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Excitation and stability of nonlinear compressible G\"ortler vortices and streaks induced by free-stream vortical disturbances

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A nonlinear receptivity framework based on the compressible boundary-region equations quantitatively links free-stream vortical disturbance level, Mach number, and wall curvature to the appearance of Görtler vortices versus streaks and to…

desk verdict Solid extension of the boundary-region program, but the experimental comparison is normalized and partly circular; the 'quantitative link' in the abstract is overstated. read the letter →

arxiv 2411.15478 v1 pith:522XHDIC submitted 2024-11-23 physics.flu-dyn

Dongdong Xu , Pierre Ricco , Elena Marensi This is my paper · ORCID
classification physics.flu-dyn
keywords Görtlervorticesboundary-layerreceptivitynonlinearstreakscompressibleboundarylayerfree-streamvorticaldisturbanceswallheattransfersecondaryinstabilityturbineblades
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to establish a quantitative link between the free-stream disturbance environment and the nonlinear development of compressible boundary layers over curved walls, treating Görtler vortices and streaks as the same receptivity phenomenon rather than separate instabilities. Solving the compressible nonlinear boundary-region equations with initial and boundary conditions that model a pair of oblique free-stream vortical modes, the authors show that the Görtler number, disturbance Reynolds number, Mach number, and free-stream turbulence level jointly decide whether mushroom-shaped Görtler vortices or bell-shaped streaks emerge. If the framework is right, the occurrence of Görtler vortices versus streaks, the enhanced skin friction, and the streamwise-elongated hot fingers on turbine-blade pressure surfaces can be predicted from measurable free-stream parameters without full-scale simulation. The paper reports agreement with experimental skin-friction and wall-heat-transfer measurements on turbine pressure surfaces, and identifies a new secondary-instability mode that may trigger transition at the stem of nonlinear streaks.

What carries the argument

The load-bearing object is the compressible nonlinear boundary-region equations, the rigorous parabolic asymptotic limit of the Navier-Stokes equations for low-frequency, long-wavelength disturbances, in which streamwise diffusion and the streamwise pressure gradient are absent but spanwise diffusion and wall curvature are retained through the Görtler number $G$ entering the wall-normal momentum equation as $G \tilde{u}^2$. They are solved as an initial-boundary-value problem: initial conditions come from a small-$\bar{x}$ expansion matched to the leading-edge region, outer boundary conditions are set by the free-stream vortical disturbance, and the perturbation is expanded in temporal and spanwise Fourier harmonics, with nonlinear terms evaluated pseudo-spectrally. The two order-one parameters $G$ and $r_t = \epsilon R_\Lambda$ carry, respectively, the centrifugal and nonlinear effects, and the secondary-instability analysis uses Floquet theory on the spanwise-periodic saturated state to find high-frequency modes.

What would settle it

Direct numerical simulation or experiment of the same turbine-blade pressure surface at the same Reynolds number, Mach number, and turbulence level, but with the actual streamwise pressure gradient and leading edge included, should reproduce the computed normalised skin-friction and Stanton-number enhancements: if the measured hot-finger spanwise spacing is not half the dominant free-stream wavelength, or if the enhancement for $T_u > 1\%$ disappears, the claimed quantitative link fails.

Watch

Extended reading notes

Core claim

The central claim is that, in the parameter regime relevant to high-pressure turbine blades, with order-one Görtler number $G$ and order-one disturbance Reynolds number $r_t = \epsilon R_\Lambda$, the compressible nonlinear boundary-region equations forced by free-stream vortical disturbances capture the route from receptivity to saturation to secondary instability. At moderate disturbance intensities, increasing wall concavity destabilises the boundary layer; increasing Mach number or frequency stabilises thermal disturbances; at high intensity the concave wall loses its grip and vortices give way to streak-like, bell-shaped structures. The resulting occurrence map in turbulence level and Görtler number separates nonlinear Görtler vortices from nonlinear streaks, with the boundary flattening near $T_u = 3\%$ for large $G$, and the authors argue this explains why mushroom-shaped structures are rarely seen over real turbine blades. The saturated mean-flow distortion produces streamwise-elongated wall-heat-transfer modulations, hot fingers, whose spanwise wavelength is half the characteristic wavelength of the free-stream disturbances, matching liquid-crystal measurements. Nonlinearly saturated states support high-frequency secondary modes whose growth rate rises with $G$, including a new varicose even mode localised at the stem of nonlinear streaks.

