Reference Direct Numerical Simulation Research Articles

Aeroacoustic Source Potential Based on Poisson’s Equation

Poisson’s equation is an important equation to postprocess the aerodynamic fields into linearized momentum modes and was recently found to be important for the computation of an isotropic pressure-like source for scalar aeroacoustic wave models, like the aeroacoustic wave equation based on Pierce’s operator (AWE-PO). Mathematically viable boundary conditions of the Poisson equation, which computes the AWE-PO source, are investigated. For the different source fields, the wave propagation is computed using the AWE-PO, and the details of the sound prediction results are compared to a reference direct numerical simulation of a mixing layer. The different boundary conditions of the Poisson equation were found to have a minor influence on the overall sound prediction characteristics of the AWE-PO equation. The AWE-PO is reformulated into a simplified version of the Phillips’s equation, which mitigates the intermediate step of computing an isotropic source potential. By doing so, a previously obtained interference radiation valley in the radiated acoustic intensity of the AWE-PO results is attributed to a missing shear-noise source term.

Revisiting Tensor Basis Neural Network for Reynolds stress modeling: Application to plane channel and square duct flows

Recent advances in computing power devoted to Computational Fluid Dynamics (CFD) have opened up a new avenue for addressing the turbulence closure problem through Machine Learning (ML). Several Tensor Basis Neural Network (TBNN) frameworks aimed at enhancing turbulence Reynolds-Averaged Navier–Stokes (RANS) modeling have recently been proposed in the literature as data-driven constitutive models for systems with known invariance properties. However, persistent ambiguities remain regarding the physical adequacy of applying the General Eddy Viscosity Model (GEVM) within these frameworks. This work aims at investigating this aspect in an a priori stage for better predictions of the Reynolds stress anisotropy tensor, while preserving the Galilean and rotational invariances. In particular, we propose a general framework providing optimal tensor basis models for two types of canonical flows with increasing complexity: Plane Channel Flow (PCF) and Square Duct Flow (SDF). Subsequently, deep neural networks based on these optimal models are trained using state-of-the-art strategies to achieve a balanced and physically sound prediction of the full anisotropy tensor. A priori results obtained by the proposed framework are in very good agreement with the reference Direct Numerical Simulations (DNS) data. Notably, our shallow network with three layers excels in providing accurate predictions of the anisotropy tensor for PCF at unobserved friction Reynolds numbers, both in interpolation and extrapolation scenarios. The learning of the SDF case is more challenging because of its physical nature but also due to a lack of training data at various regimes. We propose a numerical strategy to alleviate this problem based on Transfer Learning (TL). In order to more efficiently generalize to an unseen intermediate Reτ regime, we take advantage of our prior knowledge acquired from a training with a larger and wider dataset. Our results indicate the potential of the developed network model, and demonstrate the feasibility and efficiency of the TL process in terms of training data size and training time. Based on these results, we believe there is a promising future by integrating these neural networks into an adapted in-house RANS solver.

Machine learning-based vorticity evolution and super-resolution of homogeneous isotropic turbulence using wavelet projection

A wavelet-based machine learning method is proposed for predicting the time evolution of homogeneous isotropic turbulence where vortex tubes are preserved. Three-dimensional convolutional neural networks and long short-term memory are trained with a time series of direct numerical simulation (DNS) data of homogeneous isotropic turbulence at the Taylor microscale Reynolds number 92. The predicted results are assessed by using the flow visualization of vorticity and statistics, e.g., probability density functions of vorticity and enstrophy spectra. It is found that the predicted results are in good agreement with DNS results. The small-scale flow topology considering the second and the third invariants of the velocity gradient tensor likewise shows an approximate match. Furthermore, we apply the pre-trained neural networks to coarse-grained vorticity data using super-resolution. It is shown that the super-resolved flow field well agrees with the reference DNS field, and thus small-scale information and vortex tubes are well regenerated.

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.

