Simulation of viscous flows with undulatory boundaries: Part II. Coupling with other solvers for two-fluid computations

We extend the direct numerical simulation (DNS) capability developed in [D. Yang, L. Shen, Simulation of viscous flows with undulatory boundaries: Part I. Basic solver, J. Comput. Phys. (submitted for publication) ] to the simulation of two-fluid interaction with deformable interface. Two approaches are used to couple the DNS of one fluid with the simulation of another fluid. In the first, the DNS is coupled with a potential-flow based wave solver that uses a high-order spectral (HOS) method. This coupled method is applied to simulate the interaction of turbulent wind with surface waves, including single wave train and broadband wavefield. Validation with previous theoretical and experimental studies shows the accuracy and efficiency of this coupled DNS-HOS method for capturing the essential physics of wind–wave interaction. In the second approach, both of the fluids are simulated by the DNS and are coupled by an efficient iterative scheme, in which the continuity of velocity and the balance of stress are enforced at the interface. The performance of this coupled DNS–DNS method is demonstrated and validated by several test cases including: interfacial wave between two viscous fluids, water surface wave over highly viscous mud flow with interfacial wave, and interaction of two-phase vortex pairs with a deformable interface. Comparison with existing theoretical and numerical results confirms the accuracy of this coupled DNS–DNS method. Finally, this method is applied to study the interaction of air and water turbulence. The nonlinear development of interfacial wave by the excitation of the air and water turbulence, and the wave effect on the instantaneous and statistical characteristics of the turbulence are elucidated.     

Sharp interface immersed-boundary/level-set method for wave–body interactions

A sharp interface Cartesian grid method for the large-eddy simulation of two-phase turbulent flows interacting with moving bodies is presented. The overall approach uses a sharp interface immersed boundary formulation and a level-set/ghost–fluid method for solid–fluid and fluid–fluid interface treatments, respectively. A four-step fractional-step method is used for velocity–pressure coupling, and a Lagrangian dynamic Smagorinsky subgrid-scale model is adopted for large-eddy simulations. A simple contact angle boundary condition treatment that conforms to the immersed boundary formulation is developed. A variety of test cases of different scales ranging from bubble dynamics, water entry and exit, landslide-generated waves, to ship hydrodynamics are performed for validation. Extensions for high Reynolds number ship flows using wall-layer models are also considered.     

A kind of extended Korteweg——de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

This paper considers interfacial waves propagating along the interface between a two-dimensional two-fluid with a flat bottom and a rigid upper boundary. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. It just focuses on the weakly non-linear small amplitude waves by introducing two small independent parameters: the nonlinearity ratio $\varepsilon $, represented by the ratio of amplitude to depth, and the dispersion ratio $\mu $, represented by the square of the ratio of depth to wave length, which quantify the relative importance of nonlinearity and dispersion. It derives an extended KdV equation of the interfacial waves using the method adopted by Dullin {\it et al} in the study of the surface waves when considering the order up to $O(\mu ^2)$. As expected, the equation derived from the present work includes, as special cases, those obtained by Dullin {\it et al} for surface waves when the surface tension is neglected. The equation derived using an alternative method here is the same as the equation presented by Choi and Camassa. Also it solves the equation by borrowing the method presented by Marchant used for surface waves, and obtains its asymptotic solitary wave solutions when the weakly nonlinear and weakly dispersive terms are balanced in the extended KdV equation.     

An interaction potential based lattice Boltzmann method with adaptive mesh refinement (AMR) for two-phase flow simulation

The lattice Boltzmann method (LBM) for two-phase flow simulation is often hindered by insufficient resolution at the interface. As a result, the LBM simulation of bubbles in bubbling flows is commonly limited to spherical or slightly deformed bubble shapes. In this study, the adaptive mesh refinement method for the LBM is developed to overcome such a problem. The approach for this new method is based on the improved interaction potential model, which is able to maintain grid-independent fluid properties in the two-fluid phases and at the interface. The LBM–AMR algorithm is described, especially concerning the LBM operation on a non-uniform mesh and the improved interaction potential model. Numerical simulations have been performed to validate the method in both single phase and multiphase flows. The 2D and 3D simulations of the buoyant rise of bubbles are conducted under various conditions. The agreement between the simulated bubble shape and velocity with experiments illustrates the capability of the LBM–AMR approach in predicting bubble dynamics even under the large bubble deformation conditions. Further, the LBM–AMR technique is capable of simulating a complex topology change of the interface. Integration of LBM with AMR can significantly improve the accuracy and reduce computation cost. The method developed in this study may appreciably enhance the capability of LBM in the simulation of complex multiphase flows under realistic conditions.     