Load-bearing premise

The whole quantitative comparison with turbine-blade experiments rests on treating the blade boundary layer as zero-pressure-gradient compressible Blasius flow; real pressure surfaces have streamwise pressure gradients, leading-edge bluntness, and broadband turbulence that can change skin friction and heat transfer, and the paper explicitly leaves these out.

Editorial extensions

If this is right

  • On turbine-blade pressure surfaces, the occurrence map indicates that free-stream turbulence levels above roughly $3\%$ produce nonlinear streaks rather than Görtler vortices, so mushroom-shaped structures should not be expected there.
  • Wall-heat-transfer hot fingers have a spanwise wavelength equal to half that of the dominant free-stream mode, giving a testable signature for identifying their origin in experiments.
  • Secondary-instability growth rates increase with Görtler number, so increasing blade curvature hastens breakdown once disturbances saturate.
  • Increasing Mach number at fixed Reynolds number leaves skin friction unchanged but enhances wall heat flux, separating the two design quantities.
  • The new even varicose mode localised near the wall may drive transition at the stem of streaks, affecting skin friction and heat transfer earlier than the outer odd mode.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the paper ignores streamwise pressure gradients, a natural test is to repeat the computation in a favourable-pressure-gradient boundary layer: if the location of the occurrence-map boundary and the hot-finger wavelength persist, the map is robust, and if not, pressure-gradient corrections are needed before applying it to real blades.
  • The hot-finger wavelength being exactly half the forcing wavelength suggests the mechanism is the $(0,2)$ harmonic of the mean-flow distortion; measuring the spanwise spectrum of wall heat flux in a controlled gust experiment would isolate this harmonic directly.
  • The occurrence map is likely to depend on disturbance frequency and Reynolds number as well as on turbulence level and Görtler number; scaling these variables by the neutral-stability behaviour of the underlying linear theory could collapse the map onto a single curve.
  • If the new even mode II is confirmed in direct numerical simulations of flat-plate streaks with high free-stream turbulence, it would give a concrete path from free-stream gust intensity to a specific transition mechanism.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper presents a nonlinear receptivity framework for compressible Görtler vortices and streaks. The free-stream disturbance is modelled as a pair of oblique vortical modes of equal frequency and opposite spanwise wavenumbers, and the boundary-layer response is governed by the compressible nonlinear boundary-region equations, derived as the leading-order asymptotic limit of the compressible Navier-Stokes equations. The authors solve the resulting initial-boundary-value problem by streamwise marching and study the effects of Görtler number, free-stream disturbance level, and Mach number on disturbance amplitudes, wall-shear stress, and wall-heat transfer. Results are compared with turbine-blade experiments, an occurrence map for Görtler vortices versus streaks is constructed, and secondary instabilities of nonlinearly saturated states are analysed, including a newly reported even varicose mode.

Significance. The framework is technically substantial and original in combination: the boundary-region equations are derived without ad-hoc closures for the present curved-wall compressible setting, the numerical method is state of the art, and the parameter study covers conditions relevant to high-pressure turbine blades. If the quantitative-link claims were fully supported, the paper would provide a predictive tool for turbomachinery transition. As it stands, the main value is as a nonlinear receptivity framework with a qualitative experimental match, a potentially useful occurrence map, and a systematic secondary-instability analysis, including a new even mode. The half-wavelength hot-finger statement and the 'quantitative link' claim are overstated in their present form but appear correctable within the manuscript's scope.