Flow and thermal fields modeling in jet impingement configurations using a Reynolds stress turbulence closure within the RANS and Sensitized-RANS framework

The flow and thermal field characteristics underlying turbulent convective heat transfer in two reference configurations, representing a jet exiting a planar slot and a circular tube and impinging on a heated wall, are studied computationally. The performance of a conventional Reynolds Stress Model (RSM) and its scale-resolving variant, employed in the context of a time-accurate Reynolds-Averaged Navier–Stokes (RANS) simulation, is investigated in conjunction with different modeling approaches for the turbulent heat flux uj′θ′¯, whose gradient represents the turbulence-induced heat source in the temperature transport equation. In this so-called Sensitized-RANS modeling strategy, employing an improved eddy-resolving instability-sensitive Reynolds Stress Model (IISRSM, Jakirlić and Maduta (2015)), the unresolved ‘sub-scale’ turbulent correlations represent the solutions of a set of corresponding transport equations for the entire stress tensor ui′uj′¯. The turbulent scalar-flux models follow from the classical gradient-diffusion approach with the diffusion-like coefficients formulated as a function of both the scalar eddy-viscosity and the anisotropic Reynolds stress tensor. The latter model expressions also take into account various non-linear relationships in terms of turbulent stress components. The computational results of the fully-developed channel flow, featured only by the mean shear and constant wall heat flux, representing a preliminary investigated flow configuration, agree very well with the corresponding DNS (Direct Numerical Simulation) database for both RSM versions in conjunction with all the turbulent heat flux models adopted. However, the study of the two flow impingement cases with the different structure of the impinging jet and nozzle-to-wall distance variations, characterized by a much more complex flow straining, using the considered conventional Reynolds stress model shows significantly different results for the Reynolds stress and thermal fields compared to the available DNS data, especially in the immediate impact region manifested by a significant overprediction of the turbulence intensity. In contrast, the IISRSM-relevant results for the mean flow and temperature fields, the associated integral characteristics, and the corresponding turbulence quantities are in very good agreement with the reference DNS data, due to the much higher predictive capabilities of this eddy-resolving turbulence model compared to its baseline counterpart.

High-Order Implicit Large Eddy Simulation using Entropically Damped Artificial Compressibility

Performing industrial scale incompressible Large Eddy Simulation (LES) remains particularly challenging due to computational cost limitations. Recently, the Entropically Damped Artificial Compressibility (EDAC) method was introduced, which does not require coupled global solves or pseudo-time stepping. Parallel to this, high-order unstructured methods, such as the Flux Reconstruction (FR) approach, have been proposed for Implicit LES (ILES) of compressible turbulent flows. This work presents EDAC with FR as an ILES framework for incompressible flows. Results from a Taylor–Green vortex, turbulent channel flow, and transitional and turbulent flow over an SD7003 airfoil demonstrate the approach is accurate and stable for solutions polynomials of degree ps≤4, while some instabilities were observed beyond this. Higher-order solutions were found to be significantly more accurate than their lower-order counterparts compared to reference direct numerical simulation, and had lower divergence error even with less strict incompressibility factors. These results demonstrate that combining EDAC with FR provides a simple, accurate, efficient, and stable framework for performing ILES of incompressible flows. Additionally, this fully explicit framework does not require coupled solvers, pseudo-time stepping, or explicit subgrid scale models.

About the role of the Hanratty correction in the linear response of a turbulent flow bounded by a wavy wall

Scallop patterns forming on erodible surfaces were studied historically using a linear analysis of the inner region of a turbulent boundary layer growing on a corrugated wall. Experimental observations show a phase shift between the shear stress at the wall and the wall oscillation that depends on the wavenumber. Anad hoccorrection applied to the turbulent closure and due to Hanrattyet al.(Thorsnesset al.,Chem. Engng Sci., vol. 33, issue 5, 1978, pp. 579–592; Abrams & Hanratty,J. Fluid Mech., vol. 151, issue 1, 1985, p. 443; Frederick & Hanratty,Exp. Fluids, vol. 6, issue 7, 1988, pp. 477–486) was systematically used to recover the reference experimental results. In this study, Reynolds-averaged Navier–Stokes (RANS) and direct numerical simulations (DNS) were performed and revealed the role of the Boussinesq assumption in the results obtained. We show that the Hanratty correction acts as a palliative to the misrepresentation of Reynolds stresses due to the use of the Boussinesq hypothesis. The RANS calculations based on a turbulence model using a second-order moment closure recovered the expected results obtained in the reference DNS calculations, in particular with respect to wall heat transfer. The analysis of these results highlights the critical importance of the anisotropy of the diagonal Reynolds stresses on the prediction of wall transfer under these conditions and their implication in the occurrence of scalloping.

