Critical Reynolds Number Rec Research Articles

Sediment size effects on non-Darcy flow: insights from Izbash equation and Forchheimer inertial coefficient analysis

AbstractThe transition from Darcy to non-Darcy flow regimes was investigated using column experiments. This revealed key relationships between sediment characteristics and critical thresholds for the onset of the non-Darcy flow regime, as well as inertial flow parameters. An exponential dependence of the critical Reynolds number (Rec) on hydraulic conductivity (K) and a linear dependence on sediment size (d50) was found. The analysis revealed a potentially universal relationship between the critical hydraulic gradient (Ic) and K, with a power-law exponent of –3/2, consistent with previous investigations. Additionally, Ic was found to be inversely proportional to the power law of the square root of d50. Novel relationships are derived for estimating the Izbash equation inertial exponent (n) and the Forchheimer inertial coefficient ($$\beta$$ β ) based on sediment characteristics. The exponent (n) was found to decrease with d50 and increase with K, following power-law relationships. A new equation is proposed, capable of predicting $$\beta$$ β with slightly improved accuracy, outperforming numerous and previously proposed empirical equations. Additionally, these data validate the works of Ergun and Irmay as an alternative for $$\beta$$ β estimation using porosity and sediment size. As the attainment of statistical significance in multiparameter curve fitting can be trivial, it has led to the proliferation of empirical equations for estimating β. This study highlights the limitations of existing empirical methods in determining β and emphasizes the necessity for a universal approach to predict this critical parameter, which will facilitate broader adoption of the Forchheimer equation.

Buckled elastic sheet as a vortex generator in dual channels

This study presents a dual-channel vortex generator (VG) that leverages the snap-through behaviour of flexible sheets. The VG outperforms a similar-sized rigid VG in generating vortices within dual-channel flows while minimizing pressure loss. Numerical simulations using the immersed boundary-lattice Boltzmann method analyse the dynamics and vortex generation performance of the sheet under various system parameters. Two distinct modes are identified for the elastic sheet: a sustained snap-through mode (SSTM) and a dormant mode (DM). The sheet's mode is predominantly influenced by its length ratio (L*), bending stiffness $(K_b^\ast )$ and flow strength, with the mass ratio having a minimal impact. The sheet exhibiting regular SSTM can effectively generate vortices in both channels and the vortex generation performance can be conveniently tuned by altering the sheet's initial buckling (i.e. L*). An increase in $K_b^\ast $ results in a higher critical Reynolds number (Rec) required for mode transition. An increase in L*, however, initially raises Rec and then lowers it, suggesting an optimal length ratio (approximately 0.7 for our considered system) for minimizing the Rec necessary to trigger SSTM. Furthermore, a disparity in the flow strength between channels is found to suppress the snap-through of the sheet; a greater disparity, however, is permissible to induce the SSTM of more compliant sheets. These findings underscore the potential of snap-through behaviour for enhanced flow manipulation in dual-channel systems.

Numerical investigation of fluid flowing through rough fractures subject to shear

Comprehending fluid flow in rock masses is essential for modern underground engineering, including chemical energy extraction, nuclear pollutant remediation, and hydrocarbon utilization, complicated by shear-induced and surface roughness effects in fractures. This study employed numerical simulations to investigate the fluid flow behavior in fractures with different surface roughness under shear, where the shear direction is perpendicular to the flow direction. The nonlinear flow of the fluid is observed to have a strong correlation with the confining pressure (Pz), roughness (JRC), and shear displacement (u). The generation of eddy currents is frequently linked to the presence of flow channel intricacies and the velocity of flow at a microscopic scale. The Forchheimer equation could describe the process of nonlinear phenomena accentuation very well. The fracture under Pz caused a reduction in hydraulic transmissivity (T) due to compression. Furthermore, the T changes dramatically as the shear process progresses. Based on the analysis of the Forchheimer coefficient (β) and critical Reynolds number (Rec) parameters used to determine the response of nonlinear flow, it appears that an increase in Pz facilitates the transition of the fluid into a nonlinear flow state. Conversely, shearing has the opposite effect and reduces the tendency toward nonlinear flow.

