1. Introduction
Cavitation has been the subject of intensive research for decades due to its relevance to many engineering applications such as micro-channel flows, fuel injectors and propellants [
1,
2]. Initial stages of cavitation consist of the generation of a cavitation cloud attached to the wall that primarily develops within the shear layer of the flow and is largely unobtrusive to the main flow [
1]. As the cavitation region evolves, and once a critical cavitation number is reached, cloud break-off occurs. During this process the cavity volume oscillates in either an intermittent or periodic manner. Understanding the mechanisms that govern the cloud break-off process, as well as the frequencies at which it occurs, is important since cloud break-off is linked to effects such as loss of lift in hydrofoil applications, and more generally to unwanted vibration and erosion. We recall here that the cavitation number (Cn) characterises the onset and extent of cavitation and is the ratio between a reference pressure (
) and the vapour pressure (
),
, divided by the dynamic pressure of the free stream flow,
. Lower values indicate a higher probability of cavitation [
3]:
The occurrence of cloud cavitation break-off has been related in many previous studies to the presence of a “re-entrant jet flow” at the downstream region of the cavity which forms as the liquid flow outside the cavity reattaches to the closure region [
1]. One of the earliest studies that demonstrated this phenomenon is by Furness and Hutton (1975) [
4]. They used a potential-flow-based numerical method in order to compute the structure of the re-entrant flow on a convergent–divergent nozzle until it intersected the cavity interface. Franc [
1] described the existence of a re-entrant jet as a necessary outcome of the cavity region being the region of minimum pressure. Thus, the pressure gradient around the cavity region is directed away from its lowest point, forcing the flow along the liquid-vapour interface to curve towards the cavity. The jet originates from the cavity closure region and exists as a thin viscous film travelling upstream over a surface [
5,
6,
7].
More recently, it was suggested that with a sufficient reduction in the cavitation number (giving rise to greater cavitation propensity), large-scale, periodic cloud shedding can be observed and associated with the formation and propagation of a shock within the high void-fraction bubbly mixture in the separated cavity flow [
5,
6,
8,
9]. Jahangir et al. [
6] established within a venturi nozzle that cloud break-off was produced by the presence of the re-entrant jet for a cavitation number Cn > 0.95 while, for Cn < 0.75, cloud shedding was driven by bubbly shock effects generated by the collapse of previously shed cavities. This transitional regime was also demonstrated by Pelz et al. [
7] where a relationship was established between the ratio of the Reynolds (
) and the cavitation numbers, and the occurrence of this transitional regime. In the work of Ganesh et al. [
5,
9] with the use of high-speed imaging and time-resolved X-ray densitometry of cavitation over the apex of a wedge, the features of the observed condensation shocks were related to the average and dynamic pressure, and the void fraction. The transition to strongly shedding conditions was associated with a drop in the speed of sound in the bubbly mixture within the partial cavity. The strongest shock waves occurred for Mach numbers higher than one. More specifically, as shock fronts from collapsed cavities impinged on a growing vapour cloud, the internal pressure wave propagated at a speed much closer to, or higher than, the local speed of sound which caused an internal bubbly shock and drove cloud break-off.
Both mechanisms of cloud break-off (re-entrant jet and/or shock waves) are challenging to capture numerically. The difficulty in the numerical study of the re-entrant jet is that it is mostly a boundary layer phenomenon that requires high grid resolution close to the wall. The mechanism of cloud break-off based on shock wave propagation is also hard to predict since time steps of the order of nanoseconds are required, along with fully compressible codes, to capture local pressure peaks [
10]. An additional difficulty is that there are only a few qualitative experimental data sets available for cavitating flows (especially for micro-channel flows) and even fewer ones that clearly identify the mechanisms for cloud break-off. An important reason for this is the difficulty in obtaining adequate visualisation. For example, the work of Ganesh et al. [
5,
8,
9] mentioned above, was performed for cavitation over an apex of a wedge and thus it is indicative of the dynamics of the mechanism of the cloud break-off in external geometries, while the work of Jahangir et al. [
6] was performed in a venturi geometry that is much larger than fuel injectors. An experimental investigation that was performed for numerical validation of Diesel fuel injection processes was conducted by Winklhofer et al. [
11]. The study presented ‘line of sight’ distributions of the vapour void fraction, produced by shadowgraph imaging, in a plane micro-channel between an upstream and a downstream plenum chamber, with the flow ‘sandwiched’ between two windows. Measurements exist for several different pressure differences between the chambers and this case is commonly used for numerical validation due to the simple orthogonal geometry and realistic Diesel injection pressure conditions [
12,
13,
14,
15]. The limitations of the shadowgraph images though should be noted; the visual data provided are averaged over 20 different time-steps and are the ‘line of sight’ average over the depth of the cross-section. Mauger et al. [
16] report similar visualisations of an orthogonal geometry, providing good instantaneous images of the cavitation distribution along the centre of the nozzle. The experiments show that instabilities in the shear layers intensified as the pressure difference increased and produced vortices in the flow. Mauger et al. also showed the presence of pressure waves due to the collapse of vapour bubbles in the throat as well as at the nozzle exit. The averaged cavitation topology seen in this investigation is similar to that of Winklhofer et al. [
11]. It should be pointed out, though, that in neither of these two experimental studies the cloud break-off dynamics were investigated.
