Linear three-dimensional global and asymptotic stability analysis of incompressible open cavity flow

2015 ◽  
Vol 768 ◽  
pp. 113-140 ◽  
Author(s):  
Vincenzo Citro ◽  
Flavio Giannetti ◽  
Luca Brandt ◽  
Paolo Luchini
Keyword(s):  
Growth Rate ◽  
Reynolds Number ◽  
Stability Analysis ◽  
Closed Orbit ◽  
Three Dimensional ◽  
Base Flow ◽  
Open Cavity ◽  
Critical Conditions ◽  
Global Mode ◽  
Structural Sensitivity

The viscous and inviscid linear stability of the incompressible flow past a square open cavity is studied numerically. The analysis shows that the flow first undergoes a steady three-dimensional bifurcation at a critical Reynolds number of 1370. The critical mode is localized inside the cavity and has a flat roll structure with a spanwise wavelength of about 0.47 cavity depths. The adjoint global mode reveals that the instability is most efficiently triggered in the thin region close to the upstream tip of the cavity. The structural sensitivity analysis identifies the wavemaker as the region located inside the cavity and spatially concentrated around a closed orbit. As the flow outside the cavity plays no role in the generation mechanisms leading to the bifurcation, we confirm that an appropriate parameter to describe the critical conditions in open cavity flows is the Reynolds number based on the average velocity between the two upper edges. Stabilization is achieved by a decrease of the total momentum inside the shear layer that drives the core vortex within the cavity. The mechanism of instability is then studied by means of a short-wavelength approximation considering pressureless inviscid modes. The closed streamline related to the maximum inviscid growth rate is found to be the same as that around which the global wavemaker is concentrated. The structural sensitivity field based on direct and adjoint eigenmodes, computed at a Reynolds number far higher than that of the base flow, can predict the critical orbit on which the main instabilities inside the cavity arise. Further, we show that the sub-leading unstable time-dependent modes emerging at supercritical conditions are characterized by a period that is a multiple of the revolution time of Lagrangian particles along the orbit of maximum growth rate. The eigenfrequencies of these modes, computed by global stability analysis, are in very good agreement with the asymptotic results.

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 Triglobal infinite-wing shock-buffet study

2021 ◽  
Vol 925 ◽  
Author(s):  
Wei He ◽  
Sebastian Timme
Keyword(s):  
Stability Analysis ◽  
Three Dimensional ◽  
Base Flow ◽  
Sweep Angle ◽  
Aspect Ratios ◽  
Spatially Periodic ◽  
Time Marching ◽  
High Reynolds ◽  
Limiting Case ◽  
Base Flows

This article uses triglobal stability analysis to address the question of shock-buffet unsteadiness, and associated modal dominance, on infinite wings at high Reynolds number, expanding upon recent biglobal work, aspiring to elucidate the flow phenomenon's origin and characteristics. Infinite wings are modelled by extruding an aerofoil to finite aspect ratios and imposing a periodic boundary condition without assumptions on spanwise homogeneity. Two distinct steady base flows, spanwise uniform and non-uniform, are analysed herein on straight and swept wings. Stability analysis of straight-wing uniform flow identifies both the oscillatory aerofoil mode, linked to the chordwise shock motion synchronised with a pulsation of its downstream shear layer, and several monotone (non-oscillatory), spatially periodic shock-distortion modes. Those monotone modes become outboard travelling on the swept wing with their respective frequencies and phase speeds correlated with the sweep angle. In the limiting case of very small wavenumbers approaching zero, the effect of sweep creates branches of outboard and inboard travelling modes. Overall, triglobal results for such quasi-three-dimensional base flows agree with previous biglobal studies. On the contrary, cellular patterns form in proper three-dimensional base flow on straight wings, and we present the first triglobal study of such an equilibrium solution to the governing equations. Spanwise-irregular modes are found to be sensitive to the chosen aspect ratio. Nonlinear time-marching simulations reveal the flow evolution and distinct events to confirm the insights gained through dominant modes from routine triglobal stability analysis.

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.

Structural sensitivity of spiral vortex breakdown

2013 ◽  
Vol 720 ◽  
pp. 558-581 ◽  
Author(s):  
Ubaid Ali Qadri ◽  
Dhiren Mistry ◽  
Matthew P. Juniper
Keyword(s):  
Stability Analysis ◽  
Passive Control ◽  
Stokes Equations ◽  
Vortex Breakdown ◽  
Control Strategies ◽  
Axial Component ◽  
Azimuthal Mode ◽  
Global Mode ◽  
Structural Sensitivity ◽  
Spiral Vortex

AbstractPrevious numerical simulations have shown that vortex breakdown starts with the formation of a steady axisymmetric bubble and that an unsteady spiralling mode then develops on top of this. We investigate this spiral mode with a linear global stability analysis around the steady bubble and its wake. We obtain the linear direct and adjoint global modes of the linearized Navier–Stokes equations and overlap these to obtain the structural sensitivity of the spiral mode, which identifies the wavemaker region. We also identify regions of absolute instability with a local stability analysis. At moderate swirls, we find that the $m= - 1$ azimuthal mode is the most unstable and that the wavemaker regions of the $m= - 1$ mode lie around the bubble, which is absolutely unstable. The mode is most sensitive to feedback involving the radial and azimuthal components of momentum in the region just upstream of the bubble. To a lesser extent, the mode is also sensitive to feedback involving the axial component of momentum in regions of high shear around the bubble. At an intermediate swirl, in which the bubble and wake have similar absolute growth rates, other researchers have found that the wavemaker of the nonlinear global mode lies in the wake. We agree with their analysis but find that the regions around the bubble are more influential than the wake in determining the growth rate and frequency of the linear global mode. The results from this paper provide the first steps towards passive control strategies for spiral vortex breakdown.

The stability of the variable-density Kelvin–Helmholtz billow

2008 ◽  
Vol 612 ◽  
pp. 237-260 ◽  
Author(s):  
JÉRÔME FONTANE ◽  
LAURENT JOLY
Keyword(s):  
Stability Analysis ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Variable Density ◽  
Base Flow ◽  
Shear Instability ◽  
Numerical Continuation ◽  
Small Scale ◽  
Two Dimensional ◽  
Two Fluids

We perform a three-dimensional stability analysis of the Kelvin–Helmholtz (KH) billow, developing in a shear layer between two fluids with different density. We begin with two-dimensional simulations of the temporally evolving mixing layer, yielding the unsteady base flow fields. The Reynolds number is 1500 while the Schmidt and Froude numbers are infinite. Then exponentially unstable modes are extracted from a linear stability analysis performed at the saturation of the primary mode kinetic energy. The spectrum of the least stable modes exhibits two main classes. The first class comprises three-dimensional core-centred and braid-centred modes already present in the homogeneous case. The baroclinic vorticity concentration in the braid lying on the light side of the KH billow turns the flow into a sharp vorticity ridge holding high shear levels. The hyperbolic modes benefit from the enhanced level of shear in the braid whereas elliptic modes remain quite insensitive to the modifications of the base flow. In the second class, we found typical two-dimensional modes resulting from a shear instability of the curved vorticity-enhanced braid. For a density contrast of 0.5, the wavelength of the two-dimensional instability is about ten times shorter than that of the primary wave. Its amplification rate competes well against those of the hyperbolic three-dimensional modes. The vorticity-enhanced braid thus becomes the preferred location for the development of secondary instabilities. This stands as the key feature of the transition of the variable-density mixing layer. We carry out a fully resolved numerical continuation of the nonlinear development of the two-dimensional braid-mode. Secondary roll-ups due to a small-scale Kelvin–Helmholtz mechanism are promoted by the underlying strain field and develop rapidly in the compression part of the braid. Originally analysed by Reinoud et al. (Phys. Fluids, vol. 12, 2000, p. 2489) from two-dimensional non-viscous numerical simulations, this instability is shown to substantially increase the mixing.

Linear stability analysis of two-way coupled particle-laden density current

2017 ◽  
Vol 95 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Pouriya Amini ◽  
Ehsan Khavasi ◽  
Navid Asadizanjani
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Base Flow ◽  
Interfacial Instability ◽  
Linear Stability Theory ◽  
Flow Density ◽  
Density Current ◽  
Density Currents

Stability of two-way coupled particle-laden density current is studied with the aim of linear stability analysis. Interfacial instability can be found in density currents, which effects entrainment and the rate of effective mixing. In this paper, we investigate the density current interfacial instability using linear stability theory, considering the particles attendance. The ultimate goal is to extract the governing equation for current with particles and study the effect of different parameters on stability of such currents. Base flow has velocity and density profiles of tangent hyperbolic type. Main current and particles are studied in two separate phases. It is found that current will be more stable as M0 (M0 = S∗N∗/ρ∗ where ρ∗ is the non-dimensional flow density, S∗ is the Stokes’ drag coefficient, and N∗ is the particles’ number density) grows, this is a result of number of particles and their radius, and also viscosity effects. The current is more stable as the growth rate increases. As the Richardson number in M0 rises, the growth rate value decreases. As the slope of the river bed increases, the current is less stable.

