Numerical study of unsteady turbulent cavitating flows

Comparative study between turbulence models in unsteady cavitating flow with special emphasis on shock wave propagation

In 1873, van der Waals proposed a simple equation of state that qualitatively described the phase behavior of fluids. Since then, the principles behind the van der Waals equation have been used and refined countless times in the quest for an accurate equation of state. Despite the enormous amount of work reported, the goal of a simple and accurate equation of state for even relatively simple systems, such as monatomic or polyatomic gases, has proved elusive. Therefore, the analysis of phase equilibria with equations of state is more often than not an exercise in data fitting rather than genuine prediction. In this work, we revisit the early work of Dieterici. When coupled with modern developments in equations of state, this little used approach has the potential of generating useful equations of state. To illustrate the usefulness of the Dieterici formula, we use it to derive a simple equation of state based on the Carnahan–Starling hard-sphere term and van der Waals interactions. This simple approach yields more accurate predictions of the phase co-existence of fluids than the traditional additive approach used to develop equations of state.

Numerical simulation unsteady cloud cavitating flow with a filter-based density correction model

In this paper, the cavitation and flow characteristics of the unsteady liquid nitrogen (LN2) cavitating flow in a submersible pump are investigated through both experimental and numerical approaches. The performance curve of the LN2 submersible pump is obtained via experimental measurement. Numerical simulations are performed using a modified shear stress transport k–ω turbulence model, incorporating corrections for rotation and thermal effects as per the Schnerr–Sauer cavitation model. The numerical framework is verified by comparing the cavitation morphology features with previously reported visual data of the LN2 inducer and aligning pump performance data with those obtained from experimental tests of the LN2 submersible pump. The results indicate that cavitation at the designed flow rate predominantly manifests as tip clearance vortex cavitation in the inducer. Increased flow rates exacerbate cavitation, potentially obstructing the flow passage of the impeller. The vortex identification method and the vorticity transport equation are employed to identify the vortex structures and analyze the interaction between cavitation and vortices in the unsteady LN2 cavitating flow. The vortex structures primarily concentrate at the outlet of the impeller flow passage, largely attributed to the vortex dilation term and baroclinic torque. The influence of thermal effects on the cavitation flow of submersible pumps is analyzed. An entropy production analysis model, comprehensively involving various contributing factors, is proposed and utilized to accurately predict the entropy production rate within the pump. This study not only offers an effective numerical approach but also provides valuable insight into the cavitation flow characteristics of the LN2 submersible pump.

Unsteady cavitating flow with high Reynolds number and significant instability commonly exists in fluid machinery and engineering system. The high-resolution approaches, such as direct numerical simulation and large eddy simulation, are not practical for engineering issues due to the significant cost in the computational resource. The objective of this paper is to provide the approach with Detached-Eddy Simulation (DES) model based on the Reynolds-averaged Navier–Stokes (RANS) equations for predicting unsteady cavitating flows. The credibility of the approach is validated by a set of numerical examples of its application: the unsteady cavitating flows around the two-dimensional (2D) Clark-Y hydrofoil and the three-dimensional (3D) blunt body. It is found that the calculated cavity shapes, cavity lengths and unsteady characteristics by DES model agree well with the experimental measurements and observations. Further analysis indicates that the turbulent eddy viscosity around the cavity and wake region is well predicted by the DES model, which results in the development of large-scale vortexes, and further cavitation instability. The DES model, which exhibits a significantly unsteady 3D behavior, is a more comprehensive turbulence model for unsteady cavitating flows.

