Linear analysis of the stability of particle-laden stratified shear layers

2014 ◽  
Vol 92 (2) ◽  
pp. 103-115 ◽  
Author(s):  
Ehsan Khavasi ◽  
Bahar Firoozabadi ◽  
Hossein Afshin
Keyword(s):  
Layer Thickness ◽  
Piecewise Linear ◽  
Stability Boundary ◽  
Shear Layers ◽  
Unstable Flow ◽  
Density Profiles ◽  
Background Density ◽  
Layer Flow ◽  
Bed Slope ◽  
The Stability

Hydrodynamic instabilities at the interface of stratified shear layers could occur in various modes and have an important role in the mixing process. In this work, the linear stability analysis in the temporal framework is used to study the stability characteristics of a particle-laden stratified two-layer flow for two different background density profiles: smooth (hyperbolic tangent) and piecewise linear. The effect of parameters, such as bed slope, viscosity, and particle size, on the stability is also considered. The pseudospectral collocation method employing Chebyshev polynomials is used to solve two coupled eigenvalue equations. Based on the results, there are some differences in the stability characteristics of the two density profiles. In the case of R = 1 (R is the ratio of the shear layer thickness to the density layer thickness), the stability boundary in smooth profile is the transition from the unstable flow (where the dominant unstable mode is Kelvin–Helmholtz) to the stable one where in the piecewise linear profile this boundary is the transition from Kelvin–Helmholtz to the Holmboe mode. It is also shown that the unstable region increases with the bed slope and unstable modes amplify as the bed slope increases. For R = 5 the flow does not become stable by increasing the stratification in nonzero bed slope, and in some wavenumbers the Kelvin–Helmholtz and Holmboe modes coexist. In addition, by increasing the bed slope the growth rate of the Holmboe mode and the range of its existence decrease. As expected, the viscosity makes the current more stable, and for large values of the viscosity (small Reynolds number) the flow becomes stable at long waves (small wave numbers) for all bulk Richardson numbers. Existence of small particles does not change the instability characteristics so much, however, large particles make the flow more unstable.

Numerical studies of the stability of inviscid stratified shear flows

1972 ◽  
Vol 51 (1) ◽  
pp. 39-61 ◽  
Author(s):  
Philip Hazel
Keyword(s):  
Richardson Number ◽  
Shear Flows ◽  
Stability Boundary ◽  
Phase Speed ◽  
Neutral Curve ◽  
Free Shear Layer ◽  
Density Profiles ◽  
Complex Phase ◽  
Numerical Studies ◽  
The Stability

The infinitesimal stability of inviscid, parallel, stratified shear flows to two-dimensional disturbances is described by the Taylor-Goldstein equation. Instability can only occur when the Richardson number is less than 1/4 somewhere in the flow. We consider cases where the Richardson number is everywhere non- negative. The eigenvalue problem is expressed in terms of four parameters,Ja ‘typical’ Richardson number, α the (real) wavenumber andcthe complex phase speed of the disturbance. Two computer programs are developed to integrate the stability equation and to solve for eigenvalues: the first findscgiven α andJ, the second finds α andJwhenc≡ 0 (i.e. it computes the stationary neutral curve for the flow). This is sometimes,but not always, the stability boundary in the α,Jplane. The second program works only for cases where the velocity and density profiles are antisymmetric about the velocity inflexion point. By means of these two programs, several configurations of velocity and density have been investigated, both of the free-shear-layer type and the jet type. Calculations of temporal growth rates for particular profiles have been made.

Enhancing the absolute instability of a boundary layer by adding a far-away plate

2007 ◽  
Vol 579 ◽  
pp. 29-61 ◽  
Author(s):  
J. J. HEALEY
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Piecewise Linear ◽  
Saddle Points ◽  
Absolute Instability ◽  
Infinite Domain ◽  
Unstable Flow ◽  
Order Of Magnitude ◽  
The Absolute ◽  
Layer Flow

When a solid plate, with a boundary condition of no normal flow through it, is introduced parallel to a shear layer it is normally expected to exert a stabilizing influence on any inviscid linearly unstable waves. In this paper we present an example of an absolutely unstable boundary-layer flow that can be made more absolutely unstable by the addition of a plate parallel to the original flow and far from the boundary layer itself. In particular, the addition of the plate is found to increase the growth rate of the absolute instability of the original boundary-layer flow by an order of magnitude for long waves. This phenomenon is illustrated using piecewise-linear inviscid basic-flow profiles, for which analytical dispersion relations have been derived. Long-wave stability theories have been developed in several limits clarifying the mechanisms underlying the behaviour and establishing its generic nature. The class of flows expected to exhibit this phenomenon includes a class found recently to have an exponential growth of disturbances in the wall-normal direction, owing to the approach of certain saddle-points to certain branch-cuts in the complex-wavenumber plane. The theory also suggests that a convectively unstable flow in an infinite domain can be converted, in some circumstances, into an absolutely unstable flow when the domain is made finite by the addition of a plate, however far away the plate is.

Effects of Non-Axisymmetric Volute On Rotating Stall in the Vaneless Diffuser of Centrifugal Compressors

2022 ◽  
Author(s):  
Zitian Niu ◽  
Zhenzhong Sun ◽  
Baotong Wang ◽  
Xinqian Zheng
Keyword(s):  
Flow Field ◽  
Stability Boundary ◽  
Rotating Stall ◽  
Vaneless Diffuser ◽  
Unstable Flow ◽  
Centrifugal Compressors ◽  
Cfd Simulations ◽  
Specific Location ◽  
Stable Operating ◽  
The Stability

Abstract Rotating stall is an important unstable flow phenomenon that leads to performance degradation and limits the stability boundary in centrifugal compressors. The volute is one of the sources inducing non-axisymmetric flows in centrifugal compressors, which has an important effect on compressors' aerodynamic performance. However, the influence of volute on rotating stall is unclear. Therefore, the effects of volute on rotating stall behavior have been explored in this paper by experiments and numerical simulations. The frequency of the rotating stall captured by the experiments is 43.9% of the impeller passing frequency, while it is 44.7% of IPF calculated from the numerical results, which proves the accuracy and capability of the numerical method in this work to study the rotating stall behavior. The flow fields from CFD simulations further reveal that one stall cell initializing in a particular location deforms into several stall cells while rotating along the circumferential direction and becomes much smaller in a specific location during the evolution process, and finally, it is suppressed in another specific location as a result of the distorted flow field caused by the volute. By optimizing volute geometry to reduce the distortion of the flow field, it is expected that the rotating stall can be weakened or suppressed, which is helpful to extend the stable operating range of centrifugal compressors.

scholarly journals Earth-size planet formation in the habitable zone of circumbinary stars

2020 ◽  
Vol 494 (1) ◽  
pp. 1045-1057 ◽  
Author(s):  
G O Barbosa ◽  
O C Winter ◽  
A Amarante ◽  
A Izidoro ◽  
R C Domingos ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Planet Formation ◽  
Terrestrial Planet ◽  
Close Binary ◽  
Habitable Zone ◽  
Density Profiles ◽  
Habitable Planets ◽  
Habitable Zones ◽  
Test Particles ◽  
The Stability

ABSTRACT This work investigates the possibility of close binary (CB) star systems having Earth-size planets within their habitable zones (HZs). First, we selected all known CB systems with confirmed planets (totaling 22 systems) to calculate the boundaries of their respective HZs. However, only eight systems had all the data necessary for the computation of HZ. Then, we numerically explored the stability within HZs for each one of the eight systems using test particles. From the results, we selected five systems that have stable regions inside HZs, namely Kepler-34,35,38,413, and 453. For these five cases of systems with stable regions in HZ, we perform a series of numerical simulations for planet formation considering discs composed of planetary embryos and planetesimals, with two distinct density profiles, in addition to the stars and host planets of each system. We found that in the case of the Kepler-34 and 453 systems, no Earth-size planet is formed within HZs. Although planets with Earth-like masses were formed in Kepler-453, they were outside HZ. In contrast, for the Kepler-35 and 38 systems, the results showed that potentially habitable planets are formed in all simulations. In the case of the Kepler-413system, in just one simulation, a terrestrial planet was formed within HZ.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

Effects of Non-Axisymmetric Volute on Rotating Stall in the Vaneless Diffuser of a Centrifugal Compressor

2020 ◽  
Author(s):  
Zitian Niu ◽  
Zhenzhong Sun ◽  
Baotong Wang ◽  
Xinqian Zheng
Keyword(s):  
Numerical Simulation ◽  
Flow Field ◽  
Centrifugal Compressor ◽  
Stability Boundary ◽  
Rotating Stall ◽  
Evolution Process ◽  
Vaneless Diffuser ◽  
Unstable Flow ◽  
Centrifugal Compressors ◽  
Specific Location

Abstract Rotating stall is an important unstable flow phenomenon that leads to performance degradation and limits the stability boundary in centrifugal compressors. The volute is one of the sources to induce the non-axisymmetric flow in a centrifugal compressor, which has an important effect on the performance of compressors. However, the influence of volute on rotating stall is not clear. Therefore, the effects of volute on rotating stall by experimental and numerical simulation have been explored in this paper. It’s shown that one rotating stall cell generates in a specific location and disappears in another specific location of the vaneless diffuser as a result of the distorted flow field caused by the volute. Also, the cells cannot stably rotate in a whole circle. The frequency related to rotating stall captured in the experiment is 43.9% of the impeller passing frequency (IPF), while it is 44.7% of IPF captured by three-dimensional unsteady numerical simulation, which proves the accuracy of the numerical method in this study. The numerical simulation further reveals that the stall cell initialized in a specific location can be split into several cells during the evolution process. The reason for this is that the blockage in the vaneless diffuser induced by rotating stall is weakened by the mainstream from the impeller exit to make one initialized cell disperse into several ones. The volute has an important influence on the generation and evolution process of the rotating stall cells of compressors. By optimizing volute geometry to reduce the distortion of the flow field, it is expected that rotating stall can be weakened or suppressed, which is helpful to widen the operating range of centrifugal compressors.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

scholarly journals Flexural-Torsional Vibration of Thin-Walled Beams Subjected to Combined Initial Axial Load and End Bending Moment: Application to the Design of Saw Tooth Blades

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Van Binh Phung ◽  
Anh Tuan Nguyen ◽  
Hoang Minh Dang ◽  
Thanh-Phong Dao ◽  
V. N. Duc
Keyword(s):  
Boundary Conditions ◽  
Axial Load ◽  
Stability Boundary ◽  
Bending Moment ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Rayleigh Method ◽  
Thin Walled ◽  
The Stability ◽  
Fixed End

The present paper analyzes the vibration issue of thin-walled beams under combined initial axial load and end moment in two cases with different boundary conditions, specifically the simply supported-end and the laterally fixed-end boundary conditions. The analytical expressions for the first natural frequencies of thin-walled beams were derived by two methods that are a method based on the existence of the roots theorem of differential equation systems and the Rayleigh method. In particular, the stability boundary of a beam can be determined directly from its first natural frequency expression. The analytical results are in good agreement with those from the finite element analysis software ANSYS Mechanical APDL. The research results obtained here are useful for those creating tooth blade designs of innovative frame saw machines.

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