Boundary Layer Correction Research Articles

Collaborative attitude stabilization and distributed vibration suppression of satellite with ultra–large flexible antenna

Large flexible radar satellites are widely used in space–based remote sensing for wide–area target observation and high–capacity information transmission. To achieve accurate earth observation and structural stability of an ultra–large flexible satellite in hundred–meter length concurrently, this paper presents two–time–scale collaboration for attitude stabilization and vibration suppression. First, the coupling dynamic model of the satellite with ultra–large flexible antenna is established under space disturbances. Next, singular perturbation with boundary layer correction is employed to decompose attitude and vibration controls in different time scales, respectively. Given the respective decoupling models, the slow subsystem employs an angles–only terminal sliding–mode controller with adaptive switching and input saturation to accomplish high–robustness triaxial attitude stabilization in finite time. For the fast subsystem, distributed piezoelectric actuators with leader–follower consensus and feedback control are applicable to rapid and stable vibration suppression of the ultra–large structure. Finally, simulation results validate the effectiveness and superiority of these control methods on the satellite, where collaborative control effects on multiple key indices including attitude and vibration control accuracy and settling time are significant. This work facilitates high–accuracy and –stability collaborative control of future ultra–large satellites.

Hypervelocity imperfect gas nozzle design with shared wave-elimination contour

A hypervelocity imperfect gas nozzle with a shared wave-elimination contour is designed by the residual correction method, allowing the test Mach number to be varied by changing the throat contours. Owing to imperfect gas effects, the nozzle designed by the classical method of characteristics with boundary layer correction does not produce a uniform flow field, resulting in significant deviation from the target Mach number. In this work, the computational fluid dynamics solver is used as an independent module without being coupled to the optimization code, reducing the design complexity. Designers can choose the appropriate solver according to the specified physical characteristics to consider imperfect gas effects. The Mach 15 hypervelocity nozzle designed by the residual correction method better eliminates the Mach waves and achieves a much higher flow uniformity than the nozzle designed by the classical method. On this basis, the dependence domain of the shared wave-elimination contour and the influence domain of the transonic solution are solved by the method of characteristics, and a replaceable throat contour is rigorously designed from aerodynamics theory. Quantitative evaluations show that the nozzles with a shared wave-elimination contour have the same level of flow uniformity, achieving high flow quality at Mach 13–15. The evaluation results validate the design's feasibility, supporting the future construction of hypervelocity tunnels.

The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation

In this paper, we introduce the topological state derivative for general topological dilatations and explore its relation to standard optimal control theory. We show that for a class of partial differential equations, the shape-dependent state variable can be differentiated with respect to the topology, thus leading to a linearised system resembling those occurring in standard optimal control problems. However, a lot of care has to be taken when handling the regularity of the solutions of this linearised system. In fact, we should expect different notions of (very) weak solutions, depending on whether the main part of the operator or its lower order terms are being perturbed. We also study the relationship with the topological state derivative, usually obtained through classical topological expansions involving boundary layer correctors. A feature of the topological state derivative is that it can either be derived via Stampacchia-type regularity estimates or alternately with classical asymptotic expansions. It should be noted that our approach is flexible enough to cover more than the usual case of point perturbations of the domain. In particular, and in the line of (Delfour in SIAM J Control Optim 60(1):22–47, 2022; J Convex Anal 25(3):957–982, 2018), we deal with more general dilatations of shapes, thereby yielding topological derivatives with respect to curves, surfaces or hypersurfaces. To draw the connection to usual topological derivatives, which are typically expressed with an adjoint equation, we show how usual first-order topological derivatives of shape functionals can be easily computed using the topological state derivative.

