arXiv subjects

Shijun Liao

Publications and source records attributed to Shijun Liao.

At least 19 recordsLinked to original sources

arXiv2026

Flow reorganization and transport enhancement in two-dimensional horizontal convection near a density extremum

Horizontal convection serves as a canonical model for geophysical and industrial flows. While the Oberbeck-Boussinesq approximation is well established, the impact of a nonlinear equation of state, specifically the density extremum of water near $4^\circ\mathrm{C}$, remains underexplored. Here we investigate this effect using two-dimensional direct numerical simulations over the Rayleigh number range $10^6 \le Ra \le 5\times 10^{10}$. We examine four configurations, contrasting extremum (EXT) and monotonic (MON) buoyancy boundary conditions against linear (LENT) and nonlinear (NELT) equations of state. Our results reveal that the EXT-NELT case undergoes a pronounced reorganization of the large-scale flow, evolving from a bicellular structure to a single-roll circulation driven by central `mixing plumes'. This reorganization manifests as transitional anomalies in the $Re$ scaling, while the emergence of full-depth plumes alters the heat transport mechanism. Consequently, distinct from the Rossby scaling ($Nu \sim Ra^{1/5}$) observed in the reference cases, the EXT-NELT case exhibits an enhanced heat transport scaling ranging from $Nu \sim Ra^{1/4}$ to $Nu \sim Ra^{1/3}$. To interpret this behaviour, we examine the total energy budget and identify an additional potential-energy transfer term, $Φ_{i2}$, arising from the nonlinear equation of state. The scaling argument suggests that the magnitude of this contribution is controlled by the characteristic plume height ($\hat{z}$). Specifically, when plumes penetrate the entire cavity depth ($\hat{z} \sim H$), as observed in the EXT-NELT case, the global kinetic energy dissipation is no longer described by the standard OB HC energy closure alone. The resulting model captures the main trends of the numerical data and provides a possible energy budget interpretation of the enhanced transport observed in this two-dimensional configuration.

physics.flu-dyn↗

arXiv2026

A paradox of the Navier-Stokes turbulence

The Navier-Stokes (NS) equations as a turbulence model have been widely applied in lots of fields. The NS equations contain such a fundamental assumption that all small physical/artificial disturbances could be neglected. Is this assumption correct? In this paper a two-dimensional Rayleigh-Bénard convection governed by the NS equations is predicted by traditional direct numerical simulation (DNS) using double precision arithmetic and a range of different time-steps. It is found that the final flow type tends either to vortical flow or zonal flow, whose statistics are completely different. Notably, these two flow types frequently alternate as the time-step is reduced to a very small value, suggesting that the time-step corresponding to each turbulent flow type should be densely distributed. Thus, stochastic numerical noise exerts a huge influence on the final flow type and statistics of numerically simulated NS turbulence because the time-step has a close relationship with numerical noise. This clearly indicates that small disturbances have significant influences on the NS turbulence, which therefore should not be neglected. This leads to a logical paradox for the NS turbulence, which is a great challenge for us, although a paradox often leads to some significant breakthroughs.

physics.flu-dyn↗

arXiv2026

Spatial symmetry invariance of solution of Kolmogorov flow

We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.

physics.flu-dyn↗

arXiv2026

Non-uniqueness of smooth solutions of the Navier-Stokes equations from almost the same initial conditions

Using clean numerical simulation (CNS) which can give very accurate spatiotemporal trajectory of Navier-Stokes turbulence in a finite but long enough interval of time, we give some numerical evidences that the Navier-Stokes equations admit distinct global solutions from almost the same initial conditions whose difference is very small, i.e. even at the order $10^{-40}$ of magnitude. Hopefully these examples could provide some enlightenments for the uniqueness and existence of Navier-Stokes equations, which are related to one Millennium Prize Problem of Clay Institute.

math.AP↗

arXiv2026

Experimental investigation of wall-pressure fluctuations on a fully appended submarine model at high Reynolds numbers

This paper addresses a critical gap in hydroacoustics through a systematic wind tunnel investigation of wall-pressure fluctuations on the fully appended DARPA SUBOFF model at operationally relevant Reynolds numbers ranging from $5.6 \times 10^{6}$ to $1.4 \times 10^{7}$. The experimental campaign encompasses baseline straight-ahead flow, complex maneuvering (yaw and pitch) conditions, and a first-of-its-kind assessment of a novel vortex control baffle (VCB). To ensure benchmark-quality spectral data, rigorous signal processing techniques were applied, specifically Wiener filtering for background noise suppression and dynamic transfer function correction for pinhole sensors. Key findings indicate that while spectral self-similarity holds across Reynolds numbers, the primary finding is the critical role of appendages in noise amplification. Unstable horseshoe vortex dynamics at the sail-hull junction drive localized pressure fluctuations of up to 300%, establishing this feature as a major coherent noise source. To address this, the study provides the pioneering experimental validation of the VCB. By physically suppressing horseshoe vortex formation at the sail-hull junction, the VCB achieves a global stabilization of the downstream flow, resulting in a significant 35% reduction in root-mean-square wall-pressure fluctuations at the stern and an approximately 14% reduction along the parallel mid-body. Furthermore, maneuvering conditions are shown to fundamentally reshape the pressure field, introducing substantial crossflow effects and non-monotonic spectral behaviors. This comprehensive dataset and the demonstrated efficacy of the VCB provide essential physical insights and a critical validation benchmark for the design of next-generation quiet submarines.

physics.flu-dyn↗

arXiv2025

Experimental study on the wall-pressure fluctuations of flow over an axisymmetric hull

Wall pressure fluctuations beneath the turbulent boundary layer of high-speed underwater vehicles are crucial for hydro-acoustics and acoustic stealth. However, a comprehensive understanding remains limited due to a lack of high-quality experimental data, particularly under realistic operational conditions. To address this gap, this study establishes the first high-fidelity experimental database of wall-pressure fluctuations on an axisymmetric hull at high Reynolds numbers. The dataset's primary innovation is its systematic inclusion of complex maneuvering (yaw and pitch) conditions, providing a benchmark for validating flow noise prediction models. Analysis of this dataset yields key physical insights. The study quantifies systematic Reynolds number effects, including a spectral energy shift toward lower frequencies, and spectral scaling laws by revealing the critical influence of pressure-gradient effects. These findings provide fundamental insights into non-equilibrium 3D turbulent flows and establish an essential dataset to support the design of quieter and more effective underwater vehicles.

physics.flu-dyn↗

arXiv2025

Convergent series of Stokes wave of arbitrary height in deep water via machine learning

Permanent gravity waves propagating in deep water, spanning amplitudes from infinitesimal to their theoretical limiting values, remain a classical yet challenging problem due to its inherent nonlinear complexities. Traditional analytical and numerical methods encounter substantial difficulties near the limiting wave condition due to singularities at sharp wave crests. In this study, we propose a novel hybrid framework combining the homotopy analysis method (HAM) with machine learning (ML) to efficiently compute convergent series solutions of Stokes waves in deep water for arbitrary wave amplitudes from small to theoretical limiting values, which show excellent agreement with established benchmarks. We introduce a neural network trained using only 20 representative cases whose series solution are given by means of HAM, which can rapidly predict series solutions across arbitrary steepness levels, substantially improving computational efficiency. Additionally, we develop a neural network to gain the inverse mapping from the conformal coordinates $(θ, r)$ to the physical coordinates $(x,y)$, facilitating explicit and intuitive representations of series solutions in physical plane. This HAM-ML hybrid framework represents a powerful and efficient approach to compute convergent series in a whole range of physical parameters for water waves with arbitrary wave height including even limiting waves. In this way we establish a new paradigm to quickly obtain convergent series solutions of complex nonlinear systems for a whole range of physical parameters, thereby significantly broadening the scope of series solutions that can be easily gained by means of HAM even for highly nonlinear problems in science and engineering.

physics.flu-dyn↗

arXiv2025

Clean numerical simulation (CNS) of three-dimensional turbulent Kolmogorov flow

Turbulence holds immense importance across various scientific and engineering disciplines. The direct numerical simulation (DNS) of turbulence proposed by Orszag in 1970 is a milestone in fluid mechanics, which began an era of numerical experiment for turbulence. Many researchers have reported that turbulence should be chaotic, since spatiotemporal trajectories are very sensitive to small disturbance. Thus, due to the famous butterfly-effect of chaos, unavoidable numerical noises of DNS might have great influence on spatiotemporal trajectories of turbulence. This is indeed true for a two-dimensional (2D) Kolmogorov turbulent flow, as currently revealed by a much more accurate algorithm than DNS, namely the ``clean numerical simulation'' (CNS). Different from DNS, CNS can greatly reduce both of truncation error and round-off error to any required small level so that numerical noise can be rigorously negligible throughout a time interval long enough for calculating statistics. However, In physics, 3D turbulent flow is more important than 2D turbulence. Thus, for the first time, we solve a 3D turbulent Kolmogorov flow by means of CNS in this paper, and compare our CNS result with that given by DNS in details. It is found that the spatial-temporal trajectories of the 3D Kolmogorov turbulent flow given by DNS are indeed badly polluted by numerical noise rather quickly, and besides the DNS result has significant deviations from the CNS benchmark solution not only in the spatial symmetry of flow field and the energy cascade but also even in statistics.

