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Luigi C. Berselli

Publications and source records attributed to Luigi C. Berselli.

At least 37 records · Page 2Linked to original sources
arXiv2020

On the existence of weak solutions for the steady Baldwin-Lomax model and generalizations

In this paper we consider the steady Baldwin-Lomax model, which is a rotational model proposed to describe turbulent flows at statistical equilibrium. The Baldwin-Lomax model is specifically designed to address the problem of a turbulent motion taking place in a bounded domain, with Dirichlet boundary conditions at solid boundaries. The main features of this model are the degeneracy of the operator at the boundary and a formulation in velocity/vorticity variables. The principal part of the operator is non-linear and it is degenerate, due to the presence (as a coefficient) of a power of the distance from the boundary: This fact makes the existence theory naturally set in the framework of appropriate weighted-Sobolev spaces.

math.AP↗
arXiv2020

Space-time discretization for nonlinear parabolic systems with $p$-structure

In this paper we consider nonlinear parabolic systems with elliptic part which can be also degenerate. We prove optimal error estimates for smooth enough solutions. The main novelty, with respect to previous results, is that we obtain the estimates directlywithout introducing intermediate semi-discrete problems. In addition, we prove the existence of solutions of the continuous problem with the requested regularity, if the data of the problem are smooth enough.

math.AP↗
arXiv2019

Turbulent flows as generalized Kelvin-Voigt materials: modeling and analysis

We model a 3D turbulent fluid, evolving toward a statistical equilibrium, by adding to the equations for the mean field $(v, p)$ a term like $-α\nabla\cdot(\ell(x) D v_t)$. This is of the Kelvin-Voigt form, where the Prandtl mixing length $\ell$ is not constant and vanishes at the solid walls. We get estimates for velocity $v$ in $L^\infty_t H^1_x \cap W^{1,2}_t H^{1/2}_x$, that allow us to prove the existence and uniqueness of a regular-weak solutions $(v, p)$ to the resulting system, for a given fixed eddy viscosity. We then prove a structural compactness result that highlights the robustness of the model. This allows us to pass to the limit in the quadratic source term in the equation for the turbulent kinetic energy $k$, which yields the existence of a weak solution to the corresponding Reynolds Averaged Navier-Stokes system satisfied by $(v, p, k)$.

math.AP↗
arXiv2019

On the energy equality for the 3D Navier-Stokes equations

In this paper we study the problem of energy conservation for the solutions of the initial boundary value problem associated to the 3D Navier-Stokes equations, with Dirichlet boundary conditions. First, we consider Leray-Hopf weak solutions and we prove some new criteria, involving the gradient of the velocity. Next, we compare them with the existing literature in scaling invariant spaces and with the Onsager conjecture. Then, we consider the problem of energy conservation for very-weak solutions, proving energy equality for distributional solutions belonging to the so-called Shinbrot class. A possible explanation of the role of this classical class of solutions, which is not scaling invariant, is also given.

math.AP↗
arXiv2019

Long-Time Reynolds Averaging of Reduced Order Models for Fluid Flows: Preliminary Results

We perform a theoretical and numerical investigation of the time-average of energy exchange among modes of Reduced Order Models (ROMs) of fluid flows. We are interested in the statistical equilibrium problem, and especially in the possible forward and backward average transfer of energy among ROM basis functions (modes). We consider two types of ROM modes: eigenfunctions of the Stokes operator and Proper Orthogonal Decomposition (POD) modes. We prove analytical results for both types of ROM modes and we highlight the differences between them. We also investigate numerically whether the time-average energy exchange between POD modes is positive. To this end, we utilize the one-dimensional Burgers equation as a simplified mathematical model, which is commonly used in ROM tests. The main conclusion of our numerical study is that, for long enough time intervals, the time-average energy exchange from low index POD modes to high index POD modes is positive, as predicted by our theoretical results.

math.AP↗
arXiv2018

On the Reynolds time-averaged equations and the long-time behavior of Leray-Hopf weak solutions, with applications to ensemble averages

We consider the three dimensional incompressible Navier-Stokes equations with non stationary source terms f chosen in a suitable space. We prove the existence of Leray-Hopf weak solutions and that it is possible to characterize (up to sub-sequences) their long-time averages, which satisfy the Reynolds averaged equations, involving a Reynolds stress. Moreover, we show that the turbulent dissipation is bounded by the sum of the Reynolds stress work and of the external turbulent fluxes, without any additional assumption, than that of dealing with Leray-Hopf weak solutions. Finally, in the last section we consider ensemble averages of solutions, associated to a set of different forces and we prove that the fluctuations continue to have a dissipative effect on the mean flow.

