Using computational fluid dynamic/rigidbody dynamic results to generate aerodynamicmodels for projectile flight simulationM Costello1∗ and J Sahu2
1School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA2Weapons and Materials Research Directorate, US Army Research Laboratory, Maryland, USA
The manuscript was received on 29 October 2007 and was accepted after revision for publication on 23 June 2008.
DOI: 10.1243/09544100JAERO304
Abstract: A method to efficiently generate a complete aerodynamic description for projectileflight dynamic modelling is described. At the core of the method is an unsteady, time accuratecomputational fluid dynamic simulation that is tightly coupled to a rigid projectile flight dynamicsimulation. A set of short time snippets of simulated projectile motion at different Mach numbersis computed and employed as baseline data. For each time snippet, aerodynamic forces andmoments and the full rigid body state vector of the projectile are known. With time synchronizedair loads and state vector information, aerodynamic coefficients can be estimated with a simplefitting procedure. By inspecting the condition number of the fitting matrix, it is straightforward toassess the suitability of the time history data to predict a selected set of aerodynamic coefficients.The technique it is exercised on an exemplar fin-stabilized projectile with good results.
Four basic methods to predict aerodynamic forcesand moments on a projectile in atmospheric flight arecommonly used in practice: empirical methods, windtunnel testing, computational fluid dynamic (CFD)simulation, and spark range testing. Empirical meth-ods have been found very useful in conceptual designof projectiles where rapid and inexpensive estimatesof aerodynamic coefficients are needed. These tech-niques aerodynamically describe the projectile with aset of geometric properties (diameter, number of fins,nose type, nose radius, etc.) and catalog aerodynamiccoefficients of many different projectiles as a functionof these features. This data is fit to multi-variable equa-tions to create generic models for aerodynamic coeffi-cients as a function of these basic projectile geometricproperties. The database of aerodynamic coefficientsas a function of projectile features is typically obtainedfrom wind tunnel or spark range tests. This approach
∗Corresponding author: School of Aerospace Engineering, Geor-
gia Institute of Technology, Atlanta, Georgia 30332, USA. email:
to projectile aerodynamic coefficient estimation isused in several software packages including MissileDATCOM, PRODAS, and AP98 [1–6]. The advantage ofthis technique is that it is a general method applica-ble to any projectile. However, it is the least accuratemethod of the four methods mentioned above, par-ticularly for new configurations that fall outside therealm of projectiles used to form the basic aerody-namic database. Wind tunnel testing is often usedduring projectile development programs to convergeon fine details of the aerodynamic design of the shell[7, 8]. In wind tunnel testing, a specific projectile ismounted in a wind tunnel at various angles of attackwith aerodynamic forces and moments measured atvarious Mach numbers using a sting balance. Windtunnel testing has the obvious advantage of beingbased on direct measurement of aerodynamic forcesand moments on the projectile. It is also relatively easyto change the wind tunnel model to allow detailedparametric effects to be investigated. The main dis-advantage to wind tunnel testing is that it requires awind tunnel and as such is modestly expensive. Fur-thermore, dynamic derivatives such as pitch and rolldamping as well as Magnus force and moment coef-ficients are difficult to obtain in a wind tunnel and
require a complex physical wind tunnel model. Overthe past couple of decades, tremendous strides havebeen made in the application of CFD to predictionof aerodynamic loads on air vehicles, including pro-jectiles. These methods are increasingly being usedthroughout the weapon development cycle includ-ing early in a program to create relatively low costestimates of aerodynamic characteristics and laterin a program to supplement and reduce expensiveexperimental testing. In CFD simulation, the fun-damental fluid dynamic equations are numericallysolved for a specific configuration. The most sophis-ticated computer codes are capable of unsteady timeaccurate computations using the Navier–Stokes equa-tions. Examples of these tools include, for example,CFD++, Fluent, and Overflow-D. CFD is computa-tionally expensive, requires powerful computers toobtain results in a reasonably timely manner, andrequires dedicated engineering specialists to drivethese tools [9–24]. Spark range aerodynamic testinghas long been considered the gold standard for pro-jectile aerodynamic coefficient estimation. It is themost accurate method for obtaining aerodynamic dataon a specific projectile configuration. In spark rangeaerodynamic testing, a projectile is fired through anenclosed building. At a discrete number of points dur-ing the flight of the projectile (<30) the state of theprojectile is measured using spark shadowgraphs [25–29]. The projectile state data is subsequently fit toa rigid six-degree-of-freedom projectile model usingthe aerodynamic coefficients as the fitting parame-ters [30]. While this technique is the most accuratemethod for obtaining aerodynamic data on a specificprojectile configuration, it is usually the most expen-sive alternative, requires a spark range facility, andstrictly speaking is only valid for the specific projec-tile configuration tested. More recently, aerodynamicparameters have been estimated using a combinationof radar data and on-board instrumentation [31, 32].
