Modeling three-dimensional dynamic stall

Appl. Math. Mech. -Engl. Ed., 32(4), 393 400 (2011) DOI 10.1007/s10483-011-1424-6 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011 Applied Mathematics and Mechanics (English Edition) Modeling three-dimensional dynamic stall Chao LÜ ( ), Tong-guang WANG ( ) (Jiangsu Key Laboratory of Hi-Tech Research for Wind Turbine Design, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China) (Communicated by Wen-rui HU) Abstract The dynamic stall process in three-dimensional (3D) cases on a rectangular wing undergoing a constant rate ramp-up motion is introduced to provide a qualitative analysis about the onset and development of the stall phenomenon. Based on the enhanced understanding of the mechanism of dynamic stalls, a 3D dynamic stall model is constructed with the emphasis of the onset, the growth, and the convection of the dynamic stall vortex on the 3D wing surface. The results show that this engineering dynamic stall model can simulate the 3D unsteady aerodynamic performance appropriately. Key words unsteady flow, 3D dynamic stall, separation flow, vortex motion Chinese Library Classification V211.1 + 5 2010 Mathematics Subject Classification 76G25 1 Introduction Dynamic stalls are phenomena that affect the aerodynamic characteristics of airfoils, wings, and rotors in unsteady flows. They are due to the changes, periodic or not, in the inflow conditions and the angle of attack (AOA). For example, in case of wind turbines, the stall vortex results from the atmospheric turbulence, the wind shear, or the atmospheric boundary layer. Since the operation environment of wind turbines is particularly unsteady, dynamic stalls are very common for wind turbines. Thus, the study of dynamic stalls is of great significance to wind turbine aerodynamics. Significant progress in the study of dynamic stalls has been obtained, especially in two-dimensional (2D) cases. Whether by experiments or numerical calculations, the process and mechanism of 2D dynamic stalls have been clearly understood, and many mathematical models for its engineering applications have been provided, e.g., the well-known Leishman-Beddoes model [1 2] and the office national d etudes et de recherches aerospatiales (ONERA) model [3]. Although 2D dynamic stall models have been used for the prediction of the unsteady aerodynamics in engineering, none of them can be used to simulate three-dimensional (3D) characteristics. In 3D cases, the phenomena of dynamic stalls are far more complex than those in 2D cases. Therefore, modeling 3D dynamic stalls will face greater difficulties. Some progress has been made in the analysis of 3D dynamic stalls through theoretical research and experimental investigation. Tang and Dowell [4] extended the ONERA semi-empirical Received Jan. 15, 2011 / Revised Feb. 23, 2011 Project supported by the National Basic Research Program of China (973 Program) (No. 2007CB714600) Corresponding author Tong-guang WANG, Professor, Ph. D., E-mail: tgwang@nuaa.edu.cn

394 Chao LÜ and Tong-guang WANG theoretical model for the unsteady aerodynamics in the stall regime to 3D flows, and obtained the parameter identification for the model from the experimental dynamic stall data for a low aspect ratio rectangular wing with an NACA 0012 airfoil profile oscillating in a pitch. Coton and Galbraith [5] examined the dynamic stalling of a finite wing with an aspect ratio 3.0 when subjected to constant pitch motions up to and beyond the stall. They obtained unsteady surface pressure data at 192 locations on the wing surface. Then, they provided the information on the nature and phasing of dynamic stall events in both the chordwise and the spanwise directions by the analysis of the data. Spentzos et al. [6] obtained some results using the computational fluid dynamics (CFD) method for a series of experimental conditions, and provided that the methods for solving Navier-Stokes equations had feasibility for the numerical calculation of 3D dynamic stalls. Although the improved CFD method and the tremendous increased computing power have made it possible that the full fluid dynamic governing equations can be used for the investigation and prediction of dynamic stalls, numerical computation of 3D dynamic stalls is far from practical for the day-to-day design. This is mainly because that the required computing time is very large and the transitional and turbulent flow models need to be improved. Therefore, it is necessary