Numerical simulation of cavitating flow in 2D and 3D inducer geometries

A computational method is proposed to simulate 3D unsteady cavitating flows in spatial turbopump inducers. It is based on the code FineTurbo, adapted to take into account two‐phase flow phenomena. The initial model is a time‐marching algorithm devoted to compressible flow, associated with a low‐speed preconditioner to treat low Mach number flows. The presented work covers the 3D implementation of a physical model developed in LEGI for several years to simulate 2D unsteady cavitating flows. It is based on a barotropic state law that relates the fluid density to the pressure variations. A modification of the preconditioner is proposed to treat efficiently as well highly compressible two‐phase flow areas as weakly compressible single‐phase flow conditions. The numerical model is applied to time‐accurate simulations of cavitating flow in spatial turbopump inducers. The first geometry is a 2D Venturi type section designed to simulate an inducer blade suction side. Results obtained with this simple test case, including the study of its general cavitating behaviour, numerical tests, and precise comparisons with previous experimental measurements inside the cavity, lead to a satisfactory validation of the model. A complete three‐dimensional rotating inducer geometry is then considered, and its quasi‐static behaviour in cavitating conditions is investigated. Numerical results are compared to experimental measurements and visualizations, and a promising agreement is obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Preconditioned finite element algorithms for 3D Stokes flows

Preconditioned conjugate gradient algorithms for solving 3D Stokes problems by stable piecewise discontinuous pressure finite elements are presented. The emphasis is on the preconditioning schemes and their numerical implementation for use with Hermitian based discontinuous pressure elements. For the piecewise constant discontinuous pressure elements, a variant implementation of the preconditioner proposed by Cahouet and Chabard for the continuous pressure elements is employed. For the piecewise linear discontinuous pressure elements, a new preconditioner is presented. Numerical examples are presented for the cubic lid-driven cavity problem with two representative elements, i.e. the Q2-PO and the Q2-P1 brick elements. Numerical results show that the preconditioning schemes are very effective in reducing the number of pressure iterations at very reasonable costs. It is also shown that they are insensitive to the mesh Reynolds number except for nearly steady flows (Re_m → 0) and are almost independent of mesh sizes. It is demonstrated that the schemes perform reasonably well on non-uniform meshes.     

Conforming finite element computations applied to a spatially periodic,harmonic Navier-Stokes problem

In this paper computations in the two dimensional case of a harmonic Navier-Stokes problem with periodic boundary conditions are presented. This study of an incompressible viscous fluid leads to a non-symmetric linear problem (very low Reynolds number). Moreover unknown functions have complex values (monochromatic dynamic behaviour). Numerical treatment of the incompressibility condition is a generalization of the classical treatment of Stokes problem. A mixed formulation, where discrete pressure plays the role of Lagrange multipliers is used (Uzawa algorithm). Two conforming finite element methods are tested on different meshes. The second one uses a classical refinement in the shape function: the so-called bulb function. All computational tests show that the use of a bulb function on each element gives better results than refinement in the mesh without introducing too many degrees of freedom. Finally numerical results are compared to experimental data.     

ADAPTIVE LINEARIZATION AND GRID ITERATIONS WITH THE TRI-TREE MULTIGRID REFINEMENT–RECOARSEMENT ALGORITHM FOR THE NAVIER–STOKES EQUATIONS

The tri-tree algorithm for refinements and recoarsements of finite element grids is explored. The refinement–recoarsement algorithm not only provides an accurate solution in certain parts of the grid but also has a major influence on the finite element equation system itself. The refinements of the grid lead to a more symmetric and linear equation matrix. The recoarsements will ensure that the grid is not finer than is necessary for preventing divergence in an iterative solution procedure. The refinement–recoarsement algorithm is a dynamic procedure and the grid is adapted to the instant solution. In the tri-tree multigrid algorithm the solution from a coarser grid is scaled relatively to the increase in velocity boundary condition for the finer grid. In order to have a good start vector for the solution of the finer grid, the global Reynolds number or velocity boundary condition should not be subject to large changes. For each grid and velocity solution the element Reynolds number is computed and used as the grid adaption indicator during the refinement–recoarsement procedure. The iterative tri-tree multigrid method includes iterations with respect to the grid. At each Reynolds number the same boundary condition s are applied and the grid is adapted to the solution iteratively until the number of unknowns and elements in the grid becomes constant. In the present paper the following properties of the tri-tree algorithm are explored: the influence of the increase in boundary velocities and the size of the grid adaption indicator on the amount of work for solving the equations, the number of linear iterations and the solution error estimate between grid levels. The present work indicates that in addition to the linear and non-linear iterations, attention should also be given to grid adaption iterations. © 1997 by John Wiley & Sons, Ltd.     

高超声速低温喷管副喷管参数化设计

 通过对DF化学激光器高超声速低温(HYLTE)喷管副喷管流动与几何参数进行制约关系和敏感性分析,确定了能够对副喷管进行参数化设计的4个基本几何控制参数和2个调节参数,初步建立了HYLTE喷管副喷管的参数化设计方法。引入了相对未混合度作为衡量喷管混合效果的性能参数,建立了较完备的3维反应流数值模拟计算程序。结果表明：该方法选择的基本参数能够敏感地反映HYLTE喷管副喷管的尺寸特征和关键性能。     

Computation Models for Parameterized Complexity

A parameterized computational problem is a set of pairs (x, k), where k is a distinguished item called “parameter”. FPT is the class of fixed-parameter tractable problems: for any fixed value of k, they are solvable in time bounded by a polynomial of degree α, where α is a constant not dependent on the parameter. In order to deal with parameterized intractability, Downey and Fellows have introduced a hierarchy of classes W[l] ? W[2] ? ? containing likely intractable parameterized problems, and they have shown that such classes have many natural, complete languages. In this paper we analyze several variations of the halting problem for nondeterministic Turing machines with parameterized time, and we show that its parameterized complexity strongly depends on some resources like the number of tapes, head and internal states, and on the size of the alphabet. Notice that classical polynomial-time complexity fails in distinguishing such features. As byproducts, we show that parameterized complexity is a useful tool for the study of the intrinsic power of some computational models, and we underline the different “computational powers” of some levels of the parameterized hierarchy.     

Temporal accuracy analysis of phase change convection simulations using the JFNK‐SIMPLE algorithm

The incompressible Navier–Stokes and energy conservation equations with phase change effects are applied to two benchmark problems: (1) non‐dimensional freezing with convection; and (2) pure gallium melting. Using a Jacobian‐free Newton–Krylov (JFNK) fully implicit solution method preconditioned with the SIMPLE (Numerical Heat Transfer and Fluid Flow. Hemisphere: New York, 1980) algorithm using centred discretization in space and three‐level discretization in time converges with second‐order accuracy for these problems. In the case of non‐dimensional freezing, the temporal accuracy is sensitive to the choice of velocity attenuation parameter. By comparing to solutions with first‐order backward Euler discretization in time, it is shown that the second‐order accuracy in time is required to resolve the fine‐scale convection structure during early gallium melting. Qualitative discrepancies develop over time for both the first‐order temporal discretized simulation using the JFNK‐SIMPLE algorithm that converges the nonlinearities and a SIMPLE‐based algorithm that converges to a more common mass balance condition. The discrepancies in the JFNK‐SIMPLE simulations using only first‐order rather than second‐order accurate temporal discretization for a given time step size appear to be offset in time. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation

Based on a new global variational formulation, a spectral element approximation of the incompressible Navier–Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems. Iterative solutions of such systems are feasible and attractive for large problems. It is shown that, provided an appropriate pre‐conditioner is chosen for the pressure system, the nested conjugate gradient methods can be applied to obtain rapid convergence rates. Detailed numerical examples are given to prove the quality of the pre‐conditioner. Thanks to the rapid iterative convergence, the global Uzawa algorithm takes advantage of this as compared with the classical iteration by sub‐domain procedures. Furthermore, a generalization of the pre‐conditioned iterative algorithm to flow simulation is carried out. Comparisons of computational complexity between the Navier–Stokes/Euler coupled solution and the full Navier–Stokes solution are made. It is shown that the gain obtained by using the Navier–Stokes/Euler coupled solution is generally considerable. Copyright © 2000 John Wiley & Sons, Ltd.     

One‐shot pseudo‐time method for aerodynamic shape optimization using the Navier–Stokes equations

This paper presents a numerical method for aerodynamic shape optimization problems in compressible viscous flow. It is based on simultaneous pseudo‐time stepping in which stationary states are obtained by solving the pseudo‐stationary system of equations representing the state, costate and design equations. The main advantages of this method are that it blends in nicely with previously existing pseudo‐time‐stepping methods for the state and the costate equations, that it requires no additional globalization in the design space, and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP methods. For design examples of 2D problems, the overall cost of computation can be reduced to less than 2 times the forward simulation runs. Copyright © 2011 John Wiley & Sons, Ltd.