Benchmark problems for numerical treatment of backflow at open boundaries

Bertoglio, C; Caiazzo, Alfonso; Bazilevs, Yuri; Braack, Malte; Esmaily, Mahdi; Gravemeier, V.; Marsden, Alison L.; Pironneau, Olivier; Vignon-Clementel, Irene E.; Wall, Wolfgang A.

doi:10.1002/cnm.2918

Cited by 54 publications

(54 citation statements)

References 33 publications

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“…The jet breakdown location shifted continuously downstream of the sudden expansion and eventually reached the end of the computational domain, which caused backflow at the outlet and a diverged numerical solution. The latter is well known; however, Passerini et al still reported a lower

l_{m a x}^{+}

compared with our

l_{m a x}^{+} = 11.0 false(l_{a v e r a g e}^{+} = 1.3 false)

obtained on the 80M linear (

P_{1} - P_{1}

) element equivalent mesh with constant node spacing. This comparison may indicate that the mesh used by Passerini et al was rather refined in the high shear rate regions, and consequently equally coarser elsewhere.…”

Section: Discussioncontrasting

confidence: 51%

The FDA nozzle benchmark: “In theory there is no difference between theory and practice, but in practice there is”

Bergersen

Mortensen

Valen‐Sendstad

2018

Numer Methods Biomed Eng

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The utility of flow simulations relies on the robustness of computational fluid dynamics (CFD) solvers and reproducibility of results. The aim of this study was to validate the Oasis CFD solver against in vitro experimental measurements of jet breakdown location from the FDA nozzle benchmark at Reynolds number 3500, which is in the particularly challenging transitional regime. Simulations were performed on meshes consisting of 5, 10, 17, and 28 million (M) tetrahedra, with Δt = 10 seconds. The 5M and 10M simulation jets broke down in reasonable agreement with the experiments. However, the 17M and 28M simulation jets broke down further downstream. But which of our simulations are "correct"? From a theoretical point of view, they are all wrong because the jet should not break down in the absence of disturbances. The geometry is axisymmetric with no geometrical features that can generate angular velocities. A stable flow was supported by linear stability analysis. From a physical point of view, a finite amount of "noise" will always be present in experiments, which lowers transition point. To replicate noise numerically, we prescribed minor random angular velocities (approximately 0.31%), much smaller than the reported flow asymmetry (approximately 3%) and model accuracy (approximately 1%), at the inlet of the 17M simulation, which shifted the jet breakdown location closer to the measurements. Hence, the high-resolution simulations and "noise" experiment can potentially explain discrepancies in transition between sometimes "sterile" CFD and inherently noisy "ground truth" experiments. Thus, we have shown that numerical simulations can agree with experiments, but for the wrong reasons.

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l_{m a x}^{+}

compared with our

l_{m a x}^{+} = 11.0 false(l_{a v e r a g e}^{+} = 1.3 false)

obtained on the 80M linear (

P_{1} - P_{1}

Section: Discussioncontrasting

confidence: 51%

The FDA nozzle benchmark: “In theory there is no difference between theory and practice, but in practice there is”

Bergersen

Mortensen

Valen‐Sendstad

2018

Numer Methods Biomed Eng

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“…In this work as well as in most of related work, the error estimates for problem (1) usually are derived under the no-slip condition u u u = 0 0 0 for the velocity. This excludes, for example, channel-like problems with in-and outflow which are important, for example for biomedical flows (see the review [7]). Therefore, an extension of the error estimates to such more practically relevant flow problems is desired.…”

