Modelling of Dynamic Stability Derivatives Using Cfd

25th INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES

MODELLING OF DYNAMIC STABILITY DERIVATIVES USING CFD

Sean Tuling CSIR

Keywords: Navier-Stokes, CFD, Dynamic Derivatives

Abstract List of Symbols

c Damping force, Nm.(deg/s)−1 An exploratory investigation into the use of a CA Store axial force coefficient commercial Navier-Stokes time accurate code, Cm Moment coefficient about the store Fluent, for simulating and predicting the com- y body axis bined pitch damping derivatives, Cmq +Cmα˙ , was Cmq Moment coefficient about the store conducted. body pitch axis due to pitch veloc- The stable store, a low l/d finned missile like ∂Cm −1 ity, ∂ ql , rad configuration, was used for the investigation. The 2V C Moment coefficient about the store derivatives were extracted from the response of mα˙ the store, which was simulated by providing a ini- body pitch axis due to time rate ∂Cm change of angle of attack, ∂ α˙ l , tial disturbance and allowing the store to oscillate 2V freely at the system natural frequency. The same rad−1 setup, as used for the numerical simulations, was CN Store normal force coefficient used for the experimental simulations. F Force Static steady state runs were initially per- I Mass moment of inertia, kg.m2 formed, and compared favourably to experimen- Ksupport Support or flexure stiffness/spirng tal results providing confidence in the mesh and constant, Nm.deg−1 solver parameters. l Reference length, m The results of the numerical simulations did M Mach number not compare favourably with the experimental re- q Angular time rate of change about sults. The numerical investigations (both inviscid the store y body axis, rad.s−1 and viscous) overpredicted the derivatives com- ωl rf Reduced frequency, 2V pared to the experimental results by a factor of S Reference Area, m2 three. The time taken for the simulations was V Free-stream velocity, m.s−1 considered reasonable. α Angle of attack, degrees The two main factors that need to be inves- ρ Freestream air density, kg.m−3 tigated, which may be causing the poor correla- θ Angle about the store body pitch tion, are the time step size and the experimental axis apparatus sting effects. Nomenclature

CFD Computational Fluid Dynamics LSWT Low Speed Wind Tunnel MSWT Medium Speed Wind Tunnel

1 SEAN TULING

1 Introduction ing methodology to predict the pitch damping derivatives for a ballistic shell in the supersonic In the process of designing and engineering a flight regime. This method reduces the unsteady, flight vehicle to meet stated specifications and moving geometry problem to a static steady state objectives, the prediction of aerodynamics loads case, though the frame of reference needs to be is required for structural stressing, and the predic- transformed. tion of performance and stability and handling. Murman [5] has used the reduced frequency Of the many sub-disciplines in aeronautical engi- technique to predict the dynamic stability deriva- neering, flight load load prediction is one of the tives for the standard dynamics model for the most difficult, yet one of the primary inputs to sub- and transonic Mach numbers, the basic achieving efficient and safe flight. finner missile in the supersonic regime, and mod- Non-experimental prediction methodologies ified finner missile configuration for transonic have matured over the last 100 years of power and supersonic flight regimes. The reduced fre- manned flight, to the extent where time in- quency method represents the response of a vehi- dependent or steady state predictions compare cle with a small predictable number of frequency favourably with experimental predictions. The components. time dependent or dynamics stability derivative Green et al [1] attempted to use a time depen- predictions are, however, inadequate. Probably dent panel method code to predict the dynamic the best available prediction techniques available stability derivatives for the F16XL configuration. are semi-empirical. Characterisation of dynam- The code, however, failed to capture the basic and ics stability derivatives are, however, still ob- essential features of the experimentally measured tained experimentally, even though the experi- dynamic data for the F16XL configuration. mental techniques provide limited information. One of the unique features of the exploratory Recently, attempts using lower order computa- investigation was to perform the simulations us- tional techniques have yielded limited success ing commercially available codes such as CFD- [1], pointing towards the complex flow mecha- RC Fastran and Fluent. The simulations are thus nisms involved in time dependent responses for time dependent (in contrast to the above men- flight vehicles. It would thus be appropriate to tioned methods). attempt to use higher order techniques to develop The code chosen was Fluent. Fluent has a prediction capabilities. time accurate moving body capability, requires The modelling and simulation of dynamic an unstructured tetrahedral mesh, while the mo- stability derivatives using the Navier-Stokes tion is simulated using a user-defined function. computational fluid dynamics (CFD) approach is This paper details the attempt in simulating a current research topic. This is a natural ex- the pitch dynamic stability derivatives Cmq +Cmα˙ tension of the maturation of Navier-Stokes CFD with a Navier-Stokes CFD code, Fluent, using the as a prediction technique, firstly in in modelling time dependent free oscillation technique, of the steady state or time averaged/independent cases stable store configuration at low subsonic condi- and then unsteady or time dependent cases. tions, and comparing these to experimentally ob- To this end, an exploratory investigation of tained results. The question of whether the dy- using CFD to model dynamic stability derivatives namic stability derivatives can reasonably (with- was initiated, with a view to developing a predic- out significant computational effort) be simulated tion capability. The exploratory investigation is using the time dependent free oscillation tech- the topic of this paper. The results of the investi- nique using Fluent was being investigated. Suc- gation are compared to experimental data. cessful prediction can provide a useful prediction A number of previous attempts at predict- tool for the conceptual or initial design phase. ing dynamic stability derivatives have been attempted. Weinacht [2],[3],[4] used the lunar con-

