From Rate-Dependent to Rate-Independent Brittle Crack Propagation

On the base of many experimental results, e.g., Ravi-Chandar and Knauss (Int. J. Fract. 26:65–80, 1984), Sharon et al. (Phys. Rev. Lett. 76(12):2117–2120, 1996), Hauch and Marder (Int. J. Fract. 90:133–151, 1998), the object of our analysis is a rate-dependent model for the propagation of a crack in brittle materials. Restricting ourselves to the quasi-static framework, our goal is a mathematical study of the evolution equation in the geometries of the ‘Single Edge Notch Tension’ and of the ‘Compact Tension’. Besides existence and uniqueness, emphasis is placed on the regularity of the evolution making reference also to the ‘velocity gap’. The transition to the rate-independent model of Griffith is obtained by time rescaling, proving convergence of the rescaled evolutions and of their energies. Further, the discontinuities of the rate-independent evolution are characterized in terms of unstable points of the free energy. Results are illustrated by a couple of numerical examples in the above mentioned geometries.     

Effect of chamber pressure on spreading and splashing of liquid drops upon impact on a dry smooth stationary surface

Liquid drop impacts on a smooth surface were studied at elevated chamber pressures to characterize the effect of gas pressure on drop spreading and splashing. Five common liquids were tested at impact speeds between 1.0 and 3.5 m/s and pressure up to 12 bars. Based on experiments at atmospheric pressure, a modification to the “free spreading” model (Scheller and Bousfield in AIChE Paper 41(6):1357–1367, 1995) has been proposed that improves the prediction accuracy of maximum spread factors from an error of 15–5%. At high chamber pressures, drop spreading and maximum spread factor were found to be independent of pressure. The splash ratio (Xu et al. in Phys Rev Lett 94:184505, 2005) showed a non-constant behavior, and a power-law model was demonstrated to predict the increase in splash ratio with decreasing impact speed in the low impact speed regime. Also, drop shape was found to affect splash promotion or suppression for an asymmetry greater than 7–8% of the equivalent drop diameter. The observations of the current work could be especially useful for the study of formation of deposits and wall combustion in engine cylinders.     

A constrained theory for single crystal shape memory wires with application to restrained recovery

The theory of thin wires developed in Dret and Meunier (Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 337:143–147, 2003) is adapted to phase-transforming materials with large elastic moduli in the sense discussed in James and Rizzoni (J Elast 59:399–436, 2000). The result is a one-dimensional constitutive model for shape memory wires, characterized by a small number of material constants. The model is used to analyze self-accommodated and detwinned microstructures and to study superelasticity. It also turns out that the model successfully reproduces the behavior of shape memory wires in experiments of restrained recovery (Tsoi et al. in Mater Sci Eng A 368:299–310, 2004; Tsoi in 50:3535–3544, 2002; S̆ittner et al. in Mater Sci Eng A 286:298–311, 2000; vokoun in Smart Mater Struct 12:680–685, 2003; Zheng and Cui in Intermetallics 12:1305–1309, 2004; Zheng et al. in J Mater Sci Technol 20(4):390–394, 2004). In particular, the model is able to predict the shift to higher transformation temperatures on heating. The model also captures the effect of prestraining on the evolution of the recovery stress and of the martensite volume fraction.     

Development and characterization of a variable turbulence generation system

Experimental turbulent combustion studies require systems that can simulate the turbulence intensities [u′/U ₀ ~ 20–30% (Koutmos and McGuirk in Exp Fluids 7(5):344–354, 1989)] and operating conditions of real systems. Furthermore, it is important to have systems where turbulence intensity can be varied independently of mean flow velocity, as quantities such as turbulent flame speed and turbulent flame brush thickness exhibit complex and not yet fully understood dependencies upon both U ₀ and u′. Finally, high pressure operation in a highly pre-heated environment requires systems that can be sealed, withstand high gas temperatures, and have remotely variable turbulence intensity that does not require system shut down and disassembly. This paper describes the development and characterization of a variable turbulence generation system for turbulent combustion studies. The system is capable of a wide range of turbulence intensities (10–30%) and turbulent Reynolds numbers (140–2,200) over a range of flow velocities. An important aspect of this system is the ability to vary the turbulence intensity remotely, without changing the mean flow velocity. This system is similar to the turbulence generators described by Videto and Santavicca (Combust Sci Technol 76(1):159–164, 1991) and Coppola and Gomez (Exp Therm Fluid Sci 33(7):1037–1048, 2009), where variable blockage ratio slots are located upstream of a contoured nozzle. Vortical structures from the slots impinge on the walls of the contoured nozzle to produce fine-scale turbulence. The flow field was characterized for two nozzle diameters using three-component Laser Doppler velocimetry (LDV) and hotwire anemometry for mean flow velocities from 4 to 50 m/s. This paper describes the key design features of the system, as well as the variation of mean and RMS velocity, integral length scales, and spectra with nozzle diameter, flow velocity, and turbulence generator blockage ratio.     

