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Computational modeling of turbulent mixing in a jet in crossflow
International Journal of Heat and Fluid Flow xxx (2013) xxx–xxx
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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Computational modeling of turbulent mixing in a jet in crossflow Flavio Cesar Cunha Galeazzo a,⇑, Georg Donnert a, Camilo Cárdenas a, Julia Sedlmaier a, Peter Habisreuther a, Nikolaos Zarzalis a, Christian Beck b, Werner Krebs b a b
Karlsruhe Institute of Technology, Engler-Bunte-Institute, Division of Combustion Technology, Engler-Bunte-Ring 1, 76131 Karlsruhe, Germany Siemens AG, Mellinghofer Str. 55, 45466 Muelheim an der Ruhr, Germany
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Turbulent mixing RANS URANS LES Intermittency
a b s t r a c t The jet in crossflow is a configuration of highest theoretical and practical importance, in which the turbulent mixing plays a major role. High-resolution measurements using Particle Image Velocimetry combined with Laser Induced Fluorescence have been conducted and used to validate simulations ranging from simple steady-state Reynolds-averaged Navier Stokes to sophisticated large-eddy simulation. The reasons for the erratic behavior of steady-state simulations in the given case, in which large-scale structures dominate the turbulent mixing, have been discussed. The analysis of intermittency proved to be an appropriate framework to account for the influence of these flow structures on the jet in crossflow, contributing to the explanation of the poor performance of the steady-state simulations. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Turbulent mixing plays an important role in the dynamics of turbulent flows, with many consequences in nature and engineering. The simplest case of turbulent mixing occurs between passive scalars. Examples are mixing of fluids with the same density, or loaded with non-reacting tracers. Simulation of mixing in turbulent flows is of great importance from theoretical and practical points of view. The increasing computational resources allow the simulation of larger, more complex systems in shorter timeframes. The dissemination of this technology comes with new challenges, as the increased focus on the robustness, accuracy and predictive capabilities of the models, which are faced with more complex systems and phenomena. The jet in crossflow (JIC) is a flow configuration with features like large-scale coherent structures and recirculation regions. Due to its good mixing capability, this flow configuration is frequently found in technical devices like mixers in the process industry, plumes from chimneys, engine exhaust gas pipes and gas turbines. Investigations on the JIC, focusing in mixing of chimney plumes, started in the 1930s (Ooms, 1967; List, 1982). A more systematic analysis of the JIC started in the 1970s, culminating in the detailed measurements of Andreopoulos and Rodi (1984) using a three sensor hot-wire probe. Design correlations known for their practical relevance have been developed by Holdeman et al. (1997). Alvarez et al. (1993) has shown that steady-state simulations predict the ⇑ Corresponding author. Tel.: +49 721 608 42808 (Germany); fax: +49 721 608 43805. E-mail address: [email protected] (F.C.C. Galeazzo).
turbulent mixing of the JIC in poor agreement with experimental data. Margason (1993) reviews numerous investigations of the JIC configuration, much of them focusing on the complex system of vortices and their contribution to the stability of the flow field. Laser diagnostic tools have made tremendous progress in recent years, also benefiting studies about the JIC. Some contemporary references employing laser diagnostics on the JIC have been performed under weakly turbulent conditions, with low Reynolds numbers that do not represent the majority of the applications of the JIC (Su and Mungal, 2004; Cárdenas et al., 2007, 2010). The newly developed high-resolution laser diagnostic system using simultaneously Particle Image Velocimetry and Laser Induced Fluorescence has been realized at the Engler-Bunte-Institute and its application to the jet in crossflow configuration shows that it is able to deliver high quality correlated velocity-scalar data for this system under highly turbulent conditions (Galeazzo et al., 2011). The high-resolution measurements enabled the quantification of the known weaknesses of steady-state simulations in cases where large-scale structures dominate the turbulent mixing. 2. Experimental setup Some of the experimental results presented in this work were already published in Galeazzo et al. (2011). In contrast to other literature data which employs mostly laminar or weakly turbulent flows to study the JIC, these experiments were conducted under highly turbulent conditions. Table 1 shows the flow boundary conditions. The velocity ratio R is defined as the square root of the
0142-727X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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was used for the PIV evaluation. The statistics were evaluated from a total of 6200 samples per measuring area.
Table 1 Boundary conditions. Crossflow
Jet
Inlet Bulk velocity Turbulence intensity Re
9.08 m/s 1.5% 6.24 104
Inlet Bulk velocity Turbulence intensity Re
37.72 m/s 7% 1.92 104
momentum ratio R ¼ ðqjet U 2jet =qcross U 2cross Þ1=2 , and takes the value 4.15 for the current flow configuration. The experimental facility consists of a channel with square cross section (108 108 mm) in which a round jet (inner diameter D = 8 mm) is mounted flush to the wall. The pipe used to feed the jet is long enough to ensure a fully developed velocity profile. The center of the jet is placed 328 mm downstream of the beginning of the channel, where a plug flow velocity profile is generated by a specially built contraction nozzle. Optical access to the channel is given by four fused quartz windows placed at each side of the test section. The measurement technique is illustrated in Fig. 1. It consisted of a combination of two laser diagnostic methods: Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF). A frequency doubled Nd: YAG laser with an excited wavelength of 532 nm is used as light source. A Galilean telescope fans out the laser light beam, which is then guided through the test section. This light sheet provides two different signals in the mixing region in the JIC arrangement. The scattered light (PIV) comes from aerosol particles added to both flows at a wavelength of 532 nm, while the fluorescence signal (LIF) is generated by excited NO2 molecules added only to the jet flow at a range of wavelengths between 550 nm and 690 nm (Gulati and Warren, 1994). These two signals are spectrally separated and acquired by two different cameras. A CCD-camera (DantecÒ 80C60 HiSense PIV-Camera, 1280 1024 pixels) is used for the scattered light while an intensified CCD-camera (Roper ScientificÒ 512 512 pixels) is used for the fluorescence light. A commercial program (Dantec FlowManagerÒ Version 1.10) was used to post-process the PIV signals. The LIF signals were postprocessed using a program developed in-house. Both signals were acquired simultaneously, allowing spatial measurements of the instantaneous velocity and concentration fields and their correlation. An imaging area of 27.3 27.3 mm was used to completely resolve the flow phenomena including the high velocity gradients. An interrogation area of 32 32 pixels with an overlap of 50%
3. Numerical setup The simulations were performed using the open-source CFD toolbox OpenFOAM version 2.0 (OpenCFD, 2011). The computational domain was chosen in a way to save computational time, while capturing all important phenomena of the JIC. The domain extended 100 mm in the upstream crossflow direction and 200 mm in the downstream direction, as depicted in Fig. 2. The flow was modeled as incompressible. Three simulation approaches for turbulence have been studied: large-eddy simulation (LES), unsteady Reynolds-averaged Navier Stokes (URANS) and steady-state RANS. For the LES, a dynamic version (Lilly, 1992) of the Smagorinsky sub-grid scale turbulence model (Smagorinsky, 1963) was employed, while the SST turbulence model (Menter, 1994) was used for the RANS simulations. For the RANS simulations, the turbulent Schmidt number rt is constant and equal to 0.9, while the LES employed rt equal to 1.0, when not otherwise noted. The Reynolds stress tensor ðu0i u0i Þ of the RANS simulations has been calculated using the Boussinesq approximation (Wilcox 1998), with the Reynolds flux vector ðu0i c0 Þ being calculated in the same way, using a turbulent diffusivity Dt and the mean scalar gradient. As the SST turbulence model does not predict the root mean square value of the passive scalar crms directly, the following transport equation for the variance of the passive scalar cvar = (crms)2 has been used
@ðqcv ar Þ ~ i cv ar Þ rðqDt rcv ar Þ þ rðqu @t
e
¼ C 1 qDt rcv ar rcv ar C 2 q cv ar ; k
ð1Þ
with C1 = 2.8 and C2 = 2. Tilde denotes a Favre-averaged quantity. For the URANS simulation the contributions of the turbulence modeling have been summed up with the contribution of the velocity and passive scalar fluctuations calculated directly during the simulation. In contrast to RANS and URANS, the LES resolves the largest part of the turbulent fluctuations directly. Nevertheless, the small fraction of the turbulence being modeled by the sub-grid scale model has been added to all appropriate results. The convection terms in the RANS simulations have been discretized using the Gamma scheme developed by Jasak (1996). Being based on the central differencing scheme, it preserves overall second-order accuracy. In the LES, the filtered linear scheme
Fig. 1. Overview of the 2D-PIV/LIF measurement technique.
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
F.C.C. Galeazzo et al. / International Journal of Heat and Fluid Flow xxx (2013) xxx–xxx
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Fig. 2. Overview of the computational domain with dimensions and the coordinate system employed, top. The fine grid at the symmetry plane near the jet inlet, bottom.
(OpenCFD, 2011) has been used for the momentum equation, which is a low dissipation, second order scheme based on central differencing. The Gamma scheme was employed in the transport equation of the passive scalar in LES context, with the blending factor kept to the minimum value of 0.1 as proposed by Jasak (1996), limiting the adverse effects of the added numerical diffusion. The pressure correction was performed using the PISO algorithm proposed by Issa (1986). For the unsteady simulations the time was discretized using a second-order accurate backward method. The averaging time was 0.15 s, which translates into 170 characteristic time units D/Ucross. Two grids have been developed for the simulations, employing hexahedral-shaped elements: a coarse grid composed of 1.5 million elements, with the diameter of the jet inlet having a resolution of 27 elements, and a fine grid composed of 7.5 million elements, with an increased jet inlet resolution of 39 elements (see Fig. 2). A no-slip boundary condition was applied at the walls, with typical y+ values at the crossflow walls about 15 for the coarse and 12 for the fine grid, and at the pipe walls about 12 for the two grids. The grid dependence of the results was explored. Fig. 3 shows line plots of the mean velocity component U, mean passive scalar C, specific Reynolds stress component u0 u0 and root mean square of the passive scalar crms at the symmetry plane (y/D = 0) and axial position x/D = 1.0 for RANS simulations using the coarse and fine grids. Profiles of U and u0 u0 from Galeazzo et al. (2011) for the same conditions are also included, which employed a very fine tetrahedral grid with more than 10 million elements and element sizes in the near-jet region equivalent to 1/40D. The velocity profiles are in good agreement with each other, with a small velocity deficit in the simulation using the coarse grid. The same trend could be observed for the Reynolds stress, with good agreement between the fine grid and the results from Galeazzo et al. (2011), while the coarse grid influenced negatively the results. Analyzing the profiles of mean passive scalar and passive scalar rms showed that the coarse grid is also impacting the results of these variables. The
Fig. 3. Line plots of mean velocity component U/Ucross, mean passive scalar C, specific Reynolds stress component u0 u0 =U 2cross and passive scalar rms crms at the symmetry plane (y/D = 0) and x/D = 1.0 for RANS simulations using different grids and results from Galeazzo et al. (2011).
fine grid, producing results comparable to the grid from Galeazzo et al. (2011) with the advantage of being composed of hexahedral elements, was used in the remaining work for all simulations. The jet trajectory in the JIC is influenced by the velocity profiles of the jet and the crossflow (Muppidi and Mahesh, 2005). In this work, the JIC fills a large part of the channel cross section, which increases the importance of the description of the crossflow boundary layer, as the velocity in the middle of the channel increases as the flow develops. Unfortunately, the crossflow boundary layer thickness could not be resolved using laser diagnostics because of the interference caused by the laser light reflected by the walls. Galeazzo et al. (2011) employed an alternative method to evaluate the development of the boundary layer, using the fact that the developing velocity profile of the crossflow depends on
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 4. Profiles of mean velocity components U/Ucross and W/Ucross and specific Reynolds stress components u0 u0 =U 2cross and w0 w0 =U 2cross , 1D above the jet exit, z/D = 1.0, y/D = 0.
the growth of the boundary layer. RANS simulations over the full length of the channel were compared to measurements using Laser Doppler Velocimetry (LDV) of the crossflow velocity at the center of the channel along the x direction. The profiles are in good agreement, indicating that the growth of the boundary layer was simulated correctly. To include the correct boundary layer thickness in the current simulations using the computational domain depicted in Fig. 2, velocity profiles of RANS simulations over the full length of the channel have been applied at the inflow boundaries, naturally including the predicted boundary layer thickness. The jet velocity and turbulence profiles are depicted in Fig. 4, which shows comparisons between the measurements and simulations 1D above the jet inlet (z/D = 1) and varying ±1D along the xdirection. The measurements are consistent, showing peaks of specific Reynolds stress components in the same regions where large mean velocity gradients occur. The velocity profiles of the unsteady simulations (LES and URANS) show a good agreement with the measurements, while the RANS simulation predicted the jet to be narrower than in the measurements. The turbulence is highly anisotropic in this region, with the peak values of the w0 w0 Reynolds stress component being almost three times higher than the u0 u0 component. Anisotropic turbulence is naturally simulated using the LES formulation, which is reflected in the good agreement of the predicted Reynolds stress levels. On the other hand, the RANS and URANS simulations had difficulties predicting the w0 w0 level correctly. The description of turbulence at the inlet boundary conditions for LES is a known challenge. For that purpose a boundary condition based on the work of Klein et al. (2003) has been used. Turbulent fluctuations have been applied to the mean velocity profiles at the inflow boundaries of the LES. The spurious pressure fluctuations generated at the inlet are one drawback of this approach. The solution is to apply a correction to the velocity field, to ensure that the mass flow remains constant. In every time step, after the turbulence fluctuations are superimposed on the mean velocity profiles, the mass flow is evaluated and compared to the reference mass flow. Then the velocity field is multiplied by a factor, typically smaller than 0.5%, correcting the value of the mass flow. With this correction, the pressure fluctuations were drastically reduced, followed by a significant improvement in the stability of the simulation. The boundary conditions of the mixing were defined with the pure jet flow having a passive scalar concentration C value of unity, while the pure crossflow has a value of zero. 4. Intermittency One of the characteristics of the jet in crossflow is that it has intermittent regions, meaning that in these regions fluid from the jet and from the crossflow can be found in an alternating matter.
