Direct Numerical Simulation Results Research Articles

Direct numerical simulation of flow and heat transfer of supercritical water with different heat fluxes

The flow and heat transfer characteristics of supercritical water (SCW) are important considerations in the thermal-hydraulic design of the Supercritical Water-Cooled Reactor (SCWR). In the present study, the direct numerical simulation (DNS), developed with OpenFOAM, was utilized to investigate the effect of heat flux on the flow and heat transfer of SCW in a vertical circular pipe. The investigation involves detailed analyses of the wall temperature, velocity distribution, and turbulent statistics at various heat fluxes. It was found that buoyancy first impairs heat transfer, and then recovers heat transfer. This is attributed to the variation of buoyancy production and turbulence production, which are resulted from the buoyancy direct effect and indirect effect, respectively. As the heat flux increases, the recovery occurs earlier with higher magnitude. The vortex structures based on Q-criterion clearly illustrate the same process as well. Several heat-transfer correlations were assessed based on the DNS results. It indicates that Dittus and Boelter correlation (1930) cannot accurately predict the heat-transfer characteristics of SCW, while Chen and Fang correlation (2014) exhibits a higher level of predictive accuracy for Nusselt number. In addition, several turbulence models were evaluated against the DNS data. The results show significant deviations in predicting the heat transfer of SCW. The large discrepancy may be due to the inappropriate model coefficients, the failed predication of turbulence kinetic energy, as well as the constant turbulent Prandtl number commonly applied.

Towards stochastic subgrid‐scale modeling of turbulent thermal convection in an under‐resolved off‐lattice Boltzmann method

AbstractA characteristic‐based Off‐Lattice Boltzmann Method (OLBM) and a stochastic One‐Dimensional Turbulence (ODT) model is utilized for numerical simulation of turbulent thermal convection. Standalone ODT results for low‐order statistics are compared with those from various eddy‐viscosity‐based subgrid‐scale models utilized in Large‐Eddy Simulations (LES) with OLBM. The predictive capabilities of both approaches are discussed by comparison with available reference Direct Numerical Simulation (DNS) results. All turbulence models are able to predicted the mean temperature, but fail to fully capture fluctuations. While the OLBM aims to represent large‐scale structures, it misses some constitutional small‐scale fluctuations. By contrast, the reduced‐order ODT model captures small‐scale fluctuations in the vicinity of the wall, but cannot resolve the organized bulk flow. Here, the modeling capabilities of both OLBM and ODT as standalone tools are discussed. On this basis, a strategy for the incorporation of ODT as wall model in OLBM is suggested.

Lagrangian Supersaturation Fluctuations at the Cloud Edge.

Evaporation of cloud droplets accelerates when turbulence mixes dry air into the cloud, affecting droplet-size distributions in atmospheric clouds, combustion sprays, and jets of exhaled droplets. The challenge is to model local correlations between droplet numbers, sizes, and supersaturation, which determine supersaturation fluctuations along droplet paths (Lagrangian fluctuations). We derived a statistical model that accounts for these correlations. Its predictions are in quantitative agreement with results of direct numerical simulations, and explain the key mechanisms at play.

Direct Numerical Simulation of High-Enthalpy Turbulent Boundary-Layer Flow with Light Gas Injections

Two light gases (He and H2) are, respectively, introduced upstream of a high-enthalpy turbulent flat-plate flow with a boundary-layer edge Mach number of Maδ=2.25 and temperature of Tδ=1800 K. The flow condition refers to the after-shock wave flow on a blunt-body hypersonic vehicle (Duan and Martín, AIAA Journal, Vol. 49, No. 1, 2011, pp. 172–184). Direct numerical simulation results show that the injection of these light gases has little effect on the mean velocity profiles but significantly reduces the near-wall density and skin friction. The separation and fragmentation of near-wall vortical structures are restrained, and the Reynolds shear stress decreases. The injection of inert He inhibits the dissociation reaction of O2 and weakens the chemical nonequilibrium effect, resulting in enhanced mean and fluctuating temperatures. The injection of active H2 promotes the reaction between H2 and O2, which increases the mean temperature but inhibits its fluctuation. After decomposing the mean skin friction into physics-informed contributions, both injections largely reduce the turbulent kinetic energy production term Cf,T and the spatial growth term of the flow, Cf,G, through lowering the near-wall density and reducing vortices.

