Nonlinear stability analysis of viscoelastic Taylor–Couette flow in the presence of viscous heating

2002 ◽  
Vol 14 (3) ◽  
pp. 1056-1064 ◽  
Author(s):  
U. A. Al-Mubaiyedh ◽  
R. Sureshkumar ◽  
B. Khomami
Keyword(s):  
Stability Analysis ◽  
Couette Flow ◽  
Nonlinear Stability ◽  
Viscous Heating ◽  
Nonlinear Stability Analysis ◽  
Taylor Couette ◽  
Taylor Couette Flow

The effect of viscous heating on the stability of Taylor–Couette flow

2002 ◽  
Vol 462 ◽  
pp. 111-132 ◽  
Author(s):  
U. A. AL-MUBAIYEDH ◽  
R. SURESHKUMAR ◽  
B. KHOMAMI
Keyword(s):  
Stability Analysis ◽  
Secondary Flow ◽  
Linear Stability ◽  
Couette Flow ◽  
Temperature Difference ◽  
Time Dependent ◽  
Viscous Heating ◽  
Taylor Couette ◽  
The Stability ◽  
Taylor Couette Flow

The influence of viscous heating on the stability of Taylor–Couette flow is investigated theoretically. Based on a linear stability analysis it is shown that viscous heating leads to significant destabilization of the Taylor–Couette flow. Specifically, it is shown that in the presence of viscous dissipation the most dangerous disturbances are axisymmetric and that the temporal characteristic of the secondary flow is very sensitive to the thermal boundary conditions. If the temperature difference between the two cylinders is small, the secondary flow is stationary as in the case of isothermal Taylor–Couette flow. However, when the temperature difference between the two cylinders is large, time-dependent secondary states are predicted. These linear stability predictions are in agreement with the experimental observations of White & Muller (2000) in terms of onset conditions as well as the spatiotemporal characteristics of the secondary flow. Nonlinear stability analysis has revealed that over a broad range of operating conditions, the bifurcation to the time-dependent secondary state is subcritical, while stationary states result as a consequence of supercritical bifurcation. Moreover, the supercritically bifurcated stationary state undergoes a secondary bifurcation to a time-dependent flow. Overall, the structure of the time-dependent state predicted by the analysis compares very well with the experimental observations of White & Muller (2000) that correspond to slowly moving vortices parallel to the cylinder axis. The significant destabilization observed in the presence of viscous heating arises as the result of the coupling of the perturbation velocity and the base-state temperature gradient that gives rise to fluctuations in the radial temperature distribution. Due to the thermal sensitivity of the fluid these fluctuations greatly modify the fluid viscosity and reduce the dissipation of disturbances provided by the viscous stress terms in the momentum equation.

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

2018 ◽  
Vol 840 ◽  
pp. 5-24 ◽  
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.

Numerical stability analysis of the Taylor-Couette flow in the two-dimensional case

2009 ◽  
Vol 49 (4) ◽  
pp. 729-742 ◽  
Author(s):  
O. M. Belotserkovskii ◽  
V. V. Denisenko ◽  
A. V. Konyukhov ◽  
A. M. Oparin ◽  
O. V. Troshkin ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Couette Flow ◽  
Numerical Stability ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Numerical Stability Analysis ◽  
Taylor Couette ◽  
Taylor Couette Flow
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