Experimental investigation and tomography analysis of Darcy-Forchheimer flows in thermal protection systems

In thermal protection systems (TPS), Darcy’s law or Darcy-Forchheimer’s law is employed to model the pyrolysis gas flow within the anisotropic porous ablator depending on the flow regime considered. A key challenge with using these laws is first, the knowledge of the validity domain of each flow regime in terms of a critical Reynolds number (Rec). Secondly, the lack of data on macroscopic properties, namely, the permeability and Forchheimer tensors is particularly challenging for the relevance of the models. The objective of this work is to contribute to overcoming these challenges by performing experimental and X-ray tomographic image-based characterization of Calcarb, a commercial carbon preform used for manufacturing TPS. For this purpose, fluid flow within Calcarb was studied experimentally in the Through-Thickness (TT) and the In-Plane (IP) directions for Reynolds numbers of 0.05 to 10.46 -representative of the TPS application. Tomography image-based micro-scale simulations, involving the direct resolution of the Navier–Stokes equations under both flow regimes, were also performed. Experimental results exhibit the anisotropic nature of Calcarb, namely through Rec values, corresponding to the Darcy flow regime limit, slightly different in the two directions (Rec of 0.31 and 0.43) with measured permeability values of 1.248×10−10 m2 and 1.615×10−10 m2 for TT and IP directions respectively. In the Forchheimer regime, experimental Forchheimer coefficients β were 2.0010×105 m−1 (TT) and 1.4948×105 m−1 (IP). During the simulation process, a numerical strategy was defined to obtain the permeability tensor yielding values within 8% of the experimental ones. The comparison of experimental results vs simulation results in the Forchheimer regime was performed through the analysis of the pressure gradients as functions of Re in the x, y, and z directions. The numerical estimations were compared successfully with experimental measurements, with a discrepancy of 5.2%, for Re values up to 2.4.

Study on the Performance of a Surface with Coupled Wettability Difference and Convex-Stripe Array for Improved Air Layer Stability.

The existence of an air layer reduces friction drag on superhydrophobic surfaces. Therefore, improving the air layer stability of superhydrophobic surfaces holds immense significance in reducing both energy consumption and environmental pollution caused by friction drag. Based on the properties of mathematical discretization and the contact angle hysteresis generated by the wettability difference, a surface coupled with a wettability difference treatment and a convex-stripe array is developed by laser engraving and fluorine modification, and its performance in improving the air layer stability is experimentally studied in a von Kármán swirling flow field. The results show that the destabilization of the air layer is mainly caused by the Kelvin-Helmholtz instability, which is triggered by the density difference between gas and liquid, as well as the tangential velocity difference between gas and liquid. When the air layer is relatively thin, tangential wave destabilization occurs, whereas for larger thicknesses, the destabilization mode is coupled wave destabilization. The maximum Reynolds number that keeps the air layer fully covering the surface of the rotating disk (with drag reduction performance) during the disk rotation process is defined as the critical Reynolds number (Rec), which is 1.62 × 105 for the uniform superhydrophobic surface and 3.24 × 105 for the superhydrophobic surface with a convex stripe on the outermost ring (SCSSP). Individual treatments of wettability difference and a convex-stripe array on the SCSSP further improve the air layer stability, but Rec remains at 3.24 × 105. Finally, the coupling of the wettability difference treatment with a convex-stripe array significantly improves the air layer stability, resulting in an increase of Rec to 4.05 × 105, and the drag reduction rate stably maintained around 30%.

An experimental investigation on the settling of disk particles in annular region: Wall effect and drag coefficient

