An immersed-boundary method for compressible viscous flows

This paper combines a state-of-the-art method for solving the preconditioned compressible Navier-Stokes equations accurately and efficiently for a wide range of the Mach number with an immersed-boundary approach which allows one to use Cartesian grids for arbitrarily complex geometries. The method is validated versus well documented test problems for a wide range of the Reynolds and Mach numbers. The numerical results demonstrate the efficiency and versatility of the proposed approach as well as its accuracy, from incompressible to supersonic flow conditions, for moderate values of the Reynolds number. Further improvements, obtained via local grid refinement or non-linear wall functions, can render the proposed approach a formidable tool for studying complex three-dimensional flows of industrial interest.     

Meshless kinetic upwind method for compressible, viscous rotating flows

This paper presents the split-stencil least square kinetic upwind method for Navier–Stokes (SLKNS) solver using kinetic flux vector splitting (KFVS) scheme with Chapman-Enskog distribution. SLKNS solver operates on an arbitrary distribution of points and uses a novel least squares method which differs from the normal equations approach as it generates the non-symmetric cross-product matrix by selective splitting of the set of neighbours to avoid ill-conditioning. SLKNS also uses the axi-symmetric formulation of the Boltzmann equation and kinetic slip boundary condition. SLKNS is capable of capturing weak secondary flows as well as features of strong rotation characterized by steep density gradient and thin boundary layers towards the peripheral region with a rarefied central core.     

Numerical simulation of steady compressible flows using a zonal formulation. Part II: viscous flows

Viscous flow simulations are usually based on the Navier–Stokes equations representing the balance of mass, momentum, and energy. For many high Reynolds number flows, the viscous effects are only limited to small regions in the neighborhood of solid surfaces and in the wakes. We present here a zonal formulation with a reduced system of equations in the outer inviscid flow region. As in part I of this work, the velocity components are calculated from a generalized form of the Cauchy–Riemann equations with non-vanishing vorticity only in the inner region, where the governing equations including the viscous terms are solved. The viscous effects are transferred to the Cauchy–Riemann equations through a forcing function (vorticity), and the process is repeated until convergence. Preliminary results are presented and compared to standard Navier–Stokes calculations for two and three dimensional flow problems.     

A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids

A Cartesian cut-cell method which allows the solution of two- and three-dimensional viscous, compressible flow problems on arbitrarily refined graded meshes is presented. The finite-volume method uses cut cells at the boundaries rendering the method strictly conservative in terms of mass, momentum, and energy. For three-dimensional compressible flows, such a method has not been presented in the literature, yet. Since ghost cells can be arbitrarily positioned in space the proposed method is flexible in terms of shape and size of embedded boundaries. A key issue for Cartesian grid methods is the discretization at mesh interfaces and boundaries and the specification of boundary conditions. A linear least-squares method is used to reconstruct the cell center gradients in irregular regions of the mesh, which are used to formulate the surface flux. Expressions to impose boundary conditions and to compute the viscous terms on the boundary are derived. The overall discretization is shown to be second-order accurate in L¹. The accuracy of the method and the quality of the solutions are demonstrated in several two- and three-dimensional test cases of steady and unsteady flows.     

Computation of subcritical compressible flows

A finite element analysis and iterative solution of steady plane inviscid compressible flows is developed. Specific attention is directed to subcritical flows in which the nonlinear governing equation is elliptic, and to slightly supercritical mixed flows. The underlying variational theory for this nonlinear flow problem and a corresponding finite element formulation are presented. A Newton-Raphson iteration is introduced within this approximate analysis to provide efficient solution of subcritical flows and also slightly supercritical flows in which the nonlinear flow equation is of mixed type. The performance of the algorithm is studied with particular reference to the effect of a two-parameter scheme in which incident Mach number $\text{M}{}_{\mathrm{x}}$ and specific heat ratio γ may be varied. By thus regularizing the operator during the early iteration history, more efficient computations can be realized in the mildly-transonic flow regime of higher incident Mach numbers.     

Large stencil viscous flux linearization for the simulation of 3D compressible turbulent flows with backward-Euler schemes

The purpose of this article is to study different approximate linearizations of the RANS equations viscous fluxes, for numerical simulations of compressible turbulent flows with backward-Euler schemes. The explicit convective flux under consideration is centred with artificial dissipation. The discrete viscous flux, calculated from cell-centred evaluation of the gradients, needs less computations and memory storage than other discretizations. But, in other respects, the balance of this numerical flux has a large stencil, which is not coherent with the 3-point per mesh direction stencil of classical implicit stages. Therefore 3-point and 5-point per mesh direction approximate linearizations are built from the thin layer flux formula. The stability condition of the corresponding backward-Euler schemes is given for a scalar linear equation (for the basic non-factored version of scheme and with LU-relaxation). Multigrid and monogrid computations of turbulent flow around two external configurations are performed with Wilcox’s k-ω turbulence model. The 5-point per mesh direction linearizations, coherent with the differential of the fluxes balance of thin layer approximation of explicit viscous fluxes, leads to the most efficient implicit stages.     

The PCICE-FEM scheme for highly compressible axisymmetric flows

The recently developed PCICE-FEM scheme (Journal of Computational Physics, vol. 198, 659, 2004) is extended to two-dimensional axisymmetric geometries. The main discretization problem for nodal-based axisymmetric formulations lies in deriving a closed form as the radial coordinates approach zero along the axis of symmetry. This problem is addressed by employing the finite element piecewise linear approximations to both the flow variables and (separately) to the nodal values of the radial coordinates. The resulting formulation is an elegant treatment of the axisymmetric coordinate system with out noticeable loss of spatial accuracy and little additional cost in computational effort. An overview of the PCICE algorithm for the axisymmetric governing equations will be followed by a detailed axisymmetric finite element formulation for the PCICE-FEM scheme. The ability of the PCICE-FEM scheme to accurately and efficiently simulate highly compressible axisymmetric flows is demonstrated.     

Treatment of uncertain material interfaces in compressible flows

To treat uncertain interface position is an important issue for complex applications. In this paper, we address the characterization of randomly perturbed interfaces between fluids thanks to stochastic modeling and uncertainty quantification through the 2D Euler system. The perturbed interface is modeled as a random field and represented by a Karhunen–Loève expansion. The stochastic 2D Euler system is solved applying Polynomial Chaos theory through the Intrusive Polynomial Moment Method (IPMM). This stochastic resolution method is fully explained and studied (theoretically and numerically). Stochastic Richtmyer–Meshkov unstable flows are solved and presented for several configurations of the uncertain interface (different rugosities) between the fluids. The probability density functions of the mass density of the fluid in the vicinity of the interface are computed built and compared for the different simulations: the system exhibits strong sensitivity with respect to the stochastic initially leading modes.     

Computational challenges of viscous incompressible flows

Over the past 30 years, numerical methods and simulation tools for incompressible flows have been advanced as a subset of the computational fluid dynamics (CFD) discipline. Although incompressible flows are encountered in many areas of engineering, simulation of compressible flow has been the major driver for developing computational algorithms and tools. This is probably due to the rather stringent requirements for predicting aerodynamic performance characteristics of flight vehicles, while flow devices involving low-speed or incompressible flow could be reasonably well designed without resorting to accurate numerical simulations. As flow devices are required to be more sophisticated and highly efficient, CFD tools become increasingly important in fluid engineering for incompressible and low-speed flow. This paper reviews some of the successes made possible by advances in computational technologies during the same period, and discusses some of the current challenges faced in computing incompressible flows.     

The computation of massively separated flows using compressible vorticity confinement methods

Vorticity confinement methods have been shown to be very effective in the computation of flows involving the convection of thin vortical layers. These are the only Eulerian methods whereby simulations of these layers remain very thin and persist long distances without significant dissipation. Initially developed by Steinhoff and co-workers for incompressible flow, these methods have been used successfully to predict complex flows, particularly involving helicopter rotors. Recently, the method has been extended to a compressible finite-volume form, which will enable the methods to be used for a much broader class of problems. In this paper, we examine the ability of the compressible vortex confinement methodology to handle an important class of vortex-dominated flows involving massive separation from bluff bodies. We evaluate the effectiveness of the method by comparisons with experimental data and available state-of-the-art computations. An important conclusion of the present work is that vortex confinement applied to massively separated flows, without modeling the viscous terms and on an essentially inviscid grid, can result in a reasonable approximation to turbulent separated flows. The computed flow structures and velocity profiles were in good agreement with time-averaged values of the data and with LES simulations even though the confinement approach utilized more than a factor of 50 fewer cells in the computation (20,000 compare to more the 1 million). The success of the method for these classes of flows may be attributed to the accurate calculation of the rotational inviscid flow which dominates the convection of the large-scale flow structures.     

Nature of viscous flows near sharp corners

Explicit solutions of two-dimensional, steady-state Navier-Stokes equations are derived in the neighborhood of sharp corners where a sliding wall meets a stationary wall and causes a mathematical singularity. These solutions are valid for small Reynolds numbers. A semi-analytic technique is used to derive these solutions. Some comparisons with numerical solutions are also carried out.     

Core spreading vortex methods in two-dimensional viscous flows

The purpose of the present paper is: (1) to derive an integral equation of Fredholm type with respect to vorticity from the two-dimensional Navier-Stokes equations, (2) to derive two vortex methods which are based on the core spreading model and (3) to analyze them, by using this integral equation. The convergence and stability properties of two methods, the Gaussian core spreading model and the alternative Gaussian core spreading model, for high Reynolds number flows are analyzed under the assumption of smooth initial condition with bounded support and a free-space boundary.     

Performance of parallel AMG-preconditioners in CFD-codes for weakly compressible flows

We examine the performance in terms of computing time of different parallel AMG algorithms that are applied within the context of industrial computational fluid dynamics (CFD) problems. We give an overview over the most important classes of algorithms described in literature, pick out four fundamentally different algorithms and perform numerical experiments on up to 16 processors with two benchmarks representing an important class of CFD-problems. The results indicate that aggregation-based algorithms have advantages compared to algorithms based on the concept of C–F-splitting.     

Optimal aerodynamic design of airfoils in unsteady viscous flows

A continuous adjoint formulation is used to determine optimal airfoil shapes in unsteady viscous flows at Re = 1 × 10⁴. The Reynolds number is based on the free-stream speed and the chord length of the airfoil. A finite element method based on streamline-upwind Petrov/Galerkin (SUPG) and pressure-stabilized Petrov/Galerkin (PSPG) stabilizations is used to solve both the flow and adjoint equations. The airfoil is parametrized via a Non-Uniform Rational B-Splines (NURBS) curve. Three different objective functions are used to obtain optimal shapes: maximize lift, minimize drag and minimize ratio of drag to lift. The objective functions are formulated on the basis of time-averaged aerodynamic coefficients. The three objective functions result in diverse airfoil geometries. The resulting airfoils are thin, with the largest thickness to chord ratio being only 5.4%. The shapes obtained are further investigated for their aerodynamic performance. Maximization of time-averaged lift leads to an airfoil that produces more than six times more lift compared to the NACA 0012 airfoil. The excess lift is a consequence of the large peak and extended region of high suction on the upper surface and high pressure on the lower surface. Minimization of drag results in an airfoil with a sharp leading edge. The flow remains attached for close to 70% of the chord length. Minimization of the ratio of drag to lift results in an airfoil with a shallow dimple on the upper surface. It leads to a fairly large value of the time-averaged ratio of lift to drag (～ 17.8). The high value is mostly achieved by a 447% increase in lift and 16% reduction in drag, compared to a NACA 0012 airfoil. Imposition of volume constraint, for the cases studied, is found to result in airfoils that have lower aerodynamic performance.     

Preconditioned methods for simulations of low speed compressible flows

A time-derivative preconditioned system of equations suitable for the numerical simulation of inviscid compressible flow at low speeds is formulated. The preconditioned system of equations are hyperbolic in time and remain well-conditioned in the incompressible limit. The preconditioning formulation is easily generalized to multicomponent/multiphase mixtures. When applying conservative methods to multicomponent flows with sharp fluid interfaces, nonphysical solution behavior is observed. This stimulated the authors to develop an alternative solution method based on the nonconservative form of the equations which does not generate the aforementioned nonphysical behavior. Before the results of the application of the nonconservative method to multicomponent flow problems is reported, the accuracy of the method on single component flows will be demonstrated. In this report a series of steady and unsteady inviscid flow problems are simulated using the nonconservative method and a well-known conservative scheme. It is demonstrated that the nonconservative method is both accurate and robust for smooth low speed flows, in comparison to its conservative counterpart.     

Mathematical theory of compressible,viscous, and heat conducting fluids

We develop a mathematical theory of compressible, viscous, and heat conducting fluids based on the concept of weak solutions and the second law of thermodynamics. We derive suitable a priori estimates and show the weak sequential stability property for the complete Navier–Stokes–Fourier system.     

Design of airfoils in incompressible viscous flows by numerical optimization

A method is outlined for the design of airfoils in incompressible viscous flows by numerical optimization wherein a reduced number of design coordinates are used to define the airfoil shape. The optimization problem is formulated as a nongradient search in a finite constrained parameter space. The approach is to define the airfoil as a linear combination of basic shapes which may be analytically or numerically defined. The design problem is to determine the participation of each of these basic shapes in defining the optimum airfoil. The aerodynamic analysis program is specially developed to fit the requirements of the optimization program and is based on the vortex singularity method for inviscid flow analysis and the momentum integral method for boundary layer analysis. Four examples have been worked out to illustrate the proposed design method. In these, modifications to four different airfoil geometries are made to achieve either a minimum drag coefficient or a minimum pitching moment coefficient under prescribed constraints. The results show that significant drag or pitching moment reduction is possible through shape manipulation alone.     

The SPH technique applied to free surface flows

This paper deals with an application of the SPH (Smooth Particle Hydrodynamics) technique to treat free surface problems. The SPH technique was originally conceived and developed for treating astrophysical problems and belongs to the class of “meshless” methods that dispense with the requirement of a computational grid. Instead, a cloud of particles is used to represent the continuum, the contact interaction between them is introduced with their subsequent trajectory being computed in the Lagrangian sense. The design and implementation of the method for transport equations and the Euler inviscid equations is fairly well-documented. Applications to the treatment of free surface flows is however more recent. In this work, the computation of three-dimensional free surface flows with the method is presented. The introduction of Riemann solvers to model the breakup of the initial surface discontinuities between particles is a novel feature of this work. For purposes of illustration, a three-dimensional simulation of the Vaiont dam disaster that occurred in 1963 in northern Italy is presented. This is a case where complicated three-dimensional geometries are involved and was chosen to show-off the versatility of the technique. The results are in general agreement with the qualitative observations and reconstruction of the event as reported by experts. The SPH technique is found to be very promising and powerful for application to free surface flows. In particular, the stage is being reached where for hydraulic problems; it may be used as a powerful simulation tool to delineate high-risk zones downstream of a possible dam failure where geometries of almost arbitrary complexity are involved. At the present time, significant progress is being achieved in developing the technique for application in different domains.