Fluid Dynamics Research

We derive the incompressible Navier–Stokes equations from the lattice Boltzmann equation using the Chapman–Enskog expansion and the Sone expansion and clarify the differences between the two approaches. In the Chapman–Enskog expansion, we first derive the compressible Navier–Stokes equations on the multiple time scales (the acoustic and diffusive time scales). Then the incompressible Navier–Stokes equations are derived under the conditions of low Mach number flows and small density variations on the diffusive time scale. If the acoustic time scale remains in the analysis of the derived macroscopic equations, the incompressible Navier–Stokes equations are recovered with only the first-order spatial accuracy. On the other hand, in the Sone expansion we can derive the incompressible Navier–Stokes equations under the condition of low Mach number flows on the diffusive time scale. Despite some differences between the two approaches, we obtain the same result that the flow velocity and the pressure satisfy the incompressible Navier–Stokes equations with the second-order spatial accuracy in low Mach number flows on the diffusive time scale. The accuracy is verified through simulating a generalized Taylor–Green problem.

This paper explores the potential of a streamlined adaptive finite point method (FPM) in tackling two-dimensional coupled Burgers' equations, employing them as a testbed for further advancements. Firstly the coupled system is transformed into a two-dimensional heat equation through Cole–Hopf transformation and then this transformed equation is split into one-dimensional heat equations at intermediate temporal levels along X and Y directions and these one-dimensional equations are finally to be treated with the adaptive FPM. The distinctive feature of the adaptive FPM used here lies in employing an implicit 4-point stencil within each local cell to compute the solution at a higher temporal level through a linear combination of solutions from the preceding temporal level. The coefficients involved in this linear combination are derived via the local fundamental solutions within that cell, thereby imbuing the formulations with the intrinsic essence of the exact solution. Moreover, the separation constant utilized is tailored to consistently integrate the influence of the initial solution, independent of the temporal level. The method's theoretical underpinnings ensure its conditionally stable, consistent, and convergent behavior. The accuracy of the scheme is substantiated by its proficient handling of diverse examples, attesting to its superior cost-effectiveness and time efficiency.

The linear stability of unstably stratified shear flows between two horizontal parallel plates has been investigated. The eigenvalue problem was solved numerically by making use of the expansion method in Chebyshev polynomials, and critical Rayleigh numbers were obtained accurately in the Reynolds number range of [0.01,100]. It was found that the critical Rayleigh number for two-dimensional disturbances increases with an increase of the Reynolds number. The result strongly supports previous stability analyses except for the analysis by Makino and Ishikawa (1985) in which a decrease of the critical Rayleigh number was obtained. For some cases, a discontinuity in the critical wavenumber occurs, due to the development of two extrema in the neutral stability boundary.

An inviscid vortex dynamics simulation of interaction of several circular vortex rings produced the power spectrum which had the Kolmogorov -5/3 power law. The enstrophy spectrum, which is equivalent to the dissipation spectrum in homogeneous turbulence, had the 1/3 power law in the same wavenumber range. The wavenumber range of the -5/3 power-law spectrum slightly depends on the number of the vortex rings and their initial configurations, being wider for smaller radius of core of the vortex rings. It was suggested that a uniform spatial distribution of high-vorticity regions is essential to have the -5/3 power law. The length scale of energy-containing eddies was obtained as the reciprocal of a wavenumber at which the energy spectrum attained a dominant peak, while the Kolmogorov length scale was defined as that of a wavenumber at which the enstrophy spectrum attained a dominant peak. Five invariants of inviscid vortical motion, i.e. the total energy, total momentum, total angular momentum, total helicity and total vorticity were maintained constant within tolerable deviations from the corresponding initial values until vorticity tends to diverge.

This paper reports the mean streaming flow generated in a double bifurcation during reciprocating flow calculated using direct numerical simulations. Motivated by the medical ventilation technique of high-frequency ventilation (HFV), we investigate the potential for mean streaming to be maintained in this geometry as the frequency of reciprocation is increased while concurrently reducing the amplitude (and thereby reducing the volume per cycle). We identify four distinct regimes of flow. The first and second occur at low to moderate frequencies and generate significant streaming flows due to the interaction between Dean vortices that are generated during both the in- and out-flows. The third and fourth occur at high frequencies and produce reduced streaming, due to the reduction in formation length of the Dean vortices. Notably, the fourth regime at the highest frequencies investigated appears to show a switch in the direction of the streaming flow at the wall. Considering the motivating application of HFV, we show that currently employed frequencies are low, and much higher frequencies (and subsequently lower volumes per cycle) could potentially be employed.

The fluid dynamic phenomena of liquid drop impact are described and reviewed. These phenomena include bouncing, spreading and splashing on solid surfaces, and bouncing, coalescence and splashing on liquid surfaces. Further, cavitation and the entrainment of gas into an impacted liquid may be observed. In order to distinguish properly between the results of different experiments different impact scenarios are discussed. The specific conditions under which the above phenomena did occur in experiments are analyzed and the characteristics of drop impact phenomena are described in detail.

Analysis methods based on mode decomposition have been proposed to describe the characteristics of flow phenomena. Among them, proper orthogonal decomposition (POD), which decomposes modes into eigenvalues and basis vectors, has long been used. Many studies have shown that POD is a useful method for capturing the characteristics of unsteady flow. In particular, Snapshot POD has attracted much recent attention and has been used to solve unsteady flow problems. However, the basis vectors of the mode obtained by conventional POD is different for each condition. Therefore, whether the basis vectors of each mode are switching in the direction of parameters (e.g. different shapes or different Reynolds numbers) or whether they develop or decay is difficult to discuss. As a result, discussions on conventional POD tend to be qualitative. To address this issue, the present study uses Parametric Global POD, a method that perfectly matches basis vectors in results with different parameters (in this study, different Reynolds numbers). Parametric Global POD method was applied to the analysis of the flow field in a curved pipe and found to capture the development or decay of modes with major basis vectors in the direction of parameters, which is difficult to achieve with conventional POD methods.

Understanding the thermal conditions inside a burning cigarette is a top priority for controlling chemical emissions and cigarette design. Since experimental methods are difficult to observe in depth, this paper starts from the perspective of numerical simulation and models the structure of the tobacco distribution of the cigarette, integrating the end surface ignition model, puffing model, chemical reaction model, heat and mass transfer and diffusion model have established a three-dimensional comprehensive model that can represent the changes in combustion cone morphology during cigarette combustion. The model covers chemical reaction and mass transfer as well as generation, flow and reaction mechanism. The simulation results show that the model can better predict the temperature distribution, component distribution and combustion cone morphology changes during cigarette smoking and combustion. It provides an effective means for in-depth research on cigarette combustion.

Thin liquid film flowing down the inner concave surface of a vertical cylindrical vessel is examined. At the top of the vessel, the water is injected horizontally at high speed circumferentially along the vessel wall and flows downwards due to the action of gravity. This turbulent film flow is modeled using the large eddy simulation (LES) and Reynolds averaged Navier–Stokes (RANS) approaches combined with the volume-of-fluid method. The results of both methods are validated with direct numerical simulation. The Favre-filtered two-phase LES, which is implemented and studied in this paper, can reasonably predict the film thickness similarly to that of the RANS approach using the elliptic blending Reynolds stress model, although it requires fine resolution in the wall region. The effect of volume flow rate on the film structure and thickness is investigated. The film thickness is shown to be nearly constant when the wall is partially wetted and changes as the cubic root of the volume flow rate when the spinning film encloses the entire circumference of the vessel.

We derive the incompressible Navier–Stokes equations from the lattice Boltzmann equation using the Chapman–Enskog expansion and the Sone expansion and clarify the differences between the two approaches. In the Chapman–Enskog expansion, we first derive the compressible Navier–Stokes equations on the multiple time scales (the acoustic and diffusive time scales). Then the incompressible Navier–Stokes equations are derived under the conditions of low Mach number flows and small density variations on the diffusive time scale. If the acoustic time scale remains in the analysis of the derived macroscopic equations, the incompressible Navier–Stokes equations are recovered with only the first-order spatial accuracy. On the other hand, in the Sone expansion we can derive the incompressible Navier–Stokes equations under the condition of low Mach number flows on the diffusive time scale. Despite some differences between the two approaches, we obtain the same result that the flow velocity and the pressure satisfy the incompressible Navier–Stokes equations with the second-order spatial accuracy in low Mach number flows on the diffusive time scale. The accuracy is verified through simulating a generalized Taylor–Green problem.

In this study, the squirmer model with a prescribed tangential velocity is used as a model for swimming microorganisms where its geometric center is offset from the center of mass (bottom-heavy). The settling behavior and interactions of two bottom-heavy squirmers in a vertical channel are simulated numerically under low Reynolds number. Five settling modes, i.e. stable vertical settling, stable inclined settling, wall-attracting oscillatory, oscillatory, and chaotic motion are identified. In addition to the swimming Reynolds number Re_s [0.1,1.0], density ratio γ [1.1,2.1], and swimming strength β [−7,7], another bottom-heavy parameter ER (the ratio of the distance from the center of mass to the geometric center relative to the radius, in the range of [0a₀,0.75a₀] is introduced. The effects of these parameters on the settling modes of bottom-heavy squirmers, terminal Reynolds number Re_t, and interactions of the two bottom-heavy squirmers are discussed. The results showed that a pair of neutral bottom-heavy squirmers more easily achieved a stable structure at the channel center. In contrast, a pair of bottom-heavy pushers were more likely to be captured by the channel walls, leading to a stable structure near the walls. The stable symmetric structure of a pair of bottom-heavy pullers was disturbed, resulting in turbulence. Increasing the swimming strength β accelerates the settling of a pair of pushers. For different ER, the settling speed of two bottom-heavy pushers is greater than that of two bottom-heavy pullers. Additionally, the difference in settling speed between two bottom-heavy squirmers becomes more pronounced with an increase in Re_s. As γ increases, the settling behavior of bottom-heavy squirmers with high β differs from that of those with low β. Moreover, Re_t of a pair of pushers gradually approaches that of neutral bottom-heavy squirmers.

The Darcy–Forchheimer model extends Darcy's law to account for nonlinear inertial effects in fluid dynamics. This paper integrates fuzzy logic and variable exponent sequence spaces into the model to address the inherent uncertainties and heterogeneities in porous media. Through Fredholm operator theory and fixed-point theorems, we establish the existence and uniqueness of solutions to the nonlinear system. Numerical investigations highlight the impact of varying porous media conditions on fluid behavior. As a main outcome, we observe that solutions exhibit a smooth behavior for moderate levels of permeability and porosity in the media, while for increased fuzzy norms in both media properties, the fluid velocity profile exhibits fluctuations due to the influence of the fuzzy terms.

The flow of a film induced by gravity is not only widespread in nature but also has significant applications in various industrial technologies such as coating techniques, nanotechnology, microfluidic chips, and heat exchangers. The film exhibits various nonlinear dynamic phenomena due to the interactions between surface tension, gravity, and other forces, making the study of this type of flow of great importance. However, the theoretical derivation of thin film fluid dynamics is complex, with diverse working conditions, making numerical solutions difficult, time-consuming, and labor-intensive. Therefore, it is of significant importance to seek an approach that differs from theoretical derivation or numerical solutions for the study of thin film fluid dynamics. With the rapid development of deep learning, this paper employs physics-informed neural networks (PINNs) algorithm, in conjunction with the partial differential equation governing the fluid film thickness, to conduct research on forward prediction of the film thickness variation over time and space, and inverse problem solving to determine unknown parameters in the governing equation from data. The study investigates the governing equation for the thickness of a falling film along an inclined plane and applies the PINNs method to solve it. The research predicted the film thickness for three different characteristic waveforms and conducted a comparative analysis with solutions obtained using the commercial software COMSOL. Additionally, the unknown parameters in the governing equation were inversely solved using limited data, and the differences in prediction results with and without noisy data were compared.

In this article, three-dimensional lid-driven flows in cavities with a semicircular round bottom are studied. We have first focused on lid-driven flow in a semicircular cavity with a unit square moving lid and height 1/2. The critical Reynolds number $Re_\textrm{cr}$ for the transition from steady flow to unsteady one has been obtained. Based on the averaged velocity field in one cycle of fluid flow motion, the flow difference between the averaged one and velocity field, called oscillation mode, at several time instances in such cycle shows an almost identical pattern for several Reynolds numbers close to $Re_\textrm{cr}$. This similarity indicates the oscillation mode associated with the Hopf bifurcation originated at Re less than $Re_\textrm{cr}$. For lid-driven flow in a cavity with a semicircular round bottom and height one, its oscillation mode shows a periodic change of local secondary flows associated with Hopf bifurcation and pairs of Taylor–Görtler-like vortices are obtained.

In this paper, we present a review of featured works in the field of hydrodynamics with the main aim to clarify the ways of understanding the algorithms for solving the Navier–Stokes equations. Discussing the existing algorithms, approaches and analytical or semi-analytical methods, we especially note that important problems of stability for the exact solutions should be explored accordingly relate to this respect, e.g. exploring the case of non-stationary helical flows of the Navier–Stokes equations for incompressible fluids with variable (spatially dependent) coefficient of proportionality α between velocity and the curl field of the flow. Meanwhile, the system of Navier–Stokes equations (including continuity equation) has been successfully explored previously with respect to the existence of analytical way for presentation of non-stationary helical flows of the aforementioned type. Conditions for the stability criteria of the exact solution for such the type of flows are obtained herein in the current research, for which non-stationary helical flow with invariant Bernoulli-function is considered.

This paper reviews recent developments in the understanding of underwater bio-mimetic propulsion. Two impressive models of underwater propulsion are considered: cruise and fast-start. First, we introduce the progression of bio-mimetic propulsion, especially underwater propulsion, where some primary conceptions are touched upon. Second, the understanding of flapping foils, considered as one of the most efficient cruise styles of aquatic animals, is introduced, where the effect of kinematics and the shape and flexibility of foils on generating thrust are elucidated respectively. Fast-start propulsion is always exhibited when predator behaviour occurs, and we provide an explicit introduction of corresponding zoological experiments and numerical simulations. We also provide some predictions about underwater bio-mimetic propulsion.

Obtaining real flow information is important in various fields, but is a difficult issue because measurement data are usually limited in time and space, and computational results usually do not represent the exact state of real flows. Problems inherent in the realization of numerical simulation of real-world flows include the difficulty in representing exact initial and boundary conditions and the difficulty in representing unstable flow characteristics. This article reviews studies dealing with these problems. First, an overview of basic flow measurement methodologies and measurement data interpolation/approximation techniques is presented. Then, studies on methods of integrating numerical simulation and measurement, namely, four-dimensional variational data assimilation (4D-Var), Kalman filters (KFs), state observers, etc are discussed. The first problem is properly solved by these integration methodologies. The second problem can be partially solved with 4D-Var in which only initial and boundary conditions are control parameters. If an appropriate control parameter capable of modifying the dynamical structure of the model is included in the formulation of 4D-Var, unstable modes are properly suppressed and the second problem is solved. The state observer and KFs also solve the second problem by modifying mathematical models to stabilize the unstable modes of the original dynamical system by applying feedback signals. These integration methodologies are now applied in simulation of real-world flows in a wide variety of research fields. Examples are presented for basic fluid dynamics and applications in meteorology, aerospace, medicine, etc.

The extensive evaluation studies of the lattice Boltzmann method for micro-scale flows (μ-flow LBM) by the author's group are summarized. For the two-dimensional test cases, force-driven Poiseuille flows, Couette flows, a combined nanochannel flow, and flows in a nanochannel with a square- or triangular cylinder are discussed. The three-dimensional (3D) test cases are nano-mesh flows and a flow between 3D bumpy walls. The reference data for the complex test flow geometries are from the molecular dynamics simulations of the Lennard-Jones fluid by the author's group. The focused flows are mainly in the slip and a part of the transitional flow regimes at Kn < 1. The evaluated schemes of the μ-flow LBMs are the lattice Bhatnagar–Gross–Krook and the multiple-relaxation time LBMs with several boundary conditions and discrete velocity models. The effects of the discrete velocity models, the wall boundary conditions, the near-wall correction models of the molecular mean free path and the regularization process are discussed to confirm the applicability and the limitations of the μ-flow LBMs for complex flow geometries.

We analyze a 1 + 1-dimensional directed percolation system as a model for the spatio-temporal aspects of the turbulence transition in pipe flow and other shear flows. Space and time are discrete, and the model is characterized by two parameters: one describes the probability to remain turbulent in the next step and the other characterizes the spreading of turbulence to the neighboring cells. The transition to a persistent turbulence is evident in mean field arguments, but the actual critical values and exponents are considerably renormalized by fluctuations. Extensive numerical tests show that the model falls into the universality class of one-dimensional (1D) directed percolation. We also discuss the spreading of localized perturbations and an extension to 2D systems.

A semi-analytical study of coupled translation and rotation of a composite spherical particle (a hard sphere core coated with a permeable porous layer) in a viscous fluid inside an eccentric spherical cavity normal to their common diameter is presented in the quasi-steady limit of low Reynolds number. To solve the Stokes and Brinkman equations for the flow fields outside and inside the porous layer, respectively, a general solution is constructed from the fundamental solutions in the two spherical coordinate systems based on both the composite particle and the cavity. The boundary conditions at the cavity wall and inner and outer surfaces of the porous layer are satisfied by a collocation method. Numerical results for the force and torque exerted on the particle by the fluid are obtained with good convergence for various values of the relevant parameters in practical applications. For the translation and rotation of a composite sphere inside a concentric cavity, our force and torque results agree well with the available solutions in the literature. The force and torque on a translating and rotating particle increase monotonically with increases in the ratios of particle radius to porous layer permeation length, core-to-particle radii, and particle-to-cavity radii. In general, they also increase with an increase in the relative distance between the particle and cavity centers. The boundary effect of the cavity on the translation of the particle is much more pronounced than that on the rotation. The coupling effect in the simultaneous translation and rotation inside an eccentric spherical cavity is complicated and not a monotonic function of the particle-to-cavity radius ratio.

A semi-analytical study of coupled translation and rotation of a composite spherical particle (a hard sphere core coated with a permeable porous layer) in a viscous fluid inside an eccentric spherical cavity normal to their common diameter is presented in the quasi-steady limit of low Reynolds number. To solve the Stokes and Brinkman equations for the flow fields outside and inside the porous layer, respectively, a general solution is constructed from the fundamental solutions in the two spherical coordinate systems based on both the composite particle and the cavity. The boundary conditions at the cavity wall and inner and outer surfaces of the porous layer are satisfied by a collocation method. Numerical results for the force and torque exerted on the particle by the fluid are obtained with good convergence for various values of the relevant parameters in practical applications. For the translation and rotation of a composite sphere inside a concentric cavity, our force and torque results agree well with the available solutions in the literature. The force and torque on a translating and rotating particle increase monotonically with increases in the ratios of particle radius to porous layer permeation length, core-to-particle radii, and particle-to-cavity radii. In general, they also increase with an increase in the relative distance between the particle and cavity centers. The boundary effect of the cavity on the translation of the particle is much more pronounced than that on the rotation. The coupling effect in the simultaneous translation and rotation inside an eccentric spherical cavity is complicated and not a monotonic function of the particle-to-cavity radius ratio.

We derive the incompressible Navier–Stokes equations from the lattice Boltzmann equation using the Chapman–Enskog expansion and the Sone expansion and clarify the differences between the two approaches. In the Chapman–Enskog expansion, we first derive the compressible Navier–Stokes equations on the multiple time scales (the acoustic and diffusive time scales). Then the incompressible Navier–Stokes equations are derived under the conditions of low Mach number flows and small density variations on the diffusive time scale. If the acoustic time scale remains in the analysis of the derived macroscopic equations, the incompressible Navier–Stokes equations are recovered with only the first-order spatial accuracy. On the other hand, in the Sone expansion we can derive the incompressible Navier–Stokes equations under the condition of low Mach number flows on the diffusive time scale. Despite some differences between the two approaches, we obtain the same result that the flow velocity and the pressure satisfy the incompressible Navier–Stokes equations with the second-order spatial accuracy in low Mach number flows on the diffusive time scale. The accuracy is verified through simulating a generalized Taylor–Green problem.

This paper explores the potential of a streamlined adaptive finite point method (FPM) in tackling two-dimensional coupled Burgers' equations, employing them as a testbed for further advancements. Firstly the coupled system is transformed into a two-dimensional heat equation through Cole–Hopf transformation and then this transformed equation is split into one-dimensional heat equations at intermediate temporal levels along X and Y directions and these one-dimensional equations are finally to be treated with the adaptive FPM. The distinctive feature of the adaptive FPM used here lies in employing an implicit 4-point stencil within each local cell to compute the solution at a higher temporal level through a linear combination of solutions from the preceding temporal level. The coefficients involved in this linear combination are derived via the local fundamental solutions within that cell, thereby imbuing the formulations with the intrinsic essence of the exact solution. Moreover, the separation constant utilized is tailored to consistently integrate the influence of the initial solution, independent of the temporal level. The method's theoretical underpinnings ensure its conditionally stable, consistent, and convergent behavior. The accuracy of the scheme is substantiated by its proficient handling of diverse examples, attesting to its superior cost-effectiveness and time efficiency.

Understanding the thermal conditions inside a burning cigarette is a top priority for controlling chemical emissions and cigarette design. Since experimental methods are difficult to observe in depth, this paper starts from the perspective of numerical simulation and models the structure of the tobacco distribution of the cigarette, integrating the end surface ignition model, puffing model, chemical reaction model, heat and mass transfer and diffusion model have established a three-dimensional comprehensive model that can represent the changes in combustion cone morphology during cigarette combustion. The model covers chemical reaction and mass transfer as well as generation, flow and reaction mechanism. The simulation results show that the model can better predict the temperature distribution, component distribution and combustion cone morphology changes during cigarette smoking and combustion. It provides an effective means for in-depth research on cigarette combustion.

Thin liquid film flowing down the inner concave surface of a vertical cylindrical vessel is examined. At the top of the vessel, the water is injected horizontally at high speed circumferentially along the vessel wall and flows downwards due to the action of gravity. This turbulent film flow is modeled using the large eddy simulation (LES) and Reynolds averaged Navier–Stokes (RANS) approaches combined with the volume-of-fluid method. The results of both methods are validated with direct numerical simulation. The Favre-filtered two-phase LES, which is implemented and studied in this paper, can reasonably predict the film thickness similarly to that of the RANS approach using the elliptic blending Reynolds stress model, although it requires fine resolution in the wall region. The effect of volume flow rate on the film structure and thickness is investigated. The film thickness is shown to be nearly constant when the wall is partially wetted and changes as the cubic root of the volume flow rate when the spinning film encloses the entire circumference of the vessel.

Using methods of granular mechanics in the quasi-static limit, with inter-particle interactions derived from the lubrication limit, the intensity of velocity fluctuations in the slurry is associated with fluctuations in the local distribution of inter-particle distances. These are shown to consist of a vector intensity and a scalar intensity; the former couples to the first velocity gradient, the latter (which is associated with solidosity fluctuations) couples to the second velocity gradient. Rheologies for both are presented, as is the rheology that links the particle pressure to the intensity of the velocity fluctuations (also known as the 'granular temperature') to the dispersive pressure. The rheologies are informed by experimental results. The granular temperature profile, modified from previous work, is responsible for axial particle migration (Bagnold effect). Two broad categories are assessed: symmetrical vertical and non-symmetrical lateral flow. For the latter the roughness of the boundary walls and a non-zero density contrast are important; this case is studied for a system in which flow effects are confined to the immediate vicinity of the boundary. Sensitivity analysis reveals several key variables including the parameters that control a slipping boundary condition and the mean solidosity in the conduit. For lateral flow, a sedimentary deposit with a solidosity profile may develop near the upper or lower boundary. The theory predicts an approximate relation between the fluid-particle density contrast and sediment thickness as a function of the mean flow rate, conduit width, the mean particle diameter and fluid viscosity that has utility in a range of engineering and geological situations where particulate matter is transported in the laminar flow regime.

Jets and wakes are fundamental fluid flows that arise in a wide range of environmental and aerospace applications. They are typically studied as open systems. Here we are interested in the implications of placing the jet or wake inside of another system, as well as the implications of compliant walls. In particular, the effect of asymmetry is considered on the absolute instability properties for this internal flow, when it is transversely confined by compliant walls. Two distinct cases are considered, namely the case of two compliant walls with non-identical wall parameters and the case of identical compliant walls asymmetrically located about the fluid center line. The absolute instability characteristics are identified by following special saddle points (pinch points) of the dispersion relation in the complex wavenumber plane, and the flow's stability properties are mapped out using parameter continuation techniques. The compliant walls introduce new modes which typically dominate the stability properties of the flow, in comparison to the case of pure shear layers. In the case of symmetrically located walls with non-identical wall parameters, it was found that the absolute stability properties are dominated by the modes linked to the more flexible of the two walls. In the case of identical walls asymmetrically confining the flow, it was found that these flows exhibit smaller regions of absolute instability in parameter space, when compared to the symmetric flow configuration.

Analysis methods based on mode decomposition have been proposed to describe the characteristics of flow phenomena. Among them, proper orthogonal decomposition (POD), which decomposes modes into eigenvalues and basis vectors, has long been used. Many studies have shown that POD is a useful method for capturing the characteristics of unsteady flow. In particular, Snapshot POD has attracted much recent attention and has been used to solve unsteady flow problems. However, the basis vectors of the mode obtained by conventional POD is different for each condition. Therefore, whether the basis vectors of each mode are switching in the direction of parameters (e.g. different shapes or different Reynolds numbers) or whether they develop or decay is difficult to discuss. As a result, discussions on conventional POD tend to be qualitative. To address this issue, the present study uses Parametric Global POD, a method that perfectly matches basis vectors in results with different parameters (in this study, different Reynolds numbers). Parametric Global POD method was applied to the analysis of the flow field in a curved pipe and found to capture the development or decay of modes with major basis vectors in the direction of parameters, which is difficult to achieve with conventional POD methods.

This paper analyzes the reduction of the infinite Lundgren–Monin–Novikov (LMN) hierarchy of probability density functions (PDFs) in the statistical theory of helically symmetric turbulence. Lundgren's hierarchy is considered a complete model, i.e. fully describes the joint multi-point statistic of turbulence though at the expense of dealing with an infinite set of integro-differential equations. The LMN hierarchy and its respective side-conditions are transformed to helical coordinates and thus are dimesionally reduced. In the course of development, a number of key questions were solved, namely in particular the transformation of PDFs and sample space velocities into orthonormal coordinate systems. In a validity check it is shown, that the mean momentum equations derived from the helical LMN hierarchy via statistical moment integration are identical to the mean momentum equations derived by direct ensemble averaging the Navier–Stokes equation, in helically symmetric form. Finally, we derive the equation for the characteristic function equivalent to the PDF equation in a helically symmetric frame, which allows to generate arbitrary $n{\mathrm{^{th}}}$-order statistical moments by simple differentiation.

This paper reports the mean streaming flow generated in a double bifurcation during reciprocating flow calculated using direct numerical simulations. Motivated by the medical ventilation technique of high-frequency ventilation (HFV), we investigate the potential for mean streaming to be maintained in this geometry as the frequency of reciprocation is increased while concurrently reducing the amplitude (and thereby reducing the volume per cycle). We identify four distinct regimes of flow. The first and second occur at low to moderate frequencies and generate significant streaming flows due to the interaction between Dean vortices that are generated during both the in- and out-flows. The third and fourth occur at high frequencies and produce reduced streaming, due to the reduction in formation length of the Dean vortices. Notably, the fourth regime at the highest frequencies investigated appears to show a switch in the direction of the streaming flow at the wall. Considering the motivating application of HFV, we show that currently employed frequencies are low, and much higher frequencies (and subsequently lower volumes per cycle) could potentially be employed.