Motion of singles bubbles moving under a slightly inclined surface through stationary liquids

The influence of the liquid properties on the dynamical bubble shape and on the bubble motion has been investigated for bubbles moving under a downward facing inclined surface. The Morton number Mo varied from 2.59 × 10⁻¹¹ to 2.52 × 10⁺⁰¹. The Bond number Bo covered the range from 10 to 150 and the surface inclination angle θ was varied from 2° to 6°. To cover the wide range of Mo, several liquids such as glycerine, propanediol, water and isopropanol were used. The results have shown that the relation Fr = Fr(Bo, Mo, θ) is not adequate to describe the bubble motion, where Fr is the terminal Froude number. The choice of the terminal Reynolds number Re as the dependent parameter, allowed the clarification of the role of the Morton number on the bubble motion. At a given Bond number, the bubble Reynolds number decreases monotonously with the Morton number. Furthermore, an empirical correlation Re = Re(Bo, Mo, θ) is given that can be readily used in the mathematical modelling of bubble laden flows under solids.     

Steady rise of a small spherical gas bubble along the axis of a cylindrical pipe at high Reynolds number

Steady irrotational flow of inviscid liquid of density ρ_l around a spherical gas bubble which lies on the axis of a cylindrical pipe is investigated using the analysis of Smythe (Phys. Fluids 4 (1961) 756). The bubble radius b=qa is assumed small compared to the pipe radius a, and the interfacial tension between gas and liquid is γ. Far from the bubble, in the frame in which the bubble is at rest, the liquid velocity along the pipe is v₀, whereas the liquid velocity at points on the wall closest to the bubble is U_zw=v₀(1+1.776q³+⋯). The decrease in wall pressure as the bubble passes is therefore Δp=1.776ρ_lv₀²q³. When the Weber number W=2bv₀²ρ_l/γ is small, the bubble deforms into an oblate spheroid with aspect ratio χ=1+9W(1+1.59q³)/64. If the fluid viscosity μ is non-zero, and the Reynolds number Re=2v₀ρ_lb/μ is large, a viscous boundary layer develops on the walls of the pipe. This decays algebraically with distance downstream of the bubble, and an exponentially decaying similarity solution is found upstream. The drag D on the bubble is D=12πμv₀b(1−2.21Re^−1/2)(1+1.59q³)+7.66μv₀bRe^1/2q^9/2, larger than that given by Moore (J. Fluid Mech. 16 (1963) 161) for motion in unbounded fluid. At high Reynolds numbers the dissipation within the viscous boundary layers might dominate dissipation in the potential flow away from the pipe walls, but such high Reynolds numbers would not be achieved by a spherical air bubble rising in clean water under terrestrial gravity.     

Fluid forces on a very low Reynolds number airfoil and their prediction

This paper presents the measurements of mean and fluctuating forces on an NACA0012 airfoil over a large range of angle (α) of attack (0-90°) and low to small chord Reynolds numbers (Re_c), 5.3 × 10³-5.1 × 10⁴, which is of both fundamental and practical importance. The forces, measured using a load cell, display good agreement with the estimate from the LDA-measured cross-flow distributions of velocities in the wake based on the momentum conservation. The dependence of the forces on both α and Re_c is determined and discussed in detail. It has been found that the stall of an airfoil, characterized by a drop in the lift force and a jump in the drag force, occurs at Re_c ? 1.05 × 10⁴ but is absent at Re_c = 5.3 × 10³. A theoretical analysis is developed to predict and explain the observed dependence of the mean lift and drag on α.     

Direct numerical simulation of a turbulent boundary layer up to Reθ = 2500

Direct numerical simulations (DNSs) of a turbulent boundary layer (TBL) with Re_θ = 570-2560 were performed to investigate the spatial development of its turbulence characteristics. The inflow simulation was conducted in the range Re_θ = 570-1600 by using Lund’s method. To resolve the numerical periodicity induced by the recycling method, we adopted a sufficiently long streamwise domain of x/θ_in,i = 1000 (=125δ_0,i), where θ_in,i is the inlet momentum thickness and δ_0,i is the inlet boundary layer thickness in the inflow simulation. Furthermore, the main simulation with a length greater than 50δ₀ was carried out independently by using the inflow data, where δ₀ is the inlet boundary layer thickness of the main simulation. The integral quantities and the first-, second- and higher-order turbulence statistics were compared with those of previous data, and good agreement was found. The present study provides a useful database for the turbulence statistics of TBLs. In addition, instantaneous field and two-point correlation of the streamwise velocity fluctuations displayed the existence of the very large-scale motions (VLSMs) with the characteristic widths of 0.1-0.2δ and that the flow structure for a length of approximately ∼6δ fully occupies the streamwise domain statistically.     

