NASA/TM—2013–216601 

 

Computations on Wings with Full-Span 

Oscillating Control Surfaces using NavierStokes Equations 

 

 

Guru P. Guruswamy 

Ames Research Center, Moffett Field, California

 

 

May 2013
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NASA/TM—2013–216601 

 

Computations on Wings with Full-Span 

Oscillating Control Surfaces using NavierStokes Equations 

 

 

Guru P. Guruswamy 

Ames Research Center, Moffett Field, California 

 

 

National Aeronautics and 

Space Administration

Ames Research Center 

Moffett Field, CA 94035-1000 

May 2013
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443-757-5802 
Computations on Wings with Full-Span Oscillating 

Control Surfaces using Navier-Stokes Equations 

Guru P. Guruswamy 

NASA Advanced Supercomputing Division, Ames Research Center Moffett Field, CA

Abstract 

This paper presents a time-accurate procedure for computing unsteady aerodynamic forces 

on wings with a full-span oscillating control surface by using sheared grids. In this procedure, 

the flow is computed by solving the unsteady Reynolds-Averaged Navier-Stokes (RANS) 

equations in conjunction with the one-equation Spalart-Allmaras turbulence model. A 

module to model control surface oscillations is embedded into a general purpose RANS 

solver. Unsteady pressures and forces are validated with experimental data for wall-to-wall 

mounted and finite-tip wings with full span control surfaces. The effect of Mach number on 

unsteady aerodynamic loads is investigated for the transonic flow regime. 

I. Introduction

he use of active controls for fixed-wing aircraft [1] and rotorcraft [2] is becoming more important for modern 

configurations. Advanced transport designs, such as blended wing-bodies [3], need active controls to achieve 

safe and efficient flight. Efforts to reduce the vibrations of helicopter blades with use of active-controls are also in 

progress [2]. 

 Modeling oscillating control surfaces using linear aerodynamics theory is well established. However, higherfidelity methods are needed to account for nonlinear effects, such as those that occur in transonic flow [4]. The 

aeroelastic responses of a wing with an oscillating control surface, computed using the transonic small perturbation 

(TSP) theory, have been shown to cause important transonic flow effects [5] such as a reversal of control surface 

effectiveness that occurs as the shock wave crosses the hinge line. In order to account for flow complexities such as 

blade-vortex interactions of rotor blades [6] and leading edge vortex-induced aeroelastic oscillations of blended 

wing body transports [3, 7], higher-fidelity methods based on the Navier-Stokes equations are used. 

 Since the original development of moderate-fidelity TSP methods [5, 8], higher-fidelity flow equations have come 

into use to model oscillating control surfaces. The first successful computation using TSP for a fixed wing with an 

oscillating control surface in the transonic flow regime is reported in Ref. 8. Computations using transpiration 

boundary conditions (without physically moving the control surface) were made in Ref. 9, using the full-potential 

theory. Ref. 10 presents a procedure that uses the Navier-Stokes equations with moving-sheared grids and 

demonstrates up to 8 degrees of control-surface amplitude, using a single grid. Later, this procedure was extended to 

accommodate larger amplitudes, based on sliding grid zones [11]. Reference 12 reports an effort using a sheared 

grid to validate results for a transport wing. The sheared grid method implemented in the Euler/Navier-Stokes-based 

aeroelastic code ENSAERO [13] was successfully applied to active control design by industry [14]. 

 To date, the sheared grid technique of modeling control surfaces has been successfully applied to complex 

geometries using patched structured grids [15] and has been implemented on parallel computers [16]. The 

aeroelastic code HIMAP [17] based on the Navier-Stokes equations using patched structured grids includes a control 

surface module that has been implemented in a parallel computing environment. In addition to such patched 

structured grid codes, aeroelastic capabilities have been implemented in OVERFLOW [18], which is based on the 

RANS equations and uses overset grids, which has been well established to model flows over non-oscillating control 

surfaces [19]. Recently, an oscillating control surface capability for rotorcraft was added to OVERFLOW using 

overset grids in gaps [20] and was validated for integrated air loads. Ref. 20 reports significant differences between 

computations and measurements, particularly for flap moments. While the approach presented in Ref. 20 promises 

to be accurate, a promise that has yet to be realized in such quantities as unsteady surface pressures, it also requires 

more grid points and a more complicated grid generation process. Time step restrictions associated with tightly 

spaced grid points in gaps can also be an issue with this approach. 

T 
 

 

 

Fig. 1 Typical sheared grid. 

 

Fig. 2 Experimental set-up for NLR wall-wall model. 

