1. Introduction
Open- and deep-cavity flows with abrupt geometric boundaries often exhibit complex unsteady flow phenomena, including shear-layer separation, vortex shedding, downstream-edge impingement and strong recirculation (Morrall et al. Reference Morrall, Quayle and Campobasso2020; Li et al. Reference Li, Yang and He2024). In classical open-cavity configurations, the separated shear layer develops across the cavity mouth and convects vortical structures towards the downstream edge. The interaction of the shear layer with the downstream edge and the internal recirculating flow can generate pronounced velocity and pressure fluctuations, and strongly influence momentum exchange between the incoming flow and the cavity interior. Although the lid-driven square cavity is a canonical benchmark model with simple geometry and well-defined boundary conditions, practical engineering cavities, such as aerospace weapon bays, turbine inspection channels and nuclear reactor cooling passages, often exhibit deep-cavity configurations with
$L/D$
< 1, resulting in flow characteristics that differ substantially from ideal square cavities (Burggraf Reference Burggraf1966; Shankar & Deshpande Reference Shankar and Deshpande2000; Cornu et al. Reference Cornu, Keirsbulck, Kerhervé, Aloui and Lippert2016). Under high-Reynolds-number inflow conditions, the shear layer over the cavity mouth interacts with the internal recirculating flow and cavity walls, producing primary vortex deformation, secondary-flow development, local recirculation, turbulence-energy redistribution, and spatially non-uniform velocity responses (Sampath & Sinhamahapatra Reference Sampath and Sinhamahapatra2016; Saxena & Yaghoobian Reference Saxena and Yaghoobian2020). Low-aspect-ratio cavities in high-speed flows are also of engineering interest because of their strong unsteadiness, wall loading, and flow-control relevance (Jing et al. Reference Jing, Haiming, Guo, Song and Roy2015; Paolicchi & Santos Reference Paolicchi and Santos2018).
Research on cavity flows has evolved from standard square cavities to engineering deep cavities, from low to high Reynolds numbers, and from single-plane measurements to multi-plane measurements. Early studies mainly focused on lid-driven square cavities, analysing the effect of wall shear on the primary vortex, secondary vortices and circulation patterns (Bozeman & Dalton Reference Bozeman and Dalton1973; Koseff & Street Reference Koseff and Street1984). With the advancement of numerical simulations and experimental techniques, large-eddy simulations (LES), direct numerical simulations (DNS), and high-resolution particle image velocimetry (PIV) have been extensively used, revealing the splitting of primary vortices, thinning of boundary layers, and turbulence structure evolution at high Reynolds numbers (Lawson & Barakos Reference Lawson and Barakos2011; Wang et al. Reference Wang, Peng, Cheng, Gao, Yin, Shen and Gao2025). For inflow-driven deep cavities, previous studies primarily focused on shear-layer development, recirculation zones and vortex evolution (Wu et al. Reference Wu, Luo and Fan2008; Faure Reference Faure2014; Huang & Chung Reference Huang and Chung2023). Recent works further investigated turbulence modulation by particle-laden flows (Bluestein & Bohl Reference Bluestein and Bohl2021), shear-layer instability in rarefied hypersonic flows (Jin et al. Reference Jin, Cheng, Wang and Wang2022, Reference Jin, Cheng, Wang and Wang2023; Guo et al. Reference Guo, Luo and Wu2024), and cavity-induced boundary-layer transition (Bai et al. Reference Bai, Duan and Xiao2022), highlighting the complexity of deep-cavity flows beyond classical square cavities (Wahba Reference Wahba2014).
Despite extensive studies of cavity flows, many investigations have focused on pressure fluctuations or the symmetric mid-plane flow field. In contrast, systematic experimental measurements of turbulence statistics and instantaneous velocity extremes across different streamwise and spanwise planes remain limited. Even in nominally two-dimensional cavities, significant three-dimensional unsteady features exist, and sidewall interference can alter shear-layer development, vortex shedding, pressure fluctuations, and the spatial organisation of the flow (Basley et al. Reference Basley, Pastur, Delprat and Lusseyran2013; Di Cicca et al. Reference Di Cicca, Martinez, Haigermoser and Onorato2013; Douay et al. Reference Douay, Faure and Lusseyran2013; Turpin et al. Reference Turpin, Speth, Sherer and Granlund2021). Experimental and numerical studies of finite-span cavities show that sidewall effects significantly influence the spatial distribution of primary and secondary vortices and recirculation structures (Zhang & Naguib Reference Zhang and Naguib2011; Liu et al. Reference Liu, Gómez and Theofilis2016; Picella et al. Reference Picella, Loiseau, Lusseyran, Robinet, Cherubini and Pastur2018; Guo et al. Reference Guo, Gong and Zhang2020), leading to notable differences among streamwise planes at different spanwise locations. Therefore, two main gaps remain for high-Reynolds-number deep-cavity flows. First, experimental data for high-Reynolds-number deep cavities are still scarce, limiting the understanding of primary vortex deformation, secondary vortex suppression, shear-layer breakdown and turbulence-energy redistribution, which are often inferred from DNS or LES studies (Heinz Reference Heinz2020; Taghvaei & Amani Reference Taghvaei and Amani2023; Smith et al. Reference Smith, Lawson and Vio2024). Second, the modulation of vortical structures, turbulence statistics and instantaneous velocity extremes by sidewalls across different measurement planes lacks systematic experimental validation (Liu et al. Reference Liu, Sun, Yeh, Ukeiley, Cattafesta and Taira2021; Naseer et al. Reference Naseer, Arif, Leung and Lam2023). In addition, although classical open-cavity studies have established the importance of shear-layer feedback and self-sustained oscillations, velocity-field evidence from multi-plane PIV measurements remains valuable for clarifying how strong velocity events, fluctuation intensities and higher-order statistics vary spatially inside finite-width deep cavities.To address these gaps, an inflow-driven deep-cavity experiment with
$L/D$
=
$1/3$
was conducted at Re = 2.55 × 105. Three streamwise planes, namely the near-wall plane (
$z/W$
= 0.10), intermediate plane (
$z/W$
= 0.25) and middle plane (
$z/W$
= 0.50), together with three spanwise planes (
$x/L$
= 0.10, 0.25, 0.50), were systematically measured using two-component PIV. The present study provides a multi-plane analysis of the mean flow structures, root mean square (RMS) velocity fluctuations, in-plane turbulent kinetic energy (TKE), instantaneous maximum velocities, and higher-order statistics. Particular attention is given to the spatial variations among the measured planes, the modulation associated with sidewall confinement and downstream flow turning, and the distribution of local intermittent high-velocity events. By comparing both streamwise and spanwise measurement planes, the study aims to clarify the spatial non-uniformity and anisotropic fluctuation characteristics of high-Reynolds-number deep-cavity flow, and to provide experimental data for turbulence-model validation and engineering flow analysis.
2. Experimental set-up and methods
2.1. Experimental model and apparatus
To investigate the flow structures and spatial non-uniformity within an inflow-driven deep cavity, a laboratory experimental system was constructed. The cavity model was made of transparent acrylic with internal dimensions L × W × D = 0.1 m × 0.1 m × 0.3 m, where L is the streamwise length, W is the spanwise width, and D is the vertical depth. The coordinate system is defined hereafter with x along the streamwise direction, y along the vertical direction, and z along the spanwise direction. Accordingly, the streamwise measurement planes correspond to
$xy$
planes at different z locations, whereas the spanwise measurement planes correspond to yz planes at different x locations. The cavity has length-to-depth ratio
$L/D$
= 1/3, and width-to-depth ratio W/D = 1/3, with wall thickness 7 mm. The cavity was connected to upstream and downstream straight test sections with square cross-sections measuring 0.10 m × 0.10 m. The upstream and downstream straight test-section lengths were 0.95 m and 0.20 m, respectively. The cavity top connects smoothly to a straight test section wall, allowing water to flow continuously over the cavity mouth and form a shear layer. The upstream flow channel provides the incoming flow over the cavity mouth. No additional obstacles were placed inside the cavity, and the measurement region covers the main flow area, allowing the analysis of flow structures and turbulence statistics. Figure 1(a) shows the experimental model and the locations of three representative streamwise and spanwise measurement planes. Figure 1(b) illustrates the data acquisition and processing workflow.
