召集人：董明（中国科学院力学所，教授）、张蒙齐（新加坡国立大学，副教授）

时间：2025.02.23—2025.03.01

Tianyuan mathematical workshop

Flow Instability: Physical Mechanisms and Mathematical Descriptions

Schedule

Monday, Feb. 24^th, 2025

Large-scale mean flow of isolated turbulent band in channel flow: vortical structures, analytical solutions, and dynamic mechanisms

Jianjun Tao

Peking University

Isolated turbulent band, the typical structure of the subcritical transition in channel flows, has a downstream head and a bulk part inclined at a characteristic angle. In the large-scale mean flow, a ν-shape vortex found at the head elongates into the bulk part, forming a pair of counter-rotating vortex tube structures. It is revealed numerically and theoretically that the head and the bulk convection velocities reflect the obliquely forward and the backward self-induced velocities of the ν-shape vortex and the vortex tube pair, respectively. The difference between these convection velocities provides a restoring angular momentum to retain the characteristic inclination angle through a self-adjustment process. In addition, asymptotic analytical solutions of the large-scale circulation around the bulk part are solve to understand its upstream and downstream decay rates.

Circulation area rule for two-dimensional instability-driven turbulence

Jin-Han Xie

Peking University

Migdal showed that in three-dimensional homogeneous turbulence, the probability density function (PDF) of velocity circulation only depends on the minimum area enclosed by the loop and is independent of the specific loop shape, called the area rule of circulation. So far, the area law is numerically and experimentally justified only when the characteristic scale of the loop lies in the inertial range, and lyer et al. found that the circulation normalized by its moments better follows the area law than the circulation itself, which however is not theoretically justified. This work first proves that the PDF of normalized velocity circulation on a closed loop as a function of the area surrounded by this loop is a solution of the loop equation in two-dimensional instability-driven turbulence. We also derived and numerically justified that the normalized area law holds even when the scales are not in the inertial range when the system is instability-driven, and dissipations are in the form of viscosity, hyperviscosity, or linear resistance. After proving the area rule, we can define the characteristic scale of circulation as the square root of the circulation enclosed minimum area. And it is amazingly found that the circulation's scale-dependent statistics show a bifractal feature in 3D, 2D and quantum turbulence, which implies that circulation could be a good candidate for studying universal properties of turbulence.

LCSs-based Spatiotemporal Manifolds in Unsteady Fluid Flows and Their Properties

Jiazhong Zhang

School of Energy and Power Engineering, Xi’an Jiaotong University,

Xi’an 710049, P. R. China

E-mail: jzzhang@mail.xjtu.edu.cn

Abstract

From viewpoint of dynamic system and topological physics, unsteady flow is one dissipative dynamic system, the initial study shows that there exist intrinsic spatiotemporal structures hidden in complex unsteady flows, and their topology properties have significant influences on the flow performance. The Hyperbolic, Elliptic and Parabolic Lagrangian Coherent Structures (LCSs), with invariant spatiotemporal properties in a period, are introduced and developed to describe and analyze the complex flow structures in unsteady flows, and some numerical methods following nonlinear dynamics are given in capturing LCSs. Further, some fine functional structures, like energy sink, targeted energy transfer, are shown and studied in aeroacoustics and others. As the results, the methods based on complete intrinsic Lagrangian Coherent Structures could describe and analyze the flow structures and dynamics of complex unsteady flows quantitatively, and further control the flow with one new and accurate method.

Revisit the wrinkled flame front instability

Wang Junjie, Wang Rui, Xu Hui

School of Aeronautics and Astronautics, Shanghai Jiao Tong University, China

The flame front instability originates from the exponential growth of perturbations on the flame surface due to Darrieus-Landau and Thermo-Diffusive instabilities, as well as the nonlinear propagation of small-scale flame wrinkles. To quantitatively investigate the linear and nonlinear evolution of wrinkled flame fronts, a direct numerical simulation program is developed for the combustion simulation. Based on the single-step reaction of H2/air mixture, the effective ranges and limitations of theoretical dispersion relations are analyzed by comparing with the numerical results. It is found that the modified Clavin dispersion relation can accurately predict the linear growth rate at low-wavenumber region by considering the gravitational effect as a constant term. For TD-unstable flames where theoretical relations are not applicable, the numerical dispersion relation remains symmetric and can be accurately fitted by forth-order polynomial. In the nonlinear region, by varying the expansion ratio, gravity, domain height, and Lewis number, it is found that the propagation speed of DL-unstable flames is proportional to the linear growth rate, and increasing the domain height or decreasing the Lewis number can either lead to non-steady propagation. Three-dimensional flame fronts exhibit a characteristic ridge structure, resulting in higher propagation speeds for three-dimensional flame compared to two-dimensional flame under the same condition. The large-scale evolution of both DL-unstable and TD-unstable flames is investigated finally. For DL-unstable flames, three effects of linear instability on the large-scale flame propagation have been identified: altering the overall curvature of the flame front, changing the fusion rate of flame wrinkle, and promoting or inhibiting front splitting. Due to the combined effect of these mechanisms, the propagation speed of large-scale flame is no longer proportional to its linear stability. For TD-unstable flames, flame-fingers can be observed. The flame-finger is formed by the rapid propagation of a particular flame wrinkle compared to its surrounding wrinkles.

