Direct Numerical Simulation Data Research Articles

Validation of RANS-Based Turbulence Models Against High-Resolution Experiments and DNS for Buoyancy-Driven Flow with Stratified Fronts

To provide computational fluid dynamics (CFD)–grade experimental data for studying stratification, measurements on the High-Resolution Jet (HiRJet) facility at the University of Michigan have been conducted with density differences of 1.5 % and − 1.5 % , respectively. Fluid with a density different from the fluid initially present in the HiRJet tank was injected, and the propagation of the time-dependent density stratification was captured on a two-dimensional plane with the aid of the wire-mesh sensor technique for Reynolds numbers near 5000 and Richardson numbers near 0.29. Direct numerical simulations (DNSs) of the two cases have also been conducted to expand the multifidelity database. The novel experimental and DNS data were then used to assess the predictive capabilities of the Standard k − ε (SKE) model and the Reynolds Stress Transport (RST) model. In particular, the propagation speed and thickness of the stratification fronts were assessed by comparing the CFD results against the experimental and DNS data. It was found that the general trends of the stratified density fronts were well predicted by the CFD simulations; however, slight overprediction of the thickness of the stratification layer was found with the SKE model while the RST model gave a larger overprediction of the mixing. Examination of the turbulent statistics showed that the turbulent viscosity was largely overpredicted by the RST model compared to the SKE model.

New formulations for the mean wall-normal velocity and Reynolds shear stress in a turbulent boundary layer under zero pressure gradient

In this study, a novel analytical solution is derived rigorously for the mean wall-normal velocity in a turbulent boundary layer (TBL) under zero pressure gradient (ZPG), without relying on any a priori assumptions regarding the mean streamwise velocity. By neglecting the higher-order terms in the exact solution, an approximate formulation for the mean wall-normal velocity is obtained. The accuracy of this approximation is then validated through a comparison with data from direct numerical simulations (DNS). Drawing upon the distinct behaviours of force balance in the inner and outer regions of the ZPG TBL, simplified approximate mean momentum equations are derived by decomposing the Reynolds shear stress into inner and outer parts. The inner and outer Reynolds shear stresses are then obtained through integration of the corresponding approximate mean momentum equations. The outer Reynolds shear stress is revealed to exhibit a strong connection to the mean wall-normal velocity. Specifically, it is observed that $R_{uv-{out}}/u^2_\tau \approx 1 - UV/(U_e V_e)$ , where $u_\tau$ represents the friction velocity, and $U_e$ and $V_e$ denote the mean streamwise and wall-normal velocities at the boundary layer edge, respectively. Finally, an approximate formulation for the Reynolds shear stress across the entire TBL is developed by synthesizing the inner and outer parts. Extensive validation against both DNS and experimental data, spanning a wide range of Reynolds numbers, demonstrates the excellent agreement achieved.

An explicit representation for mean profiles and fluxes in forced passive scalar convection

We derive explicit formulae for the mean profiles of passive scalars (either temperature or concentration of a diffusing substance), and their respective wall fluxes (either heat or mass fluxes), in forced turbulent convection, as a function of the Reynolds and Prandtl numbers. Direct numerical simulation data for turbulent flow within a smooth straight pipe of circular cross-section, at friction Reynolds number ${{Re}}_{\tau }=1140$ , in the range of Prandtl numbers from ${{Pr}}=0.00625$ to ${{Pr}}=16$ , are used to infer the proper analytical form of the eddy diffusivity. This is leveraged to derive accurate predictive formulae for the mean passive scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling laws result for the thickness of the conductive (diffusive) layer, and for the Nusselt number, which significantly extend the predictive envelope of classical formulae.

A Swin-transformer-based model for efficient compression of turbulent flow data

This study proposes a novel deep-learning-based method for generating reduced representations of turbulent flows that ensures efficient storage and transfer while maintaining high accuracy during decompression. A Swin-transformer (ST) network combined with a physical constraints-based loss function is utilized to compress the turbulent flows with high compression ratios and then restore the data with underlying physical properties. The forced isotropic turbulence is used to demonstrate the ability of the ST-based model, where the instantaneous and statistical results show the excellent ability of the model to recover the flow data with a remarkable accuracy. Furthermore, the capability of the ST model is compared with a typical convolutional neural network-based auto-encoder (CNN-AE) by using the turbulent channel flow at two friction Reynolds numbers Reτ = 180 and 550. The results generated by the ST model are significantly more consistent with the direct numerical simulation data than those recovered by the CNN-AE, indicating the superior ability of the ST model to compress and restore the turbulent flow. This study also compares the compression performance of the ST model at different compression ratios (CR s) and finds that the model has low enough error even at very high CR. Additionally, the effect of transfer learning (TL) is investigated, showing that TL reduces the training time by 64% while maintaining high accuracy. The results illustrate for the first time that the Swin-transformer-based model incorporating a physically constrained loss function can compress and restore turbulent flows with the correct physics.

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.

An open-source anisotropic k−ε−v2−f model for turbulent viscoelastic duct flows

A novel open-source anisotropic k−ε−v2−f model is presented for turbulent viscoelastic duct flow with dilute polymeric solutions described by the finitely extensible nonlinear elastic-Peterlin constitutive model. The turbulence model for channel and square duct flow of Newtonian fluids is adapted to incorporate the polymeric terms within the governing equations. All the required non-linear terms are validated with simple closure models and are assessed a priori against independent direct numerical simulation data in fully developed channel flow. The NLTij term, which accounts for the interaction between fluctuating components of the conformation tensor and the velocity gradient tensor, is modeled with the mean flow direction, ti, and wall-normal, ni, present in the Newtonian model, based on the streamwise alignment of mean polymer stretch. The implicit polymer effects on pressure–strain are assessed with a simple ad hoc closure accounting for the reduced near-wall production of turbulent kinetic energy. The same closure is also adapted for the spanwise Reynolds stress predictions of polymer-enhanced secondary flow. The model performs well in channel flow and captures low, intermediate, and high drag reduction features for a wide range of rheological parameters. The capabilities are extended for square ducts (or any regular polygon) due to the symmetric modeling of the closure models, which can predict the mean streamwise and secondary flow features associated with second normal Reynolds stress differences. Accessible codes and models are crucial for the advancement and improvement of turbulent viscoelastic models, and an OpenFOAM C++ code package is developed and freely available on GitHub (https://github.com/MikeMcDermott-Code/v2f).

Rotational vector-based analysis of turbulent structures in channel flow using large eddy simulation simulation

Even with modern measurement techniques and data from direct numerical simulation (DNS), it is very difficult to identify the individual attached eddies and understand their dynamical behavior due to the multi-scale nature of the eddies in wall-bounded flows, which puts these issues at the center of the current debate. However, Liutex vector ( L for short), a rotational vector field with information on both the rotation axis and swirling strength, has recently been developed by Prof. Liu’s group as a more accurate and clear definition of a vortex. Combining conventional and L methods may provide more detailed information about the complex flow structures as well as insights into the flow’s mixing and transport features. We simulated the channel flow in large eddy simulation by implementing an inflow condition based on the box turbulence. After validating the results with DNS data, we used L isosurfaces and their vector profiles to track ordered flow structures in wall-bounded turbulence. Based on the data, we observe numerous turbulent phenomena that have been described in other works with different visualization techniques. Moreover, the shear contamination on the wall is the most severe while all the root-mean-square L component variations are negligible. Due to the presence of background shear, the peak location of vorticity fluctuation is closer to the wall than the corresponding L fluctuation, and the displacement of peak location brought on by shear contamination is greatest for the spanwise component (z-component) of the vorticity fluctuation. According to the two-point correlation of L components, the streamwise size of turbulent structures does not vary considerably with , however, the spanwise size of turbulent structures increases gradually as increases.

A priori test of perfectly stirred reactor approach to evaluating mean fuel consumption and heat release rates in highly turbulent premixed flames

