# American Institute of Mathematical Sciences

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

 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.
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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