Dynamics of unsteady compressible cavitating flows associated with the cavity shedding

Unsteady cavitating flow usually contains various vapor structures with different length scales and significant fluid compressibility, which poses great challenges to numerical simulation. In this work, we propose a compressible Eulerian–Lagrangian method to investigate cloud cavitation inside a nozzle, with particular emphasis on the effect of compressibility on multiscale structures. The macroscopic cavity evolution is dealt with using large eddy simulation and the volume of fluid method in a compressible Eulerian framework. Microscopic bubble dynamics are described using the equation of bubble motion and the compressible Rayleigh–Plesset equation. A two-way coupling algorithm is established to bridge the two frameworks. The calculated results are validated by comparison with available experimental data. The numerical results show that the compressibility exacerbates the instability of the cavitating flow field, generates more discrete bubbles, and promotes liquid–vapor mass exchange, which further increases the overall cavitation volume. Furthermore, two distinct power laws for bubble size distribution, with exponent −4/3 for small bubbles and −10/3 for large bubbles, are obtained. Owing to the influence of the high flow velocity and high ambient pressure in the nozzle, and the wall effects on bubble dynamics, the number of large-size bubbles calculated by the numerical simulation is greater than that obtained by theoretical analysis. The formation of these bubbles is affected by shear on the liquid–vapor interface, the presence of vortex structures, and turbulence, thus differing significantly from the case of cavitation flow around a hydrofoil.

We investigated the three-dimensional characteristics in the unsteady cavitating flow over an axisymmetric body with a blunt headform. A high-speed video camera was used to visualize the cavitating flow structures. A modified PANS (Partially-averaged Navier-Stokes) turbulence model coupled with Kubota cavitation model was used to simulate the time-evolution process of the unsteady cavitating flows. The finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures (LCS) methods are applied to investigate the three-dimensional flow patterns. The results show that the numerical method is capable of simulating the unsteady cavitating flow for capturing the attached cavity with maximum-length, the cavity break-off and U-type shedding, in accordance with the qualitative features observed in the experiment. Three-dimensional characteristics are extremely impressive in the unsteady cavitating flows. The re-entrant flows originate from different circumferential positions at the closure region of the attached cavity, and move upstream at different speeds. Moreover, substantial circumferential motions in the unsteady cavitating flow are found based on Lagrangian trajectories, and it is reasonably deduced that the U-type shedding cavity is primarily induced by the effect of re-entrant flow motions consisting of moving upstream and circumferential flows as well.

Considering that large eddy simulation (LES) often requires more computational resources to capture small-scale flow characteristics, while the Reynolds-Averaged Navier–Stokes (RANS) methods offers better computational efficiency at the expense of accuracy. The traditional Zwart–Gerber–Belamri cavitation model (Z–G–B) has good universality but insufficiently considers the influencing factors of cavitation. This article proposes a method combining very large eddy simulation (VLES) with the Z–G–B cavitation model (VLES_Z–G–B). However, there remains a certain discrepancies between the predictions and experimental results. To address this, on the basis of the Z–G–B cavitation model, the unsteady cavitation computation method (VLES_Water_Non) is developed, considering the weak compressibility effect of the fluid and incorporating three components: the liquid, vapor, and non-condensable gas. The results indicate that the VLES_Water_Non method compensates for the deficiencies of the VLES_Z–G–B approach, and clarifying that the combined effect of re-entrant flow, side-incident reverse flow, and mainstream are the primary causes of cavity shedding. Using a robust Ω-criterion, multiscale flow features during the cavity evolution process were identified. Additionally, it was found that the vortex stretching and dilation terms are the main reason for the flow separation and formation of complex structures around the hydrofoil. The baroclinic torque term has an significant impact on the generation of vorticity during the shedding process of the cavity. This study is expected to provide a more precise scale resolution method and predictive accuracy for analyzing the internal flow mechanisms of complex fluid machinery.

