Two-dimensional Navier Research Articles

Global classical solution to two-dimensional compressible Navier–Stokes equations with large data in [formula omitted

In this paper, we prove the global well-posedness of the classical solution to the 2D Cauchy problem of the compressible Navier–Stokes equations with arbitrarily large initial data and non-vacuum far-fields when the shear viscosity μ is a positive constant and the bulk viscosity λ(ρ)=ρβ with β>43. Note that the initial data can be arbitrarily large with or without vacuum states. For the non-vacuum initial data, our global well-posedness result implies that the classical solution to the 2D Cauchy problem will not develop the vacuum states in any finite time. Moreover, the global well-posedness result still holds true when the initial data contains the vacuum states in a subset of R2 provided some compatibility conditions are satisfied. Some new weighted estimates for the density and the velocity are obtained in this paper and these self-contained estimates reflect the fact that the weighted density and velocity can propagate along with the flow, which are intrinsic to the two-dimensional Cauchy problem with the non-vacuum far-fields.

Continuity of pullback and uniform attractors

We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterized by λ∈Λ, where Λ is a complete metric space, such that for each λ∈Λ there exists a unique pullback attractor Aλ(t). Using the theory of Baire category we show under natural conditions that there exists a residual set Λ⁎⊆Λ such that for every t∈R the function λ↦Aλ(t) is continuous at each λ∈Λ⁎ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors Aλ, there is a residual set at which the map λ↦Aλ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when Λ is compact that the continuity of pullback attractors and uniform attractors with respect to λ is equivalent to pullback equi-attraction and, respectively, uniform equi-attraction. These abstract results are then illustrated in the context of the Lorenz equations and the two-dimensional Navier–Stokes equations.

Numerical Study of Transitional SD7003 Airfoil Interacting with Canonical Upstream Flow Disturbances

The current work reports on the results of high-accuracy two-dimensional and high-fidelity three-dimensional Navier–Stokes simulations of an SD7003 airfoil interacting with canonical upstream flow disturbances at low and moderate Reynolds numbers corresponding to laminar and transitional flow regimes. Three deterministic forms of upstream flow disturbances are considered, including sharp-edge gust, time-harmonic gust, and Taylor vortex models. The results obtained for the airfoil unsteady aerodynamic responses to gust perturbations with variable amplitude and duration are discussed in comparison with results from linearized inviscid incompressible unsteady aerodynamic theory to elucidate the importance of viscous effects in the considered flow regimes. Overall, the investigation reveals that the high-amplitude gust responses correlate well with inviscid theories, which is in contrast to the low-amplitude gust responses that appear to be strongly affected by viscous effects.

Numerical simulations of porous medium with different permeabilities and positions in a laterally-heated cylindrical enclosure for crystal growth

This paper presents an investigation of flow and heat transfer in a large diameter (6.25 in) cylindrical enclosure heated laterally and containing a porous block that simulates the basket of nutrients used in a crystal growth reactor. The numerical model entails the use of a commercially available computational engine provided by ANSYS FLUENT, and based on a two-dimensional (2D) axisymmetric Reynolds-averaged Navier Stokes (RANS) equations. The porous medium is simulated using the Brinkman-extended model accounting for the Darcy and Forchheimer induced pressure drops. The porous ‘plug’ effects are analyzed as both its permeability/inertial resistance and locations in the reactor are changed on a parametric basis, while the Rayleigh number (Ra=gβΔTL3να) is kept constant at 1.98 × 109. Additionally, the effect of different ratios of the hot to the cold zone lengths are investigated as a part of the current effort. For all cases, the velocity and temperature distributions in the reactor are analyzed together with the flow patterns in, and around the porous block. A comprehensive discussion is provided with regard to the effects of the position of the porous block and its permeability on both the immediately adjacent, and far flows. The consequences on the temperature distribution in the enclosure, when the ratio of the length of the hot-to-cold zones is changed, are also analyzed.

Nonlinear Lifting Line Theory Applied to Vertical Axis Wind Turbines: Development of a Practical Design Tool

Recently, a new interest in vertical axis wind turbine (VAWT) technology is fueled by research on floating support structures for large-scale offshore wind energy application. For the application on floating structures at multimegawatt size, the VAWT concept may offer distinct advantages over the conventional horizontal axis wind turbine (HAWT) design. As an example, VAWT turbines are better suited for upscaling, and at multimegawatt size, the problem of periodic fatigue cycles reduces significantly due to a very low rotational speed. Additionally, the possibility to store the transmission and electricity generation system at the bottom, compared to the tower top as in a HAWT, can lead to a considerable reduction of material logistics costs. However, as most VAWT research stalled in the mid 1990s, no sophisticated and established tools to investigate this concept further exist today. Due to the complex interaction between unsteady aerodynamics and movement of the floating structure, fully coupled simulation tools modeling both aero and structural dynamics are needed. A nonlinear lifting line free vortex wake (LLFVW) code was recently integrated into the open source wind turbine simulation suite qblade. This paper describes some of the necessary adaptions of the algorithm, which differentiates it from the usual application in HAWT simulations. A focus is set on achieving a high robustness and computational efficiency. A short validation study compares LLFVW results with those of a two-dimensional (2D) unsteady Reynolds-averaged Navier–Stokes (URANS) simulation.

