Implicit Large Eddy Simulation of Low Reynolds ...

AIAA 2008-225

46th AIAA Aerospace Sciences Meeting and Exhibit 7 - 10 January 2008, Reno, Nevada

Implicit Large Eddy Simulation of Low Reynolds Number Flow Past the SD7003 Airfoil Marshall C. Galbraith* University of Cincinnati, Cincinnati, OH, 45221

Downloaded by MASSACHUSETTS INST OF TECHNOLOGY on November 15, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2008-225

and Miguel R. Visbal† U.S. Air Force Research Laboratory, Wright-Patterson Air Force Base, OH, 45433

The formation and burst of a laminar separation bubble has long been known to be detrimental to the performance of airfoils operating at low Reynolds numbers (Re < 105). With increasing interest on Micro Air Vehicles (MAV), further understanding of the formation and subsequent turbulent breakdown of laminar separation bubbles is required for improved handling, stability, and endurance of MAV’s. An investigation of flow past an SD7003 airfoil over the Reynolds number range 104 < Re < 9x104 is presented. This airfoil was selected due to its robust, thin laminar separating bubble, and the availability of highresolution experimental data. A high-order implicit large-eddy simulation (ILES) approach is shown capable of capturing the laminar separation, transition, and subsequent threedimensional breakdown. The ILES methodology also predicts without change in parameters the passage into full airfoil stall at high incidence. Results of the computations and comparison with experimental data are analyzed and discussed. Good agreement is generally found between experiment and computations for separation, reattachment, and transition locations, as well as aerodynamic loads.

Nomenclature C Cp

= Chord length = Time- and spanwise-mean coefficient of pressure, 2( p ∞ − p ) / ρ ∞U ∞2 µ ∂u Cf = Time- and spanwise-mean skin friction coefficient, 2 Re ∂n Et = Total specific energy FI, GI, HI = Inviscid vector fluxes Fv, Gv, Hv = Viscous vector fluxes I, J, K = Coordinate grid indices in the circumferential, surface normal, and spanwise directions J = Jacobian of coordinate transformation M = Mach number Pr = Prandtl number; 0.72 for air Q = Vector of dependent variables Re = Reference Reynolds number, ρ ∞U ∞ C / µ ∞ T = Non-dimensional static temperature U, V, W = Contravariant velocity components = Freestream reference speed U∞ X, Y, Z = Non-dimensional Cartesian coordinates hb = Maximum laminar separation bubble height *

PhD Graduate Student, Department of Aerospace Engineering & Engineering Mechanics, 745 Baldwin Hall ML 0070, PO Box 210070, Cincinnati, OH 45221-0070, Student Member AIAA. † Technical Area Leader, Computational Sciences Branch, Aeronautical Sciences Division, Associate Fellow AIAA. 1 American Institute of Aeronautics and Astronautics This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.


k = n = p = qi = t = u, v, w = u ′v′ = xs, xt, xr = ∆t = α = γ = δ = δξ2, δη2, δζ2 = δξ6, δη6, δζ6 µ = ξ, η, ζ = ξt, ξx, ξy, ξz, ηt, ηx, ηy, ηz, ζt, ζx, ζy, ζz ρ = ω =

∇2

Heat transfer coefficient Wall normal direction Non- dimensional static pressure Component of heat flux vector Non-dimensional time Non-dimensional Cartesian velocity components in x, y, z directions Reynolds shear stress uv − u v Separation, transition, and reattachment locations Non-dimensional time step Angle of attack Specific heat ratio; 1.4 for air Kronecker delta function Second- and sixth-order finite difference operators in ξ, η, and ζ directions Non-dimensional molecular viscosity coefficient Control coordinates = Metric coefficients of the coordinate transformation Non-dimensional density Non-dimensional frequency

= Laplace Operator

Subscripts ∞ = Freestream reference conditions Superscripts − = Time mean quantities ‘ = Fluctuating components

L

I. Introduction

OW Reynolds flow has been of interest for model airplane designers for decades. As a result, a large database of experimental and numerical data for airfoils for fixed wing configurations has been compiled. Several investigators, such as F. Schmitz1 and more recently R. Eppler2 and S. Selig3 to name a few, have contributed with a vast number of experimental measurements and advanced aerodynamic design methodologies. In recent years, an interest on small Unmanned Air Vehicles, including Micro Air Vehicles (MAV), capable of performing a wide range of missions has grown from the onset of newly developed micro system technologies. Due to their size and low air speed, these vehicles typically operate at Re on the order of 104 to 105. At these low Reynolds numbers, the flow may remain laminar over a significant portion of the airfoil and is unable to sustain even mild adverse pressure gradients. For moderate incidence, separation leads to the formation of a laminar separation bubble (LSB) which breakdowns into turbulence prior to reattachment. The LSB moves towards the leading edge with increasing angle of attack and becomes shorter in streamwise extent. Eventually, as the stall angle is exceeded, reattachment is no longer possible and so-called bubble bursting ensues. The onset and successive breakdown of the LSB at low Reynolds number is know to be detrimental to performance, endurance, and stability of MAV’s. As transition is affected by a wide range of parameters such as wall roughness, freestream turbulence, pressure gradient, acoustic noise, etc., a comprehensive transition model which accounts for all factors has not been developed. Instead, transition models typically focus on only one or two parameters in order to predict transition location. Such models range from simple empirical methods based on linear stability theories, linear or nonlinear parabolized stability equations, to more comprehensive Navier-Stokes models. A design-oriented approach adopted by many researchers is the eN method which is based on linear stability analysis and boundary layer theory. Here, local growth rates of unstable waves based on velocity profiles are evaluated by solving the Orr-Sommerfeld equation. Transition is said to occur when the amplification of the most unstable Tollmien-Schlichting waves reaches a specified critical threshold. This method is used, for instance, to predict transition location in the popular airfoil-design code XFOIL4 2 American Institute of Aeronautics and Astronautics


