Number Of Independent Unknowns Research Articles

Revisiting the Reynolds-averaged Navier–Stokes equations

Abstract This study revisits the Reynolds-averaged Navier–Stokes (RANS) equations and finds that the existing literature is erroneous regarding the primary unknowns and the number of independent unknowns in the RANS. The literature claims that the Reynolds stress tensor has six independent unknowns, but in fact the six unknowns can be reduced to three that are functions of the three velocity fluctuation components, because the Reynolds stress tensor is simply an integration of a second-order dyadic tensor of flow velocity fluctuations rather than a general symmetric tensor. This difficult situation is resolved by returning to the time of Reynolds in 1895 and revisiting Reynolds’ averaging formulation of turbulence. The study of turbulence modeling could focus on the velocity fluctuations instead of the Reynolds stress. An advantage of modeling the velocity fluctuations is, from both physical and experimental perspectives, that the velocity fluctuation components are observable whereas the Reynolds stress tensor is not.

Mechanical Stability Investigation of Advanced Composite Plates Resting on Elastic Foundations Using a New Four-Unknown Refined Theory

A refined and simple shear deformation theory for mechanical buckling of composite plate resting on two-parameter Pasternak’s foundations is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces.Therefore, there is no need to use shear correction factor. The number of independent unknowns of present theory is four, as against five in other shear deformation theories.It is assumed that the warping of the cross sections generated by transverse shear is presented by a hyperbolic function. The stability equations are determined using the present theory and based on the existence of material symmetry with respect to the median plane.The nonlinear strain-displacement of Von Karman relations are also taken into consideration. .The boundary conditions for the plate are assumed to be simply supported in all edges. Closed-form solutions are presented to calculate the critical load of mecanical buckling, which are useful for engineers in design. The effects of the foundation parameters,side-to-thickness ratio and modulus ratio, the isotropic and orthotropic square plates are considered in this analysis.are presented comprehensively for the mechanical buckling of rectangular composite plates.

Buckling Analysis of Anti-Symmetric Cross-Ply Laminated Composite Plates Under Different Boundary Conditions

The buckling analysis of anti-symmetric cross-ply laminated composite plates under different boundary conditions is examined by using a refined higher order exponential shear deformation theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In this investigation, the equations of motion for simply supported thick laminated rectangular plates are derived and obtained through the use of Hamilton’s principle. The closed-form solutions of anti-symmetric cross-ply and angle-ply laminates are obtained using Navier solution. Numerical results for critical buckling loads anti-symmetric cross-ply laminated composite plates are presented. The validity of the present study is demonstrated by comparison with other higher-order solutions reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the buckling behaviors of anti-symmetric cross-ply laminated composite plates under different boundary conditions

A simple higher order shear deformation theory for mechanical behavior of laminated composite plates

In the present study, the static, buckling, and free vibration of laminated composite plates is examined using a refined shear deformation theory and developed for a bending analysis of orthotropic laminated composite plates. These models take into account the parabolic distribution of transverse shear stresses and satisfy the condition of zero shear stresses on the top and bottom surfaces of the plates. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the equa- tion of motion for simply supported thick laminated rect- angular plates is obtained through the use of Hamilton's principle. The accuracy of the analysis presented is demonstrated by comparing the results with solutions derived from other higher order models and with data found in the literature. It can be concluded that the proposed theory is accurate and simple in solving the static, the buckling, and free vibration behaviors of laminated composite plates.

Geometry and Self-stress of Single-Wall Carbon Nanotubes and Graphene via a Discrete Model Based on a 2nd-Generation REBO Potential

The main purpose of this paper is to evaluate the self-stress state of single-walled Carbon NanoTubes (CNTs) and Flat Graphene Strips (FGSs) in their natural equilibrium state, that is, the state prior to the application of external loads. We model CNTs as discrete elastic structures, whose shape and volume changes are governed by a Reactive Empirical Bond-Order (REBO) interatomic potential of second generation. The kinematical variables we consider are bond lengths, bond angles, and dihedral angles; to changes of each of these variables we associate a work-conjugate nanostress. To determine the self-stress state in a given CNT, we formulate the load-free equilibrium problem as a minimum problem for the interatomic potential, whose solution yields the equilibrium nanostresses; next, by exploiting the nonlinear constitutive dependence we derive for nanostresses in terms of a list of kinematical variables, we determine the equilibrium values of the latter; finally, from the equilibrium values of the kinematical variables we deduce the natural geometry and, in particular, the natural radius. Our theoretical framework accommodates CNTs of whatever chirality. In the achiral case, the stationarity conditions implied by energy minimization are relatively easy to derive and solve numerically, because we can count on maximal intrinsic symmetries and hence the number of independent unknowns is reduced to a minimum; for chiral CNTs, we prefer to solve the minimum problem directly. The natural-radius predictions we achieve within our discrete-mechanics framework are in good agreement with the results of calculations based on Density Functional Theory (DFT) and Tight Binding (TB) theory; the same is true for our predictions of the self-energy, that is, the energy associated with self-stress (called cohesive energy in the literature); we surmise that our discrete mechanical model may serve as a source of benchmarks for Molecular-Dynamics (MD) simulation algorithms. We find that self-stress depends on changes in both bond and dihedral angles in achiral CNTs and, in addition, on changes in bond length in chiral CNTs. Our analysis applies also to FGSs, whose self-stress and self-energy we evaluate; we find that in FGSs self-stress is associated exclusively with changes in bond angle.