Experimental configuration and data-processing procedure for the deep-cavity flow measurements. (a) Cavity model and measurement planes and (b) Data acquisition and processing workflow.
2.2. Experimental conditions and plane arrangement
The flow field was measured using a time-resolved two-component PIV system. The imaging system consisted of an M220 high-speed camera equipped with a 50 mm F1.4 fixed-focus lens. The illumination source was a 10 W continuous 532 nm green laser, which was expanded into a light sheet with thickness approximately 1 mm using a cylindrical lens assembly. Hollow glass microspheres of diameter 10 μm and density 1.1 g cm−3 were used as tracer particles. Images were acquired in single-frame mode at resolution 1920 × 1080 pixels and sampling frequency 700 Hz. Each measurement plane was recorded continuously for 4 s, yielding 2800 instantaneous frames. The PIV image processing was performed using RFlow4 software with a multi-pass iterative cross-correlation algorithm. The initial interrogation-window size was 64 × 128 pixels, and was refined to final interrogation-window size 24 × 32 pixels, with 25 % overlap. The scale factor obtained from the PIV calibration was 0.313 mm pixel−1. Based on this scale factor, the final interrogation window corresponded to physical area approximately 7.51 mm × 10.02 mm, and the final velocity-vector spacing was approximately 5.63 mm × 7.51 mm.
The working fluid was water at room temperature, with kinematic viscosity
$v$
= 1.0 × 10−6 m2 s−1. The Reynolds number was defined as Re = U
lid
D/
$v$
, where U
lid
is the characteristic inflow velocity above the cavity, and D is the cavity depth. In the experiments, U
lid
= 0.85 m s−1, corresponding to Re = 2.55 × 105. Based on this coordinate system, streamwise measurement planes were located at z = 0.01, 0.025, 0.05 m, and spanwise measurement planes were located at x = 0.01, 0.025, 0.05 m. All measurement planes had field of view 0.1 m × 0.3 m.
2.3. Experimental procedure
Prior to experiments, the cavity and circulation system were cleaned. Tracer particles were added and circulated to ensure uniform distribution. The laser sheet was aligned with the measurement plane, and the camera was positioned perpendicularly to the measurement region. The pump was started, and once the flow stabilised, measurements were initiated. Each plane was recorded continuously for 4 s (2800 frames). The streamwise planes were measured in the order z = 0.05, 0.025, 0.01. For spanwise planes near the sidewall, bubbles were removed before acquisition to prevent optical interference. Processed PIV images yielded instantaneous velocity fields, from which time-averaged velocities, RMS fluctuations, TKE and higher-order statistics were calculated to characterise the flow structure and spatial non-uniformity.
2.4. Uncertainty analysis
Measurement uncertainty mainly arises from PIV image displacement accuracy, tracer particle response, inflow velocity measurement, and statistical convergence. The PIV displacement was computed using multi-pass iterative cross-correlation with sub-pixel precision 0.1 pixels. Combined with calibration factor 0.313 mm pixel−1 and frame interval 1429 μs, the instantaneous velocity uncertainty is approximately 0.0219 m s−1, corresponding to a 2.6 % relative error compared with the characteristic inflow velocity 0.85 m s−1. The tracer particle Stokes number was 5.17 × 10−5, much smaller than 0.1, indicating negligible particle inertia and good flow tracking. Multiple inflow velocity measurements above the cavity were averaged to reduce fluctuations in the Reynolds number calculation.
Each measurement plane provided 2800 samples for statistical analysis. To evaluate the convergence of the first- and second-order statistics, a sample-size convergence analysis was performed on a representative streamwise mid-plane. For a given sample size N, the first-order moments
$\overline{u}$
and
$\overline{v}$
, the RMS velocities u
rms
,
$v$
rms
, and the two-component in-plane fluctuation energy k
2C
were calculated and compared with the corresponding full-sample statistics. The relative convergence error was defined as
$E_{Q}(N)=\|Q_{N}-Q_{{N_{\textit{total}}}}\|_{2}/\|Q_{{N_{\textit{total}}}}\|_{2}\times 100\,\%$
. As shown in figure 2(a), the relative errors of all examined quantities decrease continuously with increasing sample size. At N/N
total
= 0.95, the relative errors of
$\overline{u}$
,
$\overline{v}$
, u
rms
,
$v$
rms
and k
2C
are 2.51 %, 1.22 %, 2.37 %, 1.84 % and 2.81 %, respectively. Figure 2(b) further shows that the normalised statistics at the representative high-fluctuation location remain stable over the final portion of the record. The convergence assessment demonstrates that the present sample size provides stable estimates of the time-averaged velocity, RMS velocity fluctuations, and in-plane fluctuation energy. Accordingly, the subsequent analysis focuses on the spatial distributions of the mean flow and second-order statistics, together with the time-domain characteristics and probability distributions of the instantaneous maximum velocity.
Sample-size convergence analysis of first- and second-order statistics. (a) Full-field relative convergence errors calculated with respect to the full-sample statistics. The examined quantities include the first-order moments
$\overline{u}$
and
$\overline{v}$
, the RMS velocities u
rms
and
$v$
rms
, and the two-component in-plane fluctuation energy k
2C
. (b) Pointwise convergence of normalised statistics at the representative high-fluctuation location selected from the maximum full-sample k
2C
.
2.5. Data processing and analysis methods
2.5.1. Time-averaged velocity
Time-resolved two-component PIV measurements provided 2800 instantaneous velocity fields. Each velocity field contained the spatial coordinates and the corresponding two in-plane velocity components. In the streamwise
$xy$
planes, the measured velocity components were u and
$v$
, corresponding to the streamwise and vertical directions, respectively, whereas in the spanwise yz planes, the measured velocity components were
$w$
and
$v$
, corresponding to the spanwise and vertical directions, respectively. The following definitions are written for u and
$v$
in the streamwise planes, and the same procedure was applied to the spanwise planes by replacing u with
$w$
. After interpolation onto a uniform grid, the instantaneous velocities and their squares at each grid point were accumulated for further statistical analysis. The time-averaged velocity components were calculated as
\begin{equation} \sum_{i=1}^{N}u_{i},\quad \sum _{i=1}^{N}v_{i},\quad \sum_{i=1}^{N}u_{i}^{2},\quad \sum_{i=1}^{N}v_{i}^{2}, \end{equation}
where N = 2800 is the total number of frames. The time-averaged velocity components are defined as
\begin{equation} \overline{u}=\frac{1}{N}\sum_{i=1}^{N}u_{i},\quad \overline{v}=\frac{1}{N}\sum_{i=1}^{N}v_{i}. \end{equation}
The corresponding time-averaged speed is calculated as
\begin{equation} U=\sqrt{\overline{u}^{2}+\overline{v}^{2}}. \end{equation}
Time-averaged velocity fields provide a clear visualisation of the flow patterns and spatial variations in velocity gradients (Karim & Mohammed Reference Karim and Mohammed2020).
2.5.2. The RMS velocity
The RMS velocity quantifies the intensity of velocity fluctuations, reflecting the strength of turbulent motion (Castro Reference Castro2015). For the streamwise and vertical velocity components, the RMS values were computed as
\begin{equation} u_{\textit{rms}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}{u_{i}}^{2}-\overline{u}^{2}},\quad v_{\textit{rms}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}{v_{i}}^{2}-\overline{v}^{2}}, \end{equation}
where u
rms
and
$v$
rms
represent the fluctuation intensities in the streamwise and vertical directions, respectively. These values can be used to assess turbulence anisotropy and spatial non-uniformity in the measurement plane.
