A new averaged general dynamic equation (GDE) for nanoparticles in the turbulent flow is derived by considering the combined effect of convection, Brownian diffusion, turbulent diffusion, turbulent coagulation, and fluctuating coagulation. The equation is solved with the Taylor-series expansion moment method in a turbulent pipe flow. The experiments are performed. The numerical results of particle size distribution correlate well with the experimental data. The results show that, for a turbulent nanoparticulate flow, a fluctuating coagulation term should be included in the averaged particle GDE. The larger the Schmidt number is and the lower the Reynolds number is, the smaller the value of ratio of particle diameter at the outlet to that at the inlet is. At the outlet, the particle number concentration increases from the near-wall region to the near-center region. The larger the Schmidt number is and the higher the Reynolds number is, the larger the difference in particle number concentration between the near-wall region and near-center region is. Particle polydispersity increases from the near-center region to the near-wall region. The particles with a smaller Schmidt number and the flow with a higher Reynolds number show a higher polydispersity. The degree of particle polydispersity is higher considering fluctuating coagulation than that without considering fluctuating coagulation.
A self-adaptive-grid method is applied to numerical simulation of the evolution of aircraft wake vortex with the large eddy simulation (LES). The Idaho Falls (IDF) measurement of run 9 case is simulated numerically and compared with that of the field experimental data. The comparison shows that the method is reliable in the complex atmospheric environment with crosswind and ground effect. In addition, six cases with different ambient atmospheric turbulences and Brunt Väisälä (BV) frequencies are computed with the LES. The main characteristics of vortex are appropriately simulated by the current method. The onset time of rapid decay and the descending of vortices are in agreement with the previous measurements and the numerical prediction. Also, secondary structures such as baroclinic vorticity and helical structures are also simulated. Only approximately 6 million grid points are needed in computation with the present method, while the number can be as large as 34 million when using a uniform mesh with the same core resolution. The self-adaptive-grid method is proved to be practical in the numerical research of aircraft wake vortex.
The formation and evolution of aerosol in turbulent flows are ubiquitous in both industrial processes and nature. The intricate interaction of turbulent mixing and aerosol evolution in a canonical turbulent mixing layer was investigated by a direct numerical simulation (DNS) in a recent study (Zhou, K., Attili, A., Alshaarawi, A., and Bisetti, F. Simulation of aerosol nucleation and growth in a turbulent mixing layer. Physics of Fluids, 26, 065106 (2014)). In this work, Monte Carlo (MC) simulation of aerosol evolution is carried out along Lagrangian trajectories obtained in the previous simulation, in order to quantify the error of the moment method used in the previous simulation. Moreover, the particle size distribution (PSD), not available in the previous works, is also investigated. Along a fluid parcel moving through the turbulent flow, temperature and vapor concentration exhibit complex fluctuations, triggering complicate aerosol processes and rendering complex PSD. However, the mean PSD is found to be bi-modal in most of the mixing layer except that a tri-modal distribution is found in the turbulent transition region. The simulated PSDs agree with the experiment observations available in the literature. A different explanation on the formation of such PSDs is provided.
A numerical analysis model based on two-dimensional shallow water differential equations is presented for straight open-channel flow with partial vegetation across the channel. Both the drag force acting on vegetation and the momentum exchange between the vegetation and non-vegetation zones are considered. The depth-averaged streamwise velocity is solved by the singular perturbation method, while the Reynolds stress is calculated based on the results of the streamwise velocity. Comparisons with the experimental data indicate that the accuracy of prediction is significantly improved by introducing a term for the secondary current in the model. A sensitivity analysis shows that a sound choice of the secondary current intensity coefficient is important for an accurate prediction of the depth-averaged streamwise velocity near the vegetation and non-vegetation interfaces, and the drag force coefficient is crucial for predictions in the vegetation zone.
The present paper deals with the multiple solutions and their stability analysis of non-Newtonian micropolar nanofluid slip flow past a shrinking sheet in the presence of a passively controlled nanoparticle boundary condition. The Lie group transformation is used to find the similarity transformations which transform the governing transport equations to a system of coupled ordinary differential equations with boundary conditions. These coupled set of ordinary differential equation is then solved using the RungeKutta-Fehlberg fourth-fifth order (RKF45) method and the ode15s solver in MATLAB. For stability analysis, the eigenvalue problem is solved to check the physically realizable solution. The upper branch is found to be stable, whereas the lower branch is unstable. The critical values (turning points) for suction (0< sc< s) and the shrinking parameter (χc< χ< 0) are also shown graphically for both no-slip and multiple-slip conditions. Multiple regression analysis for the stable solution is carried out to investigate the impact of various pertinent parameters on heat transfer rates. The Nusselt number is found to be a decreasing function of the thermophoresis and Brownian motion parameters.