Screw dislocations and an interfacial blunt crack with viscoelastic interface

This work deals with the influence of Kelvin-type viscoelastic interface on the generation of screw dislocations near the interfacial blunt crack tip in light of a pair of concentrated loads. The stress fields for dislocation and concentrated load have been obtained by using the integral transform and conformal mapping, the stress intensity factor have been studied, the image force acting on dislocation has been analyzed. The region r_b where n screw dislocations are generated by a pair of concentrated loads and dislocation number are obtained by displacement compatibility and stress compatibility conditions of self-consistent and self-equilibrated systems. The results show that: the force acting on dislocation starts with the value that a perfectly bonded interface, then with relaxation of the imperfect interface; the shield effect for dislocation decreases as time goes by; in addition, with time elapsing, the influence of material shear modulus rate on shielding effect becomes weaker and weaker. The scale of multiplier α(r_b/a) increases with relaxation of imperfect interface, the larger ratio of crack geometry c/a and the smaller ratio of shear modulus μ₁/μ₂ will lead the higher scale of multiplier. When μ₁/μ₂ = 1, the screw dislocations number first increases and then decreases with relaxation of imperfect interface, In addition, it possesses the highest value at t₀ ≈ 1 and tends to vanish at t₀ = ∞. When μ₁ < μ₂, the screw dislocations number increases with relaxation of imperfect interface. When μ₁ > μ₂, the screw dislocations number first increases then decreases with relaxation of imperfect interface, and possesses the highest value at t₀ ≈ 1, the negative value are exclude from the discussion.     

Temporal stability of a particle-laden jet

The temporal instability of a particle-laden jet was investigated numerically which took into consideration the parametric effects of jet parameter, B, jet Reynolds number, Re_j, particle mass loading, Z and Stokes number, St. The linear stability theory was used to derive the instability equations of a viscous particle-laden jet flow. The single-phase instability of a top-hat jet was then calculated and compared with the available analytical theories. The numerical results agree well with the analytical results for both the axisymmetric (n = 0) and first azimuthal (n = 1) modes. The results show that the first azimuthal mode disturbance is usually more unstable than that of the axisymmetric mode. But the axisymmetric mode disturbance can be more unstable when Z is high enough (i.e., Z ? 0.1). The higher B and Re_j are, the more unstable the particle-laden jet will be. The existence of particles enhances the flow stability. With the increasing of Z, the jet flow will grow more stable. The inviscid single-phase jet is the most unstable. The wave amplification, c_i first decreases with the increasing of St and then increases afterwards. There exist certain values of St, at which the jet is the most stable.     

Separation zones in the flow near a small-angle convex corner

On the basis of an asymptotic analysis of the Navier-Stokes system of equations for large Reynolds numbers (Re → ∞), the plane incompressible fluid flow near a surface having a convex corner with a small angle 2θ* is investigated. It is shown that for θ* = O(Re^?1/4), in addition to the known solution that describes a separated flow completely localized in a thin “viscous” sublayer of the interaction region near the corner point, another solution corresponding to a flow with a developed separation zone is possible. For θ ₀ = Re^1/4 θ* = O(1), the longitudinal dimension of this zone varies from finite values up to values of the order of Re^?3/8. The nonuniqueness of the solution is established on a certain range of variation of the parameter θ ₀. The dependence of the drag coefficient on the angle θ* is found.     

Direct numerical simulation of stationary particles in homogeneous turbulence decay: Application of the k–ε model

This study focuses on understanding how the presence of particles, in homogeneous turbulence decay, affects the dissipation of dissipation coefficient within the volume averaged dissipation transport equation. In developing this equation, the coefficient for dissipation of dissipation was assumed to be the sum of the single phase coefficient and an additional coefficient that is related to the effects of the dispersed phase. Direct numerical simulation was used to isolate the effect of stationary particles in homogeneous turbulent decay at low Reynolds numbers (Re_L = 3.3 and 12.5). The particles were positioned at each grid point and modeled as point forces and a comparison was made between a 64³ and 128³ domain. The results show that the dissipation of dissipation coefficient correlates well with a dimensionless parameter called the momentum coupling factor.     

