物理信息神经网络（PINNs）通过将控制物理定律嵌入神经网络训练，提供了一种无网格框架用于求解偏微分方程。近期研究表明，通过将雷诺数作为额外的网络输入，参数化PINNs可以学习跨雷诺数的Navier-Stokes解。然而，仅基于物理的PINNs在对流主导的高雷诺数流动中常因优化刚度和多尺度流动结构而失去精度。本研究提出了一种稀疏监督混合参数化PINNs框架，用于不可压缩Navier-Stokes流动，具有区域感知学习和局部雷诺数监督。该方法在二维顶盖驱动空腔流动中进行了演示，并在后向台阶流动中进一步验证。在低雷诺数下，仅使用控制方程和边界条件的物理PINNs能准确预测速度和压力场。在较高雷诺数下，引入稀疏CFD监督结合迁移学习以提高预测精度。尽管训练范围跨越（500 < Re < 1000），CFD监督仅应用于（750 < Re < 850）范围内，且仅使用（3%-20%）的计算点。结果表明，大约（5%）的监督数据足以实现准确的流动预测。与CFD模拟的比较表明，在插值和有限外推范围内，速度、压力、涡量和再附特性均具有良好的一致性。所提出的框架为跨不同雷诺数的不可压缩流动提供了一种实用且数据高效的混合策略。
英文摘要：
Physics-informed neural networks (PINNs) provide a mesh-free framework for solving partial differential equations by embedding governing physics into neural-network training. Recent studies have shown that parameterized PINNs can learn Navier-Stokes solutions across Reynolds numbers by treating Reynolds number as an additional network input. However, physics-only PINNs often lose accuracy in convection-dominated high-Reynolds-number flows because of optimization stiffness and multiscale flow structures.
This study presents a sparse-supervised hybrid parameterized PINNs framework for incompressible Navier-Stokes flows with regime-aware learning and localized Reynolds-number supervision. The approach is demonstrated for two-dimensional lid-driven cavity flow and further validated for backward-facing step flow. At low Reynolds numbers, physics-only PINNs accurately predict velocity and pressure fields using only governing equations and boundary conditions. At higher Reynolds numbers, sparse CFD supervision combined with transfer learning is introduced to improve predictive accuracy.
Although the training range spans (500 < Re < 1000), CFD supervision is applied only within (750 < Re < 850) using just (3%-20%) of computational points. Results show that approximately (5%) supervised data are sufficient for accurate flow prediction. Comparisons with CFD simulations demonstrate strong agreement in velocity, pressure, vorticity, and reattachment characteristics across interpolation and limited extrapolation regimes. The proposed framework provides a practical and data-efficient hybrid strategy for incompressible flows across varying Reynolds numbers.
Translational and rotational swimming at small Reynolds number of a planar assembly of identical spheres immersed in an incompressible viscous fluid is studied on the basis of a set of equations of motion for the individual spheres. The motion of the spheres is caused by actuating forces and forces derived from a direct interaction potential, as well as hydrodynamic forces exerted by the fluid as frictional and added mass hydrodynamic interactions. The translational and rotational swimming velocities of the assembly are deduced from momentum and angular momentum balance equations. The mean pow…
Swimming at small Reynolds number of a linear assembly of identical spheres immersed in a viscous fluid is studied on the basis of a set of equations of motion for the individual spheres. The motion of the spheres is caused by actuating forces and forces derived from a direct interaction potential, as well as hydrodynamic forces exerted by the fluid as frictional and added mass hydrodynamic interactions. The swimming velocity is deduced from the momentum balance equation for the assembly of spheres, and the mean power required during a period is calculated from an instantaneous power equation.…
Predicting the long time or late time states of two-dimensional incompressible, high Reynolds number, slowly decaying turbulence has been one of the long-standing problems. Using ``point vortices'' as ``inviscid'' building blocks, which do not respect incompressibility, statistical mechanical models conserving only total energy and zero total circulation result in the well-known sinh-Poisson relation between vorticity and stream function. On the other hand, statistical mechanics of ``inviscid patch'' vortices, which respects incompressibility by conserving regions of zero and nonzero vorticity…
High resolution simulations of incompressible flows have become routine across a range of engineering applications. Despite their routine use, due to the high dimensional parameter space present for most practical applications, a comprehensive exploration of the available parameter space is often impractical. In this work, we demonstrate the ability of physics-constrained deep learning methods to provide an efficient means of exploring high-dimensional parameter spaces with minimal amounts of data from high resolution computational fluid dynamic simulations. As a specific application, we choos…
Explicit expressions are derived for the matrices determining the mean translational and rotational swimming velocities and the mean rate of dissipation for Stokesian swimming at low Reynolds number of a distorting sphere in a viscous incompressible fluid. As an application an efficient helical propeller-type stroke is found and its properties are calculated.