Unsteady three-dimensional Direct Numerical Simulation (DNS) data obtained earlier by Dave et al. ( J Fluid Mech 2020; 884: A46) from a statistically planar and one-dimensional, highly turbulent, moderately lean hydrogen-air flame propagating in a box are processed to perform a priori test of perfectly stirred reactor model. The test aims in particularly at estimating mesh resolution (or filter width within large eddy simulation framework) required to neglect variations in the temperature and mixture composition within a computational cell when evaluating mean (or filtered) fuel consumption and heat release rates. For this purpose, fuel consumption and heat release rates sampled directly from the DNS data and averaged over a cube of width [Formula: see text] are compared with fuel consumption and heat release rates calculated using the temperature and species concentrations averaged over the same cube. Moreover, turbulent burning velocities computed by integrating the former and latter rates are compared with one another. A ratio of [Formula: see text] to a laminar flame thickness [Formula: see text] is varied from 0.44 to 1.8. The obtained results indicate that the tested simple approach performs reasonably well (poor) if [Formula: see text] ([Formula: see text], respectively). This result is further supported by directly filtering fuel consumption rate in a laminar premixed flame. The values of the thickness [Formula: see text], calculated using detail chemical mechanisms for different fuels under elevated temperatures and pressures associated with combustion in piston engines, indicate that it is difficult to satisfy the constraint of [Formula: see text] in contemporary unsteady multidimensional numerical simulations of turbulent burning in such engines.

Deep learning closure models for large-eddy simulation of flows around bluff bodies

Near-wall flow simulation remains a central challenge in aerodynamics modelling: Reynolds-averaged Navier–Stokes predictions of separated flows are often inaccurate, and large-eddy simulation (LES) can require prohibitively small near-wall mesh sizes. A deep learning (DL) closure model for LES is developed by introducing untrained neural networks into the governing equations and training in situ for incompressible flows around rectangular prisms at moderate Reynolds numbers. The DL-LES models are trained using adjoint partial differential equation (PDE) optimization methods to match, as closely as possible, direct numerical simulation (DNS) data. They are then evaluated out-of-sample – for aspect ratios, Reynolds numbers and bluff-body geometries not included in the training data – and compared with standard LES models. The DL-LES models outperform these models and are able to achieve accurate LES predictions on a relatively coarse mesh (downsampled from the DNS mesh by factors of four or eight in each Cartesian direction). We study the accuracy of the DL-LES model for predicting the drag coefficient, near-wall and far-field mean flow, and resolved Reynolds stress. A crucial challenge is that the LES quantities of interest are the steady-state flow statistics; for example, a time-averaged velocity component $\langle {u}_i\rangle (x) = \lim _{t \rightarrow \infty } ({1}/{t}) \int _0^t u_i(s,x)\, {\rm d}s$ . Calculating the steady-state flow statistics therefore requires simulating the DL-LES equations over a large number of flow times through the domain. It is a non-trivial question whether an unsteady PDE model with a functional form defined by a deep neural network can remain stable and accurate on $t \in [0, \infty )$ , especially when trained over comparatively short time intervals. Our results demonstrate that the DL-LES models are accurate and stable over long time horizons, which enables the estimation of the steady-state mean velocity, fluctuations and drag coefficient of turbulent flows around bluff bodies relevant to aerodynamics applications.

On the origin of streamwise vortices in braid regions for compressible mixing layers

The origin of initial streamwise vortices in braid regions and their relationship with deformed spanwise vortices are numerically studied via direct numerical simulation (DNS) in the compressible mixing layer with a convective Mach number (Mc) of 0.4. Through an analysis of fast Fourier transform on DNS data, two low-rank velocity models (vd and vs) are developed to demonstrate that both deformed spanwise vortices and streamwise vortices are all originated from the linear superposition of a fundamental norm mode [mode (1, 0)], a pair of fundamental oblique modes [modes (1, ±1)], and a mean mode. Further investigations reveal that, downstream of spanwise vortices, the increase in amplitude ratio (Ao/An) between modes (1, ±1) and mode (1, 0) leads to the formation of deformed spanwise vortices in vortex regions. As the amplitude ratio Ao/An further increases, reaching the threshold that the streamwise rotation motion from modes (1, ±1) exceeds the spanwise irrotational deformation from mode (1, 0), streamwise vortices are generated in braid regions. The aforementioned formation path for deformed spanwise vortices and streamwise vortices provides a mechanism support for our flow visualization results that the emergence of deformed spanwise vortices precedes that of streamwise vortices in the flow field.

On the motion of hairpin filaments in the atmospheric boundary layer

On the motion of hairpin filaments in the atmospheric boundary layer

Inflow turbulence generation using an equivalent boundary layer model

Eddy-resolved simulation of external flow usually requires inflow boundary conditions representing a turbulent boundary layer (TBL) flow, and the quality of the inflow turbulent fluctuation directly impact the accuracy and the cost of the simulation. The present study proposes a new method to generate TBL inflow turbulence, i.e., the equivalent boundary layer (EBL) model. Based on the open-channel model, EBL approximates TBL flow at a given Reynolds number by recovering the mean momentum balance with driving force. It simulates streamwise homogeneous turbulence, applying periodic boundary conditions and, thus, overcomes the complexity and artificiality incurred by the classic recycling–rescaling methods. The current paper discusses the difference between turbulent channel and boundary layer flows from the equation point of view and designs the driving force corresponding to the mean inertial force of boundary layer. Also, the total shear stress models for obtaining the driving force are validated both a priori and a posteriori. Direct numerical simulations (DNS) are carried out for EBLs at Reθ=1000,1420, and 2000 (where Reθ is the Reynolds number based on the momentum thickness), showing that the EBL model well reflects the statistical characteristics of TBL at corresponding Reynolds numbers. The application of the EBL model for the generation of inflow turbulence is also demonstrated by DNS of turbulent boundary layers with inlet Reθ=1000,1420, and 2000. The computational results agree well with generally acknowledged DNS data published in the literature, in terms of streamwise developing statistics, and profiles and energy spectra at characteristic cross sections. Judging from the mean velocity, the adjustment section is shorter than one boundary layer thickness.

Neural networks for large eddy simulations of wall-bounded turbulence: numerical experiments and challenges.

We examine the application of neural network-based methods to improve the accuracy of large eddy simulations of incompressible turbulent flows. The networks are trained to learn a mapping between flow features and the subgrid scales, and applied locally and instantaneously-in the same way as traditional physics-based subgrid closures. Models that use only the local resolved strain rate are poorly correlated with the actual subgrid forces obtained from filtering direct numerical simulation data. We see that highly accurate models in a priori testing are inaccurate in forward calculations, owing to the preponderance of numerical errors in implicitly filtered large eddy simulations. A network that accounts for the discretization errors is trained and found to be unstable in a posteriori testing. We identify a number of challenges that the approach faces, including a distribution shift that affects networks that fail to account for numerical errors.

Prandtl number effects on passive scalars in turbulent pipe flow

We study the statistics of passive scalars (either temperature or concentration of a diffusing substance) at friction Reynolds number ${Re}_{\tau }=1140$ , for turbulent flow within a smooth straight pipe of circular cross-section, in the range of Prandtl numbers from ${Pr}=0.00625$ , to ${Pr}=16$ , using direct numerical simulations (DNS) of the Navier–Stokes equations. Whereas the organization of passive scalars is similar to the axial velocity field at ${Pr} = O(1)$ , similarity is impaired at low Prandtl number, at which the buffer-layer dynamics is filtered out, and at high Prandtl number, at which the passive scalar fluctuations become confined to the near-wall layer. The mean scalar profiles at ${Pr} \gtrsim 0.0125$ are found to exhibit logarithmic overlap layers, and universal parabolic distributions in the core part of the flow. Near-universality of the eddy diffusivity is exploited to derive accurate predictive formulas for the mean scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling formulas are derived for the thickness of the conductive (diffusive) layer, for the peak scalar variance, and its production rate. The DNS data are leveraged to synthesize a modified form of the classical predictive formula of Kader & Yaglom (Intl J. Heat Mass Transfer, vol. 15, 1972, pp. 2329–2351), which is capable of accounting accurately for the dependence on both Reynolds and Prandtl numbers, for ${Pr} \gtrsim 0.25$ .

An Assessment of Second Moment Closure Modeling for Stratified Wakes Using Direct Numerical Simulations Ensembles

Abstract Buoyant wakes encountered in the ocean environment are characterized by high Reynolds (Re) and Froude (Fr) numbers, leading to significant space–time resolution requirements for turbulence resolving CFD models (i.e., direct numerical simulations (DNS), large eddy simulations (LES)). Therefore, Reynolds-averaged Navier–Stokes (RANS) based models are attractive for these configurations. The inherently complex dynamics of stratified systems render eddy-viscosity-based modeling inappropriate. RANS second-moment closure (SMC) based modeling is more suitable because it accounts for flow anisotropy by solving the transport equations of important second-moment terms. Accordingly, eleven transport equations are solved at the SMC level, and a range of submodels are implemented for diffusion, pressure strain and scrambling, and dissipation terms. This work studies nonstratified and stratified towed wakes using SMC and DNS. Submodels in the SMC are evaluated in terms of how well their exact Reynolds averaged form impacts the accuracy of the full RANS closure. An ensemble average of 40 and 80–100 DNS realizations are required and conducted for these temporally evolving nonstratified and stratified wakes, respectively, to obtain converged higher-order statistics. SMC over-predicts wake height by over a factor of 2, and under-predicts defect velocity, wake width, and turbulent kinetic and potential energies by factors ranging from 1.3 to 3.5. Also, SMC predicts a near isotropic decay of normal Reynolds stresses (a33→−0.25), in contrast to the anisotropic decay (a33→−0.64) returned by DNS. The DNS data also provide important insights related to the inaccuracy of the dissipation rate isotropy assumption and the non-negligible contribution of pressure diffusion terms. These results lead to several important recommendations for SMC modeling improvement.

Lagrangian model for passive scalar gradients in turbulence

The equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier–Stokes equation. However, such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian. A recent model handles these unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson & Meneveau, Phys. Rev. Fluids, vol. 2 (7), 2017, 072601). This model is in remarkable agreement with direct numerical simulations (DNS) data and works for arbitrary Taylor Reynolds numbers $\textit {Re}_\lambda$ . Inspired by this, we develop a Lagrangian model for passive scalar gradients in isotropic turbulence. The equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term, which we model using the RDGF approach. However, comparisons of the statistics obtained from this model with DNS data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to $\textit {Re}_\lambda \approx 500$ , beyond this, the model breaks down.

Vortex imprints on a free surface as proxy for surface divergence

In turbulence near a free surface, strong vortices attach to the surface, creating surface imprints visible as nearly circular ‘dimples’. By studying these imprints in direct numerical simulation (DNS) data we make two observations. First, the imprints of surface-attached vortices can be very effectively distinguished from other turbulent surface features using two physical features: they are nearly circular in shape, and persist for a long time compared with other pertinent time scales. Secondly, the instantaneous number of surface dimples from surface-attached vortices in an area, $N(t)$ , is intimately related to its mean-square surface divergence, $\beta ^2(t)$ . We develop a simple and physically transparent computer vision procedure which, using the properties of low eccentricity and longevity, detects and tracks vortices from their surface features only, with sensitivity and accuracy of $90\,\%$ or better. We compare $N(t)$ and $\beta ^2(t)$ , finding a normalised cross-correlation of $0.90$ , with changes in $N$ lagging around $0.8T_\infty$ behind those in $\beta ^2$ ( $T_\infty$ is an integral time scale), confirming the common observation that vortices are spawned by strong upwelling events where $\beta ^2$ is large. These findings suggest that the rate of mass flux across the surface, being closely related to surface divergence, can be estimated remotely in some natural flows using visible free-surface dimples as proxy.

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.

Drag model of finite-sized particle in turbulent wall-bound flow over sediment bed

Drag force acting on a particle is vital for the accurate simulation of turbulent multiphase flows, but the robust drag model is still an open issue. Fully resolved direct numerical simulation (DNS) with an immersed boundary method is performed to investigate the drag force on saltating particles in wall turbulence over a sediment bed. Results show that, for saltating particles, the drag force along the particle trajectories cannot be estimated accurately by traditional drag models originally developed for an isolated particle that depends on the particle-wall separation distance or local volume fraction in addition to the particle Reynolds number. The errors between the models and DNS are especially clear during the descending phase of the particles. Through simple theoretical analysis and DNS data fitting, we present a corrected factor using the classical, particle Reynolds number dependent drag force model as the benchmark model. The new drag model, which takes the particle vertical velocity into account, can reasonably predict the mean drag force obtained by DNS along a particle trajectory.

Wavelet analysis of supersonic shock-boundary-layer interaction

We present a wavelet analysis of supersonic shock-boundary-layer interaction. We have used direct numerical simulation data for supersonic flow over a compression ramp and performed orthogonal anisotropic wavelets. The wavelet-based method of extracting coherent structures is applied to the flow vorticity field, decomposed into coherent and incoherent contributions using thresholding of the wavelet coefficients. The statistics of the coherent part of vorticity are close to the statistics of the total field. The study aims to improve our understanding of the shock-boundary-layer interaction, the role of vorticity, and the relationship between the flow's coherent and incoherent vorticity components with the near-wall sound. Our analysis shows a substantial correlation between the incoherent part of vorticity components and wall-pressure fluctuations.