2.5.3. Instantaneous velocity and maximum plane velocity
Instantaneous velocity fluctuations are defined as
\begin{equation} u'_{i}=u_{i}-\overline{u},\quad v'_{i}=v_{i}-\overline{v}, \end{equation}
the instantaneous speed at any point (x i ,y j ) is given by
\begin{equation} U(x_{i},y_{\!j},t_{k})=\sqrt{u(x_{i},y_{\!j},t_{k})^{2}+v(x_{i},y_{\!j},t_{k})^{2}}, \end{equation}
and the maximum instantaneous speed over the measurement plane at each time step is
\begin{equation} U_{\textit{max}}\left(t_{k}\right)=\max _{i,j}\left[U\left(x_{i},y_{\!j},t_{k}\right)\right]\!. \end{equation}
The corresponding statistical metrics are defined as
\begin{equation} \overline{U_{\textit{max}}}=\frac{1}{N}\sum_{k=1}^{N}U_{\textit{max}}\left(t_{k}\right)\!, \end{equation}
\begin{equation} {U}_{\textit{max}}^{pp}=\max (U_{\textit{max}}(t_{k}))-\min (U_{\textit{max}}(t_{k})), \end{equation}
\begin{equation} \sigma _{{U_{\textit{max}}}}=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left(U_{\textit{max}}\left(t_{k}\right)-\overline{U_{\textit{max}}}\right)^{2}}, \end{equation}
\begin{equation} I_{{U_{\textit{max}}}}=\frac{\sigma _{{U_{\textit{max}}}}}{\overline{U_{\textit{max}}}}. \end{equation}
2.5.4. Higher-order statistics
Skewness and kurtosis describe the probability distribution of U max (Shankar & Deshpande Reference Shankar and Deshpande2000):
\begin{equation} \textit{skewness}=\frac{\dfrac{1}{N}\sum \limits_{k=1}^{N}\left(U_{\textit{max}}\left(t_{k}\right)-\overline{U_{\textit{max}}}\right)^{3}}{\left[\dfrac{1}{N}\sum\limits_{k=1}^{N}\left(U_{\textit{max}}\left(t_{k}\right)-\overline{U_{\textit{max}}}\right)^{2}\right]^{{3}/{2}}}, \end{equation}
\begin{equation} \textit{kurtosis}=\frac{\dfrac{1}{N}\sum \limits_{k=1}^{N}\left(U_{\textit{max}}\left(t_{k}\right)-\overline{U_{\textit{max}}}\right)^{4}}{\left[\dfrac{1}{N}\sum\limits_{k=1}^{N}\left(U_{\textit{max}}\left(t_{k}\right)-\overline{U_{\textit{max}}}\right)^{2}\right]^{2}}.\end{equation}
Positive skewness indicates more frequent high-speed events, while kurtosis significantly greater than 3 indicates heavy-tailed distributions and extreme events.
Time-averaged velocity fields on different streamwise planes: (a) resultant velocity
$U/U$
lid
with superimposed mean velocity vectors; (b) horizontal velocity
$u/U$
lid
; (c) vertical velocity
$v/U$
lid
.
2.5.5. The TKE
For a two-dimensional measurement plane, the TKE per unit mass is
\begin{equation} \begin{array}{l} k=\dfrac{1}{2}\left(u'^{2}+v'^{2}\right)\!.\\ \end{array} \end{equation}
This represents the kinetic energy of velocity fluctuations.
3. Results and discussion
3.1. Time-averaged velocity distribution
3.1.1. Streamwise plane time-averaged velocity distribution
Figure 3 presents the time-averaged velocity fields on three streamwise planes at
$z/W$
= 0.1, 0.25, 0.5, showing the overall distribution of the time-averaged velocity across different spanwise locations. Figure 3(a) shows the resultant velocity
$U/U$
lid
with superimposed mean velocity vectors, figure 3(b) shows the horizontal velocity component
$u/U$
lid
, and figure 3(c) shows the vertical velocity component
$v/U$
lid
. To quantitatively illustrate the variation of velocity magnitudes and components at different locations, figure 4 presents the time-averaged velocity along typical characteristic lines: figures 4(a) and 4(b) show the resultant velocity
$U/U$
lid
along vertical lines at
$x/L$
= 0.5 and
$x/L$
= 0.05, figure 4(c) shows
$U/U$
lid
along a horizontal line at
$y/D$
= 0.2, figure 4(d) shows
$u/U$
lid
along the vertical line
$x/L$
= 0.5, and figure 4(e) shows
$v/U$
lid
along the horizontal line at
$y/D$
= 0.70.
Time-averaged velocity distributions along characteristic lines on different streamwise planes: (a)
$U/U$
lid
along vertical line
$x/L$
= 0.5; (b)
$U/U$
lid
along vertical line
$x/L$
= 0.05; (c)
$U/U$
lid
along horizontal line
$y/D$
= 0.20; (d) horizontal velocity
$u/U$
lid
along vertical line
$x/L$
= 0.5; (e) vertical velocity
$v/U$
lid
along horizontal line
$y/D$
= 0.7.
Figure 3(b) indicates that
$u/U$
lid
in the upper region of the cavity is mainly negative, suggesting that the upper flow moves predominantly in the negative streamwise direction. Figure 3(c) shows negative
$v/U$
lid
near the left-hand wall, and positive
$v/U$
lid
near the right-hand wall, indicating downward motion near the left-hand wall and upward motion near the right-hand wall. A positive
$u/U$
lid
is observed in the central region, implying that part of the fluid moves from left to right in the middle of the cavity. These velocity distributions together indicate the formation of a primary recirculating structure in the measured streamwise planes, driven by the top incoming flow. This recirculating structure indicates that the incoming momentum is redirected into the cavity through a wall-attached downwash pathway rather than being uniformly transported throughout the whole cavity depth. Figure 3(a) shows that the high-value region of
$U/U$
lid
is mainly located near the left-hand wall, and extends downwards along the wall. This high-velocity band is spatially consistent with the negativev/U
lid
region shown in figure 3(c), indicating a downward in-plane motion near the left-hand wall. The mean velocity vectors superimposed on figure 3(a) further show that the flow entering from the top opening is directed downwards along the left-hand wall and then turns into the main recirculation region. This spatial correspondence indicates that the left-hand-wall high-velocity region is associated with a downwash-dominated momentum pathway from the top opening. In contrast, the mid- and lower-cavity regions away from this pathway exhibit much lower velocities, suggesting weakened momentum transport towards the cavity interior. The differences among the three streamwise planes further indicate that this pathway is modulated by spanwise non-uniformity and sidewall effects. Therefore, the mean flow cannot be regarded as purely two-dimensional; the finite cavity width modifies the penetration and redistribution of the incoming momentum at different spanwise locations. Unlike high-Re lid-driven cubic cavities, where the circulation can be maintained from the driving wall to the bottom (Wang et al. Reference Wang, Peng, Cheng, Gao, Yin, Shen and Gao2025), the deep cavity is driven by flow over the top opening with a relatively large depth, making the mid- and lower-cavity regions more prone to low-velocity zones. In this study, the low-velocity zone is defined as the region where
$U/U$
lid
< 0.05. For the mid- and lower-cavity region (
$y/D$
≤ 0.70), the low-velocity area fractions are 74.22 %, 74.00 %, 72.67 % for
$z/W$
= 0.10, 0.25, 0.50, respectively. The corresponding mean velocities within the low-velocity zone are
$\overline{U}_{\textit{LV}}$
/U
lid
= 0.0247, 0.0193, 0.0224, indicating that the low-velocity zone occupies most of the mid- and lower-cavity region in all three streamwise planes. This large low-velocity fraction suggests weak ventilation and limited momentum exchange in the lower part of the deep cavity, even under the present high-Reynolds-number condition.
Time-averaged velocity fields on different spanwise planes: (a) in-plane resultant velocity U
yz
/U
lid
with superimposed mean secondary-flow vectors; (b) spanwise velocity component
$w$
yz
/U
lid
; (c) vertical velocity component
$v$
yz
/U
lid
.
