Velocity Gradient Parameter Research Articles

Boundary Value Problems of a Kirchhoff Type Plate Model Based on the Simplified Strain Gradient Elasticity and the Application

A new type of thin plate model and the related nonclassical boundary value problems were established within the framework of strain gradient and velocity gradient elasticity. The closed-form solutions of deflections and free vibrational frequencies of a simply supported plate resting on an elastic foundation were obtained. The results of the present model agree well with those predicted by the molecular dynamics. Numerical results show that, the elastic foundation and the strain gradient parameter have a stiffness-hardening effect, while the velocity gradient parameter has a stiffness-softening effect. The proposed boundary value problems are of great significance to the study of the mechanical behaviors of plates under complex boundary conditions and external loadings. Furthermore, it will be useful for developing effective numerical methods such as the finite element method, the finite difference method and the Garlerkin method.

Global velocity gradient evaluation: An innovative approach using CFD modeling applied to water and wastewater treatment plants

This paper proposes an alternative approach for estimating the Global Velocity Gradient parameter (G) through hydrodynamic characteristics using Computational Fluid Dynamics Modeling (CFD). In the design of water and wastewater treatment plants, several hydraulic and geometric parameters are needed to guarantee the efficiency of the treatment processes. Among these parameters, G is estimated using empirical equations or prototypes. The use of empirical equations is a simple and fast way to estimate G but they are often incompatible with the geometry of the unit, leading to a high number of errors. On the other hand, the use of prototypes leads to higher accuracy, but their construction is time consuming and can be very costly. Thus, an in-depth analysis of the basic equations for obtaining G and of CFD modeling aspects can assist in obtaining this parameter without using empirical equations or prototypes. The proposed approach was tested in six flocculation units based on helical geometry (called Helically Coiled Tube Flocculators −HCTFs). As a result, a high correlation between G obtained from prototypes (used as reference) and G obtained from the corresponding CFD modeling results was verified (with a determination coefficient higher than 0.9995), indicating that this approach can be a promising tool for designing new water and wastewater treatment units and the improvement of existing ones.

Torsional vibration of size-dependent viscoelastic rods using nonlocal strain and velocity gradient theory

In this paper the torsional vibration of size-dependent viscoelastic nanorods embedded in an elastic medium with different boundary conditions is investigated. The novelty of this study consists of combining the nonlocal theory with the strain and velocity gradient theory to capture both softening and stiffening size-dependent behavior of the nanorods. The viscoelastic behavior is modeled using the so-called Kelvin–Voigt viscoelastic damping model. Three length-scale parameters are incorporated in this newly combined theory, namely, a nonlocal, a strain gradient, and a velocity gradient parameter. The governing equation of motion and its boundary conditions for the vibration analysis of nanorods are derived by employing Hamilton’s principle. It is shown that the expressions of the classical stress and the stress gradient resultants are only defined for different values of the nonlocal and strain gradient parameters. The case where these are equal may seem to result in an inconsistency to the general equation of motion and the related non-classical boundary conditions. A rigorous investigation is conducted to prove that the proposed solution is consistent with physics. Damped eigenvalue solutions are obtained both analytically and numerically using a Locally adaptive Differential Quadrature Method (LaDQM). Analytical results of linear free vibration response are obtained for various length-scales and compared with LaDQM numerical results.

Closed-form frequency solutions for simplified strain gradient beams with higher-order inertia

This paper develops an Euler−Bernoulli beam model within the context of a simplified strain gradient theory with higher-order inertia. In contrast to the classical beam models, the proposed gradient beam models can capture the size effects by introducing not only the internal length l2 related to strain gradient but also the internal length l1 related to velocity gradient. The governing equation of motion and boundary conditions are derived by using the variational principles. The closed-form solutions for free vibrations of beams with three typical boundary conditions are obtained. Numerical results show that the choices of the higher-order boundary conditions have a minor effect on the natural frequencies of beams. In addition, the inclusion of the strain gradient parameter l2 increases the effective stiffness of beams and hence it increases the natural frequencies of beams; whereas the inclusion of the velocity gradient parameter l1 acts as an equivalent compression force in the governing equation and therefore it leads to the decrease of the natural frequencies of beams. Moreover, the significant Poisson effect on the natural frequencies of beams is observed when the thickness of the beam is comparable to the strain gradient parameter l2. The closed-form solutions for natural frequencies of beams presented in this work may serve as benchmark results for other numerical methods.