Acoustic Modeling Using the Aeroacoustic Wave Equation Based on Pierce’s Operator

The capabilities of an aeroacoustic wave equation based on Pierce’s operator (AWE-PO) for modeling subsonic flow-induced sound and for sound prediction are investigated. The wave equation is applied to an isothermal two-dimensional mixing layer computed by direct numerical simulation. In contrast to a direct numerical simulation, providing the acoustic fluctuations directly, the simulations based on Lighthill’s wave equation and the AWE-PO rely on a hybrid workflow to predict the generated sound field. Special attention is put on the interpretation of the right-hand side of both wave equations. Comparing the terms on the right-hand side in Lighthill’s theory and AWE-PO suggests a source amplitude for AWE-PO that is 90% smaller. This reduction is attributed to the filtering property of the material derivative. Finally, the results of the acoustic far-field pressure are compared. It is shown that the radiated sound field’s directivity, propagation, and convection effects are well captured for both wave equations. The computations using Lighthill’s equation and AWE-PO are found to provide acoustic intensities within 1.8 dB from the reference direct numerical simulation. This error is comparable with the errors reported for Lighthill’s equation in previous studies.

High-resolution reconstruction algorithm for the three-dimensional velocity field produced by atomization of two impinging jets based on deep learning

The velocity fields measured by experiments or determined through simulations are essential in advancing our understanding of the complex atomization process of impinging jets. However, existing methods are expensive and time-consuming. In this study, we apply deep learning to the estimation of the three-dimensional velocity fields produced by the atomization of two impinging jets. Two deep learning models are developed, namely, a liquid volume fraction (LVF) estimation model based on the Swin Transformer architecture and a three-dimensional velocity field estimation model based on four-dimensional convolution (4D-Conv). The dataset for training the models is generated by direct numerical simulations (DNS). To train the LVF model, we utilize two gray images generated by a pinhole camera model, mimicking the acquisition of experimental images. We then introduce a mask generated by binocular vision techniques into the LVF model. The LVF fields estimated with the mask are in better agreement with the reference DNS data. We further utilize the estimated LVF fields to train the 4D-Conv-based model. The mean absolute percentage error compared with the results of a full-flow test is found to be less than 5%. The results indicate that the proposed approach has the potential to accurately reconstruct volume velocity data from two-dimensional images.

Direct numerical simulation of flow and heat transport in a closely-spaced bare rod bundle

Direct Numerical Simulation (DNS) of fully-developed flow and heat transfer is performed for a bare rod bundle with a low pitch-to-diameter ratio using the highly-scalabale spectral element code Nek5000. The simulation is performed at a Reynolds number, Reh, based on the bulk velocity and hydraulic diameter, of 9800, with iso-temperature and iso-flux thermal boundary conditions and Prandtl numbers 0.025, 1.0 and 2.0. The mean velocity and temperature statistics are first validated against the experimental measurements of Hooper et al. (1983) and complementary DNS results of Lai et al. (2019) which are reported in the literature at a higher Reh of 22,600. The mean flow and temperature are demonstrated a similar behaviour at the present low Renolds number and the higher Reynolds number of Lai et al. (2019), indicating the Reynolds number scaling of the mean flow. The anisotropy of the flow is illustrated using distribution of normal and Reynolds stresses, and the Lumley invariant map. It is shown that the turbulence in the subchannel region is characteristic of wall-bounded turbulent flows such as channel flows. In the narrow gap, however, turbulence is suppressed, only enhanced in the streamwise direction due to flow pulsations. Further, temperature statistics with both thermal boundary conditions are presented, demonstrating the effect of the Prandtl number on the mean and fluctuating quantities. The temperature statistics with iso-temperature wall boundary conditions, not reported in literature before, are shown to behave differently in the interstitial subchannel region and the narrow gap between two rods. The distribution of turbulent heat flux is also presented. The streamwise component of turbulent heat flux is again shown to be enhanced, in comparison to the same in the subchannel. The pulsating motion of flow across the narrow gap is discussed further. A frequency analysis of the velocity in the narrow gap elucidates the frequency of flow pulsations. Unlike mean flow statistics, the flow pulsations are shown to exhibit a low Reynolds number effect. The reference DNS data presented in the present work, complementary to that of Lai et al. (2019), serves as a benchmark against which lower order turbulence models may be further developed or improved. In particular, the newly presented anisotropy data and the turbulence heat flux data are essential for further development, improvement and validation of turbulent heat transfer models at different Prandtl numbers.

