Time-accurate Method Research Articles

Flow field saddles and their relation to vortex asymmetry

The relationship of flow field saddles to the formation of asymmetric vortices is studied by employing a conical Navier-Stokes equation solver. Local grid resolution studies were used to demonstrate the importance of capturing the leeside saddle point and the secondary separation and reattachment points. The transient solutions from the unconverged symmetric flow to the converged symmetric flow illustrate a saddle point shift mechanism providing an explanation for the necessity of adequate grid resolution in this region. Also studied were the paths and final solutions obtained with perturbed and unperturbed local time-stepping procedures, and a perturbed time accurate method. The results indicate that the final solutions are virtually identical and that the same general transient paths are followed. However, these paths are not identical to those obtained with a quasi-steady pitch up of the cone in which an abrupt shift from symmetric to asymmetric results occurs. Finally, the qualitative accuracy of the conical results is assessed by comparisons of the spherical cap streamlines with experimental results of Lowson and Ponton. Excellent agreement is obtained for the incidence ratios at which symmetric vortices are formed, a saddle point appears and when vortex asymmetry occurs, although the solver does not compute accurately the higher incidence flows which were reported to be non-conical in the experiments. The quasi-steady pitch up cases also showed remarkable qualitative similarity to the experimental data.

On the solution of the Navier-Stokes equations using Chebyshev projection schemes with third-order accuracy in time

A third-order time-accurate projection method for approximating the Navier-Stokes equations for incompressible flow is presented. In order to compute a pressure unpolluted by spurious modes, two Chebyshev collocation spatial discretizations, where the pressure is approximated by lower-order polynomials than for the velocity, are compared. Only one collocation grid is used, and no pressure boundary condition is needed. The Navier-Stokes problem is reduced to the successive solution of Helmholtz problems for the velocity and pseudo-Poisson problems for the pressure. These problems are solved by direct methods. Using an exact solution, spectral spatial accuracy, and third-order time accuracy, for both the velocity and the pressure, are checked. The stability properties are discussed by considering the regularized cavity flow at various Reynolds numbers.

Temporal, spatial and thermal features of 3-D Rayleigh-Bénard convection by a least-squares finite element method

Numerical solutions of 3-D time-dependent Rayleigh-Bénard convection are presented in this work. The temporal, spatial and thermal features of convective patterns are studied for four different geometric aspect ratios, 2:1:2, 4:1:4, 5:1:5 and 3.5:1:2.1 at supercritical Rayleigh numbers Ra = 8 × 10 3, 2.4 × 10 4 and Prandtl numbers Pr = 0.71, 2.5. Several physical phenomena, such as multicellular flow pattern, oscillatory transient solution, ‘T-shaped’ rolls at the ends of a rectangular box, and roll alignment, are observed in our simulations. The numerical technique is based on an implicit, fully coupled, and time-accurate method, which consists of the Crank-Nicolson scheme for time integration, Newton's method for the convective terms with extensive linearization steps, and a least-squares finite element method. A matrix-free algorithm of the Jacobi conjugate gradient method is implemented to solve the symmetric, positive definite linear system of equations.

Analysis of flow in a square cavity with an oscillating top wall

The flow induced by the oscillatory motion of a solid body is important in a number of practical problems. As the solid boundary oscillates harmonically, there is steady streaming motion invoked by the Reynolds stresses, which could cause extensive migration of the fluid during a period of fluid motion. We here analyzed the flow in a square cavity with an oscillating top wall for the parameters which make the time derivatives and the convective terms equally important in the entire cavity flow. The full Navier-Stokes equations are solved by the second-order time accurate Momentum Coupling Method which is devised by the authors. The particular numerical scheme does not need subiteration at each time step which is usually a required process to calculate the incompressible Navier-Stokes equations. The effect of two parameters, the Reynolds number and the frequency parameter, on the oscillatory flow has been investigated.

NUMERICAL SIMULATION OF TWO-DIMENSIONAL BUOYANCY-DRIVEN TURBULENCE IN A TALL RECTANGULAR CAVITY

A computational study of natural convection of air in a tall rectangular cavity with 4:1 aspect ratio is conducted. In an effort to investigate the applicability of the Boussinesq approximation to turbulent flow simulation, the cavity is differentially heated from the sides and is insulated at the ends at a Rayleigh number of 109. Starting from quiescent and isothermal flow conditions, the flow is driven to turbulence without any artificial perturbations. The computer programme developed integrates the two-dimensional, time-dependent Navier–Stokes equations with the Boussinesq approximation and the energy equation by a time-accurate method on a stretched, staggered grid. The simulation proceeds to a statistically steady solution in which large-scale structures are found in the mean. Both mean and fluctuating quantities provide good agreement with experimental results.

