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arxiv: 2607.01097 · v1 · pith:ZL35RTTNnew · submitted 2026-07-01 · ⚛️ physics.flu-dyn

Numerical Study of Compressibility and Velocity Parameter Effects on Spatially Evolving Supersonic Turbulent Shear Layers

M. R. B. Shahadat , Z. Li , F. A. Jaberi , D. Livescu This is my paper

Pith reviewed 2026-07-02 05:16 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords supersonic shear layersconvective Mach numberentrainment ratioself-similaritydirect numerical simulationcompressibility effectsturbulent mixingcross-stream velocity
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The pith

DNS of supersonic shear layers shows entrainment ratio increases with convective Mach number and velocity parameter, with excess on the high-speed side.

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

The paper runs direct numerical simulations of spatially developing supersonic turbulent shear layers across a range of convective Mach numbers and velocity parameters. It identifies a far-downstream self-similar regime where thickness grows linearly, peak Reynolds stresses stay constant, and normalized statistics collapse under scalings that include compressibility and centerline shifts. From the self-similar energy equation the authors derive an approximate cross-stream velocity profile that matches the DNS data. This profile supplies a closed-form expression for the entrainment ratio, which rises with both compressibility and velocity difference. A reader would care because the result supplies a practical way to estimate mixing rates in high-speed compressible flows without running full simulations for every case.

Core claim

At distant downstream locations self-similarity is attained for all cases, identified by collapse of normalized mean streamwise velocity, constant peak normalized Reynolds stresses, and linear growth of shear-layer thickness and momentum thickness. The self-similar forms of the continuity, momentum, and energy equations are written with compressibility and centerline shifts included. The normalized density distribution inside the layer explains compressibility effects on statistics and far-field cross-stream velocity; density variation is tied to dissipation in the energy equation. An approximate equation for cross-stream velocity is obtained whose profiles agree with DNS, and this equation

What carries the argument

The approximate equation for cross-stream velocity derived from the self-similar energy equation, which incorporates the normalized density distribution to account for compressibility effects on entrainment.

If this is right

  • All examined lower-order and higher-order turbulence statistics collapse inside the self-similar region when the proposed scalings are used.
  • The self-similar normalized density distribution accounts for compressibility effects on far-field cross-stream velocity.
  • The entrainment ratio increases with both convective Mach number and velocity parameter.
  • Entrainment favors the high-speed side, consistent with a geometric interpretation of the ratio.

Where Pith is reading between the lines

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

  • The derived entrainment expression could be inserted into reduced-order models for supersonic mixing layers without requiring new DNS for each parameter set.
  • If the density-dissipation link holds beyond the simulated range, similar scalings might simplify predictions for other spatially developing compressible shear flows.
  • Checking whether the approximate cross-stream velocity equation remains accurate when the layer is forced or when chemistry is added would test its broader utility.

Load-bearing premise

The proposed self-similar scalings that incorporate compressibility and centerline shifts remain valid and collapse all statistics once the layer reaches the far-downstream regime of linear thickness growth and constant peak Reynolds stresses.

What would settle it

A new DNS run at a higher convective Mach number or different velocity parameter in which the cross-stream velocity profile predicted by the approximate equation deviates measurably from the simulated profile would falsify the central claim.

Figures

Figures reproduced from arXiv: 2607.01097 by D. Livescu, F. A. Jaberi, M. R. B. Shahadat, Z. Li.

