W.Faller, D.Hess, W.Smith, T.Huang
(Naval Surface Warfare Center, Carderock Division, USA)
Recursive neural networks (RNNs) are a technique for developing timedependent, nonlinear equation systems. Simulation tools based on RNNs have been developed which may assist in addressing a wide range of submarine and ship hydrodynamic maneuvering and control challenges. Faster than realtime, six degreeoffreedom (6DOF) maneuver simulations have been developed for both full scale and model scale submarines using RNNs combined with model scale and full scale trials maneuvering data. The simulations show no appreciable loss of fidelity in predicting severe or emergency recovery procedure maneuvers which include rapid propeller rotation reversal. The results indicate that the RNN simulations provide accurate predictions for submarine maneuvers used to develop the simulations as well as for validation maneuvers. For hundreds of submarine maneuvers, full scale and model scale, the average simulation errors in depth are less than 3 m (full scale), less than 0.25 kts in speed (full scale), and less than one degree in pitch angle and roll angle. Consistent results were obtained across all maneuver types including crashbacks, rise jams, dive jams, rudder jams, turns, vertical and horizontal overshoots. Currently, these simulation tools are being implemented for surface ship maneuvers. In addition, full scale simulations provide the opportunity to eliminate Reynolds number scaling issues from the simulation of submarine maneuvers. Further, the capability to generate simulations of the equivalent full scale and model scale vehicle has provided a number of opportunities for studying the physics underlying vehicle maneuvering. For example, RNN simulations indicate that variations in roll dynamics (roll angular rate and roll angular acceleration) contribute significantly to the observed differences between model scale and full scale submarine maneuvers for emergency recovery procedures. Results from RNN simulations have also provided new insights into the physics of the threedimensional unsteady vortex dominated flow fields which govern the vehicle maneuvers. For six degreeoffreedom problems, traditionally, dynamic similarity has been obtained by nondimensionalizing all variables as a function of a characteristic length (L) and the initial speed U_{0}. However, RNN results indicate that dynamic similarity for maneuvering vehicles should be based on the time varying velocity U(t) and not U_{0}.
CB 
Center of Buoyancy 
CG 
Center of gravity 
BG 
Distance from CB to CG 
L 
Body length, LOA (m) 
L_{stn} 
Length CG to control surfaces (m) 
D_{p} 
Propeller Diameter (m) 
U 
Total velocity (m/sec) 
Euler angles (rads) 

x, y, z 
Displacements (m) 
u, v, w 
Linear velocities (m/sec) 
Linear accelerations (m/sec^{2}) 

p, q, r 
Angular velocities (rad/sec) 
Angular accelerations (rad/sec^{2}) 

α 
Angleofattack tan^{−1}(w/u) 
β 
Sideslip angle sin^{−1}(−v/U) 
δb 
Bowplane deflection angle 
δr 
Rudder deflection angle 
δs 
Sternplane deflection angle 
RPM 
Propeller RPM 
Re 
Reynolds number 
f 
frequency (Hz) 
ω 
2πf 
k 
reduced frequency, Lω/2U_{∞} 
t 
realtime 
Δt 
realtime increment (t_{j+1}−t_{j}) 
t′ 
nondimensional time 
Δt′ 
nondimensional time step (t′_{j+1}−t′_{j}) 
Results derived from a decade of freeflight model maneuvering data, on undersea vehicles, have demonstrated that time and spatially dependent unsteady fluid dynamic effects are large, extremely significant, and must be accounted for when predicting submarine maneuvers and controllability characteristics. In addition, this problem is accentuated by maneuvers which include reverse RPM
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Applications of Recursive Neural Network Technologies to Hydrodynamics
W.Faller, D.Hess, W.Smith, T.Huang
(Naval Surface Warfare Center, Carderock Division, USA)
ABSTRACT
Recursive neural networks (RNNs) are a technique for developing timedependent, nonlinear equation systems. Simulation tools based on RNNs have been developed which may assist in addressing a wide range of submarine and ship hydrodynamic maneuvering and control challenges. Faster than realtime, six degreeoffreedom (6DOF) maneuver simulations have been developed for both full scale and model scale submarines using RNNs combined with model scale and full scale trials maneuvering data. The simulations show no appreciable loss of fidelity in predicting severe or emergency recovery procedure maneuvers which include rapid propeller rotation reversal. The results indicate that the RNN simulations provide accurate predictions for submarine maneuvers used to develop the simulations as well as for validation maneuvers. For hundreds of submarine maneuvers, full scale and model scale, the average simulation errors in depth are less than 3 m (full scale), less than 0.25 kts in speed (full scale), and less than one degree in pitch angle and roll angle. Consistent results were obtained across all maneuver types including crashbacks, rise jams, dive jams, rudder jams, turns, vertical and horizontal overshoots. Currently, these simulation tools are being implemented for surface ship maneuvers. In addition, full scale simulations provide the opportunity to eliminate Reynolds number scaling issues from the simulation of submarine maneuvers. Further, the capability to generate simulations of the equivalent full scale and model scale vehicle has provided a number of opportunities for studying the physics underlying vehicle maneuvering. For example, RNN simulations indicate that variations in roll dynamics (roll angular rate and roll angular acceleration) contribute significantly to the observed differences between model scale and full scale submarine maneuvers for emergency recovery procedures. Results from RNN simulations have also provided new insights into the physics of the threedimensional unsteady vortex dominated flow fields which govern the vehicle maneuvers. For six degreeoffreedom problems, traditionally, dynamic similarity has been obtained by nondimensionalizing all variables as a function of a characteristic length (L) and the initial speed U0. However, RNN results indicate that dynamic similarity for maneuvering vehicles should be based on the time varying velocity U(t) and not U0.
NOTATION
CB
Center of Buoyancy
CG
Center of gravity
BG
Distance from CB to CG
L
Body length, LOA (m)
Lstn
Length CG to control surfaces (m)
Dp
Propeller Diameter (m)
U
Total velocity (m/sec)
Euler angles (rads)
x, y, z
Displacements (m)
u, v, w
Linear velocities (m/sec)
Linear accelerations (m/sec2)
p, q, r
Angular velocities (rad/sec)
Angular accelerations (rad/sec2)
α
Angleofattack tan−1(w/u)
β
Sideslip angle sin−1(−v/U)
δb
Bowplane deflection angle
δr
Rudder deflection angle
δs
Sternplane deflection angle
RPM
Propeller RPM
Re
Reynolds number
f
frequency (Hz)
ω
2πf
k
reduced frequency, Lω/2U∞
t
realtime
Δt
realtime increment (tj+1−tj)
t′
nondimensional time
Δt′
nondimensional time step (t′j+1−t′j)
INTRODUCTION
Results derived from a decade of freeflight model maneuvering data, on undersea vehicles, have demonstrated that time and spatially dependent unsteady fluid dynamic effects are large, extremely significant, and must be accounted for when predicting submarine maneuvers and controllability characteristics. In addition, this problem is accentuated by maneuvers which include reverse RPM
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on the propeller. Further, results from both freeflight model maneuvering data and full scale trial maneuvering data, have failed to provide a quantitative determination of the scaling differences between model and full scale maneuvers. A number of possible sources for these differences between model scale and full scale exist and include, Reynolds number scaling issues, experimental errors, differences in the wetside geometry, variations in mass and moments of inertia, as well as variations in the distance (BG) between the center of buoyancy and the center of gravity.
