Eulerian Formulation Research Articles

Comparative performance of nonlinear energy harvesters through strongly coupled fluid-structure-electrical interactive models

Micro power generators (MPGs) can harvest electrical power from ambient energy resources in natural environments to operate small-scale electronic devices. Such systems exploit structural vibrations due to cross-flow instabilities through fluid–structure interactions (FSI). Due to attractive features of piezoelectric transducers, we utilize them for vibration-to-electricity conversion in this work. We develop strongly coupled FSI-based computational models to handle large-amplitude structural motions in which unsteady forces exerted by fluid flows on structures, their dynamical responses, and harvested power are directly computed. Our mathematical model for incompressible flows over moving bodies relies on arbitrary Lagrangian–Eulerian (ALE) formulation. Our in-house FSI solver primarily works using radial basis interpolation technique for grid morphing. Using broad ranges of fluidic, structural, and electrical parameters, we investigate the power generation performance of two types of MPGs, working through vortex-induced vibrations of a circular cylinder and a fluttering NACA-0012 airfoil, under different excitation conditions. Optimum values of load resistance are found to maximize the harvested power. Furthermore, our findings support the linear analysis, which provides insights into the possible instabilities in the electro-mechanical system. Another salient feature of this research work is the inclusion of cubic nonlinear stiffness in the energy harvesting models.

A multi-moment finite volume formulation for the interaction between free surface flow and moving bodies with THINC method

A numerical model is developed for fluid-rigid body interaction involving free surfaces on moving unstructured grids using the Arbitrary-Lagrangian–Eulerian (ALE) formulation. The multi-moment finite volume method is used for the spatial discretization of Navier–Stokes equations in which volume integrated average (VIA) and point values (PVs) are defined as prognostic variables and updated by integral and differential form respectively. Given VIA and PV in each cell, a higher-order reconstruction function can be built in a compact stencil which improves the numerical accuracy and robustness on the unstructured grids. More importantly, it facilitates imposing a more accurate Dirichlet–Neumann condition for the partitioned interface between the fluid and solid region. The geometrical conservation law (GCL) is also formulated to govern the spatial volume element under an arbitrary mapping. In order to capture the free surfaces of multiphase flows, the volume-of-fluid (VOF) equation is solved by THINC/QQ (THINC method with quadratic surface representation and Gaussian quadrature) scheme in varying mesh, which achieves geometrical faithful interface representation, particularly for curved surfaces in a complex and moving domain. The present model is validated by some benchmark tests in two dimensions for fluid–structure interaction problems with free surfaces. The numerical results show closer agreement with the experimental data than the reference solution of other numerical models. It convinces that the present model provides accurate and robust solutions for the simulation of the complex interaction between the free surface flows and moving bodies.

A computational framework for crack propagation along contact interfaces and surfaces under load

We present the first implicit computational framework for simulating crack propagation along contact interfaces and surfaces under load in three-dimensional bodies, which is distinct from modelling the contact interaction associated with crack closure. We restrict ourselves to brittle fracture and frictionless contact and focus on numerical challenges associated with the coupling of unilateral constraints emerging from the Griffith’s criterion and the contact conditions. The formulation is based on the configurational mechanics framework and is solved using the finite element method. The approach utilises a monolithic Arbitrary Lagrangian–Eulerian formulation permitting simultaneous resolution of crack propagation and unilateral contact constraints. Contact is embedded in the model using the well-known mortar contact formulation. Evolving cracks are explicitly modelled as displacement discontinuities within the mesh. Heterogeneous approximation of arbitrary order is used to discretise spatial displacements, enabling hp-adaptive refinement around the crack front and the contact interfaces traversed by the crack. The result is a holistic approach which handles issues associated with thermodynamic consistency, numerical accuracy and robustness of the computational scheme. Several numerical examples are presented to verify the model formulation and implementation; they also highlight how contact pressure and load applied on surfaces traversed by cracks influence their propagation. The robustness of the approach is validated by comparison of our simulations with existing numerical results and an industrial experiment involving cracks of complex morphologies propagating along contact interfaces between multiple deformable bodies.