Investigation of the aerodynamic performance and flow physics on cross sections of Dragonfly wing on flapping and pitching motion in low Reynolds number

In this research, the flow physics and aerodynamic performance of dragonfly cross sections, used in Micro Aerial Vehicles (MAVs), in low Reynolds are investigated. The main objective of the research is to study the performance of dragonfly wing cross-sections flapping motion in Reynolds 5000 and 10,000. Pitching motion is one of the most important mechanisms in force lifting generation, and the effects of Reynolds number and mean angle of attack on aerodynamic coefficients have been extensively investigated for the pitching motion. In the present study, the geometry of two cross sections of dragonfly was extracted. Incompressible, two-dimensional and unsteady Navier–Stokes equations have been used to simulate the flow. k − ɛ RNG model was used for turbulence modeling. To simulate the wing pitching motion, the dynamic mesh method was used. The results showed that in flapping motion, pitching-up rotation has caused a rapid increase in lift coefficient. Furthermore, it was found that the absence of stall does not increase the lift and drag coefficients, while formation of new strong vorticity layers have caused an increase in lift coefficient. On the other hand, corrugations on the cross sections of the dragonfly in the pitching motion cause the delay of separation and increasing the lift coefficient. In flapping motion and the pitching motion, the lift coefficients of three cross sections were increased due to stronger vorticity layers by reducing the Reynolds number. Due to the existence of corrugations, the first and the second cross sections have good aerodynamic performance, compared to the flat plate. The comparison carried out in the current research showed that the second cross section is a proper replacement for the flat plate in MAVs.

The vanishing viscosity limit for 2D Navier–Stokes in a rough domain

We study the high Reynolds number limit of a viscous fluid in the presence of a rough boundary. We consider the two-dimensional incompressible Navier–Stokes equations with Navier slip boundary condition, in a domain whose boundaries exhibit fast oscillations in the form x2=ε1+αη(x1/ε), α>0. Under suitable conditions on the oscillating parameter ε and the viscosity ν, we show that solutions of the Navier–Stokes system converge to solutions of the Euler system in the vanishing limit of both ν and ε. The main issue is that the curvature of the boundary is unbounded as ε→0, which precludes the use of standard methods to obtain the inviscid limit. Our approach is to first construct an accurate boundary layer approximation to the Euler solution in the rough domain, and then to derive stability estimates for this approximation under the Navier–Stokes evolution.

Detailed Analysis of the Wake Structure of a Straight-Blade H-Darrieus Wind Turbine by Means of Wind Tunnel Experiments and Computational Fluid Dynamics Simulations

Darrieus vertical axis wind turbines (VAWTs) have been recently identified as the most promising solution for new types of applications, such as small-scale installations in complex terrains or offshore large floating platforms. To improve their efficiencies further and make them competitive with those of conventional horizontal axis wind turbines, a more in depth understanding of the physical phenomena that govern the aerodynamics past a rotating Darrieus turbine is needed. Within this context, computational fluid dynamics (CFD) can play a fundamental role, since it represents the only model able to provide a detailed and comprehensive representation of the flow. Due to the complexity of similar simulations, however, the possibility of having reliable and detailed experimental data to be used as validation test cases is pivotal to tune the numerical tools. In this study, a two-dimensional (2D) unsteady Reynolds-averaged Navier–Stokes (U-RANS) computational model was applied to analyze the wake characteristics on the midplane of a small-size H-shaped Darrieus VAWT. The turbine was tested in a large-scale, open-jet wind tunnel, including both performance and wake measurements. Thanks to the availability of such a unique set of experimental data, systematic comparisons between simulations and experiments were carried out for analyzing the structure of the wake and correlating the main macrostructures of the flow to the local aerodynamic features of the airfoils in cycloidal motion. In general, good agreement on the turbine performance estimation was constantly appreciated.

