Previous Direct Numerical Simulation Research Articles

Understanding Low-Speed Streaks and Their Function and Control through Movable Shark Scales Acting as a Passive Separation Control Mechanism

The passive bristling mechanism of the scales on the shortfin mako shark (Isurus oxyrinchus) is hypothesized to play a crucial role in controlling flow separation. In the hypothesized mechanism, the scales are triggered in response to patches of reversed flow at the onset of separation occurring in the low-speed streaks that form in a turbulent boundary layer. The two goals of this investigation were as follows: (1) to measure the reversing flow occurring within the low-speed streaks in a separating turbulent boundary layer; (2) to understand the passive flow control mechanism of movable shark skin scales that inhibit reversing flow within the low-speed streaks. Experiments were conducted using digital particle image velocimetry (DPIV). DPIV was used to analyze the flow in a turbulent boundary layer subjected to an adverse pressure gradient formation over both a smooth flat plate and a flat plate on which shark skin specimens were affixed. The experimental analysis of the flow over the smooth flat plate corroborated the findings of previous direct numerical simulation studies, which indicated that the average spanwise spacing of the low-speed streaks increases in the presence of adverse pressure gradients upstream of the point of separation. However, the characteristics of the flow over the shark skin specimen more closely resemble that of a zero-pressure gradient turbulent boundary layer. A comparative analysis of the width and velocity of the reversed streaks between flat plate and shark skin cases reveals that the mean spanwise spacing decreases, and thus, the number of streaks increases over the shark skin. Additionally, the reversed streaks observed over shark scales are thinner and the highest negative velocity within the streaks falls within the range required to bristle the scales.

Advanced Modeling of Flow and Heat Transfer in Rotating Disk Cavities Using Open-Source Computational Fluid Dynamics

Abstract Code_Saturne, an open-source computational fluid dynamics (CFD) code, has been applied to a range of problems related to turbomachinery internal air systems. These include a closed rotor–stator disk cavity, a co-rotating disk cavity with radial outflow and a co-rotating disk cavity with axial throughflow. Unsteady Reynolds-averaged Navier–Stokes (RANS) simulations and large eddy simulations (LES) are compared with experimental data and previous direct numerical simulation and LES results. The results demonstrate Code_Saturne's capabilities for predicting flow and heat transfer inside rotating disk cavities. The Boussinesq approximation was implemented for modeling centrifugally buoyant flow and heat transfer in the rotating cavity with axial throughflow. This is validated using recent experimental data and CFD results. Good agreement is found between LES and RANS modeling in some cases, but for the axial throughflow cases, advantages of LES compared to URANS are significant for a high Reynolds number condition. The wall-modeled large eddy simulation (WMLES) method is recommended for balancing computational accuracy and cost in engineering applications.

Direct numerical simulations and spectral proper orthogonal decomposition analysis of shocklet-containing turbulent channel counter-flows

Counter-flow configurations in a confined channel flow provide an efficient framework to study high intensity turbulent mixing processes. In a previous study (Physical Review Fluids, 6(9), p.094603), a wall-bounded counter-flow turbulent channel configuration was presented as an effective framework for addressing certain challenges related to the study of compressibility effects on turbulence as an alternative to free shear layer and Poiseuille/Couette type flows. Here, the previous direct numerical simulations are extended to a higher Mach number (M=0.7) to quantify direct and indirect effects of compressibility. It is found that the configuration is able to produce large numbers of embedded shocklets, leading to significant asymmetry in probability density functions of dilatation. Reducing the Prandtl number from 0.7 to 0.2 increases the compressibility effect further by reducing the bulk heating in the channel. A peak turbulent Mach number close to unity is obtained, for which the contribution of the dilatational dissipation to the total dissipation is nevertheless found to be limited to ∼6%. Indirect effects of compressibility are much larger, with changes of up to 40% in Favre normal stresses, despite the mean flow and shear stress being almost unaffected by compressibility in this configuration. Given the inflectional nature of the turbulent mean flow it is also interesting to identify large structures. Spectral Proper Orthogonal Decomposition (SPOD) reveals a full spectrum with a slow decay of energy with mode number. Mode shapes are three-dimensional with the low frequencies displaying elongated streaks in the velocity field at the channel centre plane.