major comments (4)
  1. [§4.3, Figs 10–11, Abstract, §5] The abstract and §5 state that the calculations 'capture well' the enhancement of skin friction and wall-heat transfer and that the framework links free-stream parameters 'quantitatively' to the transitional boundary layer. The evidence in Figs 10 and 11 does not support this level of claim. Both Cf and St are normalized by their values at x_s = 0.06, so the comparisons test only the shape of the streamwise trend, not the absolute predictive accuracy. In Fig. 10(b), the Arts et al. (1990) skin-friction data are not measured but are obtained 'via our Reynolds analogy factors' (§4.3), i.e. using the computation being validated, which makes the apparent skin-friction agreement partly circular. The Radomsky & Thole (2002) data are at different turbulence intensities and on a different blade geometry, with a pressure gradient that is small but nonzero. Although the zero-pressure-gradient limitation is acknowledged in §1.3, the abstract and conclusions should be reworded to state qualitative agreement, and an absolute comparison, or at least a report of the unnormalized values, should be provided if the quantitative claim is retained.
  2. [Abstract; §2, Eq. (2.1); §4.3, Eq. (4.5), Fig. 13] The claim that the hot fingers have spanwise wavelength half that of the free-stream disturbance is a necessary consequence of the forcing model, not an independent prediction. The input (2.1) contains the two oblique modes n = ±1, and their nonlinear product generates the steady (0,±2) harmonic, which is precisely the mode shown in Figs 12(d) and 13(d) to produce the streaky wall patterns. This half-wavelength result would hold for any real spanwise-periodic input of wavenumber k_z within this two-mode model. It should be presented as a property of the model, and ideally tested with a different spanwise spectrum or with a single oblique mode, before being listed as a standalone finding.
  3. [§3, numerical parameters] The nonlinear results, including the harmonic amplitudes in Fig. 5, the occurrence map in Fig. 16, and the hot-finger patterns in Figs 12–13, depend on the spectral truncation N_t = N_z = 17 and on the streamwise and wall-normal grids. The statement that 17 modes are 'sufficient' is not accompanied by a convergence or truncation-sensitivity study. Because the central nonlinear mechanism is the transfer of energy from the fundamental modes to the (0,0), (0,2), and (2,2) harmonics, a convergence check for at least one reference case should be reported before the quantitative aspects of the nonlinear results are accepted.
  4. [§4.4, Fig. 16] The vortex/streak boundary in the occurrence map is based on a classification criterion (positive-concavity growth followed by saturation plus mushroom-shaped cross-sections) applied at a single frequency k_x = 0.0073, R_Lambda = 1124, and M∞ = 0.69. The text presents the map as representative of subsonic turbine-blade flows, but no sensitivity to k_x, κ_y, or the assumed FVD polarization is given. Since these parameters affect receptivity amplitudes and growth rates, the boundary location may not be robust. The interpretive claim should be restricted to the computed parameter range, or supplemented by a parameter-sensitivity study.
minor comments (5)
  1. [§1.3] The phrase 'the reader is refereed to table 2 of Xu et al. (2024)' should read 'the reader is referred to table 2 of Xu et al. (2024)'.
  2. [§2, Eq. (2.1)] The notation in Eq. (2.1), in particular the left-hand side and the combination of the two oblique modes, is difficult to parse; clarifying that the two terms are the complex amplitudes of the n = +1 and n = −1 spanwise components would improve readability.
  3. [§4.3, Fig. 10(b)] The statement that this is the 'first numerical verification of the effect of FVD level in the experiments of Arts et al. (1990)' is too strong, given the normalization by the value at x_s = 0.06 and the use of computed Reynolds-analogy factors to convert the heat-transfer data.
  4. [§4.5, discussion of even mode II] The claim that even mode II 'could potentially be more critical than the more unstable odd mode I' is presented without a quantitative measure (e.g., wall-normal location weighted by amplitude or transient growth); it should be explicitly labelled as a qualitative conjecture.
  5. [§4.3, Fig. 13] The comparison with the hot-finger visualizations of Butler et al. (2001) is qualitative, as the paper notes, but the caption and text could more clearly distinguish the computed time-averaged wall-heat transfer from the experimental liquid-crystal images, which include pressure-gradient and broadband-turbulence effects absent in the model.

Circularity Check

2 steps flagged · score 6.0 of 10

Two headline results are partially circular: the hot-finger half-wavelength is an algebraic consequence of the +/-k_z forcing ansatz, and the Arts et al. skin-friction comparison is generated through the model's own Reynolds-analogy factors; the core receptivity computation itself remains self-contained.

  1. other [Abstract; §2, Eq. (2.1); §4.3, Eq. (4.5), Fig. 13(d)]
    "[FVD] consisting of a pair of vortical modes with the same frequency (and hence streamwise wavenumber), but opposite spanwise wavenumbers ±k_z. ... [S]imilarly to F, the wall-heat flux modulation is induced by the steady mode (0,2). ... The time-averaged wall-heat transfer modulations, termed hot fingers, are elongated in the streamwise direction and their spanwise wavelength is half of the characteristic wavelength of the free-stream disturbances."

