scholarly journals Partial regularisation of the incompressible 𝜇(I)-rheology for granular flow

2017 â—˝  
Vol 828 â—˝  
pp. 5-32 â—˝  
Author(s):  
T. Barker â—˝  
J. M. N. T. Gray
Keyword(s):  
High Speed â—˝  
Numerical Solutions â—˝  
Stokes Equations â—˝  
Functional Dependence â—˝  
Granular Flows â—˝  
Two Dimensions â—˝  
Navier Stokes â—˝  
Major Step â—˝  
Ill Posed â—˝  
Well Posed

In recent years considerable progress has been made in the continuum modelling of granular flows, in particular the $\unicode[STIX]{x1D707}(I)$-rheology, which links the local viscosity in a flow to the strain rate and pressure through the non-dimensional inertial number $I$. This formulation greatly benefits from its similarity to the incompressible Navier–Stokes equations as it allows many existing numerical methods to be used. Unfortunately, this system of equations is ill posed when the inertial number is too high or too low. The consequence of ill posedness is that the growth rate of small perturbations tends to infinity in the high wavenumber limit. Due to this, numerical solutions are grid dependent and cannot be taken as being physically realistic. In this paper changes to the functional form of the $\unicode[STIX]{x1D707}(I)$ curve are considered, in order to maximise the range of well-posed inertial numbers, while preserving the overall structure of the equations. It is found that when the inertial number is low there exist curves for which the equations are guaranteed to be well posed. However when the inertial number is very large the equations are found to be ill posed regardless of the functional dependence of $\unicode[STIX]{x1D707}$ on $I$. A new $\unicode[STIX]{x1D707}(I)$ curve, which is inspired by the analysis of the governing equations and by experimental data, is proposed here. In order to test this regularised rheology, transient granular flows on inclined planes are studied. It is found that simulations of flows, which show signs of ill posedness with unregularised models, are numerically stable and match key experimental observations when the regularised model is used. This paper details two-dimensional transient computations of decelerating flows where the inertial number tends to zero, high-speed flows that have large inertial numbers, and flows which develop into granular rollwaves. This is the first time that granular rollwaves have been simulated in two dimensions, which represents a major step towards the simulation of other complex granular flows.

scholarly journals Well-posed and ill-posed behaviour of the -rheology for granular flow

2015 â—˝  
Vol 779 â—˝  
pp. 794-818 â—˝  
Author(s):  
T. Barker â—˝  
D. G. Schaeffer â—˝  
P. Bohorquez â—˝  
J. M. N. T. Gray
Keyword(s):  
Friction Coefficient â—˝  
Blow Up â—˝  
Stokes Equations â—˝  
Momentum Balance â—˝  
Navier Stokes â—˝  
Major Step â—˝  
Navier Stokes Equations â—˝  
Newtonian Flows â—˝  
Ill Posed â—˝  
Well Posed

In light of the successes of the Navier–Stokes equations in the study of fluid flows, similar continuum treatment of granular materials is a long-standing ambition. This is due to their wide-ranging applications in the pharmaceutical and engineering industries as well as to geophysical phenomena such as avalanches and landslides. Historically this has been attempted through modification of the dissipation terms in the momentum balance equations, effectively introducing pressure and strain-rate dependence into the viscosity. Originally, a popular model for this granular viscosity, the Coulomb rheology, proposed rate-independent plastic behaviour scaled by a constant friction coefficient ${\it\mu}$. Unfortunately, the resultant equations are always ill-posed. Mathematically ill-posed problems suffer from unbounded growth of short-wavelength perturbations, which necessarily leads to grid-dependent numerical results that do not converge as the spatial resolution is enhanced. This is unrealistic as all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent ${\it\mu}(I)$-rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number $I$ in the friction coefficient ${\it\mu}$. In this paper it is shown that the ${\it\mu}(I)$-rheology is well-posed for intermediate values of $I$, but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain.

scholarly journals On a mixed problem for the parabolic Lamé type operator

2015 â—˝  
Vol 23 (6) â—˝  
Author(s):  
Roman Puzyrev â—˝  
Alexander Shlapunov
Keyword(s):  
Stokes Equations â—˝  
Type System â—˝  
Type Operator â—˝  
Navier Stokes â—˝  
Cylindrical Domain â—˝  
Smooth Functions â—˝  
Navier Stokes Equations â—˝  
Solvability Conditions â—˝  
Ill Posed â—˝  
Well Posed

AbstractWe consider a boundary value problem for a Lamé type operator, which corresponds to a linearisation of the Navier–Stokes' equations for compressible flow of Newtonian fluids in the case where pressure is known. It consists of recovering a vector function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding Hölder spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the integral representation's method we obtain a uniqueness theorem and solvability conditions for the problem. We also describe conditions, providing dense solvability of the problem.

