The falkneer-skan equation: Numerical solutions within group invariance theory

The iterative transformation method, defined within the framework of the group invariance theory, is applied to the numerical solution of the Falkner-Skan equation with relevant boundary conditions. In this problem a boundary condition at infinity is imposed which is not suitable for a numerical use. In order to overcome this difficulty we introduce a free boundary formulation of the problem, and we define the iterative transformation method that reducess the free boundary formulation to a sequence of initial value problems. Moreover, as far as the value of the wall shear stress is concerned we propose a numerical test of convergence. The usefulness of our approach is illustrated by considering the wall shear stress for the classical Homann and Hiemenz flows. In the Homann's case we apply the proposed numerical test of convergence, and meaningful numerical results are listed. Moreover, for both cases we compared our results with those reported in literature.     

An investigation of transonic flow over axisymmetric rigid body

The aim of this study is to investigate transonic flow over the axisymmetric rigid body of revolutions using matched asymptotic expansions of high Reynolds number flow. For this purpose the triple-deck model is employed. It allows to study the flow separation near a junction line where a circular cylinder is connected to a divergent conical body. It is found that in the axisymmetric transonic flow the interaction region is governed by the viscous-inviscid interaction process, where the axisymmetric Karman-Guderley equation in the inviscid part of the flow should be coupled with Prandtl’s boundary layer equations for the viscous sublayer. The coupled governing equations of the interaction region is solved using a semi-direct numerical method considering proper boundary conditions. Numerical results imply that incipience of separation may appear over the axisymmetric rigid body subject to body shape and transonic axisymmetric nature makes the flow much less prone to separation as compared to the two-dimensional flow.     

Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study

A numerical simulation for mixed convective three-dimensional slip flow of water-based nanofluids with temperature jump boundary condition is presented. The flow is caused by nonlinear stretching surface. Conservation of energy equation involves the radiation heat flux term. Applied transverse magnetic effect of variable kind is also incorporated. Suitable nonlinear similarity transformations are used to reduce the governing equations into a set of self-similar equations. The subsequent equations are solved numerically by using shooting method. The solutions for the velocity and temperature distributions are computed for several values of flow pertinent parameters. Further, the numerical values for skin-friction coefficients and Nusselt number in respect of different nanoparticles are tabulated. A comparison between our numerical and already existing results has also been made. It is found that the velocity and thermal slip boundary condition showed a significant effect on momentum and thermal boundary layer thickness at the wall. The presence of nanoparticles stabilizes the thermal boundary layer growth.

     

Turbulent boundary layer structure of flow over a smooth-curved ramp

The effects of a convex-curved wall followed by a recovery over a flat surface on a turbulent boundary layer structure are addressed via large-eddy simulation (LES). The curved wall constitutes a smooth ramp formed by a portion of circular arc. The statistically two-dimensional upstream boundary layer flow is realistically fed by an injected inflow boundary condition. The inflow is extracted from a simultaneously simulated flat-plate boundary layer which is computed based on a compressible rescaling method. After flowing over the curved surface the flow is allowed to recover its realistic condition by passing over a downstream flat surface. The Reynolds number introduced at the inlet section of the computational domain which starts 4 times the ramp length (L_r) upstream of the curved surface is Re_δo=U_∞δ_o/ν=9907. The Reynolds number is based on the inflow boundary layer thickness δ_o, the free-stream velocity U_∞ and the kinematic viscosity ν.Mean flow predictions obtained using the present LES with the rescaling–recycling inflow condition agree well with the available experimental data from literature. The Reynolds stress components match the experimental one. However, small deviation occurs due to the smaller-domain height used in the present simulation. The experiments showed that there is a generated pressure gradient on the upper wall and this in return affects the turbulence energy on the other wall. The numerical data as well as the experiments show an enhancement of the turbulent stresses in the adverse pressure gradient region. The increased level of turbulent stresses is accompanied with large peaks aligned with the inflection point of the velocity profiles. The high stress levels are nearly unchanged by reattachment process, decaying only after the mean velocity recovered and the high production of turbulence near the outer layer drops. The recovery of the outer layer is due to the turbulent eddies generated by the separation region. Numerical visualizations show strong elongation and lifting of eddies in the region of the adverse pressure gradient generated by the curved wall. Computations of two-point correlations are also performed to represent the formation and deformation of the turbulent eddies before, over and after the curved wall. Different effects on the eddy size and its structure angle are presented.     

Pricing American options under jump-diffusion models using local weak form meshless techniques

Recently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert, A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth, A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 (2012), pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.     

