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Simplification of the expansions of viscous terms in basic aerodynamic equations in non-orthogonal curvilinear coordinate system

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Abstract

The application of non-orthogonal curvilinear coordinate system to the calculation of the flow field inside the channel, with complex boundary geometry, can effectively simplify the work of designing the calculation program and improve the accuracy of calculation [1]. Therefore, it is obviously necessary to expand the viscous terms, i.e. viscous force, rate of work done by viscous stress and dissipation, in basic aerodynamic equations in the nonorthogonal curvilinear system [2]. However, each of these expansions consistes of tens or even hundreds of algebraic terms. The expansions of these three viscous terms discribed in this paper are considerably simplified by analysing their order of magnitude.

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Abbreviations

f :

Viscous force

f i :

Covariant components of viscous force

g ij :

Covariant components of metric tensor

g ij :

Countravariant components of metric tensor

g :

Determinant consisting ofg ij

w:

Velocity

w i :

Covariant components of w

w i :

Countravariant components of w

x i :

Curvilinear coordinates

г i jk :

Christoffel symbol of the second kind

μ :

Coefficient of viscosity

ρ :

Density of gas

Π :

Stress tensor

F :

3-D column matrix consisting off i

Φ :

Dissipation function

\(\tilde W\) :

3-D column matrix consisting ofw i

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{W}\) :

3-D column matrix consisting ofw i

D :

3-D column matrix consisting of the component ∂/∂xi of operator ∇

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{G}\) :

3-D square matrix consisting ofg ij

\(\tilde G\) :

3-D square matrix consisting ofg ij

\(\tilde G_{;j}\) :

3-D square matrix consisting of ∂gik/∂xf

i,j,k :

Indexes

T :

Superscript of transpositive matrix

References

  1. Wu, Chung-hua, Three-dimensional turbomachine flow equations expressed with respect to non-orthogonal curvilinear coordinates and non-orthogonal velocity components and methods of solution,J. of Mechanical Engineering,15, 1 (1979), 1–23. (in Chinese)

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  2. Wang, Zhong-qi and Kang Sun, The expansions of viscous terms in the basic aerodynamic equations in non-orthogonal curvilinear coordinate and method of their numerical differentiation,J. of Eng. Thermophysics,6, 2, May (1985), 142–144. (in Chinese)

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  3. Chen, Nai-xing, Problems in viscous gas flow in turbomachinery — viscous and heat-transfer terms, and methods for solution of basic equations,Scientia Sinica (Series A),27, 6 (1983), 648–661.

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Authors and Affiliations

  1. Haerbin Institute of Technology, Haerbin

    Wang Zhong-qi & Kang Shun

Authors
  1. Wang Zhong-qi
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  2. Kang Shun
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Communicated by Wu Wang-yi

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Zhong-qi, W., Shun, K. Simplification of the expansions of viscous terms in basic aerodynamic equations in non-orthogonal curvilinear coordinate system. Appl Math Mech 8, 89–96 (1987). https://doi.org/10.1007/BF02014502

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  • DOI: https://doi.org/10.1007/BF02014502

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