JFM Perspectives
Flow over natural or engineered surfaces: an adjoint homogenization perspective
- Alessandro Bottaro
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- 27 August 2019, P1
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Natural and engineered surfaces are never smooth, but irregular, rough at different scales, compliant, possibly porous, liquid impregnated or superhydrophobic. The correct numerical modelling of fluid flowing through and around them is important but poses problems. For media characterized by a periodic or quasi-periodic microstructure of characteristic dimensions smaller than the relevant scales of the flow, multiscale homogenization can be used to study the effect of the surface, avoiding the numerical resolution of small details. Here, we revisit the homogenization strategy using adjoint variables to model the interaction between a fluid in motion and regularly micro-textured, permeable or impermeable walls. The approach described allows for the easy derivation of auxiliary/adjoint systems of equations which, after averaging, yield macroscopic tensorial properties, such as permeability, elasticity, slip, transpiration, etc. When the fluid in the neighbourhood of the microstructure is in the Stokes regime, classical results are recovered. Adjoint homogenization, however, permits simple extension of the analysis to the case in which the flow displays nonlinear effects. Then, the properties extracted from the auxiliary systems take the name of effective properties and do not depend only on the geometrical details of the medium, but also on the microscopic characteristics of the fluid motion. Examples are shown to demonstrate the usefulness of adjoint homogenization to extract effective tensor properties without the need for ad hoc parameters. In particular, notable results reported herein include:
(i) an original formulation to describe filtration in porous media in the presence of inertial effects;
(ii) the microscopic and macroscopic equations needed to characterize flows through poroelastic media;
(iii) an extended Navier’s condition to be employed at the boundary between a fluid and an impermeable rough wall, with roughness elements which can be either rigid or linearly elastic;
(iv) the microscopic problems needed to define the relevant parameters for a Saffman-like condition at the interface between a fluid and a porous substrate; and
(v) the macroscopic equations which hold at the dividing surface between a free-fluid region and a fluid-saturated poroelastic domain.
JFM Rapids
Emergence of substructures inside the large-scale circulation induces transition in flow reversals in turbulent thermal convection
- Xin Chen, Shi-Di Huang, Ke-Qing Xia, Heng-Dong Xi
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- 19 August 2019, R1
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We present an experimental study of the reversal of the large-scale circulation (LSC) in quasi-two-dimensional turbulent Rayleigh–Bénard convection. It is found that there exists a transition in the Rayleigh number ($Ra$) dependence of the reversal rate $f$ with two distinct scalings: for $Ra$ less than a transitional value $Ra_{t}$, the non-dimensionalized reversal rate $ft_{E}\sim Ra^{-1.09}$; however, for higher $Ra$ the scaling changes to $ft_{E}\sim Ra^{-3.06}$, where $t_{E}$ is the turnover time of the LSC. Flow visualization shows that this regime transition originates from a change in flow topology from a single-roll state to a new, less stable, abnormal single-roll state with substructures inside the single roll. The emergence of the substructures inside the LSC lowers the energy barrier for the flow reversals to occur and leads to a slower decay of $f$ with $Ra$. Detailed analysis reveals that, although it is the corner rolls that trigger the reversal event, the probability for the occurrence of reversals mainly depends on the stability of the LSC. This is supported by a model we proposed to predict the critical condition for the transition, which agrees well with the experimental results.
The charmed string: self-supporting loops through air drag
- Adrian Daerr, Juliette Courson, Margaux Abello, Wladimir Toutain, Bruno Andreotti
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- 20 August 2019, R2
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The string shooter experiment uses counter-rotating pulleys to propel a closed string forward. Its steady state exhibits a transition from a gravity-dominated regime at low velocity towards a high-velocity regime where the string takes the form of a self-supporting loop. Here we show that this loop of light string is not suspended in the air due to inertia, but through the hydrodynamic drag exerted by the surrounding fluid, namely air. We investigate this drag experimentally and theoretically for a smooth long cylinder moving along its axis. We then derive the equations describing the shape of the string loop in the limit of vanishing string radius. The solutions present a critical point, analogous to a hydraulic jump, separating a supercritical zone where the wave velocity is smaller than the rope velocity, from a subcritical zone where waves propagate faster than the rope velocity. This property could be leveraged to create a white hole analogue similar to what has been demonstrated using surface waves on a flowing fluid. Loop solutions that are regular at the critical point are derived, discussed and compared to the experiment. In the general case, however, the critical point turns out to be the locus of a sharp turn of the string, which is modelled theoretically as a discontinuity. The hydrodynamic regularisation of this geometrical singularity, which involves non-local and added mass effects, is discussed on the basis of dimensional analysis.
Third-order structure functions for isotropic turbulence with bidirectional energy transfer
- Jin-Han Xie, Oliver Bühler
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- 02 September 2019, R3
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We derive and test a new heuristic theory for third-order structure functions that resolves the forcing scale in the scenario of simultaneous spectral energy transfer to both small and large scales, which can occur naturally, for example, in rotating stratified turbulence or magnetohydrodynamical (MHD) turbulence. The theory has three parameters – namely the upscale/downscale energy transfer rates and the forcing scale – and it includes the classic inertial-range theories as local limits. When applied to measured data, our global-in-scale theory can deduce the energy transfer rates using the full range of data, therefore it has broader applications compared with the local theories, especially in situations where the data is imperfect. In addition, because of the resolution of forcing scales, the new theory can detect the scales of energy input, which was impossible before. We test our new theory with a two-dimensional simulation of MHD turbulence.
Streamwise inclination angle of large wall-attached structures in turbulent boundary layers
- Rahul Deshpande, Jason P. Monty, Ivan Marusic
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- 02 September 2019, R4
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The streamwise inclination angle of large wall-attached structures, in the log region of a canonical turbulent boundary layer, is estimated via spectral coherence analysis, and is found to be approximately $45^{\circ }$. This is consistent with assumptions used in prior attached eddy model-based simulations. Given that the inclination angle obtained via standard two-point correlations is influenced by the range of scales in the turbulent flow (Marusic, Phys. Fluids, vol. 13 (3), 2001, pp. 735–743), the present result is obtained by isolating the large wall-attached structures from the rest of the turbulence. This is achieved by introducing a spanwise offset between two hot-wire probes, synchronously measuring the streamwise velocity at a near-wall and log-region reference location, to assess the wall coherence. The methodology is shown to be effective by applying it to data sets across Reynolds numbers, $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3})$–$O(10^{6})$.
JFM Papers
Dispersion of active particles in confined unidirectional flows
- Weiquan Jiang, Guoqian Chen
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- 16 August 2019, pp. 1-34
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Transport of micro-organisms in confined flows can be characterized by a one-dimensional overall dispersion mechanism, of importance to various biotechnological applications. Based on Brenner’s generalized Taylor dispersion theory, an overall dispersion model is analytically studied in the present work for a dilute suspension of active particles in confined unidirectional flows. With the confined section of the channel and the swimming orientation space taken together as the local space and the longitudinal coordinate standing for the one-dimensional global space, this model is analytically accurate and possessed of wide adaptability in terms of the swimming Péclet number. The Robin boundary condition is introduced to account for wall accumulation of active particles, and compared with a typical reflection boundary condition. Complications associated with the boundary conditions for analytical derivation are removed respectively by a decomposition of the distribution function and an extension of the flow field. Interesting solutions are concretely found and intensively illustrated. Detailed case studies on the transport of spherical and rod-like particles to illustrate the dispersion mechanism are presented with respect to a Couette flow and a plane Poiseuille flow. Associated with the local distribution of particles, extensive descriptions are given for the dynamical system behaviours such as accumulation near both stable points/lines and boundaries, symmetric polarization structure, closed orbits, trapping effect, nematic alignments and bimodalization of swimming direction. For spherical particles, the accumulation is shown leading to a reduction of the overall dispersivity in both of the flows, while for rod-like active particles in the Couette flow, the accumulation can result in an enhancement of dispersion, due to the nematic alignments of particles towards streamlines.