On the computation of compressible multiphase flows with heat and mass transfer in elastic pipelines

The present work is motivated by the engineering need to simulate the multiphase flows in the course of the start-up of a long-distance crude oil pipeline. We propose a model for compressible multiphase flows with heat and mass transfer and diffusion processes in elastic pipelines. The model is derived with two approaches, i.e., (a) the spatial averaging procedure of a single phase model and (b) the Arbitrary Lagrangian Eulerian (ALE) formulation of the Baer-Nunziato model with quasi-1D approximation. The diffusion processes include the viscous dissipation, wall friction, heat conduction, and heat exchange with external environment. In particular, the wall friction consists of steady friction term calculated by Darcy-Weisbach formula, and the unsteady friction determined by the instantaneous-acceleration-based (IAB) model. The unsteady friction modifies the characteristic structure of the hyperbolic part of the model. For the solution of the hyperbolic part, we propose a three-wave approximate Riemann solver incorporating the unsteady friction term. Mass and heat transfer are realized via instantaneous relaxations of the chemical potential and temperature, respectively. Efficient iterative relaxation procedures for N-phase flows have been proposed. We have validated the proposed model and numerical methods against some benchmark multiphase problems and applied the model to calculate the start-up of a realistic liquid pipeline with intermediate pump station boundary condition.

Structure-preserving discretizations of two-phase Navier–Stokes flow using fitted and unfitted approaches

We consider the numerical approximation of a sharp-interface model for two-phase flow, which is given by the incompressible Navier–Stokes equations in the bulk domain together with the classical interface conditions on the interface. We propose structure-preserving finite element methods for the model, meaning in particular that volume preservation and energy decay are satisfied on the discrete level. For the evolving fluid interface, we employ parametric finite element approximations that introduce an implicit tangential velocity to improve the quality of the interface mesh. For the two-phase Navier–Stokes equations, we consider two different approaches: an unfitted and a fitted finite element method, respectively. In the unfitted approach, the constructed method is based on an Eulerian weak formulation, while in the fitted approach a novel arbitrary Lagrangian–Eulerian (ALE) weak formulation is introduced. Using suitable discretizations of these two formulations, we introduce two finite element methods and prove their structure-preserving properties. Numerical results are presented to show the accuracy and efficiency of the introduced methods.

Optimal Backstepping Flight Controller Design Based a New Compact Gravitational Search Algorithm for Hexarotor Attitude Stabilization

In this paper, a hexarotor Unmanned Aerial Vehicle (UAV) flight dynamics is derived and its control system is designed. A complete mathematical model of the UAV is obtained using Newton–Euler formulation and it is used for simulation. However, a reduced model is derived for the control design purpose which is based on the backstepping approach for hexarotor attitude stabilization and altitude trajectory tracking. The main contribution of this paper consists of the introduction of a new compact Gravitational Search Algorithm (cGSA) and its application for improving the backstepping controller performances in terms of tracking errors. Indeed, using the compact optimization paradigm enhances the performances of classical Population-based Algorithms (PBAs). Moreover, the introduced cGSA is applied to compute the optimal gains of the considered controller. In addition, the cGSA is compared with two compact optimization algorithms that are compact Particle Swarm Optimization (cPSO) and compact Teaching Learning-Based Optimization (cTLBO). The proposed algorithm shows encouraging results.

Discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

In this paper, we present a discontinuous Galerkin (DG) method for the simulation of incompressible and immiscible two-phase flow in fractured porous media with capillary pressure. The hybrid-dimensional fracture models are considered in which fractures are treated as (d−1)-dimensional interfaces in the d-dimensional domain and fluid exchange between fractures and surrounding matrix is taken into account. The approximation schemes combining the DG method with the fully implicit backward Euler time discretization are formulated for models with a single fracture and intersecting fractures network, and the stability and unique solvability are established. Optimal error estimates in the discrete H1-norm for saturation and pressure are derived. Numerical experiments in two-dimensional geometry are conducted to verify the accuracy of our theoretical analysis. In addition to the backward Euler formulation, we also implement the algorithm with a kind of conventional IMplicit Pressure Explicit Saturation (IMPES) method.

Configurational forces and ALE formulation for geometrically exact, sliding shells in non-material domains

The paper presents a novel Arbitrary Lagrangian–Eulerian formulation for dynamic problems of geometrically exact, sliding shells. In contrast to previous researches where the sliding motions are prescribed at sliding boundaries and the existence of configurational forces is not identified, the configurational momentum equation that governs the evolution of sliding motions is derived in a consistent variational framework in the present paper. To this end, the time-varying material domain due to the presence of sliding motion is mapped onto a time-invariant mesh domain, variations at fixed material and at fixed mesh coordinates are introduced, and their relationship with variation of material coordinates at fixed mesh coordinates are derived. Hamilton’s principle of variation of action is employed to derive the strong form of mechanical and configurational momentum equations together with natural boundary conditions at the sliding boundaries. In the finite element formulation, transfinite interpolation is employed to relate material coordinates of nodes inside the domain to the values at the sliding boundaries. The discrete form of Hamilton’s variational principle leads to discrete governing equations of the proposed ALE formulation. The generalized-α scheme is adjusted to integration the resulting mechanical and configurational momentum equations. Numerical examples are presented to validate correctness and efficiency of the proposed formulation.

