A Mathematical Model for the Fire-extinguishing Rocket Flight in a Turbulent Atmosphere

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1 A Mathematical Model for the Fire-extinuishin Rocket Fliht in a Turbulent Atmosphere CRISTINA MIHAILESCU Electromecanica Ploiesti SA Soseaua Ploiesti-Tiroviste, Km 8 ROMANIA crismihailescu@yahoo.com Abstract: The paper analyses the evolution of rockets used to extinuish fires developed in open space in conditions of turbulent atmosphere. After the eneral definitions of the turbulent atmosphere and workin hypotheses, the model of and turbulent atmosphere obtained from analytical relations is presented in detail. Startin from this model, the components of velocity of as functions of time are deduced. Introducin the components of velocity into the model of simulation of the rocket fliht, a fascicle of the trajectories is obtained. The work novelty consists in the new technique used to model the turbulent atmosphere influence on unuided rocket fliht. Key-Words: turbulent atmosphere, rocket trajectory, statistical function, dynamics of fliht, models Introduction Fire-extinuishin rocket trajectory must be evaluated in the early project desin stae, when the extreme atmospheric s, the vertical air flows and the no wind must be considered as possible perturbations. The effect of lare scale perturbation is important for establishin the averae trajectory. A supplementary task would be to determine the trajectory dispersion due to perturbation radients produced by the atmospheric usts. The importance of the study lies in the results it provides: the probable trajectory, its dispersion and the rocket response to dynamic loadin. This paper tries to provide an answer to the problem of turbulent atmosphere influence on fire-extinuishin rocket fliht. Definition of Atmospheric Turbulence. Hypotheses Since is a random process that cannot be described by explicit functions of time, only a statistical, probabilistic approach can be taken. The velocity field, variable in time and space, may be considered as result of an averae value and a perturbation around it. In the case of lon time flihts, the averae value may be modelled as referencin the fliht with respect to a mobile rectanular frame which moves with the wind s averae velocity. In a point located by r x, x, the air averae velocity is: ( x3 [ u u u ] T u( r, t 3, ( It may be described properly by a correlation matrix composed of elements like this: Ri, j ( ξ, τ < ui ( r, t u j ( r+ξ, t+τ >, ( which represents the temporal and spatial averae of the product between a component ui located at a point defined by r at the time t, and the component u from the point r + ξ at the time t + τ. To this matrix we can associate a spectral function matrix whose elements are defined by the Fourier transform function: Θ i( Ωξ+ωτ j ( Ω, ω ( ξ, τ dξ dξ dξ 4 j e 3 (π R dτ(3 The inverse of these elements belon to the correlation matrix: i( Ωξ+ωτ j ( ξ, τ Θ j ( Ω, ω e dω dω dω3 R dω (4 Generally, the atmosphere s are not Gaussian. However, for practical purposes, the normal repartition may be used, producin sinificant simplifications. On the other hand, the statistical parameters R i, j and j j Θ of the are defined in each point in space, r and variable with time t. A particular aspect of the fire-extinuishin rocket is that it flies a lon time, with a sliht chane in velocity manitude, due to slow burnin of rocket propellant. This is a sinificant theoretical advantae because it allows admittin the hypothesis that the is a stationary process. Another hypothesis is that the could be considered homoenous on small intervals, which means that R i, j and Θ j do not depend on r, alonside of at least a trajectory sement. At hih altitude, the is identical in all points belonin to the same layer, which means it miht be considered homoenous. Close to the round often chanes with altitude occur. However, in the case of small launch anles, homoenous miht be considered alon the trajectories close to round. ISSN: ISBN:

