Direct Numerical Simulation Approach Research Articles

Comparison between DNS and RANS approaches for liquid metal flows around a square rod bundle

The thermal-hydraulic characteristics of liquid metal flows around rod bundles are of great interest for the research and design of fourth generation nuclear reactors. Currently, a large research effort is aimed at the development of accurate numerical models for low Prandtl number fluid flows, since the data available in the literature are quite scarce. Direct Numerical Simulation (DNS) is undoubtedly the most accurate approach, but its large requirements of computational resources and time make it less practical than other simplified methods such as the Reynolds-Average Navier Stokes (RANS) approach. The present paper provides a comparison between numerical results of a flow of liquid Lead-Bismuth Eutectic (LBE) at Pr=0.031 around four vertical cylindrical rods arranged in a square lattice, obtained by DNS and RANS. Several turbulence models are considered, including the standard k−ε, k−ω SST, and two Reynolds stress models, namely the ones by Launder, Reece and Rodi (LRR), and Speziale, Sarkar and Gatski (SSG). The accuracy of these models is assessed by comparing the mean Nusselt number, the pressure drop, and local field distributions with those obtained by DNS.

Analysis of non-uniform laminar flow past a circular cylinder on the flow and sound field evolution using direct numerical simulation approach

Analysis of non-uniform laminar flow past a circular cylinder on the flow and sound field evolution using direct numerical simulation approach

Three-dimensional wake transition of a heated square cylinder in the presence of cross-buoyancy

The three-dimensional flow transition is examined in the wake of a heated square cylinder subjected to horizontal cross-flow perpendicular to gravity utilizing a direct numerical simulation approach. The surface of the square cylinder is heated uniformly to an elevated temperature Tw, and the amount of excess temperature is represented as the over-heat ratio ε=(Tw−T∞)/T∞, where T∞ represents the surrounding temperature. The effects of large-scale heating on the transport properties and thermal straining of the fluid particles are captured using an in-house non-Oberbeck–Boussinesq compressible model. The compressible flow governing equations (in a body-fitted coordinate system) are solved using a variant of flux-based particle velocity upwind-modified+ (PVU-M+) technique [Ahmad et al., “On the formation and sustenance of the compressible vortex rings in starting axisymmetric jets: A phenomenological approach,” Phys. Fluids 32, 126114 (2020)]. In this investigation, all computations are conducted at a low Mach number (Ma = 0.1) and air (Prandtl number, Pr = 0.71) is used as the working fluid. As the heating level rises, the shape and wavelength of the vortical structure undergo significant alterations. At Re = 250, the mode-B transition with a shorter spanwise wavelength and the mode-D transition with a longer wavelength are observed, respectively, for heating levels ε=0.0−0.2 and ε=0.8−1.0. Furthermore, for heating levels in the range 0.4≤ε≤0.6, an intermediate wavelength of the mode-E transition is detected. The temporal variation of fluid properties such as the force coefficient (CL, CD) and the Nusselt number (Nu) are shown at various heating levels. In addition, surface vorticity is examined in order to comprehend the flow dynamics near the surface of a heated square cylinder.

Biorthogonal decomposition of the disturbance flow field generated by particle impingement on a hypersonic boundary layer

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.

Numerical study on compressible flow around a circular cylinder in proximity to the wall

In the present study, the compressible flow around a circular cylinder in proximity to the wall is investigated. By using direct numerical simulation approach, the flow behavior with the Reynolds number 100 and the Mach numbers 0.1–0.6 are examined. The numerical result shows that the compressibility effect would enhance the flow stability, whereas the strength of shedding vortices, the shedding frequency, and the fluctuation of aerodynamic force are reduced by increasing the Mach number. Through the examination of signed enstrophy and shear steepness in the cylinder boundary layer, it is found that the vortices shedding from the free-stream side and the wall side are concurrently inhibited by the wall when the cylinder is fully embedded in the wall boundary layer, and the strength of the wall-side vortices is relatively weaker. This imbalance would be intensified when the cylinder gets closer to the wall, and thus a reduction in the shedding frequency is produced. Based on this effect of wall proximity on frequency, the flow behavior is divided into three regimes, i.e., high-frequency, low-frequency, and completely suppression regimes. Finally, from the observation of mean forces, the trend of mean drag and lift force coefficients with regard to the Mach number and gap ratio is described as the scaling behavior in terms of two piecewise functions.