    The forcing (2.1) contains only spanwise Fourier components e^{+ik_z z} and e^{-ik_z z}. The quadratic interaction of the forced modes (1,+1) and (1,-1) necessarily generates the steady (0,2) component, whose spanwise wavenumber is 2k_z, i.e. wavelength Lambda/2. The reported 'half-wavelength' hot-finger spacing is therefore an algebraic consequence of the two-mode ansatz and the harmonic expansion (2.14); it is encoded in the input rather than being an emergent prediction that could have come out differently.

  2. fitted input called prediction [§4.3, Figs. 10(b) and 11(b)]
    "As the wall-shear stress was not measured by Arts et al. (1990), we have used their wall-heat transfer data and computed the skin-friction coefficients via our Reynolds analogy factors. Considering that the Reynolds analogy is not strictly valid in pressure-gradient and transitional flows, our estimate of the skin-friction coefficient can only be regarded as qualitative."

    The Reynolds analogy factor Ra = 2 St / C_f shown in Fig. 9 is computed from the same nonlinear boundary-region solution that is being validated. Putting the experimental Stanton numbers through that model Ra yields 'experimental' C_f values that inherit the model's own relation between heat transfer and skin friction, so the Fig. 10(b) C_f agreement is not an independent check of skin friction; it largely restates the Stanton-number comparison. The claim of capturing the enhanced skin friction is therefore partly constructed from the model's own output, even though the authors explicitly label the estimate qualitative.

full rationale

The governing nonlinear boundary-region equations (2.6)-(2.10) are not fitted to the comparison data; they are solved by direct marching with prescribed free-stream-disturbance parameters, and the Mach/curvature/FVD trends, nonlinear saturation, occurrence map and secondary-instability modes are produced by the computation itself. No load-bearing uniqueness theorem is imported from the authors' prior work; the citations to Marensi et al. (2017), Viaro & Ricco (2019a), Ricco (2007) and Xu et al. (2017) supply initial conditions and nonlinear-term definitions that are prior derivations, not the present conclusions. However, two steps in the headline claims do reduce by construction. First, the abstract's hot-finger spanwise wavelength 'half of the characteristic wavelength' is forced by the choice of two oblique FVD modes ±k_z in (2.1), because the nonlinear product of modes (1,+1) and (1,-1) generates the steady (0,2) harmonic with wavenumber 2k_z; the half-wavelength is an algebraic identity of the ansatz rather than an emergent prediction. Second, because Arts et al. did not measure wall shear, their 'skin-friction coefficients' are computed from the model's own Reynolds-analogy factors, i.e. C_f = 2 St / Ra_computed, so the C_f comparison in Fig. 10(b) largely restates the Stanton-number comparison and is not an independent skin-friction validation. The additional normalization by the value at x_s = 0.06 means absolute enhancement levels are not tested. These caveats do not invalidate the self-contained receptivity framework, but they mean the 'quantitative link' and 'capture well' claims are only partially supported, giving a score of 6 rather than a fully circular result.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central computation rests on the asymptotic boundary-region framework and on several modeling simplifications: a two-mode free-stream disturbance, no pressure gradient, and a finite Fourier truncation. None are new physical entities. The occurence-map classification and the hot-finger scaling are, to a degree, products of these assumptions rather than independent discoveries.

free parameters (3)
  • Reference flow parameters (M∞, T_w, R_Λ, G, k_x, κ_z, κ_y)
    Chosen to match high-pressure turbine blade experiments (Arts et al. 1990; Camci & Arts 1990), not fitted to the presented results.
  • FVD amplitude and polarization (u∞_x,± = u∞_y,± = 1, u∞_z,± = -1)
    Selected to satisfy the solenoidal condition and to model a particular gust orientation; spans a single disturbance configuration rather than a spectrum.
  • Numerical truncation N_t=N_z=17
    Set after stating sufficiency in §3; no convergence test is provided.
assumptions (6)
  • domain assumption Free-stream vortical disturbance is a pair of oblique modes with equal frequency and opposite spanwise wavenumbers ±k_z (2.1)
    Simplification of broadband free-stream turbulence; the paper states it is 'reasonable to study vortices excited by a pair of dominant oblique FVD components' (§2).
  • domain assumption The flow is a perfect gas with constant Prandtl number Pr=0.707 and viscosity μ=T^0.76; free stream is isentropic and shocks are weak and distant
    Standard for subsonic and transonic boundary layers; invoked in the base flow (2.13) and in equations (2.6)-(2.10).
  • domain assumption The boundary layer is zero-pressure-gradient compressible Blasius flow; curvature does not affect the base flow at leading order
    Stated in §1.3 as a limitation; base flow (2.13) neglects pressure gradient and leading-edge bluntness.
  • ad hoc to paper Fourier truncation N_t=N_z=17 modes is sufficient to capture nonlinear effects
    Stated in §3 without a convergence study.
  • ad hoc to paper Definition of Görtler vortices vs streaks based on positive-concavity growth and mushroom shape
    Introduced in §4.4 to draw the occurrence map; not derived from a physical criterion.
  • domain assumption Low-frequency, long-streamwise-wavelength disturbances are the most receptive; k_x=O(R^{-1})
    Asymptotic distinguished limit for the boundary-region equations, following Hall (1983).

