Compact High Order Research Articles

Global 2D stability analysis of the cross lid-driven cavity flow with a streamfunction-vorticity approach

ABSTRACTWe have recently analyzed the global two-dimensional (2D) stability of the staggered lid-driven cavity (LDC) flow with a higher order compact (HOC) approach. In the analysis, critical parameters are determined for both the parallel and anti-parallel motion of the lids and a detailed analysis has been carried out on either side of the critical values.In this article, we carry out an investigation of flow stabilities inside a two-sided cross lid-driven cavity with a pair of opposite lids moving in both parallel and anti-parallel directions. On discretization, the governing 2D Navier–Stokes (N–S) equations describing the steady flow and flow perturbations results in a generalized eigenvalue problem which is solved for determining the critical parameters on four different grids. Elaborate computation is performed for a wide range of Reynolds numbers (Re) on either side of the critical values in the range 200 ⩽ Re ⩽ 10000. For flows below the critical Reynolds number Rec, our numerical results are compared with established steady-state results and excellent agreement is obtained in all the cases. For Reynolds numbers above Rec, phase plane and spectral density analysis confirmed the existence of periodic, quasi-periodic, and stable flow patterns.

Compact high order finite volume method on unstructured grids I: Basic formulations and one-dimensional schemes

The large reconstruction stencil has been the major bottleneck problem in developing high order finite volume schemes on unstructured grids. This paper presents a compact reconstruction procedure for arbitrarily high order finite volume method on unstructured grids to overcome this shortcoming. In this procedure, a set of constitutive relations are constructed by requiring the reconstruction polynomial and its derivatives on the control volume of interest to conserve their averages on face-neighboring cells. These relations result in an over-determined linear equation system, which, in the sense of least-squares, can be reduced to a block-tridiagonal system in the one-dimensional case. The one-dimensional formulations of the reconstruction are discussed in detail and a Fourier analysis is presented to study the dispersion/dissipation and stability properties. The WBAP limiter based on the secondary reconstruction is used to suppress the non-physical oscillations near discontinuities while achieve high order accuracy in smooth regions of the solution. Numerical results demonstrate the method's high order accuracy, robustness and shock capturing capability. A new reconstruction procedure for high order finite volume scheme is proposed.This procedure can achieve arbitrary order of accuracy using a compact stencil consisting of only face neighboring cells.The spectral properties of the proposed finite volume schemes are analyzed.Some test cases are solved numerically to show the superior performances of the proposed schemes.

Compact high order finite volume method on unstructured grids II: Extension to two-dimensional Euler equations

Abstract In this paper, the compact least-squares finite volume method on unstructured grids proposed in our previous paper is extended to multi-dimensional systems, namely the two-dimensional Euler equations. The key element of this scheme is the compact least-squares reconstruction in which a set of constitutive relations are constructed by requiring the reconstruction polynomial and its spatial derivatives on the control volume of interest to conserve their averages on the face-neighboring cells. These relations result in an over-determined linear equation system. A large sparse system of linear equations is resulted by using the least-squares technique. An efficient solution strategy is of crucial importance for the application of the proposed scheme in multi-dimensional problems since both direct and iterative solvers for this system are computationally very expensive. In the present paper, it is found that in the cases of steady flow simulation and unsteady flow simulation using dual time stepping technique, the present reconstruction method can be coupled with temporal discretization scheme to achieve high computational efficiency. The WBAP limiter and a problem-independent shock detector are used in the simulation of flow with discontinuities. Numerical results demonstrate the high order accuracy, high computational efficiency and capability of handling both complex physics and geometries of the proposed schemes.

