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In this paper, we consider analytic initial conditions with finite energy, whose complex spatial continuation is a superposition of a smooth background flow and a singular field. Through explicit calculation in the complex plane, we show that under some assumptions, the solution to the 3-D Euler equation ceases to be analytic in the real domain in finite time.
Implicit, finite-difference, shock-capturing algorithm calculates inviscid, hypersonic flows in chemical equilibrium. Implicit formulation chosen because overcomes limitation on mathematical stability encountered in explicit formulations. For dynamical portion of problem, Euler equations written in conservation-law form in Cartesian coordinate system for two-dimensional or axisymmetric flow. For chemical portion of problem, equilibrium state of gas at each point in computational grid determined by minimizing local Gibbs free energy, subject to local conservation of molecules, atoms, ions, and total enthalpy. Major advantage: resulting algorithm naturally stable and captures strong shocks without help of artificial-dissipation terms to damp out spurious numerical oscillations.
High order essentially non-oscillatory (ENO) schemes, originally designed for compressible flow and in general for hyperbolic conservation laws, are applied to incompressible Euler and Navier-Stokes equations with periodic boundary conditions. The projection to divergence-free velocity fields is achieved by fourth-order central differences through fast Fourier transforms (FFT) and a mild high-order filtering. The objective of this work is to assess the resolution of ENO schemes for large scale features of the flow when a coarse grid is used and small scale features of the flow, such as shears and roll-ups, are not fully resolved. It is found that high-order ENO schemes remain stable under such situations and quantities related to large scale features, such as the total circulation around the roll-up region, are adequately resolved.
A procedure is developed for the numerical simulation of stagnation-free inviscid supersonic and hypersonic flows about waveriders with sharp leading edges. The numerical approach involves the development of a specialized grid generator (named HYGRID), an algebraic solution-adaptive grid scheme, and a modified flow solving method. A comparison of the results obtained for several waverider geometries with exact solutions, other numerical solutions, and experimental results demonstrated the ability of the new procedure to produce stagnation-free Euler solutions about sharp-edged configurations and to describe the physics of the flow in these regions.
An implicit, shock-capturing finite-difference code which is used to calculate two-dimensional inlet flow fields in a supersonic free stream is explained. The Euler equations are subjected to general nonorthogonal transformation and a body-fitted coordinate system is employed. The mathematical formulation of the problem is given along with the numerical algorithm. Initial and boundary conditions, numerical stability, program limitations, and accuracy is discussed. An overall program logic as well as instructions for program use and operation are also furnished.
Two types of boundaries are considered: rigid walls, and artificial (open) boundaries which were arbitrarily drawn somewhere across a wider flow field. A set of partial differential equations (typically, the Euler equations) has an infinite number of solutions, each one defined by a set of initial and boundary conditions. The initial conditions remaining the same, any change in the boundary conditions will produce a new solution. To pose the problem well, a necessary and sufficient number of boundary conditions are prescribed.
The development of an efficient model for the response of a multistage turbine to either a total pressure or total temperature distortion is discussed. Each turbine blade row is modeled as an actuator disk and is often used to describe the flow through turbine and compressor blade rows. The flow approaching the blade row is assumed to be subsonic and inviscid. The distortion occurs at upstream infinity and is in the form of total pressure or total temperature. A solution was found to unsteady Euler equations by using a finite volume method. With prescribed inlet and boundary conditions flow variables such as density, pressure, and velocities can be calculated at cell centers in the entire flow region. The inlet and exit of the blade row coincide with lines of constant.
A technique allowing time-staggered solution of partial differential equations is presented in this report. Using this technique, called time-partitioning, simulation execution speedup is proportional to the number of processors used because all processors operate simultaneously, with each updating of the solution grid at a different time point. The technique is limited by neither the number of processors available nor by the dimension of the solution grid. Time-partitioning was used to obtain the flow pattern through a cascade of airfoils, modeled by the Euler partial differential equations. An execution speedup factor of 1.77 was achieved using a two processor Cray X-MP/24 computer.
The transonic flow around a low-aspect-ratio infinite wing in a wind tunnel is investigated by means of numerical simulations, with a focus on shock-induced separated flows. A coarse global grid with far-field boundaries matching those of the test section is subdivided into zones: the flow in the clustered zones near the wing is analyzed by solving the Reynolds-averaged Navier-Stokes equations, while that farther from the wing is modeled with the Euler equations. The results are presented graphically, and it is shown that a mushroomlike separated flow with two counterrotating vortices can be simulated when the correct shock strength is imposed (by careful selection of the artificial dissipation, the boundary conditions, the grid refinement, the algebraic turbulence model, and the geometry representation).
A technique allowing time-staggered solution of partial differential equations is presented in this report. Using this technique, called time-partitioning, simulation execution speedup is proportional to the number of processors used because all processors operate simultaneously, with each updating of the solution grid at a different time point. The technique is limited by neither the number of processors available nor by the dimension of the solution grid. Time-partitioning was used to obtain the flow pattern through a cascade of airfoils, modeled by the Euler partial differential equations. An execution speedup factor of 1.77 was achieved using a two processor Cray X-MP/24 computer.