math.AP↗
arXiv2018

Suitable weak solutions of the Navier-Stokes equations constructed by a space-time numerical discretization

We prove that weak solutions obtained as limits of certain numerical space-time discretizations are suitable in the sense of Scheffer and Caffarelli-Kohn-Nirenberg. More precisely, in the space-periodic setting, we consider a full discretization in which the theta-method is used to discretize the time variable, while in the space variables we consider appropriate families of finite elements. The main result is the validity of the so-called local energy inequality.

math.AP↗
arXiv2017

On the Bardina's model in the whole space

We consider the Bardina's model for turbulent incompressible flows in the whole space with a cut-off frequency of order 1/alpha. We show that for any alpha >0 fixed, the model has a unique regular solution defined for all time.

math.AP↗
arXiv2017

On the convergence of a fully discrete scheme of LES type to physically relevant solutions of the incompressible Navier-Stokes

Obtaining reliable numerical simulations of turbulent fluids is a challenging problem in computational fluid mechanics. The Large Eddy Simulations (LES) models are efficient tools to approximate turbulent fluids and an important step in the validation of these models is the ability to reproduce relevant properties of the flow. In this paper we consider a fully discrete approximation of the Navier-Stokes-Voigt model by an implicit Euler algorithm (with respect to the time variable) and a Fourier-Galerkin method (in the space variables). We prove the convergence to weak solutions of the incompressible Navier-Stokes equations satisfying the natural local entropy condition, hence selecting the so-called physically relevant solutions

math.AP↗
arXiv2016

Global regularity properties of steady shear thinning flows

In this paper we study the regularity of weak solutions to systems of p-Stokes type, describing the motion of some shear thinning fluids in certain steady regimes. In particular we address the problem of regularity up to the boundary improving previous results especially in terms of the allowed range for the parameter p.

math.AP↗
arXiv2016

Suitable weak solutions to the 3D Navier-Stokes equations are constructed with the Voigt Approximation

In this paper we consider the Navier-Stokes equations supplemented with either the Dirichlet or vorticity-based Navier boundary conditions. We prove that weak solutions obtained as limits of solutions to the Navier-Stokes-Voigt model satisfy the local energy inequality. Moreover, in the periodic setting we prove that if the parameters are chosen in an appropriate way, then we can construct suitable weak solutions trough a Fourier-Galerkin finite-dimensional approximation in the space variables.

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arXiv2013

Optimal error estimate for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids

In this paper we study the numerical error arising in the space-time approximation of unsteady generalized Newtonian fluids which possess a stress-tensor with $(p,δ)$-structure. A semi-implicit time-discretization scheme coupled with conforming inf-sup stable finite element space discretization is analyzed. The main result, which improves previous suboptimal estimates as those in [A. Prohl, and M. Ruzicka, SIAM J. Numer. Anal., 39 (2001), pp. 214--249] is the optimal $O(k+h)$ error-estimate valid in the range $p\in (3/2,2]$, where $k$ and $h$ are the time-step and the mesh-size, respectively. Our results hold in three-dimensional domains (with periodic boundary conditions) and are uniform with respect to the degeneracy parameter $\in [0,δ_0]$ of the extra stress tensor.

math.NA↗
arXiv2012

Pulsatile Viscous Flows in Elliptical Vessels and Annuli: Solution to the Inverse Problem, with Application to Blood and Cerebrospinal Fluid Flow

We consider the fully-developed flow of an incompressible Newtonian fluid in a cylindrical vessel with elliptical cross-section (both an ellipse and the annulus between two confocal ellipses). In particular, we address an inverse problem, namely to compute the velocity field associated with a given, time-periodic flow rate. This is motivated by the fact that flow rate is the main physical quantity which can be actually measured in many practical situations. We propose a novel numerical strategy, which is nonetheless grounded on several analytical relations. The proposed method leads to the solution of some simple ordinary differential systems. It holds promise to be more amenable to implementation than previous approaches, which are substantially based on the challenging computation of Mathieu functions. Some numerical results are reported, based on measured data for human blood flow in the internal carotid artery, and cerebrospinal fluid (CSF) flow in the upper cervical region of the human spine. As expected, computational efficiency is the main asset of our solution: a speed-up factor over 10^3 was obtained, compared to more elaborate numerical approaches. The main goal of the present study is to provide an improved source of initial/boundary data for more ambitious numerical approaches, as well as a benchmark solution for pulsatile flows in elliptical sections with given flow rate. The proposed method can be effectively applied to bio-fluid dynamics investigations (possibly addressing key aspects of relevant diseases), to biomedical applications (including targeted drug delivery and energy harvesting for implantable devices), up to longer-term medical microrobotics applications.

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