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On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow
1. | Departement of Mathematics, Humboldt University Berlin, Unter den Linden 6, 10099 Berlin, Germany, Germany |
References:
[1] |
G. K. Batchelor, "An Introduction to Fluid Mechanics," Cambridge Univ. Press, Cambridge, 1967. |
[2] |
S. Clain and R. Touzani, Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients, Math. Model. Num. Anal., 31 (1977), 845-870. |
[3] |
P. Dreyfuss, Results for a turbulent system with unbounded viscosities: Weak formulations, existence of solutions, boundedness and smoothness, Nonlinear Anal., 68 (2008), 1462-1478.
doi: 10.1016/j.na.2006.12.040. |
[4] |
P.-É. Druet and J. Naumann, On the existence of weak solutions to a stationary one-equation RANS model with unbounded eddy viscosities, Ann. Univ. Ferrara, 55 (2009), 67-87.
doi: 10.1007/s11565-009-0062-8. |
[5] |
J. Fröhlich, "Large Eddy Simulation Turbulenter Strömungen," Teubner Verlag, Wiesbaden, 2006. |
[6] |
T. Gallouët, J. Lederer, R. Lewandowski, F. Murat and L. Tartar, On a turbulent system with unbounded eddy viscosities, Nonlin. Anal., 52 (2003), 1051-1068.
doi: 10.1016/S0362-546X(01)00890-2. |
[7] |
M. Jischa, "Konvektiver Impuls-, Wärme- und Stoffaustausch," Vieweg-Verlag, Braunschweig/Wiesbaden, 1982. |
[8] |
B. L. Launder and D. B. Spalding, "Lectures in Mathematical Models of Turbulence," Academic Press, London, 1972. |
[9] |
J. Lederer and R. Lewandowski, A RANS 3D model with unbounded eddy viscosities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 413-441.
doi: 10.1016/j.anihpc.2006.03.011. |
[10] |
J. Naumann, Existence of weak solutions to the equations of stationary motion of heat-conducting incompressible viscous fluids, in "Nonlinear Elliptic and Parabolic Problems," Progress Nonl. Diff. Equs. Appl., 64, Birkhäuser, (2005), 373-390.
doi: 10.1007/3-7643-7385-7_21. |
[11] |
J. Naumann, M. Pokorný and J. Wolf, On the existence of weak solutions to the equations of steady flow of heat-conducting fluids with dissipative heating, Nonlin. Anal. Real World Appl., 13 (2012), 1600-1620.
doi: 10.1016/j.nonrwa.2011.11.018. |
[12] |
H. Oertel, "Prandtl-Essentials of Fluid Mechanics," Third edition, Applied Mathematical Sciences, 158, Springer, New York, 2010. |
[13] |
S. B. Pope, "Turbulent Flows," Cambridge Univ. Press, Cambridge, 2006. |
[14] |
L. Prandtl, Bericht über Untersuchungen zur ausgebildeten Turbulenz, Zeitschr. angew. Math. Mech., 5 (1925), 136-139. |
[15] |
L. Prandtl, Über die ausgebildete Turbulenz, in "Verhandl. II," Intern. Kongress Techn. Mech., Zürich 1926, Füßli-Verlag, Zürich, (1927), 62-75. |
[16] |
L. Prandtl, Über ein neues Formelsystem für die ausgebildete Turbulenz, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., 1 (1946), 6-19. |
show all references
References:
[1] |
G. K. Batchelor, "An Introduction to Fluid Mechanics," Cambridge Univ. Press, Cambridge, 1967. |
[2] |
S. Clain and R. Touzani, Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients, Math. Model. Num. Anal., 31 (1977), 845-870. |
[3] |
P. Dreyfuss, Results for a turbulent system with unbounded viscosities: Weak formulations, existence of solutions, boundedness and smoothness, Nonlinear Anal., 68 (2008), 1462-1478.
doi: 10.1016/j.na.2006.12.040. |
[4] |
P.-É. Druet and J. Naumann, On the existence of weak solutions to a stationary one-equation RANS model with unbounded eddy viscosities, Ann. Univ. Ferrara, 55 (2009), 67-87.
doi: 10.1007/s11565-009-0062-8. |
[5] |
J. Fröhlich, "Large Eddy Simulation Turbulenter Strömungen," Teubner Verlag, Wiesbaden, 2006. |
[6] |
T. Gallouët, J. Lederer, R. Lewandowski, F. Murat and L. Tartar, On a turbulent system with unbounded eddy viscosities, Nonlin. Anal., 52 (2003), 1051-1068.
doi: 10.1016/S0362-546X(01)00890-2. |
[7] |
M. Jischa, "Konvektiver Impuls-, Wärme- und Stoffaustausch," Vieweg-Verlag, Braunschweig/Wiesbaden, 1982. |
[8] |
B. L. Launder and D. B. Spalding, "Lectures in Mathematical Models of Turbulence," Academic Press, London, 1972. |
[9] |
J. Lederer and R. Lewandowski, A RANS 3D model with unbounded eddy viscosities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 413-441.
doi: 10.1016/j.anihpc.2006.03.011. |
[10] |
J. Naumann, Existence of weak solutions to the equations of stationary motion of heat-conducting incompressible viscous fluids, in "Nonlinear Elliptic and Parabolic Problems," Progress Nonl. Diff. Equs. Appl., 64, Birkhäuser, (2005), 373-390.
doi: 10.1007/3-7643-7385-7_21. |
[11] |
J. Naumann, M. Pokorný and J. Wolf, On the existence of weak solutions to the equations of steady flow of heat-conducting fluids with dissipative heating, Nonlin. Anal. Real World Appl., 13 (2012), 1600-1620.
doi: 10.1016/j.nonrwa.2011.11.018. |
[12] |
H. Oertel, "Prandtl-Essentials of Fluid Mechanics," Third edition, Applied Mathematical Sciences, 158, Springer, New York, 2010. |
[13] |
S. B. Pope, "Turbulent Flows," Cambridge Univ. Press, Cambridge, 2006. |
[14] |
L. Prandtl, Bericht über Untersuchungen zur ausgebildeten Turbulenz, Zeitschr. angew. Math. Mech., 5 (1925), 136-139. |
[15] |
L. Prandtl, Über die ausgebildete Turbulenz, in "Verhandl. II," Intern. Kongress Techn. Mech., Zürich 1926, Füßli-Verlag, Zürich, (1927), 62-75. |
[16] |
L. Prandtl, Über ein neues Formelsystem für die ausgebildete Turbulenz, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., 1 (1946), 6-19. |
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