Figure 1
Figure 1. Figure 1: Planar view of spatially developing shear layer. The two free streams are represented by subscripts 1 and 2, respectively. Stream 1 represents the high-speed side and stream 2 represents the low-speed side. (- - - - -) line represents the upper edge and (- - - - -) line represents the lower edge of the shear layer. The difference between these two lines is the shear layer thickness (𝛿). (- - - - -) line (𝑥… view at source ↗
Figure 2
Figure 2. Figure 2: Transverse domain size independency for (a) momentum thickness (b) TKE      1 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spanwise domain size independency for (a) shear layer thickness (b) momentum thickness these three domain sizes exhibit acceptable convergence. Since the results of the domain size 𝐿𝑥2=256 and 𝐿𝑥2=296 are almost identical, 𝐿𝑥2=256 was used in most of the simulated cases. A slightly larger 𝐿𝑥2=268 was used for one of the cases to reach the same convergence level as for the rest of the cases. The domain size… view at source ↗
Figure 4
Figure 4. Figure 4: Grid convergence for (a) Momentum thickness (b) Shear Layer thickness (c) Mean streamwise velocity (d) TKE (e) Mean scalar (f) Mean dissipation simulations, a grid size of 0.2 is employed in all three directions, as it is computationally affordable and, with this grid size, the statistics considered are accurately calculated. 4.4. Cases Simulated This study systematically examines the impacts of 𝑀𝑐 and 𝜆 b… view at source ↗
Figure 5
Figure 5. Figure 5: 3D and 2D contours of the reference case: (a) 3D iso-contours of instantaneous streamwise velocity (𝑢1) (b) 3D iso-contours of Q-criterion colored by streamwise velocity (c) 2D iso-contours of instantaneous density and (d) 2D iso-contours of instantaneous pressure. Figures (c) and (d) are magnified to provide a closer examination of the core of the shear layer. Figures show that the shear layer is confined… view at source ↗
Figure 6
Figure 6. Figure 6: Streamwise evolution of mean (a) streamwise velocity, (b) density, (c) pressure, (d) transverse/cross￾stream velocity, (e) Reynolds normal stress in the streamwise direction, (f) Reynolds shear stress, and (g) Reynolds normal stress in the transverse direction. The Reynolds stress profiles have peak values around the center of the shear layer and go to zero outside the shear layer, as expected. In the next… view at source ↗
Figure 7
Figure 7. Figure 7: The evolution of mean and turbulent quantities in self-similar coordinates. Figures (a), (b), and (c) show the linear growth of the shear layer and momentum thickness. Figure (d) shows the collapse of the streamwise velocity profiles and figures (e), (f), (g), and (h) show the collapse of Reynolds stresses in the self-similar zone. Quantities like shear layer thickness, momentum thickness, and streamwise v… view at source ↗
Figure 8
Figure 8. Figure 8: The collapse of (a) scaled mean density, (b) scaled mean transverse velocity, (c) scaled mean pressure, and (d) scaled mean temperature in self-similar coordinates. Mean density becomes self-similar around 𝑥1 ≈ 407, and mean pressure and temperature become self-similar around 𝑥1 ≈ 467. Figure (e) shows the mean dissipation at different streamwise locations and figure (f) shows the collapse of scaled mean d… view at source ↗
Figure 9
Figure 9. Figure 9: 𝜖ˆ𝑡 compared to 𝑅ˆ 𝛥(𝜌𝑇𝑢) and 𝑅ˆ𝑃𝛥𝑢 for the reference case. It is clear from the figure that 𝜖ˆ𝑡 is much larger than the summation of 𝑅ˆ 𝛥(𝜌𝑇𝑢) and 𝑅ˆ𝑃𝛥𝑢. the equation can be expressed as follows: − ∫ +∞ −∞ 𝜌 𝑑𝜂 ˆ = 4(𝛾 − 1)𝑀2 𝑐 ∫ +∞ −∞ [𝜖ˆ𝑡 − (𝑅ˆ 𝛥(𝜌𝑇𝑢) + 𝑅ˆ𝑃𝛥𝑢)] 𝑑𝜂. (5.36) According to our DNS data, the 𝜖ˆ𝑡 is much larger compared to the summation of 𝑅ˆ 𝛥(𝜌𝑇𝑢) and 𝑅ˆ𝑃𝛥𝑢 ( [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 10
Figure 10. Figure 10: The evolution of shear layer thickness at different 𝑀𝑐 and showed that they depend on two distinct parameters that characterize the flow, namely 𝑀𝑐 and 𝜆. In this section, we consider the role of these two parameters on the evolution of the flow. In addition to the usual metrics such as shear layer thickness and mean streamwise velocity and density, we will also examine the transverse velocity dependence … view at source ↗
Figure 11
Figure 11. Figure 11: Normalized growth rate at different 𝑀𝑐 comparing current DNS results to previous literature (Kim et al. (2020a), Zhang et al. (2019), Papamoschou & Roshko (1988a), Pantano & Sarkar (2002), Samimy & Elliott (1990), Rossmann et al. (2002),Clemens & Mungal (1992), Fu & Li (2006), Freund et al. (2000), Barre et al. (1997), Debisschop & Bonnet (1993), Goebel & Dutton (1990), Kourta & Sauvage (2002)). growth ra… view at source ↗
Figure 12
Figure 12. Figure 12: Color contour (𝑥1–𝑥2 plane) of instantaneous density for a) M𝑐= 0.8 and b) M𝑐= 1.6 that the shear layer begins to develop most upstream for the smallest convective Mach number, 𝑀𝑐 = 0.8. The consistent trend of decreasing growth rate with increasing 𝑀𝑐 is evident, yet all mean and turbulent quantities maintain self-similarity across different 𝑀𝑐 values, as demonstrated in below in figures 13 and 15. The r… view at source ↗
Figure 13
Figure 13. Figure 13: The collapse of (a) mean streamwise velocity (b) 𝑅e11 (c) and (d) 𝑅e22 in two different scalings and (e) 𝑅e12 in self-similar coordinates for all the 𝑀𝑐 cases of 𝑅e11 has not been universally consistent (Kim et al. 2020a). The results of the present study are consistent with those of Burr & Dutton (1990), Goebel & Dutton (1991) and Kim et al. (2020a). The physical explanation for 𝑅e11 not being affected b… view at source ↗
Figure 14
Figure 14. Figure 14: Normalized turbulent intensities (a) 𝑈𝑟𝑚𝑠 /𝛥𝑈= √︁ 𝑅e11/𝛥𝑈 (b) 𝑉𝑟𝑚𝑠 /𝛥𝑈= √︁ 𝑅e22/𝛥𝑈 (c) 𝑈𝑉𝑟𝑚𝑠 /𝛥𝑈= √︁ 𝑅e12/𝛥𝑈 from previous literature. Pantano & Sarkar (2002) performed DNS of constant density (density ratio, s=1) temporal shear layer. Though the present simulations were not performed exactly under the same conditions, they agree with the previous literature. Mean density profiles at different 𝑀𝑐 values i… view at source ↗
Figure 15
Figure 15. Figure 15: The variation of mean density and mean pressure in the self-similar coordinate at different 𝑀𝑐 values. (a) Density profiles do not collapse using 𝜌01= 𝜌0 scaling (b) Collapse of density profile using 𝜌01= 𝜌0 𝜓(M𝑐) scaling (c) Collapse of pressure at different 𝑀𝑐 values.          [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The scaled mean transverse velocity in the self-similar coordinate compared with profiles from our analytical equation at different 𝑀𝑐 values. The analytical solution is reasonably close to the DNS results. velocity profile in the self-similar coordinate with our suggested approximate equation at different 𝑀𝑐 values. The profiles from the DNS results are very close to the approximate equation proposed [P… view at source ↗
Figure 17
Figure 17. Figure 17: Shear layer thickness at different 𝜆 values 6.2. Velocity Parameter As explained above and suggested in previous studies (Mehta 1991; RAGAB 1988; Wei et al. 2022a) the shear layer growth rate is significantly affected by the velocity parameter, 𝜆, but no comprehensive study was carried out for the compressible case, where the effects of 𝜆 could be separated from those of 𝑀𝑐 and density ratio of the free s… view at source ↗
Figure 18
Figure 18. Figure 18: Color contour (𝑥1-𝑥2 plot) of instantaneous density for a) 𝜆=0.2 and b) 𝜆=0.5 indicates that all the cases are statistically well converged and sufficient time and space averaging was considered [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The collapse of scaled (a) mean streamwise velocity, (b) 𝑅e11, (c) 𝑅e22 and (d) 𝑅e12 for different 𝜆 values in the self-similar coordinate          [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The variation of mean density and mean pressure in the self-similar coordinate at different 𝜆 values. (a) Collapse of scaled mean density using 𝜌01= 𝜌0 scaling (b) Collapse of scaled mean pressure at different 𝜆 values. two sides of the layer, as seen in figures 22 (a) and (b), with faster growth on the low-speed side as 𝑀𝑐 or 𝜆 increases. Additionally, the centerline of the shear layer shifts towards the… view at source ↗
Figure 21
Figure 21. Figure 21: The scaled mean transverse velocity compared with the profiles obtained from our approximate equation in the self-similar coordinate at different 𝜆 values. The approximate solution closely follow the DNS results.      1     [PITH_FULL_IMAGE:figures/full_fig_p035_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The growth of the shear layer denoted by the positions where the mean axial velocity reaches 90% of the free stream velocity and the centerline position (where the mean axial velocity is zero) at different (a) 𝑀𝑐 (b) v𝜆 values. The edges are defined following asymmetry parameter at different (c) 𝑀𝑐 and (d) 𝜆 values. The vertical lines indicate the locations where a self-similar zone starts for a particula… view at source ↗
Figure 23
Figure 23. Figure 23: Coordinate system for entrainment ratio analysis. Subscript h denotes the high-speed side and subscript l denotes the low-speed side. As previously noted, in the spatially developing shear layer, the centerline shifts towards the low-speed side, and due to this change in orientation, the shear layer entrains varying amounts of fluid from the free streams (D’Ovidio & Coats 2013, Dimotakis 1986). The entrai… view at source ↗
Figure 24
Figure 24. Figure 24: (a) Variation of 𝜙 with 𝑀𝑐 at 𝜆 = 0.4. The incompressible value at 𝑀𝑐 = 0 is taken from Mehta & Westphal (1985) at 𝜆 = 0.37. In the range of higher 𝑀𝑐 values, 𝜙 varies linearly with 𝑀𝑐. The equation of the linear fit using Least Square Error (LSE) is 𝜙 = 0.02𝑀𝑐 − 0.1886. (b) Incompressible and compressible 𝜙 values at different 𝑀𝑐 and velocity ratios. The incompressible data at 𝑟 = 0 and 0.46 have been ta… view at source ↗
Figure 25
Figure 25. Figure 25: (a) The variation in mean dissipation at different 𝜆 in the self-similar zones. (b) The collapse of mean dissipation at different 𝜆 using our suggested scaling. (c) The variation of mean dissipation at different 𝑀𝑐. (d) The collapse of mean dissipation at different 𝑀𝑐 using our suggested scaling Appendix A. The scaling for the mean dissipation In this section, the behavior of mean dissipation is presented… view at source ↗
read the original abstract

