O n T u r b u l e n c e M o d e l l i n g ( R A N S ) a n d S i m u l a t i o n ( L E S, V - LES)

Transcription

1 Report Series - Presentations TransAT CFD/CMFD O n T u r b u l e n c e M o d e l l i n g ( R A N S ) a n d S i m u l a t i o n ( L E S, V - LES) ASCOMP GmbH Edited by: Dr D. Lakehal Release Date: Sep., 2014 Reference: TRS-P/

2 Table of Contents 1. ON TRANSAT ADVANCED TURBULENCE MODELLING IN TRANSAT Algebraic stress modeling in TransAT, EASM Validation ALGEBRAIC FLUX MODELING IN TRANSAT (AFM) Mathematical Modelling: Isothermal Context Mathematical Modelling: Buoyancy Driven Context Validation: Natural Convection Closed Cavity Problem Partitioned 2D Enclosure Heated From Below Validation: Mixed Convection Steiner s reverse transition in a 2D axisymmetric pipe Validation: Forced Convection Multiple Impinging Jets on a Surface LES in TransAT The filtered equations The base Smagorinsky SGS kernel The WALE SGS variant The Dynamic SGS Approach DSM Validation: thermal mixing in the Vattenfall T-junction Validation: Thermal mixing in a cross-flow type channel T-junction V-LES in TransAT The basics Validation: Flow across a cyclic tube bundle

3 Abstract: This document describes flow compressibility features in the CMFD code TransAT, for both single and multiphase flow problems. This technology is employed for a variety of practical energy-related applications. The note introduces briefly the models and highlights selected canonical and industrial applications. 1. On Transat The CMFD code TransAT deals with complex-fluids, single and multiphase flows of industrial relevance. It adopts an original meshing technology among other existing commercial codes, known as the Immersed Surfaces Technology (IST). Mesh generation can also be achieved either using traditional Boundary Fitted Coordinates (BFC), with help of external meshing softwares like GRIDGEN, ICEM. The IST Technology combined with Block-Mesh Refinement (BMR) capability offers great advantages for complex geometries modelling. This code is particularly suited for complex fluid flows and offers powerful solution algorithms suited for parallel processing -using both MPI and OpenMP protocols, a wide portfolio of turbulence models and approaches -including LES and VLES, combustion and reactive flows, multiphase physics with conjugate heat and mass transfer. A battery of multiphase flow modelling approaches is implemented in TransAT, including Level Sets, VOF, Phase Field based on the so-called Cahn-Hilliard equations, Homogeneous Algebraic Slip, Eulerian field formulation, Lagrangian Particle Tracking up to 4-way coupling, Dense Particle (granular) Mixtures. This code is also well suited for very complex physics, including non-newtonian single and multi-fluid flows, microfluidics physics, e.g. Marangoni effects, electro-wetting, dynamic contact angle, etc. The code is best used when installed in a high performance computing cluster to perform large scale simulations". Briefly, the three segments in which the ASCOMP s code excels are: (1) advanced RANS turbulence models, (2) Multiphase flow heat transfer, and (3) Scale Resolving Turbulence strategies like LES and its sub-variants including V-LES and DES -Very Large-Eddy Simulation and Detached Eddy Simulation (Chatzikiryakou et al, 2011; Lakehal et al., 2011; Labois and Lakehal, 2011). The combination of these three advanced features makes the code TransAT well suited to deal with HVAC and pressure-loss-in-pipes problems, including in two-phase flow (Caviezel et al., 2012). 2. Advanced Turbulence Modelling in Transat 2.1 Algebraic stress modeling in TransAT, EASM In the statistical modeling approach for predicting turbulent flows the flow is described via averaging using the "Reynolds or Favre averaging concept", leading to the Reynolds Averaged Navier-Stokes equations (RANS). The flow is thus decomposed into mean and fluctuating components. While RANS provides a widely applicable tool to solve turbulence problems, it faces predictive performance issues in many configurations, including: rotation, swirling, non-homogeneity, strong body forces, curvature and secondary flow motions. To extend the applicability of RANS to internal flows with features from the list above, modifications via various sophistications are added: including solving the Reynolds stresses (RSM), or accounting for their effect in an algebraic, implicit way, EASM, short for Explicit Algebraic Stress Models. While RSM are potentially better, the models are expensive, and their use is often facing instability issues and non-convergence, which makes EASM a better candidate. The examples below show how EASM behaves in complex configurations. Prior to that, let us briefly introduce the basic differences between linear RANS and EASM. 3

4 Linear RANS are built on the idea that the Reynolds stress tensor is linearly dependent on the strain rate S ij; In the EASM variant, however, the Reynolds stress tensor is non-linearly dependent on both the strain rate S ij and rotation tensor ij: with This combination of strain and rotation tensor makes the models sensitive curvature, secondary flow motion, rotation and anisotropy of stresses, which reflects recirculation. An extension of the linear and non-linear eddy-viscosity models for buoyancy-driven flows requires adding the gravity terms, e.g. in the linear context, the stress reads: 2.2 Validation Figure 1: Sketch of the setup for the backward facing step problem. The examples discussed next highlight the advantages of EASM over linear RANS turbulence models. First, two variants of EASM (Gatski and Speziale, 1993 and Shih, Zhu and Lumely, 1993) compared to linear RANS for the flow featuring a strong recirculating motion; the flow in a backward facing step. The comparison (Fig. 2) between simulation results and experimental PTV data of Kasagi et al. (1995) shows that the EASM behaves much better than the linear RANS models employed both in terms of mean and turbulent quantities. Note that use is made of the TLV two-layer model for near-wall treatment, which avoids having to use low-re models. The results for the backward facing step in Figure 2 show the good agreement between the experimental data and the EASM results. The next exercise deals with the modelling of turbulent flow in a pipe, a necessary test prior to deal with the more complex variant of rotating pipe flow. The 2D axisymmetric simulation was conducted for a Re= using the SZL95 (Shih et al., 1995) variant of EASM s, by reference to the experiment of Zagarola (1997). The simulation results plotted 4

