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The purpose of this paper is to present the terms of grey incidence analysis models.

The definitions of basic terms about various grey incidence analysis models are presented one by one.

The reader could know the basic explanation about the important terms about various grey incidence analysis models from this paper.

Many of the authors’ colleagues thought that unified definitions of key terms would be beneficial for both the readers and the authors.

It is a fundamental work to standardize all the definitions of terms for a new discipline. It is also propitious to spread and universal of grey system theory.

The purpose of this paper is to present the terms of grey clustering evaluation models.

The definitions of basic terms about grey clustering evaluation models are presented one by one.

The reader could know the basic explanation about the important terms about various grey clustering evaluation models from this paper.

Many of the authors’ colleagues thought that unified definitions of key terms would be beneficial for both the readers and the authors.

It is a fundamental work to standardise all the definitions of terms for a new discipline. It is also propitious to spread and universal of grey system theory.

In this paper, the authors prove that the Douglas space of second kind with a generalised form of special (α, β)-metric F, is conformally invariant.

For, the authors have used the notion of conformal transformation and Douglas space.

The authors found some results to show that the Douglas space of second kind with certain (α, β)-metrics such as Randers metric, first approximate Matsumoto metric along with some special (α, β)-metrics, is invariant under a conformal change.

The authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind.

This paper aims to present a topology optimization (TopO) method for designing heat exchangers (HEx) with two working fluids to be kept apart. The introduction of cut–cells gives rise to the cut-cell TopO method, which computes the optimal distribution of an artificial impermeability field and successfully overcomes the weaknesses of the standard density-based TopO (denTopO) by computing the fluid–solid interface (FSI) at each cycle. This allows to accurately solve the flow and conjugate heat transfer (CHT) problem by imposing exact boundary conditions on the computed FSI and results to correct performances computed without the need to re-evaluate the optimized solutions on a body-fitted grid.

The elements of an artificial impermeability distribution field defined on a background grid act as the design variables and allow topological changes to take place. Post-processing them yields two fields indicating the location of the two flow streams inside the HEx. At each TopO cycle, the FSIs computed based on these two fields are used as the cutting surfaces of the cut-cell grid. On the so-computed grid, the incompressible Navier–Stokes equations, coupled with the Spalart–Allmaras turbulence model, and the temperature equation are solved. The derivatives of the objective and constraint functions with respect to the design variables of TopO are computed by the continuous adjoint method, using consistent discretization schemes devised thanks to the “Think Discrete – Do Continuous” (TDDC) adjoint methodology.

The effectiveness of the cut-cell–based TopO method for designing HEx is demonstrated in 2D parallel/counter flow and 3D counter flow HEx operating under both laminar and turbulent flow conditions. Compared to the standard denTopO, its ability to compute FSIs along which accurate boundary conditions are imposed, increases the accuracy of the flow solver, which usually leads to optimal, rather than sub-optimal, solutions that truly satisfy the imposed constraints.

This work proposes a new/complete methodology for the TopO of two-fluid systems including CHT that relies on the cut-cell method. This successfully combines aspects from both TopO and Shape Optimization (ShpO) in a single framework thus overcoming the well-known downsides of standard denTopO regarding its accuracy or the need for a follow-up ShpO after TopO. Instead of adding the well-known Brinkman penalization terms into the flow equations, it computes the FSIs at each optimization cycle allowing the solution of the CHT problem on a cut-cell grid.

We identify several theoretical shortcomings in the derivative formulas for rational Bézier curves and propose a new formula for the derivatives.

Dependence on control points gives a new recursive approach to the derivation of rational Bézier curves.

We present the new derivation formula for rational Bézier curves that overcomes this drawback and show that the kth degree derivative of a nth degree rational Bézier curve can be written in terms of a (2^kn)th degree rational Bézier curve.

This paper leads to a further complete understanding of the derivatives of rational Bézier curves.

This study aims to propose a robust total variation diminishing (TVD) weighted average flux (WAF) finite volume scheme for investigating compressible gas–liquid mixture flows.