Stability theory for metal pad roll in cylindrical liquid metal batteries

When liquid metal batteries are charged or discharged, strong electrical currents are passing through the three liquid layers that we find in their interior. This may result in the metal pad roll instability that drives gravity waves on the interfaces between the layers. In this paper, we investigate theoretically metal pad roll instability in idealised cylindrical liquid metal batteries that were simulated previously by Weber et al. (Phys. Fluids, vol. 29, no. 5, 2017b, 054101) and Horstmann et al. (J. Fluid Mech., vol. 845, 2018, pp. 1–35). Near the instability threshold, we expect weakly destabilised gravity waves, and in this parameter regime, we can use perturbation methods to find explicit formulas for the growth rate of all possible waves. This perturbative approach also allows us to include dissipative effects, hence we can locate the instability threshold with good precision. We show that our theoretical growth rates are in quantitative agreement with previous and new direct numerical simulations. We explain how our theory can be used to estimate a lower bound on cell size beneath which metal pad roll instability is unlikely.

Production and transport of turbulent kinetic energy from premixed flames subjected to mean pressure gradients

The influence of mean favorable pressure gradients on the production and transport of turbulent kinetic energy is experimentally investigated in a premixed bluff-body combustor. The combustor wall geometry is modified into converging, straight, or diverging configurations to experimentally alter the mean pressure gradient in the reacting flow field. For each configuration, turbulent kinetic energy production and transport dynamics are characterized with high-speed particle image velocimetry and simultaneous CH* chemiluminescence imaging. For all cases, the premixed reaction increases the turbulent kinetic energy and integral length scales in the flow. However, increasing the magnitude of the mean favorable pressure gradient increases the turbulent kinetic energy produced by the flame. The source of increased turbulent fluctuations from reactants to products is first investigated with a steady flow energy analysis, and is accompanied with a Reynolds decomposed investigation of the kinetic energy in the flow. As the mean pressure gradient increases, the turbulent kinetic energy in the flow increases due to stronger magnitudes of flame-generated turbulence in the form of baroclinic torque. The evolution of the flame-generated turbulence is examined by decomposing the transport equation of turbulent kinetic energy. Increasing the mean favorable pressure gradient augments the largest magnitude transport terms, and the budget of turbulent kinetic energy is primarily driven by pressure gradient work and viscous dissipation. The results here differ from previous direct numerical simulation (DNS) studies, and should ultimately motivate additional research to aid the development of computational models which can better predict turbulent flame-flow dynamics in confined environments with mean pressure gradients.

A General Single-Node Second-Order Dirichlet Boundary Condition for the Convection–Diffusion Equation Based on the Lattice Boltzmann Method

A general single-node second-order Dirichlet boundary condition for curved boundaries for the convection–diffusion equation based on the lattice Boltzmann method has been developed. The boundary condition simply utilizes the bounce back rule for the temperature distribution and linear interpolation. The developed boundary condition is quite general and stable. The asymptotic analysis indicates that the proposed boundary scheme is of second-order accuracy. The comparison in terms of L2-norm between the simulation results and the analytical solutions confirms that it is indeed second-order accurate for 2D and 3D symmetric flows comprising straight and curved boundaries. Moreover, the Nusselt numbers for a spherical particle in uniform flow obtained by current simulations agree very well with the predictions of empirical correlations and the data of previous direct numerical simulations with a maximal relative deviation within 8%.

Model for the radial distribution function of polydisperse inertial spheres settling in homogeneous, isotropic turbulence

The inertial lag in particles' response to turbulent motions drives clustering, but differential sedimentation attenuates it when sizes are unequal. We accurately model the radial distribution function of polydisperse sedimenting particles in turbulence by building on previous theory and direct numerical simulations and validate it against available experimental data. Our model suggests that inertial clustering of micron size drops in clouds promotes like-sized drop collision. This can break the bottleneck of small nearly monodisperse droplets formed by condensation and facilitate a transition to rapid differential-sedimentation-driven coalescence which, in turn, leads to precipitation.

Three-Dimensional Direct Numerical Simulations of a Yawed Square Cylinder in Steady Flow

The effects of yaw angle on wake characteristics of a stationary square cylinder were investigated in terms of the hydrodynamic forces, the vortex shedding frequency, and the vortical structures using direct numerical simulations (DNS) at a Reynolds number of 1000. In total, four yaw angles, namely, α = 0°, 15°, 30°, and 45°, were considered. The three-dimensional (3D) Navier–Stokes equations were solved directly using the finite volume method in OpenFOAM. It was found that the first-order statistics of the drag coefficient and the Strouhal number satisfied the independence principle (IP) closely. However, the second-order statistics of the drag and lift coefficients deviated apparently from the IP for α ≥ 25°. The iso-surfaces of the spanwise vorticity gradually disorganized and the magnitudes of the spanwise vorticity contour decreased as the yaw angle α was increased from 0° to 45°. By contrast, the streamwise vorticity iso-surfaces were found to become more organized and the magnitudes of the spanwise velocity contour became larger as a result of the increase in yaw angle, indicating the impairment of the quasi-two-dimensionality and the enhancement of the three-dimensionality of the wake flow. Extensive comparisons were also made with previous DNS results for a yawed circular cylinder, and both similarities and differences between these two kinds of cylinder wakes are discussed.

