Convective Inertia Research Articles

Structure of the streaming flow generated by a sphere in a fluid undergoing rectilinear oscillation

A solid body in a viscous fluid undergoing oscillatory motion naturally produces a steady secondary flow due to convective inertia. This phenomenon is embodied in the streaming flow generated by a sphere in an unbounded fluid executing rectilinear oscillations. We review the considerable literature on this canonical problem and summarise exact and asymptotic formulas in the small-amplitude limit. These analytical formulas are used to explore the characteristic flow structure of this problem and clarify previously unreported features. A single, toroidal-shaped vortex exists in each hemisphere regardless of the oscillation frequency, which can drive a counter-flow away from the sphere. The vortex centre moves monotonically away from the sphere with decreasing oscillation frequency, and engulfs the entire flow domain for $\beta \equiv \omega R^2/\nu < 16.317$ , where $\omega$ is the angular oscillation frequency, $R$ the sphere radius, and $\nu$ the fluid kinematic viscosity. This seemingly abrupt change in flow structure at the critical frequency $\beta _{critical} =16.317$ , and its quantification, appear to have not been reported previously. We perform a direct numerical simulation of the Navier–Stokes equations, to (1) confirm existence of this critical frequency at finite amplitude, and (2) examine its variation with amplitude. This reveals a universal relationship between the critical frequency and oscillation amplitude, clarifying previous reports on the structure of this streaming flow. The critical frequency is shown to be identical for the streaming flow and the cycle-averaged particle paths, establishing that the critical frequency is accessible directly using standard measurements.

Hydrodynamic force on a small squirmer moving with a time-dependent velocity at small Reynolds numbers

We calculate the hydrodynamic force on a small spherical, unsteady squirmer moving with a time-dependent velocity in a fluid at rest, taking into account convective and unsteady fluid inertia effects in perturbation theory. Our results generalise those of Lovalenti & Brady (J. Fluid Mech., vol. 256, 1993, pp. 561–605) from passive to active spherical particles. We find that convective inertia changes the history contribution to the hydrodynamic force, as it does for passive particles. We determine how the hydrodynamic force depends on the swimming gait of the unsteady squirmer. Since swimming breaks the spherical symmetry of the problem, the force is not determined completely by the outer solution of the asymptotic matching problem, as it is for passive spheres. There are additional contributions due to the inhomogeneous solution of the inner problem. We also compute the disturbance flow, illustrating convective and unsteady effects when the particle experiences a sudden start followed by a sudden stop.

Theoretical and numerical analysis of the flow through a diffuser/nozzle element in pulsatile laminar conditions

Diffuser/nozzle pipes produce a directional flow resistance that is often exploited in microcirculation to generate a pumping action. This work presents an approximate time-dependent theoretical solution based on the mechanical energy conservation equation to predict the laminar flow rate through an ideal diffuser/nozzle pump. The theoretical solution is then used to characterize the dimensionless parameters that control the dynamics of the valveless pump in the pulsatile flow regime. A suitable numerical model is also implemented to solve the flow in a parametrized two-dimensional axial-symmetric domain subjected to an oscillating pressure, and its results are used to assess the theoretical solution. The pump dynamics and the main model parameters, such as the energy-loss coefficients, result in the following dependence on the ratios between the viscous force, the advective inertia, and the temporal inertia, i.e., the Reynolds (Red), Womersley (Wod), and Strouhal (St) numbers referred to throat diameter. In particular, The Womersley number plays an essential role in controlling the global energy loss when Red < 100. The flow transition is also investigated and found when Red exceeds a critical value, which increases with Wod. Finally, the pump efficiency is found to reach its maximum when the convective and temporal inertia become comparable, i.e., St=O(1), consistent with the observed range of St in real-world diffuser/nozzle pumps. This optimum range of functioning of the pump is also observed for cerebrospinal pulsatile flow in the Sylvius aqueduct, suggesting that the modeled mechanism is used to promote or enhance cerebrospinal fluid circulation.

