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Advanced CFD Analysis of Aerodynamics Using CFX

2001-102

Jorge Carregal-Ferreira, Achim Holzwarth, Florian Menter, Thomas Esch, Antoine Luu

AEA Technology GmbH, CFX Staudenfeldweg 12, D-83624 Otterfing, Germany

Tel. +49 8024 9054 10 Info@cfx-germany.com

Abstract CFX-5 is a leading and established CFD code for fluid dynamic analysis, which uses MSC.Patran as underlying CAD engine in its pre-processor CFX-Build. A key item in all CFD analyses is to obtain accurate and reliable solutions. Most industrial flows include turbulent flow structures that cannot be resolved numerically on available computers. To overcome the resolution limitations, CFD codes usually solve the Reynolds-Averaged Navier-Stokes equations augmented by turbulence models to compute the averaged turbulent stresses. The k-ε turbulence model with logarithmic wall functions is widely used in industry to compute external and internal flows. It is well known, however, that this model has the tendency to overestimate the shear stress and heat transfer at walls. A particular problem in the aeronautics and aerospace industry is that this model also under-predicts separation and hence the prediction of the overall performance of aerodynamic devices is often too optimistic. To address these deficiencies, turbulence models based on the k-? formulation have been proposed which have shown very promising results. In particular the SST turbulence model, which combines the advantages of the k-ε and the k-ω turbulence models, has been implemented into CFX-5. In addition, two advanced near-wall treatment methods, a consistent scalable wall function and an automatic switch between wall functions and the low-Re method, are presented. This paper shows validation cases in which these models have been applied to the aerodynamic analysis of aeronautical applications. Details of the models and a comparison with experimental is shown and discussed. The results demonstrate the progress that has been made in obtaining accurate solutions of turbulent flows with CFX.

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1 Introduction Computational Fluid Dynamics (CFD) is a standard tool in industry for analyzing external and internal flows of industrial devices. Examples where CFD is being used in the aeronautics industry are the prediction of lift and drag of airfoils and wings, flow around fuselages and entire aircrafts. On the engine side, CFD is used in the prediction of the performance and heat transfer in jet engines, including intakes, compressors, combustion chambers, turbines, and nozzles. Additional areas of application are internal flows in aircraft cabins, power supply units etc. CFX-5 [1] is a leading and established general purpose CFD code for internal and external aerodynamic fluid flow analyses that can be used for all of the above applications. CFX-5’s preprocessor CFX-Build uses MSC.Patran [2], which provides a modern and user-friendly environment for setting up and preparing the geometry model for subsequent numerical analysis. In the aeronautics industry there are several key demands on a modern CFD tool which the CFX software addresses:

• Efficient geometry generation and import: The pre-processor CFX-Build, which is based on MSC.Patran, allows the user to generate the geometry from scratch using MSC.Patran powerful creation facilities such as projection, extrusion, and manifolding. In addition it has direct or native CAD import interfaces to CAD systems such as Pro/Engineer, UG, CATIA, and I-IDEAS. The native CAD interfaces allow for importing geometries directly from CAD systems as an alternative to the IGES format.

• Rapid mesh generation: Hybrid, unstructured grid technology is essential because it

allows generating grids automatically and provides the user with a high geometric flexibility. CFX-5 offers an automatic mesh generator that has been developed jointly with the General Electric Corporate R&D Centre. In addition, mesh generators provided by MSC.Patran are available and meshes from other sources can be imported.

• User-friendly handling: A complete GUI driven definition of the case and appropriate

pre-processing (CFX-Build) and post-processing (CFX-Post) tools for setting up the simulation and for qualitative and quantitative analyses of the results allows the engineer to focus on the analysis of the device.

• Advanced integration into design environments: Interactive modification of run time

parameters and batch programming capabilities allows to use CFX-5 for optimisation of engineering devices. Fluid properties, boundary conditions, solution parameters, and job control are accessible through parameter files.