Recently, Radespiel et al.5, Lian and Shyy6, and Yuan et al.7 successfully coupled Reynolds averaged NavierStokes (RANS) solvers with the eN method to predict laminar to turbulent transition on low-Re airfoils. While this has been shown to be a computationally efficient method, it is limited by its inherent assumptions of twodimensional parallel steady velocity profiles and thin boundary layers. While these limitations are acceptable in many situations, in particular for design purposes, full three-dimensional MAV configuration analysis is beyond the scope of such simple approaches. For instance, even though the eN method has been extended to three-dimensions, a physical interpretation of the method in this situation is unclear8. In addition for maneuvering airfoils, high angle of attack excursions may promote the development of leading-edge dynamic stall and leading-edge vortices whose stability falls outside of the aforementioned simple transition prediction framework. The present work investigates the feasibility of an implicit large-eddy simulation (ILES) approach to predict laminar separation bubble formation and transition for low Reynolds–number airfoil applications. The ILES approach, previously introduced in Refs. 9 and 10, is based on higher-order compact schemes for the spatial derivatives and a Pade-type low-pass filter to provide stability. The high-order scheme allows for accurate capturing of the separation and transition process, whereas the highly-discriminating low-pass filter is used in lieu of a standard sub-grid-scale (SGS) model to enforce regularization in turbulent regions. This approach is very attractive since it provides a seamless methodology for mixed laminar, transitional and turbulent three-dimensional flows. Results are presented for flow over a SD7003 airfoil section, shown in Figure 1. The SD7003 airfoil was chosen due to the existence of experimental data available for comparison, as well as for the relatively large LSB observed on the suction side of the airfoil. High resolution velocity and Reynolds stress measurements have been provided by Radespiel5. Experiments were conducted in a water channel, as well as in a low-noise wind tunnel both at the Technical University of Braunschweig (TU-BS). Freestream turbulence intensities were 0.08% and 0.8% for the wind tunnel and water channel respectively. Measurements are available for Reynolds number 6x104 at 4° angle of attack in the wind tunnel, and at 8°, and 11° in the water channel. PIV measurements for the SD7003 airfoil were also obtained by Ol et al.11 at the Wright Patterson Air Force Base (WPAFB) water channel with a freestream turbulence intensity of less than 0.1%. Aerodynamic load measurements are also available from Ol et al.11 and Selig et al.12, 13. Computations were performed for the SD7003 airfoil at several Reynolds numbers (104 < Re < 9x104) and angles of attack (2°-14°). The effects of grid resolution, spanwise extent, and freestream Mach number on the computed solution are investigated. Detailed comparison with experimental data is provided in terms of the flow structure and aerodynamic loads.

Figure 1: The SD7003 airfoil.

II. Governing Equations Present computations utilize the flow solver FDL3DI, a higher-order accurate, parallel, Chimera, Large Eddy Simulation solver from Wright Patterson Air Force Base. FDL3DI has been proven reliable for many steady and unsteady fluid flow problems.14, 15, 16, 17, 18 The FDL3DI code solves the unsteady, three-dimensional, compressible, unfiltered Navier-Stokes equations ∂Q ∂FI ∂G I ∂H I 1 ⎡ ∂Fv ∂Gv ∂H v ⎤ + + + = + + ∂t ∂ξ ∂η ∂ζ ∂η ∂ζ ⎥⎦ Re ⎢⎣ ∂ξ

(1)

Here, the vector of dependent variables is expressed as Q=

1 [ρ J

ρu ρv ρw ρEt ]T

3 American Institute of Aeronautics and Astronautics

(2)

While the inviscid flux vectors are

ρU ρV ρW ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎢ ρuU + ξ p ⎥ ⎢ ρuV + η p ⎥ ⎢ ρuW + ζ p ⎥ x ⎥ x ⎥ x ⎥ 1⎢ 1⎢ 1⎢ FI = ⎢ ρvU + ξ y p ⎥ , GI = ⎢ ρvV + η y p ⎥ , H I = ⎢ ρvW + ζ y p ⎥ J⎢ J⎢ J⎢ ⎥ ⎥ ⎥ ⎢ ρwU + ξ z p ⎥ ⎢ ρwV + η z p ⎥ ⎢ ρwW + ζ z p ⎥ ⎢ ρEtU + pU~ ⎥ ⎢ ρEtV + pV~ ⎥ ⎢ ρEtW + pW~ ⎥ ⎣ ⎣ ⎣ ⎦ ⎦ ⎦

(3)


where ~ U = ξt + ξ xu + ξ y v + ξ z w = ξt + U ~ V = ηt + η xu + η y v + η z w = ηt + V

~ W = ζ t + ζ xu + ζ y v + ζ z w = ηt + W Et =

T

(γ − 1)

M ∞2

+

(

1 2 u + v 2 + w2 2

(4)

)

And the viscous flux vectors as ⎡ 0 ⎤ ⎡ 0 ⎤ ⎡ 0 ⎤ ⎢ξ τ ⎥ ⎢η τ ⎥ ⎢ζ τ ⎥ xi i1 ⎥ xi i1 ⎥ xi i1 ⎥ ⎢ ⎢ 1⎢ 1⎢ 1 ⎢⎢ ⎥ ⎥ ξ τ η τ ζ Fv = x i 2 , Gv = x i2 , H v = x τ i2 ⎥ J⎢ i ⎥ J⎢ i ⎥ J⎢ i ⎥ ⎢ξ xi τ i 3 ⎥ ⎢η xi τ i 3 ⎥ ⎢ζ xi τ i 3 ⎥ ⎢ ξ x bi ⎥ ⎢ η x bi ⎥ ⎢ ζ x bi ⎥ ⎣ i ⎦ ⎣ i ⎦ ⎣ i ⎦

