the early stage of its destabilization. The spread rate of the spot does not appear to be significantly enhanced by the destabilization of the neighboring streak in the cases examined, as the spot spreads into the streak before the streak completely breaks down for the specific case considered; the effects of streak amplitude, spacing, and Reynolds number are yet to be examined. It is clearly observed, however, that the unsteadiness lasts a lot longer at a given streamwise location when a low-speed streak is present about the spot—the streak continues to remain unstable and generate unsteady streamwise vorticity even after the spot has passed the location. The engineering outcome of this interaction would be a larger intermittency factor and thus a greater overall skin friction and heat transfer than predictions based on spots evolving in quiescent environments. Results suggest that suppression of spot spreading and reduction of streak amplitude using shielding strips are promising strategies to mitigate this interaction.

We thank the Ohio Supercomputer Center (OSC) and the Texas Advanced Computing Center (TACC) for supercomputing resources. We thank our collaborators Professor Garry Brown (Princeton University) and Professor Edward White (Texas A & M University) and group for discussions.

• The U.S. Air Force Office of Scientific Research (AFOSR) (Grant No. FA9550-19-1-0145; Funder ID: 10.13039/100000181).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

k =

height of roughness element, m

$Reδ0*$ =

Reynolds number based on the displacement thickness

t =

time, s

u,v,w =

instantaneous streamwise, wall-normal and spanwise velocities, m/s

$U¯$ =

time and ensemble averaged streamwise velocity, m/s

$U∞$ =

free stream velocity, m/s

$urms′$ =

root-mean-squared fluctuation intensity of the streamwise velocity, m/s

x, y, z =

streamwise, wall-normal, and spanwise displacements, m

x' =

streamwise distance measured from the trailing edge of the spot generator

$δ0*$ =

displacement thickness of the boundary layer at the inlet, m

$τw$ =

wall shear stress = $μ∂U¯∂y$⁠, N/m²

$τw0$ =

wall shear stress of the undisturbed Blasius boundary layer at inlet, N/m²

$ωx, ωy, ωz$ =

streamwise, wall-normal, and spanwise vorticity, 1/s

$ωz0$ =

spanwise vorticity at the wall of the undisturbed Blasius boundary layer at the inlet = 0.56 $U∞/δ0*$⁠, 1/s