## Abstract

Laminar flow control (LFC) is one of the key enabling technologies for quiet and efficient supersonic aircraft. Recent work at Arizona State University (ASU) has led to a novel concept for passive LFC, which employs distributed leading edge roughness to limit the growth of naturally dominant crossflow instabilities in a swept-wing boundary layer. Predicated on nonlinear modification of the mean boundary-layer flow via controlled receptivity, the ASU concept requires a holistic prediction approach that accounts for all major stages within transition in an integrated manner. As a first step in developing an engineering methodology for the design and optimization of roughness-based supersonic LFC, this paper reports on canonical findings related to receptivity plus linear and nonlinear development of stationary crossflow instabilities on a Mach 2.4, 73° swept airfoil with a chord Reynolds number of 16.3 million.

## 1. Introduction

Given the recently renewed quest for long-range supersonic flight, both Defence Advanced Research Projects Agency (DARPA) and National Aeronautics and Space Administration (NASA) have partnered with industry and universities to help develop the next generation of supersonic vehicles after Concorde. Future efforts under DARPA's Quiet Supersonic Platform (QSP) programme will focus on a long-range strike military vehicle, while NASA's Supersonic Vehicle Technology (SVT) programme is geared towards key technologies that would enable the future fleet of supersonic transports to match the safety and environmental capabilities of its subsonic counterpart. Common to both programmes is the pursuit of aggressive performance goals to meet the mission requirements of a military vehicle and to ensure the economic and environmental viability of a civilian transport.

Laminar flow control (LFC) is one of the key technologies being addressed under the above programmes. Traditional LFC techniques involve either passive control via tailoring of the inviscid pressure gradient (Kroo *et al*. 2002), or active stabilization via surface suction (Anders & Fischer 1999) or surface cooling (Reshotko 1994). Researchers at Arizona State University (ASU) proposed a new LFC technique of using artificially introduced surface roughness to delay crossflow-induced transition (Saric *et al*. 1998; Saric & Reed 2002). As discussed by the National Research Council panel on Breakthrough Technology for Commercial Supersonic Aircraft (NRC 2002), stringent requirements for component performance plus economic and environmental challenges for a supersonic aircraft leave little room for inefficiencies in airframe design. Therefore, irrespective of the choice of LFC technique, it is necessary to have accurate and reliable prediction tools for the transition process (Crouch 1997). *N*-factor correlations based only on linear growth characteristics have been shown to be reasonably accurate in a broad range of flows. However, they offer rather limited scope for further refinement, especially in the context of transition prediction for crossflow-dominated flows.

Owing to preferential excitation of stationary crossflow modes in a benign unsteady environment (Choudhari 1994), linear stability alone may not predict the dominant spectrum of instability modes in swept-wing boundary layers. Furthermore, the extended length of nonlinear interactions during crossflow-dominated transition (Reibert & Saric 1997; Saric *et al*. 1998; Malik *et al*. 1999; Haynes & Reed 2000), together with a potentially longer region of laminar breakdown at higher speeds, would indicate a strong need for embedding the nonlinear stage into transition prediction methodology for supersonic swept-wing flows. In the context of low-speed laboratory experiments involving a small number of artificially excited stationary modes, an *N*-factor criterion based on the linear amplification of secondary instabilities provided more accurate transition onset predictions than either a linear *N*-factor or an absolute amplitude criterion based on the primary instability alone (Malik *et al*. 1999). Practical applications of the secondary *N*-factor criterion face two major challenges: (i) the inherent uncertainty in the initial amplitude spectrum during natural transition, and (ii) the enormous computational cost associated with the nonlinear interactions that need to be examined.

The ASU concept for roughness-based transition control (Saric *et al*. 1998; Saric & Reed 2002) underscores the emerging need for advanced transition prediction techniques (figure 1), initially as a source of supplementary information and eventually for high fidelity analysis during aerodynamic design. The ASU concept uses controlled leading edge roughness to introduce subdominant (stationary) crossflow modes that are unlikely to cause transition on their own. However, because of their stronger amplification immediately downstream of the location of excitation, these modes tend to suppress the initial growth of naturally dominant modes and, hence, delay the onset of transition. Because this concept is predicated on the process of nonlinear mode interactions induced via artificial receptivity, it inherently relies upon physical mechanisms not reflected in the predictions based on linear stability. In addition to any practical benefits derived from this mode of LFC, the ASU concept provides for a partially ‘controlled’ input disturbance spectrum; this makes it easier to apply an advanced transition prediction approach that mimics the essential details of the actual transition process. In other words, roughness-based control also provides a useful test bed, intermediate in complexity, against which to calibrate the holistic or integrated transition prediction approach as outlined in figure 1.

Much of the earlier work pertaining to roughness-based control was carried out for low-speed configurations and relatively small chord Reynolds numbers (*Re*_{c}=*O*(2–3×10^{6}); Saric *et al*. 1998). The physics of stationary crossflow modes is not particularly sensitive to Mach number effects, at least through the low supersonic range. However, higher Reynolds numbers, coupled with supersonic flow speeds, pose inherent hurdles to any form of LFC. The challenges at higher Reynolds numbers include both technical aspects (related to highly unstable boundary layers) and those related to practical implementation (because of magnified sensitivity to various factors that cannot be easily controlled in a realistic flow environment, particularly when the form of control is purely passive; D. Bushnell 2001, personal communication).

Considerable work is currently in progress under the QSP and SVT programmes to address and overcome the high Reynolds number challenges in the context of roughness-based LFC, including wind tunnel experiments at intermediate and full-scale Reynolds numbers. This effort includes the development of a modular transition analysis capability that would enable hierarchical levels of physical fidelity, depending on the relevant combination of prediction accuracy, turnaround time, computational resources and ability to specify additional details concerning the disturbance environment (figure 1). The objective behind the present work is to examine the design and optimization of roughness-based LFC within the framework of this hierarchical approach. To that end, this paper examines various aspects of crossflow instability on a Mach 2.4, 73° swept airfoil configuration at the highest *Re* condition (*Re*_{c}=16.3 million based on mid-span, streamwise chord) from the ASU Mach 2.4 wind tunnel experiments of Saric & Reed (2002). Linear and nonlinear development of crossflow instabilities over this configuration are examined in §§2 and 3, respectively. These results establish the potential significance of the initial amplitudes of the control input and the naturally occurring crossflow instabilities towards their subsequent nonlinear development and, hence, the onset of transition. Accordingly, the generation of stationary crossflow instabilities in a supersonic boundary layer by short-scale non-uniformities in surface geometry is studied in §4. Concluding remarks are presented in §5.

## 2. Linear stability characteristics

The LFC experiments reported by Saric & Reed (2002) were conducted in the Mach 2.4 ASU 0.2 m Supersonic Tunnel. The swept-wing model had a leading edge sweep of 73°, 0.3 m streamwise chord at the mid-span location and a symmetric cross-section with thickness-to-chord ratio of 4%. The cross-section of this airfoil was designed to be conducive to the passive, roughness-based control strategy. Specifically, it featured a small enough leading edge radius (to ensure laminar flow along the attachment line) and a favourable pressure gradient through at least 80% of the chord (to promote crossflow modes over first mode, i.e. streamwise instabilities). The chord *Re* was varied from *Re*_{c}=8.7 to 16.3 million during the experiments; the case with *Re*_{c}=16.3 million was selected for the analysis presented in this paper. The mean flow has been modelled as invariant along the span. Based on the surface pressure prediction reported in fig. 2 of Saric & Reed (2002) and assuming adiabatic thermal conditions at the model surface, the mean boundary-layer flow was computed by using the BLSTA code (Wie 1992) and was subsequently used during the linear and nonlinear stability calculations described in this paper.

### (a) Stationary crossflow modes

Figure 2 displays the *N*-factor predictions (i.e. integrated amplification ratios relative to an approximate neutral location) based on a linear version of the parabolized stability equations (PSE; Herbert 1997) including the effects of surface curvature (i.e. non-parallel with curvature or NPWC) for spanwise wavelengths ranging from *λ*_{z}=0.92 to 12 mm. The PSE code used in this study has been previously cross-validated against a high order direct numerical simulation (DNS) solver for a variety of canonical flow configurations (Chang 2003; Jiang *et al*. 2004). Figure 3 shows the NPWC growth rates along with those based on the quasi-parallel approximation, both with and without surface curvature effects (QPWC and QPNC, respectively), and non-parallel predictions with no curvature (NPNC) for selected wavelengths of the disturbance modes. As expected, effects of both non-parallel mean flow and the surface curvature are most significant near the leading edge. In general, the destabilizing effects of non-parallel mean flow are partially nullified by the stabilizing influence of convex curvature. This cancellation brings the growth rate predictions based on QPNC in closer alignment with the NPWC growth rates, except at larger wavelengths for which the non-parallel effects are more significant.

Results plotted in figure 2 illustrate the large magnitudes of *N*-factors involved (reaching *N*=9 by 30% chord and *N*=23 by 85% chord) and the relatively broad range of wavelengths (2–5 mm) corresponding to high amplification ratios over a dominant portion of the chord. Both of these features are symptomatic of high Reynolds number flows and are indicative of the challenges involved in successfully implementing LFC in such flows. The shorter wavelength modes with *λ*_{z}<1.5 mm have *N*-factors below nine, whereas the longer wavelength modes with *λ*_{z}>1.71 mm (which becomes unstable somewhat farther downstream) continue to amplify throughout the chordwise region plotted in figure 2.

### (b) Non-stationary crossflow modes

Illustrative predictions for linear amplification characteristics of non-stationary (i.e. travelling) crossflow modes are shown in figure 3 for a spanwise wavelength of *λ*_{z}=3 mm, which corresponds to the most unstable stationary crossflow mode. Consistent with previous experience based on other swept-wing configurations, the growth factors for travelling modes are much larger than those for the stationary modes. The disparity between the linear growth of travelling and stationary modes is particularly striking because of the already large *N*-factors for stationary crossflow modes. The difference between peak *N*-factors for the stationary and non-stationary modes is as large as 20 for *λ*_{z}=3 mm and almost 12 for *λ*_{z}*=*1.5 mm. Again, the issue of when and why such enormously large linear growth potential might be suppressed in favour of stationary modes cannot be answered until we better understand the unsteady disturbance environment and the associated receptivity mechanisms.

As demonstrated during the ASU wind tunnel experiments, linear stability results can provide useful insights during preliminary design of roughness-based LFC, particularly with respect to the location and spacing of the artificial roughness elements. Of course, purely linear tools are inadequate to determine an optimal control configuration. Furthermore, owing to the intrinsically nonlinear nature of the ASU control concept, linear theory cannot provide any clues regarding the extent of transition delay due to artificial roughness. Nonlinear modal interactions must be considered for this purpose.

## 3. Nonlinear development of stationary modes

To gain insights into the effects of disturbance nonlinearity in higher *Re* swept-wing configurations, we use nonlinear PSE to examine elementary classes of nonlinear interactions between stationary crossflow modes. For convenience, most of the nonlinear calculations described herein have been initiated with linear eigenfunctions (together with a specified amplitude spectrum) at *x*/*c*≈0.075, which is the first location at which an unstable eigenmode could be easily found for the longest wavelength mode (12 mm) included in this study. All disturbance amplitudes are quoted in terms of the peak perturbation in chordwise velocity relative to the free-stream speed and, unless stated otherwise, the phase difference between input modes is assumed to be zero in cases where multiple stationary modes were introduced at the initial location.

### (a) Single-mode development

#### (i) Most unstable stationary mode (*λ*_{z}=3 mm)

We first consider the nonlinear evolution of the linearly most unstable stationary mode with *λ*_{z}=3 mm. Chordwise evolution of the fundamental and its first harmonic for initial amplitudes ranging from *A*(*x*_{i})=10^{−7} to 10^{−4} is shown in figure 4*a*. For low initial amplitudes (cases *A*(*x*_{i})=10^{−6} and 10^{−7} in figure 4*a*), the fundamental mode amplifies monotonically until its amplitude has reached nearly 15%. At *A*(*x*_{i})=10^{−7}, the fundamental amplitude ascends from just 1 to 15% across only 12% of the overall chord. The normalized velocity contours plotted in figure 4*b* indicate a strong possibility that the onset of secondary instability will occur well before such peak amplitudes have been reached. During prior work (Chang *et al*. 1995), similar overturning of the velocity contours and the resulting regions of high shear away from the wall have been correlated with the onset of high-frequency secondary instabilities.

For cases *A*(*x*_{i})=10^{−4} and 10^{−5}, the fundamental amplitude reaches a peak at amplitudes in excess of 12%, then decreases somewhat before continuing to increase further and achieving another peak. The chordwise locations of the local minimum and the second peak during the fundamental evolution approximately coincide with a maximum and minimum, respectively, in the accompanying evolution of the first harmonic that is induced purely via nonlinear effects. This behaviour suggests a cyclic energy exchange between the fundamental and the first harmonic. Finally, we observe from figure 4*a* that each logarithmic increase in the initial vortex amplitude leads to an approximately constant upstream shift in the approach to peak fundamental amplitudes.

Detailed measurements and associated computations of nonlinear crossflow evolution for the case of a single dominant stationary mode have been reported previously for a low-speed configuration at relatively low chord Reynolds number (*Re*_{c}=2.4–3.2 million; Reibert & Saric 1997; Malik *et al*. 1999; Haynes & Reed 2000). In those experiments, an artificial array of spanwise periodic roughness elements was used to introduce a controlled initial disturbance that could dominate the natural spectrum of disturbances arising from background roughness. The computed evolution of modal amplitudes in figure 4*a* indicates qualitative similarities between crossflow evolution in the high-speed, higher *Re*_{c} configuration examined here and the low-speed, lower *Re*_{c} configuration studied in Reibert & Saric (1997). Secondary instability calculations (Malik *et al*. 1999) for a select set of test conditions from Reibert & Saric (1997) had demonstrated a small but finite lag between the onset of a high-frequency secondary instability and the measured transition locations, especially in cases that involved less pronounced variations in the region of peak vortex amplitude (i.e. a relatively extended state of quasi-saturation). Given the known sensitivity of secondary growth rates to primary vortex amplitude and the seemingly faster rise in the fundamental amplitudes as noted in figure 4*a*, the above lag might be reduced in the present case, causing transition to occur sooner after the secondary instability sets in.

The potentially shorter region of secondary instability for the present class of high *Re* configurations would have an interesting implication for transition prediction within the holistic framework. If it can be established that the primary amplitudes along the rapidly rising portion of the primary amplitude curve are large enough for the secondary instability to set in, then the complicated and time-consuming task of secondary instability predictions could be avoided in favour of a simpler transition criterion based on the amplitude of the primary disturbance alone. Of course, additional parameter studies, particularly for realistic initial amplitude spectra, are necessary to establish the generality of the strongly non-equilibrium nature of disturbance evolution noted above.

#### (ii) Subdominant modes

We now consider the isolated nonlinear development of shorter wavelength, subdominant stationary modes, which are suitable as the control input for roughness-based LFC. The Mach 2.4 ASU experiments (Saric & Reed 2002) involved a roughness spacing of *λ*_{z}=1.7 mm because the stationary mode at this wavelength (along with other modes in its vicinity) satisfied several criteria based on the low-speed experiments (Saric *et al*. 1998). Specifically, the *λ*_{z}=1.7 mm mode has a somewhat earlier onset of linear instability compared with the most unstable modes, and it has a smaller *N*-factor, which should prevent this mode from achieving amplitudes large enough to cause transition on its own. The nonlinear evolution of a single mode with *λ*_{z}=1.71 mm is presented in figure 5 for various initial amplitudes. Owing to nonlinear effects, this mode begins to decay increasingly farther upstream of its theoretical neutral location. Similar decay behaviour was found for the 1.5 mm mode (which also satisfies the above-mentioned criteria) and even at the 2 mm mode (which has a relatively large linear *N*-factor).

The results in figure 5 indicate that the peak amplitude of *λ*_{z}=1.71 mm mode never exceeds 10%, but a double peak appears for sufficiently large initial amplitudes. The lower peak amplitude and the relatively narrow peak region further support the suitability of this mode for roughness-based LFC within the upstream portion of the airfoil. Also observe that the location and streamwise extent of the peak amplitude region (eventually responsible for the maximum control action) is substantially influenced by the initial amplitude of this mode. Because of the relatively short region over which the subdominant mode appears to sustain its peak amplitudes, it is possible that a multistage control (based on additional control input via another mode that is active over the mid-chord region) would be useful for high *Re* applications of roughness-based LFC.

The longer wavelength mode at *λ*_{z}=4.5 mm (which is subdominant in the leading edge region) does sustain significantly nonlinear amplitudes over the mid-chord region for a range of initial amplitudes. The modal amplitudes at *λ*_{z}=4.5 mm display an extended region of quasi-saturated behaviour, before increasing rather rapidly to levels that are comparable to those of the 3 mm mode in figure 4*a*. Similar to the local peaks and valleys in modal amplitude curves in figure 4*a*, the region of quasi-saturation for the 4.5 mm mode is also marked by what appears to be a cyclic energy exchange between the dominant fundamental and its first harmonic.

### (b) Two-mode interaction

#### (i) Control via shorter wavelength mode

To simulate the effects of roughness-based LFC in the simplest possible framework, we consider the effect of nonlinear interaction between the subdominant 1.5 mm mode and the most unstable 3 mm mode. Figure 6 displays the effect of the magnitude of control input (i.e. initial amplitude of the 1.5 mm mode) on the subsequent evolution of the target mode with *λ*_{z}=3 mm. The initial amplitude of the naturally dominant target mode was held fixed at 10^{−5} in each of these cases. Obviously, within the range of parameters shown in figure 6, successive increases in control input produce the beneficial effect of delayed growth of the target mode. The suppression of the dominant mode appears to correlate qualitatively with the behaviour of the mean flow correction associated with the control mode. Specifically, suppression of target mode growth rates begins only after the control mode has achieved significant amplitudes, but continues past the peak of the 1.5 mm mode. The evolution of the mean flow correction associated with the control input also indicates a slight lag with respect to the amplitude of the fundamental mode.

For all cases in figure 6, a relatively disturbance-free zone follows the decay of the control input and precedes the eventual rise of the target 3 mm mode. Except for effectively reduced initial amplitude, this rise is analogous to the single-mode development in figure 4*a*. Accordingly, the speculations pertaining to transition prediction in the single-mode case are also relevant to the controlled case. Thus, as long as the peak amplitudes of the 1.5 mm mode do not lead to premature transition, then a primary amplitude criterion of the type discussed in §3*a* may be relevant to the controlled case in figure 6 as well. Specifically, the primary amplitude criterion would enable a relatively straightforward assessment of the anticipated shift in the transition onset location due to variations in the initial control input, based on accompanying shift in the primary amplitude curve. For example, increasing the control input from case (*a*) to case (*d*) in figure 6 might produce a transition delay of approximately 20% of the model chord.

Additional calculations reveal that the role of variations in the initial amplitude of the target 3 mm mode is analogous but opposite to that of variations in the control input. This condition indicates the significance of the relative initial amplitudes of the two modes involved in the interaction. The wavelength of the control input was also found to have a significant effect on the suppression of the target mode. Wavelengths somewhat larger than the 1.5 mm wavelength considered in figure 6 may provide more effective control, albeit at the risk of increasing the peak disturbance amplitudes due to the control input. As an example, for a control input in the same range of initial amplitudes as in figure 6, the 2 mm mode prevented the 3 mm target mode from attaining significant amplitude. However, the peak modal amplitude at the 2 mm control wavelength was larger in comparison to the control input at 1.5 mm. Even though the highest control amplitudes are still confined to a relatively narrow range of locations (similar to figure 5), the resulting risk of premature transition due to the control input cannot be fully assessed without computations of secondary instability and/or additional experiments.

#### (ii) Two-mode interaction: effect of relative phase

We now illustrate the effects of the relative initial phase on the downstream evolution of the two-mode interaction considered in figure 6. Modal evolution for three selected values of the phase difference parameter (measured as the phase of the peak chordwise velocity perturbation for the target mode at the initial location, relative to that of the control input at 1.5 mm wavelength) is shown in figure 7. These results suggest that intermodal phase differences can have a significant effect on the ensuing evolution of the modes participating in the nonlinear interaction. The significance of the relative phase between an energetic fundamental and its lower amplitude subharmonic has previously been documented, but only for streamwise instabilities in free-shear flows (Monkewitz 1988) and boundary layers (Saric *et al*. 1984). Of course, the phase dependency of bimodal interactions is weakened when one of the modes is not a subharmonic (or a harmonic) of the other mode, indicating that such mode pairs communicate primarily through the mean flow distortion term, which is independent of the phase.

If the above-mentioned significance of the relative phase proves to be generic to swept-wing configurations, then it could open up an additional avenue (albeit perhaps not a practical one) to influence the nonlinear crossflow evolution in cases where the naturally occurring disturbance spectrum is dominated by a single wavelength. Similarly, it will be worthwhile to investigate whether phase scrambling in a broadband spectrum of instabilities can render the nonlinear mean state less unstable to high-frequency secondary instabilities for a given level of disturbance energy. Finally, uncertainties associated with modal phases would indicate the need for a stochastic prediction approach such as that investigated by Rubinstein & Choudhari (2000) for triadic and other interactions.

#### (iii) Basic state modification due to control input

To provide further insight into the underlying physics of the nonlinear control mechanism considered in figures 6 and 7, figure 8*a* compares the spanwise-averaged crossflow velocity profile *U*_{cf}(*y*) for the case involving a specified control input at *λ*_{z}=1.5 mm with that for the unperturbed laminar boundary layer. Introduction of the control input clearly decreases the peak crossflow velocity. The shape of the crossflow profile remains unaffected by the control input; thus, the reduction in peak crossflow velocity amounts to a proportional decrease in the crossflow Reynolds number, a parameter that is sometimes used for empirical predictions of transition onset in three-dimensional boundary layers. Thus, an optimal roughness configuration may be designed to maximize the stabilizing influence of reduction in the crossflow Reynolds number, while simultaneously containing the potential for high-frequency secondary instabilities due to generalized inflexion points in the streamline-aligned velocity profile.

We also investigated the linear stability of the modified mean (i.e. spanwise averaged) profiles using both classical linear stability theory (LST) and the PSE. As indicated in figure 8*b*, introduction of the control input leads to a reduction in the growth rates of the dominant stationary crossflow mode with *λ*_{z}=3 mm. Comparable stabilization of the boundary layer is indicated by both LST and PSE. Similar reductions in growth rates were also obtained for the most unstable travelling crossflow modes at *λ*_{z}=3 mm. It must be noted that the above stability calculations neglect the spanwise variation of the modified basic flow, which may not be negligible owing to the shorter spanwise scales of the control input. However, the above linear analysis may be justified for general pairs of control and target modes that are either distantly related or not at all related harmonically and, hence, communicate with each other through the mean flow correction alone as discussed in §3*b*(ii).

#### (iv) Control via longer wavelength modes

A preliminary calculation to assess the role of control input in the form of a slower-growing, larger wavelength mode (6 mm in this case) was also carried out. Within an intermediate range of locations, a phase-locked interaction between the control input and its linearly unstable harmonic actually led to an increased growth rate of the target mode (*λ*_{z}=3 mm) above its linear value. However, after reaching sufficiently large amplitudes farther downstream, the subharmonic mode did contain the 3 mm mode to lower amplitudes than it might have achieved without any control input. Of course, further analysis is necessary to ensure that the combined flow field does not exhibit an earlier onset of secondary instabilities than might be expected on the basis of modal evolution alone.

The possibility of using a subharmonic mode for control has been noted in prior calculations of roughness-based LFC (Janke & Balakumar 1999). However, the combined effect of control input involving both shorter and longer wavelength subdominant modes needs to be investigated. Because the two classes of modes may be active in different regions of the chord, the combined approach might conceivably be more effective than using either mode alone.

### (c) Three-mode interactions

We now illustrate an additional facet of nonlinear crossflow interactions by considering quadratic interactions between two initially energetic modes with *λ*_{z}=1.5 and 2.25 mm, respectively, and investigate their effect on the evolution of the ‘difference’ mode with a wavelength of 4.5 mm. Only the shorter wavelength modes are seeded at the initial location; the longer 4.5 mm mode is initialized with a nominal amplitude of 10^{−7}. Owing to a sustained transfer of energy from the pair of shorter wavelength modes, the amplitude of the longer wavelength mode increases progressively with the magnitude of control input at *λ*_{z}=1.5 mm. Indeed, figure 9 shows that for a sufficiently large initial amplitude of the control input (*A*(*x*_{i})=0.002), the 4.5 mm mode actually becomes the dominant mode at the downstream locations.

