Direct Numerical Simulation Technique Research Articles

Study of flow resistance coefficients acting on regular non-spherical particles in simple shear flow at moderate Reynolds numbers

This contribution is devoted to the study of the aerodynamic characteristics of non-spherical particles of definite regular shape, such as prolate and oblate ellipsoids as well as cylinders, immersed in a locally linear shear flow. The flow resistance coefficients of drag, lift and pitching torque are computed by means of the Particle Resolved Direct Numerical Simulation (PR-DNS) technique for such shapes as function of the Reynolds number Re. The parameter space comprises particle aspect ratio, AR, fluid spin ratio, ζ, and particle orientation angle, α. The Reynolds numbers of interest are in the intermediate range 1≤Re≤100, common in industrial and environmental processes. To properly understand and explain the obtained results, the role of the friction and pressure contributions to total flow coefficients is analyzed in detail for the different cases, allowing the differences between the considered shapes to be pointed out. On the other hand, the behavior of the pressure and skin friction coefficients in the particle plane of symmetry parallel to the flow is investigated as a function of the previous parameters (shape, AR, ζ, α), which provides further insights into the features of the shear flow around the non-spherical particles at finite Re. Finally, the influence of the shear flow magnitude and incidence angle on the location of the center of aerodynamic force is devised for the three shapes considered as function of Reynolds number and particle aspect ratio. It is expected that the information generated in this work will be useful for researchers to enhance the modeling of non-spherical particles immersed in non-uniform flows.

Order reduction of fracture mechanics in porous microstructures: A multiscale computing framework

We present a multiscale framework to systematically reduce the order of fracture-mechanics computations in complex porous microstructures captured by a pore-scale image (e.g., X-ray μCT). The framework allows formulating approximations that are more accurate than classical discrete element methods (DEM) and faster than high-fidelity direct numerical simulation (DNS) techniques, such as phase-field models (PFM). We propose two such approximations called the pore-level multiscale method (PLMM) and the image-based discrete element method (iDEM), the latter a generalization of classical DEM wherein the solid geometry is not simplified. Unlike iDEM, PLMM is equipped with the ability to iteratively control errors away from, but not at, the cracks. We refer to the uncorrected approximation as M0, and that with ω corrections as Mω. Both methods are based on the hypotheses that (H1) cracks nucleate at geometric constrictions of the solid, and (H2) bending/torsion moments at such constrictions are negligible. iDEM additionally assumes that (H3) the geometric enlargements of the solid are rigid. Through a series of benchmarks, performed on complex geometries subject to hydraulic, tensile, shear, and compressive loading, we show that H1 is largely valid under non-compressive loading, and H2 degrades the most under shear loading when bending/torsion moments dominate. H3 has two consequences: (1) M0 predictions can be reproduced by iDEM at a fraction of the cost, (2) but to compute them, iDEM’s parameters (spring constants) must be calibrated to reference data, which may not be available. Moreover, the calibrated parameters are non-unique and depend on the reference data chosen. The foregoing benchmarks are validated against two well-known PFMs, in which inherent uncertainties and deficiencies are also identified, especially under compressive loading. These are attributed to the so-called energy splits used. Lastly, we derive and validate closed-form expressions for the tensile and shear strengths of the solid used by PLMM/iDEM and the length-scale parameter and fracture toughness in PFM. This work provides fundamental insights into how porous materials fail, and how accurate approximations based on such understanding can be constructed and refined in the future.

Numerical study of the flow and heat transfer characteristics in a scale model of the vessel cooling system for the HTTR

