HYDRODYNAMICS AND ACOUSTICS

2022 ◊ Volume 2 (92) ◊ Issue 3 ◊ p. 209-228

A. O. Borysyuk^*

^* Institute of Hydromechanics of NAS of Ukraine, Kyiv, Ukraine

Studying the flow in a straight flat rigid channel with axisymmetric expansion in terms of variables stream function-vorticity-pressure. Part 1. Theory

Gidrodin. akust. 2022, 2(3):209-228

https://doi.org/10.15407/jha2022.03.209

TEXT LANGUAGE: Ukrainian

ABSTRACT

A numerical method in the variables stream function-vorticity-pressure is developed to solve a problem of fluid motion in an infinite straight flat rigid-walled channel with a local rigid axisymmetric expansion of rectangular shape. The method has the first order of accuracy in a temporal and the second order of accuracy in spatial coordinates. In the developed method, the formulated problem is solved by means of introducing the stream function and the vorticity, and the corresponding transition from the variables velocity-pressure to the variables stream function-vorticity-pressure, subsequent non-dimensionalizing the relationships obtained on the basis of that transition, choice of both the computational domain and the corresponding spatial and temporal integration mesh, having small constant steps in the temporal and spatial coordinates, discretization of the noted non-dimensional relationships in the appropriate nodes of the chosen mesh and subsequent solving the algebraic equations obtained with the use of the indicated discretization. In making the discretization, its temporal part is carried out on the basis of the two-point upward differencing scheme, whereas the spatial one is based on the two-point adverse flow differencing schemes (for the convective term of the non-linear vorticity equation) and the fifths-point differencing schemes (for the diffusive term of the noted equation and the Poisson's equations for the stream function and the pressure) for the corresponding co-ordinates. The iterative method of successive over relaxation is applied to solve the linear algebraic equations for the stream function and the pressure (the only difference between these equations is in their right parts, which are known). As for the algebraic relationship for the vorticity (that was obtained after performing the noted discretization), it does not require application of any method for its solution, because actually it is a computational scheme to find immediately the vorticity on the basis of the known values of the corresponding magnitudes found at the previous time step (at the initial time, the values of all the magnitudes are prescribed).

KEY WORDS

flow, channel, expansion, method

REFERENCES

[1] J. C. Lasheras, "The biomechanics of arterial aneurysms," vol. 39, no. 1, pp. 293-319, 2007.
https://doi.org/10.1146/annurev.fluid.39.050905.110128

[2] J. R. Cebral, F. Mut, M. Raschi, E. Scrivano, R. Ceratto, P. Lylyk, and C. M. Putman, "Aneurysm rupture following treatment with flow-diverting stents: computational hemodynamics analysis of treatment," vol. 32, no. 1, pp. 27-33, 2010.
https://doi.org/10.3174/ajnr.A2398

[3] Z. Kulcsar, E. Houdart, A. Bonafe, G. Parker, J. Millar, A. J. P. Goddard, S. Renowden, G. Gal, B. Turowski, K. Mitchell, F. Gray, M. Rodriguez, R. van den Berg, A. Gruber, H. Desal, I. Wanke, and D. A. R¨ufenacht, "Intra-aneurysmal thrombosis as a possible cause of delayed aneurysm rupture after flow-diversion treatment," vol. 32, no. 1, pp. 20-25, 2010.
https://doi.org/10.3174/ajnr.A2370

[4] K. Rhee, M. H. Han, and S. H. Cha, "Changes of flow characteristics by stenting in aneurysm models: Influence of aneurysm geometry and stent porosity," vol. 30, no. 7, pp. 894-904, 2002.
https://doi.org/10.1114/1.1500406

[5] A. O. Borisyuk, "A method to solve a problem of flow in a channel with an axisymmetric rectangular expansion," vol. 41, no. 1, pp. 59-68, 2019.
https://doi.org/10.18372/2310-5461.41.13530

[6] B. V. Rathish Kumar and K. B. Naidu, "Hemodynamics in aneurysm," vol. 29, no. 2, pp. 119-139, 1996.
https://doi.org/10.1006/cbmr.1996.0011

[7] D. M. Sforza, C. M. Putman, and J. R. Cebral, "Hemodynamics of cerebral aneurysms," vol. 41, no. 1, pp. 91-107, 2009.
https://doi.org/10.1146/annurev.fluid.40.111406.102126

[8] L. G. Loitsyanskii, Mechanics of liquids and gases. Oxford: Pergamon Press, 2015.

[9] G. K. Batchelor, An introduction to fluid dynamics. Cambridge: Cambridge University Press, 1967.

[10] A. O. Borisyuk, "Study of the flow and acoustic fields in a rigid-walled channel of circular cross-section with a local axisymmetric narrowing. part 1. a theory," International Journal of Fluid Mechanics Research, vol. 34, no. 2, pp. 99-114, 2007.
https://doi.org/10.1615/InterJFluidMechRes.v34.i2.10

[11] A. O. Borisyuk, "Study of the flow and acoustic fields in a rigid-walled channel of circular cross-section with a local axisymmetric narrowing. part 2. numerical results," International Journal of Fluid Mechanics Research, vol. 34, no. 3, pp. 191-209, 2007.
https://doi.org/10.1615/InterJFluidMechRes.v34.i3.10

[12] V. S. Malyuga, "Numerical investigation of flow in a duct with two serial stenosis. an algorithm of the solution," Applied Hydromecanics, vol. 12 (84), no. 4, pp. 45-62, 2010.

[13] A. V. Shaldenko and A. A. Gourzhii, "Analysis of the heat-transfer processes in a straight channel with insertions at low Reynolds numbers," Applied Hydromecanics, vol. 17 (89), no. 3, pp. 55-66, 2017.

[14] E. V. Brujatskiy, A. G. Kostin, and E. I. Nikiforovich, "Numerical investigation of the velocity and pressure fields in a flat duct having a rectangular ledge on the wall," Applied Hydromecanics, vol. 13 (85), no. 3, pp. 33-47, 2011.

[15] A. O. Borisyuk, "Experimental study of wall pressure fluctuations in rigid and elastic pipes behind an axisymmetric narrowing," Journal of Fluids and Structures, vol. 26, no. 4, pp. 658-674, 2010.
https://doi.org/10.1016/j.jfluidstructs.2010.03.005

[16] J. H. Ferziger and M. Peric, Computational methods for fluid dynamics. Berlin: Springer, 2002.
https://doi.org/10.1007/978-3-642-56026-2

[17] C. Hirsch, Numerical computation of internal and external flows. Oxford: Butterworth-Heinemann, 2007.

[18] P. J. Roache, Computational fluid dynamics. Albuquerque, NM: Hermosa, 1976.