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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

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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

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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

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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

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

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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.

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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.

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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

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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

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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.

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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.

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

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

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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.

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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

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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

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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

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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

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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

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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

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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.

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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

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

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

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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

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

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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

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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

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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

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