A determination of the reasons underlying the observed maneuvering differences are complicated by the unsteady fluid dynamics which govern a majority of the vehicle motions. The fluid dynamics for a submarine are characterized by relatively large, timevarying combined anglesofattack (α) and sideslip angles (β). The maneuvers also appear to be strongly influenced by forced unsteady fluid mechanics. The maneuvering vehicle is characterized by pitch and yaw angular rates (q, r) as well as angular accelerations which yield significant values for the reduced frequency parameter (k). Consistent with standard conventions the length constant used to determine (k) was half the boat length (L/2). To analytically estimate the reduced frequency of the maneuvering submarine, representative cases were approximated as harmonic motions.1 To simplify the analysis, the period of the motion (τ) was calculated using a sawtooth wave which is dependent on only the pitch and yaw angular rates. The frequency (f) for the harmonic motion was calculated as 1/τ. For simplicity, the mean amplitude αm was taken to be zero in all cases and only pure pitching or yawing conditions were calculated. Oscillation amplitudes of 5, 10 and 20 degrees, and RCM pitch and yaw rates of 0.5, 2 and 4 deg/sec were assumed. The estimates for the pitch and yaw rates were intentionally conservative so as to underestimate, rather than overestimate, the reduced frequency (k). This approximation is shown schematically in Figure 1. As shown, if the oscillation amplitude αω is halved, the period is halved and the corresponding frequency is doubled. Figure 2 summarizes the range of reduced frequencies as a function of speed (U) obtained from this analysis.
Clearly, based on this analysis a majority of maneuvers have large reduced frequencies (k≫0.1). This indicates that the vehicle motion drives the development of the unsteady separated flow field. The vortical flow field, in turn, gives rise to the forces generated on the hull during these maneuvers. Further, under these conditions, Reynolds number is known to have only a second order effect on the flow field development. For a limited range of data, experiments support this contention.2
Figure 1. Submarine maneuvering approximated as a harmonic motion
αω=5 deg
αω=10 deg
αω=20 deg
Gentle Maneuver
0.5 deg/sec
k=0.67/U
U=7 k=0.1
k=0.33/U
U=7 k=0.05
k=0.17/U
U=7 k=0.02
Intermediate
2.0 deg/sec
k=3.34/U
U=7 k=0.48
k=1.67/U
U=7 k=0.24
k=0.84/U
U=7 k=0.12
Severe Maneuver
4.0 deg/sec
k=6.7/U
U=7 k=0.96
k=3.35/U
U=7 k=0.48
k=1.67/U
U=7 k=0.24
Figure 2. Reduced frequencies (k) for the RCM during maneuvering.
Accordingly, development of simulation methodologies capable of representing unsteady separated flows is of paramount importance if new maneuver simulation tools are to be developed. Prior work indicates that, across an extremely broad range of parameters, both steady and unsteady fluid mechanics can be modeled using recursive neural networks. Techniques have also been addressed for integrating these unsteady fluid dynamic RNN simulations with mechanical actuators to demonstrate the ease with which adaptive control systems might be produced.
Recursive neural network simulations of unsteady boundary layer development, dynamic stall and dynamic reattachment for threedimensional unsteady separated flow fields have been described.3−7 Further, techniques have been examined for integrating these recursive neural network (RNN) reference models within adaptive control systems. Overall, the results clearly demonstrated that RNNs are a highly effective technique for the timedependent simulation of threedimensional unsteady separated flow fields.3−7 Operationally, the unsteady flowfield wing interactions could be predicted for any time period over which the motion history was a known function (a few milliseconds to tens of seconds). Consistent with numerous experimental
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investigations, the simulation results clearly demonstrate that the interactions between the body and the unsteady flow field are highly repeatable. For the neural network controller, the results indicated that unsteady aerodynamic processes can be controlled across a range of unsteady motion histories.6
The successful extension of these RNN technologies to submarine six degreeoffreedom maneuver simulation is the subject of this paper. The work described, herein, demonstrates the capability to use RNN maneuver simulations to provide direct simulations of full scale submarines, to independently simulate model scale submarines, and to use the combination of simulations to facilitate a quantitative determination of the scaling differences observed between model scale and full scale submarine maneuvers.
METHODS
Recursive Neural Networks
Recursive neural networks (RNNs) are a technique for developing timedependent, nonlinear equation systems. The basic RNN architecture is comprised of an input vector or layer, hidden layer(s) and an output layer as shown in Fig. 3. Each layer is comprised of a set of units or processing elements at which information is processed. Information processing proceeds from left to right, with each input vector yielding a corresponding output vector. Recursive connections, or recurrent feedback, are utilized where the predicted outputs become part of the next input vector. By adding recurrence between the input and output layers timedependence can be incorporated within the RNN solution.
Figure 3. Overview of the recursive neural network architecture for maneuvering simulation.
Information processing within each layer of the RNN occurs at many simple processing units or elements, each of which is described by an activation function. The activation function determines the inputoutput relationships of each processing unit. The sigmoidal activation function described in Eqn. 1 is perhaps the most common nonlinear activation function, and is the activation function utilized in all the RNN maneuver simulations. As shown, the output of the activation function (y) is a function of the input (x).
(1)
Each processing unit is connected to other processing units via adjustable weights (coefficients). Typically, the number of processing units in the input and output layers are defined by the problem, whereas the number of hidden layers as well as the number of processing units per hidden layer is user defined. For a fully connected RNN, each processing unit in one layer connects to every processing element in the next layer. The input (x) to a processing element is described by Eqn. 2, and the corresponding output (y) of that processing element is given by Eqn. 1. For each unit, the output side (y) is defined by the subscript (i) and the input side (x) by the subscript (j). This process is carried forward from the input vector to the hidden layer(s) and from the hidden layer(s) to the output layer. Thus, the input vector is transformed nonlinearly into an output vector. The relative ease with which coupled timedependent, nonlinear inputoutput relationships can be calculated is one of the significant strengths of the RNN approach. However, to utilize this capability requires a method, the learning algorithm, for adjusting the weights (coefficients) connecting the processing units such that a given time history of input vector yields the correct predicted output time history.
(2)
The RNN weights are adjusted by presenting a sequence of training vectors (inputs) to the RNN and calculating the RNN outputs. Since the error is the difference between the target (measured) values and the RNN predicted values, this process requires data sets where both the input and output vector time histories are known. The calculated error is then backpropagated, using a gradient descent method, to adjust the weights such that the error will be minimized. By iteratively repeating this process over a number of inputoutput data sets the weights can gradually be adjusted such that the RNN becomes an accurate time dependent, nonlinear transfer function between the input and output time histories.
Once the RNN training is complete, the final weights (coefficients) are fixed. For a fully
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converged solution, the accuracy of the RNN simulation can then be judged in one of three ways. For a parameter space (x,y) to be simulated, training data would consist of a limited set of (x,y) pairs. Following training, the prediction accuracy can first be tested for data sets on which the neural network was trained. Second, the RNN predictions can be checked for cases which are within the parameter space, but which were not used during training. Thus, the RNN capability to generalize (interpolate) within the parameter space can be determined. Since neural networks are typically designed to operate within a predefined parameter space, this is the most critical test and is often used to validate the final RNN solution. Third, RNN predictions can also be tested against data sets which are outside of the training parameter space. Thus, the RNN capability to generalize (extrapolate) outside of the parameter space can be tested. The limitation on testing RNN extrapolation, as well as extrapolation for other simulation techniques, typically occurs because the experimental data sets required for comparison do not exist.
Recursive Neural Network Solutions
Recursive neural networks are a method for developing timedependent, nonlinear equations which describe the behavior of complex systems. As a framework within which to consider the RNN solution, assume as a starting point a linearized state equation where the output vector of the equation system might be the vehicle dynamics. If the control variables are described by a vector and the state vector by then can be calculated as given by Eqn. 3. Further, each state vector is a function of the prior states and control inputs as given in Eqn. 4.
(3)
(4)
However, for complex nonlinear problems, a solution of this form typically cannot account for all of the underlying physics which would be required to achieve an exact solution. As such, each predicted state vector will include some error. This error can be defined as the difference between what the mathematical simulation predicts and what the actual measured response of the physical system would be to the same control inputs. Not surprisingly, as the vehicle maneuvers to be simulated become increasingly more complex the magnitude of the error terms tend to increase. These error terms can be accounted for in the mathematical solution by rewriting the state vector as a function of the predicted state plus the error term ēj+1. This new state vector can be denoted as ŷj+1, Eqn. 5, and the corresponding state equation as shown in Eqns. 6 and 7. For simplicity, the assumption is that errors in the predicted states do not change the control inputs. Clearly, this would not be the case if a feedback control system was being utilized.
(5)
(6)
(7)
Further, since the error terms are not independent of each other, one difficulty encountered with such solutions is that as a first approximation the error term ēj+1 can roughly be assumed to sum or accumulate over time. As such, the error in the final state vector ŷn can be approximated by the total error (E) given in Eqn. 8. In general, this total error (E) increases proportionally as a function of both the complexity of the vehicle maneuver to be simulated and the maneuver duration (length of time).
(8)
Further, since ŷj+1 may correspond to the vehicle accelerations, the integration of the acceleration terms, which include the error ēj+1 at each time step, yield vehicle velocities which typically have correspondingly larger errors than the accelerations. In turn, the subsequent integration of the velocities to yield the vehicle trajectory and attitude can further magnify the differences which may be observed between the simulation predicted and the measured response of the vehicle. The same difficulties exist for simulations which directly calculate the forces and moments and then solve for the accelerations.
While the matrices A and Β are not separate in the RNN, for the purpose of discussion, the RNN solution can be considered to have the form shown below. The control vector ûj is also a function of the state vector ŷj and is given by Eqn. 9. For the RNN, the A and Β matrices also vary as a function of time, and are a function of both the control vector ûj and the state vector ŷj as given by Eqn. 10. The predicted state vector ŷj+1, in turn, is a nonlinear function of
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these timevarying matrices and input vectors as given by Eqn. 11.
(9)
(10)
(11)
A critical issue in simulation is that, for all practical purposes, an exact solution which would have error terms ēj+1 equal to zero does not exist. As such, simulation fidelity is dependent not only on predicting the state vector, but perhaps more importantly on effectively dealing with and minimizing the predicted error terms ēj+1. A significant advantage, which intentionally has been built into the RNN paradigm, is that the RNN equation system can be developed such that ŷj+1 is the correct predicted state. In other words, the RNN equation system can operate in a fashion where the correct predicted state is the measured value plus a small error ēj+1. The error may, or may not be, zero at any time step. However, as long as the error remains small, the RNN simulation, for all intents and purposes, predicts the state vector that would correspond to the measured response of the physical system to the same control inputs. As such, unlike other solutions, as a first approximation the predicted error terms for a converged solution do not integrate over time. The total error (E) for an RNN simulation typically has a magnitude similar to the magnitude of the predicted error at any time step ej, Eqn. 12.
(12)
This is significant because any error in the predicted state vectors is both small and roughly a constant over time. Further, in general, the errors in the state vectors do not increase as a function of the complexity of the vehicle maneuver or the maneuver duration. In addition, as discussed elsewhere1, since ŷj+1 within the RNN simulation can correspond to the six state variables (linear velocities and angular rates), the subsequent integration to yield the vehicle trajectory and attitude typically results in relatively small differences between the RNN simulation predictions and the measured response of the vehicle. As described below, across hundreds of submarine maneuvers, full scale and model scale, RNN simulations yield errors in depth which on average are less than 3 m (full scale), less than 0.25 kts in speed (full scale), and less than one degree in pitch angle and roll angle.
Experimental Data
Submarine maneuvers have been obtained from two sources; full scale submarine trials and freeflight radio controlled model (RCM) experiments. Full scale submarine trials have the advantage that no scaling issues need to be addressed. However, as described below, three of the state variables (u, v and w) must be estimated since no direct measurements are typically available. The RCM can provide a much broader set of measurements, including forces and moments on both the control surfaces and propeller, but maneuvering differences due to scaling issues between model scale and full scale must then be addressed. The Reynolds number is approximately 107 for the RCM based on total body length versus 109 for the equivalent full scale submarine.
The RCM was designed to provide an experimental testbed for determining the steady and unsteady fluid mechanics effects of vehicle maneuvering. The RCM is scaled from the full scale vehicle based on the linear scale ratio of the vehicle lengths (Lship/Lmodel). Using the RCM, a full range of motion and control measurements are available across all maneuver types. Figure 3 depicts the basic RCM functional arrangement. Data is collected from on board sensors which include vertical reference gyros, linear and angular accelerometers. All sensor data is digitized and stored either on board or telemetered to shore in real time via a Pulse Code Modulated data uplink. In addition to the on board data sensors an external acoustic tracking system is used to provide x, y, z tracking of the timevarying displacements (trajectory) of the model. A detailed explanation of the experimental system including both the onboard instrumentation as well as the acoustic tracking system has previously been provided.8 Figure 4 shows the reference coordinate system (righthand rule) with (z) being positive downwards. Table 1 summarizes the experimental and analytically derived six degreeoffreedom (6DOF) data obtained for a maneuvering vehicle.
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Figure 3. Overview of RCM operation in the Maneuvering and Seakeeping Basin (MASK).
Figure 4. Reference coordinate system
Table 1. Data obtained for six degreeoffreedom maneuvers
U
(Total velocity)
(Linear vels and accels)
(Angular vels and accels)
(Euler angles)
x, y, z
(Displacements)
To develop the RNN maneuver simulations, the 6DOF state variables [u, v, w, p, q, r] as well as the accelerations the timevarying Euler angles and the trajectories [x, y, z] were obtained from the RCM during severe maneuvers.8 In addition, maneuvering data were obtained from full scale submarine trials. To utilize the full scale trials data the three state variables (u, v, w) were estimated using a set of three feed forward neural networks.
Full scale trials conducted in the open ocean are able to accurately measure depth (z), speed (U), the Euler angles (φ, θ, ψ), and the angular velocities (p, q, r). The angular accelerations may be derived from the measured data. However, measurements of the trajectory (x, y), the linear velocities (u, v, w), and accurate measurements of the linear accelerations are unavailable. The RCM provides accurate measurements of the depth (z), speed (U), the Euler angles (φ, θ, ψ), and the angular velocities (p, q, r). In addition, the RCM is operated in a facility equipped with an acoustic tracking system which permits accurate time histories of the trajectory (x, y) to be obtained during each maneuver. Since measured trajectory data and depth are available, the linear velocities (u, v, w) as well as the linear accelerations may be derived for the RCM. This data is then used to train feed forward neural networks to predict the linear velocity components as a function of variables which are measured at both model scale and full scale. The technique results in a set of three nonlinear equations, feed forward neural networks, which can be used to decompose the total velocity (U) into the three linear velocity components.
Specifically, three neural networks are utilized for the prediction of u, v and w, respectively. In each case, the input variables are U, q, r, φ and θ which are variables that are measured for both RCM and full scale maneuvers. The predicted neural network output is u, v or w. This is shown schematically in Fig. 5. Each of the three networks is trained using RCM data for which both the inputs and outputs are known. Posttraining, the trained neural networks can then be used with full scale data as the inputs to provide time histories for the full scale linear velocities (u, v, w). In this manner, the feed forward neural networks can serve as virtual sensors for the estimation of u, v and w for the full scale trials data.
Figure 5. Feed forward neural network used to estimate the linear velocity components for full scale maneuvers. One neural network is constructed for each velocity term u, v, and w, respectively.
After predictions for u, v and w have been acquired, the velocity magnitudes can be refined using known mathematical relationships between measured and predicted values. Although no tracking system is available to provide x and y, depth, z, is measured during full scale trials by means of pressure sensors.
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Therefore, the predicted velocity components u, v, and w are transformed from the body coordinate system to the inertial coordinate system using a direction cosine matrix formed using the measured Euler angles and then integrated to obtain predictions for x, y, and z. Then, the measured time history for z is substituted in place of the predicted depth and the process is reversed to obtain revised predictions of u, v, and w. These corrections are small with the largest correction in the w component. The overall speed of the vehicle U is also a measured quantity. Since, among the velocity components the largest contributor to U is u, we can use Umeas to further improve the u component prediction as given in Eqn. 13.
(13)
Further, these steps ensure that the u, v and w estimates are mathematically consistent with those measured full scale variables that exist. The minor corrections introduced by this refinement process have enhanced the u and w predictions at the expense of the v prediction; any remaining error is expected to be in the v component. A second correction algorithm which permits the error in lateral velocity (v) to be corrected is currently under development.
