Academic literature on the topic 'Low Mach assumption'

Purpose The purpose of this paper is to develop a new approach for rapid computation of subsonic and low-transonic rotary derivatives with the available steady solutions obtained by Euler computational fluid dynamics (CFD) codes. Design/methodology/approach The approach is achieved by the perturbation on the steady-state pressure of Euler CFD codes. The resulting perturbation relation is established at a reference Mach number between rotary derivatives and normal velocity on surface due to angular velocity. The solution of the reference Mach number is generated technically by Prandtl–Glauert compressibility correction based on any Mach number of interest under the assumption of simple strip theory. Rotary derivatives of any Mach number of interest are then inversely predicted by the Prandtl–Glauert rule based on the reference Mach number aforementioned. Findings The resulting method has been verified for three typical different cases of the Basic Finner Reference Projectile, the Standard Dynamics Model Aircraft and the Orion Crew Module. In comparison with the original perturbation method, the performance at subsonic and low-transonic Mach numbers has significantly improved with satisfactory accuracy for most design efforts. Originality/value The approach presented is verified to be an efficient way for computation of subsonic and low-transonic rotary derivatives, which are performed almost at the same time as an accounting solution of steady Euler equations.

This work presents an improved artificial compressibility method that enables correct sound propagation behaviour in low Mach number flows. As an extension of incompressible approaches, it offers the advantage of low computational costs since compressible flow equations are simplified under the isentropic assumption, and the coupling of flow and sound is preserved. The non-uniform speed of sound is considered in this method, and eigenvalue analysis of the modified governing equations reveals that the propagation speed of pseudo waves restores to that of physical acoustic waves. The effect of Mach number on the dissipation and dispersion error is then studied using a two-dimentional monopole example with uniform background flow. The results suggest that the proposed method can provide a satisfactory prediction of sound propagation when the Mach number is below 0.3. Lastly, a direct noise computation of the aerofoil trailing-edge noise problem is performed to assess its capability to deal with the interaction between flow and sound. It shows that the present method can well capture the multiple-tone phenomenon.

The factors which affect the sound production of a vortex as it passes through a nozzle are investigated at both low and high Mach numbers using time-accurate inviscid-flow computations. Vortex circulation, initial position, and mean-flow Mach number are shown to be the primary factors which influence the amplitude and phase of the sound produced. Nozzle geometry and distribution of vorticity are also shown to play significant roles in determining the detailed form of the signal. Additionally, it is shown that solution bifurcations are possible at sufficiently large values of vortex circulation. Comparisons are made between sound signals computed directly using a numerical method for the Euler equations and predictions obtained using a compressible vortex-sound analogy coupled with a compact-source assumption for the computation of vorticity dynamics. The results confirm that the latter approach is accurate for a range of problems with low mean-flow Mach numbers. At higher Mach numbers, however, the non-compactness of the source becomes apparent, resulting in significant changes to the character of the signal which cannot be predicted using the analogy-based approach. Implications for the construction of simplified models of vortex sound in solid-rocket nozzles are discussed.

AbstractA large database of new direct numerical simulations of forced compressible turbulence on up to $204{8}^{3} $ grids, and a range of Reynolds (${R}_{\lambda } $) and turbulent Mach (${M}_{t} $) numbers, is analysed to study the scaling of pressure, density and temperature fluctuations. Small-perturbation analysis is used to study the scaling of variances, and different cross-correlations as well as spectra. Qualitative differences are observed between low and high ${M}_{t} $. The probability density functions (p.d.f.s) of pressure and density are negatively skewed at low ${M}_{t} $ (consistent with incompressible results) but become positively skewed at high ${M}_{t} $. The positive tails are found to follow a log-normal distribution. A new variable is introduced to quantify departures from isentropic fluctuations (an assumption commonly used in the literature) and is found to increase as ${ M}_{t}^{2} $. However, positive fluctuations of pressure and density tend to be more isentropic than negative fluctuations. In general, Reynolds number effects on single-point statistics are observed to be weak. The spectral behaviour of pressure, density and temperature is also investigated. While at low ${M}_{t} $, pressure appears to scale as ${k}^{- 7/ 3} $ ($k$ is the wavenumber) in the inertial range as in incompressible flows, a ${k}^{- 5/ 3} $ scaling also appears to be consistent with the data at a range of Mach numbers. Density and temperature spectra are found to scale as ${k}^{- 5/ 3} $ for a range of Mach numbers.

One-dimensional models for predicting the damage induced by pressure transients in piping systems conveying liquids have been proposed and analysed recently. However, such works have been concerned mainly with the adequacy of the constitutive equations adopted for different pipe materials and with the numerical techniques used for approximating the solution of the resulting mathematical problems. In the present paper the suitability of the simplifying low Mach number assumption adopted in the modeling is investigated. The analysis is carried out based on the eigenvalue problem associated to the governing equations, without appealing to any specific mechanical behavior of the pipe material. Numerical results obtained for the most used pipe materials show that this simplifying assumption is adequate for metallic tubes, but may fail when plastic tubes are considered.

The mechanisms of sound generation in a Mach 0.9, Reynolds number 3600 turbulent jet are investigated by direct numerical simulation. Details of the numerical method are briefly outlined and results are validated against an experiment at the same flow conditions (Stromberg, McLaughlin & Troutt 1980). Lighthill's theory is used to define a nominal acoustic source in the jet, and a numerical solution of Lighthill's equation is compared to the simulation to verify the computational procedures. The acoustic source is Fourier transformed in the axial coordinate and time and then filtered in order to identify and separate components capable of radiating to the far field. This procedure indicates that the peak radiating component of the source is coincident with neither the peak of the full unfiltered source nor that of the turbulent kinetic energy. The phase velocities of significant components range from approximately 5% to 50% of the ambient sound speed which calls into question the commonly made assumption that the noise sources convect at a single velocity. Space–time correlations demonstrate that the sources are not acoustically compact in the streamwise direction and that the portion of the source that radiates at angles greater than 45° is stationary. Filtering non-radiating wavenumber components of the source at single frequencies reveals that a simple modulated wave forms for the source, as might be predicted by linear stability analysis. At small angles from the jet axis the noise from these modes is highly directional, better described by an exponential than a standard Doppler factor.

