Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2020, , 549 - 560, 15.06.2020
https://doi.org/10.17798/bitlisfen.601653

Öz


Kaynakça

  • Referans1 Adams R.A. 1975. Sobolev Spaces, Academic Press, New York
  • Referans2 Akbaş Belenli M., Rebholz L.G., Tone F. 2015. A Note on the Importance of Mass Conservation in Long Time Stability of Navier-Stokes Simulations Using Finite Elements, Applied Mathematics Letters, 45: 98-102.
  • Referans3 Chen W., Gunzburger M., Sun D., Wang X. 2013. Efficient and Long Time Accurate Second-Order Methods for Stokes-Darcy System, Journal of Numerical Analysis, 51(5): 2563-2584.
  • Referans4 Ewald B., Tone F. 2013. Approximation of the Long-TermDynamics of the Dynamical System generated by the Two-Dimensional Thermohydraulies Equations, International Journal of Numerical Analysis and Modelling, 10(3): 509-535.
  • Referans5 Girault V., Raviart P.A. 1979. Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics 719, Springer-Verlag, Berlin.
  • Referans6. Gottlieb S., Tone F., Wang C., Wang X., Wirosoetisno D. 2012. Long Time Stability of A Classical Efficient Scheme for Two Dimensional Navier-Stokes Equations, SIAM Journal on Numerical Analysis, 50(1): 126-150. Referans7 Heister T., Olshanskii M.A., Rebholz L.G. 2017. Unconditional Long-Time Stability of A Velocity-Vorticity Method for 2D Navier-Stokes Equations, Numerische Mathematik, 135(1): 143-167.
  • Referans8 John V., Linke A., Merdon C., Neilan M., Rebholz L. 2017. On the Divergence Constranint in Mixed Finite Element Methods for Incompressible Flows, SIAM Review, 59(3): 492-544.
  • Referans9 9. Layton W. 2008. Introduction to Finite Element Methods for Incompressible, Viscous Flow, SIAM, Philadelphia.
  • Referans 10 Lee H.K., Olshanskii M.A., Rebholz L.G. 2011. On Error Analysis for the 3D Navier-Stokes Equations in Velocity-Vorticity-Helicity Form, SIAM Journal of Numerical Analysis, 49(2): 711-732.
  • Referans11 Linke A. 2014. On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and A New Variational Crime, Computer Methods in Applied Mechanics and Engineering, 268: 782-800.
  • Referans12 Linke A., Merdon C. 2016. Pressure-Robustness and Discrete Helmholtz Projectors in Mixed Finite Element Methods for the Incompressible Navier- Stokes Equations, Computer Methods in Applied Mechanics and Engineering, 311: 304-326.
  • Referans13 Ahmed N., Linke A., Merdon C. 2017. Towards Pressure-Robust Mixed Methods fort he Incompressible Navier-Stokes Equations, In Proceedings of the Finite Volumes for Complex Applications 8, 1, 2: 351-359.
  • Referans14 Pedlosky J. 1992. Geophysical Fluid Dynamics, Springer, New York.
  • Referans15 Tone F. 2007. On the Long-Time Stability of the Crank-Nicholson Scheme for the 2D Navier-Stokes Equations, Numerical Methods for Partial Differential Equations, 23(5): 1235-1248.
  • Referans16 Tone F. 2009. On the Long-Time H^2-Stability of the Implicit Euler Scheme for the 2D magneto-hydrodynamics Equations, Journal of Scientific Computing, 38(3): 331-348.
  • Referans17 Tone F., Wirosoetisno D. 2006. On the Long-Time Stability of the Implicit Euler Scheme for the two-dimensional Navier-Stokes Equations, SIAM Journal on Numerical Analysis, 44(1): 29-40.
  • Referans18. Wang X. 2012. An Efficient Second Order in Time Scheme for Approximating Long Time Statistical Properties of the Two Dimensional Navier-Stokes Equations, Numerische Mathematik, 121(4): 753-779.

On the long-time stability of finite element solutions of the navier-stokes equations in a rotating frame of reference

Yıl 2020, , 549 - 560, 15.06.2020
https://doi.org/10.17798/bitlisfen.601653

Öz

Bu makale dönen bir referans
sisteminde verilen Navier-Stokes denklemlerinin uzun zamanlı kararlılık davranışını
zamana göre kesin ve uyarlanabilir sonlu elemanlar yöntemi ile çalışır.
Önerilen sayısal şema iki tane ayrıştırılmış adımdan oluşur. İlk adımda,
Navier-Stokes denklemleri lineerleştirilmiş, standart geri adımlı, Euler sonlu
elemanlar metoduyla (GA-SEM) çözülür. İkinci adımda, ilk adımda elde edilen
yaklaşık hız çözümü iki adımlı, lineer zaman filtresiyle ileri işlenir.
Yaklaşık hız çözümünün
-normuna göre tüm
zamanlarda kararlı olduğu ispatlanır. Kararlılık analizinin yeniliği, yaklaşık
hız çözümü için elde edilen karalılık sınırının herhangi bir Gronwall tipi değerlendirme
gerektirmemesi ve sınırın Reynolds sayısına bağlılığının polinomsal olması
ancak üstel olmamasıdır ki bu uzun zamanlı kararlılık konusunda çok yaygın
değildir. Çalışma, algoritmayı test etmek için ayrıca iki deney sunar. İlk
deney algoritmanın çözümünü birkaç farklı, sonlu elemanlar yöntemi ile karşılaştırır.
Sonuçlar, şemanın uzun zamanlı simülasyonlarda özellikle daha küçük
 değerleri için, divergence-free SEM ile elde
edilen çözümlerinin divergence-free olmayan metotlarla elde edilen çözümlere göre
çok daha doğru sonuç verdiğini gösterir. Diğer taraftan ikinci deney ise filter
basamağının etkisinin uzun zaman aralıkları üzerinde kütle korunumunu artırmak
olduğunu gösterir.