to develop and use empirical methods for dynamic predictions for both the wind industry and the aerospace industry. As mentioned above, the development of empirical or semi-empirical engineering models for dynamic stalls has been necessitated by the fast calculations required in engineering. However, due to the extremely complicated nature of the dynamic stall phenomena, it is extremely difficult to establish an engineering model for 3D dynamic stalls. In this paper, an effort is made to construct an engineering model for 3D dynamic stalls. The dynamic stall process in 3D cases on a rectangular wing undergoing a dynamic motion is focused. Much attention is paid to the similarities and differences between 2D and 3D dynamic stalls. The 3D and 2D flows are linked so that the feature of the flow near the mid-span tends to be 2D and starts to be 3D when the section moves to the tip. Therefore, the 3D feature will be added along the span to the tip. 2 Mechanism of 3D dynamic stalls The topology of the flow visualization is shown in Fig. 1 [7]. In the figure, only two main vortical structures are identified, i.e., the dynamic stall vortex emanating from the leading edge region and the tip vortices. Actually, there is another structure, i.e., the shear layer vortex, which is a collection of the vorticity originating in the flow reversal near the trailing edge. The process of the influences among these three vortices is quite complex. The shear layer vortex has not been identified in the pictures because, from the flow visualization, it appears to have little influence on the onset of the dynamic stall vortex and its convection, which provide the main focuses for this study. In Fig. 1(a), the dynamic stall vortex is initially formed almost uniformly along the span near the quarter chord. Once formed, the vortex continues to grow and gives rise to increasing suction near the leading edge. The uniformity of the vortex growth is short-lived. As soon as the trailing edge separation begins on the wing, the vortex system starts to exhibit strong 3D features. In Fig. 1(b), the segments of the vortex near the mid-span become stronger than those on the outboard sections in particular. The vortex system then starts to convect, moving faster at the mid-span than on outboard locations. In Fig. 1(c), the weaker segments of the vortex near the tip are, at the same time, forced downwards the surface by the tip vortices, and the central section of the vortex lifts from the surface. Thus, the so-called Ω structure forms. In Fig. 1(d), the subsequent convection of the vortex system is very complicated with differential convection rates across the span. Ultimately, however, the vortex system passes the trailing edge uniformly.

Modeling three-dimensional dynamic stall 395 Fig. 1 Topology images for 3D dynamic stall vortex 3 Modeling of 3D dynamic stalls In this paper, the modeling of 3D dynamic stalls is divided into three parts, the modeling of the attached flow, the modeling of the separated flow, and the modeling of the onset of the dynamic stall vortex and its convection. 3.1 Modeling of attached flow The modeling of the unsteady attached flow is the premise of the accuracy of the dynamic stall modeling. In this paper, unsteady effects in the attached flow are simulated by the superposition of indicial aerodynamic responses, which is made up by a circulatory component and an impulsive component as done by Leishman [1] and Leishman and Beddoes [2]. In 3D cases, the normal force coefficient slope is different across the span, and less than that in 2D cases. Obviously, the normal force at the section near the tip is less than that near the mid-span. 3.2 Modeling of separated flow 3.2.1 Quasi-2D modeling at mid-span Under dynamic conditions, the flow separation location is usually delayed due to the change of the effective AOA and the delay of the boundary separation. In this paper, the effective AOA is a delayed AOA, i.e., Δα (S) =Δα(S) ( 1 e ) S Tα, (1) where T α is a time-delay constant, S =2Wt/c is a non-dimensional time representing the distance traveled at the resultant velocity W by the airfoil in semi-chords with the chord c, and { Δα (S) =α (S) α (S ΔS), (2) Δα(S) =α(s) α(s ΔS). Using the Kirchhoff flow theory depicted in Ref. [8], the trailing edge separation point associated with the delayed AOA is written as ( f C n =4 C nα (α α 0 ) 1 ) 2. (3) 2 Then, the delayed separation location f can be obtained by the delayed AOA α.

