arXiv subjects
Publications and source records attributed to Roger Lewandowski.
We consider two laminar incompressible flows coupled by the continuous law at a fixed interface. We approach the system by one that satisfies a friction Navier law, and we show that when the friction coefficient goes to infinity, the solutions converges to a solution of the initial system. We then write a numerical Schwarz-like coupling algorithm and run 2D-simulations, that yields same convergence result.
In this paper we show that the rotational Smagorinsky model for turbulent flows, can be put, for a wide range of parameters in the setting of Bochner pseudo-monotone evolution equations. This allows to prove existence of weak solutions a) identifying a proper functional setting in weighted spaces and b) checking some easily verifiable assumptions, at fixed time. We also will discuss the critical role of the exponents present in the model (power of the distance function and power of the curl) for what concerns the application of the theory of pseudo-monotone operators.
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 $-\alpha \nabla\cdot(\ell(x) D v_t)$. 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 $(v, p, k)$.
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.
To date no satisfying model exists to explain the mean velocity profile within the whole turbulent layer of canonical wall bounded flows. We propose a modification of the velocity profile expression that ensues from a recently proposed stochastic representation of fluid flows dynamics. This modeling, called modeling under location uncertainty introduces in a rigorous way a subgrid term generalizing the eddy-viscosity assumption and an eddy-induced advection term resulting from turbulence inhomogeneity. This latter term gives rise to a theoretically well-grounded model for the transitional zone between the viscous sublayer and the turbulent sublayer. An expression of the small-scale velocity component is also provided in the viscous zone. Numerical assessment of the results are provided for turbulent boundary layer flows, for pipe flows and channel flows at various Reynolds numbers.
This PhD thesis is devoted to deterministic study of the turbulence in the Navier- Stokes equations. The thesis is divided in four independent chapters.The first chapter involves a rigorous discussion about the energy's dissipation law, proposed by theory of the turbulence K41, in the deterministic setting of the homogeneous and incompressible Navier-Stokes equations, with a stationary external force (the force only depends of the spatial variable) and on the whole space R3. The energy's dissipation law, also called the Kolmogorov's dissipation law, characterizes the energy's dissipation rate (in the form of heat) of a turbulent fluid and this law was developed by A.N. Kolmogorov in 1941. However, its deduction (which uses mainly tools of statistics) is not fully understood until our days and then an active research area consists in studying this law in the rigorous framework of the Navier-Stokes equations which describe in a mathematical way the fluids motion and in particular the movement of turbulent fluids. In this setting, the purpose of this chapter is to highlight the fact that if we consider the Navier-Stokes equations on R3 then certain physical quantities, necessary for the study of the Kolmogorov's dissipation law, have no a rigorous definition and then to give a sense to these quantities we suggest to consider the Navier-Stokes equations with an additional damping term. In the framework of these damped equations, we obtain some estimates for the energy's dissipation rate according to the Kolmogorov's dissipation law.In the second chapter we are interested in study the stationary solutions of the damped Navier- Stokes introduced in the previous chapter. These stationary solutions are a particular type of solutions which do not depend of the temporal variable and their study is motivated by the fact that we always consider the Navier-Stokes equations with a stationary external force. In this chapter we study two properties of the stationary solutions : the first property concerns the stability of these solutions where we prove that if we have a control on the external force then all non stationary solution (with depends of both spatial and temporal variables) converges toward a stationary solution. The second property concerns the decay in spatial variable of the stationary solutions. These properties of stationary solutions are a consequence of the damping term introduced in the Navier-Stokes equations.In the third chapter we still study the stationary solutions of Navier-Stokes equations but now we consider the classical equations (without any additional damping term). The purpose of this chapter is to study an other problem related to the deterministic description of the turbulence : the frequency decay of the stationary solutions. Indeed, according to the K41 theory, if the fluid is in a laminar setting then the stationary solutions of the Navier-Stokes equations must exhibit a exponential frequency decay which starts at lows frequencies. But, if the fluid is in a turbulent setting then this exponential frequency decay must be observed only at highs frequencies. In this chapter, using some Fourier analysis tools, we give a precise description of this exponential frequency decay in the laminar and in the turbulent setting.In the fourth and last chapter we return to the stationary solutions of the classical Navier-Stokes equations and we study the uniqueness of these solutions in the particular case without any external force. Following some ideas of G. Seregin, we study the uniqueness of these solutions first in the framework of Lebesgue spaces of and then in the a general framework of Morrey spaces.
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.
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.
We study the Navier-Stokes equations with an extra eddy viscosity term in the whole space in three dimensions. We introduce a suitable regularized system for which we prove the existence of a regular solution defined for all time. We prove that when the regularizing parameter goes to zero, the solution of the regularized system converges to a turbulent solution of the initial system.
