Semi-Empirical Formulas of Drag/Lift Coefficients for High Speed Rigid Body Manoeuring in Water Column Peter C. Chu and Chenwu Fan Naal Ocean Analysis and Prediction Laboratory Naal Postgraduate School, Monterey, California, USA Paul R. Gefken Polter Laboratory, SRI International, Menlo Park, California, USA Abstract Falling of rigid body through water column with high speed is inestigated experimentally and theoretically. Seeral experiments were conducted to shoot rigid bodies with the density ratio higher than into the hydrographical tank. During the experiments, we carefully obsere the position and orientation of the bomb-like rigid bodies. Using the experimental data a semi-empirical formulas for the drag/lift coefficients were obtained. With the gien drag/lift coefficients, the momentum and moment of momentum equations of a fast-moing rigid body can be integrated numerically. The numerical results are well compared by the experimental data. Keywords: Body-Flow Interaction, Bomb Manoeuring, Body Trajectory and Orientation, Drag/Lift Coefficients, Inerse Modeling, Bomb Drop Experiments Introduction Study on falling rigid body through water column with high water entry speed has wide scientific significance and technical application. Scientific studies inole nonlinear dynamics, body and multi-phase fluid interaction, supercaitation, bubble dynamics, and instability theory. Technical application
of the hydrodynamics of a rigid body moing fast in fluids includes aeronautics, naigation, and ciil engineering. The rigid-body dynamics allows one to set up six nonlinear equations for the most general motion (Chu et al., 004, 005a, b): three momentum equations and three moment-of-momentum equations. These equations are ready to sole if hydrodynamic coefficients (such as drag and lift coefficients) are gien for computing hydrodynamic forces and torques. Unfortunately, these coefficients (usually Reynolds number dependent) are known only for bodies with some simple geometry such as cylindrical and spherical, not for bodies with more complicated geometry such as the Mk-84 general purpose bombs. Therefore, the key issue for soling the basic equations is to determine the hydrodynamic coefficients. In this study, an inerse model is deeloped for determining the drag/lift coefficients from the rigid-body s trajectory and orientation. Then a bomb strike experiment is conducted to collect the data for the trajectory and orientation. Using the experimental data, semi-empirical formulas are deried for the drag/lift coefficients. Hydrodynamic Force and Torque Consider an axially symmetric rigid-body with length L such as bomb falling through water column with the centers of mass (c m ) and olume (c ) on the main axis (Fig. a). The position of the body is represented by the position of c m, r(t) = xi +yj + zk, (a) which is called the translation. The positions of the two end-points (such as head and tail points) are represented by r h (t) and r t (t). The difference between the two ectors in nondimensional form r r h t e = r r, (b) h t is the unit ector representing the body s main axis direction. The translation elocity is gien by dr,. dt = u u =V e (a) where V and e are the speed and unit ector of the rigid-body elocity. Let be the water-to-body relatie elocity (called the relatie elocity). If the water elocity is much smaller than the rigid-body elocity, the water-to-body relatie elocity can be approximately gien by u = Ve. (b) Usually the two ectors (e, e ) are not parallel and their ector product leads to a unit attack ector e = e e, (3) α sinα
where α is the angle between (e, e ). For a two dimensional motion, if ( β, γ ) are the eleation angles of the rigid body and its elocity, the difference α = β γ, is the water attack angle (Fig. b). Hydrodynamic force on a rigid body consists of a drag force (F d ) F = f e, e = e, (4) d d d d and a lift force (F l ) ( e e) e F = f e, e = l l l l ( e e) e. (5) Their magnitudes are determined by the drag and lift laws, f = C ρav, f = C ρav, (6) d d w l l w where ρ is the water density; A is the under-water area of projection; and w ( C, C ) are the drag and lift coefficients. Let σ be the ector from the center d l of mass (COM) to the center of olume (COV), or σ = σe. The center of the hydrodynamic force (COF) is the location where the fulcrum is chosen with zero torque. In fact, the hydrodynamic torque is calculated from the resultant drag and * lift force exerted on COF with COM as the fulcrum ( M ). If we shift the h exerted point of the resultant drag and lift force from COF to COV, the torque * ( M ) contains two parts h ( ) M = M + M, M = σe F + F, M = M e, (7) * h h t h d l t t α where M h is the hydrodynamic torque with the resultant drag and lift force exerted on COV, and M t is the torque caused by the shift of the exerting point from COF to COV. Here, the magnitude of M t is calculated by the drag law M = C ρ A L V, (8) t m w w where C m is the moment coefficient; and L w is the under-water path length. e β M F l c c m α γ F d σ Fig.. Shift of the exerted point of the drag/lift forces from COF to COV with an extra torque M t. Here, (,, α βγ ) are the attack angle, eleation angels of the body and its elocity. 3
3 Basic Equations Let (i, j, k) be the unit ectors of the Cartesian coordinate system fixed to the Earth with (i, j) in the horizontal plan and k in the ertical (positie upward). Time deriatie of (a) gies the acceleration of COM (Chu et al., 006a, b), du dv de = e + V. (9) dt dt dt If the translation elocity has an eleation angle γ and an azimuth angleψ, the unit ector e is represented by e = cosγ cosψi + cosγ sinψj+ sin γk, (0) Substitution of (0) into (9) leads to de dγ d γ ψ ψ = e + e, () dt dt dt where γ ψ e = sin γ cosψi sinγ sinψj+ cos γk, e = cos λsinψi + cos λcos ψj, γ ψ γ ψ,, e e e e e e. (3) The momentum equation is gien by dv dγ γ dψ ψ m e + mv( e + e ) = ( ρπ m) gk f e + fe. (4) d l l dt dt dt where (m, Π ) are the mass and olume of the rigid body; (f d, f l ) are the drag and lift forces. Let Ω * be the body s angular elocity, which is decomposed into two parts with the one along the unit ector e (self-spinning or bank) and the other part Ω (azimuth and eleation) perpendicular to e, Ω* = Ω e +Ω e, e i e = 0, (5) s ω ω where e is the unit ector of Ω (perpendicular to e), ω Ω =Ωe Ω= Ω. (6) ω For axially symmetric body, J is a diagonal matrix J 0 0 J = 0 J 0, (7) 0 0 J 3 with J, J, and J 3 the moments of inertia. The moment of momentum equation for small self-spinning elocity is gien by (Chu and Fan, 007) dω J σρπ ge k + σ ( f e e + fe e ) + M + M. (8) d d l l t ae dt where M az is the torque due to the azimuth and eleation rotation, 4
L ΩL M = e C ρd V xω xdx = C DLVΩe, for V M ( ) ρ 3 ae ω L r r r r ω r V L r Ω = e C D( V x ) xdx C D( V x ) xdx ae ω ρ Ω L r r ρ Ω Vr r r Ω 3 4 V 4 V V r r r C ρdl V V L L r r r = + Ω + Ω e 4 ΩL 3 ΩL 6 ΩL 4 Hydrodynamic Coefficients Ω L for > V. r Prediction of the rigid-body s orientation and COM location is to integrate the momentum equation (4) and moment of momentum equation (8) with known coefficients: C d, C l, and C m. Inner products between equation (4) and γ ψ the unit ectors ( e, e, e ) gie dv ( ρπ m) gk e m C = dt, (9) d ρ A V V w γ ψ mv ( e dγ / dt + e dψ / dt) ie ( ρπ m) gkie l l C =. (0) l ρ A V V w Inner product of (8) by the ector e r leads to dω Ji ie σρπ g ( e k) ie r r dt σ C = ( C ( e e ) ie + C ( e e ) i e ), () m d d r l l r Lw ρ ALV w w where e = e e, V = Vi e. () r ω r r The formulas (9)-() proide the basis for the experimental determination of (C d, C l, C m ) since each item in the right-hands of (9)-() can be measured by experiment. ω 5
5 Bomb Drop Experiment Models of Mk-84 bombs with and without tail section are taken as examples to illustrate the methodology for determination of the bulk drag/lift coefficients, and in turn the prediction of location and orientation of a fastmoing rigid-body through the water column. The primary objectie is to determine the Mk84 trajectory through the ery shallow water zone to proide an estimate of the maximum bomb-to-target standoff and required fuse delay time for optimum target lethality. Because it is possible that a portion, or all, of the guidance tail section may become separated from the warhead during water entry, it is necessary to determine the Mk84 trajectory for a ariety of different tail configurations ranging from a warhead with a completely intact tail section and four fins to a warhead with the tail section completely. Using the Hopkinson scaling laws, /-scale Mk84 bomb models were designed and constructed in SRI that matched the oerall casing shape and mass inertial properties of the full-scale Mk84 prototype. To model the different possible damaged tail configurations, we fabricated models that consisted of the warhead section with a complete tail section and four fins, a complete tail section and two fins, a complete tail section and no fins, and with the tail section remoed. The models were accelerated to elocities of up to about 454 m s - using a gas gun. The gun was positioned oer a 6 m deep by 9 m diameter pool, located at SRI s Corral Hollow Experiment Site (CHES). Two orthogonal Phantom 7 high-speed ideo (HSV) cameras operating at 0,000 fps were used to record the water entry and underwater trajectory. The digital HSV data were used to generate depth ersus horizontal trajectory, position-time history, elocity-time history, deceleration-time history, and drag coefficient-time history profiles. Typically, up to three experiments were performed for each model configuration to determine the oerall reproducibility (Gefken, 006). A gas gun with 0.0 m (4 in.) diameter and.5 m (60-in.) long was positioned oer a 6.0 m (0-ft) deep by 9.4 m (30-ft) diameter pool located at SRI s Corral Hollow Experiment Site (CHES). The gas gun barrel was eacuated before launching the scale model to preent an air blast from disturbing the water surface prior to the model impacting the water surface. At the end of the gas gun there was a massie steel ring to strip the sabot from the scale model. At high elocities there is some deiation from the theoretical calibration cure, which may be attributed to gas blow by around the sabot or friction. For the maximum gun operating pressure of,500 psi, we were able to achiee a nominal water-entry elocity of about 304.80 m/s. Two orthogonal periscope housings were positioned in the pool to allow simultaneous aboe-water and below-water isualization of the model trajectory. The housings supported Phantom 7 high-speed ideo (HSV) cameras, which were run at 0,000 fps. Fie high-intensity, short duration (30 ms) flash bulbs were used to front-light the scale model as it entered the water and traeled under water. The HSV cameras and flash bulbs were triggered at the time the sabot was released within the gun. 6
A series of 9 experiments was performed with the different /-scale Mk84 bomb models described in subsection 6. with nominal water-entry elocities ranging from 9.48 m/s to 308.83 m/s. Table summarizes the oerall experimental matrix and water-entry conditions. Typically, the waterimpact angle of entry was between 88 o and 90 o. In Experiments 0,, and the sabot failed to fully support the scale model within the gun during the launch phase, resulting in the scale model impacting the sabot stripper plate before impacting the water. All the experimental data hae been conerted to full-scale alues. Table. Summary of Mk84 underwater trajectory experimental matrix. Experiment Number Model Type Water-Entry Velocity (m/s) Water-Entry Impact Angle ( o ) I 3.5 89. I 96.87 90.0 3 I 95.35 88.8 4 I 30.05 88.5 5 I 6.77 88.0 6 I 9.45 89.0 7 I 9.48 88. 8 II Model impacted sabot stripper plate 9 II Model impacted sabot stripper plate 0 II 95.04 90.0 II 89.96 90.0 II Model impacted sabot stripper plate 3 IV 96.6 85.7 4 IV 300.53 90.0 5 IV 300.53 88.7 6 III 304.9 90.0 7 III 98.39 87.0 8 III 9.08 88. 9 II 96.87 90.0 6 Semi-Empirical Formulas Upon completion of the drop phase, the ideo from each camera was conerted to digital format. For Mk84 warhead without tail section, ertical and horizontal locations of the two-end points (Fig. ) ersus time were recorded. From these data, the unit ector e can be directly determined using (b). The translation elocity u and the angular elocity Ω are measured and so as the fluid-to-body relatie elocity V since it is assumed that the water elocity is much smaller than the bomb elocity [i.e., (b) holds]. The unit ectors (e, e l ) are in turned determined since (e, e l ) represent the direction of V and its 90 o 7
shift. When the orientation of the bomb is measured, the unit ector e is known and so as e using (). ω Fig.. Trajectory of Mk84 with no tail section and water-entry elocity of 96 m/s (Exp-3) (from Gefken, 006). Using the unit ector e, we determine the eleation angle γ and the γ ψ azimuth angle ψ [see (0)] and the two other unit ectors ( e, e ) [see ()]. Using the unit ectors e and e, we determine the unit ector e ω r [see ()]. With the calculated temporally arying (C d, C l, C m ) and (α, Re) data, we obtain the following semi-empirical formulas for calculating the hydrodynamic coefficients, ( ) Recrit 8sin α + 0.0 if sin ( α) 0 and Re Recrit Re C (Re, α ) = d, Recrit 0.34 sin ( α ) + 0.0 otherwise Re (33).. Re Re.5sin ( ) crit α min, if sin( α) 0 C l (Re, α) = Re crit Re, 0.6 sin ( α) if sin( α) < 0 7 Re.5 0 crit =, (34) 8
dα C = Asinθ B, dt m 0.06 if θ 80 A =, B = 0.00065. (35) 0.006 if θ > 80 7 Verification The semi-empirical formulas of (C d, C l, C m ) were erified using the data collected from the experiments. We use the formulas (33)-(35) to compute the hydrodynamic coefficients (C d, C l, C m ), and then to predict the location and orientation of Mk-84 bomb in the water column by (4) and (8). Comparison between model predictions and experiments (Fig. 3) shows the alidity of feasibility of the semi-empirical formulas (33)-(35). Fig. 3. Comparison between modelled and obsered bomb trajectories for the experiment shown in Fig.. 8 Conclusions A new method has been deeloped to determine the hydrodynamic coefficients (C d, C l, C m ) of fast-moing rigid body in the water column. This method contains two parts: () establishment of the inerse relationship between (C d, C l, C m ) and the rigid-body s trajectory and orientation, and () experiments for collecting data of the rigid-body s trajectory and orientation. Using the experimental data, the inerse relationship leads to semi-empirical formulas of (C d, C l, C m ) ersus Reynolds number and attack angle. This method is much cheaper than the traditional one using the wind tunnel to determine (C d, C l, C m ). We also erify these formulas using the experimental data. 9
Acknowledgments The Office of Naal Research Breaching Technology Program and Naal Oceanographic Office supported this study. References Chu, P.C., C.W. Fan, A. D. Eans, and A. Gilles, 004: Triple coordinate transforms for prediction of falling cylinder through the water column. Journal of Applied Mechanics, 7, 9-98. Chu, P.C., A. Gilles, and C.W. Fan, 005a: Experiment of falling cylinder through the water column. Experimental and Thermal Fluid Sciences, 9, 555-568. Chu, P.C., and C.W. Fan, 005b: Pseudo-cylinder parameterization for mine impact burial prediction. Journal of Fluids Engineering, 7, 55-5. Chu, P.C., and C.W. Fan, 006a: Prediction of falling cylinder through air-watersediment columns. Journal of Applied Fluid Mechanics, 73, 300-34. Chu, P.C., and G. Ray, 006b: Prediction of high speed rigid body maneuering in air-water-sediment columns, Adances in Fluid Mechanics, 6, 3-3. Chu, P.C., and C.W. Fan, 007: Mine impact burial model (IMPACT35) erification and improement using sediment bearing factor method. IEEE Journal of Oceanic Engineering, 3 (), 34-48. Gefken, P., 006: Ealuation of precision-guided bomb trajectory through water using scaled-model experiments. SRI Final Technical Report-PYU-6600, Menlo Park, California, USA, -80. 0