Direct stiffness matrices of BEs in the Galerkin BEM formulation

In the analysis of an elastic two-dimensional solid body by means of the Symmetric Galerkin Boundary Element Method (SGBEM), difficulties arise in the computation of some terms of the solving system coefficients. In fact these coefficients are expressed as double integrals with singularities of order 1/r², r being the distance between the field and source points. In order to compute these coefficients a strategy based on Schwartz's distribution theory is employed. In this paper the direct stiffness matrix related to the generic node of the free boundary are computed in closed form.     

Vibration control of helicopter blade flapping via time-delay absorber

In this study, the controller is used to suppress the vibration due to rotor the helicopter blade flapping motion. The objective of this paper is to investigate the effect of time-delay absorber on the vibrating system when subjected to multi-parametric excitation forces. The equations of motion are described by coupled nonlinear differential equations. The averaging method is applied to obtain the frequency response equations near simultaneous sub-harmonic and internal resonance. The stability of the obtained nonlinear solution is studied and solved numerically. Numerical simulations show the steady state response amplitude versus the detuning parameter and the effects of the parameters system and controller. Effectiveness of the absorber E _a is about 2.7×10⁵ of the main system (X).     

Nonlinear dynamics between two differentially heated vertical plates in the presence of stratification

We consider the numerical simulation of the flow between infinite, differentially heated vertical plates with positive stratification. We use a two-dimensional Boussinesq approximation, with periodic boundary conditions in the vertical direction. The relative stratification parameter ${\gamma=(\frac{1}{4}Ra S)^{1/4}}$ , where Ra is the Rayleigh number and S the adimensional stratification, is kept constant and equal to 8. The Prandtl number is 0.71. We derive a complex Ginzburg-Landau equation from the equations of motion. Coefficients are computed analytically, but we find that the domain of validity of these coefficients is small and rely on the numerical simulation to adjust the coefficients over a wider range of Rayleigh numbers. We show that the Ginzburg-Landau equation is able to accurately predict the characteristics of the periodic solution at moderate Rayleigh numbers. Above the primary bifurcation at Ra = 1.63 × 10⁵, the Ginzburg-Landau model is found to be Benjamin-Feir unstable and to be characterized by modulated traveling waves and phase-defect chaos, which is supported by evidence from the DNS. As the Rayleigh number is increased beyond Ra = 2.7 × 10⁵, nonlinearities become strong and the flow is characterized by cnoidal waves.     

Propagation of horizontally polarized transverse waves in a solid with a periodic distribution of cracks

The propagation of time-harmonic waves in a solid containing a periodic distribution of cracks is investigated in a two-dimensional configuration. The cracks are parallel to the x-axis, and their centers are located at positions x = md, y = lh(m, l = 0, ±1, ±2,…). The wave motion is polarized in the z-direction and propagates in the y-direction (normal to the cracks). The theory of Floquet or Bloch waves, together with an appropriate Green's function and the condition of vanishing traction on the crack faces leads to a system of singular integral equations, which provides the basis for the derivation of an exact dispersion equation. Numerical results are presented for the wave number as a function of the frequency. The frequency spectrum shows a pattern of passing and stopping bands. The exact results are compared with the frequency spectrum according to a simplified theory which considers the arrays of collinear cracks in the planes y = lh (l= 0 ±1, ±2,…) as planes of homogeneous transmission and reflection. Good agreement is observed between exact and approximate results.     

Turbulence modelling of the flow past a pitching NACA0012 airfoil at and Reynolds numbers

This paper provides a study of the NACA0012 dynamic stall at Reynolds numbers 10⁵ and 10⁶ by means of two- and three-dimensional numerical simulations. The turbulence effect on the dynamic stall is studied by statistical modelling. The results are compared with experiments concerning each test case. Standard URANS turbulence modelling have shown a quite dissipative character that attenuates the instabilities and the vortex structures related to the dynamic stall. The URANS approach Organised Eddy Simulation (OES) has shown an improved behaviour at the high Reynolds number range. Emphasis is given to the physical analysis of the three-dimensional dynamic stall structure, for which there exist few numerical results in the literature, as far as the Reynolds number range is concerned. This study has shown that the downstroke phases of the pitching motion are subjected to strong three-dimensional turbulence effects along the span, whereas the flow is practically two-dimensional during the upstroke motion.     

