Elsevier

Computers & Fluids

Volume 28, Issues 4–5, May–June 1999, Pages 701-719
Computers & Fluids

Some anomalies of numerical simulation of shock waves. Part I: inviscid flows

https://doi.org/10.1016/S0045-7930(98)00051-6Get rights and content

Abstract

In this part, nonunique solutions of potential and Euler equations are discussed. Previous work related to the airfoil problem is reviewed and some new results are presented. Simple models based on Burger’s equation and quasi-one-dimensional nozzle flow are examined. An example for a three-dimensional wing admitting nonunique solutions of the potential and Euler equations is also included.

Introduction

The formulation of steady inviscid flow may have more than one solution satisfying the entropy condition. The governing equations are nonlinear and there is no reason to expect a unique solution. Similarly, several numerical solutions may satisfy the corresponding discrete equations (including the artificial viscosity terms) as well as the discrete boundary conditions.

Some of the numerical solutions may have no physical relevance, for example, negative or imaginary density. Others may be spurious because of improper discretization or numerical boundary conditions. Also, coarse mesh solutions may differ from fine mesh solutions; only in the limit of vanishing grid size are the discrete solutions related to the solutions of differential equations. Some of the solutions may not be physically stable and hence are of little interest for practical applications. However, some numerically stable algorithms can produce such solutions. Even for viscous flows, the solution of Navier–Stokes equations may not be unique, as discussed in part II of this work. Also, a steady state (independent of time) may cease to exist for a certain Reynolds number range.

For the airfoil problem, the first nonunique solution of the transonic full potential equation was computed by Steinhoff and Jameson in 1981[1]. For pure subsonic flow, the solution is unique (if the Kutta condition is imposed). However, if shocks appear and asymmetry (lift) is allowed, there is more than one solution which satisfies the discrete version of the steady equation and boundary conditions as well as the entropy inequality.

Later, Salas et al.[2] made an extensive study, including mesh refinement, and calculated for the same conditions the solution of Euler equations. Because they could not find nonunique solutions of Euler equations at that time, they made the conjecture that the nonuniqueness problem of the potential calculations is due to the irrotational and isentropic flow assumptions.

The nonunique solutions of the potential equation were obtained by others as well using different numerical methods. For example, Glowinski et al. in 1985[3], presented nonunique solutions of the potential equations based on a finite element approximation and a conjugate gradient type algorithm. For a symmetric airfoil (NACA 0012) at zero angle of attack and M=0.82 three solutions can be calculated: (i) a symmetric solution with zero lift (with negative CLα); (ii) asymmetric solution with positive lift; and (iii) asymmetric solution with negative lift. In these calculations, CL is specified rather than the angle of attack. This is in agreement with Salas’ results. On the other hand, for a slightly lower Mach number M=0.81, Glowinski et al. obtained five solutions; one symmetric with zero lift and positive CLα and two asymmetric solutions with positive lift and their mirror images, with negative lift. It is not clear, however, that these five solutions can be obtained if the mesh is excessively refined.

A few years later, and after efficient and accurate methods for the solution of Euler codes have been developed, Jameson in 1991[4] designed a series of special airfoils and he was able to compute nonunique solutions of Euler equations for these airfoils. Therefore, the nonuniqueness problem is not restricted to irrotational and isentropic flow models, as was implied by the work of Salas et al.

Recently, the first author5, 6 studied inviscid flow over a rotating cylinder and obtained nonunique solutions of both potential and Euler equations. Unlike the potential flow model, the Euler solution for a cylinder may contain vorticity, with closed streamlines, even for incompressible flows (Kirchoff’s free streamline solution for a flow normal to a plate is a special case, see also the work of Garabedian[7]). For transonic flows, a curved shock produces vorticity proportional to the rate of change of entropy normal to the streamlines (Crocco’s relation). Euler solutions for transonic flows over a fixed cylinder were calculated by Buning and Steger[8] and by Pandolfi and Larocca[9]. The solution is not stable and vorticity is shed from upper and lower shocks forming a phenomenon similar to von Karman street. For rotating cylinders, however, stable Euler solutions with positive and negative lift can be calculated. Hence, the nonuniqueness of the Euler solutions is not restricted to particular airfoil shapes (i.e. Jameson airfoil), but even for very basic shapes (cylinder) one can find multiple steady-state solutions.

Section snippets

One dimensional flows

To gain some understanding of the nonuniqueness problem, simple model problems are studied first.

Two-dimensional flows

Instead of a nozzle with a variable cross-section area distribution, one may consider a symmetric body in a channel with straight walls and let the body have wavy surfaces. On removing the channel walls, a truly two-dimensional flow is produced.

Subsonic flow over a wavy wall is periodic, and the potential and the Euler solutions should be identical. The Euler codes however may generate vorticity due to the presence of artificial viscosity and the numerical treatment of boundary conditions.

Three-dimensional wings

The author is not aware of any potential or Euler nonunique solutions for three-dimensional flows over finite aspect ratio wings. In fact, at Boeing, potential codes (for example, TRANAIR) have been heavily used for years, without any nonuniqueness problem (F. Johnson; private communication, September 1995).

Hence, one may conclude that nonuniqueness is not an issue for three-dimensional calculations, at least from a practical point of view. Nevertheless, in the following, an example using

Conclusions

Quasi-one-dimensional Euler solutions for steady compressible flows through nozzles of variable cross-section areas are studied. Examples of nonunique solutions of steady Euler equations for two-dimensional simple symmetric configurations are presented. It is also argued that, for high aspect ratio wings, steady Euler equations admit nonunique solutions. Nonunique solutions for high Reynolds number flows are presented in part II of this work. Therefore, nonuniqueness is not limited to

Acknowledgements

The calculations for the quasi-one-dimensional flows were performed by T. Matsuzawa, a graduate student at UCD.

References (27)

  • J. Steinhoff et al.

    Multiple solutions for the transonic potential flow past an airfoil

    AIAA Journal

    (1982)
  • Salas M, Jameson A, Melnik R. A comparative study of the nonuniqueness problem of the potential equation. AIAA Paper...
  • M.O. Bristeau et al.

    On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element method (II) Application to transonic flow simulations

    Computer Methods in Applied Mechanics and Engineering

    (1985)
  • Jameson A. Nonunique solutions to the Euler equations. AIAA Paper 91-1625,...
  • M. Hafez et al.

    Simulations of compressible inviscid flows over stationary and rotating cylinders

    Acta Mechanica

    (1994)
  • M. Hafez et al.

    Some anomalies of the numerical solutions of the Euler equations

    Acta Mechanica

    (1996)
  • P. Garabedian

    Nonparametric solutions of the Euler equations for steady flows

    Communications in Pure and Applied Mathematics

    (1983)
  • Buning PG, Steger JL. Solution of the two-dimensional Euler equations with generalized coordinate transformation using...
  • M. Pandolfi et al.

    Transonic flow about a circular cylinder

    Computers and Fluids

    (1989)
  • Deconinck H, Hirsch Ch. Boundary conditions for the potential equation in transonic flow calculation. SME Paper...
  • Bauer F, Garabedian P, Korn D, Jameson A. Supercritical wing sections II. In: Lecture notes in Economics and...
  • E. Murman

    Analysis of embedded shock waves calculated by relaxation methods

    AIAA Journal

    (1974)
  • B. Engquist et al.

    Stable and entropy satisfying approximations for transonic flow calculations

    Mathematics of Computing

    (1980)
  • Cited by (0)

    View full text