Barotropic State Research Articles

Structure and evolution of intense austral cut‐off lows

AbstractThis study examines the three‐dimensional structure and evolution of the 200 most intense Cut‐off Lows (COLs) in the Southern Hemisphere (SH). This is done using feature tracking and cyclone‐centred compositing based on the ERA‐Interim reanalysis. Composites confirm the existence of a well‐defined tropospheric moist cold core co‐located with warm dry air in the lower stratosphere. Such cores are surrounded by regions of strong temperature gradients (frontal zones) which move downstream throughout the life cycle. The stratospheric air intrusion into the troposphere is identified in vertical cross‐sections of potential vorticity and ozone, a process referred to as tropopause folding. Precipitation occurs ahead of the COLs because of the low (high)‐level convergence (divergence) and strong upward motion. The maximum precipitation is observed during decay, indicating a possible link between COLs and surface cyclones. Composites conditioned on relative vorticity and precipitable water suggest these variables may be related to precipitation. The COLs exhibit a westward tilt during their early stages but they change to a barotropic state in the mature stage. Finally, the main characteristics of the COLs are summarised by categories which discriminate different intensities, indicating there are differences in the structure of COLs with consequences for precipitation. These efforts aim to provide new insights into the development of COLs in the SH which could aid in identifying and forecasting their various types and associated precipitation patterns.

Experimental and numerical analysis of cavitation and pressure fluctuations in large high head propeller turbine

In the last decades, Computational Fluid Dynamics (CFD) has played an important role in the numerical prediction of cavitation characteristics of different types of hydraulic turbines. Relative results of the characteristics curves for different hydraulic shapes can be predicted very accurately. Usually the prediction of the absolute values of various characteristics is still a challenge. In the paper, we would like to present the comparison between the results of the model testing and numerical analysis of the critical cavitation coefficient for propeller turbine and the results of pressure fluctuation for various operating regimes.For the numerical analysis, the cavitation model is based on the barotropic state law which describes the density change when the pressure falls below the liquid saturation pressure. Barotropic means that the density depends only on the pressure. The model tests were performed according the IEC 60193 international standards.The results of the research contain the diagrams with sigma break curves obtained with model test and numerical analysis for different operating regimes. Furthermore, a visual observation of the cavitation on the runner blades is also presented and compared with numerically obtained cavitation regions inside the computational domain. According to the pressure fluctuations full range of operating points has been analysed and are presented.

Numerical methodology to predict and analyze cavitating flows in a Kaplan turbine

A computational methodology to predict the appearance and the evolution of the cavitation phenomena in a scale model of a 5–blades Kaplan turbine was developed. The cavitating flow was modeled by a homogeneous approach with a barotropic state law and the k − ω SST turbulence model. RANS simulations were performed, near the Best Efficiency Point, on two computational domains (D1 and D2) comprising one periodic interblades channel consisting of the guide vanes, the runner blade and the cone (domain D1) or taking into account the entire draft tube (domain D2). At first, non–cavitating simulations were carried out in domain D1 in order to match numerical conditions to the experimental ones (similar mass flow Q, head H and mainly the torque T) via an iterative procedure. From this reference operating point obtained in free–cavitation regime, cavitating simulations were performed on both computational domains by reducing the Thoma number, σ. Two different inlet boundary conditions were tested: the classical one, imposing the mass flow rate Q; the new one, fixing the total pressure and, consequently, keeping the machine head H constant during the cavitation drop, as in experiments. In all the performed calculations, the outlet boundary condition is given by a constant static pressure. Computed torque and efficiency drop curves were compared to available experimental data. The best results were obtained with the computational domain D2 applying a constant total pressure on the inlet. The torque and efficiency evolutions were well–predicted with the proposed calculation methodology, and the numerical cavitation structures agreed with experimental observations. Analyses of the blade loading during cavitation breakdown are also proposed in the article. Unsteady simulations are under investigation to improve the prediction and the analyses of more developed cavitating regimes.

Cavitation aggressiveness on a hydrofoil

To improve the analyses of the cavitation erosion phenomena, cavitating flows around an instrumented hydrofoil were studied experimentally and numerically. The experiments allowed the assessment of cavitation aggressiveness by measurements from PVDF pressure sensors. A matrix of 8 PVDF sensors was built, installed and tested for several hydrodynamic conditions corresponding to different flow velocities, angles of attack and cavitation numbers. The unsteady cavitating flows were simulated by applying the in-house 2D code, referred to as “IZ”. Unsteady Reynolds Averaged Navier-Stokes equations were solved for a homogeneous fluid with variable density. To model the cavitation phenomenon and to close the governing equation system, a barotropic state law was used associated with a modified k-ε RNG turbulence model. In order to evaluate the cavitation aggressiveness, simulations were coupled with a cavitation erosion model based on the energy balance between the cavitating flow, the vapour bubble collapses, the emitted pressure waves, and the neighbouring solid walls. The paper presents the experimental and numerical tools, the post-processing methodologies and the obtained results. A method was proposed to assess locally the cavitation aggressiveness from both numerical and experimental results and the two approaches were compared.

