Direct Numerical Simulation Approach Research Articles

Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: a direct numerical simulation approach

In this study, a newly developed direct numerical simulation (DNS) solver is utilized for the simulations of numerous stably stratified open-channel flows with bulk Reynolds number (Re b ) spanning 3400–16,900. Overall, the simulated bulk Richardson number ( $$Ri_b$$ ) ranges from 0.08 (weakly stable) to 0.49 (very stable). Thus, both continuously turbulent and (globally) intermittently turbulent cases are represented in the DNS database. Using this comprehensive database, various flux-based and gradient-based similarity relationships for energy dissipation rate (e) and temperature structure parameter ( $$C_T^2$$ ) are developed. Interestingly, these relationships exhibit only minor dependency on Re b . In order to further probe into this Re b -effect, similarity relationships are also estimated from a large-eddy simulation (LES) run of an idealized atmospheric boundary layer (very high Re b ) case study. Despite the fundamental differences in the estimation of e and $$C_T^2$$ from the DNS- and the LES-generated data, the resulting similarity relationships, especially the gradient-based ones, from these numerical approaches are found to be remarkably similar. More importantly, these simulated relationships are also comparable, at least qualitatively, to the traditional observational data-based ones. Since these simulated similarity relationships do not require Taylor’s hypothesis and do not suffer from mesoscale disturbances and/or measurement noise, they have the potential to complement the existing similarity relationships.

Filtration model for polydisperse aerosols in gas‐solid flow using granule‐resolved direct numerical simulation

An analytical framework for calculating the filtration efficiency of polydisperse aerosols in a granular bed is developed for cases where inertial impaction and interception are the principal filtration mechanisms. This framework is used to develop a model for the polydisperse single‐collector efficiency from monodisperse single‐collector efficiency correlations. Conceptually, the polydisperse model is developed by transforming the probability density of particle radius into a probability density of particle Stokes number that is then used to weight the monodisperse single‐collector efficiency at a given Stokes number. An extension of this polydisperse filtration concept results in an analytical solution for the axial variation of polydisperse particle flux in a random three‐dimensional granule configuration. In order to verify the analytical results for polydisperse particle filtration, a granule‐resolved direct numerical simulation approach is coupled with Lagrangian particle tracking to simulate filtration of polydisperse aerosols in a granular bed. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3594–3606, 2015

Direct numerical simulation of matrix diffusion across fracture/matrix interface

Accurate descriptions of matrix diffusion across the fracture/matrix interface are critical to assessing contaminant migration in fractured media. The classical transfer probability method is only applicable for relatively large diffusion coefficients and small fracture spacings, due to an intrinsic assumption of an equilibrium concentration profile in the matrix blocks. Motivated and required by practical applications, we propose a direct numerical simulation (DNS) approach without any empirical assumptions. A three-step Lagrangian algorithm was developed and validated to directly track the particle dynamics across the fracture/matrix interface, where particle's diffusive displacement across the discontinuity is controlled by an analytical, one-side reflection probability. Numerical experiments show that the DNS approach is especially efficient for small diffusion coefficients and large fracture spacings, alleviating limitations of the classical modeling approach.

3D melt convection in sapphire crystal growth: Evaluation of physical properties

We apply numerical simulation to analyze flow structure and physical properties of sapphire melt during Kyropoulos crystal growth. The Direct Numerical Simulation (DNS) approach was used to investigate 3D features of melt convection in a lab-scale furnace. The importance to account for the effect of internal radiation inside sapphire melt is discussed. For different sets of material properties, we have investigated the effect of the Marangoni and buoyancy driving forces on the melt flow. Results of 3D unsteady modeling are discussed in detail and compared with related experimental data.