A CNN-supported Lagrangian ISPH model for free surface flow

As a popular method for modeling violent free surface flow, the incompressible Smoothed Particle Hydrodynamics (ISPH) based on the Lagrangian formulation has attracted a great attention worldwide. The Lagrangian ISPH solves the unsteady Navier-Stokes and continuity equations using the projection method, in which the pressure is obtained by solving the pressure Poisson's equation (PPE) that is the most time-consuming part in the ISPH procedure. In this paper, the Convolutional Neural Network (CNN) is combined with ISPH and used to predict the fluid pressure instead of solving the PPE directly. Although limited attempts of using CNN for solving the PPE in Eulerian formulation (referred to as the Eulerian CNN framework) in mesh-based methods are found in the public domain, the present model is the first ISPH model supported by CNN in a Lagrangian formulation. The proposed model overcome several challenges associated with combining CNN with ISPH, including selecting the input parameters, formulating the objective functions, producing the training dataset and dealing with boundary conditions. Two classic free surface problems, i.e. the dam breaking and the wave propagation, are simulated to evaluate the performance of the present model. Quantitative assessments of the numerical error in terms of both the free surface profile and the pressure field are carried out. The assessments show that the new model does not only give results with satisfactory accuracy, but also requires much less computation time for estimating pressure if the number of particles is large, e.g., 100 thousands particles that is usually required in the practical ISPH simulation for free surface flow.

Proposing a Graphic Simulator for an Upper Limb Exoskeleton Robot

In this study, a graphic simulator that is used to simulate problems related to kinematics and dynamics for an exoskeleton robot arm with 5 degrees of freedom (DoF) was presented. The graphic simulator utilized the advantages of design software SolidWorks, Catia, and the computing and simulation power of SimMechanics Toolbox in Matlab. The core of the proposed graphic simulator is algorithm to solve the kinematics and dynamic problems of a developing upper-limb rehabilitation robot. The study used the proposed optimization-based algorithm to solve the inverse kinematics (IK) problem for the redundant robot model. Endpoint trajectories were imported from measurement data. The joints variable solutions obtained before entering the dynamics problem were smoothed to ensure feasibility in the later calculation process. A process to solve the inverse dynamics problem using physical model by combining the power of two software SolidWorks and SimMechanics was also proposed. This process ensured that the Robot’s design could be changed and updated to the kinematics calculation fast and easily. To evaluate this procedure, we also compared these dynamics results with results when applying the Lagrange–Euler formulation. All these calculation and simulation processes have been integrated into the graphic simulator software to show efficiency and user-friendliness.

A positivity preserving high‐order finite difference method for compressible two‐fluid flows

AbstractAny robust computational scheme for compressible flows must retain the hyperbolicity property or the real‐valued sound speed. Failure to maintain hyperbolicity, or the positivity of the square of the speed of sound, causes nonphysical distortions and the blow‐up of numerical simulations. Strong shock waves and interfacial discontinuities are ubiquitous features of the two‐fluid compressible dynamics that can potentially induce positivity‐related failure in a simulation. This article presents a positivity‐preserving algorithm for a high‐order, primitive variable‐based, weighted essentially non‐oscillatory finite difference scheme. The positivity preservation relies on a flux limiting technique that locally adapts high‐order fluxes towards the first order to retain the physical bounds of the solution without loss of high‐order convergence. This positivity preserving scheme has been devised and implemented up to eleventh order in one and two dimensions for a two‐fluid compressible model that consists of a single mass, momentum, and energy equations, as well as an advection of material parameters for capturing the interfaces. Several one‐ and two‐dimensional challenging test problems verify the performance. The scheme effectively retains high order accuracy while allowing for the simulation of several challenging problems that otherwise could not be successfully solved using the base scheme, without any penalty on the CFL condition requirement, and without any significant impact on the CPU times. The scheme represents the first high‐order (up to 11‐order) hyperbolicity‐preserving scheme for the considered two‐fluid compressible flows in the fully Eulerian formulation. The inherent efficiency of finite differences and the new robust positivity preserving quality enable modeling other challenging problems of two‐fluid and two‐phase problems.

