Residual-Based Large Eddy Simulation with Isogeometric Divergence-Conforming Discretizations

Evans, John A.; Coley, Christopher; Aronson, Ryan M.; Wetterer-Nelson, Corey L.; Bazilevs, Yuri

doi:10.1007/978-3-319-96469-0_3

Cited by 6 publications

(7 citation statements)

References 48 publications

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“…Finally, with the express goal of obtaining a residual-based stabilized formulation that maintains both strong mass conservation and pressure robustness for divergence-conforming spaces, we plan to apply the formalism presented here to the construction of VMS-based formulations yielding pointwise divergence-free coarse-and fine-scale velocities. While such formulations have been proposed in the past [63,66], an error analysis which shows robustness in the advection-dominated limit is missing for these formulations. The formalism here, however, may yield pressure-robust methodologies that are provably robust with respect to both advection and diffusion.…”

Section: Discussionmentioning

confidence: 99%

“…Before examining results obtained using our VMS-based formulation, we first examine results obtained using classical LES subgrid models [66]. In Fig.…”

Section: Application To Turbulent Fluid Flowmentioning

confidence: 99%

“…Comparison of classical LES subgrid models (no model, Static Smagorinsky, Dynamic Smagorinsky, Chollet) with DNS[64]. Results adopted from[66].…”

mentioning

confidence: 99%

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Variational multiscale modeling with discretely divergence-free subscales

Evans

Kamensky

Bazilevs

2020

Computers & Mathematics with Applications

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We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with H 1 -conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier-Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.

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Section: Discussionmentioning

confidence: 99%

“…Before examining results obtained using our VMS-based formulation, we first examine results obtained using classical LES subgrid models [66]. In Fig.…”

Section: Application To Turbulent Fluid Flowmentioning

confidence: 99%

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Variational multiscale modeling with discretely divergence-free subscales

Evans

Kamensky

Bazilevs

2020

Computers & Mathematics with Applications

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“…We compare results with DNS data presented in Reference 31 that was obtained via pseudo-spectral method on 768 3 nodes in a [0, 2 ] 3 domain. We also compare our results with other two-level VMS method that employs NURBS basis functions reported in van Opstal et al 33 and Evans et al 34 We have employed the generalized alpha method with step size Δt = 0.01 s for 33 3 mesh and Δt = 0.005 s for the 65 3 mesh to keep the same CFL number. Because of consistent tangent employed in the Newton-Raphson loop, the system of equations took two to three iterations per step to converge to a tolerance of 10 −10 .…”

Section: Structured Meshes Comprised Of Same Element Typementioning

confidence: 77%

“…In this derivation, we have employed generalized-alpha method for time discretization, however, any appropriate time integration method can also be used. In (34), m and f are the algorithmic parameters of the time integration scheme. The RHS of ( 23) is as follows…”

Section: 1mentioning

confidence: 99%

Variationally derived closure models for large eddy simulation of incompressible turbulent flows

Masud

Zhu

2021

Numerical Methods in Fluids

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We present a variationally consistent method for deriving residual-based closure models for incompressible Navier-Stokes equations. The method is based on the fine-scale variational structure facilitated by the variational multiscale framework where fine scales are driven by the residuals of the Euler-Lagrange equations of the resolved scales in the balance of momentum and conservation of mass equations. A bubble-functions based approach is applied directly to the fine-scale variational equation to derive analytical expressions for the closure model. Variational consistency of the model lends itself to rigorous linearization that results in quadratic rate of convergence of the method in the iterative solution strategy for the nonlinear equations. The method is shown to work for a family of linear and quadratic hexahedral and tetrahedral elements as well as composite discretizations that are comprised of hexahedral and tetrahedral elements in the same computational domain. Numerical tests with the proposed model are presented for various classes of turbulent flow problems to show its generality and range of applicability. The test cases investigated include Taylor-Green vortex stretching, statistically stationary wall-bounded channel flows, and modeling the effects of the geometry of the leading edge of the plate on the instability of the boundary layer that leads to flow separation and flow reversal over flat plates of finite thickness. K E Y W O R D Shexahedral and tetrahedral elements, hierarchical variational multiscale method, large eddy simulation, residual-based turbulence models, residual-free bubbles, stabilized finite elements

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A time-consistent stabilized finite element method for fluids with applications to hemodynamics

Jia,

Esmaily

2023

Sci Rep

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Several finite element methods for simulating incompressible flows rely on the streamline upwind Petrov–Galerkin stabilization (SUPG) term, which is weighted by $$\tau _{\text{SUPG}}$$ τ SUPG . The conventional formulation of $$\tau _{\text{SUPG}}$$ τ SUPG includes a constant that depends on the time step size, producing an overall method that becomes exceedingly less accurate as the time step size approaches zero. In practice, such method inconsistency introduces significant error in the solution, especially in cardiovascular simulations, where small time step sizes may be required to resolve multiple scales of the blood flow. To overcome this issue, we propose a consistent method that is based on a new definition of $$\tau _{\text{SUPG}}$$ τ SUPG . This method, which can be easily implemented on top of an existing streamline upwind Petrov–Galerkin and pressure stabilizing Petrov–Galerkin method, involves the replacement of the time step size in $$\tau _{\text{SUPG}}$$ τ SUPG with a physical time scale. This time scale is calculated in a simple operation once every time step for the entire computational domain from the ratio of the L2-norm of the acceleration and the velocity. The proposed method is compared against the conventional method using four cases: a steady pipe flow, a blood flow through vascular anatomy, an external flow over a square obstacle, and a fluid–structure interaction case involving an oscillatory flexible beam. These numerical experiments, which are performed using linear interpolation functions, show that the proposed formulation eliminates the inconsistency issue associated with the conventional formulation in all cases. While the proposed method is slightly more costly than the conventional method, it significantly reduces the error, particularly at small time step sizes. For the pipe flow where an exact solution is available, we show the conventional method can over-predict the pressure drop by a factor of three. This large error is almost completely eliminated by the proposed formulation, dropping to approximately 1% for all time step sizes and Reynolds numbers considered.

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Residual-Based Large Eddy Simulation with Isogeometric Divergence-Conforming Discretizations

Cited by 6 publications

References 48 publications

Variational multiscale modeling with discretely divergence-free subscales

Variational multiscale modeling with discretely divergence-free subscales

Variationally derived closure models for large eddy simulation of incompressible turbulent flows

A time-consistent stabilized finite element method for fluids with applications to hemodynamics

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