Fluid Dyn Research Articles

The effect of obstacle length and height in subcritical free-surface flow

Two-dimensional free-surface flow past a submerged rectangular disturbance in an open channel is considered. The forced Korteweg–de Vries model of Binder et al. (Theor Comput Fluid Dyn 20:125–144, 2006) is modified to examine the effect of varying obstacle length and height on the response of the free-surface. For a given obstacle height and flow rate in the subcritical flow regime an analysis of the steady solutions in the phase plane of the problem determines a countably infinite set of discrete obstacle lengths for which there are no waves downstream of the obstacle. A rich structure of nonlinear behaviour is also found as the height of the obstacle approaches critical values in the steady problem. The stability of the steady solutions is investigated numerically in the time-dependent problem with a pseudospectral method.

Global Well-Posedness for the Thermodynamically Refined Passively Transported Nonlinear Moisture Dynamics with Phase Changes

In this work we study the global solvability of moisture dynamics with phase changes for warm clouds. We thereby in comparison to previous studies (Hittmeir et al. in Nonlinearity 30:3676–3718, 2017) take into account the different gas constants for dry air and water vapor as well as the different heat capacities for dry air, water vapor and liquid water, which leads to a much stronger coupling of the moisture balances and the thermodynamic equation. This refined thermodynamic setting has been demonstrated to be essential, e.g. in the case of deep convective cloud columns in Hittmeir and Klein (Theoret Comput Fluid Dyn 32(2):137–164, 2017). The more complicated structure requires careful derivations of sufficient a priori estimates for proving global existence and uniqueness of solutions.

Analytical Properties for a Stochastic Rotating Shallow Water Model Under Location Uncertainty

The rotating shallow water model is a simplification of oceanic and atmospheric general circulation models that are used in many applications such as surge prediction, tsunami tracking and ocean modelling. In this paper we introduce a class of rotating shallow water models which are stochastically perturbed in order to incorporate model uncertainty into the underlying system. The stochasticity is chosen in a judicious way, by following the principles of location uncertainty, as introduced in Mémin (Geophys Astrophys Fluid Dyn 108(2):119–146, 2014). We prove that the resulting equation is part of a class of stochastic partial differential equations that have unique maximal strong solutions. The methodology is based on the construction of an approximating sequence of models taking value in an appropriately chosen finite-dimensional Littlewood-Paley space. Finally, we show that a distinguished element of this class of stochastic partial differential equations has a global weak solution.

Exact solutions for steadily travelling water waves with submerged point vortices

This paper presents a novel theoretical framework, based on the concept of the Schwarz function of a wave, for understanding water waves with vorticity in the absence of gravity and capillarity. The framework leads naturally to a taxonomy of three subcases, herein referred to as cases 1, 2 and 3, into which fall three existing studies of water waves incorporating uniform vorticity and submerged point vortices. This provides a theoretical unification of several seemingly unrelated results in the literature. It also provides a route to finding new exact solutions with this paper focussing on new solutions falling within the case 2 category. Among several presented here are a submerged point vortex pair cotravelling with a solitary deep-water wave, von Kármán point vortex streets cotravelling with a periodic deep-water wave and a point vortex row cotravelling with a wave in water of finite depth. Some other more exotic waveforms are also constructed. All these new solutions generalize those of Crowdy & Roenby (Fluid Dyn. Res., vol. 46, 2014) who found steady waves in deep water cotravelling with a submerged point vortex row for which the free surface shapes turn out to coincide with those of pure capillary waves on deep water found by Crapper (J. Fluid Mech., vol. 2, 1957). The new exact solutions are likely to provide a useful basis for asymptotic or numerical studies when additional effects such as gravity and capillarity are incorporated.