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Pith. "Pith review of Excitation and stability of nonlinear compressible G\"ortler vortices and streaks induced by free-stream vortical disturbances." pith.science (2026). https://pith.science/paper/522XHDIC

@misc{pith2026241115478,
  author       = {Pith},
  title        = {Pith review of: Excitation and stability of nonlinear compressible G\"ortler vortices and streaks induced by free-stream vortical disturbances},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/522XHDIC}},
  note         = {Machine review of arXiv:2411.15478}
}
read the original abstract

We study the generation, nonlinear development and secondary instability of unsteady G\"ortler vortices and streaks in compressible boundary layers exposed to free-stream vortical disturbances and evolving over concave, flat and convex walls. The formation and evolution of the disturbances are governed by the compressible nonlinear boundary-region equations, supplemented by initial and boundary conditions that characterise the impact of the free-stream disturbances on the boundary layer. Computations are performed for parameters typical of flows over high-pressure turbine blades, where the G\"ortler number, a measure of the curvature effects, and the disturbance Reynolds number, a measure of the nonlinear effects, are order-one quantities. At moderate intensities of the free-stream disturbances, increasing the G\"ortler number renders the boundary layer more unstable, while increasing the Mach number or the frequency stabilises the flow. As the free-stream disturbances become more intense, vortices over concave surfaces no longer develop into the characteristic mushroom-shaped structures, while the flow over convex surfaces is destabilised. An occurrence map identifies G\"ortler vortices or streaks for different levels of free-stream disturbances and G\"ortler numbers. Our calculations capture well the experimental measurements of the enhanced skin friction and wall-heat transfer over turbine-blade pressure surfaces. The time-averaged wall-heat transfer modulations, termed hot fingers, are elongated in the streamwise direction and their spanwise wavelength is half of the characteristic wavelength of the free-stream disturbances. Nonlinearly saturated disturbances are unstable to secondary high-frequencymodes, whose growth rate increases with the G\"ortler number. A new varicose even mode is reported, which may promote transition to turbulence at the stem of nonlinear streaks.

Figures

Figures reproduced from arXiv: 2411.15478 by the authors.

