Fluid Inertia Effects Research Articles

Elastic wave scattering by a fluid-saturated circular crack and effective properties of a solid with a sparse distribution of aligned cracks.

Hydraulic fractures and preexisting cracks in natural aquifers and hydrocarbon reservoirs are often saturated with fluids. Understanding the elastic wave properties in such a cracked fluid-saturated medium is of importance for many physical and engineering applications such as hydrology, petroleum engineering, oil exploration, induced seismicity, and nuclear waste disposal. In this paper, the scattering of a normally incident longitudinal (P-) wave by a fluid-saturated circular crack in an infinite elastic non-porous matrix is studied. In particular, the mechanism of hydraulic conduction (including the effects of the crack permeability and fluid inertia) inside the crack is incorporated. A semi-analytic solution for this scattering problem is derived. Based on the solution and multiple scattering theorem, an effective medium model is developed to determine the velocity dispersion and attenuation due to wave scattering in an elastic matrix with sparse distribution of aligned cracks. It is shown that the effective P-wave velocity is consistent with Gassmann's theory in the low-frequency limit. The effect of crack permeability on scattering is negligible, but the effect of fluid inertia is important. Specifically, it is found that resonance phenomena can take place inside the cracks at frequencies much lower than the scattering characteristic frequency so that rapid velocity variation can occur at relatively low frequencies. The fluid viscosity plays a damping role in weakening the resonance. The effects of crack thickness and fluid compressibility on scattering dispersion are similar to those in the case of plane-strain (two-dimensional) slit crack.

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.

Thermocapillary thin-film flows on a compliant substrate.

We study the dynamics of a thin liquid film on a compliant substrate in the presence of thermocapillary effect. A set of long-wave equations are derived to investigate the effects of fluid gravity (G), fluid inertia (Re), and Marangoni stresses (Ma) on the dynamics of the liquid film and the compliant substrate. By performing linear stability analysis and time-dependent computations of the long-wave equations, we examine two different cases: thin-film flows on a horizontally compliant substrate (β=0, where β is the inclined angle) and down a vertically compliant substrate (β=π/2), respectively. For β=0, we neglect fluid inertia and identify two different modes: (1) sinuous mode, where the deformations of liquid-air and liquid-substrate interfaces are in phase, which is induced by the fluid gravity, and (2) varicose mode, where the deformations of two interfaces are in phase opposition, which is induced by the Marangoni stresses. For β=π/2, we consider a weak fluid inertia and only observe the varicose mode driven by fluid inertia and Marangoni stresses. However, because the gravity direction is parallel to the substrate, the fluid gravity modifies the varicose mode, making the deformations of two interfaces out of phase. In particular, we also seek the nonlinear traveling-wave solutions in the case of β=π/2, revealing that fluid inertia and/or heating effect enhance the height and speed of the traveling waves. In both cases, the introduction of a strong wall heating gives rise to large deformations of both the thin liquid film and the compliant substrate.

Parametric investigation of twin tube magnetorheological dampers using a new unsteady theoretical analysis

In the present work, a new unsteady analytical model is developed for magnetorheological fluid flow through the annular gap which is opened on the piston head of twin tube magnetorheological damper, considering fluid inertia term into the momentum equation. This new unsteady model is based on Stokes’ second problem that is extended for magnetorheological fluid flow between finite oscillating parallel plates under the pressure gradient. A quasi-static analysis is also developed for magnetorheological fluid flow in twin tube damper, to compare its results with present unsteady solution and to show the effect of magnetorheological fluid inertia. The obtained results are validated experimentally and then, a parametric study is presented using both unsteady and quasi-static analysis. The effect of fluid inertia term is investigated on force–displacement and force–velocity loops, magnetorheological fluid velocity profile, pressure drop, phase difference between pressure drop and flow rate and change of plug thickness with time duration. According to the obtained results, quasi-static analysis included considerable error respect to new unsteady analysis as the gap height, magnetorheological fluid density, excitation frequencies and amplitudes are increased and yield stress is decreased. It is found that the plug thickness is considerably affected by inertia term of magnetorheological fluid.

