Dispersion Relation Equation Research Articles

Suppression of Azimuthal Combustion Instability Using Perforated Liner Under Symmetry Breaking

Symmetry breaking occurring in annular combustors shows different stability patterns due to the formation of nondegenerate modes with different frequencies. This also brings about more profound complexity for control methodology. This paper presents a three-dimensional analytical model to investigate how to control nondegenerate azimuthal modes by modifying the wall boundary conditions of a multiburner annular combustor. The stability of the system can be evaluated by solving the eigenvalue of the dispersion relation equation derived in this study. Results show that changes in the parameters of the perforated liner cause different performance in nondegenerate modes due to frequency splitting. Meanwhile, the asymmetric perforated liner alters the original asymmetry of the annular combustor, inducing a shift in the nature of nondegenerate modes when significant symmetry breaking exists. Furthermore, variations of pressure amplitude in the backing cavity lead to diverse acoustic energy dissipation and even distinct stability patterns under two types of azimuthally nonuniformly perforated liner. Overall, this paper presents a new methodology for suppressing nondegenerate azimuthal modes under symmetry breaking by utilizing a perforated liner in an annular combustor. Additionally, a clear understanding of the evolution behavior of nondegenerate modes is also provided by studying their stability behavior and energy dissipation.

Exact dispersion relation of the quantum Langmuir wave.

The normal modes, i.e., the eigensolutions to the dispersion relation equation, are the most fundamental properties of a plasma. The real part indicates the intrinsic oscillation frequency while the imaginary part the Landau damping rate. In most of the literature, the normal modes of quantum plasmas are obtained by means of small damping approximation, which is invalid for high-k modes. In this paper, we solve the exact dispersion relations via the analytical continuation scheme, and, due to the multi-value nature of the Fermi-Dirac distribution, reformation of the complex Riemann surface is required. It is found that the topological shape of the root locus in quantum plasmas is quite different from classical ones, in which both real and imaginary frequencies of high-k modes increase with k steeper than the typical linear behavior in classical plasmas. As a result, the time-evolving behavior of a high-k initial perturbation becomes ballistic-like in quantum plasmas.

Instability of viscous annular liquid film jet under high air-to-fluid velocity rate

Based on the surface wave dispersion relation equation of viscous annular liquid film jet established by the research group in the early stage, a dispersion relation equation calculation program is written according to Mathematica. A linear stability analysis is carried out for the high air-fluid velocity ratio viscous annular liquid film jet entering a compressible gas medium environment. The relationship between the surface wave growth rate of an annular liquid film jet and the surface wave number under different characteristic parameters is obtained. The connection based on the dominant wave’s growth rate or the dominant wave’s number with dimensionless parameters such as air-fluid velocity rate, air-fluid density rate, Weber number of liquid, liquid Reynolds number, and gas Mach number was analyzed. The consequences reveal that under the high air-fluid velocity ratio, the increase of the above dimensionless parameters promotes the fragmentation course of the annular liquid film jet. Moreover, the augment of the air-fluid velocity rate has the greatest influence on the stability of the viscous annular liquid film jet. In contrast, the Reynolds number of the liquid has less influence than the reliability of the annular liquid film jet. Linear stability analysis under this condition lays the foundation for the prediction model of the most unstable parameters of annular liquid film jet in practical engineering applications.

Theoretical model of azimuthal combustion instability subject to non-trivial boundary conditions

The problem of evaluating the sensitivity of non-trivial boundary conditions to the onset of azimuthal combustion instability is a longstanding challenge in the development process of modern gas turbines. The difficulty lies in how to describe three-dimensional in- and outlet boundary conditions in an artificial computational domain. To date, the existing analytical models have still failed to quantitatively explain why the features of the azimuthal combustion instability of a combustor in laboratory environment are quite different from that in a real gas turbine, making the stability control devices developed in laboratory generally lose the effectiveness in practical applications. To overcome this limitation, we provide a novel theoretical framework to directly include the effect of non-trivial boundary conditions on the azimuthal combustion instability. A key step is to take the non-trivial boundary conditions as equivalent distributed sources so as to uniformly describe the physical characteristics of the inner surface in an annular enclosure along with different in- and outlet configurations. Meanwhile, a dispersion relation equation is established by the application of three-dimensional Green’s function approach and generalized impedance concept. Results show that the effects of the generalized modal reflection coefficients on azimuthal unstable modes are extremely prominent, and even prompt the transition from stable to unstable mode, thus reasonably explaining why the thermoacoustic instability phenomena in a real gas turbine are difficult to observe in an isolated combustion chamber. Overall, this work provides an effective tool for analysis of the azimuthal combustion instability including various complicated boundary conditions.

