# Assimilation of wall-pressure measurements
in high-speed flow over a cone

David A. Buchta1 Stuart J. Laurence2  Tamer A. Zaki1<EMAIL_ADDRESS>1Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD
21218, USA 2Department of Aerospace Engineering, University of Maryland,
College Park, MD 20742, USA

###### Abstract

A nonlinear ensemble-variational (EnVar) data assimilation is performed in
order to estimate the unknown flow field over a slender cone at Mach-6, from
isolated wall-pressure measurements. The cost functional accounts for
discrepancies in wall-pressure spectra and total intensity between the
experiment and the prediction using direct numerical simulations (DNS), as
well as our relative confidence in the measurements and the estimated state.
We demonstrate the robustness of the predicted flow by direct propagation of
posterior statistics. The approach provides a unique first look at the flow
beyond the sensor data, and rigorously accounts for the role of nonlinearity
unlike previous efforts that adopted ad-hoc inflow syntheses. Away from the
wall, two- and three-dimensional assimilated states both show rope-like
structures, qualitatively similar to independent schlieren visualizations.
Despite this resemblance, and even though the planar second modes are the most
unstable upstream, three-dimensional (3D) waves must be included in the
assimilation in order to accurately reproduce the wall-pressure measurements
recorded in the Ludwieg-Tube facility. The results highlight the importance of
three-dimensionality of the field and of the base-state distortion on the
instability waves in this experiment, and motivate future measurements that
probe the 3D nature of the flow field.

##  1 Introduction

Hypersonic boundary-layer transition is extremely sensitive to environmental
disturbances. Accurate transition predictions are therefore challenging in
uncertain environments, especially when measurements are limited, for example
for flight vehicles that are commonly instrumented with isolated wall-pressure
probes. To reduce uncertainty, the current work is the first to infuse
isolated wall-pressure measurements from a physical experiment in direct
numerical simulations (DNS) of a Mach-6 boundary layer over a sharp cone. The
computation thus reproduces the measurements and provides an unprecedented
window into the transition process of the experiment.

Computational studies of transition on cones have focused on basic breakdown
mechanisms, and have sought to explore receptivity and to model the
disturbance environment in experiments. Simulations of fundamental and
subharmonic resonant instability waves initiated by a wall forcing yield
controlled breakdown scenarios that are suited for analysis (Hader & Fasel,
2019), but they can deviate significantly from measurements of surface heat
flux and wall pressure spectra (e.g. an order of magnitude difference in wall-
pressure fluctuation magnitude, Chynoweth et al., 2019). A significant
improvement was achieved by Hader & Fasel (2018) who used a simple random-
noise inflow forcing to model the receptivity processes from acoustic waves in
a Mach-6 flow. However, significant over-predictions remain: the streamwise
average heat flux by 65% and peak wall pressure power spectra by over a factor
of 20. The above studies are valuable for understanding canonical transition
scenarios and phenomenology from the experiments. The focus of the present
work is to adopt a robust approach with objective guarantees that the
simulations reproduce experimental measurements.

The herein adopted methodology systematically uses experimental measurements
to rigorously determine the disturbance environment. Our particular focus will
be the experiments by Kennedy et al. (2022), where wall-pressure measurements
were recorded on a $7^{\circ}$ straight cone with a sharp nose in the Mach-6
Ludwieg tube facility at the Air Force Research Laboratory (AFRL). The
instrumentation layout is typical for this type of research, with pressure
probes arranged in a streamwise ray to provide time-resolved information on
streamwise amplification of instabilities; in addition, three azimuthal probes
are placed to assess axi-symmetry of the pressure fluctuations. The wall-
pressure data indicate that the second mode feature prominently without full
breakdown to turbulence. Recent computations using the axisymmetric nonlinear
parabolized stability equations (NPSE) qualitatively capture the trend of the
N-factor, without quantitative agreement (Kennedy et al., 2022) or guidance on
how to objectively select the upstream disturbance spectra. Furthermore, the
role of three-dimensional waves and their relative amplitudes remains unknown
since the simulations were axisymmetric. The present approach will consider
the relevant two- and three-dimensional instability waves and optimize their
amplitudes using a variational framework so that their nonlinear evolution
best reproduces the available measurements thereby establishing confidence in
the entire reconstructed flowfield. The following section details the
experimental setup, numerical simulation, and data assimilation framework.
Section 3 presents the outcomes of the data assimilation, including an
analysis of the nonlinear dynamics from the reconstructed flow field that
faithfully reproduces the measurements. Finally, a conclusion is provided in
§4.