5 in Figure 3 agree well with the data, and are in line with previous benchmark exercises comparing OpenFoam, CCM+ and Fluent, as shown in the upper panel of the figure. Third, we plot below the comparison of the EASM results of turbulent flow in a rotating pipe. As suggested in Figure 3, only with EASM could the circumferential motion be predicted, for both low and strong swirling numbers. Linear RANS models cannot predict rotating flow motion. Figure 2: Comparison of the profiles of quantities at different axial locations for (left, top) Mean Velocity, (right, top) Kinetic Energy, (left, bottom) RMS Velocity, (right, bottom) shear stress. Figure 3: Normalized U Velocity over the radial distance. 5

6 Figure 4: (left) Normalized W Velocity (over maximum W Velocity) over the radial distance, (right) Normalized U Velocity (over mean U Velocity) over the radial distance. The next example shown in Figure 5 relates to predicting a secondary flow motion in a square duct and compared to the experiments of Kasagi et al (1995). Here too, thanks to EASM, this flow motion is well predicted, and comparison with PTV data is rather good. Note that in all bends and pipes, the flow should naturally feature secondary cross-flow motion, which is out of reach of linear RANS models, as shown by Gatski & Speziale (1993) and many others. Figure 5: Swirling pattern comparison between TransAT and the PTV data. The last example relates to the prediction of swirling flow in a combustor (Hadef and Lenze, 2005), where the fuel flow splits between an inner channel and an outer annular passage. The spray liquid sheet is injected from the fuel nozzle, which induces a small swirl. On encountering the competing the airflows, it disintegrates into smaller and larger droplets. The interaction between liquid-sheet and the air becomes unstable and disintegrates into fragments (see Figure 5). Here the flow is simulated in single phase only. The comparison between linear RANS and EASM shows again the advantage of the latter approach for this class of flows. In the graphs, one could see that a full RSM model (Fluent) compares with our EASM, with no major differences (Figures 7 and 8). EASM should thus be applied for these practical problems, since it is cheaper than RSM while it returns similar results. 6

7 Figure 6: Setup of the Hadef and Lenze (2005) experiments and swirling patterns Figure 7: Comparison between TransAT and Fluent U Velocity at different locations. 7

8 Figure 8: Comparison between TransAT and Fluent W Velocity at different locations. In summary, we could validate the EASM models implemented in TransAT for flows featuring key physics which is out of reach or linear RANS models: recirculating flows, rotating flows with body forcing, and secondary flow motion. 3. Algebraic Flux Modeling in Transat (AFM) 3.1 Mathematical Modelling: Isothermal Context In the context of linear RANS modelling for convective heat transfer, the heat flux is linked to the temperature gradient via the expression: Two equation models for thermal diffusivity have been developed in parallel with the dynamic modelling of turbulence, e.g. the k-e model. This idea was motivated by the fact that turbulent heat convection should also be characterized by a scalar (temperature) time scale that varies in space and time, just like the dynamic time scale 0 = k/. Thus, instead of tying the eddy diffusivity solely to the dynamic velocity and time scales through this quantity may contain information on the scalar time scale ( ) as well, for example Elghobashi and Launder (1983), Nagano & Kim (1988), and Abe et al. (1994) suggested models for the eddy diffusivity expressed as follows: 8

9 where denotes the dissipation rate of the temperature variance, (this implies that = / ) and is analogous to, i.e. the dissipation of TKE. Admittedly, this procedure requires development of a scalar transport equation for which is similar to the k equation in the one and two-equation turbulence models. The temperature variance could be determined from a differential convective-transport equation too: in which the first term in the R.H.S. is nothing more than a gradient-diffusion approximation to the exact triple correlation. The thermal dissipation that appears as a sink term in the R.H.S. of the equation should in principle be determined from the modelled transport equation too: where Cp1, Cp2, CD1, and CD2 are model constants. In previous proposals (Launder, 1988), however, the complications were reduced by determining from a simple algebraic relation: where R stands for the thermal-to-dynamic time-scale ratio, a measure of the relative importance of the relaxation effects of the mechanical and thermal dissipation. The importance of the values to assigned R will be discussed later when evoking buoyancy driven flows. The system of equations written above refers to high-re number flows; it therefore necessitates the Wall Functions approach much the same way as the k- model. As noted previously, this heat flux model has the capability of treating low Re number regions as well (Abe et al., 1994). There is implicitly the idea in the above relationship known as SGDH, short for standard gradient diffusion hypothesis- that the scalar flux is aligned with the temperature i gradients, which is far from being the case in many situations. In a boundary layer where significant mean temperature gradients occur only normal to the wall, the turbulent heat flux parallel to the wall was found to be twice as large as that in the normal direction. In fact, the concept of isotropy of the three flux components could be defendable in situations where the turbulent flux in the flow direction is very small compared to transport by the mean motion. Examples of flows characterized by a strong non alignment of the heat flux with its mean gradients can be found in various practical situations. Also, Experience has shown that a simple algebraic expression for the heat flux can be a defensible alternative to having to consider in detail history and relaxation effects of the heat flux by solving equation its full transport equation, provided the Reynolds stress components are determined individually, i.e. in the RSM framework. The turbulent heat-flux can then be modelled in an analogous manner to the turbulent transport term in the Reynolds stress equation, here in particular by reference to the Daly & Harlow (1970) proposal: This approach is known as the anisotropic heat flux model, or the generalized gradient 9