A methodology is presented which makes it possible to decouple a complex problem having multiple disparate length scales into problems of single length scale so that they can be solved more efficiently on a computer. The method is applied to a viscous transonic flow over an airfoil. It is found that accurate prediction of the flow over an airfoil can be obtained by solving the Euler equations on a relatively coarse global grid with viscous effects computed separately on a boundary-layer type grid and injected into the global grid solution as a combination of vorticity and trucation error.
A fast implicit upwind procedure for the two-dimensional Euler equations is described that allows accurate computations of shocked flows on nonadapted meshes. Away from shocks, the second-order accurate upwinding is based on the split-coefficient-matrix (SCM) method. In the presence of shocks, the difference stencils are modified using a floating shock fitting technique. Rapid convergence to steady-state solutions is attained with a diagonalized approximate factorization (AF) algorithm. Results are presented for Riemann's problem, for a regular shock reflection at an inviscid wall, for supersonic flow past a cylinder, and for a transonic airfoil. All computed shocks are ideally sharp and in excellent agreement with other numerical results or 'exact' solutions. Most importantly, this has been accomplished on unusually crude meshes without any attempt to align grid lines with shock fronts or to cluster grid lines around shocks.
A new and efficient method is presented for aerodynamic design optimization, which is based on a computational fluid dynamics (CFD)-sensitivity analysis algorithm. The method is applied to design a scramjet-afterbody configuration for an optimized axial thrust. The Euler equations are solved for the inviscid analysis of the flow, which in turn provides the objective function and the constraints. The CFD analysis is then coupled with the optimization procedure that uses a constrained minimization method. The sensitivity coefficients, i.e. gradients of the objective function and the constraints, needed for the optimization are obtained using a quasi-analytical method rather than the traditional brute force method of finite difference approximations. During the one-dimensional search of the optimization procedure, an approximate flow analysis (predicted flow) based on a first-order Taylor series expansion is used to reduce the computational cost. Finally, the sensitivity of the optimum objective function to various design parameters, which are kept constant during the optimization, is computed to predict new optimum solutions. The flow analysis of the demonstrative example are compared with the experimental data. It is shown that the method is more efficient than the traditional methods.
Current efforts to reduce the sonic boom of a future High Speed Civil Transport (HSCT) by careful shaping have led to the need for more accurate predictions of the near-field flow conditions of the configuration. A fully three-dimensional Euler finite volume code is used to predict sonic boom pressure signatures for two low boom concepts - one designed to cruise at Mach 2 and the other at Mach 3. Calculations were carried out using a grid topology that has been modified to reduce the inaccuracies caused by grid spreading often suffered with CFD methods when calculations several body lengths downstream become necessary. Comparisons of CFD results and experimental wind tunnel signatures are shown. Ground signatures are predicted by extrapolating the pressures predicted by the Euler code with an extrapolation method based on the Whitham theory.
The application of a multistage 3D Euler solver to the aerodynamic design of two compact radial-inflow turbines is presented, along with experimental results evaluating and validating the designs. The objectives of the program were to design, fabricate, and rig test compact radial-inflow turbines with equal or better efficiency relative to conventional designs, while having 40 percent less rotor length than current traditionally-sized radial turbines. The approach to achieving these objectives was to apply a calibrated 3D multistage Euler code to accurately predict and control the high rotor flow passage velocities and high aerodynamic loadings resulting from the reduction in rotor length. A comparison of the advanced compact designs to current state-of-the-art configurations is presented.
A three-dimensional, multi-block, multi-zone, Euler analysis has been developed and applied to analyze the flow processes induced by a lateral array of low profile vortex generators (VG). These vortex generators have been shown to alleviate boundary layer separation through the generation of streamwise vorticity. The analysis has been applied to help develop improved VG configurations in an efficient manner. Special attention has been paid to determining the accuracy requirements of the solver for calculations in which vortical mechanisms are dominant. The analysis has been used to assess the effectiveness or boundary layer energization capacity of different VG's, including the effect of scale and shape variation. Finally, the analysis has been validated through comparisons with experimental data obtained in a large-scale low-speed wind tunnel.
A new numerical method based on adding a term to the Euler/Navier-Stokes equations has been demonstrated it can effectively treat vortex-dominated flows using low-order numerical schemes and coarse grids. The numerical diffusion introduced by convection schemes can be eliminated by modifying the velocity field as a result of solving the modified flow governing equations. The modification of the velocity field, which conserves the total vorticity, essentially convects the vorticity toward its local extreme to offset the numerical diffusion. This method preserves the vortex structure even when vortices travel on coarse grid region. The method is implemented with two distinctive codes: a Navier-Stokes flow solver based on vorticity-velocity formulation and an Euler code based on velocity-pressure formulation. Problems presented in this paper include airfoil dynamic stall, vortex airfoil interaction, and vortex-fuselage interactions.
This paper presents a description and verification of RVC3D (rotor viscous code 3-D) which provides a Euler or Navier-Stokes analysis for steady three dimensional flows in turbomachinery. A motivation for this analysis is the calculation of turbine endwall heat transfer. Features of the turbulence model code include thin-layer formulation, Baldwin-Lomax or Cebeci-Smith turbulence models, node-centered finite difference formulation, and explicit four-stage Runge-Kutta time marching scheme. Results for flat plate, annular turbine cascade, turbine endwall heat transfer, and supersonic compressor blade test cases are presented.