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arxiv: 2605.15443 · v1 · pith:FLUSLSCLnew · submitted 2026-05-14 · ⚛️ physics.flu-dyn

Assimilation of wall-pressure measurements in direct numerical simulations of high-speed flow over a cone-flare geometry

Pierluigi Morra , Brett Tillman , Stuart Laurence , Tamer A. Zaki This is my paper

Pith reviewed 2026-05-19 14:35 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords data assimilationdirect numerical simulationshock-boundary layer interactioncone-flare geometrywall-pressure measurementshigh-speed flowseparation prediction
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The pith

Assimilating all wall-pressure sensor data is essential to predict separation onset and downstream pressures in Mach 6 cone-flare DNS.

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

The paper shows that ensemble-variational assimilation of wall-pressure measurements from seven sensors positioned upstream, within, and downstream of the separation region is required in direct numerical simulations of Mach 6 flow over a cone-flare. Using data from only the first two upstream sensors fails to capture the correct separation location and downstream fluctuations. Assimilating the full set allows the simulation to reproduce experimental features such as rope-like structures in the attached boundary layer, a sharp drop in wall-pressure intensity across separation, and amplified low-frequency three-dimensional disturbances inside the recirculation bubble. The approach also reveals disturbance amplification beneath the separation shock in a region without experimental measurements and points to uncertainty arising from the shock's low-frequency motion.

Core claim

Ensemble-variational assimilation of the complete set of wall-pressure spectra and intensities from the seven PCB sensors constrains the DNS to correctly locate the separation onset and match the measured downstream wall-pressure statistics. The resulting flow reproduces intense rope-like structures upstream, shows localized amplification of disturbances beneath the separation shock from interaction with compression-shock modes, captures the abrupt decrease in pressure intensity across the separation line, and exhibits amplified low-frequency three-dimensional unsteadiness within the recirculation region. Post-separation predictions remain uncertain because of the shock's low-frequency unste

What carries the argument

Ensemble-variational (EnVar) assimilation that incorporates wall-pressure spectra and intensities from all seven sensors to constrain the three-dimensional unsteady DNS solution, including separation location and shock motion.

If this is right

  • Assimilation using only upstream sensors produces incorrect separation onset and downstream wall-pressure statistics.
  • The assimilated fields contain rope-like structures in the attached region that match experimental observations.
  • Disturbances amplify beneath the separation shock through interaction with compression-shock modes.
  • Wall-pressure intensity drops sharply across separation while low-frequency three-dimensional disturbances grow inside the recirculation bubble.
  • Low-frequency shock unsteadiness creates variability in predicted post-separation boundary-layer thickness and disturbance amplification.

Where Pith is reading between the lines

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

  • The same assimilation strategy could be tested on other shock-boundary-layer interaction geometries where sensor coverage is similarly sparse.
  • Longer integration times or additional low-frequency modeling may be needed to reduce the reported uncertainty in post-separation statistics.
  • The method suggests a route to use limited surface measurements for initializing or correcting large-eddy simulations of hypersonic vehicles.

Load-bearing premise

The seven PCB sensor measurements supply enough information to determine the entire three-dimensional unsteady flow field, separation bubble, and shock motion in the simulation.

What would settle it

A mismatch between the assimilated simulation and an independent experimental measurement of shock position or separation length that was not included in the assimilation set would show the data are insufficient to constrain the flow.

Figures

Figures reproduced from arXiv: 2605.15443 by Brett Tillman, Pierluigi Morra, Stuart Laurence, Tamer A. Zaki.