The interfacial dynamics‐based cavitation model, developed in Part‐1, is further employed for unsteady flow computations. The pressure‐based operator‐splitting algorithm (PISO) is extended to handle the time‐dependent cavitating flows with particular focus on the coupling of the cavitation and turbulence models, and the large density ratio associated with cavitation. Furthermore, the compressibility effect is important for unsteady cavitating flows because in a water–vapour mixture, depending on the composition, the speed of sound inside the cavity can vary by an order of magnitude. The implications of the issue of the speed of the sound are assessed with alternative modelling approaches. Depending on the geometric confinement of the nozzle, compressibility model and cavitation numbers, either auto‐oscillation or quasi‐steady behaviour is observed. The adverse pressure gradient in the closure region is stronger at the maximum cavity size. One can also observe that the mass transfer process contributes to the cavitation dynamics. Compared to the steady flow computations, the velocity and vapour volume fraction distributions within the cavity are noticeably improved with time‐dependent computations. Copyright © 2004 John Wiley & Sons, Ltd.

We explore the properties of a simple analytic equation of state, proposed by Heyes and Woodcock [D. M. Heyes and L. V. Woodcock, Mol. Phys. 59, 1369 (1986)] which is a development of the van der Waals equation of state, improved with an analytically simple yet accurate hard-core component for the compressibility factor. We call this the hard-sphere van der Waals (HSvdW) equation of state. We show, for the first time, that the HSvdW equation of state gives analytic values for the critical parameters and that it gives significantly more realistic values for the critical density than the van der Waals and other cubic equations of state. For single-component square wells of arbitrary well width, the HSvdW equation of state also produces a critical temperature and pressure in agreement with those from a recent Gibbs Monte Carlo simulation study. The chief advantage over a similar equation of state with an alternative form for the hard-core compressibility factor, such as Carnahan–Starling, is that it leads to more tractable analytic expressions for thermodynamic properties without loss of accuracy. Extensive Monte Carlo computer simulations of square-well (SW) and square-shoulder (SS) fluids with variable well width have been performed. We have calculated the compressibility factor, internal energy, pair distribution functions, and coordination number (average number of particles within the soft-shell interaction range). We have also determined the gas–liquid coexistence curve using the particle insertion chemical potential method for the square-well case, λ=1.5σ, where λ is the diameter of the soft interaction shell. This agrees well with the Gibbs ensemble simulation coexistence curve. For the purpose of improving the attractive component of the HSvdW equation of state, we have made use of the SW and SS simulation data and that of other workers to develop another new simple equation of state based on the quasichemical local coordination number approximation (QCA) which is a lattice model of a fluid. The QCA is improved to include the local coordination of the hard-sphere fluid. Present adaptations extend the range of earlier QCA models to arbitrary well width and sign of the well (either attractive well or repulsive shoulder). The simulation coordination numbers are reproduced well by the present model, although there is little improvement in the overall equation of state over the simpler HSvdW.

The large density ratio, up to 1000 in water, between liquid and vapor, turbulence with complicated interface dynamics and fast and multiple time scales make the computation of cavitating flows difficult. Utilizing a pressure-based algorithm extended for such flows, the present study focuses on the non-equilibrium and nonstationary aspects in the turbulence model, and the compressibility effects in the cavitation model. Assessment of several modeling concepts has been made in the context of the Favre-averaged Navier-Stokes equations, along with a transport equation-based cavitation model and the e − k two-equation turbulence model. The three-dimensional turbulent cavitating flow in a hollow-jet valve is adopted as the physical focus. While the non-equilibrium and nonstationary modification to the turbulence closures do not seem to influence the qualitative characteristics of the simulation, the compressibility modeling can cause the result to vary substantially.

