Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods – Part II: Numerical solution to the backward facing step flow

The lattice Boltzmann method (LBM) and traditional finite difference methods have separate strengths when solving the incompressible Navier–Stokes equations. The LBM is an explicit method with a highly local computational nature that uses floating-point operations that involve only local data and thereby enables easy cache optimization and parallelization. However, because the LBM is an explicit method, smaller grid spacing requires smaller numerical time steps during both transient and steady state computations. Traditional implicit finite difference methods can take larger time steps as they are not limited by the CFL condition, but only by the need for time accuracy during transient computations. To take advantage of the strengths of both methods, a multiple solver, multiple grid block approach was implemented and validated for the 2-D Burgers’ equation in Part I of this work. Part II implements the multiple solver, multiple grid block approach for the 2-D backward step flow problem. The coupled LBM–VSM solver is found to be faster by a factor of 2.90 (2.87 and 2.93 for Re = 150 and Re = 500, respectively) on a single processor than the VSM for the 2-D backward step flow problem while maintaining similar accuracy.     

A variational multiscale method for incompressible turbulent flows: Bubble functions and fine scale fields

This paper presents a residual-based turbulence model for the incompressible Navier–Stokes equations. The method is derived employing the variational multiscale (VMS) framework. A multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse-scale and the fine-scale sub-problems. The fine-scale velocity field is assumed to be nonlinear and time-dependent and is modeled via the bubble functions approach applied directly to the fine-scale sub-problem. A significant contribution in this paper is a systematic and consistent derivation of the fine-scale variational operator, commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits, and variationally projects the fine-scale solution onto the coarse-scale space. A direct treatment of the fine-scale problem via bubble functions offers several fine-scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems. Numerical accuracy of the proposed method is shown on a forced-isotropic turbulence problem, statistically stationary turbulent channel flow problems at Re_T = 395 and 590, and non-equilibrium turbulent flow around a cylinder at Re = 3,900.     

Large Eddy Simulation of a turbulent non-premixed propane-air reacting flame in a cylindrical combustor

Large Eddy Simulation (LES) is applied to investigate the turbulent non-premixed combustion flow, including species concentrations and temperature, in a cylindrical combustor. Gaseous propane (C₃H₈) is injected through a circular nozzle which is attached at the centre of the combustor inlet. Preheated air with a temperature of 773 K is supplied through the annulus surrounding of this fuel nozzle. In LES a spatial filtering is applied to the governing equations to separate the flow field into large-scale and small-scale eddies. The large-scale eddies which carry most of the turbulent energy are resolved explicitly, while the unresolved small-scale eddies are modelled using the Smagorinsky model with C_s = 0.1 as well as dynamically calibrated C_s. The filtered values of the species mass fraction, temperature and density, which are the functions of the mixture fraction (conserved scalar), are determined by integration over a beta probability density function (β-PDF). The computational results are compared with those of the experimental investigation conducted by Nishida and Mukohara [1]. According to this experiment, the overall equivalence ratio of 0.6, which is calculated from the ratio of the air flow rate supplied to the combustion chamber to that of the stoichiometric reaction, is kept constant so that the turbulent combustion at the fuel nozzle exit starts under the fuel-rich conditions.     

Efficient treatment of complex geometries for large eddy simulations of turbulent flows

Incompressible turbulent flow over a backward facing step at Re_h=5100 is investigated by large eddy simulations (LES). The ratio of the oncoming boundary layer thickness δ to the step height h was set to 1.2. Additionally channel flows at various Re_τ numbers are presented for the validation of the numerical code. The results are compared with existing DNS and experimental databases. The present study focuses on different procedures for LES of engineering problems in complex geometries using structured rectangular grids. Two different methods that are able to treat complex geometrical configurations are implemented, examined and compared; namely the domain decomposition approach based on Schur’s complement and the immersed boundary method. In the present study both methods make use of a fast direct Poisson’s pressure solver based on a heavily modified version of the public domain package FISHPAK. The latter was optimised and fully parallelised for shared memory architectures, for solutions on rectangular grids stretched in one or two directions. The resulting code reaches performances of 1.0 μs/node/iter, allowing low cost computations on grids of the order of million points. The main objective of the present study was to investigate the potential of different methods for LES in complex geometrical configurations like bluff body flows and wakes. One of the main findings is that careful selection of numerical methods and implementation techniques can lead to accurate and very efficient codes, where the geometric complexity does not lead to algorithmic or numerical complexity.     