A study of initial conditions' effects on thermal properties of compressible homogeneous sheared non-isentropic turbulence using rapid distortion theory (RDT)

This paper investigates the effect of initial conditions characterized by compressibility of turbulence on the changes in scalar such as density, temperature, and pressure within the framework of rapid distortion theory (RDT) in the case of non-isentropic turbulence. This study is a follow-up of the basic work carried out by A. Simone, G.N. Coleman and C. Cambon [J. Fluid Mech. 330, 307 (1997)] in the case of quasi-isentropic turbulence and the previous work of M. Riahi and T. Lili [Fluid Dyn. Res. 52, 025501 (2020)] in the case of non-isentropic turbulence. RDT was used to examine the behavior of the root mean square (rms) fluctuations of density, temperature, and pressure. The coupling between these rms quantities, the partition factor, and the polytropic coefficient was also studied. RDT equations were solved numerically using a code that solves directly evolution equations of two-point spectral correlations for compressible homogeneous sheared non-isentropic turbulence. The RDT analysis was carried out for various initial turbulent Mach numbers Mt 0 ranging from 0.1 to 0.4, and the initial compressible turbulence is to be one of the three states concerning the fraction of kinetic energy χ0: solenoidal ( χ0 = 0), mixed ( χ0 = 0.6), and dilatational ( χ0 = 1) ( χ0 is the ratio of the initial dilatational kinetic energy to the initial total kinetic energy). It is shown from this study that the changes in scalars are strongly dependent on the initial conditions. Magnitudes and asymptotic values of rms thermodynamics fluctuations and correlations between these thermodynamics fluctuations depend of Mt 0. For large times, the isentropic state of the flow is well observed whatever Mt 0 and χ0.

Existence and Uniqueness of Isothermal, Slightly Compressible Stratified Flow

We show well-posedness for the equations describing a new model of slightly compressible fluids. This model was recently rigorously derived in Grandi and Passerini (Geophys Astrophys Fluid Dyn, 2020) from the full set of balance laws and falls in the category of anelastic Navier–Stokes fluids. In particular, we prove existence and uniqueness of global regular solutions in the two-dimensional case for initial data of arbitrary “size”, and for “small” data in three dimensions. We also show global stability of the rest state in the class of weak solutions.

Influence of Swirl Flow on Combustion and Emissions in Spark-Ignition Experimental Engine

AbstractThis paper presents a numerical study of swirl flow effects on the combustion and emissions in single-cylinder spark ignition engine. First, a three-dimensional (3D) computational fluid dyn...

Self-organizing effect of drift term against diffusion term in point vortex system evidenced by numerical simulations on PEZY-SC

We have analytically obtained a kinetic equation for a two-dimensional (2D) point vortex system with a Fokker–Planck type collision term consisting of a diffusion term and a drift term (Yatsuyanagi and Hatori 2015 Fluid Dyn. Res. 47 065506). The equation describes a mechanism of a self-organization in the system. In this paper, analytically anticipated characteristics of the collision term is evidenced by numerical simulations on PEZY-SC supercomputer system. In the previous paper, it is shown that the collision term satisfies following physically good properties: (1) charge separated, self-organized distribution typical for negative absolute temperature is achieved by the drift term, (2) when a system reaches a thermal equilibrium state, stream function ψ and vorticity ω satisfies an inequality which is important for the next property, (3) the obtained kinetic equation conserves total mean field energy and satisfies the H theorem. In this paper we will evidence the above properties numerically. The time-asymptotic distribution of the vortices corresponds to the analytically anticipated local equilibrium state in which many small regions with different local temperatures exist. Especially, the transport process by the drift term is elucidated clearly, which gives a key mechanism of the self-organization, i.e. condensation of the same-sign vortices.

Energy Partitioning Control in the PITM Hybrid RANS/LES Method for the Simulation of Turbulent Flows

The partially integrated transport modeling (PITM) method first introduced in Refs. Schiestel and Dejoan (Theor Comput Fluid Dyn 18:443, 2005) and in Chaouat and Schiestel (Phys Fluids 17:065106, 2005) provides a continuous approach for hybrid RANS-LES simulations. Inspired from the multiple scale approach, the basis of the development of the method is the spectral space in quasi-homogeneous turbulence. The PITM method embodies a partitioning control function that monitors the ratio of subfilter energy to total turbulent energy by reference to the cutoff wavenumber location. How this procedure behaves in inhomogeneous flows is an important question. The present paper demonstrates that the same control function can be used both in homogeneous and in non-homogeneous flows as well. Further on, an analysis of the effect of anisotropic filters generally used for wall flows is conducted for computing the equivalent cutoff wavenumber suitable for determining the subfilter energy of the spectrum and see how it interferes with control function. Illustrations in the turbulent plane channel flow are given that confirm the efficiency of the procedure and DNS data have been used to support and supplement the discussion.

A Reynolds shear stress model for constant-pressure boundary layers

Closure models for the Reynolds shear stress in wall-bounded flows are mostly based on various turbulence quantities like the turbulent kinetic energy (k), its dissipation rate (ϵ), the Reynolds stress (vr2) in the wall-normal direction, the mean flow velocity gradient (S), the normalized strain rate (Sk/ϵ), etc.; vr is the intensity of the fluctuating velocity component in the wall-normal direction (y). Close to the wall, (u′v′¯) is known to vary with y approximately as ∼y3. With an emphasis on this near-wall feature, a closure model is proposed here in terms of k, (vr/ur), and a weighted mean strain rate (Skvr+/ϵ); ur is the intensity of the fluctuating velocity component in the streamwise direction. The choice of ur in the present proposal is in view of the fact that, although the energy associated with ur2 in a simple flow, for example, is distributed to vr2 by fluctuating pressure, it has never been used in past closure models. In terms of the known approximate variations of k,vr,ur, etc., with y in the near-wall region, the present model suggests (u′v′¯)∼y3.25 and so nearly retains its known approximate variation ∼y3 (for small y). It is suggested here that vr+ in the weighted strain rate damps the strain rate (Sk/ϵ) significantly in the near-wall region and plays a similar role in Durbin's model [“Near-wall turbulence closure modeling without damping functions,” Theor. Comput. Fluid Dyn. 3, 1 (1991)]. It is also reported here that vr≈k1/2 and ur≈(ϵ/S)1/2 in the outer region, and these velocity scales are associated with (Sk/ϵ)∼ constant in this region; alternatively, in terms of these velocity scales, the normalized strain rate (Sk/ϵ) is the ratio of the Reynolds stresses (vr2/ur2) in the outer region.

Extension of the partially integrated transport modeling method to the simulation of passive scalar turbulent fluctuations at various Prandtl numbers

The present work aims to extend the hybrid non zonal RANS/LES partially integrated transport modeling (PITM) method to turbulent flows in the presence of passive scalar contaminant for simulating large scales of turbulent flows. Focussing on the methodological aspects, we derive the basic transport equations both for the scalar variance of fluctuations and dissipation-rate of the variance. The basis of the method was introduced in references [R. Schiestel and A. Dejoan, “Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations ”, Theor. Comput. Fluid Dyn. 18, 443 (2005)] and [B. Chaouat and R. Schiestel, “A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows ”, Phys. Fluids 17, 065106 (2005)]. It provides a continuous approach for hybrid Reynolds averaged Navier–Stokes equations-large eddy simulation (RANS-LES) with seamless coupling between RANS and LES regions. The main motivation is to simulate accurately in LES mode performed on coarse grids, scalar fluctuation fields, in addition to mean scalar fields. As known, the knowledge of the rms scalar fluctuations is often involved in handling practical engineering and geophysical flows. Like in dynamical equations, it is found that the coefficient appearing in the destruction term of the dissipation-rate of the scalar variance is a function of the cutoff-wave number and also of the Prandtl number and the Reynolds number. Depending on the value of the Prandtl-number, different expressions of this function have been derived according to the relevant physics in the wave number space. Finally, numerical simulations of fully turbulent flows including passive scalar transport fields have been performed on several meshes of medium and coarse grid resolutions at the Reynolds number Rτ=395 for the Prandtl numbers Pr=0.1, 1 and 10, respectively associated with heat transfer of liquid metals, gas and water for illustrating the theoretical development made on PITM. As expected, the PITM method provides satisfactory results in good agreement with data of direct numerical simulations. From a general point of view, this work opens new routes of modeling and simulation of turbulent flows including a passive scalar with a drastic reduction of the computational time and memory in term of number of grid points in comparison with the demanding resources of highly resolved LES.

