A note on local interior regularity of a suitable weak solution to the Navier--Stokes problem
DCDS-S Home
Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains
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,
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
G. K. Batchelor, "An Introduction to Fluid Mechanics,", Cambridge Univ. Press, (1967). Google Scholar
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. Google Scholar
P. Dreyfuss, Results for a turbulent system with unbounded viscosities: Weak formulations, existence of solutions, boundedness and smoothness,, Nonlinear Anal., 68 (2008), 1462. doi: 10.1016/j.na.2006.12.040. Google Scholar
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. doi: 10.1007/s11565-009-0062-8. Google Scholar
J. Fröhlich, "Large Eddy Simulation Turbulenter Strömungen,", Teubner Verlag, (2006). Google Scholar
T. Gallouët, J. Lederer, R. Lewandowski, F. Murat and L. Tartar, On a turbulent system with unbounded eddy viscosities,, Nonlin. Anal., 52 (2003), 1051. doi: 10.1016/S0362-546X(01)00890-2. Google Scholar
M. Jischa, "Konvektiver Impuls-, Wärme- und Stoffaustausch,", Vieweg-Verlag, (1982). Google Scholar
B. L. Launder and D. B. Spalding, "Lectures in Mathematical Models of Turbulence,", Academic Press, (1972). Google Scholar
J. Lederer and R. Lewandowski, A RANS 3D model with unbounded eddy viscosities,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 413. doi: 10.1016/j.anihpc.2006.03.011. Google Scholar
J. Naumann, Existence of weak solutions to the equations of stationary motion of heat-conducting incompressible viscous fluids,, in, 64 (2005), 373. doi: 10.1007/3-7643-7385-7_21. Google Scholar
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. doi: 10.1016/j.nonrwa.2011.11.018. Google Scholar
H. Oertel, "Prandtl-Essentials of Fluid Mechanics,", Third edition, 158 (2010). Google Scholar
S. B. Pope, "Turbulent Flows,", Cambridge Univ. Press, (2006). Google Scholar
L. Prandtl, Bericht über Untersuchungen zur ausgebildeten Turbulenz,, Zeitschr. angew. Math. Mech., 5 (1925), 136. Google Scholar
L. Prandtl, Über die ausgebildete Turbulenz,, in, (1927), 62. Google Scholar
L. Prandtl, Über ein neues Formelsystem für die ausgebildete Turbulenz,, Nachr. Akad. Wiss. Göttingen, 1 (1946), 6. Google Scholar
Xiaoxue Gong, Ying Xu, Vinay Mahadeo, Tulin Kaman, Johan Larsson, James Glimm. Mesh convergence for turbulent combustion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4383-4402. doi: 10.3934/dcds.2016.36.4383
Yong Hong Wu, B. Wiwatanapataphee. Modelling of turbulent flow and multi-phase heat transfer under electromagnetic force. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 695-706. doi: 10.3934/dcdsb.2007.8.695
Xiaoming Wang. On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 489-501. doi: 10.3934/dcdsb.2010.13.489
Baoquan Yuan, Guoquan Qin. A blowup criterion for the 2D $k$-$\varepsilon$ model equations for turbulent flows. Kinetic & Related Models, 2016, 9 (4) : 777-796. doi: 10.3934/krm.2016016
D.J. Georgiev, A. J. Roberts, D. V. Strunin. Nonlinear dynamics on centre manifolds describing turbulent floods: k-$\omega$ model. Conference Publications, 2007, 2007 (Special) : 419-428. doi: 10.3934/proc.2007.2007.419
Oleg V. Kaptsov, Alexey V. Schmidt. Reduction of three-dimensional model of the far turbulent wake to one-dimensional problem. Conference Publications, 2011, 2011 (Special) : 794-802. doi: 10.3934/proc.2011.2011.794
Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1
Andrew J. Majda, John Harlim, Boris Gershgorin. Mathematical strategies for filtering turbulent dynamical systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 441-486. doi: 10.3934/dcds.2010.27.441
Andrew J. Majda, Michal Branicki. Lessons in uncertainty quantification for turbulent dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3133-3221. doi: 10.3934/dcds.2012.32.3133
Marcel Lesieur. Two-point closure based large-eddy simulations in turbulence, Part 1: Isotropic turbulence. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 155-168. doi: 10.3934/dcdss.2011.4.155
Fredi Tröltzsch, Alberto Valli. Optimal voltage control of non-stationary eddy current problems. Mathematical Control & Related Fields, 2018, 8 (1) : 35-56. doi: 10.3934/mcrf.2018002
Marcel Lesieur. Two-point closure based large-eddy simulations in turbulence. Part 2: Inhomogeneous cases. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 227-241. doi: 10.3934/dcds.2010.28.227
Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165
Martin Oberlack, Andreas Rosteck. New statistical symmetries of the multi-point equations and its importance for turbulent scaling laws. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 451-471. doi: 10.3934/dcdss.2010.3.451
Piotr Bizoń, Dominika Hunik-Kostyra, Dmitry Pelinovsky. Stationary states of the cubic conformal flow on $ \mathbb{S}^3 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 1-32. doi: 10.3934/dcds.2020001
W. Layton, R. Lewandowski. On a well-posed turbulence model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 111-128. doi: 10.3934/dcdsb.2006.6.111
K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591
Alexey Cheskidov, Susan Friedlander, Nataša Pavlović. An inviscid dyadic model of turbulence: The global attractor. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 781-794. doi: 10.3934/dcds.2010.26.781
Peter R. Kramer, Joseph A. Biello, Yuri Lvov. Application of weak turbulence theory to FPU model. Conference Publications, 2003, 2003 (Special) : 482-491. doi: 10.3934/proc.2003.2003.482
Joachim Naumann Jörg Wolf