Showing papers on "Incompressible flow published in 2013"

A simple SPH algorithm for multi-fluid flow with high density ratios

Abstract: SUMMARY In this paper, we describe an SPH algorithm for multi-fluid flow, which is efficient, simple and robust. We derive the inviscid equations of motion from a Lagrangian together with the constraint provided by the continuity equation. The viscous flow equations then follow by adding a viscous term. Rigid boundaries are simulated using boundary force particles in a manner similar to the immersed boundary method. Each fluid is approximated as weakly compressible with a speed of sound sufficiently large to guarantee that the relative density variations are typically 1%. When the SPH force interaction is between two particles of different fluids, we increase the pressure terms. This simple procedure stabilizes the interface between the fluids. The equations of motion are integrated using a time stepping rule based on a second-order symplectic integrator. When linear and angular momentum should be conserved exactly, they are conserved to within round-off errors. We test the algorithm by simulating a variety of problems involving fluids with a density ratio in the range 1–1000. The first of these is a free surface problem with no rigid boundaries. It involves the flow of an elliptical distribution with one fluid inside the other. We show that the simulations converge as the particle spacing decreases, and the results are in good agreement with the exact inviscid, incompressible theory. The second test is similar to the first but involves the nonlinear oscillation of the fluids. As in the first test, the agreement with theory is very good, and the method converges. The third test is the simulation of waves at the interface between two fluids. The method is shown to converge, and the agreement with theory is satisfactory. The fourth test is the Rayleigh–Taylor instability for a configuration considered by other authors. Key parameters are shown to converge, and the agreement with other authors is good. The fifth and final test is how well the SPH method simulates gravity currents with density ratios in the range 2–30. The results of these simulations are in very good agreement with those of other authors and in satisfactory agreement with experimental results.Copyright © 2012 John Wiley & Sons, Ltd.

Large-Eddy Simulation in Hydraulics

Abstract: Preface 1 Introduction 1.1 The role and importance of turbulence in hydraulics 1.2 Characteristics of turbulence 1.3 Calculation approaches for turbulent flows 1.4 Scope and outline of the book 2 Basic methodology of LES 2.1 Navier-Stokes equations and Reynolds Averaging (RANS) 2.2 The idea of LES 2.3 Spatial filtering/averaging and resulting equations 2.4 Implicit filtering and Schumann's approach 2.5 Relation of LES to DNS and RANS 3 Subgrid-Scale (SGS) models 3.1 Role and desired qualities of an SGS-model 3.2 Smagorinsky model 3.3 Improved versions of eddy viscosity models 3.4 SGS models not based on the eddy viscosity concept 3.5 SGS models for the scalar transport equation 4 Numerical methods 4.1 Introduction 4.2 Discretization methods 4.3 Numerical accuracy in LES 4.4 Numerical errors 4.5 Solution methods for incompressible flow equations 4.6 LES grids 5 Implicit LES (ILES) 5.1 Introduction 5.2 Rationale for ILES and connection with LES using explicit SGS models 5.3 Adaptive Local Deconvolution Model (ALDM) 5.4 Monotonically Integrated LES (MILES) 6 Boundary and initial conditions 6.1 Periodic boundary conditions 6.2 Outflow boundary conditions 6.3 Inflow boundary conditions 6.4 Free surface boundary conditions 6.5 Smooth-wall boundary conditions 6.6 Rough-wall boundary conditions 6.7 Initial conditions 7 Hybrid RANS-LES methods 7.1 Introduction 7.2 Two-layer models 7.3 Embedded LES 7.4 Detached Eddy Simulation (DES) models 7.5 Scale-Adaptive Simulation (SAS) model 7.6 Final comments on hybrid RANS-LES models and future trends 8 Eduction of turbulence structures 8.1 Structure eduction from point signals: Two-point correlations and velocity spectra 8.2 Structure eduction from instantaneous quantities in 2D planes 8.3 Structure eduction from isosurfaces of instantaneous quantities in 3D space 9 Application examples of LES in hydraulics 9.1 Developed straight open channel flow 9.2 Flow over rough and permeable beds 9.3 Flow over bedforms 9.4 Flow through vegetation 9.5 Flow in compound channels 9.6 Flow in curved open channels 9.7 Shallow merging flows 9.8 Flow past in-stream hydraulic structures 9.9 Flow and mass exchange processes around a channel-bottom cavity 9.10 Gravity currents 9.11 Eco-hydraulics: Flow past an array of freshwater mussels 9.12 Flow in a water pump intake Appendix A - Introduction to tensor notation References Index

Isogeometric divergence-conforming b-splines for the darcy-stokes-brinkman equations

Abstract: We develop divergence-conforming B-spline discretizations for the numerical solution of the Darcy–Stokes–Brinkman equations. These discretizations are motivated by the recent theory of isogeometric discrete differential forms and may be interpreted as smooth generalizations of Raviart–Thomas elements. The new discretizations are (at least) patchwise C0 and can be directly utilized in the Galerkin solution of Darcy–Stokes–Brinkman flow for single-patch configurations. When applied to incompressible flows, these discretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. In the presence of no-slip boundary conditions and multi-patch geometries, the discontinuous Galerkin framework is invoked to enforce tangential continuity without upsetting the conservation or stability properties of the method across patch boundaries. Furthermore, as no-slip boundary conditions are enforced weakly, the method automatically defaults to a compatible discretization of Darcy flow in the limit of vanishing viscosity. The proposed discretizations are extended to general mapped geometries using divergence-preserving transformations. For sufficiently regular single-patch solutions, we prove a priori error estimates which are optimal for the discrete velocity field and suboptimal, by one order, for the discrete pressure field. Our estimates are in addition robust with respect to the parameters of the Darcy–Stokes–Brinkman problem. We present a comprehensive suite of numerical experiments which indicate optimal convergence rates for both the discrete velocity and pressure fields for general configurations, suggesting that our a priori estimates may be conservative. The focus of this paper is strictly on incompressible flows, but our theoretical results naturally extend to flows characterized by mass sources and sinks.

On Convergent Schemes for Diffuse Interface Models for Two-Phase Flow of Incompressible Fluids with General Mass Densities

Abstract: We are concerned with convergence results for fully discrete finite-element schemes suggested in [Grun and Klingbeil, J. Comput. Phys., in press, DOI: 10.1016/j.jcp.2013.10.028]. They were developed for the diffuse interface model in [H. Abels, H. Garcke, and G. Grun, Math. Models Methods Appl. Sci., 22 (2012), 1150013], which describes two-phase flow of immiscible, incompressible viscous fluids. We formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution. We identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution. In this case, our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions. Our argument is particularly based on higher regularity results ...

Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions

Abstract: In this paper, we give some new global regularity criteria for three-dimensional incompressible magnetohydrodynamics (MHD) equations. More precisely, we provide some sufficient conditions in terms of the derivatives of the velocity or pressure, for the global regularity of strong solutions to 3D incompressible MHD equations in the whole space, as well as for periodic boundary conditions. Moreover, the regularity criterion involving three of the nine components of the velocity gradient tensor is also obtained. The main results generalize the recent work by Cao and Wu (2010 Two regularity criteria for the 3D MHD equations J. Diff. Eqns 248 2263–74) and the analysis in part is based on the works by Cao C and Titi E (2008 Regularity criteria for the three-dimensional Navier–Stokes equations Indiana Univ. Math. J. 57 2643–61; 2011 Gobal regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor Arch. Rational Mech. Anal. 202 919–32) for 3D incompressible Navier–Stokes equations.

Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

Abstract: We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels et al. for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier–Stokes system with a modified convective term coupled to a Cahn–Hilliard system. The density of the mixture depends on an order parameter.

SIMPLE‐type preconditioners for cell‐centered, colocated finite volume discretization of incompressible Reynolds‐averaged Navier–Stokes equations

Abstract: SUMMARY This paper contains a comparison of four SIMPLE-type methods used as solver and as preconditioner for the iterative solution of the (Reynolds-averaged) Navier–Stokes equations, discretized with a finite volume method for cell-centered, colocated variables on unstructured grids. A matrix-free implementation is presented, and special attention is given to the treatment of the stabilization matrix to maintain a compact stencil suitable for unstructured grids. We find SIMPLER preconditioning to be robust and efficient for academic test cases and industrial test cases. Compared with the classical SIMPLE solver, SIMPLER preconditioning reduces the number of nonlinear iterations by a factor 5–20 and the CPU time by a factor 2–5 depending on the case. The flow around a ship hull at Reynolds number 2E9, for example, on a grid with cell aspect ratio up to 1:1E6, can be computed in 3 instead of 15 h.Copyright © 2012 John Wiley & Sons, Ltd.