Prediction of combustion instabilities based on the Green's function in a three-dimensional annular combustor with multiple flames

The prediction of combustion instabilities is essential for modern annular combustors fed by multiple flames. This paper develops a model based on Green's function to study the interaction of multiple flame responses with different types in a three-dimensional (3D) geometry. The flame responses are described by flame transfer functions and are recognized as monopole sources in 3D cavities. Then the dispersion relation equation in the form of matrix is performed with the condition of equality of pressure. Thermoacoustic complex frequencies, mode structure, as well as the stability behaviors in combustor can be obtained in this model and the analysis is accomplished using Rayleigh index. The results are compared to the values of existing theoretical models and a good agreement is achieved. Furthermore, the effects of non-identical flame responses on complicated slanted mode under the condition of asymmetric are studied. In addition, the combustion instabilities of radial mode can be predicted by this model when the radial position distributions of the heat sources move in the case of small ratio of inner wall to outer wall.

Temporal instability of confined three-dimensional liquid jet with heat and mass transfer under longitudinal acoustic oscillations

The temporal instability of a confined viscous liquid jet surrounded by high-speed co-flowing viscous gas phase is studied in this work. The effect of the longitudinal acoustic oscillations, which is regarded as gas axial velocity oscillations, is also considered. The heat and mass transfer is characterized by the ratio between conduction heat flux and the evaporation heat flux; then, an explicit dispersion relation equation is obtained. The results suggest that more than one unstable region appears because of the gas velocity oscillations, including Kelvin–Helmholtz (K–H) instability and parametric instability regions. Increasing the forcing frequency enhances the K–H instability, while it has a stabilizing effect on the parametric instability. In addition, the liquid jet tends to be more unstable in non-axisymmetric modes when the gas rotating strength is strong. Although the gas viscosity has a destabilizing effect on the gas–liquid interface, the destabilizing effect is weak due to the low viscosity of the gas phase. According to the linear instability theory, the dominant wavenumber will locate in the most unstable region. Moreover, the parametric instability in non-axisymmetric modes may be observable when the Weber number is large.

Modeling a Wave on Mild Sloping Bottom Topography and Its Dispersion Relation Approximation

Linear wave theory is a simple theory that researchers and engineers often use to study a wave in deep, intermediate, and shallow water regions. Many researchers mostly used it over the horizontal flat seabed, but in actual conditions, sloping seabed always exists, although mild. In this research, we try to model a wave over a mild sloping seabed by linear wave theory and analyze the influence of the seabed’s slope on the solution of the model. The model is constructed from Laplace and Bernoulli equations together with kinematic and dynamic boundary conditions. We used the result of the analytical solution to find the relation between propagation speed, wavelength, and bed slope through the dispersion relation. Because of the difference in fluid dispersive character for each water region, we also determined dispersion relation approximation by modifying the hyperbolic tangent form into hyperbolic sine-cosine and exponential form, then approximated it with Padé approximant. As the final result, exponential form modification with Padé approximant had the best agreement to exact dispersion relation equation then direct hyperbolic tangent form.

Temporal Instability of Liquid Jet in Swirling Gas with Axial Velocity Oscillation

In the present study, viscous potential flow theory is employed to investigate the temporal instability of a viscous liquid jet in swirling gas with axial velocity oscillation. The dispersion-relation equation is obtained between dimensionless growth rate and wave number. Because of the axial velocity oscillation of gas, there are more than one instability regions, including Kelvin–Helmholtz (K–H) instability region and parametric instability region. The results suggest that increasing gas swirling strength stabilizes the liquid jet in axisymmetric mode, while it enhances the instability in nonaxisymmetric modes. And the gas viscosity has a destabilizing effect. In addition, increasing the forcing frequency enhances instability in the K–H instability region. However, it has a stabilizing effect in the parametric instability region. Furthermore, increasing oscillation amplitude enhances instability in both the K–H instability region and the parametric instability region. Because of the competition between the parametric and K–H instabilities, the location of dominant wave number is uncertain.

Effect of nonlinear flame response on the design of perforated liners in suppression of combustion instability

This paper aims at investigating how to suppress combustion instability using perforated liners under nonlinear flame response. A theoretical model is thus presented to describe the interaction between the unsteady heat release, acoustics, and fluid fields, which takes both perforated liner and flame nonlinearity into account. To establish an eigenvalue problem to evaluate combustion instability, appropriate matching and boundary conditions are set up by applying conservation laws across the interfaces, consequently deriving the dispersion relation equation in the form of a matrix. In view of the nonlinearity of flame response function, it is found that the control strategy of combustion instability should be devoted to the reduction of the limit cycle amplitude to an acceptable level rather than the complete elimination of its onset. Based on the present model, we have studied the first longitudinal mode of a Rijke tube equipped with perforated liners. It is noted that the change of combustion instability evolution is sensitive to the design parameters of perforated liners, including cavity depth, installation position, and bias flow velocity. On the one hand, it is indeed possible to suppress the onset of combustion instability with elaborately designed parameters of the perforated liners. On the other hand, the limit cycle amplitude can be reduced as small as possible under various design restrictions of perforated liners in practice. Therefore, this study presents an alternative design criteria of perforated liners for the control of combustion instability.

Instability of a tangential discontinuity surface in a three-dimensional compressible medium

Compressible flows appear in many natural and technological processes, for instance, the flow of natural gases in a pipe system. Thus, a detailed study of the stability of tangential velocity discontinuity in compressible media is relevant and necessary. The first early investigation in two-dimensional (2D) media was given more than 70 years ago. In this article, we continue investigating the stability in three-dimensional (3D) media. The idealized statement of this problem in an infinite spatial space was studied by Syrovatskii in 1954. However, the omission of the absolute sign of cos θ with θ being the angle between vectors of velocity and wave number in a certain inequality produced the inaccurate conclusion that the flow is always unstable for entire values of the Mach number M. First, we revisit this case to arrive at the correct conclusion, namely that the discontinuity surface is stabilized for a large Mach number with a given value of the angle θ. Next, we introduce a real finite spatial system such that it is bounded by solid walls along the flow direction. We show that the discontinuity surface is stable if and only if the dispersion relation equation has only real roots, with a large value of the Mach number; otherwise, the surface is always unstable. In particular, we show that a smaller critical value of the Mach number is required to make the flow in a narrow channel stable.

Linear theory of current driven ion acoustic instability at the distance 0.3 to 1 AU from the Sun

Several kinds of instabilities are often observed in the solar wind. We studied the linear theory of the current-driven electrostatic ion acoustic instability at frequencies near the ion cyclotron frequency at distances from the Sun over 0.3 to 1 AU. The properties of current driven electrostatic ion-acoustic instability are investigated. The instability parameters, such as angular wave frequency, the critical drift velocity and the growth rate are calculated. We used the dispersion relation in hot magnetized plasma to calculate the current driven ion acoustic instability properties. To determine the critical drift velocity and the growth rate a full numerical solution of the dispersion relation equation is carried out. Since the growth rate of instability depends on the wave normal angle, the drift velocity of the electron along the magnetic field and the ion thermal cyclotron radius, the calculation is carried out for ranging of the wave normal angles, drift velocities and ion thermal cyclotron radius. The results show that the growth rate of instability increases with increasing distance from the Sun. The growth rate of instability increases with decreasing the wave normal angle, but the critical drift velocity decreases with decreasing the wave normal angle.

Solitary waves and double layers in an inhomogeneous electronegative plasma with heavier negative ions

The linear dispersion relation and partial differential equations are derived for the electronegative plasma with heavier negative ions embedded in an ambient magnetic field. The appropriate transformation is used to study the nonlinear structures without introducing multi-scale expansion of various quantities. The nonlinear partial differential equation obtained sustains nonlinear structures like solitons and double layers. All the linear waves and the nonlinear structures like solitons and double layers are found to be modified predominantly under the influence of superthermal particles, density inhomogeneity, electron populations, and positive to negative heavy ion ratios.

An Explosive Instability of Kelvin-Helmholtz Flows in Ferrofluids: Effect of Periodic Tangential Magnetic Field With Rotation

Objectives: The work is to learn the impingement of tangential periodic magnetic field on the interface in the middle of two superposed ferrofluids with the rotation through a nonlinear perturbation analysis. Methods or Statistical Analysis: A nonlinear instability due to streaming superposed ferrofluids stressed by rotation and a periodic tangential magnetic field is examined by employing the mode of multiple scales. The fluids are taken to be incompressible. Using the boundary conditions, the solutions of the linearized equations of motion lead to the linear dispersion equation. Findings: From this work, it is clearly seen the unfolding of amplitude of waves was influenced by an equation of nonlinear Schrodinger. Applications or Improvements: The rotation and a periodic tangential magnetic field have enormous important effects over the stability of the system. Keywords: Dispersion Relation and Schrodinger Equation, Ferrofluids, Linear Dispersion Relation, Method of Multiple Scales, Rotation

Dispersion property of electromagnetic wave in 1D magnetized ferrites photonic crystals for TE mode in longitudinal magnetization configuration

In this article, properties of dispersion characteristics and phase index of magnetized one dimensional ferrite photonic crystals are investigated for transverse electric mode when external magnetic field is along the propagation direction. The dispersion relation and phase index equation for electromagnetic wave propagating in the considered structure is obtained by using transfer matrix method. It is found that in this configuration, two Eigen modes viz. LCP and RCP waves are obtained. The effect of external magnetic field, filling factor and normalized parallel wave vector β on dispersion and phase index are investigated. It is found that filling factor, external magnetic field and β have strong influence on the dispersion and phase index for LCP wave. For RCP wave, dispersion and phase index do not show external field dependence. With increase in external magnetic fields, the first order PBGs is shifted towards high frequency domain for LCP wave whereas for RCP wave there is no effect of external magnetic field on dispersion. With increase of β at constant external magnetic field and filling factor, thickness of PBGs increases for both LCP & RCP waves. PBGs occur in the form of lobes and the number of lobes decreases with increase in β values in the same frequency domain. For same f-value, the magnitude of ɸ is lower for LCP wave as compared to RCP wave.

Kelvin–Helmholtz instability of confined Oldroyd-B liquid film with heat and mass transfer

This paper investigates the Kelvin–Helmholtz instability of an annular viscoelastic liquid film confined in a tube with a hotter, central viscous gas when the heat and mass transfer, which plays a significant role in determining flow instability and injector spraying characteristics, are taken into considered. The liquid is assumed to satisfy the Oldroyd-B model. The non-dimensional dispersion relation equation between the dimensionless growth rate and the wave number is obtained by linear stability analysis. In this work, heat and mass transfer are characterized by the heat transfer flux ratio of conduction-to-evaporation heat transfer at the interface. Maximum growth rate and range of unstable wave number both increases with an increase in heat transfer flux ratio of conduction-to-evaporation heat transfer. When the Reynolds number of the gas increases, the maximum growth rate also increases while the range of the unstable wave number is lowered. The effects of other non-dimensional parameters at the gas-to-liquid interface are also discussed.

Rigorous electromagnetic theory for waveguide evanescent field fluorescence microscopy.

Recently, waveguide evanescent field fluorescence (WEFF) microscopy was introduced and used to image and analyze cell-substrate contacts. Here, we establish a comprehensive electromagnetic theory in a seven-layer structure as a model for a typical waveguide-cell structure appropriate for WEFF microscopy and apply it to quantify cell-waveguide separation distances. First, electromagnetic fields at the various layers of a model waveguide-cell system are derived. Then, we obtain the dispersion relation or characteristic equation for TE modes with a stratified media as a cover. Waveguides supporting a defined number of modes are theoretically designed for conventional, reverse, and symmetric waveguide structures and then various waveguide parameters and the penetration depths of the evanescent fields are obtained. We show that the penetration depth of the evanescent field in a three-layer waveguide-cell structure is always lower than that of a seven-layer structure. Using the derived electromagnetic fields, the background and the excited fluorescence in the waveguide-cell gap, filled with water-soluble fluorophores, are analytically formulated. The effect of the waveguide structures on the fluorescence and the background are investigated for various modes. Numerical results are presented for the background and the stimulated fluorescence as functions of the water gap width for various waveguide structures, which can be used to find the water gap width. The results indicate that the background and excited fluorescence increase by increasing the penetration depth of the evanescent field. In addition, we show that for various guided modes of a conventional waveguide, the electric fields in the cell membrane and the cytoplasm are evanescent and they do not depend on the waveguide structure and the mode number. However, for the reverse symmetry and symmetric waveguide structures, the waves are sinusoidal in the cell membrane and the cytoplasm for the highest-order modes.

High-performance sensor achieved by hybrid guide-mode resonance/surface plasmon resonance platform.

We perform a comprehensive analysis of multiband absorption properties in a metal-dielectric-metal-dielectric (MDMD) nanostructure under TM wave illumination. The multiband absorption can be attributed to the hybridization of the surface plasmon resonance (SPR) and the guide-mode resonance (GMR), and we identify the hybrid GMR/SPR by the dispersion relation equations of the SPR and GMR, respectively. More importantly, the MDMD nanostructure is very sensitive to the change of the dielectric environment for the special hybrid structure; thus, it can function as a good candidate for ultrasensitive biochemical sensing. The highest sensitivity of the MDMD nanostructure reaches 1087nm/RIU with the figure of merit (FoM) of 23 and the new figure of merit (FoM*) of 483; it is performed by the absorption peak at 1796.1nm of the double surface plasmon polariton with the strongest field enhancement at the surface.

Wind-induced internal seiches in Vossoroca reservoir, PR, Brazil

ABSTRACT The vertical movements caused by internal waves in lakes and reservoirs have chemical and biological consequences for these ecosystems. The vast majority of studies that investigate internal waves are conducted on large lakes. There are just few researches that investigate this phenomenon on dendritic reservoirs. The purpose of this research was to identify internal waves (baroclinic mode) in the Vossoroca reservoir by using temperature time series recorded between May to November 2012. A two-layer method was used which considered rigid upper and lower boundaries. Moreover, the potential flow theory was used for both layers since the flow within each layer was considered irrotational. From the dispersion relation, we obtained the theoretical shallow internal wave period. The power spectral density (PSD) of temperature series of thermocline depth, provided by fast Fourier transform, helped in the identification on the frequency peak. Subsequently, the theoretical period was compared with the frequency spectra. Using a careful analysis (excluding the interference of solar radiation and intensity of wind), we observed a clear peak in November due to an internal wave with period around 8 hours, which matched the theoretical calculation from the dispersion relation equation for V1H1 mode. Weak winds from southwest excited a V1H1 baroclinic mode. According to spectral analysis, after the passage of this long-basin internal seiches, we identified the formation of higher vertical internal seiche modes. In addition, we observe indications of V1H1 mode degeneration.

Development of a symplectic and phase error reducing perturbation finite-difference advection scheme

ABSTRACTThe aim of this work is to develop a new scheme for solving the pure advection equation. This scheme formulated within the perturbation finite-difference context not only conserves symplecticity but also preserves the numerical dispersion relation equation. The employed symplectic integrator of second-order accuracy in time enables calculation of a long-time accurate solution in the sense that the Hamiltonian is conserved at all times. The generalized high-order spatially accurate perturbation difference scheme optimizes numerical phase accuracy through the minimization of the difference between the numerical and exact dispersion relation equations. Our proposed new class of phase error reducing perturbation difference schemes can in addition locally capture discontinuities underlying the concept of applying a shope/flux limiter. The high-order spatial accuracy can be recovered in a smooth region. Besides the Fourier analysis of the discretization errors, anisotropy and dispersion analyses are both conducted on the dispersion-relation and symplecticity-preserving pure advection scheme to shed light on the distinguished nature of the proposed scheme. Numerical tests are carried out and the results compare well with the exact solutions, demonstrating the efficiency, accuracy, and the discontinuity-resolving ability using the proposed class of high-resolution perturbation finite-difference schemes.

Dispersion relation equation preserving FDTD method for nonlinear cubic Schrödinger equation

In this study we aim to solve the cubic nonlinear Schrödinger (CNLS) equation by the method of fractional steps. Over a time step from tn to tn+1, the linear part of the Schrödinger equation is solved firstly through four time integration steps. In this part of the simulation, the explicit symplectic scheme of fourth order accuracy is adopted to approximate the time derivative term. The second-order spatial derivative term in the linear Schrödinger equation is approximated by centered scheme. The resulting symplectic and space centered difference scheme renders an optimized numerical dispersion relation equation. In the second part of the simulation, the solution of the nonlinear equation is computed exactly thanks to the embedded invariant nature within each time increment. The proposed semi-discretized difference scheme underlying the modified equation analysis of second kind and the method of dispersion error minimization has been assessed in terms of the spatial modified wavenumber or the temporal angular frequency resolution. Several problems have been solved to show that application of this new finite difference scheme for the calculation of one- and two-dimensional Schrödinger equations can deemed conserve Hamiltonian quantities and preserve dispersion relation equation (DRE).