# Effect of turbulent inflow conditions on the dynamics of a 3d supersonic cavity asymmetric flow

Jaime Vaquero, Sébastien Deck

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Jaime Vaquero, Sébastien Deck. Effect of turbulent inflow conditions on the dynamics of a 3d supersonic cavity asymmetric flow. ETMM15, Sep 2025, Dubrovnik, Croatia. pp.44-50. hal-05333711

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# EFFECT OF TURBULENT INFLOW CONDITIONS ON THE DYNAMICS OF A 3D SUPERSONIC CAVITY ASYMMETRIC FLOW

J. Vaquero¹ and S. Deck¹

¹ DAAA, ONERA, Institut Polytechnique de Paris, 92190 Meudon, France

jaime.vaquero@onera.fr

## Abstract

A supersonic three-dimensional open cavity, of naturally asymmetric flow, has been studied by means of a hybrid RANS/LES method (ZDES mode 2 (2020)) and a WMLES approach (ZDES mode 3). For the latter, turbulent inflow conditions representative of wall-bounded turbulence have been rapidly obtained thanks to a dynamic forcing method extended to compressible flows. Both simulations have been validated against experimental data and their in-depth analysis has been performed. In particular, besides the mean flow, spectral analysis of the turbulent field has been carried out in order to better assess the effect of realistic turbulent fluctuations on these kind of flows. The results suggest that the dynamics of the cavity dominate over those of the incoming turbulent boundary layer, but some small effects are still observed. Besides, the asymmetric feature has also been explored by means of the spectral POD (SPOD) analysis the velocity field.

## 1 Introduction

The study here presented focuses on supersonic three-dimensional open cavities with asymmetric mean flow. Even though three-dimensional mean flows in cavities are not a recent observation (e.g. Maull and East (1963) and Larchevêque et al (2007)), most studies have focused on flows over two-dimensional cavities. Asymmetric mean flows in long shallow cavities have been observed for instance in the experiments of Crook et al (2013) for an incompressible flow and in supersonic cavity flows by Turpin et al (2021) and more recently also studied by Baugher et al (2024) or Robison and Garmann (2024) among others. A three-dimensional geometry modifies considerably the mean flow leading to the emergence of a strong vortical structure in the fore-section of the cavity, with a subsequent change in the pressure field and spectrum. Even for cases where the mean flow does not show such an asymmetry, the three-dimensional geometry impacts the acoustic field compared to a two-dimensional cavity (Sun et al (2019)). One of the main features of three-dimensional cavities is the presence of long vortices at the side edges of the cavity, also called spillage vortices (see for instance Beresh et al (2015)). Some investigations have focused in these

vortices (e.g. DeMauro et al (2017) or Baugher et al (2024)) by modifying the geometry and observing the subsequent behaviour of cavity flow, which is of high interest for real-life applications.

Besides the geometrical considerations, other aspects have been studied in the past and in particular the impact of the incoming boundary layer. For instance, there is a clear effect depending on the laminar or turbulent state of the boundary layer and its thickness is an important factor as well (Hamilton Smith et al (2024)). When turbulence resolving approaches are employed in numerical simulations, turbulent inflow conditions are required. For subsonic cavity flows, some studies suggest that the fluctuations generated, either by a forced transition to turbulence (for instance as done by Gloerfelt and Berland (2013)) or by injection of fluctuations (e.g. Chang et al (2007)) do not significantly modify the simulation results. In supersonic cavity flows some studies used turbulent inlet conditions. However, most works seem to consider only random fluctuations superimposed to a mean boundary layer profile. As an example, Liu et al (2021) add random Fourier modes to a turbulent mean velocity profile for their Large Eddy Simulations (LES), but a properly resolved turbulent boundary layer is not observed. In the recent work of Maia et al (2023) of a Wall-modelled LES (WMLES) of a supersonic cavity, the profiles obtained of mean velocity and Reynolds stresses seem to be still quite different from canonical wall turbulence, resulting in a skin friction overestimation.

Regarding turbulence modelling approaches for supersonic cavity flows, many researches have employed approaches based on the Delayed Detached Eddy Simulation (DDES) of Spalart et al (2006) (Turpin et al (2021), Robison and Garmann (2024) or Baugher et al (2024) among others). However, using DDES in fine meshes may lead to a modelled stress depletion (Deck and Renard (2020)) resulting in a poor upstream boundary layer prediction. Moreover, the development of instabilities when switching from Reynolds-averaged Navier-Stokes (RANS) to LES treatment is often delayed which can significantly impact the development of the turbulent mixing layer (e.g. Deck (2012) for a turbulent mixing layer).

Bearing the three previous elements in mind, the
<figure><img src="image_2.png" /><figcaption>Figure 1: Instantaneous field of iso-surface of Q-criterion $Q\frac{L^2}{U_\infty^2} = 10^3$ coloured by the Mach number. The numerical Schlieren $\sqrt{\frac{\partial \rho}{\partial x_i} \frac{\partial \rho}{\partial x_i}}$ is presented in a vertical plane. The mesh employed is composed of $N_{xyz} = 110 \times 10^6$ cells.</figcaption></figure>

main goal in this work is to study the impact of inflow turbulent conditions in WMLES allowing for proper wall-bounded turbulence reproduction upstream of the cavity. A comparison is made to a hybrid RANS/LES method for which a fast development of instabilities is obtained. The Zonal Detached Eddy Simulation (ZDES) mode 2 (2020) of Deck and Renard (2020) is used for the hybrid RANS/LES simulation and ZDES mode 3 (Deck et al. (2014)) for the WMLES one. ZDES mode 2 (2020) automatically switches rapidly from RANS treatment of attached boundary layers (robust to the mesh refinement level and adverse pressure gradients) to LES in separated regions. ZDES mode 3 instead corresponds to a WMLES approach, where the inner layer of the boundary layer is treated in RANS (corresponding to the wall-model) and the outer layer fully treated in LES, and has proven its great performance in several works for high-Reynolds-number turbulent boundary layer (e.g. Vaquero et al. (2022)).

## 2 Presentation of the flow configuration

A three-dimensional cavity investigated experimentally in the ONERA continuous S8Ch wind tunnel is simulated, with an incoming flow at Mach number 1.57, and values of total pressure and total temperature of $P_t = 95.6 \text{ kPa}$ and $T_t = 298 \text{ K}$ respectively. The geometry of the cavity is $L/D = 5$ and $W/D = 1$, where $L, W$ and $D = 0.04 \text{ m}$ correspond to the length, width and depth of the cavity respectively. The Reynolds number based on the cavity length is $Re_L = \frac{\rho_\infty U_\infty L}{\mu_\infty} = 2.76 \times 10^6$, where $\rho_\infty = 0.41 \text{ kg} \cdot \text{m}^{-3}$, $U_\infty = 446 \text{ m} \cdot \text{s}^{-1}$ are the free-stream density and velocity respectively, and $\mu$ represents the dynamic viscosity. The incoming flow is at quite high Reynolds number, in particular for the boundary layer reaching $Re_\tau = \frac{\rho_w u_\tau \delta}{\mu_w} = 1373$. The symbol 'w' denotes wall quantities, $u_\tau$ is the friction

velocity (such that the wall shear stress is $\tau_w = \rho_w u_\tau^2$) and $\delta$ the boundary layer thickness.

## 3 Analysis of the mean flow

For ZDES mode 3 turbulent inflow conditions are required. Among the different inflow conditions existing in the literature, turbulent fluctuations are generated using white noise fluctuations combined with a dynamic forcing method (modified by Laraufie et al (2011)) which has been extended to compressible flows (denoted CDF). The CDF allows to significantly reduce the adaptation length for injected fluctuations to become representative of resolved wall-bounded turbulence (see Deck et al (2018)). Besides, in supersonic turbulent boundary layers using white noise is a good compromise to avoid generating too much spurious noise compared to other methods.

A first insight on the resolved turbulent flow is presented in figure 1. This figure illustrates the unsteady field of ZDES mode 2 (2020) and ZDES mode 3 simulations where the resolved turbulent fluctuations of the incoming boundary layer in ZDES mode 3 are clearly observable. In the numerical Schlieren, the white noise acoustic signature associated to the incoming turbulence is almost unnoticeable. Moreover, the quick development of resolved turbulence in the cavity shear layer for ZDES mode 2 (2020) and the fine resolution of turbulent structures in both simulations are also evidenced.

Figure 2 plots the boundary layer Mach number profile obtained at a distance of about $8.2\delta_0$ downstream of the inlet of the computational domain (which is about $1.8\delta_0$ upstream of the cavity), $\delta_0$ being the inlet boundary layer thickness. The profiles of both simulations are compared to an experimental measurement of the incoming boundary layer. One may observe that the boundary layer profile of ZDES mode
<figure><img src="image_3.png" /><figcaption>Figure 2: Mach number profile 8.2δ₀ downstream of the inlet of the computational domain. Symbols: experimental measurements, —: ZDES mode 2 (2020), —: ZDES mode 3.</figcaption></figure>