The reactor cavity cooling system (RCCS) is a passive reactor safety system commonly present in the designs of High-Temperature Gas-cooled Reactors (HTGR) that removes heat from the reactor pressure vessel by means of natural convection and radiation. It is one of the factors responsible for ensuring that the reactor does not melt down under any plausible accident scenario. For the simulation of accident scenarios, which are transient phenomena unfolding over a span of up to several days, intermediate fidelity methods and system codes must be employed to limit the models’ execution time. These models can quantify radiation heat transfer well, but heat transfer caused by natural convection must be quantified with the use of correlations for the heat transfer coefficient. It is difficult to obtain reliable correlations for HTGR RCCS heat transfer coefficients experimentally due to such a system’s size. They could, however, be obtained from high-fidelity steady-state simulations of RCCSs. The Rayleigh number in RCCSs is too high for using a Direct Numerical Simulation (DNS) technique; thus, a Reynolds-Averaged Navier–Stokes (RANS) approach must be employed. There are many RANS models, each performing best under different geometry and fluid flow conditions. To find the most suitable one for simulating an RCCS, the RANS models need to be validated. This work benchmarks various RANS models against three experiments performed on the HTTR RCCS Mockup by the Japanese Atomic Energy Agency (JAEA) in 1993. This facility is a 1/6 scale model of a vessel cooling system (VCS) for the High Temperature Engineering Test Reactor (HTTR), which is operated by JAEA. Multiple RANS models were evaluated on a simplified 2d-axisymmetric geometry. They were found to reproduce the experimental temperature profiles with errors of up to 22% for the lowest temperature benchmark and 15% for the higher temperature benchmarks. The results highlight that the pragmatic turbulence models need to be validated for high Rayleigh natural convection-driven flows and improved accordingly, more publicly available experimental data of RCCS resembling experiments is needed and indicate that a 2d-axisymmetric geometry approximation is likely insufficient to capture all the relevant phenomena in RCCS simulations.

Direct simulation of viscoelastic-viscoelastic emulsions in sliding bi-periodic frames using Cahn–Hilliard formulation

In this study, a direct numerical simulation technique is proposed for 2-D droplet emulsions in simple shear flow with viscoelastic-viscoelastic liquid systems. To treat the multi-drop problem without wall effects, the sliding bi-periodic frame is combined with the Cahn–Hilliard equation in the DEVSS/DG (Discrete Elastic-Viscous Split Stress/Discontinuous Galerkin method) finite element framework. We employ the Galerkin weak formulation for flows of an Oldroyd-B fluid in a creeping regime with the Cahn–Hilliard equations. Sliding bi-periodic frame constraints for velocity, phase variable, and chemical potential are implemented by Lagrangian multipliers. We discuss the bulk rheology, morphological development, and effect of the viscoelasticity of emulsions through numerical analysis of single-, two-, and multi-drop problems. We report that increasing Weissenberg number orients the droplet in the flow direction, and the breakup time exhibits non-monotonic behavior similar to drop deformation. The coalescence of two drops is observed to be accelerated with the presence of viscoelasticity due to an extensional flow and normal stress difference. The bulk rheology is found to be consistent with prior experimental data from multi-drop problems. For the first time, the sliding bi-periodic frame has been implemented with the combination of the DEVSS/DG finite element method and the Cahn–Hilliard equation in the absence of wall effects, which is essential for understanding industrially important droplet emulsions.

A particle-resolved heat-particle-fluid coupling model by DEM-IMB-LBM

Multifield coupling is frequently encountered and also an active area of research in geotechnical engineering. In this work, a particle-resolved direct numerical simulation (PR-DNS) technique is extended to simulate particle-fluid interaction problems involving heat transfer at the grain level. In this extended technique, an immersed moving boundary (IMB) scheme is used to couple the discrete element method (DEM) and lattice Boltzmann method (LBM), while a recently proposed Dirichlet-type thermal boundary condition is also adapted to account for heat transfer between fluid phase and solid particles. The resulting DEM-IBM-LBM model is robust to simulate moving curved boundaries with constant temperature in thermal flows. To facilitate the understanding and implementation of this coupled model for non-isothermal problems, a complete list is given for the conversion of relevant physical variables to lattice units. Then, benchmark tests, including a single-particle sedimentation and a two-particle drafting-kissing-tumbling (DKT) simulation with heat transfer, are carried out to validate the accuracy of our coupled technique. To further investigate the role of heat transfer in particle-laden flows, two multiple-particle problems with heat transfer are performed. Numerical examples demonstrate that the proposed coupling model is a promising high-resolution approach for simulating the heat-particle-fluid coupling at the grain level.

Stokes-Cahn-Hilliard formulation in sliding bi-periodic frames for the simulation of two-phase flows

A direct numerical simulation technique is proposed in this study for droplet emulsions in simple shear flow in Newtonian-Newtonian liquid systems. We combine the sliding bi-periodic frame with the diffuse interface method in the finite element framework to treat the multi-drop problem without wall effects. We employ the standard velocity-pressure formulation in a creeping regime and the diffuse interface method with the Galerkin weak formulation. Sliding bi-periodic frame constraints are implemented by Lagrangian multiplier. The results are presented for the morphological development of a single drop, two drops, and multiple drops in a sliding bi-periodic frame. To preserve the identical solution for a given relative configuration of droplets (with different droplet locations) within the context of the bi-periodic frame, a phase-translation method is proposed. The method corrects the inconsistency in the time integration, which yields discrepancy in flow solution particularly for problems with thin interfaces. By applying this procedure, accurate solutions for a thin interface are obtained regardless of location of the identical relative configuration of droplets. The effect of phase-field translation is also analyzed for the coalescence of two drops. For the first time, the sliding bi-periodic frame has been implemented with the combination of the standard finite element method and the diffuse interface method, which can be easily extended to more complex rheological liquids to solve industrially important problems.