Mass and Volume Conservation in Phase Field Models for Binary Fluids

Abstract: The commonly used incompressible phase field models for non-reactive, binary fluids, in which the Cahn-Hilliard equation is used for the transport of phase variables (volume fractions), conserve the total volume of each phase as well as the material volume, but do not conserve the mass of the fluid mixture when densities of two components are different. In this paper, we formulate the phase field theory for mixtures of two incompressible fluids, consistent with the quasi-compressible theory [28], to ensure conservation of mass and momentum for the fluid mixture in addition to conservation of volume for each fluid phase. In this formulation, the mass-average velocity is no longer divergence-free (solenoidal) when densities of two components in the mixture are not equal, making it a compressible model subject to an internal con-straint. In one formulation of the compressible models with internal constraints (model 2), energy dissipation can be clearly established. An efficient numerical method is then devised to enforce this compressible internal constraint. Numerical simulations in confined geometries for both compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions. Numerical comparisons show that (a) predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin; and (b) predictions differ significantly in binary fluid mixtures undergoing mixing with a large mixing zone. The numerical study delineates the limitation of the commonly used incompressible phase field model using volume fractions and thereby cautions its predictive value in simulating well-mixed binary fluids.

Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow

Abstract: Methods for investigating and approximating the linear dynamics of amplifier flows are examined in this paper. The procedures are derived for incompressible flow over a two-dimensional backward-facing step. First, the singular value decomposition of the resolvent is performed over a frequency range in order to identify the optimal and suboptimal harmonic forcing and responses of the flow. These forcing/responses are shown to be organized into two categories: the first accounting for the Orr and Kelvin–Helmholtz instabilities in the shear layer and the second for the advection and diffusion of perturbations in the free stream. Next, we investigate the dynamics of the flow when excited by a white in space and time noise. We compute the predominant patterns of the random flow which optimally account for the sustained variance, the empirical orthogonal functions (EOFs), as well as the predominant forcing structures which optimally contribute to the sustained variance, the stochastic optimals (SOs). The leading EOFs and SOs are expressed as a linear combination of the suboptimal forcing and responses of the flow and are related to particular instability mechanisms and/or frequency intervals. Finally, we use the leading EOFs, SOs and balanced modes (obtained from balanced truncation) to build low-order models of the flow dynamics. These models are shown to accurately recover the time propagator and resolvent of the original dynamical system. In other words, such models capture the entire flow response from any forcing and may be used in the design of efficient closed-loop controllers for amplifier flows.

Space-time aspects of a three-dimensional multi-modulated open cavity flow

Abstract: Side-band frequencies in an incompressible flow past a rectangular cavity are characterized through their space-time coherent structures. A parametric study over a range of dimensionless cavity length L/θ0 has been carried out in the incompressible regime. It yields the general evolution of self-sustained oscillations, for which primary characteristics match results in the literature. The modulating frequencies associated with side-band frequencies are usually imputed either to the two-dimensional (vortex-edge) interaction at the impingement or to three-dimensional dynamics induced by centrifugal instabilities in the inner-flow. However, secondary order features sometimes depart from commonly accepted scheme. In addition to the salient features of the flow, our observations bring to light another modulation, which may be related to the main recirculation inside the cavity. That modulation even becomes predominant for peculiar configurations. The present work focuses on such a configuration with a cavity length/depth ratio L/D = 1.5 and dimensionless cavity length L/θ0 = 76. Based on time-resolved velocity measurements, the extensive analysis is concerned with the non-linear interactions within the flow. Using laser Doppler velocimetry and time-resolved particle image velocimetry in two planes, this multi-modulated regime is so addressed through both local and global aspects. Time-resolved velocity fields provide space-time coherent data that are analysed using transfer functions, space-time diagrams, and space-extended time-Fourier decomposition.

Combined effects of compressibility and slip in flows of a Herschel–Bulkley fluid

Abstract: In this work, the combined effects of compressibility and slip in Poiseuille flows of Herschel–Bulkley fluids are investigated. The density is assumed to obey a linear equation of state, and wall slip is assumed to follow Navier’s slip condition with zero slip yield stress. The flow is considered to be weakly compressible so that the transverse velocity component is zero and the pressure is a function of the axial coordinate. Approximate semi-analytical solutions of the steady, creeping, plane and axisymmetric Poiseuille flows are derived and the effects of compressibility, slip, and the Bingham number are discussed. In the case of incompressible flow, it is shown that the velocity may become plug at a finite critical value of the slip parameter which is inversely proportional to the yield stress. In compressible flow with slip, the velocity tends to become plug upstream, which justifies the use of one-dimensional models for viscoplastic flows in long tubes. The case of pressure-dependent slip is also investigated and discussed.