Steady flow in a rapidly rotating spheroid with weak precession: Part 2

This is a continuation of our companion paper (Kida 2020b Fluid Dyn. Res.) in which the steady flow in a precessing spheroid is studied in the strong spin and weak precession limit, and the elliptic flow with uniform vorticity, the direction of which is perpendicular to the spin axis on the plane spanned by the spin and the precession axes, is shown to be excited in the interior inviscid region. In this paper we examine the boundary-layer flow connected to this elliptic flow and also the higher-order inviscid flow modified by this boundary-layer flow. The explicit expression of the velocity and pressure fields is obtained for arbitrary aspect ratio c of the polar and equatorial radii of the spheroid. The boundary-layer approximation breaks down at two critical circles on which the boundary-layer thickness as well as the normal component of velocity diverge to infinity. The flow field in the neighborhood (called the critical regions) of the critical circles is analyzed to find the same structure, up to scales, as that for a sphere. The radial component of velocity in the critical region induces the conical shear layers along the characteristic cones in the inviscid region. They reflect on the spheroid surface, and the reflection takes place endlessly in general. On the other hand, it is proved that for a spheroid of which the aspect ratio is expressed as (m/n being an irreducible fraction) the reflection terminates at (n − 2) times and the conical shear layers form a regular pattern. Since such special values of c are distributed densely over all the positive numbers, we may observe practically always a regular pattern of the conical shear layers. Such closed conical shear layers are clearly visualized by the pressure field numerically calculated in the asymptotic formulation. Furthermore, the total angular momentum of the steady flow is expressed explicitly up to the non-trivial leading order for all the three components over the whole range of the aspect ratio.

Effect of liquid elasticity on the behaviour of high-speed focused jets

We investigate the effect of highly contrasting non-Newtonian liquid properties on the formation of liquid jets with a focused shape. By using two nozzle-free ejection techniques, mechanically impact- and laser-induced, fast jets of a highly elastic (sodium polyacrylate) and weakly elastic (xanthan-gum) diluted polymer solutions are generated. A unique high-speed effect is encountered at the jet ejection onset of the highly elastic solution. Its jet-tip speed is on average 1.4 times faster in comparison to a Newtonian (glycerin/water) and the weakly elastic liquids. We explain this effect occurring as a result of the high viscoelasticity of the sodium polyacrylate solution. Additionally, a ‘bungee jumper’ jet behaviour (Morrison and Harlen in Rheol Acta 49(6):619–632, 2010) is observed in a regime of high speed (10<V_j<40 m/s) and high viscosity (mu >20 mPa s) not previously examined. We additionally characterise the viscoelastic non-breakup jet limit using the Bazilevskii et al. (Fluid Dyn 40(3):376–392, 2005) ejection criterion. Herein, the extensional rheological parameters are measured implementing a novel DoS-CaBER technique (Dinic et al. in Lab Chip 17(3):460–473, 2017). Our findings may influence results of inkjet printing technologies and recent nozzle-free ejection systems for ejecting liquids with non-Newtonian properties.Graphical abstract

Optimizing the performance of an active grid to generate high intensity isotropic free stream turbulence

We present an optimized procedure for generating isotropic, homogeneous, high intensity turbulent flows behind an active grid. The double random protocol (R. E. G. Poorte, “On the motion of bubbles in active grid generated turbulent flows,” Ph.D. thesis, University of Twente, 1998 and subsequent works) used for operating the active grid [H. Makita, “Realization of a large-scale turbulence field in a small wind tunnel,” Fluid Dyn. Res. 8, 53 (1991)] is optimized based on the tip speed ratio α ≡Vtip⟨U⟩=2πΩC⟨U⟩, where Vtip is the speed of the active grid winglet at its tip, 2C is the winglet chord length, Ω is the mean winglet rotational speed, and U is the mean flow speed. A systematic assessment of the resulting turbulent flow is carried out using both hot-wire anemometry (single point and two point) and particle image velocimetry (PIV). It is shown that just using the double random protocol does not guarantee an isotropic flow, and the tip speed ratio of the active grid elements is also important to achieve isotropy. Too high a tip speed ratio, however, results in decreased turbulence intensities. Optimal values of the tip speed ratio, given the trade-off between isotropy and turbulence intensity, are given for the double random protocol. Furthermore, our measurements clearly show that conclusions about the structure and isotropy of active-grid turbulence cannot be based on single-point measurements alone; at the least, two-point or multi-point PIV measurements are essential to evaluate the performance of active grids. This is supported by observing the behavior of longitudinal and transverse correlation functions along the arbitrary directions and the convergence of proper orthogonal decomposition reconstructions to the true value. Interestingly, a simpler test based on the ratio of longitudinal to transverse turbulence intensities appears more robust than the one based on single-point correlations.