Improvement of an LRN two-equation turbulence model reflecting multi-time scales

It is well known that turbulence energy is overpredicted by the linear k–ε model at the stagnation point. This problem causes inadequate predictions of the flow field. In this paper, to evaluate the performance of a low-Reynolds-number (LRN) type turbulence model for flows on a complex wall shape, we carried out simulations of flows around various wall shapes. Based on this evaluation, we improved the model by introducing a time scale based on the velocity gradient parameter. The proposed model is evaluated in four complex turbulent flows, i.e., forward-facing step, two-dimensional block, two-dimensional hill and three-dimensional block flows. The results, together with the proposed model, are in good agreement with the experimental data.

Blowoff Stability of Methane-Air Premixed Flame on Tube Burners

Stability of premixed flames is important in applications involving industrial and domestic impingement heat transfer processes, gas turbine combustion chamber and others. Blowoff limits of premixed methane-air flames stabilised on an uncooled Bunsen type burner is considered in the present experimental work. The effect of burner material, wall thickness and burner exit shape on the blowoff limits is presented. Burner materials covered in this study are stainless steel, brass and pyrex. Wall thicknesses considered are 1 mm, 2 mm and 3 mm for pyrex tubes of 10 mm inside diameter. The burner exit shapes covered in this study are circle, triangle, square and hexagon. The operating mixture Reynolds number range is 800 - 4000. It is found that the burners with low thermal conductivity, larger wall thickness and minimal sided polygon shapes provide better lean blowoff stability. Critical velocity gradient parameter defined on the basis of hydraulic diameter collapses the blowoff limits for all shapes covered in...

Low-Reynolds-Number One-Equation Turbulence Model with Mixed Time-Scale for Complex Flows

To perform massive and unsteady calculations of flow fields, the hybrid RANS/LES method is very useful, in which a RANS is employed near the wall, and an LES is carried out in the region away from the wall. However, to calculate adequately near-wall turbulent motions over complex terrain, a number of grids are needed near the wall. Thus, to reduce the calculation load when predicting real flow fields by computer, solving fewer transport equations is recommended. Therefore, in this study, we construct a low-Reynolds-number (LRN) one-equation model for eddy viscosity by introducing a time-scale based on the velocity gradient parameter. This time-scale has proven very effective for predicting local flows over complex terrain. The proposed model is evaluated in complex turbulent flows, such as backward-facing step flow, forward-facing step flow, flow past a two-dimensional bock and flow over a curved hill. The predicted results are compared with available experimental data. It is shown that the proposed one-equation model reproduces the results generally in good agreement with experimental data.

A New Low-Reynolds-Number Two-Equation Turbulence Model with Hybrid Time-Scales of Mean Flow and Turbulence for Complex Wall Flows

It is well known that the turbulence energy is overpredicted by the k-e model at the stagnation point. This problem causes inadequate predictions of flow field. In particular, since the wind-power energy is proportional to wind velocity cubed, it is crucial to predict a correct profile of wind in selecting the best site for a wind-power plant. The effects of the ground are often neglected owing to the very large length scale of the real environment. However, the complex terrain affects a profile of wind near the ground. Therefore, we have to consider the ground effects to obtain an exact profile of wind near the ground. In this study, to evaluate the performance of a low-Reynolds-number type turbulence model for flows on a complex terrain, we carried out a simulation of a flow around various complex terrain. From evaluation, we have improved the model introducing the time scale based on the velocity gradient parameter. The proposed model is evaluated in four complex turbulent flows, i.e., forward-facing step, two-dimensional block, two-dimensional hill and three-dimensional block flows. The proposed model indicates good agreement with the experimental data.

B110 乱流モデルによる複雑地形上の風況予測(オーガナイズドセッション18 : 環境における乱流熱・物質伝達)

Since the wind-power energy is proportional to wind velocity cubed, it is crucial to predict a correct profile of wind in selecting the best site for a wind-power plant. The effects of the ground are often neglected owing to the very large length scale of the real environment. However, the complex terrain affects a profile of wind near the ground. Therefore, we have to consider the ground effects to obtain an exact profile of wind near the ground. In this study, to evaluate the performance of a low-Reynolds-number type turbulence model in an atmospheric boundary layer, we calculated a stable boundary layer flow with a roughness surface. Also, we proposed a turbulence model including the effects of the roughness and the buoyancy. Moreover, to assess the prediction of the turbulence model on a complex terrain, we carried out a simulation of a forward-facing step flow. In this case, it is well known that the turbulence energy is overpredicted by the k-ε model at the stagnation point; thus, we have improved the model introducing the time scale based on the velocity gradient parameter. The proposed model indicates good agreement with the experimental data.

Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux

The effect of uniform suction on the steady two-dimensional laminar forced flow of a viscous incompressible fluid of temperature dependent viscosity past a wedge with uniform surface heat flux is considered. The governing equations for the flow are obtained by using suitable transformations and are solved by using an implicit finite difference method. Perturbation solutions are also obtained near the leading edge and in the downstream regime. The results are obtained in terms of the local skin friction coefficient and the rate of heat transfer for various values of the pertinent parameters, such as the Prandtl number, Pr, the velocity gradient parameter, m, the local suction parameter, ξ, and the viscosity variation parameter, ɛ. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The effect of ξ, m and ɛ on the dimensionless velocity profiles and viscosity distribution are also presented graphically for Pr = 0.7 and 7.0, which are the appropriate values for gases and water respectively.

Acceleration and Injection Effects on Longitudinal Vortices in Concave Wall Boundary Layers

Measurements of longitudinal velocity and wall temperature have been obtained in a wind tunnel with a concave wall, with momentum thickness Görtler numbers from 5 to 10 at the initial station. Vortex generators were used to enhance the vortex structure in flow with zero streamwise pressure gradient. Spanwise variations existed over a limited range which is quantified. It is shown that mild acceleration amplified the vortices and enhanced the regularity of the spanwise variation; strong acceleration, with velocity gradient parameter K near 2 times 10−6, suppressed them and limited the growth in spanwise-averaged Stanton number. The influence of favourable pressure gradients was confirmed in a curved water channel with similar Görtler numbers. Discrete hole wall jets with varying jet-to-free-stream velocity ratios were used in the water channel to examine the response of longitudinal vortices to a spanwise-periodic disturbance of different wavelengths. The wall temperature characteristic of the jets persisted for a considerable streamwise distance with velocity ratios of 0.4 and unity, while a distribution not dissimilar to that without blowing was found at all stations with a velocity ratio of 1.5.

95/00740 Applied air conditioning

It is well known that turbulence energy is overpredicted by the linear k–ε model at the stagnation point. This problem causes inadequate predictions of the flow field. In this paper, to evaluate the performance of a low-Reynolds-number (LRN) type turbulence model for flows on a complex wall shape, we carried out simulations of flows around various wall shapes. Based on this evaluation, we improved the model by introducing a time scale based on the velocity gradient parameter. The proposed model is evaluated in four complex turbulent flows, i.e., forward-facing step, two-dimensional block, two-dimensional hill and three-dimensional block flows. The results, together with the proposed model, are in good agreement with the experimental data.

Optimization of particle image velocimeters. I. Double pulsed systems

The spatial resolution, detection rate, accuracy and reliability of a particle image velocimeter (PIV) depend critically upon the careful selection of a number of parameters of the PIV system and the fluid motion. An analytical model and a Monte Carlo computer simulation have been developed to analyse the effects of experimental parameters and to optimize the system parameters. A set of six nondimensional parameters that are the most significant in optimizing PIV performance are identified. They are the data validation criterion, the particle image density, the relative in-plane image displacement, the relative out-of-plane displacement, a velocity gradient parameter, and the ratio of the mean image diameter to the interrogation spot diameter. These parameters are studied for the case of interrogation by autocorrelation analysis. By a single transformation, these results can be applied to interrogation by two-dimensional Fourier transform analysis of the Young's fringes.

Investigation of the dependence of excess attenuation of aircraft noise on distance

In previous studies of sound propagation using the Fast Field Program, one aspect of our investigation has been of the excess attenuation versus a logarithmic sound velocity gradient parameter for fixed ranges [Proc. INTER‐NOISE 86 Conf., 419–424 (1986)]. However, the dependence of excess attenuation upon distance is of greater interest for purposes of environmental assessment. This paper describes the decay of sound levels with distance for ground‐to‐ground propagation, under both upward and downward refraction conditions, and suggests how an average curve for acoustically hard and soft surfaces can be developed from this information. [Work jointly supported by the FAA and the U.S. Army Construction Engineering Research Laboratory.]

Prediction of Heat Transfer From Laminar Boundary Layers, With Emphasis on Large Free-Stream Velocity Gradients and Highly Cooled Walls

Laminar boundary-layer heat transfer and shear-stress predictions from existing similarity solutions are extended in an approximate way to perfect gas flows with a large free-stream velocity gradient parameter β and variable density-viscosity product ρμ across the boundary layer resulting from a highly cooled wall. The dimensionless enthalpy gradient at the wall gw′, to which the heat flux is related, is found not to vary appreciably with β. Thus the application of similarity solutions on a local basis to predict heat transfer from accelerated flows to an arbitrary surface may be a reasonable approximation involving a minimum amount of calculation time. Unlike gw′, the dimensionless velocity gradient at the wall fw″, to which the shear stress is related, is strongly dependent on β.