396 Chao LÜ and Tong-guang WANG 3.2.2 Modeling of 3D separation The trailing edge separation of the 3D flow is quite complicated. The more outboard the span station is, the more the separation is affected by the downwash angle and the spanwise flow. The flow separation will occur earlier at the mid-span than at the outboard sections. Therefore, there is a delay along the span. Due to the downwash influence by the tip, the flow separation location in 3D cases is delayed against that in 2D cases because of the so-called downwash angle. In this paper, the effect of the 3D downwash on the separation point is represented by a simple function as follows: f = f ( 1+ 1 ), (4) k 0 λ where k 0 is an empirical constant obtained from the unsteady experiment, and λ is the aspect ratio. Through the analysis of the experimental data [7], the separation location of each section across the 3D wing is represented by a separation function as follows: f 3D = f ( 1+ 4 3 xe 2 x λ ), (5) where x (0 x 0.5) is a dimensionless distance from the section considered to the mid-span. Obviously, this delay will decrease with the increase in the aspect ratio. 3.3 Stall-onset indication 3.3.1 Quasi-2D stall-onset indication of mid-span Clearly, due to the flow along the span and the tip vortex, 3D flows over wings are much more complex than 2D flows over airfoils, while the vortex initiation and development on the finite wing exhibit similarities qualitatively with the 2D case. Moreover, even after the stall initiation, the general qualitative features of the stall process are common to both the flows. Finally, the strong 3D effects on the wing, especially near the mid-span of the wing, begin only after the trailing edge separation becomes significant. Therefore, it is feasible that engineering models simplify the flow at the mid-span to quasi-2d flows. One of the purposes of this work is to provide an engineering 3D dynamic stall model for the prediction of the wind turbine aerodynamic performance. Owing to the wind turbines operating at low Mach numbers, the 2D stall-onset criterion presented by Sheng et al. [9] is used, α ds0, r r 0, α cr = α ss +(α ds0 α ss ) r (6), r < r 0, r 0 where a ss is a static stall-onset AOA, and the constants α ds0, α ss,andr 0 are obtained from a wind tunnel test with different values for different airfoils. At the mid-span station, the dynamic stall is initiated when α α cr. 3.3.2 3D stall-onset indication It can be seen from the flow topology images that the dynamic stall vortex onset at the leading edge is uniform and simultaneous for all sections (see Fig. 1(a)), and finally sheds from the trailing edge simultaneously, too (see Fig. 1(d)). While the vortex is moving from the leading edge to the trailing edge, the process is very complicated. This dynamic vortex remains uniform only for a very short period. The lift induced by the dynamic stall vortex at the very beginning of the initiation at the leading edge is so low that it may be negligible. With the vortex moving downstream, the dynamic vortex at the mid-span develops faster with a higher strength than the stall-onset criterion at the tip, forming an Ω-shaped vortex structure (see Figs. 1(b) and 1(c)). Predictably, the stall-onset criterion will be first achieved by the mid-span section, and then be satisfied section by section from the mid-span to the tip.

