Simulations of turbulent fluid flow around long cylindrical structures are computationally expensive because of the vast range of length scales, requiring simplifications such as dimensional reduction. Current dimensionality reduction techniques such as strip-theory and depth-averaged methods do not take into account the natural flow dissipation mechanism inherent in the small-scale three-dimensional (3-D) vortical structures. We propose a novel flow decomposition based on a local spanwise average of the flow, yielding the spanwise-averaged Navier-Stokes (SANS) equations. The SANS equations include closure terms accounting for the 3-D effects otherwise not considered in 2-D formulations. A supervised machine-learning (ML) model based on a deep convolutional neural network provides closure to the SANS system. A-priori results show up to 92% correlation between target and predicted closure terms; more than an order of magnitude better than the eddy viscosity model correlation. The trained ML model is also assessed for different Reynolds regimes and body shapes to the training case where, despite some discrepancies in the shear-layer region, high correlation values are still observed. The new SANS equations and ML closure model are also used for a-posteriori prediction. While we find evidence of known stability issues with long time ML predictions for dynamical systems, the closed SANS simulations are still capable of predicting wake metrics and induced forces with errors from 1-10%. This results in approximately an order of magnitude improvement over standard 2-D simulations while reducing the computational cost of 3-D simulations by 99.5%.
We present a derivation that begins with the Navier--Stokes equation and ends with a prediction of multiple statistically stable states identical to those observed in a spanwise rotating plane Couette flow. This derivation is able to explain the presence of multiple states in fully developed turbulence and the selection of one state over the other by differently sized computational domains and different initial conditions. According to the present derivation, two and only two statistically stable states are possible in an infinitely large plane Couette flow with spanwise rotation, and that mul…
A rapid predictive tool based on the linearised Reynolds-averaged Navier-Stokes equations is proposed in this work to investigate secondary currents generated by streamwise-independent surface topography modulations in turbulent channel flow. The tool is derived by coupling the Reynolds-averaged momentum equation to the Spalart-Allmaras transport equation for the turbulent eddy viscosity, using a nonlinear constitutive relation for the Reynolds stresses to capture correctly secondary motions. Linearised equations, describing the steady flow response to arbitrary surface modulations, are derive…
An oscillating spanwise pressure gradient is imposed numerically to control the flow separation and reduce the drag of a turbulent flow in a channel with square bars. The transverse flow produces a maximum drag reduction of 25\%, due to lower pressure and skin-friction drag forces. The pressure drag reduction reaches a maximum of 22\% and is due to a decrease of the positive high pressure in front of the bars and an increase of the low negative pressure behind the bars. The skin-friction drag reduction is caused by a lower wall-shear stress along the cavity between the bars where the flow is f…
Its is a well known fact that Turbulence exhibits non-locality, however, modeling has largely received local treatment following the work of Prandl over mixing-length model. Thus, in this article we report our findings by formulating a non-local closure model for Reynolds-averaged Navier-Stokes (RANS) equation using Fractional Calculus. Two model formulations are studied, namely one- and two-sided for Channel, Pipe and Couette flow, where the results shown have less 1% error. The motivation of two-sided model lies in recognising the fact that non-locality at a given spatial location is an aggr…
The effect of streamwise-traveling waves of spanwise wall velocity (StTW) on the quasi-streamwise vortices (QSV) populating the near-wall region of turbulent channels is studied via a conditional averaging technique applied to flow snapshots obtained via Direct Numerical Simulation. The analysis by A. Yakeno, Y. Hasegawa, N. Kasagi, "Modification of quasi-streamwise vortical structure in a drag-reduced turbulent channel flow with spanwise wall oscillation", Phys. Fluids 26, 085109 (2014), where the special case of spatially uniform wall oscillation (OW) was considered, is extended to the gener…
Using the Lagrangian transport of momentum, the Reynolds stress can be expressed in terms of basic turbulence parameters. DNS data at higher Reynolds numbers (Re= 1000 and 5200) have been used to again validate this theory, where it is the Lagrangian momentum balance between u2 and pressure fluctuation forces that determine the Reynolds stress at these conditions. This approach can be used to obtain key parameters such as the von Karman constant, inner layer thickness and the Reynolds stress itself. This expression for the Reynolds stress can be combined with RANS for solutions for turbulent c…
