Influence of localised smooth steps on the instability of a boundary layer

We consider a smooth forward facing step defined by the Gauss error function of height 4-30\% and four times the width of the local boundary layer thickness $\delta_{99}$. The boundary layer flow over a smooth forward-facing stepped plate is studied with particular emphasis on stabilisation and destabilisation of the Tollmien-Schlichting (TS) waves and subsequently on transition. The interaction between TS waves at a range of frequencies and a base flow over a single/two forward facing smooth steps is conducted by linear analysis. The results indicate that for a high frequency TS wave, the amplitude of the TS wave is attenuated in the unstable regime of the neutral stability curve corresponding to a flat plate boundary layer. Furthermore, it is observed that two smooth forward facing steps lead to a more acute reduction of the amplitude of the TS wave. When the height of a step is increased to more than 20\% of the local boundary layer thickness for a fixed width parameter, the TS wave is amplified and thereby a destabilisation effect is introduced. Therefore, stabilisation or destabilisation effect of a smooth step is typically dependent on its shape parameters. To validate the results of the linear stability analysis, where a high-frequency TS wave is damped by the forward facing smooth steps direct numerical simulation (DNS) is performed. The results of the DNS correlate favorably with the linear analysis and show that for the investigated high frequency TS wave, the K-type transition process is altered whereas the onset of the H-type transition is postponed. The results of the DNS suggest that for a high-frequency perturbation $\mathcal{F}=150$ and in the absence of other external perturbations, two forward facing steps of height 5\% and 12\% of the boundary layer thickness delayed H-type transition scenario and completely suppresses it for the K-type transition.


Introduction
In environments with low levels of disturbances, transition to turbulence is initiated by the exponential amplification of the Tollmien-Schlichting (T-S) waves followed by the growth of secondary instabilities. Breakdown to turbulence generally occurs when the amplitude of the primary instability is of the order of 10% of the free-stream velocity magnitude (Herbert 1988;Cossu & Brandt 2002). The classical process of laminarturbulent transition is subdivided into three stages: receptivity, linear eigenmode growth and non-linear breakdown to turbulence. A long-standing goal of laminar flow control (LFC) is the development of drag-reduction mechanisms for delaying transition. If the growth of the T-S waves is reduced or completely suppressed and providing no other instability mechanism comes into play, transition could be postponed or even eliminated (Davies & Carpenter 1996).
Over the past two decades, investigations into the control of the laminar to turbulent transition of the boundary layer have focused on controlling the T-S wave. The methods proposed can be classified as passive, active open-loop and active closed-loop control. Active open-loop control is achieved via powered actuation and closed-loop (feedback) control builds on any open-loop control strategy by informing actuation through flow field sensors. Passive flow control strategies do not need a power input and are achieved via physical changes to the geometry. Therefore passive control is generally considered easier and more cost-effective to implement in practice. Passive boundary layer flow control strategies are most commonly investigated by modifying the surface of a flat plate. Boiko et al. (1994) found that, in the presence of unsteady and randomly distributed streaks, the T-S waves are stabilised in a mean flow when their amplitudes are small. If the streaks reach a critical amplitude, bypass transition is triggered (Matsubara & Alfredsson 2001). Numerical simulations and stability analysis were subsequently conducted to address the effect of streaks on the T-S waves. Cossu & Brandt (2002 found that narrow span-wise modulations of the boundary layer thickness, the so-called streak, can stabilise the low amplitude T-S waves. Experimentally, Fransson et al. (2004Fransson et al. ( , 2005 observed that steady and stable streaks of moderate amplitude can be induced by generating nearly optimal vortices with cylindrical roughness elements placed near the leading edge. They also proved that these streaks have a stabilising effect on the T-S waves. Correspondingly, more evidence that well-designed roughness can reduce the viscous drag was offered by Fransson et al. (2006). For all these mechanisms, the transition delay can be attributed to the reduction of the exponential growth of the T-S waves in the presence of streaks, and to the absence of strongly destabilising non-linear interaction between two types of perturbations. It should be stressed that none of them, except Fransson et al. (2006), has successfully reported a delay in transition.