 As an alternate approach to the method of modeling gaps with overset grids, the present work presents a sheared 

grid capability [10] embedded as a module into the OVERFLOW code. In this approach the grid at the flap’s gap for 

the deflected control surface has the same topology as the grid for un-deflected control surface. Control surface 

deflections are modeled by shearing the grid at the gap. Sheared grids produce accurate results for moving control 

surfaces [10, 13, 15] and can be numerically more efficient than methods that use overset grids to model small gaps, 

which are common when active control surfaces are used. Figure 1 shows an example of a sheared grid in the 

vicinity of the flap interface. The continuity of the grid is maintained at the hinge line. 

 This work focuses on implementing and validating the sheared grid approach for use with overset grids similar to 

that developed for patched grids in Ref. 10. Grids are selected based on grid refinement studies. The effects of 

oscillating control surfaces on unsteady loads in the transonic flow regime are studied. In addition, this effort makes 

progress towards providing an efficient and robust high-fidelity analysis tool for designing active controls, 

particularly for the transonic regime. The approach presented is validated using results from an experiment 

performed at the National Aerospace Laboratory (NLR), Amsterdam [21], that the Advisory Group for Aerospace 

Research and Development (AGARD) selected as one of the detailed unsteady pressure experiments involving 

oscillating control surfaces. Another validation study is made for a finite-wing with a full-span oscillating control 

surface that was tested at the Royal Aeronautical Establishment (RAE) [22]. A search of the literature indicates that 

there has been no previous effort to utilize these data to validate flow solvers based on the Euler/Navier-Stokes 

equations. Recent papers [23, 24] that model moving control surfaces do not include any validation with 

experiment Another recent paper [25] validates only forces with experimental data at a lower oscillating frequency 

than normal operational frequencies of flaps [20]. None of the recent papers [23, 24, 25] include validation of 

unsteady-surface pressures needed before validating integrated forces [20]. This effort involves detailed validation 

of unsteady surface pressures and also integrated forces. After validation, the effects of Mach number in the 

transonic range are presented. 

II. Approach

 In this paper, the RANS equations [26] are numerically solved using the Pulliam-Chaussee diagonal form of the 

Beam-Warming central difference algorithm [27], along with the one-equation Spalart-Allmaras turbulence model 

[28]. The solutions are computed using the OVERFLOW code [18], which is based on an overset grid system. The 

second-order spatial and temporal accuracy options available in OVERFLOW 2.2c are used throughout this 

analysis. 

 In the current version of OVERFLOW, control surface motions are prescribed externally [20] and the flow 

solver needs to pause while getting the deflection input. In this work, the shearing grid capability is embedded 

within the OVERFLOW code to avoid the computer overhead time associated with input/output and pause for data 

transfers. This will facilitate embedding an active control module [5] in OVERFLOW. Grid zones in OVERFLOW 

2 

x 

y 

z 
 are assigned to different processors by a load-balancing algorithm with no user control. In the present approach it is 

assumed that the grid zone containing the control surface is assigned to a single processor without splitting the grid. 

However, this limitation could be removed if an option to assign specific processors to selected grid zones, similar 

to that implemented in HiMAP [17], were available in OVERFLOW. Confining a shearing grid module to one zone 

makes all computations within that module implicit and allows larger time steps for integration. Any impact on 

parallel efficiency can be minimized using node-filling algorithm successfully implemented in HiMAP [17]. 

 The control surface deflection at a given time t is defined as 

 δ = δ0 + δβ sin(ωt) (1) 

 

where δ0 and δβ are the mean and maximum amplitude of the oscillatory control surface deflections, respectively. 

The ω and t parameters are circular frequency of the control surface oscillation in radians per sec and time in sec, 

respectively. All unsteady components are scaled with δβ in radians. 

III. Results for Full Span Control Surface of Wall-to-Wall Mounted Wing 

A. Wind Tunnel Model 

 In this work computations are made about an AGARD configuration [21] that involves a wall-to-wall mounted 

blade with a full-span oscillating control surface. A schematic diagram of the model is shown in Fig. 2. The model 

measures 420 mm in span and 180 mm in chord, and is made of NACA64A006 airfoil sections. The leading edge of 

the full-span control surface is located at 75% chord, and from the figure in the report it is estimated that the gap is 

about 0.5% of chord. A total of 38 upper and lower surface pressure orifices, including 12 on the control surface, 

were used to measure data; the number of orifices used is relatively high compared to similar tests conducted to 

date. The measured data are given in the form of magnitude/amplitude and phase angles of unsteady pressures and 

sectional lift forces. The phase angle is the difference in period between the peaks of control surface deflection and 

the pressure/force response. The amplitude is the absolute value of the maximum response above the time-averaged 

value. The magnitude and phase angles are computed by performing a Fourier analysis [29] of time the responses. 

B. Grid for Flow Solver 

 A module was written to generate a dynamic moving grid for the oscillating control surface with a single O-H 

(wrap around airfoil in x-direction and stacked spanwise in z-direction) grid topology. This module uses the 

hyperbolic grid generator available in the Chimera Grid Tools (CGT) software package [30]. Since the gap in the 

model is very small it is not modeled in the grid. The airfoil surface grid is sheared by applying the displacements 

due to control surface deflections in the normal direction (z) and a new grid is generated. 

 The final grid selected is based on grid refinement studies for both spatial and temporal accuracies. The majority 

of results from the experiment are between free-stream Mach numbers (M∞) 0.8 and 0.9. M∞ = 0.854 for which both 

mean steady and unsteady pressure data are available is selected as a representative Mach number for grid 

refinement studies. At this M∞, the peaks in unsteady pressures due shock wave and hinge line are predominant 

compared to other lower and higher Mach numbers. The associated Reynolds number Re_c based on chord (c) is 

2.44 million. 

 First, a baseline grid with attributes based on previous experience with algebraic grids [10, 13] is selected. Nearbody grid spacing of 0.0000125 chord, near-body stretching factor of 1.125, average chordwise spacing of 0.005 and 

outer boundary (OB) grid surface location at 15c are selected. The grid spacing at the leading edge is about 0.0025c. 

With these parameters a baseline (BL) 2D O-grid with 403 points in the wrap around direction and 80 points in the 

normal direction is generated using the hyperbolic grid generator [30]. Since the wing is mounted between walls and 

measurements are made at the mid-span, 3 planes are used in the spanwise direction to run OVERFLOW in 2-D 

mode. Effect of selected parameters on sectional lift coefficient (cl) and drag coefficient (cd) are studied to determine 

the final grid. 

 Corresponding to one of the test cases steady-flow computations at M∞ = 0.854, δ0 = 1.08 deg and Re_c = 2.4 

million, each run for 10000 steps, are performed first, using a variable time-step option in OVERFLOW without 

sub-iteration. As seen in Fig. 3, cl and cd converge in about 5000 iterations. The rest of the steady-state computations 

are made using 6000 iterations. 

 To study adequacy of the grid in the chordwise direction, starting from the BL grid (403 points), computations are 

3 
made using a series of grids with increased spacing: double the spacing (203 points), quadruple the spacing (103 

points) and octuple the spacing (53 points), while keeping the remaining of the grid parameters the same. This is

 

Fig. 3 Convergence of steady cl and cd at M∞ = 0.854, 

δ0 = 1.0 deg and Re_c = 2.44 million. 

Fig. 4 Effect of chordwise spacing on cl and cd at 

M∞ = 0.854, δ0 = 1.0 deg and Re_c = 2.44 million. 

 

accomplished by modifying the baseline grid (403 points) using the cubic spline interpolations in CGT [30]. 

Figure 4 shows the effect of grid refinement in the x-direction on cl and cd. As seen, both cl and cd converge for the 

baseline grid. 

 Next normal grid spacing near the surface is varied from 0.00001 to 0.00003. As observed in Fig. 5, the initially 

selected spacing of 0.0000125 is considered adequate. Effects of decreasing stretching factor at surface in Fig. 6 

show that the selected surface stretching factor of 1.125 is adequate. These selected parameters yield an average y+ 

value of 0.9712 which is considered adequate to resolve flows at the surface. 

 Using the above selected grid parameters the effect of location of the outer boundary (OB) is studied next. From 

Fig. 7 it is seen that the outer grid needs to be at least at 25 chords which requires a grid size of 403 x 85. 

 . 

 

Fig. 5 Effect of normal spacing at surface on cl and cd

at M∞ = 0.854, δ0 = 1.0 deg and Re_c = 2.44 million. 

 

Fig. 6 Effect of normal grid stretching factor on cl

and cd at M∞ = 0.854, δ0 = 1.0 deg and Re_c = 2.44 

million. 

4 

Baseline Grid Baseline Grid 

Baseline Grid 

Baseline Grid 
Among all parameters considered for grid refinement studies it is observed that the outer boundary location has 

the strongest effect on cl and cd. As a result effect of outer boundary location is studied for unsteady cases 

 Unsteady computations are performed at M∞ = 0.854, α = 0.0 deg, Re_c = 2.44 million, k = 0.24 and δβ = 1.05 deg 

corresponding to a test case in the experiment. Computations are performed with a variable number of steps per 

cycle (NSPC). It was found that NSPC = 1200 produced a stable solution without using Newton sub-iterations 

(NWIT). The rest of the time step convergence studies are made using NSPC = 1200. 

 Since Newton iterations are required to maintain 2nd order time accuracy, the next study involved the variation of 

number of Newton iterations (NWIT). Computations are made for 4 cycles with increasing NWIT. Figure 8 shows 

plots of cl for NWIT = 5, 10 and 15. The solutions during the first cycle of the computation have not achieved 

periodic behavior and should not be quantitatively evaluated. The solutions for cycles 2-6 are periodic. The 

percentage change in 6th cycle’s peak cl from NWIT = 5 to 10 is 5% whereas from 10 to 15 is 0.5%. The responses 

are considered converged for NWIT = 15. Remaining of the unsteady computations are made using 4 cycles of 

oscillations with NSPC = 1200 and NWIT =15. 

 Figure 9 shows the effect of OB location on magnitude and phase angle of cl obtained from Fourier analysis of the 

time response. Results in Fig. 9 show that the OB location at 35 chords is adequate, which requires 90 grid points in 

the normal direction. 

 

 

Fig. 7 Effect of outer boundary location spacing on 

cl and cd at M∞ = 0.854, δ0 = 1.05 deg and Re_c = 

2.44 million. 

Fig. 8 Response of cl for M∞ = 0.854, α = 0.0 deg and 

Re_c = 2.41 million, k = 0.24 and δβ = 1.05 deg. 

C: Test Cases 

 Reference 21 reports unsteady measured results for various Mach numbers from 0.80 to 0.988 for k ranging from 

0.06 to 0.40. In this paper three cases near k = 0.25, a typical frequency for real wings, are selected for comparison. 

They are M∞ = 0.804 (k = 0.253, δβ = 1.11 deg, Re_c = 2.35 Million), M∞ = 0.854 (k = 0.240, δβ = 1.05, Re_c = 2.41 

Million) and M∞= 0.879 (k = 0.234, δβ = 1.08, Re_c = 2.44). For all three cases unsteady computations are made by 

oscillating the control surface for 4 cycles with NSPC = 1200 and NWIT = 15 using a grid of size (403 x 90). 

Fourier analyses are performed on each solution response. Comparison of time averaged pressures with experiment 

are shown in Fig. 10. Comparisons for all three cases are good. Results at M∞ = 0.804 compare better than the other 

two Mach numbers. 

 Figure 11 shows detailed comparison of in-phase and out-of-phase components of upper surface pressures with 

experiment for M∞ = 0.854. Computed trends agree well with experiment. Computed peaks near the shock-wave 

are sharper than those from measurements. The in-phase peak from the experiment is more smeared than for the outof-phase component. Out-of-phase components compare better than in-phase components near the shock-wave 

 location. Both computed and measured results show peaks near x/c = 0.75 (hinge location) for in-phase component 

5 

 

Baseline Grid 
but not for out-of-phase components. Comparisons at the leading edge for the out-of-phase components are less 

favorable than the in-phase component. 

Fig. 9 Effect of outer boundary location on 

magnitude and phase angles of unsteady cl for 

M∞ = 0.854, Re_c = 2.41 million, k = 0.24, and 

δβ = 1.05 deg. 

Fig. 10 Comparison of time averaged upper surface 

computed pressures with experiment at M∞ = 0.804 

(k = 0.253, δβ = 1.11 deg, Re_c = 2.35 Million), 

M∞ = 0.854 (k = 0.240 δβ = 1.05 deg, Re_c = 2.41 

Million) and M∞= 0.879 (k =0.234, δβ = 1.08 deg, 

Re_c = 2.44 million). 

 

 To study the effect of Mach number unsteady loads computations are made for M∞ varying from 0.80 to 0.90 in 

increments of 0.1. Average values of k = 0.242, δβ = 1.08 deg and Re_c = 2.40 million are used for these 

computations. Figure 12 shows plots of in-phase and out-of phase components for the oscillatory lift coefficient with 

increasing M∞. The in-phase component decreases with increasing M∞ and the out-of-phase component decreases up 

to about M∞ = 0.86 then increases. Results from the experiment at M∞ = 0.804, 0.822, 0.854 and 0.879 show similar 

trends. Generally the in-phase components compare better than out-of-phase components. 

Fig. 11 Comparison between computed and measured 

values of in-phase and out-of-phase components of 

upper surface oscillatory pressures for M∞ = 0.854, 

k = 0.253, δβ = 1.05 deg and Re_c = 2.41 million. 

Fig. 12 Comparison between computed and measured 

values of in-phase and out-of-phase components of 

sectional lift coefficient for various M∞ at k = 0.242, 

δβ = 1.08 deg and Re_c = 2.40 million. 

6 

Baseline Grid 

Outer Boundary Location in Chords 
VI. RAE Wing with Finite Tip 

 Reference 22 reports wind tunnel results for a low-aspect ratio (1.25) wing (NACA 64006A airfoil section) with a 

moving full-span control surface (see Fig. 13). The hinge line is located at 67% of chord. A case from experiment at 

M∞ = 0.243, k = 0.88 and δβ = 3.5 deg for which unsteady pressure results are available at 68% span station is 

selected. 

 The same chordwise O-grid topology from the NLR case is used for this wing since both have the same 

NACA64A006 airfoil section. The final grid 403 x 90 obtained after grid refinement studies for NLR case is 

selected. A H topology is used for the grid in the spanwise direction. Since details are not available, the tip is 

modeled by tapering the thickness to zero. This requires a total of 58 points in the spanwise direction with 38 points 

on the wing. Grids are clustered at the root and the tip to accommodate a smooth shearing of the grid. 

Fig. 13 RAE wind tunnel model for finite wing with 

full-span control surface. 

Fig.14 Effect of outer boundary location on cl at 68 % 

span station for M∞ = 0.243, k = 0.88, δ0 = 3.5 deg 

and Re_c = 2.0 million.

 

 Since this is a 3-D case, the adequacy of spanwise outer boundary location of the grid is verified. Figure 14 shows 

the effect of location of far span on sectional lift coefficient at 68% span for M∞ = 0.243, k = 0.88, δ0 = 3.5 deg and 

Re_c = 2.0 million. The selected outer boundary location of 10 span lengths is adequate. The remaining of the 

computations are made using a grid with 403 chordwise, 58 spanwise and 90 radial grid points. Figure 15 shows the 

3-D grid with details at the tip. 

Fig. 15 Grid for RAE wing with full-span control surface. 

7 

Hinge Line 
 To verify the adequacy of time step size, computations are made for increasing NSPC with NWIT = 15 at M∞ = 

0.243, k = 0.88 and δβ = 3.5 deg. It is found that responses become periodic in 4 cycles. Figure 16 shows the 

effects of NSPC on in- and out-of-phase components of oscillatory cl at 68% span. Results converge at NSPC = 

4800 where NWIT = 10 and 15 give almost identical results. 

 Figure 17 shows the comparison of computed results with experiment for in-phase and out-of-phase components 

of upper surface oscillatory pressure. Both components compare well in trend with the experiment. In-phase 

components compare better than out-of-phase components. Differences near the leading edge are higher than along 

the rest of the chord. Similar differences in out-of-phase values are reported in Ref. 22 in which comparisons with 

the linear theory are made. It is noted that because of the low Mach number the out-of-phase values are significantly 

smaller than in-phase values. 

Fig. 16 Effect of NSPC on in-phase values of 

oscillatory cl at 68% semi span for M∞ = 0.243, 

k = 0.88 and δβ = 3.5 deg.

Fig. 17 Comparison of upper surface oscillatory cp

between computations and experiment at 68 % span 

for M∞ = 0.243, k = 0.88 and δβ = 3.5 deg. 

V. Conclusions 

 Computations of the flows over full-span oscillating control surfaces are performed using the Reynolds-Averaged 

Navier-Stokes equations. An embedded sheared-grid approach in the context of a general purpose overset based 

CFD code is used to model the oscillating control surfaces. The procedure is validated by comparing unsteady 

pressures and integrated forces with experiments. Two test cases with the same airfoil section, one a wall-to wall 

mounted wing and another a finite-span cantilever wing, are considered. Grids are selected based on detailed grid 

refinement studies that involved effects of chordwise spacing, near surface spacing, near surface stretching factor 

and outer boundary locations. Steady computations show that results are more sensitive to outer boundary location 

than other parameters. Unsteady computations on finite tip wing need more steps per cycle than computations on 2D 

airfoil whereas the number of Newton sub-iterations is almost same for both. In general, in-phase components 

compared better with experiment than out-of-phase components. Moving control surface module developed for a 

general purpose CFD code will facilitate its application for analysis and design of active controls. The present 

method will be extended to model part-span control surfaces on parallel computers. 

Acknowledgments 

 The author thanks Henry Lee of NASA Ames Research center for providing a script for the CGT grid tool to 

generate 2-D grids. Terry Holst, chief of Fundamental Modeling and Simulation Branch at NASA Ames Research 

Center and CFD lead of Rotary wing project, made valuable suggestions on grid refinement studies. 

8

 

Number of Steps per Cycle x103 (NSPC) 
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