Modeling three-dimensional dynamic stall 397 According to the qualitative analysis that the vortical strength at the mid-span section reaches the dynamic stall-onset criterion first, and then expands along the span, an empirical formula for the 3D dynamic stall criterion can be obtained as follows: α cr 3D = α mid +Δα 3D, (7) where α mid is the AOA of the 3D wing when the effective AOA (α )atthemid-spansectionis equal to α cr,andδα 3D is empirically expressed as Δα 3D = (13x e 1 2 (5 λ2 x) 1+ 1 λ 3 x )( 100 (2.12 25re r)) 3. (8) In Eq. (8), Δα 3D is a delay in the AOA due to the 3D effect for the dynamic stall-onset at any section x, and it consists of two parts. One is 13x e 1 2 (5 λ2 x) 1+ 1 λ, which is a delay along the 3 x span depending on the aspect ratio λ and the distance x between any section and the mid-span 100 (2.12 section. The other is 25re 3 r), which is the extent of the delay depending on the reduced pitch rate r. Clearly, the larger r is, the greater the extent of the delay is. Of course, if r =0, Δα 3D = 0, which corresponds to the characteristics of static stalls. 3.4 Induction of dynamic stall 3.4.1 Quasi-2D dynamic stall simulation at mid-span section After the beginning of dynamic stalls, the normal force increases further, and thus the separation location delays further as Δf (S) =Δf (S) ( 1 e ) S Tv. (9) The general case of dynamic stalls involves the formation of a vortex near the leading edge of the airfoil, which subsequently separates from the upper surface and is transported downstream over the chord. In the Leishman-Beddoes 2D dynamic stall model, the total accumulated vortex normal force coefficient C v n under unsteady conditions is allowed to decay exponentially with time, and also updated by a new increment. For a sampled system with the sample number N, C v n(n) =C v n(n 1)e ΔS Tv +(C v (N) C v (N 1))e ΔS 2Tv, (10) in which ( 1+ f C v (N) =Cn (1 C (N) (N) ) 2 ) τ v m v, (11) 2 τ vl where τ v is denoted as the non-dimensional vortex time, i.e., τ v = 0 at the onset of dynamic stalls and τ v = τ vl when the vortex reaches the trailing edge. Both the vortex decay nondimensional time constant T v and the non-dimensional time τ vl for the vortex to traverse the chord can be obtained statistically from a variety of dynamic stall test data. The parameter m v is related to the vortical strength, and is obtained from the test data. Abrupt air loading changes occur when the critical condition for the leading edge separation is met. At this point, the accumulated vortex lift is assumed to start to convect over the chord. During the vortex convection process, the vortex force is assumed to behave according to Eqs. (10) and (11). However, the accumulation is terminated when the vortex reaches the trailing edge at τ v = τ vl. After the vortex passes the aerofoil trailing edge, the effect of the vortex-induced lift on the airfoil behavior decays rapidly, which is accomplished by halving the non-dimensional vortex decay time constant T v for the period τ vl τ 2τ vl. 3.4.2 3D dynamic stall vortex Accurate simulation of 3D vortex structures is one of the keys for modeling 3D dynamic stalls. According to the topological analysis of the flow, the vortical strength at the mid-span is larger than that near the tip, thus resulting in remarkable changes of the vortex normal force.

398 Chao LÜ and Tong-guang WANG However, there may exist interactions between the vortex and the flow field structure over the wing surface. The vertical distance of the Ω-shaped vortex from the surface is different from one spanwise section to another spanwise section. This may change the tendency that the vortical induction gradually decreases from the mid-span to both the tips, resulting in discontinuous changes in the vortex-induced force [7]. To model this effect, a variable K v 3D is used, in this paper, to control the increment of the vortex normal force along the span, namely, ( ( 1+ f C v 3D (N) =Cn C (N) ) 2 ) τ v (N) 1 m v K v 3D, (12) 2 τ vl where the empirical K v 3D is determined according to the vertical distance of the Ω-shaped vortex from the wing surface. 4 Results and discussion Example 1 The aspect ratio of the wing model is 3, the Mach number is 0.16, the pitch rate is 92.44 s 1, and the non-dimensional reduced pitch rate is 0.006 085. The experimental data come from the wind tunnel test of a rectangular wing undergoing a constant rate ramp-up motion in the Handley Page Wind Tunnel of University of Glasgow [7]. The two results are shown in Fig. 2. Fig. 2 Results of Example 1 and the experiment in Ref. [7] Figure 2 shows that the results calculated from the 3D dynamic stalls model are generally quite fine, especially at low AOAs and relatively high AOAs. At very high AOAs around and

