Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

Measurements of Transitional and Turbulent Flow in a Randomly Packed Bed of Spheres with Particle Image Velocimetry

Particle image velocimetry (PIV) has been used to investigate transitional and turbulent flow in a randomly packed bed of mono-sized transparent spheres at particle Reynolds number, \(20<{{ Re}}_{\mathrm{p}}< 3220\). The refractive index of the liquid is matched with the spheres to provide optical access to the flow within the bed without distortions. Integrated pressure drop data yield that Darcy law is valid at \({{ Re}}_{\mathrm{p}} \approx 80\). The PIV measurements show that the velocity fluctuations increase and that the time-averaged velocity distribution start to change at lower \({{ Re}}_{\mathrm{p}}\). The probability for relatively low and high velocities decreases with \({{ Re}}_{\mathrm{p}}\) and recirculation zones that appear in inertia dominated flows are suppressed by the turbulent flow at higher \({{ Re}}_{\mathrm{p}}\). Hence there is a maximum of recirculation at about \({{ Re}}_{\mathrm{p}} \approx 400\). Finally, statistical analysis of the spatial distribution of time-averaged velocities shows that the velocity distribution is clearly and weakly self-similar with respect to \({{ Re}}_{\mathrm{p}}\) for turbulent and laminar flow, respectively.     

Transitional and Turbulent Flow in a Bed of Spheres as Measured with Stereoscopic Particle Image Velocimetry

Stereoscopic particle image velocimetry has been used to investigate inertia dominated, transitional and turbulent flow in a randomly packed bed of monosized PMMA spheres. By using an index-matched fluid, the bed is optically transparent and measurements can be performed in an arbitrary position within the porous bed. The velocity field observations are carried out for particle Reynolds numbers, \({Re}_{{p}}\), between 20 and 3220, and the sampling is done at a frequency of 75 Hz. Results show that, in porous media, the dynamics of the flow can vary significantly from pore to pore. At \({Re}_{{p}}\) around 400 the spatially averaged time fluctuations of total velocity reach a maximum and the spatial variation of the time-averaged total velocity, \(u_\mathrm{tot}\) increases up to about the same \({Re}_{{p}}\) and then it decreases. Also in the studied planes, a considerable amount of the fluid moves in the perpendicular directions to the main flow direction and the time-averaged magnitude of the velocity in the main direction, \(u_{x}\), has an averaged minimum of 40% of the magnitude of \(u_\mathrm{tot}\) at \({Re}_{{p}}\) about 400. For \({Re}_{{p}} > 1600\), this ratio is nearly constant and \(u_{x}\) is on average a little bit less than 50% of \(u_\mathrm{tot}\). The importance of the results for longitudinal and transverse dispersion is discussed.     

Inertial Sensitivity of Porous Microstructures

Fluid flows through porous media are subject to different regimes, ranging from linear creeping flows to unsteady, chaotic turbulence. These different flow regimes at the pore scale have repercussions at larger scales, with the macroscale drag force experienced by a fluid moving through the medium becoming a nonlinear function of the average velocity beyond the creeping flow regime. Accurate prediction of the transition between different flow regimes is an important challenge with repercussions onto many engineering applications. Here, we are interested in the first deviation from Darcy’s law, when inertia effects become sizeable. Our goal is to define a Reynolds number, \(Re_{\mathrm{C}}\), so that the inertial deviation occurs when \(Re_{\mathrm{C}}\sim 1\) for any microstructure. The difficulty in doing so is to reduce the multiple length scales characterizing the geometry of the porous structure to a single length scale, \(\ell \). We analyze the problem using the method of volume averaging and identify a length scale in the form \(\ell =C_\lambda \sqrt{\nicefrac {K_\lambda }{\epsilon _\beta }}\), with \(C_\lambda \) a parameter that indicates the sensitivity of the microstructure to inertia. The main advantage of this definition is that an explicit formula for \(C_\lambda \) is given; \(C_\lambda \) is computed from a creeping flow simulation in the porous medium; and \(Re_{\mathrm{C}}\) can be used to predict the transition to a non-Darcian regime more accurately than by using Reynolds numbers based on alternative length scales. The theory is validated numerically with data from flow simulations for a variety of microstructures.     