Role of gravity in coagulation of colloidal particles under low-shear environments

Shear rates play a critical role in the coagulation-flocculation-sedimentation processes of colloidal particles. Under high shear environments, it is widely accepted that the median floc size at equilibrium (D) decreases with an increase in shear rates (G). For low shear conditions, however, conflicting D-G relationships were measured in previous laboratory experiments, without a clear explanation of the reasons and a reasonable reproduction of the physical processes. In this study, the direct numerical simulation technique was used to mimic the flocculation of two typical colloidal particles (i.e., latex and silica) with different densities but under similar hydrodynamic conditions. Our results show that the previous different observations at low shear are mainly caused by particle gravity, which influences the particle residence time in a device system. They also confirm previous arguments about limited residence time due to gravitational settling being the reason for the observed peak, in accordance with Winterwerp (1998) [J. Hydraul. Res. Vol 36, pp. 309–326]. This study implies that to better investigate the shear-dominated flocculation of high-density particles such as sediments, settling processes should be addressed unless particles are always in suspension under specific conditions.

Modelling non-ideal detonations in commercial explosives

Highly non-ideal explosives usually react expressively below their ideal velocities of detonation. In these cases, dimensional effects and product heterogeneities become important to proper model their respective detonation state. Although Direct Numerical Simulation (DNS) techniques can provide a complete and exact solution for this problem, their actual computation cost are still not practical for industrial applications. In order to minimize these constrains, a simplified two-dimensional steady non-ideal detonation model for cylindrical stick explosives is presented. Based on an ellipsoidal shock shape approach (ESSA), the proposed model combines the quasione-dimensional theory for the axial flow solution with the unconfined sonic post-flow conditions at the edge of the explosive. Once calibrated, the model offers the possibility to predict the non-ideal detonation state for any charge diameter, resulting in a full mapping of the diameter-effect curve of the explosive. In addition, the effect of the inert confiner on the detonation flow is calculated by coupling a mechanistic confinement approach with the ESSA model. Thus, the proposed engineering approach is used to model the main properties of one of the most common ammonium nitrate-based explosive used in mining and quarrying industries, including the complete axial flow solution.

Pore-Scale Simulation of Particle Flooding for Enhancing Oil Recovery

The particles, water and oil three-phase flow behaviors at the pore scale is significant to clarify the dynamic mechanism in the particle flooding process. In this work, a newly developed direct numerical simulation techniques, i.e., VOF-FDM-DEM method is employed to perform the simulation of several different particle flooding processes after water flooding, which are carried out with a porous structure obtained by CT scanning of a real rock. The study on the distribution of remaining oil and the displacement process of viscoelastic particles shows that the capillary barrier near the location with the abrupt change of pore radius is the main reason for the formation of remaining oil. There is a dynamic threshold in the process of producing remaining oil. Only when the displacement force exceeds this threshold, the remaining oil can be produced. The flow behavior of particle–oil–water under three different flooding modes, i.e., continuous injection, alternate injection and slug injection, is studied. It is found that the particle size and the injection mode have an important influence on the fluid flow. On this basis, the flow behavior, pressure characteristics and recovery efficiency of the three injection modes are compared. It is found that by injecting two kinds of fluids with different resistance increasing ability into the pores, they can enter into different pore channels, resulting in the imbalance of the force on the remaining oil interface and formation of different resistance between the channels, which can realize the rapid recovery of the remaining oil.

