Stability of Viscous Accretion Tori

AbstractWith respect to the stability of accretion tori the linear stability of infinitely long cylinders is investigated taking into account the effect of viscosity. Critical Reynolds numbers as a function of the Mach number and the azimuthal wavenumber of the perturbation are calculated.

Linear stability of boundary layers under solitary waves

10.1017/jfm.2014.617 ◽

2014 ◽

Vol 761 ◽

pp. 62-104 ◽

Cited By ~ 3

Author(s):

Joris C. G. Verschaeve ◽

Geir K. Pedersen

Keyword(s):

Linear Stability ◽

Solitary Waves ◽

Reynolds Numbers ◽

Quantitative Agreement ◽

Linear Instability ◽

Navier Stokes ◽

Acceleration Region ◽

Tollmien Schlichting ◽

AbstractIn the present treatise, the stability of the boundary layer under solitary waves is analysed by means of the parabolized stability equation. We investigate both surface solitary waves and internal solitary waves. The main result is that the stability of the flow is not of parametric nature as has been assumed in the literature so far. Not only does linear stability analysis highlight this misunderstanding, it also gives an explanation why Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231), Vittori & Blondeaux (Coastal Engng, vol. 58, 2011, pp. 206–213) and Ozdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) each obtained different critical Reynolds numbers in their experiments and simulations. We find that linear instability is possible in the acceleration region of the flow, leading to the question of how this relates to the observation of transition in the acceleration region in the experiments by Sumer et al. or to the conjecture of a nonlinear instability mechanism in this region by Ozdemir et al. The key concept for assessment of instabilities is the integrated amplification which has not been employed for this kind of flow before. In addition, the present analysis is not based on a uniformization of the flow but instead uses a fully nonlinear description including non-parallel effects, weakly or fully. This allows for an analysis of the sensitivity with respect to these effects. Thanks to this thorough analysis, quantitative agreement between model results and direct numerical simulation has been obtained for the problem in question. The use of a high-order accurate Navier–Stokes solver is primordial in order to obtain agreement for the accumulated amplifications of the Tollmien–Schlichting waves as revealed in this analysis. An elaborate discussion on the effects of amplitudes and water depths on the stability of the flow is presented.

On the stability of flow in an elliptic pipe which is nearly circular

10.1017/s0022112078001561 ◽

1978 ◽

Vol 87 (2) ◽

pp. 233-241 ◽

Cited By ~ 6

Author(s):

A. Davey

Keyword(s):

Reynolds Number ◽

Linear Stability ◽

Cross Section ◽

Poiseuille Flow ◽

Critical Reynolds Number ◽

Major Axis ◽

Reynolds Numbers ◽

Minor Axis ◽

The linear stability of Poiseuille flow in an elliptic pipe which is nearly circular is examined by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. We show that the temporal damping rates of non-axisymmetric infinitesimal disturbances which are concentrated near the wall of the pipe are decreased by the ellipticity. In particular we estimate that if the length of the minor axis of the cross-section of the pipe is less than about 96 ½% of that of the major axis then the flow will be unstable and a critical Reynolds number will exist. Also we calculate estimates of the ellipticities which will produce critical Reynolds numbers ranging from 1000 upwards.

On the linear stability of compressible plane Couette flow

10.1017/s0022112094003277 ◽

1994 ◽

Vol 258 ◽

pp. 131-165 ◽

Cited By ~ 24

Author(s):

Peter W. Duck ◽

Gordon Erlebacher ◽

M. Yousuff Hussaini

Keyword(s):

Linear Stability ◽

Couette Flow ◽

Small Amplitude ◽

Reynolds Numbers ◽

Inviscid Limit ◽

Plane Couette Flow ◽

Moving Wall ◽

Type Boundary ◽

The Stability ◽

Radiation Type

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

The stability of a stratified fluid

10.1017/s0022112067000771 ◽

1967 ◽

Vol 29 (2) ◽

pp. 233-240

Author(s):

J. B. Hinwood

Keyword(s):

Turbulent Flow ◽

Froude Number ◽

Reynolds Numbers ◽

Stratified Fluid ◽

Stratified Flows ◽

Rectangular Duct ◽

Interfacial Waves ◽

Homogeneous Flow ◽

For the flow of a stably-stratified fluid in the inlet region of a rectangular duct, it is shown experimentally that the upper and lower critical Reynolds numbers are functions of the interfacial Froude number F, and that if F is large they are lower than for a homogeneous flow. In stratified flows the disturbances leading to turbulent flow sometimes arise at the interface and lead to interfacial waves, whose wavelength at breaking is equal to the conduit depth.

