The reduced order NS-$\alpha$ model for incompressible flow: theory, numerical analysis and benchmark testing

Cuff, Victoria M.; Dunca, Argus A.; Manica, Carolina C.; Rebholz, Leo G.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 3, pp. 641-662.

This paper introduces a new, reduced-order NS-$\alpha$ (rNS-$\alpha$) model for the purpose of efficient, stable, and accurate simulations of incompressible flow problems on coarse meshes. We motivate the new model by discussing the difficulties in efficient and stable algorithm construction for the usual NS-$\alpha$ model, and then derive rNS-$\alpha$ by using deconvolution as an approximation to the filter inverse, which reduces the fourth order NS-$\alpha$ formulation to a second order model. After proving the new model is well-posed, we propose a $C^0$ finite element spatial discretization together with an IMEX BDF2 timestepping to create a linearized algorithm that decouples the conservation of mass and momentum equations from the filtering. We rigorously prove the algorithm is well-posed, and provided a very mild timestep restriction, is also stable and converges optimally to the model solution. Finally, we give results of several benchmark computations that confirm the theory and show the proposed model/scheme is effective at efficiently finding accurate coarse mesh solutions to flow problems.

Classification : 65N30, 76D05, 76F65
Mots clés : Large Eddy Simulation, approximate deconvolution, finite element method, turbulent channel flow

N.A. Adams and S. Stolz, On the Approximate Deconvolution procedure for LES. Phys. Fluids 2 (1999) 1699–1701. | Zbl

N. Adams and S. Stolz, A subgrid-scale deconvolution approach for shock capturing. J. Comput. Phys. 178 (2002) 391–426. | DOI | Zbl

M. Benzi and M. Olshanskii, An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28 (2006) 2095–2113. | DOI | Zbl

L.C. Berselli and L. Bisconti, On the structural stability of the Euler-Voight and Navier-Stokes-Voight models. Nonlinear Anal. 75 (2012) 117–130. | DOI | Zbl

J. Bramble, J. Pasciak and O. Steinbach, On the stability of the $L^2$ projection in $H^1\left(\Omega \right)$. Math. Comput. 71 (2002) 147–156. | DOI | Zbl

T. Chacon and R. Lewandowski, Mathematical and numerical foundations of turbulence models and applications. Springer, New York (2014). | Zbl

S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa–Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81 (1998) 5338–5341. | DOI | Zbl

S. Chen, C. Foias, E. Olson, E.S. Titi and W. Wynne, A connection between the Camassa–Holm equations and turbulent flows in channels and pipes. Phys. Fluids 11 (1999) 2343–2353. | DOI | Zbl

A. Cheskidov, Boundary layer for the Navier–Stokes-$\alpha$ model of fluid turbulence. Arch. Ration. Mech. Anal. 172 (2004) 333–362. | DOI | Zbl

A.A. Dunca, A two-level multiscale deconvolution method for the large eddy simulation of turbulent flows. Math. Models Methods Appl. Sci. 22 (2012) 1250001. | DOI | Zbl

A. Dunca and Y. Epshteyn, On the Stolz–Adams deconvolution model for the Large-Eddy simulation of turbulent flows. SIAM J. Math. Anal. 37 (2005) 1890–1902. | DOI | Zbl

V.J. Ervin and N. Heuer, Approximation of time-dependent, viscoelastic fluid flow: Crank–Nicolson, finite element approximation. Numer. Methods Partial Differ. Eq. 20 (2003) 248–283. | DOI | Zbl

C. Foias, D.D. Holm and E.S. Titi, The Navier–Stokes-alpha model of fluid turbulence. Physica D 152 (2001) 505–519. | DOI | Zbl

C. Foias, D.D. Holm and E.S. Titi, The three dimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence theory. J. Dyn. Differ. Eq. 14 (2002) 1–35. | DOI | Zbl

K. Galvin, L. Rebholz and C. Trenchea, Efficient, unconditionally stable, and optimally accurate fe algorithms for approximate deconvolution models. SIAM J. Numer. Anal. 52 (2014) 678–707. | DOI | Zbl

V. Girault and P.-A. Raviart. Finite element methods for Navier–Stokes equations: theory and algorithms. Springer-Verlag (1986). | Zbl

J.L. Guermond, J.T. Oden and S. Prudhomme, An interpretation of the Navier–Stokes-alpha model as a frame-indifferent Leray regularization. Physica D 177 (2003) 23–30. | DOI | Zbl

T. Heister and G. Rapin, Efficient augmented Lagrangian-type preconditioning for the Oseen problem using grad-div stabilization. Int. J. Numer. Meth. Fluids 71 (2013) 118–134. | DOI | Zbl

J. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. Part IV: Error analysis for the second order time discretization. SIAM J. Numer. Anal. 2 (1990) 353–384. | DOI | Zbl

D. Holm and B.T. Nadiga, Modeling mesoscale turbulence in the barotropic double-gyre circulation. J. Phys. Oceanogr. 33 (2003) 2355–2365. | DOI

