1 Introduction

Laminar-turbulent transition in boundary-layer flows is crucially affected by environmental perturbations and wall imperfections. If both factors are of low level, then transition would be triggered by the accumulation of the instability modes, e.g., the Tollmien-Schlichting (T-S) wave in the subsonic regime or the Mach mode in the supersonic regime, which is referred to as the natural route. The presence of small wall imperfections, due to engineering design or introduced as the strategy of laminar-flow control, could usually alter the amplitude of the oncoming instability mode in a local region, leading to a promotion or delay of the transition. The imperfections could be a roughness, a gap, a step, a suction slot, and a trailing edge, and they induce the ensuring scattering of the oncoming instability modes through generating certain mean-flow distortions.

Recently, Wu and Dong^[1] proposed a local scattering theory, based on the larger-Reynolds-number triple-deck formalism, to characterize the impact of a local scatter (wall imperfection) on oncoming instability modes in subsonic boundary layers. The principle of this theory, i.e., a so-called transmission coefficient, defined as the ratio of the amplitude downstream of the scatter to that upstream, was introduced. In that paper, the scatter was taken as a hump or an indentation, and their destabilizing effect on the oncoming T-S waves was systematically investigated.

Apart from the roughness cases, other forms of scatters may result in different impacts. Reynolds and Saric^[2] conducted experiments on a laminar flat-plate boundary layer with porous suction panels. The stabilizing effects of suction on the growth of T-S waves were examined. The experimental data were compared with the results from the theoretical model in Ref. [3], which predicted the amplitude of the T-S wave by solving the Orr-Sommerfeld (O-S) equation with the base flow being constructed by the linearized triple-deck theory, and good agreement was found for relatively small suction velocities. The failure of the comparison for moderate suction velocities is believed to be attributed to two reasons: (ⅰ) the linearized triple-deck solution is only valid for weak wall perturbations (suction velocities), and (ⅱ) the parallel-flow assumption used in the O-S equation does not apply in the close neighbourhood of the suction panel because of the rapidly distorting mean flow. Later, on the study of the optimal location of a suction strip for delaying transition, Masad and Nayfeh^[4] improved the calculation of the mean-flow distortion by using an interactive boundary-layer theory. However, the use of the linear stability theory with the parallel-flow assumption is still questionable. The effect of surface heating on the laminar flow control was also studied by Masad and Nayfeh^[4] and Masad^[5] with the same approach. Balakumar et al.^[6] studied the instability of a hypersonic boundary layer over a compression corner with the direct numerical simulation (DNS), and particular attention was paid on the interaction between the oncoming second Mach mode and the separation bubble formed at the corner. It was found that the second mode remained almost neutral when propagating over the corner, but recovered to the exponential-growing feature in the downstream limit, which indicated a stabilizing effect. Such phenomenon was also observed in Refs. [7] and [8].

Local steps on the wall surface are often seen in a variety of engineering applications, which also have potential effects on boundary-layer transition due to its scattering effect on the oncoming instability modes. It has been confirmed that a backward-facing step (BFS) has a destabilizing effect on the instability modes and promotes transition^[9-10]. However, the impact of a forward-facing step (FFS) is in debate. Worner et al.^[11] studied the role of a rectangular roughness on the oncoming T-S wave in an incompressible boundary layer with the DNS. The numerical results show that the amplitude of the T-S wave grows more slowly near the FFS, but a much greater amplification is seen near the BFS, resulting in an overall destabilizing effect. The conclusion that the FFS stabilizes the T-S wave is in contrast with the experiments of Wang and Gaster^[9] and Costantini et al.^[12]. A later numerical investigation on the impact of FFS in a transonic boundary layer was provided by Edelmann and Rist^[13], and the role of the FFS was found to be destabilizing. The disagreement may be attributed to the difference of the numerical and experimental configurations. In the study of Ref. [11], a rectangular obstacle with both the FFS and BFS was introduced, while the experiments of Refs. [9] and [12] only took one single step into account. In principle, if the width of the obstacle in the former setup is taken to be sufficiently large, the interaction between the FFS/BFS and the oncoming T-S wave can be considered independent. However, whether the width is sufficiently large or not needs to be examined. In this paper, we will focus on this issue by using the local scattering theory^[1].