Numerical study of large amplitude,nonsinusoidal motion and camber effects on pitching airfoil propulsion

The effects of large amplitude and nonsinusoidal motion on pitching airfoil aerodynamics for thrust generation were numerically studied with a 2-D NACA0012 airfoil used, and various 2-D NACA asymmetric airfoils were applied for camber effect study. The large amplitude effect study has been undertaken over a wide range of reduced frequency k (from 6 to 18) and pitching amplitude θ (from 5° to 30°) at Re=1.35×10⁴ with sinusoidal pitching profile used. It is shown that the large pitching amplitude results in much more thrust generated than that at low pitching amplitude and the increase of thrust with amplitude becomes slow when the amplitude reaches some degree. However, the propulsive efficiency noticeably decreases with the increase of θ at a fixed k.An adjustable parameter K was employed to realize various nonsinusoidal motions and the effect of nonsinusoidal motion was investigated with various unsteady parameters (θ, k) applied. The results reveal that nonsinusoidal motion has a noticeable effect on the aerodynamic performance, as it affects the instantaneous force coefficients, maximum thrust coefficients and flow structures. An increase in K results in a better thrust generation performance at fixed θ and k, especially for K>0. It is also shown that the larger K noticeably influences the wake pattern and induces a stronger reverse von Karman vortex street in the wake, which in turn leads to the increased thrust. The camber study was performed on various 2-D NACA airfoils with different cambers and camber locations undergoing sinusoidal pitching motion at θ=5° and Re=1.35×10⁴. It is found that varying camber offers little improvement in thrust generation performance.     

Mixed convection and conduction heat transfer in open cavities

A numerical study has been carried out on mixed convection and conduction in open cavities. The study covers the Rayleigh number from 10 to 10⁶, the Reynolds number from 1 to 10³ andk _r =k _w /k _a from 1 to 100 forPr=0.72 (air) and cavity aspect ratioB from 0.8 to 1.3. The flow is assumed to be laminar and two-dimensional. The density variation is taken into account by the Boussinesq approximation. The controlvolume approach is used for solving the governing equations of conjugate heat transfer involving conduction in the walls. Streamlines and isotherms in the system are produced and the results are represented in terms of the Nusselt number as function of other parameters. Correlations are derived to calculate heat transfer through the cavity openings.     

Motion of singles bubbles moving under a slightly inclined surface through stationary liquids

The influence of the liquid properties on the dynamical bubble shape and on the bubble motion has been investigated for bubbles moving under a downward facing inclined surface. The Morton number Mo varied from 2.59 × 10⁻¹¹ to 2.52 × 10⁺⁰¹. The Bond number Bo covered the range from 10 to 150 and the surface inclination angle θ was varied from 2° to 6°. To cover the wide range of Mo, several liquids such as glycerine, propanediol, water and isopropanol were used. The results have shown that the relation Fr = Fr(Bo, Mo, θ) is not adequate to describe the bubble motion, where Fr is the terminal Froude number. The choice of the terminal Reynolds number Re as the dependent parameter, allowed the clarification of the role of the Morton number on the bubble motion. At a given Bond number, the bubble Reynolds number decreases monotonously with the Morton number. Furthermore, an empirical correlation Re = Re(Bo, Mo, θ) is given that can be readily used in the mathematical modelling of bubble laden flows under solids.     

Blasenbewegung aufgrund von Marangonikonvektion

This is an experimental study on gas bubble motion in a vertical temperature gradient. Surface tension convection (Marangoni convection) superimposing gravity is discussed with respect to the theory of Young et al. as another cause for bubble motion. The Marangoni numbers for the convection inside the bubble (N ⁱ _Ma) and outside the bubble in the medium (N ^e _Ma) are introduced as criterion for the applicability of the theory. It is shown by the experiments that the Young theory holds up to Marangoni numbersN ⁱ _Ma=10^?2 and N^e _Ma=2, respectively.     