A resolvent-based prediction framework for incompressible turbulent channel flow with limited measurements

A new resolvent-based method is developed to predict the space–time properties of the flow field. To overcome the deterioration of the prediction accuracy with increasing distance between the measurements and predictions in the resolvent-based estimation (RBE), the newly proposed method utilizes the RBE to estimate the relative energy distribution near the wall rather than the absolute energy directly estimated from the measurements. Using this extra information from RBE, the new method modifies the energy distribution of the spatially uniform and uncorrelated forcing that drives the flow system by minimizing the norm of the cross-spectral density tensor of the error matrix in the near-wall region in comparison with the RBE-estimated one, and therefore it is named as the resolvent-informed white-noise-based estimation (RWE) method. For validation, three time-resolved direct numerical simulation (DNS) datasets with the friction Reynolds numbers $Re_\tau = 180$ , 550 and 950 are generated, with various locations of measurements ranging from the near-wall region ( $y^+ = 40$ ) to the upper bound of the logarithmic region ( $y/h \approx 0.2$ , where h is the half-channel height) for the predictions. Besides the RWE, three existing methods, i.e. the RBE, the $\lambda$ -model and the white-noise-based estimation (WBE), are also included for the validation. The performance of the RBE and scale-dependent model ( $\lambda$ -model) in predicting the energy spectra shows a strong dependence on the measurement locations. The newly proposed RWE shows a low sensitivity on $Re_{\tau }$ and the measurement locations, which may range from the near-wall region to the upper bound of the logarithmic region, and has a high accuracy in predicting the energy spectra. The RWE also performs well in predicting the space–time properties in terms of the correlation magnitude and the convection velocity. We further utilize the new method to reconstruct the instantaneous large-scale structures with measurements from the logarithmic region. Both the RWE and RBE perform well in estimating the instantaneous large-scale structure, and the RWE has smaller errors in the estimations near the wall. The structural inclination angles around $15^\circ$ are predicted by the RWE and WBE, which generally recover the DNS results.

Modelling of droplet multiple breakage accounting for entire energy spectrum bottleneck effect and eddy-droplet discrepancy

This work focused on the contribution of the two aspects of entire energy spectrum bottleneck effect and eddy-droplet discrepancy to droplet multiple breakage simulation. The droplet breakage mechanism model was closured by more accurate turbulent parameters determined by accounting for the bottleneck effect and a formula was proposed to consider the effect of the eddy-droplet discrepancy. The influences of the two aspects on collision/interaction frequency density, breakage probability, breakage frequency and daughter size distribution (DSD) were analyzed. The proposed model agreed well with the measured breakage frequency, relative proportion of multiple breakage modes and DSDs from single droplet experiments and drop swarm experiments. The predicted size distribution evolution also exhibited a better agreement with experimental data by comparing with other models. An obvious quantitative prediction difference on whether to consider the two aspects proved the necessity and advantage of model development.The bottleneck effect represents a physical phenomenon fully validated by experimental data and direct numerical simulation results. Ignoring bottleneck effect hinders the acceptable reproduction of longitudinal velocity gradient skewness, while considering it obviously increases the quantitative prediction accuracy of turbulence parameters. Considering the physical fact that turbulent eddies will disappear, the eddy-droplet discrepancy neglected by existing mechanism model is used to characterize the impact of eddy lifetime on eddy-droplet collision/interaction. To summarize, the improvements of this work are beneficial for consolidating the breakage model physical foundation.