2 R i, j and j Θ usually depend on wind referenced frame axes orientation. If the statistical functions do not depend on space, the is considered isotropic, which makes the velocity components square means to be equal: u u u3, (5 where the standard deviation will be called intensity. Due to the fact that rocket velocity is much hiher than their variation velocity, the may be rearded to as a frozen model. This allows us to nelect time in the function u ( r, t, which corresponds to Taylor s hypothesis. Consequently, the correlation and spectral functions become: R j ( ξ, τ R j ( ξ ; Θi, j ( Ω, ω Θ j ( Ω, (6 Consequently, the simplest model is the one of homoenous and isotropic s, Gaussian and frozen, a model which is used for the fliht outside the round proximity layer. Inside of this layer is necessary to consider anisotropic. For the isotropic, usin Batchelor s relation: Rij ( ξ ξξ [ f ( ξ ( ξ] ξ i j + ( ξ δ where ξ ξ ; δ j - Kroneker s symbol; j, (7 The lonitudinal correlation function f (ξ represents the correlation between velocity components alon an axis, iven in two points of that axis (Fiure.: R ξ f ξ ( lim u( x u( x+ξ x d ( (8 Because of the hypothesis reardin isotropy: f ( ξ f ( ξ f ( ξ3 f ( ξ, (9 and: R( ξ lim f ξ u( x u( x+ξ d x ( ( x ξ u x ( x + ξ u ( x Fi. Lonitudinal correlation of velocity components The other function (ξ, is the lateral correlation function and represents the correlation between the transversal velocity components with respect to an axis (Fiure, defined in two points: x 3 u 3 u x R33( ξ lim ξ u3( x u3( x+ξd x ( ( Based on the hypothesis reardin isotropy, it results: < u3 ( x u3 ( x+ξ >< u ( x u ( x+ξ >, ( < u3 ( x u3 ( x +ξ >... and: R33( ξ lim ξ u3( x u3( x+ξ d x ( (3 u 3 ( x + ξ u ( x 3 + ξ 3 u 3 ( x u (x + ξ x 3 u ( x u ( x u ( x 3 + ξ 3 u 3 ( x + ξ x Fi. Transversal correlation of velocity components 3 The Uniform Turbulence Model For the characteristic correlation functions, the followin relations from work [] may be used: / 3 3 f ( ξ ζ / K/ 3( ζ ; Γ(/ 3 /3 4/3 ( ξ f ( ξ ζ K/ 3( ζ (4 Γ(/ 3 ξ where: ζ, a, 339 and L representin the al characteristic lenth. Γ is the second deree Euler function and K υ is the second deree Bessel modified function. The Fourier transforms of the functions f (ξ and (ξ are spectral one-dimensional functions: Φ iωξ ( Ω ( ξ e dξ π iωξ Φ33( Ω ( ξ e d π f ; ξ (5 Fiures 3 and 4 illustrate raphical representation of functions defined by (4 and (5. For these spectral functions, throuh experimental measurements, several models have been established, amon which, the most commonly used are the von Karman model: L Φ( Ω ; + (8 3( alω Φ33( Ω π [+ ( alω π [+ ( alω ] 5 6 ] and the Dryden model: L Φ( Ω π + ( LΩ ; Φ 33 x L (6 L + 3( LΩ ( Ω π [+ ( LΩ ] 6 (7 ISSN: ISBN:

3 f,.5.5 f H> 65 m L 76 m a ξ/l Fi. 3 Characteristic correlation functions Φ L. Φ Φ33 4 The Linear Turbulence Method A hiher order approximation is the one of a field, the velocities bein considered as functions dependin on position. Authors of [], [] and [3] developed in case of airplane models in four points. These models also allow calculation of rotation velocity components p, q, r due to. Startin from the method proposed in [], we developed for rockets a model of in two points, which allow deducin of anular velocity of pitch and yaw motion. In the case, due to the slender axisymmetrical shape of the rocket, some radients may be nelected. We shall consider a model of the rocket in two points as shown in Fiure 5. (C.m O l v (F. H> 65 m L 76 m Fi. 5 Model of the rocket in two points 5 Ω L 5 Fi. 4 Characteristic spectral functions The established experimental relations [] are in ood accordance with relations (5 and may be successfully used in the case of the study models with. However, in the case of method, due to necessity to define cross correlation functions which show the mutual dependence between velocity components, the characteristic functions defined by (4 will be used. In case of altitudes over 65m the boundary layer influence is diminished and the atmosphere may be considered, a unique value for the characteristic lenth L bein recommended, both to lonitudinal and lateral spectral functions. For the model considered, accordin to [] and [] L76m. The standard deviation is the same alon all the three directions, dependin only on the intensity:,53 [ m / s] - low ; 3,5 [ m / s] - mean ; 6,4 [ m / s] - heiht (storm. The model obtained in this way is the simplest possible, bein known as the field model, in which only the components of velocity alon the three axes, reduced in the ravity centre of the rocket, are