For a two-dimensional (2D) flat-plate boundary layer, Wörner et al. (2003) predicted that the amplitude of the T-S wave is reduced by a forward-facing step because a thinner boundary layer evolving on the step is more stable than the thicker boundary layer on a flat plate. The development of two different boundary layers gives rise to an essential change of the boundary layer shear property compared with a flat plate boundary layer. In order to achieve this change, the height of the step normalized by the local boundarylayer displacement thickness δ * was 0.47. The existence of a sudden jump of the wall profile because of the size of the step, creates a separation zone in front of it. Because this separation zone is very small Wörner et al. (2003) assumed it has no influence on the T-S wave growth. However passive flow control methods should guarantee that bypass transition is not activated by the instability of the streak or the receptivity of an incoming T-S wave by the roughness array. Experimentally, the low-roughness surface finish found on some aircraft can be considered a simple passive drag-reducing device, as transition is significantly advanced by surface roughness in some scenarios (Saric et al. 2011). Shahinfar et al. (2012 showed that classical vortex generators, known for their efficiency in delaying or even inhibiting boundary layer separation, are really effective in delaying transition. However, in the presence of miniature vortex generators (MVGs), the T-S waves are being generated upstream of the device, leaving the full receptivity process of the incoming wave on the MVG array, suggesting this method may not delay transition. Downs & Fransson (2014) found that T-S wave amplitudes over spanwiseperiodic surface patterns can be reduced and demonstrated substantial delays in the onset of transition when T-S waves are forced with large amplitudes. Although discrete roughness elements such as these can delay transition in a boundary layer dominated by cross flow or streamwise instability (Saric et al. 2011), there exist a detrimental influence of roughness-induced disturbances on the boundary layer stability in many flows. Beyond the strategies discussed here, there exist several others for stabilising the T-S waves which have been studied in the past such as wall suction (Joslin 1988), Lorentz force actuators (Albrecht et al. 2006), plasma actuators (Riherd & Roy 2013). The interested reader can refer to papers of a theme issue compiled and edited by Leschziner et al. (2011).
Passive flow control strategies rely on the introduction of surface roughness elements or local modifications of surface geometry. Because of the disturbances present in the free-stream there exists a receptivity problem (Gaster 1965;Murdock 1980;Goldstein 1983;Kerschen 1989Kerschen , 1990Dietz 1999;Wu 2001;Saric et al. 2002). Practically, the receptivity problem cannot be avoided for these passive flow control strategies. In this paper, we firstly consider a single smooth step to stabilise the T-S waves and avoid both re-circulation and an essential change of shear property. We do not address the receptivity problem induced by the smooth steps in this paper. For the parameters studied, the single smooth step or two smooth steps are located in the unstable regime of the neutral curve of the corresponding flat plate boundary layer. Futher investigation could allow to mitigate the receptivity issue from the smooth steps, for typical free-stream conditions, by choosing appropriate parameters for their geometric definition. To validate the idea, the linear analysis of the classical Gaussian hump is investigated because it was believed that it has a stabilisation effect on the T-S waves (Wu & Hogg 2006). The linear analysis shows that in the presence of a single smooth step, there exist several free parameters which can be adjusted to gain a significant performance of attenuating the T-S waves. The single step height playing a key role in stabilising the T-S waves. Further, the investigation on two isolated smooth steps indicates that a significant reduction of T-S waves' amplitude can be obtained compared with a single smooth step when the same step parameters except for locations are used for individual steps. Finally, the effect of smooth steps on transition is studied by direct numerical simulation (DNS) for K-and H-type transition scenarios. The results show that K-transition can be stabilised and H-type transition cannot be stabilised but can be delayed.
