Finite element analysis of viscous,incompressible fluid flow: Part 1 : Basic methodology

A numerical procedure is developed for the analysis of general two-dimensional flows of viscous, incompressible fluids using the finite element method. The partial differential equations describing the continuum motion of the fluid are discretized by using an integral energy balance approach in conjunction with the finite element approximation. The nonlinear algebraic equations resulting from the discretization process are solved using a Picard iteration technique.A number of computational procedures are developed that allow significant reductions to be made in the computational effort required for the analysis of many flow problems. These techniques include a coarse-to-fine-mesh rezone procedure for the detailed study of regions of particular interest in a flow field and a special finite element to model far-field regions in external flow problems.     

Shock Capturing Artificial Dissipation for High-Order Finite Difference Schemes

We study 2nd-, 4th-, 6th- and 8th-order accurate finite difference schemes approximating systems of conservation laws. Our goal is to utilize the high order of accuracy of the schemes for approximating complicated flow structures and add suitable diffusion operators to capture shocks. We choose appropriate viscosity terms and prove non-linear entropy stability. In the scalar case, entropy stability enables us to prove convergence to the unique entropy solution. Moreover, a limiter function that localizes the effect of the dissipation around discontinuities is derived. The resulting scheme is entropy stable for systems, and also converges to the entropy solution in the scalar case. We present a number of numerical experiments in order to demonstrate the robustness and accuracy of our scheme. The set of examples consists of a moving shock solution to the Burgers’ equation, a solution to the Euler equations that consists of a rarefaction and two contact discontinuities and a shock/entropy wave solution to the Euler equations (Shu’s test problem). Furthermore, we use the limited scheme to compute the solution to the linear advection equation and demonstrate that the limiter quickly vanishes for smooth flows and design/high-order of accuracy is retained. The numerical results in all experiments were very good. We observe a remarkable gain in accuracy when the order of the scheme is increased.     

A treatment of discontinuities for nonlinear systems with linearly degenerate fields

In this paper we propose an artificial compression technique to avoid the numerical diffusion that standard numerical methods present in contact discontinuities. The main idea is to replace contact discontinuities by shocks. For nonlinear 1D systems we replace locally linearly degenerate fields by genuinely nonlinear fields, in such a way the solution does not vary. We apply this technique to a family of numerical schemes and we deduce that this can be seen as a discretization of the system modified by a new term, when we are in a jump of a contact discontinuity. We have also extended this technique for the multidimensional case. We prove by applying the artificial compression technique that the numerical scheme is stable under the same CFL condition. We also present different numerical schemes: Sod’s problem for 1D Euler equations, transport of a discontinuity, a stationary contact discontinuity and in the multidimensional case the transversal transport of two different geometries. We observe that in all cases the numerical diffusion is reduced.     

A pressure-enthalpy finite element model for simulating hydrothermal reservoirs

A numerical model is presented for simulating single or two-phase flow and energy transport in hydrothermal reservoirs. The model is formulated via two non-linear equations for fluid pressure and enthalpy. Both equations are solved simultaneously using a new finite element technique which employs asymmetric weighting functions to overcome numerical oscillation. Non-linearity is treated by a modified Newton-Raphson scheme which takes into account derivative discontinuities in the non-linear coefficients. This scheme also treats unknown flux boundary conditions inplicitly, thus allowing larger time steps to be taken without inducing instability. The proposed model is applied to two test examples involving one-dimensional flow in both hot water and steam dominated reservoirs. Results indicate that the numerical technique presented is efficient and the model can be used to simulate both types of reservoirs.     