Effect of compressible liquid on the contact between an elastic body and a rigid base with a periodic array of quasielliptic grooves

The frictionless contact between an elastic body and a rigid base in the presence of a periodically arranged quasielliptic grooves with in interface gaps in the presence of a compressible liquid is modeled. The contact problem formulated for the elastic half-space is reduced to a singular integral equation (SIE) with Hilbert kernel for a derivative of a height of the interface gaps, which is transformed to a SIE with Cauchy kernel that is solved analytically, and a transcendental equation for liquid’s pressure, which has been obtained from the equation of compressible barotropic liquid state. The dependences of the pressure of the liquid, shape of the gaps, average normal displacement and contact compliance of the bodies on the applied load and bulk modulus of the liquid are analysed.

Bianchi type-III bulk viscous string cosmological model in Brans-Dicke theory of gravitation

A spatially homogeneous and anisotropic Bianchi type-III space-time is considered in the framework of a scalar-tensor theory of gravitation proposed by Brans and Dicke (Phys. Rev. 124:925, 1961) in the presence of bulk viscous fluid containing one dimensional cosmic strings. We have found a determinate solution of the field equations using the plausible physical conditions (i) a barotropic equation state for the pressure and density, (ii) special law of variation for Hubble’s parameter proposed by Berman (Nuovo Cimento B74:182, 1983), (iii) shear scalar is proportional to scalar expansion and (iv) the trace of the energy tensor of the fluid vanishes. We have also assumed that bulk viscous pressure is proportional to energy density. Some physical and kinematical properties of the model are, also, discussed.

A time-dependent baroclinic model on NEC bifurcation

As it is well-known, the North Equatorial Current (NEC) bifurcates into the Kuroshio flowing northward and the equatorward Mindanao Current, which is well depicted by Munk’s theory in 1950 in terms of its climatology. However, Munk’s theory is unable to tell the NEC bifurcation variability with time. In the present paper, a time-dependent baroclinic model forced by wind, in which temporal and baroclinic terms are added to Munk’s equation, is proposed to examine the seasonal variability of the NEC bifurcation latitude. An analytical solution is obtained, with which the seasonal variability can be well described: NEC bifurcation reaches its northernmost position in December and its southernmost position in June with a range of about 1° in latitude, consistent with previous results with observations. The present solution will degenerate to Munk’s one in the case of steady and barotropic state.

An improved progressive preconditioning method for steady non‐cavitating and sheet‐cavitating flows

AbstractAn improved progressive preconditioning method for analyzing steady inviscid and laminar flows around fully wetted and sheet‐cavitating hydrofoils is presented. The preconditioning matrix is adapted automatically from the pressure and/or velocity flow‐field by a power‐law relation. The cavitating calculations are based on a single fluid approach. In this approach, the liquid/vapour mixture is treated as a homogeneous fluid whose density is controlled by a barotropic state law. This physical model is integrated with a numerical resolution derived from the cell‐centered Jameson's finite volume algorithm. The stabilization is achieved via the second‐and fourth‐order artificial dissipation scheme. Explicit four‐step Runge–Kutta time integration is applied to achieve the steady‐state condition. Results presented in the paper focus on the pressure distribution on hydrofoils wall, velocity profiles, lift and drag forces, length of sheet cavitation, and effect of the power‐law preconditioning method on convergence speed. The results show satisfactory agreement with numerical and experimental works of others. The scheme has a progressive effect on the convergence speed. The results indicate that using the power‐law preconditioner improves the convergence rate, significantly. Copyright © 2010 John Wiley & Sons, Ltd.

