Computations of high‐speed compressible flows with adaptive cell‐centered finite element method

Abstract

An upwind cell‐centered finite element formulation is combined with an adaptive meshing technique to solve Navier‐Stokes equations for high‐speed inviscid and viscous compressible flows. The finite element formulation and the computational procedure are described. An adaptive meshing technique is applied to increase the analysis solution accuracy, as well as to minimize the computational time and the computer memory requirement. The efficiency of the combined method is evaluated by the examples of Mach 2.6 inviscid flow in a channel with compression and expansion ramps, Mach 6.47 inviscid and viscous flows past a cylinder, and Mach 4 viscous flow over a flat plate.     

An inverse integral 3D compressible boundary layer method and coupling with transonic inviscid solution

Summary A new velocity profile, which has a simple expression and agrees well with experimental data in a wide range, is proposed in the present paper. Based on this profile, the governing equations of the 3D compressible inverse boundary layer method are deduced. The steady transonic viscous flow around a 3D wing can be calculated as follows: the inviscid flow is calculated by using nonisentropic full potential equation; the viscous flow is calculated by using present boundary layer method; the viscous and inviscid solutions are coupled by using semi-inverse method. Numerical results agree well with the experimental data and required computer resources are less, so that it has broad prospects in the engineering application.     

一种高效的准二维管道瞬变流计算方法

为了提高准二维管道瞬变流模型的计算效率,对现有的基于特征线法的准二维模型的算法进行了改进,提出了直接利用一维显式方程计算管道的流量和压力,取代了原算法中需通过数值积分求解平均速度再进一步计算节点压力的方法。该文提出的一维显式方程采用FVS （Flux Vector Splitting）方法,并在水库-管道-阀门系统中,将该方法与两个现有的准二维模型的算法在准确性和计算效率方面进行了比较。结果表明：该文所提出的方法与另外两个算法得到的计算结果基本相同,但是所花费的计算时间更少。因此,该文所提出的方法是一种计算准确并且高效的准二维瞬变流计算方法,适用于管道系统瞬变流的数值模拟分析。     

High-Fidelity Computational Optimization for 3-D Flexible Wings: Part I—Simultaneous Aero-Structural Design Optimization (SASDO)

The formulation and implementation of a multidisciplinary optimization method called Simultaneous Aerodynamic and Structural Design Optimization (SASDO) is presented and applied to a simple, isolated, 3-D wing in inviscid flow. The method aims to reduce the computational expense incurred in performing shape and sizing optimization using existing state-of-the-art Computational Fluid Dynamics (CFD) flow analysis, FEM structural analysis and sensitivity analysis tools. Results show that the method finds the same local optimum as a conventional optimization method with as much as 50% reduction in the computational cost and without significant modifications to the analysis tools.     

A thermodynamically consistent numerical method for transonic cascade flow simulation

The author's recent work on thermodynamically consistent numerical methods for the simulation of inviscid, adiabatic flow is extended to handle transonic flow problems of practical interest. The basic Singhal–Spalding scheme is developed and tested on two problems. The first one is the standard transonic nozzle flow. The computational results are shown to be in good agreement with the analytical solution. The second problem is a flow situation studied experimentlly by Sieverding. The inviscid results are in surprisingly good agreement with the principal qualitative features of the real folws. There are important quantitative discrepancies which are believed to be due to the inadequacies of the inviscid model rather than failings of the scheme itself. Arguments in support for this view are presented.     

A surrogate assisted evolutionary optimization method with application to the transonic airfoil design

A multi-layer perceptron neural network (NN) method is used for efficient estimation of the expensive objective functions in the evolutionary optimization with the genetic algorithm (GA). The estimation capability of the NN is improved by dynamic retraining using the data from successive generations. In addition, the normal distribution of the training data variables is used to determine well-trained parts of the design space for the NN approximation. The efficiency of the method is demonstrated by two transonic airfoil design problems considering inviscid and viscous flow solvers. Results are compared with those of the simple GA and an alternative surrogate method. The total number of flow solver calls is reduced by about 40% using this fitness approximation technique, which in turn reduces the total computational time without influencing the convergence rate of the optimization algorithm. The accuracy of the NN estimation is considerably improved using the normal distribution approach compared with the alternative method.     

Open boundary treatment of the finite amplitude wave using the boundary element method

In this paper the boundary element method is applied to the analysis of non-linear free-surface waves. A particular concern is the treatment of open boundary at the input flow boundary and output flow boundary, which uses the mass-flux and energy-flux because of the continuity of fluid. By assuming the fluid to be inviscid and incompressible and the flow to be irrotational, the problem is formulated mathematically as a two-dimensional non-linear problem in terms of a velocity potential. The equation (Laplace equation) and the boundary conditions are transformed into two boundary integral equations. Due to the non-linearity of the problem, the incremental method is used in the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the finite element method and also with the experimental values. Good agreements are obtained.     

Calculation of a hypersonic flow over bodies of complex configuration on unstructured tetrahedral meshes using the AUSM scheme

An approach to solving the three-dimensional Navier-Stokes equations on tetrahedral computational meshes on the basis of splitting by physical processes is considered. An algorithm of numerical implementation of the suggested method for solving three-dimensional problems of aerothermodynamics of freeconfiguration hypersonic flying vehicles (HFVs) is elaborated. The finite-volume method is applied to approximate the gasdynamics equations. The fluxes on the boundaries of the computational elements are calculated using the AUSM scheme. A computer code aimed at numerical simulation of the three-dimensional aerothermodynamics of the structural elements and the integral configurations of the HFV on the basis of the Euler and Navier-Stokes equations is developed. The algorithms developed are tested using the benchmark problem of a viscous hypersonic perfect gas flow over a sphere. The results of comparison of the computational data found using the suggested approach on the unstructured different-size meshes with the numerical solutions found on structured grids with application of the computational code NERAT are presented. The computational model of the flow of viscous and inviscid perfect gas developed is applied to investigate the aerothermodynamics of a model of an unmanned experimental aircraft X-43 of complex configuration.     

The dimension splitting reproducing kernel particle method for three-dimensional potential problems

In this paper, the dimension splitting reproducing kernel particle method (DSRKPM) for three-dimensional (3D) potential problems is presented. In the DSRKPM, a 3D potential problem can be transformed into a series of two-dimensional (2D) ones in the dimension splitting direction. The reproducing kernel particle method (RKPM) is used to solve each 2D problem, the essential boundary conditions are imposed by penalty method, and the discretized equation is obtained from Galerkin weak form of potential problems. Finite difference method is used in the dimension splitting direction. Then, by combining a series of the equations of the RKPM for solving 2D problems, the final equation of the DSRKPM for 3D potential problems is obtained. Five example problems on regular or irregular domains are selected to show that the DSRKPM has higher computational efficiency than the RKPM and the improved element-free Galerkin method for 3D potential problems.     

An updated interactive boundary layer method for high Reynolds number flows

The quasi‐simultaneous interactive boundary layer (IBL) method is improved with the iterative correction of an inviscid operator. The updated interactive boundary layer method (UIBL) presented in this work, uses the Hess–Smith panel method (HSPM) as an inviscid operator to update the outer flow calculation and the inviscid velocity in the interaction law (IL). The discretization of the Hilbert integral (HI) from the original method is modified to reduce the error introduced by the calculation of the HI in a restricted domain. The method is tested on a flat plate with a small indentation for two‐dimensional, steady, incompressible and laminar flow. The UIBL method is capable to predict the flow separation and reattachment with good accuracy. The accuracy of the results is competitive with the numerical solution of the Navier–Stokes equations (NSE). Copyright © 2005 John Wiley & Sons, Ltd.     

A GPU‐accelerated nodal discontinuous Galerkin method with high‐order absorbing boundary conditions and corner/edge compatibility

Discontinuous Galerkin finite element schemes exhibit attractive features for accurate large‐scale wave‐propagation simulations on modern parallel architectures. For many applications, these schemes must be coupled with nonreflective boundary treatments to limit the size of the computational domain without losing accuracy or computational efficiency, which remains a challenging task. In this paper, we present a combination of a nodal discontinuous Galerkin method with high‐order absorbing boundary conditions for cuboidal computational domains. Compatibility conditions are derived for high‐order absorbing boundary conditions intersecting at the edges and the corners of a cuboidal domain. We propose a GPU implementation of the computational procedure, which results in a multidimensional solver with equations to be solved on 0D, 1D, 2D, and 3D spatial regions. Numerical results demonstrate both the accuracy and the computational efficiency of our approach.     

Swirl flow studies in diffusers using the flux analysis method with orthogonal curvilinear coordinates

A computational method of flux analysis is applied to the study of the swirling flow of different regimes through a diffuser pipe. The flux analysis method is an iterative procedure to construct a system of orthogonal curvilinear co-ordinates consisting of stream-surface and normals following the flow. Examples of the inviscid swirling flow of an incompressible fluid are calculated, and deformations of vortices in a diffuser pipe are studied. For a simple vortex with rigid rotation the deceleration of flow is remarkable on the axis, where the total energy has the lowest level. For a Burgers type vortex, provided the circulation number exceeds a critical value, the rate of deceleration of flow along the axis become more pronounced for only a slight increase of swirl strength, and ultimately a stagnation point appears on the axis. The present method, however, may not be applied to the flow with stagnation point.     

Study of Subsonic Flow Over a TOW 2B Missile

The objective of this investigation is to study the subsonic flow over a missile. In this paper, a model of TOW 2B missile is studied. Two computational approaches are being explored, namely solutions based on the Reynolds-averaged compressible Navier-Stokes equations and solutions based on the inviscid flow (small disturbance theory). The simulations are performed at the Mach number of 0.6, 0.7, 0.8, 0.9 and 1.0 at four angles of attack of 2, 4, 6 and 8 degree. Results obtained from analytical simulation are compared with numerical data. It is found that lift and drag coefficients would go up by increasing of the angle of attack and the Mach number. Trend of changes of the results that obtained from the small disturbance theory is roughly as same as the numeric solution.     

The shape oscillations of a two-dimensional drop including viscous effects

The implementation of a boundary integral method for potential flow is presented for the case of a two-dimensional drop freely oscillating in a vacuum. Calculations using a standard boundary integral formulation and a double-layer potential boundary integral formulation are compared for the case of an inviscid drop with a clean interface. Additional comparisons are made using a least-squares spectral transform method for interpolating and differentiating versus more common methods using cubic splines and central differences. The boundary integral method for potential (i.e. inviscid) flow is extended in two viscous examples to approximate (i) the weak viscous effects in the bulk fluid far from the clean interface, or (ii) the surface viscous effects in an inviscid drop arising from an interface that is highly contaminated with an insoluble surfactant.The addition of an incompressibility constraint, implemented in a least-squares sense, to the standard boundary integral formulation is shown to significantly improve its ability to preserve the conserved quantities of volume and total energy. Nevertheless, the double-layer potential boundary integral formulation, despite its more complicated form, is found to be computationally more efficient than the standard formulation. The use of the least-squares spectral transform method is shown to be more accurate, and in certain conditions more efficient, than using cubic splines and sixth-order central differences for time-evolution of this system. Simulations approximating the damping effects of clean viscous drop are found to be consistent with small deformation theory while the calculations incorporating the damping effects of surface dilatational viscosity are shown to dissipate the energy of the oscillations at a rate that is neither exponential nor algebraic.     

Boundary element analysis of nonlinear water wave problems

A new boundary element technique based on Green's formula is applied to the analysis of nonlinear water wave problems. The problems are formulated mathematically as two-dimensional nonlinear initial-boundary value problems in terms of a velocity potential, assuming the fluid to be inviscid and incompressible, and the flow to be irrotational. In the present paper, two kinds of wave-making problems are analysed: (1) a tsunami generated by sea bed elevation; (2) generation, propagation and run-up on a vertical wall of a solitary wave. Numerical results obtained by the present method are compared with available experimental data and analytical solutions. Excellent agreements are obtained.     

Leading-edge separation from a thick,conical, slender wing at small angless of incidence

Summary The inviscid separated flow past slender rhombic cones at incidence is considered. A complex potential is constructed, in a suitable cross-flow plane, which satisfies the conditions on the wing, at infinity, and on the vortex system which models the separated flow. The results obtained both extend earlier results to small incidence, and explain an anomaly within those results.     

Optically Coated Mirror‐Embedded Microchannel to Measure Hydrophoretic Particle Ordering in Three Dimensions

Three‐dimensional (3D) measurement of the behavior of microfluidic particles is vital for improving their operational efficiency and characterization. In particular, it is important to measure particle motions in 3D for exact characterization of hydrophoresis, which utilizes 3D convective flows for size separation. Herein, the 3D measurement of hydrophoretic particle ordering for the exact characterization of hydrophoresis by using an optically coated mirror‐embedded microchannel is reported. The mirror, ideally at 45°, reflects the side view of the channel and enables 3D positional information to be obtained easily from two different orthogonal‐axis images. With this method, it is shown that hydrophoresis is governed by convective vortices and steric hindrance. It is also observed that hydrophoresis enables 3D particle focusing without sheath flows and accurate flow‐rate control. The mechanism of hydrophoresis is finally verified by conducting a computational simulation and comparing the simulation results with the experimental measurements. The hydrophoretic method can be straightforwardly integrated as a 3D particle‐focusing component in integrated microfluidic systems. The mirror‐embedded channel can also be readily fabricated in a single cast of polydimethylsiloxane, thus offering low‐cost, easy implementation of 3D particle measurement.     

The Cauchy surface wave problem from the viewpoint of a VOF method

Tidal waves produced in deep water by initial local disturbances of the surface are investigated by direct simulations using a volume-of-fluid method. Particular in case of finite disturbances this class of surface-waves has long been regarded as difficult, since the effort is substantial to obtain results in the framework of inviscid linearised potential flow theory. Due to the finite disturbances, the wave trains disappears. This phenomenon cannot be covered by the available closed analytical expressions, and it is hence investigated numerically in detail. The presented numerical approach employs a piecewise linear interface construction method. Cartesian and cylindrical cases are considered, and the numerical results are benchmarked with the analytical expressions and with experimental results. Based on this, a correlation for the maximum wave amplitude as function of the distance is derived.     

Uniqueness of the bounded flow solution in aerodynamics

We discuss uniqueness for steady incompressible inviscid flows past a body with a sharp trailing edge TE, with particular regard to multiconnected (toroidal) 3D wing configurations. Boundedness of the velocity field at TE is enforced by means of a singularity removal principal (Kutta condition). The resulting bounded flow solution is unique for 2D airfoils and 3D conventional wings. For toroidal bodies the flow depends on the available eigensolution which, however, has no direct influence on the lift. In this multiconnected case uniqueness of the bounded solution is shown to depend on the topology of the trailing edge.