Modified k – ω Model for Simulation of Cavitating Flows

Although cavitating flows are generally turbulent, the interaction of turbulence and cavitation dynamics is still not well understood. In general, two‐equation models are employed, which were originally developed for single‐phase flows. Therefore they fail by handling cavitation based flow phenomena with very high density variations (dependent on operating condition up to 40000:1). This sudden change of the density causes strong pressure gradients, secondary flows and local compressibility. The aim of this study is to enhance the Wilcox's k‐? model with empirical correlations in order to simulate turbulent cavitating flows more precisely and effciently.     

Comparison of Inviscid and Viscous Two‐Phase Calculations of Cavitating Pump Flow

In cavitating flows there is a strong interaction between the fine dispersed vapor bubbles and turbulence. Therefore in two‐phase flow calculations the prediction of turbulence is a matter of great difficulties and uncertainties. To get an idea of the influence of turbulence modelling on the calculated result, viscous and inviscid two‐phase calculations of cavitating pump flow were performed. In the inviscid calculations the influence of cavitation is isolated from turbulence effects. In the viscous calculations the effect of turbulence is modelled with a k – ϵ turbulence model. The results show that the influence of viscous effects on the flow field is weak in comparison to cavitation. However, in contrast to the steady cavitation behaviour predicted by the viscous calculations, the inviscid calculations show unsteady behaviour of the cavitation (as can be seen in the experiment).     

Numerical simulation of cavitation dynamics using a cavitation-induced-momentum-defect (CIMD) correction approach

A new unsteady cavitation event tracking model is developed for predicting vapor dynamics occurring in multi-dimensional incompressible flows. The procedure solves incompressible Navier–Stokes equations for the liquid phase supplemented with an additional vapor transport equation for the vapor phase. The novel cavitation-induced-momentum-defect (CIMD) correction methodology developed in this study accounts for cavitation inception and collapse events as relevant momentum-source terms in the liquid phase momentum equations. The model tracks cavitation zones and applies compressibility effects, employing homogeneous equilibrium model (HEM) assumptions, in constructing the source term of the vapor transport model. Effects of vapor phase accumulation and diffusion are incorporated by detailed relaxation models. A modified RNG k–ε model, including the effects of compressibility in the vapor regions, is employed for modeling turbulence effects. Numerical simulations are carried out using a finite volume methodology available within the framework of commercial CFD software code Fluent v.6.2. Simulation results are in good qualitative agreement with experiments for unsteady cloud cavitation behavior in planar nozzle flows. Multitude of mechanisms such as formation of vortex cavities, vapor cluster shedding and coalescence, cavity pinch off are sharply captured by the CIMD approach. Our results indicate the profound influence of re-entrant jet motion and adverse pressure gradients on the cavitation dynamics.     

Performance and limitations of the unsteady RANS approach

Turbulent flows in complex geometries often exhibit an oscillating behavior of large coherent structures, even in the case of steady state boundary conditions. Recently, numerous efforts have been made to resolve these oscillations by means of numerical simulations. Unfortunately, large-eddy simulations are often very time- and memory-consuming in the case of complex flows. Therefore, the unsteady RANS (URANS) approach is an attractive alternative, especially when numerical simulations are used as a design and optimization tool. Here, two complex flow situations are presented, the tundish flow and a jet in a crossflow. For these flows, relationships between the Strouhal number and important flow parameters are known from experiments. In the paper, URANS models are applied to resolve those relationships also numerically. The evaluation of the numerical results demonstrates the abilities and the limitations of the URANS approach when resolving the dynamics of large coherent structures in complex flows. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interactive computing methods for aeroelastic analysis of turbomachinery