Enhancing Flow Stability through Aerodynamic Shape Optimization for Airfoils and Bluff Bodies

Jiaqing Kou

1.School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China

2.International Joint Institute of Artificial Intelligence on Fluid Mechanics,

Northwestern Polytechnical University, Xi’an, 710072, China

3.National Key Laboratory of Aircraft Configuration Design, Xi’an 710072, China

Flow instability commonly exists in natural phenomena and engineering systems. Typical examples include the von Kármán vortex shedding behind bluff bodies, flow separation on airfoils at large angles of attack, and transonic buffet on airfoils, etc. Such instabilities give rise to force fluctuations, structural vibrations, and acoustic noise. To suppress flow instability, a number of flow control techniques have been developed. However, due to the lack of prior knowledge and theoretical guidance, designing flow control techniques often leads to strong disturbances and requires extensive trial and error with respect to control parameters. This results in low design efficiency and limits their applicability in engineering practice.

To enhance the stability of unsteady flow past complex shapes and develop effective flow control strategies, we propose several stability-based aerodynamic shape optimization frameworks. Our research mainly includes three parts: 1) An adjoint-based shape optimization design is developed to reduce the fluctuation of lift coefficient. The vortex shedding has been suppressed and the onset angle of attack of transonic buffet has been delayed. 2) The Jacobian-free global stability analysis is developed to formulate the shape optimization design framework considering the constraint of transonic buffet. The lift and drag characteristics of the optimized airfoil are enhanced and the buffeting is effectively suppressed, thereby expanding the flight envelope. 3) An aerodynamic shape optimization framework based on resolvent analysis is proposed to optimize the geometry while minimizing the resolvent gain of the flow past a cylinder at the subcritical Reynolds number. Through shape optimization, the flow separation is delayed and the vortex shedding are suppressed at higher Reynolds numbers.

In the future, to enhance the proposed techniques, reduced-order modeling and efficient gradient-based optimization algorithms will be further developed. The framework is expected to be applied to engineering problems with higher Reynolds number and higher complexity, such as wings and full aircraft.

Direct numerical simulations of two-dimensional elasto-inertial turbulence in channel flows

Hong-Na Zhang

Tianjin University

Recent studies have confirmed that, the novel turbulence type, namely elasto-inertial turbulence (EIT), can exist in two-dimensional channels. Now, the research on EIT has reached the stage of identifying the minimal flow unit (MFU). On this issue, direct numerical simulations (DNSs) of FENE-P fluid flow in two-dimensional channels with variable sizes are conducted in this study. In my talk, we demonstrate with the increase of channel length, the simulated flow experiences several different flow patterns and there exists a MFU for EIT to be self-sustained. Through capturing the onset process of EIT, we observed that EIT originates from the sheet-like extension structure located near the wall which maybe related with the wall mode rather than the center mode. The fracture and regeneration of this sheet-like structure is the key mechanism for the self-sustaining of EIT. Moreover, we also qualitatively analyzes the statistical characteristics and dynamic mechanisms of two-dimensional flow and explores its similarities and differences with three-dimensional flow. We find that the parametric effects on the statistical characteristics show an opposite trends in two-dimensional EIT versus three-dimensional EIT and drag-reducing turbulence (DRT). In addition, we also identify the anomalous Reynolds stress that contributes negatively to the flow resistance in two-dimensional EIT by quadrant analysis. The anomalous phenomenon can be attributed to the motions in the first and third quadrants are closely associated with the polymer sheet-like extension structures, which incline from the near-wall region towards the channel centre.

Shear-thinning: stabilising or destabilising time-periodic flows? A Floquet analysis of Stokes layers in Carreau fluids

Mengqi Zhang, Dongdong Wan

National University of Singapore

In this presentation, I will talk about the Floquet analysis of Stokes layers in Carreau fluids, a topic that appears to be unexplored in the literature. The time-dependent base flow in the stability analysis will be solved using two methods, i.e., a numerical method and an expansion method based on the binomial approximation of the Carreau model. I will then present and discuss the results of the Floquet analysis, with a particular focus on the effects of shear-thinning behaviour in a relatively large parameter space.

Linear instability in thermally stratified quasi-Keplerian flows

Dongdong Wan, Rikhi Bose, Mengqi Zhang*, Xiaojue Zhu*

National University of Singapore and Max-Planck Institute

Quasi-Keplerian flow is a specific type of Taylor–Couette co-rotating flow and it is significant in the study of angular momentum transport in astrophysical contexts, especially in accretion disks. While the magnetorotational instability (MRI) accounts for turbulence in certain accretion disks, it does not explain similar phenomena in protoplanetary disks, where magnetic fields are weak or absent. In this talk, we present our recent study of the influence of radial thermal stratification on the stability of quasi-Keplerian flows, subjected to radial gravitational forces resembling stellar gravity. Our analysis reveals the existence of thermo-hydrodynamic instabilities for both axisymmetric and non-axisymmetric modes over a wide range of parameters. We observe that a lower Richardson or Prandtl number tends to stabilize the flow, while a smaller radius ratio promotes instability. Furthermore, at low Prandtl numbers, the critical Taylor number follows a scaling law, suggesting the instability's relevance in accretion disk dynamics at high Taylor numbers and low Prandtl numbers. Moreover, even a modest degree of thermal stratification, indicated by a low Richardson number, can induce instability with short axial wavelengths. These results align qualitatively with predictions from fully local stability analyses using short-wavelength approximations. Overall, this study refines the understanding of thermally-driven instabilities in protoplanetary disks, providing an alternative mechanism for angular momentum transport in regions where MRI is not applicable.

Tuesday, Feb. 25^th, 2025

Feigenbaum cascade in a subcritical shear flow

Kengo Deguchi

Monash University

Identifying the onset of chaos in shear flows is a fundamental challenge in understanding the laminar-turbulent transition. Such identification is particularly difficult in subcritical shear flows, where turbulence can emerge abruptly from finite-amplitude disturbances, even though the laminar state remains linearly stable. In this talk, I will demonstrate that a Feigenbaum cascade occurs within the subcritical parameter regime of Taylor-Couette flow, leading to the onset of chaos. This research is a collaboration with Baoying Wang and Roger Ayats at ISTA, as well as Fernando Mellibovsky and Alvaro Meseguer at UPC.