Dynamic analysis of planar rigid-body mechanical systems with two-clearance revolute joints

In this paper, the behavior of planar rigid-body mechanical systems due to the dynamic interaction of multiple revolute clearance joints is numerically studied. One revolute clearance joint in a multibody mechanical system is characterized by three motions which are: the continuous contact, the free-flight, and the impact motion modes. Therefore, a mechanical system with n-number of revolute clearance joints will be characterized by 3 ⁿ motions. A slider-crank mechanism is used as a demonstrative example to study the nine simultaneous motion modes at two revolute clearance joints together with their effects on the dynamic performance of the system. The normal and the frictional forces in the revolute clearance joints are respectively modeled using the Lankarani–Nikravesh contact-force and LuGre friction models. The developed computational algorithm is implemented as a MATLAB code and is found to capture the dynamic behavior of the mechanism due to the motions in the revolute clearance joints. This study has shown that clearance joints in a multibody mechanical system have a strong dynamic interaction. The motion mode in one revolute clearance joint will determine the motion mode in the other clearance joints, and this will consequently affect the dynamic behavior of the system. Therefore, in order to capture accurately the dynamic behavior of a multi-body system, all the joints in it should be modeled as clearance joints.     

Sharp vorticity gradients in two-dimensional turbulence and the energy spectrum

Formation of sharp vorticity gradients in two-dimensional (2D) hydrodynamic turbulence and their influence on the turbulent spectra are considered. The analog of the vortex line representation as a transformation to the curvilinear system of coordinates moving together with the di-vorticity lines is developed and compressibility of this mapping appears as the main reason for the formation of the sharp vorticity gradients at high Reynolds numbers. In the case of strong anisotropy the sharp vorticity gradients can generate spectra which fall off as k ^?3 at large k, which appear to take the same form as the Kraichnan spectrum for the enstrophy cascade. For turbulence with weak anisotropy the k dependence of the spectrum due to the sharp gradients coincides with the Saffman spectrum: E(k) ~ k ^?4. Numerical investigations of decaying turbulence reveal exponential growth of di-vorticity with a spatial distributed along straight lines. Thus, indicating strong anisotropy and accordingly the spectrum is close to the k ^?3-spectrum.     

Global Weak Solutions¶for the Two-Dimensional Motion¶of Several Rigid Bodies¶in an Incompressible Viscous Fluid

We consider the two-dimensional motion of several non-homogeneous rigid bodies immersed in an incompressible non-homogeneous viscous fluid. The fluid, and the rigid bodies are contained in a fixed open bounded set of ?². The motion of the fluid is governed by the Navier-Stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies. The time variation of the fluid domain (due to the motion of the rigid bodies) is not known a priori, so we deal with a free boundary value problem. The main novelty here is thedemonstration of the global existence of weak solutions for this problem. More precisely, the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary. We give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero.     

Analytical and numerical study of double diffusive convection in a vertical enclosure

This paper presents an analytical and numerical study of natural convection of a double-diffusive fluid contained in a rectangular slot subject to uniform heat and mass fluxes along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number, Ra _T ; buoyancy ratio, N; Lewis number, Le; Prandtl number, Pr and aspect ratio of the cavity, A. In the first part of the analytical study a scale analysis is applied to the two extreme cases of heat-transfer and mass-transfer-driven flows. In the second part, an analytical solution, based on the parallel flow approximation, is reported for tall enclosures (A?1). Solutions for the flow fields, temperature and concentration distributions and Nusselt and Sherwood numbers are obtained in terms of the governing parameters of the problem. In the limits of heat-driven and solute-driven flows a good agreement is obtained between the prediction of the scale analysis and those of the analytical solution. The numerical solutions are based on the complete governing equations for two-dimensional flows, and cover the range 1≤Ra _T ≤10⁷, 0≤N≤10⁵, 10^-3≤Le≤10³, 1≤A≤20 and Pr=7. A good agreement is found between the analytical predictions and the numerical simulation.     

