[American Institute of Aeronautics and Astronautics 24th AIAA Applied Aerodynamics Conference - San Francisco, California ()] 24th AIAA Applied Aerodynamics Conference - Verification and Validation of the IMPNS Flow Solver Using the X-15 Flight Experiment

American Institute of Aeronautics and Astronautics 1 Verification and Validation of the IMPNS Flow Solver Using the X-15 Flight Vehicle Bernd Wagner*, Jason Bennett†, Michael Mifsud‡ and Scott Shaw§ School of Engineering, Cranfield University, Bedford, United Kingdom, MK43 0AL An extensive verification and validation study of Cranfield University's IMPNS flow solver has been performed for a complete hypersonic air-vehicle configuration. A hierarchical approach was adopted in which the vehicle aerodynamics were decomposed and related flow phenomenon studied. Using this hierarchy bench mark solutions and laboratory experiments were identified that provide the basis of the verification and validation exercises. Detailed comparisons of iterative and grid converged IMPNS computations with benchmark solutions and wind tunnel measurements are presented. Computations of the X- 15 wind tunnel and flight experiments are described and comparison is made with measured surface static pressures, off surface total pressure measurements and Schlieren flow visualization that demonstrate the reliability and capability of the IMPNS flow solver for complex configurations. I. Introduction he last decade has seen continuing development of high-performance computers and supporting infrastructure together with progress in the understanding of the application of computational fluid dynamics (CFD) to increasingly complex aerodynamic problems. This is leading to the deployment of CFD in support of the engineering design function at earlier points in the product design cycle creating a number of new challenges that have not been fully addressed by the research community. Chief amongst which is how to address analysis error and uncertainty in the design process. Computational Fluid Dynamics (CFD) is fundamentally a deterministic process, the continuum model equations have a unique solution and this solution can be approached asymptotically using numerical methods provided sufficient care is taken in the choice and application of the discretization scheme. However, errors inherent in the mathematical modeling are uncertain and depend upon the physical phenomena excited by the flow. A common practice in the engineering application of CFD is to assume a nominal value for the error and uncertainty. At the detailed design stage this poses no significant problems as the physical phenomena are unlikely to be significantly altered by the allowed design variations and consequently the level of modeling uncertainty can be considered uniform. Thus despite the presence of error and uncertainty no special treatment is required for credible design choices. Earlier in the process, at the concept stage the design choices may exhibit a wide variation of physical phenomena all of which have differing levels of modeling uncertainty and clearly the assumption of a uniform, nominal value is erroneous. Since almost 80% of lifecycle cost is determined during the initial stages of product development, accounting for error and uncertainty is the key to using high-fidelity analysis tools early in the design process where they can have maximum influence. It is in this light that a new approach is needed. Clearly when trying to establish whether a computational simulation is reliable or not we require some knowledge of the 'right answer'. Identifying what is meant by the 'right answer' is difficult, but raises a number of important and fundamental issues. The purpose of any computational simulation is to predict the behavior of a physical system and so at the most fundamental level the reliability of the simulation can be tested by comparing with the physical reality. However, real world experimental measurements are themselves subject to error and uncertainty and so may not provide a reliable basis for comparison. Instead we can consider comparison with laboratory experiments which offer a more controllable environment but may introduce new sources of error and are * M.Sc. student, Department of Aerospace Science. † Ph.D. student, Department of Aerospace Science. ‡ Ph.D. student, Department of Aerospace Science. § Lecturer, Department of Aerospace Science, AIAA Member. T 24th Applied Aerodynamics Conference 5 - 8 June 2006, San Francisco, California AIAA 2006-3870 Copyright © 2006 by Cranfield University. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. American Institute of Aeronautics and Astronautics 2 in themselves models of the physical reality. Provided that experimental error and uncertainty is quantified and understood then we may use statistical techniques to help make meaningful comparisons between the experimental and computed data. The computational simulation is a solution of an approximation to the physical reality and so we could suggest that it is the exact solution of the model equations that is the 'right answer', although clearly the exact solution may not reproduce the physical reality upon which the governing equations are based. Unfortunately the exact solution is usually unknown, at least for physically meaningful cases. We must also recognize that the numerical solution of the finite difference equations is not in itself necessarily an exact solution of the governing partial differential equations due to the presence of truncation errors in the spatial and temporal discretization, errors in the specification of the initial and boundary conditions and errors related to the finite precision of the computer hardware upon which the computations are performed. Fortunately, using the mathematical concepts of consistency, stability and convergence we can demonstrate that in the limit ∆x → 0 and ∆t → 0 the numerical solution of the discretized equations should tend towards the exact solution of the partial differential equations upon which they are based and this can be used to develop strategies and approximations to the continuum solution. Recognition that there may be more than one meaning of the phrase 'the right-solution' when dealing with computational simulations has led to the development of two specific concepts for formally assessing solution accuracy; verification and validation. Verification is a process that can be followed to demonstrate and understand the extent to which the governing equations are satisfied by the numerical solution while validation provides a measure of how well the mathematical model approximates the physical reality. In this paper we employ the verification and validation ideas of Roache(1), Oberkampf et al(2,3) to perform a rigorous assessment of the accuracy and reliability of Cranfield University's IMPNS flow solver. Computations are performed for a series of simple flows that isolate individual aspects of the expected flow physics. For these problems analytical and semi-analytical benchmark solutions are available that provide the basis of a verification study. More complex flows involving coupled physical phenomenon are addressed using comparisons with laboratory experiments. Having demonstrated the credibility of the IMPNS flow solver computations ar then performed for the X-15 flight experiment. Comparison is made with surface and off-surface pressure measurements made during wind tunnel and flight testing of the aircraft. II. Numerical Model A. Governing Equations The governing equations are the Favre-averaged Navier-Stokes equations describing the conservation of mass, momentum and energy for the steady flow of a compressible ideal Newtonian fluid. This system of equations can be written in conservative integral form as, 0F =∫ S ndS (1) in which nF is the flux through the surface of an arbitrary control volume bounded by the surface S. The flux may be conveniently split into convective, invnF , and diffusive, vis nF , terms so that, visinv Re 1 nnn FFF −= (2) The convective flux invnF may be obtained from, TTTinv ,0)ˆ (0,) ˆ( nunF pn += ΦΦΦΦ (3) where nˆ is the unit vector normal to the surface of the control volume and T ,1)0,0,0(0,p+= QΦΦΦΦ (4) American Institute of Aeronautics and Astronautics 3 The diffusive flux may be calculated from,                                 ∇ +                 = T kn 0 0 0 0 u k j i 0 nF T ˆ ˆ ˆ ˆ vis ττττ (5) where the components of the shear-stress, , are ( ) { }3,2,1, 3 2 ∈      ∇−∂+∂+= jiuuu TijijjiTij δµµτ (6) This system of equations is closed by assuming that air is an ideal Newtonian fluid for which Stokes hypothesis is valid. The coefficients of thermal conductivity and molecular viscosity are modeled using Sutherland's law. For a structured grid containing hexahedral cells a discrete form of Eq. 1 can be obtained, 0FFF =++ + − + − + − ½,, ½,, ½,, ½,, ,½, ,½, ][][][ kji kjikji kjikji kji sss ζζηηξξ (7) where ξF , ηF and ζF are the fluxes through cell faces aligned with the streamwise and cross-flow directions respectively. By neglecting the viscous flux in the streamwise direction a reduced form of the Navier-Stokes equations, the Parabolized Navier-Stokes equations (PNS), can be obtained, 0FFF =++ + − + − + − ½,, ½,, ½,, ½,, ,½, ,½, inv ][][][ kji kjikji kjikji kji sss ζζηηξξ (8) In Eq. 8 the cross-flow fluxes remain unaltered, except for neglected streamwise contributions to derivatives in the viscous fluxes. Provided a suitable discretisation of the streamwise flux is used to maintain stability, Eq. 8 may be solved using an implicit multiple-sweep space-marching methodology, where flow information is propagated in both directions by marching in both the positive and negative streamwise direction. If the flow outside of the boundary layer is supersonic in the streamwise direction and there is no streamwise separation present, an approximation due to Vigneron et al.