Accelerated offline setup of homogenized microscopic model for multi‐scale analyses using neural network with knowledge transfer

AbstractThe FE computational homogenization method is a predictive multi‐scale method without the need for constitutive assumptions and/or potential function postulates at the macro engineering scale. Instead, the effective micro‐structural responses are extracted directly from a representative volume element (RVE) underlying each macro point. However, the FE method is still computationally too expensive for most practical uses, since the micro‐macro FE coupling is done at each loading step/iteration for the entire domain. To this end, the machine learning method has been utilized in the literature for the offline training of a surrogate model to predict the RVE homogenized response for general loading conditions. In this contribution, the neural network (NN) is incorporated into the macro finite element framework in a non‐intrusive manner. This is termed as the FE‐NN framework, in analogy to the FE method. In general, online simulations in the FE‐NN method is very efficient, with predictions matching closely to those obtained from reference direct numerical simulations (DNS). A bottleneck with the FE‐NN framework, however, is the high computational cost associated with the data generation for offline NN model setup. In this paper, focusing on the FE‐NN multi‐scale framework for non‐linear elastic deformation of heterogeneous materials, a sequential training strategy with knowledge transfer is proposed, to enable an efficient offline microscopic NN model setup. For a given target RVE, we first consider a simplified source RVE, where data can be generated rapidly, for the NN pre‐training of surrogate model. The pre‐trained network parameters are next downloaded to initialize the target NN surrogate model, followed by a fine‐tuning training process, using only a small dataset generated by the computationally expensive high‐fidelity RVE. The efficiency of the proposed sequential learning method over the conventional NN training, as well as, its excellent predictive capability for multi‐scale analyses, are demonstrated for a multi‐phase composite material. The proposed FE‐NN‐KT approach can be implemented easily without complicated pre‐processing procedures, since the data generation only involves the standard extraction of effective responses from RVEs, together with a standard NN architecture.

Modeling simultaneous momentum and passive scalar transfer in turbulent annular Poiseuille flow

AbstractSimultaneous momentum and passive scalar transfer in weakly heated pressure‐driven turbulent concentric annular pipe flow is numerically investigated using the cylindrical formulation of the stochastic one‐dimensional turbulence (ODT) model, which is utilized here as standalone tool. In the present study, we focus on the model calibration for heated annular pipes based on recent reference direct numerical simulations (DNS) from Bagheri and Wang (Int. J. Heat Fluid Flow 86, 108725, 2020; Phys. Fluids 33, 055131, 2021). It is shown that the model is able to individually capture scalar and momentum transfer, but not both equally well at the same time. We attribute this to less dissimilar scalar and momentum transport in the model at the low Reynolds number investigated. It is argued that the model prefers a fully developed turbulent state due to its construction. Nevertheless, it is demonstrated that ODT is able to reasonably capture the radial inner‐outer asymmetry of the scalar and momentum boundary layers which yields better predictive capabilities than wall‐function‐based approaches.

Wavelet-based adaptive large-eddy simulation of supersonic channel flow with different thermal boundary conditions

This work investigates the effect of different thermal wall boundary conditions on the wavelet-based adaptive large-eddy simulation of supersonic turbulent channel flow. The compressible flow governing equations are expressed in terms of wavelet-based Favre-filtered variables and are supplied with the anisotropic minimum dissipation closure model. Various computations are performed, where the resolved temperature field is constrained by either Dirichlet (isothermal) or Neumann (adiabatic) boundary conditions at the walls. The turbulence diagnostics include mean flow features and turbulent fluctuations statistics. The successful comparison with reference direct numerical simulations demonstrates the validity and the efficiency of the wavelet-based adaptive approach for wall-bounded turbulent compressible flow, regardless of the thermal boundary conditions that are imposed.