(5)

⎛ ∂ξ k ∂u i ∂ξ k ∂u j ⎞ 2 ⎟ − µδ ∂ξ l ∂u k + ⎜ ∂x j ∂ξ k ∂xi ∂ξ k ⎟ 3 ij ∂x k ∂ξ l ⎝ ⎠ ∂ξ l ∂T k bi = u jτ ij + (γ − 1) Pr M ∞2 ∂xi ∂ξ l

(6)

where

τ ij = µ ⎜

Compact notation is used to represent x, y, and z coordinates as xi, i = 1, 2, 3 respectively and similarly for ξi, for ξ, η, and ζ. Dependent variables have been non-dimensionalized by their respective reference values, except for p which is non-dimensionalized by ρ ∞V∞2 . All length scales are non-dimensionalized by the reference length C, the chord length of the airfoil. Sutherland’s law is used for molecular viscosity coefficient µ as well as the perfect gas law p=

ρT γM ∞2

(7)

to close the Navier-Stokes equations. It should be noted that the above governing equations correspond to the original unfiltered Navier-Stokes equations, and are used without change in laminar, transitional, or fully turbulent regions of the flow. Unlike the standard LES approach, no additional sub-grid stress (SGS) and heat flux terms are appended. Instead, the highorder low-pass filter operator (noted later) is applied to the conserved dependent variables during the solution of the standard Navier-Stokes equations. This highly-discriminating filter selectively damps only the evolving poorly 4 American Institute of Aeronautics and Astronautics

resolved high-frequency content of the solution9, 10. This filtering regularization procedure provides an attractive alternative to the use of standard sub-grid-scale (SGS) models, and has been found to yield suitable results for several turbulent flows on LES level grids. A re-interpretation of this implicit LES (ILES) approach in the context of an Approximate Deconvolution Model19 has been provided by Mathew et al.20.

III. Numerical Method


Time accurate solutions of Eq. 1 are obtained with the implicit approximate-factorization algorithm of Beam and Warming21 employing Newton-like subiterations. A higher order discretization is implemented as p ⎡ ⎛ 2∆t ⎞ ⎛ ∂F p ⎛ ∂H Ip 1 ∂Fvp ⎞⎟⎤ ⎡ ⎛ 2∆t ⎞ ⎛⎜ ∂G I 1 ∂Gvp ⎞⎟⎤ ⎡ ⎛ 2∆t ⎞ 1 ∂H vp ⎞⎟⎤ ⎥ ⎢I + ⎜ ⎥ ∆Q ⎢I + ⎜ ⎥ ⎢I + ⎜ − − ⎟δ η 2 ⎟δ ζ 2 ⎜ ⎟δ ξ 2 ⎜ I − ⎜ ∂Q Re ∂Q ⎟⎠⎥ ⎢ ⎝ 3 ⎠ Re ∂Q ⎟⎠⎥ ⎢⎣ ⎝ 3 ⎠ ⎜⎝ ∂Q Re ∂Q ⎟⎠⎥⎦ ⎢⎣ ⎝ 3 ⎠ ⎜⎝ ∂Q ⎝ ⎦⎣ ⎦ (8) ⎤ 1 1 1 ⎛ 2∆t ⎞ ⎡⎛ 1 ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ p p p p n n −1 = −⎜ + δ ξ 6 ⎜ FI − Fvp ⎟ + δ η 6 ⎜ G I − Gvp ⎟ + δ ζ 6 ⎜ H I − H vp ⎟⎥ ⎟ ⎢⎜ ⎟ 3Q − 4Q + Q Re Re Re ⎝ 3 ⎠ ⎣⎝ 2∆t ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎦

(

)

Here, Q at the n + 1 time level Q n+1 is approximated by Q p +1 = ∆Q + Q p . For p = 1, Q p = Q n . Temporal derivative are represented by second order backward-implicit time differencing. The implicit segment of the algorithm is discritized with second-order central differencing for all spatial derivatives and utilizes non-linear artificial dissipation to augment stability.22 The efficiency of the implicit algorithm was increased by solving the factorized equations in diagonal form. In order to maintain temporal accuracy, which can be degraded by the diagonal form, three subiterations were utilized within a time step.23 This technique is commonly used to reduce errors due to linearization, factorization, and explicit application of boundary conditions. Compact differencing of sixth-order accuracy is utilized on the right-hand side of Eq. 8 is based on the pentadiagonal system of Lele24 and is capable of obtaining spectral-like resolution. A centered implicit compact difference operator is used to achieve this by reducing the associated discretization error. The sixth-order tridiagonal subset of Lele’s system is presented here with an arbitrary scalar f for one spatial direction 1 ⎛ ∂f ⎜ 3 ⎜⎝ ∂ξ

⎞ ⎛ ∂f ⎟⎟ + ⎜⎜ ⎠ i −1 ⎝ ∂ξ

⎞ 1 ⎛ ∂f ⎞ 14 ⎛ f − f i −1 ⎞ 1 ⎛ f i + 2 − f i −2 ⎞ ⎟⎟ + ⎜⎜ ⎟⎟ = ⎜ i +1 ⎟+ ⎜ ⎟ 3 ξ 2 4 ∂ ⎠ 9⎝ ⎠ ⎠i ⎝ ⎠ i +1 9 ⎝

(9)

This implicit time marching scheme has been successfully applied to unsteady vortical flow by Visbal and Gaitonde25. A 10th-order and 8th-order non-dispersive spatial filter developed by Gaitonde et al.26 was applied after each sub-iteration. This method has been shown to be more effective than explicitly added artificial dissipation to maintain both stability and accuracy on stretched curvilinear meshes.27 In addition, lower-order filtering as well as lower-order spatial differencing schemes are employed at the boundary for the same purpose. For the present simulation, the sixth-order compact differencing scheme was implemented in all spatial directions with 8th-order filtering in streamwise and surface normal directions and 10th-order filtering in the spanwise direction. Parallel execution is achieved by partitioning the mesh with a five point overlap between grids. Such an overlap is required to maintain the formal order of accuracy of the solution across grid boundaries. In addition, higher order accurate interpolation is employed to maintain spatial accuracy in overset meshes.28 Decomposition was performed with automated tools developed for FDL3DI.29

IV. Computational Mesh and Boundary Conditions All computational meshes were generated using the software package GridGen30. Initially, a single baseline Ogrid was generated about the SD7003 airfoil as depicted in Figure 2. Grid coordinates are oriented such that ξ traverses clockwise around the airfoil, η is normal to the surface, and ζ denotes the spanwise direction. The baseline mesh contained 315x151x101 points in the ξ, η, ζ directions, respectively, which corresponds to approximately 4.8 million grid points. The farfield boundary was located 30 chords away from the airfoil in order to reduce its influence on the solution near the airfoil. Unless otherwise noted, a spanwise extent z/C = 0.2 was prescribed.


Freestream 315x151x101


∂ui =0 ∂x

Figure 2: Baseline Computational mesh. Grid dimensions of 315x151x101.

Freestream 163x50x41

∂ui =0 ∂x

315x47x65

b)

a) 428x90x101 (69%)

101x74x65

c)

Figure 3: Overset computational mesh. Grid dimensions (ξ, η, ζ) are depicted for each grid. a) Background O-grid. b) Near O-grid. c) Body fitted grids

Next, a more refined overset mesh system was constructed in order to exploit the Chimera (overset) capabilities of the FDL3DI solver. The baseline mesh was used as the basis for mesh refinement and a portion of it was retained as the near O-grid shown in Figure 3b. A background circular O-grid (Figure 3a) was also generated away from the airfoil to again move the farfield boundary conditions 30 chords away from the airfoil. However, as this mesh is away from the separated flow of interest, its spanwise resolution was reduced to 41 points. Similarly, the near O-grid and pressure-side grid were reduced to 65 spanwise points, while the grid on the suction side of the airfoil maintained 101 spanwise points. The majority of the refinement took place on the suction surface of the airfoil. Here, the mesh was doubled in both the surface normal and streamwise directions compared to the baseline mesh. After these modifications, the total number of mesh points is approximately 5.7 million grid points, with 69% of the points on the suction side of the airfoil. The mesh was then decomposed into 60 blocks for parallel execution. 6 American Institute of Aeronautics and Astronautics

Boundary conditions were specified in the following manner. Freestream conditions were prescribed with fixed dependent variables on the majority of the far-field boundary. A zero velocity gradient was imposed on the wake region of the outer boundary, as depicted in Figures 2 and 3. A no-slip, 4th-order explicit zero pressure gradient, adiabatic temperature condition was used on the surface of the airfoil.


V. Results Effects of both angle of attack and Reynolds number are considered. A range of positive angles of attacks, i.e. 2°, 4°, 6°, 8°, 11°, and 14°, were computed for Reynolds number 6x104. Computations at Re 104, 3x104, 6x104, and 9x104 were also computed at 8° angle of attack. In order to limit numerical errors, several numerical parameters were investigated for the 4° angle of attack with a Reynolds number of 6x104. In particular, grid resolution, spanwise width, and freestream Mach number were considered. Time and spanwise mean surface Cp and Reynolds shear stresses were used to evaluate influence of each parameter. All mean values were calculated by averaging the spanwise averaged time accurate solution over eight characteristic times with a non-dimensional time step of ∆t = 0.00015. Before calculating mean values, the time accurate solution was allowed to develop from an initial solution over five characteristic times. In order to reduce computational cost associated with initial transients from a freestream initial solution, two-dimensional solutions were used for the initial condition of the three-dimensional mesh. The two-dimensional solution was transferred onto the three-dimensional mesh through simple extrusion. Both 6th- and 2nd- order computations were performed on the baseline mesh at 4° angle of attack. The 2nd-order solution was obtained with a 2nd-order version of the FDL3DI solver which also incorporates fourth-order spectral damping terms. Mean surface Cp and Reynolds shear stress values from these calculations, along with values from overset mesh calculations, are presented in Figure 4. In particular, note the difference between the 2nd-order and 6thorder computations on the baseline mesh. The 6th-order scheme has a sharper pressure gradient in the transition region compared to the 2nd-order computation. More importantly, absolute magnitude of the Reynolds stress is lower in the 2nd-order computations than in the 6th-order case. In fact, the values for the baseline 6th-order computation resemble those found with finer overset grid. This illustrates how the higher order spatial differencing captures finer flow structures than a 2nd-order scheme on the same mesh. A comparison of separation, transition, and reattachment locations, as well as maximum laminar separation bubble height is given in Table 1. Turbulent transition locations are determined according to Ref. 5. Transition is said to occur when the Reynolds shear stress reaches a value of 0.1% and exhibits a clear visible rise. For all the simulations, the separation location occurs at approximately the same location. However, transition occurs further upstream for both baseline mesh calculations. Differences are also observed in the reattachment location and maximum bubble height. The pressure gradient in the transition regions is also much sharper on the overset mesh as compared to the baseline mesh computations, as shown in Figure 4. These results indicate that all fine details of transition are not captured, even with the 6th-order scheme, on the coarser baseline mesh. However, as it will be shown later, the solution on the overset mesh agrees well with available experimental data, and therefore is deemed adequate to resolve all relevant flow features. Given the spanwise periodic boundary conditions, computations on the three-dimensional meshes represents an airfoil with an infinite span. Even so, spanwise extent is an important parameter in the computational setup. If the span is too small, the flow will be constrained and flow structures will not develop properly. On the other hand, while a large spanwise extent will not impose any constraints on the flow, it will incur additional computational requirements which may not be necessary. Therefore, an investigation of the effect of spanwise width was performed. In order to eliminate the influence of grid spacing, the number of grid points in the spanwise direction was modified such that the grid spacing remained near constant for all cases. Three spanwise extents were employed (namely, 0.1, 0.2, and 0.3) at an angle of attack of 4° and Reynolds number 6x104. Corresponding mean Cp values and Reynolds shear stresses are shown in Figure 5. By inspection, both surface pressure and turbulent Reynolds stresses appear to exhibit little variation with prescribed spanwise width. In addition, instantaneous contours of the Q-criterion for the tree spanwise extents are given in Figure 6. The Q-criterion provides a means of visualizing vortex cores and identifies turbulent structures; for sub-sonic flow it is simply the Laplacian of the static pressure field over the density, i.e.31 Q − criterion =

∇2P 2ρ


(10)


By inspection of Figure 6, the flow structures observed on for all spanwise extents are similar in shape and size. This indicates that the spanwise extent of 0.1 is sufficient for this angle of attack of 4°. However, considering the computational costs, and the desire to compute a full range of angles of attack, as well as an investigation of Reynolds number effect using the same mesh, the spanwise extent of 0.2 was maintained for all remaining computations. All experimental measurements were conducted under incompressible flow conditions. As FDL3DI solves the compressible Navier-Stokes equations, a reference Mach number is required. However, specifying a Mach number that is too low leads to numerical instabilities. In fact, computations at a Mach number of 0.05 lead to spurious nonphysical flow structures near the leading edge of the airfoil. Such issues could be addressed with preconditioning for low Mach number flows; however no such methods have been implemented in FDL3DI. Therefore, in order to maintain numerical stability for all angles of attach and Re numbers, computations were performed at reference Mach number 0.1. To verify that this Mach number is low enough, a solution was computed with a reference Mach number of 0.075. Results of these computations are compared in Figure 7. Both mean surface Cp and Reynolds shear stress show little difference between computations. Therefore, Mach number 0.1 is concluded a satisfactorily low Mach number. Table 1: Locations of interest on baseline and overset mesh computations (α = 4°, Re = 6x104) Computation Separation Transition Reattachment Max Bubble xs/C xt/C xr/C Height, hb/C Baseline 2nd-order 0.25 0.46 0.66 0.034 Baseline 6th-order 0.24 0.45 0.61 0.028 Overset 6th-order 0.23 0.55 0.65 0.030

a) b) Figure 4: Effect of grid resolution and numerical scheme (α = 4°, Re = 6x104): a) Mean surface Cp b) Reynolds shear stress



a)

b)

Figure 5: Effect of spanwise extent (α = 4°, Re = 6x104): a) Mean surface Cp b) Reynolds shear stress

Figure 6: Iso-surface of Q-criterion for varying spanwise extent at α = 4°

a)

b) 4

Figure 7: Effect of Mach number (α = 4°, Re = 6x10 ): a) Mean surface Cp b) Reynolds shear stress



A. Comparison with Experimental Data Computed Reynolds shear stress is compared with that of TU-SB and HFWT experimental measurements in Figures 8 and 9a for the available angles of attack, i.e. 4°, 8°, and 11°. Good agreement between computations and both experimental data sets in terms of shape, magnitude, and extent of the fluctuating region are observed for α = 4° and 11°. However, for α = 8°, absolute magnitude of computed Reynolds shear stress are lower in magnitude and occur, along with transition, further downstream than observed by experimental TU-BS measurements. Similar trends were observed in computations by Radespiel et al.5 and Yuan et al.7. These differences are attributed to higher freestream turbulence intensity, approximately 0.8%, and lower PIV imagery resolution used during an earlier campaign during which these measurements were taken. For the HFWT data, Ol et al. speculates that the lack of a recirculation region in the experimental data could come from increased freestream turbulence intensity, or a shortcoming PIV image pars for adequate convergence of flow statistics; or both. Despite the higher freestream turbulence intensity in the TU-BS water tunnel, the ILES computed Reynolds shear stress agrees well with measured values at 11° angle of attack. Reasonable agreement is also observed with the HFWT data set. While the computed LSB is slightly thicker, separation and reattachment locations agree well. The airfoil is near stall at this angle of attach and requires a greater pressure recovery in the LSB in order for it to reattach. This strong pressure gradient amplifies disturbances and which leads to a rapid transition to turbulent flow forming a short LSB. It would appear that at 11° angle of attack, as opposed to 8° angle of attack, the influence of the transition process is less influenced by freestream turbulence intensity. Table 2 compares measurements of separation, transition, reattachment, and maximum LSB height, from two facilities along with computations with XFOIL4 and present ILES computation at 4° angle of attack. Here, computed separation locations of XFOIL and ILES fall in between both experimental measurements. However, the ILES separation location of 23% chord fall in the range of values computed from LES calculations and eN methods by Yuan et al.7. The ILES computed transition location of 55% chord is in well agreement of the measured TU-SB transition location of 53% chord. Transition at the HFWT tunnel was measured at 47% chord which would be consistent assuming slightly higher freestream intensity in this facility of ~0.1% compared to 0.08% of the TU-SB low-noise wind tunnel. Reattachment locations are also consistent between ILES and TU-SB at 65% chord and 62% chord respectively. Again, reattachment measured at HFWT occurs slightly further upstream at 58%. However, the maximum height of the LSB differs little be between both experimental measurements and ILES computations. Lift and drag coefficients were computed from the ILES simulations by integrating skin friction and pressure around the airfoil. In order to accurately perform the integration in overlapping regions of the mesh, the solution was first transferred to a singe mesh, i.e. the baseline mesh. Because points did not necessarily coincide in the refined region of the suction surface of the airfoil, linear interpolation was used to transfer the solution.32 Integrated lift and drag coefficients are compared with experimental measurements of Selig et al.12, 13 and Ol et al.11 as well as values computed with XFOIL in Figure 9b. ILES lift coefficients agree well with measurements of Selig et al. and Ol et al.. Measurements by Ol et al. predict stall at 11° angle of attack which is captured by the ILES simulations. In fact, even the post stall lift coefficient of 14° agrees well with the measured lift coefficient. Computations with XFOIL generally agree well with experimental measurements up till stall. Drag coefficients are slightly over estimated by the ILES calculations compared to measurements by Selig et al. Mean surface Cp, Reynolds shear stresses, and skin friction coefficients for all angles of attack computed are presented in Figures 10 and 11. Here, even at the lowest angle of attack of 2°, an LSB forms, transitions to turbulent flow near the trailing edge, and reattaches at 93% chord. Turbulent transition is indicated by the drop in skin friction coefficient which coincides with the sharp pressure gradient observed after the flat pressure plateau typical of LSB’s. As the angle of attack increases, the adverse pressure gradient grows and the LSB shortens. This causes both the separation and turbulent transition locations to move upstream as shown in Table 3. In addition, the absolute magnitude of the Reynolds shear stresses increases as the LSB moves towards the leading edge as shown in Figure 10b. When the airfoil us fully stalled at 14° angle of attack, the mean surface Cp becomes flat across the entire suction side of the airfoil. Instantaneous iso-surfaces of the Q-criterion of all angles of attack are shown in Figure 12. These iso-surfaces represent vortex structures. As the LSB breaks down, a coherent spanwise vortex over the extent of the airfoil forms and subsequently breaks down in to turbulent structures. Note that the ILES method has seamlessly captured evolution from a closed LSB to bubble bursting and stall without modification to any parameters.