Settling non-spherical and spherical particles in the annular region is common in many industrial applications. This work shows the intensification of the performance of settling particles in an annular channel by reducing the drag coefficient. The disk/cylinder particles settle in annular channels filled with Newtonian and non-Newtonian fluids. An aqueous solution of different concentrations of glycerin and Carboxymethylcellulose (CMC) was considered as the Newtonian and non-Newtonian fluids. The experimental results reported in the present work covered the 0.17 ≤ blockage ratio ≤0.55, 0.14 ≤ k (flow consistency index) ≤1.81 (Pa.sn), 0.64 ≤ n (flow behavior index) ≤1, and annular channel Reynolds number 0.40 ≤ Re ≤63.44 and 0.05 ≤ Re ≤43.02 in Newtonian and non-Newtonian fluids respectively. The wall factor, f, was increased with the increase in Re and the decrease in blockage ratio. At higher Reynolds numbers (Re), the particle’s wall factor was only influenced by the blockage ratio. The wall factor was reduced due to the particle’s shape change from cylinder to disk, i.e., decreasing the particle’s sphericity. The wall factor of particles is observed to be higher in non-Newtonian than in Newtonian fluids. The critical Reynolds number (Rec ) decreased with the H/d (Height/diameter of disk) ratio and sphericity of the studied particles. Both the drag coefficients, in the absence ( C D ∞ ) and presence ( C D ) of the wall effect, declined with the Reynolds number. Suitable correlations were developed for CD with low values of RMSLD (root mean square logarithmic deviation). The experimentally obtained drag coefficient data were also analyzed successfully using the laminar model available in Ansys (Fluent) −18.1. The computed velocity contours analyzed the wall effect and CD variations successfully. The use of the annular channel as a drag reduction device is encouraged due to the observed higher wall factor and decreased drag coefficient for particle settling in comparison to the non-annular channel.

Thermocapillary thin film flows on a slippery substrate with odd viscosity effects

This study investigates the behavior of a thin liquid with odd viscosity effects as it flows down a uniformly heated, slippery inclined plane, where the liquid’s time-reversal symmetry is broken. The breaking of symmetry results in interesting effects, as the antisymmetric part of the fluid stress tensor does not vanish. Two models, the Benney-type equation model (BEM) and the weighted residual model (WRM), are constructed to account for the combined effects of slip length, thermocapillarity, and odd viscosity. A detailed stability analysis determines both models’ critical Reynolds numbers Rec. For small slip lengths, the first-order WRM is better at capturing the instability threshold than the first-order BEM. While both models account for the effects of odd viscosity and thermocapillarity, only the Rec of WRM incorporates the wall slip effects. Another significant finding is that the BEM effectively avoids the issue of finite-time blow-up by incorporating odd viscosity. Additionally, employing a weakly nonlinear stability analysis with multiple scales uncovers four flow regions in BEM: supercritical stable, subcritical unstable, unconditional stable, and explosive zones. Two separate bifurcation scenarios emerge for various wave numbers: supercritical within a specific range and subcritical for larger wave numbers. The presence of odd viscosity alleviates the reduction of the unconditional stable zone and the increase in the explosive zone, which are caused by the combined influence of slip and thermal effects. Numerical investigation of traveling wave solutions of WRM shows that wave height is promoted by slip and thermal effects but reduced with increasing odd viscosity coefficient. Further numerical simulations of WRM on a larger domain demonstrate the stabilizing effects of odd viscosity and its interaction with destabilizing slip and thermocapillary effects.

Hydrodynamics and instabilities of a falling liquid film with an insoluble surfactant

In this study, we investigate the linear and weakly nonlinear stability of a liquid film flowing down an inclined plane with an insoluble surfactant. First, the nonlinear evolution equations of a liquid film thickness and surfactant concentration are derived using the long-wave expansion method at a moderate Reynolds number (0 < Re ≤ 20). The linear stability of the flow is examined using the normal-mode method, and the linear stability criterion and critical Reynolds number Rec are obtained. The results reveal the destabilizing nature with increasing Reynolds number Re and the stabilizing nature with increasing Marangoni number M. Second, the nonlinear equations described by the complex Ginzburg–Landau equation are obtained using the multiple-scale method to investigate the weakly nonlinear stability of the system. The results show that a new linear instability region appears above the neutral stability curve caused by the solute-Marangoni effect, which develops into a supercritical stable zone under the influence of nonlinear factors. Increasing M generally improves the stability of the flow but continuing to increase M under the condition of M > Mc (critical Marangoni number) improves the nonlinear instability in the region and transforms part of the unconditional stability zone into a subcritical instability zone. The increase in Re extends an explosive unstable zone and reduces the threshold amplitude in the subcritical unstable zone. In contrast, the unconditional stable zone decreases and disappears after increasing Re to a specific value, which reflects the destabilizing effect of Re on the nonlinear zone of the flow.

Evolution of channel flow and Darcy’s law beyond the critical Reynolds number

For incompressible channel flow, there is a critical state, characterized by a critical Reynolds number Rec and a critical wavevector mc along the channel direction, beyond which the channel flow becomes unstable in the linear regime. In this work, we investigate the channel flow beyond the critical state and find the existence of a new fluctuating, quasi-stationary flow that comprises the laminar Poiseuille flow superposed with a counter-flow component, accompanied by vortices and anti-vortices. The net flow rate is reduced by ~ 15% from the linear, laminar regime. Our study is facilitated by the analytical solution of the linearized, incompressible, three-dimensional (3D) Navier–Stokes (NS) equation in the channel geometry, with the Navier boundary condition, alternatively denoted as the hydrodynamic modes (HMs). By using the HMs as the complete mathematical basis for expanding the velocity in the NS equation, the Rec is evaluated to 5-digit accuracy when compared to the well-known Orszag result, without invoking the standard Orr-Sommerfeld equation. Beyond Rec, the analytical solution is indispensable in offering physical insight to those features of the counter-flow component that differs from any of the pressure-driven channel flows. In particular, the counter flow is found to comprise multiple HMs, some with opposite flow direction, that can lead to a net boundary reaction force along the counter-flow direction. The latter is analyzed to be necessary for satisfying Newton’s law. Experimental verification of the predictions is discussed.Graphical

An optimized stability framework for three-dimensional Hartman flow via Chebyshev collocation simulations

Hydrodynamics instability is studied for an electrical conducting fluid against small disturbance between the channels by using normal magnetized force. Chebyshev collocation technique is used to determine the stability of three-dimensional Hartmann flow problem. The distinctive case of the perturbations is calculated. It is also considered that perturbations intensity be determined by just on quantified by different parameters. We have considered one of the flow stabilities conditions to analyze our problem. QZ (Qualitat and Zuverlassigkeit) technique is applied to investigate the problem to draw stability curves. αc,βc are critical wave numbers in streamwise and span-wise respectively and for a big range of Hartmann value (Ha), we obtained critical Reynolds number (Rec). It is found that Couette flow is destabilized for a specific value of Magnetize force while with greater or lesser magnitude than the particular one will stabilize the flow. Disturbances with particular oblique angle θ will grow while the others will decay for the three-dimensional disturbance. We observed from over results that for Ha(>2.0) Rec, increases steadily and when Ha more than 3.886, Rec decreases fast to the minimum. It is also established that for drop down from Ha = 3.886 and Rec becomes very large. For distinct Hartmann values, there exist two Rec numbers due to closed contour. The outcomes of current study are utilized in drug-delivery systems and photodynamic therapy.

Shear imposed falling film with odd viscosity effects

This study examines the behavior of a thin liquid film flowing on an inclined plane subject to imposed shear stress. The liquid’s time-reversal symmetry is broken, and we consider the odd part of the viscosity to describe the Navier–Stokes equation. To investigate the interplay between the imposed shear and odd viscosity on the flow dynamics, we develop a weighted residual model (WRM). We determine the model’s critical Reynolds number (Rec) through linear stability analysis. Our findings indicate that uphill shear tends to stabilize the flow, while downhill shear enhances the instability, albeit reduced by the presence of odd viscosity. We also construct an Orr–Sommerfeld (OS) type eigenvalue problem for normal mode analysis and derive Rec. We discover that in the longwave regime, RecWRM=RecOS. Finally, our numerical simulations of the model align well with our linear stability analysis.

Effect of micro-grooves on drag reduction in Taylor–Couette flow

Taylor–Couette flow with micro-grooves on the rotating inner cylinder is investigated to reveal the effect of surface structures on drag reduction. The Reynolds number (Re) ranges from 160 to 18 700. On the one hand, in the regimes of wave vortex flow (WVF, 160 < Re < 1010) and modulated wavy vortices (MWV, 1010 < Re < 1380) flow, the micro-grooves always reduce the torque, indicating drag reduction. Increasing either the size of micro-groove or Re, drag reduction will be enhanced. On the other hand, when the flow regime enters turbulent Taylor vortices (TTV, Re > 1380), drag reduction will be suppressed as Re increases and eventually turns to drag increase. The bigger the groove size, the smaller the critical Re where it turns from drag reduction to drag increase. To reveal the underlying mechanism of the effect of micro-grooves on drag reduction, particle image velocimetry measurements are conducted to observe the vortex flow structures, which demonstrates two aspects affecting the drag of Taylor–Couette flow over micro-grooved wall. First, the weakening of the large-scale Taylor vortex will lead to drag reduction. Second, the roughness effect will result in drag increase. In WVF/MWV, the former plays a dominant role, while in TTV, the latter dominates. In addition, a relationship between the micro-groove size and the predictive critical Reynolds number (Rec) is developed, providing a method for controlling the wall drag.

Linear stability of Poiseuille flow of viscoelastic fluid in a porous medium

We study the instability of plane Poiseuille flow of the viscoelastic second-order fluid in a homogeneous porous medium. The viscoelastic fluid between two parallel plates is driven by the pressure gradient. The effects of elasticity number E (depends on fluid properties, geometry; E is defined below) and Darcy number Da (gives the permeability of porous medium; Da is defined below) on flow stability are analyzed through the energy method that provides qualitative behavior of flow stability, and the numerical solution of generalized eigenvalue problem that gives the precise upper bound for stability. The plane Poiseuille flow of second-order fluid becomes unstable for increasing elasticity number while preserving Newtonian eigenspectrum up to a certain range of E. For large elasticity number, instability appears as a part of both wall and center modes for all Darcy numbers. We also noticed that along each neutral stability curve, the eigenfunctions are all antisymmetric with a single extremum near the channel walls. When E = 0.0011, we found an additional new elastic mode, which is unstable and also antisymmetric. For E < 0.0011, the neutral curves split into two lobes with different minima. The critical Reynolds number Rec is found to be decreasing (increasing) for higher (lower) values of fluid elasticity (Darcy number). Physical mechanisms are discussed in detail.

Experiments on the Drag and Lift Coefficients of a Spinning Sphere

The drag and lift coefficients are important parameters that affect the particle motion in a viscous fluid. In the present study, the drag and lift coefficients of a spinning sphere in a water tank were studied experimentally using a high-speed camera. To this end, 22 cases were studied to cover a wide range of dimensionless angular speeds (0.149 < Rr < 3.471) and Reynolds numbers (610 < Re < 3472). Based on the present experimental data and the results obtained from the literature, expressions were developed to calculate the lift and drag coefficients. The performed analyses on lift coefficient show that there is a critical Reynolds number (Rec) at each dimensionless angular speed. When 0 < Re < Rec, the lift coefficient decreases with increasing the Reynolds number, while it is constant when Rec < Re< 3500. The constant lift coefficient corresponding to different spin speeds was defined as the limit value of the lift coefficient. It is found that when 1 < Rr < 12, the limit value of the lift coefficient is 0.37, while the limit value of the lift coefficient increases with increasing dimensionless angular speed at 0 < Rr < 1. It is found that the spin increases the drag coefficient of a spinning sphere only when 0 < Rr < 10. Moreover, the performed analyses show that the drag coefficient of a spinning sphere is less than that of a non-spinning sphere when 10 < Rr < 25. Furthermore, the lift-to-drag ratio of a spinning sphere is discussed in this article.

Sidewall effect on turbulent band in subcritical transition of high-aspect-ratio duct flow

We numerically studied high-aspect-ratio channel flows with spanwise sidewalls, that is, wide duct flows, in its subcritical turbulent transitional regime. The infinite channel flow is known to form large-scale intermittency of turbulent–laminar coexistence and undergo two-stage transition (or crossover transition) process: a second-order phase transition with a critical Reynolds number Rec≈1000 and a deviation from it to maintain turbulence down to the global critical value Reg≈700. However, a real channel must have spatial finiteness, and its effect on transition phenomena is nontrivial. With the objective of understanding the turbulence maintenance limitations in the real channel flow, we investigated the effect of spanwise finiteness on the localized turbulence and its criticality, using direct numerical simulation. In our widest duct with an aspect ratio of 1:96 in the flow cross section, turbulent bands colliding with sidewalls above Re = 1069 often stochastically reflected or reversely traveled, keeping two-dimensional intermittencies with oblique bands, similar to the channel flow, whereas, in a narrower duct of 1:24, the critical value was higher as 1151 in the steeper transition profile, forming a quasi-one-dimensional intermittency dominantly. The transition in the high-aspect-ratio duct flow was converged to Reg≈1000 as the sidewall distance was increased. The critical phenomenon differs significantly from the channel flow for all duct flows, even for high aspect ratios. Due to spatial finiteness, the duct flows become fully laminar within a finite time for Re≲1000, unlike the channel flow. Possible causes of the difference in Reg between the two systems with fixed pressure gradient and fixed flow rate are discussed.

A linear route to elasto-inertial turbulence

Pipe flow of Newtonian fluids is well known to exhibit a transition from the laminar to turbulent regime usually at a Reynolds number ~ 2000, despite being linearly stable at all Reynolds numbers. In stark contrast, we show that pressure-driven pipe flow of a viscoelastic (Oldroyd-B) fluid is linearly unstable to axisymmetric perturbations. The dimensionless groups that govern stability are the Reynolds number Re = ρUmaxR/η, the elasticity number E = λη/(ρR2) and the ratio of solvent to solution viscosity β = ηs/η; here, R is the pipe radius, Umax is the maximum velocity of the base flow, ρ is the fluid density, and λ is the polymer relaxation time. The unstable mode has a phase speed close to Umax over the entire unstable region in the Re-E-β space. In parameter regimes accessible to experiments (e.g. β > 0.6 and E > 0.08), the critical Reynolds number Rec ~ 400, with the associated eigenfunctions spread out across the pipe cross section. In the asymptotic limit of E(1-β) << 1, but fixed E, the critical Reynolds number for instability and the critical wavenumber diverge as Rec ~ (E (1-β))-3/2 and kc ~ (E(1-β))-1/2 respectively. The unstable eigenfunction in this limit is localized near the centerline, implying that the unstable mode belongs to a class of viscoelastic ‘center modes'.The instability identified in this study comprehensively dispels the prevailing notion of pipe flow of viscoelastic fluids being linearly stable in the Re-W plane (W = Re E being the Weissenberg number). The prediction of a linear instability is consistent with several experimental studies on pipe flow of polymer solutions, ranging from reports of ‘early turbulence’ in the 1970's to the more recent discovery of ‘elasto-inertial turbulence’ (B. Hof and coworkers, Proc. Natl. Acad. Sci., 110, 10557–10562 (2013)), wherein transition is observed at Re much lower than 2000. An analogous center-mode instability is predicted for viscoelastic channel flows over a similar range of parameters. Thus, there is the suggestion of a universal linear mechanism that underlies the onset of turbulence in rectilinear viscoelastic shearing flows for sufficiently elastic dilute polymer solutions, marking a possible paradigm shift in our understanding of transition in such flows. We will end with a discussion of the possible transition scenarios in the Re-W-β space for viscoelastic shearing flows.

Lid-driven cavity flow-induced dynamics of a neutrally buoyant solid: Effect of Reynolds number, flexibility, and size

The present work is on Fluid flexible–Solid Interaction (FfSI), involving a recirculating flow-induced motion of a neutrally buoyant and deformable circular solid. For a Newtonian fluid flow and neo-Hookean flexible-solid deformation, a single FfSI solver—based on fully Eulerian and monolithic approaches—is used. The effect of Reynolds Number Re (20–500), volume fraction Φ (1%–12%) of the solid, and its non-dimensional shear modulus G*(0.02–1) on transient/periodic flow-induced solid-motion and the associated FfSI analysis is presented. The solid undergoes a transient spiraling motion before attaining a periodic orbit-based limit cycle. The flow also attains the periodic state after the initial transients. Time-averaged flow velocity-magnitude ⟨v*⟩ surrounding the limit cycle increases with increasing Re, increasing G*, and decreasing Φ. Equivalent radius req* of the limit cycle and time-averaged velocity-magnitude ⟨vc*⟩ of the centroid of the solid increase with increasing Re and decrease with decreasing G* (or increasing flexibility) and increasing volume fraction Φ (or size) of the solid. Also, frequency f* of the limit cycle decreases with increasing Re and remains almost constant with G* and Φ. With increasing Φ, the limit cycle undergoes a transition from the single loop to double loop beyond a critical volume fraction Φc=2%. A critical Reynolds number Rec, below which the periodic limit cycle collapses to a point, decreases with decreasing Φ. Our findings will help in the prediction and control of the motion of the solid in a bounded fluid flow involving solids of varying flexibility, which is relevant to a wide range of industrial and biological applications.