Computational Simulation of the Tethered Undersea Kites for Power Generation

Volume 6B: Energy â—˝  
2015 â—˝  
Author(s):  
Amirmahdi Ghasemi â—˝  
David J. Olinger â—˝  
Gretar Tryggvason
Keyword(s):  
Power Output â—˝  
High Speed â—˝  
Stokes Equations â—˝  
Computational Simulation â—˝  
Hydrodynamic Forces â—˝  
Two Dimensions â—˝  
Ocean Current â—˝  
Navier Stokes â—˝  
Immersed Boundary â—˝  
Flow Equations

The dynamic motion of tethered undersea kites (TUSK) is studied using numerical simulations. TUSK systems consist of a rigid-winged kite moving in an ocean current. The kite is connected by tethers to a platform on the ocean surface or anchored to the seabed. Hydrodynamic forces generated by the kite are transmitted through the tethers to a generator on the platform to produce electricity. TUSK systems are being considered as an alternative to marine turbines since the kite can move in high speed motions to increase power production compared to conventional marine turbines. The two-dimensional Navier-Stokes equations are solved on a regular structured grid that comprises the ocean current flow, and an immersed boundary method is used for the rigid kite. A two-step projection method along with Open Multi-Processing (OpenMP) is employed to solve the flow equations. The reel-out and reel-in velocities of the two tethers are adjusted to control the kite angle of attack and the resultant hydrodynamic forces. A baseline simulation was studied where a high net power output was achieved during successive kite power and retraction phases. System power output, vorticity flow fields, tether tensions, and hydrodynamic coefficients for the kite are determined. The power output results are in good agreement with established theoretical results for a kite moving in two dimensions.

scholarly journals Multi-block technique applied to Navier-Stokes equations in two dimensions

2018 â—˝  
Vol 39 (2) â—˝  
pp. 115
Author(s):  
Jeferson Osmar de Almeida â—˝  
Diomar Cesar Lobão â—˝  
Cleyton Senior Stampa â—˝  
Gustavo Benitez Alvarez
Keyword(s):  
Numerical Method â—˝  
Numerical Solutions â—˝  
Stokes Equations â—˝  
Fluid Flows â—˝  
Numerical Data â—˝  
Two Dimensions â—˝  
Navier Stokes â—˝  
Navier Stokes Equations â—˝  
Flow Problems â—˝  
Numerical Solver

In this work, numerical solutions of the two-dimensional Navier-Stokes and Euler equations using explicit MacCormack method on multi-block structured mesh are presented for steady state and unsteady state compressible fluid flows. The multi-block technique and generalized coordinate system are used to develop a numerical solver which can be applied for a large range of compressible flow problems on complex geometries without modifying the governing equations and numerical method. Besides that the numerical method is based on a finite difference approach and the generalized coordinates introduced allow the application of the boundary conditions easily. The subsonic flow over a backward facing step and supersonic flow over a curved ramp are presented, and the results are compared with the experimental and numerical data.

Comparison of numerical solutions of the Navier?Stokes equations and a kinetic model in the one-dimensional case

Fluid Dynamics â—˝  
10.1007/bf01089651 â—˝  
1980 â—˝  
Vol 15 (5) â—˝  
pp. 752-755 â—˝  
Author(s):  
A. A. Makhmudov â—˝  
S. P. Popov
Keyword(s):  
Kinetic Model â—˝  
Numerical Solutions â—˝  
Stokes Equations â—˝  
Dimensional Case â—˝  
Navier Stokes â—˝  
Navier Stokes Equations â—˝  
One Dimensional â—˝  
The One

A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

1989 â—˝  
Vol 209 â—˝  
pp. 285-308 â—˝  
Author(s):  
R. J. Bodonyi â—˝  
W. J. C. Welch â—˝  
P. W. Duck â—˝  
M. Tadjfar
Keyword(s):  
Numerical Solutions â—˝  
Free Stream â—˝  
Stokes Equations â—˝  
Numerical Study â—˝  
Navier Stokes â—˝  
Surface Geometry â—˝  
Navier Stokes Equations â—˝  
S Waves â—˝  
Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