10 diffusion hypothesis (GGDH), a definition that points to the fact that heat transfer is driven by an anisotropic thermal diffusion: This approach has the merit to conform to many experimental findings, including the measurements of turbulent heat transfer in pipes and boundary layer flows by Bremhorst and Bullock (1973), and by many others. Indeed, these authors demonstrated that turbulent heat flux in the flow direction are two to three times larger than in the direction normal to the wall while the streamwise temperature gradient is negligible compared to that normal to the surface. Another more sophisticated approach consists in determining the heat flow by invoking the WET (Wealth Earnings Time) theory, a syllogism applied by Launder (1988) to turbulent heat flux which lead to: (Value of Second Moment Production Rate of Second Moment Turbulent Time Scale). This, together with the turbulent time scale taken as k/ε yields: The WET model is supposed to remedy the drawback of all other variants, in which the heat flux is only generated by temperature gradients; which is not always the case, for example the mixed layer formed close to a heated a wall featuring a uniform vertical temperature gradient is not necessarily linked to turbulence, so the heat flux is actually over-represented in the relative sense. The same is true when vertical temperature gradients are small: here it is the velocity gradients that cause the wall to flow heat transfer. 3.2 Mathematical Modelling: Buoyancy Driven Context In the buoyancy driven context, too, the modelling starts from reducing the transport equation for the heat flux to an algebraic expression assuming that convection balances diffusion and production and dissipation of k and 2 are locally in balance: which is superior to the GGDH/WET formulation alone, in that it has the ability to predict a vertical heat flux with just the temperature variance actions, even in the absence of mean temperature and/or velocity gradients. These algebraic expressions can be closed by solving the transport equations (modified when needed for low-re-number situations and near wall effects) for the turbulence kinetic energy and its rate of dissipation, for the temperature variance and its dissipation, resulting in a four-equation model (k ) discussed by Hanjalic and Kenjeres (1995) and Kenjeres and Hanjalic (2000). A further simplification can be achieved by expressing in terms of the three other variables from the assumed ratio of the thermal to mechanical turbulence timescales R defined previously, with R=Const or prescribed by an algebraic function in terms of available variables. This reduces the model to a three-equation one, k-. Although in many situations R is not constant, such an assumption with the three-equation models displayed remarkable success in a number of flows. In summary, Algebraic heat Flux Models (AFM) amount to solving an additional temperature variance ( 2 ) equation with a prescribed thermal dissipation and is coupled implicitly to the momentum equations. 10

11 3.3 Validation: Natural Convection Closed Cavity Problem The accuracy of the numerical method was tested against high quality results of other investigators for the classic closed 2D cavity problem, with two opposite vertical walls kept at a temperature difference and two adiabatic horizontal walls. The accuracy of the flow and temperature fields is depending on the mesh and, in order to deliver mesh independent results, different mesh sizes are needed for different Ra numbers. Comparison between the results of TransAT and the high quality results of Le Queré (1991) for two Ra numbers are shown in table 1 (Ra=106) and table 2 (Ra=108). The parameters reported here are the most important non-dimensional integral and local parameters of the flow field, namely: The average Nusselt number on the hot wall NuRa-1/4, the vertical gradient of the thermal stratification in the centre S, the vertical non- dimensional velocity maximum at half the cavity height Vmax and the horizontal non-dimensional velocity maximum at half the cavity width Umax. mesh S NuRa -1/4 vmax umax % diff with transat result on 81x81 mesh 21x x x Absolute values 81x Le Quere % diff with Le Quere Table 1: Accuracy of the solution for Ra=106, closed cavity The results of TransAT are mesh independent already on mesh 41x41, as for this resolution differences are seen to be less than 1% compared to values on the finest mesh for all parameters. The agreement of the fine-mesh results of TransAT with the ones of Le Quere is excellent. Similar and consistent conclusions are drown for the results of TransAT for Ra=108, where the mesh independent solution is accomplished on mesh 81x81, for this closed cavity problem. It is worth noting that Umax result for TransAT is in excellent agreement with the benchmark solution of Le Quere, while Henkes and Hoogendoorn [12] reported excellent agreement for all parameters with the exception of Umax, which was found to be in more than 5% difference with the benchmark solution of Le Quere, even on their finest mesh with 120x120 points. Streamlines and temperature contours were found in excellent agreement with the ones shown in Henkes and Hoogendoorn (1993) for both Ra numbers. A sample comparison for temperature contours are shown in Fig. 9. mesh S NuRa -1/4 vmax umax % diff with transat result on 121x121 mesh 41x x x x Absolute values 81x Le Quere % diff with Le Quere Table 2: Accuracy of the solution for Ra=108, closed cavity 11

12 Figure 9: Isothermal contours at Ra=108; left Henkes and Hoogendoorn (1993), right TransAT Partitioned 2D Enclosure Heated From Below This well-known exercise (partitioned 2D enclosure heated from below at different aspect ratios: AR=1:4; 1:5; 1:8) was selected to compare the various heat transfer models: SGDH, GGDH, WET, AFM. It is postulated from earlier experiences that AFM should enhance the quality of the simulation results as to mixing under natural convection conditions. This test case was simulated by ASCOMP for the aspect ratio 1:4, and by UCL for AR=1:4. Two Rayleigh number flow conditions were simulated in 2D under steady state conditions: Ra=107, 108. It is important to note that this flow features very large coherent structures that are nominally not within reach of steady-state 2D RANS simulations, as noted by Hanjalic (2002). In these simulations, we have employed the quality assurance policy, through the adoption of the ERCOFTAC CFD Best Practice Guidelines in order to minimize user s influence on the results. Figure 10 below depicts the velocity contours inside the enclosure for the two Rayleigh numbers (Ra=107, 108). Large-scale vortices located at the corners strengthen with the Rayleigh number. The right panel shows actually that the lower corner vortices tend to wash the surface transporting more heat from the wall region to the core flow. Figure 10: (top panels) Iso-contours of the flow and temperature in the enclosure for two different Rayleigh numbers (Ra=107 and 108). 12

13 Figure 11: (top panels) profiles of temperature in the enclosure for Ra=107, comparing SGDH, GGDH, WET and AFM. (lower panel). AFM prediction of temperature profiles for different Rayleigh numbers (Ra=106, 107 and 108). The upper panel of Figure 11 compares various model variants for the non-dimensional temperature profiles in the canopy for Ra=107. It seems that only with the AFM model (red line) the results match the data for Ra=107; with all other state-of-the-art models (SGDH, GGDH, WET), the flow re-laminarizes, while it should in effect remain turbulent for this Rayleigh number. Plotting now the same temperature profiles obtained with the AFM alone, for the transitional Ra number of 106, in addition to the turbulent cases for which data are available, reveals that indeed the AFM is sensitive to this change, in that it predicts indeed the transitional case as well, where the flow is still laminar. 3.4 Validation: Mixed Convection Steiner s reverse transition in a 2D axisymmetric pipe In the experiments of Steiner (1971), a mixed convection flow regime is obtained in an ascending flow of air in a vertical pipe. In turbulent mixed convection situations, buoyancy affects in a certain measure inertia-dominated flows. As to simulations, a 2D axisymmetric 13

14 pipe of 8 cm diameter and 4m length (L/D=50) is considered, without unheated length though. The details are given in Table xx below, where ; Two test cases were simulated, purposely with the Re=5.000 case, which according to Steiner is an important one since it exhibits the so-called reverse transition mechanism, in which the flow in the boundary layer becomes laminar as well as fluctuating and that it oscillates with a predominating period. Case x x x x10 8 Table 3: Flow parameters in the Steiner case The simulation results obtained with SGDH, GGDH, WET and AFM models for Re=5.000 and are compared with the measurements of Steiner (1971). The grid consists in 141x45 cells to cover half the domain; the second-order HLPA convection scheme is employed. The influence of Buoyancy was assured via controlling the ratio of Grashof to Reynolds number. The simulated bulk and wall temperature evolutions along the pipe are shown in Figure 12 for both Reynolds numbers. Interestingly, as was to be expected, the Re=5.000 seems to be more difficult to predict than the case; the latter shows a linear evolution of the wall temperature proper to fully developed turbulent flow, while the transitional case feature a bumpy structure reflecting the fluctuating laminar-turbulent boundary layer. But all models seem to predict the same evolution, with a smoother lower-value profile in the AFM result though, for Re= in particular. For Case-I, the models show all good agreement with the data as to the Nusselt number evolution (Fig. 13). For Case 2, the simulations show an over-prediction for the Nusselt number, and surprisingly, the AFM (non-tuned) returns higher values in line with the lower wall-temperature predictions. Note that the Nusselt number plots indicate that the flow is indeed fully developed. In other words, all algebraic non-linear models show now improvement at all as compared to SGDH. The predictive performance of the AFM (ATHFM) alone is displayed in Fig

15 Figure 12: Wall and bulk temperature evolution along the pipe for all models. (upper panels) Re=5000; (lower panels); Re= Figure 13: Evolution of Nusselt number for (left) Re=5000 and Re=14900 Figure 14: Evolution of Nusselt number for Re=5000 and Re=14900 using AFM 3.5 Validation: Forced Convection Multiple Impinging Jets on a Surface We consider here a multiple jets impinging normally on a flat surface, which are used frequently to achieve efficient cooling or heating of solid walls. In contrast to single jets, turbulence structure in multiple jet configurations is more complex. Here, the additional factor is the interaction between neighbouring jets, which depending on their mutual distance can have a dominant effect on heat transfer intensity and especially on its distribution over the impingement surface. Most literature dealing with multiple jets reports flow field data in jet arrays of custom-made nozzle arrangements, but few results are available on the measurements of mean flow and turbulence characteristics. 15

16 The present validation exercise is inspired from the experiment of Geers et al. (2005). Because most linear eddy viscosity models cannot reproduce properly the stress anisotropy, they fail in reproducing heat transfer. That the stress field is closely related to heat transfer can be illustrated also indirectly: the models which reproduce well the turbulence stress field yield as a rule better predictions of wall heat transfer, even if a simple eddy-diffusivity concept is used for the turbulent heat flux. The simulations were conducted in 3D using the SZL95 EASM model combined with the GGDH for heat transfer. The simulations results were compared to CFX results using the SST model. Curious phenomena, such as symmetry breaking, which has been observed in the experiments and by other modellers of this case are also predicted here (Fig. 15, upper panel), despite forcing symmetry boundary conditions. Lower panels of Fig. 15 show the comparison between TransAT and the data as to the wall heat transfer (here the Nusselt number distribution). The comparison of the velocity profile and Nusslet number along the surface reveals a good agreement of the GGDH with the data. Figure 15: Calculated (left panel) vs. measured (right panel) Nu iso-contours on the wall surface 16

