高等学校計算数学学報 = Numerical mathematics

Turbulence, Coherent Structures, Dynamical Systems and SymmetryP. Holmes, J. L. Lumley, G. Berkooz, and C. W. Rowley, 2nd ed., Cambridge University Press, Cambridge, England, U.K., 2012, 386 pp., $90

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Numerical Approximation Of Partial Differential Equations

Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flows

For over a century, reduced order models (ROMs) have been a fundamental discipline of theoretical fluid mechanics. Early examples include Galerkin models inspired by the Orr–Sommerfeld stability equation and numerous vortex models, of which the von Karman vortex street is one of the most prominent. Subsequent ROMs typically relied on first principles, like mathematical Galerkin models, weakly nonlinear stability theory, and two- and three-dimensional vortex models. Aubry et al. [J. Fluid Mech. 192, 115–173 (1988)] pioneered the data-driven proper orthogonal decomposition (POD) modeling. In early POD modeling, available data were used to build an optimal basis, which was then utilized in a classical Galerkin procedure to construct the ROM, but data have made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we take a modest step and illustrate the impact of data-driven modeling on one significant ROM area. Specifically, we focus on ROM closures, which are correction terms that are added to the classical ROMs in order to model the effect of the discarded ROM modes in under-resolved simulations. Through simple examples, we illustrate the main modeling principles used to construct the classical ROMs, motivate and introduce modern ROM closures, and show how data-driven modeling, artificial intelligence, and machine learning have changed the standard ROM methodology over the last two decades. Finally, we outline our vision on how the state-of-the-art data-driven modeling can continue to reshape the field of reduced order modeling.

On closures for reduced order models—A spectrum of first-principle to machine-learned avenues

In this work we present a reduced basis Smagorinsky turbulence model for steady flows. We approximate the nonlinear eddy diffusion term using the empirical interpolation method (cf. [M. A. Grepl et al., ESAIM Math. Model. Numer. Anal., 41 (2007), pp. 575--605; Barrault et al., C. R. Acad. Sci. Paris Ser. I Math., 339 (2004), pp. 667--672]) and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for a Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on [S. Deparis, SIAM J. Sci. Comput., 46 (2008), pp. 2039--2067] and [A. Manzoni, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 1199--1226], according to the Brezzi--Rappaz--Raviart stability theory, and adapted for the nonlinear eddy diffusion term. We present some numerical tests, programmed in FreeFem++ (cf. [F. Hecht, J. Numer. Math., 20 (2012), pp. 251--265]), in which we show a speedup on the computation by factor larger...

On a certified Smagorinsky reduced basis turbulence model

In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.

Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height

In this work, we study the performance of some local projection-based solvers in the Large Eddy Simulation (LES) of laminar and turbulent flows governed by the incompressible Navier–Stokes Equations (NSE). On one side, we focus on a high-order term-by-term stabilization Finite Element (FE) method that has one level, in the sense that it is defined on a single mesh, and in which the projection-stabilized structure of standard Local Projection Stabilization (LPS) methods is replaced by an interpolation-stabilized structure. The interest of LPS methods is that they ensure a self-adapting high accuracy in laminar regions of turbulent flows, which turns to be of overall optimal high accuracy if the flow is fully laminar. On the other side, we propose a new Reduced Basis (RB) Variational Multi-Scale (VMS)-Smargorinsky turbulence model, based upon an empirical interpolation of the sub-grid eddy viscosity term. This method yields dramatical improvements of the computing time for benchmark flows. An overview about known results from the numerical analysis of the proposed methods is given, by highlighting the used mathematical tools. In the numerical study, we have considered two well known problems with applications in industry: the (3D) turbulent flow in a channel and the (2D/3D) recirculating flow in a lid-driven cavity.