Buckling and free vibration analysis of laminated composite plates using an efficient and simple higher order shear deformation theory

In this paper, the buckling and free vibration analysis of laminated composite plates using an efficient and simple higher order shear deformation theory are examined by using a refined shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In this analysis, the equations of motion for simply supported thick laminated rectangular plates are derived and obtained through the use of Hamilton’s principle. The closed-form solutions of anti-symmetric cross-ply and angle- ply laminates are obtained using Navier solution. Numerical results of the present study are compared with three-dimensional elasticity solutions and results of the first-order and the other higher-order theories reported in the literature. It can be concluded that the proposed theory is accurate and simple in solving the buckling and free vibration behaviors of laminated composite plates.

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.

A representer‐based inverse method for groundwater flow and transport applications

Groundwater inverse problems are concerned with the estimation of uncertain model parameters, such as hydraulic conductivity, from field or laboratory measurements. In practice, model and measurement errors compromise the ability of inverse procedures to provide accurate results. It is important to account for such errors in order to determine the proper weight to give to each source of information. Probabilistic descriptions of model and measurement errors can be incorporated into classical variational inverse procedures, but the computational demands are excessive if the model errors vary over time. An alternative approach based on representer expansions is able to efficiently accommodate time‐dependent errors for large problems. In the representer approach, unknown variables are expanded in finite series which depend on unknown functions called representers. Each representer quantifies the influence of a given measurement on the estimate of a particular variable. This procedure replaces the original inverse problem by an equivalent problem where the number of independent unknowns is proportional to the number of measurements. The representer approach is especially advantageous in groundwater problems, where the total number of measurements is often small. This approach is illustrated with a synthetic flow and transport example that includes time‐dependent model errors. The representer algorithm is able to provide good estimates of a spatially variable hydraulic conductivity field and good predictions of solute concentration. Its computational demands are reasonable, and it is relatively easy to implement. The example reveals that it is beneficial to account for model errors even when they are difficult to estimate.

Finite Element Analysis of Composite and Sandwich Plates Using a Continuous Inter-laminar Shear Stress Model

A four noded plate element based on a refined higher order shear deformation theory is developed for the analysis of composite and sandwich plates. This plate theory satisfies the conditions of inter-laminar shear stress continuity and stress free top and bottom surfaces of the plate. Moreover, the number of independent unknowns is the same as that in the first order shear deformation theory. The inter-elemental continuity, as desired by the plate theory, is fully satisfied in the present element. Numerical examples of composite and sandwich plates are solved to validate the element. Some new results are also presented.

Reduction of the number of unknowns in large stacked planar structures by a fringe aperture formulation

For scattering problems comprising a combination of planar structures, the total number of unknowns may be significantly reduced if an aperture formulation is employed rather than a patch formulation. The rationale behind using the aperture formulation is based on the recognition that the decay rate of the scattered aperture field is independent of the size of the scatterer. Therefore, any scatterer may be surrounded by an aperture of fixed width over which an integral equation is formulated. The area of this aperture is proportional to the perimeter of the scatterer rather than its area, and it becomes much smaller compared with the entire scatterer area as the size of the scatterer increases, hence the reduction in the number of independent unknowns. A truncation criterion for the finite aperture is determined via a numerical study of the aperture field behavior for various angles of incidence. In addition, the a priori knowledge of the physical optics component is also taken into account, reducing the unknown function to an aperture field component that is the outcome of the remaining fringe current only. The total current distribution can be subsequently derived from this field by adding the known physical optics field and invoking the inverse of the Green's function in the spectral domain. This analysis of isolated planar scatterers results in a spectral scattering matrix representation that is subsequently used for cascading of stacked structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>