M. Woodgate

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.

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.

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

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.

A grid deformation technique for unsteady flow computations

SUMMARY A grid deformation technique is presented here based on a transfinite interpolation algorithm applied to the grid displacements. The method, tested using a two-dimensional flow solver that uses an implicit dual-time method for the solution of the unsteady Euler equations on deforming grids, is applicable to problems with time varying geometries arising from aeroelasticity and free surface marine problems. The present work is placed into a multi-block framework and fits into the development of a generally applicable parallel multi-block flow solver. The effect of grid deformation is examined and comparison with rigidly rotated grids is made for a series of pitching aerofoil test cases selected from the AGARD aeroelastic configurations for the NACA0012 aerofoil. The effect of using a geometric conservation law is also examined. Finally, a demonstration test case for the Williams aerofoil with an oscillating flap is presented, showing the capability of the grid deformation technique. Copyright

Direct Aeroelastic Bifurcation Analysis of a Symmetric Wing Based on Euler Equations

The application of a sparse matrix solver for the direct calculation of Hopf bifurcation points for the flexible AGARD 445.6 wing in a transonic flow modeled using computational fluid dynamics is considered. The iteration scheme for solving the Hopf equations is based on a modified Newton’s method. Direct solution of the linear system for the updates has previously been restrictive for application of the method, and the sparse solver overcomes this limitation. Previous work has demonstrated the scheme for aerofoil calculations. The current paper gives the first three-dimensional results with the method, showing that an entire flutter boundary for the AGARD 445.6 wing can be traced out in a time comparable to that required for a small number of time-marching calculations, yielding two orders of magnitude improvement when compared to the time-marching approach.

Hopf Bifurcation Calculations for a Symmetric Airfoil in Transonic Flow

The application of a sparse matrix solver for the direct calculation of Hopf bifurcation points arising for an airfoil moving in pitch and plunge in a transonic flow is considered. The iteration scheme for solving the Hopf equations is based on a modified Newton’s method. Direct solution of the linear system for the updates has previously been restrictive for application of the method, and the sparse solver overcomes this limitation. Results of experiments with the approximation to the Jacobian matrix driving the iteration to convergence are presented. Finally, it is shown that an entire flutter boundary for the NACA0012 airfoil can be traced out in a time comparable to that required for a small number of time-response calculations. I. Introduction C OMPUTATIONAL fluid dynamics (CFD) has matured to the point where it is being applied to complex problems in external aerodynamics. Aeroelastic analysis relies on high-fidelity predictions of aerodynamics, particularly for phenomena associated with shock motions or separation. These two observations have motivated the development of CFD-based aeroelastic simulation, a field now being called computational aeroelasticity. Developments in computational aeroelasticity have mainly been focused on time-marching calculations, where the temporal response of a system to an initial perturbation is calculated to determine growth or decay, and from this to infer stability. This type of simulation has developed significantly in the past decade, with efforts concentrating on mesh movement, load and displacement transfer between the aerodynamic and structural grids, 1 and sequencing of solutions. 2,3 Recent and impressive example calculations have been made for complete aircraft configurations. 4,5 The time-marching method will remain a powerful tool in computational aeroelasticity because of its generality. However, the cost of these calculations motivates attempts to find quicker ways of evaluating stability while still retaining the detailed aerodynamic predictions given by CFD. One way of doing this is to boil down the CFD into a reduced-order model that still retains the essence of the aerodynamics. Various approaches have been proposed, with an expansion of the flowfield in a truncated series of modes derived from proper orthogonal decomposition currently receiving much attention. 6 A second approach proposed by Morton and Beran from the U.S. Air Force is to use dynamic systems theory to characterize the nature of the aeroelastic instability and then to use this additional information to concentrate the use of the CFD. Aeroelastic instabilities that are commonly termed flutter are of the Hopf type, where an eigenvalue of the system Jacobian matrix crosses the imaginary axis at the flutter point. A model problem was used to evaluate the approach 7 in which the main difficulties associated with the method (calculation of the Jacobian matrix, solution of the augmented system by Newton’s method, solution of a large sparse linear system) were considered. The method was applied to an aeroelastic system consisting of an airfoil moving in pitch and plunge in Ref. 8. The

A data exchange method for fluid-structure interaction problems

This paper presents and illustrates an interpolation method for the exchange of displacement data between fluid and structural meshes in a fluid-structure interaction simulation. The method is a local method where element volume conservation is central, and does not rely on information from the structural model. Results are evaluated for several two and three dimensional problems. Comparisons with the infinite plate spline method show that the new method gives a more realistic representation of the recovered surface than currently used methods.

Implicit method for the time marching analysis of flutter

This paper evaluates a time marching simulation method for flutter which is based on a solution of the Euler equations and a linear modal structural model. Jameson’s pseudo time method is used for the time stepping, allowing sequencing errors to be avoided without incurring additional computational cost. Transfinite interpolation of displacements is used for grid regeneration and a constant volume transformation for inter-grid interpolation. The flow pseudo steady state is calculated using an unfactored implicit method which features a Krylov subspace solution of an approximately linearised system. The spatial discretisation is made using Osher’s approximate Riemann solver with MUSCL interpolation. The method is evaluated against available results for the AGARD 445.6 wing. This wing, which is made of laminated mahogany, was tested at NASA Langley in the 1960s and has been the standard test case for simulation methods ever since. The structural model in the current work was built in NASTRAN using homogeneous plate elements. The comparisons show good agreement for the prediction of flutter boundaries. The solution method allows larger time steps to be taken than other methods.