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October  2013, 6(5): 1371-1390. doi: 10.3934/dcdss.2013.6.1371

On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow

Joachim Naumann 1, and  Jörg Wolf 1,

1. 

Departement of Mathematics, Humboldt University Berlin, Unter den Linden 6, 10099 Berlin, Germany, Germany

Received  November 2011 Revised  March 2012 Published  March 2013

Starting from Prandtl's (1945) turbulence model, we consider two systems of PDEs for the scalar functions $u$ and $k$ which characterize the stationary turbulent pipe-flow. This system is completed by a homogeneous Dirichlet condition on $u$, and homogeneuos Neumann or mixed boundary conditions on $k$, respectively. For these boundary value problems we prove the existence of weak solutions $(u,k)$ such that $k>0$ on a set of positive measure.
Keywords: eddy viscosity, Prandtl's turbulence model, mean turbulent kinetic energy., stationary uni-directional turbulent pipe-flow.
Mathematics Subject Classification: Primary: 35J70, 76F02; Secondary: 35Q35, 76F2.
Citation: Joachim Naumann, Jörg Wolf. On Prandtl's turbulence model: Existence of weak solutions to the equations of stationary turbulent pipe-flow. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1371-1390. doi: 10.3934/dcdss.2013.6.1371
References:
[1]

G. K. Batchelor, "An Introduction to Fluid Mechanics," Cambridge Univ. Press, Cambridge, 1967. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Fröhlich, "Large Eddy Simulation Turbulenter Strömungen," Teubner Verlag, Wiesbaden, 2006. Google Scholar

[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.  Google Scholar

[7]

M. Jischa, "Konvektiver Impuls-, Wärme- und Stoffaustausch," Vieweg-Verlag, Braunschweig/Wiesbaden, 1982. Google Scholar

[8]

B. L. Launder and D. B. Spalding, "Lectures in Mathematical Models of Turbulence," Academic Press, London, 1972. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. Oertel, "Prandtl-Essentials of Fluid Mechanics," Third edition, Applied Mathematical Sciences, 158, Springer, New York, 2010.  Google Scholar

[13]

S. B. Pope, "Turbulent Flows," Cambridge Univ. Press, Cambridge, 2006. Google Scholar

[14]

L. Prandtl, Bericht über Untersuchungen zur ausgebildeten Turbulenz, Zeitschr. angew. Math. Mech., 5 (1925), 136-139. Google Scholar

[15]

L. Prandtl, Über die ausgebildete Turbulenz, in "Verhandl. II," Intern. Kongress Techn. Mech., Zürich 1926, Füßli-Verlag, Zürich, (1927), 62-75. Google Scholar

[16]

L. Prandtl, Über ein neues Formelsystem für die ausgebildete Turbulenz, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., 1 (1946), 6-19.  Google Scholar

show all references

References:
[1]

G. K. Batchelor, "An Introduction to Fluid Mechanics," Cambridge Univ. Press, Cambridge, 1967. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Fröhlich, "Large Eddy Simulation Turbulenter Strömungen," Teubner Verlag, Wiesbaden, 2006. Google Scholar

[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.  Google Scholar

[7]

M. Jischa, "Konvektiver Impuls-, Wärme- und Stoffaustausch," Vieweg-Verlag, Braunschweig/Wiesbaden, 1982. Google Scholar

[8]

B. L. Launder and D. B. Spalding, "Lectures in Mathematical Models of Turbulence," Academic Press, London, 1972. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. Oertel, "Prandtl-Essentials of Fluid Mechanics," Third edition, Applied Mathematical Sciences, 158, Springer, New York, 2010.  Google Scholar

[13]

S. B. Pope, "Turbulent Flows," Cambridge Univ. Press, Cambridge, 2006. Google Scholar

[14]

L. Prandtl, Bericht über Untersuchungen zur ausgebildeten Turbulenz, Zeitschr. angew. Math. Mech., 5 (1925), 136-139. Google Scholar

[15]

L. Prandtl, Über die ausgebildete Turbulenz, in "Verhandl. II," Intern. Kongress Techn. Mech., Zürich 1926, Füßli-Verlag, Zürich, (1927), 62-75. Google Scholar

[16]

L. Prandtl, Über ein neues Formelsystem für die ausgebildete Turbulenz, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., 1 (1946), 6-19.  Google Scholar

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