Asymptotic-analysis-inspired boundary conditions aiming at eliminating polymer diffusive instability

Dong Ming

Chinese Academy of Science

The recent discovery of polymer diffusion instability (PDI) by Beneitez et al. (Phys. Rev. Fluids, 2023, 8: L101901) presents challenges in implementing artificial conformation diffusion (ACD) in transition simulations of viscoelastic wall-shear flows. In this paper, we begin with an asymptotic analysis of PDI within the near-wall thin diffusive layer. In the limits of low conformation diffusivity, high Weissenberg number and dilute solution, the complex instability system is significantly simplified, reducing the number of controlling parameters from five to just one. Based on this reduced asymptotic instability system, we investigate the instability properties across the entire parameter space and develop a set of proper boundary conditions for the conformation tensor to avoid unstable PDI. These newly-developed boundary conditions are further validated within the original ACD instability system, incorporating both the Oldroyd-B and FENE-P constitutive models. The results demonstrate that these conditions effectively eliminate unstable discrete modes in the eigenspectrum for a representative set of controlling parameters. Consequently, this work offers a promising approach for achieving reliable polymer-flow simulations with ACD, ensuring both numerical stability and accuracy.

Penetrative convection: stability, bifurcation and scaling laws

Zijing Ding

Harbin Institute of Technology

Penetrative convection refers to the phenomenon of an unstably stratified fluid layer entering into a stably stratified layer, which is ubiquitous in atmospheric flow, water convection beneath ice, and convections in Saturn and its icy moons. Unlike Rayleigh-Benard convection, penetrative convection is far more complicated, e.g. the subcritical instability phenomenon and multiple turbulent states are observed in penetrative convection, which are typically not presented in Rayleigh-Benard convection. In this talk, I will introduce the state-of-the-art research in penetrative convection and discuss the instability, bifurcation and scaling laws in penetrative convection due to a nonlinear density constitutive relation.

Boundary-layer stability of supercritical fluids

Jie Ren

Beijing Institute of Technology

In this presentation, we explore recent advancements in understanding the stability of boundary layers in supercritical fluids. Using CO, at 80 bar as a representative free-stream condition, we examine three distinct flow configurations: a flat plate, a swept wing, and a concave plate. The temperature gradients in the flow are designed to remain below, above, or traverse the Widom line, defined by the peak of isobaric specific heat. Crossing the Widom line introduces novel instability mechanisms, potentially altering the dominant modes and accelerating the transition to turbulence. This talk emphasizes the influence of flow configurations and highlights the sensitivity of stability characteristics to these variations.

Dynamics of ferrofluid jets: the Hamiltonian framework

Zhan Wang (zwang@imech.ac.cn)

Institute of Mechanics, Chinese Academy of Sciences

Ferrohydrodynamics deals with the mechanics of fluid motion influenced by strong forces of magnetic polarization. Developing an understanding of the consequences of these forces enjoys wide usage in industries, including magnetic resonance imaging, dynamic loudspeakers, magneto-optical sensors, and heat transfer or dissipation. This talk focuses on the stability and dynamics of solitary waves propagating along the surface of an inviscid ferrofluid jet from analytical and numerical aspects. First, we give a detailed proof of the Hamilton principle for the axisymmetric system, as well as the canonical variables. Next, the homogeneous expansion of the Dirichlet-Neumann operator (DNO), slightly differing from Guyenne & Parau (2016), is obtained. Finally, a systematic procedure is proposed to derive model equations of multiple scales in various possible limits from the full potential problem in the Hamiltonian/Lagrangian framework. In particular, based on the Hamiltonian perturbation theory, we propose a simplified model with full dispersion by truncating the DNO expansion at the cubic order for the kinetic energy. It is shown that the model agrees well with the full Euler equations for the speed-amplitude and speed-energy bifurcation curves and wave profiles. Based on the model, we examine analytically the stability properties of axisymmetric solitary waves subject to longitudinal disturbances. Our analytical result, consistent with that obtained by Saffman (1985), indicates that in the axisymmetric system, the stability exchange for solitary waves (namely, the superharmonic instability) also occurs at the stationary points of the speed-energy bifurcation curve.

Towards more physics-based transition prediction by incorporating receptivity for hypersonic boundary layers

Caihong Su

Tianjin University

Accurate prediction of the boundary layer transition is highly desirable in the design of hypersonic flight vehicles. Traditional method predicts the transition location based on the most amplified linear instability mode but neglects receptivity that initiates the instability. In this presentation, I will report on our research endeavors aimed at elucidating the receptivity process and improving the transition prediction method. Specifically, we focus on the receptivity of a blunt cone boundary layer to freestream disturbances, using direct numerical simulation and linear stability theory. We have developed a novel method identifying the key disturbances exciting the first and second modes. This method has revealed a notable consistency in the receptivity mechanisms across a wide range of nose bluntness. Building upon this insight, we established a receptivity model and incorporated it into the transition prediction method. The improved approach indicated that the first mode, which is often overlooked in transitional predictions, may contribute to transition due to its significantly effective receptivity process.

Instability of streaky hypersonic boundary layers over cooled walls

Dongdong Xu

Sheffield University

We investigate the instability of streaks induced by free-stream vortical disturbances in cooled-wall hypersonic boundary layers. Three distinct types of instability modes are identified: modified Mack second modes, newly discovered secondary instability modes, and sinuous or varicose modes akin to those in incompressible boundary layers. A unique characteristic of the newly discovered secondary-instability modes is that their eigenfunctions concentrate in the high-speed regions of streaks. The flow distortion caused by hypersonic streaks destabilizes three-dimensional secondary instability waves while suppressing the growth of modified Mack modes, both radiating and nonradiating. This suppression of radiating modes can effectively reduce the two-dimensional noise generated by the boundary layer. Numerical results indicate that, unlike streaks in boundary layers with adiabatic walls, where instability is detected only in high-speed regions, the secondary instability of streaks in cooled-wall hypersonic layers occurs in both low- and high-speed regions.

The effects of compressibility on the linear spatio-temporal stability of confined two-dimensional shear layers

Benshuai Lyu

Peking University

High-speed shear flows, such as jets commonly found in rocket launch pads or aeroacoustic facilities, are often bounded by solid walls. The effects of compressibility on the linear spatio-temporal stability characteristics of confined shear flows are not yet fully understood. In this work, a spatio-temporal linear stability analysis is performed on a confined two-dimensional jet/wake to investigate the effects of sidewalls and compressibility on its absolute instability. The flow motion is decomposed into varicose and sinuous modes. We show that at subsonic shear Mach numbers, increasing the shear Mach number stabilizes the varicose mode when the confinement is either very strong or very weak, whereas it destabilizes the mode when the confinement is moderate. Additionally, the sinuous mode of a subsonic wake also stabilizes with increasing shear Mach numbers. At low supersonic shear Mach numbers, the varicose mode of weakly confined jets or strongly confined wakes is minimally affected by the confinement ratio. Similarly, the sinuous mode of strongly confined wakes or weakly confined jets is not influenced by the confinement ratio. At high supersonic shear Mach numbers, a new absolutely unstable mode is identified when the shear Mach number exceeds a critical value. We show that this mode is caused by the interaction of acoustic and hydrodynamic modes.

Wednesday, Feb. 26^th, 2025

Concentration of weak solutions in compressible flows

Xianpeng Hu

PolyU, Hong Kong

We will discuss some recent progress in the concentration phenomenon of weak solutions for compressible flows. Some new observations will be discussed.

Three-dimensional coherent structures in a curved pipe flow

Runjie Song & Kengo Deguchi

Monash University

Dean's approximation for curved pipe flow, valid under loose coiling and high Reynolds numbers, is extended to study three-dimensional travelling waves. Two distinct types of solutions bifurcate from the Dean's classic two-vortex solution. The first type arises through a supercritical bifurcation from inviscid linear instability, and the corresponding self-consistent asymptotic structure aligns with the vortex-wave interaction theory. The second type emerges from a subcritical bifurcation by curvature-induced instabilities and satisfies the boundary region equations. A connection to the zero-curvature limit was not found. However, by continuing from known self-sustained exact coherent structures in the straight pipe flow problem, another family of three-dimensional travelling waves can be shown to exist across all Dean numbers. The self-sustained solutions also possess the two high-Reynolds-number limits. While the vortex-wave interaction type of solutions can be computed at large Dean numbers, their branch remains unconnected to the Dean vortex solution branch.

Model reduction via spectral submanifolds and its application in fluids

Mingwu Li

Southern University of Science and Technology

Model reduction has been an important topic in fluid mechanics. In this talk, I present a powerful reduction framework via spectral submanifolds (SSMs) established by the team lead by Prof. George Haller from ETH Zurich. SSMs are low-dimensional attractors and can be used to achieve exact model reduction. SSM-based reduced-order models (ROMs) can be constructed in both equation and data-driven settings. These ROMs enable effective and efficient predictions. I will review the basic concepts of SSMs and the ideas of reduction via SSMs. Then I will show several interesting applications of SSM-based reductions in fluids, including transitions in plane Couette flow with multiple equilibria, capturing edge of chaos in pipe flows, fluid sloshing in a tank, and chaotic dynamics of an inverted flag.

Influence of slip on the three-dimensional instability of flow past an elongated superhydrophobic bluff body

Yong-Liang Xiong

Department of Mechanics, Huazhong University of Science & Technology, Wuhan 430074, China

Superhydrophobic surfaces (SHSs) have been shown be capable of reducing skin friction drag as well as influencing the flow around coated bodies including cylinders and spheres. In this talk, the effects of SHSs, consisting of microgrates oriented normal to the flow direction, on the onset of three-dimensional instability of flow past a bluff body will be introduced through Floquet analysis. The SHS was modeled by shear free condition for the air-water interface. The results show that the SHSs increase the vortex shedding frequency. The Floquet analysis reveals that both mode B' and mode S' are suppressed dramatically by the partial slip condition compared to a regular no-slip body. However, mode A is less affected by the application of partial spanwise slip. Correspondingly, the critical spanwise wavelengths are not significantly affected by SHS. Similar phenomenon is observed in flow past a circular cylinder coated by SHS. The results also revealed that mode B' and S' are collapsed into mode A due to the increase of the width of the air-water region for flow past an elongated body. Surprisingly, the critical Reynold numbers of different modes are variously affected by the variation of gas fraction (GF). The unstable modes with short wavelength, such as mode B' and S', become more stable with increasing GF. On the contrary, it is opposite for the unstable mode A with longer wavelength. Aspect ratio of the elongated bluff body are found to be important on the transition of three-dimensional instability. Finally, the application of SHSs could modify the transition route from two- to three-dimensionality by alternative of different unstable modes. As the wavelength of the unstable mode decreases, the inhibition of three-dimensional instability becomes more efficient by SHSs.