• Robust and fast numerical solution methods: The simulation of complex 3-dimensional

turbulent viscous flows in engineering applications requires a solver technology that delivers accurate solutions and rapid turn-around times. The CFX solver technology employs a unique and leading combination of robust and accurate higher-order numerical discretisation schemes, algebraic multi-grid solution method and scalable parallel computing on homogenous and heterogeneous computer platforms.

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• Advanced turbulence models: In order to obtain accurate solutions it is essential to use advanced turbulence models, which reproduce the physics of turbulence. This is a key item for most engineering applications and also the main subject of this paper.

Most industrial flows include turbulent flow structures that cannot be resolved numerically on available computers. To overcome the resolution limitations, CFD codes usually solve the Reynolds-Averaged Navier-Stokes equations augmented by turbulence models to compute the averaged turbulent stresses. These models are often the limiting aspect in the accuracy of numerical simulations. Close to solid structures or walls, boundary layers exist which typically require a high grid resolution and a special treatment in the numerical methods because of the steep velocity gradients. In most CFD codes the k-e turbulence model in combination with standard or logarithmic wall functions is used. Standard wall functions assume that the first near-wall grid node in the flow domain is within the universal logarithmic law of the boundary layer. In practice it is often difficult for the user to generate a mesh that is fine enough to resolve the boundary layer and simultaneously place the first grid node within the flow region where the logarithmic scaling law is valid. Hence, the above requirement for the position of the first near-wall grid node is very often violated which leads to poor predictions of velocity distributions, aerodynamic losses, and heat transfer rates. One of the technically very important boundary layer effects is separation of the flow under the influence of adverse pressure gradients. Separation has a significant effect on the overall performance of technical devices, like diffusers, turbine blades or aerodynamic bodies. Furthermore, separation can strongly influence other effects like heat transfer. Typically, most turbulence models predict attached flow even for pressure gradients where the experiments show strong separation. This is a problematic phenomenon, as reduced separation usually results in an overly optimistic prediction of machine characteristics. In some applications, this can have dangerous consequences, like in the deferred prediction of wing stall on aeroplanes. Another important issue is heat transfer from solid walls that are heated or cooled by the fluid flow. The heat flux is controlled by the boundary layer characteristics. Typically, standard turbulence models are poor in predicting accurate heat transfer rates, which then leads to wrong design modifications or in the worst case to malfunction of the technical device in service. Because of the aforementioned drawbacks of the standard models, improved turbulence models and wall treatment methods have been developed and implemented in CFX-5. In CFX-5 a wide range of turbulence models are implemented covering k-e based models, k-? based models, second order (Reynolds stress) models, and Large Eddy Simulation (LES). In the present paper we will focus on the two-equation formulations and advanced near wall treatment methods, which have been explicitly developed to address the needs in the aeronautics industry. In the next section the k-e model, the k-? model, and the SST model are described. Section 3 describes the advanced methods for near wall treatment. In section 4 the results of several cases are presented and compared to experimental data. The results demonstrate the progress that has been made in obtaining accurate solutions with CFX.

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2 Turbulence Models 2.1 The k-ε Turbulence Model In the k-e model [3] transport equations for the turbulent kinetic energy k and its dissipation e (e is the amount of k per mass and time converted into heat by viscous action) are solved and subsequently the turbulent viscosity is calculated from the product of a turbulence velocity scale and a turbulence length scale.

The e-equation is used to determine the turbulent length scale. Experience has shown that the choice of the scale equation has a significant effect on the prediction of turbulent flows, and that the e-equation has severe limitations in the near-wall region. A major problem is that this model leads to an over-prediction of the turbulent length scale in flows with adverse pressure gradients, which results in high wall shear stress. For example, the stall characteristic of an aircraft is controlled by flow separation on the wing. Turbulence models based on the e-equation predict the onset of separation too late and under-predict the total amount of separation, which results in an optimistic and wrong prediction of the performance characteristic of the airfoil. 2.2 The k-ω Turbulence Model An alternative to the e-equation is the ?-equation in the form developed by Wilcox [4]. Instead of the equation for the turbulent dissipation rate e, an equation for the turbulent frequency ? of the large scales is used. The ? -equation has significant advantages near the surface and accurately predicts the turbulent length scale in adverse pressure gradient flows, leading to improved wall shear stress and heat transfer predictions. Furthermore, the model has a very simple low-Re formulation, which does not require additional non-linear wall damping terms. The turbulent viscosity is calculated from a turbulence velocity scale and the turbulence frequency. One of the main advantages of the k-? model is its robustness even for complex applications, and the reduced resolution demands for an integration to the wall. It was pointed out by Menter [5] that the main deficiency of the standard k-? model is the strong sensitivity of the solution to free stream values for ? outside the boundary layer. In order to avoid this problem, a combination of the k-? model near the wall and the k-e model away from the wall has been proposed, leading to the SST (Shear-Stress-Transport) model [6,7]. 2.3 The SST Turbulence Model The SST model combines the advantages of the k-e and the k-ω model to achieve an optimal model formulation for a wide range of applications. For this a blending function F1 is introduced which is equal to one near the solid surface and equal to zero for the flow domain away from the wall. It activates the k-? model in the near wall region and the k-e model for the rest of the flow. By this approach the attractive near-wall performance of the k-? model can be used without the potential errors resulting from the free stream sensitivity of that model. In addition, the SST model also features a modification of the definition of the eddy viscosity, which can be interpreted as a variable cµ, where cµ in the k-e model is constant. This modification is required to accurately capture the onset of separation under pressure gradients.

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In a recent NASA Technical Memorandum [8] the SST model was rated the most accurate model in its class. The SST model and other advanced turbulence models are implemented in CFX-5, which gives CFX a leading position in the area of turbulence modelling and accurate prediction of fluid flow. In addition to the advanced turbulence models, an advanced near wall treatment has been developed and implemented into CFX-5, which allows the user to benefit from the advantages of the model, even for grids with a reduced near wall spacing. 3 Near Wall Treatment An important issue of turbulence modelling is the numerical treatment of the equations in regions close to walls. The near-wall formulation determines the accuracy of the wall shear stress and heat transfer predictions. Also, it has an important influence on the development of boundary layers, including the onset of separation. Two approaches are commonly used to model the flow in the near-wall region:

• The wall function method. • The low-Reynolds number (low-Re) method.

In the wall function approach the viscosity affected, laminar sublayer region (the viscous sublayer) is bridged by employing empirical formulas to provide near-wall boundary conditions for the mean flow and turbulence transport equations. These formulas connect the wall conditions to the dependent variables at the near-wall grid node, which is presumed to lie in the fully turbulent region of the boundary layer where the mean velocity scales logarithmic with the distance from the wall. An alternative approach to the wall function method is to use a fine grid analysis in which computations are extended through the laminar viscous sublayer close to the wall. The low-Re approach requires a very fine grid in the near-wall region in order to resolve the complete boundary layer and no empirical formulas are used to describe the connection between the wall and near-wall grid node. 3.1 Scalable Wall Functions For the standard wall functions, which are commonly used in most CFD codes, the distance of the near-wall grid node must be such that it is within the zone where the logarithmic velocity scaling law is valid. A the same time a sufficient number of grind nodes must be placed close to the wall in order to resolve the velocity gradients of the boundary layer. The user must take care that the resolution of the grid and the distance of the near-wall grid node is set appropriately. In many technical cases this is a non-trivial procedure, which can only be achieved iteratively or is even practically impossible. Refining the mesh under such restrictions does not necessarily give a unique solution of increasing accuracy. In many technical applications the grid requirements for standard wall functions are not met which results in poor predictions.

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In order to avoid the problem of inconsistent degradation of the prediction with increased refinement of the grid, CFX has recently introduced a new formulation for wall function based models, called scalable wall functions [9]. For the moment this is the only available wall function formulation, which allows the user to apply arbitrarily fine grids without a violation of the underlying logarithmic profile assumptions. The scaleable wall function is the default setting in CFX-5 for the k-ε based models. The basic idea behind the scalable wall function approach is to assume that the solid surface coincides with the edge of the viscous sublayer, which is the intersection between the linear and the logarithmic near wall velocity profile. The computed non-dimensional wall distance, which is used in the wall functions, is not allowed to fall below the limiting value that defines the width of the viscous sublayer. Therefore, the present implementation forces all grid nodes to be outside the viscous sublayer and allows for a consistent grid refinement, i.e. the user may refine the mesh without the need to take care of the position of the near-wall grid node and still gets an improved prediction. 3.2 Automatic Selection of Wall functions or Low-Reynolds Number Formulation While the scalable wall functions presented above allow for a consistent grid refinement, they are based on physical assumptions which are critical in flows at lower Reynolds numbers, because the sublayer portion of the boundary layer is neglected in the mass and momentum equations. It is generally accepted that the integration of the boundary layer through the thin viscous sub-layer near the wall, is preferable to the use of wall function boundary conditions. The challenge is, however, that a strict low-Reynolds number model requires a very fine grid resolution. This requirement can hardly be achieved for all walls in complex geometries. It would also lead to excessive cell aspect ratios, which in most cases would require a double precision computation, thereby doubling the memory consumption of the code. If the grid is too coarse, the use of a low-Re model will result in a poor prediction. It is therefore desirable to offer the user a formulation that will automatically switch from wall functions to a low-Re formulation as the grid is refined. The k-? based models, including the SST model, have the advantage that an analytical expression is known for ? in the viscous sublayer, which can be exploited to achieve the aforementioned goal. The basic idea is to blend the wall value of ? between the logarithmic and the low-Re formulation. While in the wall function formulation the near wall grid point is treated as being outside the edge of the viscous sublayer, the location of the near wall grid node is virtually moved down through the viscous sublayer as the grid is refined in the low-Re mode. Note that the physical location of the first grid node is always at the wall. The error in the wall function formulation results from this virtual shift which amounts to a reduction in displacement thickness. This error is always present in the wall function mode, although it is negligible if the viscous sublayer is thin when compared to the complete boundary layer. This error reduces to zero as the present method shifts to the low-Re mode. This new wall boundary condition has been implemented in combination with the SST model [10]. It exploits the simple and robust near wall formulation of the underlying k-ω model and switches automatically from a low-Reynolds number formulation to a wall function treatment based on the grid density. The advantage is that the user can make optimal use of the advanced performance of the turbulence model, for a given grid. The automatic wall treatment avoids the deterioration of the results typically seen if low-Re models are applied on under-resolved grids.

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4 Validation Cases and Applications In the following subsections several numerical analyses are presented and compared to experimental data. The SST model and the k-e model, and the different wall treatments are compared and discussed. 4.1 Flat Plate: Standard vs Scalable Wall Functions The first case illustrates the different behavior of the standard and the scalable wall functions on the friction coefficient of the boundary layer flow on a flat plate. Fig. 1 shows the skin friction coefficient obtained with the k-ε model and the standard wall functions for different grid resolutions.

The results of Fig. 1 show that as the grid is refined the values for cf depart from the exact solution and, hence, the prediction becomes worse. In contrast, Fig. 2 shows that with the scalable wall functions the results consistently improve as the grid is refined. On the finest grid, Grid 5, the results are identical to the exact solution. Grid 5 is not shown in Fig. 1 because it did not converge with the standard wall functions. The scalable wall functions not only provide a consistent improvement on refined grids of the prediction but also improves the robustness of the solution on fine grids.

Figure 1: Skin Friction of flat pate for computation with k-ε model and standard wall functions. Grid 1: coarse grid resolution, Grid 4: finest grid resolution.

Figure 2: Skin friction coefficient for computation of flat plate with k-ε model with scalable wall functions. Grid 1: coarse grid resolution, Grid 5: finest grid resolution.

Refinement Refinement

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4.2 Flate Plate: Automatic Switch between Wall Functions and Low-Re Model The second case shows the impact of the automatic switch between wall functions and the low-Re method. Fig. 3 shows the smooth shift of the wall treatment from a viscous sublayer to a wall function treatment, based on the grid spacing. As the grid is refined, the model automatically shifts from the wall function method to the low-Re method. The exact velocity profile is in all cases recovered.

Fig. 4 shows the dependence of the skin friction cf on the non-dimensional near-wall grid spacing for a pure low-Re model and for the new method in comparison to the exact solution. For a very fine grid the low-Re model is in line with the exact solution. However, as the grid becomes coarser the solution of the low-Re model rapidly deviates from the exact solution. In contrast, the new method where the model switches automatically to wall function as the grid becomes coarse remains in a very narrow band around the exact solution and provides a much more robust and reliable solution procedure. 4.3 Plane Diffuser Diffusers are typical flow devices where separation may occur. The SST and the k-e model have been applied to the plane diffuser of Gersten et al [11]. Fig. 5 shows the streak lines of the diffuser for computations with the two turbulence models. With the k-e model the flow is completely attached and no separation is predicted, whereas with the SST model a strong separation and a re-circulation zone is predicted. The under-prediction of separation, as seen in this example is a typical limitation of the k-e model.

Figure 3: Automatic wall treatment for velocity profile.

Figure 4: Grid sensitivity for different near-wall for low-Re model and new automatic switch..

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Fig. 6 shows the velocity profiles in the diffuser, computed with the SST and the k-ε model and compares them with the experimental data. The velocity profile obtained with the k-e model deviates from the experiments because the model does not capture separation. On the other hand, the results clearly demonstrate the superior performance of the SST model under adverse pressure gradient conditions.

k-ε model

SST model

k-ε model

SST model

Figure 5: Plane Diffuser, calculated with k-ε model and SST model.

Figure 6: Velocity profiles for diffuser flow. Comparison of turbulence models.

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4.4 NACA 4412 – Airfoil The NACA 441 airfoil is a classical validation case for high lift devices where the adverse pressure gradient flow is dominating. Separation occurs on the suction side close to the trailing edge of the airfoil. The two-dimensional experiments [12] have been performed with air at a free stream Mach number of Ma8 = 0.2, a Reynolds number of Re = 2.8 × 105 and an angle of attack of α = 13.87. In the computation, adiabatic walls and constant fluid properties are applied, in close agreement with the experimental conditions.

Fig. 7 shows the hexahedral grid for the airfoil. Fig. 8 shows the Mach number distribution for the airfoil, which was obtained from the computation with the SST model. The stagnation point and the maximum velocity distribution close to the leading edge of the airfoil are visible. Close to the trailing edge of the suction side a region of low Mach number can be observed. This is region where separation occurs.

Fig. 10: Comparison of experimental results of the pressure coefficient Cp with different turbulence models for the NACA 4412 airfoil.

Fig. 7: Hexahedral grid for NACA 4412 Airfoil.

Fig. 11: Velocity profiles perpendicular to walls with different turbulence models for the NACA 4412 airfoil.

Fig. 8: Mach number for NACA 4412 Airfoil.

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Fig. 10 shows a comparison of the pressure coefficient Cp from the computation with different turbulence models and the experiment. The models shown in the figure are the k-e model, the SST model, and an algebraic stress model, which is not described in this paper. The results of all models are in close agreement wit the experimental data, apart from the area at the trailing edge where separation occurs. This is visible in Fig. 11, where the velocity distribution perpendicular to the wall of the airfoil is shown. The three different turbulence models are shown in comparison with the experimental data. The SST model predicts the separation zone in close agreement with the data, whereas the k-ε model fails to capture the physics of this flow. The algebraic stress model is somewhere between the two other models. 4.5 Pipe Expansion with Heat Transfer Figure (12) shows the set-up for the experiment of Baughn et al [13] of an axis-symmetric pipe expansion with heat transfer. The geometry is an axis-symmetrical pipe with a sudden expansion of D/d = 3.33. A constant heat flux is imposed on the wall of the pipe with the larger diameter, whereas all other walls are adiabatic. The Reynolds number based on the diameter of the outer pipe is ReD = 2.0 × 104. Constant fluid properties for air have been used in the computation. Figure 13 shows results of the simulation with the k-ε and the SST model. A fine grid of 100 × 240 was used for the simulation. Grid convergence was ensured by a series of simulations on 5 different grids. The simulation shows that the SST model captures the wall temperature well, even in the separated region. The computations have been performed with the low-Re number wall treatment of the model. It can be seen that the k-ε model leads to an over-prediction of the heat transfer in the region of flow reattachment. This is a result of the over-prediction of the turbulent length scale in this region. Note that in this test case the onset of separation is fixed by the sudden change in the geometry. This avoids the problems of the k-ε model to predict the onset of separation as observed in the diffuser flow of Section 4.3. Included in Fig. 13 is also a computation with the two-layer k-e turbulence model. In this formulation, the e-equation is only solved in the outer part of the boundary layer, whereas the inner portion of the logarithmic layer and the viscous sublayer is treated by a mixing length formulation. The model is commonly used

Figure 12 Geometry of the pipe expansion.

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for the prediction of heat transfer. Note that this formulation significantly under-predicts the wall heat transfer rate. This is consistent with the observations in Bredberg et al. [14]. As the high Reynolds number versions of the two k-e model simulations are identical, Fig. 13 also demonstrates the importance of the near-wall treatment of the equations.

5 Summary Standard and advanced turbulence models and new formulations for the near wall treatment have been discussed. The SST model, which combines the advantages of the k-e model and the k-? model, has been presented and applied to a variety of test cases and applications. The comparison with experimental data demonstrates the superior predictions obtained with this model in comparison with models purely based on the e-equation. This is particular true for flows with adverse pressure gradients, which can be found in many applications in the aeronautics industry, including diffusers, airfoils, and heat exchangers. In addition, two advanced near-wall treatment formulations, a consistent scalable wall function and an automatic switch between wall functions and the low-Re method, have been presented. The scalable wall function can be applied with the k-e model and allows the user to refine the mesh and obtain a consistently improved solution on the finer mesh, which is not the case if the standard wall functions are applied. The automatic

Fig. 13: Comparison of experimental Nusselt number distribution with results from the SST, the k-e model, and the k-e two-layer model for sudden pipe expansion.

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switch between wall functions and the low-Re model selects the most appropriate near wall treatment automatically by detecting the refinement of the grid. The model can be applied in combination with the SST model and allows to obtain improved prediction on different types of grids. These advanced models are implemented in CFX-5, which is thereby further enhanced as a leading CFD tool for aerodynamic simulations. The validation cases demonstrate the progress that has been made in obtaining accurate solutions of turbulent flows with CFX. 6 Acknowledgements The authors would like to thank all CFX co-workers who contributed to this paper by implementing the models in CFX-5 and by calculating the test cases. 7 References [1] CFX-5, Users Manual, CFX-International, 2001. [2] MSC.Patran Users Manual, MSC.Software Corporation, Los Angeles, CA, 2000. [3] Jones, W.P. and Launder B.E., ”The prediction of laminarization with a two-equation model

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[12] Coles,D. and Wadcock, A.J. “Flying-hot-wire study of flow past an NACA 4412 airfoil at maximum lift”, AIAA Journal, Vol. 17, No. 4 1979.

[13] Baughn, J.W., et al., ”Local heat transfer downstream of an abrupt expansion in a circular

channel with constant wall heat flux”, Journal of Heat Transfer, Vol. 106, 1984, pp. 789-796. [14] Bredberg, J., Davidson, L. and Iacovides, H., “Comparison of near-wall behavior and its

effect on heat transfer for k-? and k-e turbulence models in rib-roughened channels”. In Nagano., Y., Hanjalic, K., and Tsuji, T. eds: 3rd Int. Symposium on Turbulent Heat and Mass Transfer, 2000, pp 381-388.