Linear Spatial Stability of Developing Flow in a Parallel Plate Channel

1981 ◽  
Vol 48 (1) ◽  
pp. 192-194 ◽  
Author(s):  
S. C. Gupta ◽  
V. K. Garg
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Parallel Plate ◽  
Critical Reynolds Number ◽  
Two Dimensional ◽  
Spatial Stability ◽  
Percent Change ◽  
Developing Flow ◽  
The Stability ◽  
Plate Channel

It is found that even a 5 percent change in the velocity profile produces a 100 percent change in the critical Reynolds number for the stability of developing flow very close to the entrance of a two-dimensional channel.

Stability of developing flow in a pipe: non-axisymmetric disturbances

1981 ◽  
Vol 110 ◽  
pp. 209-216 ◽  
Author(s):  
Vijay K. Garg
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Critical Reynolds Number ◽  
Temporal Stability ◽  
Developing Flow ◽  
The Stability ◽  
Axisymmetric Disturbances ◽  
Stability Results ◽  
Boundary Layer Edge ◽  
Critical Wavenumber

Spatial stability results for the developing flow in a rigid circular pipe are presented for the velocity profile obtained by the Hornbeck (1963) method and compared with the available temporal stability results for the velocity profile obtained by Sparrow, Lin & Lundgren (1964). The disturbance is taken to be non-axisymmetric, and Gram–Schmidt orthonormalization is used to remove the parasitic errors during numerical integration.It is found that the stability characteristics are very sensitive to the velocity field in the inlet region. At all axial locations investigated the critical frequency and critical wavenumber for the Hornbeck profile are larger than the corresponding values for the Sparrow profile while the critical Reynolds number is smaller. The minimum critical Reynolds number for the Hornbeck profile is only 13250 and occurs at $\overline{X} = 0.0032$ compared with 19780 at $\overline{X} = 0.0049$ for the Sparrow profile. The maximum difference between the two velocity profiles occurs near the boundary-layer edge but is within 5%. Results for the Hornbeck profile are found to be closer to the experimental data of Sarpkaya (1975).

Stability of non-parabolic flow in a flexible tube

1999 ◽  
Vol 395 ◽  
pp. 211-236 ◽  
Author(s):  
V. SHANKAR ◽  
V. KUMARAN
Keyword(s):  
Reynolds Number ◽  
Velocity Profile ◽  
Asymptotic Analysis ◽  
High Reynolds Number ◽  
Reynolds Numbers ◽  
Flexible Tube ◽  
Developing Flow ◽  
Parabolic Flow ◽  
High Reynolds ◽  
The Stability

Flows with velocity profiles very different from the parabolic velocity profile can occur in the entrance region of a tube as well as in tubes with converging/diverging cross-sections. In this paper, asymptotic and numerical studies are undertaken to analyse the temporal stability of such ‘non-parabolic’ flows in a flexible tube in the limit of high Reynolds numbers. Two specific cases are considered: (i) developing flow in a flexible tube; (ii) flow in a slightly converging flexible tube. Though the mean velocity profile contains both axial and radial components, the flow is assumed to be locally parallel in the stability analysis. The fluid is Newtonian and incompressible, while the flexible wall is modelled as a viscoelastic solid. A high Reynolds number asymptotic analysis shows that the non-parabolic velocity profiles can become unstable in the inviscid limit. This inviscid instability is qualitatively different from that observed in previous studies on the stability of parabolic flow in a flexible tube, and from the instability of developing flow in a rigid tube. The results of the asymptotic analysis are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the developing flow could be unstable at much lower Reynolds numbers than the parabolic flow, and hence this instability can be important in destabilizing the fluid flow through flexible tubes at moderate and high Reynolds number. For flow in a slightly converging tube, even small deviations from the parabolic profile are found to be sufficient for the present instability mechanism to be operative. The dominant non-parallel effects are incorporated using an asymptotic analysis, and this indicates that non-parallel effects do not significantly affect the neutral stability curves. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

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.

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.

Flow Behavior of FENE-P Fluids in Grooved Parallel Plates With Transverse Cavities

2016 ◽  
Author(s):  
Abdelkader Filali ◽  
Lyes Khezzar ◽  
Mohamed Alshehhi
Keyword(s):  
Reynolds Number ◽  
Flow Field ◽  
Parallel Plate ◽  
Flow Behavior ◽  
Rheological Parameters ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Fluid Elasticity ◽  
Newtonian Case ◽  
Plate Channel

Numerical investigation of the flow behavior for Newtonian and viscoelastic FENE-P fluids in a parallel-plate channel with transverse rectangular cavities is carried out using ANSYS-POLYFLOW code. A two-dimensional, laminar and steady flow is considered and the flow behavior influenced by the generated vortices at the transverse rectangular cavities has been studied. The effect of Reynolds number, fluid elasticity and the rheological parameters of the FENE-P model L2, on the flow field is examined. In all non-Newtonian considered cases, different flow field were observed which shows different behavior compared to the Newtonian case.