Direct Numerical Simulation of Turbulent Heat Transfer with Slip and Temperature Jump

The fully developed turbulent flow and heat transfer in a plane channel at a nominal shear Reynolds number of 180 is studied via the direct numerical simulation technique. A linear velocity-slip boundary condition is assumed on one of the channel walls, whereas the classical no-slip condition is imposed on the other wall. The heat transfer at the slip wall is assumed to occur with and without temperature jump, whereas a classical first-kind temperature boundary condition is assumed on the no-slip wall. Various statistical quantities are shown and discussed. It is observed that the velocity fluctuations are attenuated in the proximity of the slip wall but they are amplified in the vicinity of the no-slip wall. The behavior is somewhat different for the temperature fluctuations. For the case without temperature jump, the temperature fluctuations are amplified near the slip wall, whereas they are attenuated close to the no-slip wall. For the case with temperature jump, the modifications of turbulent heat fluxes are more dramatic and they decrease by increasing the amount of wall slip. In general, the velocity slip condition will result in an enhanced heat transfer if no temperature jump is taken into account. This behavior is reversed when the temperature jump at the slip wall is taken into account. The budget of temperature variance is shown and discussed for the case with temperature jump.

Wind Reversal in Bubbly Natural Convection

The multi-phase Rayleigh–Bènard convection has been weakly investigated, even though it plays a leading role in the theoretical and applied physics of the heat transfer enhancement. For the case of moderate turbulent convection, a rather unexpected result is an unusual kind of wind reversal, in the sense that the fluid is found to be strongly influenced by the bubbles, whereas the bubbles themselves appear to be little affected by the fluid, despite the relative smallness of the Stokes numbers. The wind reversal induced by the bubbles dispersed in the fluid is a new and remarkable phenomenon in multi-phase flows that provides further perspectives in understanding the complex physics leading the enhancement of thermal convection. For this reason, the fundamental research proposed in this paper aimed to identify a space of control parameters and the physical mechanisms responsible for the wind reversal induced by dispersed bubbles in a confined convective flow. The strength of the following description lies in an innovative numerical approach, based on the multi-scale physics induced by the coupling of the local thermal and mechanical mechanisms arising between each bubble and the surrounding fluid. The continuous phase has been solved numerically using the direct numerical simulation (DNS) technique and each bubble has been tracked by means of a particle Lagrangian model.

Sedimentation of general shaped particles using a multigrid fictitious boundary method

In this paper, a direct numerical simulation technique, the Finite Element Fictitious Boundary Method (FBM), is used to simulate fluid–solid two-phase flows of different general shaped particles. The momentum interactions between solid and fluid phases are handled by using the FBM. The continuity and momentum equations are solved on a fixed Eulerian grid that is independent of flow features by using a discrete projection scheme inside a multi-grid finite element approach. A detailed description is presented for the geometric representation and modeling of two-dimensional particles of different general shapes, i.e., circular, elliptical, square, rectangular, triangular, and pentagonal shapes inside the fluid. We discussed the effects of particle shapes and the influences on the settling behavior of the particles. A comparison of the settling trajectories of the particles of the same mass but with different shapes is presented. Moreover, depending upon the particle’s shape, some interesting facts are discovered, which have a great influence on the particles’ trajectory and settling velocity. Some very important correlations between the drag force coefficient and particle’s Reynolds numbers with different density ratios of particles are obtained. Furthermore, we also studied the settling behavior of elliptical and rectangular particles with different axis ratios and a boomerang particle with different concave angles. The authors of the article agree to the retraction of the article effective AUGUST 20, 2021.

Pore-Scale Modeling of the Fuel Cell Catalyst Layer: The Role of Nafion Thin Films

In this work, a multiphysics pore-scale model of electrochemical reactive transport inside a fuel cell catalyst layer (CL) is presented and is used to elucidate the role of Nafion thin films on the overall performance of the CL. One of the main challenges of modeling transport in CL is the hierarchy of length scales and the multiphysics nature of transport. Consequently, traditional continuum methods, in which volume-averaged quantities are used to encapsulate the microstructural features, cannot give a clear picture of the multiphysics transport processes inside the CL. Alternatively, pore-scale methods, which take into account the true geometry down to the pore level, are suited to capture these processes for such systems. Previously, researchers have mainly employed direct numerical simulation (DNS) techniques for pore-scale modeling of the CL. While these techniques are highly accurate, they are very computationally expensive. This limits the domain size that can be simulated in a reasonable time. In this work, pore network modeling, a computationally cheap technique to simulate pore-scale transport, is used to study transport in the CL. Using this approach, we were able to speed up the simulation runtime up to 20-50x compared to DNS. Consequently, simulation of much larger domains of the CL was made possible, leading to more representative results. Finally, the developed multiphysics pore-scale model was used to study the role of Nafion thin films on the overall performance of the CL. It is notable that the network approach was only applied to the void phase, and the Nafion region was resolved using DNS. Therefore, yet further speed up can potentially be obtained by applying the network approach to the Nafion phase.

