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Report discusses application of upwind-differencing numerical-integration scheme in conjunction with method of pseudocompressibility to computations of flows of incompressible fluids, with emphasis on steady-state flows. Three test cases presented: driven flow in square cavity, flow over backward-facing step, and flow around circular cylinder.
We consider the nonlinear stability of a fully three-dimensional boundary layer flow in an incompressible fluid and derive an equation governing the nonlinear development of a stationary cross-flow vortex. The amplitude equation is a novel integro-differential equation which has spatial derivatives of the amplitude occurring in the kernal function. It is shown that the evolution of the cross-flow vortex is strongly coupled to the properties of an unsteady wall layer which is in fact driven by an unknown slip velocity, proportional to the amplitude of the cross-flow vortex. The work is extended to obtain the corresponding equation for rotating disk flow. A number of special cases are examined and the numerical solution for one of cases, and further analysis, demonstrates the existence of finite-distance as well as focussing type singularities. The numerical solutions also indicate the presence of a new type of nonlinear wave solution for a certain set of parameter values.
We consider a viscous incompressible fluid flow driven between two parallel plates by a constant pressure gradient. The flow is at a finite Reynolds number, with an 0(l) disturbance in the form of a traveling wave. A phase equation approach is used to discuss the evolution of slowly varying fully nonlinear two dimensional wavetrains. We consider uniform wavetrains in detail, showing that the development of a wavenumber perturbation is governed by Burgers equation in most cases. The wavenumber perturbation theory, constructed using the phase equation approach for a uniform wavetrain, is shown to be distinct from an amplitude perturbation expansion about the periodic flow. In fact we show that the amplitude equation contains only linear terms and is simply the heat equation. We review, briefly, the well known dynamics of Burgers equation, which imply that both shock structures and finite time singularities of the wavenumber perturbation can occur with respect to the slow scales. Numerical computations have been performed to identify areas of the (wavenumber, Reynolds number, energy) neutral surface for which each of these possibilities can occur. We note that the evolution equations will breakdown under certain circumstances, in particular for a weakly nonlinear secondary flow. Finally we extend the theory to three dimensions and discuss the limit of a weak spanwise dependence for uniform wavetrains, showing that two functions are required to describe the evolution. These unknowns are a phase and a pressure function which satisfy a pair of linearly coupled partial differential equations. The results obtained from applying the same analysis to the fully three dimensional problem are included as an appendix.
Speed of computer code solving three-dimensional Navier-Stokes equations of flow of incompressible fluid by fractional-step method increased significantly by use of multigrid procedures. In method, equations solved on general nonorthogonal curvilinear coordinate grid, using volume fluxes. At each time step, computations performed in two fractional steps. In first step, equations of conservation of momentum solved by use of gradient of pressure from previous time step via explicit approximate-factorization method, yielding approximate flow field that does not satisfy equation of conservation of mass. In second step, discrete Poisson-like equation with Neumann-type boundary conditions, formed by combining equations of conservation of momentum and mass, solved iteratively.
Chorin's method of artificial compressibility is extended to both compressible and incompressible fluids by using physical arguments to define artificial fluid properties that make up a local preconditioning matrix. In particular, perturbation expansions are used to provide appropriate temporal derivatives for the equations of motion at both low speeds and low Reynolds numbers. These limiting forms are then combined into a single function that smoothly merges into the physical time derivatives at high speeds so that the equations are left unchanged at transonic, high Reynolds number conditions. The effectiveness of the resulting preconditioning procedures for the Navier-Stokes equations is demonstrated for a wide speed and Reynolds number ranges by means of stability results and computational solutions. Nevertheless, the preconditioned equations sometimes fail to provide a solution for applications for which the non-preconditioned equations converge. Often this is because the reduced dissipation in the preconditioned equations results in an unsteady solution while the more dissipative non-preconditioned equations result in a steady state. Problems of this type represent a computational challenge; it is important to distinguish between non-convergence of algorithms, and the non-existence of steady state solutions.
In this work, we review some new discoveries and physical interpretations that concern the secular and dynamical instabilities and the associated structural changes in rotating, self-gravitating, incompressible fluids. For such fluids, we have been able to examine in depth and to understand physically a variety of classic results. Here we discuss the structural stability and the lambda-transitions of astrophysical fluids. We conclude with a summary of all the different types of lambda-transitions found in the course of our investigation and with a discussion of the implications of our results for superfluids, catastrophes, thermodynamical phase transitions, and the breaking of symmetry and topology.