An interaction viscous‐inviscid method for efficiently computing steady and unsteady viscous flows is presented. The inviscid domain is modeled using a finite element discretization of the full potential equation. The viscous region is modeled using a finite difference boundary layer technique. The two regions are simultaneously coupled using the transpiration approach. A time linearization technique is applied to this interactive method. For unsteady flows, the fluid is assumed to be composed of a mean or steady flow plus a harmonically varying small unsteady disturbance. Numerically exact nonreflecting boundary conditions are used for the far field conditions. Results for some steady and unsteady, laminar and turbulent flow problems are compared to linearized Navier‐Stokes or time‐marching boundary layer methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling unsteady flow characteristics using smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) method has been extensively used to simulate unsteady free surface flows. The works dedicated to simulation of unsteady internal flows have been generally performed to study the transient start up of steady flows under constant driving forces and for low Reynolds number regimes. However, most of the fluid flow phenomena are unsteady by nature and at moderate to high Reynolds numbers. In this study, first a benchmark case (transient Poiseuille flow) is simulated to evaluate the ability of SPH to simulate internal transient flows at low and moderate Reynolds numbers (Re = 0.05, 500 and 1500). For this benchmark case, the performance of the two most commonly used formulations for viscous term modeling is investigated, as well as the effect of using the XSPH variant. Some points regarding using the symmetric form for pressure gradient modeling are also briefly discussed. Then, the application of SPH is extended to oscillating flows imposed by oscillating body force (Womersley type flow) and oscillating moving boundary (Stokes’ second problem) at different frequencies and amplitudes. There is a very good agreement between SPH results and exact solution even if there is a large phase lag between the oscillating pressure difference and moving boundary and the movement of the SPH particles generated. Finally, a modified formulation for wall shear stress calculations is suggested and verified against exact solutions. In all presented cases, the spatial convergence analysis is performed.     

A Roe-type scheme with low Mach number preconditioning for a two-phase compressible flow model with pressure relaxation

We describe two-phase compressible flows by a hyperbolic six-equation single-velocity two-phase flow model with stiff mechanical relaxation. In particular, we are interested in the simulation of liquid-gas mixtures such as cavitating flows. The model equations are numerically approximated via a fractional step algorithm, which alternates between the solution of the homogeneous hyperbolic portion of the system through Godunov-type finite volume schemes, and the solution of a system of ordinary differential equations that takes into account the pressure relaxation terms. When used in this algorithm, classical schemes such as Roe’s or HLLC prove to be very efficient to simulate the dynamics of transonic and supersonic flows. Unfortunately, these methods suffer from the well known difficulties of loss of accuracy and efficiency for low Mach number regimes encountered by upwind finite volume discretizations. This issue is particularly critical for liquid-gasmixtures due to the large and rapid variation in the flow of the acoustic impedance. To cure the problem of loss of accuracy at low Mach number, in this work we apply to our original Roe-type scheme for the two-phase flow model the Turkel’s preconditioning technique studied by Guillard–Viozat [Computers & Fluids, 28, 1999] for the Roe’s scheme for the classical Euler equations.We present numerical results for a two-dimensional liquid-gas channel flow test that show the effectiveness of the resulting Roe-Turkel method for the two-phase system.     

On a fully nonlinear degenerate parabolic system modeling immiscible gas–water displacement in porous media

In this paper, we are interested in the simultaneous flow of two immiscible fluid phases within a porous medium. We consider a two-phase flow model where the fluids are immiscible and there is no mass transfer between the phases. The medium is saturated by compressible/incompressible phase flows. We study the gas–water displacement without simplified assumptions on the state law of gas density. We establish an existence result for the nonlinear degenerate parabolic system based on new energy estimate on pressures.     

