Numerical solutions for the time-dependent viscous flow between two rotating coaxial disks

1965 ◽  
Vol 21 (4) ◽  
pp. 623-633 ◽  
Author(s):  
Carl E. Pearson
Keyword(s):  
Steady State ◽  
Viscous Flow ◽  
Numerical Solutions ◽  
Preceding Paper ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Main Body ◽  
Rotating Disks ◽  
High Reynolds ◽  
Steady State Solutions

The nature of the steady-state viscous flow between two large rotating disks has often been discussed, usually qualitatively, in the literature. Using a version of the numerical method described in the preceding paper (Pearson 1965), digital computer solutions for the time-dependent case are obtained (steady-state solutions are then obtainable as limiting cases for large times). Solutions are given for impulsively started disks, and for counter-rotating disks. Of interest is the fact that, at high Reynolds numbers, the solution for the latter problem is unsymmetrical; moreover, the main body of the fluid rotates at a higher angular velocity than that of either disk.

Steady viscous flow past a sphere at high Reynolds numbers

1988 ◽  
Vol 190 ◽  
pp. 471-489 ◽  
Author(s):  
Bengt Fornberg
Keyword(s):  
Viscous Flow ◽  
Incompressible Flow ◽  
Numerical Solutions ◽  
Reynolds Numbers ◽  
High Reynolds Numbers ◽  
Flow Past ◽  
Spherical Vortex ◽  
High Reynolds ◽  
Flow Past A Sphere

Numerical solutions are presented for steady incompressible flow past a sphere. At high Reynolds numbers (results are presented up to R = 5000), the wake is found to resemble a Hill's spherical vortex.

Computation of the flow between two rotating coaxial disks

1977 ◽  
Vol 81 (4) ◽  
pp. 689-699 ◽  
Author(s):  
M. Holodniok ◽  
M. Kubicek ◽  
V. Hlavácek
Keyword(s):  
Rigid Body ◽  
Physical Interpretation ◽  
Reynolds Numbers ◽  
Main Body ◽  
Rotating Disks ◽  
Governing Equations ◽  
High Reynolds Numbers ◽  
Newton Raphson ◽  
High Reynolds ◽  
Raphson Method

A numerical investigation of the problem of rotating disks is made using the Newton–Raphson method. It is shown that the governing equations may exhibit one, three or five solutions. A physical interpretation of the calculated profiles will be presented. The results computed reveal that both Batchelor and Stewartson analysis yields for high Reynolds numbers results which are in agreement with our observations, i.e. the fluid may rotate as a rigid body or the main body of the fluid may be almost at rest, respectively. Occurrence of a two-cell situation at particular branches will be discussed.

Fourier-Bessel Series Model for the Stefan Problem With Internal Heat Generation in Cylindrical Coordinates

2018 ◽  
Author(s):  
Lyudmyla Barannyk ◽  
John Crepeau ◽  
Patrick Paulus ◽  
Ali Siahpush
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Heat Generation ◽  
Numerical Solutions ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Time Dependent ◽  
Cylindrical Coordinates ◽  
System Approaches ◽  
Melting And Solidification

A nonlinear, first-order ordinary differential equation that involves Fourier-Bessel series terms has been derived to model the time-dependent motion of the solid-liquid interface during melting and solidification of a material with constant internal heat generation in cylindrical coordinates. The model is valid for all Stefan numbers. One of the primary applications of this problem is for a nuclear fuel rod during meltdown. The numerical solutions to this differential equation are compared to the solutions of a previously derived model that was based on the quasi-steady approximation, which is valid only for Stefan numbers less than one. The model presented in this paper contains exponentially decaying terms in the form of Fourier-Bessel series for the temperature gradients in both the solid and liquid phases. The agreement between the two models is excellent in the low Stefan number regime. For higher Stefan numbers, where the quasi-steady model is not accurate, the new model differs from the approximate model since it incorporates the time-dependent terms for small times, and as the system approaches steady-state, the curves converge. At higher Stefan numbers, the system approaches steady-state faster than for lower Stefan numbers. During the transient process for both melting and solidification, the temperature profiles become parabolic.

scholarly journals Constitutive relations for compressible granular flow in the inertial regime

2019 ◽  
Vol 874 ◽  
pp. 926-951 ◽  
Author(s):  
D. G. Schaeffer ◽  
T. Barker ◽  
D. Tsuji ◽  
P. Gremaud ◽  
M. Shearer ◽  
...  
Keyword(s):  
Steady State ◽  
Granular Flow ◽  
Numerical Solutions ◽  
Volume Fraction ◽  
Constitutive Relations ◽  
Practical Interest ◽  
Time Dependent ◽  
Direct Result ◽  
Wide Range ◽  
Inertial Regime

Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology that does not suffer from such defects is proposed. In the framework of compressible $I$-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.

On the spatial instability of piecewise linear free shear layers

1987 ◽  
Vol 174 ◽  
pp. 553-563 ◽  
Author(s):  
T. F. Balsa
Keyword(s):  
Velocity Profile ◽  
Numerical Solutions ◽  
Piecewise Linear ◽  
Reynolds Numbers ◽  
Shear Layers ◽  
Spatial Instability ◽  
Free Shear Layers ◽  
Linear Shear ◽  
High Reynolds ◽  
Sommerfeld Equation

The main goal of this paper is to clarify the spatial instability of a piecewise linear free shear flow. We do this by obtaining numerical solutions to the Orr–Sommerfeld equation at high Reynolds numbers. The velocity profile chosen is very much like a piecewise linear one, with the exception that the corners have been rounded so that the entire profile is infinitely differentiable. We find that the (viscous) spatial instability of this modified profile is virtually identical to the inviscid spatial instability of the piecewise linear profile and agrees qualitatively with the inviscid results for the tanh profile when the shear layers are convectively unstable. The unphysical features, previously identified for the piecewise linear velocity profile, arise only when the flow is absolutely unstable. In a nutshell, we see nothing wrong with the inviscid spatial instability of piecewise linear shear flows.

scholarly journals Thermal ignition in a reactive slab with unsymmetric boundary temperatures

1983 ◽  
Vol 24 (3) ◽  
pp. 279-288 ◽  
Author(s):  
K. K. Tam ◽  
P. B. Chapman
Keyword(s):  
Steady State ◽  
Critical Role ◽  
Upper And Lower Solutions ◽  
Lower Solution ◽  
Time Dependent ◽  
Thermal Ignition ◽  
Time Dependent Problem ◽  
Steady State Solutions

AbstractThe problem of thermal ignition in a reactive slab with unsymmetric temperatures equal to 0 and T is considered. Steady state upper and lower solutions are constructed. It is found that T plays a critical role. Results similar to the case with symmetric boundary temperatures are expected when T is small. When T is sufficiently large, there is only one steady state upper or lower solution. The time dependent problem is then considered. Phenomena suggested by studying the upper and lower steady state solutions are confirmed.

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