Steady-State Analysis of Journal Bearing Including Inertia Effect

Classical theory of hydrodynamic journal bearing ignores the effect of fluid inertia, due to its negligible contribution as compared to viscous forces. However, there are some situations like high density / low viscosity fluids, sudden change in cross-section area, super laminar flow etc., where inertia effect is dominant. Inertia effect is noticeable for moderate range of velocity where modified Reynolds number is in between 0 to 1.4. In the present work, steady-state analysis of hydrodynamic journal bearing including the inertia effect is performed. In order to account inertia effect, modified Reynolds equation has been derived using perturbation technique. Zeroth order and first order equations are descritised using finite difference method and are solved by Gauss-Seidel iterative scheme. Effect of inertia on various parameters i.e. load, Sommerfeld number, attitude angle, friction parameter, end flow, has been studied. It is observed that the effect of inertia is prominent on end flow and there is marginal effect on load and attitude angle.

Lattice Boltzmann study of effective viscosities of porous particle suspensions

The effective viscosities of dilute and semi-dilute suspensions in a two-dimensional shear flow are studied using the lattice Boltzmann method. The suspensions contain non-Brownian hard circular buoyant porous particles. Here a more accurate formula for intrinsic viscosity as a function of Darcy number (Da) for the whole Da regime is proposed through our numerical result. The effects of fluid inertia, permeability of the particle, and confinement of the bounding walls are investigated. It is found that for the cases with a small Da, the effective viscosity significantly increases with confinement and fluid inertia. However, for the cases with a large Da, the confinement ratio and fluid inertia have very minor effect. Moreover, for semi-dilute suspensions, the permeability of the particle weakens the effect of the hydrodynamic interactions between particles on the relative viscosity ηr and makes ηr decrease. The above phenomena can be well understood through quantifying the disturbance of the porous particle to the flow.

Flow of power-law fluid past a circular cylinder in the vicinity of a moving wall

In this paper, the flow of power-law fluid over a circular cylinder near a moving wall is simulated numerically using a finite volume method for different Reynolds numbers (Re = 1, 10, 40), gap ratios (G/D = 0.2–1.0), and power-law indices (n = 0.5–1.5). The effects of fluid inertia, wall proximity, and rheological property on the drag and lift coefficient, gap flow characteristics, and recirculation modes are studied. Possible mechanisms for the variation of the drag and lift coefficient and the change in recirculation mode are addressed. The results show that the decrease in G/D or Re results in the increase in drag and lift coefficient, while the effect of n is dependent on Re and G/D. For the drag coefficient, shear-thickening fluid is more sensitive to the change of G/D. The variation of the lift coefficient can be explained by the movement of angular position of the front stagnation point. The redistribution of the flow around the cylinder results in different recirculation modes. Five distinct modes are found and the critical value of G/D for the mode change decreases with the decrease in n and increase in Re. Variation of the relative vortex intensity and the relative flow intensity nearby leads to the change of recirculation mode.

Effect of fluid inertia on swimming of a sphere in a viscous incompressible fluid

Swimming of a sphere in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations for wave-type distortions of the spherical shape. At sizable values of the dimensionless scale number the mean swimming velocity is the result of a delicate balance between the net time-averaged flow generated directly by the surface distortions and the flow generated by the mean Reynolds force density. Depending on the stroke, this can lead to a surprising dependence of the mean swimming velocity on the kinematic viscosity of the fluid. The net flow pattern is calculated as a function of kinematic viscosity for axisymmetric strokes of the swimming sphere. The calculation covers the full range of scale number, from the friction-dominated Stokes regime in the limit of vanishing scale number to the inertia-dominated regime at large scale number. The model therefore provides paradigmatic insight into the fluid dynamics of swimming or flying of a wide range of organisms.

