Showing papers on "Incompressible flow published in 2012"

Smoothed Particle Hydrodynamics and Its Diverse Applications

Abstract: This review focuses on the applications of smoothed particle hydrodynamics (SPH) to incompressible or nearly incompressible flow. In the past 17 years, the range of applications has increased as researchers have realized the ability of SPH algorithms to handle complex physical problems. These include the disruption of free surfaces when a wave hits a rocky beach, multifluid problems that may involve the motion of rigid and elastic bodies, non-Newtonian fluids, virtual surgery, and chemical precipitation from fluids moving through fractured media. SPH provides a fascinating tool that has some of the properties of molecular dynamics while retaining the attributes of the macroscopic equations of continuum mechanics.

Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities

Abstract: A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation, we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

A mixed finite element method for Darcy flow in fractured porous media with non-matching grids

Abstract: We consider an incompressible flow problem in a N -dimensional fractured porous domain (Darcy’s problem). The fracture is represented by a (N − 1)-dimensional interface, exchanging fluid with the surrounding media. In this paper we consider the lowest-order (ℝ T0 , ℙ0 ) Raviart-Thomas mixed finite element method for the approximation of the coupled Darcy’s flows in the porous media and within the fracture, with independent meshes for the respective domains. This is achieved thanks to an enrichment with discontinuous basis functions on triangles crossed by the fracture and a weak imposition of interface conditions. First, we study the stability and convergence properties of the resulting numerical scheme in the uncoupled case, when the known solution of the fracture problem provides an immersed boundary condition. We detail the implementation issues and discuss the algebraic properties of the associated linear system. Next, we focus on the coupled problem and propose an iterative porous domain/fracture domain iterative method to solve for fluid flow in both the porous media and the fracture and compare the results with those of a traditional monolithic approach. Numerical results are provided confirming convergence rates and algebraic properties predicted by the theory. In particular, we discuss preconditioning and equilibration techniques to make the condition number of the discrete problem independent of the position of the immersed interface. Finally, two and three dimensional simulations of Darcy’s flow in different configurations (highly and poorly permeable fracture) are analyzed and discussed.

Levelset based fluid topology optimization using the extended finite element method

Abstract: This study focuses on finding the optimal layout of fluidic devices subjected to incompressible flow at low Reynolds numbers. The proposed approach uses a levelset method to describe the fluid-solid interface geometry. The flow field is modeled by the incompressible Navier–Stokes equations and discretized by the extended finite element method (XFEM). The no-slip condition along the fluid-solid interface is enforced via a stabilized Lagrange multiplier method. Unlike the commonly used porosity approach, the XFEM approach does not rely on a material interpolation scheme, which allows for more flexibility in formulating the design problems. Further, it mitigates shortcomings of the porosity approach, including spurious pressure diffusion through solid material, strong dependency of the accuracy of the boundary enforcement with respect to the model parameters which may affect the optimization results, and poor boundary resolution. Numerical studies verify that the proposed method is able to recover optimization results obtained with the porosity approach. Further, it is demonstrated that the XFEM approach yields physical results for problems that cannot be solved with the porosity approach.

MultiFLIP for energetic two-phase fluid simulation

Abstract: Physically-based liquid animations often ignore the influence of air, giving up interesting behavior. We present a new method which treats both air and liquid as incompressible, more accurately reproducing the reality observed at scales relevant to computer animation. The Fluid Implicit Particle (FLIP) method, already shown to effectively simulate incompressible fluids with low numerical dissipation, is extended to two-phase flow by associating a phase bit with each particle. The liquid surface is reproduced at each time step from the particle positions, which are adjusted to prevent mixing near the surface and to allow for accurate surface tension. The liquid surface is adjusted around small-scale features so they are represented in the grid-based pressure projection, while separate, loosely coupled velocity fields reduce unwanted influence between the phases. The resulting scheme is easy to implement, requires little parameter tuning, and is shown to reproduce lively two-phase fluid phenomena.

