2.1 Model for frequency prediction in acoustics

Figure 1.

Experimental set-up (top view).

As introduced in § 1, the analytical model predicting the cavity frequency in acoustics relies on the natural frequency for low Mach numbers and on the feedback loop in the mixing layer for high Mach numbers.

The natural frequency of an acoustic cavity can be computed by simplifying it as a rectangular enclosure with fluid at rest (Plumbee, Gibson & Lassiter Reference Plumbee, Gibson and Lassiter1962). This frequency is ‘natural’ as it is an intrinsic property of the cavity, and does not depend on the adjacent flow. Cavity-type oscillations, induced by compressibility effects, can also be computed using the Helmholtz resonator theory.

For the feedback loop, the model considers the following events. As a vortex is shed (near the upstream corner of the cavity, denoted by subscript $v$, with a phase $\unicode[STIX]{x1D6F7}_{v}$), it is advected by the flow and impinges the downstream corner. There, it generates a local pressure variation (following the Bernoulli principle) inducing a pressure wave (denoted by subscript $p$, with a phase $\unicode[STIX]{x1D6F7}_{p}$) with the same frequency as that of the vortex shedding, which propagates in every direction, including upwards, i.e. towards the upstream corner of the cavity. The model assumes that the selected frequency is the one that fulfils the resonant conditions. This implies that the phase difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6F7}_{v}-\unicode[STIX]{x1D6F7}_{p}$, where $v$ and $p$ refer to vortex and pressure waves (see figure 1) at the upstream corner, satisfies

(2.1)$$\begin{eqnarray}\unicode[STIX]{x0394}\unicode[STIX]{x1D6F7}(x=0)=2N\unicode[STIX]{x03C0},\end{eqnarray}$$

with $N$ being an integer ($N=1,2,3,\ldots$). Using the definition of the wavelength $\unicode[STIX]{x1D706}=c/f$, with $c$ the celerity of waves and $f$ their frequency, this reads

(2.2)$$\begin{eqnarray}N=\frac{L}{\unicode[STIX]{x1D706}_{v}}+\frac{L}{\unicode[STIX]{x1D706}_{p}}.\end{eqnarray}$$

Integer $N$ can be seen as the number of vortices plus pressure wave wavelengths along the cavity–main flow interface ($x=0$ to $L$). Using $f=c_{v}/\unicode[STIX]{x1D706}_{v}=c_{p}/\unicode[STIX]{x1D706}_{p}$, this reads

(2.3)$$\begin{eqnarray}N=\frac{Lf}{c_{v}}+\frac{Lf}{c_{p}},\end{eqnarray}$$

or alternatively, introducing the Strouhal number

(2.4)$$\begin{eqnarray}St=\frac{f\,L}{U}=\frac{N}{\displaystyle \frac{U}{c_{v}}+\frac{U}{c_{p}}},\end{eqnarray}$$

with $U$ the velocity of the main stream. The value of the ratio $R_{v}=c_{v}/U$ is usually measured experimentally, and $R_{v}$ has been reported as ranging between $0.35$ (East Reference East1966) and $0.625$ (Rowley et al. Reference Rowley, Williams, Colonius, Murray and Macmynowski2006). Rossiter (Reference Rossiter1964) states that $R_{v}=0.57$, which is the standard value used in the literature.

For a given flow configuration, this model allows an infinity of solutions, with $N=1,2,3$, etc. Rossiter (Reference Rossiter1964) and subsequent authors compared the frequency predicted by this model with experimental data and noticed a systematic shift. Most of them thus introduce a correction factor, $\unicode[STIX]{x0394}N$ with $N^{\prime }=N+\unicode[STIX]{x0394}N$, attributed to ‘a constant phase lag corresponding to a time delay between the vortex arrival at the impingement corner and the emission of the acoustic pulse’ (Gloefert Reference Gloefert2009). The corrected model then reads

(2.5)$$\begin{eqnarray}\frac{f\,L}{U}=\frac{N+\unicode[STIX]{x0394}N}{\displaystyle \frac{1}{R_{v}}+M}.\end{eqnarray}$$

Empirical values for $\unicode[STIX]{x0394}N$, proposed by successive authors, exhibit a large variation, from $\unicode[STIX]{x0394}N=-0.58$ (Rossiter Reference Rossiter1964) to $\unicode[STIX]{x0394}N=0.5$ (Sarohia Reference Sarohia1977). A global three-dimensional stability analysis of the mixing layer (Bilanin & Covert Reference Bilanin and Covert1973; Alvarez, Kerschen & Tumin Reference Alvarez, Kerschen and Tumin2004), injecting the resonant condition, leads to a similar result. Note that the celerity of the pressure wave does not take into account the local flow velocity as it is supposed to travel in the cavity, where the mean velocity is negligible. The fact that the model applies to supersonic flows ($M>1$) supports this hypothesis, as the wave could not propagate upstream otherwise.

2.2 Transposition to open-channel configurations

We expect that the oscillation frequency of open-channel cavities is governed by the same two phenomena. They are transposed to the open-channel configuration in the following subsections.

2.2.1 Natural frequencies of open-channel cavities

The natural modes of a closed rectangular water volume at rest were initially introduced by Lamb (Reference Lamb1945) and recently reviewed by Rabinovitch (Reference Rabinovitch2009). These modes can be aligned with one or both cavity axes $x$ and $y$ (see table 1). The corresponding frequencies are denoted $f_{n_{x}n_{y}}$ with $n_{x}$ and $n_{y}$ the number of nodes along axes $x$ and $y$, respectively. In a rectangular basin of dimensions $l_{x}$ (along axis $x$) and $l_{y}$ (along axis $y$), the natural frequencies then read

(2.6)$$\begin{eqnarray}f_{n_{x}n_{y}}=\frac{c}{2}\left[\left(\frac{n_{x}}{l_{x}}\right)^{2}+\left(\frac{n_{y}}{l_{y}}\right)^{2}\right]^{1/2},\end{eqnarray}$$

with $c$ the celerity of gravity waves in still water (Stocker Reference Stocker1957):

(2.7)$$\begin{eqnarray}c=\frac{g}{2\unicode[STIX]{x03C0}f}\tanh \frac{2\unicode[STIX]{x03C0}\overline{h}f}{c},\end{eqnarray}$$

where $\overline{h}$ is the time-averaged water depth in the cavity and $f$ the frequency of the gravity wave. Note that this expression can reduce to the classical Merian (Reference Merian1828) expression for shallow conditions ($\unicode[STIX]{x1D706}\gg \overline{h}$) if $n_{x}=1$, $n_{y}=0$ and $c$ is simplified to $c=\sqrt{g\overline{h}}$. Following the notation of Wolfinger et al. (Reference Wolfinger, Ozen and Rockwell2012) and Tuna, Tinar & Rockwell (Reference Tuna, Tinar and Rockwell2013), $f$ can be normalized by $f_{10}=c/2L$, the natural frequency of the cavity corresponding to the first longitudinal mode (along $x$) with one node (see table 1):

(2.8)$$\begin{eqnarray}\frac{f_{n_{x}n_{y}}}{f_{10}}=\left[n_{x}^{2}+\left(n_{y}\frac{l_{x}}{l_{y}}\right)^{2}\right]^{1/2}.\end{eqnarray}$$

Table 1 plots schematic views of three observed modes of the cavity and indicates the link between $l_{x}$, $l_{y}$ and $W$, $b$, $L$ that depends on the cavity mode considered.

Table 1.

Three amongst the most common natural frequencies of a rectangular open-channel cavity.

2.2.2 Feedback model adapted to open-channel cavities

When applied to open-channel lateral cavities, the feedback model relies on the same processes as for acoustic cavities with a gravity wave instead of a pressure wave. In the axis frame linked to the cavity (see figure 1), this gravity wave travels upstream at a celerity $c$. As for the acoustic model, we assume here that: (1) both vortex and gravity waves are periodic with the same frequency $f$ and (2) they are in phase at both corners so that equations (2.3) and (2.5) still hold:

(2.9)$$\begin{eqnarray}N=\frac{Lf}{c_{v}}+\frac{Lf}{c}.\end{eqnarray}$$

Following the same normalization as for equation (2.8) and including the correction factor, the feedback frequency reads

(2.10)$$\begin{eqnarray}\frac{f}{f_{10}}=\frac{2F(N+\unicode[STIX]{x0394}N)}{\displaystyle \frac{1}{R_{v}}+F}.\end{eqnarray}$$

Again, equation (2.10) produces an infinity of solutions.

3.1 Experimental campaign

Measurements aiming at validating the analytical model were performed at INSA-Universite de Lyon in a set-up similar to that used by Mignot et al. (Reference Mignot, Cai, Launay and Escauriaza2016). It consists of a 4.80 m long and $b=30~\text{cm}$ wide open channel connected at mid-length to an adjacent rectangular lateral cavity. The channel and the cavity are horizontal, of rectangular section with glass bottom and sidewalls, with no step at the connection. At the channel inlet, the water overflows from the tank to the upstream branch of the channel through a converging section. Moreover, honeycomb meshes straighten the flow. At the outlet, a sharp crested weir adjusts the time-averaged water depth $\overline{h}=5.8~\text{cm}$ at the connection with the cavity. The flow finally recirculates in a closed loop and the corresponding flow rate is measured by an electromagnetic flow meter (ProMag50, Endress Hauser) with an uncertainty of $\pm 0.05~\text{l}~\text{s}^{-1}$. The length of the cavity (in the direction parallel to the channel) is fixed at $L=0.3~\text{m}$ and its width $W=0.5~\text{m}$.

According to a dimensional analysis, the oscillation frequency depends on three parameters: $W/L$, $b/L$ and $F$. The strategy proposed herein is to keep $W/L$ and $b/L$ constant while varying the Froude number from $F=0.33$ to $0.65$, corresponding to the range of Froude numbers for which Wolfinger et al. (Reference Wolfinger, Ozen and Rockwell2012) reported an increasing cavity frequency (for $0.3<F<0.6$) followed by a lock-on regime (i.e. a linearly constant oscillation frequency value for $0.6<F<0.85$). Unfortunately, due to experimental constraints, the Froude number could not be increased beyond 0.65 for the present experimental device. In order to avoid any hysteresis effect, each configuration is initiated with the cavity flow at rest and measurements start after the water-level oscillations at the downstream corner reach a steady state.

3.2 Measurement methods

Three types of measurements are considered herein, using three different methods. The oscillation of the free surface near the downstream corner of the interface (figure 1) is measured using an ultrasonic probe, UNDK 20I6912 S35A from Baumer Electric AG (resolution $\unicode[STIX]{x1D6FF}h\leqslant 0.3~\text{mm}$), with a sampling frequency equal to 200 Hz. Data are recorded during 10 min. Secondly, the two-dimensional horizontal velocity field, and associated vortex dynamics, in the region surrounding the interface is measured using particle image velocimetry. A 1 mm thick horizontal laser sheet is used to illuminate a horizontal plane across the mixing layer at $z/\overline{h}=0.8$. The images are recorded from underneath the flume, using an AV Manta G-235B camera, $1936\times 1216$ pixels with a spatial resolution of 0.3 mm per pixel. For each flow configuration, 33.3 s are recorded at a sampling frequency of 60 fps (2000 images). Finally, the two-dimensional free-surface deformation is measured using a fringe pattern projection method adapted from Takeda, Ina & Kobayashi (Reference Takeda, Ina and Kobayashi1981) and Przadka et al. (Reference Przadka, Cabane, Pagneux, Maurel and Petitjeans2011). A video projector, located 1.5 m above the free surface, projects vertically (downward) on the free surface a known fringe pattern image. The oscillation of the free surface then deforms the pattern so that the water depth displacement is computed at each pixel on each image by extracting the phase difference between the deformed and reference patterns. The post-processing finally applied is adapted from Aubourg (Reference Aubourg2006). In the end, for each flow configuration, the frequency of vortex shedding and free-surface oscillation is measured using three different methods, on three different days as these measurement techniques could not be applied simultaneously. The fair agreement between these frequencies gives confidence in the reproducibility and precision of the methods.