Simulation of hypersonic boundary layer on porous surfaces using OpenFOAM

Hypersonic boundary layer on porous surfaces is numerically investigated using the open-source flow solver OpenFOAM. Computations of hypersonic boundary layer have not been widely conducted especially using open-source flow solvers. The solver rhoEnergyFoam Modesti and Pirozzoli (2017), which has been developed in the OpenFOAM framework, is successfully validated for hypersonic boundary layer flow in the current study. Direct numerical simulations (DNS) of boundary layer at Mach 6 are pursued here with a disturbance related to the Mack second mode. The evolution of the Mack second mode is compared with the current analysis of linear stability theory (LST) and previous DNS data. The stabilization of the Mack second mode on porous surfaces is well resolved in the current computations. In addition to the stabilization of the Mack second mode with porous surfaces, aerothermodynamic features, including drag and surface heat transfer, are investigated. Without porous surfaces, the heat flux variation is even larger than the mean heat flux due to the flow instability. Porous surfaces significantly reduce the heat flux variation to the level of the mean heat flux. Mean drag and heat transfer are marginally affected by porous surfaces. Stress analysis is accompanied to investigate whether the pore structure experiences severe stress under such a high-speed condition. The current investigation on porous surfaces indicates that the surface porosity is an effective method to suppress instabilities and subsequently the heat flux variation on the wall without hampering mean aerothermodynamic characteristics.

Direct Numerical Simulation of Flow over a Triangular Airfoil Under Martian Conditions

Martian conditions present various challenges when designing rotorcraft. Specifically, the thin atmosphere and low sound speed require Martian rotor blades to operate in a low-Reynolds-number (1000–10,000) compressible regime, for which conventional airfoils are not designed. Here, we use PyFR to undertake high-order direct numerical simulations (DNS) of flow over a triangular airfoil at a Mach number of 0.15 and Reynolds number of 3000. Initially, spanwise periodic DNS are undertaken. Extending the domain-span-to-chord ratio from 0.3 to 0.6 leads to better agreement with wind-tunnel data at higher angles of attack, when the flow is separated. This is because smaller domain spans artificially suppress three-dimensional breakdown of coherent structures above the suction surface of the airfoil. Subsequently, full-span DNS in a virtual wind tunnel are undertaken, including all wind-tunnel walls. These capture blockage and wall boundary-layer effects, leading to better agreement with wind-tunnel data for all angles of attack compared to spanwise periodic DNS. The results are important in terms of understanding discrepancies between previous spanwise periodic DNS and wind-tunnel data. They also demonstrate the utility of high-order DNS as a tool for accurately resolving flow over triangular airfoils under Martian conditions.

Two-time interval method to circumvent particle image velocimetry dynamic range

In the velocity range outside the velocity-dynamic range of a PIV (particle image velocimetry) system, the velocity measurements are known to result in noisy velocity vectors due to erroneous detection of correlation peaks, especially in the low-velocity region. Here a simple method is proposed that uses two different timings of a pulsed laser, in conjunction with a median-test of residuals based on the velocity data, to resolve both low and high velocities in a flow field. This method detects the erroneous vectors and replaces them with the correct vectors via a comparison of two sets of PIV data, procured simultaneously with an appropriate time-delay. The validation of the proposed method is demonstrated via a comparison of the present experimental data with previous experiments and DNS (direct numerical simulations) of a round turbulent jet.•The method uses two-time intervals to resolve two sets of velocities such that each set falls within the dynamic range of PIV corresponding to the time interval. (Dynamic range heavily depends on time interval).•A quantity called normalised residual median is defined based on “universal outlier detector” which is used to consolidate the two sets of data.•The strength of the method lies in its ability to obtain instantaneous entrainment velocity even in an unsteady flow.

