scholarly journals Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations

2012 ◽  
Vol 700 ◽  
pp. 105-147 ◽  
Author(s):  
Saverio E. Spagnolie ◽  
Eric Lauga
Keyword(s):  
General Framework ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fundamental Solutions ◽  
Reduced Model ◽  
Far Field ◽  
Free Surfaces ◽  
Source Dipole ◽  
Swimming Trajectories ◽  
Propulsive Mechanism

AbstractThe swimming trajectories of self-propelled organisms or synthetic devices in a viscous fluid can be altered by hydrodynamic interactions with nearby boundaries. We explore a multipole description of swimming bodies and provide a general framework for studying the fluid-mediated modifications to swimming trajectories. A general axisymmetric swimmer is described as a linear combination of fundamental solutions to the Stokes equations: a Stokeslet dipole, a source dipole, a Stokeslet quadrupole, and a rotlet dipole. The effects of nearby walls or stress-free surfaces on swimming trajectories are described through the contribution of each singularity, and we address the question of how accurately this multipole approach captures the wall effects observed in full numerical solutions of the Stokes equations. The reduced model is used to provide simple but accurate predictions of the wall-induced attraction and pitching dynamics for model Janus particles, ciliated organisms, and bacteria-like polar swimmers. Transitions in attraction and pitching behaviour as functions of body geometry and propulsive mechanism are described. The reduced model may help to explain a number of recent experimental results.

Calculation of Hydrodynamic Forces for Unsteady Stokes Flows by Singularity Integral Equations Based on Fundamental Solutions

2013 ◽  
Vol 30 (2) ◽  
pp. 129-136 ◽  
Author(s):  
C. H. Hsiao ◽  
D. L. Young
Keyword(s):  
Time Domain ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fundamental Solutions ◽  
Hydrodynamic Forces ◽  
Stokes Flows ◽  
Total Strength ◽  
Singularity Method ◽  
Unsteady Stokes Flow ◽  
Convolution Quadrature Method

ABSTRACTThe attractive feature of the singularity method for steady Stokes flows is that the hydrodynamic forces acting on the particle can be calculated by the total strength of distributed singularities. For unsteady Stokes flows, however we have to derive hydrodynamic forces acting on a solid body in terms of the strengths of both unsteady Stokeslets as well as unsteady potential dipoles if mass and force sources are both taken into consideration. Since the hydrodynamic force formulation results in a Volterra integral equation of the first kind, and the strengths are numerically approximated by means of the Lubich convolution quadrature method (CQM) in this study. As far as the numerical solutions of time-domain integral formulations of the unsteady Stokes equations are concerned, this paper requires only the Laplace-domain instead of the time- domain fundamental solutions of the governing equations. The stability and accuracy of the proposed method are verified through some well selected numerical examples. In total we include two examples presenting the accuracy of Lubich CQM, and another two examples for calculating general hydrody-namic forces of a sphere in oscillating or non-oscillating unsteady Stokes flows. It is concluded that this study is able to extend the unsteady Stokes flow theory to more general transient motions instead to limit to the oscillating flow assumption.

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

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

Steady approach of unsteady low-Reynolds-number flow past two rotating circular cylinders

2013 ◽  
Vol 736 ◽  
pp. 414-443 ◽  
Author(s):  
Y. Ueda ◽  
T. Kida ◽  
M. Iguchi
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Vortex Method ◽  
The Self ◽  
Circular Cylinders ◽  
Far Field ◽  
Flow Past ◽  
Low Reynolds