A parallel multilevel strategy is developed using spectral (p) finite elements. Hierarchic bases are particularly well suited since the element matrices and vectors are nested and the multilevel projections easiliy performed. Since the basis degree is used to specify the multigrid level, an EBE strategy is natural br the multilevel technique. Results are presented for two candidate nonlinear elliptic transport problems: the augmented drift-diffusion equations of semiconductor device modeling and the stream function-vorticity equations of incompressible fluid dynamics.
This article describes the applications of the Spectrum(Tm) Solver in Multidisciplinary Analysis (MDA). Spectrum, a multiphysics simulation software based on the finite element method, addresses compressible and incompressible fluid flow, structural, and thermal modeling as well as the interaction between these disciplines. Multiphysics simulation is based on a single computational framework for the modeling of multiple interacting physical phenomena. Interaction constraints are enforced in a fully-coupled manner using the augmented-Lagrangian method. Within the multiphysics framework, the finite element treatment of fluids is based on Galerkin-Least-Squares (GLS) method with discontinuity capturing operators. The arbitrary-Lagrangian-Eulerian method is utilized to account for deformable fluid domains. The finite element treatment of solids and structures is based on the Hu-Washizu variational principle. The multiphysics architecture lends itself naturally to high-performance parallel computing. Aeroelastic, propulsion, thermal management and manufacturing applications are presented.
This article describes the application of the Multidisciplinary Analysis (MDA) solver, Spectrum, in analyzing a hydrogen-cooled hypersonic cowl leading-edge structure. Spectrum, a multiphysics simulation code based on the finite element method, addresses compressible and incompressible fluid flow, structural, and thermal modeling, as well as the interactions between these disciplines. Fluid-solid-thermal interactions in a hydrogen impingement-cooled leading edge are predicted using Spectrum. Two- and semi-three-dimensional models are considered for a leading edge impingement coolant, concept under either specified external heat flux or aerothermodynamic heating from a Mach 5 external flow interaction. The solution accuracy is demonstrated from mesh refinement analysis. With active cooling, the leading edge surface temperature is drastically reduced from 1807 K of the adiabatic condition to 418 K. The internal coolant temperature profile exhibits a sharp gradient near channel/solid interface. Results from two different cooling channel configurations are also presented to illustrate the different behavior of alternative active cooling schemes.
An alternative theoretical model of joint filtration flow of immiscible incompressible fluids is presented. The model takes into account relaxation processes due to the interchange of the fluids between pores of difference sizes which is driven by capillary forces. The fluids occupy connected regions in a four-dimensional space formed by three coordinates and the pore length scale. When the fluid exchange between pores of given sizes is effected by way of successive flow through pores of all the intermediate sizes, the pressure within each region is governed by a hyperbolic equation, the role of time being played by the pore linear scale. Pressure jumps across hypersurfaces separating the regions equal corresponding values of the capillary pressure. A supplementary condition at any such hypersurface requires the speed of its displacement in the four-dimensional space to coincide with the normal velocity components of both the adjoining fluids. As a result, a principally new statement of multiphase filtration flow problems is gained with allowance for capillary relaxation in the porous space.
An analysis of the non-modal growth of perturbations in a boundary layer in the presence of a streamwise pressure gradient is presented. The analysis is based on PSE equations for an incompressible fluid. Examples with Falkner-Skan profiles indicate that a favorable pressure gradient decreases the non-modal growth while an unfavorable pressure gradient leads to an increase of the amplification. It is suggested that the transient growth mechanism be utilized to choose optimal parameters of tripping elements on a low-pressure turbine (LPT) airfoil. As an example, a boundary layer flow with a streamwise pressure gradient corresponding to the pressure distribution over a LPT airfoil is considered. It is shown that there is an optimal spacing of the tripping elements and that the transient growth effect depends on the starting point. At very low Reynolds numbers, there is a possibility to enhance the transient energy growth by means of wall cooling.
An analysis of the non-modal growth of perturbations in a boundary layer in the presence of a streamwise pressure gradient is presented. The analysis is based on PSE equations for an incompressible fluid. Examples with Falkner- Skan profiles indicate that a favorable pressure gradient decreases the non-modal growth while an unfavorable pressure gradient leads to an increase of the amplification. It is suggested that the transient growth mechanism be utilized to choose optimal parameters of tripping elements on a low-pressure turbine (LPT) airfoil. As an example, a boundary-layer flow with a streamwise pressure gradient corresponding to the pressure distribution over a LPT airfoil is considered. It is shown that there is an optimal spacing of the tripping elements and that the transient growth effect depends on the starting point. The amplification is found to be small at the LPT s very low Reynolds numbers, but there is a possibility to enhance the transient energy growth by means of wall cooling.