Numerical simulations based on Reynolds-averaged Navier Stokes (RANS) models are still the work-horse tool in engineering design involving turbulent flows. Two decades ago, when LES started gaining popularity thanks to the increasing availability of computational resources, it was widely expected that it would have gradually replaced RANS methods in industrial CFD for decades to come. In the past two decades, however, while LES-based methods gained widespread applications and the earlier hope did not diminish, the predicted time when LES would replace RANS has been significantly delayed. Most industrial users are probably decades away from any routine use of scale resolving simulations, not to mention the cost, time and user skill it take to run these computations. In brief, RANS solvers, particularly those based on standard eddy viscosity models (e.g k-e, k-ω, S-A and k-ω SST) are expected to remain the workhorse in the CFD of high Reynolds number flows for decades. However, predictions from RANS simulations are known to have large discrepancies in many flows of engineering relevance, including those with swirl, pressure gradients, or mean streamline curvature. It is a consensus that the dominant cause for such discrepancies is the RANS-modeled Reynolds stresses. In light of the long stagnation in traditional turbulence modeling, researchers explored machine learning as an alternative to improve RANS modeling by leveraging data from highfidelity simulations. The goal is to make use of vast amounts of turbulent flows data, machine learning techniques and current understanding of turbulence physics to develop models with better predictive capabilities in the context of RANS simulations. Recently, in a seminar work, Ling et al (2016) developed a neural network architecture capable of embedding invariance properties into the Reynolds stress tensor predicted in output. Such a network, named the tensor basis neural network (TBNN), was applied to a variety of flow fields with encouraging results compared to both classical turbulence models and neural networks that do not preserve Galilean invariance. Yet, as in most data driven turbulence modelling approaches, the TBNN was used as a post-processing tool to correct the Reynolds stress tensor field predicted by a RANS simulation run with standard closure models. This means that, theoretically, the network can be applied only to correct the Reynolds stress tensor for the same RANS model on which it has been trained since, in general, different turbulence models yield different results depending on the flow type. Moreover, there is no physisical insight that suggests a relation between the RANS velocity gradients - used as inputs of the machine learning model - and the true Reynolds stress tensor. Differently, in this work a network with a similar architecture to the Ling’s one was trained and tested on a database of high-fidelity data of eight different flows to learn a functional mapping between the inputs of Pope’s General Eddy Viscosity Model and the anisotropic part of the Reynolds stress tensor. Then the network was embedded into a CFD RANS solver as a replacement of the standard closure model - and therefore called at every solver’s iteration. Lasty, the RANS solver with embedded TBNN was be tested on a canonical flow case - turbulent channel flow - to evaluate its performances. As for the organization of this work: in Chapter 1 further details on the data driven turbulence modelling will be given, RANS models and equations will be introduced and also an introduction to Neural Networks will be presented. In Chapter 2, it will be given a detailedexplanation of the RANS CFD solver and the neural network’s implementation. In Chapter 3, the method will be tested on a turbulent channel flow case and the results will be discussed. Lastly, in Chapter 4, some meaningful conclusions will be drawn.

Partially Averaged Navier-Stokes method for time-dependent turbulent cavitating flows

In the present work, a cavitation model based on a new truncated form of the full Rayleigh-Plesset (R-P) equation for the source terms controlling vapor generation and destruction has been developed and implemented in the ANSYS FLUENT platform. Coupled with a Filter-based density corrected model (FBDCM), the cavitating flow over a 2-D Clark-Y hydrofoil is investigated numerically with particular emphasis on understanding the effect of cavitation structures and the shedding dynamics on the hydrodynamics coefficients and surrounding flow velocity structures. The hydrofoil has a fixed angle of attack of α = 8° with a moderate Reynolds number of Re = 7.0×105. Simulations have been carried out for various cavitation numbers ranging from non-cavitating flows to the cloud cavitation regime (σ = 0.80). In particular, we compared the lift and drag coefficients, the cavitation dynamics and the time-averaged velocity with available experimental data for two cavitation models, i.e. the proposed model and Schnerr-Sauer model. The comparisons between the numerical and experimental results show that the proposed model has a better capability than Schnerr-Sauer model to capture the characteristics of lift and drag coefficients under cavitation conditions. Meanwhile, the proposed model is sufficiently robust to predict the initiation of the sheet/cloud cavity, growth towards the trailing edge, and subsequent shedding downstream, which is in accordance with the experimental quantitative features in literature.