LES of synthetic jets in boundary layer with laminar separation caused by adverse pressure gradient

In the development of synthetic jet actuators (SJAs) for active flow control, numerical simulation has played an important role. In controlling the boundary layer flow separation, an integrated numerical model which includes both the baseline flow and the SJA is still in its initial stage of development. This paper reports preliminary results of simulating the interaction between a synthetic jet and a laminar separation bubble caused by adverse pressure gradient in a boundary layer. The computational domain was three-dimensional and Large-eddy simulation (LES) was adopted. The initial and boundary conditions were defined using or referring to our wind tunnel experimental results. Prior to numerically simulating the interaction between the synthetic jets and the baseline flow, a numerical model for simulating the separation bubble was developed and verified. In the numerical model including the SJA, the synthetic jet velocity at the exit of the SJA was defined as an input. The numerical model was further verified by comparing the simulation with experimental results. Based on reasonable agreement between the numerical and experimental results, simulations were carried out to investigate the dependency of flow control using synthetic jets on the forcing frequency, focused on the lower frequency range of the Tollmien-Schlichting (T-S) instability, and on the forcing amplitude which was represented by the maximum jet velocity at the exit of the SJA. Supporting the hypothesis based on the experiment, LES results showed that the forcing frequency had stronger influence on SJA’s effective elimination of the separation bubble than the forcing amplitude did.     

A Fourth Order Scheme for Incompressible Boussinesq Equations

A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.     

字典神经网络方法快速湍流烟雾合成

目的 基于物理的烟雾模拟是计算机图形学的重要组成部分,渲染具有细小结构的高分辨率烟雾,需要大量的计算资源和高精度的数值求解方法。针对目前高精度湍流烟雾模拟速度慢,仿真困难的现状,提出了基于字典神经网络的方法,能够快速合成湍流烟雾,使得合成的结果增加细节的同时,保持高分辨率烟雾结果的重要结构信息。方法 使用高精度的数值仿真求解方法获得高分辨率和低分辨率的湍流烟雾数据,通过采集速度场局部块及相应的空间位置信息和时间特征生成数据集, 设计字典神经网络的网络架构,训练烟雾高频成分字典预测器,在GPU（graphic processing unit）上实现并行化,快速合成高分辨率的湍流烟雾结果。结果 实验表明,基于字典神经网络的方法能够在非常低分辨率的烟雾数据下合成空间和时间上连续的高分辨率湍流烟雾结果,效率比通过在GPU平台上直接仿真得到高分辨率湍流烟雾的结果快了一个数量级,且合成的烟雾结果与数值仿真方法得到的高分辨率湍流烟雾结果足够接近。结论 本文方法解决了烟雾的上采样问题,能够从非常低分辨率的烟雾仿真结果,通过设计基于字典神经网络结构以及特征描述符编码烟雾速度场的局部和全局信息,快速合成高分辨率湍流烟雾结果,且保持高精度烟雾的细节,与数值仿真方法的对比表明了本文方法的有效性。     

用边界粒子法求解柯西反问题的数值计算软件包

为推动无网格方法在反问题中的应用及相关计算软件的发展,介绍基于MATLAB开发的用边界粒子法(Boundary Particle Method,BPM)数值模拟柯西反问题的计算软件包;描述数值算法的理论基础,并结合标准算例介绍软件包的基本功能和使用方法.该软件包可求解含不同形式源项(多项式函数、三角函数、指数函数及其耦合函数等)的椭圆型偏微分方程,并且可用于处理任意几何区域问题.本软件包具有简单易用的图形用户界面,在数值模拟柯西反问题时仅需边界测量数据,且效率高、精度高.     

Numerical simulation of the interaction of a circular synthetic jet with a boundary layer

In this work, a 3D numerical simulation of circular synthetic jets issued into a laminar boundary layer developing over a flat plate was undertaken in a complementary manner alongside with an experimental study with the aim of achieving an improved understanding of the fluid mechanics underlying the interaction process between the synthetic jets and the boundary layer. The simulation was carried out in FLUENT at two diaphragm operating conditions, which produced two distinctly different vortical structures and shear stress footprints on the wall. The simulation results were validated using experimental data and a good agreement was achieved. The temporal evolution of coherent structures formed as the result of this interaction was examined using the Q-criterion. The hierarchy of the coherent structures was established which provided a credible explanation of the wall shear stress pattern observed in both the experiment and the simulation. The high spatial resolution in the near-wall region and 3D nature of the simulation results provide the information about the flow field which is not only consistent with but also additional to that from the experiment, leading to an improved understanding of the interaction process between the synthetic jets and the boundary layer and its resultant structures.     