3 is in excellent agreement against both the experiment and ZDES mode 2 (2020) which actually corresponds to a RANS profile with the Spalart-Allmaras turbulence model. Profiles of Reynolds stresses at this station, not shown here, are in good agreement as well with literature references at equivalent Reynolds number, thus evidencing the great performance of CDF in rapidly providing physical resolved wall-bounded turbulence and hence the good quality of resolved turbulence upstream of the cavity. The profiles of streamwise and spanwise Reynolds stresses in figure 3 (⟨u'²⟩ and ⟨v'²⟩ respectively) show that inflow conditions have a non negligible effect on these at the early stages of the mixing layer. ZDES mode 2 (2020) presents some turbulent fluctuations at the front edge of the cavity which are associated to the shock/expansion waves alternation. For ZDES mode 3, higher levels of Reynolds stresses are observable due to the resolved fluctuations of the incoming boundary layer. Slightly farther downstream, one may notice that ZDES mode 2 (2020) profiles get quite close to those from ZDES mode 3, and this becomes even more evident farther downstream (not shown). Hence, the intrinsic cavity dynamics seem to dominate the fine-scale turbulence disturbances inherited from the upstream boundary layer. Further analysis of the turbulent field is performed in the following section.

# 4 Turbulent field analysis

## Spectral analysis of the bottom wall pressure

Experimental measurements of pressure signals on the bottom wall of the cavity are available and presented at two positions, corresponding to $x/L = 0.25$ and $x/L = 0.5$, both of them located on the center plane ($y/D = 0$). Due to the three-dimensional nature of the mean flow, spanwise averaging is not possible for convergence of statistics. Besides, the flow is greatly influenced by low-frequency dynamics, which requires signal recording for a long period of time. These aspects together with the presence of Rossiter modes make the spectral estimator of Burg

<figure><img src="image_4.png" /><figcaption>Figure 3: Profiles of streamwise and spanwise Reynolds stresses at the front edge of the cavity (solid lines) and at $x/L = 0.04$ (dotted lines). —: ZDES mode 2 (2020), —: ZDES mode 3.</figcaption></figure>

(1978), which is a parametric autoregressive method, very well suited for the signal analysis of the present study. It has been satisfactorily employed previously for instance by Larchevêque et al (2004) for cavity flows or by Deck (2005) for airfoils in transonic flows. For the present simulations, signal has been recorded for a total time of $491U_∞/D$ which corresponds to about 24 times the period of the first Rossiter mode.

Figure 4a displays the power spectral density of the sound pressure level SPL = 20 log ($p_{rms}$/20 µPa) at $x/L = 0.5$ on the bottom wall of the cavity. The Strouhal number is defined as $St_D = fD/U_∞$. Both simulations reproduce accurately the main dynamics of the cavity when compared to the experimental data, which are essentially dominated by the Rossiter modes indicated by the vertical dashed lines according to the modified Rossiter's formula from Heller et al (1971). It is interesting to notice a slight difference for the low frequency content, where ZDES mode 3 is closer to the experimental data and provides a more accurate prediction of the first Rossiter mode ($n = 1$). ZDES mode 2 (2020) contains energy at even lower frequencies which are in contrast to the trend observed both for the experimental data and ZDES mode 3. However, this difference of levels does not contribute much to the total SPL presented in figure 4b (at $x/L = 0.5$). Moreover, the levels as well as the trend of the SPL evolution are in accordance with similar cavity flows in the literature (see for instance Zhuang et al (2006)). Good predictions are also observed at $x/L = 0.25$ and $x/L = 0.75$, with a slightly better agreement for ZDES mode 3 at the latter. It is also interesting to notice some discrepancies between both simulations at the two minima of the SPL evolution, which is more evident in figure 5 for the whole bottom wall.

## SPOD of the velocity field

The Spectral Proper Orthogonal Decomposition (SPOD) consists in solving:

$$ \hat{C}W\hat{\Psi} = \hat{\Psi}\hat{\Lambda} \quad (1) $$

where $\hat{C}$ is the cross-spectral density tensor of the
<figure><img src="image_5.png" /><figcaption>Figure 4: Sound pressure level at the bottom wall of the cavity. Symbols: experimental measurements from ONERA S8Ch wind tunnel, —: ZDES mode 2 (2020), -: ZDES mode 3.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: SPL at the bottom wall of the cavity ($z/D = -1$).</figcaption></figure>

state vector **q** considered, $\hat{\Psi}$ contains the eigen vectors which correspond to the SPOD modes and $\hat{\Lambda}$ the corresponding eigen values. The matrix **W** is a weight matrix that defines the inner product such that for two different states vectors, it is verified $\langle \mathbf{q}_1, \mathbf{q}_2 \rangle = \mathbf{q}_1^H \mathbf{W} \mathbf{q}_2$, where **H** denotes the Hermitian transpose. The reader may find more details in Schmidt and Colónnius (2020). Welch's method has been employed for the PSD estimations and the number of blocks for Welch's averaged periodogram method has been set to 8. It is a good compromise between the number of blocks and still capturing the lowest frequencies in the flow. In this study, the SPOD has been applied to the streamwise and spanwise velocity fields at the horizontal surface $z/D = 0$ bounded by all the edges of the cavity, the bottom wall being located at $z/D = -1$. A global exploration of the SPOD energy at different Rossiter modes has evidenced the great dominance of the leading SPOD mode for the second and third Rossiter modes of the cavity, containing about 95% of the energy and therefore leaving

very little energy for the remaining SPOD modes (not shown).

Some combinations of SPOD and Rossiter modes carefully chosen are illustrated in figures 6 and 7 for the streamwise and spanwise velocity components respectively. For the streamwise velocity component, first and second SPOD modes are plotted at the frequency corresponding to the second Rossiter mode (the most dominant one, figure 4a). For the spanwise velocity component, only the first SPOD mode is illustrated, but this is done for the second and the third Rossiter modes.

Figures 6a and 6b depict the leading SPOD mode of the second Rossiter mode for the streamwise velocity fluctuation. The SPOD mode is very close for both ZDES mode 2 (2020) and ZDES mode 3, which indicates that this mode does not seem to be very sensitive to the turbulent incoming field. In addition, there is a clear and interesting completely asymmetrical shape composed of somewhat oblique structures. When looking at the second SPOD mode in figures 6c and 6d, which contains much less energy as already mentioned, a quite different mode shape is obtained between both simulations. In particular, ZDES mode 3 shows a more important presence of this mode near the side edges of the cavity at around $x/L = 0.4$ and an alternating mode shape downstream. In contrast, ZDES mode 2 (2020) presents a mode which also shows an asymmetry of field, but seems essentially composed of two elongated oblique structures.

Regarding the SPOD analysis for the spanwise velocity fluctuation, the leading SPOD mode for the second Rossiter mode, figures 7a and 7b, appears to be very close between both simulations. Similarly to the observations of the SPOD of the streamwise velocity, this means that for this Rossiter mode, the leading SPOD mode seems quite insensitive to incoming turbulence and rather dominated by the intrinsic cavity dynamics. Nevertheless, there is a slight difference in the mode distribution, again near the side cavity edges. It is also in this case interesting to notice the highly
<figure><img src="image_7.png" /><figcaption>Figure 6: SPOD of the streamwise velocity component. (a), (b): SPOD leading mode; (c), (d): SPOD second mode.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: SPOD of the spanwise velocity component. SPOD leading mode for (a), (b) 2^nd^ and (c), (d) 3^rd^ Rossiter modes.</figcaption></figure>

non symmetric shape of the mode, featured again by
large oblique structures. These observations may also
be applied to the leading SPOD mode of the third
Rossiter mode illustrated in figures 7c and 7d. Indeed,
the modes distributions are qualitatively the same and
the highly asymmetric shape is also observed. It is
nonetheless worth mentioning some minor discrepan-
cies locally located around x/L = 0.8.

5 Conclusions

The effect of realistic turbulent boundary layer
fluctuations on a supersonic high-Reynolds number
cavity flow is rarely adressed in the literature due to its
complex implementation in an industrial solver. Tur-
bulent inflow conditions have been properly generated
in the present work thanks to the CDF for the WM-
LES (ZDES mode 3) simulation and their effect on
the cavity downstream has been studied. Compar-
isons to hybrid RANS/LES (ZDES mode 2 (2020))
have been made, where inflow fluctations are absent

since the incoming flow upstream of the cavity is com-
pletely treated in RANS approach. The available ex-
perimental data have proven that both simulations re-
produce accurately the dynamics of the cavity flow and
in particular the SPL spectral distribution and its evo-
lution over the cavity bottom wall. SPL at the bottom
wall seems nonetheless somewhat sensitive to the in-
flow conditions. The SPOD analysis of the two domi-
nant Rossiter modes has also evidenced a globally fair
agreement between both simulations. Discrepancies
appear to be confined to the second SPOD mode which
carries much less energy than the leading mode, thus
confirming the great dominance of the intrinsic cavity
dynamics.

This study shows that even when generating proper
turbulent fluctuations their effect on this cavity dynam-
ics seem to be secondary. This is however a promising
result for hybrid RANS/LES approaches that enable a
rapid development of instabilities such as ZDES mode
2 (2020). Besides, the consequent database generated
with both simulations and the differences observed are worth further studying. These, together with the asymmetric feature of the mean flow, compose an interesting aspect to keep exploring for supersonic cavity dynamics.

## Acknowledgments

The authors would like to thank all the people involved in the development of the FLU3M solver. The colleagues from the AMES, MAPE, MASH and MSAT units of the DAAA department of ONERA are also acknowledged, in particular Colin Leclercq and Lucas Boussard, for fruitful discussions, as well as the ONERA research project SUPERFLAP.

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