Pulsed Gradient Spin Echo Nuclear Magnetic Resonance Measurement and Simulation of Two-Fluid Taylor Vortex Flow in a Vertically Oriented Taylor–Couette Device

Magnetic resonance microscopy and Ansys Fluent^? computational fluid dynamics simulation have been used to classify Taylor vortex flows (TVF) for several single fluid and axially stratified two-fluid systems in a vertically oriented Taylor–Couette device. A Rheo-NMR (nuclear magnetic resonance) Couette system (Magritek Ltd, New Zealand) with a 1.05-mm gap was used to evaluate the transition from Couette flow to TVF in 1.65 cSt silicone oil, 1 cSt deionized water, and 0.65 cSt silicone oil. The rotation rate at which instability onset occurred agreed between experiment and simulation, as did the critical wavelength. Velocities were mapped for axially stratified two-fluid systems. The vortex containing the two-fluid interface was found to form with a significantly longer wavelength than that observed in the pure fluids. For experiments and simulations in the TVF regime, a region with no secondary flows was found at the interface, indicating interface stabilization by surface tension forces.     

Calculations of two-fluid magnetohydrodynamic axisymmetric steady-states

M3D-C¹$C^1$ is an implicit, high-order finite element code for the solution of the time-dependent nonlinear two-fluid magnetohydrodynamic equations [S.C. Jardin, J. Breslau, N. Ferraro, A high-order implicit finite element method for integrating the two-fluid magnetohydrodynamic equations in two dimensions, J. Comp. Phys. 226 (2) (2007) 2146–2174]. This code has now been extended to allow computations in toroidal geometry. Improvements to the spatial integration and time-stepping algorithms are discussed. Steady-states of a resistive two-fluid model, self-consistently including flows, anisotropic viscosity (including gyroviscosity) and heat flux, are calculated for diverted plasmas in geometries typical of the National Spherical Torus Experiment (NSTX) [M. Ono et al., Exploration of spherical torus physics in the NSTX device, Nucl. Fusion 40 (3Y) (2000) 557–561]. These states are found by time-integrating the dynamical equations until the steady-state is reached, and are therefore stationary or statistically steady on both magnetohydrodynamic and transport time-scales. Resistively driven cross-surface flows are found to be in close agreement with Pfirsch-Schlüter theory. Poloidally varying toroidal flows are in agreement with comparable calculations [A.Y. Aydemir, Shear flows at the tokamak edge and their interaction with edge-localized modes, Phys. Plasmas 14]. New effects on core toroidal rotation due to gyroviscosity and a local particle source are observed.     

Exact vortex solitons in a quasi-two-dimensional Bose–Einstein condensate with spatially inhomogeneous cubic–quintic nonlinearity

The exact vortex soliton solutions of the quasi-two-dimensional cubic–quintic Gross–Pitaevskii equation with spatially inhomogeneous nonlinearities are constructed by similarity transformation. It is demonstrated that spatially inhomogeneous cubic–quintic nonlinearity can support exact vortex solitons in which there are two quantum numbers S and m. The radius structures and density distributions of these vortex solitons are studied, and it is shown that the number of ring structure of the vortex solitons increases by one with increasing the “radial quantum number” m by one.     

Level Set方法及其在两相流数值模拟研究中的应用

本文介绍了一种崭新的捕捉两相流相界面的 Ｌｅｖｅｌ Ｓｅｔ方法,并应用该方法对两种互不相容流体间的平面界面波动现象进行了数值模拟,捕捉到了两相流界面波动过程中的一些复杂现象。计算结果与实验观测现象符合良好,表明这种方法是一种很有前途的两相流数值模拟方法,很值得进一步研究和推广。     

Effect of Acoustic Field on the Structure of a Round Minijet

Presented are results of visual studying the structure of a round minijet flowing into the atmosphere exposed to an acoustic field. The studies were performed with the laminar jet flow. According to the photo and video recording of the flow pattern we revealed characteristic features of the jet structure in the acoustic field. Characteristic vortex structures and zones with intensive turbulent mixing were detected in the flow.We revealed the process of formation of vortex structures in a laminar jet under the action of the acoustic field, vibrational and rotational jet flows at the outlet of the pipe 1.35 mm in diameter. The present study is a continuation of the research on a minijet structure in an acoustic field [13].     

A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks

We present a multiscale model for numerical simulations of dynamics of crystalline solids. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for multiscale modeling – the heterogeneous multiscale method (HMM). We derive an explicit coupling condition at the atomistic/continuum interface. Application to the dynamics of brittle cracks under various loading conditions is presented as test examples.     