Eddy resolving simulations in aerospace – Invited paper (Numerical Fluid 2014)

The future use of eddy resolving simulations (ERS) such as Large Eddy Simulation (LES), Direct Numerical Simulation (DNS) and related approaches in aerospace is explored. The turbulence modeling requirements with respect to aeroengines and aircraft is contrasted. For the latter, higher Reynolds numbers are more prevalent and this especially gives rise to the need for the hybridization of ERS methods with Reynolds Averaged Navier–Stokes (RANS) approaches. Zones where future use of pure ERS methods is now possible and those where hybridizations with RANS will be needed is outlined. The major focus is the aeroengine for which the component scales are much smaller. This gives rise to generally more benign Reynolds numbers. The use of eddy resolving methods in a wide range of zones in an aeroengine is discussed and the potential benefits and also cost drawbacks with such approaches noted. The tension when using such computationally intensive calculations in an area where the coupling of components and even the airframe and engine is becoming increasingly important is explored. Also, the numerical methods and meshing requirements are considered and the implications of ERS methods for future numerical algorithms. It is postulated that such simulations are ready now for niche uses in industry. However, to perform the scale of simulations that industry requires, to meet pressing environmental needs, challenges remain. For example, there is the need to develop optimal numerical methods that both map to the accuracy requirements for ERS and also future computer architectures.

Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study

AbstractWhen a turbulent flow in a porous medium is determined numerically, the crucial question is whether turbulence models should account only for turbulent structures restricted in size to the pore scale or whether the size of turbulent structures could exceed the pore scale. The latter would mean the existence of macroscopic turbulence in porous media, when turbulent eddies exceed the pore size. In order to determine the real size of turbulent structures in a porous medium, we simulated the turbulent flow by direct numerical simulation (DNS) calculations, thus avoiding turbulence modelling of any kind. With this study, which for the first time uses DNS calculations, we provide benchmark data for turbulent flow in porous media. Since perfect DNS calculations require the resolution of scales down to the Kolmogorov scale, often only approximate DNS solutions can be obtained, especially for high Reynolds numbers. This is accounted for by using and comparing two different DNS approaches, a finite volume method (FVM) with grid refinement towards the wall and a lattice Boltzmann method (LBM) with equal grid distribution. The solid matrix was simulated by a large number of rectangular bars arranged periodically. The number of bars in the solution domain with periodic boundary conditions was reduced systematically until a minimum size was found that does not suppress any large-scale turbulent structures. Two-point correlations, integral length scales and energy spectra were determined in order to answer the question of whether or not macroscopic turbulence can be found in porous media.

CFD model for pneumatic mixing with bubble chains: Application to glass melts

A relatively simple approximate single-phase model (SPM) was developed for flow simulation of viscous liquid motion generated by a chain of rising bubbles. The complete detailed information about the gas–liquid two-phase flow was obtained with help of the full multi-phase model (MPM) using the direct numerical simulation (DNS) approach of the volume of fluid method (VOF), as implemented in the Fluent software. The overall driving effect of the bubble chain was expressed with an approximate buoyancy force closure formula, obtained in form of correlations of the MPM numerical data. This formula was compared with other published closures and used as a source term in SPM. SPM was solved and the results compared with MPM. Good agreement was found. SPM worked well in a range of bubbling frequency, liquid viscosity, and with temperature gradients. The replacement of the highly demanding and complex MPM with SPM is a great simplification of the flow modelling. SPM can be readily used for simulating the flow structure in containers with pneumatic mixing and design of equipment, where several rows of bubbling nozzles are installed. In this work, our modelling was done for the case of glass melts in glass furnaces.