The strong intermodal energy transfer noted above may be explained by the fact that the triad of linear stationary modes is nearly phase locked over a significant portion of the chord, consistent with the relatively narrow range of vortex orientations for stationary crossflow modes. The magnitude of the phase detuning factor (defined as the real part of the complex detuning factor , where *α* denotes the chordwise wavenumber of the crossflow mode and the subscript indicates the spanwise wavelength in millimetres) for the above triadic interaction is less than 3% throughout *x*/*c*>0.15. Thus, when two modes from a triad have sufficiently large initial amplitudes, nonlinear interactions between the two can easily lead to significant receptivity of the third unseeded mode. For the case examined in figure 9, such triadic interactions appear to favour the excitation of the largest wavelength mode. Analogous energy transfer to the longer wavelength modes was also noted in the case of other triads (such as 1.71, 3 and 4 mm) as well as when the input spectrum had a more broadband character.

We note that the presence of multiple modes also alters the spanwise variation of the modified basic state for secondary instabilities. Depending on the nature of interference between the dominant modes, the multimode interactions could result in a smoothing of the spanwise gradients involved (and wall normal gradients as well, depending on the peak locations for the various modes) for a fixed level of disturbance energy. If this smoothing happens, then the secondary instability process might be weakened, particularly for the class of modes associated with spanwise gradients of the modified basic state.

## 4. Receptivity due to small-amplitude surface roughness

The illustrative findings presented in §3 indicate potential intricacies of nonlinear interactions between crossflow instabilities, especially in a high Reynolds number environment. All calculations were based on arbitrarily chosen initial amplitudes of the modes involved. Given the observed effect of the initial amplitude spectrum on the range of locations where disturbance amplitudes attain relatively large levels (i.e. corresponding to potential onset of secondary instabilities), it is essential to investigate the receptivity mechanisms for the dominant modes involved in those interactions. However, the finding that the shift in expected transition location scales approximately with logarithmic variation in the initial amplitude of the target mode suggests that reasonable estimates of natural receptivity could, in some cases, be an adequate substitute for higher fidelity predictions.

As discussed by Choudhari (1994), surface roughness provides a dominant and preferential source of receptivity for stationary crossflow modes. The receptivity theories of Goldstein (1985) and Ruban (1985) are easily extended to obtain the required estimates for roughness-induced excitation of stationary crossflow modes using a finite Reynolds number framework based on quasi-parallel stability equations. Here, we summarize only the pertinent findings. Receptivity is found to be most efficient for roughness array locations close to the attachment line; it becomes considerably less effective for roughness locations farther downstream. This trend appears to be generic to stationary crossflow excitation on swept-airfoil configurations (Collis & Lele 1999; Ng & Crouch 1999). Coupled with the early onset of linear instability for a broad range of wavelengths over the present configuration, such receptivity predictions indicate that the roughness array used as control input towards LFC application would be most effective when placed within first 2–3% chord on the present model. A second implication is that a majority of the natural seeding of stationary modes via uncontrolled surface roughness also occurs within the leading edge region. Therefore, it is probably adequate to maintain the surface finish qualities required for LFC applications within a narrow region close to the leading edge; that is, tolerance levels farther downstream may be allowed to be less restrictive.

The receptivity analysis also indicated that the initial amplitudes of the 1.5 mm mode are within the range of control inputs required to produce delayed growth of the dominant mode (as considered during the nonlinear PSE calculations in §3*b*). Nevertheless, we caution that curvature and non-parallel effects may significantly alter the efficiency of roughness-induced receptivity (Collis & Lele 1999; Janke 2001), especially in the leading edge region. We are currently in the process of performing more refined calculations that would enable an assessment of these effects for supersonic swept wings.

Receptivity models cannot be used to quantify the natural excitation of instabilities on a realistic LFC configuration until sufficiently detailed characterization of the natural, broadband disturbance environment is made available. The lack of adequate information concerning both free stream unsteadiness and surface disturbances (e.g. surface roughness and/or waviness) certainly represents a major obstacle in physics based predictions of laminar turbulent transition.

An initial attempt at developing a statistical characterization for surface roughness on a generic aerodynamic surface was described by Choudhari *et al*. (2003). To exploit a database of this type, deterministic receptivity models may be extended to stochastic roughness distributions, so as to characterize the receptivity in a statistical sense. There are obvious deficiencies with using statistical estimates based on ensemble averaging over a sample space of aerodynamic surfaces. However, ensemble averaging could be replaced by spanwise averaging along a simple surface if the roughness distribution is shown to be homogeneous in that direction, preferably via measurements over a sufficiently extended spanwise region. Of course, given the wide range of roughness distributions that may be encountered in practice, further work is necessary to quantify the uncertainty in the specification of the roughness and to propagate that uncertainty to establish the variability in naturally occurring instability wave amplitudes at a suitable initial location. Also desirable are two-dimensional maps of the roughness distribution that would allow direct estimates of both chordwise and spanwise spectra.

## 5. Summary and concluding remarks

In this paper, we examined various aspects of crossflow mode evolution in a supersonic, swept-wing configuration that was recently used to evaluate the feasibility of roughness-based, passive LFC in a Mach 2.4 wind tunnel experiment at ASU. The nonlinear results suggest that, even in a higher *Re* environment, the growth of the (linearly) most unstable stationary mode can be considerably delayed via nonlinear interactions with a shorter wavelength mode.

Other notable findings from the present work include a seemingly rapid rise in the amplitude of the most unstable stationary crossflow mode to levels that may well harbinger the onset of secondary instabilities, the potential significance of the relative phase between a pair of interacting modes, especially for subharmonic interactions, and quadratic interactions leading to nonlinear receptivity of larger wavelength modes. The intricacies of nonlinear crossflow interactions also indicate potential challenges in robust implementation of laminar flow technology on large *Re* configurations.

The present study also provides partial justification for the lower fidelity design practices used during the demonstration of roughness-based supersonic LFC in the ASU wind tunnel experiment (Saric & Reed 2002), especially the choice of roughness size, spacing and location. However, the findings herein also emphasize a strong need to account for the nonlinear crossflow interactions, both to minimize the risks involved and to optimize the overall performance. Fortunately, the same findings also hint at the possibility of accomplishing this ambitious goal via simpler nonlinear criteria that do not require full-fledged calculations of secondary instability. A detailed parametric study involving realistic initial spectra, together with calculations of secondary instability and additional measurements, will help establish the validity of the simpler criteria in LFC applications of interest. The role of non-stationary crossflow modes also needs to be investigated, particularly because of the huge disparity between the linear *N*-factors for the stationary and non-stationary modes. Limited calculations for a low-speed (and lower Reynolds number) configuration had suggested that the same principle, which underlies the suppression of the dominant stationary mode, might also be applicable to the travelling modes. In other words, an additional (desirable) effect of the stationary control input may be a reduced amplification of the travelling crossflow modes relative to their isolated nonlinear development. One must note, however, that the beneficial effect of the control input may be offset by the accompanying suppression of the dominant stationary modes, which may contribute to a reduced suppression of the more unstable travelling crossflow modes. A more detailed study is therefore necessary to determine the generality of this observation.

Overall, we advocate a holistic approach to transition prediction for advanced LFC configurations that are either intrinsically predicated on nonlinear transition physics (such as roughness-based control) or involve mixed mode transition. Some of the challenges in the practical implementation of the holistic approach were discussed in the paper, including potential ways to overcome those challenges. A high fidelity approach for transition prediction is particularly difficult to incorporate into design oriented computational fluid dynamics, especially in the context of multidisciplinary optimization. However, recent work (Kroo *et al*. 2002) provides an encouraging prognosis in that regard.

## Acknowledgments

The authors gratefully acknowledge useful technical discussions with Drs Craig Streett and P. Balakumar of NASA Langley Research Center and Dr Jeffrey Crouch from Boeing.

## Footnotes

One contribution of 19 to a Theme ‘New developments and applications in rapid fluid flows’.

- © 2005 The Royal Society