Transport of Pseudomonas putida in a 3-D Bench Scale Experimental Aquifer

This study is focused on the transport of Pseudomonas (P.) putida bacterial cells in a 3-D model aquifer. The pilot-scale aquifer consisted of a rectangular glass tank with internal dimensions: 120?cm length, 48?cm width, and 50?cm height, carefully packed with well-characterized quartz sand. The P. putida decay was adequately represented by a first-order model. Transport experiments with a conservative tracer and P. putida were conducted to characterize the aquifer and to investigate the bacterial behavior during transport in water saturated porous media. A 3-D, finite-difference numerical model for bacterial transport in saturated, homogeneous porous media was developed and was used to successfully fit the experimental data. Furthermore, theoretical interaction energy calculations suggested that the extended-DLVO theory seems to predict bacteria attachment onto the aquifer sand better than the classical DLVO theory.     

Preparation of slow-release regulated Rs-198 bilayer microcapsules and application of its lyophilized bacterial inoculant on Capsicum annuum L. under salt stress

The survival adaptation of bacteria in saline soil is poor. The bilayer microcapsules were prepared by secondary embedding of monolayer sodium alginate (NaAlg) - bentonite (Bent) - sodium carboxymethylcellulose (CMC) microcapsules wrapped with plant growth promoting rhizobacteria (PGPR) Pseudomonas putida Rs-198 by chitosan solution to promote the synergistic effect of bilayer microencapsulation and PGPR. The characterization of the Rs-198 bilayer microcapsules showed that the amino and carboxyl groups were cross-linked and a thin layer of chitosan was formed on the outside of the microcapsule. The bilayer microcapsule (Ch-d) with a chitosan concentration of 0.8 wt% and pH 6 showed a slow release of bacteria with a maximum release of 6.06 × 10⁹ cfu/g on the 7th day. The viable bacteria of Ch-d increased by 4.42% after 60 days of storage compared with monolayer microcapsules. The 0.9 wt% l-cysteine, 10 wt% glycerinum, 10 wt% trehalose and 12 wt% soluble starch were added as bacterial protective agents during the process of preparing the Ch-d lyophilized bacterial inoculant (Ch-d LBI). Pot experiments showed that Ch-d LBI exhibited better growth promotion of Capsicum annuum L. under salt stress. Therefore, the bilayer microcapsule as slow-release bacterial inoculant is a potential alternative for sustainable agriculture.     

Influence of Permeability Heterogeneity on Miscible CO<Subscript>2</Subscript> Flooding Efficiency in Sandstone Reservoirs: An Experimental Investigation

In this study, we systematically investigate the effect of core-scale heterogeneity on the performance of miscible CO₂ flooding under various injection modes (secondary and tertiary). Manufactured heterogeneous core plugs are used to simulate vertical and horizontal heterogeneity that may be present in a reservoir. A sample with vertical heterogeneity (i.e. a layered sample) is constructed using two axially cut half plugs each with a distinctly different permeability value. In these samples, the permeability ratio (PR) defines the ratio between the permeabilities of adjacent half plugs. Horizontal heterogeneity (i.e. a composite sample) is introduced by stacking two or three short cylindrical core segments each with a different permeability value. Our special sample construction techniques have also enabled us to investigate the effect of permeability ratio and crossflow in layered samples and axial arrangement of core segments in composite samples on the ultimate recovery of the floods. Core flooding experiments are conducted with an n-Decane–brine–CO₂ system at a pore pressure of 17.2 MPa and a temperature of 343 K. At this temperature, the minimum miscibility pressure of CO₂ with n-Decane is 12.6–12.7 MPa so it is expected that at 17.2 MPa CO₂ is fully miscible with n-Decane. The results obtained for both the composite and layered samples indicate that CO₂ injection would achieve the highest recovery factor (RF) when performed under the secondary mode (e.g. layered: 79.00%, composite: 89.83%) compared with the tertiary mode (e.g. layered: 73.2%, composite: 86.2%). This may be attributed to the effect of water shielding which impedes the access of the injected CO₂ to the residual oil under the tertiary injection mode. It is also found that the oil recovery from a layered sample decreases noticeably with an increase in the PR as higher PR makes the displacement more uneven due to CO₂ channelling. The RFs of 93.4, 87.89, 77.9 and 69.8% correspond to PRs of 1, 2.5, 5, and 12.5, respectively. In addition, for the layered samples, crossflow was found to have an important role during the recovery process; however, due to excessive channelling, this effect tends to diminish as PR increases. Compared with the layered heterogeneity, the effect of composite heterogeneity on the RF seems to be very subtle as the RF is found to be almost independent from the permeability sequence along the length of a composite sample. This outcome may have been caused by the small diameter of the plugs resulting in invariable 1-D floods.     

Pressure probe with five embedded flush-mounted sensors: unsteady pressure and velocity measurements in hydraulic turbine model

Picosecond Unstable-resonator Spatially Enhanced Detection Coherent Anti-Stokes Raman Scattering (USED-CARS) is used for the measurement of nitrogen Q-branch (ΔJ = 0) spectra in the subsonic plenum and supersonic flow of a highly nonequilibrium Mach 5 wind tunnel. Spectra are processed to infer rotational/translational (T _rot) and first-level vibrational (T _vib) temperatures in the 200–370 torr plenum simultaneously. Operation of the nominally high reduced electric field (E/n _peak ~ 500 Td), nsec pulsed discharge alone results in fairly significant vibrational loading, T _vib ~ 720 K/T _rot ~ 380 K; addition of an orthogonal low E/n (~10 Td) DC sustainer discharge produces substantial vibrational loading, T _vib ~ 2,000 K/T _rot ~ 450 K. Effects of injection of CO₂, NO, and H₂ downstream of the pulser–sustainer discharge are examined, which result in vibrational relaxation accompanied by simultaneous gas heating, T _vib ~ 800–1,000 K/T _rot ~ 600 K. CARSk measurements within very low-density flows in the Mach 5 expansion nozzle are also performed, with T _vib measured in both the supersonic free-stream and downstream of a bow shock created by a 5-mm-diameter cylindrical test object in the Mach 5 flow. Measurements within 300 μm of the cylinder leading edge show that for pure N₂, or N₂ with 0.25 torr CO₂ injection, no vibrational relaxation is observed behind the bow shock.     

Low Pressure Gas Percolation Characteristic in Ultra-low Permeability Porous Media

Low pressure gas percolation characteristic in ultra-low permeability porous media is investigated in this article through core flow experiments. The results show that the wall-slip layer covers more than 10% of the average porous channel radius on account of minimum pore size when the permeability is below 0.1 × 10^?3μ m ² order, and seepage behavior is contrasted to that in mid-high permeability pore media. When the gas pressure is not high enough, the flow regime turns into transitional flow instead of slip flow, and nonlinear relationship between the measured gas permeability and the reciprocal of average pressure exists. The gas measuring permeability experiment would be influenced by the non-linear relationship. If Klinkenberg-corrected method is applied to speculate the equivalent liquid permeability, the extrapolated value will become less or minus. Simultaneously, actual gas flow velocity at the outlet is beyond the calculated value with Klinkenberg formula. A new gas seepage model based on the general slip boundary condition is derived from the homogenization technique in this article. At last the flow model is examined to be suitable for representing the gas flow behavior in ultra-low permeability media and estimating the absolute permeability from single-point, steady-states measurements.     

Chemical Alterations Induced by Rock�CFluid Interactions When Injecting Brines in High Porosity Chalks

Effect of the aqueous chemistry on the mechanical strength of chalk has extensively been studied during the last decade. At high temperatures (~130°C), chalk exposed to seawater is significantly weaker compared to chalk exposed to distilled water when considering the hydrostatic yield strength and the following creep phase. The explanation of these experimental results must be of a chemical nature, as the density and viscosity of the aqueous phase vary little among these different brines. We present the results from simplified aqueous chemistry using MgCl₂ brines, and compare these results with seawater. Previous studies show that different ions, e.g. Ca²⁺, Mg²⁺, ${{\rm SO}{_{4}}^{2-}}$ in the injected brine, as well as the chalk mineralogy have an impact on the stability of the rock. We performed mechanical tests on chalk cores from Liège and Stevns Klint; it was found that these two outcrop chalks exhibit an unexpected difference in their mechanical responses when comparing cores flooded with NaCl and MgCl₂ at 130°C. The results of this study show that the effects of magnesium seem to be governed not only by the differences in mineralogy, but also a time dependency on chalk deformation is additionally observed. Independent of the chalk type tested, the chemical analyses performed show that when MgCl₂ is flooded through the core, a significant loss of magnesium and a considerable additional amount of calcium are detected in the effluent. The experimental observations fit very well with the time-dependent chemical changes gained from the mathematical model of this study that accounts for transport effects (convection and molecular diffusion) as well as chemical processes such as precipitation/dissolution. Based on the calculations and chemical analyses, we argue that the loss of magnesium and the production of calcium cannot solely be a consequence of a substitution process. The calculations rather indicate that magnesium is precipitated forming new mineral phases and in this process not only calcite, but also silicates are dissolved. The amount of dissolved calcium and silicon from the rock matrix is significant and could thus cause an additional deformation to take place. Both the retention of magnesium in the chalk core and the formation of newly precipitated magnesium-bearing carbonates and/or magnesium-bearing clay-like minerals after flooding with MgCl₂ brine were demonstrated using scanning electron microscopic methods. In addition, precipitation of anhydrite as a result of flooding with seawater-like brine was proven. The water-induced strain not only depends on the ion composition of the injected brine; moreover, the presence of non-carbonate minerals will most likely also have a significant influence on the mechanical behaviour of chalk.     