considered: u, v, w. The lenth l v, called aerodynamic lenth, represents the distance from the mass centre to the focal point. Based on this model, the pitch and yaw velocities, equivalent to the velocity radients, are: q ( w w ; r ( v v, (8 lv lv to which there are added the velocities due translation for the : w w ; u u ; v v (9 To calculate the velocities produced by, we expressed their correlation functions throuh the characteristic functions f and, proceedin as in the case of distribution, we calculated the spectral functions correspondin to the correlation functions and, from these, we found some possible forms of the translation and rotation velocities enerated by. For the case of the pitch velocity the followin initial relation was used: Rqq ( τ < q ( t q ( t+τ >, ( obtainin: ( lv R ( τ ( ξ ( ξ ( ξ3 qq, ( where: ξ Vτ ξ + l ξ ξ l ; ξ v 3 v ξξ+l v ξ3ξ-l v O I F I O F ξvτ Fi. 6 Correlation functions calculus schema ISSN: ISBN:

4 ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS Similarly the anular yaw velocity: Rrr( τ < r ( t r ( t+τ > < ( v v ( v' v' > ( lv ( [ < vv' >+< vv ' > < vv ' > < vv' > ] ( lv from which we obtained: Rrr ( τ [R ( Vτ,, R ( Vτ+ l, (,] v R Vτ lv ( l v ( lv R ( τ ( ξ ( ξ ( ξ3 rr, (3 From the previous relations and (3, the correlation functions for the rotation velocities were determined. The two rotation velocity components are equal, R, due to axial symmetry (Fiure 7. The qq R rr spectral function for the rotation velocity is represented in Fiure 8. We compare the velocity components calculated with the field theory to those obtained usin the field theory. These proved to be identical. Rqq lv/ H> 65 m L 76 m a ξ/l* Fi. 7 Correlation function for rotation velocities Φ qq lv /( L * H> 65 m L 76 m a Ω L/ Fi. 8 Spectral function for the rotation velocity Thus, for the lonitudinal component we obtained: Ruu ( τ < u ( t u ( t+τ >< uu' > R( Vτ,, f ( ξ (4 For the lateral component we obtained: Rvv ( τ < v ( t v ( t+τ >< vv' > R( Vτ,, ( ξ (5 Rww ( τ < w ( t w ( t+τ >< ww' > R33( Vτ,, ( ξ (6 Relations (4 (6 show that for model we found the same characteristic f and from Fiure 3 and the spectral functions shown in Fiure 4. This means that the translation velocities for the liner model are the same with those from the model and for these we can use theoretical relation (4 or the experimental relations (6 and (7. Relations (3 for the anular velocities may be used only in the case of the theoretical model defined by the characteristic functions (4. In the case of axissymmetric rockets, the only supplementary term specific to relations (3 is produced by the anular pitch/yaw velocities. If the aerodynamic lenth is comparable with respect to the characteristic lenth, the influence of this term is small and can be nelected. In the case of short rockets, like fire-extinuishin rocket, the term is important as we will see from the numerical comparison. 5 Simulation of Atmospheric Turbulence The simulation of an atmospheric is based on the method of sinusoidal sum, usin the spectral function modulus of the velocity to obtain the oscillation amplitude specific to a frequency bandwidth. So, for the spectral functions values presented in Fiure 4, we determine by interation the value of amplitude correspondin to a frequency bandwidth, which can be approximated with the frequency correspondin to the centre of the interated domain. To complete the model, to each frequency we associated an initial arbitrary phase. Usin the notation S for the portion of the area φ obtained by interatin the modulus of a spectral function from Fiure 4 and considerin that the function has been represented only for the positive values of the variable Ω, the amplitude has the value: A S (7, φ Two of the translation velocities enerated by in a point ξ are iven by: i i ( ; i u ξ A sin( Ω ξ+ φ i i w ( ξ A sin( Ωξ+ φ, (8 i where the index was iven to amplitude, pulsation and initial phase obtained from the modulus of the lonitudinal spectral function, and the index iven to the elements obtained from the modulus of the lateral spectral function. The third component of the velocity is the same as the second, exceptin the initial phase: i i v ( ξ A sin( Ωξ+ φ3 (9 i Makin the assumption that the rocket flies on a trajectory with a constant velocity V: ISSN: ISBN:

5 ξ V τ, (3 Consequently, the velocities oriinated by may be expressed as functions of time (Fiure H> 65 m L76 m V 4 m/s u/ w/ t[s] Fi. 9 The axial u and normal w velocities produced by The modulus of the spectral functions presented in Fiure 8 was used to find the anular velocities. By applyin similar relations to (7 and (8 we obtained a form of the anular velocities oriinated by with respect to time. Because the spectral functions are equal (Fi.8, the yaw and pitch velocities will be approximately equal r q, differin only by initial phases. In this case, in Fiure, we presented only the pitch velocity function q. of trajectories was obtained. The results are presented in the next tables. In order to analyse trajectory parameters, we define four check points: -Start point, -End of rocket motor burn, 3-Trajectory apex, 4-Impact point Table. Nominal trajectory parameter, Stae of fliht 3 4 T [s] V [m/s] γ [de] [m] Y [m] Table. Averae and standard deviation trajectory parameters, low. 53 Stae of Uniform model Linear model fliht T [s] V [m/s] γ [de] [m] Y [m] q*lv/ H> 65 m L76 m V 4 m/s In table for each parameter are two rows: first for averae and the second for standard deviation values. For a better analyse the main results are presented in a comparative form t[s] Fi. The pitch velocity produced by V [m/s] nominal 6 Fire-extinuishin Rocket Trajectory Startin from papers [3] and [4], some details reardin the procedure of buildin the simulation model for the fire-extinuishin rocket are presented in this section. For the turbulent atmosphere we used in order to make a comparison, the and the model. The initial values of the sinusoidal sums phases were obtained by eneratin random numbers, and the amplitude and frequency were determined startin from the spectral density functions usin the procedure described in the previous section. Because of the sequential initial random input of the phases, a fascicle Fi. Velocity diaram averae values, low ISSN: ISBN:

6 [m/s] Fi. Velocity diaram standard deviation values, low [de] γ nominal Fi. 3 Climb anle diaram, averae values, low [de].5 7 Conclusions Two models were presented in the paper. In the model, only the translation velocities of the mass centre are considered. In the model, besides the velocities, the anular velocities of pitch and yaw with respect to mass centre are added. In the case of axis-symmetric rockets, the term dependin on anular velocities is the only which couples the equations. A further simplification could be made in the case of short rockets to which the aerodynamic lenth is much smaller than the characteristic lenth. In this situation, the couplin term due to rotations is small and may be nelected, so that the model is an acceptable approximation. Considerin the model developed in our study, we have analysed in a simulation example the fliht dynamics of a rocket, in which the influence of was evaluated in four characteristic staes of the fliht. The work novelty consists in technique used in modellin the turbulent atmosphere influence on unuided rocket fliht: two point model used for the the technique used to obtain the velocity the comparison between and model Fi. 4 Climb anle diaram, standard deviation values, low Y[m] nominal 5 [m] Fi. 5 Trajectory diaram, averae values, low The averae parameters from nominal trajectory, model and model are relatively closed. Different from these, the parameters dispersion obtained with model and model are different, therefore the use of the model are rihtful. References: [] Etkin, Bernard, Dynamics of Atmospheric Fliht, John Wiley & Sons, Inc., New York, 97. [] Etkin, Bernard, Lloyd Duff Reid, Dynamics of Fliht: Stability and Control -3 rd ed, John Wiley & Sons, Inc., New York, 995. [3] Chelaru T.V., Barbu C. Mathematical model and technical solutions for multi-stae soundin rockets, usin the rotation anles, Proceedins of the 3 rd International Conference on Applied Mathematics, Simulation, Modelin (ASM 9, pp.94-99, ISBN: , NAUN, Vouliamen Athena, Greece, December 9-3, 9. [4] Chelaru T.V., Barbu C., Chelaru A., Mathematical model and technical solution for small multistae suborbital launchers, Proceedins of the 4 th International Conference on Recent Advances in Space Technoloies-RAST 9, pp. 86-9, ISBN: , Istanbul, Turkey, -3 Jun. 9. ISSN: ISBN: