Iterative numerical solutions and boundary conditions for the parabolized Navier-Stokes equations

A new numerical iteration scheme for solving the parabolized Navier-Stokes (PNS) equations is presented. This scheme has all the features and advantages of the successive line over relaxation (SLOR) technique, and thus it can be easily accelerated to get much higher rate of convergence of the global iteration scheme than previously suggested schemes. The choice of appropriate downstream boundary conditions for the PNS and Navier-Stokes equations is discussed in the context of boundary layer simulation. A critical comparison of accuracy and rate of convergence is performed for the flow over a flat plate.     

Analysis of global pressure relaxation for flows with strong interaction and separation

A global or pressure relaxation formulation for the reduced form of the Navier-Stokes equations, frequently referred to as semi-elliptic or partially parabolized or just “parabolized” Navier-Stokes (PNS),^‡ is presented. Difference procedures and relaxation solutions for the (u, v, p) system are described. The continuity equation is satisfied exactly at each grid point and a poisson pressure equation is not required explicitly. Several model problems, e.g. finite flat plate, trough, boattail and airfoil, are considered. Strong pressure interaction is evident in each case and axial flow separation occurs for several of the problems. The questions of accuracy, stability, convergence rate, and implied difference forms of the pressure and vorticity equations are addressed. Solutions for laminar and turbulent modelling are presented.     

一种新的余数系统下快速计算素域椭圆曲线点乘的方法

椭圆曲线密码运算主要是椭圆曲线点乘,后者由一系列的模乘构成。利用余数系统下的蒙哥马利模乘算法,素域中对阶取模余的模乘可以转化为对余数系统基底取模余。提出一种新的余数系统下的方法以加速计算椭圆曲线点乘。（1）与传统上取两个几乎对称的余数系统不同,该方法取了两个非对称的余数系统。其中,余数系统Γ包括两个模数{2L, 2 L-1}; 余数系统Ω包括八个模数,它们都具有如2L－2Ki+1的形式。这种选择使其模算术变得简单。（2）在上述非对称的余数系统中,大部分原来需要对椭圆曲线域特征值取模的模乘运算可以在余数系统中直接用乘法代替。此外,计算椭圆曲线点乘时用到了仅计算x坐标的蒙哥马利梯子。在每次并行的倍点和点加结束时,需要四次余数系统下的蒙哥马利模乘,以压缩中间结果的值域。因此,计算一个N位的椭圆曲线点乘,需要的时间约为55.5N·I, 其中,I是一个L/2位的乘法、一次保留进位加法、一个L/2位的加法的总延时。     

Flow characterization of valveless micropump using driving equivalent moment: theory and experiments

A valveless micropump, actuated by a PZT disk bonded to a glass plate, can generate positive flow. In order to estimate flow characteristics of micropumps, it is necessary to theoretically analyze the radial expansion (more specifically, the equivalent moment) of the PZT disk according to the voltage input. Using the equivalent moment, deflection equations are derived for the tri-layer disk (PZT, epoxy bonder and glass plate) and are confirmed to match well with experiments. The flow rate of the valveless micropump is also theoretically and experimentally investigated in terms of input voltage and oscillation frequency. The flow increased at a rate of 0.1 μL/min/V, and the maximum flow rate was obtained at the driving frequency of around 225 Hz.     

Application of the method of integral relations to boundary layer flows over blunt bodies

Calculations of boundary layer flows past blunt bodies at angles of incidence are presented. Using the method of integral relations together with the method of lines, the full three-dimensional boundary layer equations are reduced to a system of first order ordinary differential equations. The streamwise shear stress function θ and the cross-flow velocity component V are represented as suitable functions of the streamwise velocity component U. The role of the zone of dependence is automatically satisfied by the choice of differencing in the method of lines. Solutions correct to the second order are obtained in the positive shear region for flow over an ellipsoid at 30° incidence. The results are compared with corresponding finite difference solutions.     

Influences of wind flow over heritage sites: A case study of the wind environment over the Giza Plateau in Egypt

Wind influences are important environmental factors that cause deterioration of historical heritage sites. This work presents a computational framework for investigating the influences of the wind flow over such sites. The wind flow is considered to be fully turbulent, isothermal and incompressible. The present framework employs three-dimensional Reynolds Averaged Navier–Stokes (RANS) equations along with multi-block approach and non-conformal meshes to perform wind flow simulations over such sites with complex geometry. As a case study, the influences of wind flow over the “Giza Plateau”, one of the most important Egyptian historical heritage sites, were studied for the Northwest wind (at average wind speed over the year) and the Southwest windstorms. The study addresses the less understood, yet important, influences of the wind flow structure on the site and its famous monuments: the Pyramids and the Great Sphinx. Qualitative and quantitative treatments of the results are carried out for estimating the wind loading on the different monuments within the plateau. Particular attention was paid to the Great Sphinx to investigate its most vulnerable parts to the wind, as one of the critical environmental factors that cause erosion of this colossal statue.     