Context. The magnetic field strength in interstellar clouds can be estimated indirectly from measurements of dust polarization by assuming that turbulent kinetic energy is comparable to the fluctuating magnetic energy, and using the spread of polarization angles to estimate the latter. The method developed by Davis (1951, Phys. Rev., 81, 890) and by Chandrasekhar and Fermi (1953, ApJ, 118, 1137) (DCF) assumes that incompressible magnetohydrodynamic (MHD) fluctuations induce the observed dispersion of polarization angles, deriving B ∝ 1∕δθ (or, equivalently, δθ ∝ MA, in terms of the Alfvénic Mach number). However, observations show that the interstellar medium is highly compressible. Recently, two of us (ST) relaxed the incompressibility assumption and derived instead B ∝ 1/√δθ (equivalently, δθ ∝ MA2). Aims. We explored what the correct scaling is in compressible and magnetized turbulence through theoretical arguments, and tested the assumptions and the accuracy of the two methods with numerical simulations. Methods. We used 26 magnetized, isothermal, ideal-MHD numerical simulations without self-gravity and with different types of forcing. The range of MA and sonic Mach numbers Ms explored are 0.1 ≤ MA ≤ 2.0 and 0.5 ≤ Ms ≤ 20. We created synthetic polarization maps and tested the assumptions and accuracy of the two methods. Results. The synthetic data have a remarkable consistency with the δθ ∝ MA2 scaling, which is inferred by ST, while the DCF scaling failed to follow the data. Similarly, the assumption of ST that the turbulent kinetic energy is comparable to the root-mean-square (rms) of the coupling term of the magnetic energy between the mean and fluctuating magnetic field is valid within a factor of two for all MA (with the exception of solenoidally driven simulations at high MA, where the assumption fails by a factor of 10). In contrast, the assumption of DCF that the turbulent kinetic energy is comparable to the rms of the second-order fluctuating magnetic field term fails by factors of several to hundreds for sub-Alfvénic simulations. The ST method shows an accuracy better than 50% over the entire range of MA explored; DCF performs adequately only in the range of MA for which it has been optimized through the use of a “fudge factor”. For low MA, it is inaccurate by factors of tens, since it omits the magnetic energy coupling term, which is of first order and corresponds to compressible modes. We found no dependence of the accuracy of the two methods on Ms. Conclusions. The assumptions of the ST method reflect better the physical reality in clouds with compressible and magnetized turbulence, and for this reason the method provides a much better estimate of the magnetic field strength over the DCF method. Even in super-Alfvénic cases where DCF might outperform ST, the ST method still provides an adequate estimate of the magnetic field strength, while the reverse is not true.

The source locations of axisymmetric modes of screech tones are numerically investigated. Fourth-order optimized compact scheme and fourth-order Runge–Kutta method are used to solve the 2-D axisymmetric Euler equations. The screech tone is successfully reproduced, and the change in wavelength with respect to jet Mach number shows good agreement with the experimental data. At various low supersonic jet Mach numbers, the time-averaged contours of Mach number and root-mean-square pressure are investigated to identify the location of maximum interaction between shock cell structures and vortices. The source locations of two axisymmetric modes, A1 and A2 modes, are distinctly visualized and identified; the screech tones of A1 mode are generated at the apex of fifth shock cell, and the screech tones of A2 mode are generated at the apex of fourth shock cell. Based on the observation, a simple formula for the prediction of axisymmetric modes of screech tones is proposed. The formula is derived based on a form of Rossiter equation, with the assumption of different convection speeds along the jet mixing layer. The proposed formula successfully estimates the frequency of two axisymmetric modes of screech tones, which verifies that the identified source locations of the axisymmetric screech tones are reasonable.

The present work focuses on the numerical simulation of isothermal and weakly compressible Poiseuille flow in a planar channel using the Lattice Boltzmann method with multiple times of relaxation (MRT-LBE) coupled to the Finite Difference method (FDM). The active fluid considered is the air under low Mach number assumption. The flow is two-dimensional, laminar and all the physical properties are constants except the density which varies in the sense of the Boussinesq approximation. The effects of the compressibility, the inclination angle and the Reynolds number on the dynamical and thermal fields are studied numerically. The results are presented in terms of streamlines, isotherms and transverse velocity.

Direct numerical simulations of a temporally developing, low-speed, variable-density, turbulent, plane mixing layer are performed. The Navier–Stokes equations in the low-Mach-number approximation are solved using a novel algorithm based on an extended version of the velocity–vorticity formulation used by Kim et al. (J. Fluid Mech., vol 177, 1987, 133–166) for incompressible flows. Four cases with density ratios $s=1,2,4$ and 8 are considered. The simulations are run with a Prandtl number of 0.7, and achieve a $Re_{\unicode[STIX]{x1D706}}$ up to 150 during the self-similar evolution of the mixing layer. It is found that the growth rate of the mixing layer decreases with increasing density ratio, in agreement with theoretical models of this phenomenon. Comparison with high-speed data shows that the reduction of the growth rates with increasing density ratio has a weak dependence with the Mach number. In addition, the shifting of the mixing layer to the low-density stream has been characterized by analysing one-point statistics within the self-similar interval. This shifting has been quantified, and related to the growth rate of the mixing layer under the assumption that the shape of the mean velocity and density profiles do not change with the density ratio. This leads to a predictive model for the reduction of the growth rate of the momentum thickness, which agrees reasonably well with the available data. Finally, the effect of the density ratio on the turbulent structure has been analysed using flow visualizations and spectra. It is found that with increasing density ratio the longest scales in the high-density side are gradually inhibited. A gradual reduction of the energy in small scales with increasing density ratio is also observed.

AbstractThe aim of this paper is to provide a stochastic version under location uncertainty of the compressible Navier–Stokes equations. To that end, some clarifications of the stochastic Reynolds transport theorem are given when stochastic source terms are present in the right-hand side. We apply this conservation theorem to density, momentum and total energy in order to obtain a transport equation of the primitive variables, i.e. density, velocity and temperature. We show that performing low Mach and Boussinesq approximations to this more general set of equations allows us to recover the known incompressible stochastic Navier–Stokes equations and the stochastic Boussinesq equations, respectively. Finally, we provide some research directions of using this general set of equations in the perspective of relaxing the Boussinesq and hydrostatic assumptions for ocean modelling.

