2nd DA Challenge Case 2 Dataset

This test case is currently being integrated to the portal, with for now only its input data available for download.

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1. Physical scenario

This test case examines a compressible turbulent boundary layer (TBL) using data from a direct numerical simulation (DNS). The dataset corresponds to the Mach 4.9 case from Huang et al. (2022), which numerically replicates a high-speed wind tunnel experiment conducted at the National Aerothermochemistry Laboratory (NAL) at Texas A&M University. The simulation employs a quasi-adiabatic wall condition with a wall-to-recovery temperature ratio of 𝑇w/𝑇r = 0.91 and sustains a peak friction Reynolds number of Retau = 1244. In Huang et al. (2022), 𝑇w is the wall temperature, 𝑇r = 𝑇inf[1 + 𝑟(𝛾 − 1)Mainf2 /2] is the recovery temperature, 𝑟 = 0.89 is the recovery factor, 𝛾 is the ratio of specific heats, and 𝑇inf and Mainf are the free stream temperature and Mach number, respectively.

Figure 1 illustrates computational domains from Huang et al. (2022), including the subset of the TBL selected for this data assimilation (DA) challenge. Figure 2 presents cut plots of all three velocity components, along with cuts of pressure, density, and temperature within the simulated measurement domain for this challenge.

1.1. Simulation details

The DNS code solves the full three-dimensional compressible Navier-Stokes equations, assuming a perfect and ideal gas. The viscosity is determined using Sutherland’s law,

where 𝜇0 = 1.186 × 10-5 N s/m2 is the reference viscosity at the reference temperature, 𝑇0 = 66.2 K, and 𝑆 = 110.4 K is Sutherland’s constant. The thermal conductivity is 𝜅 = 𝜇cp/Pr, where cp is the heat capacity at constant pressure, and the Prandtl number is taken as a constant, Pr = 0.71.
Free stream conditions for the TBL are Mainf = 4.9, 𝑈inf = 794 m/s, 𝜌inf = 0.272 kg/m3, and 𝑇inf = 66.2 K. The wall temperature is 𝑇w = 317 K, and the wall temperature to recovery temperature ratio is 𝑇r/𝑇w = 0.91. The free stream unit Reynolds number is Reu = 48.6 × 106 m-1, and the momentum and friction Reynolds numbers are between Retheta = 13,952 and 26,762 and Retau = 726 and 1244, respectively.

1.2. Measurement domain and datasets

For the challenge, a domain of 12.24 × 2.28 × 1.16 mm3 (streamwise length by width by height) is considered, located approximately 0.74 m downstream within the DNS domain. This region corresponds to Box 3 of the M5Tw091 case in Huang et al. (2022). At this location, the boundary layer thickness is 𝛿 = 11 mm, the viscous length scale is ℓnu = 9.4 μm, and the viscous time scale is 𝜏nu = 0.26 μs. In boundary layer units, the
measurement domain size is 1.11 × 0.21 × 0.11, while in viscous units, it is 1300 × 240 × 125. The computational domain for this region consists of 200 × 50 × 110 voxels, and the DNS time step is Δ𝑡 = 0.08 μs, which is 0.31 in viscous units.

This flow case is designed to explore the limits of DA in the presence of compressibility effects and inertial particle dynamics. Despite significant advances in high-speed imaging technology and high-power, high-repetition laser sources, conducting time-resolved 3D Lagrangian particle tracking (LPT) experiments in high-Mach-number flows remains extremely challenging. The test cases in this study utilize the Shimadzu Hyper Vision HPVX2 High-Speed Video Camera, which can capture full-frame images at 400 × 250 pixels (~0.1 MP) with a frame rate of 1 MHz. Additionally, the camera supports frame rates of up to 10 MHz in a reduced-frame mode.

For the present flow, based on the results in Zhou et al. (2024), the particle separation needed for a scale-resolving reconstruction of the TBL is estimated to be Δ𝑥 = 65.8 μm (7 viscous lengths), and the time step needed to resolve ∂/ ∂𝑡 terms is estimated to be Δ𝑡 = 0.16 μs (0.6 viscous times). Therefore, fully resolving the flow in the measurement domain would require tracking roughly 120 × 103 particles at a rate of 6.25 MHz. This is outside the scope of what can be achieved using the Hyper Vision HPV-X2. Therefore, we consider four imaging scenarios, using real and ideal sensor sizes of 0.1 MP and 1 MP, respectively, and real and ideal full-frame imaging rates of 1 MHz and 6.25 MHz. Datasets for this case are
summarized below in Table 1.

