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

In this paper, we describe the use of a new nonlinear partially-averaged Navier–Stokes (PANS) model with near-wall correction for simulating the cavitating flow around a Clark-Y hydrofoil. For comparison, the standard [Formula: see text]–[Formula: see text] PANS model is also used. The results demonstrate that compared to [Formula: see text]–[Formula: see text] PANS and experiment, the new PANS model shows better performance for cavitation flow, including time-averaged velocity, root mean square (rms) velocity and cavity shedding processing. Through the calculation of the lift and drag coefficient at [Formula: see text] and [Formula: see text], it can be concluded that the cavitation will decrease the lift and increase the drag of the hydrofoil, resulting in a decrease of the lift-to-drag ratio. From the analysis of different terms in both the turbulent kinetic energy (TKE) and dissipation rate transport equations of the cloud cavitation, it is found that the production term and the dissipation term are dominant in the turbulent transport, and they are mainly distributed in the vapor–liquid interface and the trailing edge of the hydrofoil.

The turbulence closure is significant to unsteady cavitating flow computations as the flow is frequently time-dependent accompanied with multiple scales of vortex. A turbulence bridging model named as PANS (Partially-averaged Navier-Stokes) purported for any filter-width is developed recently. The model filter width is controlled through two parameters: the unresolved-to-total ratios of kinetic energy fk and dissipation rate fω. In the present paper, the PANS method based on k-ω model is used to simulate unsteady cavitating flows over a Clark-y hydrofoil. The main objective of this work is to present the characteristics of PANS k-ω model and evaluate it depending on experimental data. The PANS k-ω model is implemented with various filter parameters (fk=0.2~1, fω =1/fk). The comparisons with the experimental data show that with the decrease of the filter parameter fk, the PANS model can reasonably predict the time evolution process of cavity shapes and lift force fluctuating in time. As the PANS model with smaller fk can overcome the over-prediction of turbulent kinetic energy with original k-ω model, the time-averaged eddy viscosity at the rear of attached cavity decreases and more levels of physical turbulent fluctuations are resolved. What's more, it is found that the value of ω in the free stream significantly affects the numerical results such as time-averaged cavity and fluctuations of the lift coefficient. With decreasing fk, the sensitivity of ω-equation on free stream becomes much weaker.

A modification to the PANS (partially averaged Navier-Stokes) model is proposed to simulate unsteady cavitating flows. In the model, the parameter fk is modified to vary as a function of the ratios between the water density and the mixture density in the local flows. The objective of this study is to validate the modified model and further understand the interaction between turbulence and cavitation around a Clark-Y hydrofoil. The comparisons between the numerical and experiment results show that the modified model can be improved to predict the cavity evolution, vortex shedding frequency and the lift force fluctuating in time fairly well, as it can effectively modulate the eddy viscosity in the cavitating region and various levels of physical turbulent fluctuations are resolved. In addition, from the computational results, it is proved that cavitation phenomenon physically influences the turbulent level, especially by the vortex shedding behaviors. Also, the mean u-velocity profiles demonstrate that the attached cavity thickness can alter the local turbulent shear layer.

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.

The present work deals with validation of the PANS (Partially-Averaged Navier-Stokes) computational methodology coupled with the \(\zeta -f\) RANS model of turbulence [10], formulated and implemented into the CFD software package AVL-FIRE by Basara et al. [2], in a broad range of wall-bounded flows featured by separation, swirling and tumbling motion. The configurations considered in the part I include a fully-developed flow in a plane channel (DNS by Moser et al. [14]) and flow separating from a continuous curved surface (reference database is provided by LES, [7] and [3]). The PANS approach, whose validation represents the prime objective of the present work, provides a seamless transition from the fully-modelled Unsteady RANS to the fully-resolved direct numerical solution (DNS) as the unresolved-to-total ratios of kinetic energy and its dissipation are appropriately varied. In addition, the complementary RANS (by using the \(\zeta -f\) model) computations of the flow configurations considered are also performed. The results obtained illustrate the PANS model capability to capture the turbulence unsteadiness leading consequently to a correct prediction of time-averaged flow quantities, unlike its RANS counterpart. The companion part II [4] is concerned with the PANS model application to more complex flow configurations including a tumbling vortex generation and compression and a swirling flow in a tube, the outlet of which is designed as an orifice with eccentric opening.