##  2 Flow configuration and methodology

###  2.1 Experiment and measurements

The experimental flow configuration is shown in figure 1(a). The Mach
$M_{\infty}=6.14$ flow developed as part of the dynamics of a Ludwieg tube,
whereby heated ($T_{0}=450\,\text{K}$) and pressurized gas was contained in
the charge tube upstream of the converging-diverging nozzle. Once the fast-
acting valve was opened, the gas accelerated through the nozzle to calculated
free-stream conditions of $U_{\infty}=904\,\text{m/s}$,
$T_{\infty}=54\,\text{K}$, and $\rho_{\infty}=0.0274\,\text{kg/m}^{3}$,
yielding a unit free-stream Reynolds number $Re_{\infty}/L=7.11\times
10^{6}\text{/m}$. The total test time was approximately 0.2 seconds, which was
separated into two approximately time-stationary periods, lasting 100
milliseconds. Data from the second period is analyzed for this work (for
detailed characteristics of AFRL Ludwieg Tube, see Kimmel et al., 2017;
Kennedy et al., 2022).

Figure 1: (a) Schematic of the flow configuration. (b) Extended time series of
the PCB pressure data and (c) a detail for a 80 $\upmu\text{s}$ window.
Signals are offset by 2 kPa for clarity.

The test article was a $\theta=7^{\circ}$ half angle circular cone with a
sharp nose (tip radius $r_{n}$=0.508 mm) and a total length of 414 mm. The
model was installed at zero incidence to the streamwise direction; it was at
room temperature ($T_{w}=300$ K) prior to the start of the experiments, and
changes to the surface temperature during the brief test time are small enough
that they can be ignored. The cone was instrumented with six PCB model 132A
piezo-electric pressure sensors, positioned at four streamwise positions
$\left\\{s_{1},s_{2},s_{3},s_{4}\right\\}=\left\\{215,241,266,291,316\right\\}\,\text{mm}$,
measured downstream of the nose. In addition, the $s_{4}$ row has two probes
offset from the primary ray of sensors by $\pm 8.5^{\circ}$ in order to assess
azimuthal dependence. Simulations and test data suggest that the effective
sensing area has a diameter of approximately $d_{s}=0.97$ mm (Ort & Dosch,
2019). For the case considered, time-resolved pressure data are available for
six probes, acquired at a frequency of 5 MHz. A sample of the measurements is
shown in figures 1(b,c). In (c), the pressure signatures show the presence of
second-mode wave packets amplifying and advecting downstream. For any instance
in (b,c), the signals do not appear to be chaotic with a broad range of time
scales, suggesting the flow has not transitioned to turbulence at the sensors
locations. High-speed schlieren measurements were also acquired, which provide
additional points of validation above the wall. However, our focus is on wall
data which are considered to be the primary modality taken during flight.

An analysis of the experimental data can aid in the choice of inflow
frequencies and wavenumbers adopted in the data-assimilation simulations. The
time-resolved pressure measurements are Fourier analyzed using the Welch
method, similar to previous efforts (Casper et al., 2016; Kennedy et al.,
2022). The window size of 80 $\upmu\text{s}$ (fig 1c) furnishes $10^{3}$ non-
overlapping Hann windows, which yields converged spectral amplitudes. The
spectral resolution was halved by combining every two frequency bins, while
conserving energy, in order to reduce the dimensionality of the data-
assimilation problem. Although the flow is instantaneously three dimensional
(figure 1c), it is statistically axisymmetric under nominal conditions, i.e.,
zero incidence. Since the experimental setup was nominally axisymmetric, we
assume homogeneity in the azimuthal direction and average the spectra of the
three probes at $s_{4}$. The post-processed measurements are then concatenated
into two vectors,

$\displaystyle\mathbf{m}_{S}=\left[\hat{p}\hat{p}^{\star}(s_{1},f_{1}),\ldots,\hat{p}\hat{p}^{\star}(s_{4},f_{N_{f}})\right]^{\top}\text{and}\,\,\,\mathbf{m}_{I}=\left[{\sum}_{i}\hat{p}\hat{p}^{\star}(s_{1},f_{i}),\ldots,{\sum}_{i}\hat{p}\hat{p}^{\star}(s_{4},f_{i})\right]^{\top}\\!\\!\\!\\!,\vspace*{-12pt}$