Figure 1
Figure 1. Schematic of the physical domain for the concave-wall case. The sketches of the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Effect of Gortler number on the downstream development of ¨ 𝑢𝑟𝑚𝑠,𝑚𝑎𝑥 and 𝜏𝑟𝑚𝑠,𝑚𝑎𝑥 induced by (𝑎, 𝑏) 𝑇𝑢 = 1% and (𝑐, 𝑑) 𝑇𝑢 = 6%. same as in the flat-wall case. The convex curvature is not influential up to ¯𝑥 = 0.35 for such a higher FVD level. For the cases considered, the boundary-layer dynamics is therefore largely independent of the curvature up to 𝑥𝑠 = 1.2, i.e. for most of the extent of the turbine blade [PITH… view at source ↗
Figure 3
Figure 3. Effect of FVD level on the downstream development of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: Effect of Mach number on the downstream development of [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Development of the fundamental mode (𝑚, 𝑛) = (1, 1) and the harmonic components (𝑚, 𝑛) = (0, 0), (2, 0), (0, 2), (2, 2) of temperature disturbance for different Gortler numbers: ¨ (a,b) G = 35.2, (c,d) G = 0, (e,f) G = −281.6, and FVD levels: (a,c,e) 𝑇𝑢 = 1%, (b,d,f) 𝑇…
Figure 6
Figure 6. Figure 6: Profiles of instantaneous (𝑎, 𝑐, 𝑒) streamwise velocity and (𝑏, 𝑑, 𝑓 ) temperature at ¯𝑥 = 0.54, 𝑧 = 0 for 𝑇𝑢 = 6% and different Gortler numbers. ¨ with the phase, becoming highly inflectional at certain phases (𝜙 = 𝜋/2 and 3𝜋/4). This behaviour suggests that the flow …
Figure 7
Figure 7. Figure 7: Contours of the instantaneous (𝑎 − 𝑐) streamwise velocity and (𝑑 − 𝑓 ) temperature in the 𝑦 − 𝑧 plane for 𝑇𝑢 = 1% at ¯𝑥 = 1.5. The increment of the contour values is 0.1 for the velocity and 0.05 for the temperature. The coordinate 𝑦 is related to the similarity variab…
Figure 8
Figure 8. Figure 8: Contours of the instantaneous (𝑎 − 𝑐) streamwise velocity and (𝑑 − 𝑓 ) temperature in the 𝑦 − 𝑧 plane for 𝑇𝑢 = 6% at ¯𝑥 = 0.36. The increment of the contour values is 0.1 for the velocity and 0.05 for the temperature [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Reynolds analogy factor along the streamwise direction for different FVD levels [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Comparison of (𝑎) the computed skin-friction coefficients with (𝑏) the experimental data of Arts et al. (1990) (A) and Radomsky & Thole (2002) (RT). The coefficients are normalised by the value C𝑓 0 at 𝑥𝑠 = 0.06. The line in (𝑏) shows the skin-friction coefficient com…
Figure 11
Figure 11. Figure 11: Comparison of (𝑎) the computed Stanton numbers with (𝑏) the experimental data of Arts et al. (1990) (A). The Stanton numbers are normalised by the value S𝑡0 at 𝑥𝑠 = 0.06. The line in (𝑏) shows the Stanton number computed by large eddy simulations (LES) without inflow …
Figure 12
Figure 12. Figure 12: (𝑎 − 𝑐) Time-averaged wall-shear stress F (𝑥𝑠, 𝑧𝑠), defined in equation (4.4), for different FVD levels. Panel (𝑑) shows the contour of the timed-averaged streamwise velocity streaks, given by mode (0,2). The wall-normal coordinate is 𝑦𝑠 = 𝑦 ∗ /𝐶 ∗ 𝑎𝑥. The Gortler num…
Figure 13
Figure 13. Figure 13: (𝑎 − 𝑐) Absolute value of the time-averaged wall-heat transfer, |Q (𝑥𝑠, 𝑧𝑠)|, defined in equation (4.5), for different FVD levels. Panel (𝑑) is a contour of the timed-averaged temperature streaks, given by mode (0,2). Panel (𝑒) shows the experimental measurements of B…
Figure 14
Figure 14. Figure 14: (𝑎 − 𝑐) Time-averaged wall-shear stress, F (𝑥𝑠, 𝑧𝑠), defined in equation (4.4), and (𝑑 − 𝑓 ) absolute value of the time-averaged wall-heat transfer, |Q (𝑥𝑠, 𝑧𝑠)|, defined in equation (4.5). The numerical data are for (𝑎, 𝑑) M∞ = 0, (𝑏, 𝑒) M∞ = 0.69 and (𝑐, 𝑓 ) M∞ = 1.…
Figure 15
Figure 15. Figure 15: (𝑎) Growth of 𝑢𝑟𝑚𝑠,𝑚𝑎𝑥 for G = 35.2 and different FVD levels. The portions of the trends highlighted in red indicate where the linear and the nonlinear solutions overlap. The darker portions of the trends denote a 𝑢𝑟𝑚𝑠,𝑚𝑎𝑥 growth with positive concavity. The saturatio…
Figure 16
Figure 16. Figure 16: Occurrence map of nonlinear streaks and G [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: (𝑎, 𝑐) Temporal growth rates and (𝑏, 𝑑) phase speeds of the secondary-instability modes of Gortler vortices. Panels ( ¨ 𝑎, 𝑏) are for the concave-wall case (G = 35.2) and panels (𝑐, 𝑑) are for the flat-wall case (G = 0). The red lines represent odd mode I, the blue li…
Figure 18
Figure 18. Figure 18: Eigenfunctions of secondary unstable modes, shown by contours of the [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]
Figure 19
Figure 19. Figure 19: Characteristics of secondary instability of streaks: [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: Eigenfunctions (absolute value, black lines) of secondary unstable modes, [PITH_FULL_IMAGE:figures/full_fig_p030_20.png]

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Pith tools

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