Dynamic simulation of a coated microbubble in an unbounded flow: response to a step change in pressure

2017 ◽  
Vol 822 ◽  
pp. 717-761 ◽  
Author(s):  
M. Vlachomitrou ◽  
N. Pelekasis
Keyword(s):  
Stokes Equations ◽  
Step Change ◽  
Basis Functions ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Viscous Forces ◽  
Compressive Stresses ◽  
Post Buckling ◽  
Flow Domain

A numerical method is developed to study the dynamic behaviour of an encapsulated bubble when the viscous forces of the surrounding liquid are accounted for. The continuity and Navier–Stokes equations are solved for the liquid, whereas the coating is described as a viscoelastic shell with bending resistance. The Galerkin Finite Element Methodology is employed for the spatial discretization of the flow domain surrounding the bubble, with the standard staggered grid arrangement that uses biquadratic and bilinear Lagrangian basis functions for the velocity and pressure in the liquid, respectively, coupled with a superparametric scheme with $B$-cubic splines as basis functions pertaining to the location of the interface. The spine method and the elliptic mesh generation technique are used for updating the mesh points in the interior of the flow domain as the shape of the interface evolves with time, with the latter being distinctly superior in capturing severely distorted shapes. The stabilizing effect of the liquid viscosity is demonstrated, as it alters the amplitude of the disturbance for which a bubble deforms and/or collapses. For a step change in the far-field pressure the dynamic evolution of the microbubble is captured until a static equilibrium is achieved. Static shapes that are significantly compressed are captured in the post-buckling regime, leading to symmetric or asymmetric shapes, depending on the relative dilatation to bending stiffness ratio. As the external overpressure increases, shapes corresponding to all the solution families that were captured evolve to exhibit contact as the two poles approach each other. Shell viscosity prevents jet formation by relaxing compressive stresses and bending moments around the indentation generated at the poles due to shell buckling. This behaviour is conjectured to be the inception process leading to static shapes with contact regions.

The Effects of a Hydrostatic Pocket Aspect Ratio, Supply Orifice Position, and Attack Angle on Steady-State Flow Patterns, Pressure, and Shear Characteristics

1993 ◽  
Vol 115 (4) ◽  
pp. 678-685 ◽  
Author(s):  
M. J. Braun ◽  
F. K. Choy ◽  
Y. M. Zhou
Keyword(s):  
Stokes Equations ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Computational Error ◽  
Convergence Criteria ◽  
Steady State Flow ◽  
Shear Forces ◽  
Navier Stokes Equations ◽  
Recirculating Flow ◽  
Shear Characteristics

The flow in a hydrostatic pocket is described by a mathematical model that uses the Navier-Stokes equations written in terms of the primary variables, u, v, and p. Using the conservative formulation, a finite difference method is applied through a staggered grid. The power law scheme is applied in the treatment of the convective terms for this highly recirculating flow. The discussion pertaining to the convergence of the numerical scheme and the computational error, shows that the strict convergence criteria applied to both velocities and pressure were successfully statisfied. The numerical model is applied in a parametric mode to the study of the velocities, the pressure patterns, and shear forces that characterize the flow in a square (α = 1), deep (α>1), and shallow (α≪1) hydrostatic pocket. The effects of the variation of the location and angle of the hydrostatic jet are also investigated.

The Numerical Prediction of Viscous Flow and Heat Transfer in Tube Banks

1978 ◽  
Vol 100 (4) ◽  
pp. 565-571 ◽  
Author(s):  
B. E. Launder ◽  
T. H. Massey
Keyword(s):  
Heat Transfer ◽  
Viscous Flow ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Substantial Contribution ◽  
Navier Stokes Equations ◽  
Flow And Heat Transfer ◽  
Flow Domain ◽  
Tube Banks

A scheme for handling the numerical analysis of viscous flow and heat transfer in tube banks is presented. It involves the use of a cylindrical network of nodes in the vicinity of the tubes with a Cartesian mesh covering the remainder of the flow domain. The approach has been incorporated into the numerical solving algorithm for the Navier Stokes equations of Gasman, et al. [8]. A number of demonstration calculations is presented including a numerical simulation of the staggered square bank for which Bergelin and co-workers [4, 9] have reported experimental results for pressure drop and heat transfer rate. Agreement between predicted and measured characteristics is satisfactory when account is taken of end and entry effects that are present in the experiments but necessarily omitted from the calculations. Indeed the close agreement of the laminar predictions with measurements extends to Reynolds numbers in excess of 1000, a level at which it has hitherto been supposed that turbulent motion in the fluid made a substantial contribution to friction and heat transfer.

