Fluid dynamics in intrinsically curved geometries is encountered in many physical systems in nature, ranging from microscopic bio-membranes all the way up to general relativity at cosmological scales. Despite the diversity of applications, all of these systems share a common feature: the free motion of particles is affected by inertial forces originating from the curvature of the embedding space. Here we reveal a fundamental process underlying fluid dynamics in curved space: the free motion of fluids, in the complete absence of solid walls or obstacles, exhibits loss of energy due exclusively to the intrinsic curvature of space. We find that local sources of curvature generate viscous stresses as a result of the inertial forces. The curvature-induced viscous forces are shown to cause hitherto unnoticed and yet appreciable energy dissipation, which might play a significant role for a variety of physical systems involving fluid dynamics in curved spaces.
We investigate Poiseuille channel flow through intrinsically curved media, equipped with localized metric perturbations. To this end, we study the flux of a fluid driven through the curved channel in dependence of the spatial deformation, characterized by the parameters of the metric perturbations (amplitude, range and density). We find that the flux depends only on a specific combination of parameters, which we identify as the average metric perturbation, and derive a universal flux law for the Poiseuille flow. For the purpose of this study, we have improved and validated our recently develop…
Incompressible fluids on curved surfaces are considered with respect to the interplay between topology, geometry and fluid properties using a surface vorticity-stream function formulation, which is solved using parametric finite elements. Motivated by designed examples for superfluids, we consider the influence of a geometric potential on vortices for fluids with finite viscosity and show numerical examples in which a change in the geometry is used to manipulate the flow field.
We numerically investigate the effect of entrance condition on the spatial and temporal evolution of multiple three-dimensional vortex pairs and wall shear stress distribution in a curved artery model. We perform this study using a Newtonian blood-analog fluid subjected to a pulsatile flow with two inflow conditions. The first flow condition is fully developed while the second condition is undeveloped (i.e. uniform). We discuss the connection along the axial direction between regions of organized vorticity observed at various cross-sections of the model and compare results between the differen…
We have found that the relation between the flow through campylotic (generically curved) media, consisting of randomly located curvature perturbations, and the average Ricci scalar of the system exhibits two distinct functional expressions (hysteresis), depending on whether the typical spatial extent of the curvature perturbation lies above or below the critical value maximizing the overall Ricci curvature. Furthermore, the flow through such systems as a function of the number of curvature perturbations presents a sublinear behavior for large concentrations due to the interference between curv…
We study the Dean instability in curved channels using the lattice Boltzmann model for generalized metrics. For this purpose, we first improve and validate the method by measuring the critical Dean number at the transition from laminar to vortex flow for a streamwise curved rectangular channel, obtaining very good agreement with the literature values. Taking advantage of the easy implementation of arbitrary metrics within our model, we study the fluid flow through a double-curved channel, using ellipsoidal coordinates, and study the transition to vortex flow in dependence of the two perpendicu…
We derive an expression for the velocity profile of a pressure-driven yield-stress fluid flow-ing around a two-dimensional concentric annulus. This result allows the prediction of the effects of channel curvature on the pressure gradient required to initiate flow for given yield stress, and for the width of the plug region and the flux through the channel at different curvatures. We use it to validate numerical simulations of the flow from a straight channel into a curved channel which show how the fluid first yields everywhere before reaching the predicted velocity profile.
We numerically investigate the influence of flow development on secondary flow patterns and subsequent wall shear stress distributions in a curved artery model, and we compute vascular metrics commonly used to assess variations in blood flow characteristics as it applies to arterial disease. We model a human artery with a simple, rigid 180-degree curved tube with circular cross-section and constant curvature, neglecting effects of taper, torsion and elasticity. High-fidelity numerical results are computed from an in-house discontinuous spectral element flow solver. The flow rate used in this s…
When considering flows in biological membranes, they are usually treated as flat, though more often than not, they are curved surfaces, even extremely curved, as in the case of the endoplasmic reticulum. Here, we study the topological effects of curvature on flows in membranes. Focusing on a system of many point vortical defects, we are able to cast the viscous dynamics of the defects in terms of a geometric Hamiltonian. In contrast to the planar situation, the flows generate additional defects of positive index. For the simpler situation of two vortices, we analytically predict the location o…
This paper reports the bistability of curved compression ramp (CCR) flows. It reveals that both separation and attachment states can be stably established even for the same boundary conditions. Firstly, to investigate the effects of initial condition and evolutionary history, a thought experiment involving two processes with the same three steps but in different orders is designed, possibly constructing the two distinct-different stable CCR flows. Subsequently, three-dimensional direct numerical simulations are then performed to replicate the thought experiment, verifying the existence of the …
We consider steady gravity-driven flow of a thin layer of viscous fluid over a curved substrate. The substrate has topographical variations (`bumps') on a large scale compared to the layer thickness. Using lubrication theory, we find the velocity field in generalized curvilinear coordinates. We correct the velocity field so as to satisfy kinematic constraints, which is essential to avoid particles escaping the fluid when computing their trajectories. We then investigate the particle transport properties of flows over substrates with translational symmetry, where chaotic motion is precluded. Th…
This paper is concerned with the application of finite element methods to obtain solutions for steady fully developed second-grade flows in a curved pipe of circular cross-section and arbitrary curvature ratio, under a given axial pressure gradient. The qualitative and quantitative behavior of the secondary flows is analyzed with respect to inertia and viscoelasticity.
