Compact upwind algorithms for viscous compressible flow calculations

A family of high-order accurate compact upwind difference operators have been used together with the split fluxes of the KFVS (Kinetic Flux Vector Splitting) scheme to obtain high order semi-discretizations of nonlinear convective terms in 2-D Navier-Stokes equations governing viscous compressible flows. A TVD multi-stage Runge-Kutta time stepping scheme is used to compute steady states for selected transonic/supersonic inviscid and viscous flow problems which indicate the higher accuracy and low diffusion realizable in such schemes. This work was done as part of the Indo-Russian Long-Term integrated programme and was supported by the Department of Science & Technology, Government of India.     

Boltzmann schemes for continuum gas dynamics

Many problems arising in the aerodynamic design of aerospace vehicles require the numerical solution of the Euler equations of gas dynamics. These are nonlinear partial differential equations admitting weak solutions such as shock waves and constructing robust numerical schemes for these equations is a challenging task. A new line of research called Boltzmann or kinetic schemes discussed in the present paper exploits the connection between the Boltzmann equation of the kinetic theory of gases and the Euler equations for inviscid compressible flows. Because of this connection, a suitable moment of a numerical scheme for the Boltzmann equation yields a numerical scheme for the Euler equations. This idea called the “moment method strategy” turns out to be an extremely rich methodology for developing robust numerical schemes for the Euler equations. The richness is demonstrated by developing a variety of kinetic schemes such as kinetic numerical method, kinetic flux vector splitting method, thermal velocity based splitting, multidirectional upwind method and least squares weak upwind scheme. A 3-D time-marching Euler code calledbheema based on the kinetic flux vector splitting method and its variants involving equilibrium chemistry have been developed for computing hypersonic reentry flows. The results obtained from the codebheema demonstrate the robustness and the utility of the kinetic flux vector splitting method as a design tool in aerodynamics. The work presented in this paper is based on the research work done by several graduate students at our laboratory and collaborators from research and development organizations within the country.     

A cut‐cell non‐conforming Cartesian mesh method for compressible and incompressible flow

This paper details a multigrid‐accelerated cut‐cell non‐conforming Cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow. This is done via a single equation set that describes sub‐, trans‐, and supersonic flows. Cut‐cell technology is developed to furnish body‐fitted meshes with an overlapping mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. Spatial discretization is effected via an edge‐based vertex‐centred finite volume method. An alternative dual‐mesh construction strategy, similar to the cell‐centred method, is developed. Incompressibility is dealt with via an artificial compressibility algorithm, and stabilization achieved with artificial dissipation. In compressible flow, shocks are captured via pressure switch‐activated upwinding. The solution process is accelerated with full approximation storage (FAS) multigrid where coarse meshes are generated automatically via a volume agglomeration methodology. This is the first time that the proposed discretization and solution methods are employed to solve a single compressible–incompressible equation set on cut‐cell Cartesian meshes. The developed technology is validated by numerical experiments. The standard discretization and alternative methods were found equivalent in accuracy and computational cost. The multigrid implementation achieved decreases in CPU time of up to one order of magnitude. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical Methods for Accurate Computation of Design Sensitivities for Two-Dimensional Navier-Stokes Problems

Gradient computations can be a limiting factor in algorithm efficiency and accuracy for optimization based design. In this paper, we present three parameterized flow problems and consider the evaluation of state sensitivities both theoretically and numerically. Existence and uniqueness results are given for the sensitivities of a specific group of two-dimensional Navier-Stokes problems. We then turn our attention to obtaining numerical approximations to state sensitivities. We show convergence of our numerical sensitivities using a problem having an exact solution. Next, two problems, flow around a cylinder and flow over a bump, are used to evaluate several computational schemes. In particular, a local projection scheme for improved state derivative approximations and the use of an adaptive finite element scheme are shown to be important techniques for obtaining accurate sensitivity approximations. Lastly, we evaluate the impact of these computational techniques on cost function and gradient calculation.     