Stability and Transition Over a Low-Reynolds Number Airfoil

2016 ◽  
Author(s):  
Paul Ziadé ◽  
Pierre E. Sullivan
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Separation Bubble ◽  
Three Dimensional ◽  
Peak Frequency ◽  
Laminar Separation Bubble ◽  
Airfoil Surface ◽  
Laminar Separation

Large-eddy simulation and linear stability analysis were performed on a NACA 0025 airfoil at a chord Reynolds number of 105 and four angles of attack. The computations showed that the initial vortex roll-up quickly breaks down to three-dimensional turbulence. Flow separation was observed at all angles, whereas only the lowest angle of attack formed a laminar separation bubble due to flow transition occuring close to the airfoil surface. A Chebyshev collocation method was employed to solve the viscous and inviscid stability equations. Linear stability analysis demonstrated that high-frequency disturbances occur in the laminar separation bubble case, whereas lower frequencies are present for the fully separated angles of attack. The maximum disturbance growth rates were dampened with the addition of viscosity but negligible change in peak frequency was noted.

Channel formation by turbidity currents: Navier–Stokes-based linear stability analysis

2008 ◽  
Vol 615 ◽  
pp. 185-210 ◽  
Author(s):  
B. HALL ◽  
E. MEIBURG ◽  
B. KNELLER
Keyword(s):  
Shear Stress ◽  
Stability Analysis ◽  
Linear Stability ◽  
Suspended Sediment Concentration ◽  
Three Dimensional ◽  
Sediment Concentration ◽  
Base Flow ◽  
Navier Stokes ◽  
Streamwise Vortices ◽  
Sediment Bed

The linear stability of an erodible sediment bed beneath a turbidity current is analysed, in order to identify potential mechanisms responsible for the formation of longitudinal gullies and channels. On the basis of the three-dimensional Navier–Stokes equations, the stability analysis accounts for the coupled interaction of the three-dimensional fluid and particle motion inside the current with the erodible bed below it. For instability to occur, the suspended sediment concentration of the base flow needs to decay away from the sediment bed more slowly than does the shear stress inside the current. Under such conditions, an upward protrusion of the sediment bed will find itself in an environment where erosion decays more quickly than sedimentation, and so it will keep increasing. Conversely, a local valley in the sediment bed will see erosion increase more strongly than sedimentation, which again will amplify the initial perturbation.The destabilizing effect of the base flow is modulated by the stabilizing perturbation of the suspended sediment concentration and by the shear stress due to a secondary flow structure in the form of counter-rotating streamwise vortices. These streamwise vortices are stabilizing for small Reynolds and Péclet numbers and destabilizing for large values.For a representative current height of O(10–100m), the linear stability analysis provides the most amplified wavelength in the range of 250–2500m, which is consistent with field observations reported in the literature. In contrast to previous analyses based on depth-averaged equations, the instability mechanism identified here does not require any assumptions about sub- or supercritical flow, nor does it require the presence of a slope or a slope break.

The effect of surfactant on the stability of a fluid filament embedded in a viscous fluid

1999 ◽  
Vol 382 ◽  
pp. 331-349 ◽  
Author(s):  
S. HANSEN ◽  
G. W. M. PETERS ◽  
H. E. H. MEIJER
Keyword(s):  
Surface Tension ◽  
Growth Rate ◽  
Reynolds Number ◽  
Stability Analysis ◽  
Viscous Fluid ◽  
High Elasticity ◽  
Fluid Filament ◽  
The Stability ◽  
Elasticity Number

The effect of surfactant on the breakup of a viscous filament, initially at rest, surrounded by another viscous fluid is studied using linear stability analysis. The role of the surfactant is characterized by the elasticity number – a high elasticity number implies that surfactant is important. As expected, the surfactant slows the growth rate of disturbances. The influence of surfactant on the dominant wavenumber is less trivial. In the Stokes regime, the dominant wavenumber for most viscosity ratios increases with the elasticity number; for filament to matrix viscosity ratios ranging from about 0.03 to 0.4, the dominant wavenumber decreases when the elasticity number increases. Interestingly, a surfactant does not affect the stability of a filament when the surface tension (or Reynolds) number is very large.

scholarly journals Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls

2015 ◽  
Vol 788 ◽  
pp. 129-146 ◽  
Author(s):  
Jānis Priede ◽  
Thomas Arlt ◽  
Leo Bühler
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Strong Magnetic Field ◽  
Linear Stability ◽  
Critical Reynolds Number ◽  
Metal Flow ◽  
Three Dimensional ◽  
Transverse Magnetic ◽  
Base Flow ◽  
Square Duct

This study is concerned with the numerical linear stability analysis of liquid-metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios $c\ll 1$, an extremely strong magnetic field with Hartmann number $\mathit{Ha}\sim c^{-4}$ is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1$, 0.1 and 0.01 and Hartmann numbers up to $10^{4}$. As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness ${\it\delta}\sim \mathit{Ha}^{-1/2}$. This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $\mathit{Re}_{c}\sim 110\mathit{Ha}^{1/2}$ and $k_{c}\sim 0.5\mathit{Ha}^{1/2}$. The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll 1$ is $\overline{\mathit{Re}}_{c}\approx 520$. Although this value is somewhat larger than $\overline{\mathit{Re}}_{c}\approx 313$ found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.

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