Analysis of a filtered flamelet approach for coarse DNS of premixed turbulent combustion

Nowadays numerical simulations of premixed turbulent combustion can be performed at high spatial resolutions due to the continuously increasing computational power. At a resolution of the order of the laminar flame thickness there is negligible turbulence in the subgrid level and a proper reconstruction of the filtered flame is adequate for accurate predictions. In this scenario, which can be termed as a coarse scale Direct Numerical Simulation (DNS), reconstruction of the filtered flame using a spatially filtered flamelet (1-D laminar flame) is analysed. First, DNS of a turbulent premixed slot burner is computed to generate a reference database. This database is filtered at different length scales of the order of laminar flame thickness for an a priori assessment of turbulent flame reconstruction using a filtered flamelet. The filtered DNS source term, molecular diffusion and subgrid convection of the reaction progress variable is compared with a flamelet convoluted with top-hat profiles of the same order. The source term and molecular diffusion are approximated well from the filtered flamelet while the unresolved convection shows deviation. Finally, the reconstruction of the turbulent flame is tested a posteriori on the same filter widths (mesh size) and the results are compared. It is found that a flamelet filtered spatially at the order of the computational mesh and tabulated using a single controlling variable i.e. the reaction progress variable is suitable for a coarse scale DNS of premixed turbulent combustion.

DNS of hydrodynamically interacting droplets in turbulent clouds: Parallel implementation and scalability analysis using 2D domain decomposition

The study of turbulent collision of cloud droplets requires simultaneous considerations of the transport by background air turbulence (i.e., geometric collision rate) and influence of droplet disturbance flows (i.e., collision efficiency). In recent years, this multiscale problem has been addressed through a hybrid direct numerical simulation (HDNS) approach (Ayala et al., 2007). This approach, while currently is the only viable tool to quantify the effects of air turbulence on collision statistics, is computationally expensive. In order to extend the HDNS approach to higher flow Reynolds numbers, here we developed a highly scalable implementation of the approach using 2D domain decomposition. The scalability of the parallel implementation was studied using several parallel computers, at 5123 and 10243 grid resolutions with O(106)–O(107) droplets. It was found that the execution time scaled with number of processors almost linearly until it saturates and deteriorates due to communication latency issues.To better understand the scalability, we developed a complexity analysis by partitioning the execution tasks into computation, communication, and data copy. Using this complexity analysis, we were able to predict the scalability performance of our parallel code. Furthermore, the theory was used to estimate the maximum number of processors below which the approximately linear scalability is sustained. We theoretically showed that we could efficiently solved problems of up to 81923 with O(100,000) processors. The complexity analysis revealed that the pseudo-spectral simulation of background turbulent flow for a dilute droplet suspension typical of cloud conditions typically takes about 80% of the total execution time, except when the droplets are small (less than 5 μm in a flow with energy dissipation rate of 400 cm2/s3 and liquid water content of 1 g/m3), for which case the particle–particle hydrodynamic interactions become the bottleneck. The complexity analysis was also used to explore some alternative methods to handle FFT calculations within the flow simulation and to advance droplets less than 5 μm in radius, for better computational efficiency. Finally, preliminary results are reported to shed light on the Reynolds number-dependence of collision kernel of non-interacting droplets.

Simulations of heterogeneous propellant combustion : Effect of particle orientation and shape

In this paper we have developed a direct numerical simulation approach to study the combustion of ammonium perchlorate-based heterogeneous solid propellants. Oxidizer particles are first supposed to be spheroidal and the effect of their orientation on the burning rate is examined. Particles aligned normal to the combustion surface are found to burn faster than when aligned parallel while randomly-oriented particles generally burn at an intermediate rate, at least for moderate aspect ratio. Simulations also show that the burning rate of random spheroids is relatively independent of the aspect ratio. Finally, the effect of particle shape is investigated using superellipsoidal particles but is here found to be limited for the chosen propellant and pressure.

