Nonlinear Two-dimensional Simulation Research Articles

Linear Quadratic Guidance Laws with Intercept Angle Constraints and Varying Speed Adversaries

The paper proposes two linear quadratic guidance laws, which enable target interception while imposing a predetermined terminal intercept angle. Unlike previous solutions, which usually assume constant missile and target speeds, time-dependent known speed profiles of the adversaries are addressed. The guidance laws are derived for linear, arbitrary-order missile and target dynamics using optimal control and differential games theories. Specific closed-form results are provided for ideal missile and target dynamics with constant axial acceleration adversaries. The performance of the proposed guidance laws is investigated using a nonlinear two-dimensional simulation, and the results are compared to two state-of-the-art optimal control and differential-games-based guidance laws that can impose the intercept angle. It is shown that the proposed guidance laws can intercept the target with small miss distances and intercept angle errors and that they have better performance than the compared state-of-the-art guidance laws when the adversaries’ speeds vary. The selection of which guidance law to use depends on the scenario. When the target is predictable, the optimal-control-based guidance law is better suited; whereas when the target is intelligent, the differential-games-based guidance law is a more appropriate choice.

On the peripheral intensification of two-dimensional vortices in smaller-scale randomly forcing flow

The evolution of a monopolar vortex embedded in the field of smaller-scale randomly forced vorticity is examined using fully nonlinear two-dimensional simulations at large Reynolds numbers. The vortex is considered to be compact if its angular momentum decreases with the radius on the scale comparable to the radius of maximum azimuthal velocity. The energy decays without forcing, while the vortex remains compact despite its viscous spreading. This scenario dramatically changes in the strong forcing regime, characterized by the substantial growth of the vortex energy due to the increase in velocity and angular momentum at the vortex periphery so that ultimately, the vortex transforms into a noncompact structure. The maximum of angular momentum redistribution is found to be proportional to the enstrophy of smaller-scale vorticity field. The results have important implications for better understanding the fate of vortices and physical mechanisms of energy transfer.

Cooperative Differential Game Guidance Law for Two Pursuers and One Evader

For the nonlinear engagement between two pursuers and a single evader, in which small angle deviation assumption or other linearization conditions may not be applicable, a cooperative guidance law is analyzed and derived in the framework of a zero-sum two-person differential game. State-dependent Riccati-equation method is used to obtain the suboptimal solutions of the pursuit-evasion problem by choosing the line of sight angular rates as state variables, and closed-loop form guidance strategies that can be used online for the players are derived. Time-to-go estimations are not had to be taken into account to apply the guidance law proposed. Nonlinear two-dimensional simulations are carried out to validate the performance of the cooperative guidance law, and the comparison with a current cooperative linear quadratic differential game guidance law is made.

Smart homing guidance strategy with control saturation against a cooperative target-defender team

A smart homing guidance strategy with control saturation against a target-defender team is derived. It is noteworthy that a cooperative strategy of the target-defender team is applied, which has been proved more challenging for the homing guidance. The defender missile is launched by the target and guided by a cooperative augmented proportional navigation (APN). At the same time, the target performs a one-switch maneuver to cooperate and minimize the defender's acceleration requirement. The problem is analyzed for arbitrary-order linear dynamics of the agents in the linearized form but validated by the mathematical simulations by using nonlinear kinematics. The perfect information of three agents' states is assumed. Then, a method to deal with the target-defender team is proposed. It contains a combined performance index penalizing the miss distance relative to the target and energy consumption in the whole duration. Besides, the specific miss distance related to the defender is regarded as an inequality constraint. An analytical solution for the smart guidance strategy against the APN guided defender is derived. Meanwhile, the control saturations are introduced to get more realistic and reasonable insights to this practical target-missile-defender problem. A simple but effective iterative searching technique is proposed to determine the saturation time points. The solution provides an optimal homing strategy to evade the defender with a specific miss distance and intercept the target with the minimum miss distance in the minimum energy manner. Nonlinear two-dimensional simulation results are used to validate the theoretical analysis. By comparison with the optimal differential game guidance (ODGG) and the combined minimum effort guidance (CMEG), the superiority of this smart guidance strategy is concluded.

Cooperative Differential Games Guidance Laws for Imposing a Relative Intercept Angle

A cooperative guidance law for a team of interceptors trying to intercept, from multiple directions, an evading target is proposed. An example scenario of interest is that of intercepting an aerial target, such as a ballistic missile, from multiple directions. The engagement is analyzed in the framework of a linear quadratic zero-sum two-person differential game, where the team of interceptors constitutes one of the adversaries and the target the other. An arbitrary number of interceptors, each having linear dynamics, is considered. The obtained guidance law for the team of interceptors enables enforcement of a relative geometry in between the group of missiles and the target. Such an approach is superior (in the sense of the required control effort) to that where each interceptor independently enforces, using a one-on-one strategy, a preselected intercept angle that satisfies the relative intercept requirement. A nonlinear two-dimensional simulation is used to investigate the performance of the obtained cooperative guidance law. It is shown that the proposed guidance law is superior to the optimal-control-based cooperative guidance law when the target is unpredictable and that the interceptors’ acceleration requirements are comparable to conventional guidance laws that do not impose an angular constraint at interception.