physics.flu-dyn↗

arXiv2025

Reply to the comments of McMullen et al. (arXiv:2510.04828)

McMullen et al. [1] comment that the numerical simulations that explicitly include random velocity fluctuations ``should exhibit a thermal-fluctuation-dominated range'' consistent with the literature, so that our results (J. Fluid Mech. 1008, R2, 2025) [2] ``contradict other results in the literature''. First of all, we would give an opposite example against this viewpoint: DNS results (that are badly polluted by numerical noises quickly, as mention in Section 2) implicitly include random numerical noises, but they also DO NOT exhibit a thermal-fluctuation-dominated range. In other words, DNS results in the literature qualitatively agree with ours at this point. In addition, we highly suggest that influences of numerical noises on statistics of turbulent flows given by ALL numerical approaches should be carefully checked, since numerical noises might have huge influences on statistics of chaotic systems (including turbulence), as pointed by Lorenz [3] in 2006. Detailed replies are given below.

physics.flu-dyn↗

arXiv2025

Ultra-chaotic property of Navier-Stokes turbulence

A chaotic system is called ultra-chaos when its statistics have sensitivity dependence on initial condition and/or other small disturbances. In this paper, using two-dimensional turbulent Kolmogorov flow as an example, we illustrate that tiny variation of initial condition of Navier-Stokes equations can lead to huge differences not only in spatiotemporal trajectory but also in flow symmetry and its statistics. Here, in order to avoid the influence of artificial numerical noise, we apply ``clean numerical simulation'' (CNS) which can guarantee that the numerical noise can be reduced to such a desired low level that they are negligible in a time interval long enough for calculating statistics. This discovery highly suggests that the Navier-Stokes turbulence (i.e. turbulence governed by the Navier-Stokes equations) might be an ultra-chaos, say, small disturbances must be considered even from viewpoint of statistics. This however leads to a paradox in logic, since small disturbances, which are unavoidable in practice, are unfortunately neglected by the Navier-Stokes turbulence. Some fundamental characteristics of turbulence model are discussed and suggested in general meanings.

nlin.CD↗

arXiv2025

Discovery of 10,059 new three-dimensional periodic orbits of general three-body problem

A very few three-dimensional (3D) periodic orbits of general three-body problem (with three finite masses) have been discovered since Newton mentioned it in 1680s. Using a high-accuracy numerical strategy we discovered 10,059 three-dimensional periodic orbits of the three-body problem in the cases of $m_{1}=m_{2}=1$ and $m_{3}=0.1n$ where $1\leq n\leq 20$ is an integer, among which 1,996 (about 20\%) are linearly stable. Note that our approach is valid for arbitrary mass $m_{3}$ so that in theory we can gain an arbitrarily large amount of 3D periodic orbits of the three-body problem. In the case of three equal masses, we discovered twenty-one 3D ``choerographical'' periodic orbits whose three bodies move periodically in a single closed orbit. It is very interesting that, in the case of two equal masses, we discovered 273 three-dimensional periodic orbits with the two bodies ($m_{1}=m_{2}=1$) moving along a single closed orbit and the third ($m_{3}\neq 1$) along a different one: we name them ``piano-trio'' orbits, like a trio for two violins and one piano. To the best of our knowledge, all of these 3D periodic orbits have never been reported, indicating the novelty of this work. The large amount of these new 3D periodic orbits are helpful for us to have better understandings about chaotic properties of the famous three-body problem, which ``are, so to say, the only opening through which we can try to penetrate in a place which, up to now, was supposed to be inaccessible'', as pointed out by Poincaré, the founder of chaos theory.

nlin.CD↗

arXiv2025

Symbolic Regression-Enhanced Dynamic Wake Meandering: Fast and Physically Consistent Wind-Turbine Wake Modeling