On the prediction of noise generated by urban air mobility (UAM) vehicles. II. Implementation of the Farassat F1A formulation into a modern surface-vorticity panel solver

This study focuses on the integration of established acoustic prediction techniques directly into a surface-vorticity solver. The main objective is to enhance an aircraft designer's ability to characterize the acoustic signatures generated by urban air mobility (UAM) vehicles, in general, and distributed electric propulsion (DEP) concepts, in particular, including unmanned aerial vehicles. Our solver consists of a reliable, surface-vorticity panel code that incorporates viscous boundary-layer corrections. Thus, it offers a computationally efficient commercial tool for conceptual design and preliminary aerodynamic analysis. By implementing the Farassat F1A acoustics formulation directly into the solver, a new intuitive capability is achieved, which is both conversive with modern engineering tools and efficient in setup and speed of execution. In addition to its application to the X-57 high-lift propeller and the Revolutionary Vertical Lift Technology Tiltwing electric Vertical Take-Off and Landing (eVTOL) vehicle by the National Aeronautics and Space Administration, this capability is systematically demonstrated using three particular case studies. These consist of both single- and six-propeller Joby S4 eVTOL as well as a small eight-propeller Kittyhawk KH-H1 DEP vehicle. Although the details of this tool and underlying equations are showcased in this article, the acoustic metrics that can be effectively used to characterize the noise level generated by a UAM in flight are described in a companion article. By embedding this assortment of insightful metrics into a simple and user-friendly flow solver, a much improved flow-acoustic analysis capability is thereby provided to support the design of future aircraft.

Nozzle design optimization for supersonic wind tunnel by using surrogate-assisted evolutionary algorithms

For high-precision measurement in supersonic wind tunnel experiments, it is of crucial importance to produce a uniform flow in the measurement section downstream of the nozzle. This paper proposes and verifies a new design methodology for supersonic wind tunnel nozzles that can generate highly uniform airstream at a design Mach number at the nozzle exit. Shape design optimization has been conducted by employing surrogate-assisted evolutionary algorithms coupled with computational fluid dynamics. This approach has yielded higher flow uniformity than that of the nozzle designed by using the method of characteristics in the inviscid regime. By applying boundary layer correction to the nozzle contour obtained from the inviscid optimization, nearly uniform core flow of Mach 2.5 has been achieved at the nozzle exit in the presence of viscosity. In the viscous optimization, nozzle shape optimization has been performed by incorporating viscous simulations without using boundary layer correction to evaluate its efficacy in comparison with the former approach. It has been found that the deviation from the design Mach number and the flow deflection from the horizontal direction cannot be minimized simultaneously. This has been attributed to the constraint associated with the nozzle length. It has also been found that the result of the former approach combining the inviscid optimization and boundary layer correction can be regarded as one of the results of the viscous optimization.

Multiscale analysis of composite structures based on higher-order asymptotic homogenization with boundary layer correction

First-order homogenization generally becomes inaccurate for materials with a weak scale separation between characteristic lengths of the heterogeneities and the structural problem. It is also unable to provide a correct solution in the vicinity of the boundaries due to the loss of periodicity in these regions. In this article, we demonstrate the effectiveness of higher-order homogenization, up to the third-order, in estimating correctly the heterogeneous solution, for cases with a low scale separation in elastic composite materials. We also propose a higher-order general boundary layer method, effective for various boundary conditions (Dirichlet, Neumann or mixed), to correct the obtained estimation near the boundaries. The efficiency and accuracy of the proposed methods are studied on various numerical examples dealing with elastic laminates and fiber–matrix composites.

Solving multiscale steady radiative transfer equation using neural networks with uniform stability

This paper concerns solving the steady radiative transfer equation with diffusive scaling, using the physics informed neural networks (PINNs). The idea of PINNs is to minimize a least-square loss function, that consists of the residual from the governing equation, the mismatch from the boundary conditions, and other physical constraints such as conservation. It is advantageous of being flexible and easy to execute, and brings the potential for high dimensional problems. Nevertheless, due the presence of small scales, the vanilla PINNs can be extremely unstable for solving multiscale steady transfer equations. In this paper, we propose a new formulation of the loss based on the macro-micro decomposition. We prove that, the new loss function is uniformly stable with respect to the small Knudsen number in the sense that the \(L^2\)-error of the neural network solution is uniformly controlled by the loss. When the boundary condition is an-isotropic, a boundary layer emerges in the diffusion limit and therefore brings an additional difficulty in training the neural network. To resolve this issue, we include a boundary layer corrector that carries over the sharp transition part of the solution and leaves the rest easy to be approximated. The effectiveness of the new methodology is demonstrated in extensive numerical examples.