Parametric study on the drag and heat flux reduction mechanism of forward-facing cavity on a blunt body in supersonic flows

Drag and heat flux reduction is very important for the design of the hypersonic vehicle, and many novel strategies have been proposed recently. As one of the valid drag and heat flux reduction schemes, the forward-facing cavity has attracted an increasing attention for the scholars. In the current study, the influence of the cavity configuration on the drag and heat flux reduction mechanism of a blunt body has been investigated numerically by the two-dimensional axisymmetric Reynolds-averaged Navier–Stokes (RANS) equations coupled with the SST k–ω turbulence model, and the effects of the depth of the cavity and the angle of its trailing edge have been taken into consideration. At the same time, the numerical approaches employed in this paper have been validated against the available experimental data in the open literature, as well as the grid independency analysis. The obtained results shows that the cavity configuration has only a slight impact on the drag reduction of the blunt body in the supersonic flow with the freestream Mach number being 3.98, and the maximum drag reduction coefficient is within 1.0%. The forward-facing cavity with L/D=4.0 and θ=45° is the most beneficial configuration for the heat flux reduction of the blunt body in the range considered in the current study, and it should be validated by the wind tunnel test in the near future.

Effect of sound-absorbing coatings on the disturbance evolution in a flow of a mixture of vibrationally excited gases

The flow around a solid plate and a plate with a sound-absorbing coating at a non-zero angle of attack in a hypersonic flow (M∞=8.44) of a mixture of vibrationally excited carbon dioxide and nitrogen is considered. Numerical simulations are performed by solving two-dimensional unsteady Navier–Stokes equations with a two-temperature model of relaxing flows. The vibrational energy as a function of time is defined by the Landau–Teller equation. A skeleton model, which is a set of square elements arranged in a staggered order, is used for simulating the porous coating made of foamed nickel with a porosity coefficient of 95%. The distance between the elements is equal to the pore diameter of the real sound-absorbing material. Data on the evolution of disturbances on the solid plate and on the plate with the sound-absorbing coating are presented for various angles of attack and CO2 concentrations in the mixture. The experimental and calculated data on pressure fluctuations on the plate surfaces are found to be in good agreement. The effects of various parameters of the sound-absorbing coating (depth, length, and location at the flat plate) are considered. It is shown that the sound-absorbing coating significantly reduces the intensity of pressure fluctuations on the plate surface as compared to the solid surface (up to 50% depending on the length and location of the sound-absorbing coating).

Laser-plasma induced shock waves in micro shock tubes

Shock wave generation with help of lasers or shock tubes, for example, is a common subject at macroscopic scale. On the other hand, the current tendency towards smaller scales becomes more and more important. In particular, shock waves in small channels or tubes with sub-mm or micron-sized diameter have attracted much interest within the last years. But downscaling of shock wave effects such as pressure decrease and Mach number attenuation during propagation, viscous and heat effects, laminar and turbulent flow is not necessarily straightforward from macro to micro or even nano range. Although several theoretical investigations are available in this new field, there is a strong demand on experiments, which are mostly missing due to the lack of suitable methods. The present work introduces a novel method for the generation of shock waves at microscale, namely laser-induced micro shock waves (LIMS). The LIMS method applies a femtosecond laser to induce an optical breakdown in a thin aluminum target located at the entrance of a micro tube. Subsequently, a shock wave is launched by the high pressure aluminum plasma and starts propagating into the tube. The topical work presents, for the first time, experimental investigations on direct micro shock wave generation and propagation at well-defined conditions in micron-sized tubes. They are performed for different conditions and tubes down to 50 μm diameter. Different from previous shock wave investigations in the mm-diameter range that involve pressure transducers, the present work applies non-contact measurements by optical methods. The experiments are supported by additional simulations. A one-dimensional numerical hydrocode is applied to simulate the shock wave generation process. Further propagation of the shock in a micro tube is analyzed by solving two-dimensional Navier–Stokes equations. Both simulations agree well with the experimental results.

Hypersonic boundary layer stabilization by using a wavy surface

Numerical simulation of hypersonic (M∞=6) flow and evolution of disturbances on a smooth plate and a shallow grooved plate was performed by solving two-dimensional Navier– Stokes equations. Computational soft-ware verification was conducted by comparison with existing data of pressure pulsations on plates surface. It was showed that wavy surface significantly decrease pressure pulsations on plate surface and does not increase the value of mean heat fluxes. Data about effect of wavy surfaces with different form on the disturbances intensity in hypersonic boundary layer was obtained.

