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doi: 10.1002/fld.5406pmid: N/A
The modeling of fluids is an important field for mechanics of materials. In this work, we demonstrate that Hamilton's principle, which is well‐known for the modeling of solids, can also be formulated to derive the Navier–Stokes equations, which paves the way for easy inclusion of complex material constraints. Furthermore, we expand Hamilton's principle to enable the introduction of “internal variables”, which describe the space‐ and time‐dependent evolution of the material properties. Hereby, a novel strategy for the modeling of non‐Newtonian fluids is given. Eventually, Hamilton's principle inherently enables a space‐time formulation with the automatic derivation of the correct formal functional setting, which covers different scales of viscosity through the internal variable. The resulting system is a space‐time multiscale model for fluid flow, which is based on an additional partial differential equation. The model constitutes thus a much more adaptive description of the complex processes in non‐Newtonian fluid flow as possible for classical models based on algebraic constitutive laws. This also includes a spatially and temporally local evolution of the effective viscosity, depending on the local flow conditions rather than material parameters and resulting in both shear‐thinning and shear‐thickening behavior. Numerical examples substantiate our proposed setting by some studies from Newtonian flow to non‐Newtonian regimes with fading or increasing viscosity.
doi: 10.1002/fld.70006pmid: N/A
This paper proposes the unified interpolated‐based scheme for curved boundary treatment of the discrete unified gas kinetic scheme (DUGKS). The construction of the proposed boundary scheme is the combination of interpolation and the straight boundary condition (i.e., bounce‐back (BB), non‐equilibrium bounce‐back (NEBB), and moment‐based scheme). To note that, this paper is the first to adopt the moment‐based boundary condition to combine with the interpolation‐based scheme for curved boundary treatment. The asymptotic analysis confirms that the proposed schemes are of first‐order accuracy. Their feasibility and accuracy are examined for different spatial grid resolutions through several numerical tests. They are robust and easy to implement. The results agree well with the analytical solution and validate the first‐order accuracy. It is found that the moment‐based scheme has better accuracy than both BB and NEBB schemes.
AL Garalleh, Hakim; Rasheed, Haroon Ur; Khan, Waris; Mahmoud, Emad E.; Al Agha, Afnan
doi: 10.1002/fld.70007pmid: N/A
The induced convective flow of three‐dimensional Casson nanofluid governed by a bi‐directional stretching surface has potential practical implications in numerous engineering fields, such as heat exchangers, cooling systems for heat‐generating devices, and more. This investigation aims to analytically examine the natural convection mechanism and heat transfer analysis of a Casson nanofluid inside a porous surface exposed to a uniform magnetic field. Moreover, this research explores the physical insights of thermal characteristics by incorporating the effects of chemical reactions, velocity slip, Brownian diffusion, and heat sources/sinks on the transient magnetohydrodynamic flow of the nanofluid. The proposed flow framework is described by a system of partial differential equations, which are transformed into dimensionless ordinary differential equations using appropriate variables. The closed‐form solutions of a set of leading characteristic dimensionless equations are obtained analytically through the efficient homotopic analysis method. Furthermore, stability and convergence analyses of the series solutions are performed to validate the computational results explicitly. The computational findings reveal a significant decrease in flow velocity, temperature, and particle concentration profiles as the Casson fluid parameter increases. Additionally, the effects on skin friction, Nusselt number, and Sherwood number are discussed in detail. This study aims to enhance the understanding of flow dynamics and heat and mass transfer mechanisms across various applications, offering valuable insights for engineering and scientific advancements. The authors accept that all the computational outcomes in this research, both analytical and numerical, are authentic and not published elsewhere.
Kamal, M. A.; Hasan, Ahmed M. M.; Rashed, Youssef F.; Farid, Ahmed Fady
doi: 10.1002/fld.70005pmid: N/A
This paper presents a novel mesh‐free approach for solving the Navier–Stokes equations. The method makes use of the meshless method of fundamental solutions (MFS) and the Monte Carlo integration technique for computing the domain integral of the convective terms. No domain or boundary discretization is required. This approach facilitates numerical computation while ensuring accuracy and stability. By imposing a penalty parameter, the Navier–Stokes equations are transformed to resemble the Navier equations of elasticity. Hence, elasticity based fundamental solutions are employed. The proposed formulation is validated through numerical examples, demonstrating its efficacy in capturing steady‐state flow phenomena through several examples. This highly parallelized system is then accelerated via GPU computing. Overall, the proposed method provides a promising paradigm for advancing computational fluid mechanics, offering a versatile framework with broad applicability in engineering and scientific domains.
