1. 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 CFD Workshop

2. 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. CFD Workshop

3. 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. CFD Workshop

4. Implications To represent the damping of turbulence across the Buffer Layer and the Viscous CFD Workshop Resolve the rapid changes across the Buffer and Viscous Sub-Layers, using low-Reynolds-number models, with fine 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

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

6. 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 Conventional (Log-law based) wall function.

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. 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. CFD Workshop Alternative Strategies

9. UMIST Wall-Function Strategies CFD Workshop Common Features  Boundary conditions: At y = 0, U=0 T=T W At y = y n U n =(U P +U N )/2 T n =(T P +T N )/2  Wall Shear Stress obtained from dU/dy y=0 & Wall Heat Flux from dT/dy y=0  Average generation rate of turbulence obtained from

10. UMIST-N Numerical Wall-Function CFD Workshop - 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. UMIST-N Numerical Wall-Function CFD Workshop 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. UMIST-N Numerical Wall-Function CFD Workshop 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. UMIST-A, Analytical Wall-Function CFD Workshop 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, μ t.

15. UMIST-A, Analytical Wall-Function CFD Workshop Dissipation rate across the near-wall cell Conventional WF UMIST - A y < y v : ε = 2 ν k P / y v 2 y < y d : ε = 2 ν k P / y d 2 y > y v : ε = k P 3/2 / c ℓ y y > y d : ε = k P 3/2 / c ℓ y y v * = 20 y d * = 5.1

16. UMIST-A, Analytical Wall-Function CFD Workshop 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. UMIST-A, Analytical Wall Function Acceleration Parameter The cell-averaged dissipation rate of turbulence energy in the near-wall cell, is empirically adjusted through F ε : CFD Workshop Where F ε is an algebraic function of the acceleration parameter λ≡ τ W /τ v

18. 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, temperature- induced changes of viscosity across this layer are included. CFD Workshop

19. 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<y v, no longer applies. CFD Workshop This is corrected, through the introduction of an effective molecular Prandtl number in the enthalpy equation

20. 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. CFD Workshop Assumed variation for wall normal velocity, V. When wall normal velocity away from the wall: C Tn1 = C Tn2 = 0

21. UMIST-A, Analytical Wall Function Extension to flows over rough surfaces CFD Workshop Surface roughness affects the modelling of near-wall turbulence modifying the dimensionless thickness of the viscosity-dominated sub-layer, y v *. For a smooth surface : y vs * = 10.8 For a rough surface : y v * = y* vs [ 1 - (h*/70) m ] Where m is empirically determined.

22. UMIST-A, Analytical Wall Function Extension to 3-Dimensional Boundary Layers CFD Workshop Transport equations for wall-parallel momentum in two directions can be independently solved. U r : Wall-Parallel component of Resultant Velocity at near-wall node U t : Wall-Parallel velocity normal to U r Boundary Conditions At y = 0 U r =0 U t = 0 At y=x n U r = 0.5*(U rP +U rN ) U t =0

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

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

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

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

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

28. 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. CFD Workshop

29.  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. CFD Workshop Future Directions