The partially averaged Navier–Stokes (PANS) model, proposed in Girimaji [Partially-averaged Navier-Stokes model for turbulence: a Reynolds-averaged Navier-Stokes to direct numerical simulation bridging method. ASME J Appl Mech. 2006;73(3):413–421], can be used to simulate turbulent flows either as a RANS, LES or DNS. The PANS model includes which denotes the ratio of modelled to total kinetic energy. In RANS, , and in DNS it goes to zero. In the present study we propose a new formulation for based on the H-equivalence introduced by Friess et al. [Toward an equivalence criterion for hybrid RANS/LES methods. Int J Heat Fluid Flow. 2015;122:233–246]. In this formulation the expression of is derived to mimic DES. This new formulation behaves very much like classic DES, even though the two formulations use different mechanisms to separate modelled and resolved scales. They show very similar performance in separated flows as well as in attached boundary layers. Moreover, the new formulation exhibits similar robustness features as DES.

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.

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

Czochralski melt flow is an outcome of complex interactions of centrifugal, buoyancy, coriolis and surface tension forces, which act at different length and time scales. As a consequence, the characteristic flow structures that develop in the melt are delineated in terms of recirculating flow cells typical of rotating Bénard–Marangoni convection. In the present study, Partially Averaged Navier–Stokes (PANS) method is used for the first time to study an idealized Czochralski crystal growth set-up. It is observed that with a reduction in the PANS filter width, more turbulent scales are resolved and the present PANS model is able to resolve almost all the characteristic flow structures in the Czochralski flow at a comparatively lower computational cost compared with more advanced turbulence modelling tools, such as Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES).

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

4 Modeling, simulation and mixing time calculation of stirred tank for nanofluids using partially-averaged Navier–Stokes (PANS) ku − ϵu turbulence model was published in Basic Sciences for Sustainable Development on page 85.

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.

This paper describes a strategy for the very large eddy simulation (VLES) of complex turbulent flows, such as flows in the aircraft engine combustors or flows around the rotorcrafts. The large scales of turbulence are directly calculated; the effects of the rest unresolved scales are approximated by the eddy viscosity models evolved from the state-ofthe-art models used in the Reynolds-averaged Navier-Stokes (RANS). The dependent variables and governing equations are based on the concept of temporal filtering. The contents of both resolved and unresolved scales are regulated by the width of the filter. In this approach, the dependent variables and governing equations will naturally evolve from RANS to VLES and further towards LES (large eddy simulation) when the width of the temporal filter is decreased from the turbulent integral time scale to its fraction and all the way towards the Taylor micro time scale. We refer this strategy as the partially resolved numerical simulation (PRNS). The main feature of PRNS is that its constitutive equations for the unresolved turbulence have no relationship with the mesh size, hence formally grid independent; its subscale models have the nature of a non-linear dynamic system capable of effectively controlling the subscale stresses to perform VLES with a relatively coarse grid. In addition, its implementation to a turbulent CFD code is straightforward.

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.

The partially averaged Navier–Stokes (PANS) equations are used to predict the variable-density Rayleigh–Taylor (RT) flow at Atwood number 0.5 and maximum Reynolds number 500. This is a prototypical problem of material mixing, featuring laminar, transitional, and turbulent flow, instabilities and coherent structures, density fluctuations, and production of turbulence kinetic energy by both shear and buoyancy mechanisms. These features pose numerous challenges to modeling and simulation, making the RT flow ideal to develop the validation space of the recently proposed PANS Besnard–Harlow–Rauenzahn-linear eddy viscosity model closure. The numerical simulations are conducted at different levels of physical resolution and test three approaches to set the parameters fϕ defining the range of physically resolved scales. The computations demonstrate the efficiency (accuracy vs cost) of the PANS model predicting the spatiotemporal development of the RT flow. Results comparable to large-eddy simulations and direct numerical simulations are obtained at significantly lower physical resolution without the limitations of the Reynolds-averaged Navier–Stokes equations in these transitional flows. The data also illustrate the importance of appropriate selection of the physical resolution and the resolved fraction of each dependent quantity ϕ of the turbulent closure, fϕ. These two aspects determine the ability of the model to resolve the flow phenomena not amenable to modeling by the closure and, as such, the computations' fidelity.