American Institute of Mathematical Sciences

Advanced
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, (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.   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.  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.  doi: 10.1007/s11565-009-0062-8.  Google Scholar

[5]

J. Fröhlich, "Large Eddy Simulation Turbulenter Strömungen,", Teubner Verlag, (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.  doi: 10.1016/S0362-546X(01)00890-2.  Google Scholar

[7]

M. Jischa, "Konvektiver Impuls-, Wärme- und Stoffaustausch,", Vieweg-Verlag, (1982).   Google Scholar

[8]

B. L. Launder and D. B. Spalding, "Lectures in Mathematical Models of Turbulence,", Academic Press, (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.  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, 64 (2005), 373.  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.  doi: 10.1016/j.nonrwa.2011.11.018.  Google Scholar

[12]

H. Oertel, "Prandtl-Essentials of Fluid Mechanics,", Third edition, 158 (2010).   Google Scholar

[13]

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

[14]

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

[15]

L. Prandtl, Über die ausgebildete Turbulenz,, in, (1927), 62.   Google Scholar

[16]

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

show all references

References:
[1]

G. K. Batchelor, "An Introduction to Fluid Mechanics,", Cambridge Univ. Press, (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.   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.  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.  doi: 10.1007/s11565-009-0062-8.  Google Scholar

[5]

J. Fröhlich, "Large Eddy Simulation Turbulenter Strömungen,", Teubner Verlag, (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.  doi: 10.1016/S0362-546X(01)00890-2.  Google Scholar

[7]

M. Jischa, "Konvektiver Impuls-, Wärme- und Stoffaustausch,", Vieweg-Verlag, (1982).   Google Scholar

[8]

B. L. Launder and D. B. Spalding, "Lectures in Mathematical Models of Turbulence,", Academic Press, (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.  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, 64 (2005), 373.  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.  doi: 10.1016/j.nonrwa.2011.11.018.  Google Scholar

[12]

H. Oertel, "Prandtl-Essentials of Fluid Mechanics,", Third edition, 158 (2010).   Google Scholar

[13]

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

[14]

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

[15]

L. Prandtl, Über die ausgebildete Turbulenz,, in, (1927), 62.   Google Scholar

[16]

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

[1]

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
[2]

Fabian Rüffler, Volker Mehrmann, Falk M. Hante. Optimal model switching for gas flow in pipe networks. Networks & Heterogeneous Media, 2018, 13 (4) : 641-661. doi: 10.3934/nhm.2018029
[3]

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
[4]

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
[5]

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
[6]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

Shigeru Takata, Masanari Hattori, Takumu Miyauchi. On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020041
[20]

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

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (40)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences