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arxiv: 2604.15801 · v1 · submitted 2026-04-17 · ⚛️ physics.flu-dyn

Stabilisation of second Mack mode in hypersonic boundary layers through spanwise non-uniform surface temperature distribution

L. Boscagli , G. Rigas , O. Marxen , P. J. K. Bruce This is my paper

Pith reviewed 2026-05-10 08:20 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords hypersonic boundary layersecond Mack modeflow controlsurface temperature distributionsteady streaksdirect numerical simulationlaminar-turbulent transitionMach number effects
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The pith

Spanwise non-uniform wall temperature generates steady streaks that reduce second Mack mode energy by up to 60 percent in hypersonic boundary layers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether a spanwise variation in surface temperature can create steady streaks capable of damping the second Mack mode, the dominant instability in hypersonic boundary layers. Direct numerical simulations at Mach numbers from 4.8 to 6 demonstrate that these streaks lower disturbance energy by as much as 60 percent when the spanwise wavelength is chosen appropriately. The approach exploits the flow's own aerothermodynamic response rather than added hardware, raising the possibility of a passive means to delay transition. If the linear damping carries over to realistic conditions, it would lower peak heat loads on vehicle surfaces. The work supplies quantitative guidance on wavelength selection for both wind-tunnel and flight wall-temperature ratios.

Core claim

Imposing a steady spanwise non-uniform surface temperature distribution on a flat-plate hypersonic boundary layer produces streamwise streaks that interact with and attenuate the second Mack mode, yielding energy reductions reaching approximately 60 percent; for the most amplified frequency at Mach 6 the optimum spanwise wavelength lies between 8 and 10 local boundary-layer thicknesses.

What carries the argument

Steady streaks induced by prescribed spanwise wall-temperature variations, which alter the mean flow profile and thereby reduce the linear growth rate of the second Mack mode.

If this is right

  • Stabilization performance varies strongly with the imposed spanwise wavelength of the temperature distribution.
  • For Mach 6 flow the most effective wavelength is approximately 8 to 10 times the local boundary-layer thickness.
  • The same control works across wall-temperature ratios typical of both ground-test facilities and actual flight.
  • The resulting guidance can be used to design future passive thermal-control surfaces for hypersonic vehicles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may be realized passively by selecting surface materials with tailored thermal conductivity patterns rather than active heating.
  • If transition is postponed, integrated heat loads and skin-friction drag on entire vehicle surfaces would decrease beyond the local instability suppression shown here.
  • The streaks might interact constructively or destructively with other known hypersonic control devices such as distributed roughness.
  • Extension to three-dimensional geometries or non-zero angle-of-attack flows would require separate verification because the base-flow symmetry assumed here would no longer hold.

Load-bearing premise

The linear energy reduction measured in the simulations will continue to delay transition once nonlinear, three-dimensional, and flight-relevant disturbances are present and the temperature pattern can be sustained passively.

What would settle it

A wind-tunnel experiment at Mach 6 with a controlled spanwise wall-temperature distribution that shows no measurable delay in transition location or sustained second-Mack-mode growth would falsify the practical effectiveness of the method.

Figures

Figures reproduced from arXiv: 2604.15801 by G. Rigas, L. Boscagli, O. Marxen, P. J. K. Bruce.

Figure 1
Figure 1. Figure 1: (a) Streamwise, 𝑥, and (b) spanwise, 𝑧, 2D schematics of the computational domain, boundary conditions and initial solution. Streamwise and wall-normal, y, grid refinement displayed every 10𝑡 ℎ and 15𝑡 ℎ point, respectively. Flow is left to right, and the domain is periodic in the spanwise direction. boundary layer is thinner. In the spanwise direction, 13 points per spanwise wavelength of the streaks (𝜆𝑧)… view at source ↗
Figure 2
Figure 2. Figure 2: Second Mack mode growth rate based on linear stability analysis for a laminar, [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Second Mack mode growth rate (𝜎, black) and non-dimensional phase speed (𝑐 𝑝ℎ, red) based on (uncontrolled) DNS (lines) and LST (markers); filtered (dashed line) and unfiltered (solid line) DNS data computed from wall static pressure fluctuations. Black dot-dashed line demarcates second Mack mode stable (𝜎 < 0) and unstable(𝜎 > 0) regions, respectively; red dot-dashed lines mark the phase speed of slow… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Self-similar temperature profiles and (b) second Mack mode growth rate based on (uncontrolled) DNS (lines) and LST (markers). In (b), the DNS data are computed from the spanwise averaged wall static pressure fluctuations; the black dashed line demarcates second Mack mode stable (𝜎 < 0) and unstable(𝜎 > 0) regions, respectively. et al. (2025) using appropriately selected materials with different thermal… view at source ↗
Figure 5
Figure 5. Figure 5: Effect of non-uniform wall temperature on integrated surface heat flux. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: DNS results showing wall temperature distribution for the uncontrolled and [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: DNS results showing the effect of actuator/control overlap on ( [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: DNS results showing the effect of streak wavelength ( [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: DNS results showing the (a) influence of 𝜆𝑧 on second Mack mode stabilisation (left y-axis) and maximum streak amplitude (right y-axis); (b) non-dimensional streamwise distribution of the ratio of the base flow boundary layer thickness (𝛿99) to the fundamental spanwise wavelength of the streaks (𝜆𝑧 ). When the sum of the two terms in the equations above is negative (R𝑒𝑡 ℎ,𝜌 + R𝑒𝑡 ℎ,𝑇 < 0), the energy of th… view at source ↗
Figure 10
Figure 10. Figure 10: Spatial (x-y) distribution of the thermoacoustic Reynolds stresses for [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: DNS results showing the effect of streak wavelength on the streamwise [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: DNS results showing the effect of streak wavelength ( [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: DNS results showing the effect of wall temperature on streamwise velocity [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Local, parallel LST of the DNS base flow showing the effect of control streaks [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: DNS results showing the influence of 𝜆𝑧 on second Mack mode stabilisation (left y-axis) at (nearly) constant maximum streak amplitude (right y-axis). 𝑇˜𝑤,∞ 𝑅𝑒𝑢𝑛𝑖𝑡 𝑇𝑤,𝑏𝑎𝑠𝑒 𝑅𝑒𝑥𝑠 𝑅𝑒𝑥𝑒 𝐹 = 𝜔/(𝑀2 ∞𝑅𝑒∞) 𝑅𝑒𝑥𝑐,𝑠𝑡𝑟𝑖 𝑝 𝐿𝑠𝑡𝑟 𝑖 𝑝 𝐴𝑇𝑤 216.17K 10.9 × 1061/m 3.0 1600 3200 7.5 × 10−5 1800 4.45 0.3 216.17K 10.9 × 1061/m 3.0 600 2200 12.0 × 10−5 800 2.09 0.3 216.17K 10.9 × 1061/m 3.0 100 1800 16.0 × 10−5 350 0.97 [0.2, 0.3… view at source ↗
Figure 16
Figure 16. Figure 16: DNS results showing the sensitivity of control effectiveness to changes in [PITH_FULL_IMAGE:figures/full_fig_p021_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: DNS results showing the effect of streak wavelength ( [PITH_FULL_IMAGE:figures/full_fig_p022_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: DNS results showing the influence of ℎ˜ 0,∞ on second Mack mode stabilisation (left y-axis) and streak amplitude at the streamwise location of maximum amplification of the second Mack mode (right y-axis). 0 250 Re y h˜ 0,∞ × 10−6 [J/kg] = 0.3 0 250 Re y h˜ 0,∞ × 10−6 [J/kg] = 0.7 1500 2000 2500 3000 Rex 0 250 Re y h˜ 0,∞ × 10−6 [J/kg] = 1.8 0.0e + 00 1.5e − 02 3.0e − 02 |uˆ| f k [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 19
Figure 19. Figure 19: DNS results showing the effect of freestream total enthalpy on the spatial (x-y) [PITH_FULL_IMAGE:figures/full_fig_p024_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: DNS results showing the influence of ℎ˜ 0,∞ on the streamwise distribution of the wall-normal, maximum amplitude of the second Mack mode static pressure fluctuations for the the uncontrolled (black lines) and controlled (red lines) configurations. (𝑇˜∞) also reduces and the wall gets significantly colder in absolute terms, although its non￾dimensional ratio relative to the freestream static temperature is… view at source ↗
Figure 21
Figure 21. Figure 21: DNS results showing the effect of base flow wall temperature on the streamwise [PITH_FULL_IMAGE:figures/full_fig_p026_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: DNS results showing the influence of Mach number on ( [PITH_FULL_IMAGE:figures/full_fig_p027_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: DNS results showing the effect of Mach number on second Mack mode [PITH_FULL_IMAGE:figures/full_fig_p027_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: DNS results showing the influence of base flow wall temperature on the effect [PITH_FULL_IMAGE:figures/full_fig_p028_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: DNS results showing the influence of base flow wall temperature on the modal [PITH_FULL_IMAGE:figures/full_fig_p029_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: DNS results showing (a) control method effectiveness (left y-axis) and amplitude of the streaks (right y-axis) for the heated configurations; (b) streamwise distribution of second Mack mode energy for case 1. The inset in (b) depicts the energy of the forcing disturbance. −10 −5 0 5 |ˆ·|(0,0) Uncontrolled 0 1 2 3 4 5 y/δ99,in ρ d 2u dy2 dρ dy du dy d dy  ρ du dy 0 2 4 6 |ˆ·|(0,0) Uncontrolled dρ dy du d… view at source ↗
Figure 27
Figure 27. Figure 27: DNS base flow, wall normal profiles for case 1 at [PITH_FULL_IMAGE:figures/full_fig_p030_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: DNS cold base flow (𝑇𝑤,𝑏𝑎𝑠𝑒 = 3) configuration: 𝑀∞ = 6, ℎ˜ 0,∞ = 0.7 × 106 J/kg in table 5. Perturbation profiles at various streamwise location in the region of second Mack mode amplification for the controlled configuration relative to the uncontrolled case. amplification of the second Mack mode. This provides initial guidance for future experimental investigations. A set of parametric studies has been … view at source ↗
Figure 29
Figure 29. Figure 29: Effect of spanwise grid refinement on the streamwise distribution of ( [PITH_FULL_IMAGE:figures/full_fig_p033_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Effect of the overlap between the disturbance forcing region and the control on [PITH_FULL_IMAGE:figures/full_fig_p034_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Distribution of wall (a) temperature and (b) instantaneous static pressure fluctuations for the case with 𝜆𝑧,𝑑𝑜𝑚𝑎𝑖𝑛 = 4𝜆𝑧 . The black-dashed line marks the end of the region of blowing and suction (actuator). 2000 2500 3000 Rex 0.00 0.01 0.02 As u λz 4λz (a) 2000 2500 3000 Rex 0 50 100 150 Efk Chu/Efk Chu, 0 (f, k) = (1,0) λz 4λz (b) [PITH_FULL_IMAGE:figures/full_fig_p035_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Streamwise distribution of (a) streak amplitude and (b) second Mack mode linear amplification for the configurations with 𝜆𝑧,𝑑𝑜𝑚𝑎𝑖𝑛 = 𝜆𝑧 (red) and 4𝜆𝑧 (blue). size (𝜆𝑧,𝑑𝑜𝑚𝑎𝑖𝑛 = 4𝜆𝑧 = 4.8, figure 31a). This enabled an assessment of a possible influence of streaks sub-harmonics (𝜆 ⩾ 2𝜆𝑧) on the linear amplification of the second Mack mode. The number of grid nodes in the spanwise direction was similarly inc… view at source ↗
Figure 33
Figure 33. Figure 33: Effect of Mach number on maximum spatial growth rate of first and second [PITH_FULL_IMAGE:figures/full_fig_p036_33.png] view at source ↗
read the original abstract

