Embed Size (px)
Citation preview
IFASD-2009-050
A NONLINEAR HARMONIC BALANCE METHOD FORTHE CFD CODE OVERFLOW 2
Chad H. Custer1, Jeffrey P. Thomas2, Earl H. Dowell3, and Kenneth C. Hall4
Duke UniversityMechanical Engineering & Materials Science
144 Hudson Hall, Box 90300Durham, NC 27708
[email protected], [email protected], [email protected],[email protected]
Keywords. CFD, OVERFLOW, Harmonic Balance Solution, Overset Grid.
Abstract. A National Aeronautics and Space Administration computational fluid dy-namics code, OVERFLOW 2, was modified to utilize a harmonic balance solution method.This modification allows for the direct calculation of the nonlinear frequency-domain so-lution of a periodic unsteady flow while avoiding the time consuming calculation of longphysical transients that arise in aeroelastic applications. With the usual implementationof this harmonic balance method, converting an implicit flow solver from a time marchingsolution method to a harmonic balance solution method results in an unstable numericalscheme. However, a relatively simple and computationally inexpensive stabilization tech-nique has been developed and is utilized in this paper. With this stabilization technique,it is possible to convert an existing implicit time-domain solver to a nonlinear frequency-domain method with minimal modifications to the existing code. This new frequency-domain version of OVERFLOW 2 utilizes the many features of the original code, such asvarious discretization methods and several turbulence models. The use of Chimera over-set grids in OVERFLOW 2 requires care when implemented in the frequency-domain.This paper presents a harmonic balance version of OVERFLOW 2 capable of solving onoverset grids for sufficiently small unsteady amplitudes.
1 INTRODUCTION
Flutter and limit cycle oscillation (LCO) calculations for complex geometries of flexible,deforming structures in an aerodynamic flow are computationally expensive due to themany grid points needed to model the complex flow accurately. Flutter is the dynamicinstability of the aeroelastic (fluid-structure) system and LCO is the nonlinear oscillationthat may follow. Near flutter the aeroelastic damping is small and thus the physicaltransients can be very long. Using traditional time-domain computational fluid dynam-ics (CFD) methods, such transients must be modeled time-accurately, while only theperiodically converged solution is typically of interest.
Nonlinear frequency-domain methods offer the benefit of calculating the periodic responsedirectly, thus avoiding the need to model long physical transients. This leads to a dramaticreduction in computational cost for lightly damped and physically unstable systems.
The harmonic balance (HB) nonlinear frequency-domain solution method is highly effi-cient with the computational time being at least an order of magnitude faster than thetime-marching solution for aeroelastic analyses [1]. Also, the HB method is capable ofmodeling nonlinearities [2]. These properties make the method useful for a wide variety
1
of applications such as the modeling of limit cycle oscillations in nonlinear aeroelasticsystems [3, 4].
Many CFD codes exist with the majority of these codes modeling the flow field by time-marching a spatially and temporally discretized version of the conservation equations.This technique is highly versatile since any body motion can be considered. However,when studying aeroelasticity one is often primarily interested in the response of the flowfield to harmonic motion of the body.
While it is possible to write a nonlinear frequency-domain solver de novo, it is possi-ble to convert a time-domain flow solver to a frequency-domain method with a modestamount of CFD code modification. Basing the HB solver on a time-domain code signifi-cantly reduces code development time. By using the time-domain flow solver to drive thefrequency-domain code, we are able to utilize current state-of-the-art methods alreadyimplemented within a given time-domain solver. Specifically, we are interested in using avariety of discretization techniques and turbulence models as well as the Chimera oversetgrid method.
2 HARMONIC BALANCE METHOD
The harmonic balance method developed by Hall et al. [5] takes advantage of the tempo-rally periodic nature of the flow field by assuming the conservation variables can each beaccurately represented by a Fourier series in time. It is then possible to recast the gov-erning equations and solve for the Fourier coefficients of the conservation variables. Oneof the major features of the HB method is that the pseudo-time marching scheme usedto solve for the Fourier coefficients of the conservation variables takes the same form asthe scheme by which the original governing equations are solved for the steady state case.The only addition to the update step required by the HB method is the inclusion of anadditional term known as the source term. The fact that the equations to be solved for theHB method mimic the original governing equations allows one to utilize computationalmethods already well developed in the literature.
