Algorithm and Code Development for Unsteady Three-Dimensional Navier-Stokes Equations - Research Summary

Sep 4, 2026

Uploaded by Asher

Research develops an improved numerical algorithm and computer code to solve the unsteady three-dimensional Euler and Navier-Stokes equations for aeroelastic applications. The work updates NASA Ames’s aeroelasticity code ENSAERO by adopting a streamwise upwind algorithm with central-differencing replaced where higher accuracy is required. The method is extended for moving grids using a finite-volume conservative framework and incorporates an implicit extension for computational efficiency. Validation uses unsteady test cases over oscillating wings.

LanguageEnglish

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Introduction

Research Efforts

Results

F-5 Wing

Clipped Delta Wing

33 P.j30 # Algorithm and Code Developmentfor Unsteady Three-DimensionalNavier-Stokes Equations Dr.Shigeru Obayashi (AA一15“03-)AL. !i9M ANuC491-.503vLn T门：VT.AJY THD:“-0I*LNSI0*ALNAVF:9一5 L0AT'NS annuol²eort,! Apr., -n三r.17/1 (“CAT Inst.Unc15170CSCL 2008B/4 00151 June 1991 NCC2-605 MCAT Institute3933 Blue Gum DriveSan Jose,CA 95127 Algorithm and Code Developmentfor Unsteady Three-DimensionalNavier-Stokes Equations Dr.Shigeru Obayashi June 1991 NCC2-605 MCAT Institute3933 Blue Gum DriveSan Jose,CA 95127 Algorithm and Code Development forUnsteady Three-DimensionalNavier-Stokes Equations Shigeru Obayashi ## Introduction Research in aeroelasticity with computational methods,including the nu-merical prediction of futter boundaries,requires a sophisticated algorithmand computer code to simulate unsteady flow phenomena in three dimen-sions.The purpose of this research is to develop an improved algorithm andcomputer code for solving the unsteady three-dimensional Euler and Navier-Stokes equations.These equations accurately describe the flow phenomenafor aeroelastic applications. An algorithm that uses central differencing has being used in the com-puter code,ENSAERO,for aeroelasticity at the NASA Ames Research Cen-ter.However,modern upwind algorithms can produce more accurate solu-tions for the Euler and Navier-Stokes equations.Among upwind algorithms,a streamwise upwind algorithm has recently been developed by Obayashiand Goorjian(see Appendix A).Most multidimensional upwind algorithmsare frst constructed in one dimension and then extended to multiple di-mensions by applying the one-dimensional procedure in each coordinate di-rection.On the other hand,the present method follows the flow physicsmore closely than the coordinate upwind methods.The steady-state com-putations confirmed the higher resolution of the present algorithm over thecentral-difference method as well as over other upwind methods. The objective of this research is to update Ames's aeroelasticity code,ENSAERO,by using the improved streamwise upwind algorithm.This re-port summarizes briefly the work performed during the period,April 1,1989 through March 30,1991.Additional details on the various aspects of thestudy are given in Appendices. ## Research Efforts The following specific objectives have been performed: ·The algorithm was extended to handle the moving grid system.Thefinite-volume concept is essential to extend the algorithm.The result-ing algorithm is conservative for any motion of the coordinate system. ·Two extensions to an implicit method were considered.The implicitextension that makes the algorithm computationally efficient is imple-mented into ENSAERO. ● The new fow solver has been validated through the solution of testproblems.The test cases include three-dimensional problems with fixedand moving grids. ·Finally,the new fow solver has been implemented into NASA Ames'saeroelasticity code,ENSAERO. Results In this work,only the first-order time-accurate methods are considered be-cause of computational efficiency.However,time accuracy is an essentialrequirement for aeroelastic computations.Numerical schemes used for flowcalculations in aeroelasticity must guarantee the correct calculation of am-plitude and phase of unsteady pressures.In order to verify the time accuracyof the present code,unsteady flows over various wings undergoing prescribedoscillatory and ramp motions have been computed. F-5 Wing The first test case shown here is an unsteady viscous flow over the F-5 wingwhich has an aspect ratio of 2.98,a taper ratio of 0.31 and a leading edgesweep angle of 31.92°.The test case is chosen at M=0.9,Re=9×10⁶ and the angle of attack of 0°.Since the airfoil section is supercritical,thefow is shock-free on the most of the wing surface at this fow condition.Inthe unsteady case,the wing is pitching with a pitch amplitude of 0.11°ata reduced frequency of 0.55 as illustrated in Fig.1.The experiment

FAQs

What problem does the research address for aeroelastic computations?

It addresses the need for a sophisticated algorithm and computer code to simulate unsteady flow phenomena in three dimensions, enabling accurate prediction of unsteady pressure amplitude and phase for aeroelastic applications.

How does the new approach improve upon the previous ENSAERO central-differencing method?

The research replaces central differencing with a streamwise upwind algorithm to produce more accurate solutions for the Euler and Navier-Stokes equations, with steady-state computations confirming higher resolution.

How is the algorithm adapted to handle moving grids?

The algorithm is extended using the finite-volume concept so it remains conservative for any motion of the coordinate system, enabling implementation within the aeroelasticity code ENSAERO.

Sep 4, 2026

Uploaded by Asher