Practical implications of the erroneous treatment of exposure time in the Eulerian hemolysis power law model.

Eulerian and Lagrangian power-law formulations are both widely used for computational fluid dynamics (CFD) to predict flow-induced hemolysis in blood-contacting medical devices. Both are based on the same empirical power-law correlation between hemolysis and the shear stress and exposure time. In the Lagrangian approach, blood damage is predicted by tracking both the stress and exposure time along a finite number of pathlines in the domain. In the Eulerian approach, a scalar transport equation is solved for a time-linearized damage index within the entire domain. Previous analytical work has demonstrated that there is a fundamental problem with the treatment of exposure time in the Eulerian model formulation such that the only condition under which the model correctly represents the true exposure time is in a flow field with no streamwise velocity variation. However, the practical implications of this limitation have yet to be thoroughly investigated. In this study, we demonstrate the inaccuracy of Eulerian hemolysis power-law model predictions due to the erroneous treatment of exposure time by systematically considering four benchmark test cases with increasing degrees of flow acceleration: Poiseuille flow through a straight tube, inclined Couette flow, and flow through a converging tube with two different convergence ratios. Compared with Lagrangian predictions, we show that, as flow acceleration becomes more pronounced, the resultant inaccuracy in the Eulerian predictions increases. For the inclined Couette flow case, there is a small degree of flow acceleration that yields a discrepancy in the range of 10% between Lagrangian and Eulerian predictions. For flows with a larger degree of acceleration, as occurs in the converging tube flow cases, the discrepancy is considerably larger (up to 257%). The inaccuracy of hemolysis predictions due to the erroneous treatment of exposure time in the Eulerian power-law model can be significant when there is streamwise velocity variation in the flow field. These results may partially explain the extremely large variability in CFD hemolysis predictions reported in the literature between Lagrangian and Eulerian models.

A novel Eulerian SPH shallow water model for 2D overland flow simulations

A 2D Eulerian meshless shallow water model based on smoothed particle hydrodynamics (SPH) is proposed to study overland flow. Rain on a dry bed to generate a runoff flow is one common hydraulic phenomenon. However, it is difficult for existing Lagrangian SPH shallow water models to present the flow because no computational particle exists on a dry bed. To overcome this difficulty, the Eulerian form of the 2D shallow water equations is used to describe flow behavior in the proposed model. An HLLC approximate Riemann solver is applied to calculate fluxes between two neighboring particles to smear out the instabilities caused by the discretization of the nonlinear convective term and the discontinuity of wet-dry beds. Additionally, we use a surface reconstruction method (SRM) to ensure that the proposed model has well-balanced and positivity-preserving properties. Bed friction and infiltration effects are also analytically considered. Eight case studies involving various overland flows with wet-dry transitions, rainfall, infiltration, bed friction and complex topography are adopted to test the proposed model capacity. The first-order convergence rate is obtained in the convergence analysis. Against the exact and measured results, in the analytical case studies, the relative root-mean-square errors of depth and velocity are approximately 0.02% to 0.3% and 1.3% to 1.7%, respectively. In the experimental case studies, the relative root-mean-square errors of depth and discharge (velocity) are approximately 3.8% and 1.2% to 6.1%, respectively. The good agreement demonstrates between the results shows the reliability of the proposed 2D Eulerian SPH shallow water model for real-world overland flows.

Combined Micro-Structural Effects of Linearly Increasing Reynolds Number and Mean Inflow Velocity on Flow Fields with Mesh Independence Analysis in Non-Classical Framework

