Transition prediction for three-dimensional flows using parallel computation

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Computers & Fluids 38 (2009) 121–136 www.elsevier.com/locate/compfluid

Transition prediction for three-dimensional flows using parallel computation N. Krimmelbein *, R. Radespiel University of Braunschweig, Institute for Fluid Mechanics, Bienroder Weg 3, 38106 Braunschweig, Germany Received 2 March 2007; received in revised form 10 December 2007; accepted 28 January 2008 Available online 12 February 2008

Abstract A computational method to predict transition lines for general three-dimensional configurations is presented. The method consists of a coupled program system including a 3D Navier–Stokes solver, a transition module, a boundary layer code and a stability code. The newly developed transition module has been adapted to be used with parallel computation to account for the high computational demand for three-dimensional configurations. Detailed computations have been performed to show the ability of the Navier–Stokes code to provide three-dimensional boundary layer data of high accuracy needed for the stability analysis. A comprehensive investigation on general computational and parallel performance identifies the numerical effort for the transition prediction method. The procedure has been validated comparing the numerical results with experiments for the flow around an inclined prolate spheroid. Feasibility studies on generic transport aircraft have been performed to show the code’s capability to predict transition lines on general complex geometries. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Predicting and modelling the laminar–turbulent transition in Reynolds-averaged Navier–Stokes (RANS) solvers is a necessary requirement for the computation of flows around complex, three-dimensional geometries. Neglecting or using inaccurate transition locations may lead to significant errors of the predicted aerodynamic performance. Transition from laminar to turbulent flow is a complex phenomenon and can occur through very different mechanisms depending on on-flow parameters such as Reynolds and Mach number or free-stream turbulence, on surface properties or on the detailed development of the laminar boundary layer. Existing transition prediction methods vary in their approaches from purely empirical transition criteria over physically based stability equations that take into account non-local and non-linear effects of disturbance growth, correlation-based transition models [16,17] to direct numerical simulations (DNS). The state-of-the-art transition predic*

Corresponding author. E-mail address: [email protected] (N. Krimmelbein).

0045-7930/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2008.01.004

tion method for thin aircraft boundary layers is the so called eN-method [28,33], based on local, linear stability theory. eN-methods were thoroughly calibrated and used in many applications of wing flows that are unstable due to Tollmien–Schlichting or cross flow instabilities [24,26] and thus represent significant industrial value. Recent advances in predicting transition onset for complex flows address the prediction of unsteady transition on moving airfoils [19] and the application to 2D laminar separation bubbles [35] including unsteadiness effects [34]. Increasing focus is placed on the prediction of transition for threedimensional boundary layers on high aspect-ratio wings and high-lift configurations [4,12,13,15,32], flows around bodies of revolution [31] and general three-dimensional aircraft configurations [3,10,18]. The present work addresses the prediction of transition for flows around general three-dimensional, complex configurations. For automatic transition prediction in Navier–Stokes computations a transition prediction module is developed and tested. The module uses an hybrid approach to calculate the relevant data for transition prediction, i.e., laminar boundary layer data are either extracted from the 3D Reynolds-averaged Navier–Stokes

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Nomenclature iH Ma N n p Re u, v wct a

angle of incidence of horizontal tail plane Mach number N-factor grid points normal to wall number of processes Reynolds number velocity components in streamwise/cross flow direction wall clock time angle of attack, complex spatial wave number

solutions or calculated using a boundary layer method for swept, tapered wings. To account for the computational effort for the three-dimensional flow calculation around general three-dimensional components and configurations, the transition module is developed to be used in fully parallel Navier–Stokes computations. The objective of this paper is to give insight in the development and parallelization of the transition module and the application to general three-dimensional, complex flows. First validation calculation have been performed for the three-dimensional flow around an inclined prolate spheroid and the principle application to general three-dimensional configurations is demonstrated with a feasibility study for two generic test cases. 2. Description of methods 2.1. Linear stability theory and eN-method The classical linear stability theory evaluates the stability of a laminar boundary layer by examining the development of small disturbances. This is used for transition prediction in form of the eN-method. The principal approach of the theory is to superimpose instationary disturbances onto the stationary boundary layer flow. The assumption of a harmonic wave as disturbance is superposed to the mean flow quantities: q0 ðx; y; z; tÞ ¼ qðzÞeðaxþbyxtÞ

ð1Þ

This approach leads to a system of second order differential equations for the amplitude function. For incompressible media, the system can be rearranged and written as one fourth-order differential equation, the Orr–Sommerfeld equation [23]. 2.2. Numerical methods 2.2.1. Navier–Stokes solver Tau The DLR Tau code [29,30] is a Navier–Stokes solver for the calculation of viscous and inviscid flows around general complex geometries. The solver is based on the finite volume method and uses a dual grid approach where the flow

b d d* k x

complex spatial wave number boundary layer thickness displacement thickness sweep angle complex temporal wave number

Subscripts CF cross flow TS Tollmien—Schlichting

variables are associated with the vertices of the original grid. Tau can be used on structured and unstructured (hybrid) grids. Generally, a semi-structured grid layer above surfaces is used to resolve boundary layers, whereas the rest of the computational domain is filled with an unstructured grid. For parallel computations, a domain decomposition approach is used and the massage passing concept using MPI [36] is applied. In a preprocessing step, the grid is divided into a certain number of subdomains using a bisection algorithm [30]. After partitioning the grid, the solver computes the flow data on a single domain per process. The data is regularly updated on points lying in the overlap region between a certain domain and its neighbours. The solver computes the fluxes with a second-order central scheme or one of various upwind schemes with linear reconstruction for second-order accuracy. Time integration is performed by either applying an explicit, multistage Runge–Kutta scheme or an implicit, lower–upper symmetric Gauss–Seidel (LU-SGS) scheme. Turbulent flows are modelled using different Spalart–Allmaras or k  x turbulence models. For transitional flows, laminar regions can be designated by the definition of polygon lines on the surface of the geometry and prescribing the maximum height of the laminar region over the surface. For the transitional computation, the turbulent production terms are suppressed in the laminar flow area. For convergence acceleration residual smoothing, local time stepping and a multigrid approach can be applied. To extend the solver capability to incompressible flows, a low-Mach-number preconditioning approach is implemented.

2.2.2. Transition prediction method For automatic transition prediction in Navier–Stokes computations a coupled program system has been developed. This system consists of a transition prediction module [10,14,15] implemented directly into the DLR Tau code [30]. Specific elements of the module are the boundary layer code Coco [25] for swept, tapered wings and the linear sta-

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bility equations solver Lilo [27]. Both, Coco and Lilo are sequential programs, which are accessible from the main transition module via file I/O and system calls. The transition prediction module, which has been developed with special focus on predicting transition for flows around general complex, three-dimensional geometries supports parallel computing. It can be shown, that for general transition prediction, and especially for the prediction of cross flow type transition, a high normal-to-wall mesh density is required to resolve the boundary layer adequately [10]. To overcome this constraint, a boundary layer method is in many cases an efficient alternative to a high resolution Navier–Stokes computation. However, boundary layer methods, and in particular quasi-2D boundary layer methods, have certain limitations for complex flows, e.g., for flows around complex geometries and low aspect-ratio wings or in the presence of laminar separation bubbles. To deal with both, the use of coarse grids across boundary layers for rapid engineering applications and highly resolved boundary layers for detailed flow analysis, a hybrid approach is used here, i.e., boundary layer data can either be extracted directly from the Navier–Stokes solution or a first-order boundary layer method for swept, tapered wings is applied. If the eN-method is used for transition prediction, a suitable integration path to calculate the N-factors from the amplification rates has to be applied. The velocity and direction of the energy transport of a wave is represented by the group velocity [23] and thus, the group velocity can be taken as the amplification direction [23]. The group velocity trajectory in turn can be approximated by an edge streamline [23], and this streamline can be used as the integration path. However, the boundary layer edge is not represented by an edge (or external) streamline. Referring to edge streamline means ‘‘the local projections of the loci, where the streamlines cross the boundary layer edge” [6]. This means that for the calculation of the edge streamlines the velocity vector at the boundary layer edge is determined and projected onto the geometry for every surface grid point. The streamlines are then calculated on the surface of the geometry using a Runge–Kutta integration scheme. The integration of the amplification rates is done in two different ways, depending on the approach to calculate the boundary layer data. If the boundary layer profiles are extracted directly from the Navier–Stokes solution, the integration is performed along the edge streamlines determined by the transition module. If the boundary layer code is used, the integration path is determined inside the stability code. I.e., from infinite swept wing considerations [7] the edge streamline is determined from geometrical relations and the direction of the boundary layer edge velocity. The boundary layer code needs, together with certain onflow conditions (Re, Ma, angle of attack) and the pressure distribution, the geometrical sweep angles as input. The geometrical sweep angles are extracted from the grid topology at the leading and trailing edges, but the edge streamlines do not necessarily cross the leading edge. This means, that

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an automated determination of the sweep angle and the application of the boundary layer code along edge streamlines is not suitable. Instead, the boundary layer code is applied to line-in-flight cuts distributed along the wing span. These cutting lines are the intersections of the surface of the geometry with planes parallel to the on-flow direction. The line-in-flight cuts enclose the leading and trailing edges and the local geometrical sweep angles can be extracted from the grid. In addition to the determination of the sweep angles an alteration or detailed analysis of the pressure distribution around the attachment line to account for the use of the effective sweep angle concept [20] (not implemented yet) is simplified by the use of line-in-flight cuts. In a preprocessing step, right before the RANS solver iterations start, data not depending on the flow solution are processed (Fig. 1), i.e., calculation of wall-normal lines, line-in-flight cuts and geometrical sweep angles. During the solution process of the solver, the iteration process is interrupted in certain intervals, and the transition prediction is executed. A new transition line is calculated and applied with under-relaxation to generate new laminar and turbulent regions in the RANS solver. This procedure is repeated, until the transition lines itself are converged (Fig. 1), i.e., the transition points on the transition detection lines stay constant in terms of grid points. For each transition prediction step, the following general procedure is executed (Fig. 2). For all surface points of the geometry the boundary layer profiles are assembled along wall-normal lines, integral boundary layer data are determined and the boundary layer edge velocity is projected onto the surface. After the calculation of the 3D edge streamlines, the velocity profiles along the streamlines are extracted and passed to the stability analysis. For the application of the boundary layer code, a series of line-in-flight cutting lines at different positions of the wing is determined once in the preprocessing stage of the transition prediction module. During runtime, these

Fig. 1. Coupled program system. Navier–Stokes solver with transition module.

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Fig. 2. Determination of new transition locations with the transition module.

cutting lines are separated in an upper and lower surface part at the corresponding attachment line and the pressure distribution is extracted and passed along with the on-flow conditions and the local sweep angles to the boundary layer code. The velocity profiles along the cutting lines are then calculated and transferred to the stability analysis (Fig. 2). The stability analysis yields a series of N-factor curves, which are analyzed in the transition prediction module with appropriate N-factor criteria to give new transition locations for every streamline/cutting line. The limiting N-factors are applied using the 2N-factor strategy, treating the N-factors for Tollmien–Schlichting and cross flow instabilities independently. The interaction of Tollmien– Schlichting and cross flow waves cannot be evaluated by linear local stability theory and is instead modelled by applying a stability curve, where the critical N-factor of one instability form depends on the local N-factor of the other instability form [31]. 2.2.3. Linear stability equations solver Lilo Lilo [27] is a sequential computer code for efficient stability analysis of laminar boundary layers and the calculation of N-factor curves. It allows utilizing local as well as non-local stability theory with or without curvature effects for compressible or incompressible flows. The assumption of a harmonic wave as disturbance (Eq. (1)) leads to a system of second-order differential equations, which form a linear eigenvalue problem for the complex disturbance frequency x. I.e., the temporal stability problem is considered and the complex wave numbers a and b have to be prescribed. The parameters of the eigenvalue problem

depend on the basic flow properties (Ma, Re), on the local velocity and temperature profiles of the laminar boundary layer and on their first and second derivatives. Generally, the boundary layer flow is assumed to be a parallel flow [23]. The solution of the eigenvalue problem results in local amplification rates, which are integrated along appropriate integration paths [23] (Section 2.2.2). For this, the Gaster transformation [5] is used, to transform the temporal growth into spatial growth. Different integration strategies for the computation of the N-factor from the amplification rates are available [23]. In the coupling with the transition module described in this paper, the prescribed-frequency/ prescribed-propagation-direction integration strategy [23] is used for Tollmien–Schlichting instabilities, and the prescribed-frequency/prescribed-wavelength integration strategy [23] is used for cross flow instabilities. It is assumed, that the maximum amplification direction corresponds to the direction of the group velocity and the trajectory of the group velocity can be approximated by a streamline at the boundary layer edge [2,22] (Section 2.2.2). The integrated amplifications yield a single N-factor curve for a single frequency/wavelength. To cover the whole spectrum of amplified waves, integrations for different frequencies/wavelength have to be performed. The envelope of the N-factor then represents the maximum amplification factor and the envelope is used to predict transition by comparing the values of N with experimentally determined critical N-factors. For the analysis of Tollmien–Schlichting instabilities, the boundary layer of a single streamline is evaluated to find an amplified eigenvalue. If an amplified mode is found, the frequency of this mode is calculated, and the mode is traced up-, and downstream, until the stagnation point or the end of the laminar part of the streamline is reached or, generally, the mode is no longer amplified. Near the two bounding points of the unstable region for this particular mode with this particular frequency, the frequency is increased or decreased, respectively, until the investigated mode is damped. This is done to estimate the maximum extension of the indifference curve in the x–x plane of the stability diagram [22]. This stability diagram shows the range of amplified waves as a function of streamwise distance x [2]. The upper and lower overall limits of the frequencies give an estimate of the range of amplified frequencies to be investigated for the N-factor calculation. In the next step, the amplifications of several waves with different frequencies, within the previously defined range, are calculated. With initial values for the complex eigenvalue derived from a QR-decomposition, the eigenvalues for each streamline point are determined with an inverse Rayleigh iteration and the local amplification rates are calculated. Each wave is traced up- and downstream and the amplification rates are calculated, until the wave is no longer amplified. The procedure for cross flow instabilities, including the estimation of a wave length range and the calculation of the amplifications, is similar to that of the Tollmien–Schlichting analysis described above.

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3. Implementation and parallelization issues The DLR Tau code uses domain decomposition for parallel computation. For a given number p of processors the computational grid is divided into p subgrids (subdomains). Each of the processors computes on one of the subgrids. A continuous communication between the processes is performed on points lying in the overlap region between a certain domain and its neighbours using MPI [36], where only local data is communicated. In contrast to this, nonlocal data has to be communicated for the transition prediction process as well. The parallelization of the transition prediction module can be divided into two steps, one performed prior to the Navier–Stokes computation as a preprocessing step, and the other one executed at every transition prediction step during runtime. In the preprocessing step, all data independent of the flow solution are calculated, whereas in the transition prediction steps data of the current flow solution are needed. Parallelization by means of the transition prediction module is needed for the determination of wall-normal lines, the assembly of velocity and temperature profiles along the normals, the calculation of edge streamlines, the assembly of line-in-flight cuts and the execution of the sequential, external programs. Calculation of data in form of lines (i.e., wall-normal lines, edge streamlines, line-in-flight cuts) within the transition module is effectively an ordered assembly of a list of grid points. These points are gathered, beginning at starting points (i.e., surface points, user-defined starting points) and ending at user-defined or geometrically provided endpoints. For parallel computation, another limit is a domain boundary. In this case, the endpoints of the lines will be communicated to the neighbour domain where they serve as new start points for another loop of the assembly of the lines. Parallelization regarding the transition prediction module is considered as the ability to process partitioned Navier–Stokes solutions. E.g., for the calculation of wallnormal lines, edge streamlines or line-in-flight cuts only domains containing sections of these lines are involved in the computation. Domains not containing sections of these lines have to run idle during this calculation process. For this reason, a complete parallel execution is not possible in general but sequential execution of the transition module is kept to a minimum. 3.1. Determination of boundary layer data The basis of the determination of boundary layer data from the Navier–Stokes solution is the knowledge of the wall-normal lines corresponding to the surface grid points and the list of grid points associated with these lines. The length of the wall-normals is limited by a maximum distance or by a maximum number of grid points, both defined by the user and usually in accordance with the

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extension of the structured part of the hybrid Navier– Stokes grid. Velocity profiles inside the boundary layer are directly accessible, if the surface point and the point associated with the end of the wall-normal line are placed in the same domain. In this case, only the knowledge of the point list for the wall-normal has to be known, and the velocities at each wall-normal point is interpolated from the Navier–Stokes solution using an inverse distance weighting approach [21]. If a boundary layer profile is cut by a domain boundary, the wall-normal lines are divided at the domain boundaries into separate wall-normal parts, where each part of the wall-normal lies in a different domain. The velocities for each wall-normal part are interpolated from the Navier–Stokes solution of the corresponding domain and are communicated to the domain containing the surface point associated with the examined wall-normal. After the velocity profiles are assembled, the boundary layer edge is detected and all relevant boundary layer data are calculated and stored together with the velocity vector of the boundary layer edge at the surface grid points. 3.2. Calculation of edge streamlines and line-in-flight cuts In the transition module, a distinction between three types of transition detection lines is made: (i) attachment lines, (ii) streamlines along the boundary layer edge, and (iii) streamlines derived from line-in-flight cuts of the geometry. It has to be noted that the boundary layer edge is not a streamline. Referring to edge streamline in this context means the ‘‘local projections of the loci, where the streamlines cross the boundary layer edge” [6]. The attachment line is a particular streamline, which divides the flow into two parts, one part following the upper surface of the geometry and another part following the lower surface of the geometry [1]. The attachment lines are calculated using the surface shear stress distribution. This can be done, since the flow in the attachment line is two-dimensional and the skin friction line coincides with the projection of the edge streamline onto the surface of the geometry. This procedure reduces possible inaccuracies in the detection of the boundary layer edge in the attachment line region of the Navier– Stokes solution, as this has found to be a challenging task. The calculation is based on a multistage Runge–Kutta scheme and is started at a user-defined initial point. The integration is executed against the streamwise direction along the attachment line and is stopped, if a stagnation point is reached, or a user-defined geometrical limit is reached. After the determination of all attachment lines of the problem, the calculation of the edge streamlines is executed. The integration is based on the same multistage Runge–Kutta scheme as for the attachment lines, but uses the boundary layer edge velocities, projected onto the surface, as input. The calculation of the edge streamlines is stopped, if a previously calculated attachment line, the

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trailing edge of a geometry part, or a user-defined geometrical limit is reached. Starting point is again a user-defined initial point on the surface, but the integration is performed in both directions, upstream and downstream. Line-inflight cuts are defined in the preprocessing stage of the transition prediction process. They are assembled by storing the intersection points of lines connecting two surface points with user-defined cutting planes. During runtime, the pressure distribution along the line-in-flight cuts is interpolated from the Navier–Stokes solution and the lines are divided into upper and lower surface parts at the attachment line. 3.3. Execution of external programs The transition prediction module uses different external programs for the determination of boundary layer data and the stability analysis. These programs are sequential programs which are accessible from the main transition module via file I/O and system calls and are not explicitly written for the utilization by a parallel and automated transition prediction process within a Navier–Stokes solver. The external programs are designed to process the data of a single transition detection line (edge streamline or linein-flight cut) at one sequential run. An alternative approach, compared to the domain decomposition principle of the Tau code, is applied for the parallelization. For each process of the Navier–Stokes calculation one external program is executed sequentially. I.e., if the Navier–Stokes calculation is run on p processors, the external programs are started from each of the p processes independently. This means, that e.g., p stability analyses with the external stabiliy code run parallel on the p processes, so that p transition detection lines can be processed parallel and a parallel performance of the actually sequential external programs is achieved. The advantage of this approach is that only few modifications of the source codes had to be made, i.e., ensuring unique filenames for the communication between the external programs and the transition prediction module. A disadvantage is that no full parallelization is obtained, if the number of streamlines is not an integer multiple of the number of processes. In the case of a combined execution of Coco and Lilo, i.e., the input data for Lilo is generated by the boundary layer code instead of being extracted directly from the Navier–Stokes solution, the output files from Coco can be directly used as input files for Lilo, since Coco uses the standardized Lilo-format for output.

file at an = 4.0°, Ma = 0.23, Re = 2.39  106, k = 60.0° was investigated. Transition was prescribed at x/c = 0.03 on the upper surface and at x/c = 0.85 on the lower surface. Steady calculations were performed, using an explicit Runge–Kutta scheme, 3w multigrid cycle and low-Mach-number preconditioning. Turbulent flow was modelled using the standard Spalart–Allmaras turbulence model. For comparison, computations with the boundary layer code Coco were performed, using the pressure distribution from the Navier–Stokes solution as input. For the Navier–Stokes computations different hybrid grids with different grid resolutions were used. The structured parts of the grids, covering the boundary layer, are resolved with 128–512 grid points on the wing surface and 32–128 grid points normal to the surface. I.e., for the finest grid, approximately 60–120 points normal to the surface resolve the laminar boundary layer. An assessment of the streamwise velocity profiles shows a good agreement of all Navier–Stokes profiles with the profiles from the boundary layer code (Fig. 3). However, the cross flow velocity profiles differ significantly for the coarser grids from the reference profiles of the boundary layer code calculation (Fig. 4). The accuracy of the velocity profiles in turn has a direct influence of the computed Nfactors from the stability code (Fig. 5). From the assessment of the velocity profiles and from the resulting N-factor envelopes of the different calculations it is expected that for Tollmien–Schlichting instabilities a normal-to-wall grid resolution of 32 grid points gives appropriate results, but at least 48 grid points are necessary for an accurate transition prediction. For an accurate prediction of transition caused by cross flow instabilities a normal-to-wall resolution of up to 128 grid points is needed. These results can be related to fully three-dimensional high aspect-ratio wing flow because of the similarity of the flow field. For fuselages, these requirements are supported by the results of the validation study in Section 4.3. Additionally, the results of the validation study in this

4. Results 4.1. Code verification Specific investigations have been performed to analyze the influence of the grid resolution on the N-factor calculation with the stability solver Lilo. For this study, the flow around an infinite swept wing model with ONERA D pro-

Fig. 3. Streamwise velocity profiles. Different grid resolutions, comparison with BL-code. ONERA D, an = 4.0°, Ma = 0.23, Re = 2.39  106, k = 60.0°, lower surface, x/c = 0.65.

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Fig. 4. Crossflow velocity profiles. Different grid resolutions, comparison with BL-code. ONERA D, an = 4.0°, Ma = 0.23, Re = 2.39  106, k = 60.0°, lower surface, x/c = 0.65. Fig. 6. Edge streamlines and domain boundaries. 1, 2, 4 and 6 domains. 6:1 prolate spheroid.

Fig. 5. NTS- and NCF-envelopes for different grid resolutions. ONERA D, an = 4.0°, Ma = 0.23, Re = 2.39  106, k = 60.0°, lower surface.

paper are generally confirmed in [31] where the threedimensional flow field was calculated with a boundary layer method for the investigated configuration.

For the steady computations a hybrid grid was used, with the structured grid part covering the boundary layer around the prolate spheroid. The overall number of points of the grid is 2.8 million, with a resolution of the structured grid part of 128 grid points normal to the wall and approx. 300 grid points in flow direction. Low-Mach-number preconditioning was applied together with the implicit LUSGS time integration scheme and a 3w multigrid cycle. Turbulent flow was modelled using the standard Spalart– Allmaras turbulence model. The grid partitioning, and its effect on the domain shapes on the surface grid, is shown in Figs. 6 and 7 for a different number of domains. The computations were carried out using a cluster equipped with AMD opteron 2.2 GHz processors and 1-gigabitethernet.

4.2. Validation of code parallelization Different calculations have been performed to demonstrate the parallelization of the transition prediction module and its influence on the computational effort. The conception of the study on the parallelization of the transition prediction module is to apply one single transition prediction step to a fully converged, transitional flow solution. As basic case, the transition prediction for the flow around a 6:1 prolate spheroid according to Ref. [8] was chosen. The flow parameters, a = 5.0°, Re = 6.5  106, Ma = 0.13, were set to obtain both, Tollmien–Schlichting and cross flow instabilities for nearly the complete laminar section of the prolate spheroid. An evaluation of both types of instabilities on all streamlines leads to the maximum computational effort for the stability solver. As will be shown later, this is also the major proportion of one transition prediction step. Hence, regarding the computational time of one transition prediction step, the results will give an idea of the maximum computational demand of the transition prediction module.

Fig. 7. Edge streamlines and domain boundaries. 8, 10, 12 and 14 domains, 6:1 prolate spheroid.

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As reference, the computational demand of the Tau code itself was evaluated in terms of wall clock time. In Fig. 8, the dependence of the computational time for one multigrid iteration on the number of processors is depicted. If not otherwise noted, the times are normalized using the wall clock time of one sequential Tau solver iteration step. It is seen that the Tau solver parallels well for the processor numbers investigated here.

Fig. 8. Computational time wct vs. number of processes p. One Tau multigrid iteration, 3w cycle, implicit LU-SGS scheme.

Fig. 9. Computational time wct vs. number of processes p. Tau transition setting.

One time consuming part of the transition prediction process is the setting of the new found transition lines in the grid. Here, the fully parallelized routines of the Tau preprocessor are used. The dependence of the computational time on the number of domains of this part is shown in Fig. 9. Again, a fairly good scaling is achieved, however, the computational demand of the parallel transition setting is approximately three times higher as for one Tau multigrid cycle (cf. Fig. 8). The overall computational demand of one transition prediction step depending on number of edge streamlines and number of processors is displayed in Fig. 10. As long as the number of processors is an integer multiple of the number of streamlines processed by the transition module, a very good scaling is achieved. Additionally, it can be seen that there exists a linear dependence of the computational effort on the number of processed streamlines and that most of the computational effort is caused by the execution of the linear stability equations solver. It has to be noted however, that the computational effort to process different streamlines with the stability solver Lilo may lead to very different computational times, since not all streamlines have the same laminar length and additionally exhibit varying indifference and transition points. However, most streamlines are very similar in their properties, and thus in the computational effort needed for processing. Since the assignment of the streamlines to the different domains is done regardless of their properties (which are not known a priori anyway), there are unfavourable combinations possible, where one domain processes several streamlines with high computational demand. This may lead to increased computational time, as the overall time for the transition prediction depends

Fig. 10. Computational time wct vs. number of processes p. Transition prediction step and execution of stability code. 6, 12 and 24 streamlines.

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Fig. 11. Computational time wct vs. number of processes p. Transition prediction step w/o stability code. 6, 12 and 24 streamlines.

Fig. 12. Computational time wct vs. number of processes p. Transition prediction step, normalized with wct for one multigrid cycle of the Tau solver. 6, 12 and 24 streamlines.

on the maximum time spent on one domain. An example can be found in Fig. 10, for the case of 12 streamlines on 6 processors. If the number of domains exceeds the number of streamlines to be processed, no further benefit is gained in terms of computational time from the parallel execution of the transition module. This is due to the type of parallelization of the execution of the external programs, as described in Section 3.3. Having a closer look at the computational demand of the transition prediction module without the expensive stability analysis, it can be seen that, as long as the partitioning of the grid is rather symmetric (Figs. 6 and 7), a smooth development of the scaling is achieved (Fig. 11). A favourable partitioning of the grid can improve the computational demand significantly, see the jump in the curves of Fig. 11 when going from 6 to 8 processors. It can be seen from Fig. 6, that for 6 domains all domains are distributed serially in streamwise direction, and a parallel processing of the streamlines parts (e.g., when integrating the streamlines parts) is not effective. With 8 domains (Fig. 7) the different partitioning leads to certain domains that are parallely placed regarding the streamwise direction. Now, for streamline parts lying inside these parallel ordered domains, a parallel computational processing is possible. In Fig. 12, the computational time for one transition prediction step has been normalized with the computational time for one multigrid cycle of the Tau solver for the corresponding number of processors. It can be seen, that the computational effort of one transition prediction step compared to one Tau multigrid cycle is rather constant as long as the number of processors does not exceed the number of streamlines to be analyzed by linear stability

theory. For the here investigated case, the computational demand of processing 6 (12, 24) streamlines is approximately 25 (45, 90) times as high as the computational demand of one Tau iteration. The main reason for the relatively high computational demand for one transition prediction step is that the ratio of the average number of points on the streamlines to the overall number of grid points is rather high for this case. The mean number of points of the streamlines is approximately 300 (200 in the laminar part), giving a ratio of streamline to overall grid points of approximately 1/ 14,000 per streamline. An alternative computation was therefore performed with transition predicted on the upper and lower surfaces of the horizontal tail plane of a generic transport aircraft. For this case a hybrid grid was used, with a structured grid part covering the boundary layers on the geometry’s surfaces, with a resolution of the structured grid part normal to the wall of 48 grid points. With an overall number of grid points of 12 million and an average streamline length of 100 points (60 in the laminar part) on the horizontal tail plane, a ratio of the points of 1/ 120,000 per streamline is achieved. The relevant flow parameters are a =  4.0°, iH = 4.0°, Re = 2.3  106, Ma = 0.2. For both test cases, the prolate spheroid as described before and the generic transport aircraft, only Tollmien– Schlichting instabilities are now considered. The computations were carried out using 8 domains and 6, 12, and 24 edge streamlines. An idea of the grid partitioning for the generic transport aircraft is given in Fig. 13, which displays the location of the edge streamlines as well. The final result of the comparison is given in Fig. 14, where the computational time is normalized with the corresponding time used

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converged intermediate solutions after updating the transition location. For the flow around a prolate spheroid (discussed in Section 4.3), an average overall number of timesteps of 20.000–35.000 was needed, with 4–6 transition prediction steps to reach a well converged solution. The fraction of the overall consumed time used by the transition prediction process was then 1%. 4.3. Code validation

Fig. 13. Edge streamlines and domain boundaries. 8 domains, horizontal tail plane, generic transport aircraft.

Fig. 14. Computational time wct vs. number of streamlines n. 6:1 prolate spheroid, generic transport aircraft, 8 domains.

for one multigrid cycle (LU-SGS, 3w). For cases with a more favourable ratio of streamline size to grid size a significant reduction in relative computational time is obviously achieved. The high computational effort of one transition prediction step compared to one multigrid cycle should not be overrated when regarding the overall computational effort of a complete, fully converged calculation of a flow with transition prediction. The main part of the computational time is due to the iterative procedure of the transition prediction (Fig. 1). The high overall computational cost comes from rather large numbers of multigrid cycles needed to get

Validation of the transition prediction method is a necessary and important part of the utilization of this method in engineering applications. First validation investigations have therefore been undertaken in predicting the transition lines for the fully three-dimensional flow around a 6:1 prolate spheroid. At certain on-flow conditions transition is characterized to change from pure Tollmien–Schlichting transition to pure cross flow transition with regions where both types of waves may interact and lead to transition. Comprehensive measurements of the flow behaviour around an inclined prolate spheroid were accomplished at the DFVLR (now DLR) 3 m  3 m low speed wind tunnel, Go¨ttingen. Surface hot film probes measuring the local wall shear stress were applied for the investigation of the three-dimensional boundary layer [9]. Twelve measuring stations in streamwise direction and 30–80 measuring stations in circumferential direction were used. The evaluation of the local wall shear stress provides detailed information of the laminar–turbulent transition of the boundary layer. The measurements include a certain range of Reynolds number and angle of attack. From this repertory, five test cases have been chosen for the validation of the transition prediction module. It is well known that local, linear stability theory cannot analyze the interaction of Tollmien–Schlichting and cross flow waves from first principles. However, an empirical approach to overcome this deficiency is to reduce the critical N-values in the NTS-NCF space for simultaneously excited Tollmien–Schlichting and cross flow waves by assuming that the critical NTS-factor decays linearly with increasing NCF. Numerical investigations of the transition for the flow around the inclined prolate spheroid in [31] yield the diagram of Fig. 15, which was applied for the present validation calculations to account for the interaction of the two wave types. The steady calculations were carried out using lowMach-number preconditioning and the implicit LU-SGS time integration scheme. Turbulent flow was modelled using the standard Spalart–Allmaras turbulence model. In the structured grid part, a normal-to-wall resolution of 128 points was used, i.e., 60–100 points resolve the laminar boundary layer of the prolate spheroid. The resolution in streamwise direction is approx. 300 points, the overall number of grid points is 2.8 million. On-flow conditions varied from Ma = 0.03 to Ma = 0.13, a = 5.0° to a = 15.0° and Re = 1.5  106 to Re = 6.5  106. For the transition prediction 31 streamlines were equally distrib-

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Fig. 17. Comparison of computed and measured transition location. 6:1 Prolate spheroid, a = 10.0°, Re = 1.5  106, Ma = 0.03.

Fig. 15. Stability diagram. NTS vs. NCF.

Turning to the high Reynolds number cases (Re = 6.5  106, Ma = 0.13, a = 5.0°, 10.0° and 15.0°), transition is now caused by Tollmien–Schlichting and cross

uted over the prolate spheroid’s surface. The comparison of experimental and numerical results has been made by plotting the numerical transition line onto the contour plot of the experimental wall shear stress. From the local increase in wall shear stress transition on the prolate spheroid can be estimated. For an improved visualization of the overall information of the transition, the surface of the prolate spheroid has been mapped onto a 2D plot. The measuring stations from the experiment are given in the contour plots as small circles. For the low Reynolds number cases (Re = 1.5  106, Ma = 0.03, a = 5.0° and 10.0°), transition is triggered purely by Tollmien–Schlichting instabilities and is very well predicted by the transition module (Figs. 16, 17). Although for a = 10.0° cross flow amplifications are present during the iterative transition prediction, the final, converged transition is purely caused by Tollmien–Schlichting instabilities.

Fig. 18. Comparison of computed and measured transition location. 6:1 Prolate spheroid, a = 5.0°, Re = 6.5  106, Ma = 0.13.

Fig. 16. Comparison of computed and measured transition location. 6:1 Prolate spheroid, a = 5.0°, Re = 1.5  106, Ma = 0.03.

Fig. 19. Comparison of computed and measured transition location. 6:1 Prolate spheroid, a = 10.0°, Re = 6.5  106, Ma = 0.13.

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flow amplifications (Figs. 18–20). Generally, the transition is caused by Tollmien–Schlichting waves near the windward and leeward symmetry lines, whereas for the remaining part cross flow instabilities play a growing role with increasing angle of attack. For a = 5.0°, nearly the whole transition line is represented by simultaneously excited Tollmien–Schlichting and cross flow waves. Increasing the angle of attack to 10.0° leads to a development of a region with pure cross flow transition, that is even enlarged for a = 15.0°. In conclusion, all transition lines for the test cases investigated so far are predicted in good agreement with the experimental results. For the high Reynolds number cases, transition is predicted slightly too far upstream, but the general qualitative shape is represented fairly well. 4.4. Feasibility study 4.4.1. Generic transport aircraft The first test case of the feasibility study displays the flow around a generic, complex three-dimensional aircraft configuration. The objective was to predict transition simultaneously on all relevant surfaces of the configuration, namely body, vertical tail plane and upper and lower surfaces of main wing and horizontal tail plane. A hybrid grid was used, with a very moderate resolution of the structured part, with generally 32 grid points normal to the wall, except for the horizontal tail plane with 48 points normal to the wall. The chosen grid density in the boundary layer results in a fairly accurate prediction of boundary layer profiles while keeping the overall computational demand relatively low. This test case was run completely in parallel mode, using a partitioning of 8 domains for the grid, with an overall number of grid points of 12 million. For faster convergence, low-Mach-number preconditioning was applied, together with the implicit LU-SGS time integration scheme. Turbulent flow was modelled using the standard Spalart–Allmaras turbulence model.

The flow conditions were chosen to ensure attached flow over nearly all surfaces (a =  4.0°, iH = 4.0°, Re = 2.3  106, Ma = 0.2). With regard to the moderate resolution of the boundary layers, only Tollmien–Schlichting instabilities were considered. Previous investigations indicated that a much higher grid resolution is needed for accurate prediction of cross flow instabilities. Because of the very coarse resolution of the laminar boundary layer on the fuselage, both in wall-normal and streamwise direction, a simple transition criterion was applied that is not based on boundary layer data. I.e., a modified cp,min-criterion was used, setting the transition a short distance downstream of the pressure minimum. The main wing is equipped with a deflected flap, leading to large separated areas well before transition would have been predicted by linear stability theory. Here, the laminar separation point was used as transition point instead, in order to avoid unsteadiness effects and convergence problems of the Navier–Stokes iterations. For all other surfaces the linear stability theory in form of the eN-method was applied, with a critical N-factor of 7.5, corresponding to a turbulence level of 0.13%. Figs. 21 and 22 show the calculated edge streamlines together with the converged transition lines. For all wing like surfaces 6 streamlines have been considered to resolve the problem, however the body has been covered with 11 streamlines. The predicted transition lines are located in the adverse pressure gradient region, as expected for the investigated transition scenario. This test case shows the ability of the transition prediction module to predict transition on all relevant transitional surfaces of a generic transport aircraft configuration. It is especially demonstrated that the approach using edge streamlines from the Navier–Stokes is suitable for transition prediction for geometrically very different components (fuselage, wing) in one and the same computation. 4.4.2. 3D high-lift configuration Another test case for the transition prediction module is the flow over a generic, three-dimensional high-lift

Fig. 20. Comparison of computed and measured transition location. 6:1 Prolate spheroid, a = 15.0°, Re = 6.5  106, Ma = 0.13.

Fig. 21. Edge streamlines and transition locations. Generic transport aircraft, upper surfaces, a =  4.0°, Ma = 0.2, Re = 2.3  106, iH = 4.0°.

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Fig. 22. Edge streamlines and transition locations. Generic transport aircraft, lower surfaces, a =  4.0°, Ma = 0.2, Re = 2.3  106, iH = 4.0°.

configuration. The hybrid grid for this configuration does not permit a good resolution of the laminar boundary layer, since it has only 20 points normal to the wall in the structured grid part. Therefore this test case served as a platform to apply the boundary layer method for infinite swept, tapered wings attached to the transition module. The high-lift configuration is equipped with slat and flap (Fig. 23), the overall number of grid points for this case is 8 million and the test case was run in parallel mode using 8 domains. For the steady computations low-Mach-number preconditioning was applied, together with the implicit LU-SGS time integration scheme. Turbulent flow was modelled using the standard Spalart–Allmaras turbulence model. The flow conditions are set to match flow conditions that lead to the beginning of local separation (a = 12.0°, Ma = 0.174, Re = 1.34  106). Transition has been predicted on all wing surfaces (except for the lower surface of the slat), using the two different methods to determine the main flow properties for the linear stability analysis (Section 2.2.2): (i) using the pressure distribution and local sweep angles and applying a boundary layer code along line-in-flight cuts to generate the velocity profiles, and (ii) extraction of the velocity profiles directly from the Navier–Stokes solution along edge streamlines. As described in Section 2.2.2, for both methods the eNmethod, i.e., the integration of the amplification rates, is

Fig. 23. Geometry of 3D high-lift configuration.

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eventually applied along edge streamlines. These are generated from geometrical relations of the approach for infinite swept, tapered wings and the direction of the boundary layer edge velocity for method (i), or, for method (ii) are directly available from the transition module. One outcome of the application of both methods is reflected in the different development of the boundary layer velocity profiles. As already indicated by the grid dependency analysis in Section 4.1 a clear discrepancy in the velocity profiles can be seen (Figs. 24, 25). The disagreement is most evident for the crossflow velocity profiles and the first and second derivatives of all profiles. The main reason is the insufficient resolution of the boundary layer in the Navier–Stokes solution compared to the boundary layer method. However, uncertainties are introduced since in the current application of the boundary layer code the effect of the effective sweep angle is not considered. The effective sweep angle approach includes the displacement effects of the fuselage and the wing itself, which lead to an increase of the velocity of the flow along the attachment line [13,20]. These effects are neglected when using the geometrical sweep angle. Additionally, a bad resolution of the pressure distribution around the attachment line can lead to inaccurately determined stagnation-cp-values which may be inconsistent with the wing sweep [3,13]. This eventually can result in inaccurate boundary layer profiles from the boundary layer code. On the lower surface of the deflected flap no transition was found with method (i) (Fig. 26). This is in accordance with the appearing pressure distribution on this element (Fig. 27), which has a favourable gradient from the attachment line to the trailing edge. On the main wing’s lower surface, the predicted transition line with boundary layer data from the boundary layer code is close to the local pressure minimum (Figs. 26, 27), and is caused by Tollmien–Schlichting instabilities. For the upper surfaces of slat, main wing and flap, transition is predicted at the laminar separation location of the boundary layer method in the region of strong adverse pressure gradient (Figs. 28, 29). In the current application of method (i), the eN-method is

Fig. 24. Streamwise velocity profiles. 3D high-lift configuration, main wing, lower surface, a = 12.0°, Ma = 0.174, Re = 1.34  106, g = 0.6, x/ cloc = 0.28.

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Fig. 25. Crossflow velocity profiles. 3D high-lift configuration, main wing, lower surface, a = 12.0°, Ma = 0.174, Re = 1.34  106, g = 0.6, x/ cloc = 0.28.

Fig. 26. Edge streamlines, line-in-flight cuts, computed transition locations. 3D high-lift configuration, lower surfaces, a = 12.0°, Ma = 0.174, Re = 1.34  106.

Fig. 27. Pressure distribution. 3D high-lift configuration, lower surfaces, a = 12.0°, Ma = 0.174, Re = 1.34  106, transition prediction with boundary layer code data.

only applied, if the critical N-factor is reached before laminar separation, else the laminar separation point is used as transition point. An alternative approach would be to extrapolate the N-factor envelope into the separated flow area. For method (ii), i.e., transition prediction with boundary layer profiles extracted from the Navier–Stokes solution, considerably different transition lines are calculated (Figs. 26, 28), leading to a very different flow behaviour. In accordance with the grid dependency analysis in Section 4.1, transition is predicted to far downstream compared to the predicted transition with method (i). This is a direct cause of the insufficient resolution of the boundary layer in the Navier–Stokes solution, both in wall-normal and in streamwise direction. The inaccurate transition locations in turn result in (unphysically) large separation areas shortly after the pressure minimum (not shown here). In conclusion, the application of the boundary layer method is recommended for cases which do not meet the high grid requirements for accurate stability analyses. However, a careful revision of the extraction of the pressure distribution and the application of the effective sweep angle approach has to be made. The extraction of the boundary layer profiles from the Navier–Stokes solution is the preferred approach in the presence of laminar separation bubbles, as the stability analysis can then be extended into the separated flow regime. Additionally, the boundary layer code approach for infinite swept, tapered wings presented here is not suitable for fuselages. Here, the application of the transition criterion along edge streamlines from the Navier–Stokes solution is necessary. Generally, the application of the transition prediction module shows the promising capability of this method to predict transitional flows on general, three-dimensional high-lift configurations. However, validation against experimental data has not yet been possible due to the lack of reliable experimental data.

Fig. 28. Edge streamlines, line-in-flight cuts, computed transition locations. 3D high-lift configuration, upper surfaces, a = 12.0°, Ma = 0.174, Re = 1.34  106.

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dreas Krumbein of DLR, German Aerospace Center, Go¨ttingen, Germany, for general discussions of automatic transition prediction, testing of the transition module and his feedback as a user, Geza Schrauf of Airbus Deutschland GmbH, Bremen, Germany, for providing the codes Lilo and Coco and for guidance in their application, Hans Peter Kreplin of DLR, German Aerospace Center, Go¨ttingen, Germany, for providing the experimental data of the inclined spheroid, Thomas Gerhold of DLR, German Aerospace Center, Go¨ttingen, Germany, for discussions of the parallelization in Tau and Norbert Kroll of DLR, German Aerospace Center, Braunschweig, Germany, for general support of the project. References Fig. 29. Pressure distribution. 3D high-lift configuration, upper surfaces, a = 12.0°, Ma = 0.174, Re = 1.34  106, transition prediction with boundary layer code data.

5. Conclusions The general approach of the present work is to demonstrate the applicability of transition prediction in form of the eN-method to complex configurations and fully threedimensional boundary layers. The development of a transition prediction module attached to a Navier–Stokes solver is presented, with special emphasis on the parallelization of the transition prediction procedure for the application with domain decomposed Navier–Stokes solutions. A detailed study on the effect of the parallelization on the computational effort is accomplished and detailed results on the computational demand are given. The flow around three different configurations is investigated, where two cases are used to demonstrate the feasibility of the presented approach, and one test case was successfully used for the validation of the transition module. For the flow around an inclined prolate spheroid, it is shown that the two N-factor eN-method with boundary layer data from a Navier–Stokes solution is applicable for the prediction of transition of three-dimensional boundary layer flows. The validation study shows a good agreement of the numerically predicted transition lines with experimental data. The outcome of the feasibility studies is the demonstration of the usability of the edge streamline approach to apply the eN-method in a suitable way for transition prediction on all relevant surfaces of an aircraft configurations. However, comparison with experimental data for 3D wing flows still needs to be performed in the future to validate the presented approach.

Acknowledgements The present work was performed as part of the German research project Megadesign [11]. The authors thank An-

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