Achieving hover equilibrium in free flight with a flexible flapping wing

Recent discoveries in the fields of flapping wing aerodynamics and fluid–structure interaction have demonstrated that flexible wings can generate more lift than rigid wings. However, the implications of wing flexibility on the flight dynamics of flapping wing flyers is still an open research question. The main difficulty is that the free flight of flapping flyers with flexible wings is a result of the dynamic balance between unsteady aerodynamics, fluid–structure interaction, and flight dynamics. This study presents a fully coupled three-way flight simulator that solves the two-dimensional Navier–Stokes equations, tightly coupled to the Euler–Bernoulli beam equations of the wing and the nonlinear multi-body equations of motion for the dynamics at the fruit fly scale. A novel trim algorithm is used to determine the hover equilibrium in the longitudinal plane. The control inputs, i.e. the flapping amplitude, stroke plane angle, and flapping offset angle as well as the initial conditions are determined that effectively eliminate average body accelerations to less than 3% of gravitational acceleration. The resulting hover equilibrium control parameters flapping amplitude, stroke plane angle and the total power required agree well with the biological observation of fruit flies. Body oscillations in hovering free flight affect the flexible response of the wing compared to prescribed body motion without oscillation. The affected wing motion reduces the lift coefficient by up to 8.7% for the stiffest wing, necessitating slightly different control inputs to achieve trim. Finally, the power required to achieve hover equilibrium is 32%–94% lower for flexible wings than for rigid wings that are actively rotated to match the same passive pitch schedule.

Compressible Navier–Stokes system: Large solutions and incompressible limit

Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier–Stokes equations supplemented with arbitrary large initial velocity v0 and almost constant density ϱ0, for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical incompressible Navier–Stokes equations supplemented with the divergence-free projection of v0, is global. The systems are examined in Rd with d≥2, in the critical B˙2,1s Besov spaces framework.

Transition Prediction in Hypersonic Boundary Layers Using Receptivity and Freestream Spectra

Boundary-layer transition in hypersonic flows over a straight cone can be predicted using measured freestream spectra, receptivity, and threshold values for the wall-pressure fluctuations at the transition onset points. Simulations are performed for hypersonic boundary-layer flows over a 7 deg half-angle straight cone with varying bluntness at a freestream Mach number of 10. The steady and the unsteady flowfields are obtained by solving the two-dimensional Navier–Stokes equations in axisymmetric coordinates using a fifth-order-accurate weighted essentially nonoscillatory scheme for space discretization and using a third-order total-variation-diminishing Runge–Kutta scheme for time integration. The calculated factors at the transition onset location increase gradually with increasing unit Reynolds numbers for flow over a sharp cone and remain almost the same for flow over a blunt cone. The receptivity coefficient increases slightly with increasing unit Reynolds numbers. They are on the order of four for a sharp cone and are on the order of one for a blunt cone. The location of transition onset predicted from the simulation including the freestream spectrum, receptivity, and the linear and the weakly nonlinear evolutions yields a solution close to the measured onset location for the sharp cone. The simulations overpredict transition onset by about 20% for the blunt cone.

Computational Fluid Dynamic Analysis of Fluidic Actuator for Active Flow Control Applications

A numerical study is conducted to explore the unsteady nature of fluidic oscillators, which are responsible for generating sweeping jets for effective flow control of large-scale applications. Two- and three-dimensional unsteady Reynolds-averaged Navier–Stokes and three-dimensional improved delayed detached-eddy simulations are employed to simulate and resolve the flow structures in the internal flow passages of a fluidic oscillator and an external flowfield in a quiescent flow condition. The predicted flow generated by an enlarged actuator with an outlet diameter of 25 mm using air as a working fluid is validated against measurements at various supply rates. A comparison of current computational results using two-dimensional and three-dimensional unsteady Reynolds-averaged Navier–Stokes simulations with experiments indicates a reasonable agreement in the jet oscillation frequency. However, a significant discrepancy in the streamwise velocity field is observed between the two-dimensional unsteady Reynolds-averaged Navier–Stokes and experimental data. The three-dimensional unsteady Reynolds-averaged Navier–Stokes simulation also underpredicts the jet width, and consequently overpredicts the extension of the jet in the axial direction downstream of the actuator exit. Excellent agreement in the jet oscillation frequency and internal and external flowfields is obtained at two supply rates between the three-dimensional improved delayed detached-eddy simulation and the measurements. Simulations are also performed for a small-scaled actuator with a hydraulic outlet diameter of 1.3 mm, with water as the working fluid. The predicted oscillation frequency of the jet is compared with the experiments. A scaling analysis is also conducted by analyzing and correlating the flow characteristics and key parameters of the two actuators’ scales.

Aerodynamic Analysis of an Airfoil With Leading Edge Pitting Erosion

Leading edge erosion is a considerable threat to wind turbine performance and blade maintenance, and it is very imperative to accurately predict the influence of various degrees of erosion on wind turbine performance. In the present study, an attempt to investigate the effects of leading edge erosion on the aerodynamics of wind turbine airfoil is undertaken by using computational fluid dynamics (CFD) method. A new pitting erosion model is proposed and semicircle cavities were used to represent the erosion pits in the simulation. Two-dimensional incompressible Reynolds-averaged Navier–Stokes equation and shear stress transport (SST) k–ω turbulence model are adopted to compute the aerodynamics of a S809 airfoil with leading edge pitting erosions, where the influences of pits depth, densities, distribution area, and locations are considered. The results indicate that pitting erosion has remarkably undesirable influences on the aerodynamic performance of the airfoil, and the critical pits depth, density, and distribution area degrade the airfoil aerodynamic performance mostly were obtained. In addition, the dominant parameters are determined by the correlation coefficient path analysis method, results showed that all parameters have non-negligible effects on the aerodynamics of S809 airfoil, and the Reynolds number is of the most important, followed by pits density, pits depth, and pits distribution area. Meanwhile, the direct and indirect effects of these factors are analyzed, and it is found that the indirect effects are very small and the parameters can be considered to be independent with each other.

Experimental study on the motion of droplets excited by Lamb waves on an inclined non-piezoelectric substrate

The motion of water droplets induced by Lamb waves on an inclined glass substrate, which is excited by a single phase transducer (SPT), has been investigated in this study. There are three main factors that affect the motion, namely, input power, droplet volume, and plate inclination angle. When the droplet volume and plate inclination angle are fixed, the velocities of the two (advancing and receding) edges of the droplet are found to be strongly related to the input power. Moreover, under a certain input power and a certain droplet volume, the larger the angle, the higher the velocities of the advancing and receding edges of the droplet. However, the droplet velocity is not proportional to the droplet volume under a certain input power and a certain angle. In addition, two-dimensional time-dependent incompressible Navier–Stokes equations of the droplet on the inclined glass slope are investigated.

Conjugate-mixed convection heat transfer in a two-sided lid-driven cavity filled with nanofluid using Manninen's two phase model

A numerical study is carried out concerning conjugate mixed convection heat transfer of various type of the nanofluid in a two-dimensional square enclosure using Manninen's model (Two phase mixture model). In the both cases of internal/external heating and cooling, hot and cold surfaces are kept at different constant temperatures while other walls of the enclosure are thermally insulated. Two-dimensional Navier–Stokes, energy and volume fraction equation are solved using the finite volume method and SIMPLE algorithm. This study has been conducted to investigate the effects of the certain pertinent parameters such as: Richardson number (10 − 2 ≤ Ri ≤ 102), thermal conductivity ratio (0.1 ≤ Kr ≤ 10), presence of the conductive obstacles, number of the hot obstacles, volume fraction (0 ≤ ϕ ≤ 0.05), size (25 nm ≤ dp ≤ 145 nm) and type (Cu, Al2O3 and TiO2) of the nanoparticles on the fluid flow and heat transfer characteristics. The simulation results indicate that depending on the boundary conditions of the problem, there is an optimal volume fraction of the solid particles at which the maximum heat transfer rate can be obtained within the enclosure. It is also observed that at high Richardson numbers, the particle distribution is fairly non-uniform while by decreasing the Richardson number and nanoparticles diameter, distribution of solid particles becomes uniform. Moreover, it is found that at low values of Richardson numbers, type of the nanoparticles has nominal effect on heat transfer rate while at high Ri by increasing the thermal conductivity of the solid particles, the total Nusselt number and optimal particle loading increase. Finally, the results demonstrate that, adding a vertical conductive wall into the enclosure leads to a significant reduction of the heat transfer rate at all values of Richardson numbers.

Algebraic construction and numerical behavior of a new s-consistent difference scheme for the 2D Navier–Stokes equations

In this paper, we consider a regular grid with equal spatial spacings and construct a new finite difference approximation (difference scheme) for the system of two-dimensional Navier–Stokes equations describing the unsteady motion of an incompressible viscous liquid of constant viscosity. In so doing, we use earlier constructed discretization of the system of three equations: the continuity equation and the proper Navier–Stokes equations. Then, we compute the canonical Gröbner basis form for the obtained discrete system. It gives one more difference equation which is equivalent to the pressure Poisson equation modulo difference ideal generated by the Navier–Stokes equations, and thereby comprises a new finite difference approximation (scheme). We show that the new scheme is strongly consistent. Besides, our computational experiments demonstrate much better numerical behavior of the new scheme in comparison with the other strongly consistent schemes we constructed earlier and with the scheme which is not strongly consistent.