Fluid-inertia torque on spheroids in pseudo-plastic fluid flows: effect of shear-thinning rheology

Fluid-inertia torque remarkably affects the orientation of non-spherical particles in Newtonian flows whereas this torque induced by convective fluid inertia in particle-laden pseudo-plastic flows is still unknown. In the present study we numerically investigate the fluid-inertia torque on a neutrally buoyant spheroid in the Carreau-type pseudo-plastic fluid flows at finite Reynolds numbers with the immersed boundary method. The results show that compared with the fluid-inertia torque in Newtonian flows, the magnitude of the fluid-inertia torque on spheroids is remarkably attenuated by the shear-thinning rheology in pseudo-plastic fluid flows. The deviation of fluid-inertia torque between pseudo-plastic and Newtonian flows is more significant with decreasing Reynolds numbers, indicating the importance of the effect of shear-thinning rheology at small Reynolds numbers. Moreover, the spheroid rotation rate is reduced in pseudo-plastic fluids, and the equilibrium orientation of oblate spheroids changes non-monotonically with the shear-thinning effect in the linear shear flow of pseudo-plastic fluids. The present findings imply the importance of the effect of shear-thinning rheology on the torques of spheroids, which could be potentially applied for the control of particle orientations in pseudo-plastic fluids in the future.

Rapidly rotating self-gravitating Boussinesq fluid. II. Onset of thermal inertial convection in oblate spheroidal cavities

For decades researchers have been studying the problem of thermal instabilities in rotating spheres or spherical shells. A key assumption is that the geometrical flattening due to centrifugal force can be neglected, which seems to be a good approximation for many planets and stars. However, this has never been tested. In this paper, the linear stability onset problem at low Prandtl number is considered in oblate spheroids, whose shapes are consistent with figure theory, and it is compared with the previously known spherical system solutions.

Unsteady and inertial dynamics of a small active particle in a fluid

It is well known that the reversibility of Stokes flow makes it difficult for small microorganisms to swim. Inertial effects break this reversibility, allowing new mechanisms of propulsion and feeding. Therefore it is important to understand the effects of unsteady and fluid inertia on the dynamics of microorganisms in flow. In this work, we show how to translate known inertial effects for non-motile organisms to motile ones, from passive to active particles. The method relies on a principle used earlier by Legendre and Magnaudet (1997) to deduce inertial corrections to the lift force on a bubble from the inertial drag on a solid sphere, using the fact that small inertial effects are determined by the far field of the disturbance flow. The method allows for example to compute the inertial effect of unsteady fluid accelerations on motile organisms, and the inertial forces such organisms experience in steady shear flow. We explain why the method fails to describe the effect of convective fluid inertia.

Buoyancy effect on the stagnation point flow of a hybrid nanofluid toward a vertical plate in a saturated porous medium

The mixed convection on a stagnation-point flow of a thermo micropolar hybrid nanofluid through a vertical surface in a saturated porous medium having inertial and microstructure characteristics is investigated using Darcy–Brinkman model. The impact of convective inertia along with the porous-Forchheimer inertia is also considered. This study is designed to distinguish the influence of the two distinct nanoparticles such as magnesium oxide (MgO) nanoparticle and Silver (Ag) nanoparticle along with the regular base fluid (water) to form the MgO–Ag/water hybrid nanofluid. These hybrid nanofluids can be traced back to their unique enhancement in the rate of heat transfer and improvement in thermal performance as applicable in the dynamics of fuel and coolant in the automobile. The leading equations of the present problem are transformed into a system of ordinary differential equations by the use of appropriate similarity transformations. We employ a MATLAB solver called the boundary-value problem of fourth-order (bvp4c) to solve the resulting equations. The outcomes are unique for the buoyancy aiding flow (λc>0) while dual or multiple outcomes are marked and existed under buoyancy opposing flow (λc<0).

