Analysis of Nonlinear Spectral Eddy-Viscosity Models of Turbulence

Gunzburger, Max; Lee, Eun Jung; Saka, Yuki; Trenchea, Catalin; Wang, Xiaoming

doi:10.1007/s10915-009-9335-8

Cited by 6 publications

(7 citation statements)

References 19 publications

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“…The analysis we perform herein to prove unconditional stability and optimal convergence can be applied to other advection-diffusion models, e.g., Lions' hyperviscosity model [28], the modified NSE of Ladyzhenskaya [24] and nonlinear spectral eddyviscosity models of turbulence [11]. In particular, we note our proposed method(s) are related to one discussed by H. Johnston and J.G.…”

mentioning

confidence: 90%

An optimally accurate discrete regularization for second order timestepping methods for Navier–Stokes equations

Jiang

Mohebujjaman

Rebholz

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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We propose a new, optimally accurate numerical regularization/stabilization for (a family of) second order timestepping methods for the Navier-Stokes equations (NSE). The method combines a linear treatment of the advection term, together with a stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure. We rigorously prove that the entire new family of methods are unconditionally stable and O(∆t 2) accurate. The idea of 'curvature stabilization' is new to CFD and is intended as an improvement over the commonly used 'speed stabilization', which is only first order accurate in time and can have an adverse affect on important flow quantities such as drag coefficients. Numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows.

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mentioning

confidence: 90%

An optimally accurate discrete regularization for second order timestepping methods for Navier–Stokes equations

Jiang

Mohebujjaman

Rebholz

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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“…We denote by P N the projection onto this space. We take them from [15] (though they may have appeared elsewhere).…”

Section: Appendix a Miscellaneous Resultsmentioning

confidence: 99%

“…Such spectral viscosity (SV) methods were first proposed by Tadmor in [43] and have been shown to converge to entropy solutions of the underlying scalar conservation laws in a series of papers [43,44,45,31,37]. Spectral viscosity methods have been employed to robustly approximate turbulent flow in [18,15] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough Initial Data

Lanthaler

Mishra

2019

Found Comput Math

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We propose a spectral viscosity method to approximate the twodimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class i.e. it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method. arXiv:1903.12361v2 [math.NA] 24 Oct 2019 providing a priori control on various norms of ω, such as L p -norms [33].Global existence and uniqueness results for the two-dimensional incompressible Euler equations with smooth initial data are classical [33,7]. For non-smooth initial conditions, the work by Yudovich [49] has established existence and uniqueness for bounded initial vorticity, i.e. ω 0 ∈ L ∞ . The uniqueness result of [49] has later been extended to vorticities belonging to slightly more general spaces [47,48,50]. An existence result for vorticity ω 0 ∈ L p , 1 < p < ∞ has been obtained by Diperna and Majda [7]. It is shown in [7] that the sequence obtained by solving the Euler equations for mollified initial data is strongly compact in L 2 . The existence of a weak solution for ω 0 ∈ L p is then established by passing to the limit. Further extensions of the result of Diperna and Majda can be obtained by compensated compactness methods for initial vorticity ω 0 belonging to e.g. Orlicz spaces such as ω 0 ∈ L log(L) α , α 1/2, which are compactly embedded in H −1 [34, 3, 2, 30]. These methods break down for ω 0 ∈ L 1 .In his celebrated work [6], Delort has shown the existence of solutions to the Euler equations with initial vorticity ω 0 = ω 0 + ω 0 , where ω 0 is a finite, nonnegative Radon measure belonging also to H −1 , and ω 0 ∈ L p , for some p > 1. These initial data correspond to the interesting case of vortex sheets i.e. vorticity concentrated on curves in the two-dimensional spatial domain [33]. In [6], it is remarked that the proof can be extended to allow for ω 0 ∈ L 1 . A detailed proof of this claim has subsequently been provided by Vecchi and Wu [46]. The results of Delort [6], and Vecchi and Wu [46], remain the most general existence results for the incompressible Euler equations in two dimensions. The question of existence of solutions beyond this Delort class, for instance, when ω 0 is an arbitrary signed bounded measure, remains open. The uniqueness question also remains open, even for vorticities ω 0 ∈ L p , p < ∞.1.2. Numerical schemes. It is not possible to represent solutions of the incompressible Euler equations in terms of analytical solution formulas, even in two space dimensions. Hence, numerical approximation of (1.1) is a necessary and key ingredient in the study of the inco...

