European Conference on Computatio nal Fluid Dynamics ECCOMAS CFD 2006 P. Wesseling, E. Oñate, J. Périaux (Eds) TU Delft, The Netherlands, 2006
Numerical Simulation of Flow over an Axisymmetric Body in Free
Flight
N. Nishikawa *, K Kaita† and , T.Ishide‡
*†Chiba University, Faculty Engineering,Japan e-mail: nisikawar@faculty.chiba-u.jp
Web page: http://www.em.eng.chiba-u.jp/~lab9/* †‡
Kisarazu National.Coll.Tech Japan e-mail: ishide@m.kisarazu..ac.jp
Key words: Fluid Dynamics, Incompressible Flow, Free Flight, Moving Grid
Abstract.Flow field around a projectile after thrust has been stopped and in inertia flight is studied numerically by a finite difference scheme. This study aims at clarifying the mechanism of free flight, which is generated and developed by rotation of body and gravity. Among two types of projectiles concerned a three-dimensional flow around the slender body, such as aircraft body, rocket .causes drastic variation with high angle of attack and has considerable influence on the aerodynamic behavior. The flow over a paraboloidal-nose cylinder at pitching rotation is considered with inertia translating motion and the flow symmetry assumption . Another example is a oblate spheroid, and in these examples the initial condition is the flow at steady ‘flight’. In present numerical study coordinate system fixed on the body ,with non-inertial frame of reference, which yields additional terms in Navier Stokes equation. The dual-time pseudo compressibility code is applied for incompressible flow. The Newton’s 2nd law is used with the balance of aerodynamic force and gravity together with angular momentum equation. For the slender body the initial incidence angle is horizontal or 40deg. For the spheroid the initial motion is set either upwardi: counter to gravity-direction or downward one. For Reynolds numbers lower than 10000 ,the behavior of flow field and varying incidence angle will be discussed as well asthe trajectory of body.
1 INTRODUCTION
In aerodynamics studies , moving frame formuration has been applied for the oscillating wing 1 or coning of slender body 2 after 1970. Other than industrial problem the free falling body problems have also been paid attention since 19th century. More attention is paid recently as a targett in multiphysics or fluid –solid interaction problems, for example by colleagues of Wang3 .
3 TITLE, AUTHORS, AFFILIATION, KEY WORDS
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N. Nishikawa , K Kaita.and and , T.Ishide
3
for the paraboloidal-nose cylinder as shown in Fig.1. Since the body is undergoing coupled translating-pitching motion the grids move and rotate with the same motion as that of the body . The coordinates are transformed into a system of general-curvilinear coordinates. In addition to usual pseudo-compressibility approach, pseudo time derivative is added respectively to four equations for primitive variables such as pressure and velocity.
Equation of angular momentum and equation of motion are expressed as follows.
(3)
(4)
Here F(D,S,L),is fluid dynamical force, D, L and N denotes Drag, Lift force, and angular momentum respectively. For pitching-only-case, replacing Eq.(4), we use dUg/dt+ΩW=-D/ρ-gsinΘ,dVg/dt=gcosΘsinϕ dWg/dt- ΩUg=L/ ρ+gcosΘ cosϕ and where the side force S is assumed to be zero as well as, yawing is neglected as well as φ:angle of roll is assumed to φ =0 which yields Vg=0.Time integration of Eqs3,4 coupled with varying D and L gives Ω, Ug, and Wg respectively. We introduce dimensionless gravity g=9.8d/U*2, a,nd ρ = ρbody / ρfluid (ratio of specific weight of ‘ vehicle’ to density of ambient fluid). Time integration of Eqs. (3),(4) gives the components of grid speed and angular velocity are Ug ,Wgand Ω, respectively.
2.2 BFC Equations
To alleviate the difficulty such as lack of hyperbolic nature in the usual incompressible continuity equation, the continuity equation is modified by additional pseudo-compressibility term. Including such continuity equation expression in body-fitted-Coordinate together with Navier-Stokes Equation (2) is as follows.
K H G G F F E E t Q v v v = ∂ ∂ + ∂ − ∂ + ∂ − ∂ + ∂ − ∂ + ∂ ∂ τ ζ η ξ ) ( ) ( ) ( (5)
where Q = [p/β, u, v, w]T. is unknown variable matrix, and where β,τ pseudo-compressibility parameter, and pseudo-time, respectively. Here, source term is rewritten from Eq.2.
Among flux vectors E,Ev are expressed as follows. ξ + ξ + ξ + = p Uw p Uv p Uu U J E z y x 1 + + + + + + = ζ η ξ ζ η ξ ζ η ξ w g w g w g v g v g v g u g u g u g J Ev 13 12 11 13 12 11 13 12 11 0 1 (7)
The primitive variables vector Q and the flux vectors E. F, and G can be written as usual expressions as well as for the viscous flux vectors; Ev, Fv, Gv which includes Reynolds number Re.
2.3 Beam and Warming Method
The numerical algorithm used to advance Eq.(5) is an implicit, approximately factored finite difference scheme known as Beam and Warming scheme 10 .The time differencing used in the present code, generally known as the trapezoidal rule as follows.
(
)
(
)
(
)
(
)
(
)
(
)
∆ − + − ∂ ∂ + − ∂ ∂ + − ∂ ∂ ∆ + = ∆ − + − ∂ ∂ + − ∂ ∂ + − ∂ ∂ ∆ + + + + + + t Q Q G G F F E E H t Q Q G G F F E E H m n n v n v n v n m n n v n v n v n ζ η ξ τ ζ η ξ τ 2 2 1 1 1 1 1 (8)Thus, hereafter unknown is
∆ H n+1=(H n+1-H n)
Eq.(5 ) for ∆H n is subjected under time-iteration until the time derivative of H with respect to τ attains to be less than 10-4, simultaneously with attaining H = Q. That is, this pseudo-time-marching is an relaxation iteration. This procedure of dual time scheme is a modification to INS3D code.
Solution for H n+1 is nonlinear in nature since flux-vectors are nonlinear functions of H n+1, therefore the following linearization procedure is used. Here, flux Jacobians R n, , A n , which are derivative with respect to H , are introduced as follows.
(
H H)
R Q H R Q Qn+1 = n + n n+1− n , n =∂ ∂(
H H)
A E H A E En+1 = n + n n+1− n , n =∂ ∂Flux Jacobians with respect to H for F,G, Ev, Fv, and Gv have similar for form as that for E. When the marching for pseudo-time is converged, value Qm can be identified as solution Qm+1 for the next physical time step. ADI procedure is performed for each of three BFC coordinate as follows.
First A. Author, Second B. Author and Third C. Coauthor.
3 TITLE, AUTHORS, AFFILIATION, KEY WORDS
The first page must contain the Title, Author(s), Affiliation(s), Key words and the Abstract. The Introduction must begin immediately after the Abstract, following the format of this template.
3.1 Title
The title should be written centered, in 14pt, boldface Roman, all capital letters. It should be single spaced if the title is more than one line long.
3.2 Author
The author's name should include first name, middle initial and surname. It should be written centered, in 12pt boldface Roman, 12pt below the title.
3.3 Affiliation
Author's affiliation should be written centered, in 11pt Roma n, 12pt below the list of authors. A 12pt space should separate two different affiliations.
3.4 Key words
Please, write no more than six key words, which will be used to compile the CD-ROM index. They should be written left aligned, in 12pt Roman, and the line must begin with the words Key words: boldfaced. A 12pt space should separate the key words from the affiliations.
3.5 Abstract
Use 12pt Italic Roman for the abstract. The word Abstract must be set in boldface, not italicized, at the beginning of the first line. The abstract text should be justified and separated 12pt from the key words, as shown in the first page of these instructions. The abstract should neither be too short nor exceed the first page.
4 HEADINGS 4.1 Main headings
The main headings should be written left aligned, in 12pt, boldface and all capital Roman letters. There should be a 12pt space before and 6pt after the main headings.
4.2 Secondary headings
increment of position or variation of trajectory. Added mass effect should consider for low density case,e.g. Magnaude and Eames12.
3 RESULTS AND DISCUISSION
Numerical results discussed are mainly for Re = 7500, choosing the reference length as diameter of 7.5cm. Density parameter is in the range 10<ρ<1600 , where lower values for the motion in water and higher one for flight in air.
CASE I Paraboloid\al-Nose-Cylinder
For cases I at α = 40deg are discussed The code was verified by previous results11 in Fig.2 where static pressure and locations of separation lines are compared with a wind-tunnel test for the same body shape and almost good agreement is found, in spite of the laminar calculation. The agreement verifies that the present code predicts a flow at the range of Reynolds number shown in the figures. First , we show the results at constant angular velocity Ω=.0.1.
In Fig.3 at representative axial cross-section θ-wise(circumferential) distribution of upper-lower difference of flow variables are plotted versus the x:span position with abcissa θ :ie. x=(local body radius)sinθ; for example, pressure difference is shown by ∆P=P(x)up – P(x)low. Velocity values are 10 times of those at grid next to the body surface. The curves are for instantaneous pitching angle for α=–29deg and α=–59deg and for α=–29deg the difference in velocity smaller than α=–59deg case. The latter incidence means more negative-attack angle stage ,i.e. the nose-end is lower than tail. The higher curve of U,V are for α=–59deg, simultaneously the difference ∆P become larger. Moreover, for α=–59deg or as going rear of body, ∆P is changed to positive near leeside symmetry line , higher curve of U,V shows that relatively larger velocity fluid approaches to and away from symmetry line ,or more rapidly. Near the region adverse pressure region, ∆P >0 appears which implies that the fluid is carrried from upper – leeside to lower side of body-edge. Evidently spanwise velocity Uspan is odd function and other variables are symmetric with respect to body-symmetry plane.
In Fig.4 pressure and axial velocity Uaxx10 is plotted versus Z axial coordinate along symmetry line. For pitching motion from positive incidence α=12deg to α=–12deg; negative incidence, the suction peak moves from leeside to windward side at right end of abscissa. Evidently at zero incidence the flow variables at leeside distribution and windward side one do not have the same, however they should be the same if the steady –zero-incidence flow is realized. This implies that phase lag from incidence angle appears especially for velocity field .
N. Nishikawa , K Kaita.and and , T.Ishide
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zone in Fig.5(a) imples that the zone compressed vertically. The velocity vectors at constant angular velocity (Ω=.0.1) solution13 is illustrated for comparison and the velocity not necessarily follows incidence angle espacially at
α=213deg stage. This result imples that the present code needs some impovement for large Ω.
The solution for given angular velocity is not reffered afterward in the present paper and free flight case will be discussd . The trajectory of the body in Fig.6 is shown for density parameters ρ=50 by plotting elevation; h/d versus horizontal flight distance X/d. In Fig.7 the history of incidence angle is shown for three cases of density parameters
ρ=10,20,30 however difference due to ρ appears after U*t/d>6 .In the results shown in Fig.6 or 7, naturally as lighter body or smaller density ratio, the larger decrease in incidence angle can be observed as far as the results for the range of ρ<100 the simple falling down stage. Vanishing horizontal velocity is attained after X/d>8 or X>60cm flight distance. For large densities;ρ and initialLift-off incidence α=40deg which may yields typical ‘Launching problem’ trajectories are shown Fig.8 where body attains the larger flight distance x/d>120. Thus, as the density ratio becomes larger ,the drag forces by CFD results have smaller contribution or the coupling with Navier-Stokes and inertia equations (3 ,4) is weaker.
CASE II Oblate Spheroid
In.Fig.9 the grid near the oblate spheroid is shown, where 70x144x76 grids are distributed in perpendicular to surface,circumferential,and azumuth direction respectively. In Fig.10 the ‘streamlines’ are shown for typical stages of ‘flight’ A,B,C corresponding to the A,B,C marks in Fig 12 shown for upward start case , where gravity and drag has same sign or gF>0.
In Fig.10, at vertical cross-section(θ=90, θ=270deg plane),in which plane the spheroid has incidence α= 16deg initially. Behind edge at θ=90deg the vortex go downstream and is elongated ,at B:Ut/d=2.5. Comparison of this shape with circular cross-section of ‘vortex’ behind edge at θ=0deg at horizontal cross-section(θ=0,θ =180deg plane) implies that vortex may has banana -like shape(one end is slim). At stage C:Ut/d=4.5.another shedding of the vortex can be observed behind θ=90deg-edge
Fig.11 the trajectory of the body is shown for initial incidence 8deg,and 16deg It is natural that in 8 deg case smaller horizontal shift is observed. In Fig.12 elevation has peak like as 2.2 lower than those in Fig.11.
4 CONCLUSIONS
The flow field and flight trajectory have been predicted for paraboloidal-nose cylinder and oblate spheroid. The results clarify the flow field at free flight and trajectory of body for some initial incidence angles.
The non-inertial formulation is not necessarily advantageous for schemes which do not respond to outer boundary condition in small number of iteration for continuity.
REFERENCES
[1] MaCroskey,W.J,Ann.Rev.Fluid.Mech,Vol.14,485-311(1972) [2] Schiff ,L.B.AIAA J,10,10 (1972)
[3] Wang,,Z.J,Birch J.M. and Dickinson,M.H., J.Expe.Biology,297,449-460(2004) [4] Takakura,Y. Higashino,H. And Ogawa,Computer& Fluids,Vol.27,645-650(1998))
[5] Takeuchi,S.,Kajishima,T.and Tamazaki,T,6th World. Congr. Compu.Mech,pp177.(Beijing,2004) [6] Kandil, O. and Chuang, H., AIAA Paper-88-2280 (1988)
[7] Kandil, O. and Kandil, H., AIAA-Paper 94-1426-CP(1994)
[8] Nishikawa, N. and Mikami, F, Transacti.JSME,68,No.669,Ser. ,1407-1414(2002) or N. Nishikawa, T. Shimizu, H. Maeda and F. Mikami, :Computational Fluid Dynamics 2000 pp793-794 Springer(2001) [9]Suzuki K and Kubota, H Proc. ISCFD,Nagoya ,317-322(1989)
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-100 -80 -60 -40 -20 0 20 40 60 80 100
Lee-side Circum Ang. Windward
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -Cp RE=63000 Re=30000 Re=7500:T=66 Cpexpe. x=-0
Fig.5 (c) α = 13deg : Ω=0.1;given Fig.5 (d) α = 213deg Ω=0.1 given
N. Nishikawa , K Kaita.and and , T.Ishide
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(A ver. Hori)
(B ver. Hori)
N. Nishikawa , K Kaita.and and , T.Ishide
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(B ver. Hori)
(C ver. Hori)
