# Wall function approaches using two equation model biology essay

Module: Research Methods (MACE61078)Date of Submission: 19th April 2013. Submitted toProf Hector IacovidesSubmitted ByJahedul Islam ChowdhuryStudent ID# 8572073

## Contents

1Introduction12Analytical Wall Function Treatments22. 1UMIST-A Scheme32. 2UMIST N Scheme53Wall Function approaches using two equation model74Wall Function for oscillating flow104. 1Periodic Channel flow- UMIST-N Approach104. 2Oscillating Turbulent flow- UMIST-A Approach125Direct Numerical Simulation136Conclusions15Bibliography16

## Nomenclature

kTurbulent kinetic energy

## ????

Dissipation rate of k

## �”

specific dissipation rate of kC???? turbulent model constantCllog-law constantNuNusselt numberReReynolds numberReDhReynolds number based on hydraulic diameterReτReynolds number based on frictional velocityU+Nondimensional velocityy+Dimensionless distance

## ????

Dynamic viscosityρDensity of the fluid???? tEddy or turbulent viscosity???? t*Dimensionless eddy or turbulent viscosityy*Dimensionless distance to the nearest wallBoBuoyancy parameterτwWall shear stressPkProduction of kLxLength in x directionLyLength in y directionLzLength in z directionuτ*Instantaneous velocity fluctuationTWall temperature

## 1. Introduction

The Computational Fluid Dynamics (CFD) is the computer solutions of a number of fundamental equations to predict pretty much any type of fluid motion. CFD may sometimes be used to investigate the physical behaviour of flow, but the common goal is to provide an input to engineering analysis and design. The CFD has developed very rapidly in last two decades because of the advancement of technology and increase of computing processor capacity of computer. Considering the development of technology, Researchers focuses toward more complex calculations. With the high-speed supercomputers, better solutions of the complicated flow can be achieved. Now a days, CFD and experimental analysis are extensively used as a design tool in most of the industries. The position of CFD is currently behind the experimental analysis due to the fact that CFD does not produce absolute results. The reason for this is that the numerical methods, which govern the solutions in a CFD problem, rely on several modelling assumptions that may not have been validated to a satisfactory level. However, CFD presently offers itself as a powerful design tool and even more so in the future because of: firstly, dangerous or expensive trial and error experiments whereas CFD reduce the cost and time that usually require for the experiment. Secondly, the numerical schemes and physical models that are the building blocks of CFD are continually improving. Moreover, it can be able to produce reliable and accurate results on one particular design compare to the exact one. For this reason, most of the researchers prefer CFD rather than Experimental analysis. The CFD analysis starts with dividing or discretizing the geometry to be modelled into usually a large number of small computational cells. Discretization is the method of approximating the differential governing equations by a system of algebraic equations for the variables at some set of discrete locations in space and time. The discrete locations are normally referred to as the ” grid” or the ” mesh”. This analysis is iteratively solving equations for the conservation of mass, heat, and momentum, and the dissipation of motion into turbulence. The additional effects mentioned above are added into the solution procedure if needed. The results are often presented graphically to assist in visualizing what’s actually happening in the flow domainMost flows of relevance to engineering applications are clearly turbulent, and most CFD codes are designed to compute and predicting the behaviour of turbulent flow. Most commonly, this is accomplished by the adoption of a turbulence model that makes use of certain assumptions to approximate flow behaviour using more tractable equations. This precludes the possibility of predicting every feature of the flow, but the parameters providing the greatest engineering relevance may be estimated at a greatly reduced computational cost. CFD approximations of turbulent flow near the walls generally employ one of two broad strategies to resolve the very influential, complex, but thin near-wall viscosity-affected sub-layer. One method uses a fine numerical mesh and a turbulence model which take viscous influences into account; the other use a “ wall functions’’. However, because of the complexity of the three-dimensional time-dependent flows in engines, the use of low-Reynolds number turbulence models to resolve the near wall regions is computationally very expensive. Thus wall functions are applied in these regions, coupled to a high Reynolds number turbulence model for the bulk flow. The functions used apply the log-law for velocity variation across the near wall regions derived from simple wall shear flows. However, because of the oscillating nature of the flow in engines due to the piston motion, it is not clear that such wall functions are appropriate, as the near-wall flow will also vary in an unsteady fashion. The aim of the literature survey of this project is to study the wall function to an oscillating turbulent flow imposed on top of a steady through flow and analyse earlier works on wall function near the wall. And hence, ultimately, gather the knowledge to develop an alternative analytical wall function that does properly take the effects of oscillation into account. The scope of the literature survey is to cover the study of different turbulence modelling approaches i. e. two equation models (k-ε, k-ω), Direct Numerical Simulation (DNS) etc. However, major focus of the research would be the near wall treatment of the turbulent flow for instances, wall function, sub-grid approaches (UMIST-A and UMIST-N). Moreover, previous works of the wall function for oscillating turbulent flow will also be covered by this survey.

