American Institute of Aeronautics and Astronautics Paper 2004-05661

Grid Sensitivity Effects in Collection Efficiency Computation

Luis C. de Castro Santos , Luiz Tobaldini Neto, Ramon Papa, Guilherme L. Oliveira and Antônio B. Jesus

EMBRAER, São José dos Campos, SP, BRAZIL, 12227-901

Sutikno Wirogo

FLUENT Inc., Lebanon, NH, USA, 03766-1442

Abstract

The present work compares the effects on accuracy of different grid generation choices and practices with respect to collection efficiency calculation on a typical aircraft configuration. It is known that hexahedral meshes provide more accurate and better representation of the surface gradients, but on the other hand, requires more expertise and effort. This paper compares hexahedral, tetrahedral and hybrid (tetra + prism) meshes with respect to collection. Results indicate that the impingement region limits are rather insensitive to mesh choice, but the maximum and averaged collection can vary significantly. Some general levels of equivalency are proposed based on the experiments performed for the DLR-F4 configuration.

Introduction

An important issue in the aeronautical industry is accounting the effect of ice accretion. Accretion usually takes place during holding conditions, when the aircraft is forced to spend a considerably large amount of time, up to 45 minutes, at lower altitudes, under meteorologically favorable conditions for the accumulation of super-cooled water droplets. When those droplets collide with the airframe, heat is released and the droplets accrete as ice. The main hazard to flight comes from the change of the aerodynamic characteristics of the aircraft, especially in critical phases such as approach and landing. This problem has been extensively addressed by the installation of icing protection systems, which heat the critical surfaces preventing hazardous ice formation. Another issue of concern are non-protected areas, which are not essential to the overall flight quality, but where the accumulated ice can be shed during the aircraft transition from holding to descent and landing, presenting the risk of impact on the engine, nacelles, control surfaces, etc. A vital parameter for the evaluation of ice accretion, and consequently the need of thermal protection, is the collection efficiency parameter (β), which relates the fraction of the captured droplets to the free-stream droplet density. There are basically two methods to compute this parameter, the discrete particle Lagrangian based approach and the Eulerian approach. Discrete particle methods, although extensively used in several industry codes, are not cost effective when dealing with

complex three-dimensional shapes. The number of particle seeded in the flow has to be very high in order to provide a precise and smooth computation of β. Eulerian methods, on the other hand, eliminate the particle seeding issue since the droplets are represented as a continuous phase. But, as any scalar transport equation, it is subject to numerical issues like numerical dissipation, which can mask the actual physical results. A common requirement of both methods is a fine representation of the surface, and therefore a significant fraction of the analysis time is spent on rigorous grid generation followed by an intensive number-crunching period. Since this is the bottleneck of the analysis any effort in order to reduce analysis time can have a significant impact on the overall cost of ice formation risk assessment.

The purpose of the present work is to investigate the comparative accuracy of the different grid generation choices and practices with respect to collection efficiency calculation on a typical aircraft configuration. It is known that hexahedral meshes provide more accurate and better representation of the surface gradients. However, the corresponding grid generation process, especially for complex shapes, is much more elaborate, and time consuming, than for tetrahedral meshes. The question is then how that choice affects the computation of the collection efficiency and how fine the grid must be in each case. The tests objective is to establish minimal level of refinement requirements for different design phases.

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AIAA 2004-566

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American Institute of Aeronautics and Astronautics Paper 2004-05662

Numerical Method

For ice accretion analysis it is assumed that the volume fraction of the water droplets is small and the aerodynamic effects on the continuous gas flow field are negligible. This permits the computation of the droplet motion to be decoupled from the gas flow calculations. The flow field of the gas phase around the body is calculated using the FLUENT general-purpose solver, which is applicable to inviscid/viscous and incompressible/compressible flow fields. A description of the solver can be found in the Fluent User’s Guide [1].

The droplet motion can be solved either in the Lagrangian or the Eulerian frame. The governing equations for the particle motion in the Lagrangian frame is:

Fuuu

a +−= )(Kdt

d (1)

where u is the droplet velocity vector and ua is the gas phase velocity vector. F is the force per unit mass on the droplet due to external sources other than the drag. Such external forces can be the pressure gradients in the fluid, the added mass effect or the force due to gravity.

The corresponding droplet conservation equations in the Eulerian frame are given below:

( ) 0=⋅∇+∂∂

uαραρt

(2)

( ) Fuuuuu

a αραραραρ +−=⊗∇+∂∂

)(Kt

(3)where⊗ is the dyadic product, α �is the droplet volume fraction and K is the momentum exchange coefficient related to drag. The droplet x, y and z velocities along with its volume fraction are defined as User-Defined-Scalars in FLUENT. These are solved according to equations (2) and (3) in finite volume formulation using the User-Defined-Scalar-Transport framework in FLUENT. The boundary conditions and the relation to compute the collection coefficient (β) can be found on references [2,3,4].

Reference [4] presents several validation cases, with respect to other tools as LEWICE [5] and FENSAP-ICE [3], and also to experimental data, including full three-dimensional geometriessuch as the Boeing 737-300 engine nacelle wind tunnel model [7].

The level of accuracy of this test case supports the comparison proposed on the next item.

Preliminary Computation

The methodology of the proposed study is to analyze several types of grids each with different levels of refinement. A 3-D sphere placed in a prismatic farfield is used a trial geometry. The first case has purely hexahedral elements, such as the one presented in Figure 2(a), the second and third are purely tetrahedral meshes, with different levels of refinement, just as seen in Figure 2(b) and 2(c), and the last is a hybrid mesh, with 10 prisms layers grown out of the surface triangles transitioning to tetrahedral cells which fill up the volume. On the Figure 2(d) one notices the surface level of refinement of the hybrid mesh is the same as the first tetra mesh. The flow field of the continuous phase is computed using FLUENT, as inviscid at Mach 0.2. The droplet mass and momentum conservation equations in the Eulerian frame are solved using the User Defined Scalar Transport framework within FLUENT. At the impingement region, a boundary condition that allows the impinging droplets to escape from the domain is implemented.

The computed collection coefficient (β) for each mesh is shown in Figure 3. The effect of mesh type can be clearly seen on the pictures. The hexahedral mesh (Fig.3a) provides a smooth continuous solution for the collection coefficient with sharp impingement limits for the level of refinement chosen. The use of tetra meshes (Fig.3b and Fig.3c), independent of the discretization level, scatters the results alternating regions of with very distinct levels of β. This effect is typical of a three-dimensional calculation, in 2D, according to [1], it is not observed.

When the prism layers (Fig.3d) are added a dramatic improvement is observed, although the issue of smoothness remains the overall appearance of the plot indicates a much more reliable estimate of the collection coefficient. Analyzing the values of the collection coefficient for a mid-plane cut of the mesh, those effects are quantitatively better represented (Fig.4). The maximum level of for the hexahedral mesh is about 0.41. For both tetra meshes it remains around 0.31-0.32, a reduction of about 24%. For the prism layered mesh the maximum level is even higher than the hexa mesh, reaching 0.45. Comparing the area weighted average of the collection coefficient the discrepancy level

American Institute of Aeronautics and Astronautics Paper 2004-05663

reaches about 30% for both tetra meshes, while for the hybrid it is less than 5%. Despite diffusion the impingement limit is reasonably well captured (depending, of course, on the application need of accuracy). The reason for such behavior can be explained, with help of Fig.5, as being mostly related to the convective terms. Since the convective part of the equation is driven by the continuous phase velocity field, which has a much higher magnitude than the numerical dispersion speeds the impingement limits are less affected than the intensity of the discrete phase concentration. This observation supports the use tetra meshes, which are much easier and faster to generate, for preliminary analysis. The collection maximum magnitude is known to be higher than the computed level, as the 25% of the present example, but the impingement limits are somewhat accurate, and known to be smaller than in the refined hexa case. These findings can lead the preliminary design of the extension of protected areas. When a more refined estimate of the power requirements of the ice protection system is necessary a hybrid mesh with prism layers can provide a better estimate. Considering that in the preliminary phases the geometry changes quickly, the need of a faster analysis process, which can take advantage of previously generated meshes, such as those used for aerodynamic design, becomes more important. Finally after the geometry is “frozen” a hexa mesh can provide the final check necessary to validate the preliminary system design.

Practical Application

In order to validate the findings for the sphere, in a typical configuration, the Drag Prediction Workshop case, with the DLR-F4 wing- body geometry, is used. For an initial comparison two meshes were used: An extra coarse hexa mesh, obtained by de-refining the coarse mesh generated by ICEM, with 692558 volume cells and 12710 surface quads; and the tetra + prism coarse mesh generated by ICEM, with 1964682 volume cells and 79124 surface tri. Some sample views of the meshes are presented on Figure 6.

The flow field corresponds to average a holding pattern at 15,000 ft, ISA, at 200 knots, AOA 2°, with MVD=40 µm droplets. Both flow and droplet Eulerian phase where obtained using FLUENT, just as the previous sphere cases. Figures 7 and 8 show contours of collection coefficient on the nose and close to wing root. It is noticeable the spread on the results with the tetra

+ prism mesh. Both mesh display the characteristics radome collection and windshield concentration. A close look of the wing root on Figure 8, for the hexa mesh, clearly shows the shadowing effects due to fuselage. On the tetra + prism mesh the spread masks out such effect.

Figure 9 shows the volume fraction (α) for both the surface and some select y constant planes along the span. Since the aircraft has a small angle of attack, and generates lift, a shadow area is clearly seen on the upper surface, while on the lower surface the concentration increases. The general smearing effect is observed when comparing both meshes.

Figure 10 shows the collection coefficient at the same stations show before. The maximum value of beta is the same on both meshes when the impingement limits have a significant change. This effect was not observed on the test with the sphere and is an indication of the effect of lift. Also since the number of surface cells on each case is very different, this issue will require further study.

Conclusion

From the preliminary experiments with the sphere it was observed that the failure to accurate represent gradients, specially close to the surface where the droplets will do a very strong turn. Significantly affects the level of collection. That is why the cases which have prism layers (hex or prism) produces a better solution.

The analysis performed on the DLR-F4 geometry, following the guidelines of the sphere experiment, verify that the choice of a tetra + prism layer mesh can be effective in order to reduce mesh generation time. In terms of maximum beta values the results, for the single case tested, are essentially at the same level. In terms of the impingement limits there is a considerable spread, which can be due to different number of cells and surface elements from each case. This effect not only interferes with the solution of the scalar equations directly, but also the flowfield itself, which could enhance the discrepancies. Further studies are being conducted in order to separate these effects.

American Institute of Aeronautics and Astronautics Paper 2004-05664

References

[1] - Fluent 6.0 User’s Guide, 2001, Fluent Inc.

[2] - Y. Bourgault, Z. Boutanios and W.G. Habashi.: An Eulerian Approach to 3-D Droplet Impingement Simulation Using FENSAP-ICE. Part I: Model, Algorithm and Validation, Journal of Aircraft, 37:95-103, 2000.

[3] - Y. Bourgault and W.G. Habashi: A new PDE-based Icing Model in FENSAP-ICE. Canadian Aeronautics and Space Institute, 7th CASI Aerodynamics Symposium, Montreal, Quebec, Canada, May 2-5, 1999.

[4] – S. Wirogo, S. Srirambhatla; An Eulerian Method to Calculate the Collection Efficiency on Two and Three Dimensional Bodies. 42th AIAA

Annual Meeting and Exhibit, Reno, Nevada, 2003.

[5] - Ruff, G.A., and Berkowitz, B.M., 1990. “Users Manual for the NASA Lewis Ice Accretion Prediction Code (LEWICE).” NASA CR185129.

[6] – Bidwell, C.S., and Mohler, S.R. Jr., 1995. “Collection Efficiency and Ice Accretion Calculations for a Sphere, a Swept MS(1)-317 Wing, a Swept NACA-0012 Wing Tip, an Axisymmetric Inlet, and a Boeing 737-300 Inlet.” NASA TM 106831.

[ 7] - Papadakis, M., Elongonan, R., Freund, G.A. Jr., Breer, M., Zumwalt, G.W., and Whitmer, L., 1989. “An Experimental Method for Measuring Water Droplet Impingement Efficiency on Two-and Three-Dimensional Bodies.” NASA CR 4257.

American Institute of Aeronautics and Astronautics Paper 2004-05665

Fig. 1 Comparison of composite collection efficiency for 10.16 cm diameter sphere with free stream velocity of 75 m/s and droplet distribution with MVD=18.6µm. Reference [6].

(a) 4896 surface quad (sphere), 410720 hexa cells (b) 16502 surface tri (sphere), 394982 tetra cells

(c) 64446 surface tri (sphere), 781867 tetra cells (d) 16496 surface tri (sphere), 420441 hybrid cells

Fig. 2 Surface view of the hexahedral, tetrahedral and hybrid meshes

American Institute of Aeronautics and Astronautics Paper 2004-05666

(a) 4896 surface quad (sphere), 410720 hexa cells (b) 16502 surface tri (sphere), 394982 tetra cells

(c) 64446 surface tri (sphere), 781867 tetra cells (d) 16496 surface tri (sphere), 420441 hybrid cells

Fig. 3 Collection Coefficient (β) (UDM4) contour plot for the hexahedral, tetrahedral and hybrid meshes

Fig. 5 Local Collection Efficiency – max β = 0.45 , impingement limits z ∈ [-0.1, 0.1].

American Institute of Aeronautics and Astronautics Paper 2004-05667

(a) Hexa Mesh - General View

(b) Tetra + Prism Mesh - General View

(c) Hexa Mesh - Nose View Detail

(c) Tetra + Prism Mesh - Nose View Detail

Fig. 6 General View of the DPW Related Meshes

American Institute of Aeronautics and Astronautics Paper 2004-05668

(a) Hexa Mesh – max β = 0.85

(b) Tetra + Prism Mesh - max β = 0.85

Fig. 7 Front View of the DLR-F4 – Collection Efficiency Levels

(a) Hexa Mesh – max β = 0.8

American Institute of Aeronautics and Astronautics Paper 2004-05669

(b) Tetra + Prism Mesh - max β = 0.8

Fig. 8 Wing Root View of the DLR-F4 – Collection Efficiency Levels

(a) Hexa Mesh – UDS 3 (volume fraction)

American Institute of Aeronautics and Astronautics Paper 2004-056610

(b) Tetra + Prism Mesh – UDS 3 (volume fraction)

Fig. 9 UDS 3 (volume fraction) Contours for select cuts (y=0.8, 2.0, 4.0, 5.5)

(a) y = 0.8

American Institute of Aeronautics and Astronautics Paper 2004-056611

(b) y = 2.0

(c) y = 4.0

(d) y = 5.5

Fig. 10 – Local Collection Coefficient Comparison along y-stations