On unsteady flows of implicitly constituted incompressible fluids

Abstract: We consider unsteady flows of incompressible fluids with a general implicit con- stitutive equation relating the deviatoric part of the Cauchy stress S and the symmetric part of the velocity gradient D in such a way that it leads to a maximal monotone (possibly multivalued) graph and the rate of dissipation is characterized by the sum of a Young function depending on D and its conjugate being a function of S. Such a framework is very robust and includes, among others, classical power-law fluids, stress power-law fluids, fluids with activation criteria of Bingham or Herschel-Bulkley type, and shear rate-dependent fluids with discontinuous viscosities as special cases. The appearance of S and D in all the assumptions characterizing the implicit relationship G(D,S )= 0 is fully symmetric. We establish long-time and large-data existence of weak solution to such a system completed by the initial and the Navier slip boundary conditions in both the subcrit- ical and supercritical cases. We use tools such as Orlicz functions, properties of spatially dependent maximal monotone operators, and Lipschitz approximations of Bochner functions taking values in Orlicz-Sobolev spaces.

Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows

Abstract: We consider passive scalar mixing by a prescribed divergence-free velocity vector field in a periodic box and address the following question: Starting from a given initial inhomogeneous distribution of passive tracers, and given a certain energy budget, power budget, or finite palenstrophy budget, what incompressible flow field best mixes the scalar quantity? We focus on the optimal stirring strategy recently proposed by Lin et al. [“Optimal stirring strategies for passive scalar mixing,” J. Fluid Mech. 675, 465 (2011)]10.1017/S0022112011000292 that determines the flow field that instantaneously maximizes the depletion of the H−1 mix-norm. In this work, we bridge some of the gap between the best available a priori analysis and simulation results. After recalling some previous analysis, we present an explicit example demonstrating finite-time perfect mixing with a finite energy constraint on the stirring flow. On the other hand, using a recent result by Wirosoetisno et al. [“Long time stability of a classi...

Local Poisson SPH For Viscous Incompressible Fluids

Abstract: Enforcing fluid incompressibility is one of the time-consuming aspects in SPH. In this paper, we present a local Poisson SPH (LPSPH) method to solve incompressibility for particle based fluid simulation. Considering the pressure Poisson equation, we first convert it into an integral form, and then apply a discretization to convert the continuous integral equation to a discretized summation over all the particles in the local pressure integration domain determined by the local geometry. To control the approximation error, we further integrate our local pressure solver into the predictive-corrective framework to avoid the computational cost of solving a pressure Poisson equation globally. Our method can effectively eliminate the large density deviations mainly caused by the solid boundary treatment and free surface topological change, and show advantage of a higher convergence rate over the predictive-corrective incompressible SPH (PCISPH). © 2012 Wiley Periodicals, Inc.

Aeroacoustic power generated by a compact axisymmetric cavity: prediction of self-sustained oscillation and influence of the depth

Abstract: Aeroacoustic power generation due to a self-sustained oscillation by an axisymmetric compact cavity exposed to a low-Mach-number grazing flow is studied both experimentally and numerically. The feedback effect is produced by the velocity fluctuations resulting from a coupling with acoustic standing waves in a coaxial pipe. A numerical methodology that combines incompressible flow simulations with vortex sound theory is used to predict the time-averaged acoustic source power generated by the cavity. The effect of cavity depth on the whistling is addressed. It is observed that the whistling occurs around a peak-whistling Strouhal number which depends on the cavity depth to width ratio. The proposed numerical method provides excellent predictions of the peak-whistling Strouhal number as a function of cavity depth. Given the oscillation amplitude, the numerical method predicts the time-averaged acoustic source power within a factor of two for moderate fluctuation amplitudes. For deep cavities the time-averaged acoustic source power appears to be independent of the cavity depth