representing the individual spectral amplitudes and overall intensity of the
signal. The experimental spectra and signal intensity, used in the data
assimilation, are shown with symbols in figure 2. At position $s_{1}$, the
spectrum is dominated by a peak between $f=225$-$275$ kHz, which is expected
due to the presence of unstable second modes on straight cones (Kennedy et
al., 2022). As the boundary layer thickens downstream, the frequency for the
peak energy decreases. By $s_{4}$, the peak amplitude is near 200 kHz, and
higher-harmonics, near 400 kHz, also appear, possibly due to nonlinearity from
these high-amplitude waves. The total intensity of the signal amplifies
between $s_{1}$-$s_{3}$ and appears to saturate between $s_{3}$ and $s_{4}$,
also signaling the presence of nonlinear effects. The goal of this work then
is to identify the unknown two- and three-dimensional instability waves
upstream of $s_{1}$ that reproduce these measurements, and to study the
associated flow field beyond the scope of the sensor data.

###  2.2 Simulation configuration

The simulation domain is shown in figure 1. The flow behind the cone-generated
shock is considered in order to interpret the measured wall-pressure spectra
in terms of post-shock, boundary-layer disturbances. The base flow is
axisymmetric and obeys the Taylor-Maccoll approximation above the boundary
layer, which is a solution to the Blasius equations in coordinates parallel
($\xi$) and normal ($\eta$) to the cone surface using the Mangler-Levy-Lees
transformation. Prescribing the initial base state,
$\mathbf{q}_{B}(\xi,\eta)$, in this manner is a well-established approach
(Sivasubramanian & Fasel, 2015). The inflow Reynolds number, based on the
post-shock, boundary-layer-edge conditions ($U_{e}=880\,\text{m/s}$,
$T_{e}=65\,\text{K}$, and $\rho_{\infty}=0.0458\,\text{kg/m}^{3}$) is
$Re_{o}\equiv\rho_{e}U_{e}L_{o}/\mu_{e}=1345$ using the length scale
$L_{o}=\sqrt{\nu_{e}\xi_{o}/U_{e}}$. At this streamwise position, the inflow
condition is expressed as a superposition of the base state and instability
waves,

$\displaystyle\mathbf{q}_{o}$
$\displaystyle=\mathbf{q}_{B}(\xi_{0},\eta)+Re\left(\sum_{m}\sum_{n}c_{\scriptscriptstyle{{n,m}}}\widehat{\mathbf{q}}_{\scriptscriptstyle{n,m}}(\eta)\exp[\mathrm{i}k_{m}\phi-\mathrm{i}\omega_{n}t+\mathrm{i}\varphi_{\scriptscriptstyle{{n,m}}}]\right)\\!,\vspace*{-12pt}$
(1)

where $c_{\scriptscriptstyle{{n,m}}}$ is the amplitude,
$\widehat{\mathbf{q}}_{\scriptscriptstyle{{n,m}}}$ is the discrete, slow-mode
profile, and ($k_{m}$, $\omega_{n}$, $\varphi_{n,m}$) are the azimuthal
wavenumber, frequency, and relative phase, respectively, for each $(n,m)$
instability wave. The data assimilation attempts to identify the vector
$\mathbf{c}=[\ldots,c_{\scriptscriptstyle{{n,m}}},\ldots]^{\top}$ of
amplitudes of these waves at the inlet. We assume the average spectra is
independent of relative modal phase ($\varphi_{n,m}$), which is reasonable
given the random nature of free-stream tunnel forcing and the relatively long-
time acquisition of the spectra. To reflect this assumption and lack of
information especially for 3D waves, each phase is independently selected from
a uniform random distribution $\varphi_{n,m}=\left[-\pi,\pi\right]$ and
remains fixed during the course of the data assimilation. In contrast, if the
objective were to assimilate the instantaneous wall-pressure signals, the
relative phases of the inflow modes would need to be included in the control
vector and accurately estimated (Buchta & Zaki, 2021).

Figure 2: Schematic of the EnVar data assimilation framework.

