AN IMPLICIT-EXPLICIT UPWIND ALGORITHM FOR THE PARABOLIZED NAVIER-STOKES EQUATIONS

In this paper, a second-order implicit-explicit upwind algorithm has been developed for three-dimensional Parabolized Navier-Stokes(PNS) equations. The agreement between the results of the new upwind algorithm and those of the im- plicit upwind algorithm and its ability in marching a long distance along the stream- wise direction have been shown for the supersonic viscous flow past a sphere-cone body. The CPU time is greatly reduced.

A STABILIZED CRANK-NICOLSON MIXED FINITE VOLUME ELEMENT FORMULATION FOR THE NON-STATIONARY PARABOLIZED NAVIER-STOKES EQUATIONS

A time semi-discrete Crank-Nicolson （CN） formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite volume element （SCNMFVE） formu- lation based on two local Gaussian integrals and parameter-free with the second-order time accuracy is established directly from the time semi-discrete CN formulation so that it could avoid the discussion for semi-discrete SCNMFVE formulation with respect to spatial wriables and its theoretical analysis becomes very simple. Finally, the error estimates of SCNMFVE solutions are provided.

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.

A NEW HIGH-ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THREE-DIMENSIONAL PARABOLIC EQUATIONS

In this paper, a new three-level explicit difference scheme with high-order accuracy is proposed for solving three-dimensional parabolic equations. The stability condition is r = Delta t/Delta x(2) = Delta t/Delta gamma(2) = Delta t/Delta z(2) less than or equal to 1/4, and the truncation error is O(Delta t(2) + Delta x(4)).

Three-dimensional parabolic equation model for seismo-acoustic propagation: Theoretical development and preliminary numerical implementation

A three-dimensional（3D） parabolic equation（PE） model for sound propagation in a seismo-acoustic waveguide is developed in Cartesian coordinates, with x, y, and z representing the marching direction, the longitudinal direction, and the depth direction, respectively. Two sets of 3D PEs for horizontally homogenous media are derived by rewriting the 3D elastic motion equations and simultaneously choosing proper dependent variables. The numerical scheme is for now restricted to the y-independent bathymetry. Accuracy of the numerical scheme is validated, and its azimuthal limitation is analyzed. In addition, effects of horizontal refraction in a wedge-shaped waveguide and another waveguide with a polyline bottom are illustrated. Great efforts should be made in future to provide this model with the ability to handle arbitrarily irregular fluid-elastic interfaces.

An implicit upwind parabolized Navier-Stokes code for chemically nonequilibrium flows

The previously developed single-sweep parabolized Navier-Stokes （SSPNS） space marching code for ideal gas flows has been extended to compute chemically nonequilibrium flows. In the code, the strongly coupled set of gas dynamics, species conservation, and turbulence equations is integrated with the implicit lower-upper symmetric GaussSeidel （LU-SGS） method in the streamwise direction in a space marching manner. The AUSMPW＋ scheme is used to calculate the inviscid fluxes in the crossflow direction, while the conventional central scheme for the viscous fluxes. The k-g two-equation turbulence model is used. The revised SSPNS code is validated by computing the Burrows-Kurkov non-premixed H2/air supersonic combustion flows, premixed H2/air hypersonic combustion flows in a three-dimensional duct with a 15° compression ramp, as well as the hypersonic laminar chemically nonequilibrium air flows around two 10° half-angle cones. The results of these calculations are in good agreement with those of experiments, NASA UPS or Prabhu＇s PNS codes. It can be concluded that the SSPNS code is highly efficient for steady supersonic/ hypersonic chemically reaction flows when there is no large streamwise separation.

