Two-dimensional Unsteady Problem Research Articles

Heat and mass transfer in anisotropic heat-protective composite materials under aerodynamic heating

This article presents a mathematical model of heat and mass transfer in anisotropic heat-protective composite materials (HPCM) during phase transformations of HPCM binders with the formation of a porous coke residue and pyrolysis gases filtering through the residue to the outer boundary. Using known binder decomposition and nonlinear filtration laws for random HPCMs, the model determines the velocity and coordinates of the two-dimensional HPCM binder decomposition zone, as well as the two-dimensional regions of the porous-gas residue and the initial phase, which are unsteadily separated by a moving zone of binder decomposition. In the newly formed porous-gas region, a two-dimensional unsteady problem of anisotropic heat conduction was solved taking into account nonlinear anisotropic gas filtering. In the initial region unaffected by the binder decomposition, a two-dimensional unsteady problem of anisotropic heat conduction was solved. The mass and linear velocities of the binder decomposition (pyrolysis) zone were calculated from Stefan conditions for heat flow and temperature continuity. The complex model was solved by the previously developed effective and absolutely stable method of alternating directions with extrapolation. New results wereobtained and discussed.

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

Flow structures of strong interaction between an oblique shock wave and a supersonic streamwise vortex

The oblique shock/vortex interaction (OSVI) is numerically investigated based on the large-eddy simulation method. A Mach interaction between separated shock and incident shock can be found when the pressure at the recirculation region reaches a certain level. Based on the idea of spatial–temporal correlation, which considers the three-dimensional steady interaction as a two-dimensional unsteady problem, a qualitative analysis is conducted to explain complicated three-dimensional shock structures. The interaction can be regarded as a combination of the following events: the interaction between circular shock and normal shock, the reflection of shock wave on a subsonic interface, and the interaction between secondary circular shock and other shock structures. Though the original vortex has broken down, a pair of streamwise vortices can be observed in the downstream flow field, the formation of which is associated with the split of the recirculation region. Moreover, the recirculation region is found to act as a solid body, which means that the flow angle along a splitting curve can reflect the splitting speed. Three stages can be identified according to the change process of the flow angle along the splitting curve, which are rapid growth, linear growth, and decrease stages. Inspired by the studies on the shock-induced boundary layer separation, the flow field of the strong OSVI with a regular interaction is modeled to predict the initial flow angle of the splitting point which is the foundation of the study on other stages. The interaction type between separated shock and incident shock can also be judged according to this approach.

A half boundary method for two dimensional unsteady convection–diffusion equations

Convection diffusion equation is widely applied in many fields of science and technology. Many practical engineering problems can be expressed by this equation in unsteady state. However, it is usually difficult and time-consuming to find its solution. In this paper, a boundary type method named half boundary method (HBM) is proposed for two dimensional unsteady convection–diffusion equations. The main idea of HBM is introducing new variables to reduce the order of the equation and build relations of variables between nodes inside the area and nodes at half of the boundaries, namely the unknown variables. Since only variables at half of the boundaries are chosen as unknown variables, using HBM can realize dimensionality reduction and the maximum matrix order is less than that in finite volume method when considering large division number, which makes HBM fast and efficient. After the unknown variables are obtained by the boundary conditions, all variables at nodes can be obtained simultaneously according to the relations mentioned above. Numerical studies are carried out for convection dominated problems, problems with variable or discontinuous coefficient and problems with mixed boundary conditions, which show the validity of HBM for two-dimensional unsteady convection-diffusion problems and high accuracy for convection-domination problems.