Regimes of thermo-compositional convection and related dynamos in rotating spherical shells

Convection and magnetic field generation in the Earth and planetary interiors are driven by both thermal and compositional gradients. In this work numerical simulations of finite-amplitude double-diffusive convection and dynamo action in rapidly rotating spherical shells full of incompressible two-component electrically-conducting fluid are reported. Four distinct regimes of rotating double-diffusive convection identified in a recent linear analysis (Silva, Mather and Simitev, Geophys. Astrophys. Fluid Dyn. 2019, 113, 377) are found to persist significantly beyond the onset of instability while their regime transitions remain abrupt. In the semi-convecting and the fingering regimes characteristic flow velocities are small compared to those in the thermally- and compositionally-dominated overturning regimes, while zonal flows remain weak in all regimes apart from the thermally-dominated one. Compositionally-dominated overturning convection exhibits significantly narrower azimuthal structures compared to all other regimes while differential rotation becomes the dominant flow component in the thermally-dominated case as driving is increased. Dynamo action occurs in all regimes apart from the regime of fingering convection. While dynamos persist in the semi-convective regime they are very much impaired by small flow intensities and very weak differential rotation in this regime which makes poloidal to toroidal field conversion problematic. The dynamos in the thermally-dominated regime include oscillating dipolar, quadrupolar and multipolar cases similar to the ones known from earlier parameter studies. Dynamos in the compositionally-dominated regime exhibit subdued temporal variation and remain predominantly dipolar due to weak zonal flow in this regime. These results significantly enhance our understanding of the primary drivers of planetary core flows and magnetic fields.

Stability and evolution of two opposite-signed quasi-geostrophic shallow-water vortex patches

We examine the equilibrium forms, linear stability and nonlinear evolution of two patches having opposite-signed, uniform potential vorticity anomalies in a single-layer shallow-water flow, under the quasi-geostrophic approximation. We widely vary the vortex area ratio, the potential vorticity anomaly ratio, as well as the Rossby deformation length to unravel the full complexity of possible interactions in this system. Opposite-signed vortex interactions turn out to be far richer than their like-signed counterparts, comprehensively examined in a previous study (Jalali and Dritschel 2018, Geophys. Astrophys. Fluid Dyn. 2018, 112, 375). Unstable equilibria may evolve into a myriad of forms, many unsteady and aperiodic, and the original two vortex patches may break up into many patches which survive for long times, perhaps indefinitely.

Streaming controlled by meniscus shape

Surface waves called meniscus waves often appear in systems that are close to the capillary length scale. Since the meniscus shape determines the form of the meniscus waves, the resulting streaming circulation has features distinct from those caused by other capillary–gravity waves recently reported in the literature. In the present study, we produce symmetric and antisymmetric meniscus shapes by controlling boundary wettability and excite meniscus waves by oscillating the meniscus vertically. The symmetric and antisymmetric configurations produce different surface capillary–gravity wave modes and streaming flow structures. The root-mean-square speed of the streaming circulation increases with the second power of the forcing amplitude in both configurations. The flow symmetry of streaming circulation is retained under the symmetric meniscus, while it is lost under the antisymmetric meniscus. The streaming circulation pattern beneath the meniscus observed in our experiments is qualitatively explained using the method introduced by Nicolas & Vega (Fluid Dyn. Res., vol. 32 (4), 2003, pp. 119–139) and Gordillo & Mujica (J. Fluid Mech., vol. 754, 2014, pp. 590–604).