Direct Numerical Simulations (DNS) of a spatially developing supersonic turbulent shear layer are conducted for a range of convective Mach numbers ($M_c$) and velocity parameters ($\lambda$) to examine the effects of compressibility and advection on the growth rate, self-similarity, flow statistics, asymmetry, and entrainment of the layer. At distant downstream locations, self-similarity is attained for all cases. The self-similar region is identified by the collapse of normalized mean streamwise velocity, the constant peak of normalized Reynolds stresses, and the linear growth rate of the shear layer thickness and momentum thickness. Despite significant variations in lower-order and higher-order statistics across different $M_c$ and $\lambda$ values, profiles of all turbulence quantities examined collapse within the self-similar region using our proposed self-similar scalings. The self-similar forms of continuity, momentum, and energy equations have been formulated, incorporating compressibility and centerline shifts. The self-similar normalized density distribution inside the layer is used to explain the effects of compressibility on various flow statistics, including the far-field cross-stream velocity. The density variation is linked to dissipation effects as revealed by our analysis of the self-similar energy equation. An approximate equation for the cross-stream velocity is developed, and the profiles of cross-stream velocity obtained from this equation show good agreement with the DNS results. A geometric interpretation of the entrainment ratio is presented, and the approximate equation for the cross-stream velocity is used to provide a general closed-form expression of the entrainment ratio. The entrainment ratio increases with $M_c$ and $\lambda$, favoring excess entrainment on the high-speed side.

