Analysis and computation of non-equilibrium two-phase flows

Non-equilibrium fluid mechanics and thermodynamics of two-phase vapour-droplet and gas-particle flow are considered. Formation of the droplets as well as their subsequent interaction with the vapour are discussed. A new theory of nucleation in steam turbines is developed that reproduces many aspects of measured droplet size spectra which cannot be explained by any available steady-flow theories. (Steam turbines are responsible for 80% of global electricity production and the presence of moisture significantly reduces turbine efficiency costing 50 million pounds per annum in UK alone.) Fluid dynamic interactions discussed include flow instabilities induced by condensation, condensation wave theory, relaxation gas dynamics for vapour-droplet flow, thermal choking due to non-equilibrium condensation, the structure of shock waves and their development through unsteady processes, and jump conditions and the interpretation of total pressure in two-phase flows.     

A decoupled augmented IIM strategy for incompressible two‐phase flows with interfaces on irregular domains

A decoupled augmented immersed interface method for solving incompressible two‐phase flows involving both irregular domains and interfaces is presented. In order to impose the prescribed velocity at the boundary of the irregular domain, singular force as one set of augmented variables is introduced. The velocity components at the two‐fluid interface as another set of augmented variables are introduced to satisfy the continuity condition of the velocity across the interface so that the jump conditions for the velocity and pressure are decoupled across the interface. The augmented variables and/or the forces along the interface/boundary are related to the jumps in both pressure and velocity and the jumps in their derivatives across the interface/boundary and applied to the fluid through jump conditions. The resulting augmented equation is a couple system of these two sets of augmented variables, and the direct application of the GMRES is impractical due to larger iterations. In this work, the novel decoupling of two sets of the augmented variables is proposed, and the decoupled augmented equation is then solved by the LU or the GMRES method. The Stokes equations are discretized via the finite difference method with the incorporation of jump contributions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa‐type method. The numerical results show that second‐order accuracy for the velocity is confirmed. The present method has also been applied to solve for incompressible two‐phase Navier–Stokes flow with interfaces on irregular domains. Copyright © 2011 John Wiley & Sons, Ltd.     

A Cartesian ghost‐cell multigrid Poisson solver for incompressible flows

In this paper, a simple Cartesian ghost‐cell multigrid Poisson solver is proposed for simulating incompressible fluid flows. The flow field is discretized efficiently on a rectangular mesh, in which solid bodies are immersed. A small number of ghost mesh cells and their symmetric image cells are distributed in the vicinity of the solid boundary. With the aid of the ghost and image cells, the Dirichlet and Neumann boundary conditions can be implemented effectively. Chorin's fractional‐step projection method is adopted for the coupling of velocity and pressure for the solution of the Navier–Stokes equations. Point‐wise Gauss–Seidel iteration is used to solve the pressure Poisson equation. To speed up the convergence of the solution to the corresponding linear system, sub‐level coarse meshes embedded with ghost and image cells are also introduced and operated in a sequential V‐cycle. Several test cases including the classical ideal incompressible flow around a cylinder, a lid‐driven cavity flow and viscous flow past a fixed/rotating cylinder are presented to demonstrate the accuracy and efficiency of the current approach. Copyright © 2010 John Wiley & Sons, Ltd.     

A front remeshing technique for a Lagrangian description of moving interfaces in two‐fluid flows

The numerical analysis of two‐fluid flows involves the treatment of a discontinuity that appears at the separating interface. Classical Lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface. However, the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front. To avoid this problem, an interface remeshing algorithm based on the diffuse approximation of the interface curvature is proposed in this work. In addition, the remeshed front is enforced to preserve the global volume. These new aspects are incorporated in an existing fluid dynamics formulation for the analysis of two‐fluid flows problems. The resulting formulation is called in this work as the moving Lagrangian interface remeshing technique (MLIRT). Copyright © 2005 John Wiley & Sons, Ltd.     

A stream function implicit finite difference scheme for 2D incompressible flows of Newtonian fluids

The development of a new algorithm to solve the Navier–Stokes equations by an implicit formulation for the finite difference method is presented, that can be used to solve two‐dimensional incompressible flows by formulating the problem in terms of only one variable, the stream function. Two algebraic equations with 11 unknowns are obtained from the discretized mathematical model through the ADI method. An original algorithm is developed which allows a reduction from the original 11 unknowns to five and the use of the Pentadiagonal Matrix Algorithm (PDMA) in each one of the equations. An iterative cycle of calculations is implemented to assess the accuracy and speed of convergence of the algorithm. The relaxation parameter required is analytically obtained in terms of the size of the grid and the value of the Reynolds number by imposing the diagonal dominancy condition in the resulting pentadiagonal matrixes. The algorithm developed is tested by solving two classical steady fluid mechanics problems: cavity‐driven flow with Re=100, 400 and 1000 and flow in a sudden expansion with expansion ratio H/h=2 and Re=50, 100 and 200. The results obtained for the stream function are compared with values obtained by different available numerical methods, to evaluate the accuracy and the CPU time required by the proposed algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

Generalized-α scheme in the PFEM for velocity-pressure and displacement-pressure formulations of the incompressible Navier–Stokes equations

Despite the increasing use of the particle finite element method (PFEM) in fluid flow simulation and the outstanding success of the Generalized-$\alpha$ (GA) time integration method, very little discussion has been devoted to their combined performance. This work aims to contribute in this regard by addressing three main aspects. First, it includes a detailed implementation analysis of the GA method in PFEM. The work recognizes and compares different implementation approaches from the literature, which differ mainly in the terms that are $\alpha$-interpolated (state variables or forces of momentum equation) and the type of treatment for the pressure in the time integration scheme. Second, the work compares the performance of the GA method against the Backward Euler and Newmark schemes for the solution of the incompressible Navier–Stokes equations. Third, the study is enriched by considering not only the classical velocity-pressure formulation but also the displacement-pressure formulation that is gaining interest in the fluid-structure interaction field. The work is carried out using various 2D and 3D benchmark problems such as the fluid sloshing, the solitary wave propagation, the flow around a cylinder, and the collapse of a cylindrical water column.     

A novel RBF-FD meshless scheme in curvilinear geometry for unbounded flows

A novel local radial basis function Radial Basis Function-Finite Difference (RBF-FD) scheme has been developed in curvilinear geometry and implemented to unbounded fluid flows. The far field boundary condition that arises due to the unboundedness of the fluid was handled efficiently and achieved higher order accurate results. The RBF-FD is combined with an upwind-based scheme to handle convective terms effectively. The effect of shape parameter on the accuracy of the results and the variation of shape parameter with the number of nodes are numerically investigated. The order of accuracy of the method is found in comparison with a finite difference scheme.     

A discrete splitting finite element method for numerical simulations of incompressible Navier–Stokes flows

The presence of the pressure and the convection terms in incompressible Navier–Stokes equations makes their numerical simulation a challenging task. The indefinite system as a consequence of the absence of the pressure in continuity equation is ill‐conditioned. This difficulty has been overcome by various splitting techniques, but these techniques incur the ambiguity of numerical boundary conditions for the pressure as well as for the intermediate velocity (whenever introduced). We present a new and straightforward discrete splitting technique which never resorts to numerical boundary conditions. The non‐linear convection term can be treated by four different approaches, and here we present a new linear implicit time scheme. These two new techniques are implemented with a finite element method and numerical verifications are made. Copyright © 2005 John Wiley & Sons, Ltd.     

A projection semi‐implicit scheme for the coupling of an elastic structure with an incompressible fluid

We address the numerical simulation of fluid–structure systems involving an incompressible viscous fluid. This issue is particularly difficult to face when the fluid added‐mass acting on the structure is strong, as it happens in hemodynamics for example. Indeed, several works have shown that, in such situations, implicit coupling seems to be necessary in order to avoid numerical instabilities. Although significant improvements have been achieved during the last years, solving implicit coupling often exhibits a prohibitive computational cost. In this work, we introduce a semi‐implicit coupling scheme which remains stable for a reasonable range of the discretization parameters. The first idea consists in treating implicitly the added‐mass effect, whereas the other contributions (geometrical non‐linearities, viscous and convective effects) are treated explicitly. The second idea, relies on the fact that this kind of explicit–implicit splitting can be naturally performed using a Chorin–Temam projection scheme in the fluid. We prove (conditional) stability of the scheme for a fully discrete formulation. Several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches. Copyright © 2006 John Wiley & Sons, Ltd.     

A truly mesh-distortion-enabled implementation of cell-based smoothed finite element method for incompressible fluid flows with fixed and moving boundaries

Mesh-free properties are part of the superiority of cell-based smoothed finite element method (CS-FEM), but have yet to be fully exploited for computational fluid dynamics. A novel implementation of CS-FEM for incompressible viscous fluid flows in stationary and deforming domains discretized by severely distorted bilinear four-node quadrilateral (Q4) elements is presented in this article. The negative determinant of the Jacobian transformation from the Cartesian coordinates to the natural coordinates is intentionally stipulated for the corresponding mesh over which FEM inevitably fails in practice. It is found that, without ad hoc modifications, CS-FEM incurs unsatisfactory results and even a failure on fixed meshes. To cater for general computations on either a uniform or nonuniform mesh represented by these badly degenerated elements, four smoothing cells (SCs) are deployed in convex Q4 element whereas one SC in concave Q4 element. A simple hourglass control is introduced into those under-integrated quadrilaterals for stabilizing the one-SC quadrature in smoothed Galerkin weak form. Thanks to the adoption of characteristic-based split (CBS) scheme for the fluid solution, a byproduct is the unfolded equivalence of the CBS stabilization and balancing tensor diffusivity under the incompressibility constraint. Several benchmark problems involving incompressible fluid flow and fluid-structure interaction are solved. Numerical results show the good accuracy and robustness of the proposed approach that raises a seductive idea for resolving moving-mesh problems.     

A high-order Lie groups scheme for solving the recovery of external force in nonlinear system

In this paper, we develop the high-order Lie group scheme to estimate a real-time external force exerted on nonlinear system. For estimating a real-time data, the equations of motion and the supplemental data are combined into a set of differential algebraic equations (DAEs), which is solved by an implicit Lie group differential algebraic equations method (LGDAE) and a Newton iterative algorithm. However, an explicit LGDAE cannot work, and an implicit scheme cannot avoid a lot of iterative numbers and overcome numerical instability under a long time span and noisy-level effect. Because the original implicit LGDAE cannot satisfy the constraint of the cone structure, Lie group and Lie algebra at each time step, solutions highly dependent on ones cannot easily converge by a large time step. Therefore, we develop a high-order explicit LGDAE to avoid iterative numbers and numerical instability, and at the same time combined with higher order sliding modes to obtain real-time external force. The accuracy and efficiency of the novel scheme is validated by comparing the estimation results with the previous literature.     

A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains

In this paper, the inverse Cauchy problems for elliptic equations, including the Laplace equation, the Poisson equation, and the Helmholtz equation, defined in annular domains are investigated. When the outer boundary of an annulus is imposed by overspecified boundary data, we seek unknown data in the inner boundary through a combination of the spring-damping regularization method (SDRM) and the mixed group-preserving scheme (MGPS). Several numerical examples are examined to show that the MGPS plus the SDRM can overcome the ill-posed behavior of this highly ill-conditioned inverse Cauchy problem. The presently proposed novel algorithm has good efficiency and stability against the disturbance from large random noise even up to 50%, and the computational cost of MGPS is very time saving.     

A continued‐fraction‐based high‐order transmitting boundary for wave propagation in unbounded domains of arbitrary geometry

A high‐order local transmitting boundary is developed to model the propagation of elastic waves in unbounded domains. This transmitting boundary is applicable to scalar and vector waves, to unbounded domains of arbitrary geometry and to anisotropic materials. The formulation is based on a continued‐fraction solution of the dynamic‐stiffness matrix of an unbounded domain. The coefficient matrices of the continued fraction are determined recursively from the scaled boundary finite element equation in dynamic stiffness. The solution converges rapidly over the whole frequency range as the order of the continued fraction increases. Using the continued‐fraction solution and introducing auxiliary variables, a high‐order local transmitting boundary is formulated as an equation of motion with symmetric and frequency‐independent coefficient matrices. It can be coupled seamlessly with finite elements. Standard procedures in structural dynamics are directly applicable for evaluating the response in the frequency and time domains. Analytical and numerical examples demonstrate the high rate of convergence and efficiency of this high‐order local transmitting boundary. Copyright © 2007 John Wiley & Sons, Ltd.