Hydromagnetic instability of a free shear layer at small magnetic Reynolds numbers

1971 ◽  
Vol 49 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Kanefusa Gotoh
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Shear Layer ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Free Shear Layer ◽  
Electrically Conducting ◽  
The Stability ◽  
The Magnetic Field

The effect of a uniform and parallel magnetic field upon the stability of a free shear layer of an electrically conducting fluid is investigated. The equations of the velocity and the magnetic disturbances are solved numerically and it is shown that the flow is stabilized with increasing magnetic field. When the magnetic field is expressed in terms of the parameter N (= M2/R2), where M is the Hartmann number and R is the Reynolds number, the lowest critical Reynolds number is caused by the two-dimensional disturbances. So long as 0 [les ] N [les ] 0·0092 the flow is unstable at all R. For 0·0092 < N [les ] 0·0233 the flow is unstable at 0 < R < Ruc where Ruc decreases as N increases. For 0·0233 < N < 0·0295 the flow is unstable at Rlc < R < Ruc where Rlc increases with N. Lastly for N > 0·0295 the flow is stable at all R. When the magnetic field is measured by M, the lowest critical Reynolds number is still due to the two-dimensional disturbances provided 0 [les ] M [les ] 0·52, and Rc is given by the corresponding Rlc. For M > 0·52, Rc is expressed as Rc = 5·8M, and the responsible disturbance is the three-dimensional one which propagates at angle cos−1(0·52/M) to the direction of the basic flow.

Stability of Nonparallel Developing Flow in an Annulus to Asymmetric Disturbances

1982 ◽  
Vol 49 (2) ◽  
pp. 436-439
Author(s):  
V. K. Garg
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Flow Theory ◽  
Diameter Ratio ◽  
Parallel Flow ◽  
Wave Number ◽  
Spatial Stability ◽  
Concentric Annulus ◽  
Developing Flow ◽  
Azimuthal Wave Number

Linear spatial stability of the nonparallel developing flow in a concentric annulus shows that the asymmetric disturbance with an azimuthal wave number equal to unity is more unstable than the axisymmetric disturbance at all axial locations. Also, in the near entry region, the critical Reynolds number corresponding to the parallel flow theory is as much as three times that due to the nonparallel theory for some values of the annular diameter ratio.

On the stability of plane Poiseuille flow with a finite conductivity in an aligned magnetic field

1968 ◽  
Vol 33 (3) ◽  
pp. 433-443 ◽  
Author(s):  
Sung-Hwan Ko
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Order Differential Equation ◽  
Three Dimensional ◽  
Sixth Order ◽  
Finite Conductivity ◽  
Two Dimensional ◽  
The Stability

A study is made of the stability of a viscous, incompressible fluid with a finite conductivity flowing between parallel planes in a parallel magnetic field. The general form of the magnetohydrodynamic stability equation is a sixth-order differential equation. The complete sixth-order differential equation is solved numerically as an eigenvalue problem. Stability curves are obtained for a range of values of the magnetic Reynolds number Rm and the Alfvé n number A based on two-dimensional disturbances. It is found that the minimum critical Reynolds number is raised as Rm increases for a given A2 and as A2 increases for a given Rm, respectively. The stability curve closes and finally degenerates to a point which gives the critical value for Rm or A2. Results obtained for two-dimensional disturbances are modified to take into account three-dimensional disturbances. Then the minimum critical Reynolds number where three-dimensional disturbances become apparent is obtained, below which two-dimensional disturbances are the most unstable.

Stability of the developing laminar flow in a parallel-plate channel

1967 ◽  
Vol 30 (2) ◽  
pp. 209-224 ◽  
Author(s):  
T. S. Chen ◽  
E. M. Sparrow
Keyword(s):  
Numerical Methods ◽  
Laminar Flow ◽  
Velocity Profile ◽  
Parallel Plate ◽  
Reynolds Numbers ◽  
Neutral Stability ◽  
Wide Range ◽  
Critical Reynolds Numbers ◽  
The Stability ◽  
Plate Channel

The hydrodynamic stability of the developing laminar flow in the entrance region of a parallel-plate channel is investigated using the theory of small disturbances. The stability of the fully developed flow is also re-examined. A wide range of analytical (i.e. asymptotic) and numerical methods are employed in the stability investigation. Among the asymptotic methods, each of three viscous solutions (singular, regular and composite) is used along with the inviscid solution to provide critical Reynolds numbers and complete neutral stability curves. Two numerical methods, finite differences and stepwise integration, are applied to calculate critical Reynolds numbers. The basic flow in the development region is treated from two stand-points: as a channel velocity profile and as a boundary-layer velocity profile. Extensive comparisons among the various methods and flow models disclose their various strengths and ranges of applicability. As a general result, it is found that the critical Reynolds number decreases monotonically with increasing distance from the channel entrance, approaching the fully developed value as a limit.

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