AbstractReal-world knowledge graphs (KGs) are usually incomplete—that is, miss some facts representing valid information. So, when applied to such KGs, standard symbolic query engines fail to produce answers that are expected but not logically entailed by the KGs. To overcome this issue, state-of-the-art ML-based approaches first embed KGs and queries into a low-dimensional vector space, and then produce query answers based on the proximity of the candidate entity and the query embeddings in the embedding space. This allows embedding-based approaches to obtain expected answers that are not logically entailed. However, embedding-based approaches are not applicable in the inductive setting, where KG entities (i.e., constants) seen at runtime may differ from those seen during training. In this paper, we propose a novel neuro-symbolic approach to query answering over incomplete KGs applicable in the inductive setting. Our approach first symbolically augments the input KG with facts representing parts of the KG that match query fragments, and then applies a generalisation of the Relational Graph Convolutional Networks (RGCNs) to the augmented KG to produce the predicted query answers. We formally prove that, under reasonable assumptions, our approach can capture an approach based on vanilla RGCNs (and no KG augmentation) using a (often substantially) smaller number of layers. Finally, we empirically validate our theoretical findings by evaluating an implementation of our approach against the RGCN baseline on several dedicated benchmarks.

THE IDENTITY OF THE CONSTITUTION IN THE CONTEXT OF POLISH CONSTITUTIONAL EXPERIENCE AND TRADITIONS In the Polish doctrine of constitutional law, the issues related to the identity of the constitution have emerged primarily in the course of discussions concerning the ways of amending that act. This issue was closely related to the evolution of the notion of “constitution” and with the place that the constitution was given in the system of sources of law. Nowadays, this approach does not seem to fully reflect the meaning of the category “identity of the constitution”. The article refers to certain historical experiences related to constitutional changes, and identified in the Polish legislation, starting from the March Constitution of 1921. Polish constitutional experience points to a certain weakness of the assumption of stability of constitutional provisions in the absence of institutional guarantees corresponding to this assumption.

Chapter five explores the “aesthetic frames” that constitute the sensory configuration of security and justice in the post-9/11 “age of terror,” through Enrique Urbizu’s film No habrá paz para los malvados (2011). The film exposes what Michael J. Shapiro (2009) has called the “violent cartography” of national security in the context of the Madrid terrorist attack on March 11, 2004. The chapter shows how, while at the narrative level the film seemingly reinforces the ideological discourses of security and justice constituted by the state and the media, at the level of aesthetics the film disturbs and reconfigures the frames within which the former discourses are to be understood. The film uses a post-Western aesthetics that recycles themes, tropes, and styles of classical Western cinema to actively interrupt and modify its assumptions, ideology, and values. Against the state’s rhetoric of safety, protection, and prevention, the film makes visible both the failure of the law to respond to violence and the inability of the police to intervene in it effectively. By leaving the viewer with a disquieting feeling of uncertainty and danger, the film asks the question, is there a place for justice between violence and the law?

“A crowd of solemn-faced African villagers sat on the stone benches around the whitewashed walls of the court, squatted on the concrete floor and stood four deep outside.” In early May, the magistrate holds a preliminary inquiry (PI) in Maswa. After a three-day hearing, the magistrate orders Bill to be held for trial in the High Court in Mwanza because the prosecution has presented a prima facie case. Several eyewitnesses are called to testify about the events on Impala Hill on March 27, 1966. Although the prosecution presents its entire case, the defense offers none of its evidence because under Tanzanian law it can be reserved for later. A strategic decision is made based on the assumption that the prosecution has adequate evidence to lead the magistrate to order Bill to trial; in this way, the prosecution will be unprepared and surprised by evidence put forward at the trial.

Revolution? The Year 1848 in the Countryside. Contrary to widespread assumptions, it was not only in Vienna and other major European cities that revolutionary movements occurred in 1848. Rural areas too saw a wave of protests against existing institutions and experimentation with new forms of political activity, involving large segments of the population. This concerned not only the traditional resistance to the manorial system and its dues, which attained its goal in 1848 with the defeudalisation law, but multiple other phenomena too. This chapter discusses innovations in political communication, elections and parliamentary politics, the National Guard, and the reactions to the October Revolution in Vienna. It is reasonable to believe that these new experiences left permanent marks on the political consciousness of the rural population and had a long-term impact on the development of its relationship with the state.

This chapter addresses the uneven ecclesiastical implementation of the Usurarum voraginem. For all the heated clerical rhetoric around the evils of moneylending, and notwithstanding the threat of severe sanctions for non-compliance, many bishops were reluctant to enforce expulsion within their dioceses—a finding that challenges persistent assumptions about the dynamics of the medieval church's campaign against usury. Through a series of case studies drawn from the Low Countries, the Rhineland, and Italy, the chapter explores the reasons for this widespread episcopal reluctance, as well as the continuing secular opposition to expulsion across most of western Europe. Secular stonewalling obviously played a major role in hampering the decree's enforcement. As medieval bishops knew all too well, ecclesiastical censures were often no match for lordly obstinacy. Ecclesiastical complicity was also at work. Many bishops profited directly from the presence of foreign moneylenders in their jurisdictions, while others depended on the ready supply of credit that they provided. Despite the severe automatic sanctions imposed by the decree, only a handful of bishops suffered any formal rebuke for their inaction.

The coastal passenger transport in touristic areas like Costa Smeralda (Sardinia), Cinque terre (Liguria), Costiera Amalfitana (Campania), Venetian Lagoon (Veneto) and others, is constantly growing. At the same time, the sensitivity of authorities to the issue of environmental impact in those areas is leading the transportation companies to investigate technical solutions that can guarantee high volumes of passenger being as much as possible eco-friendly. Hybrid or full electric passenger vessels are becoming more and more popular, starting from this assumption the authors examined the possibility to combine state of art technologies, with an innovative approach to match the propulsion system with hull resistance data, in order to propose a passenger ferry capable to operate in protected areas with an extremely low impact on the environment and taking advantage also from a basic energy distribution ashore. The usage of new generation batteries, with the highest safety standards, will be also investigated. The paper starts from the determination of the operative profile for the ferry, evaluating the best solution in terms of efficiency and power management, considering the resistance data of various hulls, focusing on a traditional displacement hull, and then developing a study of the propulsion system through the usage of last generation generators with variable speed and batteries different from traditional Li-Fe-PO4, in order to achieve a high efficiency level.

Abstract Nearly two decades after the Supreme Court ruled in Roe v. Wade1 that a woman has a constitutional right to terminate her pregnancy, abortion remains one of the most divisive and emotionally charged issues in America. Pro-lifers march with posters of macerated fetuses; pro-choicers use a bloody coat hanger as their symbol of the days of illegal abortions. But behind the drama and the emotion are claims that can be subjected to philosophical scrutiny. Is the unborn a human being, with a right to life like any other human being, as pro lifers maintain? If it is, then very few abortions, if any, could be justified. For we do not generally think that it is morally permissible to kill children because they are unwanted or illegitimate or severely handicapped. On the other hand, if the fetus is not a child, but only part of the pregnant woman’s body, then restrictive abortion laws would be as difficult to justify in a pluralistic society as laws against contraception. For restrictive abortion laws impose enormous physical, emotional, and financial burdens on women. Even legal moralists, who hold that society has the right to enforce its moral beliefs through law, could not justify the imposition of such heavy burdens. Only the assumption that the unborn is a human being like any other, entitled to the law’s protection, could justify the prohibition of abortion. Thus, the moral status of the unborn is central to the abortion debate.

The present study is devoted to the modeling of mean flow effects while computing thermoacoustic modes under the zero Mach number assumption. It is first recalled that the acoustic impedance modelling a compressor or a turbine must be prescribed under an energetical form instead of the classical acoustic variables. Then we demonstrate the feasibility to take into account the coupling between acoustic and entropy waves in a zero Mach number framework to capture a family of low frequency entropic modes. The proposed approach relies on a new Delayed Entropy Coupled Boundary Condition (DECBC) and proves able to capture a family of Low frequency entropic mode even though no mean flow term is included into the fluctuating pressure equation.

Abstract Low speed and high speed flow phenomena in pillow plate channels are considered. High speed flows were investigated by means of analytical methods and fully three-dimensional computational fluid dynamics (CFD) simulations. The theoretical analysis indicated that a Fanno-type flow model described high speed flow behavior in pillow plate channels reasonably well. Since only wavy walls with smooth profiles were involved, linearized gas dynamics was applied in order to derive similarity laws for the high speed flows. The detailed CFD analysis was used to support the assumption of a Fanno-type flow. The effects of the wavy wall structures on pressure drop and Mach number distribution within the flow path were investigated in detail. The present analysis demonstrates that pillow plate heat exchangers represent promising candidates for high speed turbo machinery applications.

Abstract In recent years, Lattice Boltzmann Methods (LBM’s) have emerged as a popular class of paradigms for the simulation of multiphase flows. These methods rely on discretized Boltzmann equations to represent the individual multiphase species. Among LBM’s advantages is its ability to explicitly account for interfacial physics and its local streaming/collision operations which make it ideally suited for parallelization. However, one drawback of LBM is in the simulation of incompressible multiphase flow, whereby the density should remain constant along material characteristics. Because LBM uses a state equation to relate pressure and density, incompressibility cannot be enforced directly. This is true even for incompressible single-phase LBM calculations, in which a finite density drop is needed to drive through the flow. This is also the case for compressible Navier-Stokes algorithms when applied to low Mach number flow. To mitigate compressibility effects, LBM can be used in low Mach regimes which should keep material density variation small. In this work, we demonstrate that the assumption of low Mach number is not sufficient in multiphase internal flows. In such flows, in the absence of a Pressure Poisson constraint to enforce incompressibility, LBM predicts a compressible solution whereby a density gradient must develop to conserve mass. Imposition of inflow/outflow boundary conditions or a mean body force can ensure that mass is conserved globally, thereby quelling density variation. The primary numerical problem we study is the deformation of a liquid droplet immersed in another fluid. Though LBM is not typically conducted with a pressure Poisson equation, we incorporate one in this work and demonstrate that its inclusion can significantly lower the density variation in view of maintaining an incompressible flow.

Ship hull slamming and wave impact on wave energy converters (WEC) are an important problem in naval architecture and marine hydrokinetic for the survivability of vessels and WECs in adverse environmental conditions. An idealized canonical model for these problems can be traced back to the water entry of a wedge–shaped rigid object. Common asymptotic and empirical models are often based on the assumption that the fluid is incompressible and the initial wedge entry velocity to sound speed ratio (defined as the characterized Mach number) is low. In this paper, we present some initial results of a numerical study of the effect of fluid compressibility on the peak slamming impact pressure. The deadrise angle and entry velocity of the model are varied to examine the sensitivity of the peak and distribution of the pressure response with respect to these parameters. The fluid compressibility is controlled through the bulk modulus and manifested as various sound speeds. The normalization using this sound speed shows that the pressure coefficient changes consistently for a certain deadrise angle. The results indicates that despite a low Mach number, the fluid compressibility in the numerical model affects the peak pressure significantly, implying an overestimation in engineering application by the classical asymptotic incompressible theory.

Thermo-acoustic combustion instabilities have frequently been reported to occur when slight changes in operating conditions lead to significant and abrupt changes in flame shape or flame position, i.e. changes in the mode of flame stabilization. The present paper offers an explanation and mathematical model of this observation. The analysis rests on the assumption that changes in the mode of flame stabilization are accompanied by a significant variation of the pressure drop across burner and flame, such that the pressure drop - flow rate characteristic locally displays a negative slope. In the limit of low frequencies (Helmholtz mode), it is then straightforward to show that an oscillatory instability can result from such behavior. A stability criterion is derived, relating the non-dimensionalized gradient of the pressure drop characteristic to the Helmholtz number of the burner. The physics of the instability mechanism is explained, and it is observed that the Rayleigh criterion need not be satisfied for this kind of instability to occur. The analysis is then extended to higher frequencies and the transfer matrix for a burner with non-monotonic pressure drop is derived in the limit of low Mach number and negligible fluctuations of the rate of heat release. Experimental results obtained with an externally premixed swirl burner are presented. The pressure drop characteristic, the observed onset of instability and the instability frequency match very well the analytical predictions.