Monte Carlo Assessment of the Impact of Oscillatory and Pulsating Boundary Conditions on the Flow Through Porous Media

The linear stability analysis of vertical throughflow of power law fluid for double-diffusive convection with Soret effect in a porous channel is investigated in this study. The upper and lower boundaries are assumed to be permeable, isothermal and isosolutal. The linear stability of vertical through flow is influenced by the interactions among the non-Newtonian Rayleigh number (Ra), Buoyancy ratio (N), Lewis number (Le), Péclet number (Pe), Soret parameter (Sr) and power law index (n). The results indicate that the Soret parameter has a significant influence on convective instability of power law fluid. It has also been noticed that buoyancy ratio has a dual effect on the instability of fluid flow. Further, it is noticed that the basic temperature and concentration profiles have singularities at \(Pe = 0\) and \(Le = 1\), the convective instability is looked into for the limiting case of \(Pe\rightarrow 0\) and \(Le \rightarrow 1\). For the case of pure thermal convection with no vertical throughflow, the present numerical results coincide with the solution of standard Horton–Rogers–Lapwood problem. The present results for critical Rayleigh number obtained using bvp4c and two-term Galerkin approximation are compared with those available in the literature and are tabulated.     

Numerical investigation of mixed convection heat transfer from two isothermal circular cylinders in tandem arrangement: buoyancy,spacing ratio,and confinement effects

This paper presents a two-dimensional numerical study for mixed convection in a laminar cross-flow with a pair of stationary equal-sized isothermal cylinders in tandem arrangement confined in a channel. The governing equations are solved using the control volume method on a nonuniform orthogonal Cartesian grid, and the immersed boundary method is employed to identify the cylinders placed in the flow field. The numerical scheme is first validated against standard cases of symmetrically confined isothermal circular cylinders in plane channels, and grid convergence tests were also examined. The objective of the present study was to investigate the influence of buoyancy and the blockage ratio constraint on the flow and heat transfer characteristics of the immersed cylinder array. Using a fixed Reynolds number based on cylinder diameter of \(Re_{D} = 200\), a fixed value of the Prandtl number of \(Pr = 7\), and a blockage ratio of \(D/H = 0.2\), all possible flow regimes are considered by setting the longitudinal spacing ratio (\(\sigma = L/D\)) between the cylinder axes to 2, 3, and 5 for values of the buoyancy parameter (Richardson number) in the range \(-1\le Ri\le 4\). The interference effects and complex flow features are presented in the form of mean and instantaneous velocity, vorticity, and temperature distributions. The results demonstrate how the buoyancy, spacing ratio, and wall confinement affect the wake structure and vortex dynamics. In addition, local and average heat transfer characteristics of both cylinders are comprehensively presented for a wide range in the parametric space.     

Numerical simulation of inertial flow of heated and cooled viscoelastic fluids inside a planar sudden expansion channel: investigation of stresses effects on the total dissipation

In this paper, the inertial and non-isothermal flow of viscoelastic fluids in a planar channel with 1:3 sudden expansion has been simulated for Brinkman numbers in the range \( - \,20 \le Br \le 20 \). The mass, momentum and energy conservation equations with the non-linear form of Phan-Thien–Tanner constitutive equation are used to describe the behavior of heated and cooled viscoelastic fluids flow. The properties of fluid are assumed temperature-dependent and the viscous dissipation terms are considered in the energy equation. The object of the current paper is to investigate the stresses and their effects on heat generation via the viscous dissipation terms in the energy equation for inertial flow of heated and cooled viscoelastic fluids. Therefore, plots of streamlines, isothermal lines, normal stress (\( \tau_{xx} \)), normal-transverse stress (\( \tau_{yy} \)) and shear stress (\( \tau_{xy} \)), total dissipation, temperature and local Nusselt numbers have been drawn and examined in the channel expansion. The results show that for the asymmetric flow of heated and cooled viscoelastic fluids, the maximum values of total dissipation are located adjacent to the lower wall and at the centerline of the channel expansion. Also, by incrementing the Brinkman number in the hydrodynamically and thermally developing and fully developed zones, the values of total dissipation are increased.     

