Abstract
In this paper, the entropy preserving (EP) scheme (which is introduced recently by Jameson) has been considered deeply and compared with the other artificial viscosity and upwind schemes. The discretization of the governing equations in the EP scheme is performed in such a way that the entropy is conserved in all those points with no shock. The purpose of this study was to introduce a stable numerical method that enters a minimum artificial dissipation only in the vicinity of shocks. In this paper, an inviscid one-dimensional flow through a convergent–divergent nozzle and a viscous two-dimensional flow with axial symmetry are considered. It is shown that the EP scheme is more accurate if the number of mesh points is increased; and in contrast to other schemes, there is no limit in increasing the number of points.
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Abbreviations
- C p :
-
Specific heat capacity \({\left({\frac{J}{kg.K}} \right)}\)
- E :
-
Total energy \({\left({\frac{J}{kg}}\right)}\)
- F :
-
Energy flux \({\left({\frac{W}{kg}}\right)}\)
- f :
-
Flux vector \({\left({\frac{m^{2}}{s^{2}}}\right)}\)
- G :
-
Integral of flux vector \({\left({\frac{W}{kg}}\right)}\)
- h :
-
Entropy function \({\left({\frac{kg^{2}}{m.s^{2}.K}}\right)}\)
- Pr :
-
Prandtl number
- R :
-
Residual value \({\left({\frac{m}{s^{2}}}\right)}\)
- T :
-
Temperature (K)
- t :
-
Time (s)
- u :
-
State variable \({\left({\frac{m}{s}}\right)}\)
- w :
-
Change of variable for symmetry
- x, y :
-
Coordinates direction
- \({\Delta t}\) :
-
Time step (s)
- \({\Delta x}\) :
-
The distance between two adjacent points (m)
- \({\theta}\) :
-
Integral variable
- \({\mu}\) :
-
Viscosity (Pa.s)
- n :
-
Mesh point number
- T :
-
Transpose
- −:
-
Time averaging
- \({\wedge}\) :
-
Spatial averaging
- j :
-
Counting index (indicating the coordinate direction)
- max :
-
Maximum
- min :
-
Minimum
- n :
-
Mesh point number
- u :
-
Indicating the derivative with respect to the desired parameter
- \({\infty}\) :
-
The infinity variable
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Javadi, A., Pasandideh-Fard, M. & Malek-Jafarian, M. Analysis of One-Dimensional Inviscid and Two-Dimensional Viscous Flows Using Entropy Preserving Method. Arab J Sci Eng 39, 7315–7325 (2014). https://doi.org/10.1007/s13369-014-1300-7
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DOI: https://doi.org/10.1007/s13369-014-1300-7