Quantitative Analysis of Reynolds and Navier–Stokes Based Modeling Approaches for Isothermal Newtonian Elastohydrodynamic Lubrication

10.1115/1.4050272 â—˝  
2021 â—˝  
Vol 143 (12) â—˝  
Author(s):  
Leoluca Scurria â—˝  
Tommaso Tamarozzi â—˝  
Oleg Voronkov â—˝  
Dieter Fauconnier
Keyword(s):  
High Speed â—˝  
Stokes Equations â—˝  
Thickness Distribution â—˝  
Elastohydrodynamic Lubrication â—˝  
Lubricant Film â—˝  
Secondary Flows â—˝  
Navier Stokes â—˝  
Shear Stresses â—˝  
Low Viscosity â—˝  
Fluid Film Thickness

Abstract When simulating elastohydrodynamic lubrication, two main approaches are usually followed to predict the pressure and fluid film thickness distribution throughout the contact. The conventional approach relies on the Reynolds equation to describe the thin lubricant film, which is coupled to a Boussinesq description of the linear elastic deformation of the solids. A more accurate, yet a time-consuming method is the use of computational fluid dynamics in which the Navier–Stokes equations describe the flow of the thin lubricant film, coupled to a finite element solver for the description of the local contact deformation. This investigation aims at assessing both methods for different lubrication conditions in different elastohydrodynamic lubrication (EHL) regimes and quantify their differences to understand advantages and limitations of both methods. This investigation shows how the results from both approaches deviate for three scenarios: (1) inertial contributions (Re > 1), i.e., thick films, high speed, and low viscosity; (2) high shear stresses leading to secondary flows; and (3) large deformations of the solids leading to inaccuracies of the Boussinesq equation.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 â—˝  
Vol 153 (A2) â—˝  
Author(s):  
Q Yang â—˝  
W Qiu
Keyword(s):  
Free Surface â—˝  
Numerical Solutions â—˝  
Stokes Equations â—˝  
Type Equation â—˝  
The Body â—˝  
Navier Stokes â—˝  
Time Step â—˝  
Navier Stokes Equations â—˝  
Highly Nonlinear â—˝  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Numerical Solutions of the Viscous Flow Equations for a Class of Closed Flows

1965 â—˝  
Vol 69 (658) â—˝  
pp. 714-718 â—˝  
Author(s):  
Ronald D. Mills
Keyword(s):  
Moving Boundary â—˝  
Numerical Solutions â—˝  
Stokes Equations â—˝  
Fluid Motion â—˝  
Navier Stokes â—˝  
Steady Flows â—˝  
Flow Equations â—˝  
Steady Streaming â—˝  
Difference Formula â—˝  
Surface Passing

The Navier-Stokes equations are solved iteratively on a small digital computer for the class of flows generated within a rectangular “cavity” by a surface passing over its open end. Solutions are presented for depth/breadth ratios ƛ=0.5 (shallow), 10 (square), 20 (deep) and Reynolds number 100. Flow photographs ore obtained which largely confirm the predicted flows. The theoretical velocity profiles and pressure distributions through the centre of the vortex in the square cavity are calculated.In an appendix an improved finite difference formula is given for the vorticity generated at a moving boundary.Since Thorn began his pioneering work some thirty-five years ago the number of numerical solutions which have been obtained for the equations of incompressible viscous fluid motion remains small (see bibliographies of Thom and Apelt, Fromm). The known solutions are principally for steady streaming flows, although two methods have now been used with success for non-steady flows (Payne jets and Fromm flow past obstacles). By contrast this paper is concerned with the class of closed flows generated in a rectangular region of varying depth/breadth ratio by a surface passing over an open end. This problem has been considered for a number of reasons.

scholarly journals AN OPTIMAL DESIGN TECHNOLOGY FOR AERODYNAMIC CONFIGURATIONS BASED ON THE NUMERICAL SOLUTIONS OF THE FULL NAVIER-STOKES EQUATIONS

2017 â—˝  
pp. 90-98 â—˝  
Author(s):  
Sergey Vladimirovich PEYGIN â—˝  
â—˝  
Kirill Aleksandrovich STEPANOV â—˝  
Sergey Viktorovich TIMCHENKO â—˝  
â—˝  
...  
Keyword(s):  
Optimal Design â—˝  
Numerical Solutions â—˝  
Stokes Equations â—˝  
Navier Stokes â—˝  
Design Technology â—˝  
Navier Stokes Equations
Sign in / Sign up

Export Citation Format

Share Document