AbstractThe long-time viscous flow about two identical rotating circular cylinders in a side-by-side arrangement is investigated using an adaptive numerical scheme based on the vortex method. The Stokes solution of the steady flow about the two-cylinder cluster produces a uniform stream in the far field, which is the so-called Jeffery’s paradox. The present work first addresses the validation of the vortex method for a low-Reynolds-number computation. The unsteady flow past an abruptly started purely rotating circular cylinder is therefore computed and compared with an exact solution to the Navier–Stokes equations. The steady state is then found to be obtained for $t\gg 1$ with ${\mathit{Re}}_{\omega } {r}^{2} \ll t$, where the characteristic length and velocity are respectively normalized with the radius ${a}_{1} $ of the circular cylinder and the circumferential velocity ${\Omega }_{1} {a}_{1} $. Then, the influence of the Reynolds number ${\mathit{Re}}_{\omega } = { a}_{1}^{2} {\Omega }_{1} / \nu $ about the two-cylinder cluster is investigated in the range $0. 125\leqslant {\mathit{Re}}_{\omega } \leqslant 40$. The convection influence forms a pair of circulations (called self-induced closed streamlines) ahead of the cylinders to alter the symmetry of the streamline whereas the low-Reynolds-number computation (${\mathit{Re}}_{\omega } = 0. 125$) reaches the steady regime in a proper inner domain. The self-induced closed streamline is formed at far field due to the boundary condition being zero at infinity. When the two-cylinder cluster is immersed in a uniform flow, which is equivalent to Jeffery’s solution, the streamline behaves like excellent Jeffery’s flow at ${\mathit{Re}}_{\omega } = 1. 25$ (although the drag force is almost zero). On the other hand, the influence of the gap spacing between the cylinders is also investigated and it is shown that there are two kinds of flow regimes including Jeffery’s flow. At a proper distance from the cylinders, the self-induced far-field velocity, which is almost equivalent to Jeffery’s solution, is successfully observed in a two-cylinder arrangement.

ANALYSIS OF A HETEROGENEOUS DOMAIN DECOMPOSITION FOR COMPRESSIBLE VISCOUS FLOW

2001 ◽  
Vol 11 (04) ◽  
pp. 565-599 ◽  
Author(s):  
CRISTIAN A. COCLICI ◽  
WOLFGANG L. WENDLAND
Keyword(s):  
Domain Decomposition ◽  
Stokes Equations ◽  
Domain Decomposition Method ◽  
Near Field ◽  
Outer Region ◽  
Navier Stokes ◽  
Far Field ◽  
Computational Domain ◽  
Linearized Euler Equations ◽  
Naca0012 Airfoil

We analyze a nonoverlapping domain decomposition method for the treatment of two-dimensional compressible viscous flows around airfoils. Since at some distance to the given profile the inertial forces are strongly dominant, there the viscosity effects are neglected and the flow is assumed to be inviscid. Accordingly, we consider a decomposition of the original flow field into a bounded computational domain (near field) and a complementary outer region (far field). The compressible Navier–Stokes equations are used close to the profile and are coupled with the linearized Euler equations in the far field by appropriate transmission conditions, according to the physical properties and the mathematical type of the corresponding partial differential equations. We present some results of flow around the NACA0012 airfoil and develop an a posteriori analysis of the approximate solution, showing that conservation of mass, momentum and energy are asymptotically attained with the linear model in the far field.

Spin-up from rest of a compressible fluid in a rapidly rotating cylinder

1992 ◽  
Vol 237 ◽  
pp. 413-434 ◽  
Author(s):  
Jae Min Hyun ◽  
Jun Sang Park
Keyword(s):  
Mach Number ◽  
Compressible Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Direction ◽  
Ekman Layer ◽  
Infinite Cylinder ◽  
Full Time ◽  
Viscous Effects ◽  
Initial State

Spin-up flows of a compressible gas in a finite, closed cylinder from an initial state of rest are studied, The flow is characterized by small reference Ekman numbers, and the peripheral Mach number is O(1). Comprehensive numerical solutions have been obtained for the full, time-dependent compressible Navier-Stokes equations. The details of the flow, temperature, and density evolution are described. In the early phase of spin-up, owing to the thermoacoustic disturbances caused by the compressible Rayleigh effect, the flows are oscillatory, and this oscillatory behaviour is pronounced at higher Mach numbers. The principal dynamical role of the Ekman layer is dominant over moderate times of orders of the homogeneous spin-up timescales. Owing to the density stratification in the radial direction, the Ekman layer is thicker in the central region of the interior. The interior azimuthal flows are mainly uniform in the axial direction. As the Mach number increases, the rate of spin-up in the interior becomes slower, and the propagating shear front is more diffusive. Explicit comparisons with the results for an infinite cylinder are made to ascertain the contributions of the endwall disks. In contrast to the usual incompressible spin-up from rest, the viscous effects are relatively more important for the case of a compressible fluid.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
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 ◽  
Vol 69 (658) ◽  
pp. 714-718 ◽  
Author(s):  
Ronald D. Mills
Keyword(s):  
Moving Boundary ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Navier Stokes ◽  
Steady Flows ◽  
Flow Equations ◽  
Steady Streaming ◽  
Difference Formula ◽  
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.

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