RESEARCH OF EXTERNAL MASS TRANSFER PROCESSES FOR ADSORPTION FROM SOLUTIONS IN A APPARATUS WITH STIRRING

A study has been mode of transport-controlled mass transfer-controlled to particles suspended in a stirred vessel. The motion of particle in a fluid was examined and a method of predicting relative velocities in terms of Kolmogoroff’s theory of local isotropic turbulence for mass transfer was outlined.&#x0D; To provide a more concrete visualization of complex wave form of turbulence, the concepts of eddies, of eddy velocity, scale (or wave number) and energy spectrum, have proved convenient.&#x0D; Large scale motions of scale contain almost all of the energy and they are directly responsible for energy diffusion throughout the stirring vessel by kinetic and pressure energies. However, almost no energy is dissipated by the large-scale energy-containing eddies. A scale of motion less than is responsible for convective energy transfer to even smaller eddy sires. At still smaller eddy scales, close to a characteristic microscale, both viscous energy dissipation and convection are the rule. The last range of eddies has been termed the universal equilibrium range. It has been further divided into a low eddy size region, the viscous dissipation subrange, and a larger eddy size region, the inertial convection subrange.&#x0D; Measurements of energy spectrum in mixing vessel are shown that there is a range, where the so called -(5/3) power law is effective. Accordingly, the theory of local isotropy of Kolmogoroff can be applied because existence of the internal subrange. As the integrated value of local energy dissipation rate agrees with the power per unit mass of liquid from the impeller, almost all energy from the impeller is viscous dissipated in eddies of microscale.&#x0D; The correlation for mass transfer to particles suspended in a stirred vessel is recommended. The results of experimental study are approximately 12 % above the predicted values.

Fluid Inertia Effect on Spiral-Grooved Mechanical Face Seals Considering Cavitation Effects

An extended Reynolds equation is presented to evaluate the effects of the fluid inertia and cavitation on the performance of spiral-grooved mechanical face seals under high-speed conditions. Segregated finite element procedures are employed to solve the fluid inertia terms and the fluid film pressure distribution with the artificial viscosity stabilization strategy and streamline-upwind/Petrov-Galerkin (SUPG) finite element method. The pressure jump is captured at the sudden variation in film thickness. The fluid inertia effect on the sealing performance is determined at different rotational speeds, groove depths, and spiral angles. The results show that the inertia effect due to the film thickness discontinuities improves the load-carrying capacity and increases the leakage rate. The convective inertia forces in the sealing gap flow promote fluid cavitation. The fluid inertia effect on the sealing performance is closely dependent on the spiral angles.

Onset of Inertial Magnetoconvection in Rotating Fluid Spheres

The onset of convection in the form of magneto-inertial waves in a rotating fluid sphere permeated by a constant axial electric current is studied in this paper. Thermo-inertial convection is a distinctive flow regime on the border between rotating thermal convection and wave propagation. It occurs in astrophysical and geophysical contexts where self-sustained or external magnetic fields are commonly present. To investigate the onset of motion, a perturbation method is used here with an inviscid balance in the leading order and a buoyancy force acting against weak viscous dissipation in the next order of approximation. Analytical evaluation of constituent integral quantities is enabled by applying a Green’s function method for the exact solution of the heat equation following our earlier non-magnetic analysis. Results for the case of thermally infinitely conducting boundaries and for the case of nearly thermally insulating boundaries are obtained. In both cases, explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived in the limit of high thermal diffusivity. It is found that an imposed azimuthal magnetic field exerts a stabilizing influence on the onset of inertial convection and as a consequence magneto-inertial convection with azimuthal wave number of unity is generally preferred.

Inertial forces in the Maxey–Riley equation in nonuniform flows

Here, inertial forces on particles in nonuniform ambient flows, previously missing from the Maxey---Riley formulation of inertial particle dynamics, are identified. These forces involve both local and convective fluid inertia and depend on the curvature of the ambient flow. These new terms provide quantitative corrections and qualitative additions to the Maxey---Riley equation and are relevant when the particle size is appreciable relative to the characteristic length scale of the ambient flow.