Comparing the different streamwise planes, the high-speed region near the left-hand wall in the
$z/W$
= 0.25 and
$z/W$
= 0.5 planes is more pronounced than in the
$z/W$
= 0.1 plane (figure 3
a). Peak statistics indicate that the peak values of
$U/U$
lid
are 0.288 and 0.287 for
$z/W$
= 0.25 and
$z/W$
= 0.5, respectively, higher than 0.254 for the near-wall
$z/W$
= 0.1 plane, suggesting that the left-hand-wall downwash strength is reduced near the wall due to viscous blockage (Laha et al. Reference Laha, Fourtakas, Das and Keshmiri2025). Figure 4(a) shows that along the vertical line at
$x/L$
= 0.5,
$U/U$
lid
varies significantly with depth. The
$z/W$
= 0.1 plane exhibits local high velocity approximately 0.141 near
$y/D$
= 0.53, higher than the other two planes, indicating that the local high-speed region extends more towards the mid-lower cavity. Jin et al. (Reference Jin, Cheng, Wang and Wang2022) reported dead-water zones at the cavity bottom in non-rectangular deep cavities. The present high-Re deep cavity shows similar low-velocity zones, especially in the mid- and lower-cavity region, indicating weak momentum transport and low shear at the cavity bottom. Figure 4(d) shows multiple sign changes of
$u/U$
lid
along the vertical line at
$x/L$
= 0.5: negative in the upper cavity, positive in the middle, and near zero or negative near the bottom, demonstrating that horizontal velocity in the deep cavity is non-monotonic along depth, influenced by the primary recirculation and low-velocity regions (Samantaray et al. Reference Samantaray, Das and Patel2020). Figure 4(e) shows the distribution of
$v$
, with negative values on the left, and positive values on the right, for all three planes, indicating downward flow on the left-hand side and upward flow on the right-hand side. The
$z/W$
= 0.1 and
$z/W$
= 0.5 planes show more pronounced positive
$v$
peaks, while the
$z/W$
= 0.25 plane has lower peaks, indicating weaker vertical velocity response in the middle streamwise plane. Overall, the time-averaged velocity distribution varies significantly among the three planes, showing that spanwise location affects the downwash extent, low-velocity zones, and mean recirculation structure in the cavity.
3.1.2. Spanwise plane time-averaged velocity distribution
Figure 5 presents the time-averaged velocity fields on three spanwise planes at
$x/L$
= 0.1, 0.25, 0.5. Figure 5(a) shows the in-plane resultant velocity U
yz
/U
lid
, figure 5(b) shows the spanwise component
$w$
yz
/U
lid
, and figure 5(c) shows the vertical component
$v$
yz
/U
lid
. Here, U
yz
is defined as the resultant velocity composed of
$w$
yz
and
$v$
yz
in the spanwise plane. Figure 6 shows the time-averaged velocity along characteristic lines: figures 6(a) and 6(b) present
$v$
yz
/U
lid
along horizontal lines at
$y/D$
= 0.75 and
$y/D$
= 0.35, respectively, and figure 6(c) presents
$w$
yz
/U
lid
along the vertical line at
$z/W$
= 0.5. The three spanwise planes are near the downstream wall and flow-turning region, the intermediate region, and the upstream inflow side, respectively, to compare the velocity structure across different streamwise locations.
Time-averaged velocity along characteristic lines on different spanwise planes: (a)
$v$
yz
/U
lid
along the horizontal line at
$y/D$
= 0.75; (b)
$v$
yz
/U
lid
along the horizontal line at
$y/D$
= 0.35; (c)
$w$
yz
/U
lid
along the vertical line at
$z/W$
= 0.5.
Figure 5(a) shows that the high-value region of U
yz
/U
lid
expands from
$x/L$
= 0.50 to
$x/L$
= 0.10, with peak values 0.045, 0.082, 0.154, respectively. This indicates that the plane near the downstream wall (
$x/L$
= 0.10) has the strongest local in-plane velocity response, while the inflow-side plane (
$x/L$
= 0.50) has a lower in-plane velocity. The secondary-flow vectors superimposed on figure 5(a) show that at
$x/L$
= 0.10, the upper-cavity vectors are dense, relatively long, and predominantly oriented downwards, indicating a strong downward in-plane motion. This downward branch turns in the mid- and lower-cavity region, and connects with the upward motion in the lower-left region, forming a local secondary-flow circulation on the yz plane. This structure is associated with the progression of the mean recirculating flow towards the downstream wall, where wall confinement and the no-penetration condition restrict further streamwise motion and promote flow turning within the cavity. Thus the enhanced yz plane motion near
$x/L$
= 0.10 reflects the redirection of streamwise recirculating momentum into vertical and spanwise secondary motion near the downstream wall. Figure 5(c) shows significant differences in
$v$
yz
/U
lid
among the planes: at
$x/L$
= 0.1, a negative high-value region occurs in the upper cavity, and a positive high-value region in the lower-left region, indicating pronounced downward motion in the upper part, and upward motion in the lower part. As the measurement plane moves to
$x/L$
= 0.25 and
$x/L$
= 0.5, the intensity of positive and negative extreme regions weakens, showing notable differences in vertical flow direction and strength. Figures 6(a) and 6(b) further illustrate that along the horizontal line at
$y/D$
= 0.75, the negative peak is approximately −0.152 at
$x/L$
= 0.10, and −0.071 at
$x/L$
= 0.25, whereas the value becomes positive and relatively small at
$x/L$
= 0.50, approximately 0.033. This indicates that downward motion is strongest near the downstream wall. Along
$y/D$
= 0.35, all three planes show positive
$v$
yz
/U
lid
peaks, increasing from 0.012 at
$x/L$
= 0.50 to 0.026 at
$x/L$
= 0.25, and reaching 0.031 at
$x/L$
= 0.10, indicating stronger upward motion near the downstream wall. Figure 6(c) shows that
$w$
yz
/U
lid
along
$z/W$
= 0.50 has a single positive peak at
$x/L$
= 0.10 and
$x/L$
= 0.25, with values 0.031 and 0.027, respectively, whereas
$x/L$
= 0.50 has two positive peaks, 0.026 and 0.014. This indicates a more dispersed velocity distribution on the inflow side and a more concentrated structure near the downstream wall. Overall, figures 5 and 6 indicate that the time-averaged velocity distribution on the spanwise planes exhibits pronounced spatial non-uniformity:
$x/L$
= 0.10 has the highest in-plane velocity peak, with clear downward motion above and upward motion below;
$x/L$
= 0.25 shows an intermediate response; and
$x/L$
= 0.50 exhibits lower velocity but a more dispersed spanwise distribution. The stronger in-plane velocity response near
$x/L$
= 0.10 is mainly related to downstream-wall-induced flow turning. As the measurement plane moves away from the downstream wall, this turning effect weakens, leading to lower and more dispersed in-plane velocity responses. This streamwise variation indicates that the spanwise plane mean flow is controlled not only by the incoming shear layer, but also by the redistribution of recirculating flow as it approaches the downstream wall.
3.2. The RMS velocity fluctuation distributions
To further investigate the spatial distribution of velocity fluctuations inside the deep cavity, the RMS values of each velocity component were computed from the instantaneous velocity fluctuations relative to the time-averaged velocity. These RMS values characterise the magnitude of local deviations from the mean flow, and reveal the anisotropy and spatial non-uniformity of turbulence. The sample-size convergence analysis in § 2.4 shows that the RMS statistics approach stable values over the final portion of the record, with relative errors 2.37 % for u
rms
and 1.84 % for
$v$
rms
at
$N/N$
total
= 0.95. Therefore, the following RMS fields are interpreted as resolved-scale second-order statistics. In the streamwise planes, the streamwise velocity fluctuations u
rms
/U
lid
and vertical velocity fluctuations
$v$
rms
/U
lid
were analysed, while in the spanwise planes, the spanwise velocity fluctuations
$w$
yz-rms
/U
lid
and vertical velocity fluctuations
$v$
yz-rms
/U
lid
were examined. Because RMS quantities are sensitive to intermittent events, the following discussion focuses on robust spatial patterns rather than isolated local contour variations.
3.2.1. Streamwise plane RMS velocity distribution
Figure 7 presents the RMS distributions for three streamwise planes at
$z/W$
= 0.5, 0.25, 0.1, with figure 7(a) showing u
rms
/U
lid
, and figure 7(b) showing
$v$
rms
/U
lid
. Figure 8 further shows the RMS variations along selected characteristic lines, where figure 8(a) presents u
rms
/U
lid
along the vertical line at
$x/L$
= 0.50, and figure 8(b) presents
$v$
rms
/U
lid
along the vertical high-fluctuation line at
$x/L$
= 0.20.
Normalised RMS velocity distributions on the streamwise
$xy$
planes.
Distributions of normalised RMS velocity fluctuations along selected characteristic lines on the streamwise
$xy$
planes: (a) u
rms
/U
lid
along the vertical line at
$x/L$
= 0.50; (b)
$v$
rms
/U
lid
along the vertical line at
$x/L$
= 0.20.
The distribution of u
rms
/U
lid
indicates that the streamwise velocity fluctuations increase towards the sidewall. The intermediate plane (
$z/W$
= 0.25) exhibits slightly higher RMS levels than the middle plane (
$z/W$
= 0.50), whereas the near-wall plane (
$z/W$
= 0.10) shows a pronounced peak of approximately u
rms
/U
lid
= 0.054, and a secondary local peak of approximately u
rms
/U
lid
= 0.039 along the characteristic line. These values are higher than those observed on the other two planes, suggesting that the near-wall plane has not only stronger primary streamwise fluctuations, but also more pronounced secondary high-fluctuation regions. This enhancement trend indicates that proximity to the sidewall, combined with wall confinement and cavity shear-layer interactions, amplifies local velocity gradients and promotes the concentration of streamwise velocity fluctuations near the wall (Ferrari et al. Reference Ferrari, Franco and Hegele2024). Considering the time-averaged flow field, interactions between the primary recirculation vortex and smaller-scale vortical structures beneath it may further enhance the near-wall u
rms
/U
lid
peak. In physical terms, the sidewall can restrict spanwise motion while enhancing local shear and velocity-gradient intermittency, which helps to explain the increase in streamwise fluctuation intensity near the wall.
Unlike u
rms
/U
lid
, the normalised vertical velocity fluctuation
$v$
rms
/U
lid
is most pronounced on the intermediate plane (
$z/W$
= 0.25), where a wide high-fluctuation region with strong intensity is observed. The middle plane (
$z/W$
= 0.50) is relatively weaker, while the near-wall plane (
$z/W$
= 0.10) shows local enhancement but does not reach the level observed on the intermediate plane. Quantitative analysis along the characteristic line indicates that the peak value of
$v$
rms
/U
lid
at
$z/W$
= 0.25 reaches approximately 0.072, which is higher than those at the middle plane and the near-wall plane. The
$v$
rms
/U
lid
distribution also exhibits clear left–right asymmetry, with the high-fluctuation region concentrated on the side where the incoming flow and shear effect are stronger. This demonstrates that vertical velocity fluctuations are primarily influenced by cavity separation, shear-layer development, and interactions with internal recirculation structures. The intermediate plane, located between the near-wall and middle regions, experiences stronger coupling among the separated flow, primary recirculation vortex and secondary shear structures, forming a broad high-intensity vertical fluctuation region. The different locations of the u
rms
and
$v$
rms
maxima indicate anisotropic fluctuation behaviour: streamwise fluctuations are more sensitive to sidewall confinement, whereas vertical fluctuations are more closely related to shear-layer penetration and recirculation-induced vertical motion.
Overall, the RMS distributions in the streamwise
$xy$
planes reveal pronounced anisotropy of velocity fluctuations within the deep cavity. The normalised streamwise fluctuation u
rms
/U
lid
increases near the sidewall, reflecting the modulation of wall confinement and shear-layer interaction, whereas
$v$
rms
/U
lid
reaches its highest level at
$z/W$
= 0.25, indicating that vertical fluctuations are more strongly associated with the coupling among shear-layer development, recirculation vortices and secondary shear structures. These distribution characteristics are consistent with previous cavity-flow studies. Guan et al. (Reference Guan, Agarwal and Chiew2018) used PIV measurements in a large-aspect-ratio rectangular cavity, and found that the cavity shear layer is the main high-fluctuation region. Ozgoren (Reference Ozgoren2006) reported that flow separation and vortex shedding can amplify normal velocity fluctuations and form localised high-RMS structures. Wang et al. (Reference Wang, Hou, Zhou, Peng and Lu2026) observed in high-Reynolds-number cube-cavity near-wall turbulence that velocity fluctuations differ significantly between the incoming-impact side and the downstream side.
3.2.2. Spanwise plane RMS velocity distribution
Figure 9 shows the RMS distributions on three spanwise planes at
$x/L$
= 0.5, 0.25, 0.1, with figure 9(a) displaying
$w$
yz-rms
/U
lid
, and figure 9(b) displaying
$v$
yz-rms
/U
lid
. Figure 10 presents the corresponding characteristic-line distributions, where figure 10(a) shows
$w$
yz-rms
/U
lid
along the spanwise line at
$z/W$
= 0.50, and figure 10(b) shows
$v$
yz-rms
/U
lid
along the high-fluctuation line at
$z/W$
= 0.70.
Normalised RMS velocity distributions on different spanwise yz planes: (a)
$w$
yz-rms
/U
lid
, (b)
$v$
yz-rms
/U
lid
.
Characteristic-line distributions of normalised RMS velocity fluctuations on different spanwise yz planes: (a)
$w$
yz-rms
/U
lid
along the spanwise line at
$z/W$
= 0.50; (b)
$v$
yz-rms
/U
lid
along the high-fluctuation line at
$z/W$
= 0.70.
High
$w$
yz-rms
/U
lid
regions appear near the cavity shear layer on all three spanwise planes. Moving from the upstream side (
$x/L$
= 0.50) to the downstream bending region (
$x/L$
= 0.10), the spanwise velocity fluctuations increase. The near-downstream plane (
$x/L$
= 0.10) shows a peak of approximately 0.041, while the corresponding peaks on the
$x/L$
= 0.25 and
$x/L$
= 0.50 planes are approximately 0.021 and 0.024, respectively. This increase indicates that the downstream wall and flow-turning region significantly enhance the spanwise velocity fluctuations. The enhanced local velocity gradients near the downstream sidewall and bending region suggest that wall confinement and flow redirection strongly modulate the spanwise fluctuation intensity. This is consistent with the mean-flow result that the downstream wall redirects part of the recirculating motion into the measured yz plane, thereby strengthening spanwise velocity fluctuations near the turning region.
The normalised vertical velocity fluctuation
$v$
yz-rms
/U
lid
reaches its highest level on the intermediate plane (
$x/L$
= 0.25), with a peak of approximately 0.044. This plane exhibits a broad high-fluctuation region and a distinct fluctuation core. In contrast, the
$x/L$
= 0.10 and
$x/L$
= 0.50 planes show relatively weaker vertical fluctuation intensity. The high-fluctuation core also shifts along the spanwise direction. At
$x/L$
= 0.50, the core is located near
$z/W$
= 0.80; at
$x/L$
= 0.25, it moves to approximately
$z/W$
= 0.70; and at
$x/L$
= 0.10, it further shifts towards
$z/W$
= 0.40. This spatial migration reflects the influence of internal recirculation, local shear and flow turning at different streamwise locations. The migration of the fluctuation core indicates that the strongest vertical fluctuations are redistributed along the span as the recirculating flow evolves from the upstream side towards the downstream turning region.