Investigation of droplet collisions for solutions with different solids content

The collision behaviour of droplets and the collision outcome are investigated for high viscous polymer solutions. For that purpose, two droplet chains produced by piezoelectric droplet generators are directed towards each other at a certain angle so that individual droplet pairs collide. For recording the collision event, one double-image and one high-speed CCD camera were used. One camera is positioned perpendicular to the collision plane recording the outcome of the collision, and the second camera is aligned parallel to the collision plane to assure that the droplet chains are exactly in one plane. A new approach for tracking droplets in combination with an extended particle tracking velocimetry algorithm has been developed. Time-resolved series of pictures were used to analyse the dynamics of droplet collisions. The three different water soluble substances were saccharose and 1-Ethenyl-2-pyrrolidone (PVP) with different molecular weights (K17, K30). The solvent was demineralised water. The solids contents ranged from 20 to 60 %, 5 to 25 % and 5 to 35 %, yielding dynamic viscosities in the range of 2–60 mPa s. Results were collected for different pairs of impact angles and Weber numbers in order to establish common collision maps for characterising the outcomes. Here, relative velocities between 0.5 and 4 m/s and impact parameters in the interval from 0 to 1 for equal-sized droplets (Δ = 1) have been investigated. Additionally, satellite formation will be discussed exemplarily for K30. A comparison with common models of different authors (Ashgriz and Poo in J Fluid Mech 221:183–204, 1990; Estrade et al. in Int J Heat Fluid Flow 20:486–491, 1999) mainly derived for low viscous droplets revealed that the upper limit of their validity is given by an Ohnesorge number of Oh = 0.115 and a capillary number of Ca = 0.577. For higher values of these non-dimensional parameters and hence higher dynamic viscosities, these models are unable to predict correctly the boundaries between collision scenarios. The model proposed by Jiang et al. (J Fluid Mech 234:171–190, 1992), which includes viscous dissipation, is able to predict the boundary between coalescence and stretching separation for higher viscosities (i.e. Oh > 0.115 and Ca > 0.577). However, the model constants are not identical for different solution properties. As a conclusion, an alteration of the collision appearance takes place because of the relative importance between surface tension and viscosity.     

Perforation of 6082-T651 Aluminum Plates with 7.62 mm APM2 Bullets at Normal and Oblique Impacts

We conducted an experimental study to understand the mechanisms and dominant parameters for 7.62 mm APM2 bullets that perforate 6082-T651 aluminum armor plates at oblique impacts. The 7.62-mm-diameter, 10.7 g, APM2 bullet consists of a brass jacket, lead filler, and a 5.25 g, ogive-nose, hard steel core. The brass and lead were stripped from the APM2 bullets by the targets, so we conducted ballistic experiments with both the APM2 bullets and only the hard steel cores. These projectiles were fired from a rifle to striking velocities between 400 and 1,000 m/s into 20-mm-thick plates at normal impact (β?=?0^o) and at oblique angles of β?=?15^o, 30^o, and 45^o. Measured residual and ballistic-limit velocities for the full bullet and the hard core were within a few percent for normal impact and all oblique angles. Thus, we showed that the perforation process was dominated by the hard steel core of the bullet. In addition, we conducted large strain, compression tests on the 6082-T651 plate material for input to perforation equations derived from a cavity-expansion model for the steel core projectiles. Model predictions were shown to be in good agreement with measured ballistic-limit and residual velocity measurements for β?=?0^o, 15^o, and 30^o. We also presented a scaling law for the bullet that showed the ballistic-limit velocities were proportional to the square root of the product of plate thickness and a material strength term.     