Computational fluid dynamics simulation of buoyant mixing of miscible fluids in a tilted tube

Buoyancy-driven flows and mixing of fluids with different densities occur frequently both in nature and as part of industrial processes within chemical and petroleum engineering. This work investigates the buoyant exchange flow of two miscible fluids in a long tube with closed ends at varying tilt angles using OpenFOAM. The study focuses on the evolution of the concentration field and front velocities of the mixing zone at different inclinations. Numerical results based on a miscible solver agree with previous experiments and direct numerical simulations. Treating the fluids instead as immiscible with no surface tension leads to unrealistically high front velocities at intermediate inclinations.

Characteristics of wall-shear stress fluctuations in shock wave and turbulent boundary layer interaction

The wall-shear stress (WSS) fluctuations in the interaction of an oblique shock wave with a flat-plate turbulent boundary layer are investigated by means of direct numerical simulation (DNS) at Mach 2.25. The numerical results agree very well with previous experiments and DNS data in terms of turbulence statistics, wall pressure, and skin friction. The fluctuating WSS characteristics, including probability density function (PDF), frequency spectrum, space–time correlation, and convection velocity, are analysed systematically. It is found that the positively skewed PDF shape of the streamwise WSS fluctuations is significantly changed due to the presence of a separation bubble, while the PDF shape of the spanwise component is slightly affected, exhibiting a symmetric behaviour across the interaction. The weighted power-spectrum density map indicates that the low-frequency unsteadiness associated with the separated shock - exhibits little influence on the spectrum for either component, and no enhancement of the low-frequency energy is observed. A significant reduction in the spatial extent of the two-point correlation is observed, causing spanwise elongated coherence for the streamwise WSS fluctuations in the separation region. Moreover, the elliptic behaviour of the space–time correlations is essentially preserved throughout the interaction, and this is accompanied by a sudden reduction of the convection velocity in the separation bubble.

Large eddy simulations of high-magnetic Reynolds number magnetohydrodynamic turbulence for non-helical and helical initial conditions: A study of two sub-grid scale models

Pseudo-spectral large eddy simulation (LES) calculations of high-magnetic Reynolds number (Rem) incompressible magnetohydrodynamic (MHD) turbulence are carried out for two initial conditions, namely, the non-helical Orszag–Tang vortex and the strongly helical Arnold–Beltrami–Childress (ABC) flows using two eddy-viscosity-based sub-grid scale (SGS) approaches: the cross-helicity (CH) and dynamic Smagorinsky (DS) models. Validation is conducted through comparisons of 1923 LES calculations with in-house 5123 direct numerical simulations (DNS) at Reynolds number, Re=Rem=800. The results show that the CH model performs better than the DS model. The performance of the SGS models at higher Re is further evaluated by carrying out 3843 LES calculations at Re=Rem=7500. Various quantities including turbulent kinetic energy, turbulent magnetic energy, cross-helicity, helicity, vorticity structures, cosine of angle between velocity and magnetic field, cosine of angle between velocity and vorticity field, kinetic and magnetic energy spectra, and energy fluxes are analyzed to understand the capability of the two LES models in predicting the evolution of MHD turbulence. The higher Reynolds number flow shows a delay in the maximum dissipation with increased transfer of energy toward small scales, resulting in a −5/3 Kolmogorov inertial sub-range scaling. In addition, the effect of Reynolds number on the alignment between velocity and magnetic field, and the energy transfer between kinetic and magnetic energy, is studied. With the ABC flow having strong helicity and zero cross-helicity at low and high Reynolds numbers, a strong dynamo effect is also observed using the LES models, which is consistent with previous DNS.