Prior to the refinement procedure, the ability of the neural networks to predict the linear velocities can be judged using RCM data. The following steps were carried out for each of the three neural networks. After obtaining a predicted value from a neural network at a given time step, the predicted and measured values are scaled to full scale magnitudes and then several error measures are computed at each time step. The error measures at each time step are then averaged over all of the time steps during the course of the maneuver. Representative error measures for each neural network are given in Table 2. In each case, the average angle measure (AAM) is quite close to 1.0 indicating that the comparison of the relative shapes and magnitude of the predicted and measured time histories for each velocity component is excellent. The absolute errors are reported in m/sec and correspond to full scale values. Again, the results indicate that the neural networks provide accurate estimations of the linear velocity components u, v and w.
Table 2. Neural network predictions, prior to the refinement procedure. All error values are full scale.
AAM
Absolute Error
u
1.00
0.01 m/sec
v
0.98
0.03 m/sec
w
0.95
0.03 m/sec
Representative errors for the linear velocity components after the refinement process are given in Table 3. Again, the absolute errors are reported in m/sec and correspond to full scale values. The results indicate that the refined estimates provide improved predictions for u and w, without significantly changing the prediction of v. The post refinement linear velocity components are used to complete the full scale trials maneuvering data.
Table 3. Neural network predictions, after the refinement procedure. All error values are full scale.
AAM
Absolute Error
u
1.00
0.01 m/sec
v
0.98
0.03 m/sec
w
1.00
0.01 m/sec
Overall, the method described provides an accurate, mathematically consistent and reliable means for estimating the remaining three state variables which are required for a complete description of the six degreeoffreedom motion. The accuracy of these estimates, as judged by the ability to recover the known full scale measurements is excellent. Further, this technique enables the use of full scale trials data for the development of RNN simulations of full scale submarine maneuvers. The full scale RNN simulation results, described below, indicate that the accuracy of this method is sufficient to permit accurate simulations of full scale submarines to be developed.
Application Of RNNs To Maneuver Simulation
Figure 6 schematically shows the RNN 6DOF maneuver simulation. To function as a maneuver simulation requires that the only external inputs to the RNN be the timevarying control signals and the initial conditions at time (t0). The inputs (controls) were the propeller RPM, the rudder deflection angle, the sternplane deflection angle(s) and the bowplane deflection angle. The component force
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modules, described in more detail below, act on these controls and provide the inputs, as forces, to the RNN. The predicted outputs are the timevarying 6DOF state variables [u, v, w, p, q, r]. To model the timedependence of the vehicle dynamics, which is reflective of the timedependence of the unsteady fluid mechanics, the predicted outputs (state variables) were recursed, fed back, as inputs to the RNN throughout the vehicle motion history. The accelerations the Euler angles and the trajectories [x, y, z] are calculated internally. Further, as described below, the RNN simulations are based on a novel formulation of dynamic similarity.
Figure 6. The RNN six degreeoffreedom maneuvering simulation. The inputs to the RNN are the propeller RPM and plane deflection angles. The RNN outputs are the six state variables (u, v, w, p, q, and r) from which the displacements (trajectory) and angles can be derived.
For six degreeoffreedom maneuver simulations, traditionally, dynamic similarity has been obtained by nondimensionalizing all variables as a function of a characteristic length (L) and the initial speed U0. However, the known physics of unsteady fluid mechanics suggests that to maintain dynamic similarity for maneuvering vehicles all variables, including time, should be nondimensionalized based on the timevarying velocity U(t), rather than U0. The reasons for this can be summarized as follows. The reduced frequency for the majority of maneuvers is large (k≫0.1). The forces which, in turn, determine the accelerations predominantly result from the interaction between the far field, threedimensional vortical flow field and the surface of the hull. The vehicle can change speed drastically during a maneuver. The redistribution of vorticity in the far field is governed by convective fluid motions, and convective fluid motions scale as a function of the timevarying velocity U(t) and not the initial speed U0. The general solution for dynamic similarity can be shown to be based on U(t).1 Dynamic similarity based on U0 is a special case of the general solution, and occurs when U(t) equals a constant. As such, a new formulation of dynamic similarity was developed for the RNN simulations based on the timevarying velocity U(t).
Further, analyses based on RNN maneuver simulations support the contention that dynamic similarity should be based on U(t). If one compares the three possible forms of solutions which can be derived for RNN simulations; (1) simulations based on realtime, (2) simulations based on scaling all variables using U0, and (3) simulations based on scaling all variables using U(t) the following results are observed for the simulation predictions of the vehicle maneuvers. Simulations based on realtime are of poor fidelity. Simulations based on dimensionless variables calculated using U0 are limited to good simulation fidelity for maneuvers which all have, within some finite range, the same initial speed U0. This problem is accentuated by maneuvers which involve reverse RPM on the propeller. Simulations based on dimensionless variables calculated using U(t) show good simulation fidelity across all maneuvers, including maneuvers which have negative RPM on the propeller. The calculation of dynamic similarity, based on U(t), includes both nondimensional time as well as the nondimensionalization of all other variables and is summarized below.
For reference, nondimensional time is traditionally calculated as shown in Eqn. 14. This of course, assumes that the velocity of the vehicle does not change. While initially it may seem that the formulation for nondimensional time based on U(t) would be given by Eqn. 15, this does not turn out to be the case. Since within the simulation the nondimensional timestep Δt′ must be equal to a constant, the problem which occurs is that time, as well as nondimensional time, must sequentially march forward into the future. However, if during a maneuver the value of U(t) decreases rapidly it is quite possible to get the following situation the calculated value is smaller than the previously calculated value of nondimensional time, an unacceptable situation. Therefore, nondimensional time based on U(t) must be formulated as shown in Eqn. 16. The conversion back to real time is given by Eqn. 17. Note, the traditional form of nondimensional time given by Eqn. 14 is a special case of Eqns. 16 and 17 and occurs when U(j) equals a constant. The critical issue for dynamic similarity, in time, is that Δt′ must always be equal to a constant. Therefore, must always be equal to a constant. To reiterate, the general formulation for dynamic similarity, in time, which captures the known physics of the problem, is given in Eqns. 16 and 17. For the RNN maneuver simulations, a nondimensional step size of Δt′ equal to 0.05 yields
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accurate solutions for maneuver simulations in the speed range between 5 knots and flank speed (full scale). For lower speed maneuvers a smaller nondimensional step size, on the order of Δt′ equal to 0.0125 to 0.025, is required.
(14)
(15)
(16)
(17)
The same considerations apply to the nondimensionalization of the velocities and accelerations. To maintain dynamic similarity based on U(j) requires that the remaining variables be rendered dimensionless as shown in Eqns. 18–21. A more detailed description of this process, including the exact formulation required to implement these changes within the RNN simulation code, has previously been given.1
(18)
(19)
(20)
(21)
Control surface deflections which serve as inputs to the RNN were formulated as forces based on the effective or local angleofattack. For any control surface, the force Fj used as the input to the RNN is given in Eqn. 22. The induced angleofattack is determined from the local velocity (v or w) and the rotational rate (q or r) as shown in Eqn. 23. The full equations have previously been described in more detail.1 Critically, the coefficient values for C1C3 can be set equal to one. This formulation, in general, yields the correct force timehistory, but not necessarily the correct magnitude. If the force histories are known, then the coefficients required to match both the time history and magnitude can readily be determined. Further, this simple approximation appears to be adequate for both all moveable as well as flapped control surfaces. The RNN input term for the propeller force (RPM′) is given in Eqn. 24. The length constant (Dp) is the propeller diameter. The inputs to the RNN for the forces on the hull were based on the angleofattack and sideslip angle as given in Eqns. 25 and 26. The full details of the input terms, the RNN formulation as a 6DOF solver, the RNN outputs and subsequent reconstruction of the trajectory and attitude predicted by the simulation have previously been described in detail.1
(22)
(23)
(24)
(25)
(26)
To train the RNN a subset of the available experimental data records, encompassing all maneuver types, were used to “teach” the neural network the relationship between the timevarying control signals and the timedependent vehicle dynamics. The remaining data sets were used to validate the RNN simulation and demonstrate a general solution across all maneuvers. Following development, no additional changes were made to the RNN simulations; the weight matrices were fixed and the control inputs for both training and validation data sets were simply presented to the RNN 6DOF maneuver simulation. The complete RNN equation system as well as the learning algorithms required for training the simulation are beyond the scope of this paper, but have previously been described in detail.1 Using this paradigm, both RCM and full scale submarine simulations have been constructed. Representative results for both model scale and full scale submarine simulations are described below.