Accurately modeling wind turbine wakes is essential for optimizing wind farm performance but remains a persistent challenge. While the dynamic wake meandering (DWM) model captures unsteady wake behavior, it suffers from near-wake inaccuracies due to empirical closures. We propose a Symbolic Regression-enhanced DWM (SRDWM) framework that achieves equation-level closure by embedding symbolic expressions for volumetric forcing and boundary terms explicitly into governing equations. These physically consistent expressions are discovered from LES data using symbolic regression guided by a hierarchical, domain-informed decomposition strategy. A revised wake-added turbulence formulation is further introduced to enhance turbulence intensity predictions. Extensive validation across varying inflows shows that SRDWM accurately reproduces both mean wake characteristics and turbulent dynamics, achieving full spatiotemporal resolution with over three orders of magnitude speedup compared to LES. The results highlight symbolic regression as a bridge between data and physics, enabling interpretable and generalizable modeling.

physics.flu-dyn↗

arXiv2025

HAM-Schrödingerisation: a generic framework of quantum simulation for any nonlinear PDEs

Recently, Jin et al. proposed a quantum simulation technique for ANY linear partial differential equations (PDEs), called Schrödingerisation [1,2,3]. In this paper, the Schrödingerisation technique for quantum simulation is expanded to ANY nonlinear PDEs by combining it with the homotopy analysis method (HAM). The HAM can transfer a nonlinear PDE into a series of linear PDEs with guaranteeing convergence of the series. In this way, ANY nonlinear PDEs can be solved by quantum simulation using a quantum computer. For simplicity, we call the procedure ``HAM-Schrödingerisation quantum algorithm''. Quantum computing is a groundbreaking technique. Hopefully, the ``HAM-Schrödingerisation quantum algorithm'' can open a door to highly efficient simulation of complicated turbulent flows by means of quantum computing in future.

quant-ph↗

arXiv2025

Physical significance of artificial numerical noise in direct numerical simulation of turbulence

Using clean numerical simulation (CNS) in which artificial numerical noise is negligible over a finite, sufficiently long interval of time, we provide evidence, for the first time, that artificial numerical noise in direct numerical simulation (DNS) of turbulence is approximately equivalent to thermal fluctuation and/or stochastic environmental noise. This confers physical significance on the artificial numerical noise of DNS of the Navier-Stokes equations. As a result, DNS on a fine mesh should correspond to turbulence under small internal/external physical disturbance, whereas DNS on a sparse mesh corresponds to turbulent flow under large physical disturbance, respectively. The key point is that: all of them have physical meanings and so are correct in terms of their deterministic physics, even if their statistics are quite different. This is illustrated herein. Our paper provides a positive viewpoint regarding the presence of artificial numerical noise in DNS.

physics.flu-dyn↗

arXiv2025

Noise-expansion cascade: an origin of randomness of turbulence

Randomness is one of the most important characteristics of turbulence, but its origin remains an open question. By means of a ``thought experiment'' via several clean numerical experiments based on the Navier-Stokes equations for two-dimensional turbulent Kolmogorov flow, we reveal a new phenomenon, which we call the ``noise-expansion cascade'' whereby all micro-level noises/disturbances at different orders of magnitudes in the initial condition of Navier-Stokes equations enlarge consistently, say, one by one like an inverse cascade, to macro-level. More importantly, each noise/disturbance input may greatly change the macro-level characteristics and statistics of the resulting turbulence, clearly indicating that micro-level noise/disturbance might have great influence on macro-level characteristics and statistics of turbulence. Besides, the noise-expansion cascade closely connects randomness of micro-level noise/disturbance and macro-level disorder of turbulence, thus revealing an origin of randomness of turbulence. This also highly suggests that unavoidable thermal fluctuations must be considered when simulating turbulence, even if such fluctuations are several orders of magnitudes smaller than other external environmental disturbances. Hopefully, the ``noise-expansion cascade'' as a fundamental property of the NS equations could greatly deepen our understandings about turbulence, and besides is helpful for attacking the fourth millennium problem posed by Clay Mathematics Institute in 2000.

physics.flu-dyn↗

arXiv2025

Effects of appendages on the turbulence and flow noise of a submarine model using high-order scheme