Direct numerical simulation of heat and mass transfer of spheres in a fluidized bed

We have developed a direct numerical simulation approach combined with the immersed boundary (DNS–IB) method for studying heat transfer in particulate flows. In this method, fluid velocity and temperature fields are obtained by solving the modified momentum and heat transfer equations, which are due to the presence of heated particles in the fluid; particles are tracked individually and their velocities and positions are solved based on the equations of linear and angular motions; particle temperature is assumed to be constant. The momentum and heat exchanges between a particle and the surrounding fluid at its surface are resolved using the immersed boundary method with the direct forcing scheme. The DNS–IB method has been used to study the heat transfer of 225 heated spheres in a fluidized bed. By exploring the rich data generated from the DNS–IB simulations, we are able to obtain statistically averaged fluid and particle velocities as well as the overall heat transfer rate in the fluidized bed. Good agreement between the current study and the one by Pan et al. (2002) is found for the hydrodynamic properties of the bed such as pressure gradients within the bed and the relationship between fluidization velocity and bed solid fraction. The particle-averaged Nusselt number is found to increase as the fluidization velocity increases and the bed height rises; particles at the entrance of the bed tend to have the maximum heat transfer rate because of the higher particle–fluid temperature gradients in this region; as the fluid moves upward in the bed, it gets warmer, which reduces particle–fluid temperature gradients and decreases the transfer rate of particles.

A two‐scale failure model for heterogeneous materials: numerical implementation based on the finite element method

SUMMARYIn the first part of this contribution, a brief theoretical revision of the mechanical and variational foundations of a Failure‐Oriented Multiscale Formulation devised for modeling failure in heterogeneous materials is described.The proposed model considers two well separated physical length scales, namely: (i) the macroscale where nucleation and evolution of a cohesive surface is considered as a medium to characterize the degradation phenomenon occurring at the lower length scale, and (ii) the microscale where some mechanical processes that lead to the material failure are taking place, such as strain localization, damage, shear band formation, and so on. These processes are modeled using the concept of Representative Volume Element (RVE). On the macroscale, the traction separation response, characterizing the mechanical behavior of the cohesive interface, is a result of the failure processes simulated in the microscale. The traction separation response is obtained by a particular homogenization technique applied on specific RVE sub‐domains. Standard, as well as, Non‐Standard boundary conditions are consistently derived in order to preserve objectivity of the homogenized response with respect to the micro‐cell size.In the second part of the paper, and as an original contribution, the detailed numerical implementation of the two‐scale model based on the finite element method is presented. Special attention is devoted to the topics, which are distinctive of the Failure‐Oriented Multiscale Formulation, such as: (i) the finite element technologies adopted in each scale along with their corresponding algorithmic expressions, (ii) the generalized treatment given to the kinematical boundary conditions in the RVE, and (iii) how these kinematical restrictions affect the capturing of macroscopic material instability modes and the posterior evolution of failure at the RVE level.Finally, a set of numerical simulations is performed in order to show the potentialities of the proposed methodology, as well as, to compare and validate the numerical solutions furnished by the two‐scale model with respect to a direct numerical simulation approach. Copyright © 2013 John Wiley & Sons, Ltd.

Interaction of a Mach 2.25 turbulent boundary layer with a fluttering panel using direct numerical simulation

The interaction between a thin metallic panel and a Mach 2.25 turbulent boundary layer is investigated using a direct numerical simulation approach for coupled fluid-structure problems. The solid solution is found by integrating the finite-strain, finite-deformation equations of elasticity using a non-linear 3D finite element solver, while the direct numerical simulation of the boundary layer uses a finite-difference compressible Navier-Stokes solver. The initially laminar boundary layer contains low amplitude unstable eigenmodes that grow in time and excite traveling bending waves in the panel. As the boundary layer transitions to a fully turbulent state, with Reθ ≈ 1200, the panel's bending waves coalesce into a standing wave pattern exhibiting flutter with a final amplitude approximately 20 times the panel thickness. The corresponding panel deflection is roughly 25 wall units and reaches across the sonic line in the boundary layer profile. Once it reaches a limit cycle state, the panel/boundary layer system is examined in detail where it is found that turbulence statistics appear to be modified by the presence of the compliant panel, the effect of which is forgotten within one integral length downstream of the panel.