Thursday, Feb. 27^th, 2025

Large eddy simulation of boundary layer flow over an axisymmetric bow: transition and induced noise

Wei-Xi Huang

Tsinghua University

Large-eddy simulation of boundary layer flow over the curved edge of an axisymmetric bow is carried out to investigate the characteristics of transition and induced noise. The effects of geometric curvature and inflow turbulence intensity (ITI) are examined. With a low ITI level, natural transition takes place at the rear end of the straight section. With higher ITI levels, turbulence emerges immediately and evolves gradually following a strong favorable pressure gradient (FPG) region near the forehead, which is significantly influenced by the large streamwise curvature. Within the FPG region, the root mean square of WPF decreases rapidly, with the frequency spectra of WPF exhibiting good scalability with outer variables. Moreover, higher turbulence intensity levels lead to larger skin friction, which is related to the development of TBL. Furthermore, the structural vibration is obtained by the excitation of wall pressure fluctuation (WPF). Flow-induced and structural noises are calculated by using FW-H equation. At the sonar platform, the peak sound pressure level (SPL) exists on the sides in proximity to the bow’s symmetry axis. As the frequency increases, the location of the maximum SPL of flow-induced noise exhibits a lateral shift, aligning with the characteristics of WPF. Compared to the FPG region, the WPG region predominantly generates the flow-induced noise at the sonar platform. In the low-frequency range, the vibration mode remains unexcited, and the flow-induced noise mainly contributes to the sonar noise. However, within the mid to high-frequency ranges, the contributions from flow-induced noise and structural noise are similar. On the other hand, different with the characteristics of sonar noise, far-field structural noise achieves a SPL comparable to that of the flow-induced noise.

Zonostrophic instability and its stochasticity

Chen Wang

Beijing Normal University at Zhuhai

Zonostrophic instability refers to the phenomenon that weak zonal mean flow grows exponentially in random field of waves, and can explain the formation of strong zonal flows that are ubiquitous on planet atmosphere. In previous studies of zonostrophic instability, it was often assumed that although the waves are stochastic, the mean flow evolve deterministically, following an ergodic assumption that the zonal mean is equivalent to ensemble mean. In this study, we will demonstrate that this assumption does not hold well for general conditions, and the mean flow can be considerably stochastic. We will further evaluate the impact of mean-flow stochasticity on the zonostrophic instability, and demonstrate that it results in an under-estimation of the growth rate. An improved dispersion relation is derived based on the impact of mean-flow stochasticity.

Vortex dynamic of unsteady flow past a circular cylinder

Bofu Wang, Zifang Li, Quan Zhou

Shnaghai University

The wake of unsteady approaching flow with in-line oscillation past a circular cylinder near a parallel wall is investigated using direct numerical simulation. The vortex dynamics are studied for three gap-to-diameter ratios ($G/D = 1.0$, 0.5, and 0.2) at a fixed Reynolds number of $Re = 500$ and an oscillation amplitude of $A/D = 0.2$. Across the frequency ratio range $0 < f_d/f_{st*} < 3.0$, the vortex shedding are categorized into 'unlocked-on' and 'locked-on' regimes for each $G/D$. In the 'unlocked-on' regime, the shedding frequency $f_{sd}$ for the larger gap ratios ($G/D = 1.0$ and 0.5) is close to $f_{st*}$, which is the vortex shedding frequency for steady flow past a cylinder near the wall. For $G/D = 0.2$, vortex shedding is suppressed due to the strong wall effect. In the 'locked-on' regime, the mode transitions for the larger gap ratios follow the sequence: A mode $\rightarrow$ P mode $\rightarrow$ S mode. The A and S modes are synchronized with the oscillation. For the A mode, the vortex shedding modes present asymmetric with a 1/2 subharmonic frequency response. The shedding vortices form vortex clusters that are convected downstream and dissipated. An interesting observation for $G/D = 0.5$ is that downstream, the vortex clusters catch up with those formed earlier. In S mode, the vortex shedding modes present quasi-symmetric and the flow characteristics differ between low- and high-frequency incoming flows.  At low frequencies, binary vortices shedding from the gap side interact with those from the upper side and the secondary vortices, leading to periodic vortex pairing and a disrupted vortex street. At high frequencies, a well-formed binary vortex street develops. In P mode, vortex shedding mode alternate between A mode and S mode. For $G/D = 1.0$, this alternation is periodic, whereas for $G/D = 0.5$, it is intermittent.  For the small gap ratio $G/D = 0.2$, only S mode is observed in the 'locked-on' regime due to suppressed vortex shedding on the gap side. Additionally, the results indicate that the three-dimensional flow structures are suppressed under high-frequency oscillations.

A multi-parameter linear stability and sensitivity analysis strategy

Jun-Hua Pan

School of Engineering Science, University of Chinese Academy of Sciences, Beijing 101408, China

For a flow system governed by multiple dimensionless parameters, its stability and sensitivity equations are derived from a generalised vector-form governing equation comprised of various dimensionless parameters that represent different physical forces affecting the system instability. By introducing adjoint variables and constructing the Lagrangian identity, a differential relationship between the eigenvalue of perturbation mode and dimensionless parameters is determined and defined as the global sensitivity gradient. The global sensitivity gradient can directly and intuitively evaluate the competitive relationship among the influences of various parameters on system instability. To demonstrate the effectiveness of present method, three applications are presented: two-dimensional flow around a circular cylinder with a single dimensionless parameter Re, three-dimensional axisymmetric magnetohydrodynamics (MHD) flow around a sphere with two parameters Re and N, and two-dimensional MHD mixed convection with three parameters Re, Gr and Ha.

