Transcript Slide 1

CFD Workshop
STRATEGIES FOR VERSATILE AND
ECONOMICAL MODELLING OF NEAR-WALL
TURBULENCE
Hector Iacovides
Turbulence Mechanics Group
School of Mechanical, Aerospace & Civil Engineering,
The University of Manchester,
Co-Investigators: Brian E Launder and Tim J Craft
Researchers
A V Gerasimov and S. E. Gant
N A Mostafa, A. Omranian and A Zacharos
Introduction
CFD Workshop
• Objective:
To develop a mathematical/numerical framework to
reproduce the effects of near-wall turbulence on the flow
and thermal development.
• Motivation
Near-wall turbulence critical in determining the thermal
resistance between a surface and a moving fluid.
CFD Workshop
Boundary Layer Turbulence
• Turbulent boundary layers can divided into the four
regions, shown below.
• In high-Reynolds-number flows, the Buffer and the
Viscous Sub-Layer regions considerably thinner than
what is indicated in the diagram.
• Mean velocity, mean temperature and turbulence
properties, undergo their strongest changes across the
viscous sub-layer and buffer layer.
Implications
CFD Workshop
• To represent the damping of turbulence across the Buffer
Layer and the Viscous
Resolve the rapid changes across the
Buffer and Viscous Sub-Layers, using
low-Reynolds-number models, with
fine near-wall meshes, of about 20 gridnodes for 30<y+<0.
Use of large near-wall control volumes
with a prescribed, variation of near-wall
velocity, based on the log-law.
From Log-law and value of the wallparallel velocity at the near wall node,
the wall shear stress and the average
generation rate of turbulence over each
near-wall control volume are computed
CFD Workshop
Conventional (Log-law based) wall function.
CFD Workshop
Conventional (Log-law based) wall function.
• Highly economical and widely used
• Assumes that the near-wall velocity follows the
logarithmic profile, turbulence is in local equilibrium and
also that turbulent shear stress remains constant across
near-wall control volume.
• In complex flows these assumptions break down and
wall-function predictions become inaccurate and
unreliable.
• Examples: Accelerating, impinging, buoyant, rotating,
separated, strongly heated and three-dimensional flows.
CFD Workshop
Earlier Attempts to Refine Wall-Functions
•Chieng and Launder 1980, Numerical Heat Transfer
Linear variation of
turbulent kinetic energy, k,
outside viscous sub-layer.
Quadratic variation of k,
across sub-layer
Linear variation of
turbulent shear stress.
• Giofalo and Collins, 1989, J, Heat Mass Transfer
Extension for near-wall node in buffer layer region.
Alternative Strategies
CFD Workshop
• Unified Modelling through Integrated Sub-layer Transport
(UMIST). Manchester TM Group, 2001.
 Preserve the overall framework of the wall-function strategy.
 No log-law and the constant total shear stress assumptions.
 Produce near-wall variation of velocity and temperature, through the
integration of locally 1-D transport equations for the wall-parallel
momentum and enthalpy.
UMIST Wall-Function Strategies CFD Workshop
Common Features
 Boundary conditions:
At
y = 0,
U=0
T=TW
At
y = yn
Un=(UP+UN)/2
Tn =(TP+TN)/2
 Wall Shear Stress obtained from dU/dyy=0 & Wall Heat
Flux from dT/dyy=0
 Average generation rate of turbulence obtained from
CFD Workshop
UMIST-N Numerical Wall-Function
- Each near-wall cell is divided into
a number of sub-volumes.
- The simplified transport equations
for the wall-parallel momentum and
enthalpy are numerically solved
across the near-wall cells.
- The wall normal velocity at the sub-grid nodes is obtained
from local sub-cell continuity.
- The turbulent viscosity at the sub-grid nodes is
determined by numerically solving simplified equations of a
low-Reynolds-number model.
CFD Workshop
UMIST-N Numerical Wall-Function
For the Launder-Sharma model, for example:
Integration of the source & sink terms of the above equations provides
the average source & sink terms for k and ε over the near-wall cells.
CFD Workshop
UMIST-N Numerical Wall-Function
Axi-symmetric Impinging Jet, with non-linear k-ε
CPU Comparisons
CFD Workshop
UMIST-N, Numerical Wall-Function
Pipe Expansion, Nonlinear k-ε
CFD Workshop
UMIST-A, Analytical Wall-Function
The simplified transport equations for the wall momentum
and enthalpy are integrated analytically across the nearwall cell.
This is accomplished through the use
of a prescribed variation for the
turbulent viscosity, μt.
CFD Workshop
UMIST-A, Analytical Wall-Function
Dissipation rate across the near-wall cell
Conventional WF
y < yv : ε = 2 ν kP/ yv2
y > yv : ε = kP3/2 / cℓ y
UMIST - A
y < yd : ε = 2 ν kP/ yd2
y > yd : ε = kP3/2 / cℓ y
yv* = 20
yd* = 5.1
CFD Workshop
UMIST-A, Analytical Wall-Function
Further Extensions
- Introduction of Acceleration/Deceleration Effects
- Temperature Variation of Viscosity
- High Prandl Number Modification
- Modeling of Wall-Normal Convection in impinging
flows.