According to the vibration characteristics of the round window, a mechanical model of a round window membrane is established. The Euler equation of the round window and the complementary boundary conditions are derived by the variational principle. Combined with the Bessel function, an analytical solution of the round window displacement is obtained by MATHEMATICA. Combined with clinical characteristics of round window membrane lesion, the effect of sound transmission due to thickening of the round window membrane caused by the otitis media, shrinkage of the round window membrane area caused by otosclerosis, and hardening of the round window membrane itself is analyzed. The results show that with thickening of the round window membrane, the displacement of the round window membrane is decreased. In the meantime, with hardening of the round window membrane and shrinkage of the membrane area, the maximum displacement of the round window membrane is gradually reduced, leading to a decrease in sound transmission. Thus, the analytical analysis can avoid interference of environment and the technical level of personnel, and it can evaluate transmission performance of the round window membrane efficiently, providing a theoretical basis for the reverse excitation of artificial prosthesis.
In this work, the mechanical behavior of a block of soft material subject to large deformation from a series of wedge-shaped indenters is evaluated. Data fields acquired from digital image correlation (DIC) are compared with the existing theoretical models. The slope angles of the wedges vary from 5° to 73.5°, and the minimum measurement uncertainties of the DIC system are established in advance to define the accuracy. It is concluded that the assumptions underpinning the analytical theory make it difficult to characterize large deformation of soft materials during contact. The strain fields are also obtained from the measured displacement field and verify the previously postulated existence of two deformation sectors, namely, a so-called shrinkage sector symmetric to the loading axis and an expansion sector, which become smaller with the increasing load and decreasing wedge angle.
The paper is devoted to mathematical modelling of static and dynamic stability of a simply supported three-layered beam with a metal foam core. Mechanical properties of the core vary along the vertical direction. The field of displacements is formulated using the classical broken line hypothesis and the proposed nonlinear hypothesis that generalizes the classical one. Using both hypotheses, the strains are determined as well as the stresses of each layer. The kinetic energy, the elastic strain energy, and the work of load are also determined. The system of equations of motion is derived using Hamilton’s principle. Finally, the system of three equations is reduced to one equation of motion, in particular, the Mathieu equation. The Bubnov-Galerkin method is used to solve the system of equations of motion, and the Runge-Kutta method is used to solve the second-order differential equation. Numerical calculations are done for the chosen family of beams. The critical loads, unstable regions, angular frequencies of the beam, and the static and dynamic equilibrium paths are calculated analytically and verified numerically. The results of this study are presented in the forms of figures and tables.
In this research, vibration and wave propagation analysis of a twisted microbeam on Pasternak foundation is investigated. The strain-displacement relations (kinematic equations) are calculated by the displacement fields of the twisted micro-beam. The strain gradient theory (SGT) is used to implement the size dependent effect at microscale. Finally, using an energy method and Hamilton’s principle, the governing equations of motion for the twisted micro-beam are derived. Natural frequencies and the wave propagation speed of the twisted micro-beam are calculated with an analytical method. Also, the natural frequency, the phase speed, the cut-off frequency, and the wave number of the twisted micro-beam are obtained by considering three material length scale parameters, the rate of twist angle, the thickness, the length of twisted micro-beam, and the elastic medium. The results of this work indicate that the phase speed in a twisted micro-beam increases with an increase in the rate of twist angle. Moreover, the wave number is inversely related with the thickness of micro-beam. Meanwhile, it is directly related to the wave propagation frequency. Increasing the rate of twist angle causes the increase in the natural frequency especially with higher thickness. The effect of the twist angle rate on the group velocity is observed at a lower wave propagation frequency.
This study carries out a complete analysis of time-space solution of hydrodynamics of pentagonal/decagonal quasicrystals. The behaviors of wave propagation for phonons and diffusion for phasons and coupling between phonon-phason fields are explored explicitly. Comprehensive discussion on physical time-space variations of all hydrodynamic field variables of the alloy quasicrystals is given. The computational specimen is simple, convenient in testing computational results, and provides a possibility that is easy to test experimentally. The quantitative results of mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, viscosity stresses, and their time-space variations help us understand the motion of solid quasicrystals in a hydrodynamic condition (long-wavelength and low-frequency). The analysis presented in this paper can be used for octagonal and dodecagonal quasicrystals and is easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. Some problems explored by the computational results are also discussed.