Extension of a hybrid thermal LBE scheme for large-eddy simulations of turbulent convective flows

Following the work of Lallemand and Luo [Lallemand P, Luo L-S. Theory of the lattice Boltzmann method: acoustic and thermal properties in two and three dimensions. Phys Rev E 2003;68:036706] we validate, apply and extend the hybrid thermal lattice Boltzmann scheme (HTLBE) by a large-eddy approach to simulate turbulent convective flows. For the mass and momentum equations, a multiple-relaxation-time LBE scheme is used while the heat equation is solved numerically by a finite difference scheme. We extend the hybrid model by a Smagorinsky subgrid scale model for both the fluid flow and the heat flux. Validation studies are presented for laminar and turbulent natural convection in a cavity at various Rayleigh numbers up to 5 × 10¹⁰ for Pr = 0.71 using a serial code in 2D and a parallel code in 3D, respectively. Correlations of the Nusselt number are discussed and compared to benchmark data. As an application we simulated forced convection in a building with inner courtyard at Re = 50 000.     

Critical pressure for capillary valves in a Lab-on-a-Disk: CFD and flow visualization

Capillary valves are used as pressure barriers to control flow sequencing in microfluidic devices. Influence of valves height on liquid flow pattern and critical pressure are studied through flow visualization and CFD predictions (Gambit® 2.2.30 and FLUENT® 6.2.16). Both hydrophilic and hydrophobic walls are studied. Results show that the surface tension plays a major role in the liquid progress through the microchannel/valve and also in the valve filling process. Critical pressure varies linearly with the valve hydraulic diameter in the range 0.91 < D_h < 3.5 [mm] according to: P = 14.14 · D_h + 47.42 [Pa].     

Computational assessment of several hydrogen-free high energy compounds

Tetrazino-tetrazine-tetraoxide (TTTO) is an attractive high energy compound, but unfortunately, it is not yet experimentally synthesized so far. Isomerization of TTTO leads to its five isomers, bond-separation energies were empolyed to compare the global stability of six compounds, it is found that isomer 1 has the highest bond-separation energy (1204.6 kJ/mol), compared with TTTO (1151.2 kJ/mol); thermodynamic properties of six compounds were theoretically calculated, including standard formation enthalpies (solid and gaseous), standard fusion enthalpies, standard vaporation enthalpies, standard sublimation enthalpies, lattice energies and normal melting points, normal boiling points; their detonation performances were also computed, including detonation heat (Q, cal/g), detonation velocity (D, km/s), detonation pressure (P, GPa) and impact sensitivity (h₅₀, cm), compared with TTTO (Q = 1311.01 J/g, D = 9.228 km/s, P = 40.556 GPa, h₅₀ = 12.7 cm), isomer 5 exhibites better detonation performances (Q = 1523.74 J/g, D = 9.389 km/s, P = 41.329 GPa, h₅₀ =  28.4 cm).     

Modal analysis and modified cascade neural networks in identification of geometrical parameters of circular arches

A hybrid computational system, composed of the finite element method (FEM) and cascade neural network system (CNNs), is applied to the identification of three geometrical parameters of elastic arches, i.e. span l, height f and cross-sectional thickness h. FEM is used in the direct (forward) analysis, which corresponds to the mapping α = {l, f, h} → {ω_j}, where: α – vector of control parameters, ω_j – arch eigenfrequencies. The reverse analysis is related to the identification procedure in which the reverse mapping is performed {ω_j} → {α_i}. For the identification purposes a recurrent, three level CNNs of structure (D^k-H^k-1)_s was formulated, where: k – recurrence step, s = I, II, III-levels of cascade system. The Semi-Bayesian approach is introduced for the design of CNNs applying the MML Maximum Marginal Likelihood) criterion. The computation of hyperparameters is performed by means of the Bayesian procedure evidence. The numerical analysis proves a great numerical efficiency of the proposed hybrid approach for both the perfect (noiseless) values of eigenfrequencies and noisy ones simulated by an added artificial noise.     

Stability and convergence of sequential methods for coupled flow and geomechanics: Drained and undrained splits

We perform a stability and convergence analysis of sequential methods for coupled flow and geomechanics, in which the mechanics sub-problem is solved first. We consider slow deformations, so that inertia is negligible and the mechanical problem is governed by an elliptic equation. We use Biot’s self-consistent theory to obtain the classical parabolic-type flow problem. We use a generalized midpoint rule (parameter α between 0 and 1) time discretization, and consider two classical sequential methods: the drained and undrained splits.The von Neumann method provides sharp stability estimates for the linear poroelasticity problem. The drained split with backward Euler time discretization (α = 1) is conditionally stable, and its stability depends only on the coupling strength, and it is independent of time step size. The drained split with the midpoint rule (α = 0.5) is unconditionally unstable. The mixed time discretization, with α = 1.0 for mechanics and α = 0.5 for flow, has the same stability properties as the backward Euler scheme. The von Neumann method indicates that the undrained split is unconditionally stable when α ? 0.5.We extend the stability analysis to the nonlinear regime (poro-elastoplasticity) via the energy method. It is well known that the drained split does not inherit the contractivity property of the continuum problem, thereby precluding unconditional stability. For the undrained split we show that it is B-stable (therefore unconditionally stable at the algorithmic level) when α ? 0.5.We also analyze convergence of the drained and undrained splits, and derive the a priori error estimates from matrix algebra and spectral analysis. We show that the drained split with a fixed number of iterations is not convergent even when it is stable. The undrained split with a fixed number of iterations is convergent for a compressible system (i.e., finite Biot modulus). For a nearly-incompressible system (i.e., very large Biot modulus), the undrained split loses first-order accuracy, and becomes non-convergent in time.We also study the rate of convergence of both splits when they are used in a fully-iterated sequential scheme. When the medium permeability is high or the time step size is large, which corresponds to a high diffusion of pressure, the error amplification of the drained split is lower and therefore converges faster than the undrained split. The situation is reversed in the case of low permeability and small time step size.We provide numerical experiments supporting all the stability and convergence estimates of the drained and undrained splits, in the linear and nonlinear regimes. We also show that our spatial discretization (finite volumes for flow and finite elements for mechanics) removes the well-documented spurious instability in consolidation problems at early times.     

Simulation of constituent transport using a reduced 3D constituent transport model (CTM) driven by HF Radar: Model application and error analysis

Data-driven constituent transport models (CTM), which take surface current measurements from High Frequency (HF) Radar as input can be applied within the context of real-time environmental monitoring, oceanographic assessment, response to episodic events, as well as search and rescue in surface waters.This paper discusses a numerical method that allows for the evaluation of diffusion coefficients in anisotropic flow fields from surface current measurements using HF Radar. The numerical scheme developed was incorporated into a data-driven CTM and through model error analyses, the effects of using spatially variable transport coefficients on model predictions were examined. The error analyses were performed on the model by varying the cell Reynolds number, Re = f(u,K,Δx) between 0.15 and 100, where u is the velocity vector within the flow field, K is a diffusivity tensor and Δx is the computational grid cell size.Two instantaneous releases of conservative material were then modeled, the model being initialized at two different locations within the domain. From the two simulation runs, marked differences in the predicted spatial extent of the conservative material resulting from the spatially varying diffusivity values within the study area were observed. Model predictions in terms of variance or size estimates of a diffusing patch were found to be more affected from using inaccurate diffusivity estimates, and less affected from using inaccurate current measurements. The largest errors occurred at Re > 2 associated with changing diffusivity values, going up to as much as a 25-fold difference in variance estimates at Re = 100. Very little effect on variance estimates due to varying velocity values were observed even at Re > 2. This model was applied within the framework of constituent tracking to Corpus Christi Bay in the Texas Gulf of Mexico region.     

Evaluation of three methodologies to estimate the VO2max in people of different ages

Aging and gender are factors that affect the variation of physical work capacity. The present paper highlights the importance of the metabolism used by ergonomics to establish the appropriate limits of loads at work. This study compares the aerobic capacity of people from 20 to 71 years old split in 5 different groups. The laboratory experiment tested 33 volunteers (19 women and 14 men). A submaximal step test was used to measure the VO₂ using a portable breath by breath metabolic system and a telemetric heart rate monitor. Three methods to estimate the VO_2max were compared: 1) a direct measurement of VO₂, 2) estimation by heart rate, and 3) a step test method using predetermined charts. Significant difference was encountered among the estimation methods as well as among the age ranges (F_2,92 = 6.43, p < 0.05 y F_4,92 = 7.18, p < 0.05 respectively). The method of direct measurement and the method of predetermined charts were different for the estimation of the VO_2max with a confidence level of 95%. The method of predetermined charts is better adapted for males and people younger than 30 years. The estimation through non invasive heart rate apparatus was a good appraiser of the maximal oxygen consumption considering both genders and all the age groups.     

Bounds for the Roots of Lacunary Polynomials

A polynomial P(X)  = X^d + a_d − 1X^d − 1 + ⋯ is called lacunary when a_d − 1 =  0. We give bounds for the roots of such polynomials with complex coefficients. These bounds are much smaller than for general polynomials.     

On reliability of the folded hypercubes

In this paper, we explore the 2-extra connectivity and 2-extra-edge-connectivity of the folded hypercube FQ_n. We show that κ₂(FQ_n) = 3n − 2 for n ⩾ 8; and λ₂(FQ_n) = 3n − 1 for n ⩾ 5. That is, for n ⩾ 8 (resp. n ⩾ 5), at least 3n − 2 vertices (resp. 3n − 1 edges) of FQ_n are removed to get a disconnected graph that contains no isolated vertices (resp. edges). When the folded hypercube is used to model the topological structure of a large-scale parallel processing system, these results can provide more accurate measurements for reliability and fault tolerance of the system.     

The possibility of using C20 fullerene and graphene as semiconductor segments for detection,and destruction of cyanogen-chloride chemical agent

Detection of hazardous chemical species by changing the electrical conductivity of a semiconductor matter is a proposed and applied way for decreasing their subsequent unpleasant effects. Recently, many examples of using inorganic or organic materials, polymeric, and also nano-sized species as sensors were reported in which, in some cases, those matters were strongly affective and suitable.In this project, we have made an assessment on whether the graphene segment or C₂₀ fullerene, able to sense the existence of cyanogen chloride NCCl? In order to gain trustable results, the possible reaction pathways along with the adsorption kinetics were investigated. Moreover, the electronic density of states DOS showed that C₂₀ fullerene senses the existence of cyanogen chloride agent with a clearer signal (ΔE_g = 0.0110 eV) compared to the graphene segment (ΔE_g = 0.0001 eV). Also the adsorption energy calculations showed that cyanogen chloride could be adsorbed by the fullerene in a multi-step process (E_ads1 = −0.852 kcal mol⁻¹; E_ads2 = −0.446 kcal mol⁻¹; E_ads3 = −2.330 kcal mol⁻¹).     

