The propagation of linear waves in non-ideal compressible fluids plays a crucial role in numerous physical and engineering applications, particularly in the study of instabilities, aeroacoustics and turbulence modelling. This work investigates linear waves in viscous and heat-conducting non-ideal compressible fluids, modelled by the Navier–Stokes–Fourier equations and a fully arbitrary equation of state (EOS). The linearised governing equations are derived to analyse the dispersion relations when the EOS differs from that of an ideal gas. Special attention is given to the influence of non-ideal effects and various dimensionless numbers on wave propagation speed and attenuation. By extending classical results from Kovásznay (1953 J. Aeronaut. Sci. vol. 20, no. 10, pp. 657–674) and Chu (1965 Acta Mech. vol. 1, no. 3, pp. 215–234) obtained under the ideal gas assumption, this study highlights the modifications introduced by arbitrary EOSs to the linear wave dynamics in non-ideal compressible flows. This work paves the path for an improved understanding and modelling of wave propagation, turbulence and linear stability in arbitrary viscous and heat-conducting fluids.

We provide a rigorous analysis of the self-similar solution of the temporal turbulent boundary layer, recently proposed by Biau (2023 Comput. Fluids 254, 105795), in which a body force is used to maintain a statistically steady turbulent boundary layer with periodic boundary conditions in the streamwise direction. We derive explicit expressions for the forcing amplitudes which can maintain such flows, and identify those which can hold either the displacement thickness or the momentum thickness equal to unity. This opens the door to the first main result of the paper, which is to prove upper bounds on skin friction for the temporal turbulent boundary layer. We use the Constantin–Doering–Hopf bounding method to show, rigorously, that the skin-friction coefficient for periodic turbulent boundary layer flows is bounded above by a uniform constant which decreases asymptotically with Reynolds number. This asymptotic behaviour is within a logarithmic correction of well-known empirical scaling laws for skin friction. This gives the first evidence, applicable at asymptotically high Reynolds numbers, to suggest that Biau’s self-similar solution of the temporal turbulent boundary layer exhibits statistical similarities with canonical, spatially evolving, boundary layers. Furthermore, we show how the identified forcing formula implies an alternative, and simpler, numerical implementation of periodic boundary layer flows. We give a detailed numerical study of this scheme presenting direct numerical simulations up to a momentum Reynolds number of $\textit{Re}_\theta = 2000$ and implicit large-eddy simulations up to $\textit{Re}_\theta = 8300$, and show that these results compare well with data from canonical spatially evolving boundary layers at equivalent Reynolds numbers.

Equilibrium, travelling-wave and periodic-orbit solutions of the Navier–Stokes equations provide a promising avenue for investigating the structure, dynamics and statistics of transitional flows. Many such invariant solutions have been computed for wall-bounded shear flows, including plane Couette, plane Poiseuille and pipe flow. However, the organisation of invariant solutions is not well understood. In this paper we focus on the role of symmetries in the organisation and computation of invariant solutions of plane Poiseuille flow. We show that enforcing symmetries while computing invariant solutions increases the efficiency of the numerical methods, and that redundancies between search spaces can be eliminated by consideration of equivalence relations between symmetry subgroups. We determine all symmetry subgroups of plane Poiseuille flow in a doubly periodic domain up to translations by half the periodic lengths and classify the subgroups into equivalence classes, each of which represents a physically distinct set of symmetries and an associated set of physically distinct invariant solutions. We calculate fifteen new travelling waves of plane Poiseuille flow in seven distinct symmetry groups and discuss their relevance to the dynamics of transitional turbulence. We present a few examples of subgroups with fractional shifts other than half the periodic lengths and one travelling-wave solution whose symmetry involves shifts by one third of the periodic lengths. We conclude with a discussion and some open questions about the role of symmetry in the behaviour of shear flows.

This work experimentally explores the alignment of the vorticity vector and the strain-rate tensor eigenvectors at locations of extreme upscale and downscale energy transfer. We show that the turbulent von Kármán flow displays vorticity–strain alignment behaviour across a large range of Reynolds numbers, which is very similar to previous studies on homogeneous, isotropic turbulence. We observe that this behaviour is amplified for the largest downscale energy transfer events, which tend to be associated with sheet-like geometries. These events are also shown to have characteristics previously associated with high flow field nonlinearity and singularities. In contrast, the largest upscale energy transfer events display very different structures which showcase a strong preference for vortex compression. Notably, in both cases we find that these trends are strengthened as the probed scales approach the Kolmogorov scale. We then show further evidence for the argument that strain self-amplification is the most salient feature in characterising the cascade direction. Finally, we identify possible invariant behaviour for the largest energy transfer events, even at scales near the Kolmogorov scale.

Over the past few decades, numerous N-phase incompressible diffuse-interface flow models with non-matching densities have been proposed. Despite aiming to describe the same physics, these models are generally distinct, and an overarching modelling framework is absent. This paper provides a unified framework for N-phase incompressible Navier–Stokes Cahn–Hilliard Allen–Cahn mixture models with a single momentum equation. The framework emerges naturally from continuum mixture theory, exhibits an energy-dissipative structure, and is invariant to the choice of fundamental variables. This opens the door to exploring connections between existing N-phase models and facilitates the computation of N-phase flow models rooted in continuum mixture theory.

