基于动态近似边界条件的气动弹性数值模拟

发展了一种利用欧拉方程计算非定常气动力的数值方法,通过在固定物面边界上满足动态近似边界条件计算出非定常气动力,避免了在每个时间步重新生成网格或需用动网格技术进行网格变形处理过程,提高了计算效率。运用这种方法计算了一系列非定常气动力算例,并与非结构动网格准确边界条件下的欧拉方程解和实验数据进行了比较,进一步分析了翼型俯仰角和马赫数对非定常气动力相对误差的影响。将气动力解算器与结构方程耦合进行气动弹性数值模拟,计算了跨音速具有S型颤振边界的二元气动弹性标准算例-Isogaiwing。算例结果表明,利用动态近似边界条件的欧拉方程具有简便、高效的特点,并能在小振幅情况下得到与精确边界条件精度相当的非定常流场解,还可以用于气动弹性分析。     

Viscous oscillating cascade aerodynamics and flutter by a locally analytical method

A mathematical model is developed to analyze the viscous aerodynamics of an harmonically oscillating flat plate airfoil cascade in an incompressible laminar flow. The steady flow field is described by the Navier-Stokes equations, with the unsteady viscous flow modeled as a small perturbation to this steady flow. Solutions for both the steady and the unsteady viscous flow fields are then obtained by developing locally analytical solutions. The significant effects of Reynolds number, elastic axis, interblade phase angle and incidence angle on the oscillating cascade unsteady aerodynamics and torsional flutter characteristics are then demonstrated.List of symbols C airfoil chord - C _M unsteady moment coefficient - k reduced frequency, U/U - Re Reynolds number, U C/v - S cascade spacing - U free-stream velocity magnitude - x _ea elastic axis location - x mean flow direction coordinate - y normal flow direction coordinate - y _m mean airfoil position - x x-direction step size - y y-direction step size - (x ₀, y ₀) center of grid element - ₀ mean flow incidence angle - amplitude of airfoil oscillation - interblade phase angle - nondimensional unsteady stream function - nondimensional steady stream function - nondimensional unsteady vorticity - nondimensional steady vorticity - cascade stagger angle     

A panel method computation for oscillating aerofoil in compressible flow

A panel method using source and doublet singularity has been proposed to solve for subcritical aerodynamics of a two dimensional steady and unsteady aerofoil. The source singularities are placed on the aerofoil surface. The doublet singularity is distributed by a function along the chordline of the aerofoil; this distribution is further projected downstream into infinity. The aerodynamics of an oscillating aerofoil is investigated. The governing unsteady linearized potential equation has a Hankel function as its fundamental solution, which is a source type function. A combination of source and doublet singularity is therefore used for solving the unsteady compressible problem by means of the panel method, this methodology being an extension of a steady aerofoil formulation. Incremental effects of profile change in aerofoil and wake geometry are accounted for. A surface boundary condition is applied on the stationary mean aerofoil surface with time dependent geometrical changes accounted for. An unsteady Kutta condition of equal pressure across the trailing edge is assumed. Results are presented on the aerodynamic influence of Mach number, oscillating frequency parameter, angle of incidence and change of pivoting point. Results are also compared with linear theory, a subsonic experimental result and a subcritical solution of a transonic model.     

Oscillating cascade unsteady aerodynamics including separated flow effects

A mathematical model is developed to predict the effect of flow separation on the unsteady aerodynamic lift and moment acting on a two-dimensional flat plate cascade which is harmonically oscillating in a subsonic flow field. The unsteady flow is considered to be a small perturbation to the uniform steady flow, with the steady flow assumed to separate at a specified fixed position on the airfoil suction surface. This formulation does not require the difference in the upwash velocity across the airfoil in the separated flow region to be determined before calculating the unsteady pressure difference across the chordline of the airfoils, thereby eliminating the assumption that the upwash difference is zero at the trailing edge when the steady flow is separated. Results obtained demonstrate that although flow separation decreases bending mode stability, it does not result in bending mode flutter. However, flow separation can result in torsion mode flutter, with this instability being a function of the location of both the separation point and the elastic axis.     