In addition, for specific submarines, full scale and model scale simulations have been constructed for the same submarine. The full scale simulation(s), for these cases, were developed based on the full scale submarine trials maneuvering data. The model scale RCM simulation(s), for these cases, were developed using the RCM maneuvering data.
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Following the development of the simulations, comparisons could then be drawn between any of the RNN simulation results and any of the maneuvering data from either full scale or model scale. This is shown schematically in Fig. 7. The RNN simulation of the full scale submarine can be compared to the full scale trials data, the model scale data or the model scale RNN simulation. Similarly, the RNN simulation of the model scale submarine can be compared to the model scale data, the full scale trials data or the full scale RNN simulations. Further, within the RNN maneuver simulation any of the control sequences or state variables can be manipulated. This permits new maneuvers to be studied, the effect of errors in the timing and/or magnitude of control sequences on the ensuing submarine maneuvers to be determined, and provides the opportunity to quantitatively study the scaling differences between model scale and full scale submarines.
In particular, for one such scaling study, the normally observed differences in the steady state roll angle could be matched between the RCM and full scale by altering the roll angle at time (t0) to match either the corresponding RCM or full scale roll angle . Thus, RCM simulations could be run with the full scale steady roll angle, and full scale simulations could be run with the RCM steady roll angle. These simulation results could then be directly compared to the experimentally measured maneuvers. Similarly, for the same study, any of the vehicle dynamics could be altered by changing the values within the simulation. For example, since the roll dynamics of the model scale vehicle are known at each time step, and correspond in a onetoone fashion with the full scale roll dynamics one can readily substitute the model scale rolling behavior for the corresponding full scale values. In particular, the submarine roll angular rate and roll angular acceleration can be matched between the RCM and full scale by substituting the RCM roll dynamics into the full scale simulation at each time step, and the full scale roll dynamics into the RCM simulation at each time step. It should be noted, that this direct substitution approach is only applicable to correlation maneuvers where there is a onetoone match in the control sequences and time histories of both the model scale and full scale submarine. Critically, these changes not only alter the roll characteristics, but effect the entire submarine dynamics by acting on the other variables throughout the time history of the simulation. These simulation results can then be directly compared to the experimentally measured maneuvers. Thus, the effects of roll on full scale maneuvers as compared to RCM maneuvers can be directly determined. Similarly, the effects of roll on model scale maneuvers as compared to full scale maneuvers can be directly determined.
Figure 7. The RNN six degreeoffreedom maneuvering simulations. The full scale and model scale simulations can be compared directly to the experimental data and/or to the simulation results.
RESULTS
RNN Model Scale Simulation
The maneuvering simulation results, shown herein, were developed in two ways. First, one RNN 6DOF simulation was developed for all maneuver types. Second, a single RNN 6DOF simulation was separately developed for each individual maneuver type (crashbacks, rise jams, dive jams, rudder jams, turns, vertical and horizontal overshoots). The speed of these maneuvers ranged between 12 kts and flank speed. For each case, the results are categorized as errors for training, RNN development, data sets, and errors for the validation maneuvers which had not been previously “seen” by the simulation. As shown below, the full simulation compares favorably to the individual simulations, and indicates that only one RNN is required to provide an accurate simulation for all maneuvers.
To judge the accuracy of the simulation, all predicted and measured variables were coplotted; velocities, accelerations, Euler angles, displacements and control signals. In addition, the following error measures were calculated for all variables; the absolute error in magnitude and the average angle measure (AAM) were calculated using the measured maneuvering data and the RNN simulation predictions. The average angle measure has a value of one in the best case, a perfect simulation of the maneuver, when the predicted and measured values are
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exactly the same. Conversely, for very poor predictions, the average angle measure has a value of zero. The exception to this occurs when the signal is of small magnitude, near zero, for the average angle measure. The absolute error in magnitude was simply calculated as the absolute value of the difference between the measured and predicted values summed over each data point and then normalized by the number of data points. Since the predicted state variables are integrated to calculate the trajectory and attitude, the maximum error typically occurs at the end of each maneuver, and can be assumed to typically have a magnitude of roughly twice the absolute error in magnitude.
All of the model scale results have been scaled to the equivalent full scale distances, velocities or angles as appropriate. As such, all error measures were calculated based on the errors which would be expected for full scale submarine maneuvers. An error in depth of 3 meters, for example, is a full scale error not a model scale error. Herein, only the quantitative results based on the calculated error measures are given. All of these values are consistent with the plots, and indicate that the simulation predictions agree well with the experimentally measured data. For the roll angle, the low values obtained for the average angle measure can in part be attributed to a small offset, near zero, between the measured and predicted values. For these cases, the absolute error may be more representative of the true error observed between the predicted and measured roll angle.
Listed below are results computed over all data sets, for a typical submarine simulation. The maneuvers consisted of crashbacks, rise jams, dive jams, rudder jams, turns, vertical and horizontal overshoots. The results are summarized for training data sets in Table 4 and validation data sets in Table 5. The data set was comprised of a total of 250 maneuvers of which 184 were used to develop the RNN and 66 were used to validate the simulation. The results show that the RNN simulation accurately predicts the vehicle maneuvers. Across all 250 maneuvers, the average error in depth was less than 3 m (full scale), less than 0.25 kts in speed (full scale), and less than one degree in pitch angle and roll angle. However, the simulation is not perfect; of the 250 maneuvers roughly 5–10% of the predicted maneuvers scored measurably below the average values. Similarly, roughly 5–10% were predicted measurably better than the average values. Significantly, the results for data sets used to validate the simulation, Table 5, are consistent with the data sets used in development, Table 4. This indicates that the simulation, to the extent experimental data exists to compare with, is a general solution across all of the maneuvers.
Table 4. Simulation results for all maneuvers. Full scale error measures for the training data sets. (184 data sets)
AAM
Absolute Error
Speed (U)
1.00
0.16 kts
Roll Angle
0.58
0.52 deg
Pitch Angle (θ)
0.83
0.93 deg
Depth (z)
0.98
2.29 m
Table 5. Simulation results for all maneuvers. Full scale error measures for the validation data sets. (66 data sets)
AAM
Absolute Error
Speed (U)
0.99
0.21 kts
Roll Angle
0.53
0.69 deg
Pitch Angle (θ)
0.83
1.12 deg
Depth (z)
0.97
2.47 m
Representative results for single maneuver RNN simulations are given in Tables 6–9. The results for both controlled and uncontrolled crashbacks are given in Tables 6 and 7. The results for rise jams are shown in Tables 8 and 9. Consistent results were obtained for all other maneuver types. In each case, the number of data sets used to validate the simulation as well as the number of data sets used in development are given. These numbers vary and are a function of the available total number of data sets for each type of maneuver. The results show that each individual simulation accurately predicts the specific vehicle maneuver. In all cases, the results for data sets used to validate the simulation are consistent with the data sets used in development. As would be expected, in general, the results for the validation data sets indicate that the completely novel cases are not predicted quite as well by the RNN simulation as the training data sets. However, the differences do not appear to be significant. More importantly, the results for the full simulation, based on one RNN, compare favorably to the individual simulations. This indicates that a single RNN maneuver simulation can be used for all maneuvers.
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Fig. 11a Histogram of the ligament angle (mean value=76 deg.).
Fig. 11b Mean values of the ligament angle for various combinations of (k/ho and We).
Fig. 12 Histogram of the wave number for all tests (mean value=1.75).
Fig. 13 Mean values of the drop Reynolds number for various combinations of (k/ho and We).
Fig. 14 An actual ligament (left) and its idealized form (right).
Fig. 15 Normalized velocity and the rms values (×10) of the velocity fluctuations.
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DISCUSSION
G.Hearn
University of Newcastle Upon Tyne, United Kingdom
This is the second occasion during the conference in which Prof. Sarpkaya has provided physical insight regarding natural phenomena. Since it is from such physical insight that one ultimately seeks to provide a mathematical model, and hence engineering design capability, may I congratulate and thank you for today’s very clear, precise and fundamental explanation of the liquid wall jet phenomenon.
May I now ask a question related to the spray formation at the bow of the fast moving craft? In your experimental set up, the jet is brought into contact with flat or curved plates. Considering the bow as a complicated wedge, would you expect any of the characteristics of the jet to be radically modified? In other words, will it be necessary to modify your apparatus further?
DISCUSSION
J.W.Hoyt
San Diego State University, USA
I would like to congratulate Prof. Sarpkaya for a marvelous paper with spectacular photographs. My question is: Why do the spray ligaments lean backwards?
AUTHORS’ REPLY
The questions and kind words of Professors Grant E.Hearn and Jack W.Hoyt are very much appreciated.
Regarding the effect of bow geometry, we have carried out a number of additional experiments during the past two years with symmetrical wedges (upright or inclined). The results of these experiments were not reported in the paper partly because of their more casespecific nature and partly because there was much to be learned from experiments idealized enough to reveal physics and broad enough to be relevant. Suffice it to note that even though specific characteristics of the filaments and droplets did not significantly change, the distance at which they have occurred has changed with the angle of wedge and its forward to backward inclination. There is no doubt that the complicated geometry of a bow may affect not only the filament/droplet distribution but also the position along the bow where the wall jet becomes a free jet. The bowshape effects, the consequences of increasing or decreasing the jet thickness, and the use of sea water in lieu of tap water are now being investigated systematically.
As to the reasons for the ligaments leaning mostly backwards, one may imagine a ligament as a cylindrical eddy of vertical extent h(x) ejecting with a relative average velocity v while its submerged part is subjected to a relative average velocity of u in the streamwise direction. This should lead to mostly backward leaning filaments. If, with some poetic license, one assumes that v is about four times u, then the angle of inclination from the vertical becomes about 14 degrees. This, in fact, is the average of all the filament angles we have measured. Simplistic though the explanation may be, it is hard to find a better one. Again, we thank Professors Grant E.Hearn and Jack W.Hoyt for the opportunity they have accorded to us to expand on two very important questions.
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Internal Gravity Waves Excited by a Body Moving in a Stratified Fluid
V.Borovikov,1,2 V.Bulatov,2 M.Gilman,2 Y.Vladimirov2
(1Benemerita Universidad Autonoma de Puebla, Mexico, 2Russian Academy of Sciences, Russia)
The aim of this talk is to describe the useroriented computer code composed for the numerical calculation of the field of internal gravity waves (abbreviated as IW hereafter) excited in a layer of arbitrary stratified fluid by a movig thin solid body. The linear approximations for the equations of IW and for the boundary conditions at the body’s surface are used. The IW field is expressed in terms of integrals from the Green’s function of IW equation. The code uses the analytic and asymptotic expressions for the Green’s function obtained in (1,2,3) and summed up in (4). To obtain the soughtfor values of IW field it is sufficient to introduce the set of points in which we seek for the field; the table of density values as a function of depth, the table describing the body’s shape; and its velocity and depth. The time necessary to calculate by PC the field in fixed point has order of few seconds. Since the calculations are performed with a guaranteed accuracy, these codes can be used not only for comparing theoretical calculations with experiment but for testing more complicated codes using nonlinear equations in particular case of linear approximation. The algorithm of IW field evaluation and some examples of the IW numerical calculation are depicted below.
1. INTRODUCTION.
We study the internal gravity waves (IW) excited by the moving thin solid body. Just as for surface ship waves, the internal wave field has suffitiently far off the body a Vshaped structure, but, in contrary to ship waves, the opening angle of the V depends on the stratification and the vessel speed.
A selfpropelling moving underwater object create a turbulent wake. The wake is formed behind the body, and is expanding in time, both horizontally and vertically. Further this wake collapses at a certain distance downstream and produces a significant internal wave field. This field expands both horizontally and vertically. The numerical simulation of such wake collapse is difficult, and requires a large amount of computer capacity (5,6,7). Such a model is outside the scope of this numerical facility. The object here is to calculate the IW wave field excited in result of streamline flow around the moving solid body. The body is introduced here as a nonflow condition at the body’s surface. Works (6,7) discuss the approach in which the IW excited during the collapse of the turbulent wake are approximated by the IW excited by the streamline flow around this wake treated as a moving solid body. It ia showed in these papers that such approximation is suffitiently reasonable and can indicate, for example, under which external conditions we are most likely to find internal wave signal in the real ocean from a moving object and its turbulent wake.
The model used below is based on the Boussinesq approximation. We assume that the wave field is stationary in the moving body reference frame. By assuming stationarity we do not describe transient wave phenomena, but this restriction is not very serious since the transient field rapidly decay (see (6,8)).
There is a great number of works devoted to theoretical, numerical and experimental investigations of the internal wave fields. The reader is referred to, among others, the papers by Miles (9), Keller and Munk (10), Janowitz (11), Meng and Rottman (12), Hopfinger (5), Voisin (13). Most theoretical works deal with the motion of rigid body of specified shape (cylinder, sphere, spheroid, etc.) in uniformly stratified fluid (13); are asymptotic and
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rely on farfield or largetime hypotheses. Often, the internal wave field excited by moving body is described by a field excited by system of point sources taken with a certain weighting and then the problem for exponentially varying density distribution is solved analytically (14,11,13).
For arbitrary density distribution, it is possible to solve this problem numerically. The shortcoming of this approach is the boundness of the space region in which the problem can be solved numerically. For example, in (6) the approach is presented which is based on the separation of variables, on the numerical Fourier method over the horizontal variables and on the using of the vertical eigenfunctions. It appeared to be necessary to sum over about Fourier components for field calculations on typical scale meters, number of taken into account vertical modes and horizontal body scale meters. Moreover, the numerical methods do not readily lead to the qualitative flow description.
We calculate the IW generated by an underwater thin body horizontally moving with a constant speed in a layer of stratified fluid. The linear approximation for the IW equations and boundary conditions on the body surface are used. The problem is solved by means of Green’s function of the IW equation. The solution obtained in this way is a sum of modees. The nth mode is the trhreefold integral whose integrand is expressed in terms of nth eigenfunctions and eigenvalues of corresponding vertical spectral problems. This method makes it possible to obtain the simple forms of the solution far from and near to the body. The easytointerpret qualitative description of the calculated IW field structure is possible. These asymptotic representations provide accurate approximation to the full solution except in a narrow intermediate zone, in which triple quadratures must be numerically calculated.
2. PROBLEM STATEMENT AND ITS SOLUTION IN TERMS OF GREEN’S FUNCTION.
Consider a stationary IW field generated by a moving body in layer of stratified fluid with arbitrary BrüntVäissälä frequency N(z)=−g∂ ln ρ/∂z (here g is a freefall acceleration, ρ=ρ(z) is a fluid density). The free surface z=0 is replaced by a rigid lid, and the Cartesian reference frame is such that the plane x, y coincides with the surface. We assume that the body starts moving at t=0 along the x axis with the constant velocity V which is greater than the maximal group velocity of IW propagation. Furthermore, we assume that internal Froude number Fr=V/Nh* (where h* is a maximal body height) is much greater than unity. Then the fluid particles may overcome the buoyancy potential and rise to the body height, so the potentialflow assumption holds. Therefore the pattern of fluid particles trajectories near to the body must be qualitatively the same as in the case of body motion in nonstratified fluid (6,15).
The vertical component W of velocity perturbation in incompressible stratified fluid outside the body satisfies the following equation obtained from a linearized system of hydrodynamic equations in the Boussinesq approximation (16):
(1)
where
The initial condition is
(2)
The boundary conditions are stated at the upper surface, at the body depth and at the bottom. At the upper surface we assume the rigid lid condition
(3a)
This assumption means that we do not treat the surface wave field, but we retain the surface divergence/convergence effects because we have not assumed ∂W/∂z to be zero. At the bottom we have obviously
(3b)
where Η is the stratified layer thickness.
Consider the condition at the body horizon. The body shape is described by the two smooth functions h±: Z(x, y)=z0±h±(ξ, y), where ξ=x+Vt, z0 is the body depth, and the functions h± describe the body surfaces higher and lower than z0 respectively.
We assume the noflow condition at the body surface:
where U1, U2 are the velocity perturbations in the x and y directions, respectively. On the assumption that h±(x, y)≪1, U1,2≪1 this boundary
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condition may be linearized with respect to h± and ∂h±/∂x and transferred to the depth z=z0 (6):
(3c)
This linearization assumes both h± and ∂h±/∂x to be small. If ∂h±/∂x is of the order of unity, then the perturbation velocity W is of the same order as the speed V and the linear model breaks down. The range of the applicability of this linearization is discussed in (6).
We seek the function W, which has a discontinuity
(4)
The solution of the problem (1)–(3a,b),(4) allows us to carry out the verification of the approximation (3c). For this purpose it is necessary to solve this problem numerically and then to draw calculated streamlines at z=z0. If the calculated fluid particle trajectories have qualitatively the same shape as that of the moving body, then this approximation is valid (Section 4.1).
A function W satisfying (1)–(3a,b) with the discontinuity (4) at z=z0 can be represented in the form
Here Ω is the domain and R(ξ, y) is not equal zero if (ξ, y) ∈ Ω. The function G(x, y, z, z0, t) describes the IW field vertical velocity generated by a point mass source which is switched on at t=0 and moves with the constant speed V at the depth z0 in the stratum −H<z<0. This function is a solution of the following problem
where δ(x) is Dirac function, Θ(t) is Heaviside function.
The limit of solution as t → ∞ for fixed ξ=x+Vt is the function G describing the vertical velocity of the stationary IW from a moving point mass source. This function is defined by the equations:
The vertical velocity W of stationary IW generated by a moving body is a solution of the following problem:
(5)
(6)
(7)
The solution of (5)–(7) is expressed in terms of the Green’s function as follows:
(8)
3. GREEN’S FUNCTION AND ITS ASYMPTOTICS.
The Green’s function G(ξ, y, z, z0) (where ξ=x+Vt) was obtained in (2) by the variable separation method in a form of a sum of modes:
(9)
where for ξ<0, and the functions are defined by the integrals (2)1:
(10)
(11)
with
(12)
Here μn(ν), λn(ν) are odd functions ν, and ψn(z, ν), fn(z, ν) are even functions ν which are defined for ν>0 as eigenvalues and cigenfunctions of the corresponding spectral problems:
1
Analogous expressions in (4) contain misprints
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The same expressions are valid for the vertical displacement η excited by the moving point sourse (2) if we add factor signξ in the eqn. (10) and change by and by in the eqns. (12).
Fig. 1. First three modes of vertical displacement and their sum as functions of ξ.
Mention that in the case of vertical displacement each function ηn=Gn(ξ, y, z, z0) has a discontinuity at ξ=0 but this discontinuity must vanish after summing over n since the vertical displacement η=∑ηn is the analytic function ξ for y≠0. To illustrate it we present at taken from (2) Fig. 1 the first three modes of vertical displacement and their sum for H=600 m, distribution of N(z) shoved at Fig. 2, V=2 m/sec, y=300 m, z=200 m, and z0=22 m. We see, that each mode η1, η2, η3 is discontinuous, but their sum is continuous with graphical exactness. As was mentioned in (2), separate modes of vertical velocity are continuous at ξ=0.
The expressions (9)–(11) have the simple asymptotic forms close to and far from the point source.
In the near zone, i.e. for small y and ξ, the term makes the dominant contribution into G, and the terms are relatively small. The series is converging slowly, but it is possible to obtain the asymptotic form of as n → ∞ (3):
(13)
where r2=y2+ξ2, Ko(x) is the zeroth order MacDonald function (17). Each function In has a logarithmic singularity as r → 0 and z≠z0. The series ∑In can be summed (17):
(14)
Here z±=z±z0. The expression (14) is regular for r2+(z−z0)2≠0. It is the derivative with respect to z of the fundamental solution of Laplace equation: ΔI=−4πδ(z−z0)δ(ξ)δ(y) with boundary conditions Iz=0, −H=0. The sum (14) represents the field excited by a system of point sources placed at the depths zm=±z0±2mH, m=±0, 1,2,… in a nonstratified fluid.
Fig. 2. Distribution of N(z)
We now write G in the form of a sum of a static field I and a rapidly converging series for small y, ξ:
(15)
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At large distances from the point source G splits into individual modes with each mode propagating independently from the others within its own Mach angle (10,1,6). The terms make dominant contribution to G, since are exponentially small at large y. The asymptotical representation of Gn in the vicinity of the corresponding wave front is obtained from Taylor’s expansion of dispersion curves μn(ν) close to zero: and has the form (1):
(16)
where Ai′(x) is the first derivative of Airy function.
Far from each mode wave front and within its own Mach angle the asymptotical form of Gn is
(17)
where is a root of the equation (1).
Thus, we have the following qualitative picture of the IW field at a fixed observation point located at a distance y from the source trajectory as t increases. For ξ=x+Vt<q1y the field at the observation point is exponentially small. For the front of the first vertical mode arrives at the observation point. In the vicinity of the front the field is expressed in terms of the Airy function derivative (16). Then, as ξ increases, i.e. for ξ>q1y, the field is expressed by (17). For the second vertical mode is joined to the field, for the third mode is joined to the field, etc.
4. EXAMPLES OF CALCULATIONS AND ALGORITHMS.
There are three regions of IW field. The points in the near zone are separated form the body at the distances comparable to the thickness Η. Here the field is practically independent of the stratification, and IW amplitudes are the biggest. For the points in the far zone y, ξ≥10 H. In this zone IW field splits into individual modes, each mode is situated within corresponding Mach angle and the amplitude of the mode is small outside its Mach angle. Finally, there is an intermediate zone, where the IW field structure is more complicated. The code chooses the algorithm of field calculation in dependence of position of observation point with respect to body.
Fig. 3. The distribution of the BrüntVäissälä frequency N(z).
The work of code begins from numerical calculation of dispersion curves μn(ν), λn(ν), and eigenfunctions fn(z, ν), ψ(z, ν) for given density distribution and prescribed values n. These results are used during all subsequent calculations.
Fig. 4. The shape of body.
For depicted below and taken from (4) examples of calculations we choosed the model of moving body and the stratification of fluid which were used for calculations in (6). In this model the depth of body is equal to the thickness Η=50 m of the stratified layer, maximal height of body h*=0.2 H; its velocity V=5 m/sec. The distribution of BrüntVäissälä frequency N(z) is shown at Fig. 3 and the shape of body is shown at Fig. 4.
Some results of numerical calculations are shown at Figs. 5, 6. At Fig. 5 is depicted the vertical
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velocity W as function ξ for y=0 and different values of z. Note that our results practically coincide with results of (6); the vertical velocity rapidly decreases and perturbed region expands with decrasing depth z. The amplitude of W at z=−0.5 H is only 15% of its values at z=−H which coincides with the function R(ξ, y).
At Fig. 6 is shown the vertical displacement η as function y, ξ for z=−0.5 H.
Let us present now the algorithms of field calculation.
4.1 Field in the near zone. It is convenient to use here the representation of Green’s function in the form (15). For W we obtain:
Fig. 5. Vertical velocity W at different depth and y=0.
(18)
(19)
where Wn is the exact integral representation of the individual mode (Section 4.3).
Fig. 6. Vertical displacement n for z=−0.5 H.