On the accuracy of finite volume and discontinuous Galerkin discretizations for compressible flow on unstructured grids

This paper presents a comparison between two high‐order methods. The first one is a high‐order finite volume (FV) discretization on unstructured grids that uses a meshfree method (moving least squares (MLS)) in order to construct a piecewise polynomial reconstruction and evaluate the viscous fluxes. The second method is a discontinuous Galerkin (DG) scheme. Numerical examples of inviscid and viscous flows are presented and the solutions are compared. The accuracy of both methods, for the same grid resolution, is similar, although the DG scheme requires a larger number of degrees of freedom than the FV–MLS method. Copyright © 2009 John Wiley & Sons, Ltd.     

Stable and time-dissipative finite element methods for the incompressible Navier–Stokes equations in advection dominated flows

This paper examines a new Galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1-D scalar advection–diffusion and that leads to non-oscillatory solutions as the streamline upwinding algorithms for 2-D scalar advection–diffusion and incompressible Navier–Stokes. This method retains the satisfaction of the Babuska–Brezzi condition and, thus, leads to optimal performance in the incompressible limit. This method, when, combined with the recently proposed linear unconditionally stable algorithms of Simo and Armero (1993), yields a method for solution of the incompressible Navier–Stokes equations ideal for either diffusive or advection-dominated flows. Examples from scalar advection–diffusion and the solution of the incompressible Navier–Stokes equations are presented.     

Status and future directions for turbulence modelling

Progress in the development of a hierarchy of turbulence models for Reynolds-averaged Navier-Stokes codes used in aerodynamic applications is discussed. Although steady progress is demonstrated, transfer of the modelling technology has not kept pace with the development and demands of the CFD tools. An examination of the process of model development leads to recommendations for a better process involving close coordination between modellers, CFD developers and application engineers. In instances where the process is changed and cooperation enhanced, timely transfer is realized. A turbulence modelling information database is proposed to refine the process and open it to greater participation among modelling and CFD practitioners. The citations in this paper are not in our usual format     

Meshless point collocation for the numerical solution of Navier-Stokes flow equations inside an evaporating sessile droplet

The Navier-Stokes flow inside an evaporating sessile droplet is studied in the present paper, using sophisticated meshfree numerical methods for the computation of the flow field. This problem relates to numerous modern technological applications, and has attracted several analytical and numerical investigations that expanded our knowledge on the internal microflow during droplet evaporation. Two meshless point collocation methods are applied here to this problem and used for flow computations and for comparison with analytical and more traditional numerical solutions. Particular emphasis is placed on the implementation of the velocity-correction method within the meshless procedure, ensuring the continuity equation with increased precision. The Moving Least Squares (MLS) and the Radial Basis Function (RBF) approximations are employed for the construction of the shape functions, in conjunction with the general framework of the Point Collocation Method (MPC). An augmented linear system for imposing the coupled boundary conditions that apply at the liquid-gas interface, especially the zero shear-stress boundary condition at the interface, is presented. Computations are obtained for regular, Type-I embedded nodal distributions, stressing the positivity conditions that make the matrix of the system stable and convergent. Low Reynolds number (Stokes regime), and elevated Reynolds number (Navier-Stokes regime) conditions have been studied and the solutions are compared to those of analytical and traditional CFD methods. The meshless implementation has shown a relative ease of application, compared to traditional mesh-based methods, and high convergence rate and accuracy.     

PVU and wave-particle splitting schemes for Euler equations of gas dynamics

A new way of flux-splitting, termed as the wave-particle splitting is presented for developing upwind methods for solving Euler equations of gas dynamics. Based on this splitting, two new upwind methods termed as Acoustic Flux Vector Splitting (AFVS) and Acoustic Flux Difference Splitting (AFDS) methods are developed. A new Boltzmann scheme, which closely resembles the wave-particle splitting, is developed using the kinetic theory of gases. This method, termed as Peculiar Velocity based Upwind (PVU) method, uses the concept of peculiar velocity for upwinding. A special feature of all these methods is that the unidirectional and multidirectional parts of the flux vector are treated separately. Extensive computations done using these schemes demonstrate the soundness of the ideas.     

Analysis and control of dynamic stall phenomenon using Navier-Stokes formulation involving vorticity,stream function and circulation

An unsteady Navier-Stokes (ns) analysis is developed for studying flow past a maneuvering body. The present inclusion ofcirculation in the earlierns analysis of the authors makes it feasible,for the first time, to accurately simulate the asymptotic far-field boundary condition. In the overall analysis, a clustered conformal mesh withC-grid topology is used, with the governing differential equations being solved using the implicitadi-bge technique. The effects of grid size and clustering on the flow solution, and the effect of grid stretching on the far-field solution, are studied using the flow configuration with Re=45,000 and constantrate pitch-up motion, with . The results obtained for this case compare very satisfactorily with available experimental data. Other numerical results are also used for carefully validating the analysis developed here. Anactive control strategy, consisting of modulated suction/injection at the airfoil surface, is studied, and provides satisfactory control of the unsteady separation process and, hence, of the dynamic stall vortex. This research is supported, in part, byafosr Grant Nos. 90-0249 and F49620-92-J-0292, and Ohio Supercomputer Center Grant No.pes080.     

Generalized Robin–Neumann explicit coupling schemes for incompressible fluid‐structure interaction: Stability analysis and numerics

We introduce a new class of explicit coupling schemes for the numerical solution of fluid‐structure interaction problems involving a viscous incompressible fluid and an elastic structure. These methods generalize the arguments reported in [Comput. Methods Appl. Mech. Engrg., 267:566–593, 2013, Numer. Math., 123(1):21–65, 2013] to the case of the coupling with thick‐walled structures. The basic idea lies in the derivation of an intrinsic interface Robin consistency at the space semi‐discrete level, using a lumped‐mass approximation in the structure. The fluid–solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface. Based on these methods, a new, parameter‐free, Robin–Neumann iterative procedure is also proposed for the partitioned solution of implicit coupling. A priori energy estimates, guaranteeing the stability of the schemes and the convergence of the iterative procedure, are established within a representative linear setting. The accuracy and performance of the methods are illustrated in several numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Explicit strategies for incompressible fluid‐structure interaction problems: Nitsche type mortaring versus Robin–Robin coupling

We discuss explicit coupling schemes for fluid‐structure interaction problems where the added mass effect is important. In this paper, we show the close relation between coupling schemes by using Nitsche's method and a Robin–Robin type coupling. In the latter case, the method may be implemented either using boundary integrals of the stresses or the more conventional discrete lifting operators. Recalling the explicit method proposed in Comput. Methods Appl. Mech. Engrg. 198(5‐8):766–784, 2009, we make the observation that this scheme is stable under a hyperbolic type CFL condition, but that optimal accuracy imposes a parabolic type CFL conditions because of the splitting error. Two strategies to enhance the accuracy of the coupling scheme under the hyperbolic CFL‐condition are suggested, one using extrapolation and defect‐correction and one using a penalty‐free non‐symmetric Nitsche method. Finally, we illustrate the performance of the proposed schemes on some numerical examples in two and three space dimensions. Copyright © 2013 John Wiley & Sons, Ltd.     

A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non‐smooth boundaries

The aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1D and 2D weakly singular and singular integral equations. The introduction of a smoothing transformation, besides smoothing the solution, allows also the use of a very simple and efficient collocation method based on Chebyshev polynomials of the first kind and their zeros. Further, we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements, which are given by weakly singular, singular or nearly singular integrals. Several numerical tests are given to point out the efficiency of the numerical approach we propose. Copyright © 2003 John Wiley & Sons, Ltd.     

双金属复合管固-液铸轧复合工艺环形布流器设计及其流场模拟

为解决双金属复合管固-液铸轧复合（SLCRB）工艺覆层熔体金属的环形均匀稳定布流问题,提出并设计了一种带有三级环形阶梯分流和锥形缓冲的环形布流器,并且基于商用软件Fluent建立了稳态布流过程的流体动力学模型,分析了布流器结构和铸轧速度对系统流场的影响,给出了获得均匀、稳定出口流场的布流器内部结构和布流条件,并用水模实验进行验证。结果表明:布流器内通道高宽比不小于1有利于均匀分流,并且锥形缓冲区的均流缓冲作用至关重要。以1.2 m/min的铸轧速度成功制备了外径为38 mm、内径为26 mm的铅-铝双金属复合管,覆层金属厚度均匀、界面结合效果良好。