Noise-induced shifts in dynamics of multi-rhythmic population SIP-model

Motivated by important ecological applications, we investigate a three-dimensional population system describing mutual dynamics of predator (P) and prey divided into susceptible (S) and infected (I) species. In our study, we focus on a parameter zone with complex multi-rhythmic behavior where several oscillatory attractors coexist, both regular and chaotic. An effect of random disturbances in this SIP-model is investigated on the basis of direct numerical simulation and analytical stochastic sensitivity technique. It is revealed that random fluctuations in the rate of the infection spread result in the transitions between oscillatory modes with extinction of predator. It is shown that these noise-induced phenomena are accompanied by stochastic bifurcations and multi-stage order-chaos transitions.

Membrane filtration performance enhancement and biofouling mitigation using symmetric spacers with helical filaments

Optimization of the feed spacer geometry is one of the key challenges for improved ultrafiltration performance in water treatment and desalination. Novel feed spacers with different number of helices (1–3) along the spacer filament are proposed. To elucidate the intrinsic ability of the helical feature on filtration process improvement, experiments were conducted at two different fluid inlet velocities (U0 = 0.166 m/s and 0.182 m/s). The presence of micro-helices in the filaments aids significantly in increasing the specific permeate flux when compared to the standard spacer (without helices). The highest improvement was observed in the case of 3-helical spacer (291% specific permeate flux increase at U0 = 0.182 m/s). Furthermore, Optical Coherence Tomography (OCT) imaging demonstrated less (bio)fouling amount developed on membrane surface equipped with helical spacers, whereas, a thicker and more dense cake fouling layer appeared in the case of standard spacer. Moreover, novel helical design spacers reduce the pressure drop inside the channel. The 3-helical spacer was found to have the least feed-channel pressure drop (65% of decrease relative to standard spacer). Numerical analysis was simultaneously realized by the Direct Numerical Simulation (DNS) technique to understand the hydrodynamic behavior at an elemental level inside the filtration channel. Low shear stress and high local velocity magnitudes were observed in presence of helical spacers resulting in (bio)fouling mitigation on filtration membrane surface.

Machine Learning Regression Approach to the Nanophotonic Waveguide Analyses

Machine learning is an application of artificial intelligence that focuses on the development of computer algorithms which learn automatically by extracting patterns from the data provided. Machine learning techniques can be efficiently used for a problem with a large number of parameters to be optimized and also where it is infeasible to develop an algorithm of specific instructions for performing the task. Here, we combine the finite element simulations and machine learning techniques for the prediction of mode effective indices, power confinement, and coupling length of different integrated photonics devices. Initially, we prepare a dataset using COMSOL Multiphysics and then this data is used for training while optimizing various parameters of the machine learning model. Waveguide width, height, operating wavelength, and other device dimensions are varied to record different modal solution parameters. A detailed study has been carried out for a slot waveguide structure to evaluate different machine learning model parameters including number of layers, number of nodes, choice of activation functions, and others. After training, this model is used to predict the outputs for new input device specifications. This method predicts the output for different device parameters faster than direct numerical simulation techniques. Absolute percentage error of less than 5% in predicting an output has been obtained for slot, strip, and directional waveguide coupler designs. This study paves the step towards using machine learning based optimization techniques for integrated silicon photonics devices.

Numerical study of flexible flapping wings with an immersed boundary method: Fluid–structure–acoustics interaction