17 Figure 16: (left panel) mean vertical velocity profile, (right panel) Nusselt number distribution. 4. LES in TransAT 4.1 The filtered equations The filtered mass, momentum and temperature equations for incompressible convective fluid flow take the form: where u i is the fluid velocity vector, p is the pressure, ρ stress, λ is the thermal conductivity, C p is the heat capacity, and Q is the volumetric heat source. The source terms in the RHS of the momentum equation represents the body force, F b, and the convolution-induced terms for non-equidistant and body fitted grids, F c. Further, the filtered LES equations introduce the so-called SGS stress tensor and turbulent heat flux defined as: Only the deviatoric part of the SGS stress tensor is to be modeled using a statistical approach similar to RANS; and the same is true of the turbulent heat flux. This way, turbulent scales larger than the grid size are directly solved, whereas the effects of SGS scales are modeled. The filtered equations and SGS models are now well known; details can be found in Moin et al. (1991; Peng and Davidson, 2001). 4.2 The base Smagorinsky SGS kernel LES is based on the concept of filtering the flow field by means of a convolution product. The specific super-grid part of the flow with its turbulent fluctuating content is directly predicted whereas the sub-grid scale (SGS) part is modeled, assuming that these scales are more homogeneous and universal in behavior. For turbulent flows featuring a clear inertial subrange the modeling of the SGS terms in the statistical sense could thus safely borrow ideas from the RANS context, in particular use of the zero-equation model to mimic the momentum diffusive effects on the resolved field. Use is generally made of the Eddy 17

18 Viscosity Concept, linking linearly the SGS eddy viscosity and thermal diffusivity to the gradients of the filtered velocity and temperature, respectively: 4.3 The WALE SGS variant The closure for the eddy viscosity above follows in general the Smagorinsky kernel model, linking the eddy viscosity to the square of a length scale and a time scale (the inverse of the second invariant of the resolved rate of deformation tensor S ij). It happens actually that the eddy viscosity is more sensitive to the rate of vorticity as well, in which case the base kernel is not ideal. The WALE SGS model has been precisely proposed to remedy this limitation (Ducros and Nicoud, 1999), and has been employed rather successfully since then; Briefly, it defines the SGS eddy viscosity as follows: where and reads: 4.4 The Dynamic SGS Approach DSM- In the base SGS kernel, the model constant (Cs) is either fixed or made dependent on the flow; this latter option is precisely the spirit of the dynamic model. A damping function is often introduced for the model constant Cs to accommodate the asymptotic behavior of near-wall turbulence, namely that the SGS eddy viscosity (and heat diffusivity for Pr=1) scales with y+3. Similarly, the same strategy could be used to close the turbulent SGS heat flux, where the thermal diffusivity could be determined either based on the resolved thermal-flow field, or alternatively based on the eddy viscosity (defined dynamically) and a fixed. Using the first alternative means that the turbulent Prandtl number is not imposed but is a result of the model. The advantage of DSM compared the base model is that the model constant Cs may be negative, which does not exclude possible backscatter of energy, it returns the proper asymptotic behavior of the stresses near the wall with damping as required by the base model, and vanishes in laminar flow without ad-hoc intermittency functions. The DSM approach requires though a two-level filtering, in contrast to simple Smagorinsky model, in which filtering is actually implicit, based on the grid only. The approach is based on the application of a second larger filter on top of the filtered equations (1). Without presenting the details of the model, the dynamic length scale for both the thermal and flow field are determined as follows Moin et al. (1991) (using the least-square approach): 18

19 where <. > denotes plane-averaging in flows with a clear homogeneous direction, e.g. channel flow. Using these two length scales to determine the SGS eddy viscosity and diffusivity separately helps derive the dynamic expression for the turbulent Prandtl number Moin et al. (1991). The advantage of the DSM approach here is its capacity to sensitize the eddy diffusivity to the resolved thermal-flow field. This may sound somewhat conflicting with the analytical models linking the turbulent Prandtl number to the molecular one, (e.g. Jischa and Rieke, 1979; Kays, 1994). Be it as it may, using the DSM approach will help shed light on various issues: whether the turbulent Prandtl number Pr t is truly independent of Pr only for Pr < 0.1, and whether the sharp transition for Pr t is at Pr = 0.01, as stipulated from experimental observations (i.e. Pr t suddenly increases for Pr < 0.01, otherwise it remains in the range 2-4). Our intention is to use the model and calibrate analytical approaches. 4.5 Validation: thermal mixing in the Vattenfall T-junction Thermal mixing in T-junction configurations can be found in various industrial equipment, including chemical reactors, combustion chambers, piping systems in power plants, and HVAC (Heating, Ventilating, Air-conditioning) units used for automobile air-conditioning systems (Kitada et al. 2000). Here two streams of fluids with different velocities, temperatures, and/or concentrations are mixed by turbulence. The phenomenon could potentially lead to thermal fatigue failures in energy cooling systems when cyclic stresses are imposed on the piping system due to vigorous temperature changes in regions where cold and hot flows are intensively mixed. In HVAC systems, the evaporator reduces humidity by cooling the air taken by the fan. The heater core heats a portion of this cold air. Since the temperature of hot and cold airflows are fixed by the A/C system, the temperature of air in the cabin is controlled by mixing in the HVAC unit, and is realized by controlling the flow-rate ratio of two streams, determined by the opening of the air-mix door located separating the evaporator and the heater-core. The hot and cold airflows impinge at nearly right angles, a situation lending itself to advanced CFD analysis. Flow separation and reattachment, secondary flow, anisotropy of turbulent stresses and heat transfer (including thermal stripping) are some of the complex flow features associated with the flow in a T-junction. For such applications, past experience shows that statistical time-average models need to be replaced by more sophisticated scale-resolving strategies, including LES. A number of numerical investigations of the flow in mixing T- junctions can be found in the literature (Hu and Kazimi, 2006). This study of high cycle temperature fluctuations showed the applicability of LES for the prediction of turbulent flow features in a T-junction, showing the method to perform well in general in capturing scalar mixing, secondary flow, mean and fluctuating temperature fields. The issue of SGS modelling seems not to be an issue, in contrast to the effect of turbulent inflow conditions, as shown in this paper. 19

20 Figure 17: Instantaneous temperature Profiles (top) Cross-Sectional profiles are different locations. (bottom) 2-D profile. The results shown in Figure 17 above shows the instantaneous thermal-flow field delivered by LES for the Vattenfall benchmark (Mahaffy, 2010). Overall the averaged results produced by TransAT listed below (Figures 18-20) are in excellent agreement with the experimental results (even if the employed grid was relatively coarse compared to other codes; FLUENT used 55 million cells to obtain the same quality results in this Benchmark), underscoring the fact that LES stands a chance for accurately modelling complex, industrial turbulent flows. Figure 18: Comparison of the radial profiles of the normalised temperature at different angular locations. 20

21 Figure 19: Comparison of the (top) horizontal and (bottom) vertical profiles of the normalised velocity. Figure 20: Comparison of the (top) horizontal and (bottom) vertical profiles of the Reynolds Stresses. 4.6 Validation: Thermal mixing in a cross-flow type channel T-junction This test case was selected as a benchmark for thermal mixing in the ERCOFTAC Workshop held in EDF Chatou, France, The experiments were conducted at the Mie University, Japan (Hirota et al. 2010). The configuration consists of air flow entering at 60 C in a smaller duct of diameter B joining with air flow at 12 C entering a larger duct of diameter 2B (Fig. 21). The height of the main channel H is 60 mm and the width 2A is 120 mm. The height (X-way length) of the branch cross section B is 30 mm and the width of the branch is 120 mm. The airflows in both channels were mixed in the T-junction after flowing through the heat exchangers, settling chambers and contraction flow nozzles. This difference of the channel geometry exerts considerable influences upon the flow structure and resulting temperature field after the flow merging. The origin of the coordinate system is at the spanwise centreline of the downstream edge of the T-junction. In a real HVAC unit, the exit of the mixing zone corresponds to the location of X/B=2 3. The study was conducted to clarify how two fluids with different temperature are mixed by turbulence in a cross-flow type T-junction present in HVAC systems for automotive industry. The characteristics of the velocity and temperature fields in the channel were investigated experimentally, providing high-quality flow visualization images (Fig. 22) that help in particular understand the way the mixing layer and the shear layer interact as the two streams merge in the discharge region. 21

22 The filtered transport equations (1-3) were solved using the DSM SGS model as for the previous case, with no wall-damping function. A Cartesian multiblock mesh consisting of 1.19 million cells was used. The grid distribution was carefully controlled, in particular the maximum aspect ratio in all direction. The wall-neighboring cell was fixed such as it resulted in an average y+ value of 12, justifying the need to resort to wall functions (Werner and Wengle, 1991). Air material properties were taken overall at the cold temperature of 12 C, but the other flow parameters were adapted at the hot branch inlet such as the Reynolds number as measured there is matched (the V0 inflow velocity was reduced from 2.7m/s to 2.1m/s). An adaptive time-stepping strategy (using explicit time marching) has been chosen, controlled by a CFL < 0.3, leading to an average time step of s. The flow reaches a steady state after about time steps, after which time and space averaging of the flow has been performed, typically for time steps. High order schemes were employed for both time and space differentiation, respectively 3rd order RK scheme and 3rd order Quick scheme. The 2nd order central scheme showed instability at certain events, the simulation time for MPI parallel calculation was 66H on a 128 CPU supercomputer. The convergence criterion set for pressure was per time step. Qualitative flow features are depicted in Fig. 22, showing the heat contours developing with the flow as the cold and hot stream merge together. The comparison with the experiment shown in Fig. 21 is only qualitative. Note the upper panel depicts results obtained with steady inflow conditions whereas the lower panel corresponds to the unsteady inflow conditions. The way unsteady flow conditions were imposed follows a TransAT-specific turbulence recycling strategy, in that the flow is first calculated in each branch separately until convergence, then the resulting flow is used as an imposed inflow (one cross-flow plane) for production runs with the two branches calculated together. Without this approach the flow would not develop turbulence and remains laminar in particular in the vertical pipe with a low Reynolds number, causing higher momentum of the jet penetrating in the horizontal branch deeper than in reality, which in turn returns wrong mixing. The lower panel suggests already that the location of the maximum thermal loads is on the upper pipe side immediately downstream the junction (1-2D). The heat diffuses rather fast downstream, decaying at about 10D from the junction. As it was found in the previous case, the secondary flow motion associated with a strong turbulence activity downstream the junction should be located close to the region of maximum thermal stripping on the horizontal channel. The heat contours displayed in Fig. 13 suggest indeed the flow exhibits a thin shear layer enveloping the jet, with obviously less details as compared to the visualization reported in Fig. 21. Time averaged results obtained with LES are compared with the data in Figs. 24, 25 and 26. The agreement with the experiment is very good, for both averaged velocity components in particular at the locations neighbouring the junction of the two channels. A slight discrepancy is observed as to V profile further downstream, at X/B=2. The r.m.s. profiles of the fluctuating velocities in these two directions are compared in Fig. 25, showing a good agreement with the data, albeit at X/B=0, the LES over-predicts the v - r.m.s. The same is true for the temperature (mean and r.m.s.) profiles plotted in Fig. 26: the agreement with the data is excellent for Tmean at X/B=0, but somehow the predicted profiles for the other two locations are less accurate. In particular, the mean profiles very close to the wall is less than the experiment by about 15%, then the slope of the profiles tend to be sharper in the core flow, pointing to a possible misrepresentation of the jet as it penetrates the horizontal branch. While the latter issue is possible and could be attributed to the numeric (e.g. numerical diffusion), the grid resolution, the inflow conditions or even the SGS model employed, the first one is however difficult to explain, unless there is an isolation issue in the experiment 22