Effects of fluid inertia and elasticity and expansion angles on recirculation and thermal regions of viscoelastic flow in the symmetric planar gradual expansions

The formation and growth of symmetric and asymmetric recirculation regions play an important role in the viscous dissipations, temperature distribution, and heat transfer rate. In this study, the inertial and non-isothermal flow of viscoelastic fluid has been simulated in the planar channel with a 1:3 symmetric gradual expansion for different Reynolds and elasticity numbers at three expansion angles of 30°, 45°, and 60°. Also, the thermal boundary condition of constant temperature has been used on the walls, and the constitutive equation of exponential Phan Thien–Tanner has been employed for modeling the polymeric stresses of the viscoelastic fluid. Due to the significant effect of temperature on the properties of the viscoelastic fluid, and the role of viscous dissipation in heat generation, the fluid properties are considered temperature dependent, and the terms of viscous dissipation are considered in the energy equation. The main purpose of current study is to investigate the effects of fluid inertia and elasticity and the expansion angles on the flow pattern, heat transfer characteristics, and viscous dissipation of inertial viscoelastic flow in the planar channel with gradual expansions. Therefore, the streamlines, length of vortices, isothermal lines, total viscous dissipations, and local and mean Nusselt numbers (Nu) have been examined inside the channel expansion for different Reynolds and elasticity numbers in the range of $$ 10 \le Re \le 100 $$ and $$ 0.01 \le El \le 2 $$ . The results show that the maximum values of local Nusselt numbers are enhanced by increasing the Reynolds and elasticity numbers and the expansion angle for the hydrodynamically and thermally developing conditions.

A new modeling approach for transversely oscillating square-section cylinders

This paper proposes a new generalized semi-empirical model to simulate fluid–structure interaction forces experienced by a transversely oscillating square-section cylinder by a more comprehensive approach. The total fluctuating transverse aerodynamic forces determined by this new modeling approach are composed of three main unsteady excitation modules. The first component relates to the conventional fluid inertia effect; the second component represents the varying incidence angle effect, including the wavelength-changing effect of downstream wake undulation by introducing a circulation lag function; and the last one denotes the excitation effect due to the strong resonant interaction between wake undulation and Kármán vortex shedding. In addition, the new model establishment process follows the “simple-to-use” principle, so it can be applicable even when few experimental data are available. Good agreements are observed between the predictions by this model and unsteady force coefficient results taken from harmonically forced vibration experiments. Moreover, the new model developed in this study can not only simulate the interaction effects between the vortex-induced vibration (VIV) and galloping, but also retain the predictive capability for the cases of pure vortex resonance and pure galloping phenomena. Finally, some significant findings are also presented in the context of the aeroelastic instability phenomena.

Combined effects of fluid–solid interfacial slip and fluid inertia on the hydrodynamic performance of square shape textured parallel sliding contacts

Lubrication performances of square shape textured parallel sliding contacts are examined under the combined influence of both fluid inertia and fluid slippage at the fluid–solid interface. A two-component slip length model and first-order perturbation method are adopted to formulate pressure governing equation consisting of fluid-slip and fluid inertia terms. The effect of texture size (aspect ratio), texture height ratio, reduced Reynolds number, slip length coefficient and critical threshold shear stress on the performance parameters like load support, end flow and friction parameter of parallel sliding contacts is studied. The results indicate that effect of fluid-slip is more influential than fluid inertia; therefore, the result shows similar and closer trend to the fluid-slip results. However, the magnitude of performance parameters depends on the effect of fluid inertia. Moreover, aspect ratio of 0.3–0.5 and lower value of texture height ratios can be used to achieve better hydrodynamic lubrication performance in parallel sliding contacts.

Collision rate of ice crystals with water droplets in turbulent flows

Riming, the process whereby ice crystals get coated by impacting supercooled liquid droplets, is one of the dominant processes leading to precipitation in mixed-phase clouds. How a settling crystal collides with very small water droplets has been mostly studied in laminar conditions. The present numerical study aims at providing further insight on how turbulent flow motion affects the riming of ice crystals. We model the crystals as narrow oblate ellipsoids, smaller than the Kolmogorov elementary scale. By neglecting the effect of fluid inertia on the motion of the crystals and droplets, and using direct numerical simulations of the Navier–Stokes equations in a moderately turbulent regime, over a range of kinetic energy dissipation$1~\text{cm}^{2}~\text{s}^{-3}\lesssim \unicode[STIX]{x1D700}\lesssim 256~\text{cm}^{2}~\text{s}^{-3}$, we determine the collision rate between disk-shaped ice crystals and very small liquid water droplets. Whereas differential settling plays the dominant role in determining the collision rate at small turbulence intensity, the role of turbulence becomes more important at the large values of$\unicode[STIX]{x1D700}$simulated, an effect that can be partly attributed to the increased role of inertia. We always find that collisions occur with a large probability on the rim of the ellipsoids, a phenomenon that can be explained to a large extent by kinematic considerations. The difference in the settling velocity of crystals and droplets induces a strong asymmetry in the probability of collision between the faces of the ellipsoids. Our results shed light on the physical mechanisms involved in the riming of ice crystals in clouds.

Numerical and Experimental Analyses of Dynamic Characteristics for Liquid Annular Seals With Helical Grooves in Seal Stator

Numerical and experimental analyses were carried out to investigate the dynamic characteristics of liquid annular seals with helical grooves in the seal stator. In the numerical analysis, the governing equations were the momentum equations with turbulent coefficients and the continuity equation, all averaged across the film thickness and expressed using an oblique coordinate system in which the directions of coordinate axes coincided with the circumferential direction and the direction along the helical grooves. These governing equations were solved numerically to obtain the dynamic characteristics, such as the dynamic fluid-film forces, dynamic coefficients, and whirl-frequency ratio (WFR). The numerical analysis included the effect of both fluid inertia and energy loss at the steps between the helical groove and the land sections. In the experiments, the dynamic fluid-film pressure distributions, which were induced by a small whirling motion of the rotor about the seal center, were measured to obtain the dynamic characteristics. The equivalent numerical results reasonably agree with the experimental results, demonstrating the validity of the numerical analysis. The value of the tangential dynamic fluid force induced by the rotor whirling motion decreased with increasing the helix angle γ. Consequently, the values of the cross-coupled stiffness coefficient and WFR decreased with increasing γ and became negative for large γ. In general, pump rotors rotate with a forward whirling motion under normal operating conditions. Hence, the negative value of WFR for helically grooved seals contributes to rotor stability by suppressing the forward whirling motion of the rotor.

Fluid rheological effects on particle migration in a straight rectangular microchannel

There has recently been a significantly increasing interest in the passive manipulation of particles in the flow of non-Newtonian fluids through microchannels. However, an accurate and comprehensive understanding of the various fluid rheological effects on particle migration is still largely missing. We present in this work a systematic experimental study of both the individual and the combined effects of fluid inertia, elasticity, and shear thinning on the motion of rigid spherical particles in a straight rectangular microchannel. We first study the sole effect of each of these rheological properties in a Newtonian fluid, purely elastic (i.e., Boger) fluid, and purely shear-thinning (i.e., pseudoplastic) fluid, respectively. We then study the combined effects of two or all of these rheological properties in a pseudoplastic fluid and two types of elastic shear-thinning fluids, respectively. We find that the fluid elasticity effect directs particles toward the centerline of the channel while the fluid shear-thinning effect causes particle migration toward both the centerline and corners. These two effects are combined with the fluid inertial effect to understand the particle migration in inertial pseudoplastic and viscoelastic fluid flows.

Efficiency of disengaged wet brake packs

Key objectives in off-highway vehicular powertrain development are fuel efficiency and environmental protection. As a result, palliative measures are made to reduce parasitic frictional losses while sustaining machine operational performance and reliability. A potential key contributor to the overall power loss is the rotation of disengaged wet multi-plate pack brake friction. Despite the numerous advantages of wet brake pack design, during high-speed manoeuvre in highway travel or at start-up conditions, significant frictional power losses occur. The addition of recessed grooves on the brake friction lining is used to dissipate heat during engagement. These complicate the prediction of performance of the system, particularly when disengaged. To characterise the losses produced by these components, a combined numerical and experimental approach is required. This paper presents a Reynolds-based numerical model including the effect of fluid inertia and squeeze film transience for prediction of performance of wet brake systems. Model predictions are compared with very detailed combined Navier–Stokes and Rayleigh-Plesset fluid dynamics analysis to ascertain its degree of conformity to representative physical operating conditions, as well the use of a developed experimental rig. The combined numerical and experimental approach is used to predict significant losses produced during various operating conditions. It is shown that cavitation becomes significant at low temperatures due to micro-hydrodynamic action, enhanced by high fluid viscosity. The magnitude of the losses for these components under various operating conditions is presented. The combined numerical-experimental study of wet multi-plate brakes of off-highway vehicles with cavitation flow dynamics has not hitherto been reported in the literature.

Finite Volume Simulation of Newtonian Fluids Through Combined Converging and Diverging Channel

A robust semi-implicit pressure-based Computational Fluid Dynamics (CFD) scheme (SIMPLE) scheme is adopted for modeling of steady and viscous compressible flows through converging and diverging channel. Second-order spatial accuracy will achieved through linear unstructured finite volume cell discretization. The SIMPLE scheme characterizes a development in mass-momentum coupled, pressure based schemes. The governing equations for this scheme are the conservative form of momentum equations (Navier-Stokes) and mass conservation equation. The grid will be developed and refined through Gambit package. Flow structure will be developed with effect of fluid inertia, primary, secondary and treasury vortex may be developed in the aspect ratio 1: 4, 1: 6 and 1: 8 of the converging and diverging channels. The small vortex observed in the ratio 1: 4 in each corner of the channel and with increasing the fluid inertia the left upper vortex in size is enhanced only due to increase the Reynolds numbers. When increased the ratio 1: 6 and 1: 8 the vortex in size is enhanced at lower and upper corner of the channel and occupied the whole domain. Due to filling the Porous material all vortices left, right, upper and lower are diminished and no recirculation flow rate observed at each Reynolds numbers and on every Darcy numbers. The numerical results achieved through finite volume technique through commercial CFD package ANSYS Fluent and will be compared with analytical approach of sheikh, et al. [2012] and also other numerical results with and without use of CFD packages.

The effect of lubricant inertia on fluid cavitation for high-speed squeeze film dampers

This work studies the effect of lubricant inertia on the fluid cavitation for partially sealed high-speed squeeze film dampers (SFDs) executing small amplitude circular-centered orbits (CCOs). The lubricant cavitation is modeled by both the Elrod algorithm and the Gumbel’s cavitation boundary condition to provide the comparison between the most common lubricant cavitation models. Additionally, the fluid inertia is integrated by adapting a finite-length SFD model for partially sealed dampers. The integrated SFD model is incorporated into a numerical simulation model and the results are validated by comparison with experimental data. The results of the analysis demonstrate that the fluid inertia effects significantly extend the cavitation region and influence the cavitation onset and the film reformation.

Dynamic Characterization of an Integral Squeeze Film Bearing Support Damper for a Supercritical Co2 Expander

The present work advances experimental results and analytical predictions on the dynamic performance of an integral squeeze film damper (ISFD) for application in a high-speed super-critical CO2 (sCO2) expander. The test campaign focused on conducting controlled orbital motion mechanical impedance testing aimed at extracting stiffness and damping coefficients for varying end seal clearances, excitation frequencies, and vibration amplitudes. In addition to the measurement of stiffness and damping, the testing revealed the onset of cavitation for the ISFD. Results show damping behavior that is constant with vibratory velocity for each end seal clearance case until the onset of cavitation/air ingestion, while the direct stiffness measurement was shown to be linear. Measurable added inertia coefficients were also identified. The predictive model uses an isothermal finite element method to solve for dynamic pressures for an incompressible fluid using a modified Reynolds equation accounting for fluid inertia effects. The predictions revealed good correlation for experimentally measured direct damping, but resulted in grossly overpredicted inertia coefficients when compared to experiments.