Global stability of the two-dimensional flow over a backward-facing step

Abstract: The two-dimensional, incompressible flow over a backward-facing step is considered for a systematic variation of the geometry covering expansion ratios (step to outlet height) from 0.25 to 0.975. A global temporal linear stability analysis shows that the basic flow becomes unstable to different three-dimensional modes depending on the expansion ratio. All critical modes are essentially confined to the region behind the step extending downstream up to the reattachment point of the separated eddy. An energy-transfer analysis is applied to understand the physical nature of the instabilities. If scaled appropriately, the critical Reynolds number approaches a finite asymptotic value for very large step heights. In that case centrifugal forces destabilize the flow with respect to an oscillatory critical mode. For moderately large expansion ratios an elliptical instability mechanism is identified. If the step height is further decreased the critical mode changes from oscillatory to stationary. In addition to the elliptical mechanism, the strong shear in the layer emanating from the sharp corner of the step supports the amplification process of the critical mode. For very small step heights the basic state becomes unstable due to the lift-up mechanism, which feeds back on itself via the recirculating eddy behind the step, resulting in a steady critical mode comprising pronounced slow and fast streaks.

Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model

Abstract: In this paper, we present a numerical scheme for the diffuse-interface model in [Abels, Garcke, Gr\"un, M3AS 22(3), 2012] for two-phase flow of immiscible, incompressible fluids. As that model is in particular consistent with thermodynamics, energy estimates are expected to carry over to the discrete setting. By a subtle discretization of the convective coupling with the flux of the phase-field in the momentum equation, we prove discrete consistency with thermodynamics. Numerical experiments in two spatial dimensions -- ranging from Rayleigh-Taylor instability to a comparison with previous modeling approaches -- indicate the full practicality of our scheme and enable a first validation of the new modeling approach in [Abels, Garcke, Gr\"un, M3AS 22(3), 2012].

Closed-loop control of unsteadiness over a rounded backward-facing step

Abstract: The two-dimensional, incompressible flow over a rounded backward-facing step at Reynolds number ReD 600 is characterized by a detachment of the flow close to the step followed by a recirculation zone. Even though the flow is globally stable, perturbations are amplified as they are convected along the shear layer, and the presence of upstream random noise renders the flow unsteady, leading to a broadband spectrum of excited frequencies. This paper is aimed at suppressing this unsteadiness using a controller that converts a shear-stress measurement taken from a wall-mounted sensor into a control law that is supplied to an actuator. A comprehensive study of various components of closed-loop control design ‐ covering sensor placement, choice and influence of the cost functional, accuracy of the reduced-order model, compensator stability and performance ‐ shows that successful control of this flow requires a judicious balance between estimation speed and estimation accuracy, and between stability limits and performance requirements. The inherent amplification behaviour of the flow can be reduced by an order of magnitude if the above-mentioned constraints are observed. In particular, to achieve superior controller performance, the estimation sensor should be placed upstream near the actuator to ensure sufficient estimation speed. Also, if high-performance compensators are sought, a very accurate reduced-order model is required, especially for the dynamics between the actuator and the estimation sensor; otherwise, very minute errors even at low energies and high frequencies may render the large-scale compensated linearized simulation unstable. Finally, coupling the linear compensator to nonlinear simulations shows a gradual deterioration in control performance as the amplitude of the noise increases.

Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility

Abstract: Whether or not global solutions of the 2D Navier–Stokes–Cahn–Hilliard (NS–CH) system without full viscosity and mobility can develop finite time singularities is a difficult issue. A major result of this paper deals with global regularity of strong solutions for the NS–CH system with mixed partial viscosity and mobility. In addition, the 2D NS–CH system without viscosity but with full mobility is investigated. In this case, we also prove the global existence and uniqueness of classical solutions.

Transverse instability and low-frequency flapping in incompressible separated boundary layer flows : an experimental study

Abstract: The unstable dynamics of a transitional laminar separation bubble behind a two-dimensional bump geometry is investigated experimentally using dye visualizations and particle image velocimetry measurements. For Reynolds numbers above a critical value, the initially two-dimensional recirculation bubble is subject to modulations in the spanwise direction which can trigger vortex shedding. Increasing the Reynolds number further, the unstable behaviour is dominated by a low-frequency flapping motion, well known in transonic flows, and here investigated for the first time experimentally in an incompressible flow regime. These phenomena are characterized by non-intrusive measurements of the spatial structure and the frequencies of the unsteady motion. The results are in excellent agreement with previous numerical and theoretical predictions for the same geometry.