23 that prevents from obtaining exact isothermal wall conditions, in which case, near the wall from Y/H=0 to 0.2, the channel is not well isolated. While t - r.m.s. profile at X/B=0 is well predicted, the shift observed at the downstream locations from the Tmean profiles is again transparent here. Figure 21: Schematic diagram of the test channel Hirota et al. (2010) Figure 22: Visualized image of the interface between two flows in Z/A = 0. (The solid lines show the timeaveraged interface between the main and branch flows.) Hirota et al. (2010) Figure 23: Effect of introducing inflow perturbations (temperature iso-contours) Figure 24: Time average velocity profiles at various locations of the horizontal channel (X/B=0, 1 and 2) 23

24 Figure 25: R.M.S. profiles at various locations of the channel (X/B=0, 1 and 2) Figure 26: Time average and R.M.S. temperature profiles at various locations of the channel (X/B=0, 1, 2) 5. V-LES in TransAT 5.1 The basics V-LES is based on the concept of filtering a larger part of turbulent fluctuations as compared to LES (as the name clearly implies); Speziale (1998). This directly implies the use of a more elaborate sub-grid modelling strategy than a zero-equation model like in LES. The V-LES implemented in TransAT is based on the use of k model as a sub-filter model. The filter width is no longer related to the grid size; instead it is made proportional to a characteristics length-scale ( that is larger than the grid size (~ x), but necessarily smaller than the macro length-scale of the flow (see Fig. 1). Increasing the filter width beyond the largest length scales will lead to predictions similar to the output of RANS models, whereas in the limit of a small filter-width (approaching the grid size) the model predictions should tend towards those of LES. V-LES could thus be understood as a natural link between conventional LES and URANS. If the filter width is smaller than the length scale of turbulence provided by the RANS model, then larger turbulent flow structures will be able to develop during the simulation, provided that the grid resolution and simulation parameters are adequately set (in particular regarding time stepping and the order and accuracy of the time marching schemes employed). The V-LES theory as currently used has been proposed by Johansen et al. (2004), though the reader can refer to Labois and Lakehal (2010) for more details. The filter width is denoted as in the following text. One of the key hypotheses in V-LES is that the Kolmogorov equilibrium spectrum is supposed to apply to the sub-filter flow portion. 24

25 This function cannot be known a-priori if the entire energy spectrum is not explicitly known; thus, the simple proposal from Johansen et al. (2004) is used here: Near wall boundaries, the function is forced to be equal to unity, which means that the standard model is systematically applied in these regions. This permits the use of the standard wall-functions in the V-LES context, too. The method can also be employed under low-re flow conditions, using either a two-layer approach based on a one-equation model or a Low-Re model. Finally, the turbulent viscosity for V-LES can be written as, The difference between RANS, LES and V-LES, is that in the latter approach it is necessary to specify a filter width, which can be made proportional to a characteristics length-scale of the flow. This parameter has been the object of a systematic dependence study in Labois and Lakehal (2010). Apart from that, a lower bound must be set to ensure that the filtering process is compatible with the grid resolution. Practically we impose > 1.5 grid where grid = ( x y z)1/3 in three dimensions. The original model of Johansen et al. (2004) forces the subscale model to treat near wall regions using the standard k- actually show that at the limit of wall distances (yn) at which viscous effects become negligible, Eq. (7), which takes the form t = C C3 Δk1/2 for (Δε/k2/3<<1) should rather be re-cast in the form of a Prandtl mixing length model: t = C (Cl yn f ) k1/2, where the length scale (L = Cl yn f ) is associated with the wall distance yn, modulated by a damping function f. If in this viscosity-affected layer the transport equation of the rate of dissipation is disregarded in favour of an algebraic prescription (i.e. = k3/2/ L ), the final model degenerates to a two-layer model (Lakehal and Thiele, 2001), which in the context of coupled LES-RANS is commonly known as DES, short for Detached Eddy Simulation. This is the way mode has been implemented in the code TransAT; a coupled VLES-DES approach. 5.2 Validation: Flow across a cyclic tube bundle Our numerical simulations will be compared with the experiment of Simonin and Barcouda (1986), who studied the flow across a staggered tube bundle with diameter D = 21.7 mm. The flow Reynolds number, based on the diameter of the tube, is , and the bulk velocity is 1m/s. This flow has a complex behaviour, non-homogeneous and inherently unsteady, with a flapping effect in the wake of the bundles. Simonin and Barcouda (1986) provide mean velocities in the x and y directions, as well as the Reynolds stresses <u'u'>, <v'v'>, and shear stress <u'v'> for different locations. It should be noted that in several previous numerical simulations of this flow (e.g. Rollet-Miet et al., 1999; Benhamadouche & Laurence, 2003), smaller Reynolds number (9 000 or less) were used to perform the simulations, otherwise the grid would have been considerably fine. On the other hand, Hassan and Barsamian (2004) used a slightly higher Reynolds number (21 700), and simulated the whole geometry, without using cyclic boundary conditions. The computational domain is shown on Figure 1; it has a dimension of 45 x 45 mm. The depth in the z direction is one diameter when it is not mentioned. The origin is taken at the middle of the domain. Periodic boundary conditions are applied in the x and y directions, and in the z-direction when performing three-dimensional computations. As shown in the figure, the grid is not of BFC type but of IST class; short for Immersed Surfaces Technology. In the IST the solid is described as the second component or material, with its own thermo-mechanical properties. The technique differs substantially from the Immersed Boundaries method of Peskin (1977), in that the jump condition at the solid surface is 25