This paper presents a low-communication-overhead parallel method for solving the 3D incompressible Navier-Stokes equations. A fully-explicit projection method with second-order space-time accuracy is adopted. Combined with fast Fourier transforms, the parallel diagonal dominant (PDD) algorithm for the tridiagonal system is employed to solve the pressure Poisson equation, differing from its recent applications to compact scheme derivatives computation (Abide et al. 2017) and alternating-direction-implicit method (Moon et al. 2020). The number of all-to-all communications is decreased to only tw…
In this work, a numerical scheme based on a level-set immersed boundary method is employed for the numerical simulation of the flow around two tandem circular cylinders in the subcritical flow regimes. Three different spacing ratios $\ell/\mathcal{D}$ (where $\ell$ is the center-to-center distance between the two cylinders with $\mathcal{D}$ being the diameter of the cylinders) from $2$ to $4$ is considered. The instantaneous flow structures, pressure distributions and hydrodynamic forces on two tandem cylinders are analyzed at a Reynolds number of $\mathcal{R}\mathfrak{e}=2.2\times 10^4$. The…
In this paper, we have proposed a modified Marker-And-Cell (MAC) method to investigate the problem of an unsteady 2-D incompressible flow with heat and mass transfer at low, moderate, and high Reynolds numbers with no-slip and slip boundary conditions. We have used this method to solve the governing equations along with the boundary conditions and thereby to compute the flow variables, viz. $u$-velocity, $v$-velocity, $P$, $T$, and $C$. We have used the staggered grid approach of this method to discretize the governing equations of the problem. A modified MAC algorithm was proposed and used to…
The evolution with Reynolds number of the dissipation function, normalized by wall variables, is investigated using direct numerical simulation (DNS) databases for incompressible turbulent Poiseuille flow in a plane channel, at friction Reynolds numbers up to Reτ= 2000. DNS results show that the mean part, directly dissipated by the mean flow, reaches a constant value while the turbulent part, converted into turbulent kinetic energy before being dissipated, follows a logarithmic law. This result shows that the logarithmic law of friction can be obtained without any assumption on the mean veloc…
A Fourier/Modal-Spectral-Element Method for the Simulation of High-Reynolds Number Incompressible Stratified Flows in Domains with a Single Non-Periodic Direction
Computational fluid dynamics models based on Reynolds-averaged Navier--Stokes equations with turbulence closures still play important roles in engineering design and analysis. However, the development of turbulence models has been stagnant for decades. With recent advances in machine learning, data-driven turbulence models have become attractive alternatives worth further explorations. However, a major obstacle in the development of data-driven turbulence models is the lack of training data. In this work, we survey currently available public turbulent flow databases and conclude that they are …
This work determines the inaccuracy of using Reynolds averaged Navier Stokes (RANS) turbulence models in transition to turbulent flow regimes by predicting the model-based discrepancies between RANS and large eddy simulation (LES) models and then incorporates the capabilities of machine learning algorithms to characterize the discrepancies which are defined as a function of mean flow properties of RANS simulations. First, three-dimensional CFD simulations using k-omega Shear Stress Transport (SST) and dynamic one-equation subgrid-scale models are conducted in a wall-bounded channel containing …
We use spectral analysis of Eulerian and Lagrangian dynamics to study the advective mixing in an incompressible 2D bounded cavity flow. A significant property of such a rotational flow at high Reynolds numbers is that mixing in its core is slower than wall-adjacent areas and corner eddies. We explain this property by considering the resonance between frequencies of unsteady motion -- captured by the Koopman spectral analysis of the velocity field -- and the circulation frequency of Lagrangian tracers in the mean flow. In high-Reynolds rotational 2D flows, the vorticity in the rotational core i…