2 Mathematical descriptions 2.1 Physical model

In this paper, we consider a subsonic boundary layer with a two-dimensional step located at a distance L downstream of the leading edge of a flat plate, as shown in Fig. 1. The step has a height h^*≤O(Re^-5/8L) with h^* being positive/negative representing an FFS/BFS, where the Reynolds number Re=U_∞ L/ν_∞ with U_∞ and ν_∞ denoting the velocity and kinetic viscosity of the oncoming stream, respectively. Suppose that a T-S wave is propagating in the boundary layer, which, as passing over the step, is scattered by the rapidly distorting mean flow. This leads to the local deformation of the perturbation profiles and the variation of the downstream asymptotic amplitude.

We define the Mach number as Ma=U_∞/a_∞, where a_∞ denotes the freestream sound speed. In this paper, we focus on the subsonic regime Ma < 1, and the Reynolds number is assumed to be large, i.e., Re≫1. The two-dimensional Cartesian coordinate (x, y)=(x^*, y^*)/L is used, with the origin O located at the foot of the step, where, in what follows, the superscript * denotes the dimensional quantity. Both the mean flow and the perturbation in the O(Re^-3/8L) vicinity of an FFS or a BFS can be described by the large-Re triple-deck theory^[14]. As shown in Fig. 1, three asymptotic layers with different scalings appear in the wall-normal direction. Since the interaction between the steps and the T-S wave is restricted in the lower deck, we in the next subsection only introduce the equations in this layer, and readers can find detailed introductions on the triple-deck theory in Refs. [14]-[16].

2.2 Scaling and mean flow

For simplicity, we introduce a small parameter,

(1)

and the rescaled coordinates, time, and flow field^[17],

(2)

where λ=0.332 06 is the constant associated with the wall shear of Blasius profile, C is the constant for the Chapman viscosity law, ρ_∞ is the density of the oncoming stream, and T_w/T_∞=1+(γ-1)Ma²/2 is the wall temperature with γ the ratio of the specific heat. Accordingly, we rescale the step height as

Let the shape function of the wall be Y₁=F(X), and in order to avoid numerical difficulty around the shape corner, we smooth the step by a tanh function, i.e.,

(3)

where Δ characterizes the thickness of the round corner, and it is taken to be 0.03 in this paper.

In order to solve the flow field in a rectangular domain, the Prandtl transformation is introduced,

(4)

The velocity field and pressure of the steady mean flow are denoted as (U, V) and P, respectively. Substituting (2) and (4) into the steady Navier-Stokes (N-S) equation, we obtain the leading-order governing equations in the lower deck,

(5)

which are subject to the boundary and matching conditions,

(6)

(7)

with A the displacement function of the mean flow, and the pressure-displacement (P-D) relation is

(8)

2.3 Scattering of perturbation

The perturbations of the velocity field and pressure are assumed to be small in the amplitude and are expressed as a time periodic form,

(9)

where ω is the dimensionless frequency, and is the amplitude. Substituting this expression into the linearized N-S equation leads to

(10)

which are subject to the boundary and matching conditions,

(11)

with à the displacement function of the perturbation (amplitude), and the P-D relation is

(12)

In the upstream limit, the perturbation is the unperturbed T-S wave,

(13)

where α is the dimensionless wavenumber, the displacement function  for the T-S wave is taken to be 1 for normalization, and the expressions for the eigenfunctions û, , and can be found in Ref. [1].

A transmission coefficient , defined as the ratio of the asymptotic T-S amplitude downstream of the step to that upstream, is introduced to characterize the scattering effect. The stabilizing effect appears when , while the destabilizing effect corresponds to the cases with . With the help of , the downstream perturbation field is expressed as

(14)

where the high-order terms are driven by the re-scattering of the T-S wave by the algebraically decaying "wake" of the step. Wu and Dong^[1] proposed an approach to tackle with the re-scattering effect, and it has been confirmed that the leading-order terms are sufficient to provide an accurate solution.

The above system can be solved by a local scattering framework^[1]. We first discretize the system into a rectangular domain, X₀≤X≤X_I and Y₀≤Y≤Y_J. A uniform mesh is used in the X-direction, and a non-uniform mesh is used in the Y-direction. Each grid point is denoted as X_i or Y_j with i∈ [0, I] and j∈ [0, J].

All the unknown variables are placed into a high-dimensional vector,

(15)

where . Then, the governing equations are written as a generalized linear eigenvalue problem,

(16)

where the transmission coefficient is the eigenvalue of the whole system. The perturbation field is derived from the eigenfunctions.

3 Numerical results for scattering effect of single step 3.1 Mean flow

The steady triple-deck equations (5) with the boundary and matching conditions (6) and (7) and the P-D relation (8) are solved with the numerical scheme given by El-Mistikawa^[18]. A careful resolution study has been conducted by refining the mesh size and enlarging the computational domain in both the streamwise and wall-normal directions. Figure 2 displays the contours of the mean shear and streamlines. A high shear region appears near the corner of the FFS, while a low shear region appears immediately behind the BFS. In both regions, the streamlines curve significantly, indicating an abrupt alternation of the direction of the fluid motion. A separation bubble is observed downstream of the BFS, which is apparently larger for a steeper step.

The displacement effect of the mean flow can be measured by a total displacement A+F, which characterizes the displacement from the base transverse location Y₁=0. The streamwise distributions of the total displacement function A+F, the pressure P, and the wall shear τ_w≡U_Y(X, 0) are shown in Fig. 3. In the vicinity of an FFS, the total displacement function decreases with X and approaches constants in both the upstream and downstream limits. Meanwhile, a favourable pressure gradient emerges, implying an accelerating process, which can also be seen from the distribution of the wall shear. It is understandable that the mean-flow distortion becomes greater for a steeper step. On the other hand, the BFS produces an increase of A+F and an adverse pressure gradient. The latter leads to a separation zone immediately behind the BFS for h≥1.0, which is also observed in Fig. 2. The size of the separation zone increases with h.

3.2 Scattering of T-S modes

The dispersion relation of the T-S waves is shown in Fig. 5 of Ref. [1]. The frequencies for the most unstable and the neutral T-S waves are found to be 7.25 and 2.3, respectively, with the corresponding wavenumbers of 2.48 and 1.0, and the greatest growth rate is 0.305. In this paper, we will focus on 3.0≤ω≤9.0.

The contours of the eigenfunction in the vicinity of the steps for h=± 1.0 are shown in Fig. 4, where the most unstable frequency ω = 7.25 is chosen. The rapid deformation of the T-S wave is clearly exhibited. The detailed perturbation profiles for this case are displayed in Fig. 5. As propagating towards the FFS from upstream (see Fig. 5(a)), the eigenfunction deviates from the unperturbed shape gradually, which leads to an inward movement of the transverse location of its peak. After interacting with the FFS, the eigenprofile tends to the oncoming state, with the peak location moving outward (see Fig. 5(b)). However, it is not until X=12.0 that the perturbation profile can recover the original state, i.e., only appears when X≥12.0. An opposite deviation trend is observed for the BFS case (see Fig. 5(c)), i.e., the peak of the eigenprofile moves outward as approaching the BFS. Immediately behind the BFS (X=0.03 and 0.1), the overshooting of the eigenprofile at Y < 2 is replaced by a monotonically of u versus Y (see Fig. 5(d)). Further downstream (until X=12.0), the eigenprofile recovers the unperturbed state. It is interesting to notice that, the affected region of the T-S wave is -1≲ X < 0 in the upstream direction, but 0 < X≲12 in the downstream direction. The reason is that the "wake" of the scatter exhibits a slow algebraical decay in the downstream direction. The wavelength of the T-S wave at this frequency is about 2.48, which is about 1/5 of the distortion region of the T-S wave.

In order to demonstrate the scattering effect quantitatively, we introduce a normalized displacement function,

which characterizes the amplitude of the perturbation in the scattering process as normalized by that in the smooth plate case. It follows that or corresponds to a local enhancement or suppression of the amplification of the oncoming T-S wave, and its asymptotic behaviours are

Figure 6 shows the streamwise evolution of the normalized amplitude for different ω, where (a) and (b) are for the FFS with h=1.0 and BFS with h=-1.0, respectively. All the cases exhibit a similar trend. The amplification of the perturbations increases remarkably upstream of the FFS, then it drops to a less-than-one level as travelling away from the FFS. After reaching the minimum value, increases gradually and approaches a constant in the downstream limit, which corresponds to the transmission coefficient . For the FFS, the constant is quite close to 1, indicating a rather weak scattering effect. As ω increases, the deviation of from 1 is greater. For a BFS, the evolution trend of is in the opposite direction, i.e., reduces primarily, and rises dramatically as passing over the step, and after peaking at a downstream position, it approaches in the downstream limit. Since the downstream amplitude of the T-S wave is apparently enhanced, the destabilizing effect of the BFS is convinced, which agrees with Refs. [9] and [10], and this effect increases with ω.

Figure 7 displays the dependence of the transmission coefficient on ω. The transmission coefficients for all case studies are greater than 1, indicating the destabilizing effect for both the FFS and BFS, and the destabilizing effect is greater for a higher ω and/or |h|. A BFS produces much greater enhancement on the oncoming T-S wave than an FFS with the same height, which again agrees with the experimental observation in Ref. [9].

4 Numerical results for scattering effect of wide rectangular obstacle with two steps

So far, the numerical results of the local scattering theory have confirmed the destabilizing effect of an FFS on the oncoming T-S waves. The next question we would like to answer is why the opposite conclusion was drawn by performing the numerical investigation on another configuration. In this section, we choose the physical model from Ref. [11]. As shown in Fig. 8, we replace the single step in Fig. 1 by a rectangular obstacle with the height and width of h^* and d^*. Again, h^* is restricted in the lower deck, i.e., h^*≤O(Re^-5/8O(L)), and the width is assumed to be O(Re^-3/8L) < d^* < L such that the two steps (rising and falling edges, also referred to as the FFS and BFS) can interact with the oncoming instability modes separately.

The mathematical description on this problem is exactly the same as that in the previous section, and the only difference is that the shape function of the wall is changed from (3) to

where d is the rescaled width of the obstacle according to the triple-deck scaling (the first line of (2)), i.e.,

By choosing h=1.0 and d=10.0, Fig. 9 compares the total displacement function A(X)+F(X) and the pressure P(X) of the mean flow near the FFS and BFS with the calculations of the model in Section 4 and those in Section 3. The streamwise coordinate of the latter is shifted to align the step position. Good agreement is found, implying that the two steps in the present model distort the mean flow independently.

Figure 10 shows the streamwise evolution of the normalized amplitude for different h and ω, where the positions of the FFS and BFS are marked by thin dot-dashed lines. All the cases exhibit a similar trend. At the upstream locations, the behaviour of is similar to that in Fig. 6(a). After peaking at the FFS, drops with X and stays at a basin state until the neighbourhood of the BFS is reached. The following scenario that, in Fig. 6(b), i.e., a small decrease → a drastic increase immediately downstream of the BFS → a smooth decrease that approaches , repeats. As pointed out by Wu and Dong^[1], if two scatters are independent, then their impact on the oncoming instability modes can be expressed by an overall transmission coefficient, which is the product of those of the individual scatters. The thin dashed lines represent the overall transmission coefficient, as obtained by multiplying those of an FFS and a BFS with the same h. Indeed, the overall transmission coefficient agrees with the large-X asymptote of indicating the independent nature of the two steps.

Figure 10 can qualitatively reproduce the numerical results of Worner et al.^[11]. Since on the obstacle was less than 1, Worner et al.^[11] attributed this phenomenon to the stabilizing effect of the FFS. However, there exists an alternative factor leading to the reduction of , i.e., the upstream influence of the falling edge of the obstacle. In fact, it takes quite a long distance to allow the T-S wave to recover in the wake of an FFS, and if the BFS emerges before the upstream T-S wave arrives at the asymptotic stage, then the impact of the FFS cannot be prescribed. As indicated from Fig. 5, this distance could be five times of the wavelength of the T-S wave. Moreover, it might take an even longer distance to see the asymptotic state of , as shown in Fig. 6. In the current calculation, d is only four times of the wavelength for ω=7.25, and about twice of the wavelength for ω=3.0. In Ref. [11], the width of the widest obstacle was merely taken to be twice of the T-S wavelength. Therefore, it is understandable that the wrong conclusion for the impact of an FFS is caused by the inappropriate numerical configuration.