Spanwise effects on instabilities of compressible flow over a long rectangular cavity

The stability properties of two-dimensional (2D) and three-dimensional (3D) compressible flows over a rectangular cavity with length-to-depth ratio of \(L/D=6\) are analyzed at a free-stream Mach number of \(M_\infty =0.6\) and depth-based Reynolds number of \(Re_D=502\). In this study, we closely examine the influence of three-dimensionality on the wake mode that has been reported to exhibit high-amplitude fluctuations from the formation and ejection of large-scale spanwise vortices. Direct numerical simulation (DNS) and bi-global stability analysis are utilized to study the stability characteristics of the wake mode. Using the bi-global stability analysis with the time-averaged flow as the base state, we capture the global stability properties of the wake mode at a spanwise wavenumber of \(\beta =0\). To uncover spanwise effects on the 2D wake mode, 3D DNS are performed with cavity width-to-depth ratio of \(W/D=1\) and 2. We find that the 2D wake mode is not present in the 3D cavity flow with \(W/D=2\), in which spanwise structures are observed near the rear region of the cavity. These 3D instabilities are further investigated via bi-global stability analysis for spanwise wavelengths of \(\lambda /D=0.5{-}2.0\) to reveal the eigenspectra of the 3D eigenmodes. Based on the findings of 2D and 3D global stability analysis, we conclude that the absence of the wake mode in 3D rectangular cavity flows is due to the release of kinetic energy from the spanwise vortices to the streamwise vortical structures that develops from the spanwise instabilities.     

A Note on the Relative Equilibria Bifurcations in the $$$$-Body Problem

Consider the planar Newtonian \((2N+1)\)-body problem, \(N\ge 1,\) with \(2N\) bodies of unit mass and one body of mass \(m\). Using the discrete symmetry due to the equal masses and reducing by the rotational symmetry, we show that solutions with the \(2N\) unit mass points at the vertices of two concentric regular \(N\)-gons and \(m\) at the centre at all times form invariant manifold. We study the regular \(2N\)-gon with central mass \(m\) relative equilibria within the dynamics on the invariant manifold described above. As \(m\) varies, we identify the bifurcations, relate our results to previous work and provide the spectral picture of the linearization at the relative equilibria.     

Passive Boundary Layer Flow Control Using Porous Lamination

The flow over a porous laminated flat plate is investigated from a flow control perspective through experiments and computations. A square array of circular cylinders is used to model the porous lamination. We determine the velocities at the fluid–porous interface by solving the two-dimensional Navier–Stokes and the continuity equations using a staggered flow solver and using LDV in experiments. The control parameters for the porous region are porosity, \(\phi \) and Reynolds number, Re, based on the diameter of the circular cylinders used to model the porous lamination. Computations are conducted for \(0.4< \phi < 0.9\) and \(25< Re < 1000\), and the experiments are conducted for \(\phi = 0.65\) and 0.8 at \(Re \approx 391,\ 497\) and 803. The permeability of the porous lamination is observed to induce a slip velocity at the interface, effectively making it a slip wall. The slip velocity is seen to be increasing functions of \(\phi \) and Re. For higher porosities at higher Re, the slip velocity shows non-uniform and unsteady behavior and a breakdown Reynolds number is defined based on this characteristic. A map demarcating the two regimes of flow is drawn from the computational and experimental data. We observe that the boundary layer over the porous lamination is thinner than the Blasius boundary layer and the shear stress is higher at locations over the porous lamination. We note that the porous lamination helps maintain a favorable pressure gradient at the interface which delays separation. The suitable range of porosities for effective passive separation control is deduced from the results.     