Apart from elucidating the mechanism of cloud break-off, another important aspect is the identification of the frequencies of the pressure and vapour fluctuations associated with this phenomenon. A number of papers have studied the link between frequencies of pressure fluctuations and cavitation dynamics. Experimentally, Crua and Heikal [
17] measured the injection process in a Diesel injector, using 3D laser Doppler vibrometry, with a focus on the distribution of the flow dynamics over time. Some peak frequencies were identified to be dependent on the internal nozzle cavitation along with other peak frequencies that were dependent on the injector nozzle geometry. A more recent effort to visualise vortical cloud shedding structures within an internal nozzle set up was attempted by Mitroglou et al. [
18] who showed that an increase in velocity, and hence an increase in Reynolds number (
), led to lower cloud shedding frequencies with a Strouhal number (
) number of 3.1 at the dominant frequency. In the study of Mihatsch et al. [
19], a numerical simulation using a density-based finite volume method, taking into account the compressibility of both phases and resolving collapse-induced pressure waves (CATUM code), was performed on an airfoil geometry described in [
20]. In their study, four characteristic frequencies were identified at all operating conditions (instead of two in the case of Peters et al. [
2] for the same geometry). The first and third frequency in order of amplitude were found to increase with increasing pressure level (lowest frequencies). The frequency peak with the second highest amplitude was the shedding frequency and the fourth highest peak was found to be independent of the operating condition and related to the reflection of collapse-induced pressure waves at the radial boundary of the gap (8–9 kHz). Experimental visualisation and LES modelling of a vertical orthogonal nozzle geometry was performed by He et al. [
21]. The authors used FLUENT (pressure based code) as their multiphase solver and the Schnerr–Sauer cavitation model. Two frequency peaks were identified to be caused by cavitation. The lower peak was reported to be related to cloud shedding at 3 kHz and the higher peak was related to the collapse of small bubbles at 6 kHz.
The unsteady nature of cavitating flows (including the phenomena relevant to the cavitation cloud break-off at the macro-scale as well as bubble collapse at the micro-scale) generates pressure waves that travel throughout the domain [
22,
23]. These propagating waves will interact with the domain boundaries in different ways (varying from 100% reflection to 100% transmission) depending on the experimental set up. When performing a simulation, these waves also interact with the numerical domain outlet (which might have to be chosen deliberately to not be the same as the real geometry outlet) and may be artificially reflected back (artificial in the sense that they would be transmitted during an experiment). This is particularly the case when the full domain is not modelled to reduce computational cost. To the best of the authors’ knowledge no previous studies have been performed on the role of the boundary conditions in the prediction of cavitation dynamics, especially in terms of cloud break-off, apart from the preliminary work of Pearce et al. [
14]. In this work we adopted a non-reflective boundary condition developed by Poinsot and Lele [
24] within a compressible multiphase solver. This boundary condition attempts to resolve a far field pressure boundary condition producing a pressure gradient based on the distance to the far-field. Without the non-reflective boundary condition, propagated pressure or shock waves produced by cavitation collapse would be reflected back toward nozzle exit. In order to highlight the sensitivity of cavitation cloud-off to reflected pressure waves, a fixed value pressure condition was also adopted in our work alongside the non-reflective condition.
