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203
On The Error Estimates For The Rotational PressureCorrection Projection Methods
, 2002
"... In this paper we study the rotational form of the pressurecorrection method that was proposed in [19]. We show that the rotational form of the algorithm provides better accuracy in terms of the H norm of the velocity and of the norm of the pressure than the standard form. 1. ..."
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Cited by 35 (17 self)
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In this paper we study the rotational form of the pressurecorrection method that was proposed in [19]. We show that the rotational form of the algorithm provides better accuracy in terms of the H norm of the velocity and of the norm of the pressure than the standard form. 1.
Stability and convergence of efficient NavierStokes solvers via a commutator estimate 0
, 2005
"... For strong solutions of the incompressible NavierStokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit timestepping for pres ..."
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Cited by 32 (13 self)
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For strong solutions of the incompressible NavierStokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit timestepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an infsup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the NavierStokes equations as a perturbed diffusion equation. 1
Direct optimal growth analysis for timesteppers
 INT. J. NUMER. METH. FLUIDS
, 2008
"... Methods are described for transient growth analysis of flows with arbitrary geometric complexity, where in particular the flow is not required to vary slowly in the streamwise direction. Emphasis is on capturing the global effects arising from localized convective stability in streamwisevarying flo ..."
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Cited by 30 (13 self)
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Methods are described for transient growth analysis of flows with arbitrary geometric complexity, where in particular the flow is not required to vary slowly in the streamwise direction. Emphasis is on capturing the global effects arising from localized convective stability in streamwisevarying flows. The methods employ the ‘timestepper’s approach ’ in which a nonlinear Navier–Stokes code is modified to provide evolution operators for both the forward and adjoint linearized equations. First, the underlying mathematical treatment in primitive flow variables is presented. Then, details are given for the inner level code modifications and outer level eigenvalue and SVD algorithms in the timestepper’s approach. Finally, some examples are shown and guidance provided on practical aspects of this type of largescale stability analysis.
Modeling and numerical approximation of twophase incompressible flows by a phasefield approach
"... Abstract. We present in this note a unified approach on how to design simple, efficient and energy stable time discretization schemes for the AllenCahn or CahnHilliard NavierStokes system which models twophase incompressible flows with matching or nonmatching density. Special emphasis is placed ..."
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Cited by 29 (7 self)
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Abstract. We present in this note a unified approach on how to design simple, efficient and energy stable time discretization schemes for the AllenCahn or CahnHilliard NavierStokes system which models twophase incompressible flows with matching or nonmatching density. Special emphasis is placed on designing schemes which only require solving linear systems at each time step while satisfy discrete energy laws that mimic the continuous energy laws. We construct the time discretization schemes in weak formulations so that they can be used with any consistent Galerkin type spacial discretization schemes such as finite element methods and spectral/spectralelement methods. Contents 1
AN ADAPTIVE WAVELET COLLOCATION METHOD FOR FLUIDSTRUCTURE INTERACTION AT HIGH REYNOLDS NUMBERS
, 2005
"... Two mathematical approaches are combined to calculate high Reynolds number incompressible fluidstructure interaction: a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion, and Brinkman penalization to enforce solid boundaries of arbitrary complexit ..."
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Cited by 28 (11 self)
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Two mathematical approaches are combined to calculate high Reynolds number incompressible fluidstructure interaction: a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion, and Brinkman penalization to enforce solid boundaries of arbitrary complexity. We also implement a waveletbased multilevel solver for the Poisson problem for the pressure at each time step. The method is applied to twodimensional flow around fixed and moving cylinders for Reynolds numbers in the range 3 × 10 1 ≤ Re ≤ 10 5. The compression ratios of up to 1000 are achieved. For the first time it is demonstrated in actual dynamic simulations that the compression scales like Re 1/2 over five orders of magnitude, while computational complexity scales like Re. This represents a significant improvement over the classical complexity estimate of Re 9/4 for twodimensional turbulence.
AN UNCONDITIONNALLY STABLE PRESSURE CORRECTION SCHEME FOR COMPRESSIBLE BAROTROPIC NAVIERSTOKES EQUATIONS
, 2007
"... We present in this paper a pressure correction scheme for barotropic compressible NavierStokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximumprinciplebased a priori estimates of the continuous problem also hold for the discrete solution. The ..."
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Cited by 20 (16 self)
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We present in this paper a pressure correction scheme for barotropic compressible NavierStokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximumprinciplebased a priori estimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L 2stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential. The proposed scheme is built in order to match these theoretical results, and combines a fractionalstep time discretization of pressurecorrection type to a space discretization associating low order nonconforming mixed finite elements and finite volumes. Numerical tests with an exact smooth solution show the convergence of the scheme.
A energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges
 Terentjev (Eds.), Modeling of Soft Matter, vol. IMA 141
, 2005
"... Abstract. The use of a phase field to describe interfacial phenomena has a long and fruitful tradition. There are two key ingredients to the method: the transformation of Lagrangian description of geometric motions to Eulerian description framework, and the employment of the energetic variational pr ..."
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Cited by 18 (7 self)
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Abstract. The use of a phase field to describe interfacial phenomena has a long and fruitful tradition. There are two key ingredients to the method: the transformation of Lagrangian description of geometric motions to Eulerian description framework, and the employment of the energetic variational procedure to derive the coupled systems. Several groups have used this theoretical framework to approximate NavierStokes systems for twophase flows. Recently, we have adapted the method to simulate interfacial dynamics in blends of microstructured complex fluids. This review has two objectives. The first is to give a more or less selfcontained exposition of the method. We will briefly review the literature, present the governing equations and discuss a numerical scheme based on different numerical schemes, such as spectral methods. The second objective is to elucidate the subtleties of the model that need to be handled properly for certain applications. These points, rarely discussed in the literature, are essential for a realistic representation of the physics and a successful numerical implementation. The advantages and limitations of the method will be illustrated by numerical examples. We hope that this review will encourage readers whose applications may potentially benefit from a similar approach to explore it further. Key words. Energetic variational formulation, phase field methods, CahnHilliard equation, twophase flows, complex fluids, free interfacial motions.
An Hybrid Finite VolumeFinite Element Method for Variable Density Incompressible Flows
"... This paper is devoted to the numerical simulation of variable density incompressible flows, modeled by the NavierStokes system. We introduce an hybrid scheme which combines a Finite Volume approach for treating the mass conservation equation and a Finite Element method to deal with the momentum equ ..."
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Cited by 16 (6 self)
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This paper is devoted to the numerical simulation of variable density incompressible flows, modeled by the NavierStokes system. We introduce an hybrid scheme which combines a Finite Volume approach for treating the mass conservation equation and a Finite Element method to deal with the momentum equation and the divergence free constraint. The breakthrough relies on the definition of a suitable footbridge between the two methods, through the design of compatibility condition. In turn, the method is very flexible and allows to deal with unstructured meshes. Several numerical tests are performed to show the scheme capabilities. In particular, the viscous RayleighTaylor instability evolution is carefully investigated.