Computation of supersonic inviscid flow around wings with a detached shock wave

The numerical method for computing inviscid supersonic flow around the front part of the arbitrary planform wings was presented in [1,2]. The aim of this paper, which is the extension of the previous ones, is to present a numerical method for computing the flow field above the remaining parts of the wing—the wingtip section and the central section of the wing. Here the problems are formulated for a three-dimensional steady gas-dynamic system of equations written in special curvilinear coordinates. Complicated physical domains of the solution are mapped on simple computational domains. For approximation of the differential equations the finite-difference second-order implicit schemes are used. The approximation of the wing surfaces is made with the help of the local cubic splines. According to the obtained algorithms calculations were made for the wing with elliptical planform and thick airfoil at M_∞ = 2 and M_∞ = 3.5 with the angle of attack α = 5°.     

Computational method for the design of ducts

This paper describes an inverse method that applies to two-dimensional, irrotational and incompressible flow, based on a transformation of variables from the physical plane defined by (x, y), to a computational plane defined by the stream function ψ, and the velocity potential, ?. In this new plane, the resulting differential equation describing the flow is relatively simple, and should be solved in a domain that presents always a rectangular shape, facilitating the generation of a numerical grid appropriate to the problem. The solution of this equation gives the distribution of the x coordinate throughout the flow field, while the values of the y coordinate are calculated by integration of the previous result. Some of the numerical details of the procedure will be discussed, with special emphasis on the numerical treatment of the boundary conditions. The application of the described method to the design of a curved duct with an overall deflection of around 90° will be presented, with the aim of showing the potentialities of the method, which is computationally quite simple.     

Multiobjective evolutionary computation for supersonic wing-shapeoptimization

This paper discusses the design optimization of a wing for supersonic transport (SST) using a multiple-objective genetic algorithm (MOGA). Three objective functions are used to minimize the drag for supersonic cruise, the drag for transonic cruise, and the bending moment at the wing root for supersonic cruise. The wing shape is defined by 66 design variables. A Euler flow code is used to evaluate supersonic performance, and a potential flow code is used to evaluate transonic performance. To reduce the total computational time, flow calculations are parallelized on an NEC SX-4 computer using 32 processing elements. The detailed analysis of the resulting Pareto front suggests a renewed interest in the arrow wing planform for the supersonic wing     

A contribution to the computation of transonic supersonic flows over blunt bodies

Stationary inviscid transonic supersonic flowfield around sphrese, ellipsoids and hemisphere-cylinders are calculated. The freestream Mach numbers considered are between $\text{M}{}_{\mathrm{\infty }}\text{= 1.02}$ and $\text{M}{}_{\mathrm{\infty }}\text{= 1.1}$. A special coordinate system is created which is adjusted to the subsonic stagnation point region for transonic freestream Mach numbers. The integration of the governing equations is carried out by means of a time-dependent finite-difference procedure. All the calculations discussed were stable and converged uniquely to the asymptotic stationary state. The influence of an artificial dissipation term added to the finite-difference scheme is studied. It is further shown, that in all cases the limiting characteristics come from the front part of the bodies. The results include bow shocks, sonic lines, characteristics, lines of constant pressure, density and Mach numbers. A comparison is made between such flow quantities which are calculable by analytical functions and the predicted ones. The quality of the results is checked by considering the conservation of the total enthalpy. For some examples the present solutions are compared with other theories and experimental data.     

On the use of several compact methods for the study of unsteady incompressible viscous flow round a circular cylinder

Fourth order accurate methods of mehrstellen type are compared to second order accurate methods for the solution of the unsteady incompressible Navier-Stokes equations in their vorticity stream function formulation. These methods are applied to the study of separated flow around a circular cylinder at several Reynolds numbers. The impulsively started cylinder at Re = 200 and 550, is considered without symmetry restrictions. The features illustrated include the bulge phenomenon at Re = 200, the occurrence of secondary vortices depending on the schemes used at Re = 550, and of twin secondary vortices at Re = 3000. The Karman vortex street is investigated at Re = 200 with a uniform flow in the far field and with superimposed motions of the cylinder. In this last case, a frequency analysis has allowed a critical examination of results pertaining to locked-in situations with respect to confinement effects.     

Numerical simulation of 3D fluid–structure interaction flow using an immersed object method with overlapping grids

The newly developed immersed object method (IOM) [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids. J Comput Phys 2005; 207(1): 151–72] is extended for 3D unsteady flow simulation with fluid–structure interaction (FSI), which is made possible by combining it with a parallel unstructured multigrid Navier–Stokes solver using a matrix-free implicit dual time stepping and finite volume method [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method. In: The second M.I.T. conference on computational fluid and solid mechanics, June 17–20, MIT, Cambridge, MA 02139, USA, 2003; Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method, Special issue on “Preconditioning methods: algorithms, applications and software environments. Comput Struct 2004; 82(28): 2425–36]. This uniquely combined method is then employed to perform detailed study of 3D unsteady flows with complex FSI. In the IOM, a body force term F is introduced into the momentum equations during the artificial compressibility (AC) sub-iterations so that a desired velocity distribution V₀ can be obtained on and within the object boundary, which needs not coincide with the grid, by adopting the direct forcing method. An object mesh is immersed into the flow domain to define the boundary of the object. The advantage of this is that bodies of almost arbitrary shapes can be added without grid restructuring, a procedure which is often time-consuming and computationally expensive. It has enabled us to perform complex and detailed 3D unsteady blood flow and blood–leaflets interaction in a mechanical heart valve (MHV) under physiological conditions.     

Numerical simulation of flow-induced cavity noise in self-sustained oscillations

The generation and near-field radiation of aerodynamic sound from a low-speed unsteady flow over a two-dimensional automobile door cavity is simulated by using a source-extraction-based coupling method. In the coupling procedure, the unsteady cavity flow field is first computed solving the Reynolds- averaged Navier–Stokes (RANS) equations. The radiated sound is then calculated by using a set of acoustic perturbation equations with acoustic source terms which are extracted from the time-dependent solutions of the unsteady flow. The aerodynamic and its resulting acoustic field are computed for the Reynolds number of 53,266 based on the base length of the cavity. The free stream flow velocity is taken to be 50.9 m/s. As first stage of the numerical investigation of flow-induced cavity noise, laminar flow is assumed. The CFD solver is based on a cell-centered finite volume method. A dispersion-relation-preserving (DRP), optimized, fourth-order finite difference scheme with fully staggered-grid implementation is used in the acoustic solver.     

Numerical simulation of the flow around rows of cylinders

Two-dimensional, laminar, unsteady, water flow around cylinder arrays of unequal sizes was simulated using FLUENT™ at Reynolds numbers below 150 (based on the free-stream velocity and first row cylinder diameter). The flow pattern through two rows of inline cylinders showed incomplete vortex shedding behind the first row at a separation distance less than 2d. Karman vortices were not formed and a near-stagnant separated flow region appeared between the aligned cylinders. Cylinders in staggered arrangements shed Karman vortices regardless of the separation between the two rows. This research has shed light on the detailed flow through paper machine forming fabrics.     

Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations

In this paper, we propose a new unified family of arbitrary high order accurate explicit one-step finite volume and discontinuous Galerkin schemes on unstructured triangular and tetrahedral meshes for the solution of the compressible Navier-Stokes equations. This new family of numerical methods has first been proposed in [16] for purely hyperbolic systems and has been called P_NP_M schemes, where N indicates the polynomial degree of the test functions and M is the degree of the polynomials used for flux and source computation. A particular feature of the general P_NP_M schemes is that they contain classical high order accurate finite volume schemes (N=0) as well as standard discontinuous Galerkin methods (M=N) just as special cases, which therefore allows for a direct efficiency comparison.In the application section of this paper we first show numerical convergence results on unstructured meshes obtained for the compressible Navier-Stokes equations with Sutherland’s viscosity law, comparing all third to sixth order accurate P_NP_M schemes with each other. In order to validate the method also in practice we show several classical steady and unsteady CFD applications, such as the laminar boundary layer flow over a flat plate at high Reynolds numbers, flow past a NACA0012 airfoil, the unsteady flows past a circular cylinder and a sphere, the unsteady flows of a compressible mixing layer in two space dimensions and finally we also show applications to supersonic flows with shock Mach numbers up to M_s=10.     

Organised modes and shock-vortex interaction in unsteady viscous transonic flows around an aerofoil: Part I: Mach number effect

The identification of successive stages in the transition of unsteady viscous transonic flow around an aerofoil is carried out by solving the time-dependent Navier-Stokes equations for a compressible fluid in two-dimensional approach. The numerical simulation is carried out at the Mach number range (0.2-0.98). At a fixed Reynolds number (Re=10,000), it is found that this flow undergoes the following four transition steps: It remains steady up to the Mach number values (0.2-0.35) and afterwards it develops spontaneously, without any imposed artificial perturbation, an inherent unsteadiness corresponding to a near-wake von Kármán instability, in the Mach number range (0.35-0.9). It is found that there exists a critical Mach number between the values (0.90-0.95) for which the flow returns to a steady-state. Furthermore, the flow is found to be governed by two instability processes in the Mach number range (0.75-0.8), where, apart from the von Kármán mode (mode I), a lower frequency mode II appears, due to the formation of weakly supersonic alternating zones in the region upstream of the aerofoil, related to the buffeting phenomenon. A triple role played by the increasing compressibility effects to trigger the instability processes, to maintain and to inhibit them in the transonic flow regime is therefore analysed in detail.     

On the application of boundary conditions to time dependent computations for quasi one-dimensional fluid flows

A study is made of the influence of boundary and initial conditions on time-dependent finite-difference solutions of quasi-one-dimensional duct flows. Several questions are addressed: (1) Under what conditions will a time-dependent solution converge to a steady-state supersonic flow, (2) Under what conditions will it converge to subsonic flow and (3) What conditions are necessary to insure a particular unique solution for subsonic flows. The results provide an orientation, or way of thinking, about the role of such conditions in time-dependent solutions of steady-state flows. The results also show that supersonic solutions are readily obtained by holding only pressure and temperature fixed at the duct inlet, and allowing velocity to float. However, subsonic solutions require pressure, temperature and velocity to be fixed at both the duct inlet and exit. If no conditions are held fixed at the exit, the results always converge to the supersonic solution, even if the fixed inlet mass flow is less than critical. In such a case, the program appears to generate additional mass flow between the inlet and throat, sufficient to choke the flow. These results also have some impact on two- and three-dimensional time-dependent solutions where subsonic flow is present on some or all portions of the flow boundaries.     

Hybrid vortex method for high Reynolds number flows around three-dimensional complex boundary

The hybrid vortex method, in which the vortex particle method was combined with the vortex sheet method, was extended to flows around a three-dimensional complex geometry boundary at high Reynolds number of O(10⁶). The computing domain was decomposed into an interior domain of vortex blobs and a thin numerical boundary layer of vortex sheets. The core radii of shedding blobs were related to the size of the vortex sheets. As the result of numerical experiments on the flow over a ship propeller, the hybrid vortex method was found to be acceptable for simulations of unsteady separated flows around a solid body at high Reynolds number, since the computed results showed reasonable agreement with the measured data.     

Numerical analysis of unsteady compressible laminar boundary layer flow

The unsteady compressible laminar boundary layer flow on an arbitrary cylinder due to an incident stream whose velocity varies arbitrarily with time is considered. The method presented is based on the separation between the convective and diffusive quantities. By defining some new variables, the splitting appears rather naturally, and the initialisation problem can be solved without difficulty. The transformed equations are solved with the help of a semi-implicit finite difference scheme which is unconditionally linearly stable. The computations have been applied to flows past a cylinder with constant and fluctuating free-stream velocities.     

An experimental and numerical study of the structure and stability of laminar opposed-jet flows

Experiments, simulations, and numerical bifurcation analysis are used to study the incompressible flow between two opposed tubes with disks mounted at their exits. The experiments in this axisymmetric geometry show that for low and equal Reynolds numbers, Re, at both nozzles, the flow remains symmetric about the plane halfway through the nozzle exits and the stagnation plane is located halfway between the two jets. When Re is increased past a critical value, asymmetric flow fields are obtained even when the momentum fluxes of the two opposed streams are equal. For unequal Re at the jet exits, when the fixed velocity (and the corresponding Reynolds number, Re₁) of one stream is low, the stagnation plane location, SPL, changes smoothly with the Re₂. For high enough Re₁, a hysteretic jump of SPL is observed. Particle Image Velocimetry and flow visualization demonstrate that within the hysteretic range, the two stable flow fields are anti-symmetric. The experimental setup is also studied with transient incompressible flow simulations using a spectral element solver. It is found that to accurately model the flow, we either need to extend the domain into the nozzles, or impose experimental velocity profiles at the nozzle exits. As in the experiments asymmetric flows are obtained past a critical Re. Finally, bifurcation analysis using a Newton-Picard method shows that the transition from symmetric to asymmetric flows results from the loss of stability of the symmetric flows at a pitchfork bifurcation.     

Comparative Assessment of the Hybrid Genetic Algorithm–Artificial Neural Network and Genetic Programming Methods for the Prediction of Longitudinal Velocity Field around a Single Straight Groyne

In the present paper, three-dimensional flow fields around single straight groynes with various lengths have been discussed. The dataset of the flow field is measured in the laboratory using Acoustic Doppler Velocimeter (ADV). Then, the longitudinal velocity field is modelled using a novel hybrid method of Genetic Algorithm based artificial neural network (GAA) that has the ability to automatically adjust the number of hidden neurons. To investigate the proposed method’s performance, the results of GAA is measured and compared with one of the most common genetic algorithm based prediction method, namely genetic programming (GP). It is concluded that that GAA model successfully simulates the complex velocity field, and both the velocity magnitudes and isovel shapes are well predicted by this model. The results show that GAA with RMSE of 0.1236 in test data has a significantly better performance than the GP model with RMSE of 0.2342. In addition, it was founded that the transverse coordinate of the measuring point (Y*) is the most important input variable.     

High-speed compressible flow solutions by adaptive cell-centered upwinding algorithm with modified H-correction entropy fix

Adaptive Delaunay triangulation is combined with the cell-centered upwinding algorithm to analyze high-speed compressible flow problems. The H-correction entropy fix is modified and included in the upwinding algorithm for unstructured triangular meshes to improve the computed shock wave resolution. The solution accuracy is further improved by coupling an error estimation procedure to a remeshing algorithm. Efficiency of the combined procedure is evaluated by analyzing supersonic shocks and shock propagation behaviors for both the steady and unsteady high-speed compressible flows.     

Numerical prediction of unsteady vortex shedding for large leading-edge roughness

A full two-dimensional Navier-Stokes algorithm is used to investigate unsteady, incompressible viscous flow past an airfoil leading edge with surface roughness that is characteristic of ice accretion. The roughness is added to the surface through the use of a Prandtl transposition and can generate both small-scale and large-scale roughness. The focus of the study is a detailed flow analysis of the unsteady velocity fluctuations and vortex shedding induced by the surface roughness. The results of this study are compared to experimental data on roughness-induced transition for the same roughness geometry. A comparison is made between “fluctuation intensity” values from the current algorithm to experimentally determined turbulence intensity values. The effects of the roughness Reynolds number, Re_k, are investigated and compared to experimental values of the critical roughness Reynolds number. The authors speculate that there may be a possible correlation between unsteady roughness-induced vortex shedding and the onset of experimentally measured transitional flow downstream of large-scale roughness.     

The State of the Art in Topology‐Based Visualization of Unsteady Flow

Vector fields are a common concept for the representation of many different kinds of flow phenomena in science and engineering. Methods based on vector field topology are known for their convenience for visualizing and analysing steady flows, but a counterpart for unsteady flows is still missing. However, a lot of good and relevant work aiming at such a solution is available. We give an overview of previous research leading towards topology‐based and topology‐inspired visualization of unsteady flow, pointing out the different approaches and methodologies involved as well as their relation to each other, taking classical (i.e. steady) vector field topology as our starting point. Particularly, we focus on Lagrangian methods, space–time domain approaches, local methods and stochastic and multifield approaches. Furthermore, we illustrate our review with practical examples for the different approaches.     

Vortex visualization in ultra low Reynolds number insect flight

We present the visual analysis of a biologically inspired CFD simulation of the deformable flapping wings of a dragonfly as it takes off and begins to maneuver, using vortex detection and integration-based flow lines. The additional seed placement and perceptual challenges introduced by having multiple dynamically deforming objects in the highly unsteady 3D flow domain are addressed. A brief overview of the high speed photogrammetry setup used to capture the dragonfly takeoff, parametric surfaces used for wing reconstruction, CFD solver and underlying flapping flight theory is presented to clarify the importance of several unsteady flight mechanisms, such as the leading edge vortex, that are captured visually. A novel interactive seed placement method is used to simplify the generation of seed curves that stay in the vicinity of relevant flow phenomena as they move with the flapping wings. This method allows a user to define and evaluate the quality of a seed's trajectory over time while working with a single time step. The seed curves are then used to place particles, streamlines and generalized streak lines. The novel concept of flowing seeds is also introduced in order to add visual context about the instantaneous vector fields surrounding smoothly animate streak lines. Tests show this method to be particularly effective at visually capturing vortices that move quickly or that exist for a very brief period of time. In addition, an automatic camera animation method is used to address occlusion issues caused when animating the immersed wing boundaries alongside many geometric flow lines. Each visualization method is presented at multiple time steps during the up-stroke and down-stroke to highlight the formation, attachment and shedding of the leading edge vortices in pairs of wings. Also, the visualizations show evidence of wake capture at stroke reversal which suggests the existence of previously unknown unsteady lift generation mechanisms that are unique to quad wing insects.     

The impacts of the ALE and hydrostatic-pressure approaches on the energy budget of unsteady free-surface flows

This paper focuses on the energy budget in the calculation of unsteady free-surface flows on moving grids with and without using the ‘arbitrary Lagrangian-Eulerian’ (ALE) formulation or hydrostatic-pressure assumption. The numerical tool is an in-house general-purpose solver for the unsteady, incompressible and homogeneous Navier-Stokes equations in a Cartesian domain. An explicit fractional-step method and co-located finite-volume method are used for the second-order accurate integrations in time and space. The test cases are nonlinear and linear irrotational standing waves, which allow to characterise the impacts of an ALE or Eulerian formulation with moving grids by comparison with the anticipated energy conservation. The study is also extended to viscous waves for varying wave-height-to-water-depth and basin aspect ratios. The Eulerian viewpoint produces marked overdamping as early as in the first wave period for the range of relative wave heights η₀/h > 0.01, where η₀ is the wave semi-amplitude and h is the undisturbed water depth. The hydrostatic calculations misrepresent the evolution of the potential and kinetic energies for h/L > 0.1, where L is the basin length, with spurious modes arising from different initial conditions.