Global stability of supersonic flow over a hollow cylinder/flare

Supersonic flow over a hollow cylinder/flare with a free-stream Mach number of 2.25 is numerically investigated in this study. Axisymmetric computational fluid dynamics simulations and global stability analysis (GSA) are performed for a wide range of cylinder radii and flare deflection angles. The onset of incipient and secondary separation is delayed as the cylinder radius is decreased due to the axisymmetric effects. The GSA reveals that a decrease in cylinder radius also postpones the emergence of global instability. The GSA results agree well with the results of direct numerical simulations for a supercritical case in the linear stage. The saturated flow exhibits pairs of unsteady streamwise streaks downstream of reattachment. The criterion of the global stability boundary established for supersonic flow over a compression corner (Hao et al., J. Fluid Mech, vol. 919, 2021, A4) is extended to its axisymmetric counterpart.

Three-dimensional global instability analysis for high-speed boundary layer flow

Herein, a numerical approach for the three-dimensional global instability analysis of high-speed boundary layer is developed. The approach is based on the Krylov–Schur method for large-scale eigenproblems. A shift-and-invert spectral transformation is employed to accelerate the convergence for eigenvalues of interest. Iterative linear solver is adopted in the transformation to reduce memory usage by avoiding the fill-in problem of direct solver and the explicit construction of matrices. A Robin condition based on BiGlobal instability analysis is imposed at the inflow boundary, and a non-reflecting condition derived from inviscid flow is used for the far-field boundary. The approach is validated in a Blasius boundary layer. According to a comparison between the performances of several algorithms in this case, the conjugate gradient squared method with no preconditioning is chosen as the linear solver. The results provided by three-dimensional global instability analysis consist with those from linear stability theory and BiGlobal instability analysis. A further demonstration of our approach is provided for a three-dimensional roughness-disturbed boundary layer. The fluctuations extracted from direct numerical simulation results illustrate that the symmetric component is the dominant unstable mode in roughness wake, as it grows twice as fast as the antisymmetric component. Two eigenvalues are obtained from the three-dimensional global instability analysis, and these represent the symmetric and antisymmetric unstable modes. The wavelengths and growth rates of both modes delivered by instability analysis agree well with the direct numerical simulation results, and the profiles of disturbance amplitude match the features of the varicose and sinuous modes.

Shear-induced wrinkling in accelerating thin elastic discs

The wrinkling instabilities produced by in-plane angular accelerations in a rotating disc are discussed here in a particular limit of relevance to very thin plates. By coupling the classical linear elasticity solution for this configuration with the Föppl–von Kármán plate buckling equation, a fourth-order boundary-value problem with variable coefficients is obtained. The singular-perturbation character of the resulting problem arises from a combination of factors encompassing both the pre-stress (due to the spinning motion) and the geometry of the annular domain. With the help of a simplified multiple-scale perturbation method in conjunction with matched asymptotics, we succeed in capturing the dependence of the critical (wrinkling) acceleration on the instantaneous speed of the disc as well as other physical parameters. We show that the asymptotic predictions compare well with the results of direct numerical simulations of the original bifurcation problem. The limitations of the formulae obtained are also considered, and some practical suggestions for improving their accuracy are suggested.

Circular cumulant reductions for macroscopic dynamics of oscillator populations with non-Gaussian noise.

We employ the circular cumulant approach to construct a low dimensional description of the macroscopic dynamics of populations of phase oscillators (elements) subject to non-Gaussian white noise. Two-cumulant reduction equations for α-stable noises are derived. The implementation of the approach is demonstrated for the case of the Kuramoto ensemble with non-Gaussian noise. The results of direct numerical simulation of the ensemble of N=1500 oscillators and the "exact" numerical solution for the fractional Fokker-Planck equation in the Fourier space are found to be in good agreement with the analytical solutions for two feasible circular cumulant model reductions. We also illustrate that the two-cumulant model reduction is useful for studying the bifurcations of chimera states in hierarchical populations of coupled noisy phase oscillators.