A large eddy simulation (LES) of a rocket turbopump inducer in non-cavitating and cavitating flows is presented. The computation takes full account of the interaction between the rotating inducer and the stationary casing by using a multi-frame-of-reference dynamic overset grid approach. A streamline-upwind finite element formulation with second-order accuracy both in time and space is used to discretize the governing equation. It is implemented in parallel by a domain-decomposition-programming model. The evolution of cavitation is represented by the source/sink of vapor phase in the incompressible liquid flow. The pressure-velocity coupling is based on the fractional-step method for incompressible fluid flows, in which the compressibility is taken into account through the low Mach number assumption. The internal flow of an inducer is simulated and compared with water tunnel experiments at the design (φ = 0.078) and off-design conditions (φ = 0.05 and 0.09) in non-cavitating flows. The overall head-flow characteristics of computed results show good agreement with experiments. Such results show that the applied LES code can be used as a design tool for rocket turbopump inducers.

A numerical analysis of the evaporation process in a flat plate heat pipe (FPHP) is presented. Three different heat flux configurations were applied at the evaporator section: parabolic, linear and constant; and three different wick thicknesses were evaluated. A two-dimensional explicit Finite Difference Model was used to conduct the transient computational analysis of the heat pipe. The results indicate that the vaporization rate was more intense at the evaporator for the parabolic heat flux distribution, than for either the linear or constant heat flux distributions. The velocity field is nearly identical to the shape of the input heat flux distribution. The maximum Mach number for all the cases analyzed was low, indicating that the incompressible flow assumption in the vapor was valid. The modeling results also indicated that the thickness of the heat pipe wick was a significant factor in the determination of the overall thermal resistance, with the resulting temperature drop proportional to the thickness of the wick. Typical distributions of the field variables, particularly the temperature in the wall, and the temperature, velocity and pressure in the wick and the vapor core of the flat plate heat pipe are also presented and discussed.

In a test facility for rotating annular cascades with three conical test sections of different taper angles (0°, 30°, 45°), experiments are conducted for two geometrically different turbine cascade configurations, a hub section cascade with high deflection and a tip section cascade with low deflection. The evaluation of time averaged data derived from conventional probe measurements upstream and downstream of the test wheel in the machine-fixed absolute system is based on the assumption of axisymmetric stream surfaces. The cascade characteristics, i.e. mass flow, deflection and losses, for a wide range of inlet flow angles and outlet Mach numbers are provided in the blade-fixed relative system with respect to the influence of annulus taper. Some of the results are compared with simple theoretical calculations. To obtain some informations about the spatial structure of the flow within the cascade passages, surface pressure distributions on the profiles of the rotating test wheels are measured at three different radial blade sections. For some examples those distributions are compared with numerical results on plane cascades of the same sweep and dihedral angles and the same aspect ratios. The computer code used is based on a three-dimensional time-marching finite-volume method solving the Euler equations. Both experimental and numerical results show a fairly good qualitative agreement in the three-dimensional blade surface pressure distributions. This work will be continued with detailed investigations on the spatial flow structure.

Heat transfer characteristics of unchoked and choked gas flows in micro-tubes with constant wall temperature were numerically investigated both laminar and turbulent flow cases. The numerical methodology is based on the Arbitrary-Lagrangian-Eulerian (ALE) method. The Lam-Bremhorst Low-Reynolds number turbulence model was used for turbulent flow. The compressible momentum and energy equations with the assumption of the ideal gas were solved. The computational domain should be extended to the downstream region of the hemisphere from micro-tube outlet. The back pressure was given to the outside of the downstream region. The stagnation temperature is fixed at 300K and the computations were done for the wall temperature which ranges from 305K to 350K. The tube diameter ranges from 50 to 250 μm and tube aspect ratio is 200. The stagnation pressure is chosen in such a way that the flow at micro-tube exit is enough to be fully under-expanded. By increasing the stagnation pressure, the internal flow in the micro-tube is choked and the flow at the micro-tube outlet is under-expanded. Although the velocity remains constant, the mass flow rate (Reynolds number) increases. The results in a wide range of Reynolds number and Mach number were obtained. The bulk temperature based on the static temperature and the total temperature are compared with those of the incompressible flow. A correlation for the prediction of the heat transfer rate of the unchoked and choked gas flow in micro-tubes is proposed.

Abstract This paper explores the viability of established criteria to design centrifugal compressor stages that operate CO2 close to the critical point. The work is split into two complementary studies. At first, several stage configurations are generated by varying characteristic design parameters, such as the inlet and outlet flow coefficient, work coefficient, and stage Mach number. Then, three selected configurations are optimized with a surrogate-assisted evolutionary strategy to refine specific geometrical parameters, which control the meridional shape and splitter location. The stage aerodynamics and performance are assessed with a validated RANS solver based on the homogeneous equilibrium assumption for the two-phase mixture. One of the key findings of the parametric studies is that designing a stage with a flow coefficient smaller than that suggested by design practices prevents the occurrence of two-phase flows. The onset of a secondary phase at the impeller intake alters the fluid compressibility, producing density gradients across saturated condition that can cause flow separation. Such separation occurs for liquid-like CO2 but not for vapor-like CO2, implying that consolidated design rules can apply to the latter case notwithstanding the presence of condensing flows. Shape optimizations reveal that modest efficiency improvements can be gained by changing the meridional channel and splitter location. Moreover, the optimization problem is highly multi-modal, as more than 30 different geometries can yield equivalent design efficiency. As a side effect of the optimization, the rangeability is significantly modified. Specifically, increasing the inlet slope angle of the shroud contour extends the operating range at high flow rates but reduces it at low flow rates. Therefore, this geometrical parameter can be controlled to adjust the stage rangeability without compromising design efficiency.