The role of magnetic Reynolds number in MHD forced convection heat transfer

Higher Order Compact (HOC) Scheme combined with multigrid method is developed to investigate the role of magnetic Reynolds number in MHD heat transfer from a circular cylinder which is under the influence of an external magnetic field. The governing equations are coupled nonlinear Navier–Stokes and nonlinear Maxwell’s equations which are expressed in stream function, vorticity and magnetic stream function (ψ-ω-A) formulation and decoupled energy equation. Apart from the usual kinetic Reynolds number Re, the parameters that governs the flow are magnetic Reynolds number Rm, Alfvén number β and Prandtl number Pr. The velocities of the flow are calculated with second order accurate based finite difference scheme and heat transfer equation is solved with fourth order accurate HOC scheme. The influence of the magnetic field and magnetic Reynolds number on velocity gradients is presented. It is found that the mean Nusselt number decreases until N ≤ 1 and then increases with further increase in the interaction parameter. It is also found that fluids having higher electrical conductivity can be effectively controlled with relatively low magnetic fields. The decrease in mean Nusselt number with β and Rm is in agreement with experimental findings.

Development of a high-order level set method: Compact Conservative Level Set (CCLS)

In this paper a Compact Conservative Level Set (CCLS) method has been developed to improve the accuracy and robustness of Level Set (LS) approach in simulation of incompressible two-phase flows. Since mass conservation error is known as the evident weakness of LS approach, the main aim of this research is to introduce a novel method with an excellent mass conservation property while keeping the simplicity of the approach. Here, the hyperbolic tangent function; used as LS function; is conservatively transported through a High-Order Compact (HOC) finite difference scheme. In order to retain thickness of the coated interface while retrieving the hyperbolic tangent profile, a conservative re-initialization is adopted by introducing a novel Combined Compact Difference (CCD) method. Normals are computed from the distance function, where a fast marching method; an efficient re-distancing method; is applied to re-construct signed distance function from the hyperbolic tangent function. The developed method is employed in a wide range of test cases and its noticeable mass conservation property is revealed even for complex cases, as well as its accuracy and stability in large courant numbers, long time simulation, coarse mesh and even simulations without re-initialization.

Containing High Order Compact Scheme Source of Steady Convection-Diffusion Equation

Convection-diffusion equation have a wide range of applications in many practical engineering problems, such as magnetic confinement fusion problems, heat transfer, particle diffusion. Traditional solutionof convection-diffusion equation in magnetic confinement fusion is Crank-Nicolson scheme. This paper presents a new numerical solution of one-dimensional steady-containing source convection diffusion equation high accuracy difference schemes O(t²+h⁴), which proved to be unconditionally stable using Fourier analysis, numerical experiments show the accuracy and robustness of this format, this scheme has a higher accuracy.

Optical confinement study of different semi conductor rib waveguides using higher order compact finite difference method

Semi conductor optical wave guides with well defined refractive index profile and geometric shapes which help to understand the propagation properties are widely used in the field of integrated optics. Accordingly rib waveguides are considered, as this structure allows the greatest versatility in device design in which the basic concept employed is its compositional changes in the vertical direction where the three regions of the rib structure are divided into zones with specific refractive indices. As for these types of structures no precise analytic solution is obtained, different numerical techniques have been implemented for their study in terms of their modal analysis. So an accurate higher order compact (HOC) finite difference method (FDM) in combination with conjugate gradient method (CGM) is used here for the investigation of refractive index profile of GaAs and GeSi rib waveguide structure to show their effect on transmission properties of the guided wave using Helmholtz wave equation. The difference in surface and contour plots of the polarized E-field solution clearly shows the material dependence of transmitted waves. We have also studied the variation of optical confinement factor with a number of parameters to understand the propagation properties appropriately as per the requirement of various guided wave optoelectronic devices.

Higher-Order Compact-Flow Field-Dependent Variation (HOC-FDV) Method for Solving Two-Dimensional Navier-Stokes Equations

In this article, a new, higher-order accurate method, namely higher-order compact-flow field-dependent variation (HOC-FDV) method, has been developed to solve two-dimensional Navier-Stokes equations. The HOC-FDV scheme is of third-order accuracy in time and fourth-order in space. The spatial derivatives in the flow field-dependent variation (FDV) equations proposed by Chung are approximated using higher-order compact (HOC) Hermitian (Pade) scheme. The solution procedure at each time step consists of a system of block tri-diagonal matrices which can be solved efficiently in a standard manner. Several numerical examples are tested to examine the accuracy and capability of the new scheme to capture the shock and to simulate accurately separation and discontinuity.