A stable Gaussian radial basis function method for solving nonlinear unsteady convection–diffusion–reaction equations

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection–diffusion–reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor–corrector schemes such as Adams–Bashforth and implicit–explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.     

Rotational fluid flow using a least squares collocation technique

A potential theory approach for incompressible viscous flow which leads to the biharmonic equation is first developed. A numerical least squares collocation technique using fundamental singular solutions of the biharmonic equation is then applied to a rotational flow problem with moving boundaries that produce discontinuous boundary conditions associated with the biharmonic. It is shown that the least squares technique smoothes out local disturbances in boundary data of the type which are likely to present difficulties to the more commonly used boundary element method. A compact computer program for the method and the results for the problem of a rectangular channel with one moving boundary are included along with an experimental verification of the results using the thin plate bending analogy.     

气流组织中悬浮边界面的数值仿真技术的研究

气流组织是空调房间内空气的分布、流动形式,主要反映在气流的温度、湿度、气流速度和清洁度等指标,是否符合行业标准规定的空调房间的要求.气流组织可以通过数值计算理论、计算机仿真技术给予实现.论文以一个典型的变风量中央空调房间为研究基础,以该房间内的送、回风口边界面为研究实例,通过分析悬浮在房间立体内的送、回风口的数值仿真方法,运用实体造型技术,设计出在房间立体与悬浮边界面之间的拓扑关系,使两者成为一体.然后该处理技术经过数值仿真实例的验证证明是合理的、可行的.     

Direct numerical simulation of flow separation around a NACA 0012 airfoil

Direct numerical simulation (DNS) for the flow separation and transition around a NACA 0012 airfoil with an attack angle of 4° and Reynolds number of 10⁵ based on free-stream velocity and chord length is presented. The details of the flow separation, detached shear layer, vortex shedding, breakdown to turbulence, and re-attachment of the boundary layer are captured in the simulation. Though no external disturbances are introduced, the self-excited vortex shedding and self-sustained turbulent flow may be related to the backward effect of the disturbed flow on the separation region. The vortex shedding from the separated free shear layer is attributed to the Kelvin-Helmholtz instability.     

基于等离子流动控制的翼型减阻技术

为研究基于等离子流动控制的减阻技术,基于Langtry-Menter转捩模型提出边界层转捩数值模拟技术.该技术可有效结合转捩模型与湍流模型,用标准模型验证其精确性,为采用等离子流动控制抑制边界层分离和转捩研究提供数值模拟平台.采用基于现象学模型的等离子流动控制数值模拟技术,对流动分离以及边界层转捩抑制进行数值模拟,为基...     

The Local Artificial Boundary Conditions for Numerical Simulations of the Flow Around a Submerged Body

We consider in this work the numerical approximations of the two-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface. Several vertical segments are introduced as the upstream and the downstream artificial boundaries, where a sequence of high-order local artificial boundary conditions are proposed. Then the original problem is solved in a finite computational domain, which is equivalent to a variational problem. The numerical approximations for the original problem are obtained by solving the variational problem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.     

Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes

The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.     

聚合物熔体三维等温流动的罚有限元分析

本文针对幂律流体和非线性粘弹性PTT流体，采用罚有限元法，形成了求解聚合物熔体三维等温流动速度场和粘弹性应力场的有限元模型．对非线性粘弹性PTT流体，为了降低模拟计算对计算机硬件的要求，并使模拟计算更加稳定，采用了去耦算法，包括拟体力方法和动量方程的椭圆类方程转化方法等。文中还给出了总体有限元方程组的形成和求解过程。     

Meshless local radial point interpolation to three-dimensional wave equation with Neumann's boundary conditions

In this article, the meshless local radial point interpolation (MLRPI) method is applied to simulate three-dimensional wave equation subject to given appropriate initial and Neumann's boundary conditions. The main drawback of methods in fully 3-D problems is the large computational costs. In the MLRPI method, all integrations are carried out locally over small quadrature domains of regular shapes such as a cube or a sphere. The point interpolation method with the help of radial basis functions is proposed to form shape functions in the frame of MLRPI. The local weak formulation using Heaviside step function converts the set of governing equations into local integral equations on local subdomains where Neumann's boundary condition is imposed naturally. A two-step time discretization technique with the help of the Crank-Nicolson technique is employed to approximate the time derivatives. Convergence studies in the numerical example show that the MLRPI method possesses reliable rates of convergence.