1.1 Macroeconomic summary Economic recovery has consistently outperformed the technical staff’s expectations following a steep decline in activity in the second quarter of 2020. At the same time, total and core inflation rates have fallen and remain at low levels, suggesting that a significant element of the reactivation of Colombia’s economy has been related to recovery in potential GDP. This would support the technical staff’s diagnosis of weak aggregate demand and ample excess capacity. The most recently available data on 2020 growth suggests a contraction in economic activity of 6.8%, lower than estimates from January’s Monetary Policy Report (-7.2%). High-frequency indicators suggest that economic performance was significantly more dynamic than expected in January, despite mobility restrictions and quarantine measures. This has also come amid declines in total and core inflation, the latter of which was below January projections if controlling for certain relative price changes. This suggests that the unexpected strength of recent growth contains elements of demand, and that excess capacity, while significant, could be lower than previously estimated. Nevertheless, uncertainty over the measurement of excess capacity continues to be unusually high and marked both by variations in the way different economic sectors and spending components have been affected by the pandemic, and by uneven price behavior. The size of excess capacity, and in particular the evolution of the pandemic in forthcoming quarters, constitute substantial risks to the macroeconomic forecast presented in this report. Despite the unexpected strength of the recovery, the technical staff continues to project ample excess capacity that is expected to remain on the forecast horizon, alongside core inflation that will likely remain below the target. Domestic demand remains below 2019 levels amid unusually significant uncertainty over the size of excess capacity in the economy. High national unemployment (14.6% for February 2021) reflects a loose labor market, while observed total and core inflation continue to be below 2%. Inflationary pressures from the exchange rate are expected to continue to be low, with relatively little pass-through on inflation. This would be compatible with a negative output gap. Excess productive capacity and the expectation of core inflation below the 3% target on the forecast horizon provide a basis for an expansive monetary policy posture. The technical staff’s assessment of certain shocks and their expected effects on the economy, as well as the presence of several sources of uncertainty and related assumptions about their potential macroeconomic impacts, remain a feature of this report. The coronavirus pandemic, in particular, continues to affect the public health environment, and the reopening of Colombia’s economy remains incomplete. The technical staff’s assessment is that the COVID-19 shock has affected both aggregate demand and supply, but that the impact on demand has been deeper and more persistent. Given this persistence, the central forecast accounts for a gradual tightening of the output gap in the absence of new waves of contagion, and as vaccination campaigns progress. The central forecast continues to include an expected increase of total and core inflation rates in the second quarter of 2021, alongside the lapse of the temporary price relief measures put in place in 2020. Additional COVID-19 outbreaks (of uncertain duration and intensity) represent a significant risk factor that could affect these projections. Additionally, the forecast continues to include an upward trend in sovereign risk premiums, reflected by higher levels of public debt that in the wake of the pandemic are likely to persist on the forecast horizon, even in the context of a fiscal adjustment. At the same time, the projection accounts for the shortterm effects on private domestic demand from a fiscal adjustment along the lines of the one currently being proposed by the national government. This would be compatible with a gradual recovery of private domestic demand in 2022. The size and characteristics of the fiscal adjustment that is ultimately implemented, as well as the corresponding market response, represent another source of forecast uncertainty. Newly available information offers evidence of the potential for significant changes to the macroeconomic scenario, though without altering the general diagnosis described above. The most recent data on inflation, growth, fiscal policy, and international financial conditions suggests a more dynamic economy than previously expected. However, a third wave of the pandemic has delayed the re-opening of Colombia’s economy and brought with it a deceleration in economic activity. Detailed descriptions of these considerations and subsequent changes to the macroeconomic forecast are presented below. The expected annual decline in GDP (-0.3%) in the first quarter of 2021 appears to have been less pronounced than projected in January (-4.8%). Partial closures in January to address a second wave of COVID-19 appear to have had a less significant negative impact on the economy than previously estimated. This is reflected in figures related to mobility, energy demand, industry and retail sales, foreign trade, commercial transactions from selected banks, and the national statistics agency’s (DANE) economic tracking indicator (ISE). Output is now expected to have declined annually in the first quarter by 0.3%. Private consumption likely continued to recover, registering levels somewhat above those from the previous year, while public consumption likely increased significantly. While a recovery in investment in both housing and in other buildings and structures is expected, overall investment levels in this case likely continued to be low, and gross fixed capital formation is expected to continue to show significant annual declines. Imports likely recovered to again outpace exports, though both are expected to register significant annual declines. Economic activity that outpaced projections, an increase in oil prices and other export products, and an expected increase in public spending this year account for the upward revision to the 2021 growth forecast (from 4.6% with a range between 2% and 6% in January, to 6.0% with a range between 3% and 7% in April). As a result, the output gap is expected to be smaller and to tighten more rapidly than projected in the previous report, though it is still expected to remain in negative territory on the forecast horizon. Wide forecast intervals reflect the fact that the future evolution of the COVID-19 pandemic remains a significant source of uncertainty on these projections. The delay in the recovery of economic activity as a result of the resurgence of COVID-19 in the first quarter appears to have been less significant than projected in the January report. The central forecast scenario expects this improved performance to continue in 2021 alongside increased consumer and business confidence. Low real interest rates and an active credit supply would also support this dynamic, and the overall conditions would be expected to spur a recovery in consumption and investment. Increased growth in public spending and public works based on the national government’s spending plan (Plan Financiero del Gobierno) are other factors to consider. Additionally, an expected recovery in global demand and higher projected prices for oil and coffee would further contribute to improved external revenues and would favor investment, in particular in the oil sector. Given the above, the technical staff’s 2021 growth forecast has been revised upward from 4.6% in January (range from 2% to 6%) to 6.0% in April (range from 3% to 7%). These projections account for the potential for the third wave of COVID-19 to have a larger and more persistent effect on the economy than the previous wave, while also supposing that there will not be any additional significant waves of the pandemic and that mobility restrictions will be relaxed as a result. Economic growth in 2022 is expected to be 3%, with a range between 1% and 5%. This figure would be lower than projected in the January report (3.6% with a range between 2% and 6%), due to a higher base of comparison given the upward revision to expected GDP in 2021. This forecast also takes into account the likely effects on private demand of a fiscal adjustment of the size currently being proposed by the national government, and which would come into effect in 2022. Excess in productive capacity is now expected to be lower than estimated in January but continues to be