Kinematics of the Pitch and Plunge Maneuver
The present prescribed motion simulations were performed for a submarine model (DARPA SUBOFF) mounted in a wind tunnel with a uniform constant wind speed , and fluid properties . The model length , and maximum diameter, . The model was subjected to a pitching and plunging motion such that the fixed centre of rotation at R (reference point, measured backward from the nose) was located on the body at .
A schematic of the maneuver is shown in Figure 1. The incidence angle at the reference point, , had a pitch rate, . The simulations were conducted at a model Reynolds number, .
Time in this problem can be scaled according to the time required for the flow to traverse the length of the body at zero incidence;
Similarly, the non-dimensional pitch rate of the model is defined as;
The kinematics of the experiment are described by;
where , and are components of U along the body fixed coordinates with its origin at .
Figure 1: Schematic of the planar pitch and plunge unsteady maneuver. The point indicated by the vector R is the fixed centre of rotation of the maneuver. The origin of the body fixed coordinate system is the centre of buoyancy of the body.
is the magnitude of the wind tunnel air velocity and is the angle of wind tunnel air stream to the body axis. The pivot point on the hull is located on the body axis where giving.
In the present maneuver the pivot point begins at the reference point and then moves forward with increasing incidence, .
In the present study we performed the simulations for a family of pitch and plunge maneuvers in which the kinematics are continuous. The angle of incidence at the centre of rotation, , is described by;
where we have taken and to set the rate of pitching.
Figure 2 shows the variation of incidence for 4 values of . The derivative of the pitch curve gives the pitching rate
Using equation (10) the drift angle at the centre of buoyancy, , can be determined and the result is shown in Figure 3. The rate of change of the pitch rate, , is
and the pitch rate has a maximum value of;
which occurs at . Substituting the above values of A we get,
The pitch rate and its rate of change are plotted in Figure 4. The case was chosen because it closely matches the kinematics of the Granlund and Simpson (2004) experiment discussed in a later section of this report.
Figure 3: Drift angle, , at the centre of buoyancy for the pitch and plunge maneuver as described by equations (10). Each curve corresponds to a different maximum pitch rate according to equation (13).
Figure 4: Rate of pitch and its derivative for the pitch and plunge maneuver as described by equations (11) and (12). Each curve corresponds to a different maximum pitch rate according to equation (13).
Meshing and Time Step
EXN/Aero simulations were conducted on the half body of an unappended SUBOFF hull profile and surrounding computational domain as shown in Figure 5. The mesh included 38,882,670 control volumes with 9,314,900 computational nodes. The body profile and near body mesh are shown in Figure 6. Inlet boundary conditions were imposed on the West and South domain boundaries and an “outlet” condition with uniform static pressure were imposed on the East and North surfaces. Symmetry conditions were imposed on the front and back surfaces. Transient prescribed motion simulations were run on 4 CPU and 4GPU K80 cards for 10,000 time steps of which required 50 hours or 18 s per time step. All simulations were done using the URANS formulation with the SST turbulence model.
Surface Pressures and Vorticity Fields
Images showing the surface streamlines and surface pressure distributions at 3 states of the transient pitch and plunge with are shown in Figure 7. This corresponds to the lowest pitch rate studied where the entire maneuver is spread out over, . The 3 images correspond to the steady flow at zero incidence, time of maximum pitch rate where, , and following the pitching event where the forces and moments (described in the next section) have obtained values corresponding to steady translation at . At the time of maximum pitch rate the cross flow separation extends over most of the mid body. Figure 8 shows the longitudinal vorticity contours on cross sectional planes distributed along the body.
Images showing the surface streamlines and surface pressure distributions at 4 states of the transient pitch and plunge with are shown in Figure 9. This is the highest rate of pitch considered with the drift angle rising to its maximum value in, . The 4 states correspond to zero incidence, the time of maximum pitch rate, at the end of the maneuver but prior to the forces and moments attaining steady translation values, and once the forces and moments have settled to steady translation values. At the time of maximum pitch rate the drift angle, , the cross flow separation is only visible along the afterbody. This is in contrast to Figure 7b where the cross flow separation extended on to the midbody at the same drift angle but lower pitch rate. Figure 9 (c, d) shows the surface streamlines at the end of the maneuver where the drift angle, , but before the forces and moments have settled to steady translation values. These surface streamlines are indistinguishable in spite of the significant differences in overall forces and moments for these two conditions which are presented in the next section. The vorticity contours on cross-sectional planes are shown in Figure Figure 10. Consistent with the surface streamlines Figure Figure 10c does not show prominent leeside wake vortices along the midbody as in Figure 8b which has the same angle of drift but a higher rate of pitch.
Figure 7: Surface streamlines and pressure distributions for 4 stages of the lowest pitch rate a) zero drift prior to the start of pitching, b) near the point of maximum pitch rate , and c) after steady forces and moments have developed, .
Figure 8: Longitudinal vorticity distributions on cross sectional planes for 4 stages of the most rapid transient a) zero drift prior to the start of pitching, b) near the point of maximum pitch rate , and c) after steady forces and moments have developed .