TU-BS

α = 4°

TU-BS

HFWT

HFWT

ILES

ILES

α = 8°

Figure 8: Reynolds stress and experimental data for α = 4° and α =8° at Re = 6x104

TU-BS

α = 11°

HFWT

ILES

a) b) Figure 9: a) Reynolds shear stress and experimental data for α = 11° at Re = 6x104 b) Lift and drag coefficients Table 2: Measured and Computed SD7003 LSB properties (α = 4°, Re = 6x104) Data Set Freestream Separation Transition Reattachment Max Bubble Turbulence [%] xs/C xt/C xr/C Height, hb/C TU-SB 0.08 0.30 0.53 0.62 0.028 HFWT ~0.1 0.18 0.47 0.58 0.029 XFOIL 0.070 (N=9) 0.21 0.57 0.59 ILES 0 0.23 0.55 0.65 0.030



a) Figure 10: Full angle of attack sweep (Re = 6x104): a) Mean surface Cp b) Reynolds shear stresses

b)

Figure 11: Suction surface skin friction coefficient distribution at Re = 6x104



2°

8°

4°

6°

11°

14°

Figure 12: Instantaneous iso-surfaces of Q-criterion at Re = 6x104 Table 3: Computed SD7003 LSB properties at Re = 6x104 α Separation Transition Reattachment Max Bubble Degrees xs/C xt/C xr/C Height, hb/C

2 4 6 8 11 14

0.45 0.23 0.11 0.04 0.007 0.01

0.74 0.55 0.34 0.18 0.06 0

0.92 0.65 0.45 0.28 0.16 -

0.036 0.030 0.028 0.027 0.025 -

B. Turbulent transition Velocity probes were positioned at the mid-span over several streamwise locations along the suction surface of the airfoil at 4° angle of attack and Reynolds number 6x104. The probes recorder velocities at the third grid point above the surface, which was located approximately 10-4 non-dimensional units above the surface. From the velocity history, the spectrum of turbulent kinetic energy is plotted from two of these probes in Figure 13. For the location x/C = 0.5, there appears to be a broad peak centered around ω ≈ 9 . This may correspond to the shear layers most unstable frequency, but further stability analysis is required. Further downstream at x/C = 0.94, the spectrum resembles a turbulent boundary layer with a limited emerging inertial range. Furthermore, mean velocity profiles at three locations are plotted in Figure 14. A thin laminar boundary layer is observed at x/C = 0.2 upstream of the separation location. After the flow separates, a shear layer profile with reversed flow is observed at x/C = 0.5. Downstream of the LSB reattachment, a fuller turbulent profile occurs at x/C = 0.94 consistent with observations in the turbulent kinetic energy spectrum.


0.5


0.94

Figure 13: Turbulent-kinetic-energy frequency spectra at two locations (α = 4°, Re = 6x104)

Figure 14: Mean velocity profiles (α = 4°, Re = 6x104) C. Effect of Reynolds number A series of Reynolds numbers ranging from 104 to 9x104 were computed for 8° angle of attack. This angle was chosen as it was expected to more interesting than 4° but not as close to stall as 11°. Time and spanwise mean mean surface Cp and Reynolds shear stresses are shown in Figures 15a and b respectively. As seen in the flat surface pressure profile, the airfoil is stalled at the lowest Reynolds number of 104. As the Reynolds number increases, the LSB closes and continues to decrease in size consistently with an increasing pressure gradient downstream of the LSB. While the transition location and reattachment locations move upstream with increasing Reynolds number, the separation location remains near constant as it occurs near the leading edge as presented in Table 4. Also, as shown in Figure 16, the instantaneous turbulent flow structures decrease in size with increasing Reynolds number.