Compact high order schemes with gradient-direction derivatives for absorbing boundary conditions

We consider several compact high order absorbing boundary conditions (ABCs) for the Helmholtz equation in three dimensions. A technique called “the gradient method” (GM) for ABCs is also introduced and combined with the high order ABCs. GM is based on the principle of using directional derivatives in the direction of the wavefront propagation. The new ABCs are used together with the recently introduced compact sixth order finite difference scheme for variable wave numbers. Experiments on problems with known analytic solutions produced very accurate results, demonstrating the efficacy of the high order schemes, particularly when combined with GM. The new ABCs are then applied to the SEG/EAGE Salt model, showing the advantages of the new schemes.

Analysis of Unsteady Subsonic Flow Around Pitch Oscillating Airfoil Using an Optimized High Order Compact Scheme

Analysis of Unsteady Subsonic Flow Around Pitch Oscillating Airfoil Using an Optimized High Order Compact Scheme

HOC Simulation of Double-diffusive Natural Convection in a Rectangular Cavity

Over the last few decades, Higher Order Compact (HOC) finite difference schemes have been tremendously gaining popularity among the researchers of computational fluid dynamics community, mainly because of their higher accuracies on smaller stencils. However, in most cases, their application is mainly confined to the computation of fluid flows and may be, in some cases to the problems of heat transfer only. The present work therefore intends to propose a new HOC scheme for problems in heat and mass transfer by extending an existing HOC scheme to a problem of combined heat and mass transfer. It is very interesting to observe that on a very coarser grid of size only 21X41 the results obtained are in very excellent match with the established ones.

Higher order compact (HOC) finite difference method (FDM) to study optical confinement through semiconductor rib wave guides

Higher order compact (HOC) finite difference method (FDM) to study optical confinement through semiconductor rib wave guides

A Peclet Number Based Analysis of Mixed Convection for Lid-Driven Porous Trapezoidal Enclosure

Mixed convection flows in a lid-driven trapezoidal enclosure filled with porous medium are studied numerically using our recently proposed higher order compact (HOC) scheme presented in [1]. The top wall of the enclosure is allowed to move in its own plane at a constant speed, the bottom wall is heated uniformly while all other walls maintained at constant cold temperature. The relevant parameters in the present study are Darcy number (Da) (10–5 ≤ Da ≤ 10–3), Grashof number (Gr) = 105, Prandtl number (Pr) (0.026 ≤ Pr ≤ 10)) and Reynolds number (Re) (1 ≤ Re ≤ 100). The influence of convection is analyzed with Peclet number (Pe = Re.Pr). The flow distribution affects significantly temperature distributions at high Pe irrespective of Da. Effect of Peclet numbers has been further investigated for both natural convection and forced convection dominant regimes at high Da. Strong coupling between flow fields and temperature are observed at high Pe.

Numerical Study of Shear Flow Past a Square Cylinder at Reynolds Numbers 100, 200

A numerical study of two-dimensional incompressible shear flow past a square cylinder is presented in this paper. Numerical simulations are performed using the recently developed higher order compact (HOC) scheme for K = 0.0, 0.1 for Re = 100, 200, where Re is the Reynolds number and K is the shear rate based on cylinder side. We investigate the effect of shear rate on the vortex shedding phenomenon in terms of stream function and vorticity contours, lift and drag coefficients employed on the cylinder and corresponding power spectra. The results presented here show that the vortex shedding phenomenon strongly depends on the Reynolds number as well as on the shear rate. In general, average drag coefficient decreases with increase in the shear rate for fixed Re and for K = 0.0 − 0.1, this change is negligible. In contrary to shear flow around circular cylinder, Strouhal number for flow past square cylinder decreases with increase in K.

