Using PDF Equations to describe Rayleigh Bénard Convection

Transcription

1 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 1 M. Wilczek 21 R. Friedrich 1 R. Stevens 23 D. Lohse 3 1 University of Münster 2 University of Baltimore 3 University of Twente

2 Contents MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Introduction Temperature Statistics and PDF Equations (a) Periodic Boundary Conditions 3D (b) Periodic Boundary Conditions 2D (c) Cylindrical Vessel Summary

3 Contents MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Introduction Temperature Statistics and PDF Equations (a) Periodic Boundary Conditions 2D (b) Periodic Boundary Conditions 3D (c) Cylindrical Vessel Summary

4 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 1 /28 Phenomenon Rayleigh Bénard Convection: Heated from below cooled from above Ubiquitous in nature: Atmosphere ozeans plate tectonics... Different patterns from stable laminar to highly turbulent flows Turbulent flows common in nature and applications yet hard to handle analytically Our focus: Statistical Description that connects to the Dynamics

5 W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 2 /28 Phenomenon Turbulent Rayleigh Bénard Convection

6 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 3 /28 Phenomenon Turbulent Rayleigh Bénard Convection

7 Governing Equations MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 4 /28 Oberbeck-Boussinesq Equations in Non-dimensional Form T + u T = T t u t + u u = p + Pr u + Pr Ra T e z u = 0 Control Parameters Ra Rayleigh number ( temperature difference) kinematic viscosity Pr Prandtl number (= thermal diffusivity material parameter) Γ Aspect ratio (geometry parameter)

8 Contents MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Introduction Temperature Statistics and PDF Equations (a) Periodic Boundary Conditions 3D (b) Periodic Boundary Conditions 2D (c) Cylindrical Vessel Summary

9 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 5 /28 Goal: A Statistical Description from First Principles Laminar convection: Analytical solution of basic equations is possible Turbulent convection: Analytical solution is (up to now) impossible But: Analytical solution not necessarily needed Compare ideal gas: We can t predict particles but we still can predict temperature pressure energy... Goal: Achieve statistical description for turbulent Rayleigh Bénard convection in terms of probability density function (PDF) of temperature!

10 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 6 /28 Statistical Description in Terms of Temperature PDF Deriving an Evolution Equation for Temperature PDF: Define temperature PDF as ensemble average of δ-distribution: f (T x t) = δ ( T T (x t) ) Calculate and put together derivatives of PDF Introduce conditional averages T x t Plug in Oberbeck-Boussinesq equations Evolution Equation for Temperature PDF ( t f + u T x t ) f = ( T ( = T T t T + u T x t ) T T x t f ) f

11 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 7 /28 Method of Characteristics yields average behaviour Evolution Equation for Temperature PDF ( ) t f + u T x t f = ( ) T T x t f T Apply Method of Characteristics to evolution equation of temperature PDF One obtains characteristic curves i. e. trajectories along which the evolution equation transforms from a PDE into an ODE

12 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 8 /28 Method of Characteristics Characteristics follow the vector field that is determined by the conditional averages: Vector Field of Characteristics (Ṫ ) ( ) T T x = ẋ u T x ( ) T (t) Solutions / characteristics show average behavior of a x(t) fluid particle in T -x-phase space This general framework is now applied to 3 different RB settings with different symmetries; T x are estimated from 3 different DNS

13 Contents MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Introduction Temperature Statistics and PDF Equations (a) Periodic Boundary Conditions 3D (b) Periodic Boundary Conditions 2D (c) Cylindrical Vessel Summary

14 W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 9 /28 Periodic Boundary Conditions 3D I 3D convection homogenous in horizontal direction I No-slip bottom and top plates u = 0 I Parameters: Ra = Pr = 1 Γ = 4 I Numerics: Pseudospectral and Volume Penalization Rayleigh Bénard System

15 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 10 /28 Periodic Boundary Conditions 3D Symmetries Statistics do not depend on horizontal coordinates: f (T x) = f (T z) T x = T z Phase space becomes 2D spanned by T and z Characteristics simplify to: (Ṫ ) ż = ( ) T T z u z T z

16 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 11 /28 Periodic Boundary Conditions 3D Integrating characteristics for arb. initial positions: Convergence towards Limit Cycle! Blue red: Temperature Arrows: vector field of characteristics ( ) i.e. T T z u z T z Black white: Norm of vector field (i.e. phase space speed)

17 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 12 /28 Periodic Boundary Conditions 3D Integrating characteristics for arb. initial positions: Convergence towards Limit Cycle! Color: pdf of T Arrows: Vector field Colored Circle: Solution ( T z ) following limit cycle; Temperature color coded

18 Contents MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Introduction Temperature Statistics and PDF Equations (a) Periodic Boundary Conditions 3D (b) Periodic Boundary Conditions 2D (c) Cylindrical Vessel Summary

19 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 13 /28 Periodic Boundary Conditions 2D RB System 2D convection homogenous in horizontal direction No-slip bottom and top plates u = 0 Parameters: Ra = Pr = 1 Γ = 4 Symmetries: Identical to 3D periodic case

20 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 14 /28 Periodic Boundary Conditions 2D Integrating characteristics for arb. initial positions: Convergence towards Limit Cycle! Blue red: Temperature Arrows: vector field of characteristics Black white: Norm of vector field (i.e. phase space velocity)

21 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 15 /28 Periodic Boundary Conditions 2D Integrating characteristics for arb. initial positions: Convergence towards Limit Cycle! Color: pdf of T Arrows: Vector field Colored Circle: Solution ( T z ) following limit cycle; Temperature color coded

22 Contents MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Introduction Temperature Statistics and PDF Equations (a) Periodic Boundary Conditions 3D (b) Periodic Boundary Conditions 2D (c) Cylindrical Vessel Summary

23 W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 16 /28 I Cylindrical Vessel with insulating sidewalls I Cylindrical coordinates: x = (r ϕ z) I All surfaces are no-slip u = 0 I Parameters: Ra = Pr = 1 Γ=1 I Numerics: Finite differences w/ gridpoint clustering Cylindrical Vessel

24 W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 17 /28 Cylindrical Vessel Symmetries I Homogeneous in azimuthal (ϕ-) direction: f (T x) = f (T r z) Vector field of Characteristics h T T r zi T = r hur T r zi huz T r zi z I Additional phase space dimension (radial movement) h T xi = h T r zi

25 Cylindrical Vessel MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 18 /28 Again Limit Cycle is found! RB Cycle: Heating up / moving outwards at the bottom Moving up / inwards in the bulk Cooling down / moving outwards at the top... Cornerflows w/o need for a LSC Limit Cycle lies in outer regions 0.3 < r < 0.5 and around the mean temperature 0.47 < T < 0.53

26 Contents MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Introduction Temperature Statistics and PDF Equations (a) Periodic Boundary Conditions 3D (b) Periodic Boundary Conditions 2D (c) Cylindrical Vessel Summary

27 Summary MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 19 /28 We derived an evolution equation for the PDF of temperature from first principles Unclosed terms are expressed via conditional averages which are estimated from DNS The Method of Characteristics is used to link statistics and dynamics of the system The framework allows to identify a limit cycle which shows the average transport processes in Rayleigh Bénard convection for three different cases (2 2D / 1 3D phase space) Outlook: Further investigation of limit cycle

28 Summary MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 20 /28 Methods PDF methods conditionally averaged fields Method of Characteristics Advantages from first principles description of average temperature-resolved behavior in different parts of cell results intuitive and insightful

29 Summary MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 21 /28 Disadvantages / Problems full information u(x) T (x) needed no direct desription of heat transport i.e. in terms of Nu(Ra Pr) results could be expected a priori? limit cycle?!

30 Literature MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection 22 /28 PDF Methods in general: > S. B. Pope AIAA Journal (1984) > S. B. Pope Turbulent Flows (Cambridge Univ. Press 2000) > M. Wilczek and R. Friedrich Phys. Rev. E (2009) > M. Wilczek A. Daitche and R. Friedrich J. Fluid Mech (2011) > R. Friedrich et al. Compt. Rend. Phys (2012) PDF Methods in Rayleigh Bénard convection: > J. Lülff M. Wilczek and R. Friedrich New J. Phys (2011) > J. Lülff M. Wilczek R. Stevens R. Friedrich and D. Lohse (to be published...) Volume Penalization: > K. Schneider Comput. Fluids (2005) > G. H. Keetels et al. J. Comput. Phys (2007)

31 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Thank you for your attention! Rudolf Friedrich August 2012

32 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Supplementaries

33 W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Limit Cycle 3D

34 W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Limit Cycle 2D

35 Limit Cycle Cylinder MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28

36 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Method of Characteristics Along the characteristics the PDE transforms into an ODE: ODE along the characteristic curves d ( 1 ds f (s) = ( r ur T r z ) + r r z u z T r z + ) T T r z f (s) }{{ T } d(t rz) The divergence d(t r z) of the vectorfield describes the change of the PDF along a characteristic

37 MÜNSTER Using PDF Equations to describe Rayleigh Bénard Convection /28 Method of Characteristics The ODE can be integrated along the characteristic curves giving the following solution: Integrated ODE along the characteristics s ( T T z + T z f (s) = f (s 0 ) exp ds s 0 uz T z ) T =T (s) z=z(s) This relation describes ) the change of the PDF along a certain ) trajectory in T z-phase space starting at ( T (s) z(s) ( T (s0 ) z(s 0 )