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

Impact of droplets onto surfactant-laden thin liquid films

We study the effect of insoluble surfactants on the impact of surfactant-free droplets onto surfactant-laden thin liquid films via a fully three-dimensional direct numerical simulation approach that employs a hybrid interface-tracking/level-set method, and by taking into account surfactant-induced Marangoni stresses due to gradients in interfacial surfactant concentration. Our numerical predictions for the temporal evolution of the surfactant-free crown are validated against the experimental work by Che & Matar (Langmuir, vol. 33, 2017, pp. 12140–12148). We focus on the ‘crown-splash regime’, and we observe that the crown dynamics evolves through various stages: from the growth of linear modes (through a Rayleigh–Plateau instability) to the development of nonlinearities leading to primary and secondary breakup events (through droplet shedding modulated by an end-pinching mechanism). We show that the addition of surfactants does not affect the wave selection via the Rayleigh–Plateau instability. However, the presence of surfactants plays a key role in the late stages of the dynamics as soon as the ligaments are driven out from the rim. Surfactant-induced Marangoni stresses delay the end-pinching mechanisms to result in longer ligaments prior to their capillary singularity. Our results indicate that Marangoni stresses bridge the gap between adjacent protrusions promoting the adjacent protrusions' collision and the merging of ligaments. Finally, we demonstrate that the addition of surfactants leads to surface rigidification and consequently to the retardation of the flow dynamics.

A consistent volume-of-fluid approach for direct numerical simulation of the aerodynamic breakup of a vaporizing drop

A novel simulation framework has been developed in this study for the direct numerical simulation of the aerodynamic breakup of a vaporizing drop. The interfacial multiphase flow with phase change is resolved using a consistent geometric volume-of-fluid method. The bulk fluids are viscous and incompressible with surface tension at the interface. The newly-developed numerical methods have been implemented in the Basilisk solver, in which the adaptive octree/quadtree mesh is used for spatial discretization, allowing flexibility in dynamically refining the mesh in a user-defined region. The simulation framework is extensively validated by a series of benchmark cases, including the 1D Stefan and sucking problems, the growth of a 3D spherical bubble in a superheated liquid, and a 2D film boiling problem. The simulation results agree very well with the exact solution and previous numerical studies. 2D axisymmetric simulations were performed to resolve the vaporization of a moving drop with a low Weber number in a high-temperature free stream. The computed rate of volume loss agrees well with the empirical model of drop evaporation. Finally, the validated solver is used to simulate the aerodynamic breakup of an acetone drop at a high Weber number. A fully 3D simulation is performed and the morphological evolution of the drop is accurately resolved. The rate of vaporization is found to be significantly enhanced due to the drop deformation and breakup. The drop volume decreases nonlinearly in time and at a much higher rate than the empirical correlation for a spherical drop.

Direct numerical simulation of supersonic internal flow in a model scramjet combustor under a non-reactive condition

A Mach 1.5 non-reactive flow in a cavity-stabilized combustor of a model scramjet is studied via a direct-numerical simulation approach, and the analysis is focused on the interaction among boundary layer, free shear-layer above the cavity and shock wave. It is found that the impingement of the free shear-layer on the aft wall of the cavity leads to strong turbulence kinetic energy, high local pressure, and a fan of compression waves. The compression waves evolve into an oblique shock, which reflects between the upper and lower walls and interacts with the boundary layers attached to the two walls. The analysis of the turbulence production reveals that the amplification of turbulence in the core of the shear-layer and around the reattachment point is mainly due to the shear production, but the deceleration production mechanism presents a significant impact in the regions above the aft wall of the cavity and around the shock interaction points. The very low frequency commonly observed in shock wave/boundary layer interactions is not observed in the present research, which might be due to the low Reynolds number of the studied case.

Pore-scale analysis of two-phase nanofluid flow and heat transfer in open-cell metal foams considering Brownian motion

Simultaneous use of porous media and nanofluids will increase the convective heat transfer multiple times compared to non-porous and pure fluid conditions. Heat transfer and flow transport of nanofluids inside porous media are usually simulated in large-scale with average properties, which are typically highly uncertain. Pore-scale simulation as an alternative approach can capture the characteristics of flow and heat transfer more accurately. Only few studies have been conducted to study nanofluid flow through porous media in pore-scale, and most of them employed single-phase approach without focus on different affective forces. This paper uses a pore-scale approach to investigate the flow characteristics and convective heat transfer of two-phase nanofluid flow in open-cell metal foams (OCMFs). Simulation of fluid flow and heat transfer is achieved by Buongiorno’s model. Therefore, a computational code through the OpenFOAM library that operates by a direct numerical simulation (DNS) approach and the finite volume method (FVM) is used. The momentum, energy, continuity, and nanoparticle distribution equations are discretized, and the SIMPLE algorithm is utilized for pressure and velocity coupling. In the present study, three OCMFs with a constant porosity (0.86) and various pore densities are investigated. Also, variations of pressure gradient, Nusselt number, and Darcy velocity are investigated as a function of pore density (as a geometric parameter), nanoparticle diameter, concentration, and Brownian motion force. The results indicate that the Brownian force enhances the heat transfer in OCMFs from 2% by up to 14% for the nanofluid flowing with 3% nanoparticle concentration. Also, increasing the diameter of nanoparticles reduces the Darcy velocity and heat transfer by up to 4%. On the other hand, increasing particle concentration from 3% to 5%, increases heat transfer by up to 10% and reduces the Darcy velocity by up to 9%. Finally, doubling the pore density demonstrates the complex behavior of the flow patterns, in which the heat transfer increases by up to 30% and permeability and velocity are reduced by up to 70%.

