Open Boundary Conditions Research Articles

Effect of Outlet Boundary Condition on the Acoustic Modeshape and Flame Dynamics of a Partially Premixed Swirl Stabilized Combustor

Abstract Industrial gas turbines commonly use lean premixed swirl stabilized combustors that are prone to thermo-acoustic instabilities. Combustor testing involves several steps ranging from atmospheric to high-pressure conditions. An open outlet boundary condition is maintained for atmospheric tests commonly, whereas high-pressure testing involves complex exit boundary conditions, which change the reflection coefficient and can affect the nature of instability. Current work aims at studying how the change in outlet boundary affects the nature of instability due to changes in the exit reflection coefficient and acoustic mode shape inside the combustor. A laboratory-scale industrial swirl burner that uses partially premixed methane and air at atmospheric conditions is analyzed for this purpose. A constant area contraction ratio of 6.5:1 is maintained at the exit of the combustor while varying the inlet Reynolds number at a constant global equivalence ratio. Flame dynamics based on conditional phase averaged OH* chemiluminescence images and spatial Rayleigh index maps were used to compare different flow rate and exit boundary cases. The outlet contraction affects both the frequency and amplitude of the dominant thermoacoustic mode. The orifice plate at the exit reduces the outlet reflection coefficient leading to a change in acoustic mode shape inside the combustor. Overall, the instability amplitude is reduced considerably for cases with an orifice plates at the exit boundary compared to open exit boundary cases. The results show the importance of defining outlet boundary conditions as a parameter in combustion instability studies. Care should be taken while comparing and interpreting results from different facilities where outlet boundary condition is different.

Interfacial skin modes at a non-Hermitian heterojunction

We uncover a non-Hermitian skin mode in which the eigenmodes of a nonreciprocal topoelectrical (TE) heterojunction circuit array are localized at the heterojunction interface under both open boundary condition (OBC) and periodic boundary condition (PBC). This is in marked contrast to the conventional non-Hermitian skin effect (NHSE) in homogenous non-Hermitian systems that exists under OBC but vanishes in PBC. The interfacial non-Hermitian skin effect is induced when two non-Hermitian circuit chain segments with dissimilar NHSE decay lengths are cascaded together. It is shown analytically that the interfacial NHSE decay rates are directly associated with the asymmetric intracell couplings in the TE segments. Such nonreciprocal directional couplings can be realized in practical circuits by means of negative impedance converters with current inversion (INIC). In contrast to the conventional NHSE in homogeneous systems, the TE heterojunction circuit allows for a voltage profile localized at the interface, that is independent of the boundary conditions and whose decay length can be modulated by circuit parameters. Finally, the results presented are general and applicable to other analogous non-Hermitian platforms such as photonic and condensed matter nanoscale systems.

Nonsymmorphic chiral symmetry and solitons in the Rice-Mele model

The Rice-Mele model has two topological and spatially-inversion symmetric phases, namely the Su-Schrieffer-Heeger (SSH) phase with alternating hopping only, and the charge-density-wave (CDW) phase with alternating energies only. The chiral symmetry of the SSH phase is robust in position space, so that it is preserved in the presence of the ends of a finite system and of textures in the alternating hopping. However, the chiral symmetry of the CDW wave phase is nonsymmorphic, resulting in a breaking of the bulk topology by an end or a texture in the alternating energies. We consider the presence of solitons (textures in position space separating two degenerate ground states) in finite systems with open boundary conditions. We identify the parameter range under which an atomically-sharp soliton in the CDW phase supports a localized state which lies within the band gap, and we calculate the expectation value $p_y$ of the nonsymmorphic chiral operator for this state, and the soliton electric charge. As the spatial extent of the soliton increases beyond the atomic limit, the energy level approaches zero exponentially quickly or inversely proportionally to the width, depending on microscopic details of the soliton texture. In both cases, the difference of $p_y$ from one is inversely proportional to the soliton width, while the charge is independent of the width. We investigate the robustness of the soliton level in the presence of disorder and sample-to-sample parameter variations, comparing with a single soliton level in the SSH phase with an odd number of sites.

Role of boundary conditions in the full counting statistics of topological defects after crossing a continuous phase transition

In a scenario of spontaneous symmetry breaking in finite time, topological defects are generated at a density that scale with the driving time according to the Kibble-Zurek mechanism (KZM). Signatures of universality beyond the KZM have recently been unveiled: The number distribution of topological defects has been shown to follow a binomial distribution, in which all cumulants inherit the universal power-law scaling with the quench rate, with cumulant rations being constant. In this work, we analyze the role of boundary conditions in the statistics of topological defects. In particular, we consider a lattice system with nearest-neighbor interactions subject to soft anti-periodic, open, and periodic boundary conditions implemented by an energy penalty term. We show that for fast and moderate quenches, the cumulants of the kink number distribution present a universal scaling with the quench rate that is independent of the boundary conditions except by an additive term, that becomes prominent in the limit of slow quenches, leading to the breaking of power-law behavior. We test our theoretical predictions with a one-dimensional scalar theory on a lattice.

Bending waves localized along the edge of a semi-infinite piezoelectric plate with orthogonal symmetry

We study the propagation of bending waves along the free edge of a semi-infinite piezoelectric plate within the framework of two-variable refined plate theory (TVPT, a high-order plate theory), Reissner-Mindlin refined plate theory (RMPT, a first-order plate theory), and the classical plate theory (CPT). The piezoelectric plate has macroscopic symmetry of orthogonal mm2 The governing equations are derived using Hamilton principle. The dispersion relations for electrically open and shorted boundary conditions at the free edge are obtained analytically. The difference in dispersion property between the three plate theories is analyzed. The numerical results show that the dispersion curves predicted by TVPT and RMPT are similar and have small difference over the complete frequency range, which means both the two theories are valid for the analysis of edge waves in a piezoelectric plate. But the wave velocity calculated by CPT is much larger than the two theories above and is no longer valid for high frequency and thick plate. The electrical boundary condition at the free edge has an insignificant effect on phase velocity and group velocity which can be ignored for the analysis of edge waves in a piezoelectric plate governed by bending deformation. The velocity of bending edge waves in a semi-infinite piezoelectric plate is positively related to that of Rayleigh surface wave in a traction-free piezoelectric half-space. The edge wave velocity can be enhanced when the piezoelectric plate is considered as one with weaker anisotropy.

Manipulating the non-Hermitian skin effect via electric fields

In non-Hermitian systems, the phenomenon that the bulk-band eigenstates are accumulated at the boundaries of the systems under open boundary conditions is called non-Hermitian skin effect (NHSE), which is one of the most iconic and important features of a non-Hermitian system. In this work, we investigate the fate of NHSE in the presence of electric fields by analytically calculating the dynamical evolution of an initial bulk state and numerically computing the spectral winding number, the distributions of eigenstates, as well as the dynamical evolutions. We show abundant manipulation effects of dc and ac fields on the NHSE, and that the physical mechanism behind these effects is the interplay between the Stark localization, dynamic localization and the NHSE. In addition, the finite size analysis of the non-Hermitian system with a pure dc field shows the phenomenon of size-dependent NHSE. We further propose a scheme to realize the discussed model based on an electronic circuit. The results will help to deepen the understanding of NHSE and its manipulation.

Study of storm-induced changes in circulation and temperature over the northern South China Sea during Typhoon Linfa

Hydrodynamic responses of the northern South China Sea (nSCS) to Typhoon Linfa in 2009 are examined using observations and model results. A shelf circulation model based on the Regional Ocean Modelling System (ROMS-nSCS) is used in quantifying the important physical processes affecting the storm-induced changes in circulation and hydrography in the upper ocean during Linfa. The ROMS-nSCS is forced by atmospheric forcing taken from the Climate Forecast System Reanalysis (CFSR). A parametric vortex is inserted into the CFSR atmospheric forcing to improve winds and sea-level air pressures associated with Linfa. The lateral open boundary condition of ROMS-nSCS is specified based on the global ocean circulation reanalysis produced by the Hybrid Coordinate Ocean Model (HYCOM). The simulated upper ocean circulation produced by ROMS-nSCS is characterized by strong wind-driven currents over the area affected directly by Linfa and intense near-inertial oscillations (NIOs) in the wake of Linfa. The simulated cooling in the wake of the storm is biased to the right of the storm track and agrees well with satellite-derived analysis data. The simulated NIOs also have significant rightward biases. The rotary spectral analysis of model currents demonstrates that the peak frequency fp of NIOs is about 9% greater than the local inertial frequency, which can be explained mostly by the vorticity of the background circulation. Analysis of model results also reveal that different dynamics are responsible for storm-induced cooling at different stages of Linfa. In the early development stage of Linfa, the cooling in the wake of the storm is affected mostly by the storm-induced advection in the vertical. In the mature stage of Linfa, the cooling in the wake of the storm is dominated by the joint effect of the storm-induced mixing and advection. Storm-induced warming occurs outside the wake of Linfa, due to downwelling generated by the storm.