We consider 52 frequency and azimuthal wavenumber pairs that encompass Mack’s
first- and second-mode instabilities for the Reynolds-number range
$1345\leq\sqrt{Re_{\xi}}\leq 1872$. This size of the control vector was based
on spectral analysis of the pressure-probes data to determine the dominant
frequencies in the observations (see figure 3a) and preliminary testing. The
frequencies and integer azimuthal wavenumbers are
$f\in\left[50,75,\ldots,350\right]\,\text{kHz}$ and
$k\in\left[0,20,40,60\right]$, respectively. For each $(f,k)$-pair, we
consider the most unstable slow modes (Fedorov, 2011) over the Reynolds number
range of interest, from the discrete Orr–Sommerfeld and Squire spectra. For
the computation of the inflow instability profiles
$\widehat{\mathbf{q}}_{\scriptscriptstyle{{n,m}}}(\eta)$ only, the effect of
curvature was neglected, i.e. we adopt the wavenumber $\beta_{m}=k_{m}/r_{o}$
where $r_{o}$ is the radius of the cone at the inflow, which is a reasonable
assumption for this Reynolds number (Malik & Spall, 1991). Once the inflow
condition (1) is prescribed, the downstream evolution in the DNS/LNS accounts
for curvature fully. Since the experimental measurements lack three-
dimensional information to guide the selection of relevant oblique modes, we
consider a range of azimuthal wavenumbers that are linearly unstable ($k\leq
60$), and discretize this range in a manner that we can efficiently search and
that enables the generation of important higher harmonics by nonlinear
interactions. Beyond the inflow, the azimuthal discretization of the
computations permits the formation of waves up to $k=540$, thus providing
ample resolution for secondary instabilities that arise due to fundamental
resonance between oblique and planar instability waves.

Most of the present simulations solve the compressible, nonlinear Navier-
Stokes equations in curvilinear coordinates (termed DNS) for the simulation
domain shown in figure 1. Linearized dynamics are acquired with the exact same
solver assuming a steady base state and neglecting nonlinear interactions only
(termed LNS). LNS is used to provide an initial estimate of the unknown inflow
spectra that will be refined to account for nonlinear effects; LNS will also
be referenced in the discussion to contrast the disturbance field to the
nonlinear evolution. The gas is assumed ideal with ratio of specific heats
$\gamma=1.4$ and temperature-dependent viscosity, following a power-law
formula, $T^{n}$, where $n$ is determined from measured viscosity data between
the boundary-layer edge and wall temperatures ($T_{e}=65$ K and $T_{w}=300$
K). The flow equations are solved using a standard fourth-order Runge–Kutta
scheme and interior fourth-order finite differences. Near boundaries, stencils
are biased and accuracy is reduced to second order. Second derivatives in the
viscous terms are evaluated using repeated first derivatives, instead of
discretizing the operator directly (Mattsson & Nordström, 2004), which
requires adding high-order numerical dissipation of short waves to stabilize
the solution (for details, see Vishnampet et al., 2015). The computational
domain size $(L_{\xi},L_{\eta},L_{\phi})$=(105 mm, 17.6 mm, $36^{\circ}$) is
discretized with $(N_{s},N_{y},N_{\theta})=(751,201,108)$ grid points. The
azimuthal size was chosen to extend beyond the peripheral probes at $s_{4}$ in
order reduce any artificial correlation by the periodic boundaries. In viscous
units, the grid spacing along the wall at the inflow is ($\Delta\xi^{+}$,
$\Delta\eta^{+}$, $r_{o}\Delta\phi^{+}$) = (2.8, 0.1, 3.1). The azimuthal and
streamwise grids are uniform, and the wall-normal grid spacing is stretched
according to the transformation used in Pruett et al. (1995).

For all of the simulations, the flow develops for 1.75 flow-through times,
based on the edge velocity $U_{e}$, before data acquisition commences. During
40 $\upmu\text{s}$, wall pressure is recorded at every time step; the time
step size is chosen to ensure the $\text{CFL}\approx 0.4$ throughout the
simulated time horizon. The finite-sensing area of the PCB probes is modeled
by averaging the wall pressure, $\widetilde{p}_{j}=1/N\sum_{i=1}^{N}p_{i}$,
for $N$ grid points that satisfy the inequality
$(x_{j}-x_{i})^{2}+(y_{j}-y_{i})^{2}+(z_{j}-z_{i})^{2}\leq(d_{s}/2)^{2}$ where
$d_{s}$ is the sensing diameter. To take advantage of statistical homogeneity
in the simulation, this operation is performed at the streamwise sensor
position for all azimuthal grid points. The simple averaging used to model the
sensing area in DNS was shown to improve adherence to measurements of pressure
spectra and intensity (Huang et al., 2020). The data from the simulated probe
pressure $\widetilde{p}$ are processed and concatenated into two vectors in
the same manner as the experimental measurements,

$\displaystyle\mathbf{h}_{S}=\left[\hat{p}\hat{p}^{\star}(s_{1},f_{1}),\ldots,\hat{p}\hat{p}^{\star}(s_{4},f_{N_{f}})\right]^{\top}\text{and}\,\,\,\mathbf{h}_{I}=\left[{\sum}_{i}\hat{p}\hat{p}^{\star}(s_{1},f_{i}),\ldots,{\sum}_{i}\hat{p}\hat{p}^{\star}(s_{4},f_{i})\right]^{\top}\\!\\!\\!\\!.\vspace*{-12pt}$

These vectors are used as inputs for the following data assimilation
framework.