It is assumed that four identical cameras are available for particle tracking in each scenario. The corresponding particle-per-pixel (ppp) values are 0.12 ppp for both the 0.1 MP and 1 MP sensors, with tracking feasible for particle densities up to 0.15 ppp. These test configurations encompass both currently feasible setups and potential future implementations. The latter are designed to probe the numerical limits of DA in compressible flow, pushing the boundaries of achievable spatial and temporal resolution.

1.3. Particle dynamics

Particles in compressible flows may experience strong inertial effects, causing them to deviate from the fluid velocity field. In this challenge, we consider both inertial tracer particles and idealized tracers. Ideal tracers are assumed to be massless particles that perfectly follow the flow, providing a baseline for evaluating tracking accuracy and DA performance. The particles’ equation of motion is
𝐮 = 𝐯p,
where 𝐯p is the particle velocity. Real particles are not massless and may exhibit ballistic motion in unsteady flows. The ability of inertial particles to follow the flow can be quantified using the Stokes number, St = 𝜏p/𝜏, where 𝜏p and 𝜏 are characteristic time scales of the particle and flow, respectively. In compressible TBLs, 𝜏 can be estimated using the boundary layer thickness and free stream velocity, 𝜏 ≈ 𝛿/𝑈inf (Aultman et al., 2022).

For this challenge, inertial particles are modeled as aggregates of titanium dioxide (TiO2), a commonly used seeding material for particle-based measurements in high-speed flows. In real experiments, individual TiO2 particles form aggregates of varying sizes and densities, which must be carefully characterized (Zhou & Grauer, 2023). To simplify the modeling, we assume a uniform bulk diameter of 𝑑p = 1.5 μm and a density of 𝜌p = 800 kg/m3, which are representative values based on Williams et al. (2015). The dense and sparse seeding cases in Section 1.2 have volume fractions of 6 × 10-6 and 6 × 10-7, respectively. These values ensure that all cases remain within the one-way coupling regime, where particle
motion is influenced by the flow but does not significantly alter it (Capecelatro & Wagner, 2024).

Inertial transport of particles in compressible flows is influenced by viscous, compressibility, and rarefaction effects. These effects may be characterized in terms of three non-dimensional numbers. Viscous e[ects are typically described in terms of the particle Reynolds number,

where 𝐮 − 𝐯p is the slip velocity between the particle and carrier gas. Compressibility effects are characterized in terms of the particle Mach number,

where 𝑅 = 286.99 J/kg K is the specific gas constant. Lastly, rarefaction effects are a function of the mean free path in the carrier fluid, 𝜆, and a characteristic flow scale, which is taken to be the particle diameter in low-Rep flows. The ratio of 𝜆 to 𝑑p defines the Knudsen number,

Note that Knp may be expressed in terms of Map and Rep, as shown on the right-hand side. The motion of small heavy particles in a supersonic flow is modeled using the following equation of motion (Capecelatro & Wagner, 2024),

where 𝜏p is particle relaxation time, defined as

Here, 𝐶D is the drag coefficient, which depends on Rep, Map, and Knp. In this challenge, we use the Loth drag model (Loth, 2008) to compute 𝐶D. Note that the particle motion equation above is a simplified form of the Maxey–Riley equation. Excluded terms, including the added mass, pressure gradient, Basset history, and gravity terms, are negligible in the present scenario due to the small size of TiO2 aggregates
particles and the low air-to-TiO2 density ratio (𝜌/𝜌p ≪ 1).

To simulate particle tracks, all particles (ideal or inertial) are randomly distributed within the computational domain in the first frame and initialized with the local flow velocity. The particle motion equation is solved using a fourth-order Runge–Kutta method, with periodic
boundary conditions applied. Flow states are interpolated from the gridded DNS data using linear interpolation. To mitigate boundary effects, particle tracking is first performed in an expanded domain (~20% larger than the measurement domain in each dimension). Only
tracks within the central domain are reported. Additionally, the first 100 DNS frames are discarded to eliminate transient effects introduced by initializing particles with the flow velocity.