Apart from the boundary conditions, another consideration with the numerical representation of the propagating waves in cavitating flows is the damping effect introduced by various differencing schemes. The first order upwind scheme offers a fully bounded solution but is far too diffusive, and the second order central scheme has better accuracy but is unbounded. The central differencing scheme was used by Magagnato and Dumond [
25] to simulate cavitation within a number of different geometries, producing a flat cloud topology with re-entrant driven cloud break-off. The upwind scheme, which is first-order accurate, was selected by Salvador et al. [
26] to model cavitation in the experiment by Winklhofer et al. [
11] using the Homogeneous Equilibrium Model (HEM). This scheme was selected for its stability but also produced a cloud topology which was flat and parallel to the wall with no convergence towards the centre of the duct as is seen in the experiment. Chen and Oevermann [
27] modelled cavitation in the same geometry using a compressible multiphase stochastic framework. The differencing scheme used for the convective terms was a second-order upwind scheme which also produced a flat cavitation cloud topology and did not show any waves produced by cavity collapse. He et al. [
21] conducted simulations of the same geometry using a compressible code with the Homogeneous Relaxation Model (HRM) to model phase change. A first-order upwind scheme was used for this investigation with the cavitation cloud appearing only within the laminar layer. Kärrholm, because of this, used a Total Variable Diminishing (TVD) scheme called MUSCL in OpenFOAM (taken from the name of the general sub-class of TVD solvers, monotone upstream-centered schemes for conservation laws) which maintains the monotonicity of the flow and thus is better at reproducing local large gradients [
12]. These schemes resolve a blended scheme with a dynamic face limiter to suppress the higher order contribution at points of instability. Monotonic TVD schemes can handle discontinuities that arise from discontinuous functions or numerical inconsistencies [
28]. Kärrholm’s results show a cavitation cloud topology that was qualitatively more representative of the experimental visualisations and produced pressure and velocity profiles that were in good agreement with experiment. A three phase compressible solver with phase change across a single interface was formulated by Yu et al. [
15] who conducted internal nozzle cavitation simulations using the Winklhofer geometry as well. A Normalised Variable (NV) gamma scheme was selected for the discretisation of the volume fraction and produced centreline pressure profile in very good agreement with experiments. The cloud topology though, appeared somewhat artificial. The analysis was not extended to cloud shedding. A parametric study of a high-resolution NV differencing scheme was performed by Guedri et al. [
29] and found that, although the gamma scheme was computationally efficient, it was less accurate than other higher-order schemes such as MUSCL. Higher order schemes such as the Weight Essentially Non-Oscillatory (WENO) scheme are able to capture the large discontinuities and smooth regions [
30]. However, it has been shown that higher order schemes require high computation and memory demand that can lead to stability problems [
31].
Based on the previous observations, in the first half of our paper we present a sensitivity analysis in order to further examine the effects that the boundary conditions and the differencing schemes have on the predictions of the cavitation dynamics in a micro-channel with emphasis on the cavitation bubble cloud behaviour. We compare our numerical simulations with the experiments performed by Winklhofer et al. [
11]. This experimental set up is chosen due to the simple sharp-edged geometry of the entry of the nozzle channel as well as its frequent use in the literature as a benchmark case for computational fluid dynamics (CFD) prediction of cavitating flows. Moreover, although cloud dynamics are not explicitly discussed in this experimental work, it is one of the few experimental data sets that presents qualitative (shadowgraph based vapour fraction) as well as quantitative (pressure and velocity field) measurements. In previous CFD studies of internal geometries relevant to cloud dynamics, the focus was mostly on erosion prediction and geometries like the one from Franc [
20,
32] were used. However, in the studies of these geometries, limited quantitative comparison is provided with experiments which makes questionable the conclusions drawn only from CFD predictions for the mechanisms of cloud break-off. Moreover, in the investigated cases we used the Linear-upwind stabilised transport (LUST) and MUSCL differencing schemes (the former being a blend of the centred and linear schemes) at pressures relating to the onset of cavitation and the critical cavitation, with different numerical boundary conditions. The introduction of the wave transmissive outlet boundary condition is compared to the typical boundary condition of a fixed pressure value commonly used in previous studies. The physical length of the chamber is also varied to understand better how dependant the flow is, from a numerical point of view, on the geometry.
After having established the extent of the validity, as well as the limitations, of our framework for both cases of cavitation inception and critical cavitation conditions, in the second part of our paper we turn our attention to the mechanisms that govern the cloud break-off dynamics. Using the Winklhofer geometry we have the opportunity to examine the cloud dynamics in conditions close to “choking” that have not been investigated before by numerical studies performed in the same geometry [
12,
14,
15,
27]. Previous studies predicted the shape of the free surface of the cloud to be largely flat and generally parallel to the wall for large pressure differences, due to low-order discretisation schemes used, which is inconsistent with experiments. This result however implied a cloud break-off mechanism that was reliant on the re-entrant jet dynamics. In contrast, in our study using a higher order scheme, MUSCL, we predict a vapour cloud topology converging towards the centre of the nozzle which is more consistent with the evidence of the shadowgraphic images. Using these predictions, a detailed analysis of the mechanism of the cloud break-off is presented. We conclude that the mechanism is dominated by the underlying flow pressure dynamics rather than re-entrant jet dynamics.