###  2.3 Data assimilation procedure

The unknown amplitudes $\mathbf{c}$ of instability waves are sought using an
ensemble variational framework (Buchta & Zaki, 2021). To initiate the
nonlinear optimization, an estimate for the control vector $\mathbf{c}_{0}$ is
determined using linear dynamics of the flow. This choice provides a more
physics-based approximation than adopting white noise across the spectrum and,
as such, a more accurate starting estimate for the subsequent nonlinear
optimization. For the linear estimate, the control vector is computed by
direct differentiation of

$\mathcal{J}_{l}=\frac{1}{2}||\mathbf{m}-\mathbf{L}\mathbf{v}||^{2}+\frac{\varrho}{2}||\mathbf{v}||^{2}$
(2)

with respect to the weights and identifying $\mathbf{v}$ at a stationary point
$\nabla\mathcal{J}_{l}=0$. Each column of matrix
$\mathbf{L}=\left[\ldots~{}~{}|~{}~{}\mathbf{l}_{f,k}~{}~{}|~{}~{}\ldots\right]$
represents the wall observations acquired from an independent evolution of
$i$-th instability wave at ($f,k$) using the linearized flow equations and an
axisymmetric Blasius base state. Since the performance of the linear method is
predicated on the validity of a linear assumption, only measurements from the
first sensor position $s_{1}$, where harmonics of the large-amplitude waves
are absent, are included for the estimate of $\mathbf{c}_{0}$. The second term
in (2) represents a penalty which has two desired effects: (i) it mitigates
against high-energy inflow disturbances, and (ii) it improves the conditioning
of the inverse problem. The value of $\varrho$ is chosen such that
$\mathcal{J}_{l}/\mathcal{J}_{\mathbf{v}=0}=10^{-4}$, which ensures that the
measurements are accurately reproduced to within $1$% (see §3). If the
experimental conditions only trigger a linear boundary-layer response, then
minimizing (2) will be sufficient to accurately assimilate the measurements
and predict the inflow amplitudes.

For the experiment considered in §2.1, signatures of nonlinearity are present
in the measurements. As a result, the initial estimate $\mathbf{c}_{0}$ of the
control vector based on (2) is not sufficient, and the data assimilation
requires an iterative, nonlinear optimization. We consider the following cost,

${\mathcal{J}}={\frac{1}{2}||\log_{10}\mathbf{m}_{S}-\log_{10}\mathbf{h}_{S}||_{\Sigma_{m}^{-1}}^{2}}+{\frac{1}{2}||\mathbf{m}_{I}-\mathbf{h}_{I}||_{\Sigma_{n}^{-1}}^{2}}+\frac{1}{2}||\mathbf{c}-\mathbf{c}_{i}||_{\Sigma_{c}^{-1}}^{2},\vspace*{-4pt}$
(3)

which comprises three parts. The first term is the logarithm of the pressure
spectra. Since the spectra span several orders of magnitude, the logarithm
ensures that the cost function is not solely focused on the spectral peak but
rather targets the entire range of frequencies. The second term captures the
importance of the spectral peak in determining the overall pressure intensity.
The final term represents the degree of trust in a prior control vector
($\mathbf{c}_{i}$) to avoid large steps from the previous control vector. The
choice of the cost function can affect performance of the data assimilation,
and our experience has shown that (3) furnishes better accuracy in fewer
iterations relative to, for example, pressure spectra on a linear scale.

An ensemble variational (EnVar) technique is used to update the control vector
in order to minimize the cost function (3). Figure 2 shows a schematic of the
algorithm. An updated estimate of the control vector $\mathbf{c}$ is sought
from its previous value $\mathbf{c}_{i}$ using a weighted superposition of
ensemble members $\mathbf{c}^{(j)}$, specifically
$\mathbf{c}_{i+1}=\mathbf{c}_{i}+\mathbf{P}\mathbf{w}$, where
$\mathbf{P}=\left[\ldots~{}~{}|~{}~{}\mathbf{c}^{(j)}-\mathbf{c}_{i}~{}~{}|~{}~{}\ldots\right]$
is a matrix of the perturbations vectors. Each control vector,
$\mathbf{c}_{i}$ and $\mathbf{c}^{(i)}$, is evolved with the governing
equations, in this case the Navier–Stokes equations, to produce the spatio-
temporal representation of the state $\mathbf{q}_{i}$ and $\mathbf{q}^{(j)}$.
Model observations are then extracted from the mean and ensemble states using
$\mathbf{h}_{i}=\mathcal{M}(\mathbf{c}_{i})$ and
$\mathbf{h}^{(j)}=\mathcal{M}(\mathbf{c}^{(j)})$; they are then compared with
the experimental data in the cost function. An estimate of the local gradient
and Hessian furnish the optimal weights $\mathbf{w}$ by assuming a stationary
point ($\nabla\widetilde{\mathcal{J}}=0$). Multiple updates of the control
vector can be performed in the descent direction, until the cost stagnates or
starts to increase. At this point, the entire EnVar process is repeated,
iteratively and till convergence. The number of ensemble members in the main
algorithm was $N_{e}=10$. Each EnVar iteration thus involves $N_{e}+1=11$
simulations to acquire the local gradient, where the additional DNS
corresponds to the mean of the ensemble. Previous characterizations of the
performance of this method are provided elsewhere (Mons et al., 2021;
Jahanbakhshi & Zaki, 2019; Buchta & Zaki, 2021). The results of the linear and
nonlinear optimization are presented in the next section.

##  3 Results

Figure 3: (a) Power spectra at measurement stations for the experiment
(circles), EnVar prediction (connected squares), and linear prediction (line).
(b) Integral of power spectra. The confidence in each predicted observations
is represented by $\pm 75\sigma_{i}$ where $\sigma_{i}$ is the ensemble
standard deviation of each i-th observation. (c) Cost and gradient during data
assimilation.

The outcomes of the linear and nonlinear assimilations are shown in figure 3.
By design of (2), the linear estimate accurately approximates the experimental
measurements at $s_{1}$ in (a) and (b). However, the subsequent downstream
predictions deviate from the measurements. The linear amplification of the
second mode atop the Blasius base state is more intense than in the
experiments, and modes with $f\gtrsim 300\,\text{kHz}$ decay rather than
amplify. The linear dynamics do not reproduce the wider experimental spectra
downstream or the accumulation of energy in frequencies adjacent to the
energetic second modes. Absence of nonlinear interactions contributes to these
discrepancies, specifically due to the omitted generation of higher harmonics
and the lack of base-flow distortion, which will both be discussed below. Most
importantly, these effects were also omitted from the linear estimate of the
inflow disturbance spectra, and the disagreement with downstream measurements
underscores the importance of a nonlinear interpretation of the observations.

Nonetheless, figure 3 demonstrates that our physics-based linear estimate of
the inflow is a better initial guess for the nonlinear assimilation than an
_ad hoc_ approach to inflow synthesis. Starting from the linear estimate,
figure 3(c) shows the cost reduction by nonlinear EnVar iterations relative to
the LNS, $\mathcal{J}_{ref}$, and the decrease in the gradient magnitude.
After six iterations, the cost and gradient decrease over three and four
orders of magnitude, respectively, and the trends indicate an optimum has been
identified. Additionally, the EnVar approach enables propagated uncertainty of
the prediction. By evolving ensemble members at the final iteration, the
sample standard deviation for each observation, $\sigma_{i}$, is computed.
These error bars, $\pm 75\sigma_{i}$ in figures 3(a,b), confirm the low
uncertainty for our estimates of the measurement. A positive bias, not
explained by parametric uncertainty, remains in the lowest amplitude signals
at $s_{1}$ and $s_{2}$. Overall, however, the adherence of the DNS to the
observations in terms of spectral amplitude and the overall intensity is
encouraging. This level of agreement surpasses in accuracy what we were able
to achieve using the same optimization procedure and considering 2D modes
only. In fact, when the inflow is restricted to axisymmetric waves only (not
shown), the objective function was 60% higher, and the mismatch with the
measurements was most egregious on the final sensor position.

Figures 4 (a) and (b) show the distribution of modal amplitudes predicted by
the nonlinear data assimilation procedure, assuming 2D and 3D incoming
disturbances, respectively. The axisymmetric reconstruction, which was less
effective in reducing the cost function, requires three times more inflow
energy in the two-dimensional waves in order to approximate the measurements
with lower fidelity. The better-performing 3D interpretation prominently
features the oblique waves ($275\leq f\leq 300$ and $20\leq k\leq 40$), which
demonstrates that three-dimensionality is required to reproduce experimental
measurements even when the flow is non-turbulent within the sensing region—an
important point that is often overlooked in the interpretation of wall-
pressure data in high-Mach number transitional flows in terms of the dominant
planar instability waves.