Computer-based analysis of explicit approximations to the implicit Colebrook–White equation in turbulent flow friction factor calculation

The implicit Colebrook–White equation has been widely used to estimate the friction factor for turbulent fluid-flow in rough-pipes. In this paper, the state-of-the-art review for the most currently available explicit alternatives to the Colebrook–White equation, is presented. An extensive comparison test was established on the 20 × 500 grid, for a wide range of relative roughness (ε/D) and Reynolds number (R) values (1 × 10^?6 ? ε/D ? 5 × 10^?2; 4 × 10³ ? R ? 10⁸), covering a large portion of turbulent flow zone in Moody’s diagram. Based on the comprehensive error analysis, the magnitude points in which the maximum absolute and the maximum relative error are occurred at the pair of ε/D and R values, are observed. A limiting case of the most of these approximations provided friction factor estimates that are characterized by a mean absolute error of 5 × 10^?4, a maximum absolute error of 4 × 10^?3 whereas, a mean relative error of 1.3% and a maximum relative error of 5.8%, over the entire range of ε/D and R values, respectively. For practical purposes, the complete results for the maximum and the mean relative errors versus the 20 sets of ε/D value, are also indicated in two comparative figures. The examination results for error properties of these approximations gives one an opportunity to practically evaluate the most accurate formula among of all the previous explicit models; and showing in this way its great flexibility for estimating turbulent flow friction factor. Comparative analysis for the mean relative error profile revealed, the classification for the best-fitted six equations examined was in a good agreement with those of the best model selection criterion claimed in the recent literature, for all performed simulations.     

Calibration and validation of hyperspectral indices for the estimation of broadleaved forest leaf chlorophyll content,leaf mass per area,leaf area index and leaf canopy biomass

This article aims at finding efficient hyperspectral indices for the estimation of forest sun leaf chlorophyll content (CHL, µg cm_leaf^? 2), sun leaf mass per area (LMA, g_{dry matter} m_leaf^? 2), canopy leaf area index (LAI, m²_leaf m_soil^? 2) and leaf canopy biomass (B_leaf, g_{dry matter} m_soil^? 2). These parameters are useful inputs for forest ecosystem simulations at landscape scale. The method is based on the determination of the best vegetation indices (index form and wavelengths) using the radiative transfer model PROSAIL (formed by the newly-calibrated leaf reflectance model PROSPECT coupled with the multi-layer version of the canopy radiative transfer model SAIL). The results are tested on experimental measurements at both leaf and canopy scales. At the leaf scale, it is possible to estimate CHL with high precision using a two wavelength vegetation index after a simulation based calibration. At the leaf scale, the LMA is more difficult to estimate with indices. At the canopy scale, efficient indices were determined on a generic simulated database to estimate CHL, LMA, LAI and Bleaf in a general way. These indices were then applied to two Hyperion images (50 plots) on the Fontainebleau and Fougères forests and portable spectroradiometer measurements. They showed good results with an RMSE of 8.2 µg cm^? 2 for CHL, 9.1 g m^? 2 for LMA, 1.7 m² m^? 2 for LAI and 50.6 g m^? 2 for Bleaf. However, at the canopy scale, even if the wavelengths of the calibrated indices were accurately determined with the simulated database, the regressions between the indices and the biophysical characteristics still had to be calibrated on measurements. At the canopy scale, the best indices were: for leaf chlorophyll content: ND_chl = (ρ₉₂₅ ? ρ₇₁₀)/(ρ₉₂₅ + ρ₇₁₀), for leaf mass per area: ND_LMA = (ρ₂₂₆₀ ? ρ₁₄₉₀)/(ρ₂₂₆₀ + ρ₁₄₉₀), for leaf area index: D_LAI = ρ₁₇₂₅ ? ρ₉₇₀, and for canopy leaf biomass: ND_Bleaf = (ρ₂₁₆₀ ? ρ₁₅₄₀)/(ρ₂₁₆₀ + ρ₁₅₄₀).     

