Modeled Reynolds stress is a major source of model-form uncertainties in Reynolds-averaged Navier-Stokes (RANS) simulations. Recently, a physics-informed machine-learning (PIML) approach has been proposed for reconstructing the discrepancies in RANS-modeled Reynolds stresses. The merits of the PIML framework has been demonstrated in several canonical incompressible flows. However, its performance on high-Mach-number flows is still not clear. In this work we use the PIML approach to predict the discrepancies in RANS modeled Reynolds stresses in high-Mach-number flat-plate turbulent boundary layers by using an existing DNS database. Specifically, the discrepancy function is first constructed using a DNS training flow and then used to correct RANS-predicted Reynolds stresses under flow conditions different from the DNS. The machine-learning technique is shown to significantly improve RANS-modeled turbulent normal stresses, the turbulent kinetic energy, and the Reynolds-stress anisotropy. Improvements are consistently observed when different training datasets are used. Moreover, a high-dimensional visualization technique and distance metrics are used to provide a priori assessment of prediction confidence based only on RANS simulations. This study demonstrates that the PIML approach is a computationally affordable technique for improving the accuracy of RANS-modeled Reynolds stresses for high-Mach-number turbulent flows when there is a lack of experiments and high-fidelity simulations.
With the rising of modern data science, data--driven turbulence modeling with the aid of machine learning algorithms is becoming a new promising field. Many approaches are able to achieve better Reynolds stress prediction, with much lower modeling error ($ε_M$), than traditional RANS models but they still suffer from numerical error and stability issues when the mean velocity fields are estimated using RANS equations with the predicted Reynolds stresses, illustrating that the error of solving the RANS equations ($ε_P$) is also very important. In the present work, the error $ε_P$ is studied sep…
The Reynolds stress in Holmboe instabilities at moderate Reynolds numbers is investigated using single wavelength simulations (SWS), multiple wavelength simulations (MWS), and laboratory experiments. The rightward and leftward propagating instabilities are separated with the two-dimensional discrete Fourier transform, enabling a direct comparison of the perturbation fields between the numerical simulations and linear stability analysis. The decomposition and superposition of the perturbation fields provide a new insight into the origin of Reynolds stresses. Conventionally, only the statistics …
The ability of linear stochastic response analysis to estimate coherent motions is investigated in turbulent channel flow at friction Reynolds number Re$_τ$ = 1007. The analysis is performed for spatial scales characteristic of buffer-layer and large-scale motions by separating the contributions of different temporal frequencies. Good agreement between the measured spatio-temporal power spectral densities and those estimated by means of the resolvent is found when the effect of turbulent Reynolds stresses, modelled with an eddy-viscosity associated to the turbulent mean flow, is included in th…
The Reynolds stress, or equivalently the average of the momentum flux, is key to understanding the statistical properties of turbulent flows. Both typical and rare fluctuations of the time averaged momentum flux are needed to fully characterize the slow flow evolution. The fluctuations are described by a large deviation rate function that may be calculated either from numerical simulation, or from theory. We show that, for parameter regimes in which a quasilinear approximation is accurate, the rate function can be found by solving a matrix Riccati equation. Using this tool we compute for the f…
The turbulent flow over monochromatic waves of moderate steepness is studied by means of wall resolved LES, and cover a range of wave ages and Reynolds numbers. We compute the Fourier modes of the flow and analyse the momentum balance for the mean and fundamental mode. At low wave ages, the form drag displays a large sensitivity to changes in Reynolds number, and the interaction between turbulent and wave-induced stresses increases with Reynolds number. At higher wave ages, a quasi-laminar regime is entered, where the wave-induced stress is primarily balanced by viscous stresses. To exploit th…