An experimental study of two-phase flow in simulated reduced-gravity condition: dispersed droplet to slug flow transition and slug flow

The results from an experimental study of reduced-gravity two-phase flows are reported in this paper. The experiments were conducted in simulated reduced-gravity conditions in a ground-based test facility with a circular test section of 25 mm inner diameter. The flow conditions for which data were acquired lie in the dispersed droplet to slug flow transition and slug flow regime. Local data were acquired for 17 different flow conditions at three axial locations. The acquired data complement and extend those discussed in an earlier paper by the authors (Vasavada et al. in, Exp Fluids 43: 53–75, 2007). The radial profiles and axial changes in the local data are analyzed and discussed in this paper. The area-averaged data, in conjunction with the local data, are discussed to highlight important interaction mechanisms occurring between fluid particles, i.e., drops. The data clearly show the effect of progressive coalescence leading to formation of slug drops. Furthermore, the shape of slug drops in reduced-gravity conditions was observed to be different from that in normal-gravity case. The analyses presented here show the presence of drop coalescence mechanisms that lead to the formation of slug drops and transition from dispersed droplet flow to the slug flow regime. The most likely causes of the coalescence mechanism are random collision of drops driven by turbulence eddies in the continuous phase and wake entrainment of smaller drops that follow preceding larger drops in the wake region. Data from flow conditions in which the breakup mechanism due to impact of turbulent eddies on drops illustrate the disintegration mechanism.     

The development of rheometry for strain-sensitive gelling systems and its application in a study of fibrin–thrombin gel formation

This paper reports simulated sequential frequency sweep data which have been reconstructed from time resolved viscoelastic data obtained by Fourier transform mechanical spectroscopy. Comparisons of the results show that the recording of anomalous values of the stress relaxation power law exponent α at the Gel Point under ‘rapid’ gelling conditions may be due to inappropriate rheological techniques. An appropriate rheometrical criterion is established for the application of sequential frequency sweeps in order to obtain accurate values of α in the formation of strain-sensitive, rapidly formed gels. Furthermore, using appropriate rheometry, we report values of α for fibrin–thrombin gels formed by the addition of thrombin to a physiologically relevant level of human fibrinogen, and relate these values to the microstructure of the fibrin gel network in terms of a fractal dimension. The present study is the first to report a modification of the fractal characteristics of incipient clots in fibrin–thrombin gels due to the availability of thrombin. This work confirms the hypothesis that the self-similar (fractal) stress relaxation behaviour recorded at the Gel Point of samples of coagulating blood (Evans et al. 2010a, b) is associated with the microstructural characteristics of the incipient blood clot’s fibrin network.     

Rheology of immiscible blends with particle-induced drop clusters

We consider the effects of 2.7-μm-diameter hydrophobic silica particles added to droplet–matrix blends of polyethylene oxide (PEO) and polyisobutylene (PIB). The particles adsorb on the surface of the PEO drops but protrude considerably into the PIB phase. Hence, it is possible for a single particle to adsorb onto two PEO drops simultaneously. Such particles are called “bridging” particles, and they the glue drops into noncoalescing clusters. Flow visualization studies show that shearing the sample promotes bridging-induced clustering of drops and that the structure of the clusters depends on the shear rate. Rheologically, the most significant consequence of bridging-induced drop clustering appears to be a plateau in G′ at low frequencies characteristic of gel-like behavior. The gel-like behavior develops fully after shearing the sample, and the kinetics of gel formation are faster with increasing shear stress or increasing drop volume fraction. The gel-like behavior suggests that the bridging-induced drop clusters form a weak network. Apart from particle bridging, optical microscopy also reveals that particles can organize into a hexagonal lattice on the drops’ surfaces, a phenomenon that has only been noted in aqueous systems previously. Finally, rheology and flow visualization both suggest that particles promote coalescence of drops. This is surprising in light of much past research that shows that particles that are preferentially wetted by the continuous phase generally hinder coalescence in droplet–matrix systems.     

