In the present paper, we examine the viscous flow evolution in a square cavity. Coupled with the stream function, the initial-boundary value problem of the vorticity is numerically solved by a method of iteration. The only boundary condition is the wall velocity which in turn defines the Dirichlet value for the stream function. We assert that the corner singularity in the cavity flow is in fact a theoretical artefact. By adopting suitably regulated lid velocity, excellent comparison with experiments is found because of the converged palinstrophy field. Our calculations demonstrate that asymmetrical shears initiate the formation of the vortices with counter-rotating strained cores, while the consecutive re-birth and the subsequent disintegration of the developed eddies proliferate the shears into smaller scales. This production process is particularly intense in close proximity to the solid surfaces.
Primary instability of the lid-driven flow in a cube is studied by a comprehensive linear stability approach. Two cases, in which the lid moves parallel to the cube sidewall or parallel to the diagonal plane, are considered. The SIMPLE procedure is applied for evaluation of the Krylov vectors needed for application of the Newton and Arnoldi iteration methods. The finite volume grid is gradually refined from 1003 to 2563 nodes. The computations result in the grid convergent values of the critical Reynolds number and oscillation frequency. Patterns of the most unstable perturbation are reported.…
In this paper, we developed a mathematical model for the sedimentation process of small particles in a lid-driven cavity flow with an obstacle to model the transport of microplastic particles in rivers. A stationary incompressible Navier-Stokes simulation at moderate Reynolds-numbers provides the background flow field. Spherical particles are injected into this flow field and their equation of motion is solved to determine the number of particles that sediment on the surface of the obstacle. To capture the effect of typical biological organisms and biofilms on the bottom surface of river beds,…
Direct Numerical Simulation have been carried out for a two dimensional flow in a Lid driven cavity at Reynolds number 5000 and Prandtl number 7 with water as the working fluid. Both the side walls of the enclosure are insulated(i.e. adiabatic boundary condition), while the bottom plate is at higher temperature and the top wall is at colder temperature. Effects of heating of the bottom wall and movement of the top lid have been investigated by conducting numerical simulations at different Richardson numbers by varying from low and moderate magnitudes within the limits of Boussinesq-approximati…
The breakup of a jet of a viscous fluid with viscosity $μ_{1}$ immersed into another viscous fluid with viscosity $μ_{2}$ is considered in the limit when the viscosity ratio $λ=μ_{1}/μ_{2}$ is close to zero. We show that, in this limit, a transition from ordinary continuous selfsimilarity to discrete selfsimilarity takes place as $λ$decreases. The result being that instead of a single point breakup, the rupture of the inner jet occurs through the appearance of an infinite sequence of filaments of decreasing size that will eventually produce infinite sequences of bubbles of the inner fluid insi…
Solution to two-dimensional Incompressible Navier-Stokes Equations with SIMPLE, SIMPLER and Vorticity-Stream Function Approaches. Driven-Lid Cavity Problem: Solution and Visualization
Wall-driven flow in square cavity has been studied extensively, yet it is more frequently for the rectangular cavity flow occurring in practical problems, and some flow characteristics about rectangular cavity have not been fully investigated. As a promising numerical simulation tool, the Lattice Boltzmann Method (LBM) is employed to simulate the lid-driven flow in a two-dimensional rectangular cavity in this paper. First, the code is validated for the standard square cavity, the velocity profiles, stream function values and center positions of the primary and second vortexes at different Re a…
We propose a Finite Volume Method for the simulation of elastoviscoplastic flows, modelled after the extension to the Herschel-Bulkley model by Saramito [J. Non-Newton. Fluid Mech. 158 (2009) 154-161]. The method is akin to methods for viscoelastic flows. It is applicable to cell-centred grids, both structured and unstructured, and includes a novel pressure stabilisation technique of the "momentum interpolation" type. Stabilisation of the velocity and stresses is achieved through a "both sides diffusion" technique and the CUBISTA convection scheme, respectively. A second-order accurate tempora…
In this study, the hyperbolic method is adopted to explore the flow field states of incompressible flow in a four-sided lid-driven square cavity. In particular, we focus on the flow bifurcation obtained at the critical Reynolds number $R_e \simeq 130$. In the hyperbolic method, the diffusive term is transformed into an hyperbolic one by introducing a diffusion flux term, which is the solution of an additional equation. A classical Riemann-like solver with a finite-volume discretization is thus employed for the full flux (splitted into advective and diffusive parts), in order to solve the stead…
The proposed flow in a 3-D cubic cavity is driven by its parallel walls moving in perpendicular directions to create a genuinely three-dimensional highly separated vortical flow yet having simple single-block cubical geometry of computational domain. The elevated level of helicity is caused by motion of a wall in the direction of axis of primary vortex created by a parallel wall. The velocity vector field is obtained numerically by using second-order upwind scheme and 200^3 grid. Helicity, magnitude of normalized helicity and kinematic vorticity number are evaluated for Reynolds numbers rangin…
On the basis of the two-fluid hydrodynamics, an analogue of the famous Rayleigh-Plesse equation for the dynamics of a spherical bubble in superfluid helium is obtained. The mass flow velocity $v$ and the velocity of the normal component $v_{n}$ were chosen as independent variables. Due to the two-fluid nature of HeII, the cross terms in the evolution equation for the boundary position $\ R(t)$ appeared, which were absent in classical Rayleigh-Plesset equation in ordinary fluids. One of them renormilizes the coefficient in front of $(dR/dt)^{2}$. Another additional term formally coinciding with…
High resolution simulations of incompressible flows have become routine across a range of engineering applications. Despite their routine use, due to the high dimensional parameter space present for most practical applications, a comprehensive exploration of the available parameter space is often impractical. In this work, we demonstrate the ability of physics-constrained deep learning methods to provide an efficient means of exploring high-dimensional parameter spaces with minimal amounts of data from high resolution computational fluid dynamic simulations. As a specific application, we choos…
Electron transport in two-dimensional conducting materials such as graphene, with dominant electron-electron interaction, exhibits unusual vortex flow that leads to a nonlocal current-field relation (negative resistance), distinct from the classical Ohm's law. The transport behavior of these materials is best described by low Reynolds number hydrodynamics, where the constitutive pressure-speed relation is Stoke's law. Here we report evidence of such vortices observed in a viscous flow of Newtonian fluid in a microfluidic device consisting of a rectangular cavity$-$analogous to the electronic s…
In this work, we show that the widely used bounce-back boundary condition is an incomplete form of the diffuse reflection boundary condition at the continuum limit for lattice Boltzmann simulations. By utilizing this fact, we can force the diffuse reflection scheme to work at its continuum limit so that the no-slip boundary condition can be implemented without any non-physical slip velocity error being induced by the standard bounce-back scheme. The revised boundary formulation is validated numerically by solving a pressure-driven channel flow, a lid-driven cavity flow and channel flow around …
