Conference on Turbulence and Interactions TI2006 May June Porquerolles France
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Conference on Turbulence and Interactions TI2006, May 29 - June 2, 2006, Porquerolles, France DIRECT NUMERICAL SIMULATION IN A LID-DRIVEN CAVITY AT HIGH REYNOLDS NUMBER E. Leriche †,? † Laboratoire d'Ingenierie Numerique, Institut des Sciences de l'Energie, Section de Genie Mecanique, Faculte des Sciences et Techniques de l'Ingenieur, Ecole Polytechnique Federale de Lausanne, Station 9, CH-1015 Switzerland. ?Email: emmanuel.leriche@ep .ch ABSTRACT Direct numerical simulation of the flow in a lid-driven cubical cavity has been carried out at high Reynolds numbers (based on the maximum velocity on the lid), between 1.2 104 and 2.2 104. An efficient Chebyshev spectral method has been implemented for the solution of the incompressible Navier-Stokes equations in a cubical domain. The resolution used up to 5.0 million Chebyshev collocation nodes, which enable the detailed representation of all dynamically significant scales of motion. The mean and root-mean-square velocity statistics are briefly presented. INTRODUCTION Estimates for the attainable turbulent Reynolds number by the method of direct numerical sim- ulation (DNS) have been known for several decades. This estimate is based on the ratio be- tween the largest scales to the nest ones (i.e. Kolmogorov scales), which scales like Re3/4, where Re is the Reynolds number, and to re- solve numerically all the scales, an upper bound in term of degrees of freedom (dof) is then given by Re9/4.
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Conference on Turbulence and Interactions TI2006, May 29 - June 2, 2006, Porquerolles, France DIRECT NUMERICAL SIMULATION IN A LID-DRIVEN CAVITY AT HIGH REYNOLDS NUMBER E. Leriche †,? † Laboratoire d'Ingenierie Numerique, Institut des Sciences de l'Energie, Section de Genie Mecanique, Faculte des Sciences et Techniques de l'Ingenieur, Ecole Polytechnique Federale de Lausanne, Station 9, CH-1015 Switzerland. ?Email: emmanuel.leriche@ep .ch ABSTRACT Direct numerical simulation of the flow in a lid-driven cubical cavity has been carried out at high Reynolds numbers (based on the maximum velocity on the lid), between 1.2 104 and 2.2 104. An efficient Chebyshev spectral method has been implemented for the solution of the incompressible Navier-Stokes equations in a cubical domain. The resolution used up to 5.0 million Chebyshev collocation nodes, which enable the detailed representation of all dynamically significant scales of motion. The mean and root-mean-square velocity statistics are briefly presented. INTRODUCTION Estimates for the attainable turbulent Reynolds number by the method of direct numerical sim- ulation (DNS) have been known for several decades. This estimate is based on the ratio be- tween the largest scales to the nest ones (i.e. Kolmogorov scales), which scales like Re3/4, where Re is the Reynolds number, and to re- solve numerically all the scales, an upper bound in term of degrees of freedom (dof) is then given by Re9/4.
- stokes solvers
- integra- tion time
- time scales
- maximum velocity
- chebyshev
- order extrapolation scheme
- high-order direct
- lid driven cavity
- direct numerical
Conference on Turbulence and Interactions TI2006, May 29 June 2, 2006, Porquerolles, France
DIRECT NUMERICAL SIMULATION IN A LIDDRIVEN CAVITY AT HIGH REYNOLDS NUMBER
†,∗ E. Leriche
† Laboratoired’Inge´nierieNume´rique,InstitutdesSciencesdel’Energie,SectiondeGe´nieM´ecanique, Facult´edesSciencesetTechniquesdel’Ing´enieur,EcolePolytechniqueF´ede´raledeLausanne,Station9, ∗ CH1015 Switzerland.Email: emmanuel.leriche@ep .ch
ABSTRACT Direct numerical simulation of the flow in a liddriven cubical cavity has been carried out at high Reynolds 4 4 numbers (based on the maximum velocity on the lid), between1.2 10and2.2 10. An efficient Chebyshev spectral method has been implemented for the solution of the incompressible NavierStokes equations in a cubical domain. The resolution used up to 5.0 million Chebyshev collocation nodes, which enable the detailed representation of all dynamically significant scales of motion. The mean and rootmeansquare velocity statistics are briefly presented.
INTRODUCTION
Estimates for the attainable turbulent Reynolds number by the method of direct numerical sim ulation (DNS) have been known for several decades. This estimate is based on the ratio be tween the largest scales to thenest ones(i.e. 3/4 Kolmogorov scales), which scales likeRe, whereReis the Reynolds number, and to re solve numerically all the scales, an upper bound in term of degrees of freedom (dof) is then given 9/4 byRe. The evolution in computer hardware and algorithmic developments makes it now pos sible to extend the direct numerical simulation to transitional and turbulento ws that are inho mogeneous in all space directions. The present contribution is concerned with the numerical and physical aspects of the direct simulation of incompressibleo wwithin a liddriven cu bical cavity. Theuid isenclosed in a cubical cavity where one wall is moved with a speci ed velocity.The goal is to reach the highest Reynolds number but in a very simple domain,
a cubical cavity and then to study in detail the threedimensional transitional and turbulento w properties within the cavity by means of direct simulation at high Reynolds numbers (based on the maximum velocity on the lid), between 4 4 1.2 10and2.2 10. Thephenomena eno w countered within such systems are many and poorly understood.
THE GOVERNING EQUATIONS
The uid enclosedin the cavity is assumed to be incompressible, viscous, Newtonian and ho mogeneous. The equation of motion for theuid inside the cavity is given by the NavierStokes equations. The threedimensional domain, de 3 noted by, is the open interval(]−h,+h[)and its closure is written as. The NavierStokes equations are written in vector notation as
∂u + (u∙ r)u=−rp+νu ∂t
(1)