Kelvin–Helmholtz Instability “Tube” and “Knot” Dynamics. Part I: Expanding Observational Evidence of Occurrence and Environmental Influences

Abstract Multiple recent observations in the mesosphere have revealed large-scale Kelvin–Helmholtz instabilities (KHI) exhibiting diverse spatial features and temporal evolutions. The first event reported by Hecht et al. exhibited multiple features resembling those seen to arise in early laboratory shear-flow studies described as “tube” and “knot” (T&K) dynamics by Thorpe. The potential importance of T&K dynamics in the atmosphere, and in the oceans and other stratified and sheared fluids, is due to their accelerated turbulence transitions and elevated energy dissipation rates relative to KHI turbulence transitions occurring in their absence. Motivated by these studies, we survey recent observational evidence of multiscale Kelvin–Helmholtz instabilities throughout the atmosphere, many features of which closely resemble T&K dynamics observed in the laboratory and idealized initial modeling. These efforts will guide further modeling assessing the potential importance of these T&K dynamics in turbulence generation, energy dissipation, and mixing throughout the atmosphere and other fluids. We expect these dynamics to have implications for parameterizing mixing and transport in stratified shear flows in the atmosphere and oceans that have not been considered to date. Companion papers describe results of a multiscale gravity wave direct numerical simulation (DNS) that serendipitously exhibits a number of KHI T&K events and an idealized multiscale DNS of KHI T&K dynamics without gravity wave influences. Significance Statement Kelvin–Helmholtz instabilities (KHI) occur throughout the atmosphere and induce turbulence and mixing that need to be represented in weather prediction and other models of the atmosphere and oceans. This paper documents recent atmospheric evidence for widespread, more intense, features of KHI dynamics that arise where KH billows are initially discontinuous, misaligned, or varying along their axes. These features initiate strong local vortex interactions described as “tubes” and “knots” in early laboratory experiments, suggested by, but not recognized in, earlier atmospheric and oceanic profiling, and only recently confirmed in newer, high-resolution atmospheric imaging and idealized modeling to date.

Large-eddy simulations of compressible Rayleigh–Taylor turbulence with miscible fluids using spatial gradient model

Large-eddy simulations (LES) and implicit LES (ILES) of three-dimensional compressible Rayleigh–Taylor turbulence with miscible fluids are performed and compared with direct numerical simulation (DNS) at the Atwood number At=0.5 and stratification parameters Sr = 1.0 and 4.0. Three sub-grid-scale (SGS) models including constant-coefficient spatial gradient model (CSGM), dynamic Smagorinsky model (DSM), and dynamic mixed model (DMM) are considered. The CSGM model achieves a high accuracy by using the velocity gradients at the neighboring LES grids. The a priori tests show that the CSGM model has significantly higher correlation coefficients and lower relative errors than traditional SGS models. In the a posteriori tests, the probability density functions of SGS terms predicted by the CSGM model are consistent with the filtered DNS results. The CSGM model can accurately predict the small bubble and spike structures, resulting in good predictions of mixing heights and concentration fields. The instantaneous structures, spectra, and statistics of velocity and vorticity fields are also examined, showing the excellent performance of the CSGM model compared to the ILES, DSM, and DMM models. Moreover, the predictions of the temperature and pressure fields by the CSGM model are significantly better than the traditional SGS models and ILES.

Droplet impact dynamics on shallow pools

When a fast droplet impacts a pool of the same fluid, a thin ejecta sheet that dominates the early-time dynamics emerges within the first few microseconds. Fluid and impact properties are known to affect its evolution; we experimentally reveal that the pool depth is a critical factor too. Whilst ejecta sheets can remain separate and subsequently fold inwards on deeper pools, they instead develop into outward-propagating lamellae on sufficiently shallow pools, undergoing a transition that we delineate by comprehensively varying impact inertia and pool depth. Aided by matching direct numerical simulation results, we find that this transition stems from a confinement effect of the pool base on the impact-induced pressure, which stretches the ejecta sheet to restrict flow into it from the droplet on sufficiently shallow pools. This insight is also applied to elucidate the well-known transition due to Reynolds number.