Various researchers have used CFD to estimate aero-dynamic coefficient estimation of projectiles. Earlywork focused on Euler solvers applied to steady flowproblems while more recent work has solved theReynolds-averaged Navier–Stokes equations (RANS)and large eddy simulation Navier–Stokes equationsfor both steady and unsteady conditions [9–24]. Forexample, to predict pitch damping Weinacht pre-scribed projectile motion to mimic a typical pitchdamping wind tunnel test in a CFD simulation to esti-mate the different components of the pitch dampingcoefficient of a fin-stabilized projectile [33]. Excel-lent agreement between computed and measuredpitch damping was attained. Algorithm and comput-ing advances have also led to coupling of CFD codesto projectile rigid body dynamic (RBD) codes for sim-ulation of free flight motion of a projectile in a timeaccurate manner. Aerodynamic forces and moments
are computed with the CFD solver while the freeflight motion of the projectile is computed by inte-grating the RBD equations of motion. Sahu achievedexcellent agreement between spark range measure-ments and a coupled CFD/RBD approach for a finnedstabilized projectile [34]. Projectile position and ori-entation at down range locations consistent with aspark range test were extracted from the output of theCFD/RBD software to compute aerodynamic coeffi-cients. Standard range reduction software was utilizedfor this purpose with good agreement obtained whencontrasted against example spark range results. Theability to accurately compute projectile aerodynam-ics in highly unsteady conditions has led to the notionof “virtual wind tunnels” and “virtual fly outs” wherethe simulation tools above are used to replicate a windtunnel or spark range test.
Computation time for accurate coupled CFD/RBDsimulation remains exceedingly high and does not cur-rently represent a practical method for typical flightdynamic analysis such as impact point statistics (CEP)computation where thousands of fly outs are required.Furthermore, this type of analysis does not allow thesame level of understanding of the inherent underly-ing dynamics of the system that RBD analysis usingaerodynamic coefficients yields. However, the cou-pled CFD/RBD approach does offer an indirect way torapidly compute the aerodynamic coefficients neededfor rigid six-degree-of-freedom simulation. Duringa time accurate CFD/RBD simulation, aerodynamicforces and moments and the full rigid body state vectorof the projectile are generated at each time step in thesimulation [34]. This means that aerodynamic forces,aerodynamic moments, position of the mass center,body orientation, translational velocity, and angularvelocity of the projectile are all known at the sametime instant. With time synchronized air load andstate vector information, the aerodynamic coefficientscan be estimated with a simple fitting procedure.This paper creates a method to efficiently generate acomplete aerodynamic model for a projectile in atmo-spheric flight using four short-time histories at eachMach number of interest with an industry standardtime accurate CFD/RBD simulation. The techniqueis exercised on example CFD/RBD data for a smallfin-stabilized projectile.
2 PROJECTILE CFD/RBD SIMULATION
2.1 Rigid body dynamics
The projectile CFD/RBD algorithm employed herecombines a rigid six-degree-of-freedom projectileflight dynamic model with a three-dimensional,time accurate CFD simulation. The RBD dynamic
Aerodynamic models for projectile flight simulation 1069
Fig. 1 Reference frame and position definitions
Fig. 2 Projectile orientation definitions
equations are integrated forward in time where aero-dynamic forces and moments that drive motion ofthe projectile are computed using the CFD algorithm.The RBD projectile model allows for three translationdegrees of freedom and three rotation degrees of free-dom. As shown in Figs 1 and 2, the I frame is attachedto the ground while the B frame is fixed to the projectilewith the �IB axis pointing out the nose of the projectileand the �JB and �KB unit vectors forming a right handedtriad. The projectile state vector is comprised of theinertial position components of the projectile masscenter (x, y, z), the standard aerospace sequence Eulerangles (φ, θ , ψ), the body frame components of theprojectile mass center velocity (u, v, w), and the bodyframe components of the projectile angular velocityvector (p, q, r).
