Second-order Accurate Difference Scheme Research Articles

NUMERICAL SIMULATION OF A FLUID FLOW IN THE CYLINDRICAL DUCT CONTAINING TWO DIAPHRAGMS WITH ORIFICES OF DIFFERENT DIAMETERS

The flow of a viscous incompressible fluid in a cylindrical duct with two serial diaphragms with orifices of different diameters was studied based on the numerical solution of the unsteady Navier-Stokes equations. The solution algorithm was based on the finite volume method using second-order accurate difference schemes in both space and time. The TVD (Total-Variation Diminishing) form of a central-difference scheme with a flux limiter was used for the interpolation of the convective terms. The combined evaluation of the velocity and pressure fields was carried out using the PISO (Pressure Implicit Split Operator) procedure. The problem was solved using OpenFOAM (Open-source Field Operation And Manipulation) open-source toolkit libraries using the computing power of the cluster supercomputer of the V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine. It was shown that within a certain range of the diameter ratio of the orifices of the diaphragms, a circulation movement is established in the cavity between the diaphragms. The boundary layer breaks off from the surface of the first diaphragm and forms an annular shear layer. When approaching the second diaphragm, a sequence of ring vortices is formed in it, which interact with the surface of the diaphragm and lead to the emergence of tonal sound. When the ratio of the orifice diameter of the second diaphragm to the first decreases, the share of the jet’s kinetic energy, which participates in the circulation in the middle of the cavity between the diaphragms, increases. As a result, the amplitude of velocity fluctuations in the orifice of the second diaphragm decreases. When the ratio of the diameters of the orifices increases, the share of energy involved in the circulation decreases, and when a certain value of the ratio is reached, the interaction between the vortices in the shear layer and the surface of the diaphragm does not occur. As a result, the excitation of the tonal sound stops. The Strouhal number of self-oscillations practically does not change when the diameter ratio of the orifices changes. During the calculations, two different modes of self-oscillation were obtained, which is consistent with previous works.

FAST SECOND-ORDER ACCURATE DIFFERENCE SCHEMES FOR TIME DISTRIBUTED-ORDER AND RIESZ SPACE FRACTIONAL DIFFUSION EQUATIONS

The aim of this paper is to develop fast second-order accurate difference schemes for solving one- and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one- and two-dimensional problems as follows: we first transform the time distributed-order fractional diffusion problem into the multi-term time-space fractional diffusion problem with the composite trapezoid formula. Then, we propose a second-order accurate difference scheme based on the interpolation approximation on a special point to solve the resultant problem. Meanwhile, the unconditional stability and convergence of the new difference scheme in $L_2$-norm are proved. Furthermore, we find that the discretizations lead to a series of Toeplitz systems which can be efficiently solved by Krylov subspace methods with suitable circulant preconditioners. Finally, numerical results are presented to show the effectiveness of the proposed difference methods and demonstrate the fast convergence of our preconditioned Krylov subspace methods.

A finite difference/finite element technique with error estimate for space fractional tempered diffusion-wave equation

An efficient numerical technique is proposed to solve one- and two-dimensional space fractional tempered fractional diffusion-wave equations. The space fractional is based on the Riemann–Liouville fractional derivative. At first, the temporal direction is discretized using a second-order accurate difference scheme. Then a classic Galerkin finite element is employed to obtain a full-discrete scheme. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique.

Method of corrections by higher order differences for Poisson equation with nonlocal boundary conditions

We consider the Bitsadze–Samarskii type nonlocal boundary value problem for Poisson equation in a unit square, which is solved by a difference scheme of second-order accuracy. Using this approximate solution, we correct the right-hand side of the difference scheme. It is shown that the solution of the corrected scheme converges at the rate O(|h|s) in the discrete L2-norm provided that the solution of the original problem belongs to the Sobolev space with exponent s∈[2,4].

WITHDRAWN: Method of corrections by higher order differences for one nonlocal boundary-value problem

Abstract We consider the Bitsadze–Samarskii type nonlocal boundary value problem for Poisson equation in a unit square, which is solved by a difference scheme of second-order accuracy. Using this approximate solution, we correct the right-hand side of the difference scheme. It is shown that the solution of the corrected scheme converges at the rate O ( | h | s ) in the discrete L 2 -norm provided that the solution of the original problem belongs to the Sobolev space with exponent s ∈ [ 2 , 4 ] .

