scholarly journals An Unsteady Model for Aircraft Icing Based on Tightly-Coupled Method and Phase-Field Method

Aerospace ◽  
2021 ◽  
Vol 8 (12) ◽  
pp. 373
Author(s):  
Hao Dai ◽  
Chengxiang Zhu ◽  
Ning Zhao ◽  
Chunling Zhu ◽  
Yufei Cai
Keyword(s):  
Numerical Simulation ◽  
Resistance Coefficient ◽  
Phase Field Method ◽  
Aircraft Icing ◽  
Shadow Zone ◽  
Step Method ◽  
Time Step ◽  
Hilliard Equation ◽  
Tightly Coupled ◽  
Cahn Hilliard Equation

An unsteady tightly-coupled icing model is established in this paper to solve the numerical simulation problem of unsteady aircraft icing. The multi-media fluid of air and droplets is regarded as a single medium fluid with variable material properties. Taking the droplet concentration as the phase parameter and the droplet resistance coefficient as the interphase force, the mass concentration distribution of the droplet is obtained by solving the Cahn–Hilliard equation. Fick’s law is introduced to improve the Cahn–Hilliard equation to predict the droplet shadow zone. On this basis, the procedure of the unsteady numerical simulation method for aircraft icing is established, including grid generation, the dual-time-step method to realize the unsteady calculation of the air and droplet tightly-coupled mixed flow field, and the improved shallow water icing model. Finally, through the comparative analysis of numerical examples, the effectiveness of the new model in predicting the droplet impact characteristics and the droplet shadow zone are verified. Compared with other icing models, the ice shapes predicted by the unsteady tightly-coupled model were found to be the most consistent with the experiments. In the icing comparison conditions in this manuscript, the prediction accuracy of the ice thickness at the stagnation point of the leading edge was up to 35% higher than that of LEWICE.

A New Algorithm for the Numerical Simulation of Machined Surface Topography in Multiaxis Ball-End Milling

2008 ◽  
Vol 130 (1) ◽  
Author(s):  
Wei-Hong Zhang ◽  
Gang Tan ◽  
Min Wan ◽  
Tong Gao ◽  
David Hicham Bassir
Keyword(s):  
Numerical Simulation ◽  
Surface Topography ◽  
End Milling ◽  
Computing Time ◽  
Cutting Edge ◽  
Machining Parameters ◽  
Step Method ◽  
Time Step ◽  
Machined Surface ◽  
Ball End Milling

In milling process, surface topography is a significant factor that affects directly the surface integrity and constitutes a supplement to the form error associated with the workpiece deformation. Based on the tool machining paths and the trajectory equation of the cutting edge relative to the workpiece, a new and general iterative algorithm is developed here for the numerical simulation of the machined surface topography in multiaxis ball-end milling. The influences of machining parameters such as the milling modes, cutter runout, cutter inclination direction, and inclination angle upon the topography and surface roughness values are studied in detail. Compared with existing methods, the basic advantages and novelties of the proposed method can be resumed below. First, it is unnecessary to discretize the cutting edge and tool feed motion and rotation motion. Second, influences of cutting modes and cutter inclinations are studied systematically and explicitly for the first time. The generality of the algorithm makes it possible to calculate the pointwise topography value on any sculptured surface of the workpiece. Besides, the proposed method is proved to be more efficient in saving computing time than the time step method that is commonly used. Finally, some examples are presented and simulation results are compared with experimental ones.

scholarly journals On a fractional step-splitting scheme for the Cahn-Hilliard equation

2014 ◽  
Vol 31 (7) ◽  
pp. 1151-1168 ◽  
Author(s):  
A.A. Aderogba ◽  
M. Chapwanya ◽  
J.K. Djoko
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Computational Time ◽  
Fractional Step ◽  
Step Method ◽  
Fractional Step Method ◽  
Order Accuracy ◽  
Content Type ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. Design/methodology/approach – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Findings – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Originality/value – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation.

Investigation of Tumor Growth Based on Phase Field Model

2011 ◽  
Author(s):  
Zhi Zhu He ◽  
Jing Liu
Keyword(s):  
Cell Proliferation ◽  
Tumor Growth ◽  
Tumor Cell ◽  
Phase Field ◽  
Classical Model ◽  
Phase Field Method ◽  
Phase Method ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

This paper presents and investigates the tumor growth based on a phase model. The tumor core is necrotic and inhibitor chemical species are considered. The interface of tumor and health tissue is tracked using a phase field equation. The reformulation of a classical model, accounting for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion, is derived. The advantages of the finite difference methodology employed are generality and relative simplicity implication. We present simulations of the nonlinear evolution of growing tumors morphology and discuss the effects of tumor microenvironment. Mechanisms reflecting the tumor growth and development behavior was preliminarily interpreted. Recently numerous mathematical have been developed to investigate the growth dynamics of tumor [1–8]. One of most significant model developed by Wise [8] is based on Cahn-Hilliard equation, which is conservation phase field method. Allen-Chan nonconservation phase field has been developed to track the moving interface for multiphase simulation by Sun [9]. Allen-Chan equation is second order, while Cahn-Hilliard equation is fourth order in space. Thus, we introduce the Allen-Chan phase method [9–10] to simulate the tumor growth, which is very simple for numerical simulation The computation domain is illustrated in Fig. 1, where ΩH denotes host tissue, the tumor domains is comprised of viable tumor cell ΩV and dead tumor cell ΩD. The numerical results are presented at Fig. (2–4). One can find that the growth of tumor strongly depend on the nutrients and nonlinear unstable growth may lead to finger shaped pattern, which is in agreement with recent experimental observations [7] of in vivo tumor. In summary, a phase method has been developed to study diffusion and consumption of the nutrients and tumor cell proliferation, necrosis and migration, which discloses the evolution of complex shape of tumor.

Numerical Simulation of Solitary Wave Using the Fully Lagrangian Method of Moving Particle Semi Implicit

2014 ◽  
Author(s):  
Mohammad Amin Nabian ◽  
Leila Farhadi
Keyword(s):  
Numerical Simulation ◽  
Free Surface ◽  
Solitary Wave ◽  
Stokes Equations ◽  
Rock Avalanche ◽  
Water Tank ◽  
Step Method ◽  
Fractional Step Method ◽  
Time Step ◽  
Moving Particle

A mesh-less numerical approach, called the moving particle semi implicit method (MPS), is presented to solve inviscid Navier-Stokes equations in a fully Lagrangian form using a fractional step method. This method consists of splitting each time step in two steps. The fluid is represented with particles and the motion of each particle is calculated through interactions with neighboring particles by means of a kernel function. In this paper, the MPS method is used to simulate a dynamic system consisting of a heavy box sinking vertically into a water tank, known as Scott Russell’s wave generator problem. This problem is an example of a falling rock avalanche into natural or artificial reservoirs. The box sinks into water tank and as a result the water is heaved up to form a solitary wave and a reverse plunging wave which forms a vortex. This vortex follows the solitary wave down the water tank. The good agreement between the numerical simulation and the analytical solution confirms the accuracy of the model. This proves the applicability of the present model in simulating complex free surface problems. The number of particles on free surface is presented as an indicator of stability of the model.

Research on Numerical Simulation of Support Interference for Dynamic Derivatives at Hypersonic Flight Speeds

2014 ◽  
Vol 1016 ◽  
pp. 495-500
Author(s):  
Xu Liu ◽  
Wei Liu ◽  
Yun Fei Zhao
Keyword(s):  
Numerical Simulation ◽  
Unsteady Flow ◽  
Time Derivative ◽  
Least Square ◽  
Step Method ◽  
Time Step ◽  
Dynamic Conditions ◽  
Static Conditions ◽  
Support Interference ◽  
The Impact

The research on dynamic derivative under the dynamic unsteady condition is one of the most difficult aspects of the aircraft development process. For the special experiment of dynamic derivatives, no numerical simulation of support interference has yet been systematically studied. The numerical simulation of support interference for identification of dynamic derivatives with 7o blunted cone under the hypersonic condition was done in this paper. The 2-order Roe scheme and the dual-time-step method based on LU-SGS were respectively applied to discrete of the spatial and time derivative of the unsteady flow. A least square method based on the Etkin unsteady aerodynamic model was used for identification of dynamic derivatives. Hypersonic missile module was adopted as a verification example, and the numerical calculation results are consistent with the experimental results. For two different sting support forms with 7o blunted cone, the impact of sting support interference on dynamic derivatives was studied. Results show the moment coefficients of two support forms under static conditions are essentially the same, while there is a big difference in the pitch damping derivatives under dynamic conditions. The support interference of nonlinear aerodynamic loads resulted from the shock wave induced separation and other unsteady flow structures under dynamic conditions is far more complicated than that under static conditions, and the correction law of support interference under static conditions cannot be applied directly to the unsteady dynamic situations.

Effect of Mo and Ni on Phase Separation in Fe-Cr=Mo-Ni Quaternary Alloys

2011 ◽  
Vol 409 ◽  
pp. 449-454
Author(s):  
Ling Ling Yang ◽  
Yoshiyuki Saito
Keyword(s):  
Numerical Simulation ◽  
Phase Separation ◽  
Numerical Model ◽  
Accurate Solution ◽  
Ternary Alloys ◽  
Quaternary Alloys ◽  
Hilliard Equation ◽  
Model Based ◽  
Newton Raphson ◽  
Cahn Hilliard Equation

Numerical simulation of phase separation in Fe-Cr-Mo and Fe-Cr-Ni ternary alloys and Fe-Cr-Mo-Ni quaternary alloys were performed with use of the Cahn-Hilliard equation for ternary and quaternary alloys. A new numerical model based on the Gauss-Seidel and Newton Raphson methods was utilized to obtain efficient and accurate solution.

Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fengna Yan ◽  
Yan Xu
Keyword(s):  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Implicit Time ◽  
Time Step ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Hilliard Equation ◽  
Local Discontinuous Galerkin ◽  
Energy Stable ◽  
Cahn Hilliard Equation

Abstract In this paper, we mainly study the error analysis of an unconditionally energy stable local discontinuous Galerkin (LDG) scheme for the Cahn–Hilliard equation with concentration-dependent mobility. The time discretization is based on the invariant energy quadratization (IEQ) method. The fully discrete scheme leads to a linear algebraic system to solve at each time step. The main difficulty in the error estimates is the lack of control on some jump terms at cell boundaries in the LDG discretization. Special treatments are needed for the initial condition and the non-constant mobility term of the Cahn–Hilliard equation. For the analysis of the non-constant mobility term, we take full advantage of the semi-implicit time-discrete method and bound some numerical variables in L ∞ L^{\infty} -norm by the mathematical induction method. The optimal error results are obtained for the fully discrete scheme.

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