26 implicitly accounted for, not via direct momentum forcing (using the penalty approach). It has the major advantage to solve conjugate heat transfer problems, in that conduction inside the body is directly linked to external fluid convection. The solid is first immersed into a cubical grid covered by a Cartesian mesh. The solid is defined by its external boundaries using the solid level set function. Like in fluid-fluid flows, this function represents a distance to the wall surface; is zero at the surface, negative in the fluid and positive in the solid. The treatment of viscous shear at the solid surfaces is handled very much the same way as in all CFD codes. A mean pressure forcing is applied in the x-direction to ensure that the bulk x-velocity (defined as the average of the velocity in the whole fluid domain) is 1.06 m/s. To match the experiments, the average of the velocities in the y and z directions is set to zero. Fluid properties are set so that the Reynolds number based on this bulk velocity and tube diameter in effect equals Different mesh resolutions have been used in order to check grid dependency, with a number of cells in the x and y directions between 50 and 200. When three-dimensional computations are performed, the size of a cell in the z-direction is equal to the size in the other directions. These meshes have about the same precision as those used by Benhamadouche and Laurence (2003), who used unstructured grids of 2072 cells with refinements near the walls. The grid is coarser though than the two-dimensional meshes used by Johnson (2008), who used a grid of cells. The results of these final larger-depth simulations are presented in Figure 27, where two filter-widths have been used, namely 0.05d and 0.1d. Smaller filter-width results 0.05d provide very good agreement with both the present LES results and experiments. Results using a larger filter width of 0.1d deviate slightly from the data, but they still match the measurements when looking at the mean velocities in particular. Both the streamwise normal Reynolds stress and shear stress are remarkably well predicted by V-LES as compared to LES, although the effect of varying the filter width for this depth is noticeable for the cross-flow normal stress mainly. Comparing now the work with previous other CFD-based findings using high-order turbulence models and LES, the present results are overall very satisfactory. The current results slightly underestimate the streamwise velocity close to the walls as compared to Benhamadouche and Laurence (2003) and Johnson (2008) though performed in 2D; in both these references, this quantity is rather slightly overestimated, while in contrast Hassan & Barsamian (2004) report very good results in this zone. Our results may be explained by the fact that we do not have cell refinement near the walls. The current prediction for <v> is however very good and match that of Benhamadouche and Laurence, and are better that Johnson s (2008) 2D results. The normal stress <u'u'> is very well predicted, too, though with departures from measurements with about 10 %. The behaviour is better than in Benhamadouche and Laurence (2003) paper, who report a larger peak near the walls. <v'v'> is also well reproduced by our V-LES, although being slightly underestimated in the core-flow region. The same behaviour is reported by Johnson, in contrast to Benhamadouche and Laurence, Hassan and Barsamian who obtained results with close match to the experiment. Finally, though the shear stress shows a good behaviour overall, the quantity is overestimated near the wall and shows a shift of the peak; the same being reported in the LES results of Benhamadouche and Laurence. Hassan and Barsamian report instead a closer match to the experiment. 26

27 Figure 27: Comparison of LES and V-LES on a mesh of depth 1.5d, results at x = 0 mm. 27

28 The turbulent structures of the flow are now shown in Figure 28, highlighted by iso-values of the fluctuation velocities u', v' and w'. The figure compares LES and V-LES for a filter width of 0.1d. It can be seen that the V-LES (right panels) reproduces indeed the threedimensional structures, but they are not as fine as in the LES (left panels). This difference in size of the turbulent structures explains the different conclusions drawn between the present work and that of Rollet-Miet et al. (1999) also based on 3D LES- on the size of the domain in the spanwise direction: they found indeed that a depth of one diameter was sufficient for their LES simulations, whereas a depth of at least 1.5 diameter is shown to be needed for both the V-LES and LES to capture the three-dimensional structures. The present result suggests that the large scales are the most important contribution to turbulent kinetic energy, which makes of the V-LES a very good alternative to LES, at least for this class of flow. The benefit of the V-LES is mainly in reducing the CPU time needed for full 3D LES. The fining points out to the fact that SGS models of various sophistication will not necessarily bring a marked difference compared with a simple models. Figure 28: Flow structures in LES (left) and V-LES (right). Fluctuating velocities u', v and w'. 28