Turbulent shear-layer mixing: initial conditions, and direct-numerical and large-eddy simulations
- Nek Sharan, Georgios Matheou, Paul E. Dimotakis
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- 19 August 2019, pp. 35-81
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Aspects of turbulent shear-layer mixing are investigated over a range of shear-layer Reynolds numbers, $Re_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x0394}U\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D708}$, based on the shear-layer free-stream velocity difference, $\unicode[STIX]{x0394}U$, and mixing-zone thickness, $\unicode[STIX]{x1D6FF}$, to probe the role of initial conditions in mixing stages and the evolution of the scalar-field probability density function (p.d.f.) and variance. Scalar transport is calculated for unity Schmidt numbers, approximating gas-phase diffusion. The study is based on direct-numerical simulation (DNS) and large-eddy simulation (LES), comparing different subgrid-scale (SGS) models for incompressible, uniform-density, temporally evolving forced shear-layer flows. Moderate-Reynolds-number DNS results help assess and validate LES SGS models in terms of scalar-spectrum and mixing estimates, as well as other metrics, to $Re_{\unicode[STIX]{x1D6FF}}\lesssim 3.3\times 10^{4}$. High-Reynolds-number LES investigations to $Re_{\unicode[STIX]{x1D6FF}}\lesssim 5\times 10^{5}$ help identify flow parameters and conditions that influence the evolution of scalar variance and p.d.f., e.g. marching versus non-marching. Initial conditions that generate shear flows with different mixing behaviour elucidate flow characteristics in each flow regime and identify elements that induce p.d.f. transition and scalar-variance behaviour. P.d.f. transition is found to be largely insensitive to local flow parameters, such as $Re_{\unicode[STIX]{x1D6FF}}$, or a previously proposed vortex-pairing parameter based on downstream distance, or other equivalent criteria. The present study also allows a quantitative comparison of LES SGS models in moderate- and high-$Re_{\unicode[STIX]{x1D6FF}}$ forced shear-layer flows.
The rapid distortion of two-way coupled particle-laden turbulence
- M. Houssem Kasbaoui, Donald L. Koch, Olivier Desjardins
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- 19 August 2019, pp. 82-104
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In this study, we address the modification of sheared turbulence by dispersed inertial particles. The preferential sampling of the straining regions of the flow by inertial particles in turbulence leads to an inhomogeneous distribution of particles. The strong gravitational loading exerted by the highly concentrated regions results in anisotropic alteration of turbulence at small scales in the direction of gravity. These effects are investigated in a rapid distortion theory (RDT) extended for two-way coupled particle-laden flows. To make the analysis tractable, we assume that particles have small but non-zero inertia. In the classical results for single-phase flows, the RDT assumption of fast shearing compared to the turbulence time scales leads to the distortion and shear-induced production of turbulence. In particle-laden turbulence, the coupling between the two phases under rapid shearing induces number density fluctuations that convert gravitational potential energy to turbulent kinetic energy and modulate the turbulence spectrum in a manner that increases with mass loading. Turbulence statistics obtained from RDT are compared with Euler–Lagrange simulations of homogeneously sheared particle-laden turbulence.
