Skip to main content
SpringerLink
Log in

A variable-domain variational formulation of inverse problemI A of 2-D unsteady transonic flow around oscillating airfoils

  • Original Papers
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

The present paper carries out, for the first time, a detailed theoretical investigation on the inverse problem in unsteady aerodynamics. Special attention is paid to finding proper ways of problem-posing and mathematical formulation. To demonstrate the basic idea, only an inverse problem of typeI A of unsteady transonic flow with shocks around oscillating airfoils is studied herein. It has been formulated by a family of variational principles (VP) with variable domain, in which all unknown boundary (airfoil contour) and discontinuities (shocks and free trailing vortex sheets) are handled (captured) via the functional variation with variable domain. As a result, almost all boundary- and interface-conditions have been converted into natural ones. Thus, a rigorous theoretical basis for unsteady airfoil design and finite element (FE) applications is provided. On the basis of these variational principles developed in this paper, a method using new self-deforming finite element is suggested for the numerical realization of the variable-domain variation of the functional and a numerical example is given. Its suitability and effectiveness are demonstrated by the numerical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

A 1...A 4 :

solution domain boundaries (Fig. 1)

A 0 T :

solution domain inR 3 att=0 andT

A f, Ash :

free trailing vortex sheets and shock resp.

i * :

stagnation enthalpy

κ:

isentropic index

R 2,R 3 :

2-D spacexy and 3-D spacexyt resp.

Φ:

velocity potential

\(\vec \Lambda ,\varrho ,p\) :

fluid velocity, density and pressure resp.

P m :

time-averaged pressure nondimensionalized by a reference pressure, see the definition formulae just under Eq. (14)

t, T :

time and period resp.

x, y :

cartesian coordinates

ξ, η, ζ:

coordinates in image space

V :

solution domain inR 3

δ:

variation symbol

[|X|]:

[|X|]=X +−X − is the jump inX across discontinuity surface

n, n′ :

normal component inR 2 andR 3 resp.

pr :

prescribed

−, +:

the left and right sides of a discontinuity resp.

TE :

trailing edge

τ:

tangential component inR 2

0:

coordinates of airfoils at rest

0:

restricted variation symbol [10]

,:

parameters or operators inR 3

←:

vector

References

  1. Dulikravich, G. S. (ed.): Proc. Intl Conf. Inverse Design Concepts and Optimization in Engineering Science, 1st Conf., 1984; Second Conf., 1987; 3rd Conf., 1991.

  2. AGARD Fluid Dynamics Panel: Computational Methods for Aerodynamic Design (Inverse) and Optimization, AGARD-CP-463, 1990.

  3. Liu, G. L.: Unsteady transonic flow with shock around oscillating airfoils and cascades: A variational theory, AIAA-Paper 87-1426 (1987).

  4. Liu, G. L.: Variational formulation of 2-D unsteady transonic aerodynamics of oscillating cascades. Proc. 5th Intl Symp. Unsteady Aerodynamics and Aeroelasticity of Turbomachines and Propellers (IUTAM), pp. 76–85, Beijing, China, Sept. 1989.

  5. Guo, J. H., Liu, G. L.: A variable-domain variational finite-element solution of 2-D unsteady transonic flow around oscillating airfoil. Proc. of the Intl Conf. on Aerohydroelasticity, pp. 75–81, Oct. 1993, Beijing, China.

  6. He, H., Liu, G. L.: Variational finite element calculation of 2-D unsteady compressible flow around oscillating airfoil, Proc. Second Intl Conf. Fluid Mechanics, pp. 359–364, July 1993, Beijing, China.

  7. Poling, D. R., Telionis, D. D.: The response of airfoils to periodic disturbance — The unsteady Kutta condition, AIAA-J24, 193–199 (1986).

    Google Scholar 

  8. Liu, G. L.: A systematic approach to the search and transformation for variational principles in fluid mechanics with emphasis on inverse and hybrid problems, Experimental and Computational Aerothermodynamics of Internal Flows, (Chen, N. X., Jiang, H. D. eds.), pp. 128–135, Beijing: World Publishing Corp., Sept. 1990.

    Google Scholar 

  9. Liu, G. L.: Variational principles and generalized variational principles for three-dimensional transonic flow with shock waves in a turbo-Rotor (I). Acta Mech.95, 117–130 (1992).

    Google Scholar 

  10. Finlayson, B. A.: The Method of Weighted Residuals and Variational Principles, pp. 336–337. New York: Academic Press 1972.

    Google Scholar 

  11. Liu, G. L.: A unified theory of hybrid problems for fully 3-D incompressible rotor-flow based on variational principles with variable domain. ASME J. Eng. for Gas Turbine and Power108, 252–258 (1986).

    Google Scholar 

  12. Liu, G. L.: A unified variational formulation of aeroelasticity problem for coupled ‘Fluid-Wing’ vibration system in 3-D transonic flow. Proc. Second Intl Conf. Fluid Mechanics, pp. 438–444, Beijing, China, July 1993.

  13. Liu, G. L.: Advances in research on inverse and hybrid problems of turbomachinery aerothermodynamics in China. Inverse Probl. Eng.2, 1–27 (1995).

    Google Scholar 

  14. Liu, G. L.: Unsteady inverse problem of type B for 2-D transonic flow: A variational formulation. Proc. 3rd Intl. Conf. on Fluid Mechanics, pp. 809–814, Beijing: Beijing Institute of Technology Press 1998.

    Google Scholar 

  15. Hafez, M., South, J. C. and Murman, E. M.: Artificial compressibility methods for numerical solutions of transonic full potential equation. AIAA J.17, 838–844 (1979).

    Google Scholar 

  16. Chyu, W. J., Davis, S. S.: Calculation of unsteady transonic flow over an airfoil. AIAA J.19, 684–690 (1981).

    Google Scholar 

  17. Liu, G. L.: Variable-domain variational FEM: A general approach to free/moving boundary problems in heat and fluid flow. Nonlinear Analysis30, 5229–5239 (1997).

    Google Scholar 

Download references

Author information

Author notes

  1. G. -L. Liu &  J. -H. Guo

    Present address: Shanghai Inst. of Appl. Math. and Mech., Shanghai University, 149 Yanchang Road, 200072, Shanghai, P.R. China

Authors and Affiliations

    Authors
    1. G. -L. Liu
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. J. -H. Guo
      View author publications

      You can also search for this author in PubMed Google Scholar

    Rights and permissions

    Reprints and permissions

    About this article

    Cite this article

    Liu, G.L., Guo, J.H. A variable-domain variational formulation of inverse problemI A of 2-D unsteady transonic flow around oscillating airfoils. Acta Mechanica 137, 195–209 (1999). https://doi.org/10.1007/BF01179209

    Download citation

    • Received:

    • Revised:

    • Issue Date:

    • DOI: https://doi.org/10.1007/BF01179209

    Keywords

    Search

    Navigation