A finite element formulation for global linear stability analysis of a nominally two-dimensional base flow

2014 ◽  
Vol 75 (4) ◽  
pp. 295-312 ◽  
Author(s):  
Sanjay Mittal ◽  
Sidharth GS ◽  
Abhishek Verma
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Base Flow ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Two Dimensional

scholarly journals Linear stability of confined flow around a 180-degree sharp bend

2017 ◽  
Vol 822 ◽  
pp. 813-847 ◽  
Author(s):  
Azan M. Sapardi ◽  
Wisam K. Hussam ◽  
Alban Pothérat ◽  
Gregory J. Sheard
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Base Flow ◽  
Two Dimensional ◽  
Two Dimensional Flow ◽  
The Stability

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

scholarly journals A Discrete Linear Stability Analysis Framework for Two-Dimensional Laminar Flows

2021 ◽  
Vol 16 ◽  
pp. 34-42
Author(s):  
Nitin Kumar ◽  
Sunil Chamoli ◽  
Sachin Tejyan ◽  
Pawan Kumar Pant
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Rayleigh Number ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Laminar Flows ◽  
Base Flow ◽  
Two Dimensional ◽  
Analysis Framework ◽  
Square Cylinder

A discrete linear stability analysis framework for two-dimensional laminar flows is presented. Using two case studies involving analysis of thermal and laminar flows, the stability of flows in the discrete numerical sense is addressed. The two-dimensional base flow for various values of the controlling parameter (Reynolds number for flow past a square cylinder and Rayleigh number for double-glazing problem) is computed numerically by using the lattice Boltzmann method. The governing equations, discretized using the finitedifference method in two-dimensions and are subsequently written in the form of perturbed equations with twodimensional disturbances. These equations are linearized around the base flow and form a set of partial differential equations that govern the evolution of the perturbations. The eigenvalues, stability of the base flow and the points of bifurcations are determined using normal mode analysis. The eigenvalue spectrum predicts that the critical Reynolds number is 52 and the critical Rayleigh number is 6 1.88×10 for the square cylinder and double-glazing problem, respectively, The results are consistent with the previous numerical and experimental observations.

scholarly journals A Discrete Linear Stability Analysis of Two-dimensional Laminar Flow Past a Square Cylinder

2021 ◽  
Vol 16 ◽  
pp. 109-119
Author(s):  
Nitin Kumar ◽  
Sachin Tejyan ◽  
Sunil Chamoli ◽  
Pawan Kumar Pant
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Two Dimensions ◽  
Base Flow ◽  
Mode Analysis ◽  
Two Dimensional ◽  
Square Cylinder ◽  
Flow Past

The present study focuses on the development of a numerical framework for predicting the onset of vortex sheading due to flow past a square cylinder. For this a discrete linear stability analysis framework for two-dimensional laminar flows have used. Initially the frame work is validating by using the analysis of thermal stability of flows in the discrete numerical sense. The two-dimensional base flow for various values of the controlling parameter (Reynolds number for flow past a square cylinder and Rayleigh number for double-glazing problem) is computed numerically by using the lattice Boltzmann method. The governing equations, discretized using the finite-difference method in two-dimensions and are subsequently written in the form of perturbed equations with two-dimensional disturbances. These equations are linearized around the base flow and form a set of partial differential equations that govern the evolution of the perturbations. The eigenvalues, stability of the base flow and the points of bifurcations are determined using normal mode analysis. The eigenvalue spectrum predicts that the critical Reynolds number is 52 for the flow past a square cylinder. The results are consistent with the previous numerical and experimental observations.

scholarly journals Linear Stability of a Steady Convective Flow between Permeable Cylinders

Fluids ◽  
2021 ◽  
Vol 6 (10) ◽  
pp. 342
Author(s):  
Maksims Zigunovs ◽  
Andrei Kolyshkin ◽  
Ilmars Iltins
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Chemical Reaction ◽  
Linear Stability Analysis ◽  
Convective Flow ◽  
Base Flow ◽  
Marginal Stability ◽  
Heat Sources ◽  
Rate Of Increase

Linear stability analysis of a steady convective flow in a tall vertical annulus caused by nonlinear heat sources is conducted in the paper. Heat sources are generated as a result of a chemical reaction. The effect of radial cross-flow through permeable porous walls of the annulus is analyzed. The problem is relevant to biomass thermal conversion. The base flow solution is obtained by solving nonlinear boundary value problem. Linear stability analysis is performed, using collocation method. The calculations show that radial inward or outward flow has a stabilizing effect on the flow, while the increase in the Frank–Kamenetskii parameter (proportional to the intensity of the chemical reaction) destabilizes the flow. The increase in the Reynolds number based on the radial velocity leads to the appearance of the second minimum on the marginal stability curves. The rate of increase in the critical Grashof number with respect to the Reynolds number is different for inward and outward radial flows.

Stability and dynamics of the laminar wake past a slender blunt-based axisymmetric body

2011 ◽  
Vol 676 ◽  
pp. 110-144 ◽  
Author(s):  
P. BOHORQUEZ ◽  
E. SANMIGUEL-ROJAS ◽  
A. SEVILLA ◽  
J. I. JIMÉNEZ-GONZÁLEZ ◽  
C. MARTÍNEZ-BAZÁN
Keyword(s):  
Stability Analysis ◽  
Numerical Simulations ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Velocity Ratio ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Direct Numerical Simulations ◽  
The Body ◽  
Stream Velocity

We investigate the stability properties and flow regimes of laminar wakes behind slender cylindrical bodies, of diameter D and length L, with a blunt trailing edge at zero angle of attack, combining experiments, direct numerical simulations and local/global linear stability analyses. It has been found that the flow field is steady and axisymmetric for Reynolds numbers below a critical value, Recs (L/D), which depends on the length-to-diameter ratio of the body, L/D. However, in the range of Reynolds numbers Recs(L/D) < Re < Reco(L/D), although the flow is still steady, it is no longer axisymmetric but exhibits planar symmetry. Finally, for Re > Reco, the flow becomes unsteady due to a second oscillatory bifurcation which preserves the reflectional symmetry. In addition, as the Reynolds number increases, we report a new flow regime, characterized by the presence of a secondary, low frequency oscillation while keeping the reflectional symmetry. The results reported indicate that a global linear stability analysis is adequate to predict the first bifurcation, thereby providing values of Recs nearly identical to those given by the corresponding numerical simulations. On the other hand, experiments and direct numerical simulations give similar values of Reco for the second, oscillatory bifurcation, which are however overestimated by the linear stability analysis due to the use of an axisymmetric base flow. It is also shown that both bifurcations can be stabilized by injecting a certain amount of fluid through the base of the body, quantified here as the bleed-to-free-stream velocity ratio, Cb = Wb/W∞.

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