Velocity relaxation of a particle in a confined compressible fluid

The velocity relaxation of an impulsively forced spherical particle in a fluid confined by two parallel plane walls is studied using a direct numerical simulation approach. During the relaxation process, the momentum of the particle is transmitted in the ambient fluid by viscous diffusion and sound wave propagation, and the fluid flow accompanied by each mechanism has a different character and affects the particle motion differently. Because of the bounding walls, viscous diffusion is hampered, and the accompanying shear flow is gradually diminished. However, the sound wave is repeatedly reflected and spreads diffusely. As a result, the particle motion is governed by the sound wave and backtracks differently in a bulk fluid. The time when the backtracking of the particle occurs changes non-monotonically with respect to the compressibility factor ε = ν∕ac and is minimized at the characteristic compressibility factor. This factor depends on the wall spacing, and the dependence is different at small and large wall spacing regions based on the different mechanisms causing the backtracking.

Kinematic and dynamic collision statistics of cloud droplets from high-resolution simulations

We study the dynamic and kinematic collision statistics of cloud droplets for a range of flow Taylor microscale Reynolds numbers (up to 500), using a highly scalable hybrid direct numerical simulation approach. Accurate results of radial relative velocity (RRV) and radial distribution function (RDF) at contact have been obtained by taking advantage of their power-law scaling at short separation distances. Three specific but inter-related questions have been addressed in a systematic manner for geometric collisions of same-size droplets (of radius from 10 to 60 μm) in a typical cloud turbulence (dissipation rate at 400 cm2 s−3). Firstly, both deterministic and stochastic forcing schemes were employed to test the sensitivity of the simulation results on the large-scale driving mechanism. We found that, in general, the results are quantitatively similar, with the deterministic forcing giving a slightly larger RDF and collision kernel. This difference, however, is negligible for droplets of radius less than 30 μm. Secondly, we have shown that the dependence of pair statistics on the flow Reynolds number Rλ or larger scale fluid motion is of secondary importance, with a tendency for this effect to saturate at high enough Rλ leading to Rλ-independent results. Both DNS results and theoretical arguments show that the saturation happens at a smaller Rλ for smaller droplets. Finally, since most previous studies of turbulent collision of inertial particles concerned non-sedimenting particles, we have specifically addressed the role of gravity on collision statistics, by simultaneously simulating collision statistics with and without gravity. It is shown that the collision statistics is not affected by gravity when a < ac, where the critical droplet radius ac is found to be around 30 μm for the RRV, and around 20 μm for the RDF. For larger droplets, gravity alters the particle–eddy interaction time and significantly reduces the RRV. The effect of gravity on the RDF is rather complex: gravity reduces the RDF for intermediate-sized droplets but enhances the RDF for larger droplets. In addition, we have studied the scaling exponents of both RDF and RRV, and found that gravity modifies the RDF scaling exponents for both intermediate and large particles, in a manner very similar to the effect of gravity on the RDF at contact. Gravity is shown to cause the scaling exponents for RDF and RRV to level off for large droplets, in contrast to diminishing exponents for non-sedimenting particles.

Direct numerical simulation of turbulent flow over backward-facing at high Reynolds numbers

Direct numerical simulation (DNS) was performed for the first time to study the flow over a backward-facing step at a high Reynolds number on a coarse grid. The flow over backward-facing step is the typical turbulent flow controlled by large eddy, in which the effect of small eddy could be negligible as an approximation. The grid dimension could easily satisfy the resolution requirement to describe the characteristics of a large eddy flow. Therefore, direct numerical simulation of N-S equations to obtain the turbulent flow field on the coarse grid could be realized. Numerical simulation of a two-dimensional flow over a backward-facing step at a Reynolds number Re=44000 was conducted using Euler-Lagrange finite element scheme based on the efficient operator-splitting method (OSFEM). The flow field was descretized by triangle meshes with 16669 nodes. The overall computational time only took 150 min on a PC. Both the characteristics of time-averaged and instantaneous turbulent flow were simultaneously obtained. The analysis showed that the calculated results were in good agreement with the test data. Hence, the DNS approach could become the reality to solve the complex turbulent flow with high Reynolds numbers in practical engineering.

FMM-based vortex method for simulation of isotropic turbulence on GPUs, compared with a spectral method

The Lagrangian vortex method offers an alternative numerical approach for direct numerical simulation of turbulence. The fact that it uses the fast multipole method (FMM)—a hierarchical algorithm for N-body problems with highly scalable parallel implementations—as numerical engine makes it a potentially good candidate for exascale systems. However, there have been few validation studies of Lagrangian vortex simulations and the insufficient comparisons against standard DNS codes has left ample room for skepticism. This paper presents a comparison between a Lagrangian vortex method and a pseudo-spectral method for the simulation of decaying homogeneous isotropic turbulence. This flow field is chosen despite the fact that it is not the most favorable flow problem for particle methods (which shine in wake flows or where vorticity is compact), due to the fact that it is ideal for the quantitative validation of DNS codes. We use a 2563 grid with Reλ=50 and 100 and look at the turbulence statistics, including high-order moments. The focus is on the effect of the various parameters in the vortex method, e.g., order of FMM series expansion, frequency of reinitialization, overlap ratio and time step. The vortex method uses an FMM code (exaFMM) that runs on GPU hardware using CUDA, while the spectral code (hit3d) runs on CPU only. Results indicate that, for this application (and with the current code implementations), the spectral method is an order of magnitude faster than the vortex method when using a single GPU for the FMM and six CPU cores for the FFT.

Computational prediction of mechanical hemolysis in aortic valved prostheses

The paper reports the prediction of mechanical hemolysis by three different models for the case of blood flow through aortic valved prostheses. Two of the adopted models are based on the action of instantaneous shear stress on the blood cells (stress-based), while the third accounts for the finite response time of cell deformation and relaxation (strain-based). Two aortic Dacron grafts commonly adopted in clinical practice are considered, and both are equipped with a bileaflet mechanical valve. One of the grafts reproduces the three sinuses of Valsalva, while the other is a straight tube. A direct numerical simulation approach is utilized to solve the complex fluid-structure-interaction problem and obtain detailed information of the flow patterns. To evaluate hemolysis, a large number of Lagrangian tracer particles were released at the inlet of the computational domain (upstream of the valve), and blood damage was evaluated along each trajectory for each model. We found that stress-based models predict higher levels of blood damage than the strain-based one. The same level of blood damage is observed in the two geometric configurations we considered, indicating that the adopted mechanical valve is primary risk factor for hemolysis.

Direct Numerical Simulation on the Receptivity, Instability, and Transition of Hypersonic Boundary Layers

The prediction of the laminar-turbulent transition of boundary layers is critically important to the development of hypersonic vehicles because the transition has a first-order impact on aerodynamic heating, drag, and vehicle operation. The success of transition prediction relies on a fundamental understanding of the relevant physical mechanisms. In the 20 years since the review by Kleiser &amp; Zang (1991) on the direct numerical simulation (DNS) of the boundary-layer transition, significant progress has been made on DNS in the hypersonic flow regime and in the spatial DNS approach. Many high-order shock-capturing and shock-fitting finite-difference methods have been developed and extensively applied to numerical simulations of the hypersonic boundary-layer transition. DNS has become a powerful research tool and has led to discoveries of new transition mechanisms. This article reviews the recent progress of DNS on hypersonic boundary-layer receptivity, instability, and transition. The current status and future directions are also presented.

Euler-Lagrange modelling of melting and solidification with moving solid particles

This work is devoted to the development of a numerical model for particulate flows during phase–change processes. The model is based on the Direct Numerical Simulation (DNS) approach. The particle collisions are modelled using the hard sphere approach. The interface velocity of the melting/solidification is calculated by means of the Stefan condition for each particle. In this work we consider a two–dimensional system consisting of 40 circular ice dendrites moving up owing to the density difference in a two–dimensional channel filled with hot water (first case) and undercooled water (second case). The results of the simulations are discussed.