The stability of spatially periodic flows

1981 ◽  
Vol 108 ◽  
pp. 461-474 ◽  
Author(s):  
D. N. Beaumont
Keyword(s):  
Velocity Profile ◽  
Phase Speed ◽  
Neutral Curve ◽  
Long Waves ◽  
Inviscid Fluid ◽  
Periodic Flows ◽  
A Value ◽  
Parallel Flows ◽  
Spatially Periodic ◽  
The Stability

The stability characteristics for spatially periodic parallel flows of an incompressible fluid (both inviscid and viscous) are studied. A general formula for the determination of the stability characteristics of periodic flows to long waves is obtained, and applied to approximate numerically the stability curves for the sinusoidal velocity profile. The neutral curve for the sinusoidal velocity profile is obtained analytically. The stability of two broken-line velocity profiles in an inviscid fluid is studied and the results are used to describe the overall pattern for the sinusoidal velocity profile in the case of long waves. In an inviscid fluid it is found that all periodic flows (other than the trivial flow in which the basic velocity is constant) are unstable to long waves with a value of the phase speed determined by simple integrals of the basic flow. In a viscous fluid it is found that the sinusoidal velocity profile is very unstable with the inviscid solution being a good approximation to the solution of the viscous problem when the value of the Reynolds number is greater than about 20.

On the disturbed motion of a plane vortex sheet

1958 ◽  
Vol 4 (5) ◽  
pp. 538-552 ◽  
Author(s):  
John W. Miles
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Approximation ◽  
Formal Solution ◽  
Vortex Sheet ◽  
Inviscid Fluid ◽  
Initial Value ◽  
Spatially Periodic ◽  
The Stability ◽  
Spurious Eigenvalues ◽  
Disturbed Motion

A formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods. The eigenvalue problem is investigated and the stability criterion determined. This criterion is found to be in agreement with that obtained previously by Landau (1944), Hatanaka (1949), and Pai (1954), all of whom had included spurious eigenvalues in their analyses. It is also established that supersonic disturbances may be unstable; related investigations in hydrodynamic stability have conjectured on this possibility, but the vortex sheet appears to afford the first definite example. Finally, an asymptotic approximation is developed for the displacement of a vortex sheet following a suddenly imposed, spatially periodic velocity.

scholarly journals On the long-wave instability of natural-convection boundary layers

1997 ◽  
Vol 335 ◽  
pp. 57-73 ◽  
Author(s):  
P. G. DANIELS ◽  
JOHN C. PATTERSON
Keyword(s):  
Boundary Layer ◽  
Travelling Waves ◽  
Vertical Wall ◽  
Phase Speed ◽  
Neutral Curve ◽  
Neutral Stability ◽  
Layer Width ◽  
Dimensional Boundary ◽  
Prandtl Numbers ◽  
The Stability

This paper considers the stability of the one-dimensional boundary layer generated by sudden heating of an infinite vertical wall. A quasi-steady approximation is used to analyse the asymptotic form of the lower branch of the neutral curve, corresponding to disturbances of wavelength much greater than the boundary-layer width. This leads to predictions of the critical wavenumber for neutral stability and the maximum phase speed of the travelling waves. Results are obtained for a range of Prandtl numbers and are compared with solutions of the full stability equations and with numerical simulations and experimental observations of cavity flows driven by sudden heating of the sidewalls.

The effect of stable thermal stratification on the stability of viscous parallel flows

1971 ◽  
Vol 47 (1) ◽  
pp. 1-20 ◽  
Author(s):  
K. S. Gage
Keyword(s):  
Boundary Layer ◽  
Richardson Number ◽  
Thermal Stratification ◽  
Neutral Stability ◽  
Plane Poiseuille Flow ◽  
Boundary Layer Flows ◽  
A Value ◽  
Parallel Flows ◽  
The Stability ◽  
Asymptotic Suction

A unified linear viscous stability theory is developed for a certain class of stratified parallel channel and boundary-layer flows with Prandtl number equal to unity. Results are presented for plane Poiseuille flow and the asymptotic suction boundary-layer profile, which show that the asymptotic behaviour of both branches of the curve of neutral stability has a universal character. For velocity profiles without inflexion points it is found that a mode of instability disappears as η, the local Richardson number evaluated at the critical point, approaches 0.0554 from below. Calculations for Grohne's inflexion-point profile show both major and minor curves of neutral stability for 0 < η [les ] 0.0554; for\[ 0.0554 < \eta < 0.0773 \]there is only a single curve of neutral stability; and, for η > 0.0773, the curves of neutral stability become closed, with complete stabilization being achieved for a value of η of about 0·107.

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.

Stability of plane parallel flows of electrically conducting fluids

1953 ◽  
Vol 49 (1) ◽  
pp. 166-168 ◽  
Author(s):  
D. H. Michael
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Electrically Conducting ◽  
Plane Parallel ◽  
Parallel Flows ◽  
Lower Reynolds Number ◽  
The Stability ◽  
Conducting Fluids

The ordinary theory of stability of plane parallel flows is considerably simplified by a result due to Squire (2) which says that if a velocity profile becomes unstable to a small three-dimensional disturbance at a given Reynolds number, then it will become unstable to a small two-dimensional disturbance at a lower Reynolds number. This result enables us to restrict investigation of the stability to the cases of two-dimensional disturbances.

Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel

2001 ◽  
Vol 445 ◽  
pp. 263-283 ◽  
Author(s):  
T. FUNADA ◽  
D. D. JOSEPH
Keyword(s):  
Surface Tension ◽  
Critical Velocity ◽  
Potential Flow ◽  
Critical Value ◽  
Neutral Curve ◽  
Shear Stresses ◽  
Inviscid Fluid ◽  
Neutral Stability ◽  
Viscous Potential Flow ◽  
The Stability

We study the stability of stratified gas–liquid flow in a horizontal rectangular channel using viscous potential flow. The analysis leads to an explicit dispersion relation in which the effects of surface tension and viscosity on the normal stress are not neglected but the effect of shear stresses is. Formulas for the growth rates, wave speeds and neutral stability curve are given in general and applied to experiments in air–water flows. The effects of surface tension are always important and determine the stability limits for the cases in which the volume fraction of gas is not too small. The stability criterion for viscous potential flow is expressed by a critical value of the relative velocity. The maximum critical value is when the viscosity ratio is equal to the density ratio; surprisingly the neutral curve for this viscous fluid is the same as the neutral curve for inviscid fluids. The maximum critical value of the velocity of all viscous fluids is given by that for inviscid fluid. For air at 20°C and liquids with density ρ = 1 g cm−3 the liquid viscosity for the critical conditions is 15 cP: the critical velocity for liquids with viscosities larger than 15 cP is only slightly smaller but the critical velocity for liquids with viscosities smaller than 15 cP, like water, can be much lower. The viscosity of the liquid has a strong effect on the growth rate. The viscous potential flow theory fits the experimental data for air and water well when the gas fraction is greater than about 70%.

General stability conditions for zonal flows in a one-layer model on the β-plane or the sphere

1983 ◽  
Vol 126 ◽  
pp. 463-489 ◽  
Author(s):  
P. Ripa
Keyword(s):  
Potential Vorticity ◽  
Phase Speed ◽  
Zonal Flow ◽  
Stability Conditions ◽  
Kelvin Wave ◽  
Meridional Gradient ◽  
A Value ◽  
The Difference ◽  
Zonal Average ◽  
The Stability

Sufficient stability conditions are derived for a zonal flow on the β-plane or the sphere. Two conditions guarantee both shear stability (to perturbations with vanishing zonal average) and inertial stability (to longitude-independent perturbations). These conditions are not restricted to normal-mode disturbances, and are derived without making use of the quasi-geostrophic approximation. The main limitation of the model is to have only one layer.On the β-plane, the conditions are: (i) that the product of the meridional gradient of potential vorticity and the difference between an arbitrary constant and the zonal velocity be everywhere non-negative; and (ii) that the absolute value of this difference be nowhere larger than the local phase speed of long gravity waves. Inertial stability is independently assured if the Cariolis parameter and the potential vorticity are everywhere of the same sign (this well-known condition can be easily violated near the equator, but the flow may nonetheless be stable).If the meridional gradient of potential vorticity has everywhere the same sign, then conditions (i) and (ii) can be shown to be consequences of the conservation of a total pseudo-energy E0 and pseudomomentum P0, defined so that their lowest-order contribution is quadratic in the deviation from the fundamental state (even in the case that the perturbation is longitude-independent). Thus, if there exists a value of α such that the integral of E0 − αP0 is positive-definite, then the flow is stable. In this case, the stability conditions are valid for small, rather than infinitesimal, perturbations.The parameters of stable flows, as guaranteed by these conditions, are investigated for the family of Gaussian jets centred at the equator; both the cases of an unbounded ocean and a semi-infinite ocean, poleward from a zonal wall, are considered. Easterlies with the width of a Kelvin wave and westerlies with that width or wider may be unstable, even though the gradient of potential vorticity is positive for any strength of the jet.

scholarly journals Regional Coordination and Security of Water–Energy–Food Symbiosis in Northeastern China

2021 ◽  
Vol 13 (3) ◽  
pp. 1326
Author(s):  
Hongfang Li ◽  
Huixiao Wang ◽  
Yaxue Yang ◽  
Ruxin Zhao
Keyword(s):  
External Environment ◽  
Northeastern China ◽  
Decision Makers ◽  
Human Survival ◽  
A Value ◽  
Natural Factors ◽  
Regional Coordination ◽  
Micro Level ◽  
The Stability ◽  
Coordination Degree

The interactions of water, energy, and food, which are essential resources for human survival, livelihoods, production, and development, constitute a water–energy–food (WEF) nexus. Applying symbiosis theory, the economic, social, and natural factors were considered at the same time in the WEF system, and we conducted a micro-level investigation focusing on the stability, coordination, and sustainability of the symbiotic units (water, energy, and food), and external environment of the WEF system in 36 prefecture-level cities across three northeastern provinces of China. Finally, we analyzed the synergistic safety and coupling coordination degree of the WEF system by the combination of stability, coordination, and sustainability, attending to the coordination relationship and influences of the external environment. The results indicated that the synergistic safety of the WEF system in three northeastern provinces need to equally pay attention to the stability, coordination, and sustainability of the WEF system, since their weights were 0.32, 0.36 and 0.32, respectively. During 2010–2016, the synergistic safety indexes of the WEF system ranged between 0.40 and 0.60, which was a state of boundary safety. In the current study, the coupling coordination degree of the WEF system fluctuated around a value of 0.6, maintaining a primary coordination level; while in the future of 2021–2026, it will decline to 0.57–0.60, dropping to a weak coordinated level. The conclusion could provide effective information for decision-makers to take suitable measures for the security development of a WEF system.

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