significant and affected by high levels of uncertainty, as reflected in the wide forecast intervals. The possibility of new waves of the virus (of uncertain intensity and duration) represents a significant downward risk to projected GDP growth, and is signaled by the lower limits of the ranges provided in this report. Inflation (1.51%) and inflation excluding food and regulated items (0.94%) declined in March compared to December, continuing below the 3% target. The decline in inflation in this period was below projections, explained in large part by unanticipated increases in the costs of certain foods (3.92%) and regulated items (1.52%). An increase in international food and shipping prices, increased foreign demand for beef, and specific upward pressures on perishable food supplies appear to explain a lower-than-expected deceleration in the consumer price index (CPI) for foods. An unexpected increase in regulated items prices came amid unanticipated increases in international fuel prices, on some utilities rates, and for regulated education prices. The decline in annual inflation excluding food and regulated items between December and March was in line with projections from January, though this included downward pressure from a significant reduction in telecommunications rates due to the imminent entry of a new operator. When controlling for the effects of this relative price change, inflation excluding food and regulated items exceeds levels forecast in the previous report. Within this indicator of core inflation, the CPI for goods (1.05%) accelerated due to a reversion of the effects of the VAT-free day in November, which was largely accounted for in February, and possibly by the transmission of a recent depreciation of the peso on domestic prices for certain items (electric and household appliances). For their part, services prices decelerated and showed the lowest rate of annual growth (0.89%) among the large consumer baskets in the CPI. Within the services basket, the annual change in rental prices continued to decline, while those services that continue to experience the most significant restrictions on returning to normal operations (tourism, cinemas, nightlife, etc.) continued to register significant price declines. As previously mentioned, telephone rates also fell significantly due to increased competition in the market. Total inflation is expected to continue to be affected by ample excesses in productive capacity for the remainder of 2021 and 2022, though less so than projected in January. As a result, convergence to the inflation target is now expected to be somewhat faster than estimated in the previous report, assuming the absence of significant additional outbreaks of COVID-19. The technical staff’s year-end inflation projections for 2021 and 2022 have increased, suggesting figures around 3% due largely to variation in food and regulated items prices. The projection for inflation excluding food and regulated items also increased, but remains below 3%. Price relief measures on indirect taxes implemented in 2020 are expected to lapse in the second quarter of 2021, generating a one-off effect on prices and temporarily affecting inflation excluding food and regulated items. However, indexation to low levels of past inflation, weak demand, and ample excess productive capacity are expected to keep core inflation below the target, near 2.3% at the end of 2021 (previously 2.1%). The reversion in 2021 of the effects of some price relief measures on utility rates from 2020 should lead to an increase in the CPI for regulated items in the second half of this year. Annual price changes are now expected to be higher than estimated in the January report due to an increased expected path for fuel prices and unanticipated increases in regulated education prices. The projection for the CPI for foods has increased compared to the previous report, taking into account certain factors that were not anticipated in January (a less favorable agricultural cycle, increased pressure from international prices, and transport costs). Given the above, year-end annual inflation for 2021 and 2022 is now expected to be 3% and 2.8%, respectively, which would be above projections from January (2.3% and 2,7%). For its part, expected inflation based on analyst surveys suggests year-end inflation in 2021 and 2022 of 2.8% and 3.1%, respectively. There remains significant uncertainty surrounding the inflation forecasts included in this report due to several factors: 1) the evolution of the pandemic; 2) the difficulty in evaluating the size and persistence of excess productive capacity; 3) the timing and manner in which price relief measures will lapse; and 4) the future behavior of food prices. Projected 2021 growth in foreign demand (4.4% to 5.2%) and the supposed average oil price (USD 53 to USD 61 per Brent benchmark barrel) were both revised upward. An increase in long-term international interest rates has been reflected in a depreciation of the peso and could result in relatively tighter external financial conditions for emerging market economies, including Colombia. Average growth among Colombia’s trade partners was greater than expected in the fourth quarter of 2020. This, together with a sizable fiscal stimulus approved in the United States and the onset of a massive global vaccination campaign, largely explains the projected increase in foreign demand growth in 2021. The resilience of the goods market in the face of global crisis and an expected normalization in international trade are additional factors. These considerations and the expected continuation of a gradual reduction of mobility restrictions abroad suggest that Colombia’s trade partners could grow on average by 5.2% in 2021 and around 3.4% in 2022. The improved prospects for global economic growth have led to an increase in current and expected oil prices. Production interruptions due to a heavy winter, reduced inventories, and increased supply restrictions instituted by producing countries have also contributed to the increase. Meanwhile, market forecasts and recent Federal Reserve pronouncements suggest that the benchmark interest rate in the U.S. will remain stable for the next two years. Nevertheless, a significant increase in public spending in the country has fostered expectations for greater growth and inflation, as well as increased uncertainty over the moment in which a normalization of monetary policy might begin. This has been reflected in an increase in long-term interest rates. In this context, emerging market economies in the region, including Colombia, have registered increases in sovereign risk premiums and long-term domestic interest rates, and a depreciation of local currencies against the dollar. Recent outbreaks of COVID-19 in several of these economies; limits on vaccine supply and the slow pace of immunization campaigns in some countries; a significant increase in public debt; and tensions between the United States and China, among other factors, all add to a high level of uncertainty surrounding interest rate spreads, external financing conditions, and the future performance of risk premiums. The impact that this environment could have on the exchange rate and on domestic financing conditions represent risks to the macroeconomic and monetary policy forecasts. Domestic financial conditions continue to favor recovery in economic activity. The transmission of reductions to the policy interest rate on credit rates has been significant. The banking portfolio continues to recover amid circumstances that have affected both the supply and demand for loans, and in which some credit risks have materialized. Preferential and ordinary commercial interest rates have fallen to a similar degree as the benchmark interest rate. As is generally the case, this transmission has come at a slower pace for consumer credit rates, and has been further delayed in the case of mortgage rates. Commercial credit levels stabilized above pre-pandemic levels in March, following an increase resulting from significant liquidity requirements for businesses in the second quarter of 2020. The consumer credit portfolio continued to recover and has now surpassed February 2020 levels, though overall growth in the portfolio remains low. At the same time, portfolio projections and default indicators have increased, and credit establishment earnings have come down. Despite this, credit disbursements continue to recover and solvency indicators remain well above regulatory minimums. 1.2 Monetary policy decision In its meetings in March and April the BDBR left the benchmark interest rate unchanged at 1.75%.

