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NASA NTRS1983

Implicit conservative characteristic modeling schemes for the Euler equations - A new approach

An implicit characteristic-modeling solution scheme for the Euler equations is presented. The scheme does not require the governing equations to be written in characteristic variables or the flux terms to be split into positive and negative contributions. For the two-dimensional problem of a shock wave reflecting from a flat plate, this feature and the simple solution algorithm combine to reduce the computational work per mesh point by 40 percent from that required by a standard, central-difference, implicit, solution algorithm. Application of the method to the quasi-one-dimensional nozzle flow equations for subsonic and supersonic flows without shocks shows the method to be well-conditioned for large time steps.

Wornom, S. F.↗

NASA NTRS1983

Explicit finite difference predictor and convex corrector with applications to hyperbolic partial differential equations

An explicit finite difference scheme consisting of a predictor and a corrector has been developed and applied to solve some hyperbolic partial differential equations (PDEs). The corrector is a convex-type function which is applied at each time level and at each mesh point. It consists of a parameter which may be estimated such that for larger time steps the algorithm should remain stable and generate a fast speed of convergence to the steady-state solution. Some examples have been given.

Dey, C.↗

NASA NTRS1983

A fully vectorized numerical solution of the incompressible Navier-Stokes equations

A vectorizable algorithm is presented for the implicit finite difference solution of the incompressible Navier-Stokes equations in general curvilinear coordinates. The unsteady Reynolds averaged Navier-Stokes equations solved are in two dimension and non-conservative primitive variable form. A two-layer algebraic eddy viscosity turbulence model is used to incorporate the effects of turbulence. Two momentum equations and a Poisson pressure equation, which is obtained by taking the divergence of the momentum equations and satisfying the continuity equation, are solved simultaneously at each time step. An elliptic grid generation approach is used to generate a boundary conforming coordinate system about an airfoil. The governing equations are expressed in terms of the curvilinear coordinates and are solved on a uniform rectangular computational domain. A checkerboard SOR, which can effectively utilize the computer architectural concept of vector processing, is used for iterative solution of the governing equations.

Patel, N.↗

NASA NTRS1983

Digital computer program for generating dynamic turbofan engine models (DIGTEM)

This report describes DIGTEM, a digital computer program that simulates two spool, two-stream turbofan engines. The turbofan engine model in DIGTEM contains steady-state performance maps for all of the components and has control volumes where continuity and energy balances are maintained. Rotor dynamics and duct momentum dynamics are also included. Altogether there are 16 state variables and state equations. DIGTEM features a backward-differnce integration scheme for integrating stiff systems. It trims the model equations to match a prescribed design point by calculating correction coefficients that balance out the dynamic equations. It uses the same coefficients at off-design points and iterates to a balanced engine condition. Transients can also be run. They are generated by defining controls as a function of time (open-loop control) in a user-written subroutine (TMRSP). DIGTEM has run on the IBM 370/3033 computer using implicit integration with time steps ranging from 1.0 msec to 1.0 sec. DIGTEM is generalized in the aerothermodynamic treatment of components.

Daniele, C. J.↗

NASA NTRS1983

Three-dimensional unsteady Euler equation solutions using flux vector splitting

A method for numerically solving the three dimensional unsteady Euler equations using flux vector splitting is developed. The equations are cast in curvilinear coordinates and a finite volume discretization is used. An explicit upwind second-order predictor-corrector scheme is used to solve the discretized equations. The scheme is stable for a CFL number of two and local time stepping is used to accelerate convergence for steady-state problems. Characteristic variable boundary conditions are developed and used in the far field and at surfaces. No additional dissipation terms are included in the scheme. Numerical results are compared with results from an existing three dimensional Euler code and experimental data.

Whitfield, D. L.↗

NASA NTRS1984

High speed inviscid compressible flow by the finite element method

The finite element method and an explicit time stepping algorithm which is based on Taylor-Galerkin schemes with an appropriate artificial viscosity is combined with an automatic mesh refinement process which is designed to produce accurate steady state solutions to problems of inviscid compressible flow in two dimensions. The results of two test problems are included which demonstrate the excellent performance characteristics of the proposed procedures.