MFS with time-dependent fundamental solutions for unsteady Stokes equations

This paper describes the applications of the method of fundamental solutions (MFS) for 2D and 3D unsteady Stokes equations. The desired solutions are represented by a series of unsteady Stokeslets, which are the time-dependent fundamental solutions of the unsteady Stokes equations. To obtain the unknown intensities of the fundamental solutions, the source points are properly located in the time–space domain and then the initial and boundary conditions at the time–space field points are collocated. In the time-marching process, the prescribed collocation procedure is applied in a time–space box with suitable time increment, thus the solutions are advanced in time. Numerical experiments of unsteady Stokes problems in 2D and 3D peanut-shaped domains with unsteady analytical solutions are carried out and the effects of time increments and source points on the solution accuracy are studied. The time evolution of history of numerical results shows good agreement with the analytical solutions, so it demonstrates that the proposed meshless numerical method with the concept of space–time unification is a promising meshless numerical scheme to solve the unsteady Stokes equations. In the spirit of the method of fundamental solutions, the present meshless method is free from numerical integrations as well as singularities in the spatial variables.     

Analytical and Numerical Investigations of Unsteady Graphene Oxide Nanofluid Flow Between Two Parallel Plates

In this research, different analytical methods were applied to characterize thermal behavior of unsteady graphene oxide–water nanofluid flow between two parallel moving plates. First of all, partial differential equations (PDEs) were transformed to a system of nonlinear ordinary differential equations (ODEs) using similarity solution. Then, collocation method (CM), least square method (LSM) and Galerkin method (GM) were used to solve the system of ODEs and determine velocity and temperature distribution functions. In addition, effects of moving parameter, concentration, Eckert and Prandtl numbers on nanofluid velocity and temperature profiles were examined. Next, using numerical solution of the obtained system of differential equations, the results obtained from the analytical solutions were validated with that of the numerical solution. The validation results indicated high and appropriate accuracy of the analytical solutions compared to the numerical one.     

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

The Eulerian–Lagrangian method of fundamental solutions is proposed to solve the two-dimensional unsteady Burgers’ equations. Through the Eulerian–Lagrangian technique, the quasi-linear Burgers’ equations can be converted to the characteristic diffusion equations. The method of fundamental solutions is then adopted to solve the diffusion equation through the diffusion fundamental solution; in the meantime the convective term in the Burgers’ equations is retrieved by the back-tracking scheme along the characteristics. The proposed numerical scheme is free from mesh generation and numerical integration and is a truly meshless method. Two-dimensional Burgers’ equations of one and two unknown variables with and without considering the disturbance of noisy data are analyzed. The numerical results are compared very well with the analytical solutions as well as the results by other numerical schemes. By observing these comparisons, the proposed meshless numerical scheme is convinced to be an accurate, stable and simple method for the solutions of the Burgers’ equations with irregular domain even using very coarse collocating points.     

Boundary element analysis of wave-current interaction around a large structure

A numerical approach for wave-current interaction around a large structure is investigated, based on potential flow theory, linear waves and small current velocity approximation. The velocity potential in a wave-current coexisting field is separated into a steady current potential and an unsteady wave potential. The boundary element method was then employed to compute the unsteady wave potential with effects of both a uniform current and a large body taken into consideration. It is demonstrated that the steady current potential can be expressed as the sum of a uniform current and a steady disturbance due to the presence of the object. The variation of current velocity in the vicinity of the object is then calculated by using a surface vorticity boundary integral meethod. Boundary element analysis is also used for the numerical solutions of the surface vorticity method. Substituting both unsteady wave potential and current velocity into the first-order dynamic surface boundary condition, the water surface elevation around a large structure in a wave-current coexisting field can then be obtained. Comparisons of numerical predictions with experimental results ar also made; qualitative good agreements are obtained.     

High frequency oscillating viscous flow over elliptic cylinder at incidence

Summary The problem of high frequency oscillating viscous flow over an elliptic cylinder at incidence is investigated. The method of matched inner-outer asymptotic expansion to second order is used to solve the governing equations. The steady and the unsteady modes of flow, related to the present work, are identified and separated. Both steady and unsteady drag and lift coefficients are presented and discussed. The effect of different parameters such as the Strouhal number, Reynolds number, focal length and angle of attack are explored.     