A Study of Lubricant Inertia Effects for Squeeze Film Dampers Incorporated into High-Speed Turbomachinery

This work proposes a numerical model that incorporates the effect of lubricant inertia on the hydrodynamic pressure distribution, fluid film reaction forces, and the fluid velocity component profiles for finite-length open-ended squeeze film dampers (SFDs). Firstly, the thin film flow equations for the SFD in presence of fluid inertia effects are introduced. Furthermore, a small first-order perturbation by means of the expressions for the fluid film velocity components and the lubricant pressure distribution that are expanded in power series of the squeeze film Reynolds number is applied to the flow equations. Subsequently the developed lubricant flow equations are solved to develop expressions for the velocity component profiles and the hydrodynamic pressure distribution in SFDs. The pressure expression is numerically solved by using Gauss–Seidel method with finite difference discretization. Moreover, the fluid film reaction forces are determined by numerically integrating the hydrodynamic pressure expressions over the journal surface. Additionally, the proposed pressure distribution expression and the numerical SFD forces are incorporated into a simulation model and the simulation results are compared with the existing models in the literature under different operating conditions, including eccentricity ratios and inertia effects (i.e., Reynolds numbers). The simulation results demonstrate the significant influence of both convective and temporal (i.e., unsteady) lubricant inertia terms on the SFD hydrodynamic pressure distribution and the fluid film reaction forces. Furthermore, the proposed SFD model is incorporated into a multi-mass flexible rotordynamic model to evaluate the effect of SFD fluid inertia on the mass unbalance induced steady-state vibrations of the rotor and the nodal transient orbits by implementing finite element method and transient modal integration with predictor–corrector solver. The results of the analysis demonstrate the significant effect of fluid inertia on the resonance frequencies of the rotor and the steady-state vibration amplitudes and the transient orbits at the resonance zone.

Numerical and Experimental Analyses of Static Characteristics for Liquid Annular Seals With Helical Grooves in Seal Stator

Numerical and experimental analyses were carried out to investigate the static characteristics of liquid annular seals with helical grooves in a seal stator. In the numerical analysis, the momentum equations with turbulent coefficients and the continuity equation, which were averaged across the film thickness, were numerically solved to obtain the leakage flow rate and the pressure distributions in the seal clearance. To accurately define the location of the step between the groove and the land regions in the calculation domain, these governing equations were expressed using an oblique coordinate system in which the directions of coordinate axes coincided with the circumferential direction and the direction along the helical grooves. The numerical analysis included the effects of both fluid inertia and energy loss due to expansion during the passage of fluid from the land region to the helical groove region and that due to contraction from the groove region to the land region. In the experimental analysis, the leakage flow rate and the fluid-film pressure distributions in the seal clearance were measured for the helically grooved seals with different helix angles of the helical groove. The numerical results of leakage flow rate and pressure distributions agree reasonably with the experimental results, which demonstrates the validity of the numerical analysis. The leakage flow rate of the helically grooved seals was influenced by two factors: fluid energy loss during passage through the step between the groove and the land, and the pumping effect by which the spinning motion of the rotor pushes the flow back upstream along the helical grooves. Under a low range of rotor spinning velocity, the leakage flow rate decreased with helix angle because the effect of fluid energy loss in the steps was significant. By contrast, under a high range of spinning velocity, the quantitative difference in the leakage flow rate due to the helix angle decreased compared to that under a low range because the reduction in the leakage flow rate due to the pumping effect was pronounced for a larger helix angle. The effects of helix angle and rotor spinning velocity on the leakage flow rate are explained qualitatively using a simplified model.