Homogenization of two-phase flow: high contrast of phase permeabilityHomogénéisation d'écoulement diphasique : grand contraste de perméabilité d'une phase

The steady-state two-phase flow non-linear equation is considered in the case when one of phases has low effective permeability in some periodic set, while on the complementary set it is high; the second phase has no contrast of permeabilities in different zones. A homogenization procedure gives the homogenized model with macroscopic effective permeability of the second phase depending on the gradient and on the second order derivatives of the macroscopic pressure of the first phase. This effect cannot be obtained by classical (one small parameter) homogenization. To cite this article: G.P. Panasenko, G. Virnovsky, C. R. Mecanique 331 (2003).     

UV-mediated coalescence and mixing of inkjet printed drops

In the present study, the characteristics of supersonic rectangular microjets are investigated experimentally using molecular tagging velocimetry. The jets are discharged from a convergent–divergent rectangular nozzle whose exit height is 500 μm. The jet Mach number is set to 2.0 for all tested jets, and the Reynolds number Re is altered from 154 to 5,560 by changing the stagnation pressure. The experimental results reveal that jet velocity decays principally due to abrupt jet spreading caused by jet instability for relatively high Reynolds numbers (Re > ~450). The results also reveal that the jet rapidly decelerates to a subsonic speed near the nozzle exit for a low Reynolds number (Re = 154), although the jet does not spread abruptly; i.e., a transition in velocity decay processes occurs as the Reynolds number decreases. A supersonic core length is estimated from the streamwise distribution of the centerline velocity, and the length is then normalized by the nozzle exit height and plotted against the Reynolds number. As a result, it is found that the normalized supersonic core length attains a maximum value at a certain Reynolds number near which the transition in the velocity decay process occurs.     

Determination of liquid water content and dielectric constant in porous media by the capacitive methodDétermination,par la méthode capacitive,de la teneur en eau liquide et de la constante diélectrique dans un milieu poreux

A capacitive sensor-based apparatus has been settled to determine the liquid water amount and dielectric constant in consolidated porous media. This technique relies on the dielectric properties of water, air, and mineral substrate. The experimental procedure is described for successively oven-dried samples at 323 K. It allows us to determine the sample dielectric constant as a function of the sample water amount. For limestones from Caen region, an affine relationship is found at 293 K. This is then compared with other empirical soils data and with existing homogeneisation techniques applied to undeformable heterogeneous dielectrics. To cite this article: T. Fen-Chong et al., C. R. Mecanique 332 (2004).     

Apparent Permeability and Gas Flow Behavior in Carboniferous Shale from the Qaidam Basin,China: An Experimental Study

Shale can act as an unconventional gas reservoir with low permeability and complex seepage characteristics. Study of the apparent permeability and percolation behavior of shale gas is important in understanding the permeability of shale reservoirs, to evaluate formation damage, to develop gas reservoirs, and to design wells. This study simulated methane percolation at 298.15 K under inlet pressures ranging from 0.2 to 4 MPa and a constant outlet pressure of 0.1 MPa to investigate shale gas percolation behavior and apparent permeability. Five representative shale cores from the Carboniferous Hurleg and Huitoutala formations in the eastern Qaidam Basin, China, were analyzed. Each experiment measured the volume flow rate of methane and the inlet pressure. Pseudopressure approach was used to analyze high-velocity flow in shale samples, and apparent permeability at different pressures was calculated using the traditional method. A nonlinear apparent permeability model that considers diffusion and slippage is established from theory and experimental data fitting, and the shale gas flow characteristics affected by slippage and diffusion are analyzed. The results indicate that the pseudopressure formulation that considers the effect of gas properties on high-velocity flow produces a more accurate linear representation of the experimental data. The apparent gas permeability of shale consists of contributions from Darcy permeability, slippage, and diffusion. The apparent permeability and gas flow behavior in the studied shales strongly depended on pressure. The diffusion contribution increased greatly as pressure decreased from 2 to 0.2 MPa, and the smaller the shale permeability, the greater the relative contribution of diffusion flow. At pressures greater than 2 MPa, slip flow contributes \(\sim \)20% of the total flux, Darcy flow contributes up to 70%, and diffusion makes only a minor contribution. This study provides useful information for future studies of the mechanism of shale gas percolation and the exploration and development of Qaidam Basin shale gas specifically.     

Phénomènes de vacillation d'amplitude pour un écoulement d'air en cavité tournante différentiellement chaufféeAmplitude vacillation phenomena in an air-filled differentially heated rotating annulus

A direct numerical simulation is carried out to describe the amplitude vacillation phenomena appearing between two successive steady regular waves flows in an air-filled differentially heated rotating annulus. For a fixed temperature difference, ΔT=30 K, when varying progressively the rotation rate, we have obtained the occurrence of the two amplitude vacillation instabilities observed experimentally by Read et al. (J. Fluid Mech. 238 (1992) 599–632) with a high Prandtl number fluid. The first one, denoted AV is characterized by a doubly periodic temporal behaviour with a periodic variation of wave amplitudes, while the second one corresponds to a torus-3 quasi-periodic or chaotic motion with the presence of a modulation in the wave amplitudes evolution. To cite this article: P. Maubert, A. Randriamampianina, C. R. Mecanique 331 (2003).     

Characterization of the flow for a single fluid in an excavation damaged zoneCaractérisation de l'écoulement d'un fluide monophasique en zone endommagée

The aim of this Note is to quantify the change of characteristics of the media of an Excavated Damaged Zone (EDZ) affected by several fractures. For this, we consider Darcy flow through matrix blocks and fractures with permeability of order ε²δ^θ and 1 respectively. ε is the size of a typical porous block, δ representing the relative size of the fracture and θ is a parameter characterising the permeability ratio. We derive the global behavior from the limit as ε and δ tend to zero. The resulting homogenized equation is of dual-porosity type for θ=2, but it is a simple-porosity model with effective coefficients for θ>2, and there is no flow at the macroscopic level when 0<θ<2. To cite this article: B. Amaziane et al., C. R. Mecanique 332 (2004).     

Transition vers la turbulence géostrophique pour un écoulement d'air en cavité tournante différentiellement chaufféeTransition to geostrophic turbulence in an air-filled differentially heated rotating annulus

A direct numerical simulation is carried out to determine the transition to geostrophic turbulence in an air-filled differentially heated rotating annulus. The coupled Navier–Stokes and energy equations are solved numerically with a spectral method based on collocation Chebyshev and Fourier approximations associated with a second-order time accurate scheme. For a temperature difference fixed at ΔT=5 K, when increasing the rotation rate, we have obtained the successive occurrence of the different regimes as described in the literature. The route to turbulent flow, studied with respect to two principal dimensionless parameters, the Taylor number $\text{Ta}{}^1$ and the thermal Rossby number Θ, presents steady axisymmetric solution (S), periodic flow (P₁), quasi-periodic flow (QP), periodic flow (Pn), before transition to chaotic motion (NP). To cite this article: P. Maubert, A. Randriamampianina, C. R. Mecanique 330 (2002) 365–370.     

Fluid Penetration in a Deformable Permeable Web Moving Past a Stationary Rigid Solid Cylinder

We present an analysis for the process of fluid infiltration into a deformable, thin and permeable web that moves in close proximity over a rigid and stationary solid cylinder. While this is a process of significant interest in a range of coating, printing and composites pultrusion processes, its hydrodynamics have received limited attention in the open literature. The flow in the film separating the web from the cylinder is described by lubrication theory, while fluid transfer into the web is governed by Darcy’s law. The deformation of the web at each position is a linear function of the local gap pressure; this is consistent with the assumption of a thin and rigidly supported web. Our results indicate that the web/fluid interface is forced away from the cylinder surface as it approaches it and bounces back downstream from the minimum separation point. This behavior produces a non-symmetric gap between the adjacent surfaces, and this is shown to have critical influence on the final amount of penetrating fluid. The extent of fluid penetration is also found to be affected by the web elasticity (expressed by the dimensionless Ne number) and permeability (expressed in dimensionless form via \(\hat{{K}})\) where under a specific Ne and \(\hat{{K}}\) combination a maximum penetration depth is obtained. Finally, we derive a closed-form asymptotic solution for the final infiltration depth in the limit of Ne \(<<\) 1 and \(\hat{{K}}<<\)1 and test its predictions against the above-mentioned numerical results.     