Importance of fluid inertia for the orientation of spheroids settling in turbulent flow

How non-spherical particles orient as they settle in a flow has important practical implications in a number of scientific and engineering problems. In a quiescent fluid, a slowly settling particle orients so that it settles with its broad side first. This is an effect of the torque due to convective inertia of the fluid that is set in motion by the settling particle, which maximises the drag experienced by the particle. Turbulent fluid-velocity gradients, on the other hand, tend to randomise the particle orientation. Recently the settling of non-spherical particles in turbulence was analysed neglecting the effect of convective fluid inertia, but taking into account the effect of the turbulent fluid-velocity gradients on the particle orientation. These studies reached the opposite conclusion, namely that the particle tends to settle with its narrow edge first, therefore minimising the drag on the particle. Here, we consider both effects, the convective inertial torque as well as the torque due to fluctuating fluid-velocity gradients. We ask under which circumstances either one or the other dominates. To this end we estimate the ratio of the magnitudes of the two torques. Our estimates suggest that the fluid-inertia torque prevails in high-Reynolds-number flows. In this case non-spherical particles tend to settle with orientations maximising drag. But when the Reynolds number is small, then the torque due to fluid-velocity gradients may dominate, causing the particle to settle with its narrow edge first, minimising the drag.

Numerical issues in gas flow dynamics with hydraulic shocks using high order finite volume WENO schemes

Accurate and efficient simulation of the hydraulic shock phenomenon in pipeline systems is of paramount importance. Even though the conservation-law formulation of the governing equations is here strongly advocated, the nonconservative form is still frequently used. This also concerns its mathematical conservative form. We investigated the numerical consequences of using the compressible gas flow model in the latter form while simulating a hydraulic shock. In this context, we also solved two Riemann problems. For the investigation, we used the third-, fifth- and seventh-order accurate weighted essentially non-oscillatory (WENO) scheme along with the Lax–Friedrichs solver at the cell interfaces. Both the classical finite volume WENO scheme and its modification WENO–Z have been implemented. A procedure based on the method of manufactured solutions has been developed to verify whether the numerical code solved correctly the hyperbolic set of equations. We demonstrated that the solutions of the conservative and nonconservative formulations are similar if we have smooth variations in the solution domain. The convective inertia term in the momentum equation should not be ignored. In the presence of shocks, differences in oscillating behavior and slope steepness near the discontinuities were observed. For the hydraulic shock problem, spurious oscillations appeared while using the nonconservative formulation in combination with the WENO–Z reconstruction.

Squeeze film dampers supporting high-speed rotors: Fluid inertia effects

This work represents closed-form analytical expressions for the operating parameters for short-length open-ended squeeze film dampers, including the lubricant velocity profiles, hydrodynamic pressure distribution, and lubricant reaction forces. The proposed closed-form expressions provide an accelerated calculation of the squeeze film damper parameters, specifically for rotordynamics applications. In order to determine the analytical solutions for the squeeze film damper parameters, the thin film equations for lubricant are introduced in the presence of the influence of lubricant inertia. Subsequently, two different analytical techniques, namely the momentum approximation method, and the perturbation method are applied to the thin film equations. Moreover, the solution for the lubricant flow equations are analytically determined to represent closed-form expressions for the hydrodynamic pressure distribution and the velocity component profiles in squeeze film dampers. Additionally, the expressions for the hydrodynamic pressure distribution are integrated over the journal surface, either numerically or analytically by using Booker’s integrals, to develop expressions for the fluid film reaction forces. Lastly, the developed squeeze film damper models are incorporated into simulation models in Matlab and Simulink®, and the results are compared against a well-established force coefficient model to verify the accuracy of the calculations. The results of the simulations verify the effect of the lubricant inertia components, namely the convective and temporal (i.e., unsteady) inertia components on the squeeze film damper dynamics, including hydrodynamic pressure distribution and fluid film reaction forces. Additionally, the simulation results suggest a close agreement between the proposed models and the results in the literature.

