Compressible and nonisothermal viscoelastic flow between eccentrically rotating cylinders

A Taylor–Galerkin finite element time marching scheme is derived to numerically simulate the flow of a compressible and nonisothermal viscoelastic liquid between eccentrically rotating cylinders. Numerical approximations to the governing flow and constitutive equations are computed over a custom refined unstructured grid of piecewise linear Galerkin finite elements. An original extension to the DEVSS formulation for compressible fluids is introduced to stabilise solutions of the discrete problem. The predictions of two models: the extended White–Metzner and FENE-P-MP are presented. Comparisons between the torque and load bearing capacity predicted by both models are made over a range of viscoelastic parameters. The results highlight the significant and interacting effects of elasticity and compressibility on journal torque and resultant load, and the stability of the journal bearing system.

Modeling Transient Flows in Heterogeneous Layered Porous Media Using the Space–Time Trefftz Method

In this study, we developed a novel boundary-type meshless approach for dealing with two-dimensional transient flows in heterogeneous layered porous media. The novelty of the proposed method is that we derived the Trefftz space–time basis function for the two-dimensional diffusion equation in layered porous media in the space–time domain. The continuity conditions at the interface of the subdomains were satisfied in terms of the domain decomposition method. Numerical solutions were approximated based on the superposition principle utilizing the space–time basis functions of the governing equation. Using the space–time collocation scheme, the numerical solutions of the problem were solved with boundary and initial data assigned on the space–time boundaries, which combined spatial and temporal discretizations in the space–time manifold. Accordingly, the transient flows through the heterogeneous layered porous media in the space–time domain could be solved without using a time-marching scheme. Numerical examples and a convergence analysis were carried out to validate the accuracy and the stability of the method. The results illustrate that an excellent agreement with the analytical solution was obtained. Additionally, the proposed method was relatively simple because we only needed to deal with the boundary data, even for the problems in the heterogeneous layered porous media. Finally, when compared with the conventional time-marching scheme, highly accurate solutions were obtained and the error accumulation from the time-marching scheme was avoided.

Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method

In this article, a novel meshless method using space–time radial polynomial basis function (SRPBF) for solving backward heat conduction problems is proposed. The SRPBF is constructed by incorporating time-dependent exponential function into the radial polynomial basis function. Different from the conventional radial basis function (RBF) collocation method that applies the RBF at each center point coinciding with the inner point, an innovative source collocation scheme using the sources instead of the centers is first developed for the proposed method. The randomly unstructured source, boundary, and inner points are collocated in the space–time domain, where both boundary as well as initial data may be regarded as space–time boundary conditions. The backward heat conduction problem is converted into an inverse boundary value problem such that the conventional time–marching scheme is not required. Because the SRPBF is infinitely differentiable and the corresponding derivative is a nonsingular and smooth function, solutions can be approximated by applying the SRPBF without the shape parameter. Numerical examples including the direct and backward heat conduction problems are conducted. Results show that more accurate numerical solutions than those of the conventional methods are obtained. Additionally, it is found that the error does not propagate with time such that absent temperature on the inaccessible boundaries can be recovered with high accuracy.

Symmetric Thermo-Electro-Elastic Response of Piezoelectric Hollow Cylinder Under Thermal Shock Using Lord–Shulman Theory

The response of a long hollow cylindrical vessel made from a piezoelectric material is considered in the present investigation. The piezoelectric vessel is subjected to a thermal shock on one surface. The generalized piezo-thermo-elasticity formulation of Lord and Shulman is adopted which contains a single relaxation time to consider the finite speed of temperature wave propagation. The response of the cylinder is assumed to be axi-symmetric. Three coupled equations are established as the governing equations, which are the equation of motion, the energy equation and the Maxwell equation. These equations are transformed into the dimensionless ones. With the aid of the generalized differential quadrature method, these equations are discretized in the radial direction. After that, with the aid of the Newmark time marching scheme, the temporal evolutions of the thermo-electro-elastic parameters are obtained. Novel numerical results are presented to obtain the response of the cylinder subjected to a thermal shock using the Lord and Shulman theory of thermoelasticity.

Numerical application of the nonlinear coupled constitutive relations under three-dimensional hybrid unstructured finite-volume framework

The nonlinear coupled constitutive relations (NCCR) has been studied intensively under structured finite-volume framework. For further engineering applications, it is necessary to study the potential capability of NCCR with the use of unstructured mesh. In this paper, a three-dimensional NCCR solver with hybrid unstructured techniques is developed on the basis of our in-house code. An upwind flux-splitting scheme with LU-SGS implicit time-marching scheme is employed. The accuracy of the developed NCCR solver is validated by two hypersonic cases, such as high-speed flows around 2D cylinder and 3D blunted cone tip in rarefied regimes. Computations show that the solver using hybrid unstructured techniques yields comparable good results as the structured one. Moreover, they can get solutions in better agreement with the Direct Simulation Monte Carlo (DSMC) than the NS results.