Kaynakça

  • Referans1 Adams R.A. 1975. Sobolev Spaces, Academic Press, New York
  • Referans2 Akbaş Belenli M., Rebholz L.G., Tone F. 2015. A Note on the Importance of Mass Conservation in Long Time Stability of Navier-Stokes Simulations Using Finite Elements, Applied Mathematics Letters, 45: 98-102.
  • Referans3 Chen W., Gunzburger M., Sun D., Wang X. 2013. Efficient and Long Time Accurate Second-Order Methods for Stokes-Darcy System, Journal of Numerical Analysis, 51(5): 2563-2584.
  • Referans4 Ewald B., Tone F. 2013. Approximation of the Long-TermDynamics of the Dynamical System generated by the Two-Dimensional Thermohydraulies Equations, International Journal of Numerical Analysis and Modelling, 10(3): 509-535.
  • Referans5 Girault V., Raviart P.A. 1979. Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics 719, Springer-Verlag, Berlin.
  • Referans6. Gottlieb S., Tone F., Wang C., Wang X., Wirosoetisno D. 2012. Long Time Stability of A Classical Efficient Scheme for Two Dimensional Navier-Stokes Equations, SIAM Journal on Numerical Analysis, 50(1): 126-150. Referans7 Heister T., Olshanskii M.A., Rebholz L.G. 2017. Unconditional Long-Time Stability of A Velocity-Vorticity Method for 2D Navier-Stokes Equations, Numerische Mathematik, 135(1): 143-167.
  • Referans8 John V., Linke A., Merdon C., Neilan M., Rebholz L. 2017. On the Divergence Constranint in Mixed Finite Element Methods for Incompressible Flows, SIAM Review, 59(3): 492-544.
  • Referans9 9. Layton W. 2008. Introduction to Finite Element Methods for Incompressible, Viscous Flow, SIAM, Philadelphia.
  • Referans 10 Lee H.K., Olshanskii M.A., Rebholz L.G. 2011. On Error Analysis for the 3D Navier-Stokes Equations in Velocity-Vorticity-Helicity Form, SIAM Journal of Numerical Analysis, 49(2): 711-732.
  • Referans11 Linke A. 2014. On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and A New Variational Crime, Computer Methods in Applied Mechanics and Engineering, 268: 782-800.
  • Referans12 Linke A., Merdon C. 2016. Pressure-Robustness and Discrete Helmholtz Projectors in Mixed Finite Element Methods for the Incompressible Navier- Stokes Equations, Computer Methods in Applied Mechanics and Engineering, 311: 304-326.
  • Referans13 Ahmed N., Linke A., Merdon C. 2017. Towards Pressure-Robust Mixed Methods fort he Incompressible Navier-Stokes Equations, In Proceedings of the Finite Volumes for Complex Applications 8, 1, 2: 351-359.
  • Referans14 Pedlosky J. 1992. Geophysical Fluid Dynamics, Springer, New York.
  • Referans15 Tone F. 2007. On the Long-Time Stability of the Crank-Nicholson Scheme for the 2D Navier-Stokes Equations, Numerical Methods for Partial Differential Equations, 23(5): 1235-1248.
  • Referans16 Tone F. 2009. On the Long-Time H^2-Stability of the Implicit Euler Scheme for the 2D magneto-hydrodynamics Equations, Journal of Scientific Computing, 38(3): 331-348.
  • Referans17 Tone F., Wirosoetisno D. 2006. On the Long-Time Stability of the Implicit Euler Scheme for the two-dimensional Navier-Stokes Equations, SIAM Journal on Numerical Analysis, 44(1): 29-40.
  • Referans18. Wang X. 2012. An Efficient Second Order in Time Scheme for Approximating Long Time Statistical Properties of the Two Dimensional Navier-Stokes Equations, Numerische Mathematik, 121(4): 753-779.
Toplam 17 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Araştırma Makalesi
Yazarlar

Mine Akbaş 0000-0002-4512-4432

Yayımlanma Tarihi 15 Haziran 2020
Gönderilme Tarihi 5 Ağustos 2019
Kabul Tarihi 20 Mart 2020
Yayımlandığı Sayı Yıl 2020

Kaynak Göster

IEEE M. Akbaş, “On the long-time stability of finite element solutions of the navier-stokes equations in a rotating frame of reference”, Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, c. 9, sy. 2, ss. 549–560, 2020, doi: 10.17798/bitlisfen.601653.



Bitlis Eren Üniversitesi
Fen Bilimleri Dergisi Editörlüğü

Bitlis Eren Üniversitesi Lisansüstü Eğitim Enstitüsü        
Beş Minare Mah. Ahmet Eren Bulvarı, Merkez Kampüs, 13000 BİTLİS        
E-posta: fbe@beu.edu.tr