The paper is organised as follows. In §2, we give fundamental definitions and numerical details. In §3, we show the results of linear analysis on the reduction of the T-S wave amplitudes and subsequently verify the transition delay by DNS. A further discussion is then given in §4 and subsequently, the work is concluded.

Numerical details
Nektar++ is used to implement numerical calculations of the fully nonlinear Navier-Stokes equations and the linearised Navier-Stokes equations. A stiffly stable splitting scheme, developed by is adopted which decouples the velocity and pressure fields and time integration is achieved by a second-order accurate implicit-explicit (IMEX) scheme (Karniadakis et al. 1991;Cantwell et al. 2015). A convergence study for h-and p-type refinement was performed to guarantee DNS resolution is achieved by the mesh used throughout this study.
High order curved elements are adopted to smoothly approximate the roughness elements by means of an analytical mapping. The governing equations are then discretized in each curved element by seventh-order polynomials. Once steady base flows are generated by using the non-linear Navier-Stokes equations (NSEs), the T-S waves are simulated by the linearised Navier-Stokes equations (LNSEs). As discussed below, for base flow generation, the inlet position is located sufficiently far from the first step in order to allow the base flows to recover the Blasius profile. For simplicity, the T-S waves are excited by periodic suction and blowing on wall. The convergence tolerance of the base flow generation is defined by where · 0 means the standard L 2 norm, ∂ d t denotes the discrete temporal derivative and T c is the convective time scale. A hybrid Fourier-Spectral/hp discretisation is used to discretize the 3D incompressible Navier-Stokes equations with spectral/hp elements in the stream-wise and wall-normal direction and Fourrier modes in the spanwise, periodic, direction.

Definitions
Motivated by the triple deck scaling of an incompressible boundary layer (Stewartson & Williams 1969;Messiter 1970;Neiland 1969), we introduce the following scales for the roughness elements: where x c is the centre position of roughness elements and the Reynolds number Re xc is defined by the characteristic distance from the leading edge to x c . The classical Gaussian hump and the smooth forward facing step are defined by the following expressions, respectively, whered andĥ(> 0) are the streamwise width scale and the normal direction length scale, which are defined by the corresponding physical scales d and h as followŝ For multiple-steps, the wall profile is defined formally by where X i is the centre position of each individual step and n is the number of steps.

Mathematical formulations
In order to address growth and decaying properties of the T-S waves, we solve the following 2D LNSEs whereũ andū are the perturbed velocity vector and the base flow velocity vectors respectively andp is the perturbed kinematic pressure. The above linearised equations have been non-dimensionlised with respect to the free stream velocity U ∞ and the displacement thickness of the Blasius boundary layer, δ * . For convenience, we still useū andũ to denote the non-dimensional base flow field and the perturbed velocity field. Then, without humps or steps, under the assumption of streamwise parallel flow in two dimensions, the perturbation assumes the normal form where α denotes growth rate and ω is the frequency. By a coordinate transformation (2.1), the waves can now be described locally byũ(X, η, t). Generally, for an unstable frequency ω ∈ R + , the T-S wave envelope is defined by the absolute maximum amplitude of the T-S wave as follows With the similar definition, the distorted T-S wave envelopes for two types of imperfections are denoted by the following notations, respectively For a flat plate boundary layer, let A max f (x) denote the absolute maximum amplitude Table 1. Parameters used for the linear analysis of the Gaussian hump where Re δ i * and Re δ c * are, respectively the inlet Reynolds number and the Reynolds number at the center of the roughness element. F denotes the non-dimensional perturbation frequency,ĥ is the wall-normal length scale of the roughness with respect to an inner deck scaling and h/δ99(%) the height of the roughness element with respect to boundary layer thickness at Re δ * = 1183.33. Finally Lx and Ly denote streamwise extent and height of the domain for which the 2D base flow field u obtained was independent of domain size. of the T-S wave. In order to quantify the difference between A max f (x) and A max g (X) or A max s (X), the following formulae are introduced ("#" denotes g or e) Similarly, in order to investigate behaviours of shear stress distribution around the imperfections, we introduce the following shear stress notations (2.10) Correspondingly the deviation of shear stress from flat plate conditions can be expressed as