Flow Characterization in Triply Periodic Minimal Surface (TPMS)-Based Porous Geometries: Part 1—Hydrodynamics

The modeling of flow and heat transfer in porous media systems has always been a challenge, and the extended Darcy transport models are used for macro-level analysis. However, these models are subjected to the limitations depending upon the porous geometry such as pore size, pore type, effective porosity, tortuosity, permeability, and the flow characteristics. The forced convective flow of an incompressible viscous fluid through a channel filled with four different types of porous geometries constructed using the Triply Periodic Minimal Surface (or TPMS) model is presented in this study. Four TPMS lattice shapes, namely Diamond, I-WP, Primitive, and Gyroid, are created with same volume fraction of solid subdomain as 0.68 (or void fraction as 0.32). Using different configurations for the solid subdomain by treating it as (a) solid, (b) fluid, and (c) porous zone, three different classes of porous structures are further generated for each TPMS lattice. The present study is executed with the objective to investigate the effect of shape–morphology, tortuosity, microporosity, and effective porosity on permeability and inertial drag factor. A pore-scale direct numerical simulation approach is performed for the first two types of porous media by solving the Navier–Stokes equations. The specific microporosity is quantitatively induced in the solid subdomain where Darcy–Forchheimer equation is solved, whereas the Navier–Stokes equations are solved for the void subdomain in the third type of porous media. The results reveal that Darcy flow regime exists up to the mean velocity value of U < 0.0025 m/s (Re < 10) for all the cases discussed here, and it deviates at the higher mean velocity. The conductance to the flow shown by Darcy number has the maximum and minimum values for the Primitive Type 2 and I-WP Type 1 cases. The inertial drag coefficient is minimum in Diamond lattice and maximum in Primitive lattice at lower porosity (0.32), while Primitive lattice has minimum and I-WP lattice has maximum value of inertial drag coefficient for higher porosity (~ 1).

A numerical investigation of a liquid bridge solidifying with volume change

A liquid bridge with solidification appears in living nature and numerous technical applications. Volume change upon solidification can induce a remarkable difference in the final shape of the bridge. To better understanding this problem, we use a direct numerical simulation (DNS) approach to study the solidification process of the vertical liquid bridge. The bridge is held between two equal-sized coaxial disks (or rods) with the presence of vertical gravity and the density difference between the liquid and its corresponding solid phase of the bridge. The solidification starts from the stable shape of the DNS solution (using the same method) when the rod temperature is suddenly lowered to a value below the melting point of the bridge liquid. It is found that the upper and lower phase-change interfaces propagate at nearly the same speed, and a ring (or protrusion) is formed around the bridge at mid-plane after complete solidification because of volume expansion upon phase change. The ring becomes more asymmetric with respect to the mid-plane when gravity, in terms of the Bond number, becomes more dominant. However, volume change does not affect the asymmetry of the ring. Instead, volume change significantly influences the size of the ring. The influences of the growth angle, the supercooling degree and the bridge aspect ratio on the evolution of the solid phase, the solidified bridge shape and solidification time are also revealed.

Analysis and Discussion of Two-Way Coupling Effects in Particle-Laden Turbulent Channel Flow

This paper studies the turbulence modification caused by the presence of solid particles in fully developed channel flow by means of the point particle Direct Numerical Simulations (DNS) approach. Inertial particles much smaller than the smallest vortical flow structures are considered, maintaining a volume fraction of the order , where inter-particle collisions are rare and have nearly no influence on flow development. To avoid concurrent effects that could mask the analysis of fluid turbulence interaction, gravity is not included in the study, and particle-smooth wall collisions are modelled as ideal reflections. The alteration of fluid turbulence dynamics by the particles is illustrated and discussed, providing an overview of the fluid-particle interaction phenomena occurring at both microscopic and macroscopic flow levels. Finally, the relation of such phenomena with drag-reducing effects by particles is demonstrated.

DNS and Highly-Resolved LES of Heat and Mass Transfer in Two-Phase Counter-Current Condensing Flow

A comprehensive study of direct-contact condensation heat transfer for turbulent, counter-current, liquid/vapour flow in a nearly horizontal channel at high pressure (i.e. 5 MPa) has been carried out based on Direct Numerical Simulation (DNS) and highly-resolved Large Eddy Simulation (LES) approaches. To simulate the two-phase flow situation, driven in this case by a constant pressure gradient, a single set of Navier–Stokes equations, coupled with an enthalpy conservation equation, have been employed. The interfacial mass transfer, seen in this case to be dominated by condensation, has been calculated directly from the heat flux at the liquid/vapour interface. To investigate the effect of condensation on the turbulence phenomena, and vice versa, cases have been considered involving two friction Reynolds numbers: namely Re_{*}=u_{*}h/nu =178 and Re_{*}=u_{*}h/nu =590 (u_{*}=(hvarDelta P/rho )^{1/2}). At the lower Reynolds number, three levels of water subcooling—0 K, 10 K and 40 K—have been investigated. The use of water subcooling of 0 K has enabled the validation and verification procedures associated with the numerical approach to be compared against experimental and numerical data reported in the literature. The choice of the maximum degree of water subcooling is dictated by the need to justify the periodic boundary conditions applied in this numerical study. In the simulation for the higher Reynolds number, only the case of 10 K subcooling has been included, as a consequence of the very high computation effort involved. A detailed statistical analysis of the DNS and LES data obtained from the application of the well-known wall laws has also been assessed. In the vicinity of the liquid/vapour interface, the characteristics of the turbulent motions appear somewhat diverse, depending on whether the interface is basically flat or wavy in character. For a flat interface, some damping effect of the presence of the interface on the turbulence intensity has been observed, a feature which becomes enhanced as the level of liquid subcooling is increased. In the case of a wavy interface, the damping effect is predicted as considerably less pronounced.

An improved Eulerian method in three-dimensional direct numerical simulation on the local scour around a cylinder

An improved Eulerian method with a novel morphologic model is developed based on the direct numerical simulation (DNS) approach to study the local scour. A movable immersed boundary method is proposed to prescribe the geometry of the riverbed. Two techniques, namely the Simultaneous Bed Adjustment Method (SBAM) and (Segment Plane Method) SPM, are proposed to model the complex time changing geometry of the riverbed. The SBAM technique considers sand sliding in the model and the SPM is an algorithm calculating the geometrical parameter of the complex riverbed. A wind tunnel experiment obeying Reynolds similarity and measuring the pressure in a fixed scour hole is carried out to compare with the numerical results. The results show that the movable immersed boundary method performs well in capturing the local scour and qualitatively agrees with the experimental study in how the scour hole develops. The numerical results also show that the DNS can capture the formation of the local scour in details at the beginning and find that the Reynolds stress u'v’ plays an important role in an erosion at the leading edge of the cylinder. Therefore, the newly improved Eulerian method is a valuable tool for DNS studies of local scour with the implementation of MPI.

Compressibility effects on the linear-stability of centrifugal buoyancy-induced flow

The focus of this study is to understand the evolution of instability in centrifugal buoyancy-induced flow in a rotating system. The problem is of interest in atmospheric flows as well as in engineering applications. In this study, we perform direct numerical simulations (DNS) by solving the compressible Navier–Stokes equations and multi-dimensional stability analyses by using a forced DNS approach. We systematically and independently vary the Rayleigh and Mach numbers. The heat transfer by thermal conduction is used as a base state and maintained as a reference, upon which the growth of small perturbations is investigated. It is found that the critical wavenumber obtained from the linear stability analysis at the onset of convection has a much shorter wavelength than the one that eventually appears in the non-linear regime. Further, the investigations show that compressibility effects lead to a reduction of the growth rate of the dominant mode, and it modifies the overall formation of convection cells in the flow.