The monolithic Eulerian formulation has widely been employed for solving numerically fluid–structure interaction (FSI) problems of finite structural displacement using the same mathematical variational formulation for fluid and structural dynamics. Recently, different physical features of fluid flow have been analyzed using this approach to such coupling problems by computing the classical benchmark solutions in a non-classical framework. Despite producing decent results, the analysis of micro-structural characteristics of fluid flow by applying the classical benchmark solutions still needs to be enhanced and extended further for such coupling problems. In this paper, the classical benchmark solutions have been enhanced and extended further for analyzing the combined micro-structural effects of linearly increasing Reynolds number Re and mean inflow velocity U¯ on flow fields with mesh independence analysis by employing a monolithic Eulerian formulation in a non-classical framework. To this aim, the Cosserat fluids theory is taken into account for the present coupling problem by considering three micro-rotational degrees of freedom (dof) of fluid particles. The model equations of the proposed Cosserat fluid–structure interaction (CFSI) problem are derived using underlying laws of continuum mechanics. A numerical section presents the implementation of the benchmark problem with test examples followed by a detailed evaluation of the obtained results. The results obtained indicate that a uniform linear increase in Reynolds number Re and mean inflow velocity U¯ produce the significant combined micro-structural effects on the micro-rotation velocity field ω, and this effect is found increasing on the increase of both parameters. This combined effect of increasing Re and U¯ on the velocity field u is also observed to be very significant in a sense that horizontal and vertical flow velocity profiles experience great variation by maintaining the same pattern on each increasing value of Re and U¯ at any particular location in the computational domain. Further, the mesh independence analysis is employed to verify the convergence of obtained results. The study concludes that the linear increase in Reynolds number and mean inflow velocity affect micro-rotational velocity field significantly at the micro-structural level with mesh independence analysis. Finally, some future recommendations to enhance and extend the study with some of its limitations are presented.

Analysis of Friction Stirs Welding Using the PROMETHEE Method

In friction stir welding (FSW) near the tool Thermo-mechanical condition of the piece at work the aim of the present work is to simulate Using general purpose software for this A fully coupled three-dimensional solid mechanical model has been created. ABAQUS/Arbitrary Lagrange- Eulerian (ALE) formulation re-meshing and large Enables transparency with distortions. Of FSW One of the least understood aspects of in friction stir welding (FSW) near the tool. For different welding speeds, forces and equipment Temperatures are recorded accurately in standard welding are done. Conditions. Numerical the simulations implemented are arbitrary Based on the Lograngian Eulerian (ALE) formula have the key aspect of the numerical approach is the palette and contact and friction between the tool accurate calculations of surface area. Accurate calculation of surface area. PROMETHEE is a priority for assessment enrichment Ranking system and its interactive help Descriptive complementary geometric analysis known as Promethee methods. In this The PROMETHEE method is the best solution from the analysis the shorter the distance and the longer the negative-ideal solution Determines the solution with the distance, but of these distances the comparison was not considered significant. Alternative: Conductivity, Thermal capacity, Density, Effusively, Emissivity. Evaluation Preference: Work piece, FSW Tool, Backing plate, strength, and conductivity. As a result FSW tool is very low As a result passive tool is very low got first rank for strength. The value of the dataset for Friction stir welding in PROMETHEE method shows that it results in strength is and top ranking.

One-dimensional turbulence modeling of compressible flows. I. Conservative Eulerian formulation and application to supersonic channel flow

Accurate but economical modeling of supersonic turbulent boundary layers is a standing challenge due to the intricate entanglement of temperature, density, and velocity fluctuations on top of the mean-field variation. Application of the van Driest transformation may describe well the mean state but cannot provide detailed flow information. This lack-in modeling coarse and fine-scale variability is addressed by the present study using a stochastic one-dimensional turbulence (ODT) model. ODT is a simulation methodology that represents the evolution of turbulent flow in a low-dimensional stochastic way. In this study, ODT is extended to fully compressible flows. An Eulerian framework and a conservative form of the governing equations serve as the basis of the compressible ODT model. Computational methods for statistical properties based on ODT realizations are also extended to compressible flows, and a comprehensive way of turbulent kinetic energy budget calculation based on compressible ODT is put forward for the first time. Two canonical direct numerical simulation cases of supersonic isothermal-wall channel flow at Mach numbers 1.5 and 3.0 with bulk Reynolds numbers 3000 and 4880, respectively, are used to validate the extended model. A rigorous numerical validation is presented, including the first-order mean statistics, the second-order root mean square statistics, and higher-order turbulent fluctuation statistics. In ODT results, both mean and root mean square profiles are accurately captured in the near-wall region. Near-wall temperature spectra reveal that temperature fluctuations are amplified at all turbulent scales as the effects of compressibility increase. This phenomenon is caused by intensified viscous heating at a higher Mach number, which is indicated by the steeper profiles of viscous turbulent kinetic energy budget terms in the very near-wall region. The low computational cost and predictive capabilities of ODT suggest that it is a promising approach for detailed modeling of highly turbulent compressible boundary layers. Furthermore, it is found that the ODT model requires a Mach-number-dependent increase in a viscous penalty parameter Z in wall-bounded turbulent flows to enable accurate capture of the buffer layer.

