Numerical treatment of the energy equation in compressible flows simulations

We analyze the conservation properties of various discretizations of the system of compressible Euler equations for shock-free flows, with special focus on the treatment of the energy equation and on the induced discrete equations for other thermodynamic quantities. The analysis is conducted both theoretically and numerically and considers two important factors characterizing the various formulations, namely the choice of the energy equation and the splitting used in the discretization of the convective terms. The energy equations analyzed are total and internal energy, total enthalpy, pressure, speed of sound and entropy. In all the cases examined the discretization of the convective terms is made with locally conservative and kinetic-energy preserving schemes. Some important relations between the various formulations are highlighted and the performances of the various schemes are assessed by considering two widely used test cases. Together with some popular formulations from the literature, also new and potentially useful ones are analyzed

Asymptotically entropy-conservative and kinetic-energy preserving numerical fluxes for compressible Euler equations

This paper proposes a hierarchy of numerical fluxes for the compressible flow equations which are kinetic-energy and pressure equilibrium preserving and asymptotically entropy conservative, i.e., they are able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. The fluxes are based on the use of the harmonic mean for internal energy and only use algebraic operations, making them less computationally expensive than the entropy-conserving fluxes based on the logarithmic mean. The use of the geometric mean is also explored and identified to be well-suited to reduce errors on entropy evolution. Results of numerical tests confirmed the theoretical predictions and the entropy-conserving capabilities of a selection of schemes have been compared.Comment: 9 pages, 4 figure

Asymptotically entropy conservative discretization of convective terms in compressible Euler equations

A new class of Asymptotically Entropy Conservative schemes is proposed for the numerical simulation of compressible (shock-free) turbulent flows. These schemes consist of a suitable spatial discretization of the convective terms in the Euler equations, which retains at the discrete level many important properties of the continuous formulation, resulting in enhanced reliability and robustness of the overall numerical method. In addition to the Kinetic Energy Preserving property, the formulation guarantees the preservation of pressure equilibrium in the case of uniform pressure and velocity distributions, and arbitrarily reduces the spurious production of entropy. The main feature of the proposed schemes is that, in contrast to existing Entropy Conservative schemes, which are based on the evaluation of costly transcendental functions, they are based on the specification of numerical fluxes involving only algebraic operations, resulting in an efficient and economical procedure. Numerical tests on a highly controlled one-dimensional problem, as well as on more realistic turbulent three-dimensional cases, are shown, together with a cost-efficiency study

Imaging follow-up after liver transplantation

Liver transplantation (LT) represents the best treatment for end-stage chronic liver disease, acute liver failure and early stages of hepatocellular carcinoma. Radiologists should be aware of surgical techniques to distinguish a normal appearance from pathological findings. Imaging modalities, such as ultrasound, CT and MR, provide for rapid and reliable detection of vascular and biliary complications after LT. The role of imaging in the evaluation of rejection and primary graft dysfunction is less defined. This article illustrates the main surgical anastomoses during LT, the normal appearance and complications of the liver parenchyma and vascular and biliary structures.Liver transplantation (LT) represents the best treatment for end-stage chronic liver disease, acute liver failure and early stages of hepatocellular carcinoma. Radiologists should be aware of surgical techniques to distinguish a normal appearance from pathological findings. Imaging modalities, such as ultrasound, CT and MR, provide for rapid and reliable detection of vascular and biliary complications after LT. The role of imaging in the evaluation of rejection and primary graft dysfunction is less defined. This article illustrates the main surgical anastomoses during LT, the normal appearance and complications of the liver parenchyma and vascular and biliary structures

