STRATEGIES FOR VERSATILE AND ECONOMICAL MODELLING OF NEAR-WALL TURBULENCE

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

2. CFD Workshop Introduction • 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.

3. CFD Workshop Boundary Layer Turbulence • Turbulent boundary layers can divided into the four regions, shown below. • In high-Reynolds-number flows, theBufferand theViscous Sub-Layerregions 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.

4. CFD Workshop Implications Resolve the rapid changes across the Buffer and Viscous Sub-Layers, using low-Reynolds-number models, withfine near-wall meshes, of about 20 grid-nodes 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 wall-parallel 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 • To represent the damping of turbulence across the Buffer Layer and the Viscous

5. CFD Workshop Conventional (Log-law based) wall function.

6. 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.

7. 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.

8. 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.

9. CFD Workshop UMIST Wall-Function Strategies • 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

10. 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.

11. 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.

12. CFD Workshop UMIST-N Numerical Wall-Function Axi-symmetric Impinging Jet, with non-linear k-ε CPU Comparisons

13. CFD Workshop UMIST-N, Numerical Wall-Function Pipe Expansion, Nonlinear k-ε

14. CFD Workshop UMIST-A, Analytical Wall-Function The simplified transport equations for the wall momentum and enthalpy are integrated analytically across the near-wall cell. This is accomplished through the use of a prescribed variation for the turbulent viscosity, μt.

15. CFD Workshop UMIST-A, Analytical Wall-Function Dissipation rate across the near-wall cell Conventional WF UMIST - A y < yv : ε = 2 ν kP/ yv2y < yd : ε = 2 ν kP/ yd2 y > yv : ε = kP3/2 / cℓ y y > yd : ε = kP3/2 / cℓ y yv* = 20 yd* = 5.1

16. 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

17. CFD Workshop UMIST-A, Analytical Wall FunctionAcceleration 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

18. CFD Workshop UMIST-A, Analytical Wall FunctionTemperature 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, temperature-induced changes of viscosity across this layer are included.

19. CFD Workshop UMIST-A, Analytical Wall FunctionHigh 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

20. CFD Workshop UMIST-A, Analytical Wall FunctionTreatment 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

21. CFD Workshop UMIST-A, Analytical Wall FunctionExtension 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.

22. CFD Workshop UMIST-A, Analytical Wall FunctionExtension 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 =0Ut= 0 At y=xnUr= 0.5*(UrP+UrN)Ut=0

23. CFD Workshop UMIST-A , Analytical Wall FunctionMixed Convection Down-Flow in a Heated Vertical Annulus Up-Flow in a Heated Vertical Pipe

24. CFD Workshop UMIST-A , Analytical Wall Function Mixed Convection, Opposed Wall Jet

25. CFD Workshop UMIST-A , Analytical Wall Function Buoyant Flows in Square Cavities Local Nusselt Number Top Wall Hot Wall Cold Wall Bottom Wall Hot Wall Cold Wall Re-circulating Flow Over a Sand Dune (Rough Surface) Wall Shear Stress

26. CFD Workshop UMIST-A , Analytical Wall FunctionImpingement Cooling, Local Nusselt Number Contours

27. UMIST-A , Analytical Wall FunctionComputations of Unsteady Turbulent Flows CFD Workshop Counter-Rotating Disk Cavity Instantaneous Vorticity Fields High-Re Turbulent Flow in a 90o pipe bend with a rough inner surface. Instantaneous Turbulence Intensity Instantaneous Pressure Co-Rotating Disk CavityInstantaneous Vorticity Field Time-History & Frequency Spectrum of Axial Velocity.

28. CFD Workshop Concluding Remarks • 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 near-wall 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.

29. CFD Workshop Future Directions • 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.