The fluid–structure–acoustics interaction of flexible flapping wings is numerically studied by using an immersed boundary method at a Mach number of 0.1. In this study, a wing flapping in a uniform flow with prescribed leading edge motion is considered. Apart from rigid wings, flexible wings of various bending rigidities and mass ratios are also examined. By using a direct numerical simulation technique, the sound generation mechanism is identified. The numerical results show that the wing with combined translational and rotational motion generates smaller sounds than the translating wing, and larger rotational angles transform the dipole sound to a monopole one. Similar sound fields are observed among all these wings, except that the direction of the sound shifts to the downstream for large flexible wings. Frequency analysis shows that the sound is dominated by the flapping frequency and two times of the flapping frequency in the vertical and horizontal directions, respectively. The results also show that the thrust increases first then decreases with bending rigidity for all three mass ratios considered, and the lighter wing suffers more decrease. However, the fluctuating pressure for the three mass ratios varies significantly. Specifically, it increases significantly with the decreasing bending rigidity when the structural inertia is dominating at m∗=5.0; but it experiences a reduction contrarily when the aerodynamics becomes dominant at m∗=0.5; for the medium mass ratio at m∗=1.0, no significant effects are observed. The comparisons indicate that the flexible wing with a lower mass ratio (e.g. m∗=0.5) and a medium flexibility (e.g. ω∗=0.3) achieves lower sound generation without significant thrust decrease. In addition, the sound on the windward side is pronounced significantly when the wing is flapping with a stroke plane angle less than 90o. The present results can expand the currently limited database of fluid–structure–acoustics interaction, and also provide an insight for the optimization of flight vehicle using flapping wing.

Improving the Accuracy of Fictitious Domain Method Using Indicator Function from Volume Intersection

Fictitious domain method (FDM) is a commonly accepted direct numerical simulation technique for moving boundary problems. Indicator function used to distinguish the solid zone and the fluid zone is an essential part concerning the whole prediction accuracy of FDM. In this work, a new indicator function through volume intersection is developed for FDM. In this method, the arbitrarily polyhedral cells across the interface between fluid and solid are located and subdivided into tetrahedrons. The fraction of the solid volume in each cell is accurately computed to achieve high precision of integration calculation in the particle domain, improving the accuracy of the whole method. The quadrature over the solid domain shows that the newly developed indicator function can provide results with high accuracy for variable integration in both stationary and moving boundary problems. Several numerical tests, including flow around a circular cylinder, a single sphere in a creeping shear flow, settlement of a circular particle in a closed container, and in-line oscillation of a circular cylinder, have been performed. The results show good accuracy and feasibility in dealing with the stationary boundary problem as well as the moving boundary problem. This method is accurate and conservative, which can be a feasible tool for studying problems with moving boundaries.

Fully resolved simulations of thermal convective suspensions of elliptic particles using a multigrid fictitious boundary method

In particulate flows, particle-particle collisions play a very important role in determining the flow behavior of the fluid-particle two-phase systems. Thus, it is very important in the numerical simulations of particulate flows to treat the particle-particle interactions with a felicitous method. In this paper, a recently developed direct numerical simulation (DNS) technique, the Finite Element Fictitious Boundary Method (FEM-FBM), for thermal convective particulate flows is used for the simulations of dense particulate suspensions of elliptic shaped particles. The momentum and temperature flow fields are coupled with the aid of Boussinesq approximation. The thermal and momentum interactions between solid and fluid phases are handled by using the Fictitious boundary method (FBM). The continuity, momentum, and energy equations are solved on a fixed Eulerian mesh which is independent of flow features by using a multi-grid finite element scheme. A modified collision model is proposed that can handle not only the interactions between the circular particles but also effectively treat the collisions between the elliptic shaped particles. Firstly, we validate the newly developed collision model for isothermal circular particles together with a comparative study of “drafting, kissing and tumbling” (DKT) motion of both elliptic and circular shaped isothermal particles. Then we investigated the DKT motions of the hot and cold elliptic particles with energy exchange and studied the lateral behavior of the particles in numerous settings. Further, we studied the DKT motion of catalyst particles and investigate the particles behavior at different Grashof numbers. Numerical tests are performed to show that the present method is robust and provides an efficient approach for the simulations of particulate flows with a large number of elliptic particles.

Numerical and experimental investigation into hypersonic boundary layer transition induced by roughness elements

In this work, the Direct Numerical Simulation (DNS) and Oil-Film Interferometry (OFI) technique are used to investigate the hypersonic boundary layer transition induced by single and double roughness elements at Mach number 5. For single roughness, the DNS results showed that both horseshoe vortices and hairpin vortices caused by shear layer instability can affect the boundary layer instability. The generation of the near-wall unstable structure is the key point of boundary layer transition behind the roughness element. At the downstream of the roughness element, the interaction between horseshoe vortices and hairpin vortices will spread in the spanwise direction. For double roughness elements, the effect of the spacing between roughness elements on the transition is studied. It is found that the case of higher spacing between roughness elements is more effective for inducing transition than the lower one. The interaction between two adjacent roughness elements can suppress the evolution of horseshoe vortices in the downstream and trigger the instability of shear layer. Thus, the transition will be suppressed accordingly.