Bifurcation and chaos of a flag in an inviscid flow

A two-dimensional model is developed to study the flutter instability of a flag immersed in an inviscid flow. Two dimensionless parameters governing the system are the structure-to-fluid mass ratio M^⁎ and the dimensionless incoming flow velocity U^⁎. A transition from a static steady state to a chaotic state is investigated at a fixed M^⁎=1 with increasing U^⁎. Five single-frequency periodic flapping states are identified along the route, including four symmetrical oscillation states and one asymmetrical oscillation state. For the symmetrical states, the oscillation frequency increases with the increase of U^⁎, and the drag force on the flag changes linearly with the Strouhal number. Chaotic states are observed when U^⁎ is relatively large. Three chaotic windows are observed along the route. In addition, the system transitions from one periodic state to another through either period-doubling bifurcations or quasi-periodic bifurcations, and it transitions from a periodic state to a chaotic state through quasi-periodic bifurcations.     

Study of dynamic structure and heat and mass transfer of a vertical ceramic tiles dryer using CFD simulations

In this study, we developed a two-dimensional Computational Fluid Dynamics (CFD) model to simulate dynamic structure and heat and mass transfer of a vertical ceramic tiles dryer (EVA 702). The carrier’s motion imposed the choice of a dynamic mesh based on two methods: “spring based smoothing” and “local remeshing”. The dryer airflow is considered as turbulent (Re = 1.09 × 10⁵ at the dryer inlet), therefore the Re-Normalization Group $k - \in$ model with Enhanced Wall Treatment was used as a turbulence model. The resolution of the governing equation was performed with Fluent 6.3 whose capacities do not allow the direct resolution of drying problems. Thus, a user defined scalar equation was inserted in the CFD code to model moisture content diffusion into tiles. User-defined functions were implemented to define carriers’ motion, thermo-physical properties… etc. We adopted also a “two-step” simulation method: in the first step, we follow the heat transfer coefficient evolution (H_c). In the second step, we determine the mass transfer coefficient (H_m) and the features fields of drying air and ceramic tiles. The found results in mixed convection mode (Fr = 5.39 at the dryer inlet) were used to describe dynamic and thermal fields of airflow and heat and mass transfer close to the ceramic tiles. The response of ceramic tiles to heat and mass transfer was studied based on Biot numbers. The evolutions of averages temperature and moisture content of ceramic tiles were analyzed. Lastly, comparison between experimental and numerical results showed a good agreement.     

Aerodynamics of laminar separation flutter at a transitional Reynolds number

Experimental observations of self-sustained pitch oscillations of a NACA 0012 airfoil at transitional Reynolds numbers were recently reported. The aeroelastic limit cycle oscillations, herein labelled as laminar separation flutter, occur in the range 5.0×10⁴≤Re_c≤1.3×10⁵. They are well behaved, have a small amplitude and oscillate about θ=0°. It has been speculated that laminar separation leading to the formation of a laminar separation bubble, occurring at these Reynolds numbers, plays an essential role in these oscillations. This paper focuses on the Re_c=7.7×10⁴ case, with the elastic axis located at 18.6% chord. Considering that the experimental rig acts as a dynamic balance, the aerodynamic moment is derived and is empirically modelled as a generalized Duffing–van-der-Pol nonlinearity. As expected, it behaves nonlinearly with pitch displacement and rate. It also indicates a dynamically unstable equilibrium point, i.e. negative aerodynamic damping. In addition, large eddy simulations of the flow around the airfoil undergoing prescribed simple harmonic motion, using the same amplitude and frequency as the aeroelastic oscillations, are performed. The comparison between the experiment and simulations is conclusive. Both approaches show that the work done by the airflow on the airfoil is positive and both have the same magnitude. The large eddy simulation (LES) computations indicate that at θ=0°, the pitching motion induces a lag in the separation point on both surfaces of the airfoil resulting in negative pitching moment when pitching down, and positive moment when pitching up, thus feeding the LCO.     