Modified kinetic flux vector splitting (m-KFVS) method for compressible flows

To resolve many flow features accurately, like accurate capture of suction peak in subsonic flows and crisp shocks in flows with discontinuities, to minimise the loss in stagnation pressure in isentropic flows or even flow separation in viscous flows require an accurate and low dissipative numerical scheme. The first order kinetic flux vector splitting (KFVS) method has been found to be very robust but suffers from the problem of having much more numerical diffusion than required, resulting in inaccurate computation of the above flow features. However, numerical dissipation can be reduced by refining the grid or by using higher order kinetic schemes. In flows with strong shock waves, the higher order schemes require limiters, which reduce the local order of accuracy to first order, resulting in degradation of flow features in many cases. Further, these schemes require more points in the stencil and hence consume more computational time and memory. In this paper, we present a low dissipative modified KFVS (m-KFVS) method which leads to improved splitting of inviscid fluxes. The m-KFVS method captures the above flow features more accurately compared to first order KFVS and the results are comparable to second order accurate KFVS method, by still using the first order stencil.     

Numerical study of the flow due to a cylindrical implosion

The development of the flow, produced by the sudden sublimation of a hollow cylindrical tube, is studied numerically. All discontinuities in the field (shocks, contact and gradient discontinuities) are fitted and treated as boundaries. Special attention has been given to a proper handling of boundary conditions and to the treatment of discontinuity intersections. The singularity, occurring when a shock collapses on the symmetry axis, is avoided by assuming that the axis is surrounded by a solid core. It is found that close to r = 0, the flow obeys a similarity law. In this region, good agreement is found between the calculated shock motion and its theoretical prediction.     

Modeling of self-oscillations and a search for new self-oscillatory flows

A possible mechanism of self-oscillations in flows with shock waves and contact discontinuities is studied. Flows are investigated in the heliosphere, near a blunt cone in an inhomogeneous stream, and in the vicinity of a blunt cylinder with an outflowing supersonic jet, which is supposed to be, according to this mechanism, of a self-oscillatory nature. Two-dimensional Reynolds equations with algebraic turbulent viscosity are solved by the implicit third order Runge-Kutta scheme. The results of numerical studies are presented.     

Efficient Parallelization of a Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Sub-cells

We present a shock capturing procedure for high order Discontinuous Galerkin methods, by which shock regions are refined in sub-cells and treated by finite volume techniques. Hence, our approach combines the good properties of the Discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations. Due to the sub-cell approach the interior resolution on the Discontinuous Galerkin grid cell is nearly preserved and the number of degrees of freedom remains the same. This structure allows the interpretation of the data either as DG solution or as finite volume solution on the subgrid. In this paper we explain the efficient implementation of this coupled method on massively parallel computers and show some numerical results.     

Solution to the Riemann problem in a one-dimensional magnetogasdynamic flow

The Riemann problem for a quasilinear hyperbolic system of equations governing the one-dimensional unsteady flow of an inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field, is solved approximately. This class of equations includes as a special case the Euler equations of gasdynamics. It has been observed that in contrast to the gasdynamic case, the pressure varies across the contact discontinuity. The iterative procedure is used to find the densities between the left acoustic wave and the right contact discontinuity and between the right contact discontinuity and the right acoustic wave, respectively. All other quantities follow directly throughout the (x, t)-plane, except within rarefaction waves, where an extra iterative procedure is used along with a Gaussian quadrature rule to find particle velocity; indeed, the determination of the particle velocity involves numerical integration when the magneto-acoustic wave is a rarefaction wave. Lastly, we discuss numerical examples and study the solution influenced by the magnetic field.     