Maximal mixing by incompressible fluid flows

Abstract: We consider a model for mixing binary viscous fluids under an incompressible flow. We prove the impossibility of perfect mixing in finite time for flows with finite viscous dissipation. As measures of mixedness we consider a Monge–Kantorovich–Rubinstein transportation distance and, more classically, the H−1 norm. We derive rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time. The rate of the exponential decay is uniform in the initial data.

Coupled Level-Set/Volume-of-Fluid Method for Simulation of Injector Atomization

Abstract: This paper presents results of a multiphase computational fluid dynamics code using a coupled level-set/volume-of-fluid method to simulate liquid atomization. This interface-capturing approach combines the mass conservation properties of the volume-of-fluid method with the accurate surface reconstruction properties of the level-set method, and it includes surface tension as a volume force calculated with second-order accuracy. Developed by one of the authors, the multiphase code builds upon the combined level-set/volume-of-fluid methodology to enable bubbly flow, liquid breakup, and phase-change simulations. The extension presented in this paper couples a Lagrangian dispersed phase model for postbreakup tracking of droplets with block-structured adaptive mesh refinement on the Eulerian grid. Under an appropriate set of criteria, the transfer of droplets representation from the Eulerian to the Lagrangian discretization enables the simulation of sprays on larger domains and for longer physical times without...

Maximal mixing by incompressible fluid flows

Abstract: We consider a model for mixing binary viscous fluids under an incompressible flow. We proof the impossibility of perfect mixing in finite time for flows with finite viscous dissipation. As measures of mixedness we consider a Monge--Kantorovich--Rubinstein transportation distance and, more classically, the $H^{-1}$ norm. We derive rigorous a priori lower bounds on these mixing norms which show that mixing cannot proceed faster than exponentially in time. The rate of the exponential decay is uniform in the initial data.

Minimization of divergence error in volumetric velocity measurements and implications for turbulence statistics

Abstract: Volumetric velocity measurements taken in incompressible fluids are typically hindered by a nonzero divergence error due to experimental uncertainties. Here, we present a technique to minimize divergence error by employing continuity of mass as a constraint, with minimal change to the measured velocity field. The divergence correction scheme (DCS) is implemented using a con- straint-based nonlinear optimization. An assessment of DCS is performed using direct numerical simulations (DNS) velocity fields with random noise added to emulate experimental uncertainties, together with a Tomographic particle image velocimetry data set measured in a channel flow facility at a matched Reynolds number to the DNS data (Res & 937). Results indicate that the divergence of the corrected velocity fields is reduced to near zero, and a clear improvement is evident in flow statistics. In particu- lar, significant improvements are observed for statistics computed using spatial gradients such as the velocity gra- dient tensor, enstrophy, and dissipation, where having zero divergence is most important.

Efficient Variable-Coefficient Finite-Volume Stokes Solvers

Abstract: We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity-pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 75657595], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

An efficient and stable finite element solver of higher order in space and time for nonstationary incompressible flow

Abstract: SUMMARY In this paper, we present fully implicit continuous Galerkin–Petrov (cGP) and discontinuous Galerkin (dG) time-stepping schemes for incompressible flow problems which are, in contrast to standard approaches like for instance the Crank–Nicolson scheme, of higher order in time. In particular, we analyze numerically the higher order dG(1) and cGP(2) methods, which are super convergent of third, resp., fourth order in time, whereas for the space discretization, the well-known LBB-stable finite element pair Q2∕P1disc of third-order accuracy is used. The discretized systems of nonlinear equations are treated by using the Newton method, and the associated linear subproblems are solved by means of a monolithic (geometrical) multigrid method with a blockwise Vanka-like smoother treating all components simultaneously. We perform nonstationary simulations (in 2D) for two benchmarking configurations to analyze the temporal accuracy and efficiency of the presented time discretization schemes w.r.t. CPU and numerical costs. As a first test problem, we consider a classical ‘flow around cylinder’ benchmark. Here, we concentrate on the nonstationary behavior of the flow patterns with periodic oscillations and examine the ability of the different time discretization schemes to capture the dynamics of the flow. As a second test case, we consider the nonstationary ‘flow through a Venturi pipe’. The objective of this simulation is to control the instantaneous and mean flux through this device. Copyright © 2013 John Wiley & Sons, Ltd.