Two-Phase Turbulent Kinetic Energy Budget Computation in Co-Current Taylor Bubble Flow

Simulation of two-phase flows is relevant for reactor design and safety at normal operation or during accident scenarios. Often, the two-phase flow is in a regime in which slugs are formed or where the flow stratifies. Modeling such situations using standard single-phase Reynolds-averaged Navier-Stokes (RANS) turbulence models fails due to an overestimation of the eddy viscosity at the resolved two-phase interface. To solve this, an ad hoc turbulence damping term has been proposed in the literature that reduces the turbulence production locally at a two-phase interface, analogously to turbulence wall functions. However, this approach must be tailored to the specific setting and does not consider physical contributions such as surface tension or flow topology. Therefore, the problem of two-phase interfacial turbulence must be studied more in-depth. In this work, we consider co-current turbulent Taylor bubble flow using high-fidelity numerical simulation. The Basilisk code is used to simulate a Taylor bubble rising in a vertical pipe. By simulating the bubble in a moving frame of reference, we may study the turbulent kinetic energy (TKE) budgets ahead of the bubble, in its wake, and across the interface. The implementation of the TKE budget computation and the underlying averaging techniques are validated for the single-phase region ahead of the Taylor bubble using reference direct numerical simulation data. The analysis of the TKE budgets in the setting of Taylor bubble flow allows for the study of how turbulence behaves due to the presence of a two-phase interface and, in turn, supports the improvement of two-phase RANS models.

Comparing LES and URANS results with a reference DNS of the transitional airflow in a patient-specific larynx geometry during exhalation

Air exhalation in the laryngeal area produces a complex transitional flow. A detailed investigation of the resulting flow features is very interesting from a fundamental point of view and useful to support medical researchers regarding treatment options. In the current study, the highly complex, unsteady, transitional flow in a patient-specific geometry has been described by three different numerical approaches, (1) using the unsteady Reynolds-averaged Navier–Stokes (URANS) equations with the k−ω shear stress transport (SST) turbulence model, (2) by large-eddy simulation (LES) with the WALE subgrid-scale model, and finally (3) using direct numerical simulation (DNS) – without any model needed. These three approaches deliver different pictures of the resulting flow, obviously with noticeable differences in terms of accuracy, but also regarding the associated computational efforts. The present article is devoted to the detailed comparison of these three approaches, using the DNS results as a reference. It has been found that even though URANS is able to predict with a fair accuracy the large-scale features of velocity and pressure in an average sense, it fails to predict correctly most relevant turbulent features; large differences are found concerning fluctuations, power spectral density, or spectral entropy. On the other hand, LES is able to reproduce both small-scale and large-scale flow features in a good agreement compared to DNS; additionally, LES delivers an appropriate description of the evolution of spectral entropy and can thus be used to predict flow transition. Obviously, DNS delivers an even more detailed and accurate view of the flow properties, but this comes with a tremendous increase in computing effort. For this configuration, a high-quality LES appears as best compromise between accuracy and computational requirements and, therefore, best suitable for medical studies of transitional airflows. All data-sets are publicly available under Abdelsamie et al. [1].

Numerical Investigation of Wall-Cooling Effect on Aero-Optical Distortions for Hypersonic Boundary Layer

Compressible wall-modeled large-eddy simulations of Mach 8 turbulent boundary-layer flows over a flat plate were carried out for the conditions of the hypersonic wind tunnel at Sandia National Laboratories. The simulations provide new insight into the effect of wall cooling on the aero-optical path distortions for hypersonic turbulent boundary-layer flows. Four different wall-to-recovery temperature ratios, 0.3, 0.48, 0.71, and 0.89, are considered. Despite the much lower grid resolution, the mean velocity, temperature, and resolved Reynolds stress profiles from the simulation for a temperature ratio of 0.48 are in good agreement with those from a reference direct numerical simulation. The normalized root-mean-square optical path difference obtained from the present simulations is compared with that from reference direct numerical simulations, Sandia experiments, as well as predictions obtained with a semi-analytical model by Notre Dame University. The present analysis focuses on the effect of wall cooling on the wall-normal density correlations, on key underlying assumptions of the aforementioned model such as the strong Reynolds analogy, and on the elevation angle effect on the optical path difference. Wall cooling is found to increase the velocity fluctuations and decrease the density fluctuations, resulting in an overall reduction of the normalized optical path distortion. Compared to the simulations, the basic strong Reynolds analogy overpredicts the temperature fluctuations for cooled walls. Also different from the strong Reynolds analogy, the velocity and temperature fluctuations are not perfectly anticorrelated. Finally, as the wall temperature is raised, the density correlation length, away from the wall but inside the boundary layer, increases significantly for beam paths tilted in the downstream direction.