This study employs high-fidelity numerical simulations to investigate the influence of appendages on the turbulent flow dynamics and far-field acoustic radiation of the SUBOFF submarine model at a Reynolds number of Re = 1.2*10^7. Utilizing a third-order numerical scheme combined with wall-modeled large eddy simulation (WMLES) and the Ffowcs Williams-Hawkings (FW-H) acoustic analogy, the hydrodynamic and acoustic behaviors of an appended SUBOFF configuration are compared to those of a bare hull. A computational grid of 103 million cells resolves the intricate flow interactions, while 648 hydrophones positioned 500 diameters from the model capture far-field acoustic signatures. Key results reveal that appendages significantly amplify hydrodynamic and acoustic disturbances. Flow separations and vortex shedding at appendage junctions elevate pressure-induced drag contributions, contrasting the viscous-dominated drag of the bare hull. The sail-hull interaction intensifies local surface pressure fluctuations, increasing power spectral density (PSD) amplitudes by up to an order of magnitude. In the far field, the appended SUBOFF generates sound pressure levels approximately 20 dB higher than the bare hull, with distinct dipole directivity patterns and peak noise levels (85.10 dB) observed on the central plane. Appendages also disrupt wake symmetry, introducing complex vortical structures such as horseshoe and necklace vortices. These findings demonstrate the critical influence of appendages on hydrodynamic and acoustic behavior, filling a gap in turbulence noise research for complex underwater geometries and providing a vital foundation for the noise reduction optimization of advanced underwater vehicles.

physics.flu-dyn↗

arXiv2024

Utility of High-Order Scheme for Unsteady Flow Simulations: Comparison with Second-Order Tool

The objective of this work is to investigate the utility and effectiveness of the high-order scheme for simulating unsteady turbulent flows. To achieve it, the studies were conducted from two perspectives: (i) the ability of different numerical schemes for turbulence problems under the same set of meshes; and (ii) the accuracy and stability of higher-order schemes for solving turbulence statistics for different mesh types (hexahedral, tetrahedral, and polyhedral cells). The simulations employ the third-order scheme for spatial discretization of the governing equations, while a widely-used second-order solver, namely pisoFoam, was employed for comparison. This study considers the canonical cases of the Taylor-Green vortex (TGV) problem at Re=100, 1600 and flow past a sphere at Re=10000 to address the aforementioned two key issues. For the TGV case, the high-order model significantly improves the numerical accuracy with convergence rates and reduces the numerical dissipation of nearly 1/10 of pisoFoam. In the latter case, the high-order scheme with large-eddy simulation (LES) accurately predicts the vortex structures and the flow instability, regardless of grid type. However, pisoFoam is found to be sensitive to mesh types, which results in numerous non-physical structures in the flow field due to numerical noise rather than flow physics, particularly for tetrahedral cells. Furthermore, for the typical low- and high-order flow statistics, the numerical results predicted by the present model show better agreement with the reference data and have less dependence on the type of grids compared with the conventional scheme. In addition, the obtained energy spectrum by the high-order solver accurately captures the Kelvin-Helmholtz (K-H) instability and the vortex shedding frequency, while these important features are less pronounced by the traditional low-order model.

physics.flu-dyn↗

arXiv2024

Large-eddy simulation of hydrodynamic noise from turbulent flows past an axisymmetric hull using high-order schemes

In this paper, wall-modeled large-eddy simulation (WMLES) is carried out with Ffowcs-Williams and Hawkings (FW-H) acoustic analogy to investigate the turbulent flow and hydrodynamic noise of an axisymmetric body of revolution. We first develop the numerical model based on high-order schemes and validate it by benchmark test of the turbulent flow around a circular cylinder at Re=10000. It demonstrates the capability of the present scheme to capture the primary flow patterns and the acoustic noise in the far field. Then, we conduct the numerical simulation for the turbulent flows around the DARPA SUBOFF without appendages at the Reynolds number of Re=1.2*10^7. The numerical results such as pressure coefficients and velocity fluctuations, are accurately predicted by the present model, which shows closer agreement with the experimental data than available WMLES solutions in the literature. For the parallel midbody of the hull, the wall pressure fluctuation reveals a low-frequency broadband spectrum with the majority of signal energy. The surface fluctuating pressure spectrum scales to the power of Strouhal number at the different locations, which is consistent with those in the turbulent boundary layer of the plate flows and airfoils. Moreover, the acoustic signature in the far field is investigated where the lowest sound pressure level (SPL) occurs in the upstream and downstream directions while the highest is found in the mid-parallel plane. SPLs are relatively close in the region of high acoustic pressure at the transverse plane of x/D=0, which exhibits a maximum difference of 1.2 dB between locations at different angles. In the vertical plane at z/D=0, the directivity plot reveals a symmetrical dipole pattern with vertical fluctuations stronger than the streamwise fluctuations.

physics.flu-dyn↗