The set of partial differential equations that describe fluid flow can be expressed in theform given by Eq. 1. Here the first term clearly represents the time derivative. The secondterm, N(Q(x, t)), is the spatial operator representing the derivative operations in space.
∂Q(x, t)
∂t−N(Q(x, t)) = 0 (1)
Note that it is useful to begin the formulation for the harmonic balance method withthis general expression for Navier-Stokes equations since many conservation laws can beexpressed in this form.
Consider Eq. 1 discretized in space but continuous in time. In Eq. 2, and in those tofollow, l represents the index of the conservation variable and j is the nodal index.
∂Ql,j(t)
∂t−Nl,j(t) = 0 (2)
The first step in deriving the HB method is to utilize the fact that the body is undergoingharmonic motion with fundamental frequency ω and to expand Ql,j(t) and Nl,j(t) in
2
Fourier series as
Ql,j(t) = Q0l,j +
NH∑n=1
{QCn
l,j cos(ωnt) +QSnl,j sin(ωnt)
}(3)
Nl,j(t) = N0l,j +
NH∑n=1
{NCn
l,j cos(ωnt) +NSnl,j sin(ωnt)
}. (4)
Note that the number of harmonics retained in the Fourier series expansions (NH) mustbe sufficient such that Ql,j(t) and Nl,j(t) are represented accurately. The expansions ofQl,j(t) and Nl,j(t) are then substituted into Eq. 2, resulting in a linear system where theunknowns are the Fourier coefficients of Q and N .
It is then possible to relate the Fourier coefficients of Q and N to the conservation variablessampled over one period in time using the discrete Fourier transform. The result is asystem of equations for each conservation variable at each node as given by
ω[D]Ql,j −N l,j = 0 (5)
where Ql,j contains sampled values of the lth conservation variable at the jth node.
Ql,j =
Ql,j(t0 + ∆t)Ql,j(t0 + 2∆t)
...Ql,j(t0 + T )
, N l,j =
Nl,j(t0 + ∆t)Nl,j(t0 + 2∆t)
...Nl,j(t0 + T )
(6)
T =2π
ω, ∆t =
2π
NTω
The airfoil locations at each of these sub-time levels is shown for a simple pitching airfoilcase in Fig. 1.
-αmax
0
αmax
0 T/7 2T/7 3T/7 4T/7 5T/7 6T/7 T
Angle of Attack,
α
Time
(a) Angle of attack for each sub-time level. (b) Airfoil location for each sub-time level.
Figure 1: Sub-time level grid locations for a pitching airfoil.
Equation 5 represents the governing equations recast in the frequency-domain. There areseveral important features of this equation. First, note that the first term in the above
3
equation is a pseudo-spectral (frequency-domain) representation of the time derivativeterm. This term is referred to as the source term of the equation since it contains notime derivatives per se. The second important feature of this equation is that the spatialoperator is unchanged from the time-domain formulation, it is simply evaluated at multi-ple sub-time levels. This is critical to the goal of implementing the HB method about anexisting time-domain solver. Finally, note that the vector equation for each conservationvariable at each point is of size NT where NT = 2NH + 1.
In order to mimic the form of the time-domain equations as closely as possible, a pseudo-time (non-physical time) term is added to Eq. 5. The result is an equation that is identicalin form to Eq. 2 with one extra term, i.e. the source term.
∂Ql,j(t)
∂τ−N l,j(t) + ω[D]Ql,j(t) = 0 (7)
Since the frequency-domain representation of the governing equations is given by Eq. 5,the pseudo-time term can be used to iterate rapidly to convergence. Equation 7 can bediscretized in pseudo-time using the same methods by which the original time-domainequations are discretized in physical time. Also, recall that the spatial operator is un-changed from the time-domain solver. This results in a finite-difference formulation ofthe HB method that is formally unchanged from that of the original time-domain CFDcode except for the addition of the source term.