Improved simplified and highly stable lattice Boltzmann methods for incompressible flows

In this work, improved simplified and highly stable lattice Boltzmann methods (SHSLBMs) are developed for incompressible flows. The SHSLBM is a newly developed scheme within the lattice Boltzmann method (LBM) framework, which utilizes the fractional step technology to resolve the governing equations recovered from lattice Boltzmann equation (LBE) and reconstructs the equations in the Lattice Boltzmann frame. By this treatment, the SHSLBM directly tracks the macroscopic variables in the evolution process rather than the distribution functions of each grid node, which greatly saves virtual memories and simplifies the implementation of physical boundary conditions. However, the Chapman–Enskog expansion analysis reveals that the SHSLBM recover the weakly compressible Navier–Stokes equations with the low Mach number assumption. Therefore, the original SHSLBM can be regarded as an artificial compressible method and may cause some undesired errors. By modifying the evolution equation for the density distribution function, the improved SHSLBMs can eliminate the compressible effects. The incompressible SHSLBMs are compared with the original SHSLBM in terms of accuracy and stability by simulating several two-dimensional steady and unsteady incompressible flow problems, and the results demonstrate that the present SHSLBMs ensure the second order of accuracy and can reduce the compressible effects efficiently, especially for the incompressible flows with large pressure gradients. We then extended the present SHSLBMs to study the more complicated two-dimensional lid-driven flow and found that the present results are in good agreement with available benchmark results.

Oblique elastic plate impact on thin liquid layer

The present study is concerned with possible mechanisms of air entrainment in a thin liquid layer caused by oblique impact of a deformable body on the layer. The two-dimensional unsteady problem of oblique elastic plate impact is considered within the thin-layer approximation for the first time. The plate deflection is described by the Euler beam equation. The plate edges are free of stresses and shear forces. The plate deflections are comparable with the liquid layer thickness. It is revealed in this paper that, for a stiff plate, the initial impact by the trailing edge makes the plate rotate with the leading plate edge entering water before the wetted part of the plate arrives at this edge. The air cavity trapped in such cases can be as long as 40% of the plate length. For a flexible plate, the impact does not cause the plate rotation. However, the dry part of the plate in front of the advancing wetted region is deflected toward the liquid layer also trapping the air. The numerical results are presented for elastic and rigid motions of the plate, hydrodynamic pressure in the wetted part of the plate, position of this wetted part, and the flow beneath the plate.

Meshless Method for Nonuniform Heat-Transfer/Solidification Behavior of Continuous Casting Round Billet

Numerical simulation is the primary approach to evaluate the complex solidification behavior for the continuous casting (CC) process, in which the methods based on the mesh and topology technique have been widely used to solve field variables. For the traditional techniques, such as the finite difference, finite element, and boundary element methods, it is arduous or even impossible to deal with complicated problems that involve multiphase coupling, interface tracking/reconstruction, and self-adaptation due to the inherent weaknesses in the grid structure and mesh dependence. Hence, the present work explores a meshless calculation method for the two-dimensional unsteady heat-transfer problem and proposes an element-free Galerkin (EFG) model for solving heat transfer inside the CC mold based on the moving least-squares approximation. The temperature functions are approximated and constructed by a linear basis and cubic spline weight function over a set of rectangular supporting domain; then, the discrete heat-transfer governing equation based on the EFG method is deduced. The heat flux measured in the real casting process is set as the boundary condition to calculate the nonuniform solidification of the round billet. The calculated results demonstrate that the shell thickness is consistent with that obtained by the square root law of solidification. In addition, the high heat flux region near the meniscus directly determines the growth characteristics of the initial billet shell, which ultimately results in the overall nonuniformity of the shell. The EFG method has the characteristics of fast convergence, high computational accuracy, and great discrete flexibility; it also provides a novel and effective approach for subsequent thermomechanical coupling and crack propagation analysis in the CC process.

MATHEMATICAL MODELING OF NON-STATIONARY ELASTIC WAVES STRESSES UNDER A CONCENTRATED VERTICAL EXPOSURE IN THE FORM OF DELTA FUNCTIONS ON THE SURFACE OF THE HALF-PLANE (LAMB PROBLEM)

The problem of numerical simulation of longitudinal, transverse and surface waves on the free surface of an elastic half-plane is considered. The change of the elastic contour stress on the free surface of the half­plane is given. To solve the two-dimensional unsteady dynamic problem of the mathematical theory of elasticity with initial and boundary conditions, we use the finite element method in displacements. Using the finite element method in displacements, a linear problem with initial and boundary conditions resulted in a linear Cauchy prob­lem. Some information on the numerical simulation of elastic stress waves in an elastic half-plane under concen­trated wave action in the form of a Delta function is given. The amplitude of the surface Rayleigh waves is sig­nificantly greater than the amplitudes of longitudinal, transverse and other waves with concentrated vertical ac­tion in the form of a triangular pulse on the surface of the elastic half-plane. After the surface Rayleigh waves there is a dynamic process in the form of standing waves.

A new spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations via Chebyshev polynomials

PurposeThe purpose of this paper is to develop an efficient and reliable spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations.Design/methodology/approachIn this study, the considered two-dimensional unsteady advection-diffusion equations are transformed into the equivalent partial integro-differential equations via integrating from the considered unsteady advection-diffusion equation. After this stage, by using Chebyshev polynomials of the first kind and the operational matrix of integration, the integral equation would be transformed into the system of linear algebraic equations. Robustness and efficiency of the proposed method were illustrated by six numerical simulations experimentally. The numerical results confirm that the method is efficient, highly accurate, fast and stable for solving two-dimensional unsteady advection-diffusion equations.FindingsThe proposed method can solve the equations with discontinuity near the boundaries, the advection-dominated equations and the equations in irregular domains. One of the numerical test problems designed specially to evaluate the performance of the proposed method for discontinuity near boundaries.Originality/valueThis study extends the intention of one dimensional Chebyshev approximate approaches (Yuksel and Sezer, 2013; Yuksel et al., 2015) for two-dimensional unsteady advection-diffusion problems and the basic intention of our suggested method is quite different from the approaches for hyperbolic problems (Bulbul and Sezer, 2011).

Bulk Green’s Functions in Two-dimensional Coupled Unsteady Problems of Elastic Diffusion for Orthotropic Continuum

We consider a two-dimensional unsteady coupled problem of elasticity with diffusion and preset unsteady volumetric disturbances. The mathematical model is based on a local equilibrium model of elastic diffusion. The solution is sought in integral form. Bulk Green’s functions are found via Laplace transform and Fourier transform for unbounded medium, sine and cosine transform for semi-bounded continuum, Fourier’s series for bounded continuum.

A stabilized MFE reduced-order extrapolation model based on POD for the 2D unsteady conduction-convection problem

In this study, we devote ourselves to establishing a stabilized mixed finite element (MFE) reduced-order extrapolation (SMFEROE) model holding seldom unknowns for the two-dimensional (2D) unsteady conduction-convection problem via the proper orthogonal decomposition (POD) technique, analyzing the existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions and validating the correctness and dependability of the SMFEROE model by means of numerical simulations.

Fourth-order central compact scheme for the numerical solution of incompressible Navier–Stokes equations

ABSTRACTThis paper provides an implicit central compact scheme for the numerical solution of incompressible Navier–Stokes equations. The solution procedure is based on the artificial compressibility method that transforms the governing equations into a hyperbolic-parabolic form. A fourth-order central compact scheme with a sixth-order numerical filtering is used for the discretization of convective terms and fourth-order central compact scheme for the viscous terms. Dual-time stepping approach is applied to time discretization with backward Euler difference scheme to the pseudo-time derivative, and three point second-order backward difference scheme to the physical time derivative. An approximate factorization-based alternating direction implicit scheme is used to solve the resulting block tridiagonal system of equations. The accuracy and efficiency of the proposed numerical method is verified by simulating several two-dimensional steady and unsteady benchmark problems.

ДВУМЕРНАЯ НЕСТАЦИОНАРНАЯ ЗАДАЧА УПРУГОЙ ДИФФУЗИИ ДЛЯ ОРТОТРОПНОЙ ОДНОКОМПОНЕНТНОЙ ПОЛУПЛОСКОСТИ

The two-dimensional unsteady state problem for elastic orthotropic half-plane with diffusion effect is investigated. In order to solve the problem, the local-equilibrium model of mechanical diffusion is used. This model consists of a system of equations, which describe law of motion and mass transfer. The problem is found with help of Laplace and Fourier transformations. The inverse Laplace transformation is reduced to the calculation of the originals of rational functions and performed analytically. For inverse Fourier transformation is used quadrature formulas. Keywords: mechanical diffusion, diffusion with elasticity, unsteady state problems, half-plane, Laplace transform, Fourier transform.

Two-dimensional finite element model to study unsteady state Ca2+ diffusion in neuron involving ER LEAK and SERCA

In this paper, finite element approach using two-dimensional unsteady state problem has been developed to study radial and angular calcium diffusion problem in neurons. Calcium is responsible messenger for transmitting information in communication process between neurons. The most important Ca 2+ binding proteins for the dynamics of Ca 2+ is itself buffer and other physiological parameters are located in Ca 2+ stores. In this study, the model incorporates the physiological parameters like diffusion coefficient, receptors, exogenous buffers etc. Appropriate boundary conditions have been framed in view of the physiological conditions. Computer simulations in MATLAB 7.11 are employed to investigate mathematical models of reaction–diffusion equation, the details of the implementation can heavily affect the numerical solutions and, thus, the outcome simulated on Core(TM) i3 CPU M 330 @ 2.13 GHz processing speed and 3 GB memory.

DNS of flows with helical symmetry

Some flows such as the wakes of rotating devices often display helical symmetry. We present an original DNS code for the dynamics of such helically symmetric systems. We show that, by enforcing helical symmetry, the three-dimensional Navier–Stokes equations can be reduced to a two-dimensional unsteady problem. The numerical method is a generalisation of the vorticity/streamfunction formulation in a circular domain, with finite differences in the radial direction and spectral decomposition along the azimuth. When compared to a standard three-dimensional code, this allows us to reach larger Reynolds numbers and to compute quasi-steady patterns. We illustrate the importance of helical pitch by some physical cases: the dynamics of several helical vortices and a quasi-steady vortex flow. We also study the linear dynamics and nonlinear saturation in the Batchelor vortex basic flow, a paradigmatic example of trailing vortex instability. We retrieve the behaviour of inviscid modes and present new results concerning the saturation of viscous centre modes.

Forcing term in single-phase and Shan-Chen-type multiphase lattice Boltzmann models

Numerous schemes have been proposed to incorporate a bulk forcing term into the lattice Boltzmann equation. In this paper we present a simple and straightforward comparative analysis of five popular schemes [Shan and Chen, Phys. Rev. E 47, 1815 (1993); Phys Rev Lett. 81, 1618 (1998); He et al., Phys. Rev. E 57, R13 (1998); Guo et al., Phys. Rev. E 65, 046308 (2002); Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)] in which their differences and similarities are identified. From the analysis we classify the schemes into two groups; the behaviors of the schemes in each group are proven to be identical up to second order. Numerical test simulating the two-dimensional unsteady Taylor-Green vortex flow problem demonstrate that all five schemes are of comparable accuracy for single-phase flow. However, for two-phase flow the situation is different, which is demonstrated by incorporating these schemes into different Shan-Chen-type multiphase models. The forcing scheme in the original Shan-Chen (SC) multiphase model turns out to be inaccurate in terms of the resulting surface tension for different density ratios and relaxation times. In the numerical tests, a typical equation of state and interparticle interactions including next-nearest neighbors were incorporated into the SC model. Our results confirm that the surface-tension values obtained from the original SC lattice Boltzmann method (LBM) simulation depend on the value of the relaxation time τ. For τ<0.7Δt, the surface tension agree well with the analytical solutions. However, when τ>0.7Δt, the surface tension turns out to be systematically larger than the analytical one, exceeding it by more than a factor of 2 for τ=2Δt. In contrast, with the application of the scheme proposed by He et al., the SC LBM produces very accurate surface tensions independent of the value of τ. We also found that the densities of the coexisting liquid and gas can be adjusted to match those at thermodynamic equilibrium if the particle interaction term includes next-nearest-neighbor contributions. The obtained results will be useful for further studies of two-phase flow with high density ratios using the SC LBM approach.