Asymptotic behavior of Toeplitz matrices and determinants

We consider the inverse X _N and determinant D_N(c) of an N×N Toeplitz matrix C_N=[c_i?j] ₀ ^N?1 as N ar∞. Under the condition that there exists a monotonic decreasing summable bound b _n ≧|c _n |+|c _?n |, and that the generating function \(c(\theta ) = \sum\limits_{n = - \infty }^\infty {c_n e^{i{\text{ }}n{\text{ }}\theta } }\) does not vanish, we construct a matrix iterative process which yields (i) explicit asymptotic formulae for the elements of X_N when v(c) = (2π)^?1 [arg{c(2π)}?arg{c(0)}] is zero. Thence we obtain (ii) expressions for the constants, and bounds on the remainder, in the asymptotic formula $$\ln D_N (c) = N{\text{ }}k_0 (c) + E_0 (c) + E_{1,N} (c) + \mathcal{R}_N (c),$$ and (iii) the extension of this formula to the case of general integral v(c). Under certain further conditions the monotonicity of E_1,N+?_N is proved. We discuss various identities for D_N which apply when c(θ) is a rational function of e^iθ and mention a conjecture for D _N when c(θ) has zeros, and is discontinuous with arbitrary v(c).     

Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ₁(z) = μ₀(1 + αz)^β, where μ₀, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.     

LES of turbulent flow in a concentric annulus with rotating outer wall

Fully-developed turbulent flow in a concentric annulus, r₁/r₂ = 0.5, Re_h = 12,500, with the outer wall rotating at a range of rotation rates N = U_θ,wall/U_b from 0.5 up to 4 is studied by large-eddy simulations. The focus is on the effects of moderate to very high rotation rates on the mean flow, turbulence statistics and eddy structure. For N up to ∼2, an increase in the rotation rate dampens progressively the turbulence near the rotating outer wall, while affecting only mildly the inner-wall region. At higher rotation rates this trend is reversed: for N = 2.8 close to the inner wall turbulence is dramatically reduced while the outer wall region remains turbulent with discernible helical vortices as the dominant turbulent structure. The turbulence parameters and eddy structures differ significantly for N = 2 and 2.8. This switch is attributed to the centrifuged turbulence (generated near the inner wall) prevailing over the axial inertial force as well as over the counteracting laminarizing effects of the rotating outer wall. At still higher rotation, N = 4, the flow gets laminarized but with distinct spiralling vortices akin to the Taylor–Couette rolls found between the two counter-rotating cylinders without axial flow, which is the limiting case when N approaches to infinity. The ratio of the centrifugal to axial inertial forces, Ta/Re² ∝ N² (where Ta is the Taylor number) is considered as a possible criterion for defining the conditions for the above regime change.     

Effects of the roughness height in turbulent boundary layers over rod- and cuboid-roughened walls

Direct numerical simulations (DNSs) of spatially developing turbulent boundary layers (TBLs) over two-dimensional (2D) rod-roughened walls and three-dimensional (3D) cuboid-roughened walls are conducted to investigate the effects of the roughness height on the flow characteristics in the outer layer. The rod elements are periodically aligned along the downstream direction with a pitch of p_x/θ_in = 12, and the cuboid elements are periodically staggered with a pitch of p_x/θ_in = 12 and p_z/θ_in = 3, where p_x and p_z are correspondingly the streamwise and spanwise pitches of the roughness and θ_in is the momentum thickness at the inlet. The first surface roughness is placed 80θ_in downstream from the inlet, leading to a step change from a smooth to rough surface. The rod and cuboid roughness height (k) is varied in the range of 0.1 ≤ k/θ_in ≤ 1.8 (13 ≤ δ/k ≤ 285), respectively (δ is the boundary layer thickness), and the Reynolds number based on the momentum thickness (θ) is varied in the range of Re_θ = 300 ~ 1400. For each case, the self-preservation form of the velocity-defect and the turbulent Reynolds stresses is achieved along the downstream direction. As the roughness height increases, the roughness function (ΔU⁺) extracted from the mean velocity profiles increases, although the velocity-defect profiles for the rough-wall cases show good agreement with the profile from the smooth-wall case. The magnitude of the Reynolds stresses in the outer layer increases with an increase of k/δ. The outer layer similarity between the flows over the rough and smooth-walls is found when δ/k ≥ 250 and 100 for the 2D rod and 3D cuboid, respectively. The continuous increase of the Reynolds stresses in the outer layer with an increase of k/δ is explained by a large population of very long structures over the rough-wall flows. Because the characteristic width of the structures increases continuously with an increase of k/δ for the rod and cuboid roughness, a wide width of the structures leads to frequent spanwise merging between adjacent structures. The active spanwise merging events with an increase of k/δ increase the streamwise coherence of the structures with the appearance of significant meandering.     

