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We first define energy conditions, which are gravitational analogs of the positivity of the energy in nongravitational theories. After defining the notion of singularity more precisely, we state (without proof) the singularity theorems of Hawking and Penrose, and a “counterexample,” which evades all of their assumptions. Then we define wormholes, traversable wormholes, and give as example the Morris–Thorne wormhole, with its embedding diagram.
From particle lifting in atmospheric boundary layers to dust ingestion in jet engines, the transport and deposition of inertial particles in wall-bounded turbulent flows are prevalent in both nature and industry. Due to triboelectrification during collisions, solid particles often acquire significant charges. However, the impacts of the resulting electrostatic interaction on the particle dynamics remain less understood. In this study, we present four-way coupled simulations to investigate the deposition of charged particles onto a grounded metal substrate through a fully developed turbulent boundary layer. Our numerical method tracks the dynamics of individual particles under the influence of turbulence, electrostatic forces and collisions. We first report a more pronounced near-wall accumulation and an increased wall-normal particle velocity due to particle charging. In addition, contrary to predictions from the classic Eulerian model, the wall-normal transport rate of inertial particles is significantly enhanced by electrostatic forces. A statistical approach is then applied to quantify the contributions from turbophoresis, biased sampling and electrostatic forces. For charged particles, a sharper gradient in wall-normal particle fluctuation velocity is observed, which substantially enhances turbophoresis and serves as the primary driving force of near-wall particle accumulation. Furthermore, charged particles are found to sample upward-moving fluids less frequently than neutral particles, thereby weakening the biased-sampling effect that typically pushes particles away from the wall. Finally, the wall-normal electric field is shown to depend on the competition between particle–wall and particle–particle electrostatic interactions, which helps to identify the dominant electrostatic force across a wide range of scenarios.
We first describe the Newtonian limit for gravity, in particular in the case of spherical symmetry. We use it to help with the ansatz in the case of the vacuum solution of the Einstein’s equations with spherical symmetry. After calculating the equations of motion on the ansatz, we find the most general such solution, the Schwarzschild solution. We also define what a Schwarzschild black hole is.
We first define the notion of Wald gravitational entropy, defined in a more general setting than the Bekenstein–Hawking one. Then we define Sen’s entropy function formalism, in a general gravity theory, that defines the entropy function, whose minimization at the event horizon gives the entropy. Finally, we define the effective potential of the event horizon, defined in a theory with scalars, and show that the horizon is an attractor for the equations of motion involving scalars.
Delaying the laminar–turbulent transition of a boundary layer reduces the skin-friction drag and can thereby increase the efficiency of any aerodynamic device. A passive control strategy that has reaped success in transition delay is the introduction of boundary layer streaks. Surface-mounted vortex generators have been found to feature an unstable region right behind the devices, which can be fatal in flow control if transition is triggered, leading to an increase in drag with respect to the reference case without devices. In a previous proof of concept study, numerical simulations were employed to place artificial vortices in the free stream that interact with the boundary layer and accomplish transition delay. In the current study, we present experimental results showing the feasibility of generating free-stream vortices that interact with the boundary layer, creating high- and low-speed boundary layer streaks. This type of streaky base flow can act as stabilizing if introduced properly. We confirm the success of our flow control approach by artificially introducing two-dimensional disturbances that are strongly attenuated in the presence of streaks, leading to a transition delay with respect to the reference case of approximately 40 %.
We first describe the classification of three-dimensional Lie algebras. Then we show how that implies a classification of three-dimensional Lie invariant Riemann spaces, as Bianchi spaces, associated to cosmologies. Then, we describe examples of homogenous Bianchi cosmologies, the Kasner spacetime and the Mixmaster Universe.
