The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions

We use artificial compressibility together with Richardson extrapolation in the Mach numberM as a method for solving the time dependent Navier-Stokes equation for very low Mach number flow and for incompressible flow. The question of what boundary conditions one should use for low Mach number flow, especially at inflow and outflow boundaries, is investigated theoretically, and boundary layer suppressing boundary conditions are derived. For the case of linearization around a constant flow we show that the low Mach number solution will converge with the rateO(M²) to the true incompressible solution, provided that we choose the boundary conditions correctly. The results of numerical calculations for the time dependent, nonlinear equations and for flow situations with time dependent inflow velocity profiles are presented. The convergence rateM ² to incompressible solution is numerically confirmed. It is also shown that using Richardson extrapolation toM ²= 0 in order to derive a solution with very small divergence can with good result be carried through withM ² as large as 0.1 and 0.05. As the time step in numerical methods must be chosen approximately such thatt · (i/(M x)–v/x ²) is in the stability region of the time stepping method, and asM ²=0.05 is sufficiently small to yield good results, the restriction on the time step due to the Mach number is not serious. Therefore the equations can be integrated very fast by explicit time stepping methods. This method for solving very low Mach number flow and incompressible flow is well suited to parallel processing.     

Velocity and temperature slip effects on squeezing flow of nanofluid between parallel disks in the presence of mixed convection

Velocity and temperature slip effects on squeezing flow of nanofluid between parallel disks in the presence of mixed convection is considered. Equations that govern the flow are transformed into a set of differential equations with the help of transformations. For the purpose of solution, homotopy analysis method is used. The BVPh2.0 package is utilized for the said purpose. Deviations in the velocity, temperature and the concentration profiles are depicted graphically. Mathematical expressions for skin friction coefficient, Nusselt and the Sherwood numbers are derived and the variations in these numbers are portrayed graphically. From the results obtained, we observed that the coefficient of skin friction increases with increase in Hartmann number M for the suction flow (A > 0), while in the blowing flow (A < 0) a fall is seen with increasing M. However, for rising values of velocity parameter β the effect of skin friction coefficient is opposite to that accounted for M. Variations in thermophoresis parameter N _T and thermal slip parameter γ give rise in Nusselt number for both the suction and injection at wall. For both the suction and injection at wall, Sherwood number gets a rise with growing values of Brownian motion parameter N _B, while a drop is seen in Sherwood number for increasing values of thermophoresis parameter N _T. For the sake of comparison, the same problem is also solved by employing a numerical scheme called Runge–Kutta–Fehlberg (RKF) method. Results thus obtained are compared with existing ones and are found to be in agreement.

     

Simulation of unsteady compressible flow in a channel with vibrating walls - Influence of the frequency

This study deals with the numerical solution of a 2D unsteady flow of a compressible viscous fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The channel is a simplified model of the glottal space in the human vocal tract and the flow can represent a model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract.The flow is described by the system of Navier–Stokes equations for laminar flows. The numerical solution is implemented using the finite volume method (FVM) and the predictor–corrector MacCormack scheme with Jameson artificial viscosity using a grid of quadrilateral cells. Due to the motion of the grid, the basic system of conservation laws is considered in the Arbitrary Lagrangian–Eulerian (ALE) form.The authors present the numerical simulations of flow fields in the channel, acquired from a program developed exclusively for this purpose. The numerical results for unsteady flows in the channel are presented for inlet Mach number M_∞ = 0.012, Reynolds number Re_∞ = 4.5 × 10³ and the wall motion frequency 20 and 100 Hz.     

A conservative pressure-correction method for flow at all speeds

A new fully conservative Mach-uniform staggered scheme is discussed. With this scheme one can compute flow with a Mach number ranging from the incompressible limit M↓0 up to supersonic flow M>1, with nearly uniform efficiency and accuracy. Earlier methods are based on a nonconservative discretisation of the energy equation. This results in small discrepancies in the computed shock speed for the Euler equations. The new method has similar Mach-uniform properties as the earlier methods, but is found to converge to the correct weak solution.     