Strength criterion and elastoplastic constitutive model of frozen silt in generalized plastic mechanics

Triaxial compressive tests of frozen silt were carried out under confining pressures from 0.0 to 14.0 MPa at the temperatures of −2, −4 and −6 °C. A strength criterion based upon experimental results is presented by the combination of extended Lade–Duncan strength function f_π(θ) in π-plane and f_p–q(p) in p–q-plane. In order to describe the deformation characteristic of frozen silt, an elastoplastic constitutive model in generalized plastic mechanics has been proposed for the nonlinear behavior of frozen silt, such as the pressure melting and crushing phenomena, strain softening/hardening characteristics and dilatation, etc., by employing an elliptical yield surface, together with a non-associated flow rule for the compressive mechanism, and two parabolic yield surfaces, together with non-associated flow rules for the shear mechanism. The validity of the model is verified by comparing its modeling results with the results of triaxial compressive tests. It is found that the stress–strain curves predicted by this model agree well with the corresponding experimental results both under low and high confining pressures.     

Mathematical expression of pressure gradient in the flow of spherical capsules less dense than water

This study yielded a mathematical expression to calculate the pressure gradient (ΔP/L)_m of the flow of a spherical capsule train. An experimental investigation was carried out to determine pressure drops of two-phase mixture flow of spherical ice capsules and water inside the pipelines of cooling systems. Instead of ice capsules, spherical capsules made of polypropylene material whose density (870 kg/m³) is similar to that of ice were used in the experiments. Flow behavior of the spherical capsules, 0.08 m outer diameter, was observed in the measuring section inside plexiglass pipes, 0.1 m inner diameter (ID) and 6 m in length; pressure drops were measured on the 4 m section. The investigation was carried out in the 1.2 × 10⁴ < Re < 1.5 × 10⁵ range and under transport concentration (C_tr) by 5–30%. Dimensionless numbers of the physical event were found out by conducting a dimensional analysis, so that mixture density was expressed in terms of specific gravity and in situ concentration. After arriving at certain conclusions based on the relevant experimental findings and observations, empirical and mathematical models which can be used for calculation of the pressure gradient were developed. Comparison of the mathematical model with the experimental findings revealed that pressure drop values deviated by 2.7% on average for 2.5 × 10⁴ < Re < 1.5 × 10⁵.     

Aerodynamics of laminar separation flutter at a transitional Reynolds number

Experimental observations of self-sustained pitch oscillations of a NACA 0012 airfoil at transitional Reynolds numbers were recently reported. The aeroelastic limit cycle oscillations, herein labelled as laminar separation flutter, occur in the range 5.0×10⁴≤Re_c≤1.3×10⁵. They are well behaved, have a small amplitude and oscillate about θ=0°. It has been speculated that laminar separation leading to the formation of a laminar separation bubble, occurring at these Reynolds numbers, plays an essential role in these oscillations. This paper focuses on the Re_c=7.7×10⁴ case, with the elastic axis located at 18.6% chord. Considering that the experimental rig acts as a dynamic balance, the aerodynamic moment is derived and is empirically modelled as a generalized Duffing–van-der-Pol nonlinearity. As expected, it behaves nonlinearly with pitch displacement and rate. It also indicates a dynamically unstable equilibrium point, i.e. negative aerodynamic damping. In addition, large eddy simulations of the flow around the airfoil undergoing prescribed simple harmonic motion, using the same amplitude and frequency as the aeroelastic oscillations, are performed. The comparison between the experiment and simulations is conclusive. Both approaches show that the work done by the airflow on the airfoil is positive and both have the same magnitude. The large eddy simulation (LES) computations indicate that at θ=0°, the pitching motion induces a lag in the separation point on both surfaces of the airfoil resulting in negative pitching moment when pitching down, and positive moment when pitching up, thus feeding the LCO.     

Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres

Gas–solid momentum transfer is a fundamental problem that is characterized by the dependence of normalized average fluid–particle force F on solid volume fraction ? and the Reynolds number based on the mean slip velocity Re_m. In this work we report particle-resolved direct numerical simulation (DNS) results of interphase momentum transfer in flow past fixed random assemblies of monodisperse spheres with finite fluid inertia using a continuum Navier–Stokes solver. This solver is based on a new formulation we refer to as the Particle-resolved Uncontaminated-fluid Reconcilable Immersed Boundary Method (PUReIBM). The principal advantage of this formulation is that the fluid stress at the particle surface is calculated directly from the flow solution (velocity and pressure fields), which when integrated over the surfaces of all particles yields the average fluid–particle force. We demonstrate that PUReIBM is a consistent numerical method to study gas–solid flow because it results in a force density on particle surfaces that is reconcilable with the averaged two-fluid theory. The numerical convergence and accuracy of PUReIBM are established through a comprehensive suite of validation tests. The normalized average fluid–particle force F is obtained as a function of solid volume fraction ? (0.1 ? ? ? 0.5) and mean flow Reynolds number Re_m (0.01 ? Re_m ? 300) for random assemblies of monodisperse spheres. These results extend previously reported results of  and  to a wider range of ?, Re_m, and are more accurate than those reported by Beetstra et al. (2007). Differences between the drag values obtained from PUReIBM and the drag correlation of Beetstra et al. (2007) are as high as 30% for Re_m in the range 100–300. We take advantage of PUReIBM’s ability to directly calculate the relative contributions of pressure and viscous stress to the total fluid–particle force, which is useful in developing drag correlations. Using a scaling argument, Hill et al. (2001b) proposed that the viscous contribution is independent of Re_m but the pressure contribution is linear in Re_m (for Re_m > 50). However, from PUReIBM simulations we find that the viscous contribution is not independent of the mean flow Reynolds number, although the pressure contribution does indeed vary linearly with Re_m in accord with the analysis of Hill et al. (2001b). An improved correlation for F in terms of ? and Re_m is proposed that corrects the existing correlations in Re_m range 100–300. Since this drag correlation has been inferred from simulations of fixed particle assemblies, it does not include the effect of mobility of the particles. However, the fixed-bed simulation approach is a good approximation for high Stokes number particles, which are encountered in most gas–solid flows. This improved drag correlation can be used in CFD simulations of fluidized beds that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior.     

Detached direct numerical simulations of turbulent two-phase bubbly channel flow

DNS simulations of two-phase turbulent bubbly channel flow at Re_τ = 180 (Reynolds number based on friction velocity and channel half-width) were performed using a stabilized finite element method (FEM) and a level set approach to track the air/water interfaces.     

Effect of primary jet geometry on ejector performance: A cold-flow investigation

The following cold-flow study examines the interaction of the diffracted shock wave pattern and the resulting vortex loop emitted from a shock tube of various geometries, with an ejector having a round bell-shaped inlet. The focus of the study is to examine the performance of the ejector when using different jet geometries (primary flow) to entrain secondary flow through the ejector. These include two circular nozzles with internal diameters of 15 mm and 30 mm, two elliptical nozzles with minor to major axis ratios of a/b = 0.4 and 0.6 with b = 30 mm, a square nozzle with side lengths of 30 mm, and two exotic nozzles resembling a pair of lips with axis ratios of a/b = 0.2 and 0.5 with b = 30 mm. Shock tube driver pressures of P₄ = 4, 8, and 12 bar were studied, with the pressure of the shock tube driven section P₁ being atmospheric. High-speed schlieren photography using the Shimadzu Hypervision camera along with detailed pressure measurements along the ejector and the impulse created by the ejector were conducted.     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

New type of equation for the relation between viscosity and particle content in suspensions

Many disperse systems show a typical non-Newtonian flow at relatively high concentrations of the disperse particle. However, two Newtonian viscosities η_∞ and η₀ can be, respectively, determined at high and low rates of shear. Expect for very low particle content, η_∞/η_s is proportional to exp(mϕ), where η_s is a medium viscosity, m a constant which might reflect the particle-particle interaction and ϕ the volume fraction. In considering this relationship, a new type of equation which describes the relation between the zero shear relative viscosity η_r0( ≡ η₀/η_s) of the disperse system and ϕ is proposed as follows. ln η_r0 = A(p)ϕ + am³ϕ², where A(p), the Einstein-Simha constant, is a function of the axial ratio p of dispersing particles, and a is a constant (⋍ 0.03) which depends slightly on the particle shape.The equation has been compared with the experimental results obtained for several disperse systems. A number of disperse systems of spherical particles are described well by the choice A(p) = 2.5 and a = 0.027, and a system of rod-like particles with p = 50 by the choice A(p) = 215.6 and a = 0.033. m for rod-like particles is larger than that for spherical particles.