A new all-speed flux scheme for the Euler equations

Since being proposed, the HLLEM-type schemes have been widely used because they are with high discontinuity resolutions and can be easily applied to the other system of hyperbolic conservation law. In this paper, we conduct theoretical analyses on the HLLE-type schemes’ performances at low speeds. By realizing that the excessive numerical dissipations corresponding to the velocity-difference terms of the momentum equations make these schemes incapable of obtaining physical solutions at low speeds, we adopt the function $g$ to control such dissipation. Also, we borrow the HLLEMS scheme’s construction and damp the shear waves in the vicinity of the shock to avoid the shock anomaly’s appearance. The moving contact discontinuity case and the Sod shock tube case show that the HLLEMS-AS scheme we propose in this paper can capture contact discontinuities and shocks as sharply as HLLEMS scheme. The Quirk’s odd–even test case and the hypersonic inviscid flow over a cylinder case demonstrate that HLLEMS-AS is robust against the shock anomaly. The inviscid low-speed flow around the NACA0012 airfoil case indicates that HLLEMS-AS is with a high resolution at low speeds. The turbulent flow past a backward facing step case demonstrates the shear wave capturing ability of the HLLEMS-AS scheme. These properties suggest that HLLEMS-AS is promising to be widely used in both cases of low speed and high speed.     

Metric-discontinuous zonal grid calculations using the Osher scheme

Computations on zonal grids—in particular, grids with metric discontinuities resulting from the interspersion of highly clustered regions with coarse regions—are possible using a fully conservative form of the Osher upwind scheme. These zonal grids can result from an abrupt clustering of points near solution discontinuities or near other flow features that require improved resolution. The zonal approach is shown to capture shocks with almost “shock-fitting” quality but with minimal effort. Results for inviscid flow, including quasi-one-dimensional nozzle flow, supersonic flow over a cylinder, and blast-wave diffraction by a ramp, are presented. These calculations demonstrate the powerful capabilities of the Osher scheme used in conjunction with zonal grids in simulating flow fields with complex shock patterns.     

A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients

In this work, a model for shallow water flows that accounts for the effects of horizontal density fluctuations is presented and derived. While the density is advected by the flow, a two-way feedback between the density gradients and the time evolution of the fluid is ensured through the pressure and source terms in the momentum equations. The model can be derived by vertically averaging the Euler equations while still allowing for density fluctuations in horizontal directions. The approach differs from multi-layer shallow water flows where two or more layers are considered, each of them having their own depth, velocity and constant density. A Roe-type upwind scheme is developed and the Roe matrices are computed systematically by going from the conservative to the quasi-linear form at a discrete level. Properties of the model are analyzed. The system is hyperbolic with two shock-wave families and a contact discontinuity associated to interfaces of regions with density jumps. This new field is degenerate with pressure and velocity as the corresponding Riemann invariants. We show that in some parameter regimes numerically recognizing such invariants across contact discontinuities is important to correctly compute the flow near those interfaces. We present a numerical algorithm that correctly captures all waves with a hybrid strategy. The method integrates the Riemann invariants near contact discontinuities and switches back to the conserved variables away from it to properly resolve shock waves. This strategy can be applied to any numerical scheme. Numerical solutions for a variety of tests in one and two dimensions are shown to illustrate the advantages of the strategy and the merits of the scheme.     

Euler’s predictor–corrector technique for solving the depth-integrated shallow water equations

Euler’s predictor–corrector technique combined with finite analysis method is applied to solve 2D advection–diffusion shallow water equations. In this algorithm the momentum equations are calculated by the finite analysis method based on a single mesh, while the continuity equation is solved by Euler’s predictor–corrector technique. To verify the performance of this approach, the simulation of tidal flow in the Huangpu estuary is carried out. The numerical results are found to be consistent with the field results, implying that this proposed method is effective and applicable.     

A Runge-Kutta discontinuous Galerkin method for the Euler equations

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. Like the traditional gas-kinetic schemes, our proposed RKDG method does not need to use the characteristic decomposition or the Riemann solver in computing the numerical flux at the surface of the finite elements. The integral term containing the non-linear flux can be computed exactly at the microscopic level. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods give higher resolution in solving problems with smooth solutions. Moreover, shock and contact discontinuities can be well captured by using the proposed methods.     

Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-Dimensional Viscous Compressible Flow in a Shock Tube Using Entropy and Kinetic Energy Preserving Schemes

This paper follows up on the author’s recent paper “The Construction of Discretely Conservative Finite Volume Schemes that also Globally Conserve Energy or Enthalpy”. In the case of the gas dynamics equations the previous formulation leads to an entropy preserving (EP) scheme. It is shown in the present paper that it is also possible to construct the flux of a conservative finite volume scheme to produce a kinetic energy preserving (KEP) scheme which exactly satisfies the global conservation law for kinetic energy. A proof is presented for three dimensional discretization on arbitrary grids. Both the EP and KEP schemes have been applied to the direct numerical simulation of one-dimensional viscous flow in a shock tube. The computations verify that both schemes can be used to simulate flows with shock waves and contact discontinuities without the introduction of any artificial diffusion. The KEP scheme performed better in the tests.     

Analysis of a high-order finite difference detector for discontinuities

High-order finite difference discontinuity detectors are essential for the location of discontinuities on discretized functions, especially in the application of high-order numerical methods for high-speed compressible flows for shock detection. The detectors are used mainly for switching between numerical schemes in regions of discontinuity to include artificial dissipation and avoid spurious oscillations. In this work a discontinuity detector is analysed by the construction of a piecewise polynomial function that incorporates jump discontinuities present on the function or its derivatives (up to third order) and the discussion on the selection of a cut-off value required by the detector. The detector function is also compared with other discontinuity detectors through numerical examples.     

On the solution of the unsteady Navier- Stokes equations including multicomponent finite rate chemistry

An explicit-implicit time-dependent finite difference technique is presented which has been successfully implemented for the solution of the unsteady Navier-Stokes equations including multicomponent finite rate chemistry. Stability restrictions encountered in explicit schemes due to the variable relaxation times associated with the finite rate chemical reactions are eliminated. As a result, the stability of the coupled Navier-Stokes and species conservatiion finite difference equations is shown to be governed only by the hydrodynamic stability cri- terion. The viscous nonequilibrium flow about a blunt axisymmetric nosetip at high Mach number and low Reynolds number, where the viscous shock layer and shock transition zone are merged, was computed using this time-dependent numerical scheme. Solutions along the stagnation streamline are compared with corresponding solutions obtained by Dellinger using the thin layer approximation of the Navier-Stokes equations.     

Numerical studies of the flow around a circular cylinder by a finite element method

Numerical solutions of the steady, incompressible, viscous flow past a circular cylinder are presented for Reynolds numbers R ranging from 1 to 100. The governing Navier-Stokes equations in the form of a single, fourth order differential equation for stream function and the boundary conditions are replaced by an equivalent variational principle. The numerical method is based on a finite element approximation of this principle. The resulting non-linear system is solved by the Newton-Raphson process. The pressure field is obtained from a finite element solution of the Poisson equation once the stream function is known. The results are compared with those determined by other numerical techniques and experiments. In particular, the discussion is concerned with the development of the closed wake with Reynolds number, and the tendency of R ≥ 40 flow toward instability.     

Theory and implementation of a semi-implicit finite element method for viscous incompressible flow

A numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations is derived based on the operator-splitting technique. This operator split allows separate operations on each of the variable fields to enable pressure-velocity coupling. Discretizations of the equations are formed on a nonstaggered finite element mesh and the solutions are obtained in a time-marching fashion. Several benchmark problems, including a standing vortex problem, a lid-driven cavity and a flow around a rectangular cylinder, are studied to demonstrate the robustness and accuracy of the present algorithm.     

From static approximations to full-wave analysis: The analysis of planar line discontinuities

A comparison of different methods for the characterization of planar line discontinuities with emphasis on microstrip discontinuities is given, thereby describing the present state of the art in this area of microwave circuit CAD. Quasi-static equivalent circuit models, the magnetic wall waveguide model of microstrip discontinuities, full-wave analysis techniques like the orthogonal series expansion and mode matching techniques, the spectral domain analysis technique using roof-top functions, and the finite-difference time-domain technique are described briefly and compared with respect to their numerical efficiency and their capabilities.