Asymptotic traveling wave solutions to anisotropic higher-order traffic flow models with viscosities

The boundary layer correction method is applied to study asymptotic traveling wave solutions to anisotropic higher-order traffic flow model with viscous coefficient, which clearly indicates that the truncation error between the original and approximate equations is an equivalent infinitesimal of the coefficient. Therefore, the solution is exact in the case when the coefficient vanishes, which corresponds to a certain nonviscous model following the standard hyperbolic and viscous conservation laws. Numerical simulation is implemented to reproduce a typical traveling wave or wide moving jam consisting of a transitional layer (the downstream front) and a shock (the upstream front) profiles, of which the parameters converge not only for refined meshes but for refined viscous coefficient as well. These are highly in agreement with the analytical results.

Drop impact onto wetted walls: an unsteady analytical solution for modelling crown spreading

An unsteady analytical solution is proposed to predict the spreading rate of the crown generated by an impacting droplet onto wetted walls. The modelling strategy is based on the direct integration of the boundary layer correction into the potential flow solution that leads to the well-established square-root time dependence. The original potential flow has the structure of an unsteady, stagnation point flow with decaying strength. For initial strengths of the potential flow $a_0 \geqslant 100\ {\rm s}^{-1}$, we find that a self-similar solution can also be obtained for the boundary layer in the variables $\left (r \sqrt {a(t)/(\nu t)}, z \sqrt {a(t)/(\nu t)} \right )$. The self-similarity of the solution enables a straightforward estimation of momentum losses during the spreading of the liquid layer along the wall. The proposed modelling approach yields an excellent agreement with experiments during the entire spreading phase. Moreover, it enables a smooth transition from the inertia-driven to the shear-controlled regime of crown propagation. In general, the analysis shows that momentum losses arising from viscous effects cannot be neglected during a significant portion of crown propagation, particularly for thin wall films.

Singular Perturbation Method Applied for BVP, IVP and Optimal Control to One Parameter Armature Controlled DC Servo Motor

DC servo motor discrete model of one-parameter Singular Perturbation Method (SPM) is enlarged showing zero-order, first-order and second-order approximations. In this paper, a one parameter SPM based two time scale model is considered for evaluation. Now by applying this SPM a real time Boundary Value Problems (BVP), Initial Value Problem (IVP) and Optimal Control Problem (OCP) are premeditated. Such evaluated SPM have a boundary layer correction (BLC) solution and an outer series solution. To improve degenerate solution and to recover initial and boundary conditions a BLC solution of SPM is used. The DC servo-motor control model of second order approximation is carried out for BVP, IVP and OCP. The results thus obtained are presented in comparison to the precise solution. The efficacy of the present model is evaluated in MATLAB environment.

A general boundary layer corrector for the asymptotic homogenization of elastic linear composite structures

Asymptotic homogenization method is often used in multiscale analysis of periodic structures instead of conducting a full field heterogeneous analysis, in order to achieve computational feasibility and efficiency. When completed with a relocalization process, this method may provide relevant estimates to microscale fields within the material. Nevertheless, the construction of a solution near the boundaries remains beyond the capabilities of classical relocalization schemes due to the loss of periodicity in the vicinity of the boundaries. This paper proposes a post-processing scheme in order to conduct the relocalization step within a finite element framework for periodic linear elastic composite materials. It also assesses the boundary layer effect and a new general method, effective for various boundary conditions (Dirichlet, Neumann or mixed), is proposed based on the idea of computing corrective terms as solution of auxiliary problems on the unit-cell. These terms are finally added to the usual fields obtained from the relocalization process to obtain the corrected solution near the boundaries. The efficiency, accuracy and limitation of the proposed approach are studied on various numerical examples.