Comprehensive Investigation of the Petrophysical and Two-Phase Flow Properties of the Tight Sandstone in Yanchang Formation, Ordos Basin, China: Insights from Computed Tomography Imaging and Pore Scale Modelling

Abstract Accurate determination of petrophysical and fluid transporting properties of rocks is essential for many engineering applications. In this paper, microcomputed tomography (CT) imaging technique is adopted to image the microstructure of tight sandstones drilled from Chang-7 member in Yanchang Formation. The pore geometry, pore-throat size distribution, pore connectivity, and tortuosity of the pore-throat structure are quantitatively characterized by extracting the pore network model (PNM). Direct numerical simulation (DNS) approach is applied on the segmented CT images to investigate the anisotropic permeability of tight sandstones. In addition, the unstructured mesh model of the pore space is reconstructed, and the pore scale immiscible two-phase flow is simulated by computational fluid dynamics (CFD) using the volume of fluid (VOF) model. The results indicate that the pore-throat system of tight sandstone reservoirs is mainly composed of discontinuous intergranular pores and grain margin microfractures with poor connectivity and diverse pore morphology. The two-phase flow simulations indicate that the strong heterogeneity of pore-throat structure can lead to obvious fingering phenomenon of the injected fluid, which reduces the sweep efficiency and the oil recovery as well. The residual fluids are mainly trapped in tiny throats and dead-end pores. It is found that the digital rock analysis based on CT imaging can be a cost-effective and time-saving alternative to routine core analysis of tight sandstone reservoirs.

Effects of periodic suction-blowing excitation on the aerodynamic sound generated by a laminar flow past a square cylinder using the direct numerical simulation approach

This paper analyzes the effects of a periodic suction-blowing excitation on the aerodynamic sound generated by a laminar flow past a square cylinder using the direct numerical simulation approach. The periodic suction-blowing excitation has been prescribed on the top and bottom surfaces of the square cylinder. The proper orthogonal mode decomposition (POD) technique has been used to find information about important modes associated with disturbance pressure fields. The POD technique separated the contribution of the dominant lift dipole equivalent sources and the drag dipole equivalent sources to the disturbance pressure field for the no-excitation case. The POD technique also revealed that the periodic suction-blowing excitation introduced an additional monopole equivalent sound source and a drag dipole equivalent sound source due to periodic enhancement and reduction of the body’s effective cross-sectional area. Modifications in the sound field due to changes in excitation amplitude, forcing frequency, and the phase delay between the excitation and vortex shedding process have been studied in detail. Although no significant changes in the flow field were noticed due to a small amplitude of excitation, the directivity of the sound field was significantly altered. The sound fields have been classified into five distinct zones for different periodic suction-blowing excitation frequencies. The beats of sounds were noted when the forcing frequency of excitation and the Strouhal frequency associated with vortex shedding were sufficiently close. It is observed that the in-phase excitation in which either blowing or suction is applied on both surfaces of a cylinder at a particular instant introduces a significant bias in the sound field directivity. The interaction between the lift dipole equivalent sources due to vortex shedding and the monopole and the drag dipole equivalent sources due to excitation introduces a bias in the sound field directivity. As a result, a dominant sound field is observed either in the top-left or in the bottom-left parts of the domain.

Combined Large-Eddy and Direct Numerical Simulations of a Planar Jet with Heated Co-Flow with Medium and Low Prandtl Fluids

In the present work, we have applied a combined dynamic large-eddy simulation (LES) and direct numerical simulation (DNS) approach for a three-dimensional planar jet in a turbulent forced convection regime (Re = 18000) with a heated co-flow. Results from LES are compared with Reynolds Averaged Navier-Stokes (RANS) simulations and experimental data. We have analyzed flow and heat transfer features for four values of the characteristic Prandtl numbers (Pr = 0.71, 0.2, 0.025, and 0.006), which are representatives of air, He-Xe gas mixture, Lead-Bismuth Eutectic (LBE), and sodium, respectively. The latter two low-Prandtl fluids have been considered because of their role as primary coolants in advanced fast pool-type reactor prototypes (such as the Multi-purpose Hybrid Research Reactor for High-tech Applications (MYRRHA) at SCK•CEN, Belgium). We have provided detailed insights into instantaneous and long-term time-averaged behavior of the velocity and temperature fields (the first- and second-order moments). Furthermore, we have analyzed profiles of characteristic velocity and temperature time scales and dissipation rates, as well as the power spectra of the streamwise velocity component and temperature at several characteristic locations. The mean temperature profiles demonstrated rather low sensitivity for various values of the Prandtl number. In contrast, profiles of the temperature standard deviation exhibited larger variations, decreasing in magnitude with lower Prandtl values. Here presented results of the high fidelity numerical simulations (dynamic LES/DNS) for the low-Prandtl working fluids can be used for further development, testing, and validation of the advanced RANS-type turbulence models.

A Consistent Volume-of-Fluid Approach for Direct Numerical Simulation of the Aerodynamic Breakup of a Vaporizing Drop

A Consistent Volume-of-Fluid Approach for Direct Numerical Simulation of the Aerodynamic Breakup of a Vaporizing Drop