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

3 major / 2 minor

Summary. The paper performs DNS of spatially developing supersonic turbulent shear layers over ranges of convective Mach number Mc and velocity parameter λ. It reports that self-similarity is reached far downstream, identified by linear growth of shear-layer and momentum thickness together with constant peak normalized Reynolds stresses. Using proposed self-similar scalings that incorporate compressibility effects and centerline shifts, the authors claim collapse of mean velocity, Reynolds stresses, density, and higher-order statistics. Self-similar forms of the continuity, momentum, and energy equations are written; an approximate cross-stream velocity profile derived from the self-similar energy equation is shown to agree with the DNS, and this agreement is used to obtain a closed-form expression for the entrainment ratio, which increases with both Mc and λ and favors the high-speed side.

Significance. If the collapse under the proposed scalings and the agreement of the approximate velocity equation are robust, the work supplies a concrete link between density variation, dissipation, and entrainment asymmetry in compressible shear layers, together with an explicit entrainment-ratio formula that could be tested against other datasets. The parametric DNS database would also be a useful reference for model development, provided the numerical evidence is placed on a firmer footing.

major comments (3)
  1. [Abstract / self-similarity identification] Abstract and self-similarity section: the criteria used to declare the self-similar regime (linear δ(x) growth and constant peak Reynolds stresses) are stated as sufficient for collapse of all statistics under the compressibility-adjusted normalization plus centerline shift, yet no quantitative metric (e.g., L2 residual of normalized profiles versus downstream distance or Mc) is supplied to demonstrate that these criteria guarantee the claimed universality once Mc-dependent shifts are introduced. This assumption is load-bearing for both the collapse claim and the subsequent derivation of the cross-stream velocity equation.
  2. [Section deriving approximate cross-stream velocity] Approximate cross-stream velocity equation (derived from self-similar energy equation): because the equation inherits the self-similarity assumption directly, any residual non-self-similar transients that survive the identification criteria would appear in the reported agreement with DNS; the manuscript provides no separate test (e.g., sensitivity to the downstream station chosen or to the precise form of the centerline shift) that isolates the validity of the approximation itself.
  3. [Numerical methods / results] Numerical validation: the abstract asserts that DNS results support the collapse and the velocity-equation agreement, but no grid-convergence data, resolution criteria, or comparison against established benchmarks (e.g., known growth-rate curves or low-Mc limits) are referenced. Without these, the quantitative statements about entrainment-ratio trends with Mc and λ rest on unverified numerical evidence.
minor comments (2)
  1. [Abstract] The abstract refers to “our proposed self-similar scalings” without giving their explicit functional form; a concise statement of the normalization (including the Mc-dependent centerline shift) should appear in the abstract or early in the results section.
  2. [Introduction / nomenclature] Notation for the velocity parameter λ and the convective Mach number Mc should be defined at first use and kept consistent with standard definitions in the compressible shear-layer literature.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments that help improve the clarity and robustness of our manuscript. We provide point-by-point responses to the major comments below.

read point-by-point responses

  1. Referee: [Abstract / self-similarity identification] Abstract and self-similarity section: the criteria used to declare the self-similar regime (linear δ(x) growth and constant peak Reynolds stresses) are stated as sufficient for collapse of all statistics under the compressibility-adjusted normalization plus centerline shift, yet no quantitative metric (e.g., L2 residual of normalized profiles versus downstream distance or Mc) is supplied to demonstrate that these criteria guarantee the claimed universality once Mc-dependent shifts are introduced. This assumption is load-bearing for both the collapse claim and the subsequent derivation of the cross-stream velocity equation.