Section: Practically Relevant Boundary Conditionsmentioning

confidence: 99%

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Schroeder

Lehrenfeld

Linke

et al. 2018

SeMA

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Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption ∇u ∈ L 1 (0, T ; L ∞ (Ω )) which is discussed in detail. In the sense of best practice, we review and establish pressure-and Re-semi-robust estimates for pointwise divergence-free H 1 -conforming FEM (like Scott-Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.Keywords time-dependent incompressible flow · Re-semi-robust error estimates · pressure-robustness · inf-sup stable methods · exactly divergence-free FEMA relatively new aspect in the FE analysis applied to incompressible flows is 'pressurerobustness' [41]. In its most general form, pressure-robustness of a numerical method is defined by its ability to fulfil the following requirement: if the exact solution u u u of (1) belongs to the approximation space V V V h , i.e. if u u u ∈ V V V h , then the discrete solution u u u h coincides with the exact one, that is, u u u h = u u u. In certain physical regimes of the incompressible Navier-Stokes equations -i.e., in certain benchmarks -pressure-robustness allows to use less expensive discretisation schemes without losing accuracy [49,1]. As a consequence, the following fundamental invariance principle transfers from the continuous level to the discretised case: Replacing the source term f f f by f f f + ∇ψ changes the solution (u u u, p) to (u u u, p + ψ). For example, in a potential flow, (u u u · · · ∇)u u u can be very large but it is a gradient and therefore balanced by the pressure gradient and thus does not have any impact on the velocity field. Only recently it has been shown that high Reynolds number potential flows are really challenging for the numerical solution with standard low-order Galerkin-FEM [50,41].A well-known important consequence for methods which are not pressure-robust is that already for the steady incompressible Stokes problem the velocity error estimates for kinetic and dissipation energies are corrupted by the pressure approximability multiplied by ν −1/2 [41,49]. Note that the mechanism responsible for the excitation of this kind of numerical error is a completely linear phenomenon. Exactly divergence-free FEM are naturally pressure-robust, but classical inf-sup stable velocity-pressure pairs like Taylor-Hood FEM are usually not pressure-robust. In fact, ...

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“…This indicates that when a ij (i, j = 1, 2) are chosen to be sufficiently small the matrix R will guarantee the positive semi-definiteness of the matrix Q and the non-positivity of the surface integral term in (4).…”

Section: Energy-stable Boundary Conditions Based On a Quadratic Formmentioning

confidence: 99%

“…We refer the reader to e.g. those of [2,3] which are given based on a weak formulation of the Navier-Stokes equations, and also to [4] for a recent study of several methods in the context of physiological flows. We also refer to [9,15,38] for methods dealing with two-phase and multiphase outflows and open boundaries and related issues.…”

Section: Introductionmentioning

confidence: 99%

Energy-stable boundary conditions based on a quadratic form: Applications to outflow/open-boundary problems in incompressible flows

Yang

Dong

2019

Journal of Computational Physics

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We present a set of new energy-stable open boundary conditions for tackling the backflow instability in simulations of outflow/open boundary problems for incompressible flows. These boundary conditions are developed through two steps: (i) devise a general form of boundary conditions that ensure the energy stability by re-formulating the boundary contribution into a quadratic form in terms of a symmetric matrix and computing an associated eigen problem; and (ii) require that, upon imposing the boundary conditions from the previous step, the scale of boundary dissipation should match a physical scale. These open boundary conditions can be re-cast into the form of a traction-type condition, and therefore they can be implemented numerically using the splitting-type algorithm from a previous work. The current boundary conditions can effectively overcome the backflow instability typically encountered at moderate and high Reynolds numbers. These boundary conditions in general give rise to a non-zero traction on the entire open boundary, unlike previous related methods which only take effect in the backflow regions of the boundary. Extensive numerical experiments in two and three dimensions are presented to test the effectiveness and performance of the presented methods, and simulation results are compared with the available experimental data to demonstrate their accuracy.

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Benchmark problems for numerical treatment of backflow at open boundaries

Cited by 54 publications

References 33 publications

The FDA nozzle benchmark: “In theory there is no difference between theory and practice, but in practice there is”

The FDA nozzle benchmark: “In theory there is no difference between theory and practice, but in practice there is”

Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations

Energy-stable boundary conditions based on a quadratic form: Applications to outflow/open-boundary problems in incompressible flows

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