Topology optimization of unsteady incompressible Navier–Stokes flows

This paper discusses the topology optimization of unsteady incompressible Navier–Stokes flows. An optimization problem is formulated by adding the artificial Darcy frictional force into the incompressible Navier–Stokes equations. The optimization procedure is implemented using the continuous adjoint method and the finite element method. The effects of dynamic inflow, Reynolds number and target flux on specified boundaries for the optimal topology of unsteady Navier–Stokes flows are presented. Numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady Navier–Stokes flows.     

The Local Front Reconstruction Method for direct simulation of two- and three-dimensional multiphase flows

We present a new interface reconstruction technique, the Local Front Reconstruction Method (LFRM), for incompressible multiphase flows. This new method falls in the category of Front Tracking methods but it shares automatic topology handling characteristics of the previously proposed Level Contour Reconstruction Method (LCRM). The LFRM tracks the phase interface explicitly as in Front Tracking but there is no logical connectivity between interface elements thus greatly easing the algorithmic complexity. Topological changes such as interfacial merging or pinch off are dealt with automatically and naturally as in the Level Contour Reconstruction Method. Here the method is described for both two- and three-dimensional flow geometries. The interfacial reconstruction technique in the LFRM differs from that in the LCRM formulation by foregoing using an Eulerian distance field function. Instead, the LFRM uses information from the original interface elements directly to generate the new interface in a mass conservative way thus showing significantly improved local mass conservation. Because the reconstruction procedure is independently carried out in each individual reconstruction cell after an initial localization process, an adaptive reconstruction procedure can be easily implemented to increase the accuracy while at the same time significantly decreasing the computational time required to perform the reconstruction. Several benchmarking tests are performed to validate the improved accuracy and computational efficiency as compared to the LCRM. The results demonstrate superior performance of the LFRM in maintaining detailed interfacial shapes and good local mass conservation especially when using low-resolution Eulerian grids.     

A method to simulate linear stability of impulsively accelerated density interfaces in ideal-MHD and gas dynamics

We present a numerical method to solve the linear stability of impulsively accelerated density interfaces in two dimensions such as those arising in the Richtmyer–Meshkov instability. The method uses an Eulerian approach, and is based on an upwind method to compute the temporally evolving base state and a flux vector splitting method for the perturbations. The method is applicable to either gas dynamics or magnetohydrodynamics. Numerical examples are presented for cases in which a hydrodynamic shock interacts with a single or double density interface, and a doubly shocked single density interface. Convergence tests show that the method is spatially second-order accurate for smooth flows, and between first and second-order accurate for flows with shocks.     

Nonlinear interactions between drift waves and zonal flows

Phase coherent interactions between drift waves and zonal flows are considered. For this purpose, mode coupling equations are derived by using a two-fluid model and the guiding center drifts. The equations are then Fourier analyzed to deduce the nonlinear dispersion relations. The latter depict the excitation of zonal flows due to the ponderomotive forces of drift waves. The flute-like zonal flows with insignificant density fluctuations have faster growth rates than those which have a finite wavelength along the magnetic field direction. The relevance of our investigation to drift wave driven zonal flows in computer simulations and laboratory plasmas is discussed. Received 5 April 2002 Published online 28 June 2002     

A direct-forcing embedded-boundary method with adaptive mesh refinement for fluid–structure interaction problems

In the present work we developed a structured adaptive mesh refinement (S-AMR) strategy for fluid–structure interaction problems in laminar and turbulent incompressible flows. The computational grid consists of a number of nested grid blocks at different refinement levels. The coarsest grid blocks always cover the entire computational domain, and local refinement is achieved by the bisection of selected blocks in every coordinate direction. The grid topology and data-structure is managed using the Paramesh toolkit. The filtered Navier–Stokes equations for incompressible flow are advanced in time using an explicit second-order projection scheme, where all spatial derivatives are approximated using second-order central differences on a staggered grid. For transitional and turbulent flow regimes the large-eddy simulation (LES) approach is used, where special attention is paid on the discontinuities introduced by the local refinement. For all the fluid–structure interaction problems reported in this study the complete set of equations governing the dynamics of the flow and the structure are simultaneously advanced in time using a predictor–corrector strategy. An embedded-boundary method is utilized to enforce the boundary conditions on a complex moving body which is not aligned with the grid lines. Several examples of increasing complexity are given to demonstrate the robustness and accuracy of the proposed formulation.     