Numerical study of viscoelastic flow around an oscillating circular cylinder

The viscoelasticity-induced fluid–structure interaction studies have a significant influence on practical applications. To clarify the lock-in phenomenon and the wake topology of the vibrating cylinder placed in the viscoelastic flow, the Oldroyd-B fluid flows around an oscillating circular cylinder have been numerically investigated at Re = 10 and Re = 60, respectively. The governing equations are solved by the coupling of the square-root-conformation representation approach and the discontinuous Galerkin method in framework of the high-order dual splitting scheme. In addition, the arbitrary Lagrangian–Eulerian formulation is implemented in the coupling procedure in order to account for the interaction between the fluid and the oscillating body in the flow field. With this, complex boundary movements can be tackled simply and efficiently. In numerical simulation, the force coefficients and the wake structures of vortex and stress are discussed in some detail. At Re = 10, when the frequency of cylinder is small, it is obvious that the vortex shedding takes place in the wake. As the frequency increases, almost no obvious vortex shedding is observed. Also, the wake still oscillates at the same frequency of the cylinder for all cases, even for high Wi numbers. However, different wake modes of vortex and stress are found for various frequencies at Re = 60 and Wi = 0.1. In the lock-in region, the 2S mode of wake type are observed. Beyond the lock-in region, the wake type is no longer 2S, but the formation of vortex shedding and stress distribution in the far wake recovers to its natural mode. These numerical results open up a new field of study for viscoelastic fluids.

Frequency-Domain Estimation of a Transfer Matrix of an Uncommon Quadrotor in Hover

In this article, system identification of an uncommon quadrotor in hover is discussed. Two counter-rotating big rotors on the longitudinal axis and two counter-rotating small tilt rotors on the lateral axis form this quadrotor configuration. First, the nonlinear dynamic model of this vehicle is derived from Newton–Euler formulation. Next, using the approximate linear hover model and the suitable rotor mixing matrix for axes decoupling, simplified attitude dynamics of this quadrotor are obtained. However, the effects of sensor delays, flexible modes of the airframe, and inexact decoupling are not visible in this diagonal model, which is mainly based on physical principles. Therefore, a system identification method is used to obtain a more accurate model, which is required for control design. Hence, during parametric identification of nominal model coprime factors, a small robust control criterion is targeted. Then, the frequency response function of the proposed quadrotor prototype in hover is obtained. Next, the linear parametric model of the vehicle is estimated by solving optimally conditioned least squares and subsequent Gauss–Newton problems. After that, using validation-based uncertainty quantification, the uncertain model set is constructed around the estimated coprime factors. The resulting model set and its robust control relevance are depicted.

A Simplified Eulerian Formulation of a Multi-Phase Soft Tissue Model with Homeostasis and Phase Transformation

A Simplified Eulerian Formulation of a Multi-Phase Soft Tissue Model with Homeostasis and Phase Transformation

The planetary boundary layer physical processes, the secondary thermal baroclinic circulation and inertial oscillation contribution to diurnal variation of the Etesian winds over the Aegean Sea

Etesian winds constitute an important climatological phenomenon, which does not only moderate the heat during the summer in the Aegean Sea, but provides a source of clean renewable energy as well. Even though several papers have attempted to explain their dynamical and physical characteristics, the respective processes that drive the diurnal variation of the wind speed are not fully understood. The objective of this paper is to identify the processes responsible for diurnal variation with observed maximum wind speed around noon and minimum around midnight. Analytical solutions of a primitive equation set in Eulerian form, after introducing suitable conditions and approximations, reveal an inertial oscillation over the Aegean Sea. Data based on direct observations, ECMWF IFS high resolution analyses and high-resolution simulations with the Weather Research and Forecasting (WRF) model are utilized to find out the type and structure of the planetary boundary layer (PBL) over the Aegean Sea. This PBL appears to be of a marine character and turbulent mostly during the day but less during the night. The direct impact of local and regional thermally-driven circulations is found to be the main cause of the diurnal variation of the observed wind and partly the inertial oscillation. Results from numerical simulations certify these findings. Furthermore, the momentum and Newtonian heating exchanges by the physical processes inside the PBL, where the gradient wind together with smaller scales of atmospheric motions exist, are also necessary for explaining the variability of the Etesian winds.