    Authors: We agree that a quantitative metric would strengthen the evidence for collapse under the proposed scalings. In the revised manuscript we will add L2 residuals of the normalized mean-velocity and Reynolds-stress profiles computed at successive downstream stations (and across Mc) to quantify the degree of self-similarity and the quality of the collapse once the Mc-dependent centerline shifts are applied. revision: yes

  2. Referee: [Section deriving approximate cross-stream velocity] Approximate cross-stream velocity equation (derived from self-similar energy equation): because the equation inherits the self-similarity assumption directly, any residual non-self-similar transients that survive the identification criteria would appear in the reported agreement with DNS; the manuscript provides no separate test (e.g., sensitivity to the downstream station chosen or to the precise form of the centerline shift) that isolates the validity of the approximation itself.

    Authors: The manuscript already presents the approximate cross-stream velocity at several stations inside the identified self-similar region. To isolate the approximation itself we will add a dedicated sensitivity study that varies both the chosen downstream station and the precise functional form of the centerline shift, reporting the resulting changes in agreement with the DNS data. revision: yes

  3. Referee: [Numerical methods / results] Numerical validation: the abstract asserts that DNS results support the collapse and the velocity-equation agreement, but no grid-convergence data, resolution criteria, or comparison against established benchmarks (e.g., known growth-rate curves or low-Mc limits) are referenced. Without these, the quantitative statements about entrainment-ratio trends with Mc and λ rest on unverified numerical evidence.

    Authors: We acknowledge that explicit numerical-validation details were omitted from the original submission. The revised manuscript will include grid-convergence tests for representative (Mc, λ) cases, resolution criteria based on local Kolmogorov scales, and direct comparisons of growth rates against established low-Mc benchmarks and literature data for supersonic shear layers. revision: yes

Circularity Check

0 steps flagged

No circularity: results generated by DNS of Navier-Stokes equations; approximate cross-stream velocity equation validated against simulation data rather than reducing to fitted inputs by construction.

full rationale

This is a direct numerical simulation study whose statistics are obtained by solving the compressible Navier-Stokes equations on a grid. The self-similar region is identified by independent criteria (linear thickness growth, constant peak Reynolds stresses, collapse of normalized mean velocity). Proposed scalings are then applied to demonstrate collapse of additional quantities, and an approximate cross-stream velocity relation is derived from the self-similar energy equation and shown to match the DNS profiles. No step reduces a claimed prediction to a fitted parameter or self-citation by construction; the entrainment-ratio expression is obtained from the approximate equation and DNS density field rather than being tautological. The derivation chain is therefore self-contained against external benchmarks (the underlying flow solver).

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the compressible Navier-Stokes equations solved by DNS; no new free parameters are introduced beyond the varied Mc and λ, and no new physical entities are postulated.

axioms (2)
  • standard math Compressible Navier-Stokes equations govern the flow
    Invoked implicitly as the basis for all DNS runs described in the abstract.
  • domain assumption Self-similarity is attained at sufficiently large downstream distances
    Stated as the regime in which normalized profiles collapse and thickness grows linearly.

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Reference graph

Works this paper leans on

130 extracted references

  1. [1]

    AIAA journal , volume=

    Development of a two-stream mixing layer from tripped and untripped boundary layers , author=. AIAA journal , volume=

  2. [2]

    An experimental study of plane mixing layer development , author=

  3. [3]

    AIAA AVIATION FORUM AND ASCEND 2024 , pages=

    Entrainment and Asymmetry in Spatially Developing Supersonic Turbulent Shear Layers , author=. AIAA AVIATION FORUM AND ASCEND 2024 , pages=

    2024

  4. [4]

    Desalination , volume=

    Modeling and experimental validation of direct contact crossflow packed beds condenser used in HDH desalination systems , author=. Desalination , volume=. 2023 , publisher=

    2023

  5. [5]

    Experiments in fluids , volume=

    Effect of velocity ratio on plane mixing layer development: Influence of the splitter plate wake , author=. Experiments in fluids , volume=. 1991 , publisher=

    1991

  6. [6]

    Physics of Fluids , year=

    Scaling patch analysis of planar turbulent mixing layers , author=. Physics of Fluids , year=

  7. [7]