A massively parallel hybrid scheme for direct numerical simulation of turbulent viscoelastic channel flow

This paper describes in detail a numerical scheme designed for direct numerical simulation (DNS) of turbulent drag reduction. The hybrid spatial scheme includes Fourier spectral accuracy in two directions and sixth-order compact finite differences for first and second-order wall-normal derivatives, while time marching can be up to fourth-order accurate. High-resolution and high-drag reduction viscoelastic DNS are made possible through domain decomposition with a two-dimensional MPI Cartesian grid alternatively splitting two directions of space (‘pencil’ decomposition). The resulting algorithm has been shown to scale properly up to 16384 cores on the Blue Gene/P at IDRIS–CNRS, France.Drag reduction is modeled for the three-dimensional wall-bounded channel flow of a FENE-P dilute polymer solution which mimics injection of heavy-weight flexible polymers in a Newtonian solvent. We present results for four high-drag reduction viscoelastic flows with friction Reynolds numbers Re_τ0 = 180, 395, 590 and 1000, all of them sharing the same friction Weissenberg number We_τ0 = 115 and the same rheological parameters. A primary analysis of the DNS database indicates that turbulence modification by the presence of polymers is Reynolds-number dependent. This translates into a smaller percent drag reduction with increasing Reynolds number, from 64% at Re_τ0 = 180 down to 59% at Re_τ0 = 1000, and a steeper mean current at small Reynolds number. The Reynolds number dependence is also visible in second-order statistics and in the vortex structures visualized with iso-surfaces of the Q-criterion.     

Analytical solution of gas lubricated slider microbearing

The analytical solution of a two-dimensional, isothermal, compressible gas flow in a slider microbearing is presented. A higher order accuracy of the solution is achieved by applying the boundary condition of Kn ² order for the velocity slip on the wall, together with the momentum equation of the same order (known as the Burnett equation). The analytical solution is obtained by the perturbation analysis. The order of all terms in continuum and momentum equations and in boundary conditions is evaluated by incorporating the exact relation between the Mach, Reynolds and Knudsen numbers in the modelling procedure. Low Mach number flows in microbearing with slowly varying cross-sections are considered, and it is shown that under these conditions the Burnett equation has the same form as the Navier–Stokes equation. Obtained analytical results for pressure distribution, load capacity and velocity field are compared with numerical solutions of the Boltzmann equation and some semi-analytical results, and excellent agreement is achieved. The model presented in this paper is a useful tool for the prediction of flow conditions in the microbearings. Also, its results are the benchmark test for the verifications of various numerical procedures.     

Maximum <Emphasis Type="Italic">k</Emphasis>-Splittable <Emphasis Type="Italic">s</Emphasis>,<Emphasis Type="Italic">t</Emphasis>-Flows

Given a graph with a source and a sink node, the NP-hard maximum k-splittable s,t-flow (M k SF) problem is to find a flow of maximum value from s to t with a flow decomposition using at most k paths. The multicommodity variant of this problem is a natural generalization of disjoint paths and unsplittable flow problems. Constructing a k-splittable flow requires two interdepending decisions. One has to decide on k paths (routing) and on the flow values for the paths (packing). We give efficient algorithms for computing exact and approximate solutions by decoupling the two decisions into a first packing step and a second routing step. Usually the routing is considered before the packing. Our main contributions are as follows: (i) We show that for constant k a polynomial number of packing alternatives containing at least one packing used by an optimal M k SF solution can be constructed in polynomial time. If k is part of the input, we obtain a slightly weaker result. In this case we can guarantee that, for any fixed ε>0, the computed set of alternatives contains a packing used by a (1−ε)-approximate solution. The latter result is based on the observation that (1−ε)-approximate flows only require constantly many different flow values. We believe that this observation is of interest in its own right. (ii) Based on (i), we prove that, for constant k, the M k SF problem can be solved in polynomial time on graphs of bounded treewidth. If k is part of the input, this problem is still NP-hard and we present a polynomial time approximation scheme for it.     