scholarly journals The Ocean–Land–Atmosphere Model (OLAM). Part II: Formulation and Tests of the Nonhydrostatic Dynamic Core

2008 ◽  
Vol 136 (11) ◽  
pp. 4045-4062 ◽  
Author(s):  
Robert L. Walko ◽  
Roni Avissar
Keyword(s):  
Stokes Equations ◽  
Vertical Motion ◽  
Triangular Mesh ◽  
Atmospheric Modeling ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Wave Flow ◽  
Navier Stokes Equations ◽  
Cell Method ◽  
Atmosphere Model

Abstract The dynamic core of the Ocean–Land–Atmosphere Model (OLAM), which is a new global model that is partly based on the Regional Atmospheric Modeling System (RAMS), is described and tested. OLAM adopts many features of its predecessor, but its dynamic core is new and incorporates a global geodesic grid with triangular mesh cells and a finite-volume discretization of the nonhydrostatic compressible Navier–Stokes equations. The spatial discretization of horizontal momentum is based on a C-staggered grid and uses a method that has not been previously applied in atmospheric modeling. The temporal discretization uses a unique form of time splitting that enforces consistency of advecting mass flux among all conservation equations. OLAM grid levels are horizontal, and topography is represented by the shaved-cell method. Aspects of the shaved-cell method that pertain to the OLAM discretization on the triangular mesh are described, and a method of conserving momentum in shaved cells on a C-staggered grid is presented. The dynamic core was tested in simulations with multiple vertical model levels and significant vertical motion. The tests include an idealized global circulation simulation, a cold density current, and mountain-wave flow over an orographic barrier, all of which are well-known standard benchmark experiments. OLAM gave acceptable results in all tests, demonstrating that its dynamic core produces accurate and robust solutions.

Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations

2013 ◽  
Vol 13 (4) ◽  
pp. 1013-1044 ◽  
Author(s):  
Haiyang Gao ◽  
Z. J. Wang ◽  
H. T. Huynh
Keyword(s):  
Discontinuous Galerkin ◽  
Stokes Equations ◽  
High Order Accuracy ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Correction Procedure ◽  
Spectral Difference ◽  
Navier Stokes Equations ◽  
Differential Formulation ◽  
Continuous Scheme

AbstractA new approach to high-order accuracy for the numerical solution of conservation laws introduced by Huynh and extended to simplexes by Wang and Gao is renamed CPR (correction procedure or collocation penalty via reconstruction). The CPR approach employs the differential form of the equation and accounts for the jumps in flux values at the cell boundaries by a correction procedure. In addition to being simple and economical, it unifies several existing methods including discontinuous Galerkin, staggered grid, spectral volume, and spectral difference. To discretize the dif-fusion terms, we use the BR2 (Bassi and Rebay), interior penalty, compact DG (CDG), and I-continuous approaches. The first three of these approaches, originally derived using the integral formulation, were recast here in the CPR framework, whereas the I-continuous scheme, originally derived for a quadrilateral mesh, was extended to a triangular mesh. Fourier stability and accuracy analyses for these schemes on quadrilateral and triangular meshes are carried out. Finally, results for the Navier-Stokes equations are shown to compare the various schemes as well as to demonstrate the capability of the CPR approach.

Dynamic and Stability Response of a Cylindrical Shell Subjected to Viscous Annular Flow and Thermal Load

2016 ◽  
Vol 16 (10) ◽  
pp. 1550072 ◽  
Author(s):  
W.-B. Ning ◽  
D.-Z. Wang
Keyword(s):  
Cylindrical Shell ◽  
Thermal Load ◽  
Annular Flow ◽  
Stokes Equations ◽  
Contour Method ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Critical Flow Velocity ◽  
Fluid Forces ◽  
Viscous Forces

This paper presents an analytical approach for investigating the dynamics and stability of an outer cylindrical shell conveying viscous fluid (i.e. water) in the annulus between the inner shell-type body and the outer shell with thermal load. The steady viscous forces that induce prestress on the shells are determined based on the time–mean Navier–Stokes equations. The shell motions are described by Flügge’s shell equations incorporating the prestress arising from the viscous effect. The shell-vibration-induced fluid forces are described by means of the potential flow theory, and the thermal loads are determined by the thermoelastic theory. The analytical model is conducted by the zero-level contour method with the aid of the weighted residual technology. The present study shows that the effect of viscosity in the annular flow renders the system more unstable. Moreover, the thermal load tends to reduce the critical flow velocity pronouncedly, for which there exists a critical temperature rise.

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