a)

b)

Figure 15: Comparison of Reynolds number (α = 8°): a) Mean Surface Cp b) Reynolds shear stresses

Re 104 3x104 6x104 9x104 Re 104

Table 4: Computed SD7003 LSB properties at α = 8° Separation Transition Reattachment Max Bubble xs/C xt/C xr/C Height, hb/C

0.09 0.05 0.04 0.04

0.45 0.25 0.18 0.14 Re 3x104

0.53 0.28 0.20

0.073 0.027 0.014

Re 6x104

Figure 16: Instantaneous iso-surfaces of Q-criterion at α = 8°


Re 9x104


VI. Conclusion An implicit large eddy simulation (ILES) technique has been used to predict the formation of a laminar separation bubble (LSB) and its subsequent burst and transition to turbulent structures on the SD7003 airfoil. Flow solutions were obtained with a validated Navier-Stokes solver based on high-order compact schemes with a Padefilter to remove poorly resolved high wavenumbers in the mesh in lieu of an explicit SGS model. Unlike transition models which rely on a limited number of parameters to determine transition locations, the ILES methods solves the unfiltered Navier-Stokes equations without change in the laminar, transitional, and turbulent region of the flow. In addition, the ILES method captured the shift from a closed LSB to bubble burst and stall without modification of any parameters. Computations compared qualitatively well with measured Reynolds shear stresses for available angles of attack, 4°, 8°, and 11°, at Reynolds number 6x104. Computed separation, transition, and reattachment locations were also in agreement with measured values. The transitional nature of the flow was indicated by turbulent kinetic-energy wavenumber spectra, and a fuller turbulent velocity profile was observed downstream of the reattachment location. Computed lift and drag polar (2°-14°) captures the stall angle and as well as measured lift coefficient post stall. While drag is over predicted, it also compares well with measured values. As expected, with increasing angle of attack, ILES predicts the LSB decreasing in size and moved toward the leading edge until post stall where the bubble has burst and the flow is fully separated. Reynolds shear stresses also increased consistently with increased adverse pressure gradients of higher angles of attack. Effects of Reynolds number was inspected for 8° angle of attack. Again, as expected, computations predict LSB to decrease in size and move towards the leading edge with increasing Reynolds number. Also, pressure gradients and Reynolds shear stresses increase with higher Reynolds numbers. In addition, instantaneous turbulent structures decreased in size with increased Reynolds number. Future work will investigate the effects of upstream disturbances on the stability of the LSB as well as large unsteady motions of the airfoil.

Acknowledgments The authors are grateful for AFOSR sponsorship under tasks monitored by Lt.Col R. Jefferies. This work was also supported in part by a grant of HPC time from the DoD HPC Shared Resource Centers at ASC. Much appreciation is given to M. Ol, R. Radespiel and J. Windte for providing experimental data. Gratitude is given towards D. Rizzetta and S. Sherer for their assistance with FDL3DI and pre-processing tools. The authors are also grateful to M. List, D. Galbraith, and J. Nimersheim of the University of Cincinnati for their work with visualization.

References 1

Schmitz, F.W., Aerodynamik des Fluges, Verlag Carl Lange, Duisburg, 1960. Eppler, R., Airfoil Design and Data, Springer Verlag, ISBN 3-540-52505-X, 1990. 3 Selig, M.S., Donovan, J. F., Fraser, D. B., Airfoils at Low Speeds, SoartechTech Publications. H.A. Stokely, Virginia Beach, VA, USA, 1989. 4 Drela, M., XFOIL Users Guide, Version 6.94 , MIT Aero. and Astro. Department, 2002. 5 Radespiel, R., Windte, J., and Scholz, U., “Numerical and Experimental Flow Analysis of Moving Airfoils with Laminar Separation Bubbles,” AIAA Paper 2006-501, Jan. 2006. 6 Lian, Y., and Shyy, W., “Laminar-Turbulent Transition of a Low Reynolds Number Rigid or Flexible Airfoil,” AIAA paper 06-3051, Jun 2006. 7 Yuan, W., Khalid, M., Windte J., Scholz, U., and Radespiel, R., “An Investigation of Low-Reynolds-number Flows past Airfoils,” AIAA Paper 05-4607, Jun 2005. 8 Arnal, D., Casalis, G., and Juillen, J. C., “A Survey of the Transition Prediction Methods: from Analytical Criteria to PSE and DNS,” in M. Hallback (ed.) Turbulence and Transition Modeling, Lecture Notes from the ERCOFTAC/IUTAM Summer School held in Stockholm, June 12-20, 1995, Kluwer Academic Publishers, Dordrecht, pp. 3-14, 1996. 9 Visbal, M. R. and Rizzetta, D. P., “Large-Eddy Simulation on Curvilinear Grids Using Compact Differencing and Filtering Schemes. Journal of Fluids Engineering,” 124:836–847, 2002. 10 Visbal, M. R., Morgan, P. E., and Rizzetta, D. P., “An Implicit LES Approach Based on High-Order Compact Differencing and Filtering Schemes,” AIAA Paper 2003-4098, 2003. 11 Ol, M. V., McAuliffe, B. R., Hanff, E. S., Scholz, U., and Kähler, C., “Comparison of Laminar Separation Bubble Measurements on a Low Reynolds Number Airfoil in Three Facilities,” AIAA Paper 2005-5149, Jun. 2005. 12 Selig, M. S., Guglielmo, J. J., Groeren, A. P., G. Giguere, P., “Summery of Low-Speed Airfoil Data,” SoarTech Aero Publications, H. A. Stokely, Virginia Beach, VA, USA, 1995. 2