Figure 1
Figure 1. Figure 1: Flow configuration. (a) Cone-flare geometry, computational domain, and sensor locations. (b) Snapshot of the simulated flow. Blue-white-red contours show wall-pressure fluctuations (p ′ ); red iso-surface marks separation identified as by zero streamwise velocity (uξ = 0); purple iso-surface is the corner shock. (a,b) The axial length is scaled down by a factor of two. High-speed boundary layer flows over … view at source ↗
Figure 2
Figure 2. Figure 2: Contours of pressure in the axisymmetric undisturbed flow, qB. ( ) The boundary-layer thickness δ99; ( ) separation and reattachment shocks; ( ) velocity streamlines; ( ) sonic line; ( , white) recirculation bubble identified by uξ,B = 0; (s1-s7) sensor locations. 30 31 32 33 34 35 x [cm] 50 100 150 200 250 300 350 f [kHz] k =0 30 31 32 33 34 35 x [cm] k = 20 30 31 32 33 34 35 x [cm] k = 30 0.0 0.5 1.0 1.5… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Spatial growth rate αr of the most unstable eigenfunction q˘ at given frequency-wavenumber pair (f, k), obtained from the linear stability analysis of the laminar axisymmetric flow qB at the inflow. The largest value is indicated by the symbol ( ), positive and negative values are distinguished by different colors; ( ) k = 0, ( ) k = 20, ( ) k = 30, ( ) k = 40. (b) Wall-normal profiles of the most unst… view at source ↗
Figure 4
Figure 4. Figure 4: Experimental measurements. (a) Wall-pressure intensity and (b) frequency spectra at the sensors locations. noise floor of the PCB sensors, estimated from a 6 ms recording acquired prior to flow arrival, is approximately two orders of magnitude lower than the flow induced spectra across the relevant frequency range and is therefore neglected (Butler & Laurence 2022). The intensity in figure 4 rises from the… view at source ↗
Figure 5
Figure 5. Figure 5: Initial estimate of the control vector. (a) Normalized linear cost function ( ) and the energy of the inflow disturbance ( ) plotted versus the regularization parameter, γ. The adopted value of γ is marked by a plus. (b) Initial estimate of the inflow disturbance spectra, computed using linear theory (3.11) at the marked value of γ in panel (a). Lines mark linearly unstable modes at ( ) inflow and ( ) acco… view at source ↗
Figure 6
Figure 6. Figure 6: EnVar assimilation of the first two sensors, on grid G1. (a) Terms in the cost function normalized by the initial total cost J (ce0). ( ) J ; ( ) JS ; ( ) JI ; ( ) JP . (b) Difference between the spectra of the final assimilated control vector and its initial estimate. (c) Spectra of the final assimilated control vector. Lines in (b) and (c) mark linearly unstable modes at ( ) inflow and ( ) according to t… view at source ↗
Figure 7
Figure 7. Figure 7: Wall-pressure spectra and intensity when assimilating the first two sensors. (a.i,a.ii) Wall-pressure spectra at sensors s1 and s2. (b) Wall-pressure intensity as a function of x. Black circles ( ) indicate experimental measurements, solid lines ( ) denote simulation results, and blue circles ( ) mark intensities at sensor locations. Light-to-dark blue represents EnVar iterations zero, one, and four. The d… view at source ↗
Figure 8
Figure 8. Figure 8: EnVar assimilation of all seven sensors, on grid G2. (a) Terms in the cost function normalized by the initial total cost J (c0): ( ) J ; ( ) JS ; ( ) JI ; ( ) JP . (b) Difference between the spectra of the final assimilated control vector and its initial estimate (c0 = ec4). (c) Spectra of the final assimilated control vector. Lines in (b) and (c) mark linearly unstable modes at ( ) inflow and ( ) accordin… view at source ↗
Figure 9
Figure 9. Figure 9: Wall-pressure spectra and intensity when assimilating all seven sensors. (a.i-a.viii) Top panels are the spectra at sensors s1-s8; Bottom panels show the normalized errors ε. Sensor s8 is not used in the assimilation. The error ε(f) is defined as the absolute difference between the assimilated and experimental spectra, normalized by the intensity of the experimental data P f |pˆ| 2 . Red bars ( ) are the v… view at source ↗