2 MODELLING OF DYNAMIC STABILITY DERIVATIVES USING CFD

2 Dynamic Stability Derivatives ond system from which the derivatives of interest are extracted. Varying the frequency to char- The aerodynamic loads on a flight vehicle are tra- acterize the derivative as a function of reduced ditionally expressed as a first order Taylor series frequency is, however, more difficult, requiring of derivatives [6]. For example, in coefficient changes in the system stiffness or model inertia. form, the pitching moment, Cm can be expressed For most free oscillation tests, the support system as: has a stiffness (in the form of a flexure) which can be used to control the natural frequency, in combination with the model inertia, to the desired ∂C ∂C ∂C ∂F C = m α + m β + ... + m q + α˙ + ... value. m ∂α ∂β ∂q ∂α˙ (1) The forced oscillation technique forces the The derivatives of interest are the time rate model to oscillate at the required frequency, from which derivatives can be extracted from the time dependent derivatives, namely Cmq and Cmα˙ and so forth. response of the system and model. The experi- The two longitudinal dynamic stability mental apparatus is, however, an order of magnitude more complex than for a free oscillation test, derivatives that are of greatest interest are Cmq and C . Experimental apparatus require com- since the apparatus needs to control the model mα˙ oscillation amplitude and frequency, whereas the plex mechanisms to measure Cmq alone. The most common technique employed is to measure free oscillation technique only requires a trigger mechanism. the combined derivative Cmq +Cmα˙ by simply os- cillating the model about the centre of gravity i.e. 2.2 Governing Equations a single rotational degree of freedom. The derivative Cmα˙ can be obtained from a second test (util- The equation of motion describing the single de- ising different apparatus) by heaving the model gree of freedom is expressed as: up and down. The derivative Cmq is then simply the difference between the combined deriva- Iθ¨ + cθ˙ + kθ = F (2) tive and Cmα˙ . For the purposes of the exploratory investigation only the combined derivative was For the method used is this investigation, F = simulated. 0 i.e. free oscillation, and c ≡ Cmq +Cmα˙ . I is the moment of inertia of the store and K is the spring 2.1 Experimental Prediction of Dynamic constant of the system. Stability Derivatives The spring constant of the system is algebraic sum of the support stiffness and pitching moment

Since the most accurate simulation method of ob- due to angle of attack variation, Cmα i.e. K ≡ taining dynamic stability derivatives has been ex- Ksupport +Cmα . perimental, a large number of techniques have The system response, or solution to the equa- been developed. This has further been exacer- tion of motion 2 can be written as: bated by the limited capabilities of experimental c t techniques, simply due to mechanical and mea- θ = θ0e 2I cos(ωdt + φ) (3) surement constraints. The most common tech- θ φ niques used the free and forced oscillation meth- where 0 and are arbitrary constants, and ω ods [7]. d is the damped natural frequency. The free oscillation technique essentially pro- The combined damping derivative Cmq +Cmα˙ vides an initial disturbance to the model in the is thus obtained by non-dimensionalised the 1 ρV 2Sl2 2 1 ρ 2 plane or about the axis of interest and allows the damping force c by 2V , where 2 V is the model to respond without interference. The re- freestream dynamic pressure and S is the refer- sponse of the model is easily modelled as a sec- ence area.

3 SEAN TULING

Dynamic stability derivatives have also been shown to be dependent on frequency, this rep- resented by the non-dimensional reduced frequency parameter, rf. The reduced frequency is deﬁned as:

ωl rf = (4) 2V where ω is the frequency of oscillation, l is the reference length and V is the freestream velocity. Thus simulations would be required to show the dependence of the derivatives with reduced frequency. The amplitude of the oscillations conformed to the small amplitude criteria i.e. less than 3 degrees.

3 Research Programme

To minimise the variations between the numer- Fig. 1 Stable Store configuration ical and experimental simulations, the numerical simulations were performed under similar freestream conditions as for the experimental simulations. Furthermore, the same model iner- 4 CFD Modelling tias, support system stiffness and similar oscil- lating amplitudes were used, thus also removing 4.1 Modelling Process any scaling inaccuracies. The experimental method chosen for this in- A two step modelling process was followed vestigation is the free oscillation technique with whereby the mesh was first validated for the a support system stiffness in the form of a flexure steady state case and then the same mesh was that is used to also measure the model response. used for the transient or moving mesh simula- The configuration chosen to perform the nu- tions. The effects of viscosity were also inves- merical and experimental simulations was the tigated by comparing results for the dynamic sta- stable store. This is a simple body finned mis- bility derivative simulations between the inviscid sile type configuration, has a low length to diam- case and a viscous case using a turbulence model. κ ω eter ratio and is illustrated in Figure 3. The cen- The turbulence model used was the standard - tre of gravity is mid centre of the length of the as implemented in Fluent. store (or 215.9mm from the store nose). The fins No mesh size dependency was considered for configuration used was in the cross and not the the unsteady runs. The mesh that was used was, plus configuration. All dynamic derivative sim- however, determined to be mesh independent in ulations were performed around the store steady steady state by running cases with more refined state angle of attack of 0◦. The initial pertubation meshes without any appreciable changes in the angle was 0.3◦. The store was oscillated about results. Additionally, no grid adaption was ap- the centre of gravity, this being the mid centre of plied to the mesh for the steady state cases. the length of the store.

4 MODELLING OF DYNAMIC STABILITY DERIVATIVES USING CFD

4.2 Mesh Generation

The mesh, being unstructured and using tetrahedral cells, was automatically generated in Gambit using the following parameters

• Fin edge curvature maximum angle of 20◦

• Body and in face cell size of 4mm

• Growth rate of 1.2

Y X This resulted in a mesh of approximately Z 610000 cells for the inviscid simulations. The mesh domain is illustrated in ﬁgure 4.2, while ﬁg- Fig. 3 Mesh around the store ure 4.2 shows the mesh around the store.

+ Table 1 Turbulence Parameters

Parameter Value

+ Turbulence Intensity 0.1% Turbulence Length Scale 0.0005m

++ Gz++ Gx 4.3 Steady State Validation

The mesh was validated in steady state against experimental data for both the inviscid and vis- Y Z X cous cases. The coefﬁcients of interest are the normal force, CN, and the pitching moment, Cm. Fig. 2 Fluid Domain Correlation of the axial force, CA, was not considered. The experimental data were obtained, in the A boundary layer was attached for the vis- Medium Speed Wind Tunnel (MSWT) of the cous runs using the following parameters CSIR, under the following conditions: • Mach number, M=0.4 • Cell size of 0.4mm

• Total pressure, PT = 150kPa • Growth rate 1.1 • Total temperature, TT = 310K The resulting mesh was approximately 1130000 cells. Because the flow physics over the model is The turbulence parameters used for the simu- dominated by potential flow, little differences lations are listed in the table 1. were expected between the inviscid solution and The far field boundary conditions used were those using the turbulence model. pressure-far-field for the inlet and pressure-outlet Figures 4.3 and 4.3 show the comparison of for the outlet. the viscous and inviscid solutions against the ex-

5 SEAN TULING perimental data. As expected, the viscous and in- 4.4 Dynamic Stability Derivative Simula- viscid solutions are similar. tions For the viscous runs the y+ values varied between 50 and 250. The dynamic derivative simulations were performed as for the experimental simulations.

Normal Force Comparison The moving mesh parameters used for the 3 simulation were as follows:

2 • Segregated solver

1 • Time step of 5 × 10−5 seconds (or 0.00005s)

N 0 C • PISO pressure-velocity coupling −1 Experimental CFD Inviscid CFD Viscous The simulations were performed for at least −2 one and a half cycles, from which Cmq +Cmα˙ were extracted by curve ﬁtting the response to the so- −3 −15 −10 −5 0 5 10 15 20 Angle of Attack [deg] lution of the equations of motion (as described in 3). Fig. 4 Steady state case comparison of normal force coefﬁcient, CN, for inviscid, viscous and ex- 4.5 Time Step Sizing perimental data The determination of time step size was dictated by the compromise between capturing time dependent phenomena, time step independent results and run time. The time step size determined Pitching Moment Comparison 4 for the simulations was 0.00005 seconds.

3 This time step size is such that time dependent phenomena varying at the same frequency 2 Experimental CFD Inviscid as the natural frequency would propagate through 1 CFD Viscous no more than four average sized tetrahedral cells 0 between each time step. The average cell size

m −1 C was for cells at the surface of the store for the

−2 inviscid simulations and for cells adjacent the

−3 boundary layer for the viscous simulations. Flow phenomena at the fin leading edges would thus −4 propagate through more than four cells between −5 each time step because of the cell refinement. −6 −15 −10 −5 0 5 10 15 20 The dependence of the results on time step Angle of Attack [deg] was also assessed by performing simulations for Fig. 5 Steady state case comparison of pitching varying time steps at M=0.1 for the inviscid case. This was not performed for the viscous case. The moment coefficient, Cm, for inviscid, viscous and experimental data derivative for three time steps are shown in table 2. From the table above, no significant change in results were expected for a time step below 0.00005 seconds.

6 MODELLING OF DYNAMIC STABILITY DERIVATIVES USING CFD

−3 x 10 Table 2 Time Step Size Comparison 6

4 Time [sec] Cmq +Cmα˙ 0.001 -321.9 0.0001 -290.6 2 [rad] 0.00005 -282.6 θ 0 Store Pitch Angle −2 4.6 Experimental Programme

−4 The experimental simulations were performed in the Low Speed Wind Tunnel (LSWT) of the −6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 CSIR. The freestream conditions for the dynamic Time [sec] derivative simulations are low subsonic, where the Mach number varies from about M=0.05 to Fig. 6 Typical store response M=0.13. The static pressure was at atmospheric or about 87kPa, while the static temperature was 0 Experimental approximately 300K. Actual tunnel conditions CFD Invscid CFD Viscous were, however, used for the data processing and −50 reduction. −100 At least 10 simulations were obtained for each tunnel and store conﬁguration to ascertain −150 α m

the uncertainty in the measured derivative. The +C −200 q m ∆ C maximum uncertainty was estimated at (Cmq + −250 Cmα˙ ) = 10.

−300 4.7 Observations and Results −350 A typical time response for the oscillations is −400 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 shown in figure 4.7. Reduced Frequency The results for the dynamic derivative simulations are shown in figure 4.7. The figure Fig. 7 Comparison of the numerical and experi- shows the comparison of the inviscid, viscous mental pitch damping derivative,Cmq +Cmα˙ and experimental pitch dynamic stability derivatives, Cmq +Cmα˙ , as a function of the reduced frequency. ter, where each simulation was executed across 4 The numerical simulations show no discern- nodes. able dependence on the reduced frequency, rf. Additionally the effect of viscosity also seems 5 Discussion to be negligible from the results. Of particular concern, however, is that the numerical simula- 5.1 Steady State tions overpredict the damping, by approximately 3 times. The static steady state numerical solutions corre- The calendar time taken to perform a numer- lated well with the experimental data. This pro- ical simulation for 2 cycles was approximately 3 vided confidence in the mesh being able to cap- days for the inviscid runs, and 6 days for the vis- ture the dominant flow characteristics of the con- cous runs. The runs were performed on a clus- figuration. The inviscid solution, as expected,