Identification of flow regimes around two staggered square cylinders by a numerical study

The flow over two square cylinders in staggered arrangement is simulated numerically at a fixed Reynolds number (\(Re =150\)) for different gap spacing between cylinders from 0.1 to 6 times a cylinder side to understand the flow structures. The non-inclined square cylinders are located on a line with a staggered angle of \(45^{\circ }\) to the oncoming velocity vector. All numerical simulations are carried out with a finite-volume code based on a collocated grid arrangement. The effects of vortex shedding on the various features of the flow field are numerically visualized using different flow contours such as \(\lambda _{2}\) criterion, vorticity, pressure and magnitudes of velocity to distinguish the distinctive flow patterns. By changing the gap spacing between cylinders, five different flow regimes are identified and classified as single body, periodic gap flow, aperiodic, modulated periodic and synchronized vortex shedding regimes. This study revealed that the observed multiple frequencies in global forces of the downstream cylinder in the modulated periodic regime are more properly associated with differences in vortex shedding frequencies of individual cylinders than individual shear layers reported in some previous works; particularly, both shear layers from the downstream cylinder often shed vortices at the same multiple frequencies. The maximum Strouhal number for the upstream cylinder is also identified at \({G}^{*}=1\) for aperiodic flow pattern. Furthermore, for most cases studied, the downstream cylinder experiences larger drag force than the upstream cylinder.     

Direct Numerical Simulation of Flow over Periodic Hills up to $\text {Re}_{H}= 10{,}595$

We present fully resolved computations of flow over periodic hills at the hill-Reynolds numbers \(\text {Re}_{H}=?5{,}600\) and \(\text {Re}_{H}=?10{,}595\) with the highest fidelity to date. The calculations are performed using spectral incompressible discontinuous Galerkin schemes of \(8^{\text {th}}\) and \(7^{\text {th}}\) order spatial accuracy, \(3^{\text {rd}}\) order temporal accuracy, as well as 34 and 180 million grid points, respectively. We show that the remaining discretization error is small by comparing the results to h- and p-coarsened simulations. We quantify the statistical averaging error of the reattachment length, as this quantity is widely used as an ‘error norm’ in comparing numerical schemes. The results exhibit good agreement with the experimental and numerical reference data, but the reattachment length at \(\text {Re}_{H}=?10{,}595\) is predicted slightly shorter than in the most widely used LES references. In the second part of this paper, we show the broad range of capabilities of the numerical method by assessing the scheme for underresolved simulations (implicit large-eddy simulation) of the higher Reynolds number in a detailed h/p convergence study.     

The flow past a freely rotating sphere

We consider the flow past a sphere held at a fixed position in a uniform incoming flow but free to rotate around a transverse axis. A steady pitchfork bifurcation is reported to take place at a threshold \(Re^\mathrm{OS}=206\) leading to a state with zero torque but nonzero lift. Numerical simulations allow to characterize this state up to \(Re\approx 270\) and confirm that it substantially differs from the steady-state solution which exists in the wake of a fixed, non-rotating sphere beyond the threshold \(Re^\mathrm{SS}=212\). A weakly nonlinear analysis is carried out and is shown to successfully reproduce the results and to give substantial improvement over a previous analysis (Fabre et al. in J Fluid Mech 707:24–36, 2012). The connection between the present problem and that of a sphere in free fall following an oblique, steady (OS) path is also discussed.     

Homoclinic Orbits and Entropy for Three-Dimensional Flows

We prove that every \(C^1\) three-dimensional flow with positive topological entropy can be \(C^1\) approximated by flows with homoclinic orbits.     

Effective Rheology of Two-Phase Flow in Three-Dimensional Porous Media: Experiment and Simulation

We present an experimental and numerical study of immiscible two-phase flow of Newtonian fluids in three-dimensional (3D) porous media to find the relationship between the volumetric flow rate (Q) and the total pressure difference (\(\Delta P\)) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold (\(P_t\)), making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium. In this regime, Q depends quadratically on an excess pressure drop (\(\Delta P-P_t\)). While increasing the flow rate, there is a transition, beyond which the overall flow is Newtonian and the relationship is linear. In our experiments, we build a model porous medium using a column of glass beads transporting two fluids, deionized water and air. For the numerical study, reconstructed 3D pore networks from real core samples are considered and the transport of wetting and non-wetting fluids through the network is modeled by tracking the fluid interfaces with time. We find agreement between our numerical and experimental results. Our results match with the mean-field results reported earlier.     

Glimpses of Flow Development and Degradation During Type B Drag Reduction by Aqueous Solutions of Polyacrylamide B1120

Flow development and degradation during Type B turbulent drag reduction by 0.10 to 10 wppm solutions of a partially-hydrolysed polyacrylamide B1120 of MW \(=\) 18x10⁶ was studied in a smooth pipe of ID \(=\) 4.60 mm and L/D \(=\) 210 at Reynolds numbers from 10000 to 80000 and wall shear stresses Tw from 8 to 600 Pa. B1120 solutions exhibited facets of a Type B ladder, including segments roughly parallel to, but displaced upward from, the P-K line; those that attained asymptotic maximum drag reduction at low Re√ f but departed downwards into the polymeric regime at a higher retro-onset Re√ f; and segments at MDR for all Re√ f. Axial flow enhancement profiles of S\(^{\prime }\) vs L/D reflected a superposition of flow development and polymer degradation effects, the former increasing and the latter diminishing S\(^{\prime }\) with increasing distance downstream. Solutions that induced normalized flow enhancements S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_e,p/D \(=\) L_e,n/D \(<\) 42.3, while those at maximum drag reduction showed entrance lengths L_e,m/D \(\sim \) 117, roughly 3 times the solvent L_e,n/D. Degradation kinetics were inferred by first detecting a falloff point (Re√f^∧, S\(^{{\prime }\wedge }\)), of maximum observed flow enhancement, for each polymer solution. A plot of S\(^{{\prime }\wedge }\)vs C revealed S\(^{{\prime }\wedge }\)linear in C at low C, with lower bound [S\(^{\prime }\)] \(=\) 5.0 wppm^??1, and S\(^{{\prime }\wedge }\) independent of C at high C, with upper bound S\(^{\prime }_{\mathrm {m}} =\) 15.9. The ratio S\(^{\prime }\)/S\(^{{\prime }\wedge }\) in any pipe section was interpreted to be the undegraded fraction of original polymer therein. Semi-log plots of (S\(^{\prime }\)/S\(^{{\prime }\wedge }\)) at a section vs transit time from pipe entrance thereto revealed first order kinetics, from which apparent degradation rate constants k_deg s^??1 and entrance severities ?ln(S\(^{\prime }\)/S\(^{{\prime }\wedge }\))₀ were extracted. At constant C, k_deg increased linearly with increasing wall shear stress Tw, and at constant Tw, k_deg was independent of C, providing a B1120 degradation modulus (k_deg/Tw) \(=\) (0.012 \(\pm \) 0.001) (Pa s)^??1 for 8 \(<\) Tw Pa \(<\) 600, 0.30 \(<\) C wppm \(<\) 10. Entrance severities were negligible below a threshold Twe \(\sim \) 30 Pa and increased linearly with increasing Tw for Tw \(>\) Twe. The foregoing methods were applied to Type A drag reduction by 0.10 to 10 wppm solutions of a polyethyleneoxide PEO P309, MW \(=\) 11x10⁶, in a smooth pipe of ID \(=\) 7.77 mm and L/D \(=\) 220 at Re from 4000 to 115000. P309 solutions that induced S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_e,p/D \(=\) L_e,n/D \(<\) 23, while those at MDR had entrance lengths L_e,m/D \(\sim \) 93, roughly 4 times the solvent L_e,n/D. P309 solutions described a Type A fan distorted by polymer degradation. A typical trajectory departed the P-K line at an onset point Re√ f* followed by ascending and descending polymeric regime segments separated by a falloff point Re√f^∧, of maximum flow enhancement; for all P309 solutions, onset Re√ f* = 550 \(\pm \) 100 and falloff Re√f^∧ = 2550 \(\pm \) 250, the interval between them delineating Type A drag reduction unaffected by degradation. A plot of falloff S\(^{{\prime }\wedge }\) vs C for PEO P309 solutions bore a striking resemblance to the analogous S\(^{{\prime }\wedge }\) vs C plot for solutions of PAMH B1120, indicating that the initial Type A drag reduction by P309 after onset at Re√ f* had evolved to Type B drag reduction by falloff at Re√f^∧. Presuming that Type B behaviour persisted past falloff permitted inference of P309 degradation kinetics; k_deg was found to increase linearly with increasing Tw at constant C and was independent of C at constant Tw, providing a P309 degradation modulus (k_deg/Tw) \(=\) (0.011 \(\pm \) 0.002) (Pa s)^??1 for 4 \(<\) Tw Pa \(<\) 400, 0.10 \(<\) C wppm < 5.0. Comparisons between the present degradation kinetics and previous literature showed (k_deg/Tw) data from laboratory pipes of D \(\sim \) 0.01 m to lie on a simple extension of (k_deg/Tw) data from pipelines of D \(\sim \) 0.1 m and 1.0 m, along a power-law relation (k_deg/Tw) \(=\) 10^??5.4.D^??1.6. Intrinsic slips derived from PAMH B1120 and PEO P309-at-falloff experiments were compared with previous examples from Type B drag reduction by polymers with vinylic and glycosidic backbones, showing: (i) For a given polymer, [S\(^{\prime }\)] was independent of Re√ f and pipe ID, implying insensitivity to both micro- and macro-scales of turbulence; and (ii) [S\(^{\prime }\)] increased linearly with increasing polymer chain contour length Lc, the proportionality constant \(\beta =\) 0.053 \(\pm \) 0.036 enabling estimation of flow enhancement S\(^{\prime } =\) C.Lc.β for all Type B drag reduction by polymers.     

Flow field in the wake of a bluff body driven through a steady recirculating flow

The wake produced by a bluff body driven through a steady recirculating flow is studied experimentally in a water facility using particle image velocimetry. The bluff body has a rectangular cross section of height, \(H\), and width, \(D\), such that the aspect ratio, AR = H/D, is equal to 3. The motion of the bluff body is uniform and rectilinear, and corresponds to a Reynolds number based on width, Re _D  = 9,600. The recirculating flow is confined within a hemicylindrical enclosure and is generated by planar jets emanating from slots of width, \(h\), such that \(Re_h=500\). Under these conditions, experiments are performed in a closed-loop facility that enables complete optical access to the near-wake. Velocity fields are obtained up to a distance of \(13D\) downstream of the moving body. Data include a selection of phase-averaged velocity fields representative of the wake for a baseline case (no recirculation) and an interaction case (with recirculation). Results indicate that the transient downwash flow typically observed in wakes behind finite bodies of small aspect ratio is significantly perturbed by the recirculating flow. The wake is displaced from the ground plane and exhibits a shorter recirculation zone downstream of the body. In summary, it was found that the interaction between a bluff body wake and a recirculating flow pattern alters profoundly the dynamics of the wake, which has implications on scalar transport in the wake.     

Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

An innovative tooth root profile for spur gears and its effect on service life

An innovative approach to the design of the gear-tooth root-profile, and its effects on the service life is reported in this paper. In comparison with the widely used trochoidal and the recently proposed circular-filleted root profiles, the optimum profile proposed here is a \(G^2\)-continuous curve that blends smoothly with both the involute of the tooth profile and the dedendum circle. Following the AGMA and ISO standards for fatigue loading, the von Mises stress at the critical section and stress distribution along the gear tooth root are studied. The process leading to gear-tooth failure is composed of the crack initiation phase, in number of cycles \(N_i\), and the crack propagation phase, in \(N_p\) cycles. The strain-life (\(\epsilon\)-N) method is employed to determine \(N_i\), where the crack is assumed to initiate at the critical section. Based on the ANSYS crack-analysis module, the effects of \(G^2\)-continuous blending on the stress intensity factor (SIF) are investigated for different crack sizes. Paris’ law, within the framework of Linear Elastic Fracture Mechanics, is used to correlate the SIF with crack size and, further, to determine \(N_p\). The optimum profile provides a significant reduction in SIF and improvement in both \(N_i\) and \(N_p\). Spur gears are made of high-strength steel alloy 42CrMo4, the effects of its properties and surface treatment on service life improvement not being included in this study.