具有结构非线性的二元翼面颤振研究

该文分别研究了俯仰方向带中心间隙和初偏间隙的二元翼面,并提出通过加入摩擦力矩来减弱间隙非线性影响的方案。基于简谐条件下求得的非定常空气动力矩阵,采用Roger有理函数拟合,将其转为时域下的气动力的近似表达式,之后采用四阶Runge-Kutta法求解其时域内的气动弹性响应。结果显示：中心和初偏间隙对机翼产生的影响类似,都能使其在一定的速度范围内产生复杂的极限环振荡,但两者对间隙大小的敏感度有所不同。加入的摩擦能有效抑制间隙非线性的影响,使翼面的运动在低于线性颤振速度下能够衰减。且当加入的摩擦类型确定,初始扭矩和摩擦刚度的值比较大的情况下,其值和摩擦非线性在初偏间隙中加入位置的变化都对系统的颤振速度不会有明显的影响。     

On the purely analytic computation of laminar boundary layer flow over a rotating cone

The fundamental purpose of the present research is to obtain analytical expressions for the solution of the steady laminar flow of an incompressible viscous Newtonian fluid over a rotating cone. Using a proper similarity transformation akin to the classical one of Von Karman the nonlinear equations of motion are reduced to a boundary value problem whose solution is then derived in terms of a series of exponentially-decaying functions for the full range of cone half-angle ? characterizing the conical flow structure. The exact numerical method is found to improve as the cone half-angle is decreased. The effects of the cone half-angle on the physically significant relevant parameters, such as the wall shears, the torque and the vertical suction are clarified. Purely explicit analytical expressions for the solution of governing equations to support the numerically evaluated solutions are also obtained via the homotopy analysis method.     

Some anomalies of the numerical solutions of the Euler equations

Summary The numerical solution of the unsteady Euler equations for compressible flow over a circular cylinder is obtained using standard numerical techniques. The equations, written in cylindrical coordinates, are discretized on an orthogonal grid via central differences for spatial derivatives, using a simple second order artificial viscosity form and a special treatment of the boundary conditions. Backward differences in time are employed resulting in a large system of nonlinear difference equations at each step. A direct solver (LAPACK), based on an efficient Gaussian elimination procedure for banded matrices, is used to solve the linearized system of equations. The stability of the nonunique solutions of the steady Euler equations is investigated. It is demonstrated that the symmetric solutions, with zero circulation, are not stable. For a certain Mach number range, a periodic solution is obtained where the shock oscillation persists. If a periodic circulation (within a certain frequency range) is enforced in the far field, an irregular solution emerges with unpredictable shock motions. For such a solution, the Lyapunov exponent is shown to be greater than zero, indicating the appearance of chaos.     

A locally analytic technique applied to grid generation by elliptic equations

One technique for obtaining grids for irregular geometries is to solve sets of elliptic partial differential equations. The solution of the partial differential equations yields a grid which discretizes the physical solution domain and also a transformation for the irregular physical domain to a regular computational domain. Expressing the governing equation of interest in the computational domain requires the derivatives of the physical to computational domain transformation, i.e., the metrics. These metrics are typically determined by numerical differentiation, which is a potential source of error. The locally analytic method uses the analytic solution of the locally linearized equation to develop numerical stencils. Thus, the locally analytic method allows numerical differentiation with no loss of accuracy. In this paper, the locally analytic method is applied to the solution of the Poisson and Brackbill–Saltzman equations. Comparison with an exact solution shows the locally analytic method to be more accurate than the finite difference method, both in solving the partial differential equation and evaluating the metrics. However, it is more computationally expensive.     

An order-N complexity meshless algorithm for transport-type PDEs,based on local Hermitian interpolation

This work describes a new numerical method utilising radial basis function interpolants. Based on local Hermitian interpolation of function values and boundary operators, and using an explicit time advancement formulation, the method is of order-N complexity. Computational cost to advance the solution in time is minimal, and is largely dependent on local system support size. The explicit time advancement formulation allows a novel solution technique for many nonlinear partial differential equations.The performance of the method is examined for a variety of linear convection–diffusion–reaction problems, featuring both steady and unsteady solutions. The method is also demonstrated with a nonlinear Richards’ equation model, solving an unsaturated flow in porous media problem. The technique is named the local Hermitian interpolation (LHI) method.     

A fast integral approach for drag force calculation due to oscillatory slip stokes flows

In this paper, we describe an efficient numerical method for modelling oscillatory incompressible slip Stokes flows in three dimensions. The efficiency is achieved by employing an integral approach combined with an accelerated boundary‐element‐method (BEM) solver. First the integral representations for slip flows with two different slip models are formulated. The resulting integral equations are then solved using the BEM combined with the precorrected‐FFT accelerated technique. 3D numerical codes have been developed based on the method described above. These codes are then used to calculate the drag forces on oscillating objects immersed in an unbounded slip flow. Three objects are considered, namely a sphere, a pair of plates and a comb structure. The simulated drag forces on these objects obtained from the two slip models are compared. In the sphere case, the simulated results are also compared with the analytical solutions for both the steady‐state case and the no‐slip oscillatory case and are found to be in good agreement. In addition, qualitative comparison of the simulation results with the experimental results in the plate problem is also presented in this paper. Copyright © 2004 John Wiley & Sons, Ltd.     