The function S satisfies the Laplace equation ΔS=0 and the boundary conditions (6)–(7), i.e., describes the flow of the nonstratified fluid around the body. To calculate S we use eqn. (14) for ∑In. As the numerical calculations show, at the distance of the order of H from the body, the flow is almost potential and depends only on the body shape.
However, as the distance from the body increases, it becomes necessary to take into account the fluid stratification. Then a procedure for calculating the corrections to the potential flow at the point ξ, y, z in order to take stratification into account is as follows: (1) the function S is calculated from (18); (2) the term Δ1 is determined, if Δ1≪S then next terms in (18) need not be found; (3) if Δ1 and S are of the same order, then the term Δ2 is calculated; (4) if S+Δ1≫Δ2 then the summation in (18) is interrupted, and so on.
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The procedure described makes it possible not only to calculate the IW near field accurately, but also to estimate the errors in the asymptotic expressions. At Fig. 7 are depicted for y=0.5 H, z=−0.5 H the exact value W of vertical velocity and the functions S, S Δ1, S+Δ1+Δ2. These results show that at the distances from trajectory of the order of thickness Η in is suffitient to take into account only two corrections.
Fig. 7. Vertical velocity in near zone y=0.5 H, z=−0.5 H: exact solution W (solid line), functions S (dashed line), S+Δ1 (dotand dashed line), S+Δ1+Δ2 (dotted line).
The trajectories of fluid particles for z=z0 calculated for our model are close with sufficient accuracy to the prescribed shape of moving body. It confirms the validity of linear approximation (3c) for the boundary conditions.
4.2 Field in the far zone. As was mentioned above, at large distances from the moving body the IW field splits into individual modes. Moreover, it is possible to show from (15)–(17) that only the longwave components are essential at these distances (1).
So, we disregard the exact body shape and replace it by a suitable system of point sources. For the present case the function R(ξ, y) can be written approximately in the form
(20)
where coordinates of the source ξ+, sink ξ− and constant R* are determined by the body shape.
Then, using the asymptotic representations of Green’s function (16),(17), we can write the expression for the individual mode Wn in the far zone:
(21)
The results of calculations in accordance with (21) for intermediate (y=2 H) and far (y=8 H) zones are shown at Fig. 8. Numerical estimations show that in far zone (for y>10 Η) the use of the asymptotic of the Green function and replacement of body by a suitable system “source+sink” maked it possible to calculate IW field by simple explicit expressions instead of numerical calculation of the exact integrals.
Fig. 8. The first mode of vertical velocity for z=−0.5 H in intermediate (y=2 H) and far (y=8 H) zones: exact mode W1 (solid lines), approximation (21) (dotted lines).
4.3 Field in the intermediate zone. In order to simplify the exact calculations in intermediate zone we consider here the body shape with vertical crosssection orthogonal to the x axis in the form of a semiellipse (6):
where arbitrary smooth functions σ(ξ), d(ξ) describe the height and the width of the body crosssection. In this case, using (9)–(11), it is possible to integrate in (8) over y and to obtain the expression for W in the form:
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where τ=νd(ξ)/2, J0,1(x) are the zeroth and the first order Bessel functions.
ACKNOWLEDGEMENTS.
The results described above were made possible in part by grant N M31000 from the International Science Foundation.
References
[1] Borovikov, V.A., Vladimirov, Yu.V. and Kel’bert, M.Ya. Field of internal gravity waves excited by localized sources. Izv.Atm.Ocean.Phys., No. 6, 1984, pp. 494–498.
[2] Borovikov, V.A., Bulatov, V.V., Vladimirov, Yu.V. and Levchenko, E.S. Internal wave field generated by a source resting in a moving stratified fluid. J.Appl.Mech.Tech.Phys., No. 4, 1989. pp. 563–566.
[3] Bulatov, V .V. and Vladimirov, Yu.V. Near field of internal waves exci0ted by a source in a moving stratified liquid. J.Appl.Mech.Tech.Phys., No. 1, 1991. pp. 21–25.
[4] Borovikov,V.A., Bulatov, V.V., and Vladimirov, Yu.V. Internal gravity waves excited by a body moving in a stratified fluid. Fluid Dynamics Research, 1995, Vol. 15, pp. 325–336.
[5] Hopfinger, E.J. Turbulence in stratified fluids: a review. J.Geophys.Res., Vol. 92, 1987. pp. 5287–5303;
[6] Kallen, E. Surface effects of vertically propagating waves in stratified fluid. J.Fluid Mech., Vol. 182, 1987. pp. 111–125.
[7] Lin, J.T. and Pao, Y.H. Wakes in stratified fluids. Ann.Rev.Fluid Mech., Vol. 11, 1979. pp. 317–338.
[8] Sharman, R.D. and Wurtele, M.G. Ship waves and Lee waves. J.Atmosph.Sci., Vol. 40, 1983. pp. 396–427.
[9] Miles, J.W. Internal waves, generated by a horizontally moving source. Geoph.Fluid Dyn., Vol. 2, 1971. pp. 63–87.
[10] Keller, J.B. and Munk,W.H. Internal wave wakes of a body moving in a stratified fluid. Phys.Fluids, Vol. 13, 1970. pp. 1425–1431.
[11] Janowitz, G.S. Lee waves in threedimensional stratified flow. J.Fluid Mech., Vol. 148, 1984. pp. 97–108.
[12] Meng, J.C.S. and Rottman, J.W. Linear internal waves generated by density and velocity perturbations in a linearly stratified fluid. J.Fluid Mech., Vol. 186, 1988. pp. 419–444.
[13] Voisin, B. Internal wave generation in uniformly stratified fluid. Part 1. Green’s function and point sources. J.Fluid Mech., Vol. 231, 1991. pp. 439–480.
[14] Gray, E.P., Hart, R.W. and Farrel, R.A. 1983. The structure of the internal wave Mach front generated by a point source moving in a stratified fluid. Phys.Fluids, Vol. 26, 1983. pp. 2919–2931.
[15] Snyder, W.H., Thompson, R.S., Eskridge, R.E., et al. The structure of strongly stratified flow over hills dividingstreamline concept. J.Fluid Mech., 152, 1985. pp. 249–289.
[16] Gill, A.E. AtmosphereOcean Dynamics, Academic Press, 1982.
[17] Gradstein, I.S. and Ryshik, I.M. Tables of Integrals, Series and Products. Academic Press, 1980.
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DISCUSSION
I.Sturova
Lavrentyev Institute of Hydrodynamics, Russia
The given report is devoted to the important problem of internal gravity wave generation in a stratified fluid resulting from the horizontal motion of a submerged body and its associated turbulent wake. Unfortunately, at present there is no definite understanding of all the sources responsible for a generation of internal waves by a towed or selfpropelled body. Models used in the 70s and 80s are under reconsideration at the present time (see e.g. (1)) especially at high speed of body motion.
The authors used the model proposed by Kallen (2), where a body together with a turbulent wake is modeled by an infinitely long thin body. Despite the disputability of the model used, the undoubted advantage of the given paper is the proposed effective method of determination of the Green function of a point mass source moving uniformly in a fluid with an arbitrary stable stratification. The asymptotic estimations for near and far field are obtained.
To my point of view, this method will be extremely useful for computation of the flow around the moving solid body of an arbitrary form with a nonflow condition at the body’s surface. Analogously to the computation of the flow around a body submerged under the free surface of homogeneous fluid, it is possible to use the boundary element method demanding a determination of the Green function for the problem and work out of algorithm of its effective calculation. If the problem is solved successfully together with the determination of wave field, the effect of stratification on hydrodynamics loads acting on a body can be estimated.
References
1. Robey, H.F., “The Generation of Internal Waves by a Towed Sphere and Its Wake in a Thermocline,” Physics of Fluids, Vol. 9, No. 11, 1997, pp. 3353–3367.
2. Kallen, E., “Surface Effects of Vertically Propagating Waves in Stratified Fluid,” Journal of Fluid Mechanics, Vol. 182, 1987, pp. 111–125.
AUTHORS’ REPLY
The authors agree with the discusser’s comments.