STATISTICAL CHARACTERISTICS OF PRESSURE FLUCTUATION IN SHOCK WAVE AND TURBULENT BOUNDARY LAYER INTERACTION

Shock wave and turbulent boundary layer interaction widely exists in the internal and external flow of high-speed aircraft. The aerodynamic performance and flight safety of aircraft are seriously affected by the strong pressure fluctuation in the interaction region. To investigate statistical characteristics of fluctuating pressure, the interaction between an incident shock of 33.2° and a spatially developed Mach 2.25 turbulent boundary layer is analyzed by means of direct numerical simulation (DNS). The numerical results have been carefully validated against with previous experiment and DNS at similar flow conditions in terms of mean velocity profile, turbulence intensity and wall pressure distribution. Statistics at the wall and in the outer layer, including fluctuation intensity, power spectral density, two-point correlation and space-time correlation, are quantitatively compared. The differences between them are analyzed in detail. It is found that the effect of the shock interaction on the wall-pressure fluctuation and the fluctuating pressure in the outer layer are utterly different. Based on the analysis of the power spectra density, the fluctuations in the separated region are both characterized by the low-frequency content, but in the reattachment region, the peak frequency of outer pressure fluctuations quickly shifts to higher frequency, with the low-frequency energy of wall-pressure fluctuation still being predominant. It is identified that the two-point correlations of pressure fluctuation at the wall and in the outer layer are both more elongated in the spanwise direction than that in the streamwise direction. The integral scale at the wall is generally increased, while the one in the outer layer increases sharply after passing the shock and then gradually decreases. The analysis of space-time correlation indicates that the iso-correlation contours are similar to the elliptical distribution and the convection velocity deduced by the correlation is dramatically decreased. Downstream of the interaction, the convection velocity in the outer layer is higher than that of wall-pressure fluctuation.

Evolution and Features of Dust Devil‐Like Vortices in Turbulent Rayleigh‐Bénard Convection—A Numerical Study Using Direct Numerical Simulation

AbstractDust devils are convective vortices with a vertical axis of rotation that are made visible by entrained soil particles. These soil particles contribute to the atmospheric aerosol input, influencing the Earth radiation budget. Quantifying this contribution requires reliable information about the statistics of dust devils, their formation process, and how they are maintained. In the past, this information was mainly derived from field experiments and large‐eddy simulations (LESs). Field experiments suffer from the erratic occurrence of dust devils and the limited area that can be monitored reliably. In LESs, dust devils cannot be resolved completely, especially close to the ground. Additionally, they are affected by numerical features of surface boundary conditions, as well as subgrid‐scale models in an unknown way. To mitigate these limitations, we employ direct numerical simulations (DNSs) to improve our understanding of dust devils. We comprehensively investigate the statistics and structure of dust devils for Rayleigh numbers up to 1011using DNS of Rayleigh‐Bénard convection between two plates for the first time. We find that dust devil‐like structures occur in DNS with Rayleigh numbers much lower than in the atmosphere (≥107). These results support previous DNS studies in which vortices with vertical axes were observed but not further investigated. The dust devil statistics strongly depend on the Rayleigh number and velocity boundary conditions, but depend little on the aspect ratio of the model domain. Simulated dust devils show very similar properties to convective vortices analyzed in LESs of the atmospheric boundary layer.

Energy dissipation and enstrophy production/destruction at very low Reynolds numbers in the final stage of the transition period of decay in grid turbulence

Decay characteristics of turbulent kinetic energy and enstrophy in grid turbulence have been investigated in the far downstream region (x/M∼103: x is the downstream distance from the grid, M is the mesh size of the grid) through wind tunnel experiments using hot-wire anemometry, with the lowest turbulent Reynolds number Reλ≈5. The non-dimensional dissipation rate Cε increases rapidly toward the final stage of the transition period of decay and the profile agrees well with previous direct numerical simulation [W. D. McComb et al., “Taylor's (1935) dissipation surrogate reinterpreted,” Phys. Fluids 22, 061704 (2010)] and theoretical estimation [D. Lohse, “Crossover from high to low Reynolds number turbulence,” Phys. Rev. Lett. 73, 3223 (1994)] at very low Reλ in decaying and stationary isotropic turbulence. The present result of Cε is an update on the experimental data in grid turbulence toward a very low Reλ, where measurements have been absent. The energy spectrum in the dissipation range at very low Reλ deviates from a universal form observed at high Reynolds numbers. The decay rate of enstrophy is proportional to S+2G/Reλ (S is the skewness of the longitudinal velocity derivative and G is the destruction coefficient). It is shown that G and S+2G/Reλ increase rapidly with decreasing Reλ at very low Reλ, indicating that the effect of enstrophy destruction is dominant in the final stage of the transition period of decay. The profiles of S+2G/Reλ against Reλ is well fitted by a power-law function even in the final stage of the transition period of decay.

A novel model for incorporation of differential diffusion effects in PDF simulations of non-premixed turbulent flames based on reaction-diffusion manifolds (REDIM)

In this work, reaction-diffusion manifold (REDIM) reduced chemistry is used in the simulation of turbulent non-premixed flames based on a transported-probability density function model. Differential molecular diffusion is applied in the generation of the manifolds. This is the first work to consider the gradients of the reduced variables as additional parameters in the REDIM model, and one-directional gradients are utilized to generate the REDIM reduced chemistry. Hereby, the influence of turbulence on differential molecular diffusion is automatically considered in terms of reduced variable gradients, and the physical transport properties (e.g., diffusion coefficients) are used in a detailed way, without any additional modeling (e.g., unity-Lewis number assumption). Although the scalar gradients appear as multi-directional in a general turbulent reacting flow, previous direct numerical simulation analysis reveals that REDIMs generated from one-directional gradients can accurately describe the system featuring multi-directional gradients, if this one-directional gradient has a major effect on the chemistry. Here, it is proposed to obtain such gradients under the hypothesis that the flame structure is locally one-dimensional at each spatial position. In order to retrieve the gradients of the reduced variables for the interpolation of the thermo-kinetic states from the REDIM table, the sub-grid gradient is evaluated here from the particle fields. The well-known Sandia series of flames is selected to validate the proposed algorithm. The results show that the new algorithm can reproduce the thermo-kinetic quantities with high accuracy for all investigated flames.

Numerical study on spatial scale characteristics of sound scattering by a static isentropic vortex

The scattering of acoustic waves by a vortex is a fundamental problem of the acoustic waves propagation in complex flow field, which plays an important role in academic research and engineering application for sound source localization, acoustic target recognition and detection, the far field noise prediction, such as aircraft wake vortex identification, detection and ranging, acoustic target forecasting in turbulent shear flow, acoustic measurement and sound source localization in wind tunnel test, etc. The nonlinear scattering phenomenon occurs when acoustic wave passes through the vortex, which is mainly related to the length-scale ratio between the wavelength of acoustic wave and the core radius of the vortex. In this paper, a plane acoustic wave passing through a stationary isentropic vortex is numerically simulated by solving a two-dimensional compressible, unsteady Euler equation. A sixth-order linear compact finite difference scheme is employed for spatial discretization. Time integration is performed by a four-stage fourth-order Runge-Kutta method. The eighth-order spatial compact filter scheme is adopted to suppress high frequency errors. At the far field boundaries, buffer layer is applied to handle the outgoing acoustic wave. Under the matching condition, the accuracy of the numerical results is verified by comparing with the previous direct numerical simulation results. The acoustic scattering cross-section method is introduced to analyze the effects of different length-scale ratio on the acoustic pulsation pressure, acoustic scattering effective sound pressure and acoustic scattering energy. Scattering occurs when sound waves pass through the vortex, the acoustic field in front of the vortex is basically unaffected, and the acoustic wave front remains intact. A “vacuum” region is formed slightly below the acoustic field directly behind the vortex, and two primary interference bands and several secondary interference bands are formed on the upper and lower sides of the vortex. As the length-scale ratio increases, the sound scattering decreases and the influence of the vortex flow field on the acoustic field gradually weakens. The influence region of effective sound pressure of acoustic scattering is mainly concentrated behind the vortex. With the increase of the length scale ratio, the influence gradually increases and extends to the upstream, and then the influence region gradually decreases to the vicinity of the vortex. When the length scale ratio is greater than or equal to 6, the location of the maximum effective sound pressure of sound scattering jumps from the upper right to the lower right of the vortex. The influence of acoustic wave wavelength change on the acoustic scattering energy can be divided into three parts. With the increase of the length scale ratio, the maximum sound scattering energy presents four different stages.

Cost-effective and high-fidelity method for turbulent transition in compressible boundary layer

An efficient high-fidelity simulation approach is investigated for laminar-to-turbulent transition in compressible boundary layer flow. This approach combines large-eddy simulation (LES) with the parabolized stability equations (PSE) analysis. LES is chosen for high-fidelity simulation of the transitional flow, whereas the PSE analysis is used for an efficient treatment of instability modes in the flow. Instability modes from the PSE analysis are assigned as forcing at the inlet of the LES computational domain. A framework of LES combined with PSE (PSE+LES) for the turbulent transition is demonstrated in supersonic boundary layer flow at Mach 3. Complete transition to turbulence via oblique-mode breakdown is well resolved in the current study. Detailed flow features associated with the turbulent transition are compared well with previous direct-numerical simulation (DNS) studies, including the growth of instability modes in the pre-transition regime, the transition range shown in the evolution of the skin friction, and the turbulent boundary layer. It is shown that the current PSE+LES approach can reduce the computational cost by two orders of magnitude, compared to the relevant DNS cost. The current study for supersonic flow and the authors' previous study for subsonic flow imply that the PSE+LES approach can be used in boundary layer flows at various speeds.