Stochastic Cooperative Interception Using Information Sharing Based on Engagement Staggering

A novel cooperative tracking and interception strategy, which exploits information sharing and missile staggering, is presented. The key idea underlying the approach is to exploit the superior information collected by the leading missile to improve the interception performance of the trailing missiles. For tracking a maneuvering target, the paper derives a nonlinear adaptation of an interacting multiple model filter in cooperative and noncooperative estimation modes. The optimal staggering between the missiles is derived based on a linear model and a deterministic approximation of the stochastic estimation process. An extensive Monte Carlo study, in a nonlinear two-dimensional simulation of a ballistic missile defense scenario, is used to demonstrate the viability of the proposed strategy. It is shown that, for a two-on-one interception engagement, the trailing missile’s estimation performance, in the information-sharing mode, substantially improves, when compared to that of noncooperating missiles. Combining this estimation improvement with guidance laws that use target acceleration yields a dramatic improvement in the team’s closed-loop interception performance.

Two-dimensional instability of the bottom boundary layer under a solitary wave

The objective of this paper is to establish a detailed map for the temporal instability of the bottom boundary layer (BBL) flow driven by a soliton-like wave induced pressure gradient in a U-tube-shaped tunnel, which serves as an approximation to the BBL under surface solitary waves. Both linear stability analysis and fully nonlinear two-dimensional simulations using high-order numerical methods have been carried out. The process of delineation of the stability regions as a function of boundary layer thickness-based Reynolds number of the temporally evolving base flow, Reδ, consists of two parts. In the first part, we assess the lower limit of the Reδ range within which the standard, quasi-steady, linear stability analysis is applicable when considering individual base flow profiles sampled during its transient evolution. Below this limit, transient linear stability analysis serves as a more accurate predictor of the stability properties of the base flow. In the second step, above the Reδ limit where the BBL is determined to be linearly unstable, the base flow is further classified as unconditionally stable, conditionally unstable, or unconditionally unstable in terms of its sensitivity to the amplitude and the insertion time of perturbations. Two distinct modes of instability exist in this case: post- and pre-flow-reversal modes. At a moderate value of Reδ, both modes are first observed in the wave deceleration phase. The post flow reversal mode dominates for relatively low Re and it is the one observed in Sumer et al. [“Coherent structures in wave boundary layers. Part 2. Solitary motion,” J. Fluid Mech. 646, 207 (2010)]. For Re above a threshold value of the base flow in the unconditionally unstable regime, the pre-flow reversal mode, which is longer in wavelength than its post-reversal counterpart, becomes dominant. In the same regime, the threshold Reδ value above which instability is observed in the acceleration phase of the wave is also identified. In this case, the base flow velocity profiles lack any inflection point, suggesting that the origin of such an instability is viscous. Finally, the lower Reδ limit above which quasi-steady linear stability analysis is valid may be independently obtained by adapting to the surface solitary wave BBL, an instability criterion which links the average growth rate and wave event timescale, as previously proposed in studies of the instability of the interior of progressive and solitary internal waves.

Cooperative Optimal Guidance Laws for Imposing a Relative Intercept Angle

Optimal control-based cooperative guidance laws, which enforce at intercept a relative geometry in between a group of missiles and a single maneuvering target, are presented. An example scenario of interest is that of intercepting a high-value target (such as a ballistic missile) by a team of cooperating interceptors arriving from different directions. The problem is posed in the linear quadratic framework, and closed-form analytic solutions are obtained for any team size with any linear missile dynamics. The performance of the cooperative guidance laws is investigated using a nonlinear two-dimensional simulation of the missiles’ lateral dynamics and relative kinematics. It is shown that cooperatively imposing a relative intercept angle between the missiles provides substantially better results than when each missile independently enforces, using a one-on-one strategy, a preselected intercept angle that satisfies the relative intercept requirement. It is also shown that the missiles’ acceleration requirements are comparable to conventional guidance laws, which do not impose an angular constraint at interception.

Optimal Guidance-to-Collision Law for an Accelerating Exoatmospheric Interceptor Missile

A guidance-to-collision law is presented. The guidance law is derived, using the optimal control methodology, for an accelerating exoatmospheric interceptor. It is dependent on a unique zero-effort heading angle error and on a variable denoted as the zero-effort angle of attack. The guidance law enables an accelerating interceptor to fly toward the interception point along a straight line, after the initial heading error is nulled. In addition, the guidance-to-collision law enables interception of the target with a terminal body angle. Moreover, the trajectory imposed by this guidance law enables an accelerating interceptor to estimate the target’s acceleration by using bearings-only measurements. This is possible due to the imposed line-of-sight rotation, making the range between the two vehicles observable. Using a nonlinear two-dimensional simulation, the performance of the guidance law is analyzed, showing superior performance compared to that of classical proportional navigation.

Cooperative Differential Games Strategies for Active Aircraft Protection from a Homing Missile

Cooperative pursuit―evasion strategies are derived for a team composed of two agents. The specific problem of interest is that of protecting a target aircraft from a homing missile. The target aircraft performs evasive maneuvers and launches a defending missile to intercept the homing missile. The problem is analyzed using a linear quadratic differential game formulation for arbitrary-order linear players' dynamics in the continuous and discrete domains. Perfect information is assumed. The analytic continuous and numeric discrete solutions are presented for zero-lag adversaries' dynamics. The solution of the game provides 1) the optimal cooperative evasion strategy for the target aircraft, 2) the optimal cooperative pursuit strategy for the defending missile, and 3) the optimal strategy of the homing missile for pursuing the target aircraft and for evading the defender missile. The obtained guidance laws are dependent on the zero-effort miss distances of two pursuer―evader pairs: homing missile with target aircraft and defender missile with homing missile. Conditions for the existence of a saddle-point solution are derived and the navigation gains are analyzed for various limiting cases. Nonlinear two-dimensional simulation results are used to validate the theoretical analysis. The advantages of cooperation are shown. Compared with a conventional one-on-one guidance law, cooperation significantly reduces the maneuverability requirements from the defending missile.

Linear Quadratic Guidance Laws for Imposing a TerminalIntercept Angle

Linear quadratic guidance laws that explicitly enable imposing a predetermined intercept angle are presented. Two such guidance laws are derived, using optimal control and differential game theories, for arbitrary-order linear missile dynamics. The obtained guidance laws are dependent on the well-known zero-effort miss distance and on a new variable denoted zero-effort angle. It is shown that imposing the terminal angle constraint raises considerably the gains of the guidance laws. Theoretic conditions for the existence of a saddle-point solution in the differential game are also derived. These conditions show that imposing the terminal angle constraint requires a higher maneuverability advantage from the missile. The performance of the proposed guidance laws is investigated using a nonlinear two-dimensional simulation of the missile's lateral dynamics and relative kinematics, while assuming first-order dynamics for the target's evasive maneuvers. Using a Monte Carlo study, it is shown that, for the investigated problem, a target can be intercepted with a negligible miss distance and intercept angle error even when the target maneuvers and there are large initial heading errors.

A note on the coating of an inclined plane in the presence of soluble surfactant

We consider the flow of a thin liquid film coating an inclined plane in the presence of a soluble surfactant. A two-dimensional three-equation model is derived using lubrication theory in the rapid diffusion limit and then used to investigate the stability of the fluid height and the surfactant surface and bulk concentrations. We present solutions for an insoluble surfactant system, which are then contrasted with those obtained for a system containing a soluble surfactant; both transient growth and fully nonlinear two-dimensional simulation results are discussed. Our results indicate that the characteristics of the fingering phenomena which accompany the flow are altered by the effects of solubility. In particular, we find that these effects destabilise the system further over an intermediate range of surfactant solubility.

Nonlinear capillary wave distortion and disintegration of thin planar liquid sheets

Linear and nonlinear dilational and sinuous capillary waves on thin inviscid infinite and semi-infinite planar liquid sheets in a void are analysed in a unified manner by means of a method that reduces the two-dimensional unsteady problem to a one-dimensional unsteady problem. For nonlinear dilational waves on infinite sheets, the accuracy of the numerical solutions is verified by comparing with an analytical solution. The nonlinear dilational wave maintains a reciprocal relationship between wavelength and wave speed modified from the linear theory prediction by a dependence of the product of wavelength and wave speed on the wave amplitude. For the general dilational case, nonlinear numerical simulations show that the sheet is unstable to superimposed subharmonic disturbances on the infinite sheet. Agreement for both sinuous and dilational waves is demonstrated for the infinite case between nonlinear simulations using the reduced one-dimensional approach, and nonlinear two-dimensional simulations using a discrete-vortex method. For semi-infinite dilational and sinuous distorting sheets that are periodically forced at the nozzle exit, linear and nonlinear analyses predict the appearance of two constant-amplitude waves of nearly equal wavelengths, resulting in a sheet disturbance characterized by a long-wavelength envelope of a short-wavelength oscillation. For semi-infinite sheets with sinuous waves, qualitative agreement between the dimensionally reduced analysis and experimental results is found. For example, a half-wave thinning and a sawtooth wave shape is found for the nonlinear sinuous mode. For the semi-infinite dilational case, a critical frequency-dependent Weber number is found below which one component of the disturbances decays with downstream distance. For the semi-infinite sinuous case, a critical Weber number equal to 2 is found; below this value, only one characteristic is emitted in the positive time direction from the nozzle exit.

Compressible Convection with Ionization. I. Stability, Flow Asymmetries, and Energy Transport

The influence of nonideal effects associated with ionization upon the dynamics and thermodynamics of compressible convection is studied. Linear and finite-amplitude analyses and fully nonlinear two-dimensional simulations of a plane-parallel layer of single-atomic-level hydrogen fluid are undertaken. Ionization significantly influences both the global transport properties and the local dynamics of convective flows by modifying the particle number density, specific heat, and internal energy content of the fluid. Strong temperature fluctuations and corresponding buoyancy forces develop locally in the fluid wherever rapid changes in ionization state occur. These can result in narrow regions of intense vertical flow