scholarly journals Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders

2014 ◽  
Vol 740 ◽  
pp. 278-311 ◽  
Author(s):  
Joël Tchoufag ◽  
David Fabre ◽  
Jacques Magnaudet
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Stokes Equations ◽  
Coupled System ◽  
The Body ◽  
Navier Stokes ◽  
Fluid Inertia ◽  
Stability Diagrams ◽  
Wake Instability ◽  
The Stability

AbstractThe stability of the vertical path of a gravity- or buoyancy-driven disk of arbitrary thickness falling or rising in a viscous fluid, recently studied through direct numerical simulation by Auguste, Magnaudet & Fabre (J. Fluid Mech., vol. 719, 2013, pp. 388–405), is investigated numerically in the framework of global linear stability. The disk is allowed to translate and rotate arbitrarily and the stability analysis is carried out on the fully coupled system obtained by linearizing the Navier–Stokes equations for the fluid and Newton’s equations for the body. Three disks with different diameter-to-thickness ratios are considered: one is assumed to be infinitely thin, the other two are selected as archetypes of thin and thick cylindrical bodies, respectively. The analysis spans the whole range of body-to-fluid inertia ratios and considers Reynolds numbers (based on the fall/rise velocity and body diameter) up to $350$. It reveals that four unstable modes with an azimuthal wavenumber $m=\pm 1$ exist in each case. Three of these modes result from a Hopf bifurcation while the fourth is associated with a stationary bifurcation. Varying the body-to-fluid inertia ratio yields rich and complex stability diagrams with several branch crossings resulting in frequency jumps; destabilization/restabilization sequences are also found to take place in some subdomains. The spatial structure of the unstable modes is also examined. Analyzing differences between their real and imaginary parts (which virtually correspond to two different instants of time in the dynamics of a given mode) allows us to assess qualitatively the strength of the mutual coupling between the body and fluid. Qualitative and quantitative differences between present predictions and known results for wake instability past a fixed disk enlighten the fact that the first non-vertical regimes generally result from an intrinsic coupling between the body and fluid and not merely from the instability of the sole wake.

Numerical Modelling of Hydrodynamic Instabilities in Supercritical Fluids

2021 ◽  
pp. 32-54
Author(s):  
Sakir Amiroudine
Keyword(s):  
Boundary Layer ◽  
Thermal Boundary Layer ◽  
Stokes Equations ◽  
Gravitational Instability ◽  
Heat Diffusion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stability Diagrams ◽  
The Stability ◽  
Rayleigh Benard

The case of a supercritical fluid heated from below (Rayleigh-Bénard) in a rectangular cavity is first presented. The stability of the two boundary layers (hot and cold) is analyzed by numerically solving the Navier-Stokes equations with a van der Waals gas and stability diagrams are derived. The very large compressibility and the very low heat diffusivity of near critical pure fluids induce very large density gradients which lead to a Rayleigh–Taylor-like gravitational instability of the heat diffusion layer and results in terms of growth rates and wave numbers are presented. Depending on the relative direction of the interface or the boundary layer with respect to vibration, vibrational forces can destabilize a thermal boundary layer, resulting in parametric/Rayleigh vibrational instabilities. This has recently been achieved by using a numerical model which does not require any equation of state and directly calculates properties from NIST data base, for instance.

2D and 3D Stability of Cavity Flows in High Mach Number Regimes

2019 ◽  
Author(s):  
Parshwanath S. Doshi ◽  
Rajesh Ranjan ◽  
Datta V. Gaitonde
Keyword(s):  
Stability Analysis ◽  
Mach Number ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Cavity Flows ◽  
Flame Holder ◽  
High Mach Number ◽  
Eddy Simulation ◽  
High Mach ◽  
The Stability

Abstract The stability characteristics of an open cavity flow at very high Mach number are examined with BiGlobal stability analysis based on the eigenvalues of the linearized Navier-Stokes equations. During linearization, all possible first-order terms are retained without any approximation, with particular emphasis on extracting the effects of compressibility on the flowfield. The method leverages sparse linear algebra and the implicitly restarted shift-invert Arnoldi algorithm to extract eigenvalues of practical physical consequence. The stability dynamics of cavity flows at four Mach numbers between 1.4 and 4 are considered at a Reynolds number of 502. The basic states are obtained through Large Eddy Simulation (LES). Frequency results from the stability analysis show good agreement when compared to the theoretical values using Rossiter’s formula. An examination of the stability modes reveals that the shear layer is increasingly decoupled from the cavity as the Mach number is increased. Additionally, the outer lobes of the Rossiter modes are observed to get stretched and tilted in the direction of the freestream. Future efforts will extend the present analysis to examine current and potential cavity flame holder configurations, which often have downstream walls inclined to the vertical.

Non-parallel linear stability analysis of the vertical boundary layer in a differentially heated cavity

1997 ◽  
Vol 352 ◽  
pp. 265-281 ◽  
Author(s):  
A. M. H. BROOKER ◽  
J. C. PATTERSON ◽  
S. W. ARMFIELD
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Navier Stokes ◽  
Parabolized Stability Equations ◽  
The Stability ◽  
Flow Quantity ◽  
Energy Equations ◽  
Made In

A non-parallel linear stability analysis which utilizes the assumptions made in the parabolized stability equations is applied to the buoyancy-driven flow in a differentially heated cavity. Numerical integration of the complete Navier–Stokes and energy equations is used to validate the non-parallel theory by introducing an oscillatory heat input at the upstream end of the boundary layer. In this way the stability properties are obtained by analysing the evolution of the resulting disturbances. The solutions show that the spatial growth rate and wavenumber are highly dependent on the transverse location and the disturbance flow quantity under consideration. The local solution to the parabolized stability equations accurately predicts the wave properties observed in the direct simulation whereas conventional parallel stability analysis overpredicts the spatial amplification and the wavenumber.

scholarly journals Stability Analysis of Symmetrical Rotors Partially Filled with a Viscous Incompressible Fluid

2001 ◽  
Vol 7 (5) ◽  
pp. 301-310 ◽  
Author(s):  
Zhu Changsheng
Keyword(s):  
Stability Analysis ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Damping Ratio ◽  
Navier Stokes ◽  
Fluid Viscosity ◽  
Navier Stokes Equations ◽  
Rotor Systems ◽  
The Stability

On the basis of the linearized fluid forces acting on the rotor obtained directly by using the two-dimensional Navier-Stokes equations, the stability of symmetrical rotors with a cylindrical chamber partially filled with a viscous incompressible fluid is investigated in this paper. The effects of the parameters of rotor system, such as external damping ratio, fluid fill ratio, Reynolds number and mass ratio, on the unstable regions are analyzed. It is shown that for the stability analysis of fluid filled rotor systems with external damping, the effect of the fluid viscosity on the stability should be considered. When the fluid viscosity is included, the adding external damping will make the system more stable and two unstable regions may exist even if rotors are isotropic in some casIs.

Analysis of the dripping–jetting transition in compound capillary jets

2010 ◽  
Vol 649 ◽  
pp. 523-536 ◽  
Author(s):  
M. A. HERRADA ◽  
J. M. MONTANERO ◽  
C. FERRERA ◽  
A. M. GAÑÁN-CALVO
Keyword(s):  
Stability Analysis ◽  
Convective Instability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Outer Interface ◽  
Spatio Temporal ◽  
Capillary Jets ◽  
The Stability

We examine the behaviour of a compound capillary jet from the spatio-temporal linear stability analysis of the Navier–Stokes equations. We map the jetting–dripping transition in the parameter space by calculating the Weber numbers for which the convective/absolute instability transition occurs. If the remaining dimensionless parameters are set, there are two critical Weber numbers that verify Brigg's pinch criterion. The region of absolute (convective) instability corresponds to Weber numbers smaller (larger) than the highest value of those two Weber numbers. The stability map is affected significantly by the presence of the outer interface, especially for compound jets with highly viscous cores, in which the outer interface may play an important role even though it is located very far from the core. Full numerical simulations of the Navier–Stokes equations confirm the predictions of the stability analysis.

Stability Analysis for Complex Rotational Flow

2021 ◽  
Vol 16 ◽  
pp. 62-72
Author(s):  
Ivan V. Kazachkov
Keyword(s):  
Stability Analysis ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Rotational Flow ◽  
Complex Flow ◽  
Navier Stokes Equations ◽  
Coriolis Forces ◽  
Cylindrical Coordinate ◽  
The Stability ◽  
Varying Mass

Based on the earlier developed mathematical model of the complex flow due to the double rotations in two perpendicular directions, the stability analysis is performed in the paper. The Navier-Stokes equations are derived in the coordinate system rotating around the two perpendicular different axes, the vertical one of them is arranged on some distance from the other axis of rotation, the horizontal axis is directed along the tangential line to the circle of the vertical rotation. The two centrifugal and Coriolis forces create the unique features in high oscillating flow, with localities of the stretched liquid, due to their action varying by the circumferential cylindrical coordinate in the channel flow. Stability analysis for the complex rotational flow under double rotations creating strongly varying mass forces and stretching of the liquid is considered at first