Figure 4: (a,b) Amplitudes of the optimized inflow spectra for the 2D and 3D
assimilation. (c,d) Reconstructed wall quantities: instantaneous $p^{\prime}$,
time-averaged near-wall temperature and pressure intensity. (e) Spanwise
variation of time-averaged wall pressure r.m.s and streamwise vorticity at
sensors $s_{1}$-$s_{4}$. (f) Experimental schlieren and (g,h) numerical
schlieren $\partial_{y}\rho^{\prime}$.

In addition to discovering the unknown inflow that best reproduces the
measurements, figure 4 shows that the data assimilations identify the entire
time-dependent and three-dimensional flow field, far beyond the measurements
probes. The instantaneous pressure fluctuations in (c,d) along the wall reveal
long-streamwise-wavelength, two-dimensional envelopes of instability waves
distributed along the cone. Despite its appearance, we verified using spectral
analysis (c.f. figure 6) that this pattern is not an amplitude modulation.
Instead, the pattern of repeated amplification and decay is due to an
interference of waves with approximately the same advection speed. Atop this
pattern, the signature of oblique fluctuations is prominent in (d), where the
instantaneous pressure fluctuations are more realistic than in the 2D
assimilation since they more closely reproduced the measured wall-pressure
intensity. Figure 4(d) also shows the time-averaged near-wall temperature and
squared-pressure fluctuations, both of which feature streamwise-elongated,
intense streaks extending across multiple sensor positions.

Figure 5: Two-dimensional spectra of wall pressure at the inflow ($s_{0}$) and
probe locations ($s_{1}-s_{4}$). Curves are integrals of energy over
frequency, $E_{f}(k)=\int\hat{p}\hat{p}^{\star}\,df$.

Besides the fact that the 3D interpretation is most accurate, the three-
dimensional characteristics of the flow share common features observed in
previous DNS of fundamental resonance breakdown (Sivasubramanian & Fasel,
2015; Hader & Fasel, 2019) and random-noise forcing (Hader & Fasel, 2018),
e.g. steady streamwise elongated near-wall hot streaks and pairs of counter-
rotating vortical structures. Thus, the reconstructed 3D inflow produces
previously realizable flow fields, but most importantly quantitatively agrees
with measurements. Figure 4(e) focuses on the three-dimensionality of the
wall-pressure intensity and the streamwise vorticity above the sensor
positions. At $s_{3}$ and $s_{4}$, prominent wall-pressure intensity streaks,
with peak amplitude nearly twice the mean, fall between pairs of intense
counter-rotating vortices above which fluid is ejected upward. These
structures correspond to a distortion $(f,k)=(0,20)$ to the base flow, which
can support parametric resonance (subharmonic, fundamental or detune) with the
instability waves—a point that we will revisit below.

We recall that independent schlieren measurements were performed in the
experiment, and were kept for blind comparison to the outcome of assimilating
the wall-pressure data. Figure 4(f) shows an instantaneous realization from
the experimentally acquired schlieren data, and captures the rope-like
structures near the boundary-layer edge. Figures 4(g) and (h) show numerical
schlieren from both the 2D and 3D assimilated flows. Both (g,h) feature the
rope-like structures in the wall-normal density gradient, bear clear
similarity to the experimental schlieren images in that region, and are
similar to one another near the wall. This qualitative agreement serves as
another note of caution against the often-adopted two-dimensional
interpretation of pre-transitional experimental schlieren measurements. In the
present case, only the three-dimensional inflow disturbance field can justify,
and reproduce, the measured wall-pressure data.

We initially argued that the experimental measurements show symptoms of
nonlinearity since harmonics of the dominant frequencies were observed in the
spectra (figure 3), and subsequently that three-dimensionality is important
for successful interpretation of the data. In order to assess the extent of
nonlinearity and three-dimensionality in the 3D assimilated state, in figure 5
we report the wall-pressure spectra as a function of $(f,k)$, evaluated at the
inflow ($s_{0}$) and at the four streamwise sensor positions
($s_{1}$-$s_{4}$). The broadening of the spectra starting at $s_{3}$ coincides
with the prominent base-state distortion in figure 4(d), which is an important
nonlinear effect. The figure also shows the energy integrated over the entire
frequency range and plotted versus the azimuthal wavenumber. The energy in the
oblique waves is commensurate with that in planar ones at $s_{\\{1,3,4\\}}$,
i.e. throughout the majority of the domain.