/pdf/assessment-of-self-adapting-local-projection-based-solvers-1fyc67q5q8.pdf

Assessment of self-adapting local projection-based solvers for laminar and turbulent industrial flows

Esta tesis se enmarca dentro de la resoluci_on num_erica de modelos que simulan el
comportamiento de ujos turbulentos mediante t_ecnicas de orden reducido y bajo coste
computacional. En particular, desarrollamos t_ecnicas de bases reducidas que permiten
reducir dr_asticamente el c_alculo de una soluci_on a estos modelos. El objetivo es desarrollar
modelos matem_aticos orientados al dise~no de edi_cios eco-e_cientes, lo que conlleva a la
resoluci_on de modelos complejos donde las inc_ognitas del problema aparecen acopladas. La
modelizaci_on de orden reducido proporciona reducciones de varios _ordenes de magnitud en
el coste computacional de la simulaci_on num_erica de estos procesos y problemas de dise~no,
haciendo cada vez m_as abordable su resoluci_on efectiva en tiempo real. Normalmente los
modelos de orden reducido requieren de cientos de grados de libertad en lugar de millones
como frecuentemente necesita el modelo de orden completo.
En este trabajo consideramos diferentes modelos de complejidad creciente, desarrollando
las t_ecnicas de orden reducido aplicadas a dichos modelos. Realizamos un estudio
de estabilidad para dichos m_etodos num_ericos y se completa con simulaciones num_ericas
que permiten validar los resultados te_oricos obtenidos.
En primer lugar consideramos el modelo de turbulencia para ujos de aire conocido
como modelo de Smagorinsky. Se trata de un modelo b_asico de turbulencia, que corresponde
a las ecuaciones de Navier-Stokes donde la viscosidad es una viscosidad turbulenta,
que matem_atica es una funci_on no lineal de la inc_ognita. Para la aproximaci_on de este
t_ermino utilizamos t_ecnicas de Interpolaci_on Emp__rica, desarrollando un estimador de
error a posteriori de acuerdo con la Teor__a de Brezzi-Rappaz-Raviart. Para este modelo
en su versi_on bidimensional, realizamos distintos test num_ericos obteniendo que el tiempo
de c_alculo para la velocidad del ujo se divide por mil cuando utilizamos t_ecnicas de
orden reducido. A continuacion nos ocupamos de una modi_caci_on del modelo de Smagorinsky
donde consideramos que la viscosidad turbulenta act_ua s_olo sobre las peque~nas
escalas resueltas, y adem_as consideramos una estabilizaci_on local de proyecci_on para el
c_alculo de la presi_on. El considerar esta estabilizaci_on de la presion nos permite evitar
el enriquecimiento del espacio de velocidades para obtener un m_etodo estable. Para este
modelo hemos comprobado num_ericamente que el tiempo de c_alculo se reduce m_as que
en el modelo original de Smagorinsky.
Por _ultimo consideramos un modelo acoplado de tipo Boussinesq obtenido mediante
t_ecnicas de multiescala variacional. El modelo est_a formado por las ecuaciones del modelo
de Smagorinsky junto a la ecuaci_on de la temperatura. Estas ecuaciones est_an acopladas
mediante los t_erminos de otabilidad. El estudio realizado para este modelo se centra
en aplicar t_ecnicas de orden reducido para dos tipos de par_ametros: f__sico y geom_etrico.
El tratamiento para cada uno de estos par_ametros es distinto desde el punto de vista
matem_atico. Para este problema desarrollamos de nuevo un estimador de error a posteriori
y lo validamos mediante simulaciones num_ericas sencillas que representan el estudio del
ujo de aire y la temperatura en habitaciones de geometr__a sencilla.

Development of reduced numeric models to aero-thermal flows in buildings

In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that let us obtain an affine decomposition with respect to the parameters. We construct an a posterior error estimator, based upon the Brezzi-Rappaz-Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.

Enrique Delgado Ávila

Papers

On a certified Smagorinsky reduced basis turbulence model

Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height

Assessment of self-adapting local projection-based solvers for laminar and turbulent industrial flows

Development of reduced numeric models to aero-thermal flows in buildings

Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height