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“…6. Nonlinear spectral eddy-viscosity models of turbulence [8] Algebraic turbulence models, i.e., models in which an added viscosity is introcuded that is algebraically related to the stress tensor or the velocity gradient, are among the simplest ones developed. The most common such model was developed independently by Ladyzhenskaya [20] and Smagorinsky [23] in which the added eddy viscosity ν t is proportional to |∇u| p , where u denotes the velocity; the Smagorinsky case is p = 1.…”

Section: Parameter Identification and Control Of Elliptic Equations Wmentioning

confidence: 99%

Advanced Numerical Methods for Computing Statistical Quantities of Interest from Solutions of SPDES

Gunzburger¹

2012

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The public reportmg burd en for th1s collection of mform ation is estimated to average 1 hour per response. includmg the tim e for rev>ewing instruct>ons. searclhing exi sting data so urces. gathering and mmnta1mng the data needed . and co mpleting and rev1ew 1 ng the collection of 1nforma t1 on. Send co mments regarding th1s burden est1m ate or any other aspect o fth1s collection of mformat1on . 1nclud1 ng suggesti ons for reduc1 ng the burden. to the Department of De fense. ExecutiV e Serv1ce D>rectorate (0704 -0 188) Respo ndents should be aware that notwithstanding any other prov1 s1 on of law. no person shall be sub;ect to any penalty for falling to comply w1th a collect> on of 1 nform at1on if 1 t does not display a currently v alid OMB co ntrol number PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Department of Scientific Computing REPORT NUMBERFlorida State University SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) AFOSR SPONSOR/MONITOR'S REPORT NUMBER(S)AFRL-OSR-VA-TR-2012-0282 DISTRIBUTION/AVAILABILITY STATEMENT A SUPPLEMENTARY NOTES ABSTRACTComputational simulation-based predictions are central to science and engineering and to risk assessment and decision making in economics, public policy, and military venues, including several of importance to Air Force missions. Unfortunately, predictions are often fraught with uncertainty so that effective means for quantifying that uncertainty are of pararuount importance. The research effort investigates and resolves important algorithmic, mathematical, and practical issues related to the efficient, accurate, and robust computational determination of the quantities of interest used by engineers and decision makers that are determined from solutions of partial differential equations. 1S. SUBJECT TERMS ADVANCED NUMERICAL METHODS FOR COMPUTING STATISTICAL QUANTITIES OF INTEREST FROM SOLUTIONS OF SPDESFINAL REPORT -FA9550-08-1-0415 Max Gunzburger Department of Scientific ComputingFlorida State University gunzburg@fsu.edu AbstractComputational simulation-based predictions are central to science and engineering and to risk assessment and decision making in economics, public policy, and military venues, including several of importance to Air Force missions. Unfortunately, predictions are often fraught with uncertainty so that effective means for quantifying that uncertainty are of paramount importance. The research effort investigates and resolves important algorithmic, mathematical, and practical issues related to the efficient, accurate, and robust computational determination of the quantities of interest used by engineers and decision makers that are determined from solutions of partial differential equations having random inputs. Notable accomplishments include the development of a novel approach to discretizing white noise and its application to the stochastic Navier-Stokes-Boussinesq system; development of efficient methods for solving high-dimensional backward stochastic differential equations;...

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Analysis of Nonlinear Spectral Eddy-Viscosity Models of Turbulence

Cited by 6 publications

References 19 publications

An optimally accurate discrete regularization for second order timestepping methods for Navier–Stokes equations

An optimally accurate discrete regularization for second order timestepping methods for Navier–Stokes equations

On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough Initial Data

Advanced Numerical Methods for Computing Statistical Quantities of Interest from Solutions of SPDES

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