Modeling three-dimensional dynamic stall 399 above 30, there exists big discrepancy between the calculated results and the experimental data, demonstrating that the modeling of the induction of the dynamic stalls vortex needs to be improved further when the vortex convects downstream and after the vortex sheds from the trailing edge. Nevertheless, as an engineering model for 3D dynamic stalls, the calculated results are acceptable. It is worthy to note that the dynamic stall model can capture the dynamic vortex initiation accurately. Example 2 On the basis of Example 1, the reduced pitch rate is changed to 0.015 982. The calculated results and the experimental data are shown in Fig. 3. Fig. 3 Results of Example 2 and the experiment in Ref. [7] Due to the the high pitch rate in this example, the dynamic stall vortex enhances the normal force observably. The 3D dynamic stall model provides good results in the whole AOA range compared with the experimental data. According to the topology of the 3D dynamic stall, the vorticity decreases from the mid-span to the tip. On the other hand, the vertical distance of the Ω-shaped vortex from the wing surface varies along the span. There may exist a compromise between the vorticity and the vertical distance, resulting in a discontinuous change of the vortical induction along the span. This phenomenon is represented by the experimental data in Figs. 2 and 3, where the normal force at x =0.18 is greater than those at x =0.07 and x =0.3. The 3D dynamic model appropriately captures this change of the force induced by the Ω-shaped vortex. Example 3 Tang and Dowell [4] used the ONERA model expanded into 3D cases to simulate the unsteady flows around a pitching wing of low aspect ratios, whose section is NACA0012 airfoil, and compared their calculated results with the experiment data. To validate the present

400 Chao LÜ and Tong-guang WANG Fig. 4 Results of calculations and the experiment in Ref. [4] for the pitching wing 3D dynamic stall model, the experiment of Tang and Dowell [4]. ischosenasacalculation example. In the experiment, the wing with aspect ratio of 2 is pitched between 15 27,and the reduced frequency is 0.35. The calculated results from the present model and the ON- ERA model and the experimental data for the section x =0.25 are shown in Fig. 4. The figure shows that the calculation results using the method presented in this paper agree well with the experimental results in general, and are more favorable than those of Tang and Dowell [4]. obtained by the ONERA model. Although there are certain differences in the hysteresis loop between the model and experiment, the calculated results are encouraging by consideration of the difficulties in modeling 3D dynamic stalls. 5 Conclusions An engineering model for 3D dynamic stalls has been developed based on the physical features of dynamic stalls with empirical hypothesis. This model links the onset, development, and convection of the 3D dynamic stall vortex along the span with the quasi-2d aerodynamic characteristics at the middle section of the wing undergoing dynamic motion. The calculated results from the model coincide well with the experimental data in general. Clearly, there are still areas of improvement within the modeling strategy, particularly in relation to high angles of attack. This will only be achieved through much more experimental and theoretical investigations of dynamic effects on 3D wings or blades. References [1] Leishman, J. G. Practical Modeling of Unsteady Airfoil Behaviour in Nominally Attached Two- Dimensional Compressible Flow, UM-AERO-87-6, University of Maryland (1987) [2] Leishman, J. G. and Beddoes, T. S. A semi-empirical model for dynamic stall. Journal of the American Helicopter Society, 34(3), 3 17 (1989) [3] Petot, D. Differential equation modeling of dynamic stall. La Recherche Aerospatible (English Edition), 5, 59 72 (1989) [4] Tang, D. M. and Dowell, E. H. Experimental investigation of three-dimensional dynamic stall model oscillating in pitch. Journal of Aircraft, 32(5), 163 186 (1995) [5] Coton, F. N. and Galbraith, R. A. An experimental study of dynamic stall on a finite wing. The Aeronautical Journal, 103(1023), 229 236 (1999) [6] Spentzos, A., Barakos, G. N., Badcock, K. J., Richards, B. E., Coton, F. N., and Galbraith, R. A. Computational fluid dynamic study of three-dimensional dynamic stall of various planform shapes. Journal of Aircraft, 44(4), 1118 1128 (2007) [7] Ferrechia, A. Analysis of Three Dimensional Dynamic Stall, Ph. D. dissertation, University of Glasgow (2000) [8] Bisplinghoff, R. L., Ashley, H., and Halfman, R. L. Aeroelasticity, Dover Publications, Inc., New York (1996) [9] Sheng, W., Galbraith, R. A., and Coton, F. N. A modified dynamic stall model for low Mach numbers. Journal of Solar Energy Engineering, 130(3), 031013 (2007)