Comparison between the streamwise and spanwise planes reveals clear three-dimensional differences in the RMS velocity distributions. In the streamwise
$xy$
planes, u
rms
/U
lid
is strongest near the sidewall (
$z/W$
= 0.10), whereas
$v$
rms
/U
lid
is most pronounced on the intermediate plane (
$z/W$
= 0.25). In the spanwise yz planes,
$w$
yz-rms
/U
lid
reaches its highest level in the downstream bending region (
$x/L$
= 0.10), while
$v$
yz-rms
/U
lid
is strongest on the intermediate plane (
$x/L$
= 0.25). These results demonstrate that the internal cavity turbulence exhibits strong anisotropy and spatial non-uniformity. Sidewall confinement, recirculation vortices, shear-layer development and flow-turning effects jointly modulate the distribution of velocity fluctuations. Previous studies have reported similar three-dimensional effects in cavity flows. Koseff & Street (Reference Koseff and Street1984) showed that transitional spanwise sections in three-dimensional cavities tend to develop coupled recirculation and shear vortices, leading to concentrated high-fluctuation regions between primary and secondary shear layers. Samantaray et al. (Reference Samantaray, Das and Patel2020) demonstrated through LES of three-dimensional cube-cavity flow that endwall vortices can influence the development of secondary vortical structures. The enhancement of
$v$
yz-rms
/U
lid
in the transition region and the migration of high-fluctuation cores observed in the present study are consistent with these prior observations.
3.3. Time-domain characteristics of instantaneous maximum velocity
The instantaneous maximum velocity was used to characterise the strongest local velocity response within each measurement plane at every recorded instant. The following analysis compares its time-domain evolution, mean level, fluctuation intensity and intermittent excursions among the streamwise and spanwise measurement planes.
Instantaneous maximum velocity sequences U
max
/U
lid
on different streamwise
$xy$
planes: (a) near-wall plane (
$z/W$
= 0.10); (b) intermediate plane (
$z/W$
= 0.25); (c) middle plane (
$z/W$
= 0.50). The dashed line denotes the time-averaged value
$\overline{U}_{\textit{max}}$
/U
lid
.
3.3.1. Time-domain statistics on streamwise planes
To quantify the unsteady velocity response on different streamwise planes, the instantaneous maximum velocity sequences were extracted from three streamwise
$xy$
planes located at
$z/W$
= 0.50, 0.25, 0.10. Since these planes are separated in the spanwise direction, their differences were used to quantify the spanwise modulation of the maximum velocity. The instantaneous maximum velocity represents the strongest local velocity response at each instant within the measured plane, which is more suitable for characterising local extreme events and highly unsteady regions rather than representing the mean velocity of the entire cross-section. Figures 11(a–c) illustrate the time series of the instantaneous maximum velocity U
max
/U
lid
on the three streamwise planes: near-wall plane, intermediate plane and middle plane, respectively.
The results show significant spatial differences in the time-averaged maximum velocity among the planes. The mean values of U
max
/U
lid
are 0.3340, 0.3224, 0.2918 for the middle plane (
$z/W$
= 0.50), intermediate plane (
$z/W$
= 0.25) and near-wall plane (
$z/W$
= 0.10), respectively. These observations indicate that as the measurement location moves from the near-wall plane to the middle plane, the suppression effect of the sidewall on the average intensity of local high-speed regions gradually weakens, leading to enhanced local high-speed responses. To further isolate the sidewall modulation, the mean maximum velocity was normalised by the middle-plane value R
max
=
$\overline{U}_{\textit{max}}(z)/\overline{U}_{\textit{max}}(z/W=0.50)$
. The resulting R
max
values are 1.000, 0.965, 0.874 for
$z/W$
= 0.50, 0.25, 0.10, respectively. Thus the near-wall plane retains only 87.4 % of the middle-plane maximum-velocity level, corresponding to a reduction of approximately 12.6 %. The finite-difference spanwise gradient of
$\overline{U}_{\textit{max}}$
/U
lid
is 0.204 between
$z/W$
= 0.10 and 0.25, but decreases to 0.046 between
$z/W$
= 0.25 and 0.50, indicating that the recovery of the maximum velocity is strongest near the sidewall, and becomes weaker towards the cavity middle plane. Sun et al. (Reference Sun, Liu, Cattafesta, Ukeiley and Taira2019) observed in finite-width rectangular cavity experiments that sidewalls disturb the roll-up of spanwise vortices in the shear layer, and alter the velocity and pressure fluctuation distributions within the cavity. Similarly, Kok (Reference Kok2025) demonstrated using LES that sidewall effects introduce significant three-dimensional disturbances, causing non-uniform local velocity responses along the span. The differences in the normalised maximum velocity observed here are consistent with these sidewall modulation effects. These results show that sidewall influence does not simply reduce velocity uniformly; rather, it modifies the spatial distribution of local high-speed events across the measured spanwise positions.
Over the full sampling period, the extreme values of the instantaneous maximum velocity are not consistent with the time-averaged values. The near-wall plane captures the highest instantaneous maximum velocity, with U
max
/U
lid
= 0.6079, exceeding values for the intermediate plane and middle plane. Furthermore, the near-wall plane shows a larger instantaneous range, indicating that although its mean maximum-velocity level is lower, local intermittent high-speed events can still occur. The standard deviation and RMS of the instantaneous maximum velocity further show that the intermediate plane exhibits the strongest fluctuation intensity, with σ(U
max
/U
lid
) = 0.0528, higher than the middle plane (0.0465) and near-wall plane (0.0424). The relative fluctuation intensity also reaches its maximum on the intermediate plane (0.1638), whereas the middle plane and near-wall plane correspond to 0.1392 and 0.1452, respectively. These results indicate that the velocity deviation from the mean is most pronounced on the intermediate plane, reflecting the strongest unsteady behaviour. In contrast, the middle plane, despite having the highest mean maximum velocity, exhibits more concentrated fluctuations, with instantaneous velocities statistically distributed closer to the mean. This non-monotonic behaviour indicates that the mean maximum-velocity level and the intermittency of extreme events are not controlled by exactly the same mechanism; the former is related to the averaged momentum level, whereas the latter is more sensitive to transient local fluctuation events. Overall, along the transition from the near-wall plane to the middle plane, the time-averaged level of instantaneous maximum velocity gradually increases, while the fluctuation amplitude and intensity vary non-monotonically. The normalised maximum-velocity level and its finite-difference spanwise gradient indicate that sidewall-induced suppression is strongest near
$z/W$
= 0.10, and weakens towards the cavity middle plane. The near-wall plane exhibits lower time-averaged velocity but higher instantaneous extremes, indicating stronger local intermittency; the intermediate plane shows the highest RMS and relative fluctuation intensity, suggesting the most pronounced velocity fluctuations; the middle plane demonstrates higher time-averaged maximum velocity and relatively lower fluctuation intensity, with more concentrated extreme events.
3.3.2. Time-domain statistics on spanwise planes
To compare the time-domain characteristics of strong velocity responses at different streamwise locations, the instantaneous maximum velocity sequences were extracted from the spanwise yz planes at
$x/L$
= 0.10, 0.25, 0.50. The instantaneous maximum velocity U
max
/U
lid
represents the strongest local velocity response within each measured yz plane at a given instant. Figures 12(a–c) present the temporal variations of U
max
/U
lid
on the three spanwise planes
$x/L$
= 0.10, 0.25, 0.50, respectively.
Temporal variations of the instantaneous maximum velocity U
max
/U
lid
on different spanwise yz planes: (a)
$x/L$
= 0.10, (b)
$x/L$
= 0.25, (c)
$x/L$
= 0.50.
The time-domain results indicate that the average level of instantaneous maximum velocity increases as the measurement location moves from
$x/L$
= 0.50 towards
$x/L$
= 0.10. The time-averaged maximum velocity is lowest on the
$x/L$
= 0.50 plane, with
$\overline{U}_{\textit{max}}$
/U
lid
= 0.0827. It increases to 0.1257 on the
$x/L$
= 0.25 plane, and reaches the highest value 0.1916 on the
$x/L$
= 0.10 plane, which is close to the downstream wall and flow-turning region. This indicates that the local strong velocity response within the spanwise plane gradually intensifies near the downstream turning region. The RMS values of the fluctuation component are 0.0168, 0.0132, 0.0374 for
$x/L$
= 0.10, 0.25, 0.50, respectively. This indicates that the mean level of strong velocity response increases towards the downstream wall, whereas the fluctuation intensity is strongest on the upstream inflow-side plane (
$x/L$
= 0.50). Therefore, the unsteady velocity response within the spanwise planes is influenced by flow turning, recirculation structures and wall constraints. The increase in the mean maximum velocity near
$x/L$
= 0.10 is consistent with the downstream-wall-induced concentration and redirection of the recirculating flow, whereas the non-monotonic RMS response indicates that intermittent fluctuations are also affected by upstream shear-layer variability and local recirculation dynamics.
3.4. The TKE
In the present two-component PIV measurements, the TKE discussed here refers to the in-plane TKE. For the streamwise
$xy$
planes, it is computed from the streamwise and vertical velocity fluctuations, while for the spanwise yz planes, it is computed from the spanwise and vertical velocity fluctuations. The TKE values are normalised by U
lid
2 .
3.4.1. Mechanisms of TKE distribution on streamwise planes
Figure 13 shows the normalised time-averaged in-plane TKE distributions TKE/U
lid
2 on the streamwise
$xy$
planes at
$z/W$
= 0.10, 0.25, 0.50. On the wall-adjacent plane (
$z/W$
= 0.10), the high-TKE region is mainly concentrated near the core of the primary vortex in the upper-middle part of the cavity, with maximum approximately 0.0025. On the intermediate plane (
$z/W$
= 0.25), TKE is further enhanced, with the high-TKE region expanding in area, and additional regions appearing, reaching maximum approximately 0.0029. On the middle plane (
$z/W$
= 0.50), the overall TKE decreases, and the high-TKE regions contract, with maximum value approximately 0.0019. Overall, the distribution exhibits the pattern of lower TKE near the sidewall, the highest TKE on the intermediate plane, and reduced TKE towards the cavity middle plane.
Normalised in-plane TKE distributions TKE/U
lid
2 on the streamwise
$xy$
planes.
Boundary constraints have a significant modulation effect on TKE distribution. On the near-wall plane, the high-TKE regions near
$x/L$
= 0.10 are spatially limited. The intermediate plane (
$z/W$
= 0.25), located between the middle plane and the sidewall, allows vortical structures more space for development, leading to an increased area of high-TKE regions. This indicates that the interactions among the shear layer, cavity recirculation and the primary vortex are more pronounced on this plane. On the middle plane (
$z/W$
= 0.50), located near the cavity centre and far from the sidewall, the TKE distribution is more uniform with no pronounced high-value regions. High-TKE regions are mainly located in the upper-middle cavity and areas with strong shear-layer activity, suggesting that the interaction between the inlet shear layer and the cavity recirculation is the key mechanism for TKE generation and concentration. The enhanced TKE on the intermediate plane can therefore be interpreted as the combined effect of shear-layer fluctuation, recirculation-induced velocity gradients, and partial release from direct sidewall damping. This distribution pattern is consistent with previous studies on sidewall effects in finite-width cavities: Sun et al. (Reference Sun2016) demonstrated through DNS, LES and PIV experiments that sidewalls significantly influence the three-dimensional flow characteristics, causing notable differences in flow structures and turbulent statistics at different spanwise locations. Flow near sidewalls is more affected by wall constraints and overflow vortices, while flow in central regions reflects the three-dimensional interactions between shear layers and recirculation structures.
3.4.2. Characteristics of TKE distribution on spanwise planes
Figure 14 shows the normalised time-averaged in-plane TKE distributions TKE/U
lid
2 on spanwise yz planes at
$x/L$
= 0.10, 0.25, 0.50, with cross-sections perpendicular to the incoming flow direction, revealing the evolution of turbulence along the streamwise direction. High-TKE regions are mainly concentrated on the right-hand side, due to the interaction between primary vortex rotation and cavity geometry forming a locally recirculating zone with enhanced shear, where the velocity fluctuations in the upper-right part are strongest, resulting in peak TKE in this region. This indicates that the in-plane fluctuation energy is concentrated where local shear and recirculating motion reinforce each other within the measured yz sections.
Normalised in-plane TKE distributions TKE/U lid 2 on the spanwise yz planes.
At the upstream plane (
$x/L$
= 0.50), the high-TKE region is scattered, with peak approximately 0.0010, mainly in the upper-middle part of the cavity and right-hand-side local regions. On the intermediate plane (
$x/L$
= 0.25), the high-TKE region becomes more concentrated, with peak close to approximately 0.0010 and an obvious increase in area, indicating that the influence of velocity fluctuations is strongest here. On the downstream plane (
$x/L$
= 0.10), the overall TKE decreases, and high-value regions vanish. The overall trend follows ‘upstream scattered – midstream concentrated – downstream decayed’. This spatial distribution is closely related to the interaction of the shear layer and the recirculation structures. On the upstream
$x/L$
= 0.50 plane, the high-TKE regions are scattered, indicating that the local fluctuations are not yet concentrated; in the intermediate
$x/L$
= 0.25 plane, local fluctuations are concentrated, with the largest area of high-TKE regions; downstream
$x/L$
= 0.10 exhibits significant reduction in TKE, and high-value regions disappear, indicating weakened local velocity fluctuations at this plane. Jiménez (Reference Jiménez2018) pointed out that momentum transport induced by coherent structures is an important mechanism for forming highly active turbulent regions. Beresh et al. (Reference Beresh, Wagner, Pruett, Henfling and Spillers2015) used PIV to study supersonic finite-width rectangular cavity flows, and demonstrated that sidewall effects and spanwise velocity gradients significantly influence the three-dimensional flow structures and local turbulence intensity. In the present study, the concentrated high-TKE region on the
$x/L$
= 0.25 plane can be attributed to interactions between the shear layer and recirculation structures, while the dispersed distribution upstream and significant decay downstream reflect the three-dimensional evolution characteristics of finite-span cavity flows. In summary, the TKE distribution on spanwise planes exhibits clear spatial variation along the streamwise direction: scattered upstream, concentrated and strongest in the middle, and significantly weakened downstream. This pattern provides experimental evidence for understanding the spatial distribution of local velocity fluctuations and turbulent energy transport within the deep cavity, consistent with wall turbulence coherent-structure theory and previous experimental observations in finite-width cavity flows.
3.5. Probability distribution and higher-order statistical features
3.5.1. Statistical distribution of instantaneous maximum velocity on streamwise planes
Figure 15 presents the probability density functions (PDFs) of the instantaneous maximum velocity on streamwise
$xy$
planes
$z/W$
= 0.50, 0.25, 0.10. Skewness and kurtosis were calculated for each sequence of instantaneous maximum velocities to characterise the asymmetry and the sharpness/heavy-tail features of the distributions. When the kurtosis is close to 3, the velocity distribution approximates a Gaussian distribution; significantly higher values indicate a higher likelihood of extreme events deviating from the mean.
The PDFs of instantaneous maximum velocity on streamwise
$xy$
planes: (a)
$z/W$
= 0.50, (b)
$z/W$
= 0.25, (c)
$z/W$
= 0.10.