Non-modal growth analysis of high-speed flows over an inclined cone

Xi Chen

Peking University

Spatial optimal responses to both inlet disturbances and harmonic external forcing for hypersonic flows over a blunt cone at nonzero angles of attack are obtained by efficiently solving the direct-adjoint equations with a parabolic approach. In either case, the most amplified disturbances initially take the form of localized streamwise vortices on the windward side and will undergo a two-stage evolution process when propagating downstream: they first experience a substantial algebraic growth by exploiting the Orr and lift-up mechanisms, and then smoothly transition to a quasi exponential-growth stage driven by the crossflow-instability mechanism, accompanied by an azimuthal advection of the disturbance structure towards the leeward side. The algebraic-growth phase is most receptive to the external forcing, whereas the exponential-growth stage relies on the disturbance frequency and can be significantly strengthened by increasing the angle of attack. The wavemaker delineating the structural sensitivity region for the optimal gain is shown to lie on the windward side immediately downstream of the inlet, implying a potent control strategy. Additionally, considerable non-modal growth is also observed for broadband high-frequency disturbances residing in the entropy layer.

Excitation of non-modal perturbations in hypersonic boundary layers by freestream forcing: shock-fitting harmonic linearised Navier-Stokes approach

Lei Zhao

Tianjin University

In this paper, we study the receptivity of non-modal perturbations in hypersonic boundary layers over a blunt wedge subject to freestream vortical, entropy and acoustic perturbations. Due to the absence of the Mack-mode instability and the rather weak growth of the entropy-layer instability within the domain under consideration, the nonmodal perturbation is considered as the dominant factor triggering laminarturbulent transition. This is a highly intricate problem, given the complexities arising from the presence of the bow shock, the entropy layer, and their interactions with oncoming disturbances. To tackle this challenge, we develop a novel, highly efficient numerical tool, the shockfitting harmonic linearized Navier-Stokes (SF-HLNS) approach, which offers a comprehensive investigation on the dependence of the receptivity efficiency on the nose bluntness and properties of the freestream forcing. The numerical findings suggest that the non-modal perturbations are more susceptible to freestream acoustic and entropy perturbations compared to the vortical perturbations, with the optimal spanwise length scale being comparable with the downstream boundary-layer thickness. Notably, as the nose bluntness increases, the receptivity to the acoustic and entropy perturbations intensifies, reflecting the transition reversal phenomenon observed experimentally in configurations with relatively large bluntness. Additionally, through the SF-HLNS calculations, we examine the credibility of the optimal growth theory (OGT) on describing the evolution of non-modal perturbations. While the OGT is able to predict the overall streaky structure in the downstream region, its accuracy in predicting the early-stage evolution and the energy amplification proves to be unreliable.

Stability of the k-e model in Kolmogorov flow with periodic condition

Le Fang

Beihang University

This study aims at theoretically understanding the model behaviors of using engineering RANS models. A question arises when examining the k-w SST model in an LES channel flow database. The values of k and w, according to their definitions and caculated directly from the LES database (denoted as M1), obviously differ from what RANS equations suggest (denoted as M2). Then, statistical analysis and symbolic regression method indicate that the physical quantities in RANS models, such as k and w, should be defined as new forms. From these apriori studies, a natural idea is that some RANS models might not have stable fixed point as designed under some boundary conditions. As a first simple model, we then perform investigations on the performance of the k-e model under periodic boundary conditions using Kolmogorov flow. The velocity field is frozen to allow the study of model stalibity. Two fixed points exist in the model equation, and theoretical analysis demonstrates their instability, leading to divergence in unsteady RANS computations. It is further illustrated in real numerical tests that we cannot obtain correct RANS solutions under periodic boundary conditions. This study is then expected to inspire future considerations on RANS modeling from the insight of apriori stability analysis.

Inner sound field generated by large-scale coherent structures of ring mode in the near-nozzle region of a subsonic circular jet: an asymptotic description of aeroacoustics

Zhongyu Zhang

Tianjin University

Coherent structures (CS) are a significant sound source for jet noise. In the near-nozzle region of a jet, where the nozzle diameter is comparable with the envelope length scale of the wavepacket representing the CS, the CS resides in the thin shear layer developing from the lip line. One of the significant new features in this region is the presence of an inner acoustic field in the potential core. Sound waves with a frequency much lower than that of the CS are emitted via the mechanism of envelope radiation. Similar to that on a planar mixing layer, the outer and inner acoustic fields interact with each other, and the equivalent sound sources have to be determined along with the sound fields. It is shown for the first time that nonlinear interactions of CS in the thin shear layer excite, via receptivity, the so-called discrete ‘trapped acoustic modes’ as well as low-frequency Kelvin–Helmholtz (K–H) modes, which may evolve into preferred modes in the developed region of the jet. The envelope radiation provides a key mechanism coupling the dynamics in the thin shear layer with the dynamics in the jet column regions. In the incompressible limit, this mechanism of excitation also operates, although the pressure decays exponentially towards the axis rather than behaves as sound waves in the potential core.

Friday, Feb. 28^th, 2025

流体不稳定性机理研究：DBM动理学

许爱国 ^1,2

(1.北京应用物理与计算数学研究所邓稼先创新研究中心，北京100088；

2. 北京大学应用物理与技术研究中心，北京 100871 )

摘 要：

流体不稳定性广泛存在于自然界与工程技术领域。不稳定性，给流体研究带来挑战，同时也让流体研究充满魅力。宏观连续描述开始出问题，而微观分子动力学方法又在时空尺度上无能为力的“介尺度”行为，由于模型和方法的缺乏而认知尤其薄弱。这类问题常出现在WQ物理、惯性约束聚变、航空航天、微流控等领域。这极大制约了“介尺度”行为各种效应的评估和调控技术的发展。相对于宏观流动，“介尺度”流动离散性更强，非平衡程度更高，必定表现出与离散性、非平衡程度对应的新的状态/结构和特征。除了如何模拟出这类新的动理学行为，复杂物理场分析技术的严重不足也导致了数据信息的严重浪费。

