The stability of a two-dimensional laminar jet

1958 ◽  
Vol 4 (3) ◽  
pp. 261-275 ◽  
Author(s):  
T. Tatsumi ◽  
T. Kakutani
Keyword(s):  
Reynolds Number ◽  
Basic Assumption ◽  
Wave Number ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Lower Branch ◽  
Laminar Jet ◽  
Numerical Result ◽  
The Stability ◽  
Different Parts

This paper deals with the stability of a two-dimensional laminar jet against the infinitesimal antisymmetric disturbance. The curve of the neutral stability in the (α, R)-plane (α, the wave-number; R, Reynolds number) is calculated using two different methods for the different parts of the curve; the solution is developed in powers of (αR)−1 for obtaining the upper branch of the curve and in powers of αR for the lower branch.The asymptotic behaviour of these branches is that for branch I,$\alpha \rightarrow 2, \;\; c \rightarrow \frac{2}{3}$ for $R \rightarrow \infty$; and for branch II, $R \sim 1\cdot12\alpha^{-1|2},\; c \sim 1\cdot 20 \alpha^2$ for α → 0. Some discussion is given on the validity of the basic assumption of the stability theory in relation to the numerical result obtained here.

Experimental investigations of the stability of channel flows. Part 2. Two-layered co-current flow in a rectangular channel

1972 ◽  
Vol 52 (3) ◽  
pp. 401-423 ◽  
Author(s):  
Timothy W. Kao ◽  
Cheol Park
Keyword(s):  
Reynolds Number ◽  
Current Flow ◽  
Rectangular Channel ◽  
Shear Waves ◽  
Transition To Turbulence ◽  
Wave Number ◽  
Neutral Stability ◽  
Mean Flow ◽  
Transition Range ◽  
The Stability

The stability of the laminar co-current flow of two fluids, oil and water, in a rectangular channel was investigated experimentally, with and without artificial excitation. For the ratio of viscosity explored, only the disturbances in water grew in the beginning stages of transition to turbulence. The critical water Reynolds number, based upon the hydraulic diameter of the channel and the superficial velocity defined by the ratio of flow rate of water to total cross-sectional area of the channel, was found to be 2300. The behaviour of damped and growing shear waves in water was examined in detail using artificial excitation and briefly compared with that observed in Part 1. Mean flow profiles, the amplitude distribution of disturbances in water, the amplification rate, wave speed and wavenumbers were obtained. A neutral stability boundary in the wave-number, water Reynolds number plane was also obtained experimentally.It was found that in natural transition the interfacial mode was not excited. The first appearance of interfacial waves was actually a manifestation of the shear waves in water. The role of the interface in the transition range from laminar to turbulent flow in water was to introduce and enhance spanwise oscillation in the water phase and to hasten the process of breakdown for growing disturbances.

An experimental investigation of the instability of a two-dimensional jet at low Reynolds numbers

1964 ◽  
Vol 20 (2) ◽  
pp. 337-352 ◽  
Author(s):  
Hiroshi Sato ◽  
Fujihiko Sakao
Keyword(s):  
Experimental Investigation ◽  
Reynolds Number ◽  
Maximum Velocity ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Low Reynolds Numbers ◽  
Laminar Jet ◽  
Artificial Disturbance ◽  
Low Reynolds ◽  
The Stability

An experimental investigation was made of the stability of a two-dimensional jet at low Reynolds numbers with extremely small residual disturbances both in and around the jet. The velocity distribution of a laminar jet is in agreement with Bickley's theoretical result. The stability and transition of a laminar jet are characterized by the Reynolds number based on the slit width and the maximum velocity of the jet. When the Reynolds number is less than 10, the whole jet is laminar. When the Reynolds number is between 10 and around 50, periodic velocity fluctuations are found in the jet. They die out as they travel downstream without developing into irregular fluctuations. When the Reynolds number exceeds about 50, periodic fluctuations develop into irregular, turbulent fluctuations. The frequency of the periodic fluctuation is roughly proportional to the square of the jet velocity.The stability of the jet against an artificially imposed disturbance was also investigated. Sound was used as an artificial disturbance. The disturbance is either amplified or damped in the jet depending on its frequency. The conventional stability theory was modified by considering the streamwise increase of Reynolds number. The experimental results are in agreement with the theoretical results.

On the stability of viscous flow between parallel planes in the presence of a co-planar magnetic field

1954 ◽  
Vol 221 (1145) ◽  
pp. 189-206 ◽  
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Wave Number ◽  
Neutral Stability ◽  
Energy Relation ◽  
Geometrical Properties ◽  
Electrically Conducting ◽  
Neutral Curves ◽  
The Stability ◽  
The Magnetic Field

Linearized equations are derived which govern the stability of a viscous, electrically conducting fluid in motion between two parallel planes in the presence of a co-planar magnetic field. With one suitable approximation, which restricts the valid range of Reynolds number of the theory, the problem of stability is reduced to the solution of a fourth-order ordinary differential equation. The disturbances considered are neither amplified nor damped, but are neutral. Curves of wave number against Reynolds number for neutral stability are calculated for a range of values of a certain parameter, q , which represents the magnetic effects. For given physical and geometrical properties, the critical Reynolds number above which the flow is unstable rises with the strength of the magnetic field. These results are completely within the range of the approximation mentioned. In addition, an energy relation is derived which illustrates the balance between energy transferred from the basic flow to the disturbances, and that dissipated by viscosity and by the magnetic field perturbations.

The Hydrodynamic Stability of Two Viscous Incompressible Fluids in Parallel Uniform Shearing Motion

1979 ◽  
Vol 46 (3) ◽  
pp. 499-504 ◽  
Author(s):  
D. T. Tsahalis
Keyword(s):  
Reynolds Number ◽  
Incompressible Fluid ◽  
Viscosity Ratio ◽  
Density Ratio ◽  
Steady Motion ◽  
Wave Number ◽  
Neutral Stability ◽  
The Family ◽  
The Stability ◽  
Shearing Motion

The stability problem of a thin film of a viscous incompressible fluid bounded on one side by another more viscous and less dense incompressible fluid of semi-infinite extent and on the other side by a fixed wall, where both fluids are in steady motion parallel to their interface and each fluid has a linear velocity profile, is solved for large values of the Reynolds number and small values of the viscosity ratio. Neutral stability curves of the Reynolds number versus the wave number are presented, parametrized with either the density ratio or the viscosity ratio as the family parameters.

On hydrodynamic stability in unlimited fields of viscous flow

1957 ◽  
Vol 238 (1215) ◽  
pp. 489-501 ◽  
Keyword(s):  
Reynolds Number ◽  
Hydrodynamic Stability ◽  
Critical Reynolds Number ◽  
Integral Form ◽  
Neutral Curve ◽  
Wave Number ◽  
Neutral Stability ◽  
Laminar Jet ◽  
Fourth Derivative ◽  
Sommerfeld Equation

This paper considers the hydrodynamic stability of flows in which there are no solid boundaries in the field of flow. The method used is an extension of that initiated by McKoen (1957), in which the fourth derivative, 0 iv , is assumed to be significant only near to the singular layer, but otherwise the complete fourth-order Orr—Sommerfeld equation is considered. An alternative derivation is given for McKoen’s integral form of the boundary condition for an antisymmetrical perturbation. In this integral it is necessary to approximate for (j) but not for any of its derivatives. It is shown that the present method will always lead to a neutral stability curve of wave number against Reynolds number, having two branches as R ->oo and hence a least critical R . The case of the plane laminar jet is considered, and a critical Reynolds number of 4 is obtained, which does not compare unreasonably with experiment in which unsteadiness is first detected at a Reynolds number of about 10. The lower branch of the neutral curve is found to be almost coincident with the R -axis.

The stability of the flow of an electrically conducting fluid between parallel planes under a transverse magnetic field

1955 ◽  
Vol 233 (1192) ◽  
pp. 105-125 ◽  
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Transverse Magnetic Field ◽  
Transverse Magnetic ◽  
Conducting Fluid ◽  
Wave Number ◽  
Neutral Stability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
The Stability

The stability under small disturbances is investigated of the two-dimensional laminar motion of an electrically conducting fluid under a transverse magnetic field. It is found that the dominating factor is the change in shape of the undisturbed velocity profile caused by the magnetic field, which depends only on the Hartmann number M . Curves of wave number against Reynolds number for neutral stability are calculated for a range of values of M ; for large values of M the calculations are similar to those which determine the stability of ordinary boundary-layer flow. The critical Reynolds number is found to rise very rapidly with increasing M , so that a transverse magnetic field has a powerful stabilizing influence on this type of flow.

Three-dimensional disturbances in flow between parallel planes

1960 ◽  
Vol 254 (1279) ◽  
pp. 562-569 ◽  
Keyword(s):  
Reynolds Number ◽  
Dimensional Stability ◽  
Flow Direction ◽  
Simple Procedure ◽  
Three Dimensional ◽  
Stability Diagram ◽  
Wave Number ◽  
Two Dimensional ◽  
General Answer ◽  
The Stability

The close connexion between the stability of three-dimensional and two-dimensional disturbances in flow between parallel walls has been examined and this has led to the formation of a three-dimensional stability diagram where ‘stability surfaces’ replace stability curves. The problem which has been investigated is whether the most highly amplifying disturbance at any given Reynolds number above the minimum critical Reynolds number is a two-dimensional or a three-dimensional disturbance. It has been shown that the most unstable disturbance is a two-dimensional one for a certain definite range of Reynolds number above the critical. For Reynolds numbers greater than this no definite general answer has been found; each basic undisturbed flow must be treated separately and a simple procedure has been given which, in principle, determines the type of disturbance which is most unstable. Difficulty arises in following this procedure because it requires knowledge of the two-dimensional stability curves in a certain region where this knowledge is very scanty at the moment. Althoughth is difficulty arises, in Poiseuille flow the calculations available indicate very strongly that the most unstable disturbance at any given Reynolds number above the critical is two-dimensional. Further, it is believed that this result holds for all other basic flows. A second result is that if the wave number (a) in the flow direction is specified, as well as the Reynolds number, then for a in a certain range, the most unstable disturbance is three-dimensional.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

2019 ◽  
Vol 14 (1) ◽  
pp. 52-58 ◽  
Author(s):  
A.D. Nizamova ◽  
V.N. Kireev ◽  
S.F. Urmancheev
Keyword(s):  
Reynolds Number ◽  
Spectral Characteristics ◽  
Wave Number ◽  
Critical Parameters ◽  
Fluid Viscosity ◽  
Flat Channel ◽  
Stability Equation ◽  
The Stability ◽  
Thermoviscous Fluid ◽  
Sommerfeld Equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

scholarly journals Two-Dimensional Convection Induced by Gravity and Centrifugal Forces in a Rotating Porous Layer Far Away from the Axis of Rotation

1998 ◽  
Vol 4 (2) ◽  
pp. 73-90 ◽  
Author(s):  
Peter Vadasz ◽  
Saneshan Govender
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Temperature Fields ◽  
Wave Number ◽  
Marginal Stability ◽  
Two Dimensional ◽  
Wide Range ◽  
Gravity Driven ◽  
Gravity Driven Flow ◽  
The Stability

The stability and onset of two-dimensional convection in a rotating fluid saturated porous layer subject to gravity and centrifugal body forces is investigated analytically. The problem corresponding to a layer placed far away from the centre of rotation was identified as a distinct case and therefore justifying special attention. The stability of a basic gravity driven convection is analysed. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number for different values of the gravity related Rayleigh number. For any given value of the gravity related Rayleigh number there is a transitional value of the wave number, beyond which the basic gravity driven flow is stable. The results provide the stability map for a wide range of values of the gravity related Rayleigh number, as well as the corresponding flow and temperature fields.

A two-dimensional-three-component model for spanwise rotating plane Poiseuille flow

2019 ◽  
Vol 880 ◽  
pp. 478-496 ◽  
Author(s):  
Shengqi Zhang ◽  
Zhenhua Xia ◽  
Yipeng Shi ◽  
Shiyi Chen
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Reynolds Shear Stress ◽  
Streamwise Velocity ◽  
Channel Height ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Plane Poiseuille Flow ◽  
Rotation Numbers ◽  
3C Model

Spanwise rotating plane Poiseuille flow (RPPF) is one of the canonical flow problems to study the effect of system rotation on wall-bounded shear flows and has been studied a lot in the past. In the present work, a two-dimensional-three-component (2D/3C) model for RPPF is introduced and it is shown that the present model is equivalent to a thermal convection problem with unit Prandtl number. For low Reynolds number cases, the model can be used to study the stability behaviour of the roll cells. It is found that the neutral stability curves, critical eigensolutions and critical streamfunctions of RPPF at different rotation numbers ($Ro$) almost collapse with the help of a rescaling with a newly defined Rayleigh number $Ra$ and channel height $H$. Analytic expressions for the critical Reynolds number and critical wavenumber at different $Ro$ can be obtained. For a turbulent state with high Reynolds number, the 2D/3C model for RPPF is self-sustained even without extra excitations. Simulation results also show that the profiles of mean streamwise velocity and Reynolds shear stress from the 2D/3C model share the same linear laws as the fully three-dimensional cases, although differences on the intercepts can be observed. The contours of streamwise velocity fluctuations behave like plumes in the linear law region. We also provide an explanation to the linear mean velocity profiles observed at high rotation numbers.

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