A fully implicit time accurate method for hypersonic combustion: application to shock-induced combustion instability

A fully implicit time accurate method for hypersonic combustion: application to shock-induced combustion instability

Numerical simulation of time-dependent thermocapillary convection in layered fluid systems

We consider steady and time-dependent thermocapillary convection in encapsulated layers of moderate Prandtl number fluids. Assuming flat free surfaces and fluid/fluid interfaces, the two-dimensional, time-dependent Navier-Stokes equations and the energy equation are integrated by a time-accurate method on a stretched, staggered mesh. Particular attention is focused on the nature of time dependence in water encapsulation of Fluorinert FC-75 and in ethylene glycol encapsulated by FC-75 and hexadecane. We show that similar mechanisms for time-dependent thermocapillary convection exist for single-layer and multiple-layer fluid systems, and that shear effects at a thermocapillary interface can reduce convection in an encapsulated fluid layer. Based on our estimates for the thermal coefficients of interface tension, an apparent benefit of fluid encapsulation is to lessen the tendency toward time dependence. Nomenclature AR = aspect ratio Cp = specific heat at constant pressure k = thermal conductivity Lx, Ly = x and y dimensions of the cavity Ma = Marangoni number, crTATLx/iJLK Nu(x) = Nusselt number, Eq. (14) Pr = Prandtl number, V!K p =

A fully implicit time accurate method for hypersonic combustion: application to shock-induced combustion instability

A new fully implicit, time accurate algorithm suitable for chemically reacting, viscous flows in the transonic-to-hypersonic regime is described. The method is based on a class of Total Variation Diminishing (TVD) schemes and uses successive Gauss-Seidel relaxation sweeps. The inversion of large matrices is avoided by partitioning the system into reacting and nonreacting parts, but a fully coupled interaction is still maintained. As a result, the matrices that have to be inverted are of the same size as those obtained with the commonly used point implicit methods. In this paper we illustrate applicability of the new algorithm to hypervelocity unsteady combustion applications. We present a series of numerical simulations of the periodic combustion instabilities observed in ballistic-range experiments of blunt projectiles flying at subdetonative speeds through hydrogenair mixtures. The computed frequencies of oscillation are in excellent agreement with experimental data.

Time-dependent thermocapillary convection in a rectangular cavity: numerical results for a moderate Prandtl number fluid

The present numerical simulation explores a thermal–convective mechanism for oscillatory thermocapillary convection in a shallow rectangular cavity for a Prandtl number 6.78 fluid. The computer program developed for this simulation integrates the two-dimensional, time-dependent Navier–Stokes equations and the energy equation by a time-accurate method on a stretched, staggered mesh. Flat free surfaces are assumed. The instability is shown to depend upon temporal coupling between large-scale thermal structures within the flow field and the temperature sensitive free surface. A primary result of this study is the development of a stability diagram presenting the critical Marangoni number separating the steady from the time-dependent flow states as a function of aspect ratio for the range of values between 2.3 and 3.8. Within this range, a minimum critical aspect ratio near 2.3 and a minimum critical Marangoni number near 20 000 are predicted, below which steady convection is found.

Explicit Time Accurate Pseudo-Compressibility Method for Numerical Solutions of Unsteady Incompressible Viscous Flows.

An explicit time accurate pseudo compressibility metod has been proposed for numerical solutions of unsteady incompressible viscous flows. It is based on the method of lines approach using an explicit time integration scheme for both physical and pseudo-time. The one-stage three-level Runge-Kutta scheme is applied for time integration in physical time and the rational Runge-Kutta scheme is applied for that in pseudo-time. The multigrid scheme, residual averaging, and local time stepping are incorporated in order to accelerate convergence to steady state in pseudo-time. In this paper, numerical solutions of the simple Poisson equation, one-dimensional oscillating channel flow and two-dimensional flow around a cylinder at Re=200 are presented. It is shown that the present scheme is reliable in accuracy and requires less computing cost than a conventional scheme using the SOR method.

Unsteady Rotor Dynamics in Cascade

A time-accurate potential-flow calculation method has been developed for unsteady incompressible flows through two-dimensional multi-blade-row linear cascades. The method represents the boundary surfaces by distributing piecewise linear-vortex and constant-source singularities on discrete panels. A local coordinate is assigned to each independently moving object. Blade-shed vorticity is traced at each time step. The unsteady Kutta condition applied is nonlinear and requires zero blade trailing-edge loading at each time. Its influence on the solutions depends on the blade trailing-edge shapes. Steady biplane and cascade solutions are presented and compared to exact solutions and experimental data. Unsteady solutions are validated with the Wagner function for an airfoil moving impulsively from rest and the Theodorsen function for an oscillating airfoil. The shed vortex motion and its interaction with blades are calculated and compared to an analytic solution. For a multi-blade-row cascade, the potential effect between blade rows is predicted using steady and quasi-unsteady calculations. The accuracy of the predictions is demonstrated using experimental results for a one-stage turbine stator-rotor.

Temporal and acoustic accuracy of an implicit upwind method for ducted flows

An implicit, time-accurate method for ducted-flow aeroacoustics is presented. The method is unfactored, using line Gauss-Seidel relaxation and multiple axial sweeps for convergence of each time step.

X. On an easy and at the same time accurate method of de­termining the ratio of the dispersions of glasses intended for objectives

In examining the dispersive powers of a great variety of glasses prepared hy the late Rev. W. Vernon Harcourt, I had occasion to examine several prisms which were too much striated to show clearly even the boldest dark lines of the solar spectrum. I found that I was able to get a fair measure of the dispersive powers even of these by a method depending on the achromatizing of one prism by another. If the method succeeded even with such prisms, it stands to reason that it would be still more successful with prisms of good glass. For the construction of an objective we require but one datum as regards the dispersions, namely, the ratio of the dispersions, or rather, the ratio which on being treated as if it were the ratio of the disper­sions gives the best results in practice.