A Relaxed Splitting Preconditioner for the Incompressible Navier-Stokes Equations

Abstract: A relaxed splitting preconditioner based on matrix splitting is introduced in this paper for linear systems of saddle point problem arising from numerical solution of the incompressible Navier-Stokes equations. Spectral analysis of the preconditioned matrix is presented, and numerical experiments are carried out to illustrate the convergence behavior of the preconditioner for solving both steady and unsteady incompressible flow problems.

MHD flow and heat transfer of micropolar fluid between two porous disks

Abstract: A numerical study is carried out for the axisymmetric steady laminar incompressible flow of an electrically conducting micropolar fluid between two infinite parallel porous disks with the constant uniform injection through the surface of the disks. The fluid is subjected to an external transverse magnetic field. The governing nonlinear equations of motion are transformed into a dimensionless form through von Karman’s similarity transformation. An algorithm based on a finite difference scheme is used to solve the reduced coupled ordinary differential equations under associated boundary conditions. The effects of the Reynolds number, the magnetic parameter, the micropolar parameter, and the Prandtl number on the flow velocity and temperature distributions are discussed. The results agree well with those of the previously published work for special cases. The investigation predicts that the heat transfer rate at the surfaces of the disks increases with the increases in the Reynolds number, the magnetic parameter, and the Prandtl number. The shear stresses decrease with the increase in the injection while increase with the increase in the applied magnetic field. The shear stress factor is lower for micropolar fluids than for Newtonian fluids, which may be beneficial in the flow and thermal control in the polymeric processing.

Remarks on the lifespan of the solutions to some models of incompressible fluid mechanics

Abstract: We give lower bounds for the lifespan of a solution to the inviscid Boussinesq system. In dimension two, we point out that it tends to infinity when the initial (relative) temperature tends to zero. This is, to the best of our knowledge, the first result of this kind for the inviscid Boussinesq system. In passing, we provide continuation criteria (of independent interest) in the N -dimensional case. In the second part of the paper, our method is adapted to handle the axisymmetric incompressible Euler equations with swirl. Introduction The evolution of the velocity u = u(t, x) and pressure P = P (t, x) fields of a perfect homogeneous incompressible fluid is governed by the following Euler equations: (0.1) { ∂tu+ u · ∇u+∇P = 0, div u = 0. There is a huge literature concerning the well-posedness issue for Euler equations. Roughly, they may be solved locally in time in any reasonable Banach space embedded in the set C0,1 of bounded Lipschitz functions (see e.g. [1, 4, 6, 12, 13, 17, 19, 22]). In the two-dimensional case, it is well known that Euler equations are globally well-posed for sufficiently smooth initial data. This noticeable fact relies on the conservation of the vorticity ω := ∂1u 2 − ∂2u 1 along the flow of the velocity field, and has been first proved rigorously in the pioneering works by W. Wolibner [20] and V. Yudovich [21]. This conservation property is no longer true, however, in more physically relevant contexts such as (1) the three-dimensional setting for (0.1), (2) nonhomogeneous incompressible perfect fluids, (3) inviscid fluids subjected to a buoyancy force which is advected by the velocity fluid (the so-called inviscid Boussinesq system below). As a consequence, the problem of global existence for general (even smooth or small) data is still open for the above three cases. In a recent work [9], it has been shown that for slightly nonhomogeneous two-dimensional incompressible fluids, the lifespan tends to infinity when the nonhomogeneity tends to zero. The present paper is mainly dedicated to the study of the lifespan for the first and third item. More precisely, in the first section of the paper, we shall consider the inviscid Boussinesq system: (0.2)    ∂tθ + u · ∇θ = 0, ∂tu+ u · ∇u+∇P = θeN , div u = 0. Here the relative temperature θ = θ(t, x) is a real valued function and eN stands for the unit vertical vector. 2010 Mathematics Subject Classification. 35Q35,76B03. 1It need not be nonnegative as it designates the discrepancy to some reference temperature. 1

Strong Well-Posedness of a Diffuse Interface Model for a Viscous, Quasi-Incompressible Two-Phase Flow

Abstract: In this article we discuss the existence of strong solutions locally in time for a model of a binary mixture of viscous incompressible fluids in a bounded domain. The model was derived by Lowengrub and Truskinovski. It is used to describe a diffuse interface model for a two-phase flow of two viscous incompressible Newtonian fluids with different densities. The fluids are macroscopically immiscible but partially mix in a small interfacial region. The model leads to a system of Navier–Stokes/Cahn–Hilliard type. Using a suitable result on maximal $L^2$-regularity for the linearized system, the existence of strong solutions is shown with the aid of the contraction mapping principle. The analysis shows that in the case of different densities the system is coupled in highest order and the principal part of the linearized system is of very different structure compared to the case of same densities. The linear system is solved with the aid of a general result on an abstract damped wave equation by Chen and Triggiani.

Hybrid algorithm for modeling of fluid-structure interaction in incompressible, viscous flows

Abstract: The objective of this paper is to present and to validate a new hybrid coupling (HC) algorithm for modeling of fluid-structure interaction (FSI) in incompressible, viscous flows. The HC algorithm is able to avoid numerical instability issues associated with artificial added mass effects, which are often encountered by standard loosely coupled (LC) and tightly coupled (TC) algorithms, when modeling the FSI response of flexible structures in incompressible flow. The artificial added mass effect is caused by the lag in exchange of interfacial displacements and forces between the fluid and solid solvers in partitioned algorithms. The artificial added mass effect is much more prominent for light/flexible structures moving in water, because the fluid forces are in the same order of magnitude as the solid forces, and because the speed at which numerical errors propagate in an incompressible fluid. The new HC algorithm avoids numerical instability issues associated with artificial added mass effects by embedding Theodorsen’s analytical approximation of the hydroelastic forces in the solution process to obtain better initial estimates of the displacements. Details of the new HC algorithm are presented. Numerical validation studies are shown for the forced pitching response of a steel and a plastic hydrofoil. The results show that the HC algorithm is able to converge faster, and is able to avoid numerical instability issues, compared to standard LC and TC algorithms, when modeling the transient FSI response of a plastic hydrofoil. Although the HC algorithm is only demonstrated for a NACA0009 hydrofoil subject to pure pitching motion, the method can be easily extended to model general 3-D FSI response and stability of complex, flexible structures in turbulent, incompressible, multiphase flows.

Simulation of three-dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method

Abstract: An improved immersed boundary–lattice Boltzmann method (IB–LBM) developed recently [28] was applied in this work to simulate three-dimensional (3D) flows over moving objects. By enforcing the non-slip boundary condition, the method could avoid any flow penetration to the wall. In the developed IB–LBM solver, the flow field is obtained on the non-uniform mesh by the efficient LBM that is based on the second-order one-dimensional interpolation. As a consequence, its coefficients could be computed simply. By simulating flows over a stationary sphere and torus [28] accurately and efficiently, the proposed IB–LBM showed its ability to handle 3D flow problems with curved boundaries. In this paper, we further applied this method to simulate 3D flows around moving boundaries. As a first example, the flow over a rotating sphere was simulated. The obtained results agreed very well with the previous data in the literature. Then, simulation of flow over a rotating torus was conducted. The capability of the improved IB–LBM for solving 3D flows over moving objects with complex geometries was demonstrated via the simulations of fish swimming and dragonfly flight. The numerical results displayed quantitative and qualitative agreement with the date in the literature. Copyright © 2011 John Wiley & Sons, Ltd.