Both the translational and rotational dynamicequations are expressed in the projectile body refer-ence frame. The standard rigid projectile, body frameequations of motion are given by equations (1) through(4) [35]
⎧⎨⎩
xyz
⎫⎬⎭ =
⎡⎣cθ cψ sφsθ cψ − cφsψ cφsθ cψ + sφsψ
cθ sψ sφsθ sψ + cφcψ cφsθ sψ − sφcψ
−sθ sφcθ cφcθ
⎤⎦
×⎧⎨⎩
uvw
⎫⎬⎭ (1)
⎧⎨⎩
φ
θ
ψ
⎫⎬⎭ =
⎡⎣1 sφtθ cφtθ
0 cφ −sφ
0 sφ/cθ cφ/cθ
⎤⎦
⎧⎨⎩
pqr
⎫⎬⎭ (2)
⎧⎨⎩
uvw
⎫⎬⎭ =
⎧⎨⎩
X/mY /mZ/m
⎫⎬⎭ −
⎡⎣ 0 −r q
r 0 −p−q p 0
⎤⎦
⎧⎨⎩
uvw
⎫⎬⎭ (3)
⎧⎨⎩
pqr
⎫⎬⎭ = [I ]−1
⎡⎣
⎧⎨⎩
LMN
⎫⎬⎭
−⎡⎣ 0 −r q
r 0 −p−q p 0
⎤⎦ [I ]
⎧⎨⎩
pqr
⎫⎬⎭
⎤⎦ (4)
In equations (1) and (2), the shorthand notationsα = sin(α), cα = cos(α), and tα = tan(α). Note thatthe total applied force components (X , Y , Z ) andmoment components (L, M , N ) contain contributionsfrom weight and aerodynamics. The aerodynamic por-tion of the applied loads in equations (3) and (4) iscomputed using the CFD simulation and passed to theRBD simulation.
2.2 CFD solution technique
On the other hand, the CFD flow equations areintegrated forward in time where the motion ofthe projectile that drives flow dynamics are com-puted using the RBD algorithm. The complete setof three-dimensional time-dependent Navier–Stokesequations is solved in a time-accurate manner for sim-ulation of free flight. The commercially available code,CFD++ is used for the time-accurate unsteady CFDsimulations [36, 37]. The basic numerical frameworkin the code contains unified-grid, unified-physics, andunified-computing features. The three-dimensionaltime-dependent RANS equations are solved using thefollowing finite volume equation
where W is the vector of conservative variables, F andG are the inviscid and viscous flux vectors, respectively,H is the vector of source terms, V is the cell volume,and A is the surface area of the cell face. A second-order discretization is used for the flow variablesand the turbulent viscosity equation. The turbulenceclosure is based on topology-parameter-free formu-lations and turbulence modelling thus, becomes acritical element in the calculation of turbulent flowsthat are of interest here. Two-equation higher-orderRANS turbulence models are used for the compu-tation of turbulent flows. These models are ideallysuited to unstructured book-keeping and massivelyparallel processing due to their independence fromconstraints related to the placement of boundariesand/or zonal interfaces. Higher order turbulence mod-els are generally more accurate and are widely used.A widely used turbulence model for practical appli-cations is the two-equation k–ε model [38] shownbelow
d(ρk)
dt= ∇ ·
[(μ + μt
σk
)∇k
]+ Pk − ρε (6)
d(ρε)
dt= ∇ ·
[(μ + μt
σε
)∇ε
]+ (Cε1Pk − Cε2ρε + E) T −1
t (7)
where k is the turbulence kinetic energy, ε is theturbulence dissipation rate, and μt is the turbulenceeddy viscosity which is a function of k and ε; Pk
is a production term, E is a source term, and Tt
is a realizable time scale. Even more sophisticatedturbulence modelling such as the ‘large eddy simula-tion’ (LES) exists but are prohibitively expensive forpractical problems. Recently, hybrid models knownas hybrid RANS/LES [39] have been developed thatcombine the best features of both RANS and LESmethods based on local gird resolution. The RANSpart is still based on the k–ε model and is used forthe most of the flow domain including the turbulentboundary layer region and LES is used in the wake,for example. The hybrid RANS/LES model is gener-ally more suitable for computation of unsteady flowfields especially at transonic and subsonic speeds.For the computation of supersonic flows that are ofinterest in this research, the two-equation k–ε modeldescribed above is adequate. It has been success-fully used with good results on a number of pro-jectile aerodynamic applications [19, 24, 34]. Theseturbulence equations are solved all the way to thewall and generally require fine meshes near the wallsurface.
