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Existence theory of abstract approximate deconvolution models of turbulence. (English) Zbl 1191.76058

By using approximate deconvolution, the author develops an abstract approach to modeling the motion of large eddies in a turbulent flow. The first part represents approximate deconvolution models (ADM) and an approximate deconvolution or approximate/asymptotic inverse of the filtering operator $D$. For large eddy simulation (LES), the author examines only two operators which have been studied earlier: the van Cittert deconvolution operator, and the Geurt’s approximate inverse filter. The second part reviews the averaging/filtering in LES and defines the function spaces and norms needed for variational formulation of the scale similarity model. In the third part the author finds conditions on the approximate deconvolution operator $D$ that guarantee that ADM has a weak solution. Thus, the operator $D:{L}^{2}\left(Q\right)\to {L}^{2}\left(Q\right)$, $Q={\left(0,L\right)}^{3},$ $L$ being the period, has to be a self-adjoint positive definite bounded linear operator which commutes with differentiation. The fourth part demonstrates that the weak solution is really a unique strong solution, and that the model satisfies an energy equality rather than inequality and correctly captures the global energy balance of large scales.

A few examples of deconvolution operators and their properties are reviewed in the fifth part: the van Cittert deconvolution operator, the accelerated van Cittert deconvolution operator, Tikhonov regularization of a deconvolution operator, the Geurt’s approximate filter inverse and its variation. The main conclusions are given in the sixth part.

##### MSC:
 76F02 Fundamentals of turbulence 76F65 Direct numerical and large eddy simulation of turbulence
##### References:
 [1] Adams, N.A., Stolz, S.: Deconvolution methods for subgrid-scale approximation in large eddy simulation. Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, Zurich (2001) [2] Bertero M. and Boccacci B. (1998). Introduction to Inverse Problems in Imaging. IOP Publishing Ltd., Bristol [3] Berselli L.C., Iliescu T. and Layton W. (2006). Mathematics of Large Eddy Simulation of Turbulent Flows. Springer, Berlin [4] Dunca, A., John, V., Layton, W.: The Commutation Error of the Space Averaged Navier-Stokes Equations on a Bounded Domain. Advances in Mathematical Fluid Mechanics 3, pp. 53–78. Birkhaüser Verlag, Basel (2004) [5] Dunca, A.: Space avereged Navier-Stokes equations in the presence of walls. Phd Thesis, University of Pittsburgh (2004) [6] Dunca, A., Epshteyn, Y.: On the Stolz-Adams deconvolution model for the large-eddy simulation of turbulent flows. SIAM J. Math. Anal. 1890–1902 (2006) [7] Foias C., Holm D.D. and Titi E.S. (2001). The Navier-Stokes-alpha model of fluid turbulence. Physica D. 152–153: 505–519 · Zbl 1037.76022 · doi:10.1016/S0167-2789(01)00191-9 [8] Galdi, G.P.: An Introduction to the Navier stokes initial-boundary value problem. Fundamental Directions in Mathematical Fluid Mechanics, pp. 1–70. Birkhäuser, Basel (2000) [9] Galdi G.P. and Layton W.J. (2000). Approximation of the large eddies in fluid motion II: a model for space-filtered flow. Math. Models Methods Appl. Sci. 10: 343–350 [10] Germano M. (1986). Differential filters of elliptic type. Phys. Fluids 29: 1757–1758 · Zbl 0647.76042 · doi:10.1063/1.865650 [11] Geurts B.J. (1997). Inverse modeling for large-eddy simulation. Phys. Fluids 9: 3585–3587 · doi:10.1063/1.869495 [12] John V., Layton W. and Sahin N. (2004). Derivation and analysis of near wall models for channel and recirculating flows. Comput. Math. Appl. 48: 1135–1151 · Zbl 1059.76030 · doi:10.1016/j.camwa.2004.10.011 [13] Kuerten J.G.M., Geurts B.J., Vreman A.W. and Germano M. (1999). Dynamic inverse modelling and its testing in LES of the mixing layer. Phys. Fluids 11: 3778–3785 · Zbl 1149.76442 · doi:10.1063/1.870238 [14] Layton W. and Lewandowski R. (2003). A simple and stable scale similarity model for large eddy simulation: energy balance and existence of weak solutions. Appl. Math. Lett. 16: 1205–1209 · Zbl 1039.76027 · doi:10.1016/S0893-9659(03)90118-2 [15] Layton W. and Lewandowski R. (2006). On a well-posed turbulence model. Discret. Contin. Dyn. Syst. Ser. B 6: 111–128 [16] Layton W. and Neda M. (2006). Truncation of scales by time relaxation. JMAA 325: 788–807 [17] Layton W. and Stanculescu I. (2007). K-41 optimized approximate deconvolution models. IJCSM 1: 396–411 · Zbl 1185.76707 · doi:10.1504/IJCSM.2007.016554 [18] Leray, J.: Sur le movement d’un fluide visqueux emplissant l’espace. Acta Math, vol. 63, pp. 193–248 (1934). Kluwer, Dordrecht (1997) [19] Sagaut P. (2001). Large Eddy Simulation for Incompressible Flows. Springer, Berlin [20] Stolz S., Adams N.A. and Kleiser L. (2001). The approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction. Phys. Fluids 13: 997–1015 · Zbl 1184.76530 · doi:10.1063/1.1350896 [21] Temam R. (1995). Navier-Stokes equations and nonlinear functional analysis. SIAM, Philadelphia