When considering the finite-difference representation of Eq. 7, the source term can beincluded on the n pseudo-time level or the n+ 1 pseudo-time level. Each of these optionshave serious limitations.
The original time-domain linear system of equations is of size NTD ×NTD where NTD =Nnodes ·Nvars. If included on the n+ 1 pseudo-time level, a large linear system of size
NTD ·NT ×NTD ·NT (8)
would need to be created. Not only would this increase the computational cost associatedwith solving the linear system, but also it would greatly deviate from the structure of thetime-domain solver. This method has recently been used with success [6], however it isunsuitable for the goals of this research, i.e. to construct a harmonic balance solver for atime-domain CFD code with minimal modifications.
Alternatively, if the source term is included with the n pseudo-time level, the form ofthe linear system is unchanged from the time-domain system. Instead of constructing alinear system that is scaled by a factor of NT , each of the sub-time levels can be solvedindependently. However, if implemented as described above, the scheme is numericallyunstable when used in conjunction with implicit methods.
Fortunately a relatively simple and computationally inexpensive stabilization techniquehas been developed, which allows for the source term to be evaluated at the n pseudo-time level. Thomas, et al. [7] describe the stabilization method used in the present workin considerable depth, and the reader is referred this paper for this key element of thestabilized harmonic balance method.
4
2.1 Implementation of the Harmonic Balance Method
Using the present method, few changes are needed to convert OVERFLOW 2 from itstime-domain solution method to a frequency-domain harmonic balance method. Mostimportantly, no changes are required for the handling of the temporal and spatial deriva-tive operators. The conversion of a time-domain solver to the frequency-domain harmonicbalance solution method requires modifications in three main areas of the code: initializa-tion, iteration, and post-processing. These changes are outlined below and illustrated inFig. 2, where the white boxes represent steps that are unchanged from the time-domaincode. Shaded elements represent steps that require modification.
Initialize Variables
Initialize HB Variables
Enter Pseudo-time Loop
Enter Sub-time Loop
Calculate ΔQ
Exit Sub-time Loop
Stabilize ΔQ
Exit Pseudo-time Loop
Update Q
Deform Body Grids
Post-processing
Initialization
Iteration
Post-processing
Figure 2: Harmonic balance method flow chart.
The first set of modifications is in the initialization phase. Recall that with the harmonicbalance method the solution is sampled over one period of motion, meaning that thenumber of sample points scales with the number of harmonics retained. This requiresincreased memory allocation to allow for the storage of these sub-time level solutions.Also in this phase, the computational grid is modified to correspond to the shape of thebody at each of the given sub-time levels.
The iteration section of the code is where the most significant changes are made. Theoriginal temporal and spatial operators are used to iterate each of the sub-time levelsolutions, which means that an additional sub-time loop is added. However, as notedpreviously, the spatial and temporal finite-difference operators are unchanged. Afteran update is calculated for each of the sub-time levels, the update is stabilized and thesolution is advanced. Modifications are also necessary to properly account for grid motion.Since the solution being calculated is unsteady, the time metrics must be modified to
5
account for the unsteady motion.
Finally, in the post-processing phase, routines are necessary to output useful informationsuch as the Fourier coefficients of surface pressure.
3 CHIMERA OVERSET GRIDS
When modeling a complex, three-dimensional body and the surrounding fluid domain,it is often difficult to create a well behaved single-block grid. The Chimera overset gridtechnique was developed to alleviate some of these challenges [8,9]. The general approachof the method is to create a patchwork of grids that collectively describe the surface ofthe body and the fluid domain.
The key concept of the Chimera overset grid technique is that individual grids are createdfor each component of the body and the fluid domain. Since the component grids arecreated without regard for the other portions of the body, two steps are required after theindividual grids are produced. These steps are illustrated for a two grid system where ano-grid describes the surface and near-body of a NACA 0012 airfoil and a Cartesian griddescribes the far-field as shown in Fig. 3.