The extreme heat fluxes characteristic of hypersonic flows significantly limit the flight envelope of hypersonic vehicles. The role of hydrodynamic instability and the onset of laminar to turbulent boundary layer transition is of notable importance. The effect of streaks on the suppression of planar (second Mack mode) instabilities has been previously investigated, but a potentially passive and non-intrusive control method has not been established yet. Recent work shows that streaks can be generated through a spanwise variation in surface temperature. This method exploits the aerothermodynamic characteristics of the flow, and therefore promises to be robust. This work uses direct numerical simulations to determine and quantify the effectiveness of this novel control method in the suppression of second Mack mode instability for a hypersonic boundary layer over a flat plate. The computational analyses cover a range of Mach numbers 4.8 to 6 and wall temperature ratios representative of both wind tunnel testing and flight scenarios. Among the range of configurations investigated the energy of the second Mack mode is reduced by up to approximately 60% by the steady streaks. The streak wavelength parameter plays a significant role in the stabilisation benefits. For a Mach 6 configuration, for the most linearly amplified second Mack mode disturbance frequency, nearly optimum performance is achieved for a spanwise wavelength of approximately 8 to 10 times the local boundary layer thickness. These findings open new avenues for controlling hypersonic boundary layers and offer valuable guidance for future experimental campaigns aimed at validating this novel control strategy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. This paper uses direct numerical simulations of hypersonic flat-plate boundary layers (Mach 4.8–6) to show that spanwise non-uniform wall temperature distributions generate steady streaks that reduce second-Mack-mode energy by up to ~60 %. An optimal spanwise wavelength of 8–10 local boundary-layer thicknesses is identified for the most amplified frequency at Mach 6. The work positions the temperature-induced streaks as a potentially passive control strategy.

Significance. If the reported linear energy reductions are robust and extend to nonlinear regimes, the method could supply a practical, aerothermodynamically driven route to boundary-layer stabilization. The DNS parameter study provides concrete guidance on wavelength selection. However, the absence of grid-convergence data, baseline validation, and any demonstration of delayed transition under finite-amplitude or broadband disturbances limits the immediate engineering impact.

major comments (2)
  1. [Numerical methods / results] Numerical methods / results sections: the 60 % energy-reduction figures are presented without grid-convergence studies, disturbance-initialization details, or direct comparison of baseline second-Mack-mode growth rates against established literature values, rendering the quantitative stabilization claim difficult to assess.
  2. [Results and discussion] Results and discussion: the manuscript tracks only the linear energy evolution of the primary second Mack mode; no simulations of secondary instability, nonlinear breakdown, or skin-friction rise are reported, so the central claim that the streaks produce stabilization (and ultimately delayed transition) rests on an untested extrapolation from linear modal damping.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'up to approximately 60 %' would be clearer if accompanied by the specific Mach number, wall-temperature ratio, and frequency at which the maximum occurs.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their constructive comments, which help clarify the scope and limitations of our study. We respond to each major comment below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [Numerical methods / results] Numerical methods / results sections: the 60 % energy-reduction figures are presented without grid-convergence studies, disturbance-initialization details, or direct comparison of baseline second-Mack-mode growth rates against established literature values, rendering the quantitative stabilization claim difficult to assess.

    Authors: We agree that these elements are necessary to substantiate the reported energy reductions. In the revised manuscript we will incorporate a dedicated grid-convergence study for the Mach 6 cases that yield the largest stabilization, explicit details on the initial disturbance (amplitude, frequency, and spanwise structure of the second Mack mode), and direct comparisons of the baseline growth rates against established literature values for hypersonic flat-plate boundary layers at Mach 4.8–6. These additions will allow readers to evaluate the quantitative claims more rigorously. revision: yes

  2. Referee: [Results and discussion] Results and discussion: the manuscript tracks only the linear energy evolution of the primary second Mack mode; no simulations of secondary instability, nonlinear breakdown, or skin-friction rise are reported, so the central claim that the streaks produce stabilization (and ultimately delayed transition) rests on an untested extrapolation from linear modal damping.

    Authors: The manuscript deliberately restricts attention to the linear regime, showing that spanwise temperature-induced streaks damp the primary second Mack mode energy. We will revise the discussion to state more explicitly that the observed linear damping constitutes a necessary first step toward stabilization and that any inference about delayed transition remains suggestive. The central claim is framed as suppression of the linear instability rather than a complete demonstration of transition control. While we acknowledge the value of nonlinear simulations, they lie outside the present scope. revision: partial

standing simulated objections not resolved
  • Demonstration of transition delay under finite-amplitude or broadband disturbances, which would require additional nonlinear direct numerical simulations not performed in this work.

Circularity Check

0 steps flagged

No circularity: results are direct outputs of DNS with no self-referential derivations or fitted predictions

full rationale

The paper reports stabilization effects (up to ~60% energy reduction and optimal spanwise wavelengths of 8-10 boundary-layer thicknesses) exclusively from time-marching Navier-Stokes simulations with imposed steady temperature streaks. No analytical derivation chain, parameter fitting presented as prediction, or self-citation that bears the load of the quantitative claims is present in the provided text. The central results are therefore independent computational outputs rather than reductions to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard compressible Navier-Stokes equations solved by DNS, the assumption that prescribed wall-temperature boundary conditions accurately represent a realizable passive surface, and the interpretation that linear-mode energy reduction implies transition delay. No new physical constants or entities are introduced.

axioms (2)
  • standard math Compressible Navier-Stokes equations govern the hypersonic boundary-layer flow
    Invoked implicitly by the use of DNS for the reported instability suppression
  • domain assumption Steady spanwise temperature distribution can be imposed without feedback on the flow or additional energy cost
    Required for the method to be passive and robust as claimed

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