Asymptotic Numerical Solutions for Second-Order Quasilinear Singularly Perturbed Problems

For a second-order quasilinear singularly perturbed problem under the Dirichlet boundary conditions, we propose a new asymptotic numerical method, which involves two problems: a reduced problem with a one-side boundary condition and a novel boundary layer correction problem with a two-sided boundary condition. Through the introduction of two new variables, both problems are transformed to a set of three first-order initial value problems with zero initial conditions. The Runge–Kutta method is then applied to integrate the differential equations and to determine two unknown terminal values of the new variables until they converge. The modified asymptotic numerical solution satisfies the Dirichlet boundary conditions. Some examples confirm that the newly proposed method can achieve a better asymptotic solution to the quasilinear singularly perturbed problem. For most values of the perturbing parameter, the present method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy within the entire domain.

Singularly Perturbed Solutions for a Class of Thermoelastic Weakly Coupled Problems

Based on the basic equation of Green Lindsay (G-L) theory, the thermoelastic weak coupling problem under the basic equation is discussed, that is, two thermal relaxation parameters are added to the constitutive equation, the influence of the coupling term on the temperature field and elastic field is considered, and the asymptotic solution of the governing equation is constructed. Firstly, in order to obtain the asymptotic solution, the singularly perturbed expansion method is used.Then,combined with the corresponding boundary conditions, the partial differential equation method is used to solve the external solution and the boundary layer correction term. Secondly, in the case of weak coupling, the uniformly efficient estimation of the remainder of the asymptotic solution is obtained by using Gronwall inequality, so as to obtain the uniformly efficient of the formal asymptotic solution. Finally, the first term of the asymptotic solution is numerically analyzed by using the singularly perturbed numerical method. The present work will be conducive to the analysis of thermoelastic processes and numerical simulation of different materials in the case of weak coupling.

Constructing and analyzing mathematical model of plasma characteristics in the active region of integrated p-i-n-structures by the methods of perturbation theory and conformal mappings

The results of mathematical modeling of stationary physical processes in the electron-hole plasma of the active region (i-region) of integral p-i-n-structures are presented. The mathematical model is written in the framework of the hydrodynamic thermal approximation, taking into account the phenomenological data on the effect on the dynamic characteristics of charge carriers of heating of the electron-hole plasma as a result of the release of Joule heat in the volume of the i-th region and the release of recombination energy. The model is based on a nonlinear boundary value problem on a given spatial domain with curvilinear sections of the boundary for the system of equations for the continuity of the current of charge carriers, Poisson, and thermal conductivity. The statement of the problem contains a naturally formed small parameter, which made it possible to use asymptotic methods for its analytical-numerical solution. A model nonlinear boundary value problem with a small parameter is reduced to a sequence of linear boundary value problems by the methods of perturbation theory, and the physical domain of the problem with curvilinear sections of the boundary is reduced to the canonical form by the method of conformal mappings. Stationary distributions of charge carrier concentrations and the corresponding temperature field in the active region of p-i-n-structures are obtained in the form of asymptotic series in powers of a small parameter. The process of refining solutions is iterative, with the alternate fixation of unknown tasks at different stages of the iterative process. The asymptotic series describing the behavior of the plasma concentration and potential in the region under study, in contrast to the classical ones, contain boundary layer corrections. It was found that boundary functions play a key role in describing the electrostatic plasma field. The proposed approach to solving the corresponding nonlinear problem can significantly save computing resources

Singular Perturbation Method Applied to Power Factor Correction Converter Application

A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Initial Value Problem (IVP) and Boundary Value Problem (BVP). It is applied to two state switching power converters to provide rigorous justification of the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for IVP and BVP. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

Design of a Mach 2 shape transition nozzle for lab-scale direct-connect supersonic combustor

To maintain flow quality of MOC designed nozzle, 3-dimensional shape transition nozzle which directly connects a rectangular supersonic combustor and circular vitiated air heater was designed by applying the 2-dimensional nozzle contour as the longitudinal area distribution of the nozzle. The boundary layer correction for the 3-dimensional nozzle was applied by correcting the area distribution of the axisymmetric nozzle and examining several shape transition functions through the 3-dimensional computation. The 3-dimensional nozzle with the shape transition exhibited a large effective area with a uniformly distributed Mach number, despite the presence of rectangular nozzle flow characteristics.

Control-oriented quasi-one dimensional modeling method for scramjet

Abstract The scramjet model used in the control system needs to consider both accuracy and real-time requirements. In this paper, a quasi-one dimensional component level model method for scramjet is proposed. Under the premise of meeting the real-time requirement, this modeling method has higher calculation accuracy, and can obtain higher thrust performance without exceeding the limits. In order to improve the calculation accuracy of the scramjet model, the boundary layer correction and the interaction between shock wave and boundary layer are considered in the inlet and isolator modeling process by using reference temperature method. For the scramjet compression components, the inlet and isolator models are simplified based on shock train theory to improve real-time performance. Meanwhile, the combustor model and nozzle model are established and the combustor model is established with the combination of equal area and isothermal based on temperature limitation. Considering the boundary layer correction, the accuracy of the inlet and isolator model is improved by 8.79 and 7.9% respectively. Compared with the equal area combustion mode, the combined combustion mode can get higher thrust. In the C + + simulation environment, the average calculation time of each state point is about 25 ms which meets the real-time requirements.

The Thermal Effect on Discharge Coefficient of Sonic Nozzle Based on Reconstructed Body Temperature Distribution by Improved SOR Method

For the sonic nozzles, the discharge coefficient ( $C_{d}$ ) is an essential parameter of flow characteristics, which is obviously influenced by the thermal effect determined by the nozzle body temperature distribution. In this study, two types of thermocouple arrays for temperature acquisition of the nozzle body were designed to investigate the thermal effect. The temperature isothermal maps of the nozzle body were reconstructed by improved successive over-relaxation (SOR) method combining the simple Kriging interpolation method and particle swarm optimization (SKIM-PSO), which can effectively eliminate the phenomenon of bull’s eye and nonsmoothness of the SKIM. And then, the accuracy of the new algorithm was checked by cross validation with test sets of $d = 5.25$ -mm nozzles with different structures. It is found that the maximum error $\Delta e_{\mathrm {max}}$ of the new algorithm is less than 0.2 °C. The thermal effect on discharge coefficient was analyzed with the above-reconstructed temperature distribution, which is classified into two aspects, the thermal boundary layer correction factor $C_{\mathrm{ T}}$ and the constrained thermal deformation correction factor $C_{\mathrm{ a}}$ related to the throat area. The $C_{\mathrm{ T}}$ is $1- 3.845\text {Re}_{\mathrm{ t}}^{-0.5}\,\,\Delta T/T$ from a similar solution to the boundary layer. Unlike the free expansion, the slopes of the $C_{\mathrm{ a}}$ of constrained expansion for three AISI304 nozzles with $d = 2.00$ , 5.25, and 7.45 mm are $- 1.24 \times 10^{-5}$ , $+ 1.98 \times 10^{-5}$ , and $+ 2.49 \times 10^{-5}$ , respectively. With the correction of $C_{\mathrm{ T}}$ and $C_{\mathrm{ a}}$ , the prediction error of discharge coefficient is reduced from 0.14%–0.15% to 0.08%–0.1% for $d = 2.00$ - and 7.45-mm two nozzles.

Singular Perturbation Method for Boundary Value and Optimal Problems to Power Factor Correction Converter Application

A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Boundary Value Problem (BVP) and Optimal Problem. It is applied to two state switching power converters to provide rigorous justification of\ the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for BVP and optimal control problems. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities

AbstractThe existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. The numerical results suggest that there exist three solutions designated as type $I$, type $II$ and type $III$ for the asymmetric case, type $I$ solution exists for all non-negative Reynolds number and types $II$ and $III$ solutions appear simultaneously at a common Reynolds number that depends on the value of asymmetric parameter $a$ and with the increase of $a$ the common Reynolds numbers are decreasing. We also theoretically show that there exist three solutions. The corresponding asymptotic solution for each of the multiple solutions is constructed by the method of boundary layer correction or matched asymptotic expansion for the most difficult high Reynolds number case. These asymptotic solutions are all verified by their corresponding numerical solutions.