NEW METHOD FOR IMPROVED CALCULATIONS OF UNSTEADY COMPLEX FLOWS IN LARGE ARTERIES

Using an improved computational fluid dynamics (CFD) method developed for highly unsteady three-dimensional flows, numerical simulations for oscillating flow cycles and detailed unsteady simulations of the flow and forces on the aortic vessels at the iliac bifurcation, for both healthy and diseased patients, are analyzed. Improvements in computational efficiency and acceleration in convergence are achieved by calculating both an unsteady pressure gradient which is due to fluid acceleration and a good global pressure field correction based on mass flow for the pressure Poisson equation. Applications of the enhanced method to oscillatory flow in curved pipes yield an order of magnitude increase in speed and efficiency, thus allowing the study of more complex flow problems such as flow through the mammalian abdominal aorta at the iliac arteries bifurcation. To analyze the large forces which can exist on stent graft of patients with abdominal aortic aneurysm (AAA) disease, a complete derivation of the force equations is presented. The accelerated numerical algorithm and the force equations derived are used to calculate flow and forces for two individuals whose geometry is obtained from CT data and whose respective blood pressure measurements are obtained experimentally. Although the use of endovascular stent grafts in diseased patients can alter vessel geometries, the physical characteristics of stents are still very different when compared to native blood vessels of healthy subjects. The geometry for the AAA stent graph patient studied in this investigation induced flows that resulted in large forces that are primarily caused by the blood pressure. These forces are also directly related to the flow cross-sectional area and the angle of the iliac arteries relative to the main descending aorta. Furthermore, the fluid flow is significantly disturbed in the diseased patient with large flow recirculation and stagnant regions which are not present for healthy subjects.     

Numerical Investigation of Cavitation Interacting with Pressure Wave

A computational fluid dynamics solver based on homogeneous cavitation model is employed to compute the two-phase cavitating flow. The model treats the two-phase regime as the homogeneous mixture of liquid and vapour which are locally assumed to be under both kinetic and thermodynamic equilibrium. As our focus is on pressure wave formation, propagation and its impact on cavitation bubble, the compressibility effects of liquid water have to be accounted for and hence the flow is considered to be compressible. The cavitating flow disturbed by the introduced pressure wave is simulated to investigate the unsteady features of cavitation due to the external perturbations. It is observed that the cavity becomes unstable, locally experiencing deformation or collapse, which depends on the shock wave intensity and freestream flow speed.     

Two‐phase flows in karstic geometry

Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil, and gas), and cloud and fog (water vapor, water, and air). Multiphase flows also play an important role in many engineering and environmental science applications. In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits, and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. Despite the importance of the subject, little work has been performed on multiphase flows in karstic geometry. In this paper, we present a family of phase–field (diffusive interface) models for two‐phase flow in karstic geometry. These models together with the associated interface boundary conditions are derived utilizing Onsager's extremum principle. The models derived enjoy physically important energy laws. A uniquely solvable numerical scheme that preserves the associated energy law is presented as well. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Voitkunskii Amfilokhiev Pavlovskii Model of Motion of Aqueous Polymer Solutions

We study the mathematical properties of the model of motion of aqueous polymer solutions (Voitkunskii, Amfilokhiev, Pavlovskii, 1970) and its modifications in the limiting case of small relaxation times (Pavlovskii, 1971). In both cases, we examine plane unsteady laminar flows. In the first case, the properties of the flows are similar to those of the flow of an ordinary viscous fluid. In the second case, there may exist weak discontinuities that are preserved during the motion. We also address the steady flow problem for a dilute aqueous polymer solution moving in a cylindrical tube under a longitudinal pressure gradient. In this case, a flow with rectilinear trajectories (an analog of the classical Poiseuille flow) is possible. However, in contrast to the latter, the pressure in this flow depends on all three spatial variables.     

A fast non-iterative numerical algorithm to predict unsteady partial cavitation on hydrofoils

A new algorithm to predict partial sheet cavity behavior on hydrofoils is proposed. The proposed algorithm models the unsteady partial cavitation using Boundary Element Method (BEM). In the proposed method the spatial iterative scheme is removed by means of a new approach determining the instantaneous cavity length. This iterative scheme is required in conventional algorithms to obtain the cavity length at each time step. Performance of the new algorithm for various unsteady cavitating flows with different reduced frequencies, cavitation numbers, hydrofoil geometries and inflow conditions are investigated. Comparison between the obtained results using the proposed method and those of conventional ones indicates that the present algorithm works well with sufficient accuracy. Moreover, it is shown that the proposed method is computationally more efficient than the conventional one for unsteady sheet cavitation analysis on hydrofoils.     