We first review general concepts of quantum field theory, like Feynman diagrams and path integrals. Then we define the worldline formalism for quantum field theory, and apply it to gravity, in the case of classical gravity with sources. We describe first an electromagnetic analogy of how to write down the worldline action and obtain physics from it, and then use it in the gravity case, for a black hole or neutron star. We then consider tidal forces in the nonrelativistic regime and find how to describe Love numbers and radiated power, for instance. As a simple example, we consider scalar gravity interacting with a source current and how to obtain information from Feynman diagrams. Then, define Non-Relativistic General Relativity (NRGR), a formalism for the gravitational inspiral of two bodies (e.g., two black holes) and show how to calculate the two-body (or many-body) action from Feynman diagrams and how to obtain the radiated power via gravitational waves. We find the Einstein–Infeld–Hofman Lagrangian for many bodies in the NRGR case.
We define the ADM parametrization of gravitational actions. Then, we define extrinsic curvature for both a spacelike hypersurface in the ADM parametrization and for a general surface embedded in a larger space. The Gauss–Codazzi equations for the embedding of a surface are defined and explained, as well as the Israel junction conditions. After defining the notion of Killing vectors, we consider asymptotically flat spacetimes, with the possible definitions of mass (Kumar, ADM, linearized) and the BMS group of asymptotic symmetries. Finally, we calculate and define the boundary term of the gravitational action, the Gibbons–Hawking–York term.
We describe the geometric formulation for gravity, without using an inverse metric, only the vielbein and spin connection as gauge fields. We define the general Lanczos–Lovelock Lagrangian, in particular Chern–Simons (in odd dimensions) and Born–Infeld (in even dimensions) gravity, and the topological gravity terms and their dimensional extensions. We end by an application to anomalies, specifically gravitational anomalies.
We define Penrose diagrams, which keep the causal and topological properties of gravitational spacetimes, while moving infinity to a finite distance on the diagram. We use the examples of Minkowski space, in two dimensions and dimensions greater than two, then describe Anti-de Sitter spacetime in Poincaré coordinates (the Poincaré patch), and finally consider the Schwarzschild black hole.
We define the vielbein–spin connection formulation of general relativity and describe what happens in the presence of fermions (which can only be described in this formulation). Then, we see how close is general relativity to being described by a gauge theory, in three dimensions, which is special, and in four or higher dimensions.
We describe the parametrized post-Newtonian (PPN) formalism for expansion around the Newtonian limit. First we describe the formalism for generic gravity theories, for the equations of motion and the energy-momentum tensor. Then we consider it for many-body systems (like binary inspirals) and the coefficients for PPN in general relativity in this case. We describe the effective field theory approach in this case, show how to calculate the post-Newtonian (PN) order, and how it fits with the EIH Lagrangian. Finally, we consider metric frames, the usual Einstein frame versus the generic Jordan frame, and the original Brans–Dicke theory.
Wall turbulence consists of various sizes of vortical structures that induce flow circulation around a wide range of closed Eulerian loops. Here we investigate the multiscale properties of circulation around such loops in statistically homogeneous planes parallel to the wall. Using a high-resolution direct numerical simulation database of turbulent channels at Reynolds numbers of $Re_\tau =180$, 550, 1000 and 5200, circulation statistics are obtained in planes at different wall-normal heights. Intermittency of circulation in the planes of the outer flow ($y^+ \gtrsim 0.1Re_\tau$) takes the form of universal bifractality as in homogeneous and isotropic turbulence. The bifractal character simplifies to space-filling character close to the wall, with scaling exponents that are linear in the moment order, and lower than those given by the Kolmogorov paradigm. The probability density functions of circulation are long-tailed in the outer bifractal region, with evidence showing their invariance with respect to the loop aspect ratio, while those in the inner region are closely Gaussian. The unifractality near the wall implies that the circulation there is not intermittent in character.
The aspect ratio effect on side and basal melting in fresh water is systematically investigated across a range of Rayleigh numbers and ambient temperatures using direct numerical simulations. The side mean melt rate follows a ${Ra}^{1/4}\,\gamma ^{-3/8}$ scaling relation in the side-melting dominant regime, where ${Ra}$ is the Rayleigh number, and $\gamma$ is the width-to-height aspect ratio of the ice block. In the basal-melting dominant regime, the basal mean melt rate follows a ${Ra}^{1/4}\gamma ^{3/8}$ scaling relation at low Rayleigh numbers, but transitions to a ${Ra}^{1/3}\gamma ^{1/2}$ scaling relation at higher Rayleigh numbers. This scaling transition is attributed to the formation of a bottom cavity resulting from flow separation at high Rayleigh numbers. The overall mean melt rate exhibits a non-monotonic dependence on the aspect ratio, driven by the competition between side and basal melting. The proposed theoretical model successfully captures the observed non-monotonic behaviour, and accurately predicts the overall mean melt rate over the considered range of Rayleigh numbers and ambient temperatures, especially in the side- and basal-melting dominant regimes. More specifically, the side, basal and overall mean melt rates follow a linear ${St}$ scaling relation for ambient temperatures $T_{w}\geqslant 15^{\,\circ }\textrm {C}$, with ${St}$ being the Stefan number (the ratio between sensible heat and latent heat), but deviations from this scaling relation and a non-monotonic dependence on the ambient temperature are observed at lower ambient temperatures, which can be attributed to the density anomaly effect.
This paper investigates the behaviour of turbulence production in adverse pressure gradient (APG) turbulent boundary layers (TBLs), including the range of pressure gradients from zero-pressure-gradient (ZPG) to separation, moderate and high Reynolds numbers, and equilibrium and non-equilibrium flows. The main focus is on predicting the values and positions of turbulence production peaks. Based on the unique ability of turbulence production to describe energy exchange, the idea that the ratios of the mean flow length scales to the turbulence length scales are locally smallest near peaks is proposed. Thereby, the ratios of length scales are defined for the inner and outer regions, respectively, as well as the ratios of time scales for further consideration of local information. The ratios in the inner region are found to reach the same constant value in different APG TBLs. Like turbulence production in the ZPG TBL, turbulence production in APG TBLs is shown to have a certain invariance of the inner peak. The value and position of the inner peak can also be predicted quantitatively. In contrast, the ratios in the outer region cannot be determined with unique coefficients, which accounts for the different self-similarity properties of the inner and outer regions. The outer time scale ratios establish a link between mean flow and turbulence, thus participating in the discussion on half-power laws. The present results support the existence of a half-power-law region that is not immediately adjacent to the overlapping region.
This article delves into the dynamics of inviscid annular supersonic jets, akin to those exiting converging–diverging nozzles in over-expanded regimes. It focuses on the first azimuthal Fourier mode of flow fluctuations and examines their behaviour with varying mixing layer parameters and expansion regimes. The study reveals that two unstable Kelvin–Helmholtz waves exist in all cases, with the outer-layer wave being more unstable due to differences in the velocity gradient. The inner-layer wave is more sensitive to changes in base flow and extends beyond the jet, potentially contributing to nozzle resonances. The article also investigates upstream propagating guided-jet modes, which are found to be robust and not highly sensitive to changes in base flow, which makes them essential for understanding jet dynamics. A simplified model is used to obtain ideal base flows but with realistic shape in order to study the effects of varying nozzle pressure ratios on the dynamics of the waves supported by the jet.
In this work, we describe the use of a 1D-2V quasi-neutral hybrid electrostatic PIC with Monte-Carlo Coulomb collisions and non-uniform magnetic field to model the parallel transport and confinement in an axisymmetric tandem mirror device. End-plugs, based on simple-mirrors, are positioned at each end of the device and fueled with neutral beams (25 and 100 keV) to produce a sloshing ion population and increase the density of the end-plugs relative to the central cell. Results show the formation of a potential difference barrier between the central cell and the end-plugs. This potential confines a large fraction of the low energy thermal ions in the central cell which would otherwise be lost in a simple mirror, demonstrating the advantage of the beam-driven tandem mirror configuration relative to simple mirrors. In addition, we explore the effect of end-plug electron temperature on the confinement time of the device and compare it with theoretical estimates. Finally, we discuss the limitations of the code in its present form and describe the next logical steps to improve its predictive capability such as a fully nonlinear Fokker–Planck collision operator, multiply nested flux surface solutions and modeling the exhaust region up to the wall.