Flow around two tandem square cylinders near a plane wall

An experimental study has been conducted to investigate the flow around two identical square cylinders in tandem arrangement and placed near a plane wall at a Reynolds number of 6,300. The inter-cylinder spacing ratio was varied from S ^* = 0.5 to 6, and the cylinder-to-wall gap ratio from G ^* = 0.25 to 2. Totally, 42 cases were considered to systematically examine the effects of wall proximity and the mutual interference between the two cylinders in the normalized gap–spacing (G ^*–S ^*) plane. The flow fields were captured using digital particle image velocimetry, in conjunction with measurements of the fluid forces (drag and lift) acting on the downstream cylinder using a piezoelectric load cell. The results show that the flow is highly dependent on the combined values of G ^* and S ^*. Categories relating to G ^* could be broadly classified as small-gap regime (G ^* < 0.5) at which periodic vortex shedding from the cylinders is suppressed, intermediate-gap regime (0.5 < G ^* < 1) where vortex shedding occurs but is under the influence of the wall proximity, and large-gap regime (G ^* > 1) where the wall effects become negligible. Similarly, the flow interference between the two cylinders can be divided into three basic categories as a function of S ^*, namely, shielding regime at S ^* < 1, reattachment regime at 1 < S ^* < 3, and impinging regime at S ^* > 3. Variations of force coefficients, amplitude spectra, Strouhal numbers, and Reynolds shear stress with G ^* and S ^* are presented to characterize the different flow regimes.     

Mixed convection heat transfer from three heated square cylinders in cross-flow at low Reynolds numbers

This paper presents the effects of cross buoyancy and Prandtl number on the flow and heat transfer characteristics around three equal isothermal square cylinders arranged in a staggered configuration within an unconfined medium. Transient two-dimensional numerical simulations are performed with a finite volume code based on the SIMPLEC algorithm in a collocated grid system. The pertinent dimensionless parameters, such as Reynolds, Prandtl and Richardson numbers are considered in the range of 1 ≤ Re ≤ 30, 0.7 ≤ Pr ≤ 100 and 0 ≤ Ri ≤ 1. The representative streamlines, vortex structures and isotherm patterns are presented and discussed. In addition, the overall drag and lift coefficients and average Nusselt numbers are determined to elucidate the effects of Reynolds, Prandtl and Richardson numbers on flow and heat transfer. The flow is observed to be steady for all the ranges of parameters considered. The drag coefficient is found to decrease with Re (for Ri = 0) and Ri at low Pr, whereas it increases with Pr at higher Ri. The lift coefficient decreases with Ri at low Pr and increases with Pr at higher Ri. The time and surface average cylinder Nusselt number is found to increase monotonically with Re as well as Pr while it remains almost insensitive to Ri at low Pr.     

Flow visualization of supersonic laminar flow over a backward-facing step via NPLS

An experimental study on a supersonic laminar flow over a backward-facing step of 5 mm height was undertaken in a low-noise indraft wind tunnel. To investigate the fine structures of Ma = 3.0 and 3.8 laminar flow over a backward-facing step, nanotracer planar laser scattering was adopted for flow visualization. Flow structures, including supersonic laminar boundary layer, separation, reattachment, redeveloping turbulent boundary layer, expansion wave fan and reattachment shock, were revealed in the transient flow fields. In the Ma = 3.0 BFS (backward-facing step) flow, by measuring four typical regions, it could be found that the emergence of weak shock waves was related to the K–H (Kelvin–Helmholtz) vortex which appeared in the free shear layer and that the convergence of these waves into a reattachment shock was distinct. Based on large numbers of measurements, the structure of time-averaging flow field could be gained. Reattachment occurred at the location downstream from the step, about 7–7.5 h distance. After reattachment, the recovery boundary layer developed into turbulence quickly and its thickness increased at an angle of 4.6°. At the location of X = 14h, the redeveloping boundary layer was about ten times thicker than its original thickness, but it still had not changed into fully developed turbulence. However, in the Ma = 3.8 flow, the emergence of weak shock waves could be seen seldom, due to the decrease of expansion. The reattachment point was thought to be near X = 15h according to the averaging result. The reattachment shock was not legible, which meant the expansion and compression effects were not intensive.     

On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System

In this paper we extend and complement the results in Chiodaroli et al. (Global ill-posedness of the isentropic system of gas dynamics, 2014) on the well-posedness issue for weak solutions of the compressible isentropic Euler system in 2 space dimensions with pressure law p(ρ) = ρ ^γ , γ ≥ 1. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi (Ann Math 170:1417–1436, 2009), (Arch Ration Mech Anal 195:225–260, 2010). Moreover we prove that for some of these Riemann problems and for 1 ≤ γ < 3 such solutions have a greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos in (J Diff Equ 14:202–212, 1973) does not favour the classical self-similar solutions.