- Extension to flows over rough surfaces.
- Extension to 3-dimensional boundary layers
CFD Workshop
UMIST-A, Analytical Wall Function
Acceleration Parameter
• The cell-averaged dissipation rate of turbulence energy
in the near-wall cell, is empirically adjusted through Fε:
Where Fε is an algebraic function of the acceleration
parameter λ≡τW/τv
CFD Workshop
UMIST-A, Analytical Wall Function
Temperature Variation of Viscosity
• In strongly heated flows, changes in temperature
cause variations in fluid properties (viscosity and
thermal conductivity) across the near-wall cells
• Most of the change in temperature is across the
zero-viscosity layer.
• In the Analytical integration, temperatureinduced changes of viscosity across this layer
are included.
CFD Workshop
UMIST-A, Analytical Wall Function
High Prandlt Number Modification
• At high Prandtl numbers the sub-layer, across which turbulent
transport of thermal energy is negligible, becomes thinner than the
viscous sub-layer.
• Thus, the assumption that the turbulent heat flux becomes
negligible when y<yv, no longer applies.
This is corrected, through the
introduction of an effective
molecular Prandtl number in
the enthalpy equation
CFD Workshop
UMIST-A, Analytical Wall Function
Treatment of Convection
• For Flow Impingement, a more refined approach to the inclusion of
convection becomes necessary
• Wall normal and wall parallel convection are separately evaluated
over each layer, through numerical integration.
• Wall parallel velocity U and wall normal gradient, ∂T/∂y from the
analytical solutions.
• Assumed variation for wall
normal velocity, V.
•
When wall normal velocity away from the wall:
CTn1 = CTn2 = 0
CFD Workshop
UMIST-A, Analytical Wall Function
Extension to flows over rough surfaces
Surface roughness affects the modelling of near-wall
turbulence modifying the dimensionless thickness of the
viscosity-dominated sub-layer, yv*.
For a smooth surface : yvs* = 10.8
For a rough surface :
yv* = y*vs [ 1 - (h*/70)m ]
Where m is empirically determined.
CFD Workshop
UMIST-A, Analytical Wall Function
Extension to 3-Dimensional Boundary Layers
Transport equations for wall-parallel momentum in two directions can
be independently solved.
Ur: Wall-Parallel component of Resultant Velocity at near-wall node
Ut: Wall-Parallel velocity normal to Ur
Boundary Conditions
At
y=0
Ur =0
At
y=xn
Ur= 0.5*(UrP+UrN)
Ut= 0
Ut=0
CFD Workshop
UMIST-A , Analytical Wall Function
Mixed Convection
Down-Flow
in a Heated
Vertical
Annulus
qwall
qwall
Up-Flow in a
Heated
Vertical Pipe
(a)
(b)
Inlet: Re=15023, Gr=2.163*108, Bo=0.1124
100
80
Nu
60
Exp.data of Li (1994)
y*n=50
y*n=75
*
yn=100
*
yn=150
0.333
0.8
Nu=0.023 Pr
Re
LRN Calculation
40
20
LRN terms included
0
50
100
x/d
(a)
150
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UMIST-A , Analytical Wall Function
Mixed
Convection, Opposed Wall Jet
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UMIST-A , Analytical Wall Function
Cold Wall
Hot Wall
Buoyant Flows in Square Cavities
Local Nusselt Number
Top Wall
Bottom Wall
Hot Wall
Wall Shear Stress
Re-circulating Flow
Over a Sand Dune
(Rough Surface)
Cold Wall
CFD Workshop
UMIST-A , Analytical Wall Function
Impingement Cooling, Local Nusselt Number Contours
UMIST-A , Analytical Wall FunctionCFD Workshop
Computations of Unsteady Turbulent Flows
Counter-Rotating Disk Cavity
Instantaneous Vorticity Fields
High-Re Turbulent Flow in a 90o pipe
bend with a rough inner surface.
Instantaneous Turbulence
Intensity
Co-Rotating Disk Cavity
Instantaneous Vorticity Field
Instantaneous Pressure
Time-History & Frequency Spectrum
of Axial Velocity.
Concluding Remarks
CFD Workshop
• A framework has been developed within which advanced
wall-function strategies of more general applicability can
be developed.
• The two routes followed so far, are that of an analytical
integration of the flow transport equation over the nearwall cell and one of fully numerical integration of the
simplified Transport equations for the mean and
turbulent motion.
• Both strategies improve flow and thermal predictions,
over a range of complex flows, at the cost of only modest
rise in CPU requirements.
• The analytical wall-function strategy, has been shown to
be especially versatile, but some of the extensions make
the analytical solution clomplex.
Future Directions
CFD Workshop
 One possible further development will be to develop a third
alternative which combines features from the Analytical and the
Numerical UMIST versions.
-
The turbulent viscosity is prescribed as in the Analytical wall
function, removing the need to solve transport equations for
the turbulence parameters, over the near-wall control
volume.
-
The mean flow transport equations are then solved
numerically, removing the need for special treatment for
convection or for temperature dependent fluid properties.