Two-fluid model simulation of two-phase flow problems using a conservative finite-volume method

A conservative finite-volume method is employed for simulations of two-phase flows with a two-fluid model using separate conservation equations for each fluid. The mathematical model is based on ensemble- and phase-averaged transport equations and a set of simple constitutive laws for interfacial phenomena. The equation system is solved iteratively and in a segregated manner with the IPSA and PEA algorithms developed by Spalding [7]. The method is verified by comparison with analytical solutions and experimental data for one-dimensional steady and unsteady two-phase flows. A formal solution error estimation procedure based on extrapolation techniques is employed to assess the numerical accuracy of the results.     

不可压缩流中二元翼运动的余维二分叉

考虑二元机翼在不可压缩流作用下运动的余维二分叉问题,弹簧的非线性刚度考虑为立方型。化方程为四维一阶微分方程,解析求出了系统发生叉式分叉与HOPF分叉的临界流速的表达式和发生余维二分叉的临界流速和一次刚度。在两个参数平面上研究了系统在余维二分叉点附近的不同区域内平衡点和极限环的稳定性及其个数。最后用数值积分方法给出相应的结果。     

Three-dimensional wall-bounded laminar boundary layer with span-wise cross free stream and moving boundary

In the current work, the 3D boundary layers of wall-bounded flow configurations were extended to the situations with span-wise cross moving boundary and free stream. The unsteady boundary layer is also addressed for the Falkner–Skan wedge flow with a span-wise oscillating wall or oscillating free stream. The span-wise secondary boundary layer equation is obtained using similarity transformation technique and solved analytically in terms of the primary stream-wise boundary layer flow solutions. Different fluid motion behaviors are found for these new solutions. It is found that for the span-wise secondary boundary layer flow there is no flow separation for any wall cross moving velocity, which is different from the primary stream-wise boundary layers with a reverse flow. For the unsteady boundary layer with an oscillating wall or free stream, it is seen that the solution is different from the Stokes oscillating plate or free stream problem. The unsteady wall drag increases with the increase in the oscillating frequency and decreases with increasing the primary span-wise free stream magnitude. The velocity overshooting near the wall is also seen for an oscillating free stream for a large oscillating frequency or a lower primary stream-wise free stream magnitude.     

叶栅内非定常不可压流动的数值模拟

基于SMAC(Simplified Marker and Cell)方法推导出直接求解二维非定常、不可压N-S方程的隐式数值方法.求解的基本方程是任意曲线坐标系中以逆变速度为变量的N-S方程和椭圆型的压力Poisson方程.采用该方法,对二维叶栅非定常分离流场进行了数值模拟,叶栅表面压力的计算结果与试验结果相比比较吻合,从而验证了这种方法的可靠性.同时对叶栅非定常流场的流场结构和流动机理做了初步的探讨.在均匀来流和定常边界条件下,叶栅内部流动表现出强烈的非定常性;在小冲角和高雷诺数时,叶栅尾部产生类似卡门涡街的周期性流动.     

Finite element computation of unsteady viscous transonic flows past stationary airfoils

Results are presented for computations of unsteady viscous transonic flows past a stationary NACA0012 airfoil at various angles of attack. The Reynolds number, based on the chord-length of the airfoil, is 10,000 and the Mach number is 0.85. Stabilized finite-element formulations are employed to solve the compressible Navier-Stokes equations. The equation systems, resulting from the discretization, are solved iteratively by using the preconditioned GMRES technique. Time integration of the governing equations is carried out for large values of the non-dimensional time to understand the unsteady dynamics and long-term behavior of the flows. The results show interesting flow patterns and a complex interaction between the boundary/shear layers, shock/expansion waves and the lateral boundaries of the computational domain. For transonic flow past an airfoil at various angles of attack in a narrow channel/wind-tunnel one can observe solutions that are qualitatively different from each other. At low angles of attack an unsteady wake is observed. At moderate angles of attack the interaction between the shock system and the lateral walls becomes significant and the temporal activity in the wake decreases and eventually disappears. At high angles of attack a reflection shock is formed. Hysteresis is observed at an angle of attack 8^∘. For the flow in a domain with the lateral boundaries located far away, the unsteadiness in the flow increases with an increase in the angle of attack. Computations for a Mach 2, Re 106 flow past an airfoil at 10^∘ angle of attack compare well with numerical and experimental results from other researchers