The results indicate that the statistical characteristics of the middle plane (
$z/W$
= 0.50) and the intermediate plane (
$z/W$
= 0.25) are similar. On the middle plane, skewness and kurtosis are 0.5549 and 3.1541, respectively, while on the intermediate plane, they are 0.5366 and 3.1575. The kurtosis slightly exceeds 3, and the skewness is mildly positive, indicating that the instantaneous maximum velocity distributions on these planes are approximately Gaussian, with slight right-skewness and weak heavy-tail features. This suggests that velocity extreme fluctuations are relatively continuous in the middle and intermediate planes, with intermittent extreme high-velocity events occurring occasionally but not dominating. In contrast, the near-wall plane (
$z/W$
= 0.10) shows a pronounced change: skewness increases to 1.2297, and kurtosis reaches 7.4691. The large positive skew indicates a long right tail in the distribution of instantaneous maximum velocity, meaning that events with velocities higher than the mean occur more frequently. The significantly high kurtosis reflects a strong heavy-tail characteristic, with extreme velocity events being more prominent. Thus although the time-averaged maximum velocity on the near-wall plane is lower than that on the middle plane, the instantaneous velocity sequence exhibits stronger intermittency and non-Gaussian features. Overall, the distribution of instantaneous maximum velocity on streamwise planes varies clearly with spanwise position: moving from the middle plane towards the near-wall plane, the velocity extreme distributions gradually transition from near-Gaussian to strongly right-skewed and heavy-tailed. This indicates that sidewall constraints not only affect mean velocity and fluctuation intensity, but also modulate the probability distribution of extreme velocity events. Combined with previous time-domain statistics, it can be concluded that local instantaneous high-velocity events are more likely to occur near the sidewall, whereas velocity extreme distributions on the middle and intermediate planes remain relatively stable.
These observations are consistent with common intermittent statistics reported in deep-cavity shear layers and near-wall regions. El Hassan et al. (Reference El Hassan, Keirsbulck and Labraga2009) conducted hot-wire and time-resolved PIV measurements on a large deep-cavity shear layer, revealing that velocity fluctuations near the wall and downstream regions can exhibit significant non-Gaussian characteristics, with substantially increased kurtosis, associated with local jet-like events and shear-layer intermittency. Although the sign of skewness in their study depends on the velocity component, coordinate direction and statistical object, the high kurtosis reflecting heavy-tailed distributions and enhanced intermittency is comparable to the high kurtosis and increased extreme velocity events observed in the present study on the near-wall plane (
$z/W$
= 0.10). This indicates that in confined deep-cavity flows, near-wall constraints, shear-layer disturbances and local recirculation interact to enhance the non-Gaussian characteristics of velocity extreme events.
3.5.2. Statistical distribution of instantaneous maximum velocity on spanwise planes
Figure 16 shows the PDFs of instantaneous maximum velocity on spanwise yz planes
$x/L$
= 0.50 near the upstream inflow side,
$x/L$
= 0.25 in the intermediate region, and
$x/L$
= 0.10 near the downstream wall and flow-turning region. The results reveal clear differences among the spanwise planes. On the upstream plane (
$x/L$
= 0.50), skewness and kurtosis reach 2.1288 and 10.4012, respectively, significantly higher than the other two planes. The large positive skew and high kurtosis indicate a strongly right-skewed, heavy-tailed distribution, where most of the time velocities remain at relatively low levels, but intermittent events with velocities far exceeding the mean can occur. On the intermediate plane (
$x/L$
= 0.25), skewness decreases to 0.4798 and kurtosis is 3.2049, showing that the instantaneous maximum velocity distribution is closer to Gaussian, with extreme events significantly reduced, and fluctuations concentrated around the mean. On the downstream plane (
$x/L$
= 0.10), skewness is −0.0314 and kurtosis is 2.8167. The near-zero skew indicates a nearly symmetric velocity distribution, and the kurtosis slightly below 3 shows that heavy-tailed characteristics are weak at this location. This indicates that although local maximum velocities are higher near the downstream wall and flow-turning region, extreme deviations from the mean occur less frequently. Thus the downstream plane is characterised by stronger organised mean turning motion, whereas the upstream plane retains stronger intermittent extreme events. Overall, the non-Gaussian characteristics of instantaneous maximum velocity on spanwise planes gradually decrease from
$x/L$
= 0.50 towards
$x/L$
= 0.10. The upstream plane exhibits strong skewness and heavy tails, indicating intermittent high-velocity events; the intermediate plane approaches Gaussian behaviour, and the downstream plane shows weak non-Gaussian characteristics. Comparison between streamwise and spanwise planes shows that the PDFs of instantaneous maximum velocity within the cavity exhibit pronounced spatial anisotropy. On streamwise planes, the near-wall plane (
$z/W$
= 0.10) shows more pronounced right-skewed, heavy-tailed intermittent velocity events. On spanwise planes, the upstream inflow-side plane (
$x/L$
= 0.50) exhibits the strongest right-skew and heavy-tailed distribution, while the downstream plane (
$x/L$
= 0.10) displays weakened non-Gaussian features. These differences indicate that velocity extreme events in the deep cavity are influenced not only by sidewall constraints but also by the streamwise location of the measurement plane and local flow-turning processes. This confirms that the statistics of extreme velocity events are controlled by both spanwise confinement and streamwise evolution of the recirculating flow, rather than by a single geometric effect alone.
The PDFs of instantaneous maximum velocity on spanwise yz planes: (a)
$x/L$
= 0.50, (b)
$x/L$
= 0.25, (c)
$x/L$
= 0.10.
4. Conclusions
In this study, a high-resolution PIV system was employed to systematically investigate a high-Reynolds-number (Re = 2.55 × 105) incoming-flow-driven deep cavity (
$L/D$
= 1/3) at three streamwise
$xy$
planes (
$z/W$
= 0.10, 0.25, 0.50) and three spanwise yz planes (
$x/L$
= 0.10, 0.25, 0.50). Time-averaged velocity, velocity RMS, in-plane TKE, instantaneous maximum velocity and higher-order statistics including skewness and kurtosis were analysed. The main conclusions are summarised as follows.
On streamwise
$xy$
planes, the vertical velocity fluctuation is most pronounced on the intermediate plane (
$z/W$
= 0.25), with
$v$
rms
/U
lid
reaching approximately 0.072. On the near-wall plane (
$z/W$
= 0.10), the instantaneous maximum velocity exhibits skewness 1.23, indicating significant local intermittent extreme events. In contrast, extreme events on the middle plane (
$z/W$
= 0.50) are weaker and more Gaussian-like, suggesting that the interaction between sidewall constraints and flow structures significantly regulates local intermittency.
On spanwise yz planes, the instantaneous maximum velocity shows pronounced non-Gaussian features. High-fluctuation and high-TKE regions are primarily concentrated in areas of interaction between the shear layer and internal recirculation, indicating substantial spatial anisotropy in deep-cavity flow. This demonstrates that the development of the cavity shear layer, the primary vortex and local recirculation collectively modulate the spatial distribution of local extreme events.
Overall, the present study provides experimental evidence for understanding local intermittency and spatial non-uniformity in high-Reynolds-number deep-cavity flows, offering valuable data for turbulence model validation and the control of complex cavity flows. However, some limitations remain: the experiments only cover three streamwise and three spanwise planes, leaving the full three-dimensional cavity flow uncharted; and PIV measurements capture only two velocity components, preventing direct characterisation of complete three-dimensional vortex structures and full turbulence stress fields. Future work could incorporate multi-plane or fully three-dimensional measurements, complemented by numerical simulations, to provide a more comprehensive understanding.
Funding
This study is supported by the Qin Chuang Yuan high-level innovation and entrepreneurship talent project (no. QCYRCXM2023-099), the 2024 major projects to prevent and control sand demonstration ‘Unveiled Marshal’ project from the Inner Mongolia Department of Science and Technology (no. 2024JBGS0016), the National Foreign Experts Individual Programme (Category H) under national funding (no. H20240398), and the Xianyang Science and Technology Bureau (no. L2024-ZDYF-ZDYF-SF-0042).
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Author contributions
P.W. was responsible for the conceptualisation of the study and drafting the original manuscript. P.R., Y.Y. and S.Y. participated in the research implementation. Y.P., as the corresponding author, provided financial support through funding acquisition and conducted the feasibility analysis of the research. F.L. conceived the experimental aspects of the study.
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