As the distance d between the FFS and BFS reduces, the two scatters may interact with each other in a stronger sense. The dependence of the transmission coefficient on the distance d for different frequencies is shown in Fig. 11. For h=1.0 (shown as the solid lines with filled circles), the FFS is upstream of the BFS. If d is comparable with the wavelength of the T-S wave, then the combined scatter appears as an isolated hump. The relation between and d has a similar trend as that in Ref. [1], i.e., the overall destabilizing effect is observed for all d and ω, and the maximum appears at d≈ 2.5 for ω=7.25/5.0 and d≈3.0 for ω=3.0, respectively, which are comparable with the wavelength of the T-S wave (2.48, 3.14, and 4.8 for ω=7.25, 5.0, and 3.0). Further increase or decrease of d would lead to less enhancement on the T-S wave. As expected, when d is sufficiently large, the transmission coefficient approaches a constant, which can be regarded as the combination of those of the two separated steps. Since they do not interact with each other, the consequent combined value is independent of d. Again, the destabilizing effect is greater for a higher frequency.

For h=-1.0 (shown in Fig. 11 as the dashed lines with unfilled circles), the two steps swap the positions with each other, and the combined scatter, for a relatively small d, appears as an indentation. Similar to that found by Wu and Dong^[1], the destabilizing effect for an indentation is always weaker than that for a hump with the same size provided that d is comparable with the wavelength of the T-S wave. As d increases, the transmission coefficient for each frequency again approaches a constant, which is exactly the same as that for h=1.0, indicating the independent nature of the two steps.

5 Conclusions and discussion

This paper focuses on the impact of a two-dimensional FFS/BFS step on oncoming instability (T-S) waves in a subsonic boundary layer. The height of the step is assumed to be of O(Re^-5/8L) such that the triple-deck formalism is used to describe both the mean-flow distortion and the perturbation field. The interaction between the steps and the oncoming T-S waves is quantitatively described by a local scattering framework^[1].

In the vicinity of the FFS, a favourable pressure gradient is generated, leading to an acceleration of the flow motion, while the BFS induces an adverse pressure gradient, and a separation zone appears immediately downstream of the BFS. The mean-flow distortion is stronger when the step is steeper. When a T-S wave propagates over a step, the scattering effect happens due to the rapidly distorting mean flow, and the deformation of the T-S wave is clearly exhibited in this paper. The impact of both the FFS and the BFS on the oncoming T-S wave is destabilizing, and the effect increases with the step height and the frequency. A BFS produces a much stronger destabilizing effect than an FFS. This study confirms the previous observations^{[9-10, 12]}, and sheds lights on the systematical description of the scattering effect of an individual step.

In order to reveal the reason that causes the confliction between conclusions from Worner et al.^[11] and Wang and Gaster^[9], we have also studied the scattering process under the former's configuration, i.e., a wide rectangular obstacle with both an FFS and a BBS. The distance between the two steps is chosen to be large, such that the two individuals can be considered independent. It has been checked that the calculated mean flow in the vicinity of the steps agrees with that by considering the distortion of the FFS and BFS separately. In the scattering process, an oncoming T-S wave is scattered by the two steps in sequence, and the overall transmission coefficient is equal to the product of the two individual transmission coefficients, which again confirms our scattering calculation. However, there exists a misleading phenomenon, i.e., in the region between the two steps, the amplitude of the T-S wave is suppressed, which was recognized by Worner et al.^[11] as a stabilizing effect of the FFS. As a matter of fact, such conclusion is not true. A more reasonable explanation is that the distance between the two steps is not sufficiently long, and the upstream impact of the BFS overwhelms the weakly destabilizing effect of the FFS. Additionally, it can be expected that, if we take d to be sufficiently large, then the destabilizing effect for the FFS can be observed. If the distance between the two steps is shortened, then the roughness case as studied by Wu and Dong^[1] is recovered. The transmission coefficient peaks when the distance d is comparable with the wavelength of the T-S mode, and approaches a constant for a sufficiently large d. For a moderate d, a hump produces a greater destabilizing effect than an indentation with the same size.

It is worth mentioning that, in the O(Re^-3/8L) neighbourhood of the steps, the scattering system is elliptic, and hence the parallel-flow assumption and the parabolized stability equation do not apply. The local scattering framework is, to the authors' knowledge, the most efficient approach to accommodate the elliptic nature. Although one can alternatively use DNS, yet its time-consuming property makes the systematical investigation almost impossible. Admittedly, the local scattering theory has a certain restriction, i.e., the height of the steps cannot exceed the lower-deck region, whose thickness is of O(Re^-5/8L). A higher step would generate a nonlinear distortion in the main deck, leading to the breakdown of the triple-deck scaling. Such problem requires the employment of solving the full N-S equation directly (for example, see Ref. [11]).