## 2. Analytical Wall function Treatments

Two separate projects have done on developing more general wall-functions at UMIST. The pathways of schemes are different with each other. One is UMIST�” A which is an analytical scheme, (Unified Methodology for Integrated Sub-layer Transport�” Analytical), the other is UMIST�” N, based on numerical approach .

## 2. 2 UMIST-N Scheme:

The analytical approach is not enough equipped in case where the velocity profile parallel with the wall undergoes strong skewing across the sub-layer, for instances, non-orthogonal impingement of flow on a bank of heat-exchanger tubes, for flow with strong streamline curvature where the strain stress relation provided by eddy-viscosity model does not depict the turbulence generation process etc. Considering above difficulties, a numerical scheme, UMIST�” N, has been developed by Gant  and Craft et al. . The scheme is much more similar to the low-Reynolds-number models in that the wall-function cell is itself sub-divided into, typically, 30 thin slices as shown in figure 2. With suitable simplifications, the mean flow and turbulence differential equations are solved effectively as a one-dimensional problem across this fine grid. Which ultimately generate data required as of wall function quantities such as wall shear stress, averaged source term etc. and supply appropriate boundary conditions for the whole-field solution carried out on the primary grid. Figure 2. Treatment of near wall control volume in UMIST- N There were several simplifications made by the Gant  in order to reduction in computer time that one looks form the wall function. The pressure gradient parallel to the wall is considered uniform across all the sub-grids, equal to the pressure gradient across the near-wall cell of the primary grid. The velocity component normal to the wall can be obtained by continuity rather than by solving the momentum equation normal to the wall. The paper shows that two alternative turbulence models have been employed within the above numerical treatment. One is the LRN k�” ε model of Launder and Sharma  and other is cubic non-linear eddy viscosity model of Craft et al. . However, result shows that the cubic terms in the latter model make it far more sensitive to streamline curvature than a linear EVM. Figure 3. Nusselt number variation on a flat plate beneath an axisymmetric impinging jet. (a) Linear k�” ε EVM. (b) Cubic non-linear EVM (Craft et al ). Symbols: experiments of Baughn et al ; heavy line: full LRN treatment; other lines: UMIST�” N, with different near-wall cell sizes The paper shows the first test case is for a turbulent jet impinging orthogonally onto a uniformly heated flat plate. The jet discharges from a long smooth pipe whose exit is four diameters above the plate. Figure 3 Shows the resultant variation of Nusselt number over the plate from the stagnation point (r= 0) outwards. As is seen in Figure 3(a), the computed Nusselt number at the stagnation point is more than twice the measured value. However, the computation results of wall function and full LRN are close enough. Again same test was done for the non-linear EVM (Craft et al ) instead of Linear k�” ε EVM. Now the results from computation and experiment are much closer. The only major difference between the UMIST�” N results and those of the complete LRN treatment is in the computer time required. It is suggested from the paper that the wall-function result with the same grid density takes less than one eighth of the time required for the complete low-Reynolds-number model. Another test example considered heat transfer from a mildly heated disc spinning about its own axis. The disc’s rotation induces a radially outward motion that peaks outside the VSL, the tangential velocity increases rapidly. Figure 4. Radial velocity profile for spinning disc in wall-layer coordinates. Solid line: LRN calculation; broken line with symbols: UMIST�” N; chain line: log-law; other lines: standard wall-function treatments . Result shows that the induced radial velocity predicted with UMIST-N (using the linear EVM of Launder and Sharma ) agrees very closely with the results of the corresponding LRN computation. However, plot of integral Nusselt numbers shows the negligible difference among the treatments except the computation time. Eventually, the complete low-Reynolds-number model required thirteen times more computation time than UMIST�” N. In closing to this paper, UMIST�” N is based on a local one-dimensional numerical solution of the governing equations. It is clearly perform badly for the standard log-law based wall functions. This scheme is, in principle, the more general approach, but is inevitably more computationally expensive terms of CPU time and storage requirements.