A dual time-stepping approach is used to integratethe flow equations to achieve the desired time-accuracy. The first is an ‘outer’ or global (and physical)time step that corresponds to the time discretization
of the physical time variation term. This time step canbe chosen directly by the user and is typically set toa value to represent one-hundredth of the period ofoscillation expected or forced in the transient flow. Itis also applied to every cell and is not spatially varying.An artificial or ‘inner’ or ‘local’ time variation term isadded to the basic physical equations. This time stepand corresponding ‘inner-iteration’ strategy is chosento help satisfy the physical transient equations to thedesired degree. For the inner iterations, the time stepis allowed to vary spatially. Also, relaxation with multi-grid (algebraic) acceleration is employed to reduce theresidues of the physical transient equations. It is foundthat an order of magnitude reduction in the residues isusually sufficient to produce a good transient iteration.
2.3 CFD/RBD coupling and initial conditions
The projectile in the coupled CFD/RBD simulationalong with its grid moves and rotates as the pro-jectile flies downrange. Grid velocity is assigned toeach mesh point. This general capability can be tai-lored for many specific situations. For example, thegrid point velocities can be specified to correspondto a spinning projectile. In this case, the grid speedsare assigned as if the grid is attached to the pro-jectile and spinning with it. Similarly, to account forRBDs, the grid point velocities can be set as if thegrid is attached to the rigid body with six degreesof freedom. As shown in Fig. 2, the six degrees offreedom comprises of the inertial position compo-nents of the projectile mass center (x, y, z) and thethree standard Euler angles (φ, θ , ψ), roll angle, pitchangle, and yaw angle, respectively. For the RBDs, thecoupling refers to the interaction between the aero-dynamic forces/moments and the dynamic responseof the projectile/body to these forces and moments.The forces and moments are computed every CFDtime step and transferred to a six-degree-of-freedommodule which computes the body’s response to theforces and moments. The response is converted intotranslational and rotational accelerations that are inte-grated to obtain translational and rotational velocitiesand integrated once more to obtain linear positionand angular orientation. From the dynamic response,the grid point locations and grid point velocities areset.
In order to properly initialize the CFD simula-tion, two modes of operation for the CFD codeare utilized, namely, an uncoupled and a coupledmode. The uncoupled mode is used to initializethe CFD flow solution while the coupled mode rep-resents the final time accurate coupled CFD/RBDsolution. In the uncoupled mode, the RBDs are spec-ified. The uncoupled mode begins with a computa-tion performed in ‘steady state mode’ with the grid
Aerodynamic models for projectile flight simulation 1071
velocities prescribed to account for the proper ini-tial position (x0, y0, z0), orientation (φ0, θ0, ψ0), andtranslational velocity (u0, v0, w0) components of thecomplete set of initial conditions to be prescribed.After the steady state solution is converged, the ini-tial spin rate (p0) is included and a new quasi-steadystate solution is obtained using time-accurate CFD.A sufficient number of time steps are performed sothat the angular orientation for the spin axis cor-responds to the prescribed initial conditions. Thisquasi-steady state flow solution is the starting point forthe time-accurate coupled solution. For the coupledsolution, the mesh is translated back to the desiredinitial position (x0, y0, z0) and the remaining angu-lar velocity initial conditions (q0, r0) are then added.In the coupled mode, the aerodynamic forces andmoments are passed to the RBD simulation whichpropagates the rigid state of the projectile forward intime.
3 FLIGHT DYNAMIC PROJECTILE AERODYNAMICMODEL
The applied loads in equations (3) and (4) con-tain contributions from projectile weight and bodyaerodynamic forces and moments as shown below
⎧⎨⎩
XYZ
⎫⎬⎭ = W
⎧⎨⎩
−sθ
sφcθ
cφcθ
⎫⎬⎭ − π
8ρV 2D2
×
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
CX 0 + CX 2(v2 + w2)/V 2
CNAv/V − pD2V
CYPAw/V
CNAw/V + pD2V
CYPAv/V
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
(8)
⎧⎨⎩
LMN
⎫⎬⎭ = π
8ρV 2D3
×
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
CLDD + pD2V
CLP
CMAwV
+ qD2V
CMQ + pD2V
CNPAvV
−CMAvV
+ rD2V
CMQ + pD2V
CNPAwV
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
(9)
The terms containing CYPA constitute the Magnus airload acting at the Magnus center of pressure while theterms containing CX 0, CX 2, CNA define the steady loadacting at the center of pressure. The externally appliedmoment about the projectile mass center is composedof an unsteady aerodynamic moment along with termsdue to the fact that the center of pressure and center ofMagnus are not located at the mass center. The termsinvolving CMA accounts for the center of pressure beinglocated off the mass center while the terms involving
CNPA accounts for the center of Magnus being locatedoff the mass center. The aerodynamic coefficients areall a function of local Mach number which are typicallyhandled through a table look-up scheme in projec-tile flight simulation codes. The aerodynamic modelpresented in equations (8) and (9) is the standardaerodynamic expansion for symmetric projectiles.