With the proliferation of Free Trade Agreements (FTAs) over the past decade, demand for quantitative analysis of their likely impacts has surged. The main quantitative tool for performing such analysis is Computable General Equilibrium (CGE) modeling. Yet these models have been widely criticized for performing poorly (Kehoe, 2002) and having weak econometric foundations (McKitrick, 1998; Jorgenson, 1984). FTA results have been shown to be particularly sensitive to the trade elasticities, with small trade elasticities generating large terms of trade effects and relatively modest efficiency gains, whereas large trade elasticities lead to the opposite result. Critics are understandably wary of results being determined largely by the authors’ choice of trade elasticities. Where do these trade elasticities come from? CGE modelers typically draw these elasticities from econometric work that uses time series price variation to identify an elasticity of substitution between domestic goods and composite imports (Alaouze, 1977; Alaouze, et al., 1977; Stern et al., 1976; Gallaway, McDaniel and Rivera, 2003). This approach has three problems: the use of point estimates as “truth”, the magnitude of the point estimates, and estimating the relevant elasticity. First, modelers take point estimates drawn from the econometric literature, while ignoring the precision of these estimates. As we will make clear below, the confidence one has in various CGE conclusions depends critically on the size of the confidence interval around parameter estimates. Standard “robustness checks” such as systematically raising or lowering the substitution parameters does not properly address this problem because it ignores information about which parameters we know with some precision and which we do not. A second problem with most existing studies derives from the use of import price series to identify home vs. foreign substitution, for example, tends to systematically understate the true elasticity. This is because these estimates take price variation as exogenous when estimating the import demand functions, and ignore quality variation. When quality is high, import demand and prices will be jointly high. This biases estimated elasticities toward zero. A related point is that the fixed-weight import price series used by most authors are theoretically inappropriate for estimating the elasticities of interest. CGE modelers generally examine a nested utility structure, with domestic production substitution for a CES composite import bundle. The appropriate price series is then the corresponding CES price index among foreign varieties. Constructing such an index requires knowledge of the elasticity of substitution among foreign varieties (see below). By using a fixed-weight import price series, previous estimates place too much weight on high foreign prices, and too small a weight on low foreign prices. In other words, they overstate the degree of price variation that exists, relative to a CES price index. Reconciling small trade volume movements with large import price series movements requires a small elasticity of substitution. This problem, and that of unmeasured quality variation, helps explain why typical estimated elasticities are very small. The third problem with the existing literature is that estimates taken from other researchers’ studies typically employ different levels of aggregation, and exploit different sources of price variation, from what policy modelers have in mind. Employment of elasticities in experiments ill-matched to their original estimation can be problematic. For example, estimates may be calculated at a higher or lower level of aggregation than the level of analysis than the modeler wants to examine. Estimating substitutability across sources for paddy rice gives one a quite different answer than estimates that look at agriculture as a whole. When analyzing Free Trade Agreements, the principle policy experiment is a change in relative prices among foreign suppliers caused by lowering tariffs within the FTA. Understanding the substitution this will induce across those suppliers is critical to gauging the FTA’s real effects. Using home v. foreign elasticities rather than elasticities of substitution among imports supplied from different countries may be quite misleading. Moreover, these “sourcing” elasticities are critical for constructing composite import price series to appropriate estimate home v. foreign substitutability. In summary, the history of estimating the substitution elasticities governing trade flows in CGE models has been checkered at best. Clearly there is a need for improved econometric estimation of these trade elasticities that is well-integrated into the CGE modeling framework. This paper provides such estimation and integration, and has several significant merits. First, we choose our experiment carefully. Our CGE analysis focuses on the prospective Free Trade Agreement of the Americas (FTAA) currently under negotiation. This is one of the most important FTAs currently “in play” in international negotiations. It also fits nicely with the source data used to estimate the trade elasticities, which is largely based on imports into North and South America. Our assessment is done in a perfectly competitive, comparative static setting in order to emphasize the role of the trade elasticities in determining the conventional gains/losses from such an FTA. This type of model is still widely used by government agencies for the evaluation of such agreements. Extensions to incorporate imperfect competition are straightforward, but involve the introduction of additional parameters (markups, extent of unexploited scale economies) as well as structural assumptions (entry/no-entry, nature of inter-firm rivalry) that introduce further uncertainty. Since our focus is on the effects of a PTA we estimate elasticities of substitution across multiple foreign supply sources. We do not use cross-exporter variation in prices or tariffs alone. Exporter price series exhibit a high degree of multicolinearity, and in any case, would be subject to unmeasured quality variation as described previously. Similarly, tariff variation by itself is typically unhelpful because by their very nature, Most Favored Nation (MFN) tariffs are non-discriminatory in nature, affecting all suppliers in the same way. Tariff preferences, where they exist, are often difficult to measure – sometimes being confounded by quantitative barriers, restrictive rules of origin, and other restrictions. Instead we employ a unique methodology and data set drawing on not only tariffs, but also bilateral transportation costs for goods traded internationally (Hummels, 1999). Transportation costs vary much more widely than do tariffs, allowing much more precise estimation of the trade elasticities that are central to CGE analysis of FTAs. We have highly disaggregated commodity trade flow data, and are therefore able to provide estimates that precisely match the commodity aggregation scheme employed in the subsequent CGE model. We follow the GTAP Version 5.0 aggregation scheme which includes 42 merchandise trade commodities covering food products, natural resources and manufactured goods. With the exception of two primary commodities that are not traded, we are able to estimate trade elasticities for all merchandise commodities that are significantly different form zero at the 95% confidence level. Rather than producing point estimates of the resulting welfare, export and employment effects, we report confidence intervals instead. These are based on repeated solution of the model, drawing from a distribution of trade elasticity estimates constructed based on the econometrically estimated standard errors. There is now a long history of CGE studies based on SSA: Systematic Sensitivity Analysis (Harrison and Vinod, 1992; Wigle, 1991; Pagon and Shannon, 1987) Ho