This study considers a two-phase flow composed of a liquid containing dispersed gas bubbles. To model this two-phase mixture, this paper uses a homogeneous equilibrium model (HEM) defined by two mass conservation laws for the two phases and a momentum conservation equation for the mixture. It is assumed that the velocity is the same for the two phases, and the density of phases is governed by barotropic laws. By applying the theory of hyperbolic equations, this study establishes an exact solution of the Riemann problem associated with the model equations, which allows to construct an exact Riemann solver within the first-order upwind Godunov scheme as well as a robust TVD WAF scheme.

The ability and robustness of the proposed TVD WAF scheme is validated by testing several two-phase flow problems involving different wave structures of the Riemann problem. Simulation results are compared against analytical solutions and other available numerical methods as well as experimental data in the literature. The proposed approach is much superior to other strategies in terms of the accuracy and ability of reconstruction.

The novelty of this work lies in its methodical extension of a TVD WAF scheme implementing an exact Riemann solver developed for compressible two-phase flows. Furthermore, other novelty lies on the quantitative calculation of different Riemann problem two-phase flows. Simulation results involve the verification of the constructed methods on the exact solutions of HEM without any restriction of variables.

The present study scrutinizes the relative performance of various near-wall treatments coupled with two-equation RANS models to explore the turbulence transport mechanism in terms of the kinetic energy budget in a plane wall jet and the significance of the near-wall molecular and turbulent shear, to select the best combination among the models which reveals wall jet characteristics most efficiently.

A two-dimensional steady incompressible plane wall jet in a quiescent surrounding is simulated using ANSYS-Fluent solver. Three near-wall treatments, namely the Standard Wall Function (SWF), Enhanced Wall Treatment (EWT) and Menter-Lechner (ML) treatment coupled with Realisable, RNG and Standard k-e models and also the Standard and Shear-Stress Transport (SST) k-ω models are employed for this investigation.

The ML treatment slightly overestimated the budget components on an outer scale, whereas the k-ω models strikingly underestimated them. In the buffer layer at the inner scale, the SWF highly over-predicts turbulent production and dissipation and k-ω models over-predict dissipation. Appreciably accurate inner and outer scale k-budgets are observed with the EWT schemes. With a sufficiently resolved near-wall mesh, the Realisable model with EWT exhibits the mean flow, turbulence characteristics and turbulence energy transport even better than the SST k-ω model.

Three distinct near-wall strategies are chosen for comparative performance analysis, focusing not only on the mean flow and turbulence characteristics but the turbulence energy budget as well, for finding the best combination, having potential as a viable and low-cost alternative to LES and DNS for wall jet simulation in industrial application.

The main objectives of this paper are to develop a novel perturbation method (PM) to solve the complex-orthogonal eigenvalue problem and further propose an exact complex mode superposition method (CMSM) for the non-proportionally rate-independent damped systems.

A novel PM is developed to solve the eigenvalue problem. The PM reduced the N-order generalized complex eigenvalue problem into a set of n algebraic equations by the perturbation theory. The convergence and accuracy of the PM are demonstrated by several numerical examples. Further, an exact CMSM is presented. The influences of the imaginary part response of the modal coordinate and the off-diagonal elements of the damping matrix as well as the modal truncation on the solution by CMSM are discussed to illustrate the effectiveness of the developed CMSM.

The eigenvalues obtained by PM would converge to the exact ones with the increase of the modal numbers. For seismic response, the influence of the imaginary part solutions of the modal coordinate would increase with the increase of the coupling factor. The contribution of higher modes to acceleration response is greater than that to the displacement. The cumulative mode contribution coefficient of acceleration is developed to estimate the numbers of the complex modes for the acceleration seismic response by the CMSM.

1. An eigenvalue perturbation method for a rate-independent damped system is proposed. 2. PM is carried out by the real mode and accomplishes the reduction of the matrix. 3. CMSM is established for rate-independent damped systems. 4. CMSM considers the effect of imaginary part solutions of the modal coordinate. 5. Modal truncation index is developed to estimate the complex mode number for CMSM.