4 AERODYNAMIC COEFFICIENT ESTIMATION
The time accurate coupled CFD/RBD simulationprovides a full flow solution including the aerody-namic portion of the total applied force and moment(X , Y , Z , L, M , N ) along with the full state of the rigidprojectile (x, y, z, φ, θ , ψ , u, v, w, p, q, r) at every timestep in the solution for each time snippet. Given a setof n short time histories (snippets) that each contain mtime points yields a total of h = m ∗ n time history datapoints for use in estimating the aerodynamic coef-ficients: CX 0, CX 2, CNA, CYPA, CLDD, CLP, CMA, CMQ, CNPA.Note that for fin-stabilized projectile configurations,the Magnus force and moment are usually suffi-ciently small so that CYPA and CNPA are set to zero andremoved from the fitting procedure to be describedbelow.
Equations (8) and (9) represent the applied air loadson the projectile expressed in the projectile bodyframe. Computation of the aerodynamic coefficientsis aided by transforming these equations to the instan-taneous aerodynamic angle of attack reference framethat rotates the projectile body frame about the �IB axisby the angle γ = tan−1(w/v).
− 8πρV 2D2
⎡⎣1 0 0
0 cγ sγ
0 −sγ cγ
⎤⎦
⎡⎣
⎛⎝X
YZ
⎞⎠ − W
⎛⎝−sθ
sφcθ
cφcθ
⎞⎠
⎤⎦
=
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
CX 0 + CX 2(v2 + w2)/V 2
CNA
√v2 + w2
V
pD2V
√v2 + w2
VCYPA
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
(10)
8πρV 2D3
⎡⎣1 0 0
0 cγ sγ
0 −sγ cγ
⎤⎦
⎛⎝ L
MN
⎞⎠
=
⎡⎢⎢⎢⎢⎢⎢⎢⎣
CLDD + pD2V
CLP
(vq + wr)D
2√
v2 + w2VCMQ + pD
2V
√v2 + w2
VCNPA
(vr − wq)D
2√
v2 + w2VCMQ −
√v2 + w2
VCMA
⎤⎥⎥⎥⎥⎥⎥⎥⎦
(11)
Each time history data point provides a total of sixequations given by the components of equations (10)and (11). The first component of equation (10) is
gathered together for all time history data points toform equation (12). Likewise, the second and thirdcomponents of equation (10) generate equations (13)and (14), respectively, while the first component ofequation (11) constructs equation (15). Finally, thesecond and third components of equation (11) aregathered together to form equation (16). Subscripts onthe projectile state vector and aerodynamic force andmoment components represent the time history datapoint.
⎡⎢⎢⎣
1 (v21 + w2
1)/V 21
......
1 (v2h + w2
h)/V 2h
⎤⎥⎥⎦
(CX 0
CX 2
)
=
⎡⎢⎢⎢⎢⎢⎣
− 8πρV 2
1 D2(X1 + W sin θ1)
...
− 8πρV 2
h D2(Xh + W sin θh)
⎤⎥⎥⎥⎥⎥⎦ (12)
⎡⎢⎢⎢⎣
√v2
1 + w21/V1
...√v2
h + w2h/Vh
⎤⎥⎥⎥⎦ (CNA)
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
− 8πρV 2
1 D2(Y1 cos γ1 + Z1 sin γ1
−W sin φ1 cos θ1)
...
− 8πρV 2
h D2(Yh cos γh + Zh sin γh
−W sin φh cos θh)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(13)
⎡⎢⎢⎢⎢⎢⎢⎢⎣
p1D√
v21 + w2
1
2V 21
...
phD√
v2h + w2
h
2V 2h
⎤⎥⎥⎥⎥⎥⎥⎥⎦
(CYPA)
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
− 8πρV 2
1 D2(−Y1 sin γ1 + Z1 cos γ1
−W cos φ1 cos θ1)
...