An adjoint compressible linearised Navier–Stokes approach to model generation of Tollmien–Schlichting waves by sound
- Henrique Raposo, Shahid Mughal, Richard Ashworth
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- 19 August 2019, pp. 105-129
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The generation of the first-mode instability through scattering of an acoustic wave by localised surface roughness, suction or heating is studied with a time-harmonic compressible adjoint linearised Navier–Stokes (AHLNS) approach for subsonic flow conditions. High Strouhal number analytical solutions to the compressible Stokes layer problem are deduced and shown to be in better agreement with numerical solutions compared to previous works. The adjoint methodology of Hill in the context of acoustic receptivity is extended to the compressible flow regime and an alternative formulation to predict sensitivity to the angle of incidence of an acoustic wave is proposed. Good agreement of the acoustic AHLNS receptivity model is found with published direct numerical simulations and the simpler finite Reynolds number approach. Parametric investigations of the influence of the acoustic wave angle on receptivity amplitudes reveal that the linearised unsteady boundary layer equations are a valid model of the acoustic signature for a large range of acoustic wave obliqueness values, failing only where the wave is highly oblique and travels upstream. An extensive parametric study of the influence of frequency, spanwise wavenumber, local Reynolds number and free-stream Mach number over the efficiency function for the different types of wall perturbation mechanisms is undertaken.
Nonlinear behaviour of convergent Richtmyer–Meshkov instability
- Xisheng Luo, Ming Li, Juchun Ding, Zhigang Zhai, Ting Si
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- 19 August 2019, pp. 130-141
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A novel shock tube is designed to investigate the nonlinear feature of convergent Richtmyer–Meshkov instability on a single-mode interface formed by a soap film technique. The shock tube employs a concave–oblique–convex wall profile which first transforms a planar shock into a cylindrical arc, then gradually strengthens the cylindrical shock along the oblique wall, and finally converts it back into a planar one. Therefore, the new facility can realize analysis on compressibility and nonlinearity of convergent Richtmyer–Meshkov instability by eliminating the interface deceleration and reshock. Five sinusoidal $\text{air}{-}\text{SF}_{6}$ interfaces with different amplitudes and wavelengths are considered. For all cases, the perturbation amplitude experiences a linear growth much longer than that in the planar geometry. A compressible linear model is derived by considering a constant uniform fluid compression, which shows a slight difference to the incompressible theory. However, both the linear models overestimate the perturbation growth from a very early stage due to the presence of strong nonlinearity. The nonlinear model of Wang et al. (Phys. Plasmas, vol. 22, 2015, 082702) is demonstrated to predict well the amplitude growth up to a normalized time of 1.0. The prolongation of the linear increment is mainly ascribed to the counteraction between the promotion by geometric convergence and the suppression by nonlinearity. Growths of the first three harmonics, obtained by a Fourier analysis of the interface contour, provide a first thorough validation of the nonlinear theory.
A two-dimensional numerical and experimental study of piston and sloshing resonance in moonpools with recess
- Senthuran Ravinthrakumar, Trygve Kristiansen, Bernard Molin, Babak Ommani
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- 19 August 2019, pp. 142-166
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The piston and first sloshing modes of two-dimensional moonpools with recess are investigated. Dedicated forced heave experiments are carried out. Different recess lengths are tested from $1/4$ to $1/2$ of the length of the moonpool at the mean waterline. A theoretical model to calculate the natural frequencies is developed based on linearized potential flow theory and eigenfunction expansion. Two numerical methods are implemented: a boundary element method (BEM) and a Navier–Stokes solver (CFD). Both the BEM and CFD have linearized free-surface and body-boundary conditions. As expected, the BEM over-predicts the moonpool response significantly, in particular at the first sloshing mode. The CFD is in general able to predict the maximum moonpool response adequately, both at the piston and first sloshing modes. Both numerical methods fail to predict the Duffing-type behaviour at the first sloshing mode, due to the linearized free-surface conditions. The Duffing behaviour is more pronounced for the largest recess. The main source of damping in the proximity of the first sloshing mode is discussed.