An implicit low-diffusive HLL scheme with complete time linearization: Application to cavitating barotropic flows

A numerical method for generic barotropic flows is presented, together with its application to the simulation of cavitating flows. A homogeneous-flow cavitation model is indeed considered, which leads to a barotropic state equation. The continuity and momentum equations for compressible flows are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. P1 finite elements are used for the viscous terms, while finite volumes for the convective ones. The numerical fluxes are computed by shock-capturing schemes and ad-hoc preconditioning is used to avoid accuracy problems in the low-Mach regime. A HLL flux function for barotropic flows is proposed, in which an anti-diffusive term is introduced to counteract accuracy problems for contact discontinuities and viscous flows typical of this class of schemes, while maintaining its simplicity. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. For this HLL-like flux function two different time linearizations are considered; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account. The proposed numerical ingredients are validated through the simulations of different flow configurations, viz. the Blasius boundary layer, a Riemann problem, the quasi-1D cavitating flow in a nozzle and the flow around a hydrofoil mounted in a tunnel, both in cavitating and non-cavitating conditions. The Roe flux function is also considered for comparison. It is shown that the anti-diffusive term introduced in the HLL scheme is actually effective to obtain good accuracy (similar to the one of the Roe scheme) for viscous flows and contact discontinuities. Moreover, the more complete time linearization is a key ingredient to largely improve numerical stability and efficiency in cavitating conditions.

The deep wind structure of the giant planets: Results from an anelastic general circulation model

The giant gas planets have hot convective interiors, and therefore a common assumption is that these deep atmospheres are close to a barotropic state. Here we show using a new anelastic general circulation model that baroclinic vorticity contributions are not negligible, and drive the system away from an isentropic and therefore barotropic state. The motion is still aligned with the direction of the axis of rotation as in a barotropic rotating fluid, but the wind structure has a vertical shear with stronger winds in the atmosphere than in the interior. This shear is associated with baroclinic compressibility effects. Most previous convection models of giant planets have used the Boussinesq approximation, which assumes the density is constant in depth; however, Jupiter's actual density varies by four orders of magnitude through its deep molecular envelope. We therefore developed a new general circulation model (based on the MITgcm) that is anelastic and thereby incorporates this density variation. The model's geometry is a full 3D sphere down to a small inner core. It is nonhydrostatic, uses an equation of state suitable for hydrogen–helium mixtures (SCVH), and is driven by an internal heating profile. We demonstrate the effect of compressibility by comparing anelastic and Boussinesq cases. The simulations develop a mean state that is geostrophic and hydrostatic including the often neglected, but significant, vertical Coriolis contribution. This leads to modification of the standard thermal wind relation for a deep compressible atmosphere. The interior flow organizes in large cyclonically rotating columnar eddies parallel to the rotation axis, which drive upgradient angular momentum eddy fluxes, generating the observed equatorial superrotation. Heat fluxes align with the axis of rotation, and provide a mechanism for the transport of heat poleward, which can cause the observed flat meridional emission. We address the issue of over-forcing which is common in such convection models and analyze the dependence of our results on this; showing that the vertical wind structure is not very sensitive to the Rayleigh number. We also study the effect of rotation, showing how the transition from a rapidly to a slowly rotating system affects the dynamics.

Head Drop of a Spatial Turbopump Inducer

A computational fluid dynamics model for cavitation simulation was investigated and compared with experimental results in the case of a three-blade industrial inducer. The model is based on a homogeneous approach of the multiphase flow coupled with a barotropic state law for the cool water vapor/liquid mixture. The numerical results showed a good prediction of the head drop for three flow rates. The hydrodynamic mechanism of the head drop was investigated through a global and local study of the flow fields. The evolution of power, efficiency, and the blade loading during the head drop were analyzed and correlated with the visualizations of the vapor/liquid structures. The local flow analysis was made mainly by studying the relative helicity and the axial velocity fields. A first analysis of numerical results showed the high influence of the cavitation on the backflow structure.

An analysis of the environmental energetics associated with the transition of the first South Atlantic hurricane

This study presents the first analysis of the energetics associated with a hybrid cyclone's transition in the Southern Hemisphere, Hurricane Catarina (March 2004). Catarina has earned a place in history as the first documented South Atlantic hurricane, but its unusual tropical transition is still poorly understood. Here we show that Catarina's transition was preceded by marked environmental changes in the Lorenz energy cycle, with an abrupt shift from a baroclinic to a predominantly barotropic state. Such changes help to explain the unusual vortex's growth until its transition was completed. Although the vortex's energy flux is not explicitly calculated, a likely mechanism linking the environmental energetics with Catarina is the extraction of eddy kinetic energy from horizontal momentum and heat transfers within the through component of the blocking. The results advance the understanding of this rare event and suggest that the technique has a great potential to study transitioning systems in general.

Beltrami states for compressible barotropic flows

Beltrami states for compressible barotropic flows are deduced by minimizing the total kinetic energy while keeping the total helicity constant. A Hamiltonian basis for these Beltrami states is sketched. An interesting physical application of the compressible barotropic Beltrami state arises with the Kuzmin–Oseledets formulation of compressible Euler equations. Further, Ertel's invariant is shown to become degenerate in the compressible barotropic Beltrami state.

