A mixed-time-scale low-Reynolds-number one-equation turbulence model

In this paper, a new low-Reynolds-number (LRN) one-equation turbulence model for eddy viscosity is proposed. A mixed time scale, representing a combination of three time scales: two time scales made of strain-rate parameter S and vorticity parameter Ω and the turbulent time scale k/?, is introduced into this model. The proposed model is derived from an LRN k?? two-equation model where the mixed time scale has been proved to be very effective for predicting local flows over complex terrains. In the transport equation of the model, the mixed time scale is included in the production and the dissipation terms. The new model is evaluated in channel flows at various Reynolds numbers, boundary layer flows with or without pressure gradient and backward-facing step flows with different expansion ratios and Reynolds numbers. Then the grid convergence of the model is investigated. Finally, the model performance for different values of the weighting constant C_s in the mixed time scale is assessed. The results show that the proposed model reproduces the correct wall-limiting behaviour of turbulent quantities and performs well in the near-wall region of turbulent flows. The model could be expected to be adopted in hybrid Reynolds averaged Navier–Stokes/large eddy simulation methodology for complex wall-bounded flows at high Reynolds numbers.     

An immersed boundary–lattice-Boltzmann method for the simulation of the flow past an impulsively started cylinder

We present a lattice-Boltzmann method coupled with an immersed boundary technique for the simulation of bluff body flows. The lattice-Boltzmann method for the modeling of the Navier–Stokes equations, is enhanced by a forcing term to account for the no-slip boundary condition on a non-grid conforming boundary. We investigate two alternatives of coupling the boundary forcing term with the grid nodes, namely the direct and the interpolated forcing techniques. The present LB–IB methods are validated in simulations of the incompressible flow past an impulsively started cylinder at low and moderate Reynolds numbers. We present diagnostics such as the near wall vorticity field and the drag coefficient and comparisons with previous computational and experimental works and assess the advantages and drawbacks of the two techniques.     

Entropic, LES and boundary conditions in lattice Boltzmann simulations of turbulence

Large scale (1600³-grid) entropic lattice Boltzmann (ELB) simulations are performed on the 27-bit model at sufficiently high Reynolds numbers to find intermittency corrections to the Kolmogorov k ^-5/3 inertial spectrum. Even though the transport coefficients in ELB and in the Large Eddy Simulation (LES) lattice Boltzmann schemes have very different origins, there are strong similarities in their turbulence statistics from 512³-grid simulations. A new LB moment-space boundary condition algorithm is tested on the 2D backstep problem, with excellent agreement with experimental data even up to a Reynolds number of 800.     

Stability and angular-momentum transport of fluid flows between corotating cylinders

Turbulent transport of angular momentum is a necessary process to explain accretion in astrophysical disks. Although the hydrodynamic stability of disklike flows has been tested in experiments, results are contradictory and suggest either laminar or turbulent flow. Direct numerical simulations reported here show that currently investigated laboratory flows are hydrodynamically unstable and become turbulent at low Reynolds numbers. The underlying instabilities stem from the axial boundary conditions, affect the flow globally, and enhance angular-momentum transport.     

A simplified model for flows with eddies in symmetrically expanding channels

We derive a simplified model for two-dimensional (2D) channel flows with recirculated regions at moderate Reynolds numbers based on an extension of the boundary layer (BL) theory and averaging across the channel. The model reproduces symmetry-breaking bifurcations and resulting flow structures accurately. Analytical estimates for the decay rates toward the parabolic profile before and after a sudden change in the walls agree well with the full numerical simulations. A seemingly chaotic steady flow is also discovered in a channel with periodic expansions and contractions.     

Kolmogorov scaling of turbulent flow in the vicinity of the wall

How to scale even the simplest of turbulent flows continues to be a cause for considerable controversy. In the present research, a data base compiling results from channel flow direct numerical simulations and turbulent boundary layer experiments is employed to investigate the properties of shear and normal Reynolds stresses very close to the wall. Two types of scaling based on Kolmogorov length and velocity scales are analyzed. It is shown that it is highly likely that large length scales of the order of the channel half-width or the boundary layer thickness play an important role even in the innermost regions of wall-bounded turbulent flows, which hints at the persistence of Reynolds number effects in even high Reynolds number flows.     

Lattice Boltzmann simulations of drops colliding with solid surfaces

Video images of drops colliding with solid surfaces shown by Rioboo et al. (2002) reveal that, for large drop velocities, the drops flatten and form a ring structure before receding and, in some cases, rebounding from the surface. They described the sequence of events in terms of four distinct regimes. During the initial kinematic phase, the dimensionless wetting radius of the drop follows a universal form if the drop Weber and Reynolds numbers are sufficiently large. In the second phase, the drop becomes highly flattened and the values of the Weber and Reynolds numbers influence the time evolution of the dimensionless wetting radius and its maximum value. This is followed by a third phase in which the wetting radius begins to decrease with time and the wettability of the surface influences the dynamics. This paper presents simulation results for the early stages of drop impact and spreading on a partially wetting solid surface. The simulations were performed with a modified version of the lattice Boltzmann method (LBM) developed by Inamuro et al. (2004) for a liquid-gas density ratio of 1000. The Inamuro et al. version of the LBM was modified by incorporating rigid, no-slip boundary conditions and incorporating a boundary condition on the normal derivative of the order parameter to impose the desired equilibrium contact angle.     

GPU accelerated simulations of bluff body flows using vortex particle methods

We present a GPU accelerated solver for simulations of bluff body flows in 2D using a remeshed vortex particle method and the vorticity formulation of the Brinkman penalization technique to enforce boundary conditions. The efficiency of the method relies on fast and accurate particle-grid interpolations on GPUs for the remeshing of the particles and the computation of the field operators. The GPU implementation uses OpenGL so as to perform efficient particle-grid operations and a CUFFT-based solver for the Poisson equation with unbounded boundary conditions. The accuracy and performance of the GPU simulations and their relative advantages/drawbacks over CPU based computations are reported in simulations of flows past an impulsively started circular cylinder from Reynolds numbers between 40 and 9500. The results indicate up to two orders of magnitude speed up of the GPU implementation over the respective CPU implementations. The accuracy of the GPU computations depends on the Re number of the flow. For Re up to 1000 there is little difference between GPU and CPU calculations but this agreement deteriorates (albeit remaining to within 5% in drag calculations) for higher Re numbers as the single precision of the GPU adversely affects the accuracy of the simulations.     

Modeling and simulation of thermocapillary flows using lattice Boltzmann method

To understand how thermocapillary forces manipulate droplet motion in microfluidic channels, we develop a lattice Boltzmann (LB) multiphase model to simulate thermocapillary flows. The complex hydrodynamic interactions are described by an improved color-fluid LB model, in which the interfacial tension forces and the Marangoni stresses are modeled in a consistent manner using the concept of a continuum surface force. An additional convection–diffusion equation is solved in the LB framework to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. A stress-free boundary condition is also introduced to treat outflow boundary, which can conserve the total mass of an incompressible system, thus improving the numerical stability for creeping flows.The model is firstly validated against the analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of three-dimensional deformable droplet at various Marangoni numbers, and its accuracy is once again verified against the theoretical prediction in the limit of zero Marangoni number. Finally, we numerically investigate how the localized heating from a laser can block the microfluidic droplet motion through the induced thermocapillary forces. The droplet motion can be completely blocked provided that the intensity of laser exceeds a threshold value, below which the droplet motion successively undergoes four stages: constant velocity, deceleration, acceleration, and constant velocity. When the droplet motion is completely blocked, four steady vortices are clearly visible, and the droplet is fully filled by two internal vortices. The external vortices diminish when the intensity of laser increases.     

An Immersed Boundary-Lattice Boltzmann Prediction for Particle Hydrodynamic Focusing in Annular Microchannels

We numerically study the dynamics of particle crystals in annular microchannels by the immersed-boundary(IB)lattice Boltzmann(LB) coupled model, analyze the fluid-particle interactions during the migration of particles,and reveal the underlying mechanism of a particle focusing on the presence of fluid flows. The results show that the Reynolds and Dean numbers are key factors influencing the hydrodynamics of particles. The particles migrate onto their equilibrium tracks by adjusting the Reynolds and Dean numbers. Elliptical tracks of particles during hydrodynamic focusing can be predicted by the IB-LB model. Both the small Dean number and the small particle can lead to a small size of the focusing track. This work would possibly facilitate the utilization of annular microchannel flows to obtain microfluidic flowing crystals for advanced applications in biomedicine and materials synthesis.     

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar–Gross–Krook (BGK) equation. Therefore, when the same Gauss–Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss–Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss–Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss–Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier–Stokes level and rarefied gas flows at the linearized Boltzmann model equation level.     

Large-eddy simulation of flows past a flapping airfoil using immersed boundary method

The numerical simulation of flows past flapping foils at moderate Reynolds numbers presents two challenges to computational fluid dynamics: turbulent flows and moving boundaries. The direct forcing immersed boundary(IB) method has been developed to simulate laminar flows. However,its performance in simulating turbulent flows and transitional flows with moving boundaries has not been fully evaluated. In the present work,we use the IB method to simulate fully developed turbulent channel flows and transitional flows past a stationary/plunging SD7003 airfoil. To suppress the non-physical force oscillations in the plunging case,we use the smoothed discrete delta function for interpolation in the IB method. The results of the present work demonstrate that the IB method can be used to simulate turbulent flows and transitional flows with moving boundaries.     

Flame-acoustic coupling of combustion instability in a non-premixed backward-facing step combustor: the role of acoustic-Reynolds stress

Combustion instability in a laboratory scale backward-facing step combustor is numerically investigated by carrying out an acoustically coupled incompressible large eddy simulation of turbulent reacting flow for various Reynolds numbers with fuel injection at the step. The problem is mathematically formulated as a decomposition of the full compressible Navier–Stokes equations using multi-scale analysis by recognising the small length scale and large time scale of the flow field relative to a longitudinal mode acoustic field for low mean Mach numbers. The equations are decomposed into those for an incompressible flow with temperature-dependent density to zeroth order and linearised Euler equations for acoustics as a first order compressibility correction. Explicit coupling terms between the two equation sets are identified to be the flow dilatation as a source of acoustic energy and the acoustic Reynolds stress (ARS) as a source of flow momentum. The numerical simulations are able to capture the experimentally observed flow–acoustic lock-on that signifies the onset of combustion instability, marked by a shift in the dominant frequency from an acoustic to a hydrodynamic mode and accompanied by a nonlinear variation of pressure amplitude. Attention is devoted to flow conditions at two Reynolds numbers before and after lock-on to show that, after lock-on, the ARS causes large-scale vortical rollup resulting in the evolution of a compact flame. As compared to acoustically uncoupled simulations at these Reynolds numbers that show an elongated flame with no significant roll up and disturbance in the upstream flow field, the ARS is seen to alter the shear layer dynamics by affecting the flow field upstream of the step as well, when acoustically coupled.     

Numerical Stability Analysis of FDLBM

We analyze the numerical stability of Finite Difference Lattice Boltzmann Method (FDLBM) by means of von Neumann stability analysis. The stability boundary of the FDLBM depends on the BGK relaxation time, the CFL number, the mean flow velocity, and the wavenumber. As the BGK relaxation time is increased at constant CFL number, the stability of the central difference LB scheme may not be ensured. The limits of maximum stable velocity are obtained around 0.39, 0.43, and 0.43 for the central difference, for the explicit upwind difference, and for the semi-implicit upwind difference schemes, respectively. We derive artificial viscosities for every difference scheme and investigate their influence on numerical stability. The requirements for artificial viscosity is consistent with the conditions derived from von Neumann stability analysis. This analysis elucidates that the upwind difference schemes are suitable for simulation of high Reynolds number flows.     

Large scale structures in Rayleigh-Bénard convection at high Rayleigh numbers

Direct numerical simulations of Rayleigh-Bénard convection in a plane layer with periodic boundary conditions at Rayleigh numbers up to 10(7) show that flow structures can be objectively classified as large or small scale structures because of a gap in spatial spectra. The typical size of the large scale structures does not always vary monotonically as a function of the Rayleigh number but broadly increases with increasing Rayleigh number. A mean flow (whose average over horizontal planes differs from zero) is also excited but is weak in comparison with the large scale structures. The large scale circulation observed in experiments should therefore be a manifestation of the large scale structures identified here.     

Numerical simulations of gas-liquid two-phase flows in a micro porous structure

The lattice Boltzmann method for two-phase fluid flows is applied to the simulations of gas-liquid two-phase flows in a micro porous structure for various capillary numbers at low Reynolds numbers. The behaviors of the gas-liquid interface and the velocities of the two-phase fluid in the structure are simulated, and the permeability of gas and liquid through the structure are estimated from the calculated results. By changing the void fraction, the contact angle of the interface on walls, and the surface tension, the effect of these properties on the behaviors and the permeability of the two-phase flows in the micro porous structure is investigated. It is found that the permeability of liquid flows depends on the contact angle and it increases for hydrophobic walls. It is also seen that liquid flows are choked in pores for large void fractions and low capillary numbers.     

An immersed boundary method based on discrete stream function formulation for two- and three-dimensional incompressible flows

An immersed boundary method is proposed in the framework of discrete stream function formulation for incompressible flows. In order to impose the non-slip boundary condition, the forcing term is determined implicitly by solving a linear system. The number of unknowns of the linear system is the same as that of the Lagrangian points representing the body surface. Thus the extra cost in force calculation is negligible if compared with that in the basic flow solver. In order to handle three-dimensional flows at moderate Reynolds numbers, a parallelized flow solver based on the present method is developed using the domain decomposition strategy. To verify the accuracy of the immersed-boundary method proposed in this work, flow problems of different complexity (decaying vortices, flows over stationary and oscillating cylinders and a stationary sphere, and flow over low-aspect-ratio flat-plate) are simulated and the results are in good agreement with the experimental or computational data in previously published literatures.     

On mean flow universality of turbulent wall flows. II. Asymptotic flow analysis

Understanding of the structure of turbulent flows at extreme Reynolds numbers (Re) is relevant because of several reasons: almost all turbulence theories are only valid in the high Re limit, and most turbulent flows of practical relevance are characterized by very high Re. Specific questions about wall-bounded turbulent flows at extreme Re concern the asymptotic laws of the mean velocity and turbulence statistics, their universality, the convergence of statistics towards their asymptotic profiles, and the overall physical flow organization. In extension of recent studies focusing on the mean flow at moderate and relatively high Re, the latter questions are addressed with respect to three canonical wall-bounded flows (channel flow, pipe flow, and the zero-pressure gradient turbulent boundary layer). Main results reported here are the asymptotic logarithmic law for the mean velocity and corresponding scale-separation laws for bulk flow properties, the Reynolds shear stress, the turbulence production and turbulent viscosity. A scaling analysis indicates that the establishment of a self-similar turbulence state is the condition for the development of a strict logarithmic velocity profile. The resulting overall physical flow structure at extreme Re is discussed.     

Lattice Boltzmann simulation of fluid flows in two-dimensional channel with complex geometries

Boundary conditions (BCs) play an essential role in lattice Boltzmann (LB) simulations. This paper investigates several most commonly applied BCs by evaluating the relative L₂-norm errors of the LB simulations for two-dimensional (2-D) Poiseuille flow. It is found that the relative L₂-norm error resulting from FHML's BC is smaller than that from other BCs as a whole. Then, based on the FHML's BC, it formulates an LB model for simulating fluid flows in 2-D channel with complex geometries. Afterwards, the flows between two inclined plates, in a pulmonary blood vessel and in a blood vessel with local expansion region, are simulated. The numerical results are in good agreement with the analytical predictions and clearly show that the model is effective. It is expected that the model can be extended to simulate some real biologic flows, such as blood flows in arteries, vessels with stenosises, aneurysms and bifurcations, etc.     

Numerical demonstration of fluctuation dynamo at low magnetic Prandtl numbers

Direct numerical simulations of incompressible nonhelical randomly forced MHD turbulence are used to demonstrate for the first time that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm>1 and small magnetic Prandtl number Pm<1. The dependence of the critical Rmc for dynamo on the hydrodynamic Reynolds number Re is obtained for 1 less than or similar Re less than or similar 6700. In the limit Pm<1, Rmc is about 3 times larger than for the previously well-established dynamo at large and moderate Prandtl numbers: Rmc less than or similar 200 for Re greater than or similar 6000 compared to Rmc approximately 60 for Pm>or=1. It is not yet possible to determine numerically whether the growth rate of the magnetic energy is proportional, Rm1/2 in the limit Rm-->infinity, as it should be if the dynamo is driven by the inertial-range motions at the resistive scale.