## 3. Wall function approaches using two equation model

The paper titled ‘‘ The analysis of wall function approaches using two equations turbulence model’’ was done by the Pe´rez-Segarra et al . The objective of this paper was to obtain tools which were able to approximate both fluid flow and heat transfer with less time and storage consumption, reduction of grid-sensitivity and a relatively good accuracy. However, considering the scope of literature review, fluid flow sections are going to consider only. Authors focused on WF approaches, using for inner nodes different high Reynolds number two-equation models, depending on the dissipative variable (k�” ε or k�” ω treatments). Several assumptions and numerical corrections were taken into consideration for near wall treatments. The governing equations of the time-averaged continuity, momentum and energy of the fluid flow were considered. A low- Reynolds �” number model was tested and compared to the Ince�” Launder  k�” ε LRN model and with Wilcox  k�” ω LRN model in order to find out the limitation and improvements of the numerical and computational phenomena of the wall functions. However, the governing equations were solved using finite volume techniques over a staggered discretization. A structured grid, fully implicit time integration and SIMPLE method were employed to the numerical approach. Diffusion and convection terms were calculated with the central different and UPWIND and SMART scheme for the channel flow and UPWIND and Power law for backward flow respectively. All computations were carried out using a pseudo-transient iterative algorithm, applying the biggest time-step which ensures convergence. Post processing was done using generalized Richardson extrapolation for h-refinement studies and the grid convergence index (GCI) method due to verify the numerical results provided by the continuous (LRN) models analysed in his work. Different cases were tested shown in this paper. For instances, A channel flow at moderate Reynolds number (Reτ = 590, ReDh ≈ 43, 000), A backward facing step flow (BFS) at Reynolds number of ReH= 37, 500. The inlet conditions used for test cases were identical. For channel flow, , in addition to a turbulence intensity of 7% (kin= 0. 072 ) and a specific dissipation rate and a fixed inlet temperature of 300K. 8H(a)(b)Figure 5. (a) Channel and (b) backward facing step test cases 

## Channel Flow case:

The results were plotted as U+ against y+ for different mashes and eventually compared with the DNS data by Moser et al. . The computation results of wall function showed that standard approach (SWF1: wall function treatment proposed bay Launder  and k�” ϵ modification (SWF2: modification of SWF1) were failed when near-wall node was placed at viscous regions. Though, improvement achieved when near-wall node was place at buffer layer and sublayer, above two prediction was not appropriate below y+<10. However, results from k-ω models were acceptable manner. Meanwhile, plot of k+ against y+ showed that turbulent kinetic energy predictions was comparatively good than the DNS data, especially with the WWF2 treatment. It was surprisingly found that turbulent suddenly increased at near-wall nodes placed between 5 < y+ < 20 for the second wall cell for WWF1 (ω wall function 1), computations. Overall, All treatments agreed with DNS data by Moser et al  for near-wall cells placed at logarithmic region since they are using pure log-law for y+ > 30. On the other hand, LRN computation was done with the finest mesh. Due to finer grid approach, it was found the CPU time consumption was much higher than the WF approach, though the accuracy was more or less the same for all treatments.

## Backward Facing step (BFS) Flow Case:

This type of flow require complex and related to different wall mesh grid. However, for WF treatments, seven different meshes were used to test grid-sensitivity of models whereas for LRN, it was five. It was found that using coarsest meshes, WWF1 and WWF2 treatments present a better behaviour than SWF1 and SWF2 approaches. In case of Wall function computation, If meshes refined, k�” ω approaches tend to the asymptotic results provided by the high Reynolds number k�” ω model by Wilcox et al. whereas k�” ϵ predictions were becoming less accurate. However, LRN model provides different result. In order to achieve asymptotic results, near-wall cells placed at y+= 0. 5 in both walls (development channel and downstream of the step). This is the main reason to explain the extremely dense mesh used and the high CPU time needed for both simulations. In conclusion of this paper it can be said that, researcher proposed different strategies for the improvement of the grid sensitivity for standard SWF1 treatment  which were based on the enhancing ϵ profile using a two-layer integration (SWF2 approach); changing dissipative variable from ϵ to ω and also from k�” ϵ (Launder, D. Spalding ) to k�” ω (Wilcox ) for HRN model would be used for inner nodes (WWF1 approach) and, finally, designing a blending function to reduce grid-sensitivity on an ω platform (WWF2 approach).

## 4. Wall Functions for Oscillating Flow

Many Researchers were seeking for resolving near wall region by wall function for oscillation flow. Among of which UMIST-N and UMIST-A wall functions for periodic flow application will be considered only.