On increasing the convergence rate of difference solution to the third boundary value problem of elasticity theory

We consider the third boundary value problem of static elasticity theory (stiff contact problem) in a rectangle, for solution of which we use a difference scheme of second-order accuracy. Using this approximate solution, we correct the right-hand side of the difference scheme. It is shown that the solution of the corrected scheme is convergent at the rate $O(|h|^{m})$ in the discrete $L_{2}$ -norm, provided that the solution of the original problem belongs to the Sobolev space with exponent $m\in[2, 4]$ .

Compatible convergence estimates in the method of refinement by higher-order differences

We consider the Dirichlet problem for an elliptic equation with constant coefficients, which is solved by a difference scheme of second-order accuracy. By using the approximate solution, we correct the right-hand side of the difference scheme. We show that the solution of the corrected scheme is convergent at the rate O(|h|m) in the discrete L2-norm provided that the solution of the original problem belongs to the Sobolev space with exponent m ∈ [2, 4].

A second-order linearized finite difference scheme for the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation

The purpose of this paper is to study the numerical simulation of the generalized Fisher–Kolmogorov–Petrovskii–Piskunov equation. After introducing a new variable, the integro-differential equation is transformed into an equivalent coupled system of first-order differential equations. A second-order accurate difference scheme is constructed for the new system of equations, which is proved to be local uncoupled by separation of variables. It is also proved that the scheme is uniquely solvable and second-order convergent in both time and space in L 2-norm. A numerical example is given to demonstrate the theoretical results.

A second-order accurate difference scheme for an extended Fisher–Kolmogorov equation

In this paper, we present a Crank–Nicolson type finite difference scheme to approximate the nonlinear evolutionary Extended Fisher–Kolmogorov (EFK) equation. We prove the existence of the solution by using the well-known Browder fixed-point theorem. The stability of this scheme is established in L∞-norm. The uniqueness and convergence of the solution are analyzed. We discuss an iterative algorithm for solving the difference scheme and prove its convergence.

A finite difference scheme for an interface problem

We propose a new finite difference scheme for an interface problem with piecewise constant coefficients. We use the Newton polynomial and continuity of flux to obtain finite difference scheme of second-order accuracy at the interface. We give several examples which show that the numerical solutions have O(h2) accuracy.

Optimally-stable second-order accurate difference schemes for non-linear conservation laws in 3D

In one and two spatial dimensions, Lax–Wendroff schemes provide second-order accurate optimally-stable dispersive conservation-form approximations to non-linear conservation laws. These approximations are an important ingredient in sophisticated simulation algorithms for conservation laws whose solutions are discontinuous. Straightforward generalization of these Lax–Wendroff schemes to three dimensions produces an approximation that is unconditionally unstable. However, some dimensionally-split schemes do provide second-order accurate optimally-stable approximations in 3D (and 2D), and there are sub-optimally-stable non-split Lax–Wendroff-type schemes in 3D. The main result of this paper is the creation of new Lax–Wendroff-type second-order accurate optimally-stable dispersive non-split scheme that is in conservation form. The scheme is created by using linear equivalence to transform a symmetrized dimensionally-split scheme (based on a one-dimensional Lax–Wendroff scheme) to conservation form. We then create both composite and hybrid schemes by combining the new scheme with the diffusive first-order accurate Lax–Friedrichs scheme. Codes based on these schemes perform well on difficult fluid flow problems.

A second order accurate difference scheme for the heat equation with concentrated capacity

A numerical solution to the one-dimensional heat equation with concentrated capacity is considered. A second-order accurate difference scheme is derived by the method of reduction of order on non-uniform meshes. The solvability, stability and second order L∞ convergence are proved. A numerical example demonstrates the theoretical results.

A Second-Order Accurate Capturing Scheme for 1D Inviscid Flows of Gas and Water with Vacuum Zones

A second-order accurate difference scheme is developed to study cavitation in unsteady, one-dimensional, inviscid, compressible flows of water with gas. The scheme can capture shock waves, interfaces separating gas and water, as well as cavitation zones that are modelled as vacuum states, and it takes into account water's capability to resist tensile stresses. As an extended version of the standard MUSCL scheme, this scheme is based on the solutions of local gas–water–vacuum initial value problems. In order to prevent the computed water density from becoming lower than its minimum bound, additional techniques are introduced. Numerical results are presented with gas–water Riemann problems to demonstrate the performance of the scheme. The scheme is also applied to simulate the cavitation process of the flow in a water shock tube.

A Numerical Study of Confined Turbulent Jets

A numerical investigation is reported of turbulent incompressible jets confined in two ducts, one cylindrical and the other conical with a 5 deg divergence. In each case, three Craya-Curtet numbers are considered which correspond, respectively, to flow situations with no, moderate, and strong recirculation. Turbulence closure is achieved by using the K-ε model and a recently proposed realizable Reynolds stress algebraic equation model. Calculations are carried out with a finite-volume procedure. Second-order accurate differencing schemes and sufficiently fine grids are used to ensure numerical accuracy. The calculated results are compared with experimental data. It is shown that the numerical methods presented are capable of capturing the essential flow features observed in the experiments and that the realizable Reynolds stress algebraic equation model performs better than the K-ε model for this class of flows.

Effects of shear thinning on laminar heat transfer behavior in a rectangular duct

The fluid dynamic and heat transfer behavior of laminar non-Newtonian flow through noncircular ducts is of special interest because of their wide range of potential application to compact heat exchangers and in electronics cooling. The present study investigates the heat transfer characteristics for laminar forced convection of inelastic non-Newtonian fluids in a nonuniformly heated rectangular duct where the flow is hydrodynamically developed but thermally developing. The governing equations are solved by a finite volume method. Second-order accurate differencing schemes are employed for both the diffusion and convective terms. The effects of shear thinning, given by the Carreau equation, and the viscous dissipation, characterized by the Brinkman number, are examined via the friction factor, the Nusselt number, and the bulk fluid temperature. The results indicate that when viscous dissipation is present, the heat transfer from the heated surface of the duct is greatly enhanced with increased shear thinning of the fluid while the rate of increase of the bulk fluid temperature with axial distance will significantly decrease. Consequently, the use of a shear-thinning non-Newtonian fluid for heat transfer enhancement appears to be a very promising concept that is worthy of further study.

Application of an implicit iterative difference scheme for the solution of nonstationary Navier-Stokes equations

An implicit iterative difference scheme of second-order accuracy in space and in time is proposed for numerical solution of nonstationary Navier—Stokes equations. The scheme has been applied to model a nonstationary axisymmetric flow in the near wake produced by injection through an annular nozzle from the tail of a supersonic body.

Second-Order Accurate Boundary Conditions for Compressible Flows

The accuracy and the consistency of the methods commonly used for the numerical application of the boundary conditions on the surface of solid bodies immerged in an inviscid compressible flow is examined. A new method is proposed for the application of second-order accurate boundary conditions in the case of isentropic irrotational compressible flows. The method is applied to a circular cylinder in uniform subsonic and low transonic flows. The conservation law equations are solved using the second-order accurate difference scheme of MacCormack. The results indicate a definite improvement over lower-accuracy methods. The method can be easily generalized to apply on three-dimensional solid surfaces.

Nonstationary loads in shock diffraction

The purpose of the paper is to analyze the nonlinear process of plane shock reflection from a cylinder or sphere. In contrast to [1], where the S. K. Godunov method of first-order accuracy is used, the solution is carried out by using an explicit difference scheme of second-order accuracy [2, 3]. Curves of the pressure distribution over the surface are presented and a comparison is given with the results of experiments and the theory of regular (I) and Mach (II) reflections [4], which assume rectilinearity of the shocks and perpendicularity of the Mach wave to the reflecting surface. A formula describing the maximum reflection pressure on the body surface is presented. The results of computations supplement the known information about the singularities of the reflection process obtained experimentally and theoretically.

A monotonic difference scheme of second-order accuracy

A monotonic difference scheme of second-order accuracy

Numerical solution of a problem involving the interaction of a shock wave with a cylinder in a supersonic flow

We describe a difference scheme of second-order accuracy for calculating axially symmetric flows containing shock waves, and we apply our computational results to the interaction of a shock wave with the end of a cylinder moving at supersonic speed.