High Accuracy Schemes for DNS and Acoustics

High-accuracy schemes have been proposed here to solve computational acoustics and DNS problems. This is made possible for spatial discretization by optimizing explicit and compact differencing procedures that minimize numerical error in the spectral plane. While zero-diffusion nine point explicit scheme has been proposed for the interior, additional high accuracy one-sided stencils have also been developed for ghost cells near the boundary. A new compact scheme has also been proposed for non-periodic problems—obtained by using multivariate optimization technique. Unlike DNS, the magnitude of acoustic solutions are similar to numerical noise and that rules out dissipation that is otherwise introduced via spatial and temporal discretizations. Acoustics problems are wave propagation problems and hence require Dispersion Relation Preservation (DRP) schemes that simultaneously meet high accuracy requirements and keeping numerical and physical dispersion relation identical. Emphasis is on high accuracy than high order for both DNS and acoustics. While higher order implies higher accuracy for spatial discretization, it is shown here not to be the same for time discretization. Specifically it is shown that the 2nd order accurate Adams-Bashforth (AB)—scheme produces unphysical results compared to first order accurate Euler scheme. This occurs, as the AB-scheme introduces a spurious computational mode in addition to the physical mode that apportions to itself a significant part of the initial condition that is subsequently heavily damped. Additionally, AB-scheme has poor DRP property making it a poor method for DNS and acoustics. These issues are highlighted here with the help of a solution for (a) Navier–Stokes equation for the temporal instability problem of flow past a rotating cylinder and (b) the inviscid response of a fluid dynamical system excited by simultaneous application of acoustic, vortical and entropic pulses in an uniform flow. The last problem admits analytic solution for small amplitude pulses and can be used to calibrate different methods for the treatment of non-reflecting boundary conditions as well.     

Solution domain decomposition method for viscous incompressible flow

A solution domain decomposition method is developed for steady state solution of the biharmonic-based Navier-Stokes equations. It consists of a domain decomposition in conjunction with Chebyshev collocation for spatial discretization. The interactions between subdomains are effectively decoupled by means of a superposition of auxiliary solutions to yield a set of independent elementary problems which can be solved concurrently on multiprocessor computers. Assessments are carried out to a number of test problems including the two-dimensional steady flow in a driven square cavity. Illustrative examples indicate a good performance of the proposed methodology which does not affect the convergence and stability of the discretization scheme. Spectral accuracy is retained with absolute error decaying in an exponential fashion. The numerical solutions for the driven cavity compare favorably against previously published numerical results except for a slight overprediction in the vertical velocity component at Reynolds number of 400. TheC ³ continuity is speculated to be its cause.     

Computation of subcritical compressible flows

A finite element analysis and iterative solution of steady plane inviscid compressible flows is developed. Specific attention is directed to subcritical flows in which the nonlinear governing equation is elliptic, and to slightly supercritical mixed flows. The underlying variational theory for this nonlinear flow problem and a corresponding finite element formulation are presented. A Newton-Raphson iteration is introduced within this approximate analysis to provide efficient solution of subcritical flows and also slightly supercritical flows in which the nonlinear flow equation is of mixed type. The performance of the algorithm is studied with particular reference to the effect of a two-parameter scheme in which incident Mach number $\text{M}{}_{\mathrm{x}}$ and specific heat ratio γ may be varied. By thus regularizing the operator during the early iteration history, more efficient computations can be realized in the mildly-transonic flow regime of higher incident Mach numbers.     

Fast iterative algorithms for solving the minimization of curvature-related functionals in surface fairing

A number of successful variational models for processing planar images have recently been generalized to three-dimensional (3D) surface processing. With this new dimensionality, the amount of numerical computations to solve the minimization of such new 3D formulations naturally grows up dramatically. Though the need of computationally fast and efficient numerical algorithms able to process high resolution surfaces is high, much less work has been done in this area. Recently, a two-step algorithm for the fast solution of the total curvature model was introduced in Tasdizen, Whitaker, Burchard and Osher [Geometric surface processing via normal maps, ACM Trans. Graph. 22(4) (2003), pp. 1012–1033]. In this paper, we generalize and modify this algorithm to the solution of analogues of the mean curvature model of Droske and Martin Rumpf [A level set formulation for Willmore flow, Interfaces Free Bound. 6(3) (2004), pp. 361–378] and the Gaussian curvature model of Elsey and Esedo?lu [Analogue of the total variation denoising model in the context of geometry processing, Multiscale Model. Simul. 7(4) (2009), pp. 1549–1573]. Numerical experiments are shown to illustrate the good performance of the algorithms and test results.     

High Accuracy Compact Schemes and Gibbs' Phenomenon

Compact difference schemes have been investigated for their ability to capture discontinuities. A new proposed scheme (Sengupta, Ganerwal and De (2003). J. Comp. Phys. 192(2), 677.) is compared with another from the literature Zhong (1998). J. Comp. Phys. 144, 622 that was developed for hypersonic transitional flows for their property related to spectral resolution and numerical stability. Solution of the linear convection equation is obtained that requires capturing discontinuities. We have also studied the performance of the new scheme in capturing discontinuous solution for the Burgers equation. A very simple but an effective method is proposed here in early diagnosis for evanescent discontinuities. At the discontinuity, we switch to a third order one-sided stencil, thereby retaining the high accuracy of solution. This produces solution with vastly reduced Gibbs' phenomenon of the solution. The essential causes behind Gibbs' phenomenon is also explained.     

Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines

In this work, we propose a numerical scheme to obtain approximate solutions of generalized Burgers–Fisher and Burgers–Huxley equations. The scheme is based on collocation of modified cubic B-spline functions and is applicable for a class of similar diffusion–convection–reaction equations. We use modified cubic B-spline functions for space variable and for its derivatives to obtain a system of first-order ordinary differential equations in time. We solve this system by using SSP-RK54 scheme. The stability of the method has been discussed and it is shown that the method is unconditionally stable. The approximate solutions have been computed without using any transformation or linearization. The proposed scheme needs less storage space and execution time. The test problems considered by the different researchers have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in the literature. The scheme is simple as well as computationally efficient. The scheme provides approximate solution not only at the grid points but also at any point in the solution range.     

Implementation of quadratic upstream interpolation schemes for solute transport into HYDRUS-1D

Numerical solution of the advection–dispersion equation, used to evaluate transport of solutes in porous media, requires discretization schemes for space and time stepping. We examine use of quadratic upstream interpolation schemes QUICK, QUICKEST, and the total variation diminution scheme ULTIMATE, and compare these with UPSTREAM and CENTRAL schemes in the HYDRUS-1D model. Results for purely convective transport show that quadratic schemes can reduce the oscillations compared to the CENTRAL scheme and numerical dispersion compared to the UPSTREAM scheme. When dispersion is introduced all schemes give similar results for Peclet number Pe < 2. All schemes show similar behavior for non-uniform grids that become finer in the direction of flow. When grids become coarser in the direction of flow, some schemes produce considerable oscillations, with all schemes showing significant clipping of the peak, but quadratic schemes extending the range of stability tenfold to Pe < 20. Similar results were also obtained for transport of a non-linear retarded solute transport (except the QUICK scheme) and for reactive transport (except the UPSTREAM scheme). Analysis of transient solute transport show that all schemes produce similar results for the position of the infiltration front for Pe = 2. When Pe = 10, the CENTRAL scheme produced significant oscillations near the infiltration front, compared to only minor oscillations for QUICKEST and no oscillations for the ULTIMATE scheme. These comparisons show that quadratic schemes have promise for extending the range of stability in numerical solutions of solute transport in porous media and allowing coarser grids.     

Lattice Boltzmann simulation of lid-driven flow in trapezoidal cavities

In this paper, an incompressible lattice Bhatnagar–Gross–Krook (LBGK) model proposed by Guo et al. is used to simulate lid-driven flow in a two-dimensional isosceles trapezoidal cavity. Due to the complex boundary of the trapezoidal cavity, here the extrapolation scheme proposed by Guo et al. is used to treat curved boundary. In our numerical simulations, the effects of the Reynolds number (Re) and the top angle θ on the strength, center position and number of vortices in the isosceles trapezoidal cavities are studied. Re is varied from 100 to 15,000, and the top angle θ ranges from 50 to 90. Numerical results show that, as Re increases, the phenomena in the cavity become more and more complex, and the number of the vortexes increases. We also found that the vortex near the bottom wall breaks up into two smaller vortices as θ increases up to a critical value. Furthermore, as Re is increased, the flow in the cavity undergoes a complex transition (from steady to the periodic flow, and finally to the chaotic flow). At last, the scope of critical Re for flow transition from steady to periodic state, and from periodic to chaotic state is presented for different top angles θ.     

Improved PID controller design for unstable time delay processes based on direct synthesis method and maximum sensitivity

In this paper, a proportional-integral-derivative controller in series with a lead-lag filter is designed for control of the open-loop unstable processes with time delay based on direct synthesis method. Study of the performance of the designed controllers has been carried out on various unstable processes. Set-point weighting is considered to reduce the undesirable overshoot. The proposed scheme consists of only one tuning parameter, and systematic guidelines are provided for selection of the tuning parameter based on the peak value of the sensitivity function (M_s). Robustness analysis has been carried out based on sensitivity and complementary sensitivity functions. Nominal and robust control performances are achieved with the proposed method and improved closed-loop performances are obtained when compared to the recently reported methods in the literature.     

On the Performance of Permutation Codes for Multi-User Communication

Permutation coding for multi-user communication schemes that originate from the Fast Frequency Hopping/Multiple Frequency Shift Keying modulation is investigated. Each sender is either passive or sends some signal formed as the concatenation of M elementary signals having M different specified frequencies. There is also a jammer, who can introduce disturbances. A single disturbance is either sending the signal that contains all M frequencies at a certain time instant or sending some elementary signal at all time instants. Each receiver receives a vector of M sets, where a set at each time instant contains a fixed frequency if and only if the corresponding elementary signal was sent by either some sender or the jammer. The task of the receiver is to uniquely decode the message of his sender. We present regular constructions of permutation codes for this scheme given the following parameters: the number of frequencies, number of pairs (sender, receiver), number of messages per sender, and maximum number of disturbances of the jammer.     

Optimal transfer,ordering and payment policies for joint supplier–buyer inventory model with price-sensitive trapezoidal demand and net credit

This study aims at formulating an integrated supplier–buyer inventory model when market demand is variable price-sensitive trapezoidal and the supplier offers a choice between discount in unit price and permissible delay period for settling the accounts due against the purchases made. This type of trade credit is termed as ‘net credit’. In this policy, if the buyer pays within offered time M₁, then the buyer is entitled for a cash discount; otherwise the full account must be settled by the time M₂; where M₂ > M₁ ? 0. The goal is to determine the optimal selling price, procurement quantity, number of transfers from the supplier to the buyer and payment time to maximise the joint profit per unit time. An algorithm is worked out to obtain the optimal solution. A numerical example is given to validate the proposed model. The managerial insights based on sensitivity analysis are deduced.     

Polynomial time-marching for nonperiodic boundary value problems

A polynomial interpolation time-marching technique can efficiently provide balanced spectral accuracy in both the space and time dimensions for some PDEs. The Newton-form interpolation based on Fejér points has been successfully implemented to march the periodic Fourier pseudospectral solution in time. In this paper, this spectrally accurate time-stepping technique will be extended to solve some typical nonperiodic initial boundary value problems by the Chebyshev collocation spatial approximation. Both homogeneous Neumann and Dirichlet boundary conditions will be incorporated into the time-marching scheme. For the second order wave equation, besides more accurate timemarching, the new scheme numerically has anO(1/N ²) time step size limitation of stability, much larger thanO(1/N ⁴) stability limitation in conventional finitedifference time-stepping, Chebyshev space collocation methods.     

Curvature Based Image Registration

A fully automated, non-rigid image registration algorithm is presented. The deformation field is found by minimizing a suitable measure subject to a curvature based constraint. It is a well-known fact that non-rigid image registration techniques may converge poorly if the initial position is not sufficiently near to the solution. A common approach to address this problem is to perform a time consuming rigid pre-registration step. In this paper we show that the new curvature registration not only produces accurate and smooth solutions but also allows for an automatic rigid alignment. Thus, in contrast to other popular registration schemes, the new method no longer requires a pre-registration step. Furthermore, we present an implementation of the new scheme based on the numerical solution of the underlying Euler-Lagrange equations. The real discrete cosine transform is the backbone of our implementation and leads to a stable and fast O(N log N) algorithm, where N denotes the number of voxels. Finally, we report on some numerical test runs.