Optical Properties of Cylindrical Quantum Dots with Hyperbolic-Type Axial Potential under Applied Electric Field.

In this paper, we have researched the electronic and optical properties of cylindrical quantum dot structures by selecting four different hyperbolic-type potentials in the axial direction under an axially-applied electric field. We have considered a position-dependent effective mass model in which both the smooth variation of the effective mass in the axial direction adjusted to the way the confining potentials change and its abrupt change in the radial direction have been considered in solving the eigenvalue differential equation. The calculations of the eigenvalue equation have been implemented considering both the Dirichlet conditions (zero flux) and the open boundary conditions (non-zero flux) in the planes perpendicular to the direction of the applied electric field, which guarantees the validity of the results presented in this study for quasi-steady states with extremely high lifetimes. We have used the diagonalization method combined with the finite element method to find the eigenvalues and eigenfunction of the confined electron in the cylindrical quantum dots. The numerical strategies that have been used for the solution of the differential equations allowed us to overcome the multiple problems that the boundary conditions present in the region of intersection of the flat and cylindrical faces that form the boundary of the heterostructure. To calculate the linear and third-order nonlinear optical absorption coefficients and relative changes in the refractive index, a two-level approach in the density matrix expansion is used. Our results show that the electronic and, therefore, optical properties of the structures focused on can be adjusted to obtain a suitable response for specific studies or goals by changing structural parameters such as the widths and depths of the potentials in the axial direction, as well as the electric field intensity.

Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system that is redundant at the continuous level but eliminates the need of mimicking the Hodge star operator at the discrete level. By considering the Stokes-Dirac structure representing the system together with its adjoint, which embeds the metric information directly in the codifferential, the need for an explicit discrete Hodge star is avoided altogether. By imposing the boundary conditions in a strong manner, the power balance characterizing the Stokes-Dirac structure is then retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. Numerical experiments validate the convergence of the method and the conservation properties in terms of energy balance both for the wave and Maxwell equations in a three dimensional domain. For the latter example, the magnetic and electric fields preserve their divergence free nature at the discrete level.

Long-distance entanglement in antiferromagnetic XXZ spin chain with alternating interactions

In this paper, the quantum entanglement (concurrence) of one-dimensional antiferromagnetic XXZ spin system with alternating nearest-neighbor interactions is studied by means of the density matrix renormalization group method with matrix product state form. The ground-state energy, long-distance entanglement and entanglement distribution are calculated, and the effects of alternating interactions, anisotropy and open boundary condition on them are discussed. It is found that the alternating interactions favor the generation and development of long-distance entanglement, while anisotropy inhibits that. For the entanglement distribution of the system, it is also found that the alternating interactions and open boundary condition induce the dimerization on odd and even bonds, respectively, while anisotropy always suppresses this behavior. Furthermore, both nearest-neighbor and long-distance entanglements are converged to certain values with increasing system size.

Stability of invertible, frustration-free ground states against large perturbations

A gapped ground state of a quantum spin system has a natural length scale set by the gap. This length scale governs the decay of correlations. A common intuition is that this length scale also controls the spatial relaxation towards the ground state away from impurities or boundaries. The aim of this article is to take a step towards a proof of this intuition. We assume that the ground state is frustration-free and invertible, i.e. it has no long-range entanglement. Moreover, we assume the property that we are aiming to prove for one specific kind of boundary condition; namely open boundary conditions. This assumption is also known as the "local topological quantum order" (LTQO) condition. With these assumptions we can prove stretched exponential decay away from boundaries or impurities, for any of the ground states of the perturbed system. In contrast to most earlier results, we do not assume that the perturbations at the boundary or the impurity are small. In particular, the perturbed system itself can have long-range entanglement.

Sensitivity of non-Hermitian systems

Understanding the extreme sensitivity of the eigenvalues of non-Hermitian Hamiltonians to the boundary conditions is of great importance when analyzing non-Hermitian systems, as it appears generically and is intimately connected to the skin effect and the breakdown of the conventional bulk boundary correspondence. Here we describe a method to find the eigenvalues of one-dimensional one-band models with arbitrary boundary conditions. We use this method on several systems to find analytical expressions for the eigenvalues, which give us conditions on the parameter values in the system for when we can expect the spectrum to be insensitive to a change in boundary conditions. By stacking one-dimensional chains, we use the derived results to find corresponding conditions for insensitivity for some two-dimensional systems with periodic boundary conditions in one direction. This would be hard by using other methods to detect skin effect, such as the winding of the determinant of the Bloch Hamiltonian. Finally, we use these results to make predictions about the (dis)appearance of the skin effect in purely two-dimensional systems with open boundary conditions in both directions.

Symmetry breaking and spectral structure of the interacting Hatano-Nelson model

We study the Hatano-Nelson model, i.e., a one-dimensional non-Hermitian chain of spinless fermions with nearest-neighbour nonreciprocal hopping, in the presence of repulsive nearest-neighbour interactions. At half filling, we find two $\mathcal{PT}$ transitions, as the interaction strength increases. The first transition is marked by an exceptional point between the first and the second excited state in a finite-size system and is a first-order symmetry-breaking transition into a charge-density wave regime. Persistent currents characteristic of the Hatano-Nelson model abruptly vanish at the transition. The second transition happens at a critical interaction strength that scales with the system size and can thus only be observed in finite-size systems. It is characterized by a collapse of all energy eigenvalues onto the real axis. We further show that in a strong interaction regime, but away from half filling, the many-body spectrum shows point gaps with nontrivial winding numbers, akin to the topological properties of the single-particle spectrum of the Hatano-Nelson chain, which indicates the skin effect of extensive many-body eigenstates under open boundary conditions. Our results can be applied to other models such as the non-Hermitian Su-Schrieffer-Heeger-type model and contribute to an understanding of fermionic many-body systems with non-Hermitian Hamiltonians.

System size dependent topological zero modes in coupled topolectrical chains

In this paper, we demonstrate the emergence and disappearance of topological zero modes (TZMs) in a coupled topolectrical (TE) circuit lattice. Specifically, we consider non-Hermitian TE chains in which TZMs do not occur in the individual uncoupled chains, but emerge when these chains are coupled by inter-chain capacitors. The coupled system hosts TZMs which show size-dependent behaviours and vanish beyond a certain critical size. In addition, the emergence or disappearance of the TZMs in the open boundary condition (OBC) spectra for a given size of the coupled system can be controlled by modulating the signs of its inverse decay length. Analytically, trivial and non-trivial phases of the coupled system can be distinguished by the differing ranks of their corresponding Laplacian matrix. The TE circuit framework enables the physical detection of the TZMs via electrical impedance measurements. Our work establishes the conditions for inducing TZMs and modulating their behavior in coupled TE chains.

Dephasing-enhanced Majorana zero modes in two-dimensional and three-dimensional higher-order topological superconductors

The 1D Kitaev model in the topological phase, with open boundary conditions, hosts strong Majorana zero modes. These are fermion parity-odd operators that almost commute with the Hamiltonian and manifest in long coherence times for edge degrees of freedom. We obtain higher-dimensional counterparts of such Majorana operators by explicitly computing their closed form expressions in models describing 2D and 3D higher-order superconductors. Due to the existence of such strong Majorana zero modes, the coherence time of the infinite temperature autocorrelation function of the corner Majorana operators in these models diverges with the linear system size. In the presence of a certain class of orbital-selective dissipative dynamics, the coherence times of half of the corner Majorana operators is enhanced, while the time correlations corresponding to the remaining corner Majoranas decay much faster as compared with the unitary case. We numerically demonstrate robustness of the coherence times to the presence of disorder.

Chiral fixed point in a junction of critical spin-1 chains

Junctions of one-dimensional systems are of great interest to the development of synthetic materials that harbor topological phases. We study a junction of three gapless spin-1 chains described by the $\mathrm{SU}{(2)}_{2}$ Wess-Zumino-Wittten model and coupled by exchange and chiral three-spin interactions. We show that a chiral fixed point appears as a special point on the transition line separating two regimes described by open boundary conditions, corresponding to decoupled chains and the formation of a boundary spin singlet state. Along this transition line, the junction behaves as a tunable spin circulator as the spin conductance varies continuously with the coupling constant of a marginal boundary operator. Since the spectrum of the junction contains fractional excitations such as Majorana fermions, in this paper, we set the stage for network constructions of non-Abelian chiral spin liquids.

Exact surface energy of the [formula omitted] spin chain with generic non-diagonal boundary reflections

The exact solution of the D2(1) quantum spin chain with generic non-diagonal boundary reflections is obtained. It is found that the generating functional of conserved quantities of the system can be factorized as the product of transfer matrices of two anisotropic XXZ spin chains with open boundary conditions. By using the factorization identities and the fusion technique, the eigenvalues and the Bethe ansatz equations of the model are obtained. The eigenvalues are also parameterized by the zero roots of the transfer matrix, and the patterns of root distributions are obtained. Based on them, ground states energy and the surface energies induced by the twisted boundary magnetic fields in the thermodynamic limit are obtained. These results are checked by the numerical calculations. The corresponding isotropic limit is also discussed. The results given in this paper are the foundation to study the exact physical properties of high rank Dn(1) model by using the nested processes.

Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of a crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a $2\pi$-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in the bulk, three-dimensional Brillouin zone; because this change is reversed on the complementary section of the Brillouin zone, we refer to this new invariant as a returning Thouless pump (RTP). We find that the RTP is associated to anomalous values for the angular momentum of surface states, which guarantees metallic in-gap states for open boundary condition with sharply terminated hoppings; more generally for arbitrarily terminated hoppings, surface states are characterized by Berry-Zak phases that are quantized to a rational multiple of $2\pi$. The RTP adds to the family of topological invariants (the Hopf and Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning that they can be trivialized by adding a particular trivial band to either the valence or the conduction subspace. Not all trivial band additions will nullify the RTP invariant, which allows its generalization beyond two-band Hamiltonians to arbitrarily many bands; such generalization is a hallmark of symmetry-protected delicate topology.

Wave-induced fluid pressure diffusion and anelasticity in partially saturated rocks: The influences of boundary conditions

Open or partially open boundary conditions in fully or partially saturated rocks possibly occur in many realistic geologic scenarios, and it is critical to understand the influences of wave-induced fluid pressure diffusion and anelasticity for many fields of earth science and engineering applications. Under the framework of Biot’s poroelasticity, by proposing a scale factor to represent the opening degree of the boundary, we have investigated the influences of boundary conditions on the seismic wave dispersion and attenuation in partially saturated rocks. It is found that there exist two types of fluid pressure diffusion taking place simultaneously: the mesoscopic fluid pressure diffusion between the two immiscible fluids (e.g., gas and brine) and the fluid flow associated with the outer boundary. The boundary conditions significantly control the seismic wave dispersion and attenuation signatures. The coupled fluid pressure diffusion of mesoscopic flow and outer boundary flow expedite the pore pressure equilibration processes and make the characteristic frequency one order of magnitude higher than the undrained scenario. The drained and partially drained conditions also significantly decrease the velocity of partially saturated rocks, which can be even lower than that of Gassmann’s prediction and dry rock. The reliability of the theoretical model also is validated by comparing it with the numerical simulation for drained and undrained conditions. We have determined that the proposed model is capable of better interpreting the logging data of velocity-saturation relationship for a partially saturated carbonate reservoir and experimental data of wave dispersion for a fully saturated sandstone, both of which are possibly exposed to the open boundary conditions.

A comparative study of two different shallow water formulations using stable summation by parts schemes

This study provides numerical solutions to the two-dimensional linearized shallow water equations (SWE) using a high-order finite difference scheme in Summation By Parts (SBP) form. In addition to the SBP operators for the discretizations, penalty terms, Simultaneous Approximation Terms (SAT) are applied to impose well-posed open boundary conditions. The conventional SWE with height and velocities as the prognostic variables, and a new type of the vorticity–divergence SWE with wave height gradients, vorticity and divergence as the prognostic variables were investigated. It was shown that the solution in all numerical tests enter and exit the domain without instabilities. The convergence rates were correct for all orders of the SBP operators in both the entrance and exit tests. Interestingly, the error norm of the wave height were orders of magnitude lower in the vorticity–divergence solutions compared to the conventional SWE solutions.