Haas Thermal Mixing.ppt - cds.comsol.com A. Haas, M. Scholle, R.W. Hewson, H.M. Thompson, N. Aksel, P.H. Gaskell 1) Isothermal flow 2) Heat conduction with convection. Stokes flow

LAMINAR THERMAL MIXING IN PLANE SHEAR FLOWLAMINAR THERMAL MIXING IN PLANE SHEAR FLOW

A. Haas1, M. Scholle1, H.M. Thompson2, R.W. Hewson2, N. Aksel1, P.H. Gaskell2

1University of BayreuthDepartment of Applied Mechanics and Fluid Dynamics

D-95440, Bayreuth, Germany

2University of LeedsEngineering Fluid Mechanics Research GroupEngineering Fluid Mechanics Research Group

School of Mechanical EngineeringLeeds, LS2 9JT, UK

Presented at the COMSOL Conference 2009 Milan

Problem formulationProblem formulation

• Two-dimensional • Sinusoidally corrugated substrateTwo dimensional• Steady• Periodic BC

Sinusoidally corrugated substrate• Constant , cp and

Parameters:• Wavelength:

T0• Wavelength: • Amplitude: • Mean gap width: • Lid velocity: u0

A

H T1y 0

• Temperature: T1>T0• Viscosity:

(T ) 0 1

*T

A. Haas, M. Scholle, R.W. Hewson, H.M. Thompson, N. Aksel, P.H. Gaskell

Non-dimensional model equationsNon-dimensional model equations

Continuity equation:

No buoyancy and no dissipationSimplifications:

0x zu w

* *Re 2 1 1x z x x x z z xuu wu p T u T u w

y q

Navier-Stokes equations:

x z

* *Re 2 1 1x z z z z x z xuw ww p T w T u w

Pe 0x z xx zzuT wT T T Temperature equation: x z xx zz

0 00 0Re Pe pu cu

p q

Dimensionless numbers:


0

Re Pe2 2

Methods of solutionMethods of solution

• Semi-analytical approach• Based on a variational formulation• Stokes flow limit Re 0• neglect of thermoviscosity coupling term

• Finite element methodsComsol® Multiphysics modules• Incompressible Navier-Stokesp• Heat transfer/convection


Stokes flow – equation setStokes flow – equation set

U il t ll l d t f tiUnilaterally coupled set of equations:

• Hydrodynamic field:

000

x z

x xx zz

u wp u u

u(x,a cos x) 0w(x,a cos x) 0

u(x, h) 10z xx zzp w w

( )w(x, h) 0

( , cos ) 1T x a x • Temperature field

di i l Pe 0x z xx zzuT wT T T

( , cos ) 1( , ) 0

T x a xT x h

Decoupling:

a: non-dimensional amplitude

h: non-dimensional gap-width


1) Isothermal flow 2) Heat conduction with convection

Stokes flow - HydrodynamicsStokes flow - Hydrodynamics

Lubrication approximation

Analytic solution for the assumption a 1

Finite element solution Lubrication approximation

Reasonable results even for moderate amplitudes 1a


p 1a

Stokes flow - Heat conduction & convectionStokes flow - Heat conduction & convection

Variational formulation: with non-local functional:0I

cos

: Pe ( , ) ( , ) ( , ) ( , ) d dh

a x

I T x z u T z T x z T z zx x x

Reproduces the heat conduction equation with convection:

Pe 0x z xx zzuT wT T T

Solution by Ritz‘s direct method with appropriate base functions.


Stokes flow - Temperature fieldStokes flow - Temperature fieldSemi-analytical solution Finite element solution

a 1 / 2h 3 / 4

Streamlines:


Stokes flow - Global heat transferStokes flow - Global heat transfer

Nusselt number vs. Peclét number

2 1Nu hQ

T T

0Pe2

pc u

a 1 / 2h 3 / 5h 3 / 5

Nu a 1 / 2h 3 / 4

a 1 / 2h 1

a 1 / 2


Pe

h 7 / 4

Stokes flow - Thermal feedbackStokes flow - Thermal feedback

*0( ) 1T T Thermoviscosity:

* 0

Hydrodynamic streamlines: Temperature field:

* 1 3

* 1 3

a 1 / 2


h 3 / 4Pe 100

Inertial effects – Flow structureInertial effects – Flow structureRe 0 Re = 100

a 1 / 2h 7 / 4

a 1 / 2h 1

a 1 / 2h 3 / 4

a 1 / 2h 3 / 5


h 3 / 5

Pe 100

Inertial Effects – Global Heat TransportInertial Effects – Global Heat Transport

a 1 / 2a 1 / 2h 3 / 5

Nu a 1 / 2Nu a 1 / 2h 3 / 4

a 1 / 2

a 1 / 2h 1


Pe

h 7 / 4

ConclusionsConclusions

• Stokes flow limit: Re 0 Global Heat transfer: Comparison between semi-analytical and finite element solution Influence of thermoviscosity

• Inertial effects on global heat transfer Competing kinematical and inertial effects.


PublicationsPublications

Available literature:

M. Scholle, A. Haas, R.W. Hewson, H.M. Thompson, P.H. Gaskell, N. Aksel, The effect of locally induced flow structure on global heat transfer for plane laminar shear flow. Int. J. Heat & Fluid Flow 30 175-185 (2009)& Fluid Flow 30, 175-185 (2009).

M. Scholle, A. Haas, M.C.T. Wilson, H.M. Thompson, P.H. Gaskell, N. Aksel, Eddy genesis and manipulation in plane shear flow. Phys. Fluids 21, 073602 (2009).


Documents

Haas Thermal Mixing.ppt - cds.comsol.com A. Haas, M. Scholle, R.W. Hewson, H.M. Thompson, N. Aksel, P.H. Gaskell 1) Isothermal flow 2) Heat conduction with convection. Stokes flow