scholarly journals A review of one-phase Hele-Shaw flows and a level-set method for nonstandard configurations

2021 ◽  
Vol 63 ◽  
pp. 269-307
Author(s):  
Liam C. Morrow ◽  
Timothy J. Moroney ◽  
Michael C. Dallaston ◽  
Scott W. McCue
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Moving Boundary ◽  
Classical Model ◽  
Fluid Velocity ◽  
Boundary Integral ◽  
Taylor Instability ◽  
Integral Methods ◽  
Highly Nonlinear ◽  
Hele Shaw Cell

The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.   doi:10.1017/S144618112100033X

scholarly journals A REVIEW OF ONE-PHASE HELE-SHAW FLOWS AND A LEVEL-SET METHOD FOR NONSTANDARD CONFIGURATIONS

2021 ◽  
Vol 63 (3) ◽  
pp. 269-307
Author(s):  
LIAM C. MORROW ◽  
TIMOTHY J. MORONEY ◽  
MICHAEL C. DALLASTON ◽  
SCOTT W. MCCUE
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Moving Boundary ◽  
Classical Model ◽  
Fluid Velocity ◽  
Boundary Integral ◽  
Taylor Instability ◽  
Integral Methods ◽  
Highly Nonlinear ◽  
Hele Shaw Cell

AbstractThe classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.

scholarly journals A coupled level set-boundary integral method for moving boundary simulations

2005 ◽  
pp. 277-302 ◽  
Author(s):  
M. Garzon ◽  
David Adalsteinsson ◽  
L. Gray ◽  
James Sethian
Keyword(s):  
Level Set ◽  
Integral Method ◽  
Moving Boundary ◽  
Boundary Integral ◽  
Boundary Integral Method

NUMERICAL SIMULATION OF RAYLEIGH–TAYLOR INSTABILITY BASED ON AN IMPROVED PARTICLE LEVEL SET METHOD

2014 ◽  
Vol 11 (04) ◽  
pp. 1350094 ◽  
Author(s):  
HUI TIAN ◽  
GUOJUN LI ◽  
XIONGWEN ZHANG
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Stokes Equations ◽  
Density Ratio ◽  
Immiscible Fluids ◽  
Taylor Instability ◽  
Wide Range ◽  
Particle Level ◽  
Rayleigh Taylor Instability ◽  
Particle Level Set Method

An improved particle correction procedure for particle level set method is proposed and applied to the simulation of Rayleigh–Taylor instability (RTI) of the incompressible two-phase immiscible fluids. In the proposed method, an improved particle correction method is developed to deal with all the relative positions between escaped particles and cell corners, which can reduce the disturbance arising in the distance function correction process due to the non-normal direction movement of escaped particles. The improved method is validated through accurately capturing the moving interface of the Zalesak's disk. Furthermore, coupled with the projection method for solving the Navier–Stokes equations, the time-dependent evolution of the RTI interface over a wide range of Reynolds numbers, Atwood numbers and Weber numbers are numerically investigated. A good agreement between the present results and the existing analytical solutions is obtained and the accuracy of the proposed method is further verified. Moreover, the effects of control parameters including viscosity, density ratio, and surface tension coefficient on the evolution of RTI are analyzed in detail, and a critical Weber number for the development of RTI is found.

scholarly journals Wave breaking over sloping beaches using a coupled boundary integral-level set method

2003 ◽  
Author(s):  
M. Garzon ◽  
D. Adalsteinsson ◽  
L. Gray ◽  
J.A. Sethian
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Wave Breaking ◽  
Boundary Integral

scholarly journals Numerical and Experimental Analysis of the Growth of Gravitational Interfacial Instability Generated by Two Viscous Fluids of Different Densities

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Snehamoy Majumder ◽  
Debajit Saha ◽  
Partha Mishra
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Gravitational Instability ◽  
Physical Phenomenon ◽  
Taylor Instability ◽  
Interfacial Instability ◽  
Heaviside Function ◽  
Two Fluids ◽  
Gravity Driven ◽  
Rayleigh Taylor Instability

In the geophysical context, there are a wide variety of mechanisms which may lead to the formation of unstable density stratification, leading in turn to the development of the Rayleigh-Taylor instability and, more generally, interfacial gravity-driven instabilities, which involves moving boundaries and interfaces. The purpose of this work is to study the level set method and to apply the process to study the Rayleigh-Taylor instability experimentally and numerically. With the help of a simple, inexpensive experimental arrangement, the R-T instability has been visualized with moderate accuracy for real fluids. The same physical phenomenon has been investigated numerically to track the interface of two fluids of different densities to observe the gravitational instability with the application of level set method coupled with volume of fraction replacing the Heaviside function. Good agreement between theory and experimental results was found and growth of instability for both of the methods has been plotted.

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