Turbulent burning velocity and thermodiffusive instability of premixed flames.

Reported in the paper are results of unsteady three-dimensional direct numerical simulations of laminar and turbulent, lean hydrogen-air, complex-chemistry flames propagating in forced turbulence in a box. To explore the eventual influence of thermodiffusive instability of laminar flames on turbulent burning velocity, (i) a critical length scale Λ_{n} that bounds regimes of unstable and stable laminar combustion is numerically determined by gradually decreasing the width Λ of computational domain until a stable laminar flame is obtained, and (ii) simulations of turbulent flames are performed by varying the width from Λ<Λ_{n} (in this case, the instability is suppressed) to Λ>Λ_{n} (in this case, the instability may grow). Moreover, simulations are performed either using mixture-averaged transport properties (low Lewis number flames) or setting diffusivities of all species equal to heat diffusivity of the mixture (equidiffusive flames), with all other things being equal. Obtained results show a significant increase in turbulent burning velocity U_{T} when the boundary Λ=Λ_{n} is crossed in weak turbulence, but almost equal values of U_{T} are computed at Λ<Λ_{n} and Λ>Λ_{n} in moderately turbulent flames characterized by a Karlovitz number equal to 3.4 or larger. These results imply that thermodiffusive instability of laminar premixed flames substantially affects burning velocity in weak turbulence only, in line with a simple criterion proposed by Chomiak and Lipatnikov (Phys. Rev. E 107, 015102, (2023)10.1103/PhysRevE.107.015102).

LES of premixed jet flames subjected to extreme turbulence using flamelet-generated manifolds: a comparison of unstrained and strained flamelets

Two piloted premixed CH 4 /air jet flames with extreme turbulence are modelled using the Flamelet-Generated Manifold (FGM) method in the context of Large Eddy Simulation (LES). Free-propagating and reactant-to-product laminar flames are adopted to build unstrained and strained chemical databases, respectively. Detailed comparison of the LES-FGM results using the unstrained/strained databases is conducted which is validated against the direct numerical simulation results. The results show that whether including straining effects in the database will not influence the modelling of temperature and major species (e.g. CH 4 , O 2 and CO 2 ), but using a strained database is essential for predictions of intermediate minor species (e.g. H 2 , CH 2 O, OH and CO). The main reason is that the major species and temperature are almost insensitive to the straining due to the supportive effects from the pilot flame, which also tentatively explains why the unstrained database can be employed to model these strong turbulent flames. In contrast, the intermediate minor species are more prone to the straining, so that using a strained database is crucial. The preferential diffusion effects on the species and temperature predictions are examined to be negligible. It is proposed that adding extra transport equations for these minor species also leads to satisfactory predictions if using an unstrained database, which helps to include the straining effects on the minor species. This study is expected to be significant in high-fidelity simulations of strong turbulent flames using flamelet-tabulated combustion models.

Ensemble data assimilation-based mixed subgrid-scale model for large-eddy simulations

An ensemble Kalman filter (EnKF)-based mixed model (EnKF-MM) is proposed for the subgrid-scale (SGS) closure in the large-eddy simulation (LES) of turbulence. The model coefficients are determined through the EnKF-based data assimilation technique. The direct numerical simulation (DNS) results are filtered to obtain the benchmark data for the LES. Reconstructing the correct kinetic energy spectrum of the filtered DNS (fDNS) data has been adopted as the target for the EnKF to optimize the coefficient of the functional part in the mixed model. The proposed EnKF-MM framework is subsequently tested in the LES of both the incompressible homogeneous isotropic turbulence and turbulent mixing layer. The performance of the LES is comprehensively examined through the predictions of the flow statistics including the velocity spectrum, the probability density functions (PDFs) of the SGS stress, the PDF of the strain rate, and the PDF of the SGS energy flux. The structure functions, the evolution of turbulent kinetic energy, the mean flow, the Reynolds stress profile, and the iso-surface of the Q-criterion are also examined to evaluate the spatial–temporal predictions by different SGS models. The results of the EnKF-MM framework are consistently more satisfying compared to the traditional SGS models, including the dynamic Smagorinsky model, the dynamic mixed model, and the velocity gradient model, demonstrating its great potential in the optimization of SGS models for the LES of turbulence.