    Physical Review E , volume=

    Late-time quadratic growth in single-mode Rayleigh-Taylor instability , author=. Physical Review E , volume=. 2012 , publisher=

    2012

  8. [8]

    APS Division of Fluid Dynamics Meeting Abstracts , pages=

    Entrainment in a Reacting Compressible Shear Layer , author=. APS Division of Fluid Dynamics Meeting Abstracts , pages=

  9. [9]

    Physics of Fluids , volume=

    Scaling patch analysis of planar turbulent wakes , author=. Physics of Fluids , volume=. 2022 , publisher=

    2022

  10. [10]

    Physics of Fluids , volume=

    Scaling of the mean transverse flow and Reynolds shear stress in turbulent plane jet , author=. Physics of Fluids , volume=. 2021 , publisher=

    2021

  11. [11]

    International journal of heat and mass transfer , volume=

    Scaling heat transfer in fully developed turbulent channel flow , author=. International journal of heat and mass transfer , volume=. 2005 , publisher=

    2005

  12. [12]

    Physics of Fluids , volume=

    Scaling patch analysis of turbulent planar plume , author=. Physics of Fluids , volume=. 2021 , publisher=

    2021

  13. [13]

    Journal of Fluid Mechanics , volume=

    Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows , author=. Journal of Fluid Mechanics , volume=. 2005 , publisher=

    2005

  14. [14]

    Journal of Fluid Mechanics , volume=

    Variable-density effects in incompressible non-buoyant shear-driven turbulent mixing layers , author=. Journal of Fluid Mechanics , volume=. 2020 , publisher=

    2020

  15. [15]

    Journal of Turbulence , volume =

    High-Reynolds number Rayleigh--Taylor turbulence , author=. Journal of Turbulence , volume =. 2009 , publisher=

    2009

  16. [16]

    Bulletin of the American Physical Society , year=

    Numerical Study of Variable Density Effects on a Spatially Developing Supersonic Turbulent Shear Layer , author=. Bulletin of the American Physical Society , year=

  17. [17]

    Bulletin of the American Physical Society , volume=

    Numerical Study of Spatially-Developing Supersonic Turbulent Shear Layers , author=. Bulletin of the American Physical Society , volume=. 2022 , publisher=

    2022

  18. [18]

    Annual Review of Fluid Mechanics , volume=

    Turbulence with large thermal and compositional density fluctuations , author=. Annual Review of Fluid Mechanics , volume=

  19. [19]

    48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition , pages=

    Numerical investigations of shock-turbulence interaction in a planar mixing layer , author=. 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition , pages=

  20. [20]

    47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition , pages=

    Large-Scale Simulations of High Speed Turbulent Flows , author=. 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition , pages=

  21. [21]

    1922 , publisher=

    Weather prediction by numerical process , author=. 1922 , publisher=

    1922

  22. [22]

    Physics of Fluids , volume=

    Interscale dynamics and local isotropy in high Reynolds number turbulence within triadic interactions , author=. Physics of Fluids , volume=. 1994 , publisher=

    1994

  23. [23]

    Physics of Fluids A: Fluid Dynamics , volume=

    Subgrid-scale backscatter in turbulent and transitional flows , author=. Physics of Fluids A: Fluid Dynamics , volume=. 1991 , publisher=

    1991

  24. [24]

    Journal of Fluid Mechanics , volume=

    Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetry measurements , author=. Journal of Fluid Mechanics , volume=. 2002 , publisher=

    2002

  25. [25]

    Annual review of fluid mechanics , volume=

    Compressibility effects on turbulence , author=. Annual review of fluid mechanics , volume=. 1994 , publisher=

    1994

  26. [26]

    AIP Conference Proceedings , volume=

    Subgrid-scale backscatter after the shock-turbulence interaction , author=. AIP Conference Proceedings , volume=. 2017 , organization=

    2017

  27. [27]

    Computers & fluids , volume=

    Large eddy simulation of shock/homogeneous turbulence interaction , author=. Computers & fluids , volume=. 2002 , publisher=

    2002

  28. [28]

    Journal of Fluid Mechanics , volume=

    Turbulence structure behind the shock in canonical shock--vortical turbulence interaction , author=. Journal of Fluid Mechanics , volume=. 2014 , publisher=

    2014

  29. [29]

    Journal of fluid mechanics , volume=

    Subgrid-scale backscatter in reacting and inert supersonic hydrogen--air turbulent mixing layers , author=. Journal of fluid mechanics , volume=. 2014 , publisher=

    2014

  30. [30]

    Journal of Fluid Mechanics , volume=

    Numerical study of variable density turbulence interaction with a normal shock wave , author=. Journal of Fluid Mechanics , volume=. 2017 , publisher=

    2017

  31. [31]

    Journal of fluid mechanics , volume=

    A study of compressibility effects in the high-speed turbulent shear layer using direct simulation , author=. Journal of fluid mechanics , volume=. 2002 , publisher=

    2002

  32. [32]

    Physics of Fluids , volume=

    Direct numerical simulation of spatially developing highly compressible mixing layer: Structural evolution and turbulent statistics , author=. Physics of Fluids , volume=. 2019 , publisher=

    2019

  33. [33]

    Applied Physics Letters , volume=

    Structural characteristics of supersonic mixing enhanced by introducing streamwise vortices , author=. Applied Physics Letters , volume=. 2017 , publisher=

    2017

  34. [34]

    AIAA journal , volume=

    Three-dimensional experimental study of compressibility effects on turbulent free shear layers , author=. AIAA journal , volume=. 2020 , publisher=

    2020

  35. [35]

    AIAA journal , volume=

    Effects of compressibility on the characteristics of free shear layers , author=. AIAA journal , volume=

  36. [36]

    AIAA journal , volume=

    Two-dimensional shear-layer entrainment , author=. AIAA journal , volume=

  37. [37]

    Journal of Fluid Mechanics , volume=

    The mixing layer at high Reynolds number: large-structure dynamics and entrainment , author=. Journal of Fluid Mechanics , volume=. 1976 , publisher=

    1976

  38. [38]

    Physics of Fluids A: Fluid Dynamics , volume=

    The effect of walls on a spatially growing supersonic shear layer , author=. Physics of Fluids A: Fluid Dynamics , volume=. 1990 , publisher=

    1990

  39. [39]

    AIAA journal , volume=

    Effects of a downstream disturbance on the structure of a turbulent plane mixing layer , author=. AIAA journal , volume=

  40. [40]

    Annual Review of Fluid Mechanics , volume=

    Turbulent mixing , author=. Annual Review of Fluid Mechanics , volume=. 2005 , publisher=

    2005

  41. [41]

    AIAA journal , volume=

    Instability of inviscid, compressible free shear layers , author=. AIAA journal , volume=

  42. [42]

    Journal of Fluid Mechanics , volume=

    Turbulent shear-layer mixing: growth-rate compressibility scaling , author=. Journal of Fluid Mechanics , volume=. 2000 , publisher=

    2000

  43. [43]

    Berechnung von Aufgaben der freien Turbulenz auf Grund eines neuen N

    G. Berechnung von Aufgaben der freien Turbulenz auf Grund eines neuen N. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift f. 1942 , publisher=

    1942

  44. [44]

    AIAA journal , volume=

    Experimental study of compressible turbulent mixing layers , author=. AIAA journal , volume=

  45. [45]

    AIAA journal , volume=

    Three-dimensional velocity field in a compressible mixing layer , author=. AIAA journal , volume=

  46. [46]

    Journal of Fluid Mechanics , volume=

    Planar velocity measurements in a weakly compressible mixing layer , author=. Journal of Fluid Mechanics , volume=. 2003 , publisher=

    2003

  47. [47]

    AIAA journal , volume=

    Compressibility effects in turbulent shear layers , author=. AIAA journal , volume=

  48. [48]

    Journal of Fluid Mechanics , volume=

    On density effects and large structure in turbulent mixing layers , author=. Journal of Fluid Mechanics , volume=. 1974 , publisher=

    1974

  49. [49]

    Journal of fluid Mechanics , volume=

    The compressible turbulent shear layer: an experimental study , author=. Journal of fluid Mechanics , volume=. 1988 , publisher=

    1988

  50. [50]

    Sadhana , volume=

    Observations of supersonic free shear layers , author=. Sadhana , volume=. 1988 , publisher=

    1988

  51. [51]

    Physics of Fluids A: Fluid Dynamics , volume=

    Absolute/convective instabilities and the convective Mach number in a compressible mixing layer , author=. Physics of Fluids A: Fluid Dynamics , volume=. 1990 , publisher=

    1990

  52. [52]

    26th Aerospace Sciences Meeting , pages=

    Instabilities in the free shear layer formed by two supersonic streams , author=. 26th Aerospace Sciences Meeting , pages=

  53. [53]

    AIAA journal , volume=

    A study of inviscid, supersonic mixing layers using a second-order total variational diminishing scheme , author=. AIAA journal , volume=

  54. [54]

    1970 , publisher=

    Statistical investigation of turbulent velocity and pressure fields in a two-stream mixing layer , author=. 1970 , publisher=

    1970

  55. [55]

    Journal of Fluid Mechanics , volume=

    Direct numerical simulation of a spatially developing compressible plane mixing layer: flow structures and mean flow properties , author=. Journal of Fluid Mechanics , volume=. 2012 , publisher=

    2012

  56. [56]

    Journal of Fluid Mechanics , volume=

    On the two-dimensional mixing region , author=. Journal of Fluid Mechanics , volume=. 1976 , publisher=

    1976

  57. [57]