Intermittency is present in all turbulent free shear flows (Libby 1996). The quantitative description of the intermittency is possible using the intermittency function I(xi, t), which assumes value unity in the jet flow, and zero in the crossflow. The first applications of the intermittency function were to distinguish between non-turbulent and turbulent flows, as in a free jet that flows into quiescent surroundings (Libby 1996). In this case, the intermittency function could be determined from a careful interpretation of velocity signals, as velocity fluctuations could be related to the jet flow. In contrast, the jet in crossflow is composed of two turbulent flows, making the discrimination strategy based on velocity fluctuations more subject to ambiguity. An alternative definition of the intermittency function is possible, as the jet flow can be associated with fluid having finite values of tracer concentration. Defining also a small threshold value for the concentration, an intermittency function I(xi, t) has been evaluated without ambiguity. The function I(xi, t) assumes value unity when the tracer value is higher than the threshold, and zero otherwise. As observed by Anderson et al. (1979), the value of the intermittency function is found to be relatively insensitive to the precise value of the threshold value within a reasonable range. They have studied a turbulent helium jet in an air coflow, with threshold values varying between C = 0.0075 and 0.026, and found a very limited effect on the evaluation of the intermittency. For simplicity, a fixed threshold value of C = 0.01 has been used in this work. The intermittency factor c(xi) represents the fraction of time a point is in the jet fluid, and is defined as the time average of I(xi, t)
cðxi Þ ¼ Iðxi ; tÞ:
ð2Þ
5. Results The ability of the different simulation strategies in predicting the formation and evolution of coherent structures is depicted in Fig. 5, showing the isosurface of C = 0.01 in a single time step of the LES and of the URANS and for the steady-state result of the RANS. The isosurface of the LES is very contorted, indicating a high turbulence level. However, it is difficult to discern any specific coherent structure in this picture. Large-scale structures have been also predicted by the URANS, resulting in the sinuous isosurface observed in Fig. 5. It is evident, however, that the fluctuation level is much lower in comparison to the LES. The steady-state RANS simulation does not predict large-scale structures at all, which results in a smooth passive scalar distribution. The two dimensional plots in Fig. 6 are a good starting point for the discussion of the results. In this figure, PIV measurements and the results of the simulations for the mean velocity component U and the specific Reynolds stress component u0 u0 at the symmetry plane (y/D = 0) are shown. The jet is mounted in the z-direction, so the jet fluid that enters in the channel has a very high velocity component (W) in this direction and a nearly zero velocity component in crossflow direction (U). As the jet flows into the crossflow, it is bent in the crossflow direction and U increases, creating the region of high U situated at x/D = 1 and z/D = 4. With the development of the jet, crossflow fluid is entrained, and U decreases continuously. On the lee side of the jet, a recirculation zone is formed. This region has negative U values, which is represented by the blue color in the contours. It is interesting to note that the recirculation region and the region with maximum U are very close to each other. The velocity gradient in this region is therefore very high, and very high production of turbulence is expected. The contour of u0 u0 confirms this expectation, with the locus of maximum u0 u0 situated in the region of high velocity gradient. Comparing the results of the simulations with the measurements in Fig. 6, it is clear that the LES has the better agreement. Both velocity and turbulence fields are very well reproduced,
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 5. Isosurface of passive scalar, C = 0.01. From top to bottom: LES, URANS and RANS simulations.
having the field maxima at the same position and having almost the same intensity as acquired in the measurements. Although the position of the jet is also reproduced well by the other type of simulations, the maximum values of U are slightly underpredicted. The predictions of u0 u0 , on the other hand, differ substantially between the different simulations. While the measurements show a maximum value of u0 u0 =U 2cross at the symmetry plane of 1.89, the prediction of the LES is 1.60, of the URANS is 1.03 and of the RANS simulation is 1.02. This fact can be explained by the presence of large-scale coherent structures in the jet in crossflow. The measurements were evaluated using simple time averages, so that the Reynolds stress tensor represents the whole velocity fluctuation about the time averaged mean velocity field. This implies that these fluctuations contain the contribution of two different phenomena: the turbulent fluctuations and the unsteadiness of the mean flow about the time-average created by the coherent structures, if such structures exist. The very good agreement of the fluctuation level of the LES with the measurements confirms the ability of the LES of solving the major coherent structures of the jet in crossflow. It is argued that the reason for the low u0 u0 levels predicted by the URANS simulation is the insufficient description of the coherent structures. This simulation could resolve a fraction of the unsteady character of the flow, however a significant fraction remained unresolved. It is the contribution of these unresolved coherent structures that makes the predicted fluctuation level to be lower than the measurements. As the coherent struc-