Application of spectral methods to the solution of Navier-Stokes equations

In this paper a number of topics that arise in application of Chebyshev expansion methods to the solution of the Navier-Stokes equations are addressed. These include the equivalence of finite differences and finite elements approaches, a new velocity-pressure formulation that permits easy extension to three dimensions, evaluation of the importance of pressure boundary conditions and the virtues of collocation over the tau method for satisfying boundary conditions. Example results from the velocity-pressure formulation which eliminate a non-linear momentum equation in favor of the linear continuity equation are presented. The results are for a 2-D unsteady flow on a flat plate at large Reynolds numbers. The behavior of an unsteady disturbance in such a flow is examined and compared with previous stream-function vorticity results of Murdock.     

A new unconstrained third-order plate theory for Navier solutions of symmetrically laminated plates

Among the many higher order plate theories, the third-order shear deformation plate theory (TSDT) of Reddy is, perhaps, the most widely adopted model in the study of laminated plates. It, however, imposes a restriction that transverse shear stress vanishes on the top and bottom surfaces of the plate. Such requirement, although reasonable in many engineering applications, is not valid when the plate is subject to shear traction parallel to the surface. To account for such problems, the present plate model releases the constraints of vanishing transverse shear stress on the top and bottom plate surfaces. This unconstrained third-order shear deformation plate theory (UTSDT) is particularly useful for the study of a plate with contact friction or present in a flow field where the boundary layer shear stress is significant. The governing differential equations of UTSDT are of similar complexity as that of TSDT but it yields more accurate solutions. In addition, it is more flexible as it can be degenerated to the first-order shear deformation plate theory (FSDT) of Reissner and Mindlin if the higher-order rotation coefficients are neglected and a shear correction factor is considered, or to the TSDT if the relevant rotation coefficients are constrained. The present study further develops the unconstrained theory in composite laminates. Navier solutions for bending and stress analysis of multilayered and symmetrically laminated composite plates are presented. It is concluded that the present plate model provides more accurate solutions than that of TSDT, with similar level of analytical complexity, when compared with the 3D elasticity exact solutions.     

The investigation of a compressible laminar boundary layer past an elliptical cone

The self-similar boundary layer on a sharp circular cone was calculated first by Vvedenskaya[1]. The boundary layer equations were solved in a plane containing an outflow line which lies in this case in the symmetry plane at the leeward side, and after that a solution was constructed by using a marching method along a circumferential coordinate. The calculation results for the boundary layer on an elliptic cone were presented in Bashkin's papers [2–4]. However, only the middle angles of attack (30–50°) were considered. where the outflow line of an external stream is located in the windward symmetry plane, and the flow pattern in the boundary layer is analogous to that of the circular cone. In the present the laminar boundary layer on an elliptic cone is studied for a wide range of angles of attack. The boundary layer has been calculated at small incidence when the outflow of an external flow were located out of the symmetry plane. In this case the equations are solved first in the plane containing the outflow line and then the solutions were constructed by a marching method along a circumferentialcoordinate to the windward and leeward symmetry planes. The distribution of the skin-friction coefficients and the Stanton's numbers on a cone surface was given. The similarity solution of a set of boundary layer equations was obtained for thin cones at large incidence when the stream on the windward side of the cone was directed to the cone nose. The calculations of the laminar boundary layer at hypersonic velocities were carried out to include the real equilibrium properties of the air.     

Numerical prediction of a laminar boundary layer flow on a rotating sphere

Numerical solutions to a laminar boundary layer flow past a sphere are considered. The solutions are presented using the procedure of Gosman et al. [1] with appropriate modifications. Successful numerical solution procedures have been devised for the solution of flow problems, see [5]. The SOR method is chosen as a method of solution. Although it looks like a simple method, application of such a method to nonlinear Navier-Stokes equations is highly nontrivial. The matrix method is not used because convergence was not a problem for the type of flow considered in this paper. The governing nonlinear differential equations are converted into finite difference equations by integrating the equations over a control volume and are then solved by an iterative procedure. The numerical results predict that the transverse velocity v_θ is positive in the upper hemisphere, goes to zero in the equitorial plane and becomes negative in the lower hemisphere.     

