Finite Reynolds Number Research Articles

Freely decaying turbulence in a finite domain at finite Reynolds number

We perform direct numerical simulations to study the effects of the finite Reynolds number and domain size on the decay law of Saffman turbulence. We observe that the invariant for Saffman turbulence, u2ℓ3, and non-dimensional dissipation coefficient, Cϵ = ϵ/(u3/ℓ), are sensitive to finite domain size; here, u is the rms velocity, ℓ is the integral length scale, and ϵ is the energy dissipation rate. Consequently, the exponent n in the decay law u2 ∼ t−n for Saffman turbulence deviates from 6/5. Due to the finite Reynolds number and the domain size, Saffman turbulence decays at a faster rate (i.e., n > 6/5). However, the exponent n = 6/5 is more sensitive to the domain size than to the Reynolds number. From the simulations, we find that n remains close to 6/5 as long as Rλ ≳ 10 and ℓ ≲ 0.3Lbox; here, Rλ is the Reynolds number based on the Taylor microscale and Lbox is the domain size. We also notice that n becomes slightly lower than 6/5 for a part of the decay period. Interestingly, this trend n < 6/5 is also observed earlier in freely decaying grid-generated turbulence.

Euler-Lagrange modelling of dilute particle-laden flows with arbitrary particle-size to mesh-spacing ratio

This paper addresses the two-way coupled Euler-Lagrange modelling of dilute particle-laden flows, with arbitrary particle-size to mesh-spacing ratio. Two-way coupled Euler-Lagrange methods classically require particles to be much smaller than the computational mesh cells for them to be accurately tracked. Particles that do not satisfy this requirement can be considered by introducing a source term regularisation operator that typically consists in convoluting the point-wise particle momentum sources with a smooth kernel. Particles that are larger than the mesh cells, however, generate a significant local flow disturbance, which, in turn, results in poor estimates of the fluid forces acting on them.To circumvent this issue, this paper proposes a new framework to recover the local undisturbed velocity at the location of a given particle, that is the local flow velocity from which the disturbance due to the presence of the particle is subtracted. It relies upon the solution of the Stokes flow through a regularised momentum source and is extended to finite Reynolds numbers based on the Oseen flow solution. Owing to the polynomial nature of the regularisation kernel considered in this paper, a correction for the averaged local flow disturbance can be analytically derived, allowing to filter out scales of the flow motion that are smaller than the particle, which should not be taken into account to compute the interaction/drag forces acting on the particle. The proposed correction scheme is applied to the simulation of a particle settling under the influence of gravity, for varying particle-size to mesh-spacing ratios and varying Reynolds numbers. The method is shown to nearly eliminate any impact of the underlying mesh resolution on the modelling of a particle's trajectory. Finally, optimal values for the scale of the regularisation kernel are provided and their impact on the flow is discussed.

Modelling Lagrangian velocity and acceleration in turbulent flows as infinitely differentiable stochastic processes

Abstract

Multi-scale physical model simulation of particle filtration using computational fluid dynamics

Clarifiers integrating radial cartridge filtration (RCF) are a combined unit operation variant of millennia-old sedimentation-filtration systems. Similarly, RCF is a primarily horizontal flow variant with flow orthogonal to gravity and a radial velocity gradient, in contrast to traditional deep-bed vertical filtration. These granular filters function at lower finite granular Reynolds numbers. A proposed computational fluid dynamics framework, implementing the Navier-Stokes equations, couples a pore-scale filter model with a macroscopic scale sedimentation-filtration model to create a tool examining non-Brownian particle separation. Validation is conducted using previous physical testing from a full-scale sedimentation-filtration system under steady flow and particulate loads. Model results illustrate a two-zone filtration structure with respect to particle diameter, similar to vertical filtration. The computational tool predicts particulate matter separation of 86.1% compared to 87.8% for physical testing. The physical-based computational framework does not need high-level calibration as compared to analytical, lumped, or empirical models; conferring direct extensibility to similar unit operation systems. The novel multi-scale tool simulates particulate matter fate in a modern re-incarnation of a sedimentation-filtration unit operation. The tool functions as an adjuvant that complements regulatory or certification testing. The tool can provide guidance for design or maintenance as well as system management with respect to particle fate in, and breakthrough from, granular filters in a combined unit operation.

Unsteady fluid-structure interactions in a soft-walled microchannel: A one-dimensional lubrication model for finite Reynolds number

We develop a one-dimensional model for the unsteady fluid--structure interaction (FSI) between a soft-walled microchannel and viscous fluid flow within it. A beam equation, which accounts for both transverse bending rigidity and nonlinear axial tension, is coupled to a one-dimensional fluid model obtained from depth-averaging the two-dimensional incompressible Navier--Stokes equations across the channel height. Specifically, the Navier--Stokes equations are scaled in the viscous lubrication limit relevant to microfluidics. The resulting set of coupled nonlinear partial differential equations is solved numerically through a segregated approach employing fully-implicit time stepping. We explore both the static and dynamic FSI behavior of this example microchannel system by varying a reduced Reynolds number $Re$, which necessarily changes the Strouhal number $St$, while we keep the geometry and a modified dimensionless Young's modulus $\Sigma$ fixed. At steady state, an order-of-magnitude analysis (balancing argument) shows that the axially-averaged pressure in the flow, $\langle P\rangle$, exhibits two different scaling regimes, while the maximum deformation of the top wall of the channel, $H_{\mathrm{max}}$, can fall into four different regimes, depending on the magnitudes of $Re$ and $\Sigma$. These regimes are physically explained as resulting from the competition between the inertial and viscous forces in the fluid flow as well as the bending resistance and tension in the elastic wall. Finally, the linear stability of the steady inflated microchannel shape is assessed via a modal analysis, showing the existence of many highly oscillatory but stable modes, which further highlights the computational challenge of simulating unsteady FSIs.

Simulation of sedimentation of two spheres with different densities in a square tube

We simulated the sedimentations of two unequal spheres with different densities in a square tube for Galileo numbers (Ga) from 5 to 25, resulting in a Reynolds number range of based on the terminal settling velocity. The sedimentation of spheres with different densities is dynamically more complex than that of identical spheres. At high Ga the spheres oscillate in the centreline plane of the tube, where they are initially released from rest. By contrast, the spheres move to the diagonal or reverse–diagonal plane of the tube at low Ga, reaching a steady or periodic state depending on the density difference between them. A phase diagram illustrates the transitions between different sedimentation behaviours depending on Ga and the density difference. A possible mechanism for these behaviours is also presented. Furthermore, we compare two-dimensional (2-D) and three-dimensional computations for our system to attain a better understanding of the hydrodynamic interactions between two unequal spheres at low but finite Reynolds number. Comparing relative trajectories, periods of oscillation and flow features shows that 2-D circular cylinders oscillate much more strongly and frequently than spheres under the same flow conditions. In particular, spheres do not have the discontinuity in period that arises in the 2-D case from the change in rotation sign of a heavy particle.

Collector efficiency for hydrosol deep-bed filtration of non-Brownian particles at low and finite Reynolds numbers

Aqueous filtration systems with granular media are increasingly implemented as a unit operation for the treatment of urban waters. Many of these aqueous filtration systems are designed with coarse granular media and are therefore subject to finite granular Reynolds numbers (Reg). In contrast to the Reg conditions generated by such designs, current hydrosol filtration models, such as the Yao and RT models, rely on a flow solution that is derived within the Stokes limit at low Reg. In systems that are subject to these finite and higher Reg regimes, the collector efficiency has not been examined. Therefore, in this study, we develop a 3D periodic porosity-compensated face-centered cubic sphere (PCFCC) computational fluid dynamics (CFD) model, with the surface interactions incorporated, to investigate the collector efficiency for Reg ranging from 0.01 to 20. Particle filtration induced by interception and sedimentation is examined for non-Brownian particles ranging from 1 to 100 μm under favorable surface interactions for particle adhesion. The results from the CFD-based PCFCC model agreed well with those of the classical RT and Yao models for Reg < 1. Based on 3150 simulations from the PCFCC model, we developed a new correlation for vertical aqueous filtration based on a modified gravitation number, NG*, for the initial deep-bed filtration efficiency at lower yet finite (0.01 to 20) Reg. The proposed PCFCC model has low computational cost and is extensibile from vertical to horizontal filtration at low and finite Reg.

Assessment of force models on finite-sized particles at finite Reynolds numbers

Finite-sized inertial spherical particles are fully-resolved with the immersed boundary projection method (IBPM) in the turbulent open-channel flow by direct numerical simulation (DNS). The accuracy of the particle surface force models is investigated in comparison with the total force obtained via the fully-resolved method. The results show that the steady-state resistance only performs well in the streamwise direction, while the fluid acceleration force, the added-mass force, and the shear-induced Saffman lift can effectively compensate for the large-amplitude and high-frequency characteristics of the particle surface forces, especially for the wall-normal and spanwise components. The modified steady-state resistance with the correction effects of the acceleration and the fluid shear can better represent the overall forces imposed on the particles, and it is a preferable choice of the surface force model in the Lagrangian point-particle method.

Effect of suction on laminar-flow control in subsonic boundary layers with forward-/backward-facing steps

In a subsonic boundary layer, a forward-facing step (FFS) or a backward-facing step (BFS) usually destabilizes the oncoming Tollmien–Schlichting (T-S) waves, leading to a promotion of laminar–turbulent transition. This paper studies a laminar-flow control strategy by introducing wall suction immediately ahead of the FFS or behind the BFS. The impact of the step–suction combination on an oncoming T-S wave is quantified by a transmission coefficient, defined as the ratio of the asymptotic amplitude downstream of the step and suction to that upstream. In order to solve this problem, a local scattering theory based on the large-Reynolds number (large-R) asymptotic framework and a Harmonic linearized Navier–Stokes approach, that calculates the perturbation field at finite Reynolds numbers, are employed. The latter approach is confirmed to be accurate by comparing its results with direct numerical simulations, and the results given by the two approaches agree when the Reynolds number is asymptotically large. According to the large-R triple-deck formulism, a few control parameters, such as the Mach number, Reynolds number, and wall temperature, disappear, which makes a systematical parametric study possible. The destabilizing effect of a step increases with its height, while the stabilizing effect of suction increases with its flux. For a step with a moderate height, suction with a small flux is sufficient to compensate the destabilizing effect of the step.

Transient electrohydrodynamics of a liquid drop at finite Reynolds numbers

Transient electrohydrodynamics of a liquid drop at finite Reynolds numbers is studied by computer simulations. The governing equations of the problem are solved using a parallelized front tracking/finite difference method in the framework of the Taylor–Melcher leaky dielectric theory. The effect of the dielectric properties of the fluids on the dynamic response of the drop is studied by considering three representative fluid systems, which encompass all the regions of the deformation–circulation map. A parametric study is performed over a range of capillary number , the Ohnesorge number squared and the viscosity ratio to explore the effect of these parameters on the dynamic response. At low to intermediate capillary numbers, the dynamic response resembles that of a first- or second-order mechanical system, where the deformation settles to a steady state monotonically or through oscillations. However, beyond a threshold capillary number, the dynamic response becomes nonlinear. It is shown that the Ohnesorge number squared and the viscosity ratio are the key parameters that determine the transition from a monotonic response to an oscillatory one (and vice versa) in density-matched fluid systems. To predict the dynamic response of nearly spherical drops at arbitrary , an analytical equation is developed, which results in a critical Ohnesorge number squared that demarks a monotonic response from an oscillatory one. Investigations on the effect of the viscosity ratio on the dynamic response shows that the relaxation time (for monotonic response) increases with an increase in and that the steady state deformation correlates positively with for oblate drops; however, depending on the corresponding regions of prolate drops on the deformation–circulation map, correlates positively or negatively with . A long-standing misconception regarding the oblate region is also rectified by development of a normal stress map, which determines the relative importance of the electric pressure and the normal hydrodynamic stress in setting the sense of the deformation of the drop.

Multiscale Simulation of Non-Metallic Inclusion Aggregation in a Fully Resolved Bubble Swarm in Liquid Steel

Removing inclusions from the melt is an important task in metallurgy with critical impact on the quality of the final alloy. Processes employed with this purpose, such as flotation, crucially depend on the particle size. For small inclusions, the aggregation kinetics constitute the bottleneck and, hence, determine the efficiency of the entire process. If particles smaller than all flow scales are considered, the flow can locally be replaced by a plane shear flow. In this contribution, particle interactions in plane shear flow are investigated, computing the fully resolved hydrodynamics at finite Reynolds numbers, using a lattice Boltzmann method with an immersed boundary method. Investigations with various initial conditions, several shear values and several inclusion sizes are conducted to determine collision efficiencies. It is observed that although finite Reynolds hydrodynamics play a significant role in particle collision, statistical collision efficiency barely depends on the Reynolds number. Indeed, the particle size ratio is found to be the prevalent parameter. In a second step, modeled collision dynamics are applied to particles tracked in a fully resolved bubbly flow, and collision frequencies at larger flow scale are derived.

Effects of velocity second slip model and induced magnetic field on peristaltic transport of non-Newtonian fluid in the presence of double-diffusivity convection in nanofluids

The significance of velocity second slip model of non-Newtonian fluid on peristaltic pumping in existence of double-diffusivity convection in nanofluids and induced magnetic field is deliberated. Mathematical modelling of current problem is defined in fixed frame of reference and then abridges under well- known conjecture of long wavelength and low but finite Reynolds number approximation. Precise results of coupled nonlinear partial differential equations are presented. Graphical results exhibit the performance of various supportive parameters. The phenomenon of stream functions with different wave forms is also discussed in detail. The effects of thermal energy, solute concentration, and nanoparticle fraction are also described using graphical representation.

Distributed flexibility in inertial swimmers

We study a linear inviscid model of a passively flexible swimmer with distributed flexibility, calculating its propulsive performance and optimal distributions of flexibility. The frequencies of actuation and mean stiffness ratios we consider span a large range, while the mass ratio is fixed to a low value representative of swimmers. We present results showing how the trailing edge deflection, thrust coefficient, power coefficient and efficiency vary with frequency, mean stiffness and stiffness distribution. Swimmers with distributed flexibility have the same qualitative features as those with uniform flexibility. Significant gains in thrust can be made, however, by tuning the stiffness such that a resonant response is triggered, or by concentrating stiffness towards the leading edge if resonance cannot be triggered. To minimize power, the opposite is true. Meaningful gains in efficiency can be made at low frequencies by concentrating stiffness away from the leading edge, since doing so induces efficient travelling wave kinematics. We also speculate on the effects of a finite Reynolds number in the form of streamwise drag. The drag adds an offset to the net thrust produced by the swimmer, causing efficiency-maximizing distributions of flexibility to tend towards thrust-maximizing ones, representative of what is found in nature.

Settling disks in a linearly stratified fluid

We consider the unbounded settling dynamics of a circular disk of diameter and finite thickness evolving with a vertical speed in a linearly stratified fluid of kinematic viscosity and diffusivity of the stratifying agent, at moderate Reynolds numbers ( ). The influence of the disk geometry (diameter and aspect ratio ) and of the stratified environment (buoyancy frequency , viscosity and diffusivity) are experimentally and numerically investigated. Three regimes for the settling dynamics have been identified for a disk reaching its gravitational equilibrium level. The disk first falls broadside-on, experiencing an enhanced drag force that can be linked to the stratification. A second regime corresponds to a change of stability for the disk orientation, from broadside-on to edgewise settling. This occurs when the non-dimensional velocity becomes smaller than some threshold value. Uncertainties in identifying the threshold value is discussed in terms of disk quality. It differs from the same problem in a homogeneous fluid which is associated with a fixed orientation (at its initial value) in the Stokes regime and a broadside-on settling orientation at low, but finite Reynolds numbers. Finally, the third regime corresponds to the disk returning to its broadside orientation after stopping at its neutrally buoyant level.

Asymptotic modelling and direct numerical simulations of multilayer pressure-driven flows

Abstract The nonlinear dynamics of two immiscible superposed viscous fluid layers in a channel is examined using asymptotic modelling and direct numerical simulations (DNS). The flow is driven by an imposed axial pressure gradient. Working on the assumption that one of the layers is thin, a weakly-nonlinear evolution equation for the interfacial shape is derived that couples the dynamics in the two layers via a nonlocal integral term whose kernel is determined by solving the linearised Navier–Stokes equations in the thicker fluid. The model equation incorporates salient physical effects including inertia, gravity, and surface tension, and allows for comparison with DNS at finite Reynolds numbers. Direct comparison of travelling-wave solutions obtained from the model equation and from DNS show good agreement for both stably and unstably stratified flows. Both the model and the DNS indicate regions in parameter space where unimodal, bimodal and trimodal waves co-exist. Nevertheless, the asymptotic model cannot capture the dynamics for a sufficiently strong unstable density stratification when interfacial break-up and eventual dripping occurs. In this case, complicated interfacial dynamics arise from the dominance of the gravitational force over the shear force due to the underlying flow, and this is investigated in detail using DNS.

Reynolds number dependence of velocity gradient skewness in a renormalized scheme of fluid turbulence

We obtain the behavior of skewness of the longitudinal velocity gradient with respect to the Reynolds number of a turbulent fluid from the incompressible Navier–Stokes dynamics. This is achieved in a renormalized scheme incorporating a von Kàrmàn interpolation function between the production and inertial ranges together with an interpolation between the inertial and dissipation ranges. The corresponding second- and third-order renormalized moments of the longitudinal velocity gradient and the resulting skewness values are found to depend on the Reynolds number in nontrivial fashions. For intermediate and moderately high Reynolds numbers, the skewness value does not exhibit any scaling behavior with respect to the Reynolds number. Instead, the skewness value remains bounded for any Reynolds number and it approaches a constant value for infinitely high Reynolds numbers. In the same scheme, we also incorporate another interpolation function between the production and inertial ranges coming from energy balance equation and inputs from renormalization-group analysis. We find exact agreement between the asymptotic values of skewness at high Reynolds numbers while the skewness values are in the same neighborhood in the finite Reynolds number (non-universal) regime.

Electrified cone formation in perfectly conducting viscous liquids: Self-similar growth irrespective of Reynolds number

Above a critical field strength, the free surface of an electrified, perfectly conducting viscous liquid, such as a liquid metal, is known to develop an accelerating protrusion resembling a cusp with a conic tip. Field self-enhancement from tip sharpening is reported to generate divergent power law growth in finite time of the forces acting in that region. Previous studies have established that tip sharpening proceeds via a self-similar process in two distinguished limits—the Stokes regime and the inviscid regime. Using finite element simulations to track the shape and forces acting at the tip of an electrified protrusion in a perfectly conducting Newtonian liquid, we demonstrate that the conic tip always undergoes self-similar growth irrespective of the Reynolds number. The blowup exponents at the conic apex for all terms in the Navier-Stokes equation and the normal stress boundary condition at the moving interface reveal the dominant forces at play as the Reynolds number increases. Rescaling of the tip shape by the power law representing the divergence in capillary stress at the apex yields an excellent collapse onto a universal cone shape with an interior half-angle dependent on the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin interfacial boundary layer characterized by a significant rate of strain. Additional details of the modeled flow, applicable to cone growth in systems such as liquid metal ion sources, help dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at a finite Reynolds number can be neglected.

Lamb’s solution and the stress moments for a sphere in Stokes flow

Abstract This paper describes a method to generate expressions for the moments of the fluid stress integrated over a sphere in a general Stokes flow based on Lamb’s general solution of the Stokes equations. Explicit results up to moments of order four are given together with a general recurrence relation for moments of higher order. The connection of the results to earlier expressions for the stress moments leading, in particular, to generalized Faxen-type relations is demonstrated and the connection with representations of the rotation group is addressed. This study is motivated by the central role played by the Lamb coefficient in the physalis method for the resolved simulation of particulate flows. An example of the application of this method to the shear flow of a suspension of 3200 equal spheres at finite Reynolds number is given. In addition, the present theory is applied to the analysis of the particle contribution to the mixture stress in particulate flows at both finite and vanishing Reynolds numbers. It is shown that, in the case of spatial non-uniformity, the mixture stress acquires new contributions, including a new viscosity parameter, which are explicitly calculated for the case of a dilute suspension with negligible inertia.

Centre modes in pipe flow

Abstract In two previous papers, Ozcakir, Tanveer, Hall, &amp; Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328) and Ozcakir, Hall &amp; Tanveer (2019, Nonlinear exact coherent structures in pipe flow and their instabilities, J. Fluid Mech., 868, 341–368) investigated numerically and asymptotically high Reynolds number exact coherent structures in pipe flow. It was found that, in addition to the structures described by the vortex–wave interaction theory by Hall &amp; Smith (1991, On strongly nonlinear vortex/wave interactions in boundary layer transition. J. Fluid Mech., 227, 641–666), there exists vortical structures localized near the centre of the pipe with a core of size $O(Re^{-1/4})$ convected downstream at a speed that deviates from the pipe centreline speed by $O(Re^{-1/2})$, where $Re$ is the Reynolds number. In the finite Reynolds number calculations by Ozcakir, Tanveer, Hall &amp; Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328), asymptotic state was referred to as a nonlinear viscous core state (NVC). However the reduced asymptotic equations were not solved and only limited confirmation of the theory was found numerically. Here, in order to conclusively confirm the existence of the NVC state we first describe direct numerical calculations on the asymptotically reduced $Re&gt;&gt;1$ equations for such state states. The results are then compared in detail to the finite $Re$ calculations up-to $Re=10^6$; the latter regime is at much higher values of the Reynolds number than those reported in Ozcakir, Tanveer, Hall &amp; Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328). The results are found to be in excellent agreement with the finite $Re$ calculations in a region between $Re=10^5$ and $10^6$, thereby confirming that the structure observed by Ozcakir, Tanveer, Hall &amp; Overman (2016, Travelling waves in pipe flow, J. Fluid Mech., 791, 284–328) is indeed a finite Reynolds number realization of an asymptotic NVC state.

Kelvin–Helmholtz billows above Richardson number

We study the dynamical system of a two-dimensional, forced, stratified mixing layer at finite Reynolds number $Re$, and Prandtl number $Pr=1$. We consider a hyperbolic tangent background velocity profile in the two cases of hyperbolic tangent and uniform background buoyancy stratifications, in a domain of fixed, finite width and height. The system is forced in such a way that these background profiles are a steady solution of the governing equations. As is well known, if the minimum gradient Richardson number of the flow, $Ri_{m}$, is less than a certain critical value $Ri_{c}$, the flow is linearly unstable to Kelvin–Helmholtz instability in both cases. Using Newton–Krylov iteration, we find steady, two-dimensional, finite-amplitude elliptical vortex structures – i.e. ‘Kelvin–Helmholtz billows’ – existing above $Ri_{c}$. Bifurcation diagrams are produced using branch continuation, and we explore how these diagrams change with varying $Re$. In particular, when $Re$ is sufficiently high we find that finite-amplitude Kelvin–Helmholtz billows exist when $Ri_{m}&gt;1/4$ for the background flow, which is linearly stable by the Miles–Howard theorem. For the uniform background stratification, we give a simple explanation of the dynamical system, showing the dynamics can be understood on a two-dimensional manifold embedded in state space, and demonstrate the cases in which the system is bistable. In the case of a hyperbolic tangent stratification, we also describe a new, slow-growing, linear instability of the background profiles at finite $Re$, which complicates the dynamics.