Recent simulations indicate that streamwise-preferential porous materials have the potential to reduce drag in wall-bounded turbulent flows(Gomez-de-Segura & Garcia-Mayoral 2019). This paper extends the resolvent formulation to study the effect of such anisotropic permeable substrates on turbulent channel flow. Under the resolvent formulation, the Fourier-transformed Navier-Stokes equations are interpreted as a linear forcing-response system. The nonlinear terms are considered the endogenous forcing in the system that gives rise to a velocity and pressure response. A gain-based decomposition o…
A numerical method, based on the discrete lattice Boltzmann equation, is presented for solving the volume-averaged Navier-Stokes equations. With a modified equilibrium distribution and an additional forcing term, the volume-averaged Navier-Stokes equations can be recovered from the lattice Boltzmann equation in the limit of small Mach number by the Chapman-Enskog analysis and Taylor expansion. Due to its advantages such as explicit solver and inherent parallelism, the method appears to be more competitive with traditional numerical techniques. Numerical simulations show that the proposed model…
Modelling hydrodynamic lubrication is crucial in the design of engineering components as well as for a fundamental understanding of friction mechanisms. The cornerstone of thin-film flow modelling is the Reynolds equation -- a lower-dimensional representation of the Stokes equation. However, the derivation of the Reynolds equation is based on assumptions and fixed form constitutive relations, that may not generally be valid, especially when studying systems under extreme conditions. Furthermore, these explicit assumptions about the constitutive behaviour of the fluid prohibit applications in a…
Turbulent characteristics within shear layer have been studied, recently, within vegetative canopy, buildings, dunes. Kevin-Helmholtz instability triggered hairpin vortex shedding has been widely concluded as the "signature" of mixing layer analogy. However, convoluted roughness types complicate the observing practice of turbulent evolving progress. To simplify that, spanwise homogeneous roughness has been adopted in this work to capture the streamwise flow alternations under different streamwise roughness element distances. Streamwise successive object distance $λ_x$ is used to show the rough…
We present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier-Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions caus…
The basic concepts and hypotheses of Newtonian Cosmology necessary for a consistent treatment of the averaged cosmological dynamics are formulated and discussed in details. The space-time, space, time and ensemble averages for the cosmological fluid fields are defined and analyzed with a special attention paid to their analytic properties. It is shown that all averaging procedures require an arrangement for a standard measurement device with the same measurement time interval and the same space region determined by the measurement device resolution to be prescribed to each position and each mo…
A dynamic procedure for the Lagrangian Averaged Navier-Stokes-$α$ (LANS-$α$) equations is developed where the variation in the parameter $α$ in the direction of anisotropy is determined in a self-consistent way from data contained in the simulation itself. The dynamic model is initially tested in forced and decaying isotropic turbulent flows where $α$ is constant in space but it is allowed to vary in time. It is observed that by using the dynamic LANS-$α$ procedure a more accurate simulation of the isotropic homogeneous turbulence is achieved. The energy spectra and the total kinetic energy de…
According to statistical turbulence theory, the ensemble averaged squared vorticity rho_E is expected to grow not faster than drho_E/dt ~ rho_E^{3/2}. Solving a variational problem for maximal bulk enstrophy (E) growth, velocity fields were found for which the growth rate is as large as dE/dt ~ E^3. Using numerical simulations with well resolved small scales and a quasi-Lagrangian advection to track fluid subvolumes with rapidly growing vorticity, we study spatially resolved statistics of vorticity growth. We find that the volume ensemble averaged growth bound is satisfied locally to a remarka…
In order to obtain insights into dynamics of developed plane Couette turbulence, this paper considers bifurcation structure of unstable periodic orbits (UPOs) in the large-eddy-simulation (LES) system with the Smagorinsky-type eddy viscosity model. Treating the Smagorinsky constant as a bifurcation parameter, we detect the bifurcation points connecting two known UPOs which were separately discovered in the Navier--Stokes system [G. Kawahara and S. Kida, J. Fluid Mech., 449, 291--300 (2001)]. At the moderately high Reynolds number, the LES UPO of the present study possesses the spanwise vortice…