Boundary characteristic orthogonal polynomials method in the vibration analysis of multi-span plates acting upon a moving mass

The Boundary Characteristic Orthogonal Polynomials (BCOP) method is used in this study in order to analyze multi-span plates traversed by a moving inertia load traveling on an arbitrary path with constant velocity. The plate is assumed to be free from any support at the longitudinal edges and the spans are made by simply supported constraints at width, i.e. SFSF. The plate's mode shapes are generated by the BCOP method while the boundary condition is satisfied over all computational modes. A free vibration analysis is done in order to find natural frequency. The governing differential equations of motion are derived by Hamilton's principle and the solution in the time domain is found by using the Matrix Exponential method after modeling the problem in state space. All of the convective inertia terms are included in the acceleration derivatives and the responses are presented both for the load moving on the plate's surface ignoring/including the mass inertia effect. A comprehensive parametric study on the plate's mid-spans is carried out for the single, two- and three-span plates, investigating Dynamic Amplification Factor (DAF) versus non-dimensional velocity (V). The effect of mass and aspect ratio along with the location of reference point of calculation on the dynamic behavior of a multi-span plate is investigated and many graphs are generated as spectra. One can easily find the critical velocity as well as the peak deflection for each case study by introducing a corrective factor. The solution under moving mass excitation is obtained by the factor if the same response for moving load is known.

A new model for textured surface lubrication based on a modified Reynolds equation including inertia effects

For textured surface lubrication problems, a small texture aspect ratio and a large Reynolds number can lead to strong recirculation in the textures and significant convective inertia effects. When these effects cannot be neglected, the Stokes system and the Reynolds equation are unsuitable and the Navier-Stokes equations should be considered instead. Particularly compared with the Reynolds equation, solving the Navier-Stokes system is computationally much more demanding due to its higher spatial dimension and nonlinearity. In this work, a new model is presented based on a modified Reynolds equation taking into account convective inertia and recirculation effects while so far neglecting cavitation. The model provides an accurate and considerably less expensive alternative to Navier-Stokes, demonstrated for two- and three-dimensional numerical configurations.

Heat or mass transport from drops in shearing flows. Part 2. Inertial effects on transport

We analyse the singular effects of weak inertia on the heat (or equivalently mass) transport problem from drops in linear shearing flows. For small spherical drops embedded in hyperbolic planar linear flows, which constitute a one-parameter family (the parameter being $\unicode[STIX]{x1D6FC}$ with $0\leqslant \unicode[STIX]{x1D6FC}\leqslant 1$, and whose extremal members are simple shear ($\unicode[STIX]{x1D6FC}=0$) and planar extension ($\unicode[STIX]{x1D6FC}=1$)), there are two distinct regimes for scalar (heat or mass) transport at large Péclet numbers ($Pe$) depending on the exterior streamline topology (Krishnamurthy &amp; Subramanian, J. Fluid Mech., vol. 850, 2018, pp. 439–483). When the drop-to-medium viscosity ratio ($\unicode[STIX]{x1D706}$) is larger than a critical value, $\unicode[STIX]{x1D706}_{c}=2\unicode[STIX]{x1D6FC}/(1-\unicode[STIX]{x1D6FC})$, the drop is surrounded by a region of closed streamlines in the inertialess limit ($Re=0$, $Re$ being the drop Reynolds number). Convection is incapable of transporting heat away on account of the near-field closed streamline topology, and the transport remains diffusion limited even for $Pe\rightarrow \infty$. However, weak inertia breaks open the closed streamline region, giving way to finite-$Re$ spiralling streamlines and convectively enhanced transport. For $Re=0$ the closed streamlines on the drop surface, for $\unicode[STIX]{x1D706}&gt;\unicode[STIX]{x1D706}_{c}$, are Jeffery orbits, a terminology originally used to describe the trajectories of an axisymmetric rigid particle in a simple shear flow. Based on this identification, a novel boundary layer analysis that employs a surface-flow-aligned non-orthogonal coordinate system, is used to solve the transport problem in the dual asymptotic limit $Re\ll 1$, $RePe\gg 1$, corresponding to the regime where inertial convection balances diffusion in an $O(RePe)^{-1/2}$ boundary layer. Further, the separation of time scales in the aforementioned limit, between rapid convection due to the Stokesian velocity field and the slower convection by the $O(Re)$ inertial velocity field, allows one to average the convection–diffusion equation over the phase of the Stokesian surface streamlines (Jeffery orbits), allowing a simplification of the original three-dimensional non-axisymmetric transport problem to a form resembling a much simpler axisymmetric one. A self-similar ansatz then leads to the boundary layer temperature field, and the resulting Nusselt number is given by $Nu={\mathcal{H}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})(RePe)^{1/2}$ with ${\mathcal{H}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ given in terms of a one-dimensional integral; the prefactor ${\mathcal{H}}(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D706})$ diverges for $\unicode[STIX]{x1D706}\rightarrow \unicode[STIX]{x1D706}_{c}^{+}$ due to assumptions underlying the Jeffery-orbit-averaged analysis breaking down. Although the separation of time scales necessary for the validity of the analysis no longer exists in the transition regime ($\unicode[STIX]{x1D706}$ in the neighbourhood of $\unicode[STIX]{x1D706}_{c}$), scaling arguments nevertheless highlight the manner in which the Nusselt number function connects smoothly across the open and closed streamline regimes for any finite $Pe$.

