scholarly journals Edge state modulation by mean viscosity gradients

2018 ◽  
Vol 838 ◽  
pp. 379-403 ◽  
Author(s):  
Enrico Rinaldi ◽  
Philipp Schlatter ◽  
Shervin Bagheri
Keyword(s):  
Basin Of Attraction ◽  
Reynolds Numbers ◽  
Edge State ◽  
Transition To Turbulence ◽  
Temperature Dependent Viscosity ◽  
Theoretical Support ◽  
Constant Viscosity ◽  
Streamwise Streaks ◽  
Local Shift ◽  
Dependent Viscosity

Motivated by the relevance of edge state solutions as mediators of transition, we use direct numerical simulations to study the effect of spatially non-uniform viscosity on their energy and stability in minimal channel flows. What we seek is a theoretical support rooted in a fully nonlinear framework that explains the modified threshold for transition to turbulence in flows with temperature-dependent viscosity. Consistently over a range of subcritical Reynolds numbers, we find that decreasing viscosity away from the walls weakens the streamwise streaks and the vortical structures responsible for their regeneration. The entire self-sustained cycle of the edge state is maintained on a lower kinetic energy level with a smaller driving force, compared to a flow with constant viscosity. Increasing viscosity away from the walls has the opposite effect. In both cases, the effect is proportional to the strength of the viscosity gradient. The results presented highlight a local shift in the state space of the position of the edge state relative to the laminar attractor with the consequent modulation of its basin of attraction in the proximity of the edge state and of the surrounding manifold. The implication is that the threshold for transition is reduced for perturbations evolving in the neighbourhood of the edge state in the case that viscosity decreases away from the walls, and vice versa.

Heat Transfer in Turbulent Pipe Flow for Liquids Having a Temperature-Dependent Viscosity

1978 ◽  
Vol 100 (2) ◽  
pp. 224-229 ◽  
Author(s):  
O. T. Hanna ◽  
O. C. Sandall
Keyword(s):  
Heat Transfer ◽  
Friction Factor ◽  
Pipe Flow ◽  
Turbulent Pipe Flow ◽  
Fluid Viscosity ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Analytical Approximations ◽  
Constant Viscosity ◽  
Dependent Viscosity

Analytical approximations are developed to predict the effect of a temperature-dependent viscosity on convective heat transfer through liquids in fully developed turbulent pipe flow. The analysis expresses the heat transfer coefficient ratio for variable to constant viscosity in terms of the friction factor ratio for variable to constant viscosity, Tw, Tb, and a fluid viscosity-temperature parameter β. The results are independent of any particular eddy diffusivity distribution. The formulas developed here represent an analytical approximation to the model developed by Goldmann. These approximations are in good agreement with numerical solutions of the model nonlinear differential equation. To compare the results of these calculations with experimental data, a knowledge of the effect of variable viscosity on the friction factor is required. When available correlations for the friction factor are used, the results given here are seen to agree well with experimental heat transfer coefficients over a considerable range of μw/μb.

Theoretical Solutions for Combined Forced and Free Convection in Horizontal Tubes with Temperature-Dependent Viscosity

1976 ◽  
Vol 98 (3) ◽  
pp. 459-465 ◽  
Author(s):  
S. W. Hong ◽  
A. E. Bergles
Keyword(s):  
Circular Tube ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Flux Boundary Condition ◽  
Viscosity Parameter ◽  
Horizontal Tubes ◽  
Constant Viscosity ◽  
Heat Flux Boundary Condition ◽  
Dependent Viscosity ◽  
Good Agreement

A boundary layer solution is presented for fully developed laminar flow in a horizontal circular tube, assuming large Prandtl number and temperature-dependent viscosity and density. The solution is given by Nu = C1 Ra1/4, where C1 is a function of a nondimensional viscosity parameter and the heat flux boundary condition. The heat transfer predictions for large values of the viscosity parameter are 50 percent above the constant viscosity predictions. The present analysis is in good agreement with experimental data for water and ethylene glycol flowing in electrically heated tubes which approximate the boundary conditions assumed in the analysis.

scholarly journals Analytical shock solutions at large and small Prandtl number

2013 ◽  
Vol 726 ◽  
Author(s):  
B. M. Johnson
Keyword(s):  
Prandtl Number ◽  
Maximum Error ◽  
System Of Equations ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
One Dimensional ◽  
Ideal Gases ◽  
Constant Viscosity ◽  
Dependent Viscosity ◽  
Shock Fronts

AbstractExact one-dimensional solutions to the equations of fluid dynamics are derived in the $\mathit{Pr}\rightarrow \infty $ and $\mathit{Pr}\rightarrow 0$ limits (where $\mathit{Pr}$ is the Prandtl number). The solutions are analogous to the $\mathit{Pr}= 3/ 4$ solution discovered by Becker and analytically capture the profile of shock fronts in ideal gases. The large-$\mathit{Pr}$ solution is very similar to Becker’s solution, differing only by a scale factor. The small-$\mathit{Pr}$ solution is qualitatively different, with an embedded isothermal shock occurring above a critical Mach number. Solutions are derived for constant viscosity and conductivity as well as for the case in which conduction is provided by a radiation field. For a completely general density- and temperature-dependent viscosity and conductivity, the system of equations in all three limits can be reduced to quadrature. The maximum error in the analytical solutions when compared to a numerical integration of the finite-$\mathit{Pr}$ equations is $\mathit{O}({\mathit{Pr}}^{- 1} )$ as $\mathit{Pr}\rightarrow \infty $ and $\mathit{O}(\mathit{Pr})$ as $\mathit{Pr}\rightarrow 0$.

Numerical simulations of three-dimensional thermal convection in a fluid with strongly temperature-dependent viscosity

1991 ◽  
Vol 233 ◽  
pp. 299-328 ◽  
Author(s):  
Masaki Ogawa ◽  
Gerald Schubert ◽  
Abdelfattah Zebib
Keyword(s):  
Thermal Convection ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Rectangular Pattern ◽  
Four Corners ◽  
Layer Mode ◽  
Constant Viscosity ◽  
Dependent Viscosity

Numerical calculations are presented for the steady three-dimensional structure of thermal convection of a fluid with strongly temperature-dependent viscosity in a bottom-heated rectangular box. Viscosity is assumed to depend on temperature T as exp (− ET), where E is a constant; viscosity variations across the box r (= exp (E)) as large as 105 are considered. A stagnant layer or lid of highly viscous fluid develops in the uppermost coldest part of the top cold thermal boundary layer when r > rc1, where r = rc1 ≡ 1.18 × 103Rt0.309 and Rt is the Rayleigh number based on the viscosity at the top boundary. Three-dimensional convection occurs in a rectangular pattern beneath this stagnant lid. The planform consists of hot upwelling plumes at or near the centre of a rectangle, sheets of cold sinking fluid on the four sides, and cold sinking plume concentrations immersed in the sheets. A stagnant lid does not develop, i.e. convection involves all of the fluid in the box when r < rc1. The whole-layer mode of convection occurs in a three-dimensional bimodal pattern when r > rc2 = 3.84 × 106Rt−1.35. The planform of the convection is rectangular with the coldest parts of the sinking fluid and the hottest part of the upwelling fluid occurring as plumes at the four corners and at the centre of the rectangle, respectively. Both hot uprising plumes and cold sinking plumes have sheet-like extensions, which become more well-developed as r increases. The whole-layer mode of convection occurs as two-dimensional rolls when r < min (rc1, rc2). The Nusselt number Nu depends on the viscosity at the top surface more strongly in the regime of whole-layer convection than in the regime of stagnant-lid convection. In the whole-layer convective regime, Nu depends more strongly on the viscosity at the top surface than on the viscosity at the bottom boundary.

scholarly journals Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1300
Author(s):  
Evgenii S. Baranovskii ◽  
Vyacheslav V. Provotorov ◽  
Mikhail A. Artemov ◽  
Alexey P. Zhabko
Keyword(s):  
Nonlinear Boundary ◽  
Galerkin Approximation ◽  
Large Data ◽  
Temperature Dependent ◽  
Weak Formulation ◽  
Temperature Dependent Viscosity ◽  
Pipeline Network ◽  
Flux Boundary Conditions ◽  
Creeping Flows ◽  
Dependent Viscosity

This paper deals with a 3D mathematical model for the non-isothermal steady-state flow of an incompressible fluid with temperature-dependent viscosity in a pipeline network. Using the pressure and heat flux boundary conditions, as well as the conjugation conditions to satisfy the mass balance in interior junctions of the network, we propose the weak formulation of the nonlinear boundary value problem that arises in the framework of this model. The main result of our work is an existence theorem (in the class of weak solutions) for large data. The proof of this theorem is based on a combination of the Galerkin approximation scheme with one result from the field of topological degrees for odd mappings defined on symmetric domains.

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