− 8πρV 2
h D2(−Yh sin γh + Zh cos γh
−W cos φh cos θh)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(14)
⎡⎢⎢⎢⎢⎢⎣
1p1D2V1
......
1phD2Vh
⎤⎥⎥⎥⎥⎥⎦
(CLDD
CLP
)=
⎛⎜⎜⎜⎜⎜⎝
8L1
πρV 21 D3
...8Lh
πρV 2h D3
⎞⎟⎟⎟⎟⎟⎠ (15)
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
0(v1q1 + w1r1)D
2V1
√v2
1 + w21
p1D√
v21 + w2
1
2V 21
−√
v21 + w2
1
V1
(v1r1 − w1q1)D
2V1
√v2
1 + w21
0
......
...
0(vhqh + whrh)D
2Vh
√v2
h + w2h
phD√
v2h + w2
h
2V 2h
−√
v2h + w2
h
Vh
(vhrh − whqh)D
2Vh
√v2
h + w2h
0
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
×⎛⎝CMA
CMQ
CNPA
⎞⎠
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
8πρV 2
1 D3(M1 cos γ1 + N1 sin γ1)
8πρV 2
1 D3(−M1 sin γ1 + N1 cos γ1)
...8
πρV 2h D3
(Mh cos γh + Nh sin γh)
8πρV 2
h D3(−Mh sin γh + Nh cos γh)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
(16)
Equations (12) to (16) represent a set of five uncou-pled problems to solve for the different aerodynamiccoefficients. To estimate the aerodynamic coefficientsnear a particular Mach number, a set of n time accu-rate coupled CFD/RBD simulations are created over arelatively short time period. Since an individual timesnippet is over a short time period where the projectilestate variables do not change appreciably, it is criticalthat initial conditions for the different time snippetbe selected in an informed way so that the rank ofeach of the fitting matrices above is maximal. Proper-ties of the fitting matrices above, such as the rank orcondition number, can be used as an indicator of thesuitability of the CFD/RBD simulation data to estimatethe aerodynamic coefficients at the target Mach num-ber. Equation (12) is employed to estimate the zeroyaw drag coefficient (CX 0) and the yaw drag coefficient(CX 2). To minimize the condition number of this fit-ting matrix, both low and high aerodynamic angle ofattack time snippets are required. Equation (13) is usedto compute the normal force coefficient (CNA) and itrequires time history data with a nonzero aerodynamicangle of attack. Equation (14) is used to compute theMagnus force coefficient (CYPA) and it requires timehistory data with both low and high roll rate and aero-dynamic angle of attack. Equation (15) is employed toestimate the fin cant roll coefficient (CLDD) along withthe roll damping coefficient (CLP). To minimize thecondition number of this fitting matrix, both low and
high roll rate time snippets are required. Equation (16)is employed to estimate the pitching moment coeffi-cient (CMA), the pitch damping coefficient (CMQ), andthe Magnus moment coefficient (CNPA). For successfulestimation of these coefficients, time history data withboth low and high roll rate and aerodynamic angleof attack as well as low and high aerodynamic angleof attack are required. To meet the requirements forsuccessful estimation of all five sets of aerodynamiccoefficients, four time snippets are used all with differ-ent initial conditions. Table 1 lists the four cases withlaunch conditions. Notice that the set of time snippetscontain a diverse set of initial conditions: zero aero-dynamic angle of attack and angular rates; high angleof attack and zero angular rates; low angle of attack,high roll rate with other angular rates zero; zero angleof attack, high pitch rate with other angular rates zero.
For flight dynamic simulation, aerodynamic coef-ficients are required at a set of Mach numbers thatcovers the intended spectrum of flight conditionsfor the round. If aerodynamic coefficients are esti-mated at k different Mach numbers then a total ofl = k × n CFD/RBD time snippets must be generatedto construct the entire aerodynamic database for flightsimulation purposes.
5 RESULTS
In order to exercise the method developed above, ageneric finned projectile is considered. A sketch ofthe projectile is shown in Fig. 3. The projectile hasthe following geometric and mass properties: length =0.1259 m, reference diameter = 0.013194 m, mass =0.0484 kg, mass center location from base = 0.0686 m,roll inertia = 0.74e − 06 kg m−2, pitch inertia = 0.484e−04 kg m−2.
As part of a validation of the coupled Navier-Stokesand six-degree-of-freedom method, time-accurateunsteady numerical computations were performed topredict the flow field, aerodynamic coefficients, and
Fig. 3 Generic finned projectile
Fig. 4 Unstructured mesh near the finned body
the flight paths of this fin-stabilized projectile at an ini-tial supersonic speed, M = 3. Full three-dimensionalcomputations were performed and no symmetry wasused.
An unstructured computational mesh was gener-ated for the generic finned projectile (Fig. 4). Ingeneral, most of the grid points are clustered in theboundary-layer as well as near the afterbody fin andthe wake regions. Three different grids were usedand the total number of grid points varied from 2to 6 million points for the full grid. For the largermeshes, additional grid points were clustered in theboundary-layer as well as near the afterbody fin andthe wake regions. The first spacing away from thewall was selected to yield a y+ value of 1.0 in eachcase. The projectile configuration has a base cavityand was included in the mesh generation process.The unstructured mesh also included the base cav-ity region that was present in the actual model testedand was generated using the multi-purpose intelli-gent meshing environment grid-generation softwarerecently developed by Metacomp Technologies.
Here, the primary interest is in the validation ofcoupled CFD/RBD techniques for accurate simula-tion of free flight aerodynamics and flight dynamicsof a projectile in supersonic flight. Numerical com-putations were made for the generic finned projectileconfiguration at an initial velocity of 1032 m/s. The ini-tial angle of attack was, α = 4.9◦ and initial spin ratewas 2500 rad/s. Figure 5 shows the computed pres-sure contours at a given time or at a given locationin the trajectory. It clearly shows the orientation of thebody at that instant in time and the resulting asym-metric flow field due to the body at angle of attack.
The orientation of the projectile of course changesfrom one instant in time to another as the projectileflies down range. Figure 6 shows the variation of theEuler pitch angle with distance traveled. As seen in thisfigure, both the amplitude and frequency in the Eulerangle variation are predicted very well by the com-puted results and match extremely well with the datafrom the flight tests. One can also clearly see thatthe amplitude damps out as the projectile flies downrange, i.e. with the increasing x-distance. Althoughnot shown here, similar behavior is observed with theEuler yaw angle and it damps out with the increasingx-distance. Computed results again compare very wellwith measured data from flight tests. As stated earlier,different computational meshes were used to obtainthe numerical results. Grid sizes varied from 2 to 6million total number of points. The effect of the gridsizes on the computed Euler pitch angle is also shownin Fig. 6. The computed results are grid-independent;
the computed pitch angles obtained with 4 and 6million mesh are essentially the same as those resultsobtained with the 2 million point mesh. In all sub-sequent simulations, the 4 million grid point meshhas been used. Additional validation results showingother state variables and more details can be found inreference [34].
Figures 7 to 12 present projectile state trajectoriesfor each of the four time snippets. Each time snippetis 0.023 s and contains 50 points, leading to an averageoutput time step of 0.0004. The initial conditions foreach of the time snippets is shown in Table 1. Thesefour snippets create time history data at low and highangle of attack, roll rate, and pitch rate needed foraccurate aerodynamic coefficient estimation. Noticethat cases 2 and 3 have notably more drag down dueto the high angle of attack launch conditions. Case 3 islaunched with relatively high roll rate compared to all
Fig. 7 Velocity for the time snippets.
Fig. 6 Effect of mesh size on the Euler pitch angle
Aerodynamic models for projectile flight simulation 1075
Fig. 8 Aerodynamic angle of attack for the timesnippets
Fig. 9 Roll rate for the time snippets
Fig. 10 Pitch rate for the time snippets
Fig. 11 Euler pitch angle for the time snippets
Fig. 12 Euler yaw angle for the time snippets
other cases. Case 2 generates roll rate toward the endof the time snippet due to high angle of attack roll-pitch coupling. Significant oscillations in Euler pitchangle are created in case 2 with some cross couplingresponse exhibited in Euler yaw angle. Figures 13 to16 plot aerodynamic forces and moments in the localangle of attack reference frame defined above for cases1, 3, and 4 since these cases are the primary cases usedto estimate the coefficients. For all cases, the axial forceoscillates from −20 to −25 N. There exists a slight biasbetween the CFD/RBD and estimated data of about0.5 N for low angle of attack time snippets. For mod-erately high angles of attack (Case 3), the estimateddata also oscillates with a much higher amplitude thanthe CFD/RBD data indicating that CX 2 is estimatedlarger than the CFD/RBD suggests. The normal forcetime snippets agree well between the CFD/RBD andestimated data for all time snippets. For the exam-ple finned projectile, side force (FZ ) and out-of-planemoment (MY ) are generally small (<0.5 N, 0.05 Nm)due to a negligibly small Magnus force and moment.