Squeeze film characteristics of Ferro-coupled stress inertial fluids in conical plates

PurposeThe purpose of this paper is to investigate squeezing motion between conical plates lubricated by ferro-fluid couple stress lubricants considering convective fluid inertia effects.Design/methodology/approachBased upon the Stokes couple stress theory, Ferro-hydrodynamic model of Shliomis and averaged inertia principle, squeeze film characteristics between conical plates are obtained.FindingsAccording to the results, it is found that couple stress ferro-fluid lubricants increase squeeze film characteristics. Moreover, with increasing convective fluid inertia parameter, the squeeze film characteristics are increased. In contrast, the dimensionless load-carrying capacity diminishes when half cone angle of conical plate increases.Originality/valueThis paper is relatively original and it describes the squeeze film characteristics between conical plates with ferro-fluid, convective inertia, couple stresses and half cone angle of conical plate effects.

On the Approximation of Two-Dimensional Transient Pipe Flow Using a Modified Wave Propagation Algorithm

In this study, a second-order accurate Godunov-type finite volume method is used for the solution of the two-dimensional (2D) water hammer problem. The numerical scheme applied here is well balanced and is able to treat the unsteady friction terms, together with the convective terms, within the differences between fluxes of neighboring computational cells. In order to consider the effect of unsteady friction terms during the water hammer process, k−ε and k−ω turbulence models are employed. The performance of the proposed method with the choice of different turbulence models is evaluated using experimental data obtained from one low and one high Reynolds-number turbulent test cases. In addition to velocity and pressure distributions, the turbulence characteristics of each variant of the model, including eddy viscosity, dissipation rate, and turbulent kinetic energy during the water hammer process are fully analyzed. It is found that the inclusion of the convective inertia terms leads to more accurate pressure profiles. The results also show that using a relatively high Courant–Friedrichs–Lewy (CFL) number close to unity, the introduced numerical solver with both choices of turbulence models provides reasonable and acceptable predictions for the studied flows.

Pr > 1 unsteady thermal flows and heat transfer in a finned cavity with a uniform heat flux

Unsteady thermal flows in a differentially heated cavity with a fin on the sidewall are extensively present in industrial systems. In this study, Pr > 1 unsteady thermal flows in a finned cavity with a uniform heat flux are investigated using scaling analysis and numerical simulation. The dynamics and heat transfer of unsteady thermal flows on the finned sidewall are analyzed. The scaling analysis indicates that there are four flow scenarios for the intrusion but three for the starting plume around a fin, depending on the Rayleigh number and the Prandtl number. In a typical scenario, the intrusion can move under an unsteady viscous conduction regime, an unsteady inertial conduction regime, an unsteady inertial convection regime, a steady inertial convection regime, a steady inertial conduction regime or a steady viscous conduction regime, but the starting plume can ascend under an unsteady and a steady inertial conduction regime or under a steady viscous conduction regime. Additionally, the scaling laws of the intrusion and the starting plume under different regimes are presented and verified by numerical results. There is good agreement between numerical results and the scaling predictions.