First, it is possible that nodes describing the fluid domain of one component grid maylie within another portion of the body as is the case with the Cartesian grid points lyingwithin the airfoil in Fig. 3(a). These points, such as the point labeled “A”, are blankedand a collection of these blanked points make up a hole in the grid. The result, as shownin Fig. 3(b), is a set of overlapping grids describing the body surface and fluid domain.
Second, since the domain is no longer a single-block grid with clear computational di-rections, a method for inter-grid communication must be established. Specifically, holeboundaries and grid boundaries on the interior of the domain must be treated specially.The structured grid finite differencing stencil cannot be applied across grid boundaries.Instead, the solution at these fringe points is interpolated from donor points within theneighboring grid. For example, in Fig. 3(b) the solution on the outer most points of theairfoil grid is interpolated from the Cartesian grid. One of these airfoil fringe points islabeled “B”. Likewise, the Cartesian grid points nearest to the airfoil hole (such as thepoint labeled “C”) receive data from the airfoil grid.
Since each component grid or region is created independently, the gridding process is muchsimpler than if one were to create a single-block grid describing the entire domain. Infact, for most geometries of interest it would not be possible to create a single structuredgrid. Also, since component grids are created individually, it is relatively easy to modifythe geometry of interest. For example, adding an under-wing store to a fighter aircraftwould not require the regeneration of the grids for the fighter body or wing. Instead, agrid is created for the store and added to the set of grids describing the wing and body.
There are two options for creating the domain connectivity database for use with OVER-FLOW 2. The Pegasus 5 package [10] is NASA developed software that is run as apre-processing step where the domain connectivity does not change throughout the CFDcalculation. This option is appropriate for cases without grid motion. The second optionis to use the domain connectivity function (DCF) within OVERFLOW 2 [11]. The DCFperforms hole cutting and determines domain connectivity at each physical time step of
6
(a) Near-body and far-fields grid prior to hole cutting.
(b) Fringe points for a NACA 0012 two-grid system.
Figure 3: Set of Chimera overset grids describing the NACA 0012 airfoil.
the CFD calculation. DCF is the appropriate choice when performing calculations in thetime-domain on moving grids.
3.1 Time Domain Aerodynamics of the AIM-120 Missile
The flutter and LCO response of a fighter aircraft is highly dependent on the store con-figuration. The underwing and wingtip positioning of fuel tanks, bombs and missilesdramatically influences the aeroelastic response of the fighter. One configuration of par-ticular interest consists of the AIM-120 missile, shown in Fig. 4, mounted on the wingtipsof the F-16. Although the AIM-120 is small in size relative to the F-16, modeling themissile requires a large number of grid points due to the many fins. The grid systemdescribing the isolated AIM-120 used for inviscid calculations consists of 19 near-bodygrids and one far-field grid. In all, the set of grids total three million points.
The desire to calculate the aerodynamics of the AIM-120 highlights two important points.First, the fact that this relatively simple geometry requires three million points for an in-viscid calculation shows that computational requirements are very large when performingaeroelastic calculations on complex bodies. Iterating this system through long physical
7
Figure 4: Surface grids describing the AIM-120 missile.
transients would be computationally expensive. Second, the AIM-120 calculation showsthe need for overset grid methods. It would be virtually impossible to construct a single-block, structured grid that would accurately describe this geometry. These two pointsillustrate the need for an HB overset grid solver. With this capability it will be possible toavoid calculating long physical transients in the time-domain by instead solving directlyfor the periodic response using the HB method.
Even with the capability of solving in the frequency-domain, it is desirable to reduce thesize of the computational domain when possible. Here we will consider three models ofthe isolated AIM-120 at a steady angle of attack. The first and most expensive modelsolves the Euler equations on the full three million node grid system described above.
The simplest model employs slender body theory [12,13] as described by Eq. 9
dL
dx= −ρ∞
dA
dx
[U2∂za
∂x+ U
∂za
∂t
]+ ρ∞A
[U2∂
2za
∂x2+ 2U
∂2za
∂x∂t+∂2za
∂t2
](9)
A = π
(s2 −R2 +
R4
s2
)where s the semi-span of the fins, R is the radius of the missile body, and za is thedeflection of the body. It will be shown that slender body theory (SBT) performs wellaway from the fins, however overpredicts the lift due to the fins.
As such, the final model is a SBD/CFD hybrid model. Slender body theory is used tomodel the forces on the body of the missile, while the Euler equations are solved on gridsthat describe a single forward and single aft fin. Each fin is modeled as being cantileveredfrom a wall and isolated from the other.
Clearly, each of the latter two models includes substantial assumptions. Slender bodytheory assumes that the body (including the fins) is sufficiently slender such that the flowis effectively two-dimensional in the cross-flow plane. This assumption is violated in theregion of the fins. The hybrid model assumes that the aerodynamic influence of eachcomponent on the others is small.
8
Figure 5(a) shows the coefficient of lift versus angle of attack for each of the three models.If the three million node calculation is to be considered the benchmark, slender bodytheory overpredicts the lift by nearly a factor of two while the hybrid model agrees moreclosely with the three million node calculation. Figure 5(b) displays the coefficient oflift per chord-inch along the AIM-120 missile. By studying this figure, it becomes clear
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4
Coe
ffic
ient
of
Lif
t, C
L
Angle of Attack, α [degrees]
Slender Body TheoryFull GeometryHybrid Model
(a) Coefficient of lift as a function of angle of attack.
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0 20 40 60 80 100 120 140
Coe
ffic
ient
of
Lif
t per
Cho
rd-i
nch
per
degr
ee A
oA
Chord Position from Nose [inches]
Slender Body Theory1.0o
2.0o
4.0o
(b) Coefficient of lift per chord-inch per degree angle of attack alongthe AIM-120 missile.
Figure 5: Steady aerodynamic response of the AIM-120. In Fig. 5(b) solid lines correspond to calculationsperformed on the full geometry and symbols correspond to hybrid model calculations.
that the slender body theory model of the AIM-120 performs well away from the fins,however the model drastically overpredicts the lift due to the fins. The hybrid modelmore accurately predicts the lift due to the fins, however it neglects the lift due to theaerodynamic influence of the fins on the body.
The hybrid model produces acceptable preliminary results for steady flow trials while dra-matically reducing computational cost. The entire AIM-120 is modeled by three million
9
grid points and each fin grid consists of about 50,000 nodes. This means that the hybridmodel can produce results about 30 times faster than calculations performed on the fullgeometry.
When considering the aeroelastic calculation, the reduced frequencies of interest based onfin chord of the motion are small. Since for steady trials it has been shown that the finsare primarily responsible for the lift produced, it is believed that the hybrid model willalso work well for preliminary aeroelastic calculation. However, as illustrated, in order toaccurately calculate the aerodynamic forces on a complex body such as the AIM-120, thefull geometry must be represented accurately.
3.2 Overset Grids with the Harmonic Balance Method
With the need for overset grid capabilities with the HB version of OVERFLOW 2 demon-strated, we will consider the modifications necessary for the frequency-domain imple-mentation of overset grids. Since the sub-time levels of the harmonic balance methodcorrespond to the time-domain solution sampled within one period of motion, the bodygrid is in physically different positions for each sub-time level. For example, consider theunsteady motion to be a pitching airfoil. With three harmonics retained, there are sevensub-time level grids on which the solution is calculated, as illustrated by Fig. 1.
For large amplitude body motion, each of the sub-time level grid configurations needto be treated separately. Holes must be cut in the far-field grid and the connectivitydatabase must be created for each configuration. It would be possible to modify thedomain connectivity function within OVERFLOW 2 to perform these tasks and is thesubject of continuing work. However, the scenario is greatly simplified if only smallamplitude motions are considered.
This paper presents a frequency-domain implementation of overset grids that is valid forsmall amplitude motion. This assumption of small amplitudes allows for the grid holesand connectivity to remain unchanged between sub-time levels. Thus, Pegasus 5 canbe used as a pre-processing step to blank hole points and create a single connectivitydatabase to be used for all sub-time levels.
Under these assumptions very few further modifications need to be made to the harmonicbalance capable version of OVERFLOW 2. The details of how the near-body grid isdeformed for each sub-time level and the post-processing of data is slightly modified,however the concepts are unchanged. Time domain OVERFLOW 2, and therefore theHB version of OVERFLOW 2, stores all solution variables for all grids in a single arrayusing pointers to access the proper block of data for a given grid. As long as the HBadditions to the code follow the same principles, overset grids are handled in the samemanner as a single-block grid.
4 HARMONIC BALANCE OVERSET GRID RESULTS
A goal of this research program is to perform flutter and LCO calculations in the frequency-domain on complex geometries modeled by overset grids. Toward that end, the implemen-tation of overset grids with the frequency-domain HB version of OVERFLOW 2 is firstvalidated using a simple 2-D pitching airfoil case. The overset grid trial will be validatedagainst calculations performed on a standard c-grid.
10
For consistency between the overset and single-block grid trials, the overset grid is createdfrom the c-grid and consists of three grid blocks. The first grid block is the near-bodygrid. Since this grid is simply the inner portion of the original c-grid, it too is a c-gridtopology. The other two grid blocks make up the far-field. One grid consists of the outerlayers of the c-grid while the final grid block describes the far-field aft of the airfoil. Theoverset grids are shown in Fig. 6.
Figure 6: Set of overset grids describing the NACA 0012 airfoil.
The validation trials consist of the NACA 0012 airfoil pitching about the quarter-chordat a reduced frequency based on airfoil chord of 0.5. Pitch amplitudes vary from 0.5◦ to4.0◦. The RANS model employs the Spalart-Allmaras one-equation turbulence model andis run at Mach 0.5. The pentadiagonal Beam and Warming scheme is used to discretizethe pseudo-time derivative terms while the spatial operator is discretized using the cen-tral difference method. The original c-grid consists of 401 nodes in the circumferentialdirection and 75 nodes in the radial direction; the overset version of the grid containsslightly more nodes due to the overlapping regions as shown in Fig. 6. Three harmonicsare retained in the Fourier series.
The major assumption of this implementation of overset grids for use with the harmonicbalance method is that the relative grid motion is small. This assumption allows for thegrid holes and connectivity to remain unchanged between sub-time levels. This assump-tion, and its limitations, are illustrated below. Consider the case with the unsteady pitchamplitude of 2.0◦. It is possible to study the converged Mach contours for each of thesub-time levels of the HB solution method to determine the effects of the small amplitudeassumption. As illustrated in Fig. 1, for the three harmonic case, the final sub-time levellocation corresponds to the largest displacement from the mean location. Since the inter-grid communication is established for the mean position, Fig. 7(a) shows that the Machcontours transition smoothly from the near-body grid to the far-field grid when in thisconfiguration. However, with the near-body grid in the physical location of the final sub-time level there is a discontinuity in the solution when transitioning from the near-bodyto the far-field grid, which is most apparent at the wake. Clearly this discontinuity is notphysical and illustrates the need for the unsteady motion to be sufficiently small. How-ever, it is important to note that this technique is appropriate for many applications such
11
(a) Grid configuration used for connectivity.
(b) Physical grid configuration.
Figure 7: Mach contours of the seventh sub-time level solution for a three-harmonic calculation.
as flutter point and limit cycle oscillation calculations where the motion is substantiallyless than 2.0◦.
Figure 8 displays the zeroth and first harmonic of the unsteady pressure on the airfoil forvarious unsteady pitch amplitudes. The symbols correspond to the single-grid solutionwhile the solid lines correspond to the solution calculated on the overset grid. Clearly theagreement is quite good, even for the large amplitude cases.
Next, consider a transonic case with a free stream Mach number of 0.8. For this case thepseudo-time term is discretized using the SSOR algorithm [14] and the spatial derivativesare discretized using the HLLC method [15]. For a steady flow at this Mach number, theNACA 0012 experiences a shock near the mid-chord. It might be expected that due tothe large gradients in the vicinity of the shock, the single connection method outlinedabove would break down. However, Fig. 9 shows that for unsteady amplitudes of up to1.0◦, the method in fact performs very well.
The agreement of the overset grid results with the single-block trial may be better thanexpected given the discontinuity in the flow field shown in Fig. 7(b). However, it isimportant to recall that the flow solver is unaware of this discontinuity since the grids
12
5
5.5
6
6.5
7
0 0.2 0.4 0.6 0.8 1
Zer
oth
Har
mon
ic o
f U
nste
ady
Pres
sure
, p
0 / q
∞
Airfoil Surface Location, x/c
α1 = 0.5α1 = 1.0α1 = 2.0α1 = 4.0
(a) Zeroth harmonic of unsteady pressure.
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
Rea
l Par
t of
Nor
mal
ized
Uns
tead
y Pr
essu
re,
Re(
p 1)
/ (q ∞
α1)
Airfoil Surface Location, x/c
α1 = 0.5α1 = 1.0α1 = 2.0α1 = 4.0
(b) Real part of the unsteady pressure.
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0 0.2 0.4 0.6 0.8 1
Imag
inar
y Pa
rt o
f N
orm
aliz
ed U
nste
ady
Pres
sure
, I
m(p
1) /
(q∞
α1)
Airfoil Surface Location, x/c
α1 = 0.5α1 = 1.0α1 = 2.0α1 = 4.0
(c) Imaginary part of the unsteady pressure.
Figure 8: Comparison of single- and multi-block grid solutions for a pitching airfoil at M = 0.5. Symbolscorrespond to single-block grid calculations and lines correspond to multi-block grid calcula-tions.
communicate as shown in Fig. 7(a).
5 DISCUSSION
A method has been presented that allows for the use of overset grids with a new nonlinearfrequency-domain harmonic balance solver for the NASA CFD code OVERFLOW 2. Thisoverset grid method currently assumes small amplitude motion, which allows for holecutting and grid connectivity to be performed using Pegasus 5 as a pre-processing step.The computational results validate this technique using a single- and multi-block airfoilgrid simulating a pitching airfoil. The results show very good agreement for sufficientlymodest, but realistic, unsteady amplitudes for both subsonic and transonic flow regimes.
The next phase of this research will modify the domain connectivity function withinOVERFLOW 2 to perform hole cutting and determine grid connectivity for each of thesub-time levels separately. This will remove the sufficiently small amplitude requirementand allow for the study of larger motions.
13
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1
Zer
oth
Har
mon
ic o
f U
nste
ady
Pres
sure
, p
0 / q
∞
Airfoil Surface Location, x/c
α1 = 0.5α1 = 1.0
(a) Zeroth harmonic of unsteady pressure.
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 0.2 0.4 0.6 0.8 1
Rea
l Par
t of
Nor
mal
ized
Uns
tead
y Pr
essu
re,
Re(
p 1)
/ (q ∞
α1)
Airfoil Surface Location, x/c
α1 = 0.5α1 = 1.0
(b) Real part of the unsteady pressure.
-0.4
-0.2
0
0.2
0.4
0 0.2 0.4 0.6 0.8 1
Imag
inar
y Pa
rt o
f N
orm
aliz
ed U
nste
ady
Pres
sure
, I
m(p
1) /
(q∞
α1)
Airfoil Surface Location, x/c
α1 = 0.5α1 = 1.0
(c) Imaginary part of the unsteady pressure.
Figure 9: Comparison of single- and multi-block grid solutions for a pitching airfoil at M = 0.8. Symbolscorrespond to single-block grid calculations and lines correspond to multi-block grid calcula-tions.