A Critical Comparison of DNS Versus RANS Model Results on an HPT Blade for a Small Engine

In this work, we present preliminary Direct Numerical Simulation (DNS) results of compressible flow past a High-Pressure Turbine (HPT) blade of an experimental Small Turbofan Engine (STFE) designed by Defence Research and Development Organisation (DRDO), Gas Turbine Research Establishment (GTRE) [1, 2]. These DNS results are then compared with the results obtained by two well-known industry standard RANS models, namely the Realizable k-e (RKE) and the Shear Stress Transport (SST) models along with a hybrid LES-RANS (LNS) model. All the simulations are carried out at inlet Reynolds number of 152,000 (based on chord length of the blade), and inlet Mach number of 0.16 with zero degree of angle of incidence. DNS results are obtained with approximately 94 and 129 million mesh cells. For RANS and LNS calculations, we have used 0.16 and 3.54 million mesh cells respectively. Surface pressure distributions obtained from DNS are closer to experimental values compared to RANS and LNS. Also, DNS predicts flow separation near the trailing edge when a grid is refined from 94 x 106 to 129 x 106 cells, whereas present RANS and LNS simulations fail to capture separation. Interestingly, mean surface pressure obtained from DNS with two different mesh sizes vary mostly near the separation and trailing edge region, where the experimental data points are not available.

On the emergence of secondary tones in airfoil noise

We present the results of direct numerical simulations of a NACA 0012 airfoil, with Mach number 0.3 and angle of attack of $3^\circ$ , examining the dynamics of the flow with increasing Reynolds numbers. Two-dimensional simulation results are obtained with chord-based Reynolds numbers in the range $3.2 \times 10^3 \leq Re \leq 2.70 \times 10^4$ , where each simulation uses the last time step of the previous one as a starting point, to capture the evolution of dynamics as a function of $Re$ . The development of the pressure fluctuations with time shows a transition from periodic to quasi-periodic attractor for $2.38 \times 10^4 \leq Re \leq 2.42 \times 10^4$ , leading to the emergence of secondary tones in the wall and acoustic field pressure spectra, different from peaks related to the fundamental frequency $f_1$ and the respective harmonics; a second, incommensurate frequency $f_2$ appears, leading to several secondary tones with frequency $af_1 + bf_2$ , with $a$ and $b$ integers. Further increase of the Reynolds number leads to the emergence of a tertiary frequency, $f_3$ , indicating a route to chaos of the Ruelle–Takens–Newhouse type. Such a mechanism is related to the ladder-type characteristic structure of the tones, indicating that dynamic systems theory is an important tool for understanding airfoil tonal noise.

The Whitham Modulation Solution of the Complex Modified KdV Equation

This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler–Poisson–Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory.

Comparative analysis on gas–solid drag models in MFIX-DEM simulations of bubbling fluidized bed

In this study, the open-source software MFIX-DEM simulations of a bubbling fluidized bed (BFB) are applied to assess nine drag models according to experimental and direct numerical simulation (DNS) results. The influence of superficial gas velocity on gas–solid flow is also examined. The results show that according to the distribution of time-averaged particle axial velocity in y direction, except for Wen–Yu and Tenneti–Garg–Subramaniam (TGS), other drag models are consistent with the experimental and DNS results. For the TGS drag model, the layer-by-layer movement of particles is observed, which indicates the particle velocity is not correctly predicted. The time domain and frequency domain analysis results of pressure drop of each drag model are similar. It is recommended to use the drag model derived from DNS or fine grid computational fluid dynamics–discrete element method (CFD-DEM) data first for CFD-DEM simulations. For the investigated BFB, the superficial gas velocity less than 0.9 m·s−1 should be adopted to obtain normal hydrodynamics.