7 SEAN TULING overpredicted the normal force more than the vis- step reduction. This will, however, increase the cous solution. time for the simulations by two orders of magnitude. 5.2 Dynamic Simulations 5.2.2 Mesh Dependency From figure 4.7 it is quite clear that the numerical simulations are overpredicting the damping. No mesh dependency studies were performed for No obvious cause for this has been identified. the dynamic simulations. Even though the static The same overprediction is, however, observed steady simulations were acceptable for the mesh by Murman [5] in their comparison of the basic implemented, its effect on the dynamic simula- finner configuration with wind tunnel data (over- tion remains unknown. It is recommended that prediction by a factor of 2), where they attributed this aspect also be investigated. their overprediction to sting effects. No sting effects were measured or considered for the ex- 6 Conclusions and Recommendations perimental simulations, and thus the experimental simulations could be underpredicting the dy- An exploratory investigation into the use of a namic stability derivatives. commercial Navier-Stokes time accurate code, The dependence of the dynamic derivatives Fluent, for simulating and predicting the com- on the reduced frequency was not negligible bined pitch damping derivatives, C +C , was or insignificant in the experimental simulations. mq mα˙ conducted. The study simulated the response of The numerical results seem to show the same the stable store using the free oscillation tech- trend, though the dependency is of secondary nique both numerically and experimentally. Ad- concern especially considering the magnitude of ditionally for the numerical and experimental the overprediction. simulations, the store size was identical and the freestream flight conditions similar. The re- 5.2.1 Time Step Size and Simulation Time sults showed an overprediction of the numerical The time step size was calculated such that flow derivatives by a factor of three compared to the phenomena of the same frequency as the system experimental results. natural frequency would not propagate through It can be concluded that the numerical sim- the mesh faster than three or four average cells at ulations do not compare favourably with the ex- a time for each time step. perimental simulations. The use of Fluent and the The time taken per simulation is two orders free-oscillation test technique does not yield suf- of magnitude higher than for static steady state ficiently good results, for the stable store config- simulations. The cost is, thus not negligible or uration with the applied simulation parameters, trivial and was considered when performing the to be used for even the initial design process. simulations. Increased simulation time (beyond that which is The time step size could be higher than that reasonable) may be required to capture the time required to capture the time dependent effects, dependent effects for the simulations to be us- and is definitely not capturing any effects whose able. frequencies are higher than that of the system nat- A number of aspects need to be investigated ural frequency. Additionally phenomena around to improve the accuracy of the numerical simula- the fin leading edges would be less well captured tions and correlation between the numerical and that in other areas because of the cell refinement experimental . These are recommended below. in that area. A reduction of the time step to at least 1 × 6.1 Recommendations 10−6 seconds is thus recommended and the simulations rerun to ascertain the effects of the time The following are recommended:

8 MODELLING OF DYNAMIC STABILITY DERIVATIVES USING CFD

• Investigate the effect of sting interference on the experimental investigations

• Reduce the time step to at least 1 × 10−6 seconds and rerun the simulations

• Determine the effect of mesh size on the dynamic stability derivative predictions

References

[1] Green L.L, Spence A.M. and Murphy P.C. Com- putational Methods for Dynamic Stability and Control Derivatives. 42nd AIAA Aerospace Sci- ences Meeting, 2004. AIAA 2004-0015. [2] Weinacht P, Sturek W.B. and Schiff L.B. Navier- Stokes Predictions of Pitch Damping for Axisym- metric Projectiles. Journal of Spacecraft and Rockets, Vol. 34, No. 6, 1997. [3] Weinacht P. Navier-Stokes Predictions of the In- dividual Components of the Pitch-Damping Sum. Journal of Spacecraft and Rockets, Vol.35, No. 5, 1998. [4] Weinacht P. Projectile Performance, Stability, and Free-Flight Motion Prediction Using Com- putational Fluid Dynamics. Journal of Spacecraft and Rockets, Vol. 41, No. 2, 1998. [5] Murman S.M. A Reduced-Frequency Approach for Calculating Dynamic Derivatives. 43rd AIAA Aerospace Sciences Meeting, January 10- 13, 2005. AIAA 2005-0840. [6] Bryan G.H. Stability in Aviation. MacMillian, 1911. [7] Schueler C.J, Ward L.K. and Hodapp A.E. Jr. Techniques for Measurement of Dynamic Stabil- ity Derivatives in Ground test Facilities. Agardo- graph 121, October 1967.