In situ observations of unusual drop deformation and wobbling in simple shear flow

Deformation and wobbling of a liquid drop immersed in a liquid matrix were studied under mild shear conditions for various viscosity ratios. In situ visualization experiments were conducted on a homemade transparent Couette cell incorporated to the Paar Physica MCR500 shear rheometer. The effect of drop or matrix elasticity was examined and was found to play a major role in both deformation and wobbling processes. Experimental results were compared to Jackson and Tucker (J Rheol 47:659–682, 2003), Maffettone and Minale (J Non-Newton Fluid Mech 78:227–241, 1998) and Yu and Bousmina (J Rheol 47:1011–1039, 2003) ellipsoidal models. It was found that the agreement between the Newtonian models and the experimental results required an increase in the drop viscosity. Such increment in viscosity was found to scale with the first normal stress difference.     

A Zonal RANS/LES Approach for Noise Sources Prediction

Hybrid CFD/CAA methods have generally to be used for the numerical simulation of trailing-edge noise (see [9, 20] for instance). This study focuses on the first step of such hybrid methods, which is to predict the unsteady aerodynamic sources by the mean of a 3D unsteady simulation of the flow. Such a simulation is however generally still away from the numerical capabilities of ‘usual’ supercomputers. This paper investigates the use of a zonal LES method (based on the NLDE – Non-Linear Disturbance Equations – technique) for the numerical prediction of the aerodynamic noise sources. This method makes it possible to perform only zonal LES close to the main elements responsible of sound generation, while the overall configuration is only treated by a RANS approach. Attention will be paid to the specific boundary treatment at the interface between the RANS and LES regions. More precisely, the problem of the generation of turbulent inflow conditions for the LES region will be carefully addressed. The method is first assessed in the simulation of a flat plate ended by a blunted trailing-edge, and then applied to the simulation of the flow over a NACA0012 airfoil with blunted trailing-edge.     

On the Amplitude Equations for Weakly Nonlinear Surface Waves

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185–202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3–4):303–320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220–2240, 2011). In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.     

Use of particle image velocimetry to study heterogeneous drag reduction

Large polymer filaments can form when drag reducing polymers are injected through wall slots. The presence of these structures enhances the performance of the drag reducing function by mechanisms which are not understood. This paper shows how particle image velocimetry (PIV) techniques can be used to study changes in the configuration of the injected polymer and in the structure of the velocity field with increasing drag reduction. The filaments are found to behave as solid bodies which break up in high shear regions close to a boundary. The breakup process provides an explanation of why the filaments are not observed close to a wall and offers the possibility of providing a heterogeneous distribution of small aggregates of polymers which could be more effective than uniformly distributed molecules as suggested by Hoyer and Gyr (J Non-Newton Fluid Mech 65:221–240, 1996; J Fluids Eng 120:818–823, 1998), Dunlop and Cox (Phys Fluids 20:203–213, 1977) and Vlachogiannis et al. (Phys Fluid 15:3786–3794, 2004). PIV measurements show dramatic qualitative changes in the velocity patterns at maximum drag reduction.     

Very high Reynolds number boundary layers over 3D sparse roughness and obstacles: the mean flow

Hot-wire and oil-film interferometry measurements are taken for 3D rough wall boundary layers at very high Reynolds numbers (61,000 < Re θ < 120,000) with low blockage ratios, 10 < δ/H < 135, and high roughness, 100 < H ⁺ < 4,900. The results cover flows over both rough walls and over obstacles and are compared with and provide extension to recent lower Reynolds number results. The validity of the Townsend ‘wall similarity hypothesis’ in relation to consistently increasing 3D roughness is interrogated. In agreement with recent work, Schultz and Flack (J Fluid Mech 580:381–405, 2007) and Castro (J Fluid Mech 585:469–485, 2007) found that, for relatively low roughness, Townsend’s hypothesis holds for the mean velocity field. With increasing roughness, the equilibrium layer diminishes and gradually vanishes. The viscous component of the wall shear stress decreases, while the turbulent component increases as the roughness effects extend across the boundary layer.     