Chaotic mixing simulations

Numerical simulations of high Reynolds number flows in the unit driven cavity have been performed. The system is shown to become unsteady at Re=8125 and chaotic at Re=17,000. In between this range the system switches between periodic and quasi-periodic states with step-wise changes in period. A passive concentration field and passive tracer particles are introduced into the flow at its asymptotic state to show the effects of chaos on mixing.     

带激波的一维非定常两相流动的数值模拟

本文将处理带激波的单相气体非定常流动问题的两种高分辨数值方法(随机取样法和二阶GRP有限差分法)推广应用于气固悬浮体(亦称含灰气体)两相情况,计算了含灰气体激波管中两相激波特性、波后流场结构及气固两相流动参数随时间的变化.数值结果表明:这两种方法均能给出带有尖锐间断阵面的两相激波松弛结构.二阶GRP方法在计算精度和机时耗用等方面优于随机取样法.     

Modeling the Effect of Supply Pressure Fluctuations on Cavitation Processes in Injection Nozzles

In this study cavitating flows in an injection nozzle under time‐depended inlet pressure are calculated using a dispersed bubble dynamics model. The calculations confirm that a transient rise of the nozzle inlet pressure forces cavitation to recede and a sharp decrease of the injection pressure enhances cavitation.     

Phase Field Approach to Multiphase Flow Modeling

We review the phase field (otherwise called diffuse interface) model for fluid flows, where all quantities, such as density and composition, are assumed to vary continuously in space. This approach is the natural extension of van der Waals?? theory of critical phenomena both for one-component, two-phase fluids and for partially miscible liquid mixtures. The equations of motion are derived, assuming a simple expression for the pairwise interaction potential. In particular, we see that a non-equilibrium, reversible body force appears in the Navier-Stokes equation, that is proportional to the gradient of the generalized chemical potential. This, so called Korteweg, force is responsible for the convection that is observed in otherwise quiescent systems during phase change. In addition, in binary mixtures, the diffusive flux is modeled using a Cahn-Hilliard constitutive law with a composition-dependent diffusivity, showing that it reduces to Fick??s law in the dilute limit case. Finally, the results of several numerical simulations are described, modeling, in particular, a) mixing, b) spinodal decomposition, c) nucleation, d) enhanced heat transport, e) liquid-vapor phase separation.     

Flow over a profile in a channel with dynamical effects

The work deals with a numerical solution of 2D inviscid incompressible flow over the profile NACA 0012 in a channel. The finite volume method in a form of cell‐centered scheme at quadrilateral C‐mesh is used. Governing system of equations is the system of Euler equations. Numerical results are partially compared with experimental data. Steady state solutions of the flow as well unsteady flows caused by prescribed oscillation of the profile were computed. The method of artificial compressibility and the time dependent method are used for computation of the steady state solution. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

浅水流动与污染物扩散的高分辨率计算模型

将组合型TVD格式应用于守恒型的浅水方程和污染物扩散方程，建立了二者耦合求解的高分辨率有限体积计算模型。给出了溃坝水流、明渠突扩流和污染物输运计算的典型算例，并与实验数据或其它数值结果进行了比较，证实了该模型的有效性，表明它不但能处理有激波的非恒定流问题，也能较好地计算具有任意边界的一般的浅水流动和污染物扩散问题，为浅水流动和水环境模拟提供了精度高、稳定性好、普适性强的数值方法。     

Novel energy stable schemes for Swift-Hohenberg model with quadratic-cubic nonlinearity based on the H−1-gradient flow approach

The Swift-Hohenberg model is a very important phase field crystal model which can be described many crystal phenomena. This model with quadratic-cubic nonlinearity based on the H^??1-gradient flow approach is a sixth-order system which satisfies mass conservation and energy dissipation law. The negative energy of this model will bring huge difficulties to energy stability for many existing approaches. In this paper, we consider two linear, second-order and unconditionally energy stable schemes by linear invariant energy quadratization (LIEQ) and modified scalar auxiliary variable (MSAV) approaches. These two schemes will be effective for all negative E₁. Furthermore, we proved that all the semi-discrete schemes are unconditionally energy stable with respect to a modified energy. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.