Zienkiewicz, O. C.↗

NASA NTRS1984

Comparison of the full potential and Euler formulations for computing transonic airfoil flows

A quantitative comparison between the Euler and full potential formulations with respect to speed and accuracy is presented. The robustness of the codes used is tested by a number of transonic airfoil cases. The computed results are from four transonic airfoil computer codes. The full potential codes use fully implicit iteration algorithms. The first Euler code uses a fully implicit ADI iteration scheme. The second Euler code uses an explicit Runge Kutta time stepping algorithm which is enhanced by a multigrid convergence acceleration scheme. Quantitative comparisons are made using various plots of lift coefficient versus the average mesh spacing along the airfoil. Besides yielding an asymptotic limit to the lift coefficient, these results also demonstrate the truncation error behavior of the various codes. Quantitative conclusions regarding the full potential and Euler formulations with respect to accuracy, speed, and robustness can be presented.

Flores, J.↗

NASA NTRS1983

Navier-Stokes solution on the CYBER-203 by a pseudospectral technique

A three-level, time-split, mixed spectral/finite difference method for the numerical solution of the three-dimensional, compressible Navier-Stokes equations has been developed and implemented on the Control Data Corporation (CDC) CYBER-203. This method uses a spectral representation for the flow variables in the streamwise and spanwise coordinates, and central differences in the normal direction. The five dependent variables are interleaved one horizontal plane at a time and the array of their values at the grid points of each horizontal plane is a typical vector in the computation. The code is organized so as to require, per time step, a single forward-backward pass through the entire data base. The one-and two-dimensional Fast Fourier Transforms are performed using software especially developed for the CYBER-203.

Lambiotte, J. J.↗

NASA NTRS1983

A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow

A new spectral method for solving the incompressible Navier-Stokes equations in a plane channel and between concentric cylinders is presented. The method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step. In addition, the number of dependent variables is reduced, resulting in a reduction in computer memory requirements. Several test problems have been computed for the channel flow and for flow between concentric cylinders, including Taylor-Couette flow with axisymmetric Taylor vortices and wavy vortices. In all cases, agreement with available experimental and theoretical results is very good.

Moser, R. D.↗

NASA NTRS1984

An incompressible Navier-Stokes flow solver in three-dimensional curvilinear coordinate systems using primitive variables

An implicit, finite-difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional, curvilinear coordinate system. The pressure-field solution is based on the pseudo compressibility approach in which the time derivative pressure term is introduced into the mass conservation equation to form a set of hyperbolic equations. The solution procedure employs an implicit, approximate factorization scheme. The Reynolds stresses, that are uncoupled from the implicit scheme, are lagged by one time-step to facilitate implementing various levels of the turbulence model. Test problems for external and internal flows are computed, and the results are compared with existing experimental data. The application of this technique for general three-dimensional problems is then demonstrated.

Kwak, D.↗

NASA NTRS1984

Calculation of steady and oscillating airfoil flow fields via the Navier Stokes equations

A Navier-Stokes calculation procedure is applied to high Reynolds number flow about steady and unsteady airfoils. The procedure solves the ensemble averaged governing equations via a linearized block implicit (LBI) technique which in general converges to a nominal steady flow within 120 time steps. The grid used in the computation is highly stretched thus resolving the turbulent boundary layers. Calculations have been compared with data for both NACA 4412 airfoil at high incidence and an NACA 0012 airfoil oscillating in dynamic stall. In both cases, good agreement is noted between measured and calculated surface pressure distributions.

Shamroth, S. J.↗

NASA NTRS1982

A new numerical method for the simulation of three-dimensional flow in a pipe

A new numerical technique for simulating three dimensional, unsteady, incompressible pipe flows is presented and its utility and accuracy is shown. Each vector function in the expansion of the velocity field is divergence free and satisfies the boundary conditions for viscous flow. Some of the benefits of the expansion technique are that pressure is eliminated from the dynamics, only two unknowns per mesh point are required, implicit treatment of the viscous terms is provided at no extra computational cost, and no fractional time steps are required. The method uses spectral expansions: Fourier series in the azimuthal and streamwise directions, and Jacobi polynominals in the radial direction. Previously announced in STAR as N82-31644

Leonard, A.↗

NASA NTRS1982

Shock-fitted Euler solutions to shock-vortex interactions

The interaction of a planar shock wave with one or more vortexes is computed using a pseudospectral method and a finite difference method. The development of the spectral method is emphasized. In both methods the shock wave is fitted as a boundary of the computational domain. The results show good agreement between both computational methods. The spectral method is, however, restricted to smaller time steps and requires use of filtering techniques. Previously announced in STAR as N82-28061

Salas, M. D.↗

NASA NTRS1984

Three-dimensional unsteady Euler equations solution using flux vector splitting

A method for numerically solving the three-dimensional unsteady Euler equations using flux vector splitting is developed. The equations are cast in curvilinear coordinates and a finite volume discretization is used. An explicit upwind second-order predictor-corrector scheme is used to solve the discretized equations. The scheme is stable for a CFL number of 2 and local time stepping is used to accelerate convergence for steady-state problems. Characteristic variable boundary conditions are developed and used in the far-field and at surfaces. No additional dissipation terms are included in the scheme. Numerical results are compared with results from an existing three-dimensional Euler code and experimental data.

Whitfield, D. L.↗

NASA NTRS1984

Fourier-Legendre spectral methods for incompressible channel flow

An iterative collocation technique is described for modeling implicit viscosity in three-dimensional incompressible wall bounded shear flow. The viscosity can vary temporally and in the vertical direction. Channel flow is modeled with a Fourier-Legendre approximation and the mean streamwise advection is treated implicitly. Explicit terms are handled with an Adams-Bashforth method to increase the allowable time-step for calculation of the implicit terms. The algorithm is applied to low amplitude unstable waves in a plane Poiseuille flow at an Re of 7500. Comparisons are made between results using the Legendre method and with Chebyshev polynomials. Comparable accuracy is obtained for the perturbation kinetic energy predicted using both discretizations.

Zang, T. A.↗

NASA NTRS1983

A new explicit method for the numerical solution of parabolic differential equations

A new method is derived for solving parabolic partial differential equations arising in transient heat conduction or in boundary-layer flows. The method is based on a combination of the modified differential quadrature (MDQ) method with the rational Runge-Kutta time-integration scheme. It is fully explicit, requires no matrix inversion, and is stable for any time-step for the heat equations. Burgers equation and the one- and two-dimensional heat equations are solved to demonstrate the accuracy and efficiency of the proposed algorithm. The present method is found to be very accurate and efficient when results are compared with analytic solutions.

Satofuka, N.↗

NASA NTRS1984

A preliminary study of numerical simulation of thermosolutal convection of interest to crystal growth

Calculations were performed with computer models using three types of finite difference methods of thermosolutal convection: horizontal heating of a container filled with a stably stratified solution, finger convection in a container, and finger convection in a horizontally infinite channel. The importance of including thermosolutal convection in models of crystal growth is emphasized, and the difficulties in doing so are demonstrated. It is pointed out that these difficulties, due primarily to the fine structure of the convection, may be partly overcome by the use of fine grids and implicit time stepping methods.

Miller, T. L.↗

NASA NTRS1984

Multiple steady states for characteristic initial value problems

The time dependent, isentropic, quasi-one-dimensional equations of gas dynamics and other model equations are considered under the constraint of characteristic boundary conditions. Analysis of the time evolution shows how different initial data may lead to different steady states and how seemingly anamolous behavior of the solution may be resolved. Numerical experimentation using time consistent explicit algorithms verifies the conclusions of the analysis. The use of implicit schemes with very large time steps leads to erroneous results.

Salas, M. D.↗