Direct numerical simulation of conical shock wave–turbulent boundary layer interaction
- Feng-Yuan Zuo, Antonio Memmolo, Guo-ping Huang, Sergio Pirozzoli
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- 19 August 2019, pp. 167-195
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Direct numerical simulation of the Navier–Stokes equations is carried out to investigate the interaction of a conical shock wave with a turbulent boundary layer developing over a flat plate at free-stream Mach number $M_{\infty }=2.05$ and Reynolds number $Re_{\unicode[STIX]{x1D703}}\approx 630$, based on the upstream boundary layer momentum thickness. The shock is generated by a circular cone with half opening angle $\unicode[STIX]{x1D703}_{c}=25^{\circ }$. As found in experiments, the wall pressure exhibits a distinctive N-wave signature, with a sharp peak right past the precursor shock generated at the cone apex, followed by an extended zone with favourable pressure gradient, and terminated by the trailing shock associated with recompression in the wake of the cone. The boundary layer behaviour is strongly affected by the imposed pressure gradient. Streaks are suppressed in adverse pressure gradient (APG) zones, but re-form rapidly in downstream favourable pressure gradient (FPG) zones. Three-dimensional mean flow separation is only observed in the first APG region associated with the formation of a horseshoe vortex, whereas the second APG region features an incipient detachment state, with scattered spots of instantaneous reversed flow. As found in canonical geometrically two-dimensional wedge-generated shock–boundary layer interactions, different amplification of the turbulent stress components is observed through the interacting shock system, with approach to an isotropic state in APG regions, and to a two-component anisotropic state in FPG. The general adequacy of the Boussinesq hypothesis is found to predict the spatial organization of the turbulent shear stresses, although different eddy viscosities should be used for each component, as in tensor eddy-viscosity models, or in full Reynolds stress closures.
Data assimilation method to de-noise and de-filter particle image velocimetry data
- Jurriaan J. J. Gillissen, Roland Bouffanais, Dick K. P. Yue
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- 19 August 2019, pp. 196-213
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We present a variational data assimilation method in order to improve the accuracy of velocity fields $\tilde{\boldsymbol{v}}$, that are measured using particle image velocimetry (PIV). The method minimises the space–time integral of the difference between the reconstruction $\boldsymbol{u}$ and $\tilde{\boldsymbol{v}}$, under the constraint, that $\boldsymbol{u}$ satisfies conservation of mass and momentum. We apply the method to synthetic velocimetry data, in a two-dimensional turbulent flow, where realistic PIV noise is generated by computationally mimicking the PIV measurement process. The method performs optimally when the assimilation integration time is of the order of the flow correlation time. We interpret these results by comparing them to one-dimensional diffusion and advection problems, for which we derive analytical expressions for the reconstruction error.
Impact of pressure dissipation on fluid injection into layered aquifers
- Luke T. Jenkins, Martino Foschi, Christopher W. MacMinn
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- 19 August 2019, pp. 214-238
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Carbon dioxide ($\text{CO}_{2}$) capture and subsurface storage is one method for reducing anthropogenic $\text{CO}_{2}$ emissions to mitigate climate change. It is well known that large-scale fluid injection into the subsurface leads to a buildup in pressure that gradually spreads and dissipates through lateral and vertical migration of water. This dissipation can have an important feedback on the shape of the $\text{CO}_{2}$ plume during injection, but the impact of vertical pressure dissipation, in particular, remains poorly understood. Here, we investigate the impact of lateral and vertical pressure dissipation on the injection of $\text{CO}_{2}$ into a layered aquifer system. We develop a compressible, two-phase model that couples pressure dissipation to the propagation of a $\text{CO}_{2}$ gravity current. We show that our vertically integrated, sharp-interface model is capable of efficiently and accurately capturing water migration in a layered aquifer system with an arbitrary number of aquifers. We identify two limiting cases – ‘no leakage’ and ‘strong leakage’ – in which we derive analytical expressions for the water pressure field for the corresponding single-phase injection problem. We demonstrate that pressure dissipation acts to suppress the formation of an advancing $\text{CO}_{2}$ tongue during injection, reducing the lateral extent of the plume. The properties of the seals and the number of aquifers determine the strength of pressure dissipation and subsequent coupling with the $\text{CO}_{2}$ plume. The impact of pressure dissipation on the shape of the $\text{CO}_{2}$ plume is likely to be important for storage efficiency and security.
Large-scale motions in a plane wall jet
- Ebenezer P. Gnanamanickam, Shibani Bhatt, Sravan Artham, Zheng Zhang
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- 19 August 2019, pp. 239-281
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The plane wall jet (PWJ) is a wall-bounded flow in which a wall shear layer develops in the presence of extremely energetic flow structures of the outer free-shear layer. The structure of a PWJ, developing in still air, was studied with the focus on the large scales in the flow. Wall-normal hot-wire anemometry (HWA) measurements along with double-frame particle image velocimetry (PIV) measurements (wall-normal–streamwise plane) were carried out at streamwise distances up to $162b$, where $b$ is the slot width of the PWJ exit. The nominal PWJ Reynolds number based on exit parameters was $Re_{j}\approx 5940$. Comparisons with a zero-pressure-gradient boundary layer (ZPGBL) at nominally matched friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ were also carried out as appropriate, to highlight key features of the PWJ structure. Consistent with previous work, the PWJ showed a dependence of the peak turbulent stresses on the jet exit Reynolds number. The turbulent production showed a peak corresponding to the near-wall cycle similar to the peak seen in the ZPGBL. However, another turbulent production peak was observed in the outer free-shear layer that was an order of magnitude larger than the inner one. Along with the change in sign of the viscous and Reynolds shear stresses, the PWJ was shown to have a region of very low turbulent production between these two peaks. The dissipation rate increased over the PWJ layer with a peak also in the outer region. Visualizations of the flow and two-point correlations reveal that the most energetic large-scale structures within a PWJ are vortical motions in the wall-normal–streamwise plane similar to those structures seen in free-shear layers. These structures are referred to as J (for jet) type structures. In addition two-point correlations reveal the existence of large-scale structures in the wall region which have a signature similar to those structures seen in canonical boundary layers. These structures are referred to as W (for wall) type structures. Instantaneous PIV realizations and flow visualizations reveal that these W type large-scale features are consistent with the paradigm of hairpin vortex packets in the wall region. The J type structures were seen to intrude well into the wall region while the W type structures were also seen to extend into the outer shear layer. Further, these large-scale structures were shown to modulate the amplitude of the finer scales of the flow.
A kinetic-based hyperbolic two-fluid model for binary hard-sphere mixtures
- Rodney O. Fox
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- 19 August 2019, pp. 282-329
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Starting from coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures is derived with separate mean velocities and energies for each phase. In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, using an elastic hard-sphere collision model, and the Archimedes (buoyancy) force due to spatial gradients of the pressure in each phase, as well as other forces involving spatial gradients (e.g. lift). In the derivation, the particles in a given phase have identical mass and volume, and have no internal degrees of freedom (i.e. the particles are adiabatic). The ‘hard-sphere-fluid’ phase is obtained in the limit where the particle diameter in one phase tends to zero with fixed phase density so that the number of fluid particles tends to infinity. The moment system resulting from the two BE kinetic equations is closed at second order by invoking the anisotropic Gaussian closure. The resulting two-fluid model for a binary, hard-sphere mixture therefore consists (for each phase $\unicode[STIX]{x1D6FC}=1,2$) of transport equations for the mass $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}$, mean momentum $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}$ (where $\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}$ is the velocity) and a symmetric, second-order, kinetic energy tensor $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D640}_{\unicode[STIX]{x1D6FC}}=\frac{1}{2}\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}\otimes \boldsymbol{u}_{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}})$. The trace of the fluctuating energy tensor $\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}}$ is $\text{tr}(\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}})=3\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FC}}$ where $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FC}}$ is the phase temperature (or granular