多尺度模拟，躲不开“介尺度”这道坎。“进军”介尺度，有两条路：从大往小和从小往大。鉴于需要与目前力学工程界主要采用的宏观连续建模进行对接，我们对“介尺度”行为的研究从靠近宏观连续的一侧开始，即从大往小。与重心在离散格式环节的一系列动理学方法不同，离散玻尔兹曼方法（Discrete Boltzmann Method，DBM）针对的是数值实验研究的两端：离散格式之前的物理建模和数值模拟之后的复杂物理场分析，是动理学建模方法 + 分析方法，在模拟前提供模型方程，在模拟后提供复杂物理场分析方法。DBM将传统Navier-Stokes(NS)作为自己在准连续近平衡情形的特例。但，其目的不是取代NS，而是揭示NS所遗漏的动理学行为特征，提供更多的复杂物理场分析方法，探索揭秘 NS不再准确及不再有效的离散、非平衡复杂流动，开发离散、非平衡行为特征的物理功能，研究相应的非平衡调控方法和技术。本报告针对各种不同形式的流体不稳定性研究中遇到的一些基础科学问题，介绍DBM的研究进展和相关机理、行为特征物理功能的开发。

关键词：多尺度；介尺度；非平衡；动理学；离散玻尔兹曼方法

参考文献：

[1] 许爱国，张玉东. 复杂介质动理学，科学出版社，北京：2022.

[2] Aiguo Xu, Dejia Zhang, Yanbiao Gan. Advances in the kinetics of heat and mass transfer in near-continuous complex flows[J]. Frontiers of Physics 19(4), 42500 (2024); DOI: https://doi.org/10.1007/s11467-023-1353-8 .

报告人简介：

许爱国，北京应用物理与计算数学研究所研究员；北京大学工学院兼职教授；中国工程物理研究院物理力学专家组成员，博士生导师；全国统计物理与复杂系统会议学术委员会委员；曾任邓稼先创新研究中心学术委员、中国力学大会学术委员会委员。1998年6月于中国工程物理研究院研究生院获博士学位，导师为陈式刚院士; 1998年7月- 2006年4月，先后工作于北京师范大学、韩国首尔大学、意大利巴里大学、日本京都大学。目前主要研究任务为：针对惯性约束聚变（ICF）等相关的复杂介质动理学，发展物理模型、模拟方法和分析方法，通过数值实验和理论分析，研究其中的微介观结构与非平衡行为及其对介质宏观响应特性的影响，为工程设计提供理论参考。承担ICF相关的科研项目20多项。著有《复杂介质动理学》（科学出版社，2022）。被收录“爱思唯尔 2020 中国高被引学者”、“2023 全球前 2% 顶尖科学家 (年度影响力) 榜单”、2024全球前2%顶尖科学家（生涯影响力）榜单。

Thermoacoustic instability and vortex-sound coupling in acoustic dampers

Dong Yang

Southern University of Science and Technology, Shenzhen

Abstract

Thermoacoustic instabilities (or “combustion instabilities”) come from the positive feedback between acoustics and unsteady combustion; unsteady combustion generates acoustic waves which propagate within the combustor, being reflected by the boundaries, and further disturb the flame to generate more acoustics. This coupling can generate large pressure and heat release oscillations which could reduce the lifetime of the combustion system or even lead to catastrophic damages. New thermoacoustic problems are likely to arise in the future gas turbine combustors burning carbon-free fuels such as hydrogen, so good predicting and controlling tools could be very helpful in the design stage of these combustors. Predicting it is very difficult since it requires resolving couplings across very different scales (acoustics, turbulence, and combustion). Full-scale experiments are extremely expensive, but lab-scale rig does not capture all key physics. This talk presents the state-of-the-art treatments in predicting and damping of this instability. More specifically, the state-of-the-art low-order network modelling methodology, and relevant vortex-sound coupling theories will be presented.

Stabilization Mechanisms of Traveling Crossflow Mode in Hypersonic Swept Wing Flows

Rui Zhao

Beijing Institute of Technology

Momentum potential theory (MPT) is employed to establish a physics-based interpretation of the traveling crossflow mode and analyze the underlying stabilization mechanisms of associated control methods, such as wall cooling, wall suction, and grooves. The traveling crossflow mode over a swept wing with a Mach number of 6 is first solved using direct numerical simulations. The MPT decomposition illustrates the vortical nature of the traveling crossflow mode, with the vortical component having a higher magnitude than the acoustic and thermal components. The vortical source makes the greatest contribution to the mode instability, while the response of the source terms depends on the control method used. Wall cooling primarily impacts the thermal component, thus decreasing the thermal source and subsequently changing the vortical source. Wall suction influences the vortical component directly, and only the vortical source undergoes a small reduction. The acoustic and vortical components are sensitive to grooves. The compression waves induced by grooves are identified as the source of the stabilization effect.

Recent studies on amplification factor transport transition model and eigenvalue prediction method based on convolutional neural network

Jiakuan Xu

School of Aeronautics, Northwestern Polytechnical University

Abstract：

Drag reduction has always been the pursuit of aircraft design, among which laminar design technique is one of the important approaches. In order to develop digital laminar optimization design technique, accurate transition prediction methods play a crucial role. Through verifications by numerous wind tunnel and flight experiments, the semi-empirical e-N method based on linear stability theory is considered to be a reliable transition prediction method. This report introduces two types of modeling ideas for LST-based transition prediction. One is the amplification factor transport transition model based on local variables. Since it is simple to solve, easy to operate, contains stability theory laws, and can directly give the envelope N value, it has become a prediction method trusted by aircraft design engineers. In response to the difficulties of being unable to predict crossflow induced transition and predicting failures under transonic high Reynolds number conditions, this report derives an amplification factor transport equation for stationary crossflow instabilities and proposes a new disturbance amplification factor transport transition model for T-S waves in compressible boundary layers, which significantly improve the transition prediction accuracy. The other one is the transition modeling technique based on convolutional neural networks. Our construction idea is to conduct stability analysis of boundary layer similarity solutions to obtain training samples, and finally perform mapping and eigenvalue prediction at different frequencies and wavelengths in the real boundary layer. Taking the prediction of stationary crossflow instability over a three-dimensional wing as an example, it is confirmed that the convolutional neural network established based on the similarity solution only needs one round of sample training, after which the eigenvalue prediction can be performed well once and for all. Finally, the report looks forward to the difficulties that need to be overcome in the next step.