Dynamo action in sliding plates of anisotropic electrical conductivity.

With materials of anisotropic electrical conductivity, it is possible to generate a dynamo with a simple velocity field, of the type precluded by Cowling's theorems with isotropic materials. Following a previous study by Ruderman and Ruzmaikin [M. S. Ruderman and A. A. Ruzmaikin, Magnetic field generation in an anisotropically conducting fluid, Geophys. Astrophys. Fluid Dyn. 28, 77 (1984)GAFDD30309-192910.1080/03091928408210135], who considered the dynamo effect induced by a uniform shear flow, we determine the conditions for the dynamo threshold when a solid plate is sliding over another one, both with anisotropic electrical conductivity. We obtain numerical solutions for a general class of anisotropy and obtain the conditions for the lowest magnetic Reynolds number, using a collocation Chebyshev method. In a particular geometry of anisotropy and wave number, we also derive an analytical solution, where the eigenvectors are just combinations of four exponential functions. An explicit analytical expression is obtained for the critical magnetic Reynolds number. Above the critical magnetic Reynolds number, we have also derived an analytical expression for the growth rate showing that this is a "very fast" dynamo, extrapolating on the "slow" and "fast" terminology introduced by Vainshtein and Zeldovich [S. I. Vainshtein and Y. B. Zeldovich, Reviews of topical problems: Origin of magnetic fields in astrophysics (turbulent "dynamo" mechanisms), Sov. Phys. Usp. 15, 159 (1972)SOPUAP0038-567010.1070/PU1972v015n02ABEH004960].

Sediment erosion in zero-mean-shear turbulence

Turbulence plays an evident role in particle erosion that in many practical situations superimposes with the action of a mean flow. In this paper, the turbulence effect on particle erosion is studied under zero-mean flow conditions, by using the turbulence generated by an oscillating grid. The stirring grid is located more than two mesh size away from the particle layer. The zero-mean flow below the grid has been qualified by revisiting the k–ε model of Matsunaga et al. [Fluid Dyn. Res. 25, 147–165 (1999)]. The turbulence efficiency on the settling/resuspension of the particles is quantified for various turbulence intensities, varying the size, the nature of the particles, and their buoyancy relative to the fluid. We find that the concentrations C of eroded particles collapse fairly well onto a single trend for C ≤ 5 × 10−2, when plotted as a function of the ratio between the flux of turbulent kinetic energy at the particle bed location and the particle settling flux. Above, the concentrations saturate, thus forming a plateau. Particle erosion mechanisms have been investigated in terms of competing forces within an “impulse approach.” Horizontal drag vs friction first leads to a horizontal motion followed by a vertical motion, resulting from vertical drag and lift vs buoyancy. Particle erosion occurs when both force balances are in favor of motion for a duration of 0.1–0.3 Kolmogorov time scale.

Nonnegative weak solutions of thin-film equations related to viscous flows in cylindrical geometries

Motivated by models for thin films coating cylinders in two physical cases proposed in Kerchman (J Fluid Mech 290:131–166, 1994) and Kerchman and Frenkel (Theor Comput Fluid Dyn 6:235–254, 1994), we analyze the dynamics of corresponding thin-film models. The models are governed by nonlinear, fourth-order, degenerate, parabolic PDEs. We prove, given positive and suitably regular initial data, the existence of weak solutions in all length scales of the cylinder, where all solutions are only local in time. We also prove that given a length constraint on the cylinder, long time and global in time weak solutions exist. This analytical result is motivated by numerical work on related models of Reed Ogrosky (Modeling liquid film flow inside a vertical tube, Ph.D. thesis, The University of North Carolina at Chapel Hill, 2013) in conjunction with the works (Camassa et al. in Phys Rev E 86(6):066305, 2012; Physica D Nonlinear Phenom 333:254–265, 2016; J Fluid Mech 745:682–715, 2014; J Fluid Mech 825:1056–1090, 2017).