We develop and implement a novel lattice Boltzmann scheme to study multicomponent flows on curved surfaces, coupling the continuity and Navier-Stokes equations with the Cahn-Hilliard equation to track the evolution of the binary fluid interfaces. Standard lattice Boltzmann method relies on regular Cartesian grids, which makes it generally unsuitable to study flow problems on curved surfaces. To alleviate this limitation, we use a vielbein formalism to write down the Boltzmann equation on an arbitrary geometry, and solve the evolution of the fluid distribution functions using a finite differenc…
We investigate the dynamics of flows past a stationary circular cylinder embedded on spherical and cylindrical surfaces at a fixed Reynolds number of 100. For flows on surfaces, it is convenient to express the Navier-Stokes equations in exterior calculus notation. We employ the discrete exterior calculus (DEC) method (Mohamed et al. (2016)) to numerically solve the equations. We explore the role played by curvature on dynamical quantities of interest: the drag coefficient (Cd), the lift coefficient (Cl), and the Strouhal number (St). Our simulations indicate that the effect of surface curvatur…
This paper aims at extending the code verification methodology to high-order accurate scheme implementations on curved wall-bounded domains as well as for realistic turbulent flows such as the flat plate boundary layer modelled by the Reynolds-averaged Navier-Stokes (RANS) equations. Two new manufactured solutions (MSs) are introduced with demonstrated ability to verify the treatment of slip and no-slip boundary conditions in high-order frameworks on curved domains. These MSs serve as well to discuss the impact of the method of computation of boundary normals on the order of accuracy (OOA) of …
The dispersion and dissipation properties of a scheme are important to realize high-fidelity simulations of the compressible flow, especially the cases with broadband length scales. It has been recognized that the minimization of dispersion error is an effective method to improve the precision. In addition, the proper dissipation of a scheme is important to restrain the non-physics oscillations and reserve details of flows simultaneously. The authors have previously proposed a scale sensor to adjust the numerical dissipation of a fourth-order finite difference scheme according to the local sca…
The effects of global flow rotation and curvature on the subcritical transition to turbulence in shear flows are examined. The relevant time-scales of the problem are identified by a decomposition of the flow into a laminar and a deviation from laminar parts, which is performed for rotating plane Couette and Taylor-Couette flows. The usefulness and relevance of this procedure are discussed at the same time. By comparing the self-sustaining process time-scale to the time-scales previously identified, an interpretation is brought to light for the behavior of the transition Reynolds number with t…
The bound derived in the submission (subsequently published in J. Fluid Mechanics vol 808 p 562-575, 2016) with the above title is incorrect. This corrigendum explains why and also why there can not be any quick fix.
The purpose of this note is to present a mathematical evidence of dissipation anomaly in 3D turbulent flows within a general setting for the study of energy cascade in physical scales of 3D incompressible flows recently introduced by the authors.
We examine the changes in kinetic energy dissipation of a turbulent channel flow caused by a spanwise magnetic field. The numerical study is based on our simulation data from [Krasnov, Zikanov, Schumacher, Boeck, Phys. Fluids 20, 095105 (2008)] obtained by direct and large eddy simulations. We find that the Joule dissipation can exceed the viscous dissipation in the weakly dissipative bulk region, but remains comparatively small in the turbulence-generating near-wall region.
We use a database of direct numerical simulations to evaluate parametrizations for energy dissipation rate in stably stratified flows. We show that shear-based formulations are more appropriate for stable boundary layers than commonly used buoyancy-based formulations. As part of the derivations, we explore several length scales of turbulence and investigate their dependence on local stability.
We consider the problem of existence of certain symmetrical solutions of Stokes equation on a three-dimensional manifold $M$ with a general metric possessing symmetry. These solutions correspond to unidirectional flows. We have been able to determine necessary and sufficient conditions for their existence. Symmetric unidirectional flows are fundamental for deducing the so-called Darcy's law, which is the law governing fluid flow in a Hele-Shaw cell embedded in the environment $M$. Our main interest is to depart from the usual, flat background environment, and consider the possibility of an env…
The boundary treatment is fundamental for modeling fluid flows; especially, in the lattice Boltzmann method, the curved boundary conditions effectively improve the accuracy of single-phase simulations with complex-geometry boundaries. However, the conventional curved boundary conditions usually cause dramatic mass leakage or increase when they are directly used for multiphase flow simulations. We find that the principal reason is the absence of nonideal effect in the curved boundary conditions, followed by the calculation error. In this paper, incorporating the nonideal effect into the linear …
Direct numerical simulations (DNS) of particle-laden turbulent flow in straight, mildly curved and strongly bent pipes are performed in which the solid phase is modelled as small heavy spherical particles. A total of seven populations of dilute particles with different Stokes numbers, one-way coupled with their carrier phase, are simulated. The objective is to examine the effect of the curvature on micro-particle transport and accumulation. It is shown that even a slight non-zero curvature in the flow configuration strongly impact the particle concentration map such that the concentration of i…
A new variational problem for upper bounds on the rate of energy dissipation in body-forced shear flows is formulated by including a balance parameter in the derivation from the Navier-Stokes equations. The resulting min-max problem is investigated computationally, producing new estimates that quantitatively improve previously obtained rigorous bounds. The results are compared with data from direct numerical simulations.
We present a minimal model of an incompressible flow in square duct subject to a slight curvature. Using a Poincaré-like section we identify stationary, periodic, aperiodic and chaotic regimes, depending on the unique control parameter of the problem: the Dean number (De). Aside from representing a simple, yet rich, dynamical system the present simplified model is also representative of the full problem, reproducing quite accurately the bifurcation points observed in the literature. We analyse the bifurcation diagram from De = 0 (no curvature) to De = 500, observing a periodic segment followed…
Adopting the setting for the study of existence and scale locality of the energy cascade in 3D viscous flows in physical space recently introduced by the authors to 3D inviscid flows, it is shown that the anomalous dissipation is -- in the case of decaying turbulence -- indeed capable of triggering the cascade which then continues ad infinitum, confirming Onsager's predictions.
We associate the following physical co-mover conditions of to different frame choices: i) Eckart: particle flow, ii) Landau-Lifshitz: energy flow, iii) Jüttner: moving thermometer frame. The role of fixing a flow-frame is analysed with respect to local equilibrium concentrating on dissipative currents and forces in single component relativistic fluids. The special role of a "Jüttner frame" is explored and contrasted to the more common Eckart and Landau-Lifshitz choices.
The swirling secondary flow in curved pipes is studied in three-space dimensions using a weakly compressible Smoothed Particle Hydrodynamics (WCSPH) formulation coupled to new non-reflecting outflow boundary conditions. A large-eddy simulation (LES) model for turbulence is benchmarked with existing experimental data. After validation of the present model against experimental results for a 90 grades pipe bend, a detailed numerical study aimed at reproducing experimental flow measurements for a wide range of Reynolds numbers has been performed for different pipe geometries, including U pipe bend…
The bistable states and separation hysteresis in curved compression ramp (CCR) flows, and the corresponding aerothermal characteristics (including wall friction, pressure and heat flux), are studied numerically and theoretically. Direct numerical simulations of separation hysteresis induced by variation of turning angle, as well as the influence of inflow Mach number and wall temperature on hysteresis loops, are carried out. Distributions of wall friction, pressure and heat flux are analyzed. Further, emergence of wall friction's first and second minima in the separation bubble is interpreted,…
Stokes' equations model microscale fluid flows including the flows of nanoliter-sized fluid samples in lab-on-a-chip systems. Helmholtz's dissipation theorem guarantees that the solution of Stokes' equations in a given domain minimizes viscous dissipation among all incompressible vector fields that are compatible with the velocities imposed at the domain boundaries. Helmholtz's dissipation theorem directly guarantees the uniqueness of solutions of Stokes flow, and provides a practical method for calculating approximate solutions for flow around moving bodies. However, although generalization o…
This work investigates the motion of neutrally-buoyant, slightly deformable straight and curved prolate capsules in unbounded simple shear flow at zero Reynolds number using direct simulations. The curved capsules serve as a model for the typical crescent-shaped sickle red blood cells in sickle cell disease (SCD). The effects of deformability and curvature on the dynamics are revealed. We show that with low deformability, straight prolate spheroidal capsules tumble in the shear plane as their unique asymptotically stable orbit, which contrasts with that for rigid spheroids where infinitely man…
