Primary instability of a shear-thinning film flow down an incline: experimental study

2017 ◽  
Vol 821 ◽  
Author(s):  
M. H. Allouche ◽  
V. Botton ◽  
S. Millet ◽  
D. Henry ◽  
S. Dagois-Bohy ◽  
...  
Keyword(s):  
Experimental Study ◽  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Primary Stability ◽  
Shear Thinning ◽  
Driving Frequency ◽  
Neutral Curves ◽  
Theoretical Predictions ◽  
Sommerfeld Equation ◽  
Primary Instability

The main objective of this work is to study experimentally the primary instability of non-Newtonian film flows down an inclined plane. We focus on low-concentration shear-thinning aqueous solutions obeying the Carreau law. The experimental study essentially consists of measuring wavelengths in marginal conditions, which yields the primary stability threshold for a given slope. The experimental results for neutral curves presented in the $(Re,f_{c})$ and $(Re,k)$ planes (where $f_{c}$ is the driving frequency, $k$ is the wavenumber and $Re$ is the Reynolds number) are in good agreement with the numerical results obtained by a resolution of the generalized Orr–Sommerfeld equation. The long-wave asymptotic extension of our results is consistent with former theoretical predictions of the critical Reynolds number. This is the first experimental evidence of the destabilizing effect of the shear-thinning behaviour in comparison with the Newtonian case: the critical Reynolds number is smaller, and the ratio between the critical wave celerity and the flow velocity at the free surface is larger.

Stability of thermocapillary flows in non-cylindrical liquid bridges

2002 ◽  
Vol 458 ◽  
pp. 35-73 ◽  
Author(s):  
CH. NIENHÜSER ◽  
H. C. KUHLMANN
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Critical Reynolds Number ◽  
Volume Fraction ◽  
Thermocapillary Flow ◽  
Surface Shape ◽  
Liquid Bridges ◽  
Gravity Level ◽  
Neutral Curves ◽  
Prandtl Numbers

The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

The Stability of Water Flow Over Heated and Cooled Flat Plates

1968 ◽  
Vol 90 (1) ◽  
pp. 109-114 ◽  
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 ◽  
Critical Reynolds Numbers ◽  
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.

Accurate solution of the Orr–Sommerfeld stability equation

1971 ◽  
Vol 50 (4) ◽  
pp. 689-703 ◽  
Author(s):  
Steven A. Orszag
Keyword(s):  
Reynolds Number ◽  
Chebyshev Polynomials ◽  
Critical Reynolds Number ◽  
Great Accuracy ◽  
Accurate Solution ◽  
Plane Poiseuille Flow ◽  
Stability Equation ◽  
Orthogonal Functions ◽  
The Stability ◽  
Sommerfeld Equation

The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.

Flow instabilities in the wake of a rounded square cylinder

2016 ◽  
Vol 793 ◽  
pp. 915-932 ◽  
Author(s):  
Doohyun Park ◽  
Kyung-Soo Yang
Keyword(s):  
Reynolds Number ◽  
Critical Reynolds Number ◽  
Landau Equation ◽  
Secondary Instability ◽  
Immersed Boundary ◽  
Neutral Stability ◽  
Square Cylinder ◽  
Flow Topology ◽  
Sharp Edges ◽  
Primary Instability

Instabilities in the flow past a rounded square cylinder have been numerically studied in order to clarify the effects of rounding the sharp edges of a square cylinder on the primary and secondary instabilities associated with the flow. Rounding the edges was done by inscribing a quarter circle of radius $r$ in each edge of a square cylinder of height $d$. Nine cases of rounding were considered, ranging from a square cylinder ($r/d=0$) to a circular cylinder ($r/d=0.5$) with an increment of 0.0625. Each cross-section was numerically implemented in a Cartesian grid system by using an immersed boundary method. The key parameters are the Reynolds number (Re) and the edge-radius ratio ($r/d$). For low Re, the flow is steady and symmetric with respect to the centreline. Over the first critical Reynolds number ($Re_{c1}$), the flow undergoes a Hopf bifurcation to a time-periodic flow, termed the primary instability. As Re further increases, the onset of the secondary instability of three-dimensional (3D) nature is detected beyond the second critical Reynolds number ($Re_{c2}$). Rounding the sharp edges of a square cylinder significantly affects the flow topology, leading to noticeable changes in both instabilities. By employing the Stuart–Landau equation, we investigated the criticality of the primary instability depending upon $r/d$. The onset of the 3D secondary instability was detected by using Floquet stability analysis. The temporal and spatial characteristics of the dominant modes (A, B, QP) were described. The neutral stability curves of each mode were computed depending upon $r/d$.