The problem of the motion of an elongated body of revolution in an incompressible fluid may, as is known, be solved approximately with the aid of the distribution of sources along the axis of the body. In determining the velocity field, the question of whether the body moves uniformly or with an acceleration is no factor in the problem. The presence of acceleration must be taken into account in determining the pressures acting on the body. The resistance of the body arising from the accelerated motion may be computed either directly on the basis of these pressures or with the aid of the so-called associated masses (inertia coefficients). A different condition holds in the case of the motion of bodies in a compressible gas. In this case the finite velocity of sound must be taken into account.
The "general Prandtl number" Pr(exp 1) - A(sub q)/A Pr, aside from the Reynolds number determines the ratio of turbulent to molecular heat transfer, and the temperature distribution in turbulent friction layers. A(sub q) = exchange coefficient for heat; A = exchange coefficient for momentum transfer. A formula is derived from the equation defining the general Prandtl number which describes the temperature as a function of the velocity. For fully developed thermal boundary layers all questions relating to heat transfer to and from incompressible fluids can be treated in a simple manner if the ratio of the turbulent shear stress to the total stress T(sub t)/T in the layers near the wall is known, and if the A(sub q)/A can be regarded as independent of the distance from the wall. The velocity distribution across a flat smooth channel and deep into the laminar sublayer was measured for isothermal flow to establish the shear stress ratio T(sub t)/T and to extend the universal wall friction law. The values of T(sub t)/T which resulted from these measurements can be approximately represented by a linear function of the velocity in the laminar-turbulent transition zone. The effect of the temperature relationship of the material values on the flow near the wall is briefly analyzed. It was found that the velocity at the laminar boundary (in contrast to the thickness of the laminar layer) is approximately independent of the temperature distribution. The temperature gradient at the wall and the distribution of temperature and heat flow in the turbulent friction layers were calculated on the basis of the data under two equations. The derived formulas and the figures reveal the effects of the Prandtl number, the Reynolds number, the exchange quantities and the temperature relationship of the material values.
An analysis of the optimal non-modal growth of perturbations in a boundary layer in the presence of a streamwise pressure gradient is presented. The analysis is based on PSE equations for an incompressible fluid. Examples with Falkner-Scan profiles indicate that a favorable pressure gradient decreases the non-modal growth, while an unfavorable pressure gradient leads to an increase of the amplification. It is suggested that the transient growth mechanism be utilized to choose optimal parameters of tripping elements on a low-pressure turbine (LPT) airfoil. As an example, a boundary layer flow with a streamwise pressure gradient corresponding to the pressure distribution over a LPT airfoil is considered. It is shown that there is an optimal spacing of the tripping elements and that the transient growth effect depends on the starting point.
Water ices in surface crusts of Europa, Enceladus, Saturn's main rings, and Kuiper Belt Objects can become heavily oxidized from radiolytic chemical alteration of near-surface water ice by space environment irradiation. Oxidant accumulations and gas production are manifested in part through observed H2O2 on Europa. tentatively also on Enceladus, and found elsewhere in gaseous or condensed phases at moons and rings of Jupiter and Saturn. On subsequent chemical contact in sub-surface environments with significant concentrations of primordially abundant reductants such as NH3 and CH4, oxidants of radiolytic origin can react exothermically to power gas-driven cryovolcanism. The gas-piston effect enormously amplifies the mass flow output in the case of gas formation at basal thermal margins of incompressible fluid reservoirs. Surface irradiation, H2O2 production, NH3 oxidation, and resultant heat, gas, and gas-driven mass flow rates are computed in the fluid reservoir case for selected bodies. At Enceladus the oxidant power inputs are comparable to limits on nonthermal kinetic power for the south polar plumes. Total heat output and plume gas abundance may be accounted for at Enceladus if plume activity is cyclic in high and low "Old Faithful" phases, so that oxidants can accumulate during low activity phases. Interior upwelling of primordially abundant NH3 and CH4 hydrates is assumed to resupply the reductant fuels. Much lower irradiation fluxes on Kuiper Belt Objects require correspondingly larger times for accumulation of oxidants to produce comparable resurfacing, but brightness and surface composition of some objects suggest that such activity may be ongoing.
Oscillations of sphere in incompressible viscous fluid rotating with constant angular velocity
The numerical simulation of the transonic flows of idealized fluids and of incompressible viscous fluids, by the nonlinear least squares methods is presented. The nonlinear equations, the boundary conditions, and the various constraints controlling the two types of flow are described. The standard iterative methods for solving a quasi elliptical nonlinear equation with partial derivatives are reviewed with emphasis placed on two examples: the fixed point method applied to the Gelder functional in the case of compressible subsonic flows and the Newton method used in the technique of decomposition of the lifting potential. The new abstract least squares method is discussed. It consists of substituting the nonlinear equation by a problem of minimization in a H to the minus 1 type Sobolev functional space.