Swimming of a sphere in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations for wave-type distortions of the spherical shape. At sizable values of the dimensionless scale number the mean swimming velocity is the result of a delicate balance between the net time-averaged flow generated directly by the surface distortions and the flow generated by the mean Reynolds force density. Depending on the stroke, this can lead to a surprising dependence of the mean swimming velocity on the kinematic viscosity of the fluid. The net flow pattern is calculated as a function…
We investigate properties of the scale dependence and cross-scale transfer of kinetic energy in compressible three-dimensional hydrodynamic turbulence, by means of two direct numerical simulations of decaying turbulence with initial Mach numbers M = 1/3 and M = 1, and with moderate Reynolds numbers, R_lambda ~ 100. The turbulent dynamics is analyzed using compressible and incompressible versions of the dynamic spectral transfer (ST) and the Karman-Howarth-Monin (KHM) equations. We find that the nonlinear coupling leads to a flux of the kinetic energy to small scales where it is dissipated; at …
We revisit the issue of whether thermal fluctuations are relevant for incompressible fluid turbulence, and estimate the scale at which they become important. As anticipated by Betchov in a prescient series of works more than six decades ago, this scale is about equal to the Kolmogorov length, even though that is several orders of magnitude above the mean free path. This result implies that the deterministic version of the incompressible Navier-Stokes equation is inadequate to describe the dissipation range of turbulence in molecular fluids. Within this range, the fluctuating hydrodynamics equa…
We analyze the data stemming from a forced incompressible hydrodynamic simulation on a grid of 2048^3 regularly spaced points, with a Taylor Reynolds number of Re~1300. The forcing is given by the Taylor-Green flow, which shares similarities with the flow in several laboratory experiments, and the computation is run for ten turnover times in the turbulent steady state. At this Reynolds number the anisotropic large scale flow pattern, the inertial range, the bottleneck, and the dissipative range are clearly visible, thus providing a good test case for the study of turbulence as it appears in na…
In this work, numerical solutions of the two dimensional time dependent incompressible flow, in a driven cavity at high Reynolds number Re, are presented. At high Re, there is a controversy. Some studies predicted that the flow is steady, others found time dependent non-steady flow, either periodic or aperiodic. In this study, the driven lid cavity is successfully solved using a very fine grid mesh, for Re up to $30 000$. We discretize the Vorticity-Stream formulation of the Navier-Stokes equation with the SSPRK(5,4) scheme in a $1024\times1024$ grid. Using this very fine grid, the results obt…
Using high-resolution direct numerical simulations, the height and Reynolds number dependence of higher-order statistics of the energy dissipation rate and local enstrophy are examined in incompressible, fully-developed turbulent channel flow. The statistics are studied over a range of wall distances, spanning the viscous sublayer to the channel flow centerline, for friction Reynolds numbers $Re_τ= 180$ and $Re_τ= 381$. The high resolution of the simulations allows dissipation and enstrophy moments up to fourth order to be calculated. These moments show a dependence on wall distance, and Reyno…
The complex wake modifications produced by a Gurney flap on symmetric NACA airfoils at low Reynolds number are investigated. Two-dimensional incompressible flows over NACA 0000 (flat plate), 0006, 0012 and 0018 airfoils at a Reynolds number of $Re = 1000$ are analyzed numerically to examine the flow modifications generated by the flaps for achieving lift enhancement. While high lift can be attained by the Gurney flap on airfoils at high angles of attack, highly unsteady nature of the aerodynamic forces are also observed. Analysis of the wake structures along with the lift spectra reveals four …
Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier--Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier--Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow re…