Development of an Analytical Time-Dependent Matrix/Fracture Shape Factor for Countercurrent Imbibition in Simulation of Fractured Reservoirs

In dual porosity modeling of naturally fractured reservoirs, fluids exchange between the high porous matrix blocks and high permeable fracture systems is governed by transfer function. Therefore, transfer function, and specially shape factor as the main part of it, control fluids flow behavior, which certainly have significant effects on development and management plan of naturally fractured reservoirs. Also several formulations have been proposed for shape factor by a number of researchers, nearly all of them derived for expansion mechanism. But, shape factor is a phase sensitive parameter that can greatly affect results of simulation. Moreover, several shortcomings are inherent in the derived expressions of shape factor for imbibition process. The main aim of this work is to develop a new time-dependent matrix–fracture shape factor specific to countercurrent imbibition. In this study, fluid saturation distribution within a matrix block is analytically derived by solving capillary–diffusion equation under different imposed boundary conditions for the process where countercurrent imbibition is the dominant oil drive mechanism. The validity of the solutions is checked against literature experimental data (Bourbiaux and Kalaydjian, SPERE 5, 361–368, SPE 18283, 1990) and also by performing single porosity fine grid simulations. Then, the concept of analogy between the transport phenomena is employed to propose a new expression for matrix–fracture transfer function that is used to derive transient shape factor. It is illustrated in this article that time variation of imbibtion rate and shape factor can be used to diagnose different states of imbibition process. Although, the displacement process and employed approaches are completely different in this and other studies (Chang, Technical report, 1993; Kazemi and Gilman (eds.) Flow and contaminant transport in fractured rock. Academic Press, San Dieg, 1993; Zimmerman et al., Water Resour Res, 29, 2127–2137, 1993; Lim and Aziz, J Pet Sci Eng 13, 169–178, 1995), but we arrived at the consistent values of shape factor under limiting condition of pseudo steady state flow. This means that after establishment of pseudo steady state, shape factor is only controlled by matrix geometry regardless of the displacement process, i.e., expansion or imbibition mechanism, However, shape factor is completely phase sensitive and process dependent during unsteady and late-transient states. Finally, boundary condition dependency of shape factor is investigated.     

Simplified model for mass diffusivity estimation based on dynamic pendant drop volume analysis

We present a simplified correlation for calculating the dissolved gas moles in a pendant drop during the diffusion time, for several drop shapes. After this correlation is determined, the Yang and Gu (Ind Eng Chem Res 44:4474–4483, 2005) dynamic pendant drop volume analysis (DPDVA) method for calculation of mass diffusivity from the pendant drop volume variation against time can be used. We solved the differential equation in cylindrical coordinates for the mass transfer model of the gas diffusion into the liquid inside the pendant drop, using a different characteristic length (L_C), instead of the outer radius of the syringe needle (r_n) used in Yang and Gu (Ind Eng Chem Res 44:4474–4483, 2005) for defining the dimensionless variables. L_C is the relationship between the pendant drop volume and its mass transfer surface area at the initial conditions. The generalized correlation saves time, simplifies the method application and the deviations in the diffusion coefficient calculation respect to the complete Yang and Gu model are below 6%.     

Marcus Integration Method for Shearable Plates

A method proposed by Marcus [5] to integrate the classical biharmonic equation of simply supported, unshearable plates with polygonal contour is extended to apply to shearable plates as well, provided the supporting device is of the ‘hard’ type.     

Characterizing developing adverse pressure gradient flows subject to surface roughness