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tures are not resolved at all by the steady-state RANS simulation, its fluctuation level contains only the contribution of the turbulence and is consequently lower than both unsteady simulations and measurements. Line plots allow the quantitative comparison between measurements and simulations. Fig. 7 shows line plots of the mean velocity component U/Ucross and the specific Reynolds stress component u0 u0 =U 2cross at the symmetry plane, y/D = 0. One of the most important parameters of the jet in crossflow for engineering applications is the jet penetration, which is defined in this work as the locus of maximum mean velocity component U. The jet penetration is well represented by all simulations. Comparing the profiles for U in more detail, the LES shows very good agreement in all positions, even near the bottom wall, a region known to be difficult to simulate accurately due to the recirculating flow present there. The two RANS simulations show similar results, with velocity magnitudes that are slightly lower than the measured ones in the region where the jet is located. The specific Reynolds stress component u0 u0 , on the other hand, shows a different picture. The agreement of the LES results with the measurements is very good in both magnitude and location, with only a slightly overpredicted u0 u0 level at x/D = 0. The RANS and URANS simulations show values that are consistently lower than both measurements and LES. Fig. 8 shows two dimensional plots of the passive scalar C, the intermittency factor c and the root mean square value of the passive scalar crms for the measurements and simulations at a horizontal plane, z/D = 1.5. Unfortunately, experimental data about the intermittency factor c is not available. The agreement of C predicted by the LES is very good, both in shape and level. The inner side of the kidney shape fits very well to the experimental data. Considering the URANS simulation, the gradients of the scalar are steeper than in the LES, which makes the agreement with the measurements slightly worse. The kidney shape of the jet agrees well to the experimental data. One part of the differences in the prediction of C can be explained by the contours of c. The c function represents the fraction of time in which each point remains in the jet fluid, and its distribution is closely related to the amount of largescale coherent structures which have been resolved by the simulation (Libby, 1996; Pope, 2000). Steady-state RANS simulations do not resolve coherent structures at all, and the c profiles assume the shape of a step function, because in this case the intermittency factor c is equal to the intermittency function I, assuming the value zero when C is smaller than 0.01 and the value unity otherwise. The LES is able to resolve a larger fraction of the large-scale coherent structures than the URANS, which results in c contours that are distributed over a larger area than the contours of the URANS simulation; in other words, the LES predicts less steep gradients of c than the URANS, which can also be seen in the profiles in Fig. 9. The agreement between the predictions of the LES and the URANS is worse exactly in the regions with poor agreement of c. The results of the steady-state RANS simulation are close to the results of the URANS, with the agreement of the scalar field C being not as favorable as was shown for the velocity fields. The core flow is reasonably well predicted, however with a more pronounced inner side of the kidney shape. This difficulty is expected, as the steadystate RANS simulation does not resolve coherent structures and the influence of the intermittency is more pronounced in this region. The crms is predicted by the LES in very good agreement with the measurements in both level and shape. The agreement of the URANS and the RANS results is satisfactory, however with very low levels of crms in the jet core, which does not correspond to the measurements. The peak levels predicted by the URANS are slightly higher than the RANS, which can also be seen in Fig. 9 at z/D = 1.5. The turbulent mixing can be analyzed in more detail using the line plots at the symmetry plane (y/D = 0) of the mean passive
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 6. Two dimensional maps of mean velocity component U/Ucross and specific Reynolds stress component u0 u0 =U 2cross at the symmetry plane, y/D = 0. From top to bottom: PIV measurements, LES, URANS and steady-state RANS.
scalar C and the intermittency factor c, the root mean square value of the passive scalar crms and the u0 c0 component of the specific Reynolds flux vector depicted in Fig. 9 for three different z/D positions (1.5, 3 and 4.5). The C profiles show more pronounced differences than the mean velocity depicted in Fig. 7 for the same locations. The LES simulation shows good agreement with the measurements, with slightly underpredicted peak scalar values at z/D = 3 and 4.5. The C profiles of the RANS and URANS simulations agree well, although not as good as the profiles of the mean velocity in Fig. 7 do. The position of the jet shows good agreement, however the profiles of the simulations are slightly narrower than the one of the measurements. The LES shows less steep gradients of c than the URANS, which is particularly evident at z/D = 4.5. Comparing the c and C profiles, it can be seen that the broader C distribution of the LES correlates very well with the also broader c distribution. Comparing the RANS simulations, the C profile show good agreement with the measurements at z/D = 1.5, while at z/ D = 3 and 4.5 it had problems predicting the jet borders correctly. The LES predicts crms in good agreement especially at z/D = 1.5, with the two peaks at the jet boundaries being clearly discerned. However, the overall level is lower than in the measurements,
despite the fact that the Reynolds stresses and fluxes are well represented. The peak levels of crms predicted by the URANS and RANS simulations agree well with the measurements, while the agreement of the shape of the profiles is not as favorable. The LES slightly overpredicts u0 c0 at z/D = 1.5, while for the other two locations the agreement is good. For the other simulations the agreement of u0 c0 is good, however the location of the peaks at z/D = 3 and 4.5 are slightly moved in the downstream direction in the steady-state simulation. The comparison with the experimental data has shown that, while the mean velocity is well predicted by all simulations, the prediction of the mean passive scalar C depends heavily on the level of complexity of the simulation. The agreement of C is very good using the LES, and degrades significantly when moving to URANS and RANS simulations. This different behavior is attributed to the different way that the velocity and the passive scalar react to the presence of the jet. When portions of ambient fluid are entrained by the jet flow, they are accelerated before effectively meeting the jet flow, leading to velocity fluctuations. These fluctuations, that are not turbulent in nature, are induced by the pressure fluctuations associated with the turbulence (Libby, 1996), which
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 7. Line plots of mean velocity component U/Ucross and specific Reynolds stress component u0 u0 =U 2cross at the symmetry plane, y/D = 0.
explains their different behavior in flows dominated by large-scale structures. Fig. 10 shows a planar cut along the symmetry plane of a single time step of the LES. The jet flow region is marked with gray color, while the color of the ambient fluid is white. The boundary between the jet flow and the ambient fluid is marked by the isocontour of C = 0.01, the same definition used to calculate the intermittency function. The boundary is highly convoluted, which is a result of the various coherent structures of the flow. A small region is shown in more detail, with arrows representing the local velocity vector. In the top right hand corner of the detail, a region of ambient fluid (white) is being entrained by the jet flow (gray). The entrained region is characterized by the velocity vectors, which have a component in the vertical direction. In contrast, in other areas away from the entrained region, the velocity vectors of the ambient fluid have components mainly in the horizontal direction. Although the difference in the velocity magnitude is not very pronounced, the velocity vectors of the ambient fluid in the entrained region show that the fluid has been accelerated in the vertical direction. This finding corroborates with the results of Anderson et al. (1979) and Pope (2000), which show that the entrained fluid develops higher velocity than the surrounding fluid, even before being effectively entrained. It follows that the velocity gradient across the interface between the two flows is smaller than the scalar gradient, which explains their different behavior in flows dominated by large-scale structures. The difference between the gradients of the velocity and of the passive scalar can be quantified analyzing the diffusion across the superlayer, the instantaneous interface between the two flows, as the diffusion is directly related to the gradients. In order to calculate the diffusion from the LES, the superlayer was approximated by the surface determined by the isosurface of 1% passive scalar, which is the same used to define the intermittency function (see Fig. 5). To allow a direct comparison, the instantaneous velocity component w, non-dimensioned by the bulk jet velocity Ujet, is employed along the instantaneous passive scalar c, as both are transported into the crossflow by the jet and have values spanning between zero and one. The diffusion flux across the superlayer of velocity Jw and of the passive scalar Jc are defined by the dot product of the gradients and the normal vector ~ n
@ðw=U jet Þ ~ n; @xi
ð3Þ
@c ~ n: @xi
ð4Þ
Jw ¼ m
Jc ¼ m
Integrating over the whole isosurface of a single time step of the LES resulted in a diffusion flux of velocity Jw of 2.64 106 m3/s and a diffusion flux of passive scalar Jc of 26.59 106 m3/s, a clear indicator that the passive scalar gradient is bigger than the velocity gradient across the superlayer. Another important parameter in the analysis of turbulent mixing is the turbulent Schmidt number rt. Within the RANS framework, there are two quantities that describe the turbulent diffusivity of momentum and passive scalar: the eddy viscosity mt and the turbulent diffusivity Dt, respectively. The eddy viscosity is the factor of proportionality between the specific Reynolds stress tensor u0i u0j and the mean strain-rate tensor Sij through the Boussinesq approximation (Wilcox, 1998)
2 u0i u0j ¼ 2mt Sij kdij ; 3
ð5Þ
where k is the turbulent kinetic energy and dij is the Kronecker delta. In the same way, the turbulent diffusivity Dt is the factor of proportionality between the specific Reynolds flux vector u0i c0 and the mean scalar gradient
u0i c0 ¼ Dt
@C ; @xi
ð6Þ
The ratio between the two diffusivities is the turbulent Schmidt number
rt ¼ mt =Dt :
ð7Þ
One approach often used to enhance the mixing in RANS simulations is to decrease the standard value of the turbulent Schmidt number from 0.9 and consequently scale the Reynolds flux vector with it. Fig. 11 shows the results of three simulations using the SST turbulence model, with rt of 0.3, 0.5 and 0.7. The passive scalar field of the simulation using rt = 0.3 does not have the steep
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 8. Two dimensional maps of mean passive scalar C and the intermittency factor c, z/D = 1.5. From top to bottom: LIF measurements, LES, URANS and RANS.
gradients of the standard simulation using rt = 0.9, which is consistent with the increased turbulent mixing. The above changes to rt affect the whole field simultaneously. The mixing in the jet core is well represented using the value of 0.9 while the borders appear to be better predicted using a lower value. This finding is consistent to the fact that the intermittency is higher at the jet borders; the deviations are expected to be higher in this region. It is tempting to apply the definitions of Eqs. (5) and (6) to the experimental data and to the LES simulation and calculate rt, as both evaluate the Reynolds stresses and fluxes along with the gradients of the mean quantities. However, the application of these definitions is not straightforward. Within the RANS framework the eddy viscosity is a scalar quantity, defined in Eq. (5). The turbulent diffusivity (Eq. (6)) and the turbulent Schmidt number (Eq. (7)) are also scalar quantities. On the other hand, when evaluating the results of the measurements, it becomes clear that the eddy viscosity assumes a different value for each component of the Reynolds stress tensor, being in fact a tensor quantity. The same applies to the turbulent diffusivity and to the turbulent Schmidt number, which are in fact vector quantities. The comparison of these tensor and vector quantities with the scalar results of RANS models demands great care.
Despite the shortcomings, much can be learned when analyzing the results of the measurements and the LES. The two-dimensional experimental data allows the calculation of one pair of eddy viscosity and turbulent diffusivity. For example, at the symmetry plane (y/D = 0), the Reynolds stress component u0 w0 and the Reynolds flux component u0 c0 , together with the gradients of the mean quantities, can be evaluated and the turbulent Schmidt number in the xdirection can be calculated as
rt;x
u0 w0 ¼ @U @W þ @x @z
,
u0 c0 @C @x
:
ð8Þ
Fig. 12 shows the component of rt,x obtained from the measurements and from the LES results at the symmetry plane together with the axial evolution at three vertical positions. To improve the readability of the picture, the plots were limited to regions with values of C between 0.05 and 0.95 and with absolute values of oC/ox above 0.04. The major difficulty in comparing the turbulent Schmidt numbers from the measurements or the LES with the definition from the RANS framework is actually the fact that the turbulence is not the only source of fluctuation in the jet in crossflow. Eqs. (5)
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Fig. 9. Line plots of passive scalar concentration C, intermittency factor c, passive scalar rms crms and specific Reynolds flux component u0 u0 =U 2cross at the symmetry plane, y/ D = 0.