Figure 9: Surface streamlines and pressure distributions for 4 stages of the most rapid transient zero drift prior to the start of pitching, b) near the point of maximum pitch rate c) at the end of the pitch but before the forces and moment have stabilized, , and d) after steady forces and moments have developed, .
Figure 10: Longitudinal vorticity distributions on cross sectional planes for 4 stages of the most rapid transient a) zero drift prior to the start of pitching, b) near the point of maximum pitch rate c) at the end of the pitch but forces and moment have come into equilibrium, and d) after steady forces and moments have developed, .
Hydrodynamic Forces and Moments
The forces and moments acting on an axisymmetric bare hull during a very rapid unsteady pitching and plunging maneuver starting from rest in viscous flow can be calculated from the pressure distribution corresponding to the unique velocity potential. In the case of unsteady maneuvers with finite rates of rotation the viscous forces act to produce flow separation which has a dramatic effect on the forces and moments which act on the body. The extent to which forces and moments of an unsteady maneuver can be determined from the underlying potential flow is an important practical question and likely depends on the body geometry and flow boundary conditions.
For the present pitch and plunge maneuver of the SUBOFF unappended hull, the incremental “unsteady” contributions to the potential flow force and moment for the pitching and plunging maneuver are Ref 
Note that we have dropped the Munk moment from equation (16) as this is not considered an unsteady term. In this section we will compare these to the forces and moments determined for viscous flow from CFD simulations.
The longitudinal and lateral forces corresponding to each value of are shown in Figure 11 and Figure 12. At the lowest pitch rate, , the forces make a monotonic transition from purely longitudinal flow to steady translation at, . At and the duration of the pitch maneuver is shorter but there is settling period, , before the forces attain steady translation values. At the highest pitch rate, , the forces overshoot and undershoot the steady translation values followed by a settling period similar to that at lower values of .
The yawing moment for each value of is shown in Figure 13. The moment shows similar trends to the forces at the lower pitch rates although the settling time seems less significant than observed for the forces. At the highest pitch rate the yawing moment develops an oscillation which appears well resolved by the simulation. This oscillation arises in spite of the fact that the kinematics are smooth and include no oscillations of pitch or pitch rate. In the absence of any oscillation in the maneuver one would conclude that the oscillations must originate with the viscous flow.
It is worth noting that the simulated force and moment are temporally smooth at low angles and that high frequency oscillations do not develop in the solution at high angles until these angles have been maintained for a period of time that is dependent on history of maneuver.
The lateral force and the yawing moment are plotted versus the drift angle, , in Figure 14 and Figure 15. This allows a comparison of the transient forces and moments to the steady values in pure translation at the corresponding instantaneous drift angle. The lowest pitching rate case exhibits a difference of the lateral force between the unsteady maneuver and steady translation but no significant difference in the yawing moment. At the higher pitch rates the deviation of both the lateral force and yawing moment from steady translation values become more significant. At the highest pitch rate the lateral force is increased by and order of magnitude and the yawing moment is reduced to the point where the sign is reversed.
To investigate these unsteady effects on the lateral force in viscous flow we will consider the unsteady force that would occur for the same maneuver in potential flow. This will be done by calculating the difference between the CFD obtained lateral force for viscous flow and the unsteady force of potential flow, , as calculated from equation (15). The results for, , are shown in Figure 16. The remainder is surprisingly close to the lateral forces for pure translation at the instantaneous drift angle. This is especially true at small angles of drift during the early portion of the maneuver. At higher angles of drift the force is lower than occurs in a steady translation.
Figure 17 shows difference, ,between the CFD obtained yawing moment for viscous flow and the unsteady moment of potential flow as calculated from equation (16). At low instantaneous angles of drift the yawing moment is very close to that of pure translation. At values of the moment experienced during the transient is lower than the steady translation curve with the adjustment period following the maneuver being significant.
The prescribed motion capability of EXN/Aero allows as input a table of kinematic states comprised of the velocities and rotation rates about the 3 body axis: , at a series of times. EXN/Aero then interpolates the prescribed motion states at each integration time step using cubic splines. A comparison of the forces and moments computed for a prescribed motion table and that of an exact functional expression of the kinematics according to equation (10) is shown in Figure 18.
Figure 14: Lateral forces for the pitch and plunge maneuver as described by equation (10) plotted against the instantaneous drift angle at the centre of buoyancy. Each curve corresponds to a different maximum pitch rate according to equation (13). Circles represent results for steady translation.
Figure 15: Yawing moment for the pitch and plunge maneuver as described by equation (10) plotted against the instantaneous drift angle at the centre of buoyancy. Each curve corresponds to a different maximum pitch rate according to equation (13). Circles represent results for steady translation. Chain line represents Munk’s moment for steady translation and rotation in potential flow.
Figure 16: The difference between the lateral force obtained from CFD simulation of viscous flow and the lateral force for unsteady potential flow according to equation (15). Pitch and plunge maneuver as described by equation (10), Each curve corresponds to a different maximum pitch rate according to equation (13). Circles represent results for steady translation.
Figure 17: The difference between the yawing moment obtained from CFD simulation of viscous flow and the lateral force for unsteady potential flow according to equation (16). Each curve corresponds to a different maximum pitch rate according to equation (13). Circles represent results for steady translation.
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