Experimental study on water-wave trapped modes

2011 ◽  
Vol 666 ◽  
pp. 445-476 ◽  
Author(s):  
P. J. COBELLI ◽  
V. PAGNEUX ◽  
A. MAUREL ◽  
P. PETITJEANS
Keyword(s):  
Experimental Study ◽  
Surface Deformation ◽  
Expansion Method ◽  
Linear Wave ◽  
Single Shot ◽  
Driving Frequency ◽  
Trapped Modes ◽  
Aspect Ratios ◽  
Multipole Expansion Method ◽  
Theoretical Predictions

We present an experimental study on the trapped modes occurring around a vertical surface-piercing circular cylinder of radius a placed symmetrically between the parallel walls of a long but finite water waveguide of width 2d. A wavemaker placed near the entrance of the waveguide is used to force an asymmetric perturbation into the guide, and the free-surface deformation field is measured using a global single-shot optical profilometric technique. In this configuration, several values of the aspect ratio a/d were explored for a range of driving frequencies below the waveguide's cutoff. Decomposition of the obtained fields in harmonics of the driving frequency allowed for the isolation of the linear contribution, which was subsequently separated according to the symmetries of the problem. For each of the aspect ratios considered, the spatial structure of the trapped mode was obtained and compared to the theoretical predictions given by a multipole expansion method. The waveguide–obstacle system was further characterized in terms of reflection and transmission coefficients, which led to the construction of resonance curves showing the presence of one or two trapped modes (depending on the value of a/d), a result that is consistent with the theoretical predictions available in the literature. The frequency dependency of the trapped modes with the geometrical parameter a/d was determined from these curves and successfully compared to the theoretical predictions available within the frame of linear wave theory.

scholarly journals An experimental study of the motion of a light sphere in a rotating viscous fluid

2018 ◽  
Vol 847 ◽  
pp. 119-133 ◽  
Author(s):  
T. Sauma-Pérez ◽  
C. G. Johnson ◽  
L. Yang ◽  
T. Mullin
Keyword(s):  
Experimental Study ◽  
Experimental Investigation ◽  
Reynolds Number ◽  
Viscous Fluid ◽  
Fixed Points ◽  
Critical Reynolds Number ◽  
Solid Sphere ◽  
Rotating Cylinder ◽  
Quantitative Agreement ◽  
Rotation Rates

We present the results of an experimental investigation of the motion of a light, solid sphere in a horizontal rotating cylinder filled with viscous fluid. At high rotation rates, the sphere sits near the axis of the cylinder. At lower rotation rates, a set of off-axis fixed points are observed for a range of sphere radii. The locations of these fixed points are in quantitative agreement with the predictions of a model based on available theory. The fixed points are observed to become unstable to periodic orbits below a critical Reynolds number $Re_{c}$. The radius of the observed orbits increases with Reynolds number more slowly than a typical Hopf bifurcation, in this case, growing as $1/Re^{2}$.

Experimental Study of Wake Shielding Effects of Three Cylinders in Currents

2004 ◽  
Author(s):  
Shan Huang ◽  
Jarle Bolstad ◽  
Adolfo Maro´n
Keyword(s):  
Experimental Study ◽  
Reynolds Number ◽  
Drag Reduction ◽  
Experimental Work ◽  
Critical Reynolds Number ◽  
Flow Direction ◽  
Towing Tank ◽  
The Past ◽  
Number Of Publications ◽  
Shielding Effects

A large amount of experimental work has been carried out in the past to understand forces on a single cylinder in cross flows. In comparison substantially less work can be found on the interference between multiple cylinders and the number of publications available becomes drastically smaller as the number of cylinders involved increases. The current experimental work systematically examines the wake shielding effects of three cylinders by testing in a towing tank. Though staggered, the three cylinders are arranged mainly along the flow direction. The tests are carried out in the subcritical and critical Reynolds number regimes. Significant drag reduction on downstream cylinders is identified.

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.

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