Research on the optimal dynamical systems of three-dimensional Navier-Stokes equations based on weighted residual

In this paper, the theory of constructing optimal dynamical systems based on weighted residual presented by Wu & Sha is applied to three-dimensional Navier-Stokes equations, and the optimal dynamical system modeling equations are derived. Then the multiscale global optimization method based on coarse graining analysis is presented, by which a set of approximate global optimal bases is directly obtained from Navier-Stokes equations and the construction of optimal dynamical systems is realized. The optimal bases show good properties, such as showing the physical properties of complex flows and the turbulent vortex structures, being intrinsic to real physical problem and dynamical systems, and having scaling symmetry in mathematics, etc.. In conclusion, using fewer terms of optimal bases will approach the exact solutions of Navier-Stokes equations, and the dynamical systems based on them show the most optimal behavior.

Three-Dimensional Numerical Model for Tsunami Propagation Passing Through An Obstacle

A three-dimensional numerical tsunami model is developed to analyze the nonlinear behavior of flow around obstacles with the Marker and Cell （MAC） method based on the Navier-Stokes equations. Tnrough a comparison with experimental data for the cases of dam break and solitary wave propagation, verification of the three-dimensional numerical model is given. Numerical experiment is performed for the analysis of the nonlinear behavior of flow around obstacles and compared with experimental data. The velocity and pressure around obstacles are presented with sufficient accuracy for tstmami propagation passing through an obstacle.

A Stabilized Crank-Nicolson Mixed Finite Element Method for Non-stationary Parabolized Navier-Stokes Equations

In this study, a time semi-discrete Crank-Nicolson （CN） formulation with second-order time accu- racy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite element （SCNMFE） formulation based on two local Gauss integrals and parameter- free with the second-order time accuracy is established directly from the time semi-discrete CN formulation. Thus, it could avoid the discussion for semi-discrete SCNMFE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finaly, the error estimates of SCNMFE solutions are provided.

Developments of parabolic equation method in the period of 2000–2016

Parabolic equation （PE） method is an efficient tool for modelling underwater sound propagation, particularly for problems involving range dependence. Since the PE method was first introduced into the field of underwater acoustics, it has been about 40 years, during which contributions to extending its capability has been continuously made. The most recent review paper surveyed the contributions made before 1999. In the period of 2000-2016, the development of PE method basically focuses on seismo-acoustic problems, three-dimensional problems, and realistic applications. In this paper, a review covering the contribution from 2000 to 2016 is given, and what should be done in future work is also discussed.

NUMERICAL SIMULATION OF THREE-DIMENSIONAL VORTEX DYNAMICS IN WAKE OF A CIRCULAR CYLINDER

The studies concerning the wake transition regime of the flow around a circular cylinder have drawn much attention in these years. Many experiments have been conducted for this problem but no accurate three-dimensional numerical simulations have hitherto been made. In this paper, a parallel procedure was developed to solve the three-dimensional Navier-Stokes equations on an SGI Origin3900 machine. Two different parallel strategies on this application were analyzed about their efficiency. It is found that the critical Reynolds number is 195, and the wake flow below this Reynolds number is purely two-dimensional one, while the Reynolds number goes beyond this critical point, the wake flow becomes unstable under three-dimensional small disturbances. The transition regime involves two modes of small-scale three-dimensional instability (modes A and B), depending on the regime of Reynolds number (Re). It is also found that the two different modes A and B exhibit different physical features of the flow. And many other important questions were addressed in this paper.

THREE-DIMENSIONAL FLOW AROUND TWO TANDEM CIRCULAR CYLINDERS WITH VARIOUS SPACING AT Re=220

The flow around two tandem circular cylinders was studied by a three-dimensional numerical simulation of the Navier-Stokes equations at Re=220 . The improved virtual boundary method was applied to model the no-slip boundary condition of the cylinders. The results show that as the spac ing ratio L/D≥4 , the three dimensionality occurs in the wake. When L/D≤3.5 the wake keeps a two-dimensional state at the Reynolds number Re=220 . The critical spacing for the appearance of three-dimensional instability obtained is at the range 3.5〈 L/D 〈 4, similar to the critical spacing found in two-dimensional case. Two sources of instability from upstream and downstream cylinder generate a complicat ed vortex structures in the wake, investigated by streamlines topology analysis in the streamwise plane. Many other interesting problems were also addressed in this paper.

Linear spatial instability analysis in 3D boundary layers using plane-marching 3D-LPSE

It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional （3D） boundary layers to predict the transition with the e^N method, especially when the boundary layer varies significantly in the spanwise direction. The 3D-linear parabolized stability equation （3D- LPSE） approach, a 3D extension of the two-dimensional LPSE （2D-LPSE）, is developed with a plane-marching procedure for investigating the instability of a 3D boundary layer with a significant spanwise variation. The method is suitable for a full Mach number region, and is validated by computing the unstable modes in 2D and 3D boundary layers, in both global and local instability problems. The predictions are in better agreement with the ones of the direct numerical simulation （DNS） rather than a 2D-eigenvalue problem （EVP） procedure. These results suggest that the plane-marching 3D-LPSE approach is a robust, efficient, and accurate choice for the local and global instability analysis in 2D and 3D boundary layers for all free-stream Mach numbers.

Incompressible Laminar Flow Over a Three-Dimensional Rectangular Cavity

This paper investigates unsteady incompressible flow over cavities. Previous research in incompressible cavity-flow has included flow inside and past a 2-dimensional cavity, and flow inside a 3-dimensional cavity, driven by a moving lid. The present research is focused on incompressible flow past a 3-dimensional open shallow cavity. This involves the complex interaction between the external flow and the re-circulating flow within the cavity. In particular, computation was performed on a 3-dimensional shallow rectangular cavity with a laminar boundary layer at the cavity and a Reynolds number of 5,000 and 10,000, respectively. A CFD approach, based on the unsteady Navier-Stokes equations for 3-dimensional incompressible flow, was used in the study. Typical results of the computation are presented. These results reveal the highly unsteady and complex vortical structures at high Reynolds numbers.

A Scalable Numerical Method for Simulating Flows Around High-Speed Train Under Crosswind Conditions

This paper presents a parallel Newton-Krylov-Schwarz method for the numerical simulation of unsteady flows at high Reynolds number around a high-speed train under crosswind.With a realistic train geometry,a realistic Reynolds number,and a realistic wind speed,this is a very challenging computational problem.Because of the limited parallel scalability,commercial CFD software is not suitable for supercomputers with a large number of processors.We develop a Newton-KrylovSchwarz based fully implicit method,and the corresponding parallel software,for the 3D unsteady incompressible Navier-Stokes equations discretized with a stabilized finite element method on very fine unstructured meshes.We test the algorithm and software for flows passing a train modeled after China’s high-speed train CRH380B,and we also compare our results with results obtained from commercial CFD software.Our algorithm shows very good parallel scalability on a supercomputer with over one thousand processors.

NUMERICAL SIMULATION OF TURBULENT SPOTS IN INCLINED OPEN-CHANNEL FLOW

The generation and evolution of turbulent spots in the open-channel flow are simulated numerically by using the Navier-Stokes equations. An effective numerical method with high accuracy and high resolution is developed. The fourth-order time splitting methods with high accuracy is proposed. Three-dimensional coupling difference methods are presented for the spatial discretization of the Poisson equation of pressure and Hemholtz equations of velocity, therefore, the fourth-order three-dimensional coupling central difference schemes are constituted. The fourth-order explicit upwind-biased compact difference schemes are designed to overcome the difficulty for the general higher-order central difference scheme which is inadaptable in the boundary neighborhood. The iterative algorithm and overall time marching is used to enhance efficiency. The method is applied in the numerical simulation of turbulent spots at various complex boundary conditions and flow domains. The generation and the developing process of turbulent spots are given, and the basic characteristics of turbulent spots are shown by simulating the evolution of the wall pulse in inclined open-channel flow.