On a Certified Smagorinsky Reduced Basis Turbulence Model

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...

[1]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[2]  Gianluigi Rozza,et al.  Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations , 2015, Comput. Math. Appl..

[3]  Gianluigi Rozza,et al.  Certified reduced basis approximation for parametrized partial differential equations and applications , 2011 .

[4]  Traian Iliescu,et al.  Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison , 2011, 1106.3585.

[5]  Andrea Manzoni,et al.  An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows , 2014 .

[6]  Samuele Rubino.  Numerical modelling of turbulence by Richardson number-based and VMS models , 2014 .

[7]  Yvon Maday,et al.  RB (Reduced basis) for RB (Rayleigh–Bénard) , 2013 .

[8]  Earl H. Dowell,et al.  Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation , 2013, Journal of Fluid Mechanics.

[9]  B. R. Noack.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 2013 .

[10]  Roger Lewandowski,et al.  Mathematical and Numerical Foundations of Turbulence Models and Applications , 2014 .

[11]  Traian Iliescu,et al.  Variational multiscale proper orthogonal decomposition: Navier‐stokes equations , 2012, 1210.7389.

[12]  Pierre Sagaut,et al.  Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow , 2003, Journal of Fluid Mechanics.

[13]  Annalisa Quaini,et al.  Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology , 2017, J. Comput. Phys..

[14]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[15]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[16]  Gianluigi Rozza,et al.  Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants , 2013, Numerische Mathematik.

[17]  P. Sagaut.  BOOK REVIEW: Large Eddy Simulation for Incompressible Flows. An Introduction , 2001 .

[18]  J. Smagorinsky,et al.  GENERAL CIRCULATION EXPERIMENTS WITH THE PRIMITIVE EQUATIONS , 1963 .

[19]  A. Quarteroni,et al.  Reduced Basis Methods for Partial Differential Equations: An Introduction , 2015 .

[20]  G. Rozza,et al.  On the stability of the reduced basis method for Stokes equations in parametrized domains , 2007 .

[21]  Tomás Chacón Rebollo.  An analysis technique for stabilized finite element solution of incompressible flows , 2001 .

[22]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[23]  Christine Bernardi,et al.  Discr'etisations variationnelles de probl`emes aux limites elliptiques , 2004 .

[24]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[25]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[26]  Christian B Allen,et al.  Investigation of an adaptive sampling method for data interpolation using radial basis functions , 2010 .

[27]  Gianluigi Rozza,et al.  Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations , 2015 .

[28]  Gianluigi Rozza,et al.  On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics , 2017, J. Sci. Comput..

[29]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[30]  Anthony T. Patera,et al.  A space–time variational approach to hydrodynamic stability theory , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[31]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[32]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[33]  Ramon Codina,et al.  Explicit reduced‐order models for the stabilized finite element approximation of the incompressible Navier–Stokes equations , 2013 .

[34]  Max D. Gunzburger,et al.  An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier-Stokes Equations , 2016, SIAM J. Numer. Anal..

[35]  J. Rappaz,et al.  Numerical analysis for nonlinear and bifurcation problems , 1997 .

[36]  B. R. Noack,et al.  A Novel Model Order Reduction Approach for Navier-Stokes Equations at High Reynolds Number , 2012, 1211.1720.

[37]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .

[38]  B. Armaly,et al.  Experimental and theoretical investigation of backward-facing step flow , 1983, Journal of Fluid Mechanics.

[39]  Gianluigi Rozza,et al.  Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity , 2009, J. Comput. Phys..

[40]  Simone Deparis,et al.  Reduced Basis Error Bound Computation of Parameter-Dependent Navier-Stokes Equations by the Natural Norm Approach , 2008, SIAM J. Numer. Anal..