temperature). Thus, $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}E_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\text{tr}(\unicode[STIX]{x1D640}_{\unicode[STIX]{x1D6FC}})$ is the total kinetic energy, the sum over $\unicode[STIX]{x1D6FC}$ of which is the total kinetic energy of the system, a conserved quantity. From the analysis, it is found that the BE finite-size correction leads to exact phase pressure (or stress) tensors that depend on the mean-slip velocity $\boldsymbol{u}_{12}=\boldsymbol{u}_{1}-\boldsymbol{u}_{2}$, as well as the phase temperatures for both phases. These pressure tensors also appear in the momentum-exchange terms in the mean momentum equations that produce the Archimedes force, as well as drag contributions due to fluid compressibility and a lift force due to mean fluid-velocity gradients. The closed BE energy flux tensors show a similar dependence on the mean-slip velocity. The characteristic polynomial of the flux matrix from the one-dimensional model is computed symbolically and depends on five parameters: the particle volume fractions $\unicode[STIX]{x1D711}_{1}$, $\unicode[STIX]{x1D711}_{2}$, the phase density ratio ${\mathcal{Z}}=\unicode[STIX]{x1D70C}_{f}/\unicode[STIX]{x1D70C}_{p}$, the phase temperature ratio $\unicode[STIX]{x1D6E9}_{r}=\unicode[STIX]{x1D6E9}_{2}/\unicode[STIX]{x1D6E9}_{1}$ and the mean-slip Mach number $Ma_{s}=\boldsymbol{u}_{12}/\sqrt{5\unicode[STIX]{x1D6E9}_{1}/3}$. By applying Sturm’s Theorem to the characteristic polynomial, it is demonstrated that the model is hyperbolic over a wide range of these parameters, in particular, for the physically most relevant values.
Adjoint sensitivity and optimal perturbations of the low-speed jet in cross-flow
- Marc A. Regan, Krishnan Mahesh
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- 22 August 2019, pp. 330-372
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The tri-global stability and sensitivity of the low-speed jet in cross-flow are studied using the adjoint equations and finite-time horizon optimal disturbance analysis at Reynolds number $Re=2000$, based on the average velocity at the jet exit, the jet nozzle exit diameter and the kinematic viscosity of the jet, for two jet-to-cross-flow velocity ratios $R=2$ and $4$. A novel capability is developed on unstructured grids and parallel platforms for this purpose. Asymmetric modes are more important to the overall dynamics at $R=4$, suggesting increased sensitivity to experimental asymmetries at higher $R$. Low-frequency modes show a connection to wake vortices. Adjoint modes show that the upstream shear layer is most sensitive to perturbations along the upstream side of the jet nozzle. Lower frequency downstream modes are sensitive in the cross-flow boundary layer. For $R=2$, optimal analysis reveals that for short time horizons, asymmetric perturbations dominate and grow along the counter-rotating vortex pair observed in the cross-section. However, as the time horizon increases, large transient growth is observed along the upstream shear layer. When $R=4$, the optimal perturbations for short time scales grow along the downstream shear layer. For long time horizons, they become hybrid modes that grow along both the upstream and downstream shear layers.
Consistent nonlinear deterministic and stochastic wave evolution equations from deep water to the breaking region
- T. Vrecica, Y. Toledo
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- 22 August 2019, pp. 373-404
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Modelling the evolution of the wave field in coastal waters is a complicated task, partly due to triad nonlinear wave interactions, which are one of the dominant mechanisms in this area. Stochastic formulations already implemented into large-scale operational wave models, whilst very efficient, are one-dimensional in nature and fail to account for the majority of the physical properties of the wave field evolution. This paper presents new two-dimensional (2-D) formulations for the triad interactions source term. A quasi-two-dimensional deterministic mild slope equation is improved by including dissipation and first-order spatial derivatives in the nonlinear part of equation, significantly enhancing the accuracy in the breaking zone. The newly defined deterministic model is used to derive an updated stochastic model consistent from deep waters to the breaking region. It is localized following the approach derived in Vrecica & Toledo (J. Fluid Mech., vol. 794, 2016, pp. 310–342), to which several improvements are also presented. The model is compared to measurements of breaking and non-breaking spectral evolution, showing good agreement in both cases. Finally, the model is used to analyse several interesting 2-D properties of the shoaling wave field including the evolution of directionally spread seas.
Common features between the Newtonian laminar–turbulent transition and the viscoelastic drag-reducing turbulence
- Anselmo S. Pereira, Roney L. Thompson, Gilmar Mompean
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- 27 August 2019, pp. 405-428
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The transition from laminar to turbulent flows has challenged the scientific community since the seminal work of Reynolds (Phil. Trans. R. Soc. Lond. A, vol. 174, 1883, pp. 935–982). Recently, experimental and numerical investigations on this matter have demonstrated that the spatio-temporal dynamics that are associated with transitional flows belong to the directed percolation class. In the present work, we explore the analysis of laminar–turbulent transition from the perspective of the recent theoretical development that concerns viscoelastic turbulence, i.e. the drag-reducing turbulent flow obtained from adding polymers to a Newtonian fluid. We found remarkable fingerprints of the variety of states that are present in both types of flows, as captured by a series of features that are known to be present in drag-reducing viscoelastic turbulence. In particular, when compared to a Newtonian fully turbulent flow, the universal nature of these flows includes: (i) the statistical dynamics of the alternation between active and hibernating turbulence; (ii) the weakening of elliptical and hyperbolic structures; (iii) the existence of high and low drag reduction regimes with the same boundary; (iv) the relative enhancement of the streamwise-normal stress; and (v) the slope of the energy spectrum decay with respect to the wavenumber. The maximum drag reduction profile was attained in a Newtonian flow with a Reynolds number near the boundary of the laminar regime and in a hibernating state. It is generally conjectured that, as the Reynolds number increases, the dynamics of the intermittency that characterises transitional flows migrate from a situation where heteroclinic connections between the upper and the lower branches of solutions are more frequent to another where homoclinic orbits around the upper solution become the general rule.
Vorticity dynamics in a spatially developing liquid jet inside a co-flowing gas
- A. Zandian, W. A. Sirignano, F. Hussain
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- Published online by Cambridge University Press:
- 27 August 2019, pp. 429-470
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A three-dimensional transient round liquid jet within a low-speed coaxial outer gas flow is numerically simulated and analysed via vortex dynamics ($\unicode[STIX]{x1D706}_{2}$ analysis). Two types of surface deformations are distinguished, which are separated by a large indentation on the jet stem. First, there are those inside the recirculation zone behind the leading cap – directly affecting the cap dynamics and well explained by the local vortices. Second, deformations upstream of the cap are mainly driven by the Kelvin–Helmholtz (KH) instability, unaffected by the vortices in the behind-the-cap region (BCR), and are important in the eventual atomization process. Different atomization mechanisms are identified and are delineated on a gas Weber number ($We_{g}$) versus liquid Reynolds number ($Re_{l}$) map based on the relative gas–liquid velocity. In a frame moving with the liquid velocity, this result is consistent with prior temporal studies. A simpler and clearer portrait of similarity of the atomization domains is shown by using the relative gas–liquid axial velocity, i.e. $We_{r}$ and $Re_{r}$, and avoiding the widely used velocity ratio as a third key parameter. A detailed comparison of vorticity along the axis in an Eulerian frame versus a frame fixed to a surface wave reveals that the vortex development and surface deformations are periodic in the upstream region, but this periodicity is lost closer to the BCR. In the practical range of the density ratio and for early times in the process, axial vorticity is mainly generated by baroclinicity while streamwise vortex stretching becomes more important at later times and only at lower relative velocities when pressure gradients are reduced. The inertia, vortex, pressure, viscous and surface tension forces are analysed to delineate the dominant causes of the three-dimensional instability of the axisymmetric KH structure due to surface acceleration in the axial, radial and azimuthal directions. The inertia force related to the axial gradient of kinetic energy is the main cause of the axial acceleration of the waves, while the azimuthal acceleration is mainly caused by the pressure and viscous forces. The viscous forces are negligible in the radial direction and away from the nozzle exit in the axial direction. It is interesting to note that azimuthal viscous forces are important even at high $Re_{l}$, indicating that inertia is not totally dominant in this instability occurring early in the atomization cascade.