Although an increased availability of computational resources has enabled high-fidelity simulations of turbulent flows, the RANS models are still the dominant tools for industrial applications. However, the predictive capabilities of RANS models are limited by potential inaccuracy driven by hypotheses in the Reynolds stress closure. Recently, a Physics-Informed Machine Learning (PIML) approach has been proposed to learn the functional form of Reynolds stress discrepancy in RANS simulations based on available data. It has been demonstrated that the learned discrepancy function can be used to im…
Pore-resolved direct numerical simulations of turbulent open channel flow are performed comparing the structure and dynamics of turbulence over impermeable rough and smooth walls to a porous sediment bed at permeability Reynolds number ($Re_K$) of 2.6, representative of aquatic beds. Four configurations are investigated; namely, (i) permeable bed with randomly packed sediment grains, (ii) an impermeable-wall with full layer of roughness elements matching the top layer of the sediment bed, (iii) an impermeable-wall with half layer of roughness elements , and (iv) a smooth wall. A double-averagi…
The interaction between small- and large-scale structures, and the coexisting bottom-up and top-down processes are studied in a turbulent plane Couette flow, where space-filling longitudinal rolls appear at relatively low values of the Reynolds number $Re$. A DNS database at $Re_τ=101$ is built to replicate the highest $Re$ considered in a recent experimental work by Kawata and Alfredsson (Phys. Rev. Lett., vol.120, 2018, 244501). Our study is based on the exact budget equations for the second-order structure function tensor $\langle δu_i δu_j \rangle$, i.e. the Anisotropic Generalized Kolmogo…
Reynolds-averaged Navier-Stokes (RANS) equations are widely used in engineering turbulent flow simulations. However, RANS predictions may have large discrepancies due to the uncertainties in modeled Reynolds stresses. Recently, Wang et al. demonstrated that machine learning can be used to improve the RANS modeled Reynolds stresses by leveraging data from high fidelity simulations (Physics informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data. Physical Review Fluids. 2, 034603, 2017). However, solving for mean flows from the improved Reyn…
High-spatial-resolution (HSR) two-component, two-dimensional particle-image-velocimetry (2C-2D PIV) measurements of a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) and an adverse-pressure-gradient (APG)-TBL were taken in the LMFL High Reynolds number Boundary Layer Wind Tunnel. The ZPG-TBL has a momentum-thickness based Reynolds number $Re_{δ_2} = δ_2 U_e/ν= 7,750$ while the APG-TBL has a $Re_{δ_2} = 16,240$ and a Clauser's pressure gradient parameter $β= δ_1 P_x/τ_w = 2.27$ After analysing the single-exposed PIV image data using a multigrid/multipass digital PIV (Soria, 1996) wi…
We study the laminar and turbulent channel flow over a viscous hyper-elastic wall and show that it is possible to sustain an unsteady chaotic turbulent-like flow at any Reynolds number by properly choosing the wall elastic modulus. We propose a physical explanation for this effect by evaluating the shear stress and the turbulent kinetic energy budget in the fluid and elastic layer. We vary the bulk Reynolds number from 2800 to 10 and identify two distinct mechanisms for turbulence production. At moderate and high Reynolds numbers, turbulent fluctuations activate the wall oscillations, which, i…
University of Notre Dame（圣母大学）；University of Oxford（牛津大学）；University of Illinois at Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）
The weights of a deep neural network model are optimized in conjunction with the governing flow equations to provide a model for sub-grid-scale stresses in a temporally developing plane turbulent jet at Reynolds number $Re_0=6\,000$. The objective function for training is first based on the instantaneous filtered velocity fields from a corresponding direct numerical simulation, and the training is by a stochastic gradient descent method, which uses the adjoint Navier--Stokes equations to provide the end-to-end sensitivities of the model weights to the velocity fields. In-sample and out-of-samp…
A theoretical analysis is presented for turbulent flows, applicable for canonical (channel, boundary-layer and free jet) geometries. Momentum and energy balance for a control volume moving at the local mean velocity decouples the fluctuation from the mean velocities, resulting in a symmetric set of transport equations for the Reynolds normal and shear stresses. In this formalism, gradients of the fluctuating velocities represent flux vectors, easily verifiable using the available DNS data. A derivative of this transport concept is the scaling for the Reynolds stresses in the dissipation space.…