We study the effect of acceleration and deceleration on the stability of channel flows. To do so, we derive an exact solution for laminar profiles of channel flows with an arbitrary, time-varying wall motion and pressure gradient. This solution then allows us to investigate the stability of any unsteady channel flow. In particular, we restrict our investigation to the non-normal growth of perturbations about time-varying base flows with exponentially decaying acceleration and deceleration, with comparisons to growth about a constant base flow (i.e. the time-invariant simple shear or parabolic profile). We apply this acceleration and deceleration through the velocity of the walls and through the flow rate. For accelerating base flows, perturbations never grow larger than perturbations about a constant base flow, while decelerating flows show massive amplification of perturbations – at a Reynolds number of $500$, properly timed perturbations about the decelerating base flow grow $ {O}(10^5)$ times larger than perturbations grow about a constant base flow. This amplification increases as we raise the rate of deceleration and the Reynolds number. We find that this amplification arises due to a transition from spanwise perturbations leading to the largest amplification to streamwise perturbations leading to the largest amplification that only occurs in the decelerating base flow. By evolving the optimal perturbations through the linearized equations of motion, we reveal that the decelerating base flow achieves this massive amplification through the Orr mechanism, or the down-gradient Reynolds stress mechanism, which accelerating and constant base flows cannot maintain.

In this paper, we study the hydrostatic approximation for the Navier-Stokes system in a thin domain. When we have convex initial data with Gevrey regularity of optimal index $\frac {3}{2}$ in the x variable and Sobolev regularity in the y variable, we justify the limit from the anisotropic Navier-Stokes system to the hydrostatic Navier-Stokes/Prandtl system. Due to our method in the paper being independent of $\varepsilon $, by the same argument, we also obtain the well-posedness of the hydrostatic Navier-Stokes/Prandtl system in the optimal Gevrey space. Our results improve upon the Gevrey index of $\frac {9}{8}$ found in [15, 35].

Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the Navier–Stokes equations, have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purposes, a sustained effort has been devoted to obtaining an effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the renormalisation group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier–Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier–Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in this JFM Perspectives article a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, we expound the derivation of the spatio-temporal behaviour of $n$-point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.

We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi =\{x=(x_{h}, x_3)\ \vert \vert x_{h} \vert \gt 1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi )$ satisfying the decay condition of the swirl component $ru^{\theta }_{0}\in L^{\infty }(\Pi )$.

We present a moving-least-square immersed boundary method for solving viscous incompressible flow involving deformable and rigid boundaries on a uniform Cartesian grid. For rigid boundaries, noslip conditions at the rigid interfaces are enforced using the immersed-boundary direct-forcing method. We propose a reconstruction approach that utilizes moving least squares (MLS) method to reconstruct the velocity at the forcing points in the vicinity of the rigid boundaries. For deformable boundaries, MLS method is employed to construct the interpolation and distribution operators for the immersed boundary points in the vicinity of the rigid boundaries instead of using discrete delta functions. The MLS approach allows us to avoid distributing the Lagrangian forces into the solid domains as well as to avoid using the velocity of points inside the solid domains to compute the velocity of the deformable boundaries. The present numerical technique has been validated by several examples including a Poiseuille flow in a tube, deformations of elastic capsules in shear flow and dynamics of red-blood cell in microfluidic devices.

In this paper, we study an exponential time differencing method for solving the gauge system of incompressible viscous flows governed by Stokes or Navier-Stokes equations. The momentum equation is decoupled from the kinematic equation at a discrete level and is then solved by exponential time stepping multistep schemes in our approach. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes problem. We also present a compact representation of the algorithm for problems on rectangular domains, which makes FFT-based solvers available for the resulting fully discretized system. Various numerical experiments in two and three dimensional spaces are carried out to demonstrate the accuracy and stability of the proposed method.

By combining the characteristic method and the local discontinuous Galerkin method with carefully constructing numerical fluxes, variational formulations are established for time-dependent incompressible Navier-Stokes equations in ℝ 2. The nonlinear stability is proved for the proposed symmetric variational formulation. Moreover, for general triangulations the priori estimates for the L 2–norm of the errors in both velocity and pressure are derived. Some numerical experiments are performed to verify theoretical results.

In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

In this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the H 1 norm and the pressure in the L 2 norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.

Two-grid finite element methods for the steady Navier-Stokes/Darcy model are considered. Stability and optimal error estimates in the H1-norm for velocity and piezometric approximations and the L2-norm for pressure are established under mesh sizes satisfying h = H2. A modified decoupled and linearised two-grid algorithm is developed, together with some associated optimal error estimates. Our method and results extend and improve an earlier investigation, and some numerical computations illustrate the efficiency and effectiveness of the new algorithm.

A new methodology via using an adaptive fuzzy algorithm to obtain solutions of “Two-dimensional Navier-Stokes equations” (2-D NSE) is presented in this investigation. The design objective is to find two fuzzy solutions to satisfy precisely the 2-D NSE frequently encountered in practical applications. In this study, a rough fuzzy solution is formulated with adjustable parameters firstly, and then, a set of adaptive laws for optimally tuning the free parameters in the consequent parts of the proposed fuzzy solutions are derived from minimizing an error cost function which is the square summation of approximation errors of boundary conditions, continuum equation and Navier-Stokes equations. In addition, elegant approximated error bounds between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors have also been proven. Furthermore, the error equations in mesh points can be proven to converge to zero for the 2-D NSE with two sufficient conditions.