We introduce a new Large Eddy Simulation model in a channel, based on the projection on finite element spaces as filtering operation in its variational form, for a given triangulation $\{{\cal T}_h \}_{h>0}$. The eddy viscosity is expressed in terms of the friction velocity in the boundary layer due to the wall, and is of a standard sub grid-model form outside the boundary layer. The mixing length scale is locally equal to the grid size. The computational domain is the channel without the linear sub-layer of the boundary layer. The no slip boundary condition (BC) is replaced by a Navier (BC) at the computational wall. Considering the steady state case, we show that the variational finite element model we have introduced, has a solution $(\vv_h, p_h)_{h>0}$ that converges to a solution of the steady state Navier-Stokes Equation with Navier BC.
We consider two Large Eddy Simulation (LES) models for the approximation of large scales of the equations of Magnetohydrodynamics (MHD in the sequel). We study two $\alpha$-models, which are obtained adapting to the MHD the approach by Stolz and Adams with van Cittert approximate deconvolution operators. First, we prove existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations. Then we study the behavior of solutions as the deconvolution parameter goes to infinity. The main result of this paper is the convergence to a solution of the filtered MHD equations. In the final section we study also the problem with filtering acting only on the velocity equation.
We investigate the assymptotic behaviour of the modeling error in approximate deconvolution model in the 3D periodic case, when the order $N$ of deconvolution goes to $\infty$. We consider successively the generalised Helmholz filters of order $p$ and the Gaussian filter. For Helmholz filters, we estimate the rate of convergence to zero thanks to energy budgets, Gronwall's Lemma and sharp inequalities about Fouriers coefficients of the residual stress. We next show why the same analysis does not allow to conclude convergence to zero of the error modeling in the case of Gaussian filter, leaving open issues.
We consider a 3D Approximate Deconvolution Model (ADM) which belongs to the class of Large Eddy Simulation (LES) models. We work with periodic boundary conditions and the filter that is used to average the fluid equations is the Helmholtz one. We prove existence and uniqueness of what we call a "regular weak" solution $(\wit_N,q_N)$ to the model, for any fixed order $N\in\N$ of deconvolution. Then, we prove that the sequence $\{(\wit_N,q_N)\}_{N \in \N}$ converges -in some sense- to a solution of the filtered Navier-Stokes equations, as $N$ goes to infinity. This rigorously shows that the class of ADM models we consider have the most meaningful approximation property for averages of solutions of the Navier-Stokes equations.
We display a continous equation for the deconvolution process that generalizes the Van Cittert algorithm in the case of oceanic boundary conditions for a given fixed wind. We deduce a LES model for which we have existence and uniqueness of a strong solution. Finally, we display several numerical simulations showing the practical interest of the model.
We consider a deconvolution model for 3D periodic flows. We show the existence of a global attractor for the model.
In this work, we compare three turbulence models used to parameterize the oceanic boundary layer. These three models depend on the bulk Richardson number, which is coherent with the studied region, the West Pacific Warm Pool, because of the large mean shear associated with the equatorial undercurrent. One of these models, called R224, is new and the others are Pacanowski and Philander's model (R213 model) and Gent's model (R23 model). The numerical implementation is based on a non-conservative numerical scheme. The following (three criteria) are used to compare the models: the surface current intensity, the pycnocline's form and the mixed layer depth. We initialize the code with realistic velocity and density profiles thanks the TOGA-TAO array (McPhaden, 1995). In case of static instability zone on the initial density profile, only the R224 model gives realistic results. Afterwards, we study a mixed layer induced by the wind stress. In this case, the R224 results and the Pacanowski and Philander's results are similar. Furthermore, we simulate a long time case. We obtain a linear solution for all models that is in agreement with Bennis and al.
In 1934 J. Leray proposed a regularization of the Navier-Stokes equations whose limits were weak solutions of the NSE. Recently, a modification of the Leray model, called the Leray-alpha model, has atracted study for turbulent flow simulation. One common drawback of Leray type regularizations is their low accuracy. Increasing the accuracy of a simulation based on a Leray regularization requires cutting the averaging radius, i.e., remeshing and resolving on finer meshes. This report analyzes a family of Leray type models of arbitrarily high orders of accuracy for fixed averaging radius. We establish the basic theory of the entire family including limiting behavior as the averaging radius decreases to zero, (a simple extension of results known for the Leray model). We also give a more technically interesting result on the limit as the order of the models increases with fixed averaging radius. Because of this property, increasing accuracy of the model is potentially cheaper than decreasing the averaging radius (or meshwidth) and high order models are doubly interesting.
This paper is devoted to the simulation of the flow around and inside a rigid axisymmetric net. We describe first how experimental data have been obtained. We show in detail the modelization. The model is based on a Reynolds Averaged Navier-Stokes turbulence model penalized by a term based on the Brinkman law. At the out-boundary of the computational box, we have used a "ghost" boundary condition. We show that the corresponding variational problem has a solution. Then the numerical scheme is given and the paper finishes with numerical simulations compared with the experimental data.