It is well known that Boussinesq turbulent-viscosity hypothesis can introduce uncertainty in predictions for complex flow features such as separation, reattachment, and laminar-turbulent transition. This study adopts a recent physics-based uncertainty quantification (UQ) approach to address such model form uncertainty in Reynolds-averaged Naiver- Stokes (RANS) simulations. Thus far, almost all UQ studies have focused on quantifying the model form uncertainty in turbulent flow scenarios. The focus of the study is to advance our understanding of the performance of the UQ approach on two differen…
Impact of perturbed eddy-viscosity modeling on stability and shape sensitivity of the hydro-turbine vortex rope using linearized Reynolds-averaged Navier-Stokes equations
This work extends the framework of the partially-averaged Navier-Stokes (PANS) equations to variable-density flow, \text{i.e.}, multi-material and/or compressible mixing problems with density variations and production of turbulence kinetic energy by both shear and buoyancy mechanisms. The proposed methodology is utilized to derive the PANS BHR-LEVM closure. This includes \textit{a-priori} testing to analyze and develop guidelines toward the efficient selection of the parameters controlling the physical resolution and, consequently, the range of resolved scales of PANS. Two archetypal test-case…
The goal of this dissertation is to investigate the PANS model capabilities in providing significant improvement over RANS predictions at slightly higher computational expense and producing LES quality results at significantly lower computational cost. The objectives of this study are: (i) investigate the model fidelity at a fixed level of scale resolution (Generation1-PANS/G1-PANS) for smooth surface separation, (ii) Derive the PANS closure model in regions of resolution variation (Generation2-PANS/G2-PANS), and (iii) Validate G2-PANS model for attached and separated flows. The separated flow…
Following the findings in \cite{wangsawijaya2020}, we re-examine the turbulent boundary layers developing over surfaces with spanwise heterogeneous roughness of various roughness wavelengths $0.32 \leq S/\overlineδ \leq 3.63$, where $S$ is the width of the roughness strips and $\overlineδ$ is the spanwise-averaged boundary-layer thickness. The heterogeneous cases induce counter-rotating secondary flows, and these are compared to the large-scale turbulent structures that occur naturally over the smooth wall. Both appear as meandering elongated high- and low-momentum streaks in the instantaneous…
Finding an appropriate turbulence model for a given flow case usually calls for extensive experimentation with both models and numerical solution methods. This work presents the design and implementation of a flexible, programmable software framework for assisting with numerical experiments in computational turbulence. The framework targets Reynolds-averaged Navier-Stokes models, discretized by finite element methods. The novel implementation makes use of Python and the FEniCS package, the combination of which leads to compact and reusable code, where model- and solver-specific code resemble c…
This paper considers plane channel flow modified by waves of spanwise velocity applied at the wall and travelling along the streamwise direction. Laminar and turbulent regimes for the streamwise flow are both studied. When the streamwise flow is laminar, it is unaffected by the spanwise flow induced by the waves. This flow is a thin, unsteady and streamwise-modulated boundary layer that can be expressed in terms of the Airy function of the first kind. We name it the generalized Stokes layer because it reduces to the classical oscillating Stokes layer in the limit of infinite wave speed. When t…
The partially-averaged Navier-Stokes (PANS) equations are used to predict the variable-density Rayleigh-Taylor (RT) flow at Atwood number 0.5 and maximum Reynolds number $500$. This is a prototypical problem of material mixing featuring laminar, transitional, and turbulent flow, instabilities and coherent structures, density fluctuations, and production of turbulence kinetic energy by both shear and buoyancy mechanisms. These features pose numerous challenges to modeling and simulation, making the RT flow ideal to develop the validation space of the recently proposed PANS BHR-LEVM closure. The…
It is well known that the Boussinesq turbulent viscosity hypothesis can yield inaccurate predictions when complex f low features are involved, e.g. laminar-turbulent transition. The focus of the study is to explore the capability of a physics-based uncertainty quantification (UQ) approach to quantify the model-form uncertainty in Reynolds-averaged Naiver-Stokes (RANS) simulations of laminar-turbulent transitional flows over an Selig-Donovan (SD) 7003 airfoil. This methodology perturbs the modeled Reynolds stress tensor in the momentum equations; perturbations are injected into the amplitude, e…