Viscous flow between two moving parallel disks: exact solutions and stability analysis

2002 ◽  
Vol 464 ◽  
pp. 209-215 ◽  
Author(s):  
S. N. ARISTOV ◽  
I. M. GITMAN
Keyword(s):  
Exact Solution ◽  
Stability Analysis ◽  
Incompressible Liquid ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Viscous Incompressible Liquid ◽  
Navier Stokes Equations ◽  
Parallel Disks ◽  
Different Types ◽  
The Stability

The motion of a viscous incompressible liquid between two parallel disks, moving towards each other or in opposite directions, is considered. The description of possible conditions of motion is based on the exact solution of the Navier–Stokes equations. Both stationary and transient cases have been considered. The stability of the motion is analysed for different initial perturbations. Different types of stability were found according to whether the disks moved towards or away from each other.

scholarly journals Stability of bedforms in laminar flows with free surface: from bars to ripples

2009 ◽  
Vol 642 ◽  
pp. 329-348 ◽  
Author(s):  
O. DEVAUCHELLE ◽  
L. MALVERTI ◽  
É. LAJEUNESSE ◽  
P.-Y. LAGRÉE ◽  
C. JOSSERAND ◽  
...  
Keyword(s):  
Free Surface ◽  
Stability Analysis ◽  
Stokes Equations ◽  
Transport Model ◽  
Bedload Transport ◽  
Laminar Flows ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Stability

The present paper is devoted to the formation of sand patterns by laminar flows. It focuses on the rhomboid beach pattern, formed during the backswash. A recent bedload transport model, based on a moving-grains balance, is generalized in three dimensions for viscous flows. The water flow is modelled by the full incompressible Navier–Stokes equations with a free surface. A linear stability analysis then shows the simultaneous existence of two distinct instabilities, namely ripples and bars. The comparison of the bar instability characteristics with laboratory rhomboid patterns indicates that the latter could result from the nonlinear evolution of unstable bars. This result, together with the sensibility of the stability analysis with respect to the parameters of the transport law, suggests that the rhomboid pattern could help improving sediment transport models, so critical to geomorphologists.

Numerical Modelling of Hydrodynamic Instabilities in Supercritical Fluids

2017 ◽  
pp. 33-54
Author(s):  
Sakir Amiroudine
Keyword(s):  
Boundary Layer ◽  
Thermal Boundary Layer ◽  
Stokes Equations ◽  
Gravitational Instability ◽  
Heat Diffusion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stability Diagrams ◽  
The Stability ◽  
Rayleigh Benard

The case of a supercritical fluid heated from below (Rayleigh-Bénard) in a rectangular cavity is first presented. The stability of the two boundary layers (hot and cold) is analyzed by numerically solving the Navier-Stokes equations with a van der Waals gas and stability diagrams are derived. The very large compressibility and the very low heat diffusivity of near critical pure fluids induce very large density gradients which lead to a Rayleigh–Taylor-like gravitational instability of the heat diffusion layer and results in terms of growth rates and wave numbers are presented. Depending on the relative direction of the interface or the boundary layer with respect to vibration, vibrational forces can destabilize a thermal boundary layer, resulting in parametric / Rayleigh vibrational instabilities. This has recently been achieved by using a numerical model which does not require any equation of state and directly calculates properties from NIST data base (NIST, 2000) for instance.

Advanced Numerical Methods for the Prediction of Tonal Noise in Turbomachinery—Part I: Implicit Runge–Kutta Schemes

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Graham Ashcroft ◽  
Christian Frey ◽  
Kathrin Heitkamp ◽  
Christian Weckmüller
Keyword(s):  
Stokes Equations ◽  
Temporal Integration ◽  
Aerodynamic Noise ◽  
Navier Stokes ◽  
Noise Generation ◽  
Academic Problems ◽  
Tonal Noise ◽  
Navier Stokes Equations ◽  
Runge Kutta ◽  
The Stability

This is the first part of a series of two papers on unsteady computational fluid dynamics (CFD) methods for the numerical simulation of aerodynamic noise generation and propagation. In this part, the stability, accuracy, and efficiency of implicit Runge–Kutta schemes for the temporal integration of the compressible Navier–Stokes equations are investigated in the context of a CFD code for turbomachinery applications. Using two model academic problems, the properties of two explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) schemes of second- and third-order accuracy are quantified and compared with more conventional second-order multistep methods. Finally, to assess the ESDIRK schemes in the context of an industrially relevant configuration, the schemes are applied to predict the tonal noise generation and transmission in a modern high bypass ratio fan stage and comparisons with the corresponding experimental data are provided.

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