钝舵绕流不同湍流模型数值模拟比较研究

１．引言钝舵所诱导的激波／边界层干扰流场十分复杂,其中不仅含有复杂的波系结构,而且还包含有涡系结构．对此人们已开展了很多实验研究［‘－＼揭示了流场的一些基本特征,对称面流场结构如图１所示．但是受实验条件的限制,流场的详细结构尚未得到充分研究．数值模拟的方法可以提供流场的详细结构,但当流动出现湍流时,数值模拟将遇到新的困难．目前较多的计算方法是基于雷诺平均Ｎ－Ｓ方程,需补充新的湍流模型方程以封闭方程组．在应用研究上主要以一些工程模型为主,其中代数模型是一种平衡模型,虽然仅考虑了流场的局部特性,但在…     

Rapid heating vibrations of FGM annular sector plates

In this paper, thermally induced vibration of annular sector plate made of functionally graded materials is analyzed. All of the thermomechanical properties of the FGM media are considered to be temperature dependent. Based on the uncoupled linear thermoelasticity theory, the one-dimensional transient Fourier type of heat conduction equation is established. The top and bottom surfaces of the plate are under various types of rapid heating boundary conditions. Due to the temperature dependency of the material properties, heat conduction equation becomes nonlinear. Therefore, a numerical method should be adopted. First, the generalized differential quadrature method (GDQM) is implemented to discretize the heat conduction equation across the plate thickness. Next, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson time marching technique. The obtained thermal force and thermal moment resultants at each time step from temperature profile are applied to the equations of motion. The equations of motion, based on the first-order shear deformation theory (FSDT), are derived with the aid of the Hamilton principle. Using the GDQM, two-dimensional domain of the sector plate and suitable boundary conditions are divided into a number of nodal points and differential equations are turned into a system of ordinary differential equations. To obtain the unknown displacement vector at any time, a direct integration method based on the Newmark time marching scheme is utilized. Comparison investigations are performed to validate the formulation and solution method of the present research. Various examples are demonstrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the plate, temperature dependency, sector opening angle, values of the radius, in-plane boundary conditions, and type of rapid heating boundary conditions on thermally induced response of the FGM plate under thermal shock.

     

Boundary formulations for high-order finite differences on staggered meshes

Discretizations of the compressible flow equations with flow variables defined at different staggered positions on a regular mesh have accuracy and stability advantages over standard collocated discretization, but implementation of boundary conditions is hampered without all flow variables available at any boundary point. Boundary schemes for implementation of the boundary conditions compatible with staggered mesh discretizations are considered in this study. We focus on a combination of fourth- and fifth-order schemes near the boundary that are stable when sixth-order centered schemes are used for the interior points. Characteristics-based formulations provide a physically meaningful treatment for all the variables, avoiding the use of extrapolation. However its application on staggered meshes has not been systematically studied and its implementation is unclear since this method requires collocation of all the variables at the boundary, which is not natural for standard staggered mesh formulations. We show that including all the flow variables at the boundary can be done in a way that does not affect resolution or accuracy of the formulation. Predictions based upon analysis with model equations are verified with a staggered mesh flow solver.     

A Transformation-free HOC Scheme for Incompressible Viscous Flow Past a Rotating and Translating Circular Cylinder

In the present work, a numerical study is made using a recently developed Higher Order Compact (HOC) finite difference scheme to test its capacity in capturing the very complex flow phenomenon of unsteady flow past a rotating and translating circular cylinder. The streamfunction-vorticity formulation of the Navier-stokes equations in cylindrical polar coordinate are considered as the governing equations. In the present investigation, flow is computed for a fixed Reynolds number (Re) 200 and rotational parameter values 0.5, 1.0, 2.07 and 3.25 are considered. Firstly, the flow patterns for different α values and for long time range are computed and qualitative comparisons are made with existing experimental and numerical results. Then, as a further check on the consistency of the experimental and present numerical results, quantitative comparisons are made for the velocity profiles at several locations. All these qualitative and quantitative comparisons show excellent agreements with existing experimental and numerical results.     

A new analytical study of MHD stagnation-point flow in porous media with heat transfer

The similarity solution for the MHD Hiemenz flow against a flat plate with variable wall temperature in a porous medium gives a system of nonlinear partial differential equations. These equations are solved analytically by using a novel analytical method (DTM-Padé technique which is a combination of the differential transform method and the Padé approximation). This method is applied to give solutions of nonlinear differential equations with boundary conditions at infinity. Graphical results are presented to investigate influence of the Prandtl number, permeability parameter, Hartmann number and suction/blowing parameter on the velocity and temperature profiles.     