A method to find optimal 2nd-order perturbations is presented, and applied to find the optimal spanwise-wavy surface for suppression of cylinder wake instability. Second-order perturbations are required to capture the stabilizing effect of spanwise waviness, which is ignored by standard adjoint-based sensitivity analyses. Here, previous methods are extended so that (i) 2nd-order sensitivity is formulated for base flow changes satisfying linearised Navier-Stokes, and (ii) the resulting method is applicable to a 2D global instability problem. This makes it possible to formulate 2nd-order sensiti…
The Volume-Averaged Navier-Stokes equations are used to study fluid flow in the presence of fixed or moving solids such as packed or fluidized beds. We develop a high-order finite element solver using both forms A and B of these equations. We introduce tailored stabilization techniques to prevent oscillations in regions of sharp gradients, to relax the Ladyzhenskaya-Babuska-Brezzi inf-sup condition, and to enhance the local mass conservation and the robustness of the formulation. We calculate the void fraction using the Particle Centroid Method. Using different drag models, we calculate the dr…
Rotating turbulent flows form a challenging test case for large-eddy simulation (LES). We, therefore, propose and validate a new subgrid-scale (SGS) model for such flows. The proposed SGS model consists of a dissipative eddy viscosity term as well as a nondissipative term that is nonlinear in the rate-of-strain and rate-of-rotation tensors. The two corresponding model coefficients are a function of the vortex stretching magnitude. Therefore, the model is consistent with many physical and mathematical properties of the Navier-Stokes equations and turbulent stresses, and is easy to implement. We…
University of Tehran（德黑兰大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）
Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations (PDEs). We employ PINNs for solving the Reynolds-averaged Navier$\unicode{x2013}$Stokes (RANS) equations for incompressible turbulent flows without any specific model or assumption for turbulence, and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier$\unicode{x2013}$Stokes equations for laminar flows by solving the Falkner$\unicode{x2013}$Skan boundary layer. We then apply PINNs for…
High-order methods and hybrid turbulence models have independently shown promise as means of decreasing the computational cost of scale-resolving simulations. The objective of this work is to develop the combination of these methods and analyze the effects of high-order discretizations on hybrid turbulence models, particularly with respect the optimal model parameters and the relative accuracy benefits compared to approaches such as under-resolved direct numerical simulation (URDNS). We employ the Partially-Averaged Navier-Stokes (PANS) approach using the flux reconstruction scheme on the flow…
With this study we investigate the accuracy of deep learning models for the inference of Reynolds-Averaged Navier-Stokes solutions. We focus on a modernized U-net architecture, and evaluate a large number of trained neural networks with respect to their accuracy for the calculation of pressure and velocity distributions. In particular, we illustrate how training data size and the number of weights influence the accuracy of the solutions. With our best models we arrive at a mean relative pressure and velocity error of less than 3% across a range of previously unseen airfoil shapes. In addition …
In the Large Eddy Simulation (LES) framework for modeling a turbulent flow, when the large scale velocity field is defined by low-pass filtering the full velocity field, a Taylor series expansion of the full velocity field in terms of the large scale velocity field leads (at the leading order) to the nonlinear gradient model for the subfilter stresses. Motivated by the fact that while the nonlinear gradient model shows excellent a priori agreement in resolved simulations, the use of this model by itself is problematic, we consider two models that are related, but better behaved: The Rational L…
The Reynolds-averaged Navier-Stokes (RANS) equations for steady-state assessment of incompressible turbulent flows remain the workhorse for practical computational fluid dynamics (CFD) applications. Consequently, improvements in speed or accuracy have the potential to affect a diverse range of applications. We introduce a machine learning framework for the {surrogate modeling of steady-state turbulent eddy viscosities for RANS simulations, given the initial conditions. This modeling strategy} is assessed for parametric interpolation, while numerically solving for the pressure and velocity equa…