Asymmetric dependence in hydrological extremes

Extremal dependence describes the strength of correlation between the largest observations of two variables. It is usually measured with symmetric dependence coefficients that do not depend on the order of the variables. In many cases, there is a natural asymmetry between extreme observations that can not be captured by such coefficients. An example for such asymmetry are large discharges at an upstream and a downstream stations on a river network: an extreme discharge at the upstream station will directly influence the discharge at the downstream station, but not vice versa. Simple measures for asymmetric dependence in extreme events have not yet been investigated. We propose the asymmetric tail Kendall's $\tau$ as a measure for extremal dependence that is sensitive to asymmetric behaviour in the largest observations. It essentially computes the classical Kendall's $\tau$ but conditioned on the extreme observations of one of the two variables. We show theoretical properties of this new coefficient and derive a formula to compute it for existing copula models. We further study its effectiveness and connections to causality in simulation experiments. We apply our methodology to a case study on river networks in the United Kingdom to illustrate the importance of measuring asymmetric extremal dependence in hydrology. Our results show that there is important structural information in the asymmetry that would have been missed by a symmetric measure. Our methodology is an easy but effective tool that can be applied in exploratory analysis for understanding the connections among variables and to detect possible asymmetric dependencies

Fast-projection methods for the incompressible navier–stokes equations

An analysis of existing and newly derived fast-projection methods for the numerical integration of incompressible Navier–Stokes equations is proposed. Fast-projection methods are based on the explicit time integration of the semi-discretized Navier–Stokes equations with a Runge–Kutta (RK) method, in which only one Pressure Poisson Equation is solved at each time step. The methods are based on a class of interpolation formulas for the pseudo-pressure computed inside the stages of the RK procedure to enforce the divergence-free constraint on the velocity field. The procedure is independent of the particular multi-stage method, and numerical tests are performed on some of the most commonly employed RK schemes. The proposed methodology includes, as special cases, some fast-projection schemes already presented in the literature. An order-of-accuracy analysis of the family of interpolations here presented reveals that the method generally has second-order accuracy, though it is able to attain third-order accuracy only for specific interpolation schemes. Applications to wall-bounded 2D (driven cavity) and 3D (turbulent channel flow) cases are presented to assess the performances of the schemes in more realistic configurations.Peer ReviewedPostprint (published version

Long-term radiographic and clinical-functional outcomes of isolated, displaced, closed talar neck and body fractures treated by ORIF: the timing of surgical management

Background: The main purpose of this retrospective case series study was to evaluate long-term radiographic and clinical outcomes of a consecutive series of patients diagnosed with isolated, displaced, closed talar neck or body fractures treated by open reduction and internal fixation (ORIF). Secondly, the aim was to verify the influence of the location of talar fractures on the outcomes, the prognostic value of the Hawkins sign, whether operative delays promote avascular necrosis (AVN) and if the fractures require emergent surgical management. Methods: From January 2007 to December 2012, at our institution, 31 patients underwent ORIF through the use of screws. On the basis of Inokuchi criteria, the injuries were divided between neck and body fractures, which were classified according to Hawkins and Sneppen, respectively. The patients included were divided into two groups in relation to fracture location and complexity. Radiographic assessment focused on reduction quality, bone healing, the Hawkins sign and post-traumatic arthritis (PTA) development. For the clinical evaluation, clinical-functional scores (AOFAS Ankle-Hindfoot Score; MFS; FFI-17; SF-36) and VAS were determined, and statistical analysis was performed. Results: 27 patients, 19 males and 8 females, mean age 38.3 years, were included with an average follow-up period of 83.2 months (range 49\u2013119). There were 9 neck and 19 body fractures; their reduction was anatomical or nearly anatomical in 22 cases, and all reached radiographic consolidation after a mean period of 3.4 months (range 1.7\u20137). The Hawkins sign was observed in 9 cases, in which necrosis did not develop. With a 0\u201311 day surgical timing interval, more than 60% of the patients obtained good or fair results with different scores, while 18 (66.7%) were completely satisfied (VAS: 9\u201310). The early complications included malunions (21.4%) and wound problems (25%); the late complications involved AVN (25%) and PTA (78.6%). Conclusions: Despite a high rate of long-term complications, satisfactory clinical results were achieved. Talar fracture location did not influence the outcomes, the Hawkins sign was confirmed as a positive prognostic factor, and operation timing did not influence AVN development. Hence, these injuries do not require emergent surgical management by ORIF