Numerical Simulation of 3D Cavitating Flows: Analysis of Cavitation Head Drop in Turbomachinery

The numerical simulation of cavitating flows in turbomachinery is studied at the Turbomachinery and Cavitation team of Laboratoire des Ecoulements Géophysiques et Industriels (LEGI), Grenoble, France in collaboration with the French space agency (Centre National d’Etudes Spatiales, CNES), the rocket engine division of Snecma and Numeca International. A barotropic state law is proposed to model the cavitation phenomenon and this model has been integrated in the CFD code FINE/TURBO™. An analysis methodology allowing the numerical simulation of the head drop induced by the development of cavitation in cold water was proposed and applied in the case of two four-bladed inducers and one centrifugal pump. Global results were compared to available experimental results. Internal flows in turbomachinery were investigated in depth. Numerical simulations enabled the characterization of the mechanisms leading to the head drop and the visualization of the effects of the development of cavitation on internal flows.

Numerical Analysis of Cavitation Instabilities in Inducer Blade Cascade

The cavitation behavior of a four-blade rocket engine turbopump inducer was simulated by the computational fluid dynamics (CFD) code FINE∕TURBO™. The code was modified to take into account a cavitation model based on a homogeneous approach of cavitation, coupled with a barotropic state law for the liquid∕vapor mixture. In the present study, the numerical model of unsteady cavitation was applied to a four-blade cascade drawn from the inducer geometry. Unsteady behavior of cavitation sheets attached to the inducer blade suction side depends on the flow rate and cavitation number σ. Numerical simulations of the transient evolution of cavitation on the blade cascade were performed for the nominal flow rate and different cavitation numbers, taking into account simultaneously the four blade-to-blade channels. Depending on the flow parameters, steady or unsteady behaviors spontaneously take place. In unsteady cases, subsynchronous or supersynchronous regimes were observed. Some mechanisms responsible for the development of these instabilities are proposed and discussed.

Analysis of cavitating flow structure by experimental and numerical investigations

The unsteady structure of cavitating flows is investigated by coupled experimental and numerical means. Experiments focus on the structure and dynamics of sheet cavitation on the upper side of a two-dimensional foil section in the ENSTA cavitation tunnel. Various flow conditions are investigated by varying the pressure, the flow velocity, and the incidence of the foil section. High-frequency local measurements of volume fractions of the vapour phase are performed inside the liquid/vapour mixture by a X-ray absorption method. The numerical approach is based on a macroscopic formulation of the balance equations for a two-phase flow. The assumptions required by this formulation are detailed and they are shown to be common to almost all the models used to simulate cavitating flows. In the present case we apply a single-fluid model associated with a barotropic state law that governs the mixture density evolution. Numerical simulations are performed at the experimental conditions and the results are compared to the experimental data. A reliable agreement is obtained for the internal structure of the cavity for incidence varying between 3° and 6°. Special attention is paid to the mechanisms of partial and transitional instabilities, and to the effects of the interaction between the two sides of the foil section.

Phase transitions of barotropic flow coupled to a massive rotating sphere—Derivation of a fixed point equation by the Bragg method

The kinetic energy of barotropic flow coupled to an infinitely massive rotating sphere by an unresolved complex torque mechanism is approximated by a discrete spin–lattice model of fluid vorticity on a rotating sphere, analogous to a one-step renormalized Ising model on a sphere with global interactions. The constrained energy functional is a function of spin–spin coupling and spin coupling with the rotation of the sphere. A mean field approximation similar to the Curie–Weiss theory, modeled after that used by Bragg and Williams to treat a 2D Ising model of ferromagnetism, is used to find the barotropic vorticity states at thermal equilibrium for a given temperature and rotational frequency of the sphere. A fixed point equation for the most probable barotropic flow state is one of the main results. This provides a crude model of super- and sub-rotating planetary atmospheres in which the barotropic flow can be considered to be the vertically averaged rotating stratified atmosphere and where a key order parameter is the changeable amount of angular momentum in the barotropic fluid. Using the crudest two domains partition of the resulting fixed point equation, we find that in positive temperatures associated with low-energy flows, for fixed planetary spin larger than Ω c > 0 there is a continuous transition from a disordered state in higher temperatures to a counter-rotating solid-body flow state in lower positive temperatures. The most probable state is a weakly counter-rotating mixed state for all positive temperatures when planetary spin is smaller than Ω c . For sufficiently large spins Ω > 2 Ω c , there is a single smooth change from slightly pro-rotating mixed states to a strongly pro-rotating ordered state as the negative value of T increases (or decreases in absolute value). But for smaller spins Ω < 2 Ω c there is a transition from a predominantly mixed state (for T ⪡ - 1 ) to a pro-rotating state at β - ( Ω ) < 0 plus a second β c Ω , for which the fixed point equation has three fixed points when β < β c Ω instead of just the pro-rotating one when β c Ω < β . An argument based on comparing free energy shows to that the pro-rotating state is preferred when there are three fixed points because it has the highest free energy—at negative temperatures the thermodynamically stable state is the one with the maximum free energy. In the non-rotating case ( Ω = 0 ) the most probable state changes from a mixed state for all positive and large absolute-valued negative temperatures to an ordered state of solid-body flow at small absolute-valued negative temperatures through a standard symmetry-breaking second-order phase transition. The predictions of this model for the non-rotating problem and the rotating problem agree with the predictions of the simple mean field model and the spherical model. This model differs from previous mean field theories for quasi-2D turbulence in not fixing angular momentum nor relative enstrophy—a property which increases its applicability to coupled fluid–sphere systems and by extension to 2D turbulent flows in complex domains such as no-slip square boundaries where only the total circulation is fixed—as opposed to classical statistical equilibrium models such as the vortex gas model and Miller–Robert theories that fix all the vorticity moments. Furthermore, this Bragg mean field theory is well-defined for all positive and negative temperatures unlike the classical energy–enstrophy theories.

Numerical Prediction of Cavitating Flow on a Two-Dimensional Symmetrical Hydrofoil and Comparison to Experiments

This paper presents comparisons between two-dimensional (2D) CFD simulations and experimental investigations of the cavitating flow around a symmetrical 2D hydrofoil. This configuration was proposed as a test case in the “Workshop on physical models and CFD tools for computation of cavitating flows” at the 5th International Symposium on cavitation, which was held in Osaka in November 2003. The calculations were carried out in the ENSTA laboratory (Palaiseau, France), and the experimental visualizations and measurements were performed in the IRENav cavitation tunnel (Brest, France). The calculations are based on a single-fluid approach of the cavitating flow: the liquid/vapor mixture is treated as a homogeneous fluid whose density is controlled by a barotropic state law. Results presented in the paper focus on cavitation inception, the shape and the general behavior of the sheet cavity, lift and drag forces without and with cavitation, wall pressure signals around the foil, and the frequency of the oscillations in the case of unsteady sheet cavitation. The ability of the numerical model to predict successively the noncavitating flow field, nearly steady sheet cavitation, unsteady cloud cavitation, and finally nearly supercavitating flow is discussed. It is shown that the unsteady features of the flow are correctly predicted by the model, while some subtle arrangements of the two-phase flow during the condensation process are not reproduced. A comparison between the peer numerical results obtained by several authors in the same flow configuration is also performed. Not only the cavitation model and the turbulence model, but also the numerical treatment of the equations, are found to have a strong influence on the results.

A numerical method for 3D barotropic flows in turbomachinery

A numerical method for the simulation of 3D inviscid barotropic flows in rotating frames is presented. A barotropic state law incorporating a homogeneous-flow cavitation model is considered. The discretisation is based on a finite-volume formulation applicable to unstructured grids. A shock-capturing Roe-type upwind scheme is proposed for barotropic flows. The accuracy of the proposed method at low Mach numbers is ensured by ad-hoc preconditioning, preserving time consistency. An implicit time advancing only relying on the algebraic properties of the Roe flux function, and thus applicable to a variety of problems, is presented. The proposed numerical ingredients, already validated in a 1D context and applied to 3D non-rotating computations, are then applied to the 3D water flow around a typical turbopump inducer.

Numerical Model to Predict Unsteady Cavitating Flow Behavior in Inducer Blade Cascades

Abstract The cavitation behavior of a four-blade rocket engine turbopump inducer is simulated. A two-dimensional numerical model of unsteady cavitation was applied to a blade cascade drawn from an inducer geometry. The physical model is based on a homogeneous approach of cavitation, coupled with a barotropic state law for the liquid/vapor mixture. The numerical resolution uses a pressure-correction method derived from the SIMPLE algorithm and a finite volume discretization. Unsteady behavior of sheet cavities attached to the blade suction side depends on the flow rate and cavitation number. Two different unstable configurations of cavitation are identified. The mechanisms that are responsible for these unstable behaviors are discussed, and the stress fluctuations induced on the blade by cavitation instabilities are estimated.