In this paper we consider the Navier-Stokes-$α$ (NS-$α$) model within a large-eddy simulation framework. An investigation is carried out using fully-developed turbulent channel flow at a fairly low Reynolds number. This is a flow where diffusion plays a prominent role, and presents a challenge to the nonlinear model investigated here. It is found that when $α^{2}_{k}$ is based on the mesh spacing, the NS-$α$ model has a tendency to tilt spanwise vorticity in the streamwise direction, leading to high skin friction. This is due to interaction between the spanwise vorticity, the model, and the st…
Recent progress indicates that highly symmetric recurring solutions of the Navier-Stokes equations, such as equilibria and periodic orbits, provide a skeleton for turbulence dynamics in state-space. Many of these solutions have been found for flat-walled plane Couette, channel, and pipe flows. Rough-walled flows are of great practical significance, yet no recurring solutions are known for these flows. We present a numerical homotopy method to continue solutions from flat-walled plane Couette flow (PCF) to grooved PCF, demonstrated here at a Reynolds number of 400, to act as a starting point fo…
This paper extends the resolvent formalism for wall turbulence proposed by McKeon and Sharma(2010) to account for the effect of streamwise-constant riblets. Under the resolvent formulation, the Navier-Stokes equations are interpreted as a forcing-response system: the nonlinear convective term is interpreted as a feedback forcing on the remaining linear terms, which generates a velocity and pressure response. A gain-based decomposition of the linear forcing-response transfer function --- the resolvent operator --- yields highly amplified velocity and pressure modes, which can be considered key …
Structure tensor Reynolds-averaged Navier-Stokes turbulence models with equivariant neural networks
Computational Science and Engineering Laboratory, Harvard University（哈佛大学计算科学与工程实验室）；Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Walker Department of Mechanical Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校沃克机械工程系）
Despite their well-known limitations, Reynolds-Averaged Navier-Stokes (RANS) models are still the workhorse tools for turbulent flow simulations in today's engineering application. For many practical flows, the turbulence models are by far the largest source of uncertainty. In this work we develop an open-box, physics-informed Bayesian framework for quantifying model-form uncertainties in RANS simulations. Uncertainties are introduced directly to the Reynolds stresses and are represented with compact parameterization accounting for empirical prior knowledge and physical constraints (e.g., real…
AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutions
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；ISIR（智能系统与机器人研究所）；Extrality（Extrality公司）；Institut Jean Le Rond d’Alembert（让·勒朗·达朗贝尔研究所）；Criteo AI Lab（Criteo人工智能实验室）
The basic concepts and equations of classical fluid mechanics are presented in the form necessary for the formulation of Newtonian cosmology and for derivation and analysis of a system of the averaged Navier-Stokes-Poisson equations. A special attention is paid to the analytic formulation of the definitions and equations of moving fluids and to their physical content.
The basic concepts and equations of Newtonian Cosmology are presented in the form necessary for the derivation and analysis of the averaged Navier-Stokes-Poisson equations. A particular attention is paid to the physical and cosmological hypotheses about the structure of Newtonian universes. The system of the Navier-Stokes-Poisson equations governing the cosmological dynamics of Newtonian universes is presented and discussed. A reformulation of the Navier-Stokes-Poisson equations in terms of the fluid kinematic quantities is given and the structure of this system of equations is analyzed.
Using the scale invariance of the Navier-Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the $3D$ Navier-Stokes equations, an infinite `chessboard' of estimates for these inverse length scales is displayed in terms of labels $(n,\,m)$ corresponding to $n$ derivatives of the velocity field in $L^{2m}$. The $(1,\,1)$ position corresponds to the inverse Kolmogorov length $Re^{3/4}$. These estimates ultimately converge to a finite limit, $Re^3$, as $n,\,m\to \infty$, although this limit is too large to lie within the physical v…
In the present technical note, we establish that the setting of the primitive variables of the unsteady incompressible fluid dynamics is ill-formulated in spatially periodic domains as the specification of the boundary velocity is too broad to sidestep time-dependency and approximation errors. As an illustration, we show that the Taylor-Green solution in planes suffers from the Hadamard-divergence, and the ABC flow in cubes is non-unique. In direct numerical simulations of homogeneous turbulence with no corrective precautions on the boundary values, our assertion helps us understand the well-e…