Faugeras, Blaise; Guillard, Hervé; Nkonga, Boniface; Rapetti, Francesca
doi: 10.1002/fld.70009pmid: N/A
Heat transfer in magnetically confined plasmas is characterized by extremely high anisotropic diffusion phenomena. At the core of a magnetized plasma, the heat conductivity coefficients in the parallel and perpendicular directions of the induction field can be very different. Their ratio can exceed 108$$ 1{0}^8 $$, and the pollution by purely numerical errors can make the simulation of the heat transport in the perpendicular direction very difficult. Standard numerical methods, generally used in the discretization of classical diffusion problems, are rather inefficient. The present paper analyzes a finite element approach for the solution of a highly anisotropic diffusion equation. Two families of finite elements of class 𝒞1, namely bi‐cubic Hermite‐Bézier and reduced cubic Hsieh‐Clough‐Tocher finite elements, are compared. Their performances are tested numerically for various ratios of the diffusion coefficients, on different mesh configurations, even aligned with the induction field. The time stepping is realized by an implicit high‐order Gear finite difference scheme. An example of a reduced model is also provided to comment on some obtained results.
Avazzadeh, Zakieh; Turan‐Dincel, Arzu; Hassani, Hossein
doi: 10.1002/fld.70010pmid: N/A
This paper presents an optimization algorithm designed to effectively handle a new general class of the nonlinear variable‐order fractional partial differential equations (GCNV‐OFPDEs) with nonlocal boundary conditions. Our approach involves utilizing a novel variant of the polynomials, namely generalized Abel polynomials (GAPs), and also new operational matrices to approximate the solution of the GCNV‐OFPDEs. A key aspect of our algorithm is the transformation of GCNV‐OFPDEs, along with their respective nonlocal boundary conditions, into systems of nonlinear algebraic equations. By solving these systems, we can determine the unknown coefficients and parameters. To address the nonlinear system, we employ the Lagrange multipliers to achieve optimal approximations. The convergence analysis of the approach is discussed. To validate the effectiveness of our algorithm, we conducted numerous experiments using various examples. The results obtained demonstrate the exceptional accuracy of our approach and its potential for extension to more complex problems in the future.
Costanzo, Eduardo Di; Kühl, Niklas; Marongiu, Jean‐Christophe; Rung, Thomas
doi: 10.1002/fld.70012pmid: N/A
Engineering simulations are usually based on complex, grid‐based, or mesh‐free methods for solving partial differential equations. The results of these methods cover large fields of physical quantities at very many discrete spatial locations and temporal points. Efficient compression methods can be helpful for processing and reusing such large amounts of data. A compression technique is attractive if it causes only a small additional effort and the loss of information with strong compression is low. The paper presents the development of an incremental singular value decomposition (SVD) strategy for compressing time‐dependent particle simulation results. The approach is based on an algorithm that was previously developed for grid‐based, regular snapshot data matrices. It is further developed here to process highly irregular data matrices generated by particle simulation methods during simulation. Various aspects important for information loss, computational effort, and storage requirements are discussed, and corresponding solution techniques are investigated. These include the development of an adaptive rank truncation approach, the assessment of imputation strategies to close snapshot matrix gaps caused by temporarily inactive particles, a suggestion for sequencing the data history into temporal windows as well as bundling the SVD updates. The simulation‐accompanying method is embedded in a parallel, industrialized smoothed‐particle hydrodynamics software and applied to several 2D and 3D test cases. The proposed approach reduces the memory requirement by about 90% and increases the computational effort by about 10%, while preserving the required accuracy. For the final application of a water turbine, the temporal evolution of the force and torque values for the compressed and simulated data is in excellent agreement.
Han, Ruoyu; Zhang, Meina; Zhang, Jiexing; Ni, Guoxi
doi: 10.1002/fld.70011pmid: N/A
In this article, a cell‐centered discontinuous Galerkin (DG) method is presented for solving Lagrangian radiation hydrodynamic equations (RHE). The equations are separated into a hydrodynamic part and a radiation diffusion part. These two parts are written in Lagrangian forms. The hydrodynamic part is discretized by a cell‐centered DG scheme in reference space using Taylor basis functions. An approximate Riemann solver is used for the velocity of vertices, and the radiation diffusion is solved using an interior penalty method. Due to the deformation of the basis functions in physical space, curvilinear mesh is formed. Numerical tests are presented to show its accuracy and robustness.
doi: 10.1002/fld.70008pmid: N/A
A geometric multigrid solution technique for the incompressible Navier–Stokes equations in three dimensions is presented, utilizing the concept of discretely divergence‐free finite elements without requiring the explicit construction of a basis on each mesh level. For this purpose, functions are constructed in an a priori manner spanning the subspace of discretely divergence‐free functions for the Rannacher–Turek finite element pair under consideration. Compared to mixed formulations, this approach yields smaller system matrices with no saddle point structure. This prevents the use of complex Schur complement solution techniques, and more general preconditioners can be employed. While constructing a basis for discretely divergence‐free finite elements may pose significant challenges and prevent the use of a structured assembly routine, a basis is utilized only on the coarsest mesh level of the multigrid algorithm. On finer grids, this information is extrapolated to prescribe boundary conditions efficiently. Here, special attention is required for geometries introducing bifurcations in the flow. In such cases, so‐called “global” functions with an extended support are defined, which can be used to prescribe the net flux through different branches. Various numerical examples for meshes with different shapes and boundary conditions illustrate the strengths, limitations, and future challenges of this solution concept.
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