Numerical studies in high Reynolds number aerodynamics

Two numerical approaches are presented for the computation of viscous compressible flows at high Reynolds' numbers. In the first approach, named global approach, the whole flow field, which includes viscous and inviscid regions, is determined as the solution of a single set of equations, which may be the full Navier-Stokes equations, or some approximate form of these equations. The second approach, named coupling approach, consists in solving two different sets of equations in their respective domains simultaneously; one of the two sets governs an inviscid flow whose boundary conditions are provided by the viscous effects, determined by the other set.The discussion of the global approach is centred on two particular features of the finite-difference method used: a discretization technique, directly in the physical plane with arbitrary meshes: and a mesh adaptation technique, which combines a coordinate transformation to fit the mesh system to particular lines in the flow, and a technique of dichotomy for mesh refinement. Numerical results are presented for an axisymmetric compression corner and a shock-boundary layer interaction on a flat plate, both in supersonic regime, and for a transonic nozzle flow.For the coupling approach, emphasis is given firstly to the improvement resulting from an interacting analysis where the viscous and inviscid computations are matched, and not only patched. It is shown that the parabolic problems associated with simple viscous theories are always replaced by elliptic problems, even for supersonic flows, and that “supercritical interactions” or “critical points”, as defined by Crocco-Lees, are removed. Secondly, a new coupling method, fully automatized and capable of solving directly a well-posed problem for supersonic flow, is illustrated by examples involving shock wave-boundary layer interactions and reverse flow bubbles; they concern flows over symmetrical transonic airfoils and supersonic compression ramps.     

Steady Homann flow and heat transfer of an electrically conducting second grade fluid

The steady axisymmetric flow and heat transfer of an incompressible, electrically conducting non-Newtonian second grade fluid impinging on a flat plate is investigated. An external uniform, transverse magnetic field is applied at the surface of the plate. Similarity transformation is used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. An effective numerical scheme has been adopted to solve the nonlinear ordinary differential equations. The effects of non-Newtonian flow parameters and the magnetic field on the momentum and thermal boundary layers are discussed in detail and shown graphically. It is interesting to find that the non-Newtonian parameter and the magnetic parameter have opposite effects on the momentum and thermal boundary layers. The skin friction coefficient decreases exponentially with an increase in the non-Newtonian viscoelastic parameter and increases linearly with an increase in the magnetic parameter.     

Structural response of composite plates equipped with piezoelectric actuators

We propose the modelling of piezoelectric elements perfectly bonded on an elastic structure. The study aims at predicting the static and dynamic (vibration) electromechanical responses of the structure. The model is mostly based on the kinematic assumption of the Love–Kirchhoff thin plate theory including shear function with a quadratic variation of the electric potential along the thickness direction of the piezoelectric parts. A variational formulation of piezoelectricity leads to the equations of motion for an elastic plate equipped with piezoelectric elements. An important feature of the present investigation is that the stiffness and inertial contributions of the piezoelectric patch is not neglected. Moreover, the numerical simulations demonstrate the influence of the actuator position on the global and local responses of the elastic plate for two situations (i) bilayer and (ii) sandwich configurations. A number of benchmark tests are considered in order to characterize the plate deformation when applying an electric potential to the piezoelectric patch faces. Plate vibration problem is also presented and the frequencies for the axial and flexural modes are obtained. The spectra of vibration for the plate with a time-dependent electric potential are computed.     

Piezoelectric control of composite plate vibration: Effect of electric potential distribution

The paper deals with the vibration analysis of active rectangular plates. The plates considered are composites containing piezoelectric sensor/actuator layers, which operate in a velocity feedback control to achieve transverse vibration suppression. The piezoelectric layers are poled through the thickness and equipped with traditional surface electrodes. In order to satisfy the Maxwell electrostatics equation the widely used simplification of the electric potential distribution in the actuator layer (linear across the thickness) is replaced by a combination of a half-cosine and linear distribution in the transverse direction. The in-plane spatial variation of the potential instead of applying uniform distribution is determined by the solution of the coupled electromechanical governing equations with the natural boundary conditions corresponding to both the flexural and electric potential fields. The analysis is performed for simply supported plates. Two models of the plate are considered. In the first case the displacement field is based on the Kirchhoff hypothesis. For the second the Mindlin plate model is applied. The governing coupled equations describing the active plate behaviour are derived. The influence of the electric potential distribution and also the thickness of piezoelectric layers on the plate dynamics including the natural frequencies modification is numerically investigated and discussed.