(4) enables the PNS equations to be solved by a well- defined (stable) single-sweep space-marching procedure in the streamwise direction. Vigneron obtained such a procedure by modifying the pressure gradient in the subsonic portion of the boundary layer to suppress the elliptic character of Eq. (8); this can be done by replacing the streamwise convective term invξF by, TTinvvig ,0)ξˆ(0,])ξˆ,(1[)( pmFQF ωξξ −−= (9) where         −+ = 2T 2T )ξˆ)(1(1 )ξˆ(0.91,Min)ˆ,( m m m γ γ ω ξξξξ (10) B. Numerical Method In order to obtain an efficient solution of the flow governing Eq. 9 without stability restriction on the maximum size of the space step a pseudo time term is introduced, American Institute of Aeronautics and Astronautics 4 0FFFQ =+++ + − + − + − ½,, ½,, ½,, ½,, ,½, ,½, inv ,, ][][][ kji kjikji kjikji kji kji sssV ζζηηξξ∂τ ∂ (11) As τ→∞ the pseudo-time term vanishes and the governing equations are recovered. 1. Spatial Discretization The streamwise convective flux is evaluated using a simple first order upwind scheme for single sweep calculations. For multi-sweep calculations in which flow disturbances are permitted to propagate in both upstream and downstream directions the flux vector splitting proposed by Steger and Warming(5) is employed. A high-resolution finite volume scheme based on Osher and Solomon's(6) upwind flux difference splitting is employed for the spatial discretisation of the convective flux terms in the cross flow direction. To enhance the spatial resolution in the cross-flow plane, a nominally third-order accurate slope limited MUSCL interpolation of the primitive variables is employed. The viscous fluxes are evaluated using a second-order finite volume scheme. Flow gradients required in the evaluation of the stress tensor and heat flux are obtained using Gauss' theorem and auxiliary cells constructed around the face for which the data is required. 2. Pseudo-time discretization Following spatial discretization a semi-discrete system of ordinary differential equations is obtained, 0Q =+ kji kji RV ,, ,, ∂τ ∂ (12) Eq. 12 is solved using an efficient implicit method that employs a BILU pre-conditioned Krylov sub-space method(7). The multigrid Full Approximation Scheme (FAS) is used to further accelerate convergence to the steady state(8). Further details of the numerical method and its applications can be found in references 9-15. III. Verification and Validation Hierarchy The fundamental strategy of validation is to assess how accurately the computational results compare with the experimental data, with quantified error and uncertainty estimates for both. This strategy employs a hierarchical methodology that segregates and simplifies the physical and coupling phenomena involved in the complex engineering system of interest. In the current work a hierarchy containing three distinct levels is employed. At the top of the hierarchy is the system of interest, in this case the X-15 flight vehicle, Fig. 1. This vehicle can be decomposed into a number of physical components; forebody, fuselage, wing, tail-plane, tail-fin each of which has related flow physics. The flows related to the individual components provide the second tier of the validation hierarchy, while the final tier is provided by simplifying the physical coupling further by considering individual aspects of the flow physics in isolation. Having developed a suitable hierarchy upon which to base the verification and validation study related test cases are identified. For verification we require the identification of bench mark solutions. These solutions can be obtained in a number of ways ranging from analytical solutions of the governing partial differential equations to numerical solutions obtained by independent workers, see Fig. 2. Validation is achieved by comparison of computed data with experimental measurements. As was the case with verification, there is a hierarchy of comparisons. At the top of the hierarchy is comparison with flight measurements. While such data provides comparison with the real physical system there is clearly difficulty in establishing initial Figure 1. The X-15 Flight Vehicle American Institute of Aeronautics and Astronautics 5 and boundary conditions with certainty and in making accurate measurements. This uncertainty can lead to significant differences between the computed and measured data. Laboratory experiments generally represent a move away from the physical flight conditions, typically there are problems in obtaining the physical similarity parameters such as Reynold's number, however there is generally much greater control over the physical environment and so uncertainty can be reduced significantly. This typically leads to opportunity for improved correlation of measured and computed data. In the present work we have employed a range of bench-mark solutions, wind-tunnel and flight test data to verify and validate the IMPNS solver. A summary of the computations that have been performed is presented in Fig. 3. IV. Results A. Verification using bench mark solutions A number of simple inviscid and viscous flows for which analytical and semi-analytical benchmark solutions can be obtained were computed to demonstrate the capability of the IMPNS flow solver for flows exhibiting isolated flow physics. The individual calculations reflect the hierarchy of benchmark solutions illustrated in Fig. 2. Figure 2. Hierarchy of benchmark solutions Figure 3. Verification and Validation Cases Easier to find Increasingly Accurate Analytical Solution of the PDE's Analytical Solutions of related ODE's Highly accurate numerical solutions of the PDE's Similar numerical solutions of the PDE's American Institute of Aeronautics and Astronautics 6 1. Oblique Shock Wave The first verification case concerns the computation of a two-dimensional inviscid supersonic flow over a compression surface set at an angle of 15° to the M∞ = 2.5 free stream and demonstrates the reliability of the software for oblique shock waves. Extensive iterative and grid convergence studies were undertaken before data was compared with the theoretical result. The finest grid considered contained approximately 80,000 nodes. The main results are summarised in Table 1, while a visualization of the computed flow is shown in Fig. 4. The visualization shows the expected result, two regions if uniform flow separated by a discontinuity. In the present computations the shock is relatively thick, no effort has been made to align species of grid lines with the flow features. This leads to numerical artefacts, non uniform pressure and temperature, in the solution near the solid surface. These artefacts can be eliminated by improving shock resolution or by starting the simulation downstream of the leading edge. The computed forces have been extrapolated from the two finest grid levels to the continuum solution using Richardson extrapolation, see Roache(1). Agreement with the exact solution is considered to be excellent. Physical Property Theory IMPNS % Error Local Mach Number 1.873526 1.87368 0.00822 Normalised Pressure 2.4675 2.46757 0.00284 Normalised Density 1.866549 1.86679 0.01291 Normalised Temperature 1.321958 1.32183 0.00968 Axial Force Coefficient 0.089878 0.089875 0.003 Normal Force Coefficient -0.33543 -0.33542 0.003 Table (1) Comparison of computed and theoretical results (Compression Surface) 2. Prandtl-Meyer Expansion The second verification case concerns a corner centred Prandtl-Meyer expansion in two-dimensional inviscid supersonic flow. The computation corresponds to the M∞ = 2.5 flow of air over an expansion corner set at 15° to the flow. Extensive iterative and grid convergence studies were performed. The finest grid considered contained approximately 80,000 nodes. This computation demonstrates the reliability of the software for the computation of expansions fans. The main results are compared with the exact solution in Table 2. Comparison with the exact solution is considered to be good. Physical Property Theory IMPNS % Error Mach No. after expansion 3.2368 3.23679 0.00031 Normalised Pressure 0.3274 0.327442 0.0128 Normalised Density 0.4505 0.450466 0.00755 Normalised Temperature 0.7269 0.726895 0.00069 Table (2) Comparison of computed and theoretical results (Expansion Corner) 3. Conical Shock The final inviscid benchmark solution concerns the inviscid three-dimensional flow over a cone with semi-vertex angle of 10° in a M∞ = 2.35 flow. An analytical result can be obtained for this problem through the solution of the ordinary differential equation derived by Taylor and Maccoll, see Anderson(16) for further details. Both full three- dimensional and two-dimensional axi-symmetric forms of the Euler equations were solved for this configuration. The computed data are compared with the benchmark solution in Table 3. Agreement with the exact solution is considered to be acceptable. Figure 4. Flowfield Visualisation (Static Pressure) American Institute of Aeronautics and Astronautics 7 Physical Property Theory IMPNS % Error Local Mach Number 2.2677 2.25491 0.564 Normalised Pressure 1.1781 1.16016 1.523 Normalised Density 1.1240 1.11186 1.080 Normalised Temperature 1.0481 1.04344 0.445 Table (3) Comparison of computed and theoretical results (Conical Shock) 4. Laminar Boundary Layer The final verification case is related to the prediction of a laminar boundary layer with zero pressure gradient. The problem considered involves the computation of a steady, viscous, compressible, two-dimensional supersonic flow at Mach number 2.0 over a smooth flat plate. The computations correspond to a Reynolds number of 5 330 000 based on the length of the plate. Comparisons were made between the results of the IMPNS code and those obtained using the boundary layer code described by Wilcox(17). Extensive iterative and grid convergence studies were undertaken. The final grid converged computations were performed on a structured grid containing 241 points along the plate and 161 normal to the plate. Stretching was employed to position points close to the solid surface where the solution activity was expected to be high. Computations were made with two treatments of the Vigneron approximation. In the first approximation [FCS] the Vigneron parameter ω is assumed to be constant during the transformation between the PDE based formulation for which stability was demonstrated and the integral formulation used in the present work. The error introduced by this transformation results in an apparent pressure gradient resulting in a less developed boundary layer, see Fig. 5 which compares the present computations with the result of Wilcox. In the second the approach due to Morrison and Korte(18) [MKS] a rigorous mathematical treatment of the Vigneron parameter is employed. This results in an additional source term that must be evaluated but allows the boundary layer to be correctly resolved. B. Verification using the method of manufactured solutions Benchmark simulations, such as those discussed above, have the advantage that they can be readily related to flow physics but are deficient in that they require significant physical or geometric simplification. While this simplification results in problems that are analytically tractable it can also mean that the full non-linear behavior of the governing equations is not exercised by the test cases. Recognizing that verification is a purely mathematical procedure with no regard to the detailed physics that are being modeled suggests an alternate strategy to those discussed thus far, the method of manufactured solutions. In this procedure we ignore the physical problem at hand and instead simply pick a continuum solution. This solution need not, and most probably does not, satisfy the original governing equations. To illustrate this procedure consider the following equation, 0= ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ z w y v x u ϕϕϕϕ t (13) which describes the convection of a scalar quantity φ and is typical of the equations found in the one- and two- equation turbulence models employed in IMPNS. Substituting assumed functions for the scalar and velocity fields a modified differential equation can be obtained, 0 5 10 0.0 0.5 1.0U/Ue y (m m ) Wilcox [17] FCS MKS Figure 5. Comparison of IMPNS with benchmark solution for laminar flow over a flat plate American Institute of Aeronautics and Astronautics 8 ( )tzyxR z w y v x u ,,,= ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ϕϕϕϕ t (14) where ( )tzyxR ,,, is a residual term reflecting the fact that the assumed solution is not an exact solution of the original partial differential equation. While the choice of continuum solution can be completely arbitrary, it is perhaps best to choose a solution that does not produce a trivial result in the context of the discretization scheme. We therefore require a continuum solution function that can be defined for the leading terms in the expansion of the truncation error and provides the required solution or solution gradient data at the domain boundaries. The method of manufactured solutions has been used to investigate various aspects of the algorithms contained in the IMPNS software. In the example below the method of manufactured solutions was used to investigate the discretization of the convective terms appearing in the scalar convection routines used in the turbulence model. The velocity field was frozen and the scalar field was described using, ),( yxf=ϕ (15) Computations were performed on a sequence of progressively finer two-dimensional grids until a grid converged solution was obtained. A comparison of the computed and assumed scalar fields is presented in Fig. 6. Differences between the computed and assumed solutions are small indicating that the implementation of the numerical scheme is correct. C. Validation Having established the capability of the IMPNS solver for simple flows involving isolated physical phenomenon calculations were then performed for configurations that exhibit some coupling between the physical phenomena that are expected to be of importance for the flows of interest. Three cases are presented here. The first two correspond to the experimental study of the ONERA B1 projectile(19) ajnd involve coupling between shock, boundary layers and vortical flows. The second case corresponds to the flow around the HB2 blunt projectile forebody(20). This configuration involves interactions between a detached shock wave and a boundary layer. Figure 6. Comparison of computed and assumed scalar fields -0.1 0.0 0.1 0.2 0.3 0.4 0 2 4 6 8x/D Cp Experimental PNS Figure 7. Comparison of Computed and Measured Surface Pressure Coefficients ONERA B1 (αααα = 0°°°°) -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 x/D Cp Experiment Phi=0 PNS Phi=0 Experiment Phi=180 PNS Phi=180 Figure 8. Comparison of Computed and Measured Surface Pressure Coefficients ONERA B1 (αααα = 15°°°°) American Institute of Aeronautics and Astronautics 9 1. ONERA B1 Tangent Ogive The ONERA B1 tangent ogive-cylinder body geometry consists of a 3 calibre ogive followed by a cylindrical extension providing a total length of 9 calibres. In the current work the flow around the B1 was computed for a Mach number of M = 2 and a Reynolds number of 160,000 based upon the body diameter. Two incidence angles were considered 0° and 15°. These conditions correspond to the experimental test of Barberis(18). Computations were performed on 4 grids. The finest grid contained 121 points along the body axis and 181 in both cross flow directions providing a grid with approximately 4 million cells. The grid points were distributed to fully resolve the boundary layer and other flow gradients. Iterative convergence was demonstrated. The computed surface pressure coefficients are compared with measured data for the 0° case in Fig. 7. Very good agreement is observed between the computed and the measured values. Similar agreement is observed for the axial distributions of pressure coefficient along the wind-ward and lee-ward symmetry planes at 15° incidence, see Fig. 8. Detailed comparisons of the computed and measured surface pressure distributions are made in Fig. 9. Measured and computed primary separation locations are presented in Fig. 10. Very good agreement is observed between the computed and measured data. 2. Hyper-velocity Ballistic Model HB-2 The HB-2 Hyper Velocity Ballistic model(20) is a research configuration intended to provide detailed understanding of flare stabilized projectiles in hypersonic flows. The configuration consists of a blunted cone followed by a 2.5 caliber cylinder and a 1.7 caliber flare, see Fig. 11(a). The blunt nose produces a detached shock Figure 9. Comparison of Computed and Measured Surface Pressure Coefficients ONERA B1 (αααα = 15°°°°) Figure 10. Comparison of Computed and Measured Primary Separation Points (αααα = 15°°°°) Figure 11. HB-2 geometry and Computed Flow American Institute of Aeronautics and Astronautics 10 wave with a subsonic region behind the shock, Fig. 11(b). The flow exhibits a strong elliptic characteristic in this sub-sonic region with significant upstream influence. As a consequence a single sweep space marching approach is no longer appropriate. This configuration provides an opportunity to demonstrate the multiple sweep capability of the IMPNS solver. Computations were performed at an incidence of α = 0° using an axis-symmetric formulation of the governing equations. The computed case corresponds to a Mach number M∞ = 5 and a Reynolds number based upon the cylinder diameter of 2,320,000. The flow was assumed to be fully laminar. An extensive iterative and grid convergence study was performed. A multi-sweep region extending over the sub-sonic region was employed and the remaining geometry was computed using a single-sweep approach. Results of the present computations are compared with the experimental data in Fig. 12. Agreement over the blunted cylinder is excellent. Over the cylinder- flare junction comparison is poor. It is thought the flow may have separated during the experiment, behavior that is suppressed by the use of a single-sweep in the current computation. 3. Double Fin Inlet The double fin inlet configuration studied in the present work is considered to be representative of the expected wing-fuselage and ventral fin-tail plane-fuselage interactions. The geometry consists of two double wedges mounted on a flat plate and was studied experimentally by Garrison et al26. The resulting flow is complex exhibiting a series of shock, shock-shock interactions and vortices along the wedge-plate corners and centre-line. Computations were performed at a Mach number of 3.85 and a Reynolds number of 760 million/metre. Comparisons of the computed and measured pitot pressure ratio are presented in Fig. 13. While acceptable agreement is observed between computation and experiment, there are clear differences in shock-structure. C. Comparisons for the X-15 Flight Vehicle Having established the credibility of the IMPNS solver for isolated and coupled flow phenomena of interest computations were then performed for the X-15 research aircraft. The X- 15 was the worlds first manned hypersonic research aircraft setting a series of speed records in the Mach 4-6 range between 1959 and 1967(21). Uniquely, the aircraft was extensively instrumented providing a wealth of data on supersonic and hypersonic air flows, aerodynamic heating and stability and control. The geometric complexity of the aircraft coupled with the availability of extensive flight and wind tunnel test data provide an ideal opportunity to demonstrate credible, rapid CFD. Figure 12. Comparison of computed and measured surface pressures for the HB-2 geometry (a) y = 0mm (b) y = 5mm Figure 13. Comparison of computed and measured pressure ratio American Institute of Aeronautics and Astronautics 11 1. Grid Generation A CAD database of the X-15 geometry based upon published drawings(22,23,24) forms the basis of the geometry used in the current study. Details of the surface model can be seen in Fig. 14. A structured multi-block mesh was developed using the CAD definition. A view of the grid over the forward fuselage and wing is shown in Fig. 15. An O-type topology was used over the forward body. Care was taken to ensure resolution of the boundary layer through the use of grid stretching to accommodate the expected regions of high-solution activity. Over the wings a H-type topology was employed. The multi-block mesh made extensive use of the ability of the IMPNS solver to use non-matching grids in the marching direction. This allows the resolution of the grid to be changed to accommodate new geometric features, such as the wings, without the need to propagate the fine grid into regions where it was not required. Over the mid-fuselage between the wing and tail plane an O-type mesh is employed while a H-type topology was employed over the tail plane and fins. 2. Comparison with Wind Tunnel Data Initially computations were performed for the nose geometries investigated by Franklin(23). Two configurations were computed; a sharp nose geometry for which single sweep computations were performed and a geometry with a spherically blunted nose for which multi-sweep computations were required. The computations were performed for a Mach number of M∞ = 4.7, an incidence of α = 0° and a Reynolds number of 10,500,000 per metre. Figure 14. Surface Model of the X-15 Air Vehicle Figure 15. Detail of the grid over the forward fuselage and wing (a) x/L = 0.045 (b) x/L = 0.1246 Figure 17. Comparison of computed and measured pressures for the blunt nose configuration (a) x/L = 0.045 (b) x/L = 0.1246 Figure 16. Comparison of computed and measured pressures for the sharp nose configuration American Institute of Aeronautics and Astronautics 12 Results for the sharp nose geometry are presented in Fig. 16 together with the experimental data. As the transition location was not documented two computations were performed; the first assumes that the flow is laminar while the second assumes transition at the nose. The results suggest that the pressure distribution is insensitive to transition location. Fig. 17 presents corresponding comparisons for the blunt nose configuration while Fig. 18 and Fig.19 compare the computed flow structure with wind tunnel observations. Also shown are independent computations performed by Hawkins and Dilley(25). The comparisons with the wind tunnel data are disappointing, but are in good agreement with those of other investigators. Surface pressure distributions are compared with the measured data in Fig. 20. (a) Wind Tunnel (b) Computed Figure 18. Comparison of Computed and Measured Flow Structures Figure 19. Comparison of Computed and Observed Shock Structure Figure 20. Comparison of surface pressures for the pointed and blunt nose configurations. Figure 21. Comparison of computed and measured pressure for blunt nose configurations x/L = 0.327 American Institute of Aeronautics and Astronautics 13 Considerable improvement in the comparison with the off-surface measurements is obtained in the region of the canopy, Fig. 21. The azimuthal variation of the surface pressure distributions is also well resolved, Fig. 22. The final comparisons with the experimental data are made over the wing, Fig. 23. Comparisons are provided at incidence angles of α = 0° and α = 15°. Generally the agreement between the computed surface pressure coefficients and the measured data is good. Comparison of the off surface data is less good. The experimental model is small and it is likely that probe interference effects between the pitot pressure rake and the model will be significant. (a) αααα = 0°°°° (b) αααα = 15°°°° Fig. 23 Comparison of measured and computed surface pressure distributions over the wing (mid-span) (a) x/L = 0.223 (b) x/L = 0.297 (c) x/L = 0.501 Figure 22. Comparison of surface pressures for the blunt nose configuration. American Institute of Aeronautics and Astronautics 14 3. Comparison with Flight Test Data Computations for the X-15 flight experiment are performed at a Mach number of M∞ = 4.7, at incidences of α = 0° and α = 15° and a Reynolds number of 2,900,500 per metre. The flow is assumed to be fully turbulent. Fig. 24 compares the measured and computed surface pressure distributions at x/L = 0.0337. The agreement is considered to be excellent considering the uncertainties when matching the flight condition. As with the wind tunnel experiments the surface pressure distribution along the forebody is in much better agreement with the measured data than the off surface data, Fig. 25. V. Conclusions An extensive verification and validation study of Cranfield University's IMPNS flow solver has been performed for a complete hypersonic air-vehicle configuration. A hierarchical approach was adopted in which the vehicle aerodynamics were decomposed using vehicle sub-systems and related flow phenomenon. Using this hierarchy bench mark solutions and laboratory experiments were identified that provide the basis of the verification and validation exercises. Detailed comparisons of iterative and grid converged IMPNS computations with benchmark solutions and wind tunnel measurements are presented. Verification computations demonstrating the capability of the IMPNS solver for flows involving isolated shock waves, rarefaction fans and boundary layers were presented. Comparison was made with a range of benchmark solutions including analytical solutions of the governing equations, closed-form solutions of related ordinary differential equations and numerical solutions of related ordinary differential equations. Comparison between the computed data and benchmark solutions was generally excellent. Validation computations were performed for flows that exhibit coupling of two or more of the isolated physical phenomena. For these cases comparison was made with well documented wind tunnel laboratory experiments. In Figure 24. Comparison of off surface pressures for the flight test configuration x/L = 0.0337. Figure 25. Comparison of surface pressures for the flight test configuration. American Institute of Aeronautics and Astronautics 15 addition to flows related to the verification study computations were also performed for a detached shock wave. Agreement between the IMPNS computations and the experimental data was generally good. Finally computations of the X-15 flight experiment were performed and comparison was made with measured surface static pressures and off surface total pressure measurements. The comparisons demonstrate the reliability and capability of the IMPNS flow solver for complex configurations. Acknowledgments The first author was supported under the European Union's SOCRATES exchange programme. The second author was supported by the Engineering and Physical Sciences Research Council (EPSRC) and QinetiQ Ltd. The third author was supported by the Defence Science Technology Laboratory (DSTL). The authors acknowledge the contributions of Trevor Birch (DSTL) and the assistance of NASA Geo-Lab in obtaining a CAD description of the flight vehicle. References 1. Roache, P.J., ' Verification and Validation in Computational Science and Engineering', Hermosa Publishers, 1998. 2. Oberkampf, W.L. and F.G. Blottner, "Issues in Computational Fluid Dynamics Code Verification and Validation", AIAA Journal, Vol. 36, No. 5, May 1998, pp. 687-695. 3. Oberkampf, W.L., Trucano, T.G., "Verification and validation in computational fluid dynamics", Progress in Aerospace Sciences, Vol. 38 (3), April 2002, pp. 209-272. 4. Vigneron YC, Rakich JV and Tannehill JC, “Calculation of supersonic viscous flows over delta wings with sharp leading edges”, AIAA Paper 78–1137, July (1978). 5. Chang C-L and Merkle CL, “The relation between flux vector splitting and parabolized schemes”, J. Comp. Phys. 80, (1989). 6. Osher S and Solomon F, ‘Upwind Difference Schemes for Hyperbolic Systems of Conservative Laws’, Math. of Comp. 38, 339–374 (1992). 7. Saad Y and Schultz MH, “GMRES: A generalised minimal residual algorithm for solving non-symmetric linear systems”, SIAM J. Sci. Stat. Comput. 7, pp. 856–869 (1986). 8. Brandt A, “Multi-Level Adaptive Solutions to Boundary-Value Problems”, Math. of Comp. 31, 333–390 (1977). 9. Qin N and Richards BE, “Finite volume 3DNS and PNS solutions of hypersonic viscous flows around a delta wing using Osher’s flux difference splitting”, in Proceedings of a Workshop on Hypersonic Flows for Re-entry Problems, (1990). 10. Birch TJ, Qin N and Jin X, “Computation of supersonic viscous flows around a slender body at incidence”, AIAA Paper 94-1938, (1994). 11. Shaw S and Qin N, “A matrix-free preconditioned Krylov subspace method for the PNS equations”, AIAA Paper 98-111, (1998). 12. Qin N, Ludlow DK, Zhong B, Shaw S and Birch TJ, “Multigrid acceleration of a preconditioned GMRES implicit PNS solver”, AIAA Paper 99-0779, (1999). 13. Birch TJ, Ludlow DK and Qin N, “Towards an efficient, robust and accurate solver for supersonic viscous flows”, in Proceedings of the ICAS 2000 Congress, Harrogate, UK, (2000). 14. Birch TJ, Prince SA, Ludlow DK and Qin N, “The application of a parabolized Navier-Stokes solver to some hypersonic flow problems”, AIAA Paper 2001-1753, (2001). 15. Qin N and Ludlow DK, “A cure for anomalies of Osher and AUSM+ schemes for hypersonic viscous flows around swept cylinders”, in Proceedings of the 22nd International Symposium on Shock Waves, Imperial College, London, UK, July 18-23, (Editors: Ball GJ, Hillier R and Roberts GT), pp. 635-640 (1999). 16. Anderson, J. D. ‘Modern Compressible Flow’ Second Edition Mc-Graw Hill International Editions 17. Wilcox, D. C. ‘Turbulence Modelling for CFD’ Second Edition, DCW Industries, 1998 18. Morrison JH and Korte JJ, “Implementation of Vigneron’s streamwise pressure gradient approximation in the PNS equations”, AIAA Paper 92-0189 (1992) (see also AIAA Journal 30(11) (1993)) 19. Barberis D, “Supersonic vortex flow around a missile body”, AGARD Advisory Report 303, Vol. II, (1994). 20. Jones JH, “Pressure tests on the standard hypervelocity ballistic model HB-2 at Mach 1.5 to 5”, Arnold Engineering Development Center, Technical Documentary Report, AEDC-TDR-64-246, (1964). 21. Stillwell WH, "X-15 research results - with a selected bibliography", NASA SP-60. NASA, Washington, D.C., (1965). American Institute of Aeronautics and Astronautics 16 22. Franklin AE and Lust RM, "Investigation of the aerodynamic characteristic of a 0.067-scale model of the X- 15 airplane (configuration 3) at Mach numbers of 2.29, 2.98 and 4.65", NASA TM X-38, Washington, D.C., (1959). 23. Palitz M, "Measured and calculated flow conditions on the forward fuselage of the X-15 airplane and model at mach numbers from 3.0 to 8.0", NASA TN D-3447, Washington, D.C., (1966). 24. Hodge L and Burbank PB, "Pressure distribution of a 0.0667-scale model of the X-15 airplane for an angle- of-attack range of 0 to 28 at mach number of 2.30, 2.88 and 4.65", NASA TM X-275, Washington, D.C., (1960). 25. Hawkins R and Dilley A, "CFD comparisons with wind tunnel and flight data for the X-15". In AIAA 4th International Aerospace Planes Conference, AIAA 92-5047, Orlando, US-FL, (1992). 26. T. J. Garrison et al. Flow eld surveys and computations of a crossing-shock wave/boundary-layer interaction. AIAA Journal, 34(1):50-56, January 1996.