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

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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

Abstract
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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 and 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.
[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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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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