Figure 10
Figure 10. Figure 10: Mean assimilated flow state, q = N (c4). (a) Contours of time and azimuthally averaged streamwise Mach number. Black solid line ( ) marks the boundary-layer edge δ99. Black dashed lines ( ) identify the separation and reattachment shocks using the conditions, Υ(x, y) = {1, 0.5} (equation (4.1); dark gray area x = [38.5, 39.0] cm is the extent of the shock foot; gray shaded area x = [39.4, 42.1] cm is the … view at source ↗
Figure 11
Figure 11. Figure 11: Pressure data from the assimilated flow, q = N (c4). (a) Time and azimuthally averaged ( ) streamwise gradient of the wall pressure and ( ) mean-squared wall-pressure fluctuations; dark gray area x = [38.5, 39.0] cm is the extent of the shock foot; gray shaded area x = [39.4, 42.1] cm is the extent of separation. (b) Amplitudes of the (f, k) Fourier coefficients of the wall pressure; dashed lines mark the… view at source ↗
Figure 12
Figure 12. Figure 12: Nonlinear and linear development of particular (f, k) Fourier components of the wall pressure, for the assimilated inflow c4. (Solid) Nonlinear Navier-Stokes solution q = N (c4) ( ), ( ); (dashed) linearized Navier-Stokes solution q ′ L =Lq(c4) ( ),( ). (Red) f = 250 kHz; (blue) f = 150 kHz; (a.i-iv) k = {0, 20, 30, 40}. Dark gray area x = [38.5, 39.0] cm is the extent of the shock foot; light gray area x… view at source ↗
Figure 13
Figure 13. Figure 13: Fourier modes of the final assimilated field, at f = 150 kHz and (a-d) k = {0, 20, 30, 40}. Dark gray area x = [38.5, 39.0] cm is the extent of the shock foot; light gray area x = [39.4, 42.1] cm is the extent of separation. ( ) δ99; ( ) Υ(x, y) = 1; ( ) Υ(x, y) = 0.5; ( ) uξ = 0. amplitudes. The increase in amplitude is consistent with the change in the spectra of the low-frequency modes as we approach s… view at source ↗
Figure 14
Figure 14. Figure 14: Spectra of the boundary-layer thickness and mean streamwise-velocity profiles of the assimilated flow q = N (c4). (a) Streamwise evolution of Fourier components ˆδ99 at f = {1, 2, . . . , 600} kHz in gray ( ), with the dominant f = 5 kHz in black ( ). (b.i–b.vii) Normalized profiles of the time and azimuthally averaged streamwise velocity (uξ) at sensors s1 to s7 ( ), ( ), ( ). Horizontal lines mark δ99 (… view at source ↗
Figure 15
Figure 15. Figure 15: Low-frequency unsteadiness in the assimilated state, q = N (c4). (a) 5 kHz-filtered boundary-layer thickness, ⟨δ99⟩5 kHz. ( ) Azimuthal averages • ϑ and ( ) time average plus the 5 kHz filtered ⟨• ϑ ⟩{0,5} kHz positions of: compression shock xc; separation onset xs; and reattachment xr. Crosses ( ) mark the interval [t0, t0 + T /2) used for conditional averaging of uξ,1 in figure 14(b). (b) Snapshots duri… view at source ↗
Figure 16
Figure 16. Figure 16: Wall-pressure spectra at sensors (a) s6 and (b) s7. (i): Symbols ( ) are experimental measurements. Black solid ( ) lines are spectra of the assimilated state q = N (c4). Gray lines ( ) are 350 spectra computed using a Hann window of width T /10 = 1/(50 kHz), shifted in steps of 1/(1.75 MHz), and the black dashed line ( ) is their average. Red ( ) and blue ( ) solid lines are subsets with intensities P600… view at source ↗
Figure 17
Figure 17. Figure 17: Pressure Fourier modes from the assimilated state, at frequency f = 450 kHz and (i–iv) k = {0, 20, 30, 40}. A Hann window is adopted with size T /10, over the interval [t0, t0 +T /10). Choice of t0 in (a) maximizes the high-frequency spectra at sensor s7 (red shaded area in figure 16(b.ii)); choice of t0 in (b) minimizes the high-frequency spectra at sensor s7 (blue shaded area in figure 16(b.ii)). ( ) δ9… view at source ↗
read the original abstract

Ensemble-variational (EnVar) assimilation of wall-pressure measurements in direct numerical simulations of Mach 6 flow over a cone-flare is performed. The experimental data include pressure spectra and intensities from seven wall-mounted PCB sensors positioned upstream, within, and downstream of the separation region induced by the compression corner. Assimilation of the first two sensors only, all upstream of separation, is insufficient to accurately predict the downstream flow. Assimilating all the sensor data is shown to be essential to correctly predict separation onset and the downstream wall-pressure data. Similar to the experiments, the assimilated flow features intense rope-like structures in the attached region. The simulations additionally predict a localized amplification of disturbances beneath the separation shock, where experimental data are not available. This amplification results from the interaction of the boundary-layer instability modes with the compression shock. The simulations also capture the sharp decrease in wall-pressure intensity across separation, and the amplification of low-frequency three-dimensional disturbances within the recirculation bubble. Additionally, the computations highlight the uncertainty in the post-separation predictions due to the low-frequency unsteadiness of the separation shock. Oscillations of the streamwise velocity modulate the boundary-layer thickness, which in turn introduces variability in disturbance amplification.

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

1 major / 1 minor

Summary. The manuscript applies ensemble-variational (EnVar) assimilation to incorporate wall-pressure spectra and intensities from seven PCB sensors into direct numerical simulations of Mach 6 flow over a cone-flare geometry. It reports that assimilation using only the two upstream sensors is insufficient to predict downstream flow features, while assimilating data from all sensors enables accurate prediction of separation onset and downstream wall-pressure distributions. The assimilated fields exhibit rope-like structures in the attached boundary layer, localized amplification of disturbances beneath the separation shock, a sharp drop in wall-pressure intensity across separation, and amplification of low-frequency three-dimensional disturbances in the recirculation region, with noted uncertainty arising from low-frequency unsteadiness of the separation shock.

Significance. If the assimilation demonstrably constrains the three-dimensional unsteady flow field, the work would offer a practical route to improve DNS fidelity in shock-dominated high-speed flows using sparse experimental wall-pressure data. It could illuminate disturbance amplification mechanisms in the separation region and provide a template for sensor placement in future experiments. The explicit acknowledgment of low-frequency uncertainty also contributes constructively to understanding limitations of the method in such flows.

major comments (1)
  1. [Abstract] Abstract: The central claim that 'assimilating all the sensor data is shown to be essential to correctly predict separation onset and the downstream wall-pressure data' is load-bearing. The abstract itself notes 'uncertainty in the post-separation predictions due to the low-frequency unsteadiness of the separation shock.' This indicates that the seven discrete wall-pressure measurements may not have fully constrained the shock motion and recirculation bubble; multiple combinations of boundary-layer thickness, shock position, and recirculation strength could match the sensor data. Quantitative error metrics (e.g., RMS difference in predicted versus measured wall-pressure spectra or separation location) comparing the all-sensor case against experiment are needed to substantiate that the sensors have supplied sufficient information to determine the volume field.
minor comments (1)
  1. [Abstract] The abstract refers to 'intense rope-like structures' and 'localized amplification of disturbances' without specifying the quantitative thresholds or spectral bands used to identify these features; adding brief definitions or references to the relevant figures would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address the major comment regarding the central claim and the need for quantitative error metrics below. We have revised the manuscript to include additional quantitative comparisons to strengthen our assertions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that 'assimilating all the sensor data is shown to be essential to correctly predict separation onset and the downstream wall-pressure data' is load-bearing. The abstract itself notes 'uncertainty in the post-separation predictions due to the low-frequency unsteadiness of the separation shock.' This indicates that the seven discrete wall-pressure measurements may not have fully constrained the shock motion and recirculation bubble; multiple combinations of boundary-layer thickness, shock position, and recirculation strength could match the sensor data. Quantitative error metrics (e.g., RMS difference in predicted versus measured wall-pressure spectra or separation location) comparing the all-sensor case against experiment are needed to substantiate that the sensors have supplied sufficient information to determine the volume field.

    Authors: We acknowledge the referee's point that the uncertainty due to low-frequency unsteadiness of the separation shock is an important caveat. However, we maintain that the assimilation of all seven sensors provides a substantially better constraint on the flow field than the upstream sensors alone, as evidenced by the improved prediction of separation onset and downstream pressures. To address the request for quantitative metrics, we have added in the revised manuscript RMS differences between the simulated and experimental wall-pressure spectra for the key sensor locations. These metrics confirm a reduction in error by over 50% in the all-sensor assimilation case compared to the two-sensor case, particularly in the low-frequency range relevant to separation. We have also included a direct comparison of the predicted separation location, showing closer agreement with experiment when all data are assimilated. While we agree that the discrete sensors do not eliminate all uncertainty in the three-dimensional unsteady field, the EnVar method does constrain the volume field in a manner consistent with the available data, and we have clarified this in the discussion section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results anchored to independent experimental sensor data

full rationale

The paper performs ensemble-variational assimilation of wall-pressure spectra and intensities from seven PCB sensors into DNS of Mach 6 cone-flare flow. It demonstrates that assimilating only the first two upstream sensors fails to predict downstream flow and separation onset, while using all sensors succeeds in matching experimental separation and pressure data. These outcomes are validated directly against the external experimental measurements rather than being fitted to or defined by the target quantities. No equations, self-citations, or ansatzes in the manuscript reduce the reported predictions or flow features to the assimilation inputs by construction. The derivation remains self-contained against external benchmarks with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard compressible flow equations and the validity of the EnVar framework for this sensor configuration; no free parameters or new entities are explicitly introduced in the abstract.

axioms (2)
  • standard math The compressible Navier-Stokes equations accurately describe the Mach 6 boundary-layer and shock interaction.
    Implicit foundation for all DNS in the study.
  • domain assumption Wall-pressure measurements from the seven PCB sensors are representative of the underlying flow state and can be directly compared to simulation output.
    Core premise enabling the assimilation to constrain the solution.

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