13 Selig, M. S., Donovan, J. F., Fraser, D. B., ”Airfoils at Low Speeds,” Soartech 8, Soartech Publications, H. A. Stokely, Virginia Beach, VA, USA, 1989. 14 Rizzetta, D. P., and Visbal M. R., "Numerical Investigation of Transitional Flow Through a Low-Pressure Turbine Cascade," AIAA Paper 2003-3587, Jun. 2003. 15 Gordnier, R.E. and Visbal, M.R., "Numerical Simulation of Delta-Wing Roll," AIAA Paper 93-0554, Jan. 1993. 16 Visbal, M. R., "Computational Study of Vortex Breakdown on a Pitching Delta Wing," AIAA Paper 93-2974, Jul. 1993. 17 Visbal, M. R., Gaitonde, D., and Gogineni, S., "Direct Numerical Simulation of a Forced Transitional Plane Wall Jet," AIAA Paper 98-2643, Jun. 1998. 18 Rizetta, D. P., Visbal, M.R., and Blaisdell, G.A., "A Time-Implicit High Order Compact Differencing and Filtering Scheme for Large-Eddy Simulation," International Journal for Numerical Methods in Fluids, Vol. 42, No. 6, Jun. 2003, pp. 665-693. 19 Stolz. S., and Adams, N., “An Approximate Deconvolution Procedure for Large-Eddy Simulation,” Physics of Fluids, Vol. 11, No. 7, 1999, pp. 1699–1701. 20 Mathew, J., Lechner, R., Foysi, H., Sesterhenn, J., and Friedrich, R., “An explicit filtering method for LES of compressible flows,” Physics of Fluids, Vol. 15, No. 8, 2003, pp. 2279–2289. 21 Beam, R. and Warming, R., "An Implicit Factored Scheme for the Compressible Navier-Stokes Equations," AIAA Journal, Vol. 16, No. 4, Apr. 1978, pp. 393-402. 22 Jameson, A., Schmidt, W., and Turkel, E., “Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge–Kutta Time Stepping Schemes,” AIAA Paper 81-1259, June 1981. 23 Pulliam, T.H. and Chaussee, D.S., "A Diagonal Form of an Implicit Approximate-Factorization Algorithm," Journal of Computational Physics, Vol. 39, No. 2, Feb. 1981, pp. 347-363. 24 Lele, S. A., “Compact Finite Difference Schemes with Spectral-Like Resolution,” Journal of Computational Physics, Vol. 103, No. 1, 1992, pp. 16–42. 25 Visbal,M. R., and Gaitonde, D. V., “High-Order-Accurate Methods for Complex Unsteady Subsonic Flows,” AIAA Journal, Vol. 37, No. 10, 1999, pp. 1231–1239. 26 Gaitonde, D., Shang, J. S., andYoung, J. L., “Practical Aspects of High-Order Accurate Finite-Volume Schemes for Electromagnetics,”AIAA Paper 97-0363, Jan. 1997. 27 Visbal,M. R., and Gaitonde, D. V., “High-Order-AccurateMethods for Complex Unsteady Subsonic Flows,” AIAA Journal, Vol. 37, No. 10, 1999, pp. 1231–1239. 28 Sherer, S. E., “Investigation of High-Order and Optimized Interpolation Methods with Implementation in a High-Order Overset Fluid Dynamics Solver,” Ph.D. Dissertation, Aeronautical and Astronautical Engineering Dept., Ohio State Univ., Columbus, OH, 2002. 29 Sherer, S. E., Visbal, M. R., and Galbraith, M. C., “Automated Preprocessing Tools for Use with a High-Order OversetGrid Algorithm,” AIAA Paper 06-1147, Jan 2006. 30 Steinbrenner, J. P., Chawner J. P., and Fouts, C. L., “The GRIDGEN 3D Multiple Block Grid Generation System, Volume II: User’s Manual,” Technical Report WRDC-TR-90-3022, Wright Research and Development Center, Wright Patterson AFB, OH, Feb. 1991. 31 Dubeif, Y., and Delcayre, F., “On Coherent-vortex Identification in Turbulence,” Journal of Turbulence, Vol. 1, No. 11, 2000, pp. 1-22. 32 Galbraith, M. C., and Miller, J., “Development and Application of a General Interpolation Algorithm,” AIAA Paper 063854, Jun 2006.


Implicit Large Eddy Simulation of Low Reynolds ...

Implicit Large Eddy Simulation of Low Reynolds ...

Suggest Documents

Large Eddy Simulation of Low Subcritical Reynolds ... - WSEAS

Explicit and Implicit Large Eddy Simulation of

Large-eddy simulation of very large kinetic and magnetic Reynolds ...

Estimating the effective Reynolds number in implicit large-eddy

Large-eddy simulation of a high Reynolds number flow ... - CiteSeerX

implicit large eddy simulation of premixed turbulent ...

Implicit Large Eddy Simulation of Transitional ... - Per-Olof Persson

LARGE EDDY SIMULATION USING

High-Order Implicit Large-Eddy Simulation for ... - Semantic Scholar

Large-Eddy Simulation - Sintef

Implicit Turbulence Modeling for Large-Eddy Simulation - mediaTUM

LARGE-EDDY SIMULATION OF COMBUSTION

TOWARDS LARGE EDDY SIMULATION OF

Large eddy simulation of the low temperature ignition

Large-Eddy Simulation for Acoustics

Large Eddy Simulation Applications - Cerfacs

LARGE EDDY SIMULATION USING ...

Large-Eddy Simulation for Acoustics

Large-eddy simulation of turbulent collision of

LARGE-EDDY SIMULATION OF SHOCK-WAVE ...

large-eddy simulation of isotropic homogeneous

Large-eddy simulation of organized trade wind

large-eddy simulation of neutral atmospheric

Large eddy simulation of urban features - ACPD