V. John, Reference values for drag and lift of a two dimensional time-dependent flow around a cylinder. Int. J. Numer. Meth. Fluids 44 (2004) 777–788. | DOI | Zbl

V. John and M. Roland, Simulations of the turbulent channel flow at $R{e}_{\tau }=180$ with projection-based finite element variational multiscale methods. Int. J. Numer. Meth. Fluids 55 (2007) 407–429. | DOI | Zbl

V.K. Kalantarov and E.S. Titi, Global attractors and determining modes for the 3D Navier- Stokes-Voight equations. Chin. Ann. Math. Ser. B 30 (2009) 697–714. | DOI | Zbl

J. Kim, P. Moin and R. Moser, Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177 (1987) 133–166. | DOI | Zbl

A. Larios and E.S. Titi, On the higher-order global regularity of the inviscid Voight regularization of the three-dimensional hydrodynamic models. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 603–627. | Zbl

W. Layton, Introduction to the Numerical Analysis of Incompressible Viscous Flows. SIAM (2008). | Zbl

W. Layton, On Taylor/eddy solutions of approximate deconvolution models of turbulence. Appl. Math. Lett. 24 (2011) 23–26. | DOI | Zbl

W. Layton and L. Rebholz, Approximate Deconvolution Models of Turbulence: Analysis, Phenomenology and Numerical Analysis. Springer-Verlag (2012). | Zbl

W. Layton, C. Manica, M. Neda and L. Rebholz, Numerical analysis and computational testing of a high accuracy Leray-deconvolution model of turbulence. Numer. Methods Partial Differ. Eq. 24 (2008) 555–582. | DOI | Zbl

W. Layton, C. Manica, M. Neda, M.A. Olshanskii and L. Rebholz, On the accuracy of the rotation form in simulations of the Navier–Stokes equations. J. Comput. Phys. 228 (2009) 3433–3447. | DOI | Zbl

E. Lunasin, S. Kurien, M. Taylor and E.S. Titi. A study of the Navier–Stokes-alpha model for two-dimensional turbulence. J. Turbulence 8 (2007) 751–778. | DOI

C. Manica and I. Stanculescu, Numerical analysis of Leray-Tikhonov deconvolution models of fluid motion. Comput. Math. Appl. 60 (2010) 1440–1456. | DOI | Zbl

M. Marion and R. Temam, Navier–Stokes equations: Theory and approximation. Handb. Numer. Anal. VI (1998) 503–688. | Zbl

C. Manica, M. Neda, M.A. Olshanskii and L. Rebholz, Enabling accuracy of Navier-Stokes-alpha through deconvolution and enhanced stability. ESAIM: M2AN 45 (2011) 277–308. | DOI | Numdam | Zbl

P. Mininni, D. Montgomery and A. Pouquet, Numerical solutions of the three-dimensional magnetohydrodynamic $\alpha$ model. Phys. Rev. E 71 (2005) 1–11. | DOI

R. Moser, J. Kim and N. Mansour, Direct numerical simulation of turbulent channel flow up to $R{e}_{\tau }=590$. Phys. Fluids 11 (1999) 943–945. | DOI | Zbl

A.P. Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973) 98–136. | Zbl

A.P. Oskolkov, On the theory of unsteady flows of kelvin-voigt fluids. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982) 191–202. Boundary value problems of mathematical physics and related questions in the theory of functions, 14. | Zbl

L.G. Rebholz and M. Sussman, On the high accuracy NS-$\alpha$-deconvolution model of turbulence. Math. Models Methods Appl. Sci. 20 (2010) 611–633. | DOI | Zbl

L. Rebholz and S. Watro, A note on Taylor-eddy and Kavosnay solutions of NS-$\alpha$-deconvolution and Leray-$\alpha$-deconvolution models. J. Nonlinear Dyn. 2014 (2014) 1–5. | DOI | Zbl

M. Schäfer and S. Turek, The benchmark problem ‘flow around a cylinder’ flow simulation with high performance computers II. In vol. 52 of Notes on Numerical Fluid Mechanics. Edited by E.H. Hirschel. Braunschweig, Vieweg (1996) 547–566. | Zbl

I. Stanculescu, Existence theory of abstract approximate deconvolution models of turbulence. Ann. Univ. Ferrara Sez. VII Sci. Mat. 54 (2008) 145–168. | DOI | Zbl

S. Stolz, N. Adams and L. Kleiser, The approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction. Phys. Fluids 13 (2001) 2985–3001. | DOI | Zbl

S. Stolz, N. Adams and L. Kleiser, An approximate deconvolution model for large-eddy simulations with application to incompressible wall-bounded flows. Phys. Fluids 13 (2001) 997–1015. | DOI | Zbl

S. Zhang, A new family of stable mixed finite elements for the 3d Stokes equations. Math. Comput. 74 (2005) 543–554. | DOI | Zbl