Select modes of the wall-pressure spectra from DNS are reported in figures
6(a.i, a.ii), separated into unsteady waves $(f,k)$ and base-state distortions
$(0,k)$. For comparison, figure 6(b.i) shows the linear evolution of the
unsteady modes; For the base flow in the LNS, we adopted the time-averaged,
three-dimensional, distorted state computed in the DNS, and as such we
reproduce figure 6(a.ii) in (b.ii). The spectra highlight the necessity of
nonlinear assimilation of the measurements. Between the inflow and $s_{1}$,
nonlinearity causes the spectra in (a.i) to already differ from the linearized
case (b.i). For example, linear theory predicts that $(f,k)=(250,0)$ is
important at $s_{1}$, but this interpretation would lead to an under-
prediction at the inflow; the nonlinear evolution of the mode, which is
relatively muted than amplified, would not match the measurements. Another
example relates to the dominant planar waves (225,0) and (200,0) from the DNS,
which are respectively delayed and under-predicted in LNS. To compensate, a
linear interpretation of the measurements would require large, practically
nonphysical, inflow amplitudes for these second-mode instabilities; such
prediction would lead to drastically poor nonlinear flow response.

As for the the three-dimensional waves, the most energetic mode near $s_{3}$
is $(f,k)=(225,40)$. This instability has similar amplification in subfigures
(a.i) and (b.i), and this agreement was only possible by adopting the
nonlinearly distorted base flow in the LNS. The growth of this mode also
parallels the trend of the base-state distortion $(0,20)$. The latter
corresponds to the streaks in wall-pressure intensity in figure 4(d), and can
support parametric resonance leading to the amplification of $(f,20m)$, where
$m=1,2,...$ for fundamental resonance. The most dominant of these waves,
namely $(f,k)=(225,40)$, would elude detection by the experimental azimuthal
probes at $s_{4}$, whose Nyquist wavenumber is $\approx 21$.

In summary, for the considered experiment, a linear interpretation even of the
most upstream data does not reproduce the downstream measurements, and a focus
on two-dimensional instability waves is fraught with uncertainty, even within
the non-turbulent region. Only nonlinear assimilation of the wall-pressure
sensor data provides rigorous prediction of the inflow amplitudes, and the
present work demonstrated the importance of three-dimensionality.

Figure 6: Streamwise evolution of wall-pressure spectra. Dominant unsteady
modes in (a.i) DNS and (b.i) LNS with three-dimensional distorted base state
$\mathbf{q}_{B}=\langle\mathbf{q}\rangle_{t}$. (a.ii) Dominant steady modes
from DNS, which are part of the distorted base state used in LNS (reproduced
in (b.ii)).

##  4 Conclusion

The unknown upstream disturbances for a Mach-6 flow over a sharp cone were
reconstructed by assimilation of spectral and statistical data from discrete
wall-pressure probes. The data assimilation considered a multi-objective cost
functional: (i) the logarithm of the pressure spectra promoted matching the
data across a wide range of measured frequencies and (ii) the integral of the
spectra promoted the prediction accuracy of modes comprising most of the
intensity. Fidelity of reproducing the measurements from the present Ludwieg-
tube experiment is possible only when three-dimensional nonlinear interactions
are incorporated in the assimilation framework. The need to reproduce these
interactions accurately when considering more consequential and sensitive
dynamics, e.g., transition to turbulence, remains of high importance. For the
present configuration, future measurements must probe three-dimensionality
because the nonlinear optimization showed that only with 3D effects that we
can reproduce the data.

The data assimilation approach is robust, and can be applied to other
experimental conditions. If the dynamics are linear, the initial linear
estimate of the control vector is sufficient to reproduce the measurements.
When important effects are not simulated, the quantitative deviation from the
measurements can guide the improvement of the computational model, in our case
by taking into account nonlinearity and three-dimensional instability waves.
While in this work we focused on predicting the upstream instability waves in
the boundary layer, our methodology can also be applied to estimate incident
free-stream disturbances, either post- or pre- leading-edge shock, including
in the considerably uncertain disturbance environment faced in flight.

Acknowledgements The authors are grateful to the Air Force Research Laboratory
(AFRL) for providing the measurements that were used in the present study.

Distribution Statement A: Approved for Public Release; Distribution is
Unlimited. PA# AFRL-2022-2226.

Funding. This work was supported in part by the US Air Force Office of
Scientific Research (grant no. FA9550-19-1-0230) and by the Office of Naval
Research (grant no. N00014-21-1-2148). Computational resources were provided
by the Maryland Advanced Research Computing Center (MARCC).

Declaration of interests. The authors report no conflict of interest.

Author ORCIDs.
Tamer A. Zaki, https://orcid.org/0000-0002-1979-7748

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