Nonlinear stage of the Benjamin-Feir instability: three-dimensional coherent structures and rogue waves

A specific, genuinely three-dimensional mechanism of rogue wave formation, in a late stage of the modulational instability of a perturbed Stokes deep-water wave, is recognized through numerical experiments. The simulations are based on fully nonlinear equations describing weakly three-dimensional potential flows of an ideal fluid with a free surface in terms of conformal variables. Spontaneous formation of zigzag patterns for wave amplitude is observed in a nonlinear stage of the instability. If initial wave steepness is sufficiently high (ka>0.06), these coherent structures produce rogue waves. The most tall waves appear in turns of the zigzags. For ka<0.06, the structures decay typically without formation of steep waves.     

Simulation of viscous flows with undulatory boundaries. Part I: Basic solver

A numerical method for the simulation of viscous flows with undulatory walls and free surfaces is presented. The simulation domain is discretized by a boundary-fitted and time-dependent grid. The Navier–Stokes equations, subject to fully nonlinear kinematic and dynamic boundary conditions at the free surface and no-slip boundary condition at the wall, are simulated by a hybrid pseudo-spectral and finite difference method in space and a semi-implicit fractional-step method in time. The performance of the method is demonstrated by a series of test cases including flows over wavy boundaries, various surface waves, and interaction between vortices and free surfaces. Validation by convergence test and extensive comparisons with previous theoretical, experimental, and numerical studies indicate the accuracy and efficiency of the method. Finally, a simulation example of turbulence and free surface interaction is presented. Results show that the rich features of the free surface such as surface waves, splats, anti-splats, dimples, and scars are captured accurately. Characteristic vortical structures and variation of turbulence statistics in the near-surface region are also elucidated.     

激波冲击V形界面重气体导致的壁面与旋涡作用及其对湍流混合的影响

基于多组分混合物质量分数模型,采用色散最小耗散可控的高分辨率有限体积方法,数值模拟了弱激波冲击V形空气/SF_6界面后,界面不稳定性生成的旋涡与固体壁面作用问题.激波冲击V形界面之后,因斜压效应诱导涡量沉积在界面附近,形成沿界面规则排列的多个涡对结构.旋涡的诱导作用使界面不断变形和卷起,同时旋涡之间不断发生相互并对,诱导更多更小尺度的旋涡产生.旋涡诱导作用的叠加效应,使界面尖端处的初始涡对向上下壁面发展.随后,涡结构开始与壁面发生复杂的相互作用.旋涡与壁面作用后沿壁面加速,使得物质界面沿壁面伸展,随后,旋涡从壁面回弹,并诱导二次旋涡产生.旋涡与壁面相互作用的过程,能够明显加剧物质混合.本文从物质混合的角度研究了该过程的机理,分析了旋涡与壁面作用对物质混合的影响.     

Nested Cartesian grid method in incompressible viscous fluid flow

In this work, the local grid refinement procedure is focused by using a nested Cartesian grid formulation. The method is developed for simulating unsteady viscous incompressible flows with complex immersed boundaries. A finite-volume formulation based on globally second-order accurate central-difference schemes is adopted here in conjunction with a two-step fractional-step procedure. The key aspects that needed to be considered in developing such a nested grid solver are proper imposition of interface conditions on the nested-block boundaries, and accurate discretization of the governing equations in cells that are with block-interface as a control-surface. The interpolation procedure adopted in the study allows systematic development of a discretization scheme that preserves global second-order spatial accuracy of the underlying solver, and as a result high efficiency/accuracy nested grid discretization method is developed. Herein the proposed nested grid method has been widely tested through effective simulation of four different classes of unsteady incompressible viscous flows, thereby demonstrating its performance in the solution of various complex flow–structure interactions. The numerical examples include a lid-driven cavity flow and Pearson vortex problems, flow past a circular cylinder symmetrically installed in a channel, flow past an elliptic cylinder at an angle of attack, and flow past two tandem circular cylinders of unequal diameters. For the numerical simulations of flows past bluff bodies an immersed boundary (IB) method has been implemented in which the solid object is represented by a distributed body force in the Navier–Stokes equations. The main advantages of the implemented immersed boundary method are that the simulations could be performed on a regular Cartesian grid and applied to multiple nested-block (Cartesian) structured grids without any difficulty. Through the numerical experiments the strength of the solver in effectively/accurately simulating various complex flows past different forms of immersed boundaries is extensively demonstrated, in which the nested Cartesian grid method was suitably combined together with the fractional-step algorithm to speed up the solution procedure.     

A High-Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows

We present a numerical method for computing solutions of the incompressible Euler or Navier–Stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties. The method is based on a second-order projection method for variable density flows using an “approximate projection” formulation. The boundary between the fluids is tracked with a second-order, volume-of-fluid interface tracking algorithm. We present results for viscious Rayleigh–Taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm. We also present computational results for the Rayleigh–Taylor instability in air-helium and for bubbles and drops in an air–water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts.