# Theoretical and Computational Fluid Dynamics

Online ISSN: 1432-2250
Print ISSN: 0935-4964
Recent publications
In the case of microscopic particles, the momentum exchange between the particle and the gas flow starts to deviate from the standard macroscopic particle case, i.e. the no-slip case, with slip flow occurring in the case of low to moderate particle Knudsen numbers. In order to derive new drag force models that are valid also in the slip flow regime for the case of non-spherical particles of arbitrary shapes using computational fluid dynamics, the no-slip conditions at the particle surface have to be modified in order to account for the velocity slip at the surface, mostly in the form of the Maxwell’s slip model. To allow a continuous transition in the boundary condition at the wall from the no-slip case to the slip cases for various Knudsen (Kn) number value flow regimes, a novel specific slip length model for the use with the Maxwell boundary conditions is proposed. The model is derived based on the data from the published experimental studies on spherical microparticle drag force correlations and Cunningham-based slip correction factors at standard conditions and uses a detailed CFD study on microparticle fluid dynamics to determine the correct values of the specific slip length at selected Kn number conditions. The obtained data on specific slip length are correlated using a polynomial function, resulting in the specific slip length model for the no-slip and slip flow regimes that can be applied to arbitrary convex particle shapes. Graphic abstract

Direct numerical simulation and theoretical analysis of acoustic receptivity are performed for the boundary layer on a flat plate in Mach 6 flow at various angles of attack (AoA). Slow or fast acoustic wave passes through: a bow shock at AoA =-5∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=-5^{\circ }$$\end{document}, a weak shock induced by the viscous–inviscid interaction at AoA =0∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=0^{\circ }$$\end{document} or an expansion fan emanating from the plate leading edge at AoA =5∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=5^{\circ }$$\end{document}. The study is focused on cases where the integral amplification of unstable mode S (or Mack second mode) is sufficiently large (N≈8.4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N\approx 8.4)$$\end{document} to be relevant to transition in low-disturbance environments. It is shown that excitation of dominant modes F and S occurs in a small vicinity of the plate leading edge. The initial disturbance propagates further downstream in accord with the two-mode approximation model accounting for the mean-flow nonparallel effects and the intermodal exchange mechanism. This computationally economical model can be useful for predictions of the second mode dominated transition onset using the physics-based amplitude method. Graphic abstract

The use of multitaper estimates for spectral proper orthogonal decomposition (SPOD) is explored. Multitaper and multitaper-Welch estimators that use discrete prolate spheroidal sequences (DPSS) as orthogonal data windows are compared to the standard SPOD algorithm that exclusively relies on weighted overlapped segment averaging, or Welch’s method, to estimate the cross-spectral density matrix. Two sets of turbulent flow data, one experimental and the other numerical, are used to discuss the choice of resolution bandwidth and the bias-variance tradeoff. Multitaper-Welch estimators that combine both approaches by applying orthogonal tapers to overlapping segments allow for flexible control of resolution, variance, and bias. At additional computational cost but for the same data, multitaper-Welch estimators provide lower variance estimates at fixed frequency resolution or higher frequency resolution at similar variance compared to the standard algorithm. Graphic abstract

The influence of turbulence inflow generation on direct numerical simulations (DNS) of high-speed turbulent boundary layers at Mach numbers of 2 and 5.84 is investigated. Two main classes of inflow conditions are considered, based on the recycling/rescaling (RR) and the digital filtering (DF) approach, along with suitably modified versions. A series of DNS using very long streamwise domains is first carried out to provide reliable data for the subsequent investigation. A set of diagnostic parameters is then selected to verify achievement of an equilibrium state, and correlation laws for those quantities are obtained based on benchmark cases. Simulations using shorter domains, with extent comparable with that used in the current literature, are then carried out and compared with the benchmark data. Significant deviations from equilibrium conditions are found, to a different extent for the various flow properties, and depending on the inflow turbulence seeding. We find that the RR method yields superior performance in the evaluation of the inner-scaled wall pressure fluctuations and the turbulent shear stress. DF methods instead yield quicker adjustment and better accuracy in the prediction of wall friction and of the streamwise Reynolds stress in supersonic cases. Unrealistically high values of the wall pressure variance are obtained by the baseline DF method, while the proposed DF alternatives recover a closer agreement with respect to the benchmark. The hypersonic test case highlights that similar distribution of wall friction and heat transfer are obtained by both RR and DF baseline methods. Graphical abstract

Determining the behavior of the leading-edge suction force, represented non-dimensionally by the leading-edge suction parameter (LESP), can reliably help indicate the state of flow over the airfoil and therefore the force and moment characteristics. The current work aims at studying the variations in the LESP, forces, and pitching moment with freestream Reynolds number and airfoil thickness in unsteady flows. Computational data for the NACA 0012, 0015, and 0018 airfoils undergoing a baseline pitching motion over a range of freestream Reynolds number conditions are analyzed. The critical LESP, which is the instantaneous value of LESP at leading-edge vortex initiation, is observed to first decrease and subsequently increase with Reynolds number. This behavior can be correlated to the rate at which leading-edge flow curvature increases with Reynolds number. Thicker airfoils are observed to sustain a larger amount of suction force prior to breakdown and ensuing leading-edge vortex (LEV) shedding. Lift, drag, and moment are found to be dependent on thickness and Reynolds number prior to LEV shedding due to differences in the boundary layer characteristics, but independent after suction breakdown due to the similarity in LEV dynamics. These findings serve to support the development of a more generalized definition of a suction-force parameter that is independent of flow conditions and airfoil geometry. Graphical abstract

Motivated by recent advances in the development of the numerical calculation of fine flow in liquid film, the thermocapillary convection in thin liquid film (1mm) due to temperature difference is studied in this paper. To describe the formation of the thermocapillary convection on gas-liquid interface, a two-phase system was designed, in which the momentum and energy interact directly through the free surface. The finite volume method is used to solve the N-S equation in gas phase and liquid phase, respectively, and the velocity and temperature information are exchanged on the free surface in each time step. The results show that a thermocapillary wave appears in the liquid film when the temperature difference exceeds a certain value. The temperature and velocity fluctuations on the free surface show a radiation shape. The flow field structure is completely symmetrical in the basic state, but it is axisymmetric in the case of oscillation state. The propagation direction of thermocapillary wave is affected by many factors (ambient temperature or inner wall rotation). The wave propagation direction is consistent with the rotation direction when the inner wall rotates. When the angular velocity of inner wall rotation is 8 rad/s, the wave number of thermocapillary wave will be reduced to 3, which is independent of the rotation direction. Graphical abstract

A volume of fluid method combined with an adaptive grid method was used to study the influence of Galilei (Ga) and Eötvös (Eo) numbers and characteristic parameters (such as rheological index (n) and characteristic time (λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document})) of shear-thinning liquids on the hydrodynamics of two types of unsteady bubbles. One is the bubble with central breakup behaviors, of which the rise trajectory is a straight line and the shape is symmetrical; however, the shape and centroid velocity cannot reach a steady state. Bubble shape becomes annular after radial expansion, and the centroid velocity has two peaks. The other is the unsteady bubble, of which the rise trajectory is zigzag, but both the shape and rise velocity cannot reach a steady state. The shape of this unstable bubble is flat, which causes periodic vortex shedding at the tail of a bubble. Thus, bubble rise velocity cannot reach a steady state. When the influence of viscous force is relatively weak and Eo is in the range of 50–55, a bubble shows central breakup behaviors. When Eo is low (Eo<10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<10$$\end{document}), effective Morton numbers (Moeff\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\mathrm{eff}}$$\end{document}) decrease to the magnitude of 10-7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-7}$$\end{document} and effective Reynolds numbers meet the condition of Reeff≥125.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\mathrm{eff}}\ge 125.2$$\end{document}, a bubble shows the second type of unsteady characteristics. Graphic abstract

The problem of a solitary surface gravity wave in a flow of an inviscid incompressible fluid in a channel of constant depth is considered. The problem is solved in two-dimensional formulation. The wave moves at a constant speed. In a coordinate system moving along with the wave, the flow is stationary. Its mathematical model is reduced to a boundary value problem for a strip in the complex potential plane. This is converted to a boundary value problem for a half-plane by conformal mapping. The solution is obtained using a Cauchy-type integral for the density of which a nonlinear integral equation is derived. Its solution is found with the Galerkin method and the Newton–Raphson technique. The calculated results are compared with the experimental data and the calculations by other researchers. The lower limit of the speed of a solitary wave is found. The advantage of the proposed method is the simplicity of the resulting integral equation, which makes it possible to effectively apply numerical methods of solution. Graphic abstract

In this paper, we aim to determine the effective permeability of biporous solids containing fractures in a computational homogenization framework. Precisely, at the local scale, we use the Brinkman law for the porous solids and the Stokes equations within fractures. The resolution of the Brinkman/Stokes coupled equations is performed with the fast Fourier transform iterative scheme on periodic unit cells. The role of the dimensions and orientation of the cracks is investigated. The simulation results are also compared with an analytical formulation based on Mori–Tanaka estimates. The findings indicate that, depending on the considered flow direction, the dimensions and orientation of cracks strongly affect the effective permeability of fractured porous media. Besides, we also determine the macroscopic permeability for a population of fractures with regular and random orientations, dimensions and shapes. The results are then given and discussed. Graphic Abstract

Dynamic response characteristics of five tandem circular cylinders in laminar uniform flow are studied numerically by fluid–structure interaction (FSI) computation. The Reynolds number of the incoming flow is fixed at Re =100\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=100$$\end{document}. The five cylinders are elastically mounted in both transverse and streamwise directions with an even center-to-center distance of 4, 6 and 8 times of the cylinder diameter. The non-dimensional mass of each cylinder is m∗=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m^{*}=5$$\end{document}, 10 and 15, while the reduced velocity varies in the range of Ur=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\mathrm{r}}=$$\end{document} 2–18. An FSI solver based on a modified characteristic-based split finite element method is developed for computation, and its accuracy is validated by evaluating the flow around five stationary circular cylinder and flow-induced vibrations (FIVs) of the one-cylinder and two-tandem-cylinder models against benchmark solutions. By numerical experiments, dynamic behaviors of five tandem cylinders as well as the underlying mechanisms are investigated by analyzing the generated vibration amplitude, frequency, fluid load and vortex pattern in the flow field. Sub-harmonic wake-induced vibration that has not been revealed by the existing two-cylinder and three-cylinder models is observed, and the underlying physics is discussed in detail. The results obtained are insightful into the understanding and control of FIVs of an array of cylindrical structures encountered frequently in various engineering applications. Graphical abstract

A matrix normalization scheme based on thermodynamic entropy is derived for modal decomposition techniques applied compressible flows. It is demonstrated that this normalization scheme is consistent with the scalar form of entropy. Analysis based in this consistency is performed to demonstrate the theoretical underpinnings of the Chu energy norm, which is the industry standard for compressible modal decompositions. The entropy normalization is shown mathematically to converge to the Chu normalization in the absence of strong temperature gradients. It is then compared to the Chu normalization by analyzing transient growth calculated through a linear stability analysis of a self-similar compressible boundary layer profile. The entropy norm is shown to be more sensitive to temperature fluctuations around the boundary layer than the Chu norm. These observations are further validated in a POD implementation of the entropy normalization, contracted about the conservative and primitive variables. The trends observed in transient growth analysis are observed in full-scale POD. The potential for the entropy normalization to be applied to flows with additional relevant physics, such as as thermal and chemical nonequilibrium, is explored. Graphical abstract

The local thermal effect of a flame front is simulated by a model for a mass density front by specifying a likely expansion rate. This model problem includes two independent parameters, namely the heat release parameter and a parameter akin to a Karlovitz number. The analysis is focused on the influence of the Karlovitz number on the evolution of strain properties at the crossing of the front. The latter are derived from an equation system for the velocity gradient tensor and the pressure Hessian tensor undergoing the forcing of the expansion rate. Strain eigenvalues, orientation of strain principal axes, and stretching in the direction of forcing are especially scrutinized. Furthermore, the model shows that, when approaching a flame front, the special alignments of strain are mostly caused by anisotropy of pressure Hessian resulting from forcing by expansion. Graphic abstract

The choice and placement of sensors and actuators is an essential factor determining the performance that can be realized using feedback control. This determination is especially important, but difficult, in the context of controlling transitional flows. The highly non-normal nature of the linearized Navier–Stokes equations makes the flow sensitive to small perturbations, with potentially drastic performance consequences on closed-loop flow control. Full-information controllers, such as the linear quadratic regulator (LQR), have demonstrated some success in reducing transient energy growth and suppressing transition; however, sensor-based output feedback controllers with comparable performance have been difficult to realize. In this study, we propose two methods for sensor selection that enable sensor-based output feedback controllers to recover full-information control performance: one based on a sparse controller synthesis approach, and one based on a balanced truncation procedure for model reduction. Both approaches are investigated within linear and nonlinear simulations of a sub-critical channel flow with blowing and suction actuation at the walls. We find that sensor configurations identified by both approaches allow sensor-based static output feedback LQR controllers to recover full-information LQR control performance, both in reducing transient energy growth and suppressing transition. Further, our results indicate that both the sensor selection methods and the resulting controllers exhibit robustness to Reynolds number variations. Graphic abstract

The problem of the linear stability of the stratified Kolmogorov flow driven by a sinusoidal in space force in a viscous and diffusive Boussinesq fluid is re-visited using the Floquet theory, Galerkin approximations and the method of (generalized) continued fractions. Numerical and analytical arguments are provided in favor of a conjecture that an ideal stratified Kolmogorov flow is prone to short-wave instability for Richardson numbers markedly greater than the critical Richardson number Ri =\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$=$$\end{document} ¼ that appears in the Miles–Howard theorem. The short-wave instability of the stratified Kolmogorov flow is conjectured to be due to a resonance amplification of the Doppler-shifted internal gravity wave modes, in the presence of critical levels of the main flow that are ignored in the proof of the Miles–Howard theorem, but it is emphasized that the complete resolution of the above paradox is a task for future research. Graphical abstract

A compressible boundary-layer flow over a flat plate with sharp leading edge is studied in the hypersonic limit. The interaction between the shock wave and boundary layer is characterized using the hypersonic interaction parameter χ=M∞3C/Re∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi = M_{\infty }^3\sqrt{C/\mathrm{Re}_{\infty }}$$\end{document} where M∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} and Re∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Re}_{\infty }$$\end{document} are the free-stream Mach and Reynolds numbers, respectively, and C is the Chapman–Rubesin constant. The flow is studied for Prandtl number Pr=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Pr}=1$$\end{document} using a shock-layer analysis of the equations of motion governing high-speed compressible flow. In the strong interaction limit the value for χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi$$\end{document} approaches infinity, χ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi \rightarrow \infty$$\end{document}, and there exists coupling between the shock wave and boundary layer that extends the plate length. For finite interaction, 1<χ<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< \chi < \infty$$\end{document}, there is coupling between the shock wave and boundary layer that can extend well-beyond the plate’s leading edge. To study this transition, from χ∼O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi \sim O$$\end{document}(1) to χ≫1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi \gg 1$$\end{document}, we solved the Prandtl boundary-layer equations that represent the viscous-layer flow (Region I) at non-adiabatic wall conditions using a standard line relaxation method. For the inviscid layer (Region II), we reduced the governing thin inviscid-layer equations to ordinary differential equations by using the method of characteristics. We then matched values for flow variables of similar order computed in the viscous-(Region I) and inviscid-(Region II) layers at the boundary-layer’s edge by using a minimization algorithm. Thus, solutions produced using the technique, denoted asymptotic matching Technique A, required only a single streamwise sweep to achieve convergence between these flow variables computed in the viscous and inviscid layers and matched at the boundary-layer edge. Solutions for flow variables found using Technique A are then compared with solutions for similar χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi$$\end{document} and wall enthalpy values found using a separate shock-layer analysis, denoted matching Technique B, that utilized a tangent-wedge approximation for the inviscid layer. Technique B required successive streamwise sweeps such that initial pressure conditions upstream and downstream are both satisfied at each sweep. The converged solution was obtained during the final sweep based on a preset convergence criteria. Shock-wave and boundary-layer profiles; wall pressure and shear stress computed using both Techniques A and B are compared with values computed using computational fluid dynamics (CFD). The results show good agreement. Graphical abstract

This study investigates the stability of compressible swirling wake flows including the viscous effects using linear stability theory. A spatial stability analysis is performed to evaluate the influence of the axial velocity deficit and circulation as well as the Reynolds number and Mach number as the main parameters that affect the instability. The growth rates of the unstable modes at several azimuthal wavenumbers are compared. The maximum growth rates and their dependency with respect to each parameter are analyzed. It is confirmed that the instability monotonically increases as the axial velocity deficit increases. For small axial velocity deficit, characteristics that are different from the results reported using inviscid analysis are identified and analyzed. Additionally, a decrease in instability is observed as the viscous and compressibility effects become stronger. In terms of circulation, it is confirmed that there is a certain region of circulation that exhibits maximum instability. The stability analysis is expected to serve as a part of a useful methodology for preliminary design and parametric study for engineering problems such as vortex generators in high-speed flows, owing to both efficiency and accuracy. Graphical abstract

This work presents a robust method that minimises the impact of user-selected parameter on the identification of generic models to study the coherent dynamics in turbulent flows. The objective is to gain insight into the flow dynamics from a data-driven reduced order model (ROM) that is developed from measurement data of the respective flow. For an efficient separation of the coherent dynamics, spectral proper orthogonal decomposition (SPOD) is used, projecting the flow field onto a low-dimensional subspace, so that the dominating dynamics can be represented with a minimal number of modes. A function library is defined using polynomial combinations of the temporal modal coefficients to describe the flow dynamics with a system of nonlinear ordinary differential equations. The most important library functions are identified in a two-stage cross-validation procedure (conservative and restrictive sparsification) and combined in the final model. In the first stage, the process uses a simple approximation of the derivative to match the model with the data. This stage delivers a reduced set of possible library function candidates for the model. In the second, more complex stage, the model of the entire flow is integrated over a short time and compared with the progression of the measured data. This restrictive stage allows a robust identification of nonlinearities and modal interactions in the data and their representation in the model. The method is demonstrated using data from particle image velocimetry (PIV) measurements of a circular cylinder undergoing vortex-induced vibration (VIV) at $$\mathrm{Re}=4000$$ Re = 4000 . It delivers a reduced order model that reproduces the average dynamics of the flow and reveals the interaction of coexisting flow dynamics by the model structure.

Understanding the dynamics of a water droplet after impacting on a moving wall is significant for many applications such as repelling rain droplets from a vehicle. In this paper, a water droplet impacting on a moving hydrophobic wall is studied numerically using a 3D lattice Boltzmann method (LBM). The accuracy of the present model is validated by comparing with existing correlation equations for the maximum spread factor and the contact time. It is found that the droplet spreads into an asymmetric shape after impacting on the moving wall owing to the momentum transfer from the wall to the droplet. The droplet deformation increases with the increasing of the wall velocity. Because of different bouncing behaviors of the droplet, the effect of the wall velocity on the droplet contact time varies with contact angles: the droplet contact time decreases with the increasing of the wall velocity for θ = 156°, while the droplet contact time increases with the increasing of the wall velocity for θ = 130°. It is also found that the droplet bouncing motion will be suppressed at a high wall velocity for θ = 130°. Finally, a map in terms of the Weber (We) number versus the contact angle (θ) is obtained, showing that a larger critical contact angle is required for droplet rebounding from a moving wall. This work provides a guidance that a moving wall needs to be more hydrophobic than a stationary wall to repel water droplets. Graphical abstract

The lateral thermal plume discharge in the deep cross-flow has been investigated by numerical simulation using the open-source Open FOAM code. Adaptive mesh refinement method has been applied to reduce the computational cost. The numerical simulation results show a good agreement with the previous experimental data. Three-dimensional structures illustrated by instantaneous velocity fields indicate shear layer roll-up vortices around the discharged plume. As the densimetric Froude number (Fr0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\hbox {Fr}_{0})$$\end{document} is increased, the buoyant plume penetrates more in the depth of the channel and fewer spreads in the free surface. As the Fr0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Fr}_{{0}}$$\end{document} decreases, the coherent structures become increasingly weak, with the faster breakdown of the shear layer roll-up. The instantaneous temperature contours near the free surface exhibit a vortex shedding phenomenon. Three-dimensional streamlines based on the instantaneous velocity vectors illustrate a swirl flow pattern downstream of the main channel. Increasing the Fr0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Fr}_{{0}}$$\end{document} results in weakening the swirl flow around the discharged jet core. The mixing ability of discharged plume is investigated by temporal mixing deficiency (TMD) and spatial mixing deficiency (SMD) indices. The statistical analysis of the TMD and SMD reveals the direct relationship between the mixing efficiency and reduced gravity. Graphic abstract

We consider estimation and control of the cylinder wake at low Reynolds numbers. A particular focus is on the development of efficient numerical algorithms to design optimal linear feedback controllers when there are many inputs (disturbances applied everywhere) and many outputs (perturbations measured everywhere). We propose a resolvent-based iterative algorithm to perform (i) optimal estimation of the flow using a limited number of sensors, and (ii) optimal control of the flow when the entire flow is known but only a limited number of actuators are available for control. The method takes advantage of the low-rank characteristics of the cylinder wake and provides full-dimensional solutions by implementing a terminal reduction technique based on resolvent analysis. Optimal feedback controllers are also obtained by combining the solutions of the estimation and control problems. We show that the performance of the estimators and controllers converges to the true global optima, indicating that the important physical mechanisms for estimation and control are of low rank. Graphic abstract

This paper investigates the problem of electrophoretic motion of a polyelectrolyte capsule with a porous arbitrary charged conducting shell in an electrolyte (of the same type as the one inside the capsule’s cavity) under the action of an external electric field. The corresponding boundary value problem for the velocity components and pressure in the case of small electrical potentials is analytically solved in quadratures. The solution is analyzed numerically for different values of the specific permeability of the capsule, and the thickness of the porous and the electric double layers. The minimum of electrophoretic velocity dependence on the inverse permeability of the porous layer has been found. It is shown that the electrophoretic mobility decreases upon decrease in the conductivity of the material constituting the porous layer. This means that a dielectric capsule can be used for electrophoresis as well. Moreover, its velocity will be even greater than that of a conducting capsule, all other conditions being equal. Graphic abstract

Machine learning (ML) techniques for turbulence modeling are becoming an important tool to build the bridge between low-cost-low-accurate turbulence models (like RANS) and high-cost-high-accurate procedures to represent turbulence (like DNS). In recent studies, however, it was observed that the DNS data for the Reynolds stress tensor (RST) do not satisfactorily recover the mean velocity field. This fact has two rooting sources, the lack of convergence of statistical fields, and the ill-conditioning of the RANS equations. To address these two aspects, we employ two remedies in the turbulent flow through a square duct (SD). On the one side, we applied symmetry filters on the flow data to provide more converged statistical quantities. On the other side, we contrast the traditional approach where the model target is the Reynolds stress tensor with a recent approach where the Reynolds force vector (RFV) is the target. We also provide a comparison between two ML techniques commonly used in the literature, neural network and random forest, in an invariant formulation recently proposed. The results have shown that there is a direct relation between the convergence of DNS data and the performance of data-driven turbulence models. The models obtained from symmetrical data presented lower error propagation to the mean velocity field. The Reynolds force vector is shown to be a target that can produce more accurate results, corroborating recent findings of the literature. The performance of the two ML techniques was equivalent, with small differences depending on the target quantity (RST or RFV) and the velocity component considered (main flow or secondary flow).

The paper is concerned with a doubly infinite vortex array introduced by Weihs (Nature 241:290–291, 1973; in: Wu, Brokaw, Brennen (eds) Swimming and flying in nature, Vol. 2, Plenum Press, New York, 1975) as a model problem employed in order to understand the hydrodynamic and energetic benefits of fish schooling. Weihs considered two different ‘modes’ of swimming: one where the fish swim in anti-phase and one where they swim in phase. The stability properties for the vortex array corresponding to the anti-phase mode of swimming are well understood; but this is not the case for the in-phase mode. A normal mode analysis of perturbations applied to the corresponding vortex array is carried out. The array is found to be always unstable when subjected to general perturbations, but stable solutions exist if all consecutive vortex streets are subjected to the same perturbation.

In this work, using a mathematical model and numerical simulation, we investigate the effect of time-dependent evaporation rates on stripe formation inside containers. This pattern formation is driven by the coffee-ring effect. The coffee particles inside a container move according to random walk and under the gravitational force. Because of the time-dependent evaporation rate, we can observe stripe formation inside a container after evaporation of the coffee particle-laden liquid. Various numerical experiments are performed to demonstrate the proposed model can simulate the stripe formation in a container.

Most numerical studies of airflow resistance in the lungs have presented flow dynamics in different stenosis tracheobronchial airway generations. However, not much is known about the airflow characteristics in conditions where stenoses are present at both the airflow inlet and outlet regions, especially in truncated tracheobronchial airway models. In this study, we constructed a healthy adult airway model and two stenosed airway models made up of generation six to nine (G6-G9). Computational Fluid Dynamics (CFD) simulations were performed in all airway models to investigate the airflow characteristics. We made comparison between the inlet and outlet air flow dynamics in the healthy and stenosis models during inspiration and expiration to understand if airflow resistance was higher in the upper or lower stenosis airway. Airflow velocity and wall shear stress were greater in the upper airway (G6) for both inspiration and expiration, while wall pressure was highest in the lower airway (G9). The airflow velocities in obstructed upper tracheobronchial airways during inspiration and expiration were about 3.84 m/ sand 1.49 m/s greater than in the lower airways during same respiratory cycle, respectively. The quantitative information in this study shows that stenosis at the lower airway generation has more effect on respiratory airflow than in the upper airway generations.

The nonlinear development of Görtler instability over a concave surface gives rise to a highly distorted inflectional flow field in the boundary layer that exhibits strong velocity gradients in the spanwise direction as well as in the wall-normal direction. Such a flow field is susceptible to strong, high frequency secondary instability that can lead to the onset of transition. The present numerical study uses direct numerical simulations and linear secondary instability theory to investigate finite amplitude Görtler vortices and their secondary instability characteristics, respectively, in a hypersonic flow over an axisymmetric cone with a concave aft body. The Görtler modes are excited via azimuthally periodic deformations of the surface geometry and, hence, are fully realizable. For sufficiently small initial amplitudes, the computed growth of the roughness- induced Görtler vortices is shown to agree with the predictions of optimal growth theory. Earlier work on nonlinear Görtler vortices had focused on vortex structures with intermediate amplitudes that resembled bell shaped structures, unlike the mushroom structures with thin stems encountered in lower speed flows. The present results corroborate the findings of other recent studies that fully developed mushroom structures can also exist in the hypersonic regime when the Görtler vortex amplitude is sufficiently large. Computations also reveal that the dominant modes of secondary instability correspond to an antisymmetric “stem” mode associated with the strong, nearly wall-normal shear layers bounding the stem underneath the mushroom structure. The dominant stem modes have supersonic phase velocities, resulting in acoustic radiation to the flow just outside of the boundary layer. To our knowledge, this is the first work documenting the existence of supersonic secondary instabilities in the context of stationary Görtler modes.

Hypersonic boundary-layer flows over a circular cone at a moderate yaw angle can support strong crossflow instability away from the windward and leeward rays on the plane of symmetry. Due to the more efficient excitation of stationary crossflow vortices by surface roughness, a possible path to transition in such flows corresponds to rapid amplification of the high-frequency instabilities sustained in the presence of finite amplitude stationary crossflow vortices. This paper presents a computational analysis of crossflow instability over a 7-degree half-angle, yawed circular cone in a Mach 6 free stream. Specifically, the nonlinear evolution of an azimuthally localized crossflow vortex pattern and the linear amplification characteristics of high-frequency instabilities evolving in the presence of that pattern are described for the first time. Focusing on the azimuthally compact vortex pattern allows us to overcome significant limitations of the prior secondary instability analyses of azimuthally inhomogeneous boundary layer flows. A comparison between plane-marching parabolized stability equations and direct numerical simulations (DNS) reveals favorable agreement in regard to mode shapes, most amplified disturbance frequencies, and the N-factor evolution. In contrast, the quasiparallel predictions are found to result in a severe underprediction of the N-factors. The most amplified high-frequency instabilities are found to originate from Mack’s second mode waves sustained within the upstream region of nearly unperturbed, quasi-homogeneous boundary layer.

A high-fidelity simulation of the massively separated shock/transitional boundary layer interaction caused by a 15-degrees axisymmetrical compression ramp is performed at a free stream Mach number of 6 and a transitional Reynolds number. The chosen configuration yields a strongly multiscale dynamics of the flow as the separated region oscillates at low-frequency, and high-frequency transitional instabilities are triggered by the injection of a generic noise at the inlet of the simulation. The simulation is post-processed using Proper Orthogonal Decomposition to extract the large scale low-frequency dynamics of the recirculation region. The bubble dynamics from the simulation is then compared to the results of a global linear stability analysis about the mean flow. A critical interpretation of the eigenspectrum of the linearized Navier–Stokes operator is presented. The recirculation region dynamics is found to be dominated by two coexisting modes, a quasi-steady one that expresses itself mainly in the reattachment region and that is caused by the interaction of two self-sustained instabilities, and an unsteady one linked with the separation shock-wave and the mixing layer. The unsteady mode is driven by a feedback loop in the recirculation region, which may also be relevant for other unsteady shock-motion already documented for shock-wave/turbulent boundary layer interaction. The impact of the large-scale dynamics on the transitional one is then assessed through the numerical filtering of those low wavenumber modes; they are found to have no impact on the transitional dynamics.

Shock/turbulent-boundary-layer interactions (STBLIs) are ubiquitous in high-speed flight and propulsion applications. Experimental and computational investigations of swept, three-dimensional (3-D) interactions, which exhibit quasi-conical mean-flow symmetry in the limit of infinite span, have demonstrated key differences in unsteadiness from their analogous, two-dimensional (2-D), spanwise-homogeneous counterparts. For swept interactions, represented by the swept–fin-on-plate and swept–compression–ramp-on-plate configurations, differences associated with the separated shear layers may be traced to the intermixing of 2-D (spanwise independent) and 3-D (spanwise dependent) scaling laws for the separated mean flow. This results in a broader spectrum of unsteadiness that includes relatively lower frequencies associated with the separated shear layers in 3-D interactions. However, lower frequency ranges associated with the global “breathing” of strongly separated 2-D interactions are significantly less prominent in these simple, swept 3-D interactions. A logical extension of 3-D interaction complexity is the compound interaction formed by the merging of two simple interactions. The first objective of this work is therefore to analyze the more complex picture of the dynamics of such interactions, by considering as an exemplar, wall-resolved simulations of the double-fin-on-plate configuration. We show that in the region of interaction merging, new flow scales, changes in separation topology, and the emergence of lower-frequency phenomena are observed, whereas the dynamics of the interaction near the fin leading edges are similar to those of the simple, swept interactions. The second objective is to evolve a unified understanding of the dynamics of STBLIs associated with complex configurations relevant to actual propulsion systems, which involve the coupling between multiple shock systems and multiple flow separation and attachment events. For this, we revisit the salient aspects of scaling phenomena in a manner that aids in assimilating the double-fin flow with simpler swept interactions. The emphasis is on the influence of the underlying structure of the separated flow on the dynamics. The distinct features of the compound interactions manifest in a centerline symmetry pattern that replaces the quasi-conical symmetry of simple interactions. The primary separation displays topological closure to reveal new length scales, associated unsteadiness bands, and secondary flow separation.

With an interest in developing and studying the stability of laminar undisturbed basic-state solutions, this work is focused on accurately modeling the laminar flowfield of the boundary layer transition (BOLT) geometry under nominal and off-nominal conditions (i.e., nonzero angles of pitch and yaw). The BOLT flowfield is studied using the DPLR flow solver with MUSCL Steger–Warming fluxes using a set of five grids at different resolutions and identical grid topologies. A total of three different sets of conditions are studied: two flight conditions and one wind-tunnel-scale (33%) condition. (1) For the two sets of nominal flight operating conditions, it is found that the flow structures in the centerline region of BOLT are similar to those found in prior studies including in shape, location, and extent both vertically and spanwise, but a detailed comparison of velocity contours shows that further quantitative convergence studies are warranted. The centerline region, however, extends to at most 4 cm in semi-span at the aft end of the geometry (20% of the semi-span). Away from the centerline and where wind-tunnel-scale results have observed regions of possibly transitional behavior, the laminar flowfield converges with high accuracy. (2) For nominal wind-tunnel operating conditions, all grid resolutions simulated show good agreement in most regions as compared with prior results, with any differences falling within the scatter of existing experimental and DNS results. Aside from this focus, boundary-layer stability is examined outboard of the centerline region at nonzero pitch and yaw for a flight case, and second mode and stationary crossflow instabilities are considered. Second-mode instability is found to be locally significant at certain pitch and yaw angles particularly downstream of the swept leading edges. In addition, stationary crossflow is found to become highly amplified in significant wedges extending to the aft end of the BOLT geometry, with N-factors consistent with those found for HIFiRE-5b associated with transitional flow. The reasons for amplification of these different instabilities are also investigated from a physics-based perspective.

Accurate prediction of aerothermal surface loading is of paramount importance for the design of high speed flight vehicles. In this work, we consider the numerical solution of hypersonic flow over a double-finned geometry, representative of the inlet of an air-breathing flight vehicle, characterized by three-dimensional intersecting shock-wave/turbulent boundary-layer interaction at Mach 8.3. High Reynolds numbers ($Re_L \approx 11.6 \times 10^6$ based on free-stream conditions) and the presence of cold walls ($T_w/T_\circ \approx 0.26$) leading to large near-wall temperature gradients necessitate the use of wall-modeled large-eddy simulation (WMLES) in order to make calculations computationally tractable. The comparison of the WMLES results with experimental measurements shows good agreement in the time-averaged surface heat flux and wall pressure distributions, and the WMLES predictions show reduced errors with respect to the experimental measurements than prior RANS calculations. The favorable comparisons are obtained using a standard LES wall model based on equilibrium boundary layer approximations despite the presence of numerous non-equilibrium conditions including three dimensionality in the mean, shock-boundary layer interactions, and flow separation. We demonstrate that the use of semi-local eddy viscosity scaling (in lieu of the commonly used van Driest scaling) in the LES wall model is necessary to accurately predict the surface pressure loading and heat fluxes.

Streamwise vortices and the associated streaks evolve in boundary layers over flat or concave surfaces due to disturbances initiated upstream or triggered by the wall surface. Following the transient growth phase, the fully developed vortex structures become susceptible to inviscid secondary instabilities resulting in early transition to turbulence via ‘bursting’ processes. In high-speed boundary layers, more complications arise due to compressibility and thermal effects, which become more significant for higher Mach numbers. In this paper, we study Görtler vortices developing in high-speed boundary layers using the boundary region equations (BRE) formalism, which we solve using an efficient numerical algorithm. Streaks are excited using a small transpiration velocity at the wall. Our BRE-based algorithm is found to be superior to direct numerical simulation (DNS) and ad hoc nonlinear parabolized stability equation (PSE) models. BRE solutions are less computationally costly than a full DNS and have a more rigorous theoretical foundation than PSE-based models. For example, the full development of a Görtler vortex system in high-speed boundary layers can be predicted in a matter of minutes using a single processor via the BRE approach. This substantial reduction in calculation time is one of the major achievements of this work. We show, among other things, that it allows investigation into feedback control in reasonable total computational times. We investigate the development of the Görtler vortex system via the BRE solution with feedback control parametrically at various freestream Mach numbers M∞ and spanwise separations λ of the inflow disturbances.

The effects of crossflow on the interaction between an impinging shock wave and a high-speed turbulent boundary layer are investigated using direct numerical simulations of statistically two-dimensional, three-component flow. The leading-order effect of crossflow is increased size and strength of the separation bubble, with upstream and downstream displacement of the separation and reattachment points, respectively. This effect is traced to retarded growth of the shear layer surrounding the separation bubble, with associated reduction of the turbulent shear stress. Genuinely, three-dimensional effects are observed in the interaction and in the downstream recovery zone, with mean flow direction changing both in the longitudinal and wall-normal directions. Three-dimensional, non-equilibrium effects yield substantial misalignment between turbulent stresses and mean strain rate, thus providing a challenging benchmark for the development and validation of turbulence models for compressible flows.

This article wanna discuss the very beginning point of arithmetic that no one notices in the frame of the noncommutative principle, multiplication, division, and factorial under boundary conditions. Their explanation was carried out based on the Early Mongolian calculus. New evidence has been found: Some mistakes of arithmetic influence in all sciences.

Sustained flight at hypersonic speeds is characterized by high pressure and aerothermal loads imposed on the structure of the aerodynamic vehicle. A consequence of lightening the structural design permits fluid–structure interaction phenomena that can significantly alter the flow and initiate unsteady structural responses. We investigate the coupling between high-speed laminar boundary layer flows over a mechanically compliant panel and analyze the dynamic system response of the coupled system to boundary layer instabilities by means of local convective linear stability analysis. The resulting non-dimensional interaction parameters describing the compliant system are shown to affect the boundary layer instabilities in the infinitely thin panel limit, and the transition from the rigid limit is described by two distinctly different responses: (a) a piston-like, one-dimensional panel deflection, or (b) a synchronization with flexural waves. Compliance is shown to non-monotonically change convective wave growth rates and induce uncertainty in the integrated N-factors.

Linear stability analysis is performed using a combination of two-dimensional direct simulation Monte Carlo (DSMC) (Bird in Molecular gas dynamics and the direct simulation of gas flows, Oxford University Press, Oxford, 1994) method for the computation of the basic state and solution of the pertinent eigenvalue problem, as applied to the canonical boundary layer on a semi-infinite flat plate. Three different gases are monitored, namely nitrogen, argon and air, the latter as a mixture of 79% N2 and 21% O2 at a range of free-stream Mach numbers corresponding to flight at an altitude of ∼55km. A neural network has been utilized to predict and smooth the raw DSMC data; the steady laminar profiles obtained are in very good agreement with those computed by (self-similar) boundary layer theory, under isothermal or adiabatic wall conditions, subject to the appropriate slip corrections computed in the DSMC method (Beskok and Karniadakis in Microscale Thermophys Eng 3(1):43–77, 1999; Beskok et al. in J Fluids Eng 118(3):448–456, 1996). The leading eigenmode results pertaining to the unsmoothed DSMC profiles are compared against those of the classic boundary layer theory (Mack in Boundary layer stability theory, Jet Propulsion Laboratory, Pasadena, 1969). Small quantitative, but no significant qualitative differences between the results of the two classes of steady base flows have been found at all parameters examined. The frequencies of the leading eigenmodes at all conditions examined are practically identical, while perturbations corresponding to the DSMC profiles are found to be systematically more damped than their counterparts arising in the boundary layer at the conditions examined, when the correct velocity slip and temperature jump boundary conditions are imposed in the base flow profiles; by contrast, when the classic no-slip boundary conditions are used, less damped/more unstable profiles are obtained, which would lead the flow to earlier transition. On the other hand, the DSMC profiles smoothed by the neural network are marginally more stable than their unsmoothed counterparts. A vortex generator (VG) introduced into the boundary layer downstream of the leading edge and pulsed at rather large momentum coefficient, Cμ=0.27, and scaled frequency F+≈0.98 (Greenblatt and Wygnanski in Prog Aerosp Sci 36:487–545, 2000), is used to generate linear perturbations that decay along the plate, as expected from the low value of the Reynolds number, Reδ=290, in this numerical experiment. The damping rate diminishes monotonically as the VG is placed at successive downstream positions along the plate. The characteristics of the oscillation generated in the boundary layer are predicted accurately by linear stability analysis of the undisturbed profile at the location of VG placement. Most interestingly, the effect of the generated perturbation is felt well outside of the boundary layer, generating oscillations of the leading edge shock that synchronize with linear perturbations inside the boundary layer.

The goal of this work is to build up the capability of quasi-particle simulation (QuiPS), a novel flow solver, such that it can adequately model the rarefied portion of an atmospheric reentry trajectory. Direct simulation Monte Carlo (DSMC) is the conventional solver for such conditions, but struggles to resolve transient flows, trace species, and high-level internal energy states due to stochastic noise. Quasi-particle simulation (QuiPS) is a novel Boltzmann solver that describes a system with a discretized, truncated velocity distribution function. The resulting fixed-velocity, variable weight quasi-particles enable smooth variation of macroscopic properties. The distribution function description enables the use of a variance-reduced collision model, greatly minimizing expense near equilibrium. This work presents the addition of a neutral air chemistry model to QuiPS and some demonstrative 0D simulations. The explicit representation of internal distributions in QuiPS reveals some of the flaws in existing physics models. Variance reduction, a key feature of QuiPS, can greatly reduce expense of multi-dimensional calculations, but is only cheaper when the gas composition is near chemical equilibrium.

We present direct molecular simulations (DMSs) of rovibrational excitation and dissociation of oxygen across normal shock waves. These are the first atomistic simulations of normal shock waves to rely exclusively on ab initio potential energy surfaces to describe the full collision dynamics (accounting for elastic, inelastic and reactive processes) in a dilute gas mixture of molecular and atomic oxygen. The simulated setup aims to reproduce two of the experimental conditions in the shock tube tests of Ibraguimova et al. (J Chem Phys 139:034317, 2013). We compare mixture composition and vibrational temperatures extracted from our simulations with those inferred from the shock tube tests and observe good agreement. In addition to this, we report macroscopic moments due to dissipative transport across the shock, i.e., species diffusion fluxes, viscous stresses and heat flux. Furthermore, we examine the distributions of vibrational and total internal energy of the $$\hbox {O}_{2}$$ molecules at several locations across the shock wave. We are able to follow the gradual transition from pre-shock to post-shock population distributions and, in both cases studied, find depletion of the high-energy tail of the internal energy distributions due to preferential dissociation from states close to the dissociation energy $$D_0$$. Finally, we extract molecular velocity distributions functions (VDF) of $$\hbox {O}_{2}$$ and O at selected locations across the shock to delimit the region where continuum breakdown occurs.

Traditional stability tools have done much in the last few decades to demonstrate the significance of modal instabilities as a pathway for laminar to turbulent transition in hypersonic flows, but are less effective at predicting transition in flows with significant streamwise variation and strong shock waves. Because of this, most stability analyses over blunt cones tend to focus on the growth of instabilities in regions of the flow away from the blunt tip and downstream of any strong shock waves. We develop a new shock-kinematic boundary condition which is compatible with both the finite-volume method and input–output analysis. This boundary condition enables analysis of the receptivity of blunt cones to disturbances in the free stream by careful treatment of linear interactions of small disturbances with the shock. In particular, a Mach 5.8 flow over a 7$$^{\circ }$$ half-angle cone with a 0.15" nose radius is analyzed, showing significant amplification of disturbances along the cone frustum in a 5–15 kHz bandwidth due to the destabilization of a slow acoustic boundary layer mode, and significant amplification of entropy layer instabilities between 100 and 180 kHz due to rotation/deceleration of entropy/vorticity waves. These mechanisms are receptive to free-stream disturbances in very localized positions upstream of the bow shock.

Linear instability of high-speed boundary layers is routinely examined assuming quiescent edge conditions, without reference to the internal structure of shocks or to instabilities potentially generated in them. Our recent work has shown that the kinetically modeled internal nonequilibrium zone of straight shocks away from solid boundaries exhibits low-frequency molecular fluctuations. The presence of the dominant low frequencies observed using the direct simulation Monte Carlo (DSMC) method has been explained as a consequence of the well-known bimodal probability density function (PDF) of the energy of particles inside a shock. Here, PDFs of particle energies are derived in the upstream and downstream equilibrium regions, as well as inside shocks, and it is shown for the first time that they have the form of the noncentral Chi-squared (NCCS) distributions. A linear correlation is proposed to relate the change in the shape of the analytical PDFs at a specified upstream number density and temperature as a function of Mach number, within the range $$3 \le M \le 10$$, with the DSMC-derived average characteristic low-frequency of shocks, as computed in our earlier work. At a given Mach number $$M=7.2$$ and upstream number density $$n_1=10^{22}\,\hbox {m}^{-3}$$, it is shown that the variation in DSMC-derived low frequencies is correlated with the change in most-probable-speed inside shocks at the location of maximum bulk velocity gradient for upstream translational temperature in the range $$\sim 90 \le T_{tr,1}/(K) \le 1420$$. Using the proposed linear functions, average low frequencies are estimated within the examined ranges of Mach number and input temperature and a semi-empirical relationship is derived to predict low-frequency oscillations in shocks. Our model can be used to provide realistic physics-based boundary conditions in receptivity and linear stability analysis studies of laminar-turbulent transition in high-speed flows.

Mack (1977) criticized methods referring to a single frequency perturbation for correlation of transition prediction because the external disturbance source (like free stream turbulence) should have a broadband spectrum. Delta-correlated perturbations are characterized by the mean square of physical amplitude, which is expressed as a double integral of the power spectral density in frequency and the spanwise wave number. It is suggested to evaluate this integral asymptotically. The results obtained using the asymptotic method and direct numerical integration are compared with ad hoc approaches for high speed and moderate supersonic boundary layers. This allows us to suggest recommendations on rational usage of the amplitude method with avoiding unconfirmed simplifications while reducing the computational effort to the level affordable for engineering practice.

Stochastic subgrid-scale parametrizations aim to incorporate effects of unresolved processes in an effective model by sampling from a distribution usually described in terms of resolved modes. This is an active research area in fluid dynamics where processes evolve on a wide range of spatial and temporal scales. We propose a data-driven framework where resolved modes are defined as local spatial averages and deviations from these averages are the unresolved degrees of freedom. The proposed approach is applicable to a wide range of finite volume and finite difference numerical schemes commonly used to discretize many realistic problems in fluid dynamics. In this study, we evaluate the performance of conditional generative adversarial network (GAN) in parametrizing subgrid-scale effects in a finite difference discretization of stochastically forced Burgers equation. We train a Wasserstein GAN (WGAN) conditioned on the resolved variables to learn the distribution of the subgrid flux and, thus, represent the effect of unresolved scales. The resulting WGAN is then used in an effective model to reproduce the statistical features of resolved modes. We demonstrate that various stationary statistical quantities such as spectrum, moments and autocorrelation are well approximated by this effective model.

Päschke et al. (J Fluid Mech, 2012) studied the nonlinear dynamics of strongly tilted vortices subject to asymmetric diabatic heating by asymptotic methods. They found, inter alia , that an azimuthal Fourier mode 1 heating pattern can intensify or attenuate such a vortex depending on the relative orientation of the tilt and the heating asymmetries. The theory originally addressed the gradient wind regime which, asymptotically speaking, corresponds to vortex Rossby numbers of order unity in the limit. Formally, this restricts the applicability of the theory to rather weak vortices. It is shown below that said theory is, in contrast, uniformly valid for vanishing Coriolis parameter and thus applicable to vortices up to low hurricane strengths. An extended discussion of the asymptotics as regards their physical interpretation and their implications for the overall vortex dynamics is also provided in this context. The paper’s second contribution is a series of three-dimensional numerical simulations examining the effect of different orientations of dipolar diabatic heating on idealized tropical cyclones. Comparisons with numerical solutions of the asymptotic equations yield evidence that supports the original theoretical predictions of Päschke et al. In addition, the influence of asymmetric diabatic heating on the time evolution of the vortex centerline is further analyzed, and a steering mechanism that depends on the orientation of the heating dipole is revealed. Finally, the steering mechanism is traced back to the correlation of dipolar perturbations of potential temperature, induced by the vortex tilt, and vertical velocity, for which diabatic heating not necessarily needs to be responsible, but which may have other origins.

Fluid flow around a random distribution of stationary spherical particles is a problem of substantial importance in the study of dispersed multiphase flows. In this paper, we present a machine learning methodology using generative adversarial network framework and convolutional neural network architecture to recreate particle-resolved fluid flow around a random distribution of monodispersed particles. The model was applied to various Reynolds number and particle volume fraction combinations spanning over a range of [2.69, 172.96] and [0.11, 0.45], respectively. Test performance of the model for the studied cases is very promising.

In this paper, we consider a viscous droplet migrating in a viscous fluid of a different viscosity. Further, we assume that the surface of the droplet is partially contaminated with a stagnant layer of surfactant (surface active agent which reduces the interfacial tension). We analyze the effects of the following phenomena associated with the thermocapillary migration of a droplet in a transient Stokes flow. The first is the influence of surfactant cap for an arbitrary cap angle which is partially coated on the droplet surface for both high and low surface Péclet number cases. The second is the influence of the energy changes associated with stretching and shrinkage of the interfacial area elements, when the droplet is in motion. It can be noted that for the vanishing cap angle, both high and low surface Péclet number limits reduce to the case of a pure thermocapillary migration of a droplet in a transient Stokes flow. For a given ambient flow, the migration of the droplet is controlled by the magnitude of the ambient velocity and the surface tension variations due to temperature and surfactant concentration. In particular, these surface tension variations balance the tangential stress balance. Considering axisymmetric transient Stokes flow, we obtain analytical solutions in two limiting cases, namely low and high surface Péclet number. This work considers linear variation of interfacial tension on both thermal and surfactant gradients. The main contribution is pertaining to the capillary drift and the corresponding surfactant transport on the droplet for an axisymmetric hydrodynamic as well as thermal and surfactant fields. We have analyzed the level curves corresponding to stream function and temperature fields, i.e., streamlines and isotherms for various parameters in order to develop a realistic picture of the migration pattern and the influence of thermal fields. We observe that the streamlines in the vicinity of rear end of the droplet show asymmetry due to the surfactant accumulation at that region. Increasing cap angle breaks the symmetry of the induced stream. It is seen that increasing values of nondimensional parameter that accounts for the stretching and shrinkage of the droplet surface immobilizes the surface and offers retardation to the migrating droplet. The variation of migration velocity with time suggests a control mechanism for the migration of the drop under external/surface gradients and hence may serve as a useful tool in applications like targeted drug delivery systems.

The use of spectral proper orthogonal decomposition (SPOD) to construct low-order models for broadband turbulent flows is explored. The choice of SPOD modes as basis vectors is motivated by their optimality and space-time coherence properties for statistically stationary flows. This work follows the modeling paradigm that complex nonlinear fluid dynamics can be approximated as stochastically forced linear systems. The proposed stochastic two-level SPOD-Galerkin model governs a compound state consisting of the modal expansion coefficients and forcing coefficients. In the first level, the modal expansion coefficients are advanced by the forced linearized Navier-Stokes operator under the linear time-invariant assumption. The second level governs the forcing coefficients, which compensate for the offset between the linear approximation and the true state. At this level, least squares regression is used to achieve closure by modeling nonlinear interactions between modes. The statistics of the remaining residue are used to construct a dewhitening filter that facilitates the use of white noise to drive the model. If the data residue is used as the sole input, the model accurately recovers the original flow trajectory for all times. If the residue is modeled as stochastic input, then the model generates surrogate data that accurately reproduces the second-order statistics and dynamics of the original data. The stochastic model uncertainty, predictability, and stability are quantified analytically and through Monte Carlo simulations. The model is demonstrated on large eddy simulation data of a turbulent jet at Mach number $$M=0.9$$ and Reynolds number $$\mathrm {Re}_D\approx 10^6$$.

A Taylor–Galerkin finite element time marching scheme is derived to numerically simulate the flow of a compressible and nonisothermal viscoelastic liquid between eccentrically rotating cylinders. Numerical approximations to the governing flow and constitutive equations are computed over a custom refined unstructured grid of piecewise linear Galerkin finite elements. An original extension to the DEVSS formulation for compressible fluids is introduced to stabilise solutions of the discrete problem. The predictions of two models: the extended White–Metzner and FENE-P-MP are presented. Comparisons between the torque and load bearing capacity predicted by both models are made over a range of viscoelastic parameters. The results highlight the significant and interacting effects of elasticity and compressibility on journal torque and resultant load, and the stability of the journal bearing system.

A Correction to this paper has been published: 10.1007/s00162-021-00578-8

Optimal sensor placement for fluid flows is an important and challenging problem. In this study, we propose a completely data-driven and computationally efficient method for sensor placement. We use adjoint-based gradient descent to find the sensor location that minimizes the trace of an approximation of the estimation error covariance matrix. The proposed methodology can be used in conjunction with any reduced-order modeling technique that provides a linear approximation of the fluid dynamics. Moreover, the objective function can be augmented for different applications, which we illustrate by proposing a control-oriented objective function. We demonstrate the performance of our method for reconstruction and prediction of the complex linearized Ginzburg–Landau equation in the globally unstable regime. We also construct a low-dimensional observer-based feedback controller for the flow over an inclined flat plate that is able to suppress the wake vortex shedding in the presence of system and measurement noise.

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