An experimental study was conducted to examine the effects of surface roughness and adverse pressure gradient (APG) on the development of a turbulent boundary layer. Hot-wire anemometry measurements were carried out using single and X-wire probes in all regions of a developing APG flow in an open return wind tunnel test section. The same experimental conditions (i.e., T _∞, U _ref, and C _p) were maintained for smooth, k ⁺ = 0, and rough, k ⁺ = 41–60, surfaces with Reynolds number based on momentum thickness, 3,000 < Re _θ < 40,000. The experiment was carefully designed such that the x-dependence in the flow field was known. Despite this fact, only a very small region of the boundary layer showed a balance of the various terms in the integrated boundary layer equation. The skin friction computed from this technique showed up to a 58% increase due to the surface roughness. Various equilibrium parameters were studied and the effect of roughness was investigated. The generated flow was not in equilibrium according to the Clauser (J Aero Sci 21:91–108, 1954) definition due to its developing nature. After a development region, the flow reached the equilibrium condition as defined by Castillo and George (2001), where Λ = const, is the pressure gradient parameter. Moreover, it was found that this equilibrium condition can be used to classify developing APG flows. Furthermore, the Zagarola and Smits (J Fluid Mech 373:33–79, 1998a) scaling of the mean velocity deficit, U _∞δ*/δ, can also be used as a criteria to classify developing APG flows which supports the equilibrium condition of Castillo and George (2001). With this information a ‘full APG region’ was defined.     

Strain recovery of model immiscible blends without compatibilizer

Strain recovery after the cessation of shear was studied in model immiscible blends composed of polyisobutylene drops (10–30% by weight) in a polydimethylsiloxane matrix. Blends of viscosity ratio (viscosity of the drops relative to the matrix viscosity) ranging from 0.3 to 1.7 were studied. Most of the strain recovery was attributable to interfacial tension, and could be well-described by just two parameters: the ultimate recovery and a single retardation time. Both these parameters were found to increase with the capillary number of the drops prior to cessation of shear. For blends that had reached steady shear conditions, the ultimate recovery decreased with increasing viscosity ratio, whereas the retardation time increased with increasing viscosity ratio. The retardation time was well-predicted, but the ultimate recovery was over-predicted by a linear viscoelastic model developed previously by Vinckier et al. (Rheol Acta 38:65–72, 1999).     

On Mathematical Modeling and Simulation of the Pressing Section of a Paper Machine Including Dynamic Capillary Effects: One-Dimensional Model

This study presents the dynamic capillary pressure model (Hassanizadeh and Gray, Adv Water Resour 13:169–186, 1990; Water Resour Res 29:3389–3405, 1993) adapted for the needs of paper manufacturing process simulations. The dynamic capillary pressure–saturation relation is included in a one-dimensional simulation model for the pressing section of a paper machine. The one-dimensional model is derived from a two-dimensional model by averaging with respect to the vertical direction. Then, the model is discretized by the finite volume method and solved by Newton’s method. The numerical experiments are carried out for parameters typical for the paper layer. The dynamic capillary pressure–saturation relation shows significant influence on the distribution of water pressure. The behavior of the solution agrees with laboratory experiments (Beck, Fluid pressure in a press nip: measurements and conclusions, 1983).     

Three-dimensional wake dynamics of a blunt and divergent trailing edge airfoil

The wake dynamics of an airfoil with a blunt and divergent trailing edge is investigated experimentally at relatively high Reynolds. The near wake topology is examined versus different levels of free stream turbulence FST and angles of attack, while the downstream wake evolution is characterized at various levels of FST. The FST is found to have a significant effect on the shapes of turbulence profiles and on the downstream location where the flow reaches its quasi-asymptotic behavior. Streamwise vortices (ribs) corresponding to spanwise variations of turbulence quantities are identified in the near wake region. Simultaneous multi-point hot-wire measurements indicate that their spatial arrangement is similar to Williamson’s (Ann Rev Fluid Mech 29:477–539, 1996) mode B laminar wake flow topology. The results suggest that the statistical spanwise distribution of ribs is independent of FST effects and angle of attack as long as the vortex shedding Strouhal number remains approximately similar.     

Numerical simulation of a drop undergoing large amplitude oscillatory shear

Direct numerical simulations are conducted for a Newtonian drop in a Newtonian matrix subjected to large amplitude oscillatory shear flows. In the experimental study of Guido et al. (in Rheol Acta 43:575–583, 2004), the drop shape is found to oscillate at higher harmonics of the forcing frequency when the capillary number is increased. Their phenomenological model requires a much smaller capillary number for predicting the harmonic nature of the experimental data. In this paper, computational results on the evolution of drop length and inclination angle are obtained at the same fluid and flow properties as the experiments, and are shown to reasonably reproduce the experimental data. In particular, the computed velocity fields around the drop are shown to elucidate the over-rotation, which is a mechanism for the experimentally observed harmonics.