Large eddy simulation on unstructured grids using explicit differential filtering: A case study of Taylor-Green vortex

The application of an approximate deconvolution model was extended for large eddy simulation (LES) on unstructured grids by further developing a three dimensional explicit differential filter. Decaying isotropic homogeneous turbulence at a Reynolds number 3,400 was modeled on both structured and unstructured grids. Results were compared to a reference direct numerical simulation and other results reported in the literature. A Taylor-Green vortex at a Reynolds number of 1,600 was also studied as it is a canonical problem for laminar to turbulent transitional flow. Investigative studies on the ADM order and under-relaxation coefficient, the degree of anisotropy of the grid, and the grid resolution were performed to benchmark the filter performance in conjunction with ADM. Finally, a large eddy simulation on a fully unstructured grid was performed to demonstrate the range of applications of the proposed filter. All numerical results showed strong LES numerical stability when the explicit differential filter was used.

Three-dimensional ESRGAN for super-resolution reconstruction of turbulent flows with tricubic interpolation-based transfer learning

Turbulence is a complicated phenomenon because of its chaotic behavior with multiple spatiotemporal scales. Turbulence also has irregularity and diffusivity, making predicting and reconstructing turbulence more challenging. This study proposes a deep-learning approach to reconstruct three-dimensional (3D) high-resolution turbulent flows from spatially limited data using a 3D enhanced super-resolution generative adversarial networks (3D-ESRGAN). In addition, a novel transfer-learning method based on tricubic interpolation is employed. Turbulent channel flow data at friction Reynolds numbers Reτ = 180 and Reτ = 500 were generated by direct numerical simulation (DNS) and used to estimate the performance of the deep-learning model as well as that of tricubic interpolation-based transfer learning. The results, including instantaneous velocity fields and turbulence statistics, show that the reconstructed high-resolution data agree well with the reference DNS data. The findings also indicate that the proposed 3D-ESRGAN can reconstruct 3D high-resolution turbulent flows even with limited training data.

Implementation and Validation of an Algebraic Wall Model for LES in Nek5000

Turbulent flows are most often wall-bounded, rendering the treatment of the wall essential. In this work, the near-wall layer is modelled in large eddy simulations which enables simulating high Reynolds number flows. An algebraic wall model has been implemented in the spectral element code Nek5000. It consists of an approximate boundary condition that relates the wall shear stress to the velocity measured close to the wall, on the upper edge of the first spectral element. The wall shear stress model approximates the law of the wall for hydraulically smooth cases. The model is applied to channel flow cases at Re_{tau }=1000 and at Re_{tau }=5200. The wmLES results obtained with the present implementation are seen to compare very well with those of reference direct numerical simulations in the resolved region. They also remain remarkably close to the reference results for a large part of the under-resolved region; which is not necessarily the case when using low order implementations and even other types of high order discretizations, as found in the literature. Various parameters are studied such as the time averaging, the height of the near wall under-resolved element, and the mesh requirements. The obtained results indicate that accurate results can be obtained with Nek5000 at a reduced cost thanks to this newly implemented wmLES model. This work provides the necessary guidelines for simple flows, and it will serve as a first basis for simulating more complex flows at high Reynolds numbers.

Applications of wall-models to implicit large eddy simulations in the spectral/hp element method

An implicit large eddy simulation (iLES) method based on a spectral vanishing viscosity (SVV) method has been well established in the framework of the spectral/hp element method (which is called the SVV-iLES method), and the SVV regularization introduces proper dissipation to the numerical scheme, which in turn reduces the truncation error and the uncertainty in computation. However, SVV-iLES is still limited to moderate Reynolds numbers due to the huge computational cost in resolving the inner part of the boundary layer. By modeling the near-wall layer, a wall-model coupling strategy, tailored to the features of the SVV-iLES, is proposed to substantially reduce the computational cost. This work is the first to construct the wall model of SVV-iLES based on the spectral/hp element method, and the practice rules can be regarded as guidelines for the application of the wall model to iLES with high order schemes. Numerical investigations for a well-established benchmark problem of the turbulent channel flow are implemented by the proposed wall-modeled SVV-iLES. The results obtained are consistent with the reference direct numerical simulation (DNS) data at different Reynolds numbers considered. The effect of the time-averaged velocity at the matching interface is examined, and the results suggest that the instantaneous velocity can be directly used in the algebraic wall-model, which in turn could speed up the computation. Additionally, a comparison between the classical log-law and Reichardt's law models reveals that there are no significant differences in statistical quantities in the logarithmic region. Finally, it was found that, at similar grid resolution, the p-type refinement shows better agreement with the reference DNS results.