The Stability of Water Flow Over Heated and Cooled Flat Plates

Journal of Heat Transfer ◽

10.1115/1.3597439 ◽

1968 ◽

Vol 90 (1) ◽

pp. 109-114 ◽

Cited By ~ 52

Author(s):

Ahmed R. Wazzan ◽

T. Okamura ◽

A. M. O. Smith

Keyword(s):

Reynolds Number ◽

Critical Reynolds Number ◽

Free Stream ◽

Reynolds Numbers ◽

Stream Temperature ◽

Heated Wall ◽

Flat Plates ◽

The Stability ◽

Sommerfeld Equation

The theory of two-dimensional instability of laminar flow of water over solid surfaces is extended to include the effects of heat transfer. The equation that governs the stability of these flows to Tollmien-Schlichting disturbances is the Orr-Sommerfeld equation “modified” to include the effect of viscosity variation with temperature. Numerical solutions to this equation at high Reynolds numbers are obtained using a new method of integration. The method makes use of the Gram-Schmidt orthogonalization technique to obtain linearly independent solutions upon numerically integrating the “modified Orr-Sommerfeld” equation using single precision arithmetic. The method leads to satisfactory answers for Reynolds numbers as high as Rδ* = 100,000. The analysis is applied to the case of flow over both heated and cooled flat plates. The results indicate that heating and cooling of the wall have a large influence on the stability of boundary-layer flow in water. At a free-stream temperature of 60 deg F and wall temperatures of 60, 90, 120, 135, 150, 200, and 300deg F, the critical Reynolds numbers Rδ* are 520, 7200, 15200, 15600, 14800, 10250, and 4600, respectively. At a free-stream temperature of 200F and wall temperature of 60 deg F (cooled case), the critical Reynolds number is 151. Therefore, it is evident that a heated wall has a stabilizing effect, whereas a cooled wall has a destabilizing effect. These stability calculations show that heating increases the critical Reynolds number to a maximum value (Rδ* max = 15,700 at a temperature of TW = 130 deg F) but that further heating decreases the critical Reynolds number. In order to determine the influence of the viscosity derivatives upon the results, the critical Reynolds number for the heated case of T∞ = 40 and TW = 130 deg F was determined using (a) the Orr-Sommerfeld equation and (b) the present governing equation. The resulting critical Reynolds numbers are Rδ* = 140,000 and 16,200, respectively. Therefore, it is concluded that the terms pertaining to the first and second derivatives of the viscosity have a considerable destabilizing influence.

Proceedings of the Institution of Mechanical Engineers Part B Journal of Engineering Manufacture ◽

Theoretical and Experimental Investigations of Coolant Flow in Inlet Channels of the BTA and Ejector Drills

10.1243/pime_proc_1995_209_075_02 ◽

1995 ◽

Vol 209 (3) ◽

pp. 211-220 ◽

Cited By ~ 1

Author(s):

V P Astakhov ◽

P S Subramanya ◽

M O M Osman

Keyword(s):

Axial Flow ◽

Annular Channel ◽

Reynolds Numbers ◽

Coolant Flow ◽

Experimental Investigations ◽

Annular Channels ◽

Flow Through ◽

The Stability ◽

The Mathematical Model

The coolant flow through inlet annular channels in BTA and ejector drills is investigated. The study was conducted in order to understand the influence of the channel's parameters (the channel's clearance variation along its length and eccentricity) on the coolant pressure distribution and hydraulic resistance. A new design of the ejector drill with the eccentrical location on the inner tube is proposed. A study is made of the stability in the coolant flow in the inlet annular channels. The appearance of instability is explained by the presence of Taylor macrovortices in these channels under certain combinations of boring bar rotating velocity and axial flow velocity. In order to define the unstable regimes (the critical Reynolds numbers), the mathematical model for non-isothermal flow through the annular channel is solved. The heat transfer from the swarf to the incoming coolant is investigated under different flow conditions.

Linear stability theory of oscillatory Stokes layers

10.1017/s0022112074000929 ◽

1974 ◽

Vol 62 (4) ◽

pp. 753-773 ◽

Cited By ~ 115

Author(s):

Christian Von Kerczek ◽

Stephen H. Davis

Keyword(s):

Reynolds Number ◽

Linear Stability ◽

Stability Theory ◽

Absolute Stability ◽

Reynolds Numbers ◽

Time Dependent ◽

Linear Stability Theory ◽

Full Time ◽

Full Theory ◽

The stability of the oscillatory Stokes layers is examined using two quasi-static linear theories and an integration of the full time-dependent linearized disturbance equations. The full theory predicts absolute stability within the investigated range and perhaps for all the Reynolds numbers. A given wavenumber disturbance of a Stokes layer is found to bemore stablethan that of the motionless state (zero Reynolds number). The quasi-static theories predict strong inflexional instabilities. The failure of the quasi-static theories is discussed.

Competition between the centrifugal and strato-rotational instabilities in the stratified Taylor–Couette flow

10.1017/jfm.2018.15 ◽

2018 ◽

Vol 840 ◽

pp. 5-24 ◽

Cited By ~ 6

Author(s):

Junho Park ◽

Paul Billant ◽

Jong-Jin Baik ◽

Jaemyeong Mango Seo

Keyword(s):

Stability Analysis ◽

Linear Stability ◽

Couette Flow ◽

Linear Stability Analysis ◽

Reynolds Numbers ◽

Axisymmetric Mode ◽

Stability Threshold ◽

Taylor Couette ◽

The Stability ◽

Taylor Couette Flow

The stably stratified Taylor–Couette flow is investigated experimentally and numerically through linear stability analysis. In the experiments, the stability threshold and flow regimes have been mapped over the ranges of outer and inner Reynolds numbers: $-2000<Re_{o}<2000$ and $0<Re_{i}<3000$, for the radius ratio $r_{i}/r_{o}=0.9$ and the Brunt–Väisälä frequency $N\approx 3.2~\text{rad}~\text{s}^{-1}$. The corresponding Froude numbers $F_{o}$ and $F_{i}$ are always much smaller than unity. Depending on $Re_{o}$ (or equivalently on the angular velocity ratio $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}_{o}/\unicode[STIX]{x1D6FA}_{i}$), three different regimes have been identified above instability onset: a weakly non-axisymmetric mode with low azimuthal wavenumber $m=O(1)$ is observed for $Re_{o}<0$ ($\unicode[STIX]{x1D707}<0$), a highly non-axisymmetric mode with $m\sim 12$ occurs for $Re_{o}>840$ ($\unicode[STIX]{x1D707}>0.57$) while both modes are present simultaneously in the lower and upper parts of the flow for $0\leqslant Re_{o}\leqslant 840$ ($0\leqslant \unicode[STIX]{x1D707}\leqslant 0.57$). The destabilization of these primary modes and the transition to turbulence as $Re_{i}$ increases have been also studied. The linear stability analysis proves that the weakly non-axisymmetric mode is due to the centrifugal instability while the highly non-axisymmetric mode comes from the strato-rotational instability. These two instabilities can be clearly distinguished because of their distinct dominant azimuthal wavenumber and frequency, in agreement with the recent results of Park et al. (J. Fluid Mech., vol. 822, 2017, pp. 80–108). The stability threshold and the characteristics of the primary modes observed in the experiments are in very good agreement with the numerical predictions. Moreover, we show that the centrifugal and strato-rotational instabilities are observed simultaneously for $0\leqslant Re_{o}\leqslant 840$ in the lower and upper parts of the flow, respectively, because of the variations of the local Reynolds numbers along the vertical due to the salinity gradient.

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

Fluid friction between rotating cylinders I—Torque measurements

10.1098/rspa.1936.0215 ◽

1936 ◽

Vol 157 (892) ◽

pp. 546-564 ◽

Cited By ~ 192

Keyword(s):

Reynolds Number ◽

Turbulent Flow ◽

Centrifugal Force ◽

Outer Cylinder ◽

Reynolds Numbers ◽

Stream Line ◽

Rotating Cylinders ◽

Torque Measurements ◽

The stability of fluid contained between concentric rotating cylinders has been investigated and it has been shown that, when only the inner cylinder rotates, the flow becomes unstable when a certain Reynolds number of the flow is exceeded. When the outer cylinder only is rotated, the flow is stable so far as disturbances of the type produced in the former case are concerned, but provided the Reynolds number of the flow exceeds a certain value, turbulence sets in. The object of the present experiments was partly to measure the torque reaction between two cylinders in the two cases in order to find the effect of centrifugal force on the turbulence, and partly to find the critical Reynolds numbers for the transition from stream-line to turbulent flow. The apparatus is shown diagrammatically in fig. 1.

The stability of free boundary layers between two uniform streams

10.1017/s0022112060000190 ◽

1960 ◽

Vol 7 (3) ◽

pp. 433-441 ◽

Cited By ~ 37

Author(s):

T. Tatsumi ◽

K. Gotoh

Keyword(s):

Boundary Layer ◽

Free Boundary ◽

Reynolds Numbers ◽

Velocity Profiles ◽

Point Of View ◽

Boundary Layer Flows ◽

Type Curves ◽

Neutral Curves ◽

Hydrodynamic stability of free boundary-layer flows is treated in general. It is found that the situations at low Reynolds numbers are universal for all velocity profiles of free boundary-layer type. Curves of constant amplification are calculated as far as O(R3). In particular, the asymptotic form of the neutral curves for R [eDot ] 0 is found to be α = R/(4√3), so that the critical Reynolds numbers of these flows are identically zero. The phase velocity of the disturbance is also found to be zero, for all disturbances, up to the second approximation.A method of normalizing the velocity profiles is suggested, and existing results for the stability of various profiles at large Reynolds numbers are discussed from a new point of view.