Stokes numbers of the inertial tracers range between 1 and 1.6, indicating non-negligible inertial effects. The particle Mach number remains mostly below 0.6, ensuring that shocks do not form about the particles. The particle Knudsen number falls between 0.2 and 1, significantly exceeding 0.01, which suggests weak rarefaction effects. These values of Map and Knp highlight the complex environment experienced by tracer particles in compressible TBLs.

2. Input data

2.1. Input files

Particle positions in 3D space and their corresponding IDs are provided in .dat files. Each file contains the following columns:

  1. X, Y, Z – Instantaneous particle coordinates in millimeters, where X is the streamwise direction, Y is the spanwise direction, and Z is the wall-normal direction.
  2. PartID – Unique identifier for each particle track.

Each data file includes a three-line header:

  1. Title = Snapshot #
  2. Variables = X Y Z PartID
  3. Zone = I, F = POINT

where the parameter I specifies the number of particles present in Snapshot #𝑛.

2.2. Ideal and inertial tracer cases

For each seeding density and frame rate case, a sequence of 51 consecutive frames (starting from 0) is provided. The time separation between snapshots is 0.16 μs for the highframe-rate case and 0.96 μs for the low-frame-rate case. File names follow a strict naming convention,

DA_CASE02_TR_ppp_0_AAA_FR_C_IN_PartFile_BBBB.dat

where the variable elements are:

  • AAA – fractional value of seeding density in ppp (e.g., 012 for 0.012 ppp)
  • C – integer frame rate in MHz (e.g., 6 for 6.25 MHz)
  • BBBB – snapshot number (0000 to 0050)

Note that inertial and ideal tracer cases are distinguished by the IN tag. I0 indicates an ideal tracer case and I1 indicates an inertial tracer case.

3. Requested output

Participants will be required to reconstruct velocity fields and associated flow properties on a structured Cartesian grid, ensuring compatibility with the provided highfidelity direct DNS data. Only the main information is presented here; full details will be provided when the evaluation becomes available. 

The output must be provided on a Cartesian grid with the following spacing:

  • 𝑥-direction: 205 grid points, from 𝑥 = 0 mm to 12.24 mm, with Δ𝑥 = 0.06 mm
  • 𝑦-direction: 58 grid points, from 𝑦 = 0 mm to 2.28 mm, with Δ𝑥 = 0.04 mm
  • 𝑧-direction: 59 grid points, from 𝑧 = 0 mm to 1.16 mm, with Δ𝑧 = 0.02 mm

The resultant grid has a total of 701,510 nodes. The following variables must be provided for each node:

  • Grid coordinates reported in mm
    • 𝑥, 𝑦, 𝑧
  • Velocity vector elements reported in m/s 
    • 𝑢, 𝑣, 𝑤
  • Velocity gradient tensor elements reported in 1/s

  • Thermodynamic properties
    • Static pressure, 𝑝, reported in Pa
    • Density, 𝜌, reported in kg/m3

Note that reported pressure should be reported relative to a tare pressure of 𝑝 = 0 Pa at the grid origin, 𝐱 = (0, 0, 0) mm.

References

Aultman, M.T., Disotell, K. and Duan, L., 2022. The effect of particle lag on statistics of hypersonic turbulent boundary layers subject to pressure gradients. In AIAA SciTech 2022 Forum (p. 1062).

Capecelatro, J. and Wagner, J.L., 2024. Gas–particle dynamics in high-speed flows. Annual Review of Fluid Mechanics, 56(1), pp.379–403.

Huang, J., Duan, L. and Choudhari, M.M., 2022. Direct numerical simulation of hypersonic turbulent boundary layers: effect of spatial evolution and Reynolds number. Journal of Fluid Mechanics, 937, p.A3.

Loth, E., 2008. Drag of non-spherical solid particles of regular and irregular shape. Powder Technology, 182(3), pp.342–353.

Williams, O.J., Nguyen, T., Schreyer, A.M. and Smits, A.J., 2015. Particle response analysis for particle image velocimetry in supersonic flows. Physics of Fluids, 27(7), p. 076101.

Zhou, K. and Grauer, S.J., 2023. Flow reconstruction and particle characterization from inertial Lagrangian tracks. arXiv preprint arXiv:2311.09076.

Zhou, K., Grauer, S.J., Schanz, D., Godbersen, P., Schröder, A., Rockstroh, T., Jeon, Y.J. and Wieneke, B., 2024. Benchmarking data assimilation algorithms for 3D Lagrangian particle tracking. In 21st International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics, Lisbon, Portugal (pp. 1–22).