An HOC Based Immersed Interface Scheme for Uniform Flow Past a Circular Cylinder

In this paper, we develop a new HOC based immersed interface scheme in cylindrical polar coordinates for solving viscous, incompressible, laminar flow past a circular cylinder. This new scheme is developed by clubbing the recently developed higher order compact (HOC) methodology with with special interface treatment for the points just next to the points of discontinuity. Non uniform grid in polar coordinates (r, θ) is used for the discretization, where the surface of the cylinder is considered as an interface immersed in the fluid. Streamfunction-Vorticity formulation of Navier-Stokes equation in cylindrical polar coordinates is considered with an assumption that solution inside the immersed boundary is zero. Quantitative and qualitative results are shown for various values of Reynolds number Re ≤ 100 and an excellent agreement with existing numerical and experimental results is found in each case.

Comparison of Finite Difference Schemes for the Wave Equation Based on Dispersion

Finite difference techniques are widely used for the numerical simulation of time-dependent partial differential equations. In order to get better accuracy at low computational cost, researchers have attempted to develop higher order methods by improving other lower order methods. However, these types of methods usually suffer from a high degree of numerical dispersion. In this paper, we review three higher order finite difference methods, higher order compact (HOC), compact Padé based (CPD) and non-compact Padé based (NCPD) schemes for the acoustic wave equation. We present the stability analysis of the three schemes and derive dispersion characteristics for these schemes. The effects of Courant Friedrichs Lewy (CFL) number, propagation angle and number of cells per wavelength on dispersion are studied.

A high order compact finite difference scheme for time fractional Fokker–Planck equations

In this paper, a high order compact (HOC) scheme for time fractional Fokker–Planck equations with variable convection is constructed. The scheme is studied using its matrix form by the energy method. We find that the difficulty arising from the variable coefficient can be overcome by simple modifications of the coefficient matrices. The scheme is shown to be stable and convergent with order τ2−α+h4 which is higher than some recently studied schemes. Numerical examples are given to justify the theoretical analysis.

Efficient high-order immersed interface methods for heat equations with interfaces

An efficient high-order immersed interface method (IIM) is proposed to solve two-dimensional (2D) heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact (HOC) difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit (ADI) scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.

Numerical Analysis of the Unsteady Subsonic Flow around a Plunging Airfoil

Much numerical and experimental research has been done for the flow around an oscillating airfoil. The main research topics are vortex shedding, dynamic stall phenomenon, MAVs lift and thrust generation. Until now, researches mainly have been concentrated on analyzing the wake flow for the variation of frequency and amplitude at a low angle of attack. In this study, wake structures and acoustic wave propagation characteristics were studied for a plunging airfoil at high angle of attack. The governing equations are the Navier-Stokes equation with LES turbulence model. OHOC (Optimized High-Order Compact) scheme and 4th order Runge-Kutta method were used. The Mach number is 0.3, the Reynolds number is , and the angle of attack is from 20° to 50°. The plunging frequency and the amplitude are from 0.05 to 0.15, and from 0.1 to 0.2, respectively. Due to the high resolution numerical method, wake vortex shedding and pressure wave propagation process, as well as the propagation characteristics of acoustic waves can be simulated. The results of frequency analysis show that the flow has the mixed characteristics of the forced plunging frequency and the vortex shedding frequency at high angle of attack.

A new compact high order off-step discretization for the system of 2D quasi-linear elliptic partial differential equations

A new fourth-order difference method for solving the system of two-dimensional quasi-linear elliptic equations is proposed. The difference scheme referred to as off-step discretization is applicable directly to the singular problems and problems in polar coordinates. Also, new fourth-order methods for obtaining the first-order normal derivatives of the solution are developed. The convergence analysis of the proposed method is discussed in details. The methods are applied to many physical problems to illustrate their accuracy and efficiency. MSC: 65N06.