A numerical study of stably stratified flows past spheres at Reynolds numbers $Re=200$ and $Re=300$ is reported. In these flow regimes, a neutrally stratified laminar flow induces distinctly different near-wake features. However, the flow behaviour changes significantly as the stratification increases and suppresses the scale of vertical displacements of fluid parcels. Computations for a range of Froude numbers $Fr\in [0.1,\infty]$ show that as Froude number decreases, the flow patterns for both Reynolds numbers become similar. The representative simulations of the lee-wave instability at $Fr=…
We study compressible turbulent flow in a circular pipe, at computationally high Reynolds number. Classical related issues are addressed and discussed in light of the DNS data, including validity of compressibility transformations, velocity/temperature relations, passive scalar statistics, and size of turbulent eddies.Regarding velocity statistics, we find that Huang's transformation yields excellent universality of the scaled Reynolds stresses distributions, whereas the transformation proposed by Trettel and Larsson (2016) yields better representation of the effects of strong variation of den…
The results studying various laminar flow regimes in diverging and converging plain channels (diffuser and confusor) with a small opening angle of channels (diverging and converging angles) are presented. The results are obtained for a viscous incompressible fluid by numerical simulation based on solving the Navier-Stokes equations. The paper presents the results concerning the change in the nature of flows from stationary - symmetric to stationary - asymmetric and to non-stationary in the diffuser and confusor in dependence on the Reynolds number. The ranges of existence of these flow regimes…
We introduce two new singularity detection criteria based on the work of Duchon-Robert (DR) [J. Duchon and R. Robert, Nonlinearity, 13, 249 (2000)], and Eyink [G.L. Eyink, Phys. Rev. E, 74 (2006)] which allow for the local detection of singularities with scaling exponent $h\leqslant1/2$ in experimental flows, using PIV measurements. We show that in order to detect such singularities, one does not need to have access to the whole velocity field inside a volume but can instead look for them from stereoscopic particle image velocimetry (SPIV) data on a plane. We discuss the link with the Beale-Ka…
The large structures in the outer layer of turbulent wall flows are of great physical importance, because they contain a substantial fraction of the streamwise kinetic energy and of the Reynolds stresses. Nevertheless, the organization of the outer region of wall turbulence has historically received less attention than that of the inner region, and as a consequence, it is still the subject of many open experimental and theoretical questions. In order to address some of those questions, we have performed direct numerical simulations of the turbulent incompressible flow in plane channels at Reyn…
Multiscale statistical analyses of inertial particle distributions are presented to investigate the statistical signature of clustering and void regions in particle-laden incompressible isotropic turbulence. Three-dimensional direct numerical simulations of homogeneous isotropic turbulence at high Reynolds number ($Re_λ\gtrsim 200$) with up to $10^9$ inertial particles are performed for Stokes numbers ranging from $0.05$ to $5.0$. Orthogonal wavelet analysis is then applied to the computed particle number density fields. Scale-dependent skewness and flatness values of the particle number densi…
We discuss chaotic advection in three-dimensional unsteady incompressible laminar flow, and analyse in detail the most important novel advection phenomenon in these flows; the global dispersion of passive scalars in flows with two slow and one fast velocity components. We make a comprehensive study of the first model of an experimentally realizable flow to exhibit this {\it resonance-induced\/} dispersion: biaxial unsteady spherical Couette flow is a three-dimensional incompressible laminar flow with periodic time dependence derived analytically from the Navier--Stokes equations in the low-Rey…
Energy theory for incompressible Newtonian fluids is, in many cases, capable of producing strong absolute stability criteria for steady flows. In those fluids the kinetic energy naturally defines a norm in which perturbations decay monotonically in time at sufficiently low (but non-zero) Reynolds numbers. There are, however, at least two obstructions to the generalization of such methods to Oldroyd-B fluids. One previously recognized problem is the fact that the natural energy does not correspond to a proper functional norm on perturbations. Another problem, original to this work, is the fact …