Fig. 13 Estimated (dashed) and CFD/RBD (solid) bodyaxis axial force (Fx) versus time
Fig. 14 Estimated (dashed) and CFD/RBD (solid) nor-mal force (Fy ) versus time
The CFD/RBD and estimated data agree reasonablywell, but certainly do not overlay one another. Theonly time snippet that creates notable rolling momentis case 3 which is launched with an initial roll rateof 377 rad/s. Notice that the estimated data smoothlygoes through the CFD/RBD data which oscillates ina slightly erratic manner. The in-plane moment (Mz)agrees reasonably well for both the CFD/RBD and esti-mated data. The results shown in Figs 4 to 15 are typicalfor all Mach numbers. The overall observation fromthe data is that the estimated aerodynamic model fitsthe CFD/RBD data well, with the notable exception ofaxial force where a bias is exhibited.
The example projectile investigated in this paperhas been fired in a spark range at Mach 3.0 withaerodynamic coefficients computed via conventionalaerodynamic range reduction. Table 2 presents acomparison of aerodynamic coefficients obtainedfrom spark range testing and subsequent coefficients
Fig. 15 Estimated (dashed) and CFD/RBD (solid) bodyaxis rolling moment (Mx) versus time
Fig. 16 Estimated (dashed) and CFD/RBD (solid) yaw-ing moment versus time
obtained using the method described here. Noticethat most aerodynamic coefficients such as CX0, CNA,and CMA are in reasonably good agreement with sparkrange reduced data. Axial force yaw drag and rolldamping are both different by around 20 per cent whilepitch damping is different by around 40 per cent. Withthe exception of CMQ, aerodynamic coefficients arenearly estimated to within the accuracy that can beexpected from a spark range test firing between twosets of firings. The relatively larger errors in CMQ aremore than likely due to the set of initial conditions thatcreate a large condition number for the fitting matrix.
CFD/RBD data was generated at six different Machnumbers ranging from 1.5 to 4.0. The estimationalgorithm discussed above was used to compute acomplete set of aerodynamic coefficients across itsMach range. These results are provided in Table 3.With the exception of CX2, the steady aerodynamiccoefficients are smooth and follow typical trends for
variation in Mach number. The yaw drag coefficient,CX 2, however, is somewhat erratic with a low valueof 0.21 at Mach 1.5 followed by a steady rise untilMach 3.5. Pitch damping decreases with Mach num-ber as would be expected for a fin-stabilized projectilebeyond Mach 1.0. Roll damping steadily increasesuntil Mach 4.0 when in drops off notably.
6 CONCLUSIONS
Using a time-accurate CFD simulation that is tightlycoupled to a RBDs simulation, a method to efficientlygenerate a complete aerodynamic description for pro-jectile flight dynamic modelling is described. A set of nshort-time snippets of simulated projectile motion atm different Mach numbers is computed and employedas baseline data. The combined CFD/RBD analysiscomputes time synchronized air loads and projectilestate vector information, leading to a straightforwardfitting procedure to obtain the aerodynamic coeffi-cients. The estimation procedure decouples into fivesub problems that are each solved via linear leastsquares. The method has been applied to an examplesupersonic finned projectile. A comparison of sparkrange obtained aerodynamic coefficients with the esti-mation method presented here at Mach 3 exhibitsagreement within 10 per cent for CX0, CNA, and CMA;agreement within 20 per cent for CX2 and CLP; andagreement within 40 per cent for CMQ. This techniquereported here provides a promising new means forthe CFD analyst to predict aerodynamic coefficientsfor flight dynamic simulation purposes. It can easilybe extended to flight dynamic modelling of differ-ent control effectors provided accurate CFD/RBD timesimulation is possible and an aerodynamic coefficient
expansion is defined which includes the effect of thecontrol mechanism.
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APPENDIX
Notation
CLDD fin cant aerodynamic coefficientCLP roll damping aerodynamic coefficientCMQ pitch damping moment aerodynamic
coefficientCNA normal force due to angle of attack
aerodynamic coefficientCX 0 zero yaw drag aerodynamic
coefficientCX 2 yaw drag aerodynamic coefficientCYPA Magnus force aerodynamic
