Numerical investigation of micro shock waves generation

The shock wave propagation in the micro channel of the different sizes is studied numerically in order to estimate the possibility of the experimental apparatus development. The full compressible Navier–Stokes equations are used for the numerical simulation. The shock wave velocity attenuation is found for the channel height smaller than H = 200 μm. The influence of the channel size and of the diaphragm pressure ratio on the shock wave velocity is considered. The considerable influence of the viscous effects on the shock propagation is shown.

Numerical simulation of the viscous shock tube problem by using a high resolution monotonicity-preserving scheme

This work deals with the flow generated in a shock tube after the shock wave has reflected at the end wall. For a viscous fluid, a complex unsteady interaction takes place between the incident boundary layer and reflected shock wave. The numerical simulation of this complex flow requires both robust and accurate numerical schemes. In this work, we rely on the one-step high-order scheme recently proposed by Daru and Tenaud [Daru V, Tenaud C. High order one-step monotonicity preserving schemes for unsteady flow calculations. J Comput Phys 2004;193:563-94]. With this scheme, converged results are obtained for Reynolds numbers in the range 200-1000. The interaction mechanisms are carefully analyzed as well as the flow dynamics.     

Experimental and numerical study of the recirculation flow inside a liquid meniscus focused by air

The internal motions inside a liquid meniscus in the so-called liquid cone-jet mode, which can occur upon stimulation by a coflowing gas sheath in flow focusing, are explored by both numerical simulation and experimental visualization. The results for low viscosity liquids show that, as in previous numerical simulations, a recirculating cell inside the meniscus appears when the injected liquid flow rate is reduced. Thus, as the flow rate is reduced not only the average residence time of particles in the meniscus becomes longer, but the appearance of a recirculation cell provides a natural platform for the efficient micro-mixing of different species before they are ejected through the issuing jet. The numerical results were confirmed with experimental visualization of the flow inside the meniscus using a dyed liquid. However, when the viscosity of the liquid is increased the recirculating cell disappears. In this case, viscous stresses organize the streamlines and direct the flow to the meniscus tip, which prevents the recirculating cell from being formed even for very small injected rate of liquid flow.     

Concentration-dependent viscous mixing in microfluidics: modelings and experiments

In microfluidic mixing, great attention has been devoted to the structural design to enhance mixing efficiency. However, the influence of the variant viscosity in the mixing process is rarely discussed due to the practical challenges originated from the strong and complex couplings between species concentration and other fluid properties such as density, viscosity and diffusion coefficient. In this work, a group of coupling relationships among concentration, density, viscosity, as well as diffusion coefficient are introduced to accurately simulate the mixing process with a viscous flow involved. Compared with the traditional linear approximation, the new approach is more suitable to simulate the concentration-dependent viscous mixing in microfluidics. Furthermore, a planar passive micromixer is designed to validate the coupling approach from both modeling and experiment perspectives. By comparing experimental and numerical results, it turns out that the coupling approach achieves higher accuracy than the traditional linear approximation. In addition, four derived models are experimentally tested and numerically simulated by adopting the new method. The results of each model reach a good agreement between modeling and experiment.     

An Eulerian-Lagrangian model for dense particle clouds

The interaction of a shock wave with a layer of particles, a topic important for the formation of dust clouds in air, has been analyzed by numerical simulation. The modelling method used are the Eulerian-Lagrangian and the Eulerian-Eulerian approaches, the latter is traditionally used for this type of simulation. Using Eulerian-Lagrangian simulation, it has been possible to include the effects of particle-particle and particle-wall collisions in a realistic and direct manner. Results are mainly shown as snap-shots of particle positions during the simulations and statistics for the particle displacement and collisions. The results show that collisions influence the process of particle cloud formation significantly.     

Shock-Vortex Interactions at High Mach Numbers

We perform a computational study of the interaction of a planar shock wave with a cylindrical vortex. We use a particularly robust High Resolution Shock Capturing scheme, Marquina's scheme, to obtain high quality, high resolution numerical simulations of the interaction. In the case of a very-strong shock/vortex encounter, we observe a severe reorganization of the flow field in the downstream region, which seems to be due mainly to the strength of the shock. The numerical data is analyzed to study the driving mechanisms for the production of vorticity in the interaction.     

A numerical method for a kinetic equation and its application to propagating shock waves

An efficient numerical method for kinetic equations and its application to analyses of moving shock wave problems are presented. The present study aims to give an efficient scheme for two-dimensional unsteady gas flows. An explicit MacCormack difference method is applied to solve a BGK-model equation. The efficiency and accuracy of the scheme are examined in an application to one-dimensional shock structure problems. Furthermore, the scheme is applied to a two-dimensional flow problem: nonstationary reflection of a shock wave at a wedge. The present scheme is found to be useful and efficient for the analyses of two-dimensional unsteady rarefied gas flows.     

Computational study of an incident shock wave into a Helmholtz resonator

The behavior of an incident shock wave into a Helmholtz resonator is very important from the acoustical point of view as well as the fundamental researches of shock wave dynamics. When a shock wave propagates into a Helmholtz resonator, complicated wave phenomena are formed both inside and outside the resonator. Shock wave reflections, shock wave focusing phenomena, and shock–vortex interactions cause strong pressure fluctuations inside the resonator, consequently leading to powerful sound emission. The wave phenomena inside the resonator are influenced by detailed configuration of the resonator. It is well known that the gas inside the resonator strongly oscillates at a resonance frequency, as the incident wavelength is larger, compared with the geometrical length scale of the resonator, but there are only a few works regarding a shock wave that has an extremely short wavelength. Meanwhile, the discharge process of the incident shock wave from the resonator is another interest with regard to an impulse wave generation that is a source of serious noise and vibration problems of the resonator. In the present study, the wave phenomena inside and outside the Helmholtz resonator are, in detail, investigated with a help of a computational fluid dynamics method. The incident shock Mach number is varied below 2.0, and many different types of the resonators are explored to investigate the influence of the resonator geometry on the wave phenomena. A total variation diminishing (TVD) scheme is employed to solve two-dimensional, unsteady, compressible Euler equations. The computational results are compared with existing experimental data to ensure that the present computations are valid to predict the resonator wave phenomena. Based upon the results obtained, the shock wave focusing and discharge processes, which are important in determining the resonator flow characteristics, are discussed in detail.     

Starting jet flows in a three-dimensional channel with larynx-shaped constriction

A numerical model for the three-dimensional starting jet flow in a channel with a static larynx-shaped constriction is presented. Detailed resolution of this kind of jet flow is necessary in order to understand the complex coupling between flow and acoustics in the process of human phonation. The numerical model is based on the equation of continuity and the Navier–Stokes equations. The investigations are done with the open source CFD package OpenFOAM. Numerical simulations are performed for a square-sectioned channel geometry, which is constricted with a fixed shape conforming to the fully opened human glottis. Time-dependent inflow boundary conditions are applied in order to model transient glottal flow rates. The setup of the numerical simulations corresponds to the configuration of a model experiment in order to allow detailed validation. The numerical results are in good agreement with the experimental data, when the near-wall region in the glottal gap is adequately resolved by the numerical grid. The results illustrate the complex interactions between the jet flow and the surrounding vortices.     

Accelerated numerical simulations of a heaving floating body by coupling a motion solver with a two-phase fluid solver

This paper presents a study on the coupling between a fluid solver and a motion solver to perform fluid–structure interaction (FSI) simulations of floating bodies such as point absorber wave energy converters heaving under wave loading. The two-phase fluid solver with dynamic mesh handling, interDyMFoam, is a part of the Computational Fluid Dynamics (CFD) toolbox OpenFOAM. The incompressible Navier–Stokes (NS) equations are solved together with a conservation equation for the Volume of Fluid (VoF). The motion solver is computing the kinematic body motion induced by the fluid flow. A coupling algorithm is needed between the fluid solver and the motion solver to obtain a converged solution between the hydrodynamic flow field around and the kinematic motion of the body during each time step in the transient simulation. For body geometries with a significant added mass effect, simple coupling algorithms show slow convergence or even instabilities. In this paper, we identify the mechanism for the numerical instability and we derive an accelerated coupling algorithm (based on a Jacobian) to enhance the convergence speed between the fluid and motion solver. Secondly, we illustrate the coupling algorithm by presenting a free decay test of a heaving wave energy converter. Thirdly and most challenging, a water impact test of a free falling wedge with a significant added mass effect is successfully simulated. For both test cases, the numerical results obtained by using the accelerated coupling algorithm are in a very good agreement with the experimental measurements.     

A numerical investigation of the stability of expanding liquid sheets and the influence of boundary conditions

Incompressible flow solutions are found numerically for a radially expanding liquid sheet in order to confirm analytical results for inviscid flow and to investigate viscous and non-linear effects. An hp-finite element method is used to perform the numerical simulations. In our unsteady simulations, we observe that forced sinuous pulses cause two different speed waves to travel downstream for Weber numbers greater than one. We also witness wave deceleration for Weber numbers approaching one, confirming the predictions of inviscid linear stability analysis. Comparisons are also made to theoretical predictions of the radius where the sheet becomes unstable. To determine the critical radius, the inlet Weber number is reduced until the theoretical critical radius is within the simulated domain. Surprisingly, instead of leading to breakup, this causes the sheet to change from a stable symmetric shape to a stable asymmetric shape. The transition between these shapes occurs by both supercritical and subcritical bifurcations when the Weber number based on the sheet thickness approaches one, in agreement with the theoretical work of Taylor. The absence of breakup in our simulations appears to be a direct result of allowing the interface to span the entire domain. To verify this, we examine the dependence of the solutions on domain aspect ratio, shape, and exit boundary conditions. No boundary conditions are found that allows the sheet to break-up.     

Numerical Methods with Adaptive Artificial Viscosity for Solving Navier−Stokes Equations

A numerical method for solving two-dimensional problems of a viscous compressible gas based on Navier–Stokes equations with the introduction of adaptive artificial viscosity is presented. The proposed method is implemented for areas of the general form on triangular grids. The method of the adaptive artificial viscosity is taken as the basis of the proposed numerical method and ensures the monotonicity of the solutions, even in the presence of shock waves. The artificial viscosity (introduced into the difference scheme) is constructed in such a way that it is absent in the boundary layer where the dynamic viscosity acts. The viscosity is determined from the conditions of the fulfillment of the maximum principle. The series of calculations of an external flow around a cylinder for various Reynolds and Mach numbers is described.     

The influence of the computational mesh on accuracy for initial value problems with discontinuous or nonunique solutions

Discontinuous, or weak, solutions of the wave equation, the inviscid form of Burgers equation, and the tine-dependent, two-dimensional Euler equations are studied. A numerical method of second-order accuracy in two forms, differential and integral, is used to calculate the weak solutions of these equations for several initial value problems, including supersonic flow past a wedge, a double symmetric wedge, and a sphere. The effect of the computational mesh on the accuracy of computed weak solutions including shock waves and expansion phenomena is studied. Modifications to the finite-difference method are presented which aid in obtaining desired solutions for initial value problems in which the solutions are nonunique.     

Experimental and numerical investigation of capillary flow in SU8 and PDMS microchannels with integrated pillars

Microfluidic channels with integrated pillars are fabricated on SU8 and PDMS substrates to understand the capillary flow. Microscope in conjunction with high-speed camera is used to capture the meniscus front movement through these channels for ethanol and isopropyl alcohol, respectively. In parallel, numerical simulations are conducted, using volume of fluid method, to predict the capillary flow through the microchannels with different pillar diameter to height ratio, ranging from 2.19 to 8.75 and pillar diameter to pitch ratio, ranging from 1.44 to 2.6. The pillar size (diameter, pitch and height) and the physical properties of the fluid (surface tension and viscosity) are found to have significant influence on the capillary phenomena in the microchannel. The meniscus displacement is non-uniform due to the presence of pillars and the non-uniformity in meniscus displacement is observed to increase with decrease in pitch to diameter ratio. The surface area to volume ratio is observed to play major roles in the velocity of the capillary meniscus of the devices. The filling speed is observed to change more dramatically under different pillar heights upto 120 μm and the change is slow with further increase in the pillar height. The details pertaining to the fluid distribution (meniscus front shapes) are obtained from the numerical results as well as from experiments. Numerical predictions for meniscus front shapes agree well with the experimental observations for both SU8 and PDMS microchannels. It is observed that the filling time obtained experimentally matches very well with the simulated filling time. The presence of pillars creates uniform meniscus front in the microchannel for both ethanol and isopropyl alcohol. Generalized plots in terms of dimensionless variables are also presented to predict the performance parameters for the design of these microfluidic devices. The flow is observed to have a very low Capillary number, which signifies the relative importance of surface tension to viscous effects in the present study.     

Numerical simulations of shear dependent viscoelastic flows with a combined finite element–finite volume method

A hybrid combined finite element–finite volume method has been developed for the numerical simulation of shear-dependent viscoelastic flow problems governed by a generalized Oldroyd-B model with a non-constant viscosity function. The method is applied to the 4:1 planar contraction benchmark problem, to investigate the influence of the viscosity effects on the flow and results are compared with those found in the literature for creeping Oldroyd-B flows, for a range of Weissenberg numbers. The method is also applied to flow in a smooth stenosed channel. It is shown that the qualitative behavior of the flow is influenced by the rheological properties of the fluid, namely its viscoelastic and inertial effects, as well as the shear-thinning viscosity.These results appear in the framework of a preliminary study of the numerical simulation of steady and pulsatile blood flows in two-dimensional stenotic vessels, using this hybrid finite element–finite volume method.     

Three-dimensional simulations of viscous folding in diverging microchannels

Three-dimensional simulations on the viscous folding in diverging microchannels reported by Cubaud and Mason (Phys Rev Lett 96(11):114,501, 2006a) are performed using the parallel code BLUE for multiphase flows (Shin et al. in A solver for massively parallel direct numerical simulation of three-dimensional multiphase flows. arXiv:1410.8568). The more viscous liquid \(L_1\) is injected into the channel from the center inlet, and the less viscous liquid \(L_2\) from two side inlets. Liquid \(L_1\) takes the form of a thin filament due to hydrodynamic focusing in the long channel that leads to the diverging region. The thread then becomes unstable to a folding instability, due to the longitudinal compressive stress applied to it by the diverging flow of liquid \(L_2\). Given the long computation time, we were limited to a parameter study comprising five simulations in which the flow rate ratio, the viscosity ratio, the Reynolds number, and the shape of the channel were varied relative to a reference model. In our simulations, the cross section of the thread produced by focusing is elliptical rather than circular. The initial folding axis can be either parallel or perpendicular to the narrow dimension of the chamber. In the former case, the folding slowly transforms via twisting to perpendicular folding, or it may remain parallel. The direction of folding onset is determined by the velocity profile and the elliptical shape of the thread cross section in the channel that feeds the diverging part of the cell. Due to the high viscosity contrast and very low Reynolds numbers, direct numerical simulations of this two-phase flow are very challenging and to our knowledge these are the first three-dimensional direct parallel numerical simulations of viscous threads in microchannels. Our simulations provide good qualitative comparison of the early time onset of the folding instability, however, since the computational time for these simulations is quite long, especially for such viscous threads, long-time comparisons with experiments for quantities such as folding amplitude and frequency are limited.     

Classical and variational multiscale LES of the flow around a circular cylinder on unstructured grids

The effects of numerical viscosity, subgrid scale (SGS) viscosity and grid resolution are investigated in LES and VMS-LES simulations of the flow around a circular cylinder at Re=3900 on unstructured grids. The separation between the largest and the smallest resolved scales in the VMS formulation is obtained through a variational projection operator and finite-volume cell agglomeration. Three different non-dynamic eddy-viscosity SGS models are used both in classical and in VMS-LES. The so-called small-small formulation is used in VMS-LES, i.e. the SGS viscosity is computed as a function of the smallest resolved scales. Two different grid resolutions are considered. It is found that, for each considered SGS model, the amount of SGS viscosity introduced in the VMS-LES formulation is significantly lower than in classical LES. This, together with the fact that in the VMS formulation the SGS viscosity only acts on the smallest resolved scales, has a strong impact on the results. However, a significant sensitivity of the results to the considered SGS model remains also in the VMS-LES formulation. Moreover, passing from classical LES to VMS-LES does not systematically lead to an improvement of the quality of the numerical predictions.     

Interaction of a shock wave with a boundary layer in a micro channel

The paper describes a numerical experiment simulating interaction between a shock wave and a boundary layer in a microchannel whose width is comparable with the mean molecular free path. The shock wave front structure and deceleration process are investigated. The simulation is done using the conservative projection method to solve Boltzmann kinetic equation.     

An Auto-Adaptive Multidomain Spectral Technique for Linear Stability Analysis: Application to Viscous Compressible Flows

An auto-adaptive multidomain pseudo-spectral technique is considered in order to solve the linear stability problem of viscous compressible flows. Both the locations of the interfaces and the parameters of the mappings in each subdomain are adapted by minimizing the H ² -norm of the calculated solution. Such method provides automatically—this is the key point—the best polynomial interpolation of the basic state the stability of which is studied. It turns out that the whole procedure is needed to obtain reliable results. The method is first validated against results available in the literature (both viscous incompressible and inviscid compressible Rayleigh–Taylor configurations). The efficiency of the numerical method is illustrated with results on the linear stability of the compressible viscous diffusive Rayleigh–Taylor flow where no analytical or numerical results are available. New results showing the influence of stratification, viscosity, diffusity between species and thermal diffusivity are presented.     

Asymptotic model of a thin shock layer near a wedge/cone for the Burnett approximation

Asymptotic estimations on the order of the terms of the two-dimensional Burnett equations for the characteristic regions (layers) of a hypersonic rarefied gas flow are performed. A simplified Burnett model of the flow near nonthin bodies is formulated (in accordance with the concept of a viscous shock layer) on the basis of the asymptotic analysis founded on physical and theoretical assumptions. The simplified Burnett model is presented for the surfaces in the form of a wedge/cone.