6 ACKNOWLEDGMENTS
Chad Custer is funded by a NASA Graduate Student Research Program (GSRP) fellow-ship. He would like to acknowledge the generous support and very helpful advice of hisNASA mentor, Russ Rausch, of the Aeroelasticity Branch at the NASA Langley ResearchCenter.
7 REFERENCES
[1] Hall, K. C., Thomas, J. P., Ekici, K., et al. (2003). Frequency domain techniquesfor complex and nonlinear flows in turbomachinery. In 33rd AIAA Fluid DynamicsConference and Exhibit. Orlando, FL: AIAA Paper 2003-3998.
[2] Thomas, J. P., Dowell, E. H., and Hall, K. C. (2002). A harmonic balance approachfor modeling three-dimensional nonlinear unsteady aerodynamics and aeroelastic-ity. In ASME International Mechanical Engineering Conference. New Orleans, LA:ASME Paper IMECE-2002-32532.
[3] Thomas, J. P., Dowell, E. H., and Hall, K. C. (2002). Modeling viscous tran-sonic limit cycle oscillation behavior using a harmonic balance approach. In 43rd
14
AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and MaterialsConference and Exhibit. Denver, CO: AIAA Paper 2002-1414.
[4] Thomas, J. P., Dowell, E. H., and Hall, K. C. (2003). Modeling limit cycle oscil-lations of an NLR 7301 airfoil aeroelastic configuration including correlation withexperiment. In 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynam-ics, and Materials Conference and Exhibit. Norfolk, VA: AIAA Paper 2003-1429.
[5] Hall, K. C., Thomas, J. P., and Clark, W. S. (2002). Computation of unsteadynonlinear flows in cascades using a harmonic balance technique. AIAA Journal,40(5), 879–886.
[6] Woodgate, M. A. and Badcock, K. J. (2009). Implicit harmonic balance solver fortransonic flow with forced motions. AIAA Journal, 47(4), 893–901.
[7] Thomas, J. P., Custer, C. H., Dowell, E. H., et al. (2009). Unsteady flow computationusing a harmonic balance approach implemented about the OVERFLOW 2 flowsolver. In 19th AIAA Computational Fluid Dynamics Conference. San Antonio, TX:AIAA Paper 2009-4270.
[8] Benek, J. A., Buning, P. G., and Steger, J. L. (1985). A 3-D Chimera grid embeddingtechnique. In 7th Computational Fluid Dynamics Conference. Cincinnati, OH: AIAAPaper 1985-1523, pp. 322–331.
[9] Benek, J. A., Donegan, T. L., and Suhs, N. E. (1987). Extended Chimera gridembedding scheme with application to viscous flows. In 8th Computational FluidDynamics Conference. Honolulu, HI: AIAA Paper 1987-1126, pp. 283–291.
[10] Rogers, S. E., Suhs, N. E., and Dietz, W. E. (2003). PEGASUS 5: An automatedpreprocessor for overset-grid computational fluid dynamics. AIAA Journal, 41(6),1037–1045.
[11] Nichols, R. H. and Buning, P. G. (2008). User’s Manual for OVERFLOW 2.1, 2.1ted.
[12] Bisplinghoff, R. L., Ashley, H., and Halfman, R. L. (1996). Aeroelasticity. Mineola,NY: Dover Publications.
[13] Ashley, H. and Landahl, M. (1985). Aerodynamics of Wings and Bodies. Reading,MA: Dover Publications.
[14] Nichols, R. H., Tramel, R. W., and Buning, P. G. (2006). Solver and turbulencemodel upgrades to OVERFLOW 2 for unsteady and high-speed applications. In 25thApplied Aerodynamics Conference. San Francisco, CA: AIAA Paper 2006-2824.
[15] Toro, E. F., Spruce, M., and Speares, W. (1994). Restoration of the contact surfacein the HLL-Riemann solver. Shock Waves, 4(1), 25–34.
15