Onset of vortex shedding from a bluff body modified from square cylinder to normal flat plate

This study numerically investigates the time-resolved laminar flow over rectangular cylinders with cross-sectional aspect ratio Ar varying from square cylinder (Ar = 1) to normal flat plate (Ar = 0.1) for Reynolds number Re = 40–100. The focus is given on how Ar influences the wake structure, fluid force, St, and critical Reynolds number Rec associated with the onset of vortex shedding. The Stuart-Landau equation is employed to determine Rec. With increasing Re, time-mean drag coefficient C‾D increases and decreases for Ar < 0.5 and Ar > 0.5, respectively. The Re effect on C‾D is strongest for Ar = 0.1 and weakest for Ar = 0.5. On the other hand, St monotonically increases with Re for all Ar values examined. Both C‾D and St at a given Re diminish with increasing Ar. A decrease in Ar from 1.0 to 0.1 destabilizes the flow, reducing the Rec from 47.8 to 29.8. The dependence of Rec on the flow attack angle α = (0° - 45°) for the square cylinder shows Rec declining from Rec = 47.8 at α = 0° to Rec = 40.2 at α = 45°. A comparison of Ar effects on Rec is made between rectangular (present study) and elliptical cylinders, both changing toward the normal flat plate. The rectangular cylinders have a smaller Rec than the elliptical cylinders for Ar < 1 while the opposite is the case for Ar = 1.

Experimental and numerical study for the inertial dependence of non-Darcy coefficient in rough single fractures

This study experimentally and numerically investigated the influence of the inertial effect of fluid flow on the non-Darcy coefficient (β). The results showed that the non-Darcy coefficient (β), apparent permeability (ka), and hydraulic aperture (eh) decreased with the increase of Reynolds number (Re), which indicated that the non-Darcy coefficient depended on the geometric properties of a single fracture and the fluid inertial effect. The quantification model of the non-Darcy coefficient was improved. Under the guidance of fluid inertial effect, the β performed a positive power-law relationship with eh; under the coupling effect of fluid inertial effect and fracture geometric properties, the β behaved negative power-law relationship with eh, which was similar to the non-Darcy coefficient quantitative model that ignored the fluid inertial effect in the previous study. Therefore, the geometric properties were the dominant ones for fracture flow when it was affected by the coupling effect of geometric properties and inertial effects. Then, the quantitative model of the non-Darcy coefficient was improved. Based on the improved non-Darcy coefficient quantitative model, a new critical Reynolds number prediction model (CRN model) was constructed. Compared with other CRN models, the new CRN model considering the coupling effect of fluid and media properties could more accurately predict the critical Reynolds number (Rec) in rough single fractures, which further proved the existence and importance of the inertial dependence of the non-Darcy coefficient.

Weakly viscoelastic film on a slippery slope

We study the stability of weakly viscoelastic film (Walter's B″) flowing down under gravity along a slippery inclined plane. The focus is to investigate the interaction of the bottom slip with the viscoelastic parameter as well as the influence of the other flow parameters on the stability of the flow. For the slippery substrate, we use the Navier-slip boundary condition at the solid–liquid interface. The dimensionless slip length β is first assumed to be small and its order is considered same as the order of the film aspect ratio ϵ=H/L, where H is the mean film thickness and L is a typical wavelength. To discuss the coupled effect of slip length β and viscoelastic parameter γ, we have used the classical Benney equation model (BEM) as well as the weighted residual method (WRM). For linear stability, the normal mode analysis is carried out to capture the instability threshold. The critical Reynolds numbers (Rec) are obtained for BEM and WRM separately for the system. We found that the first-order WRM is a better choice to capture the instability threshold in comparison with a first-order BEM when β is small. Another noteworthy result we obtain is that, in the absence of β, WRM and BEM yield the same expression for the critical Reynolds number. Further, we have extended the study for moderate values of β, that is, β of order unity and it is found that both BEM and WRM are able to capture the effects of β and γ. We derive the Orr–Sommerfeld (OS) type eigenvalue problem and an asymptotic analysis is performed for small perturbation wavenumbers, which gives an expression for the critical Reynolds number for the instability of very long perturbations. The critical Reynolds number obtained by the OS eigenvalue problem exactly matches with that obtained by BEM. Finally, we validate our analytical predictions by performing a direct numerical simulation of the WRM and good agreement between the results of the linear stability analysis, weighted residual model, and the numerical simulations is found.