    Journal of Fluid Mechanics , volume=

    A numerical study of a variable-density low-speed turbulent mixing layer , author=. Journal of Fluid Mechanics , volume=. 2017 , publisher=

    2017

  58. [58]

    Compressibility effects in a turbulent annular mixing layer. Part 1. Turbulence and growth rate , author=. Journal of Fluid Mechanics , volume=. 2000 , publisher=

    2000

  59. [59]

    Journal of Computational Physics , volume=

    A critical analysis of Rayleigh--Taylor growth rates , author=. Journal of Computational Physics , volume=. 2001 , publisher=

    2001

  60. [60]

    Moscow Izdatel Nauka , year=

    The theory of turbulent jets , author=. Moscow Izdatel Nauka , year=

  61. [61]

    Journal of Basic Engineering , volume =

    An analytical and experimental study of the plane, incompressible, turbulent free-shear layer with arbitrary velocity ratio and pressure gradient , author=. Journal of Basic Engineering , volume =

  62. [62]

    Free Turbulent Shear Flows, Vol

    A critical review of the experimental data for developed free turbulent shear layers , author=. Free Turbulent Shear Flows, Vol. 1 , year=

  63. [63]

    The Physics of fluids , volume=

    Spreading of two-stream supersonic turbulent mixing layers , author=. The Physics of fluids , volume=. 1986 , publisher=

    1986

  64. [64]

    The Physics of fluids , volume=

    Direct numerical simulations of chemically reacting turbulent mixing layers , author=. The Physics of fluids , volume=. 1986 , publisher=

    1986

  65. [65]

    Physics of Fluids , volume=

    Nonlinear effects in the combined Rayleigh-Taylor/Kelvin-Helmholtz instability , author=. Physics of Fluids , volume=. 2011 , publisher=

    2011

  66. [66]

    Journal of Turbulence , volume=

    Sensitivity analysis of initial condition parameters on the transitional temporal turbulent mixing layer , author=. Journal of Turbulence , volume=. 2008 , publisher=

    2008

  67. [67]

    Journal of computational physics , volume=

    Boundary conditions for direct simulations of compressible viscous flows , author=. Journal of computational physics , volume=. 1992 , publisher=

    1992

  68. [68]

    Journal of computational physics , volume=

    Efficient implementation of essentially non-oscillatory shock-capturing schemes , author=. Journal of computational physics , volume=. 1988 , publisher=

    1988

  69. [69]

    Journal of Fluid Science and Technology , volume=

    Redefining mixing length in turbulent mixing layer in terms of shear-induced vorticity , author=. Journal of Fluid Science and Technology , volume=. 2011 , publisher=

    2011

  70. [70]

    International Journal of Modern Physics: Conference Series , volume=

    Re-evaluating mixing length in turbulent mixing layer by means of high-order statistics of velocity field , author=. International Journal of Modern Physics: Conference Series , volume=. 2012 , organization=

    2012

  71. [71]

    1988 , publisher=

    Experimental and analytical investigations of supersonic mixing layers , author=. 1988 , publisher=

    1988

  72. [72]

    21st Fluid Dynamics, Plasma Dynamics and Lasers Conference , pages=

    Numerical modeling of compressible reacting turbulent shear layers , author=. 21st Fluid Dynamics, Plasma Dynamics and Lasers Conference , pages=

  73. [73]

    Journal of Turbulence , volume=

    Evolution and growth of large-scale structures in high compressibility mixing layers , author=. Journal of Turbulence , volume=. 2002 , publisher=

    2002

  74. [74]

    AIAA journal , volume=

    Experiments in nonreacting compressible shear layers , author=. AIAA journal , volume=

  75. [75]

    AIAA journal , volume=

    Two-and three-dimensional effects in the supersonic mixing layer , author=. AIAA journal , volume=

  76. [76]

    AIAA journal , volume=

    Compressibility effects on large structures and entrainment length scales in mixing layers , author=. AIAA journal , volume=. 2020 , publisher=

    2020

  77. [77]

    Organized large structure in the post-transition mixing layer. Part 1. Experimental evidence , author=. Journal of fluid mechanics , volume=. 2013 , publisher=

    2013

  78. [78]

    5th Australasian Conference on Hydraulics and Fluid Mechanics, Volume 1 , volume=

    The entrainment and large structure in turbulent mixing layers , author=. 5th Australasian Conference on Hydraulics and Fluid Mechanics, Volume 1 , volume=

  79. [79]

    Journal of Fluid Mechanics , volume=

    Turbulent boundary layer to single-stream shear layer: the transition region , author=. Journal of Fluid Mechanics , volume=. 2003 , publisher=

    2003

  80. [80]

    Physics of Fluids , volume=

    Deflection, drift, and advective growth in variable-density, laminar mixing layers , author=. Physics of Fluids , volume=. 2007 , publisher=

    2007

Showing first 80 references.