Motion of a vertical wall fixed on springs under the action of surface waves

A two-dimensional unsteady hydroelastic problem of interaction between surface waves and a moving vertical wall fixed on springs is studied. An analytical solution of the problem is constructed using a linear approximation, and a numerical solution within the framework of a nonlinear model of a potential fluid flow is found by a complex boundary element method. By means of analysis of the linear and nonlinear solutions, it is found that the linear solution can be used to predict some important characteristics of the wall motion and the fluid flow in the case of moderate wave amplitudes.

Realization and modeling of continuous spin detonation of a hydrogen-oxygen mixture in flow-type combustors. 1. Combustors of cylindrical annular geometry

A comprehensive numerical and experimental study of continuous spin detonation of a hydrogen-oxygen mixture in flow-type cylindrical annular combustors 4 and 10 cm in diameter is performed. Hydrogen is injected through injectors, and oxygen is supplied as a continuous flow through an annular slot. The flow structure is studied with variations of the flow rates of the components of the mixture and the width of the slot for oxygen supply. The region of existence of continuous spin detonation is determined as a function of the fuel-to-air equivalence ratio and specific flow rates of the components with variations of the relative width of the slot and combustor diameter. A two-dimensional unsteady gas-dynamic problem of rotation dynamics of a transverse detonation wave with geometric parameters of the combustor corresponding to those used in experiments is solved numerically. A comparison with experiments is performed, and reasonable agreement is reached for the detonation velocity and mean pressure in the combustor. It is shown that the geometric size of the transverse detonation waves is underestimated because the gas-dynamic model does not involve the mixing process, and the number of waves is almost doubled.

Unsteady Discrete Adjoint Formulation for Two-Dimensional Flow Problems with Deforming Meshes

A method to apply the discrete adjoint for computing sensitivity derivatives in two-dimensional unsteady flow problems is presented. The approach is to first develop a forward or tangent linearization of the nonlinear flow problem in which each individual component building up the complete flow solution is differentiated against the design variables using the chain rule. The reverse or adjoint linearization is then constructed by transposing and reversing the order of multiplication of the forward problem. The developed algorithm is very general in that it applies directly to the arbitrary Lagrangian-Eulerian form of the governing equations and includes the effect of deforming meshes in unsteady flows. It is shown that an unsteady adjoint formulation is essentially a single backward integration in time and that the cost of constructing the final sensitivity vector is close to that of solving the unsteady flow problem integrated forward in time. It is also shown that the unsteady adjoint formulation can be applied to time-integration schemes of different orders of accuracy with minimal changes to the base formulation. The developed technique is then applied to three optimization examples, the first in which the shape of a pitching airfoil is morphed to match a target time-dependent load profile, the second in which the shape is optimized to match a target time-dependent pressure profile, and the last in which the time-dependent drag profile is minimized without any loss in lift.

On the consistency of the direct forcing method in the fractional step solution of the Navier–Stokes equations

In the present work the application of the direct forcing method to the fractional step solution of the Navier–Stokes equations is analyzed in details. The central topic of the study is the satisfaction of the boundary conditions in the presence of immersed bodies. To this aim, simple two-dimensional problems with steady forcing and sharp-edged surfaces are solved with a standard fractional step method. It is shown that the direct forcing scheme is not able to satisfy the impenetrability condition. This fact appears to be strictly related to the use of fractional step methods. A mathematically based explanation is given, suggesting that improvements can be obtained with the iterative solution of the irrotational part of the flow, when spectral methods are used and local modification of the discrete differential operators are difficult to be implemented. The results are then verified in two-dimensional unsteady problems and three-dimensional flows with moving smooth boundaries.