Natural convection in an inclined quadrantal cavity heated and cooled on adjacent walls

Natural convective flow and heat transfer in an inclined quadrantal cavity is studied experimentally and numerically. The particle tracing method is used to visualize the fluid motion in the enclosure. Numerical solutions are obtained via a commercial CFD package, Fluent. The working fluid is distilled water. The effects of the inclination angle, ? and the Rayleigh number, Ra on fluid flow and heat transfer are investigated for the range of angle of inclination between 0° ? ? ? 360°, and Ra from 10⁵ to 10⁷. It is disclosed that heat transfer changes dramatically according to the inclination angle which affects convection currents inside, i.e. flow physics inside. A fairly good agreement is observed between the experimental and numerical results.     

Experimental analysis of thermal fields in horizontally eccentric cylindrical annuli

This paper presents new experimental results on thermal field and heat transfer in a two-dimensional annulus between horizontally eccentric cylinders. The study is conducted by means of optical techniques, for 1.07×10⁴≤Ra _L≤8.27×10⁴ and a wide eccentricity range. The horizontal eccentricity of the inner cylinder substantially alters the thermal field and the geometry of the plume, but, in analogy to the behaviour for vertical eccentricity, the average Nu is slightly affected in the investigated range of eccentricity. The concentric geometry is also considered mainly to validate the experimental technique and evaluate the accuracy of the adopted methodology by comparison with available results. Both shearing interferometer and reference beam interferometer are obtained by means of Wollaston prisms with appropriate splitting angles, so that the temperature and local Nu distributions may be evaluated quantitatively from the original pictures via digital image processing.     

Numerical investigations into the asymmetric effects on the aerodynamic response of a pitching airfoil

The effects of asymmetric sinusoidal motion on pitching airfoil aerodynamics were studied by numerical simulations for 2-D flow around a NACA0012 airfoil at Re=1.35×10⁵. Various unsteady parameters (amplitude of oscillation, d; reduced frequency, k) were applied to investigate the effect of asymmetry parameter S on the instantaneous force coefficients and flow patterns. The results reveal that S has a noticeable effect on the aerodynamic performance, as it affects the instantaneous force coefficient, maximum lift and drag coefficient, hysteresis loops and the flow structures.     

PIV measurement of separated flow over a blunt plate with different chord-to-thickness ratios

The influence of the chord-to-thickness ratio (c/t) on the spatial characteristics of the separated shear layer over a blunt plate and the leading-edge vortices embedded in the separated shear layer was studied extensively using planar particle image velocimetry (PIV). Three systems corresponding to different shedding modes were chosen for the comparative study: c/t=3, 6 and 9. The Reynolds number based on the plate's thickness (t) was Re_t=1×10³. A gigapixel CCD camera was used to acquire images with a spatial resolution of 0.06t×0.06t in the measurement range of 9.5t×4.5t. Distributions of statistical quantities, such as the streamline pattern, streamwise velocity fluctuation intensity, shear stress and reverse flow intermittency, showed that the separated shear layer in the system with c/t=3 did not reattach to the plate's surface, while the near‐wake behind the trailing edge was highly unstable because the energetic leading-edge vortices were shed into the wake. The separated shear layer of the system with c/t=6 periodically reattached to the plate's surface, which resulted in intensified fluctuations of the near wake behind the trailing edge. In the longest system (c/t=9), the separated shear layer always reattached to the plate's surface far upstream from the trailing edge, which did not induce large fluctuations of the near wake. Furthermore, the proper orthogonal decomposition (POD) was extensively employed to filter the original velocity fields spatially to identify the large-scale vortices immersed in the separated shear layer easily. The distribution of the v-v correlation coefficients of the spatially filtered flow fields reflected the organized large-scale vortices in the three systems. The number of alternations of the positive and negative correlation coefficients across the flow field were determined to be 1, 2 and 3 for the systems with c/t=3, 6 and 9, respectively; this is in agreement with the shedding mode of each system. The distribution of the swirling strength of the separated shear layer accurately determined the positions and structures of the large-scale vortices formed above the plate surface.