Dielectric and piezoelectric properties of [0 0 1] and [0 1 1]-poled relaxor ferroelectric PZN–PT and PMN–PT single crystals

The properties of PZN–PT and PMN–PT single crystals of varying compositions and orientations have been investigated. Among the various compositions studied, [0 0 1]-optimally poled PZN-(6–7)%PT and PMN-30%PT exhibit superior dielectric and piezoelectric properties, with K^T ≈ 6800–8000, d₃₃ ≈ 2800 pC/N, d₃₁ ≈ −(1200–1800) pC/N for PZN-(6–7)%PT; and K^T = 7500–9000, d₃₃ = 2200–2500 pC/N and d₃₁ = −(1100–1400) pC/N for PMN-30%PT. These two compositions are also fairly resistant to over-poling. The [0 0 1]-poled electromechanical coupling factors (k₃₃, k₃₁ and k_t) are relatively insensitive to crystal composition. [0 1 1]-optimally poled PZN-7%PT single crystal also exhibits extremely high d₃₁ values of up to −4000 pC/N with k₃₁ ≈ 0.90–0.96. While [0 1 1]-poled PZN-7%PT single crystal becomes over-poled with much degraded properties when poled at and above 0.6 kV/mm, PZN-6%PT crystal shows no signs of over-poling even when poled to 2.0 kV/mm. The presence of a certain amount (i.e., 10–15%) of orthorhombic phase in a rhombohedral matrix has been found to be responsible for the superior transverse piezoelectric properties of [0 1 1]-optimally poled PZN-(6–7)%PT. The present work shows that flux-grown PZN–PT crystals exhibit superior and consistent properties and improved over-poling resistance to flux-grown PMN–PT crystals and that, for or a given crystal composition, flux-grown PMN–PT crystals exhibit superior over-poling resistance to their melt-grown counterparts.     

A zonal RANS-LES method to determine the flow over a high-lift configuration

High Reynolds number flows are particularly challenging problems for large-eddy simulations (LES) since small-scale structures in thin and often transitional boundary layers are to be resolved. The range of the turbulent scales is enormous, especially when high-lift configuration flows are considered. For this reason, the prediction of high Reynolds number flow over the entire airfoil using LES requires huge computer resources. To remedy this problem a zonal RANS-LES method for the flow over an airfoil in high-lift configuration at Re_c=1.0×10⁶ is presented. In a first step, a 2D RANS solution is sought, from which boundary conditions are formulated for an embedded LES domain, which comprises the flap and a sub-part of the main airfoil. The turbulent fluctuations in the boundary layers at the inflow region of the LES domain are generated by controlled forcing terms, which use the turbulent shear stress profiles obtained from the RANS solution. The comparison with an LES solution for the full domain and with experimental data shows likewise results for the velocity profiles and wall pressure distributions. The zonal RANS-LES method reduces the computational effort of a full domain LES by approx. 50%.     

Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits

We analyze stability and convergence of sequential implicit methods for coupled flow and geomechanics, in which the flow problem is solved first. We employ the von Neumann and energy methods for linear and nonlinear problems, respectively. We consider two sequential methods with the generalized midpoint rule for t_n+α, where α is the parameter of time discretization: namely, the fixed-strain and fixed-stress splits. The von Neumann method indicates that the fixed-strain split is only conditionally stable, and that its stability limit is a coupling strength less than unity if α ? 0.5. On the other hand, the fixed-stress split is unconditionally stable when α ? 0.5, the amplification factors of the fixed-stress split are different from those of the undrained split and are identical to the fully coupled method. Unconditional stability of the fixed-stress split is also obtained from the energy method for poroelastoplasticity. We show that the fixed-stress split is contractive and B-stable when α ? 0.5.We also estimate the convergence behaviors for the two sequential methods by the matrix based and spectral analyses for the backward Euler method in time. From the estimates, the fixed-strain split may not be convergent with a fixed number of iterations particularly around the stability limit even though it is stable. The fixed-stress split, however, is convergent for a fixed number of iterations, showing better accuracy than the undrained split. Even when we cannot obtain the exact local bulk modulus (or exact rock compressibility) at the flow step a priori due to complex boundary conditions or the nonlinearity of the materials, the fixed-stress split can still provide stability and convergence by an appropriate estimation of the local bulk modulus, such as the dimension-based estimation, by which the employed local bulk modulus is less stiff than the exact local bulk modulus.We provide numerical examples supporting all the estimates of stability and convergence for the fixed-strain and fixed-stress splits.