Behaviour of shear stress distributions
In figure 1, the profiles of τ * g (X) and τ * g (X)/ĥ are shown against X for different values of h and the parameters in the calculations are given in Table 1. It is observed that higher h induces larger deviation of τ * g (X) from zero in figure 1(a) promoting up to a 20% deviation from the flat plate conditions. Further, in figure 1(b), the collapse of the curves τ * g (X)/ĥ indicates the linear relation between τ * g (X) andĥ for this value of d = λ T S /2. It is worth pointing out that for short humps, the linear relation does not hold exactly . For steps, the similar phenomena can be observed for the profiles of τ * s (X) and τ * s (X)/ĥ in figure 2. With the same scalesĥ and d, figure 3(a) indicates that a Gaussian hump has a more significant effect on the boundary layer shear stress profile and behind the hump, there exists a regime where τ * g (X) has a reduction. For the smooth forward facing step, the shear stress does not have a significant reduction. In figure 3(b), we compare τ * s (X) for cases with fixed heightĥ and varying step width It is worth mentioning that the boundary layer over a smooth forward-facing step can recover the flat plate boundary layer further down stream. Therefore, contrary to a sharp forward facing step (Wörner et al. 2003), the smooth forward facing step does not create a new boundary layer but rather locally decreases the thickness of the boundary layer.

Comparison between single hump and single step
The analysis of Wu & Hogg (2006) predicts that a surface hump may 'stabilise' the T-S waves. In figure 4, we shows the profiles of T * g (X) and T * s (X). We observe in figure 4(a) that although the T-S waves are stabilised within a small regime behind humps for small h, the amplitudes of the T-S waves are amplified further downstream and the overall effect is dominated by the destabilisation. Reducing hump height scaleĥ can produce stabilisation effect, but the stabilisation effect is almost negligible . Figure  4(b) shows that the T-S waves are stabilised for a large downstream distance when considering the smooth steps. Surprisingly, largeĥ results in significant reduction of the T-S waves on the step. In figure 5(a), a comparison of the envelope of the T-S waves for a hump and a step are also given. We also observe that the stabilisation effect of a step is more significant than that of a hump. A further parametric study on varying width scale d in figure 5(b) indicates that with fixed height scaleĥ, increasing width scale d gives rise to an extension of the downstream stabilisation regime and there exists a significant reduction of the amplitudes of the locally amplified T-S waves in front of the step.

Stabilisation effect of two steps
The results of the preceding section are focused on a low frequency, which give us a fundamental understanding of the stabilisation property of a step in a wide unstable regime of the T-S wave with regard to the neutral stability curve. With the understanding of the step stabilisation property, we consider comparisons between one step and two steps for a higher frequency where the corresponding parameters are given in Table 2. In figure  6, the normalised T-S waves profiles are shown with respect to different step height scales for single-step and the two-step configuration. We observe in figure 6(a) that for a single step located at Re δ c 1 * , increasing height of the step results in a notable attenuation of the T-S wave amplitudes on the step. Figure 6 Table 2. Parameters used for the linear analysis of the two forward facing smooth steps with Re δ i * the inlet Reynolds number. F denotes the non-dimensional perturbation frequency, Re δ c 1 * and Re δ c 2 * denote centre Reynolds numbers of the steps and δ99 is defined at the centre position of the first step at Re δ c 1 * . Lx and Ly denote streamwise extent of the domain and its height, respectively. For single step, the step is located at Re δ c 1 * . Corresponding to Re δ c 1 * and Re δ c 2 * , Rex 1 /10 5 and Rex 2 /10 5 are equal to 1.56 and 2.08, respectively. on two steps, the amplitudes of the T-S waves have a large reduction on the second step compared with a single step. A clear comparison is given in figure 7. From figures 6 and 7, we also observe that the maximum position of the T-S wave envelops has a slight shift to downstream.
By linear analyses, stabilisation effect of two steps is potentially significant which provides a possible strategy to stabilise the boundary layers and delay transition. In next section, the steps' effect on the boundary layer stabilisation and the transition delay is validated by DNS.

Destabilisation effect induced by steps
For a single step (figures 4 and 5) there exists a small destabilisation effect which makes T * s (X) locally positive. However, this destabilisation effect is negligible in the linear growth regime of the T-S wave, because downstream of the step the amplitude of the envelope of the T-S wave for the smooth steps is smaller than that of the flat plate boundary layer (see figures 6(a) and 7). For two smooth steps, an appropriate choice of the position of the second step prevents any extra destabilisation (see figures 6(b) and 7). Therefore, in a linear growth regime of the T-S wave, the amplitude of the T-S wave can always be attenuated by smooth steps provided a proper design of steps is adopted.

Numerical configurations
A hybrid Fourier-Spectral/hp discretisation was used to solve 3D Incompressible Navier-Stokes equations. The spanwise direction was assumed to be periodic and discretized by 64 Fourier modes and the streamwise and wall normal plane was discretized using 5576 elements (quad and triangle) within which a polynomial expansion of degree seven was imposed. K-and H-type transitions are investigated for the flow over a flat plate and with smooth steps. A Blasius profile was imposed at the inflow and, for both transition scenarios, a wall-normal velocity along the disturbance strip is prescribed by the blowing and suction boundary condition (Huai et al. 1997) ,  Table 3. Parameters used for the DNS simulation of both K-and H-type transition scenarios with the inlet Reynolds number Re δ i * , the Reynolds number at the center of each step Re δ c 1,2 * , the non-dimensional perturbation frequencies of the disturbance strip FA and FB, the relative amplitudes of the disturbance amplitudes of the fundamental A/U∞ and oblique waves B/U∞, the height scaleĥ of the steps and their heigh with respect to the boundary layer thickness h/δ99(%). The streamwise Lx, height Ly and and spanwise Lz extent of the domain is expressed as function of the boundary layer thickness δ99 defined at the centre position of the first step at Re δ c 1 * .
where ω A and ω B are the frequencies of the 2D T-S wave and the oblique waves, respectively. A and B are the disturbance amplitudes of the fundamental and the oblique waves. The function f (x) is defined by (Fasel & Konzelmann 1990) f (x) = 15.1875ξ 5 − 35.4375ξ 4 + 20.25ξ 3 , where x m = (x 1 + x 2 )/2 and g(z) = cos(2πz/λ z ) with the spanwise wavelength λ z . At x 1 and x 2 , Re δ x 1 * and Re δ x 2 * are equal to 591.37 and 608.51, respectively. For Ktype transition, the oblique waves have the same frequency as the two dimensional wave (ω A = ω B ). Also, for H-type transition, the oblique waves are sub-harmonic (ω B = ω A /2). All parameters used in the investigation are given in Table 3. Note that the frequency used for the perturbation for the forward facing smooth steps was higher than that for the linear analysis of the Gaussian hump so that the transition occurred sooner resulting in a more tractable mesh. The schematic computational domain is illustrated in figure 8.

Inhibition of the K-and delay of the H-type transition.
The validation of the DNS results for both K-and H-type transitions is corroborated by recovering the aligned arrangement of the Λ vortices for the K-type transition (see figure 9 (a)) and staggered arrangement for the H-type transition (see figure 9 (b)) as experimentally observed by Berlin et al. (1999) for the flat plate boundary layer. Figure 10 shows the evolution of the skin-friction coefficient versus Re x for two different normalized heightsĥ = 1.2 andĥ = 2.8 of the smooth steps. We observe that, for a flat plate boundary layer, the skin friction coefficient diverges from that of the Blasius boundary layer at the location where Λ vortices appear, as illustrated in figure 9. Additionally, figure 9 indicates that the streamwise passing period of Λ vortices in H-type transitions is twice that observed in K-type transitions (Herbert 1988). The streamwise evolution of the skin-friction (see figure 10) shows the K-type transition is fully stabilised by the two smooth steps whereas the H-type transition is only delayed. Additionally, increasing the height h of the smooth steps further reduces the skin friction coefficient C f in both transition scenarios. The observation of transition delay by the two smooth steps is supported by the results from the linear analysis.
To gain further insight into the differences between the two transition scenarios we consider the notation (ω, β) (Berlin et al. 1999), where ω and β are respectively, the frequency and spanwise wavenumber each normalized by the corresponding fundamental frequency/wavenumber. It has been observed that the K-type transition scenario has the main initial energy in the (1, 0) mode. The (1, ±1) mode also generates the (0, ±2) mode with a small amplitude through non-linear interaction (Berlin et al. 1999). At the late stage, the (0, ±2) mode can grow to an amplitude comparable to that of the (0, ±1) mode. Laurien & Kleiser (1989) and Berlin et al. (1999) have shown that the initial conditions for the H-type transition has the main energy in the (1, 0) mode with a small amount in the oblique subharmonic (1/2, ±1) mode. The important mode is the vortex-streak (0, ±2) mode, which is nonlinearly generated by the subharmonic mode and vital in the transition process. Linear analysis supports the concept that the Ktype transitions can be stabilised on smooth steps because the T-S wave, (1, 0) mode, amplitude is strongly damped. For the H-type transition scenario, although the T-S wave amplitude is attenuated by the two smooth steps, the (0, ±2) mode still plays a key role in initialising transition. Figure 11 shows the time-averaged streamwise velocity structures for K-and H-type scenarios with two smooth steps. We observe the different time-averaged flow structure patterns. The difference of the energy in the (0,1) mode for both the K-and H-type scenarios leads to these different patterns. As illustrated in figure 12, the (0, 2) mode plays a significant role in the late stages of transition for both transition scenarios with the two smooth steps. For the K-type scenario, in the transition regime, the energy of mode (0, 2) grows and exceeds the energy of mode (0, 1). For the flat plate, the energy of mode (0,1) finally grows again until turbulence occurs. The energy of mode (0, 2) with the two smooth steps is less than that of mode (0, 2) for the flat-plate and larger normalised smooth step heightĥ gives rise to stronger reduction of the energy. A similar reduction in energy is also observed for mode (0,1). Furthermore, the energy of mode (0,2) exceeds that of mode (0,1) and from this points onwards the energy of mode (0,1) decays. Based on the results presented in figure 12 (a), we observe that the spanwise modulation induced by mode (0,1) with energy decaying on the smooth steps is attributed to the stabilisation of the boundary layer. For the H-type scenario, the energy of mode (0, 2) grows until turbulence occurs. Transition delay for the H-type scenario is due to the energy reduction of mode (1,0) and the subsequent energy reduction of modes (0,2). Figure 13 shows the slices of streamwise velocity in yz-planes at different streamwise locations. We observe that on the two smooth steps, streamwise streaks for the H-type transition scenario become unstable but those for the K-type transition scenario are stable. Figure 14 indicates the shapes and magnitudes for streaks for the K-type scenario do not have a significant variation downstream of the two smooth steps. In figures 15 and 16, the instantaneous contours of streamwise velocity in yz-planes are compared against the flat plate conditions for the K-and H-type scenarios. For the boundary layer on a flat plate, with the same input disturbance energy, figures 15(a f − a f ) and 16(a f − a f ) show the H-type transition occurs earlier than the K-type transition. For the two smooth steps, because of the energy reduction of (1, 0) mode, figure 15(d s ) shows that the non-linear breakdown of streaks is inhibited and figure 16(d s ) shows that the non-linear interaction is weakened downstream but not inhibited. In summary, the DNS simulation shows that transition to turbulence can be avoided in the K-type scenario and delayed in the H-type scenario. In both cases, the overall effect of the two smooth steps is a net reduction in drag.

Discussion and conclusion
In this paper, we have investigated the delay of the onset of transition by passively controlling the boundary layer with smooth steps. For the K-type transition scenarios, the reduction of the primary T-S mode lead to a significant impact on the laminar-turbulent   Figure 13. Time-averaged contours of stream-wise velocity in yz-planes at streamwise locations Rex/10 5 = {3(a), 4(b), 5(c), 6(d)} for the K-(a k -d k ) and H-type transition scenario (a h -d h ) for a step height ofĥ = 1.2. Figure 14. Time-averaged contours of stream-wise velocity in xz-plane at height y = 2h + 0.6δ i 99 and Rex/10 5 ∈ [4.9, 6.3] for the K-type transition scenario with step height h = 1.2.
(as) (bs) (cs) (ds) Figure 15. Comparison of instantaneous contours of streamwise velocity in yz-planes at streamwise locations Rex/10 5 = {3(a), 4(b), 5(c), 6(d)} for the K-type transition scenario between the flat plate (a f -d f ) and two forward facing smooth steps (as-ds) of height scaleĥ = 1.2. transition. For the H-type transition scenarios, the onset of transition is dominated by (0, ±2) modes and the reduction of the primary T-S modes gives rise to a weakening of the non-linear interaction and the energy reduction of (0, ±2) modes, thereby delaying transition.
The three main findings of the present investigation are: a)Despite the mechanisms being different to that of a sharp step Wörner et al. (2003), linear stability analysis shows the unstable T-S waves are stabilised when a smooth step is introduced in the boundary layer. For a single step, the reduction of the amplitudes of the T-S wave depends on its width and height scales. Higher steps result in a significant reduction of the amplitude of the T-S waves and larger step width enlarges the extend of the stabilisation region downstream of the smooth step. b)For two smooth steps, the overall stabilisation effect is significantly greater than that for a single smooth step. The maximum amplitude of the TS wave is reduced by more than 15% with respect to the flat plate (Fig. 7).
c)The boundary layer stabilisation and transition delay for both K-and H-type transition scenarios, investigated by DNS, support the linear analyses. For the parameters used in our investigations, the DNS results show the smooth steps completely stabilise the K-type transition whereas the H-type transition is only delayed (Figs. 9-11).
In view of these three findings, we believe smooth steps could be leveraged as a passive flow control strategy to delay natural transition. However, in future studies, it will be necessary to address the receptivity issue induced by the steps and adjust the flexible parameters of their geometry to reduce or eliminate the receptivity problem in typical free-stream conditions.
Smooth steps are not the only means of stabilising unstable modes in the boundary layer. The streamwise wall curvature variation could give rise to an important impact on instability/stability of the boundary layer. On a wavy wall, Hall (2013) predicts theoretically for inviscid modes that "In the inviscid limit all boundary layers are unstable in the presence of arbitrary small modulation of the curvature". However, the effect of wall waviness on other modes , for example, the T-S and Klebanoff modes, remains unknown, especially in the transition regime. Downs & Fransson (2014) found that for a wavy wall, formed by streamwise-elongated humps which are regularly spaced in the spanwise direction, growth rates of the forced disturbances are amplified or damped depending on the shape of the humps-"Substantial delays in the onset of the transition are demonstrated when T-S waves are forced with large amplitudes".
These two findings therefore raise an important question for a future study: could a flexible hybrid stabilisation strategy, leveraging more than one stabilisation mechanism, be used to further delay natural transition of the boundary layer?