We observe the emergence of strong vertical drafts in direct numerical simulations of the Boussinesq equations in a range of parameters of geophysical interest. These structures, which appear intermittently in space and time, generate turbulence and enhance kinetic and potential energy dissipation, providing a possible explanation for the observed variability of the local energy dissipation in the bulk of oceanic flows, and the modulation of its probability distribution function. We show how, due to the extreme drafts, in runs with Froude numbers observable in geophysical scenarios, roughly 10…
Very high-order accurate finite volume scheme for the streamfunction-vorticity formulation of incompressible fluid flows with polygonal meshes on arbitrary curved boundaries
We consider a numerical approach for a covariant generalised Navier-Stokes equation on general surfaces and study the influence of varying Gaussian curvature on anomalous vortex-network active turbulence. This regime is characterised by self-assembly of finite-size vortices into linked chains of anti-ferromagnet order, which percolate through the entire surface. The simulation results reveal an alignment of these chains with minimal curvature lines of the surface and indicate a dependency of this turbulence regime on the sign and the gradient in local Gaussian curvature. While these results re…
An original experimental setup has been elaborated in order to get a better view of turbulent flows in a von Karman geometry. The availability of a very fast camera allowed to follow in time the evolution of the flows. A surprising finding is that the development of smaller whorls ceases earlier than expected and the aspect of the flows remains the same above Reynolds number of a few thousand. This fact provides an explanation of the constancy of the reduced dissipation in the same range without the need of singularity. Its cause could be in relation with the same type of behavior observed in …
In this study, we explore several integral and outer length scales of turbulence which can be formulated by using the dissipation of temperature fluctuations ($χ$) and other relevant variables. Our analyses directly lead to simple yet non-trivial parameterizations for both $χ$ and the structure parameter of temperature ($C_T^2$). For our purposes, we make use of high-fidelity data from direct numerical simulations of stratified channel flows.
The evolution with Reynolds number of the dissipation function, normalized by wall variables, is investigated using direct numerical simulation (DNS) databases for incompressible turbulent Poiseuille flow in a plane channel, at friction Reynolds numbers up to Reτ= 2000. DNS results show that the mean part, directly dissipated by the mean flow, reaches a constant value while the turbulent part, converted into turbulent kinetic energy before being dissipated, follows a logarithmic law. This result shows that the logarithmic law of friction can be obtained without any assumption on the mean veloc…
We derive an expression for the velocity profile of a pressure-driven yield-stress Bingham fluid flowing around a 2D concentric annulus. The formula requires the numerical solution of a nonlinear equation for the positions of the yield surfaces. The results allow the prediction of the effects of channel curvature on the pressure gradient required to initiate flow for given yield stress, and for the width of the plug region and the flux through the channel at different curvatures.
We derive here Lagrangian fluctuation-dissipation relations for advected scalars in wall-bounded flows. The relations equate the dissipation rate for either passive or active scalars to the variance of scalar inputs from the initial values, boundary values, and internal sources, as those are sampled backward in time by stochastic Lagrangian trajectories. New probabilistic concepts are required to represent scalar boundary conditions at the walls: the boundary local-time density at points on the wall where scalar fluxes are imposed and the boundary first hitting-time at points where scalar valu…
Using high-resolution direct numerical simulations, the height and Reynolds number dependence of higher-order statistics of the energy dissipation rate and local enstrophy are examined in incompressible, fully-developed turbulent channel flow. The statistics are studied over a range of wall distances, spanning the viscous sublayer to the channel flow centerline, for friction Reynolds numbers $Re_τ= 180$ and $Re_τ= 381$. The high resolution of the simulations allows dissipation and enstrophy moments up to fourth order to be calculated. These moments show a dependence on wall distance, and Reyno…
In the above paper an analysis has been carried out to obtain results in the Couette flow of a Newtonian fluid with viscous dissipation and temperature dependent viscosity. The fluid viscosity is an exponential function of temperature. Exact analytical solutions are obtained for the calculation of velocity and temperature. However, there are some fundamental errors in this paper which are presented herein.
Local dissipation scales are a manifestation of the intermittent small-scale nature of turbulence. We report the first experimental evaluation of the distribution of local dissipation scales in turbulent pipe flows for a range of Reynolds numbers, 2.4x10^4<=Re_D<=7.0x10^4. Our measurements at the nearly isotropic pipe centerline and within the anisotropic logarithmic layer show excellent agreement with distributions that were previously calculated from numerical simulations of homogeneous isotropic box turbulence and with those predicted by theory. The reported results suggest a universality o…
A numerical method for simulating three-phase flows with moving contact lines on arbitrarily complex surfaces is developed in the framework of lattice Boltzmann method. In this method, the immiscible three-phase flow is modeled through a multiple-relaxation-time color-gradient model, which not only allows for a full range of interfacial tensions but also produces stable outcomes for a wide range of viscosity ratios. A characteristic line model is introduced to implement the wetting boundary condition, which is not only easy to implement but also able to handle arbitrarily complex boundaries wi…