In wall-bounded shear flow the primary coherent structure is the streamwise roll and streak (R-S). Absent of an associated instability the R-S has been ascribed to non-normality mediated interaction between the mean flow and perturbations. This interaction may occur either directly due to excitation of a transiently growing perturbation or indirectly due to destabilization of the R-S by turbulent Reynolds stresses. A fundamental distinction between the direct and the indirect mechanisms, which is central to understanding the physics of turbulence, is that in the direct mechanism the R-S is its…
We study the nonlinear evolution of the centrifugal instability developing on a columnar anticyclone with a Gaussian angular velocity using a semi-linear approach. The model consists in two coupled equations: one for the linear evolution of the most unstable perturbation on the axially averaged mean flow and another for the evolution of the mean flow under the effect of the axially averaged Reynolds stresses due to the perturbation. Such model is similar to the self-consistent model of \cite{Vlado14} except that the time averaging is replaced by a spatial averaging. The non-linear evolutions o…
We study the aerodynamic response of a pre-stressed curved aileron. Whilst the fluid flow is standard (high Reynolds air flow undisturbed at infinity), the structure is designed to have a peculiar nonlinear behavior. Specifically, the aileron has only one stable equilibrium when the external forces are vanishing, but it is bistable when relevant aerodynamic loads are applied. Hence, for sufficiently high fluid velocities, another equilibrium branch is possible. We test a prototype of such an aileron in a wind tunnel; a sudden change (snap) of the shell configuration is observed when the fluid …
Technische Universität Ilmenau（伊尔梅瑙工业大学）；Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）
In this paper, we develop a multivariate regression model and a neural network model to predict the Reynolds number (Re) and Nusselt number in turbulent thermal convection. We compare their predictions with those of earlier models of convection: Grossmann-Lohse~[Phys. Rev. Lett. \textbf{86}, 3316 (2001)], revised Grossmann-Lohse~[Phys. Fluids \textbf{33}, 015113 (2021)], and Pandey-Verma [Phys. Rev. E \textbf{94}, 053106 (2016)] models. We observe that although the predictions of all the models are quite close to each other, the machine learning models developed in this work provide the best m…
Fluid turbulence is an important problem for physics and engineering. Turbulence modeling deals with the development of simplified models that can act as surrogates for representing the effects of turbulence on flow evolution. Such models correspond to a range of different fidelities, from simple eddy-viscosity-based closures to Reynolds Stress Models. Till now the focus of the data-driven turbulence modeling efforts has focused on Machine Learning augmented eddy-viscosity models. In this communication, we illustrate the manner in which the eddy-viscosity framework delimits the efficacy and pe…
We investigate the mechanisms by which inertial particles dispersed at semi-dilute conditions cause significant drag-reduction in a turbulent channel flow at $\mathrm{Re}_τ= 180$. We consider a series of four-way coupled Euler-Lagrange simulations where particles having friction Stokes number $\mathrm{St}^+ = 6$ or 30 are introduced at progressively increasing mass loading from $M=0.2$ to 1.0. The simulations show that $\mathrm{St}^+ = 30$ particles cause large drag-reduction by up to 19.74\% at $M=1.0$, whereas $\mathrm{St}^+ = 6$ particles cause large drag increase by up to 16.92\% at $M=1.0…
Deep learning (DL)-based Reynolds stress with its capability to leverage values of large data can be used to close Reynolds-averaged Navier-Stoke (RANS) equations. Type I and Type II machine learning (ML) frameworks are studied to investigate data and flow feature requirements while training DL-based Reynolds stress. The paper presents a method, flow features coverage mapping (FFCM), to quantify the physics coverage of DL-based closures that can be used to examine the sufficiency of training data points as well as input flow features for data-driven turbulence models. Three case studies are fo…
The application machine learning (ML) algorithms to turbulence modeling has shown promise over the last few years, but their application has been restricted to eddy viscosity based closure approaches. In this article we discuss rationale for the application of machine learning with high-fidelity turbulence data to develop models at the level of Reynolds stress transport modeling. Based on these rationale we compare different machine learning algorithms to determine their efficacy and robustness at modeling the different transport processes in the Reynolds Stress Transport Equations. Those data…
This paper proposes a new data assimilation method for recovering high fidelity turbulent flow field around airfoil at high Reynolds numbers based on experimental data, which is called Proper Orthogonal Decomposition Inversion (POD-Inversion) data assimilation method. Aiming at the flows including shock wave discontinuities or separated flows at high angle of attack, the proposed method can reconstruct high-fidelity turbulent flow field combining with experimental distributed force coefficients. We firstly perform the POD analysis to the turbulent eddy viscosity fields computed by SA model and…
We study the effect of particle shape on the turbulence in suspensions of spheroidal particles at volume fraction $ϕ= 10\%$ and show how the near-wall particle dynamics deeply changes with the particle aspect ratio and how this affects the global suspension behavior. The turbulence reduces with the aspect ratio of oblate particles, leading to drag reduction with respect to the single phase flow for particles with aspect ratio $\mathcal{AR}\leq1/3$, when the significant reduction in Reynolds shear stress is more than the compensation by the additional stresses, induced by the solid phase. Oblat…
We propose a 3D meshless method to compute mean pressure fields in turbulent flows from image velocimetry. The method is an extension of the constrained Radial Basis Function (RBF) formulation by \citet{Sperotto2022} to a Reynolds Averaged Navier Stokes (RANS) framework. This is designed to handle both scattered data as in Particle Tracking Velocimetry (PTV) and data in uniform grids as in correlation-based Particle Image Velocimetry (PIV). The RANS extension includes the Reynolds stresses into the constrained least square problem. We test the approach on a numerical database featuring a Backw…
Non-equilibrium wall turbulence with mean-flow three-dimensionality is ubiquitous in geophysical and engineering flows. Under these conditions, turbulence may experience a counter-intuitive depletion of the turbulent stresses, which has important implications for modelling and control. Yet, current turbulence theories have been established mainly for statistically two-dimensional equilibrium flows and are unable to predict the reduction in the Reynolds stress magnitude. In the present work, we propose a multiscale model which explains the response of non-equilibrium wall-bounded turbulence und…
In this article, we provide a methodology to reconstruct high-Reynolds number turbulent mean-flows from few time-averaged measurements. A turbulent flow over a backward-facing step at Re = 28275 is considered to illustrate the potential of the approach. The data-assimilation procedure, based on a variational approach, consists in correcting a given baseline model by tuning space-dependent source terms such that the corresponding solution matches available measurements (obtained here from direct-numerical simulations). The baseline model chosen here consists in Reynolds-Averaged-Navier-Stokes (…
This article aims to make a detailed analysis of co-flowing plane Couette flows. Particularly, the variation of flow quantities from the turbulent to non-turbulent region is studied. While the enstrophy exhibits a sharp jump, the other quantities (e.g., mean velocity, Reynolds normal stress, and kinetic energy) show a continuous variation across the interface. The budget analysis of Reynolds normal stresses reveals that the terms playing a key role in turbulence transportation vary depending on the Reynolds normal stress under study. The terms production, diffusion, and redistribution play an …
The Variable Density and Speed of Sound Vessel (VDSSV) produces subsonic turbulent flows that are both compressible and observable at all scales with existing instrumentation including hot wires and particle tracking. We realize this objective by looking at the flow of a heavy gas (sulfur hexafluoride $SF_6$), with a speed of sound almost three times lower than for air. By switching between air and $SF_6$, we isolate the influence of the turbulent Mach number (up to $M_t$ = 0.17) on turbulence statistics from the influences of changes in the Reynolds number (up to $R_λ$ = 1600), and boundary c…
The effects of density and viscosity fluctuations on the total stress balance are identified and used to create a new mean velocity transformation for compressible boundary layers. This work is enabled by an extensive database of direct numerical simulations that incorporate wall-cooling, semi-local Reynolds numbers ranging from 800 to 34000, and Mach numbers up to 12. The role, significance and physical mechanisms connecting density and viscosity fluctuations to the momentum balance and to the viscous, turbulent and total stresses are presented,allowing the creation of generalized formulation…
