K. J. Badcock

Elements of computational fluid dynamics on block structured grids using implicit solvers

Abstract This paper reviews computational fluid dynamics (CFD) for aerodynamic applications. The key elements of a rigorous CFD analysis are discussed. Modelling issues are summarised and the state of modern discretisation schemes considered. Implicit solution schemes are discussed in some detail, as is multiblock grid generation. The cost and availability of computing power is described in the context of cluster computing and its importance for CFD. Several complex applications are then considered in light of these simulation components. Verification and validation is presented for each application and the important flow mechanisms are shown through the use of the simulation results. The applications considered are: cavity flow, spiked body supersonic flow, underexpanded jet shock wave hysteresis, slender body aerodynamics and wing flutter. As a whole the paper aims to show the current strengths and limitations of CFD and the conclusions suggest a way of enhancing the usefulness of flow simulation for industrial class problems.

Investigation of three-dimensional dynamic stall using computational fluid dynamics

Numerical simulation of three-dimensional dynamic stall has been undertaken using computational fluid dynamics. The full Navier–Stokes equations, coupled with a two-equation turbulence model, where appropriate, have been solved on multiblock strucured grids in a time-accurate fashion. Results have neen obtained for wings of square planform and of NACA 0012 section. Efforts have been devoted to the accurate modeling of the flow near the wing tips, which, for this case, were sharp without tip caps. The obtained results revealed the time evolution of the dynamic stall vortex, which, for this case, takes the shape of a capital omega+spanning the wing. The obtained results compare well against experimental data both for the surface pressure distribution on the wing and the flow topology. Of significant importance is the interaction between the three-dimensional dynamic stall vortex and the tip vortex. The present results indicate that once the two vortices are formed both appear to originate from the same region, which is located near the leading edge of the tip. During the ramping of the wing, the two vortices grow significantly in size. The dynamic stall vortex dettaches from the wing in the inboard region but remains close to the wing’s leading edge near the tip. The overall configuration of the developed vortical system takes a form. To our knowledge, this is the first detailed numerical study of three-dimensional dynamic stall appearing in the literature.

Accelerating the Numerical Generation of Aerodynamic Models for Flight Simulation

The generation of a tabular aerodynamic model for design related flight dynamics studies, based on simulation generated data, is considered. The framework described accommodates two design scenarios. The first emphasizes the representation of the aerodynamic nonlinearities, and is based on sampling. The second scenario assumes incremental change from an initial geometry, for which a hi-fidelity model from the first scenario is available. In this case data fusion is used to update the model. In both cases, Kriging is used to interpolate the samples computed using simulation. A commercial jet test case, using DATCOM as a source of data, is computed to illustrate the sampling and fusion. Future application using Computational Fluid Dynamics as the source of data is considered.

Comparison of Measured and Block Structured Simulation Results for the F-16XL Aircraft

This paper presents a comparison of the predictions of three Reynolds-averaged Navier-Stokes codes for flight conditions of the F-16XL aircraft that feature vortical flow. The three codes, ENSOLV, parallel multiblock, and propulsion aerodynamics branch 3-D unsteady Reynolds-averaged Navier-Stokes, solve on structured multiblock grids. Flight data for comparison were available in the form of surface pressures, skin friction, boundary-layer data, and photographs of tufts. The three codes provided predictions that were consistent with expectations based on the turbulence modelling used, which was k-e, k-w with vortex corrections, and an algebraic stress model. The agreement with flight data was good, with the exception of the outer wing primary vortex strength. The confidence in the application of the computational fluid dynamics codes to complex fighter configurations increased significantly through this study.

Evaluation of Dynamic Derivatives Using Computational Fluid Dynamics

This paper focuses on the evaluation of the dynamic stability derivative formulation. The derivatives are calculated using the Euler and Reynolds-averaged Navier–Stokes equations, and a time-domain solver was used for the computation of aerodynamic loads for forced periodic motions. To validate the predictions, two test cases are used. For the standard dynamic model geometry, a database of dynamic simulations illustrates the effects of the systematic variation of motion and fluid parameters involved. A satisfactory agreement was observed with available experimental data, and the dependency of dynamic derivatives on a number of parameters, such as Mach number, mean angle of attack, frequency, and amplitude, was assessed. For the transonic cruiser wind-tunnel geometry, static and unsteady aerodynamic characteristics were validated against experimental measurements. The ability of models based on the dynamic derivatives to predict large-amplitude motion forces and moments was assessed. It was demonstrated that the dynamic derivative model does not represent all of the important effects due to aerodynamics.

Linear frequency domain and harmonic balance predictions of dynamic derivatives

Dynamic derivatives are used to represent the influence of the aircraft motion rates on the aerodynamic forces and moments needed for studies of flight dynamics. The use of computational fluid dynamics has potential to supplement costly wind-tunnel testing. The paper considers the problem of the fast computation of forced periodic motions using the Euler equations. Three methods are evaluated. The first is computation in the time domain, which provides the benchmark solution in the sense that the time-accurate solution is obtained. Two acceleration techniques in the frequency domain are compared. The first uses a harmonic solution of the linearized problem, referred to as the linear frequency-domain approach. The second uses the harmonic balance method, which approximates the nonlinear problem using a number of Fourier modes. These approaches are compared for the ability to predict dynamic derivatives and for computational cost. The NACA 0012 aerofoil and the DLR-F12 passenger jet wind-tunnel model are the test cases. Compared to time-domain simulations, an order of magnitude reduction in computational costs is achieved and satisfactory predictions are obtained for cases with a narrow frequency spectrum and moderate amplitudes using the frequency-domain methods.

Fast Prediction of Transonic Aeroelastic Stability and Limit Cycles

The exploitation of computational fluid dynamics for aeroelastic simulations is mainly based on time-domain simulations. There is an intense research effort to overcome the computational cost of this approach. Significant aeroelasticeffectsdrivenbynonlinearaerodynamicsincludethetransonic flutterdipandlimit-cycleoscillations.The paper describes the use of Hopf bifurcation and center manifold theory to compute flutter speeds and limit-cycle responses of wings in transonic flow when the aerodynamics are modeled by the Euler equations. The cost of the calculations is comparable to steady-state calculations based on computational fluid dynamics. The paper describes twomethodsfor findingstabilityboundariesandthenanapproachtoreducingthefull-ordersystemtotwodegreesof freedom in the critical mode. Details of the three methods are given, including the calculation of first, second, and thirdJacobiansandthesolutionofsparselinearsystems.ResultsfortheAGARDwing,asupercriticaltransporttype of wing, and the limit-cycle response of the Goland wing are given. Nomenclature A = Jacobian matrix of R with respect to w B, C = second and third Jacobian operators h = (scalar) increment for finite differences F = quadratic and higher terms in R G = Taylor coefficients of the residual restricted to the critical eigenspace H = Taylor coefficients of the residual restricted to the noncritical eigenspace f = convective flux discretization kij = coefficients in center manifold expansion of y P = right eigenvector of A, P1 � iP2 Q = left eigenvector of A, Q1 � iQ2 qs = constant scaling vector for the augmented system R = residual vector v = vector for the matrix-free product w = vector of unknowns y = part of w in the noncritical eigenspace z = part of w in the critical eigenspace � t = time step � i = sequence of eigenvalues in the inverse power method � = bifurcation parameter (dynamic pressure) ! = frequency of critical eigenvalue or shift for the inverse power method Subscripts

Driving Mechanisms of High-Speed Unsteady Spiked Body Flows, Part 2: Oscillation Mode

The driving mechanism of the unsteady e ow mode pulsation arising over axisymmetric spiked bodies has been analyzed by using computational e uid dynamics as a tool. Laminar, axisymmetric e ow at Mach 2.21 and Reynolds number (based on the blunt-body diameter) of 0.12 £106 was simulated by a spatially and temporally second-order-accurate e nite volume method. The model geometry was a forward facing cylinder of diameter D equipped with a spike of length L/D=1.00. After reviewing previous pulsation hypotheses, the numerical results were analyzed in detail. A new driving mechanism was proposed, its main features being the creation of a vortical region in the vicinity of the foreshock-aftershock intersection causing mass ine ux into the dead-air region, the existence of supersonic e ow within the dead-air region, the liftoff of the shear layer from the spike tip, and the collision of the recirculated and penetrating e ows within the expanded separated region.

Implicit Harmonic Balance Solver for Transonic Flow with Forced Motions

The computation of the aerodynamic forces arising from forced periodic motions is required for the generation of dynamic terms in models for flight simulation. The periodicity can be used to avoid using fully unsteady calculations by using the harmonic balance method. The current paper develops an implicit solver for the harmonic balance equations. The method is tested on two transonic test cases and evaluation is made against the unsteady simulation results. The first caseis for the pitching NACA 0012aerofoil. The second is for forced pitching of the F-5 wing with a wing tip launcher and missile. A reduction in computational time by one order of magnitude compared with the unsteady solver is obtained. Nomenclature A = matrix in frequency domain equation c = chord D = matrix in harmonic balance equation E = transformation matrix between frequency and time domains e = energy F, G, H = convective fluxes I = residual of semidiscrete system I = identity matrix k = reduced frequency nH = number of harmonics p = pressure R = residual vector T = period t = time u, v, w = Cartesian velocity components W = conserved variables � = angle of attack � t = pseudo time step

Solution of the unsteady Euler equations using an implicit dual-time method

An unfactored implicit time-marching method for the solution of the unsteady two-dimensional Euler equations on deforming grids is described. The present work is placed into a multiblock framework and e ts into the development of a generally applicable parallel multiblock e ow solver. The convective terms are discretized using an upwind total variation diminishing scheme, whereas the unsteady governing equations are discretized using an implicit dual-time approach. The large sparse linear system arising from the implicit time discretization at each pseudotime step is solved efe ciently by using a conjugate-gradient-type method with a preconditioning based on a block incomplete lower-upper factorization. Results are shown for a series of pitching airfoil test cases selected from the AGARD aeroelastic cone gurations for the NACA 0012 airfoil. Comparisons with experimental data and previous published results are presented. The efe ciency of the method is demonstrated by looking at the effect of a number of numerical parameters, such as the conjugate gradient tolerance and the size of the global time step and by carrying out a grid ree nement study. Finally, a demonstration test case forthe Williamsairfoil (Williams, B. R., “ An Exact Test Case for the Plane Potential Flow About Two Adjacent Lifting Aerofoils,” National Physical Lab., Aeronautical Research Council, Research Memorandum 3717, London, 1973 )with an oscillating e ap is presented, highlighting the capability of the grid deformation technique.