Study on Subcritical Transition and Turbulence Characteristics of Three-Dimensional Boundary Layers at Hypersonic Swept Wing Leading Edges

Youcheng Xi

Tsinghua University

Three-dimensional (3D) boundary layers are widely encountered in various engineering applications, with their flow characteristics playing a crucial role in the design of hypersonic vehicles and boundary layer flow control. This talk focuses on the 3D boundary layer along the leading-edge attachment line, exploring the key mechanisms underpinning the transition from laminar flow to fully developed turbulence. Combining wind tunnel experiments, stability theory, and numerical simulations, this study delves into the flow behavior and intrinsic mechanisms across different stages of the leading-edge 3D boundary layer.

The analysis begins with the instability mechanisms and the complete subcritical transition process of the boundary layer along the attachment line at the leading edge of blunt hypersonic swept bodies, with a particular emphasis on the nonlinear dynamic mechanisms triggered by roughness elements of varied heights. The study further investigates the characteristics of the fully developed 3D turbulent boundary layer post-transition. Through a systematic analysis of the entire process, this research reveals the fundamental principles of flow instability, transition, and turbulence evolution in 3D boundary layers. The findings offer valuable theoretical foundations and technical references for aerodynamic heating design, flow control, and stability enhancement of hypersonic vehicles.

Excitation and evolution of radiating modes in a supersonic boundary layer in the presence of impinging sound waves: fundamental and subharmonic resonance

Fufeng Qin

School of Mathematics and Statistics, Fujian Normal University, Fuzhou, China

Abstract. The present work investigates linear and nonlinear evolution of radiating modes in a supersonic boundary layer under the influence of impinging sound waves. Two scenarios are considered. The first is a fundamental resonance between the incident sound and the radiating mode, where the sound wave has wavenumber and frequency the same as those of the radiating mode. In this case the sound wave directly excites the radiating mode and/or acts on the pre-existing one. The second scenario is concerned with the sound wave influencing the development of the radiating mode through the mechanism of subharmonic resonance, in which case the sound wavenumber and frequency are twice those of the radiating mode. For both scenarios, a composite amplitude equation is constructed and solved to quantify the impact of the impinging sound wave on the linear and nonlinear instability characteristics of the radiating mode. The Mach-wave field of the radiating mode is changed significantly due to the incident sound.

Stochastic Resonance as a transition route in viscoelastic channel fluid flow

Yuke Li

Harbin Engineering University

Recent studies have shown that viscoelastic channel flows undergo a direct transition from laminar to chaotic flow above the instability threshold (Wic), both numerically and experimentally. In this work, we quantitatively characterize the transition flow regime above Wic, where high intensity peaks unexpectedly emerge at low frequencies in stream-wise velocity power spectra, coined stochastic resonance (SR). The key elements of SR in viscoelastic flow are identified as follows: (1) chaotic flow dynamics as the autonomous chaotic system; (2) low-intensity elastic waves as weak periodic forcing; and (3) flat, white noise dynamics of span-wise velocity as noise. A phase portrait presented by using intensities of stream-wise and span-wise velocity fluctuations also confirms as the necessary conditions of the SR emergence. Varying inlet perturbations introduces different levels of white noise, which influences the range of SR existence downstream and the ratios of synchronized frequencies. Then at the significantly reduced inlet noise intensity, SR is found at larger Wi in “elastic turbulence” and further on in drag reduction flow regimes, if stream-wise elastic wave intensity remains low and span-wise velocity power spectrum flat.

Excitation and Linear Development of Travelling Crossflow Waves to Free-Stream Vortices: The Role of Viscosity

Luyu Shen

Nanjing University of Information Science and Technology

Abstract: This study investigates the excitation and subsequent linear evolution of travelling crossflow waves induced solely by free-stream vortices, without the presence of surface roughness. The excitation occurs near the leading edge of a swept plate, where the boundary layer thickness is significantly smaller than the wavelength of the forcing disturbances. To describe the excitation process, an initial-boundary-value problem is formulated using the classical boundary-layer equations. The far-field condition, comprising gust velocity and scattered potential, is derived through rapid-distortion theory and solved using the Wiener-Hopf technique. The initial condition is constructed as a similarity solution, subject to the far-field condition in the leading-edge limit. Once excited, the travelling waves rapidly amplify and evolve into a viscous mode (Phys. Fluids, vol. 28, 1985, pp. 441–443), which is associated with a critical layer located at the peak of the effective base flow profile. As the boundary layer displacement thickness increases downstream, self-induced pressure fluctuations arise, necessitating the transition to the multiple-tired structure, replacing the classical boundary-layer equations. Simultaneously, the viscous mode transitions to a viscous–inviscid interactive mode, where the pressure fluctuations are balanced by the viscous term at the leading order of the critical-layer equation. The coupling coefficient between the initial viscous mode and the amplitude of the excited travelling waves is computed to assess receptivity. This analysis provides valuable insights for improving current understanding of the receptivity in a crossflow boundary layer.