Ann L Gaitonde

A Dual Time Method for the Solution of the Unsteady Euler Equations

A “dual-time” method for solving the three-dimensional Euler equations describing the compressible flow about wings undergoing arbitrary motions and deformations is presented. A finite-volume formulation is chosen where the volumes distort as the wing moves or deforms. Independent motion of the inner and outer boundaries of the grid is permitted with a sequence of grids generated using transfinite interpolation. An implicit real-time discretisation is used, and the equations are integrated in a fictitious pseudo time. This approach allows the real-time step to be chosen on the basis of accuracy rather than stability. It also permits the acceleration techniques commonly used to speed up steady flow calculations to be used when marching in pseudo time, without compromising real-time accuracy. A two-dimensional version of the method has also been developed and results for both two and three-dimensional transonic flows are presented and compared with experimental data where available.

A dual-time method for two-dimensional unsteady incompressible flow calculations

A method for computing unsteady incompressible viscous flows on moving or deforming meshes is described. It uses a well-established time-marching finite-volume flow solver, developed for steady compressible flows past rigid bodies. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. Such methods can be extended to steady incompressible flows using an artificial compressibility scheme. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real-time discretization and a dual-time approach, which uses a technique similar to the artificial compressibility scheme. Results are presented for test cases on both fixed and deforming meshes. Experimental, numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved.

A structure-coupled CFD method for time-marching flutter analysis

Aeroelastic analysis is a critical area of the aircraft design process, as a good understanding of the dynamic behaviour of the wing structure is essential to safe operation of the vehicle. The inevitable inaccuracies present in the modelling of such phenomena impose mass penalties, as large safety margins are necessitated, which in turn lead to overly stiff designs. In an effort to reduce the uncertainty in analysis methods, fully coupled CFD and structural models are under widespread development. We describe the results produced by such a system for a series of test cases based on the AGARD445.6 and MDO wings

AN UNFACTORED IMPLICIT MOVING MESH METHOD FOR THE TWO‐DIMENSIONAL UNSTEADY N–S EQUATIONS

An unfactored implicit time-marching method for the solution of the unsteady two-dimensional Reynolds-averaged thin layer Navier–Stokes equations is presented. The linear system arising from each implicit step is solved by the conjugate gradient squared (CGS) method with preconditioning based on an ADI factorization. The time-marching procedure has been used with a fast transfinite interpolation method to regenerate the mesh at each time step in response to the motion of the aerofoil. The main test cases examined are from the AGARD aeroelastic configurations and involve aerofoils oscillating rigidly in pitch. These test cases have been used to investigate the effect of various parameters, such as CGS tolerance and laminar/turbulent transition location, on the accuracy and efficiency of the method. Comparisons with available experimental data have been made for these cases. In order to illustrate the application of the mesh generator and flow solver to more general flows where the aerofoil deforms, results for an NACA 0012 aerofoil with an oscillating trailing edge flap are also shown.

Prescribed velocity method for simulation of aerofoil gust responses

A new method for modeling the interaction of an aerofoil with a gust using a prescribed velocity approach, called the split velocity method is presented. This approach effectively rearranges the governing equations into a form that allows for more efficient calculation and includes both the effect of the gust on the aerofoil and the effect of the aerofoil on the gust. The convection of gusts, through the domain from the far field, is investigated using the new method for a range of 1-cosine gusts. The results obtained are compared to an existing prescribed velocity approach called the field velocity method, which neglects the effect of the aerofoil on the gust. The two prescribed velocity approaches agree well for longer gusts. For shorter gusts where the gust length is close to the chord of the aerofoil, the new approach produces better results. Details of a linearized version of the split velocity method are also given. The linearized version is shown to agree well with the full method for cases when th...

Aeroelastic analysis through linear and non-linear methods: a summary of flutter prediction in the PUMA DARP

This paper presents a comparison of linear and non-linear methods for the analysis of aeroelastic behaviour and flutter boundary prediction. The methods in question include NASTRAN and ZAERO, based on linear aerodynamics, and the non-linear coupled CFD-CSD methods RANSMB and PMB, developed at the Universities of Bristol and Glasgow respectively. The test cases used for this comparison are the MDO and AGARD 445.6 weakened wing. In general, it was found that the non-linear methods demonstrate excellent agreement with respect to pressure distributions, deflections, dynamic behaviour, and flutter boundary locations for both cases. This is in contrast to previous studies involving similar methods, where notable differences across the MDO span were found, and is taken to imply good performance of the CFD-CSD interpolation schemes employed here. While the linear methods produce similar flutter boundaries to the coupled codes for the aerodynamically simple AGARD 445.6 wing, results for the transonic ‘rooftop’ MDO wing design did not agree as well.

A 2D Navier-Stokes method for unsteady compressible flow calculations on moving meshes

A moving mesh method for the computation of compressible viscous flow past deforming and moving aerofoils is described. It is based on a well established time-marching finite-volume scheme, which has been widely used for steady compressible flows. An implicit real-time discretisation is used and the equations are integrated via a dual-time scheme. This involves the introduction of derivatives of a fictitious pseudo time. The solution at each real-time step involves seeking solutions which are steady-state solutions in pseudo time. This approach decouples the stability and accuracy limitations of the scheme and permits large real-time steps to be chosen. Also well-proven convergence acceleration techniques developed for steady flows such as local-time stepping, residual averaging and multigrid may be used in the pseudo-time stepping scheme without compromising real-time accuracy. A sequence of body-conforming grids and corresponding grid speeds is required, where the inner and outer boundaries of the grid move independently. The required grids and speeds are found using a transfinite interpolation technique. Applications of the method to external compressible flows are shown, including results from a parallel version of the computer program

Reduced order modelling for aeroelastic aerofoil response to a gust

Results for the gust response of the FFAST aerofoil calculated using computational fluid dynamics (CFD) and reduced order models (ROM) constructed using the eigenvalue realization algorithm (ERA) are presented. The CFD results are calculated using the Split Velocity Method (SVM). The SVM method rearranges the Euler equations retaining all terms after splitting the velocity into a prescribed gust and the remainder. SVM allows for the capture of the effects of the gust on the aerofoil and the aerofoil on the gust. In addition to the full SVM method a linearized version, used to generate the data for producing the ROM, is also presented. For larger gusts with over 5 ◦ changes in effective angle of attack it was found that the non-linearities due to large shock motions, which are not captured by the standard ROM, became important. So in addition a version of the ROM corrected with steady state information is presented, this gives more accurate results for longer gusts particularly when used in aeroelastic simulations.

A two-dimensional linearized unsteady Euler scheme for pulse response calculations

Abstract An unsteady linearized Euler scheme for use on moving meshes is presented. This is derived from a scheme for the full non-linear unsteady Euler equations. This scheme is based on the Jameson cell-centred scheme, but is time-accurate and includes the necessary terms to account for grid motion. It is assumed that the unsteadiness in the flow and mesh is small. Using this assumption the discrete unsteady Euler equations are linearized about the full, non-linear, steady mean flow. The resulting equations are solved in the work presented here using a dual-time scheme. In the basic scheme no assumptions are made about the form of the perturbations other than that they are small. This permits the direct calculation of non-periodic flows, e.g. pulse responses. Linear pulse responses are a useful tool as they can be used to calculate the flow due to general inputs. The equations that would result from the assumption of harmonic flow are also derived. Results are presented for heave, pitch and ramp test cases and compared to full non-linear Euler results calculated using a dual-time scheme.

Investigation of Structural Modelling Methods for Aeroelastic Calculations

Flutter clearance and aeroelastic analysis of aircraft is of vital importance, as avoidance of the non-linear phenomena involved is critical to the safe operation of an aircraft. However, the complexity of the behaviour involved makes detailed analysis extremely dicult and, hence, in practise simplied linear methods have been used for all previous and current designs. However, the non-linear nature of the actual responses means that such methods have inherent inaccuracies, which requires the use of large safety margins, in turn imposing signican t weight penalties. Increasing computational power has allowed more complicated forms of analysis and, hence, fully coupled CFD and CSD solution methods are currently under development, allowing more accurate results to be generated. This paper considers a coupled full non-linear CFD solver and structural dynamic solver, and compares dieren t approaches to modelling the structural behaviour in this method. Specically mode extraction and mass and stiness matrix analysis are considered. The former is more commonly used, the latter more complex and computationally intensive. However, it is demonstrated here that for small to moderate structural grid sizes, the latter has signican t advantages in generality of approach and response accuracy and consistency, at negligible computational expense. The prediction and analysis of the aeroelastic behaviour of an aircraft is of critical importance, as any unstable response to time varying or impulsive loads (gusts, etc.) may rapidly lead to disastrous structural failure. This danger requires signican t safety margins, proportionate to the level of inaccuracy and uncertainty inherent in the prediction method used. This increased safety margin results in unnecessary stiness, and hence a higher structural weight. Methods currently common for aeroelastic analysis in industry involve the use of linear techniques for both utter and aeroservoelastic problems, allowing uncoupling of the aerodynamic and structural equations. This reduces the accuracy of the methods, particularly in the transonic igh t regime where the aerofoil thickness (usually neglected) plays a crucial role in the development and movement of shock waves. They are also inapplicable to non-linear instabilities such as control surface buzz and limit cycle oscillation. A new approach is therefore under widespread development, consisting of simulation by non-linear CFD analysis coupled to a dynamic structural representation of the body under investigation (e.g. references 1{14). This allows time-accurate non-linear analysis of dynamic behaviour, leading to much more accurate methods for utter investigation. The majority of such methods rely for prediction of structural motion on a summation of the displacement produced by a limited number of modes derived from an analysis of the structure by a commercial FE solver. Whilst this provides a simple method for producing a structural model within the main CFD algorithm that may be rapidly solved, it does mean that the delit y of the model is compromised, particularly at high frequencies. As an alternative, the full mass and stiness matrices may be extracted from the FE analysis, and once inverted, used to evolve the structure forward in time (this method is henceforth referred to as ‘MK’ for brevity). This increases the complexity of the analysis, but retains the higher frequency properties.