Fig. 10. Symmetry plane (y/D = 0) of one realization of the LES. The solid line represents the boundary between jet flow and the ambient fluid. The detail shows also arrows representing the local velocity vector.
and (6) are derived assuming that turbulence is the only source of fluctuation in the flow, as all equations in RANS context. This is clearly not the case in the jet in crossflow, especially in regions where the coherent structures have already developed. The exper-
imental results support this argument. At the jet root, near the nozzle, the coherent structures have not evolved sufficiently to affect the flow. In this region rt,x assumes values between 0.3 and 1.3, which agrees well to the values recommended in the literature.
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Fig. 11. Two dimensional maps of mean passive scalar C, z/D = 1.5, for simulations using the SST turbulence model and turbulent Schmidt numbers rt of 0.3, 0.5 and 0.7.
Fig. 12. Two dimensional maps and line plots of turbulent Schmidt number component rt,x at the symmetry plane, y/D = 0, for the measurements (a) and the LES (b).
Further downstream the influence of the coherent structures is more pronounced, and the values of rt,x begin to fluctuate between very high and very low values, which is indicated by the sudden appearance of red and blue regions in the contour plots from z/D = 3 and downstream. This finding leads to the conclusion that, in the presence of coherent structures, Eqs. (5) and (6) do not adequately represent the phenomena present there.
6. Conclusions The results of different simulations, ranging from simple steadystate Reynolds-averaged Navier Stokes (RANS) to more sophisticated unsteady RANS and large-eddy simulation (LES), have been
compared to high quality experimental data acquired using Particle Image Velocimetry combined with Laser Induced Fluorescence. The quality of the agreement of the simulation results with the measurements is strongly coupled with the description of the coherent structures. The agreement of the LES results with the measurements is very good for both mean variables and fluctuation values. The unsteady RANS simulation using the SST turbulence model could resolve only a fraction of the coherent structures. The resulting fluctuation levels are systematically lower than the measurements, even though the agreement of the mean variables is good. The analysis of the results shows that the agreement of the LES results is better than the URANS results not only because the turbulence is better reproduced, but also because the LES simulated the coherent structures in much more detail. The steady-state RANS
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simulations do not resolve the coherent structures at all, neglecting this important source of unsteadiness. It follows that the fluctuation levels predicted by the steady-state simulations are significantly lower than the measurements. The agreement of the mean velocity is better than the mean passive scalar for the steady-state simulations. This fact could be explained by the intermittency, which has a more pronounced influence on the velocity than in the passive scalar. A detailed analysis has shown that in the flow regions dominated by coherent structures, the turbulent Schmidt number rt do not adequately represent the mixing phenomena present there. Different values of rt are needed to represent well the mixing in the jet core and along the jet borders, which is consistent to the fact that the intermittency is higher at the borders. The results have shown that steady-state RANS simulations provide good quantitative and qualitative agreement with experimental data when the flow is statistically stationary, i.e., when the influence of large-scale coherent structures is negligible. However, when the flow is not statistically stationary, with pronounced large-scale coherent structures, Reynolds averaged values do not converge to their time-averages. As the coherent structures are not turbulent in nature, their influence on the mean flow is not modeled by the turbulence models. Hence, to achieve high-fidelity results for all variables, time dependent simulations are mandatory, which increases the computational cost substantially. Fortunately, the increase in the computational cost of the simulations translates into higher accuracy, especially for the turbulent mixing. Acknowledgements This project is a research commissioned by the DLR for the state of Baden-Württemberg, Germany, which is funded by the Landesstiftung Baden-Württemberg gGmbH. One part of the computation time was kindly provided by the Steinbuch Centre for Computing (SCC) of the Karlsruhe Institute of Technology (KIT). References Alvarez, J., Jones, W.P., Seoud, R., 1993. Predictions of momentum and scalar fields in a jet in cross-flow using first and second order turbulence closures. In:
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AGARD, Computational and Experimental Assessment of Jets in Cross Flow. AGARD CP-534, Winchester, UK, 24. Anderson, P., LaRue, J.C., Libby, P.A., 1979. Preferential entrainment in a 2dimensional turbulent jet in a moving stream. Phys. Fluids 22 (10), 1857– 1861. Andreopoulos, J., Rodi, W., 1984. Experimental investigation of jets in a crossflow. J. Fluid Eng. 138, 93–127. Cárdenas, C., Suntz, R., Denev, J., Bockhorn, H., 2007. Two-dimensional estimation of Reynolds-fluxes and – stresses in a jet-in-crossflow arrangement by simultaneous 2D-LIF and PIV. Appl. Phys. B-Lasers Opt. 88 (4), 581–591. Cárdenas, C., Suntz, R., Bockhorn, H., 2010. Experimental investigation of the mixing-process in a jet-in-crossflow arrangement by simultaneous 2d-LIF and PIV. In: Mewes, D. et al. (Eds.), Springer series on Heat and Mass Transfer, Micro and Macro Mixing. Springer Verlag, Berlin, Heidelberg, pp. 87–103. Galeazzo, F.C.C., Donnert, G., Habisreuther, P., Zarzalis, N., Valdes, R.J., Krebs, W., 2011. Measurement and simulation of turbulent mixing in a jet in crossflow. J. Eng. Gas. Turb. Power 133 (6), 061504.1–061504.10. Gulati, A., Warren, R., 1994. NO2-based laser induced fluorescence technique to measure cold-flow mixing. J. Propul. Power 10 (1), 54–61. Holdeman, J.D., Liscinsky, D.S., Oechsle, V.L., Samuelsen, G.S., Smith, C.E., 1997. Mixing of multiple jets with a confined subsonic crossflow: Part I cylindrical ducts. J. Eng. Gas. Turb. Power 119 (4), 852–862. Issa, R.I., 1986. Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62, 40–65. Jasak, H., 1996. Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flows. PhD Thesis. Imperial College of Science, Technology and Medicine. Klein, M., Sadiki, A., Janicka, J., 2003. A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations. J. Comput. Phys. 186, 652–665. Libby, P.A., 1996. Introduction to Turbulence. Taylor & Francis. Lilly, D.K., 1992. A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids A 4 (3), 633–635. List, E.J., 1982. Turbulent jets and plumes. Annu. Rev. Fluid Mech. 14, 18–212. Margason, R.J., 1993. Fifty years of jet in cross flow research. In: AGARD, Computational and Experimental Assessment of Jets in Cross Flow. AGARD CP-534, Winchester, UK. Menter, F.R., 1994. Two-equation eddy-viscosity models for engineering applications. AIAA J. 32, 1598–1605. Muppidi, S., Mahesh, K., 2005. Study of trajectories of jets in crossflow using direct numerical simulations. J. Fluid Mech. 530, 81–100. Ooms, G., 1967. A new method for the calculation of the plume of gases emitted by a stack. Atmos. Environ. 6 (12), 899–909. OpenCFD Ltd., 2011. OpenFOAM User Guide, Version 2.0.0. Pope, S.B., 2000. Turbulent Flows. Cambridge University Press. Smagorinsky, J., 1963. General circulation experiments with the primitive equations: I. The basic equations. Mon. Weather Rev. 91, 99–164. Su, L.K., Mungal, M.G., 2004. Simultaneous measurements of scalar and velocity field evolution in turbulent crossflowing jets. J. Fluid Mech. 513, 1–45. Wilcox, D.C., 1998. Turbulence Modeling for CFD. DCW Industries, Inc.
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Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Computational modeling of turbulent mixing in a jet in crossflow Flavio Cesar Cunha Galeazzo a,⇑, Georg Donnert a, Camilo Cárdenas a, Julia Sedlmaier a, Peter Habisreuther a, Nikolaos Zarzalis a, Christian Beck b, Werner Krebs b a b
Karlsruhe Institute of Technology, Engler-Bunte-Institute, Division of Combustion Technology, Engler-Bunte-Ring 1, 76131 Karlsruhe, Germany Siemens AG, Mellinghofer Str. 55, 45466 Muelheim an der Ruhr, Germany
a r t i c l e
i n f o
Article history: Available online xxxx Keywords: Turbulent mixing RANS URANS LES Intermittency
a b s t r a c t The jet in crossflow is a configuration of highest theoretical and practical importance, in which the turbulent mixing plays a major role. High-resolution measurements using Particle Image Velocimetry combined with Laser Induced Fluorescence have been conducted and used to validate simulations ranging from simple steady-state Reynolds-averaged Navier Stokes to sophisticated large-eddy simulation. The reasons for the erratic behavior of steady-state simulations in the given case, in which large-scale structures dominate the turbulent mixing, have been discussed. The analysis of intermittency proved to be an appropriate framework to account for the influence of these flow structures on the jet in crossflow, contributing to the explanation of the poor performance of the steady-state simulations. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Turbulent mixing plays an important role in the dynamics of turbulent flows, with many consequences in nature and engineering. The simplest case of turbulent mixing occurs between passive scalars. Examples are mixing of fluids with the same density, or loaded with non-reacting tracers. Simulation of mixing in turbulent flows is of great importance from theoretical and practical points of view. The increasing computational resources allow the simulation of larger, more complex systems in shorter timeframes. The dissemination of this technology comes with new challenges, as the increased focus on the robustness, accuracy and predictive capabilities of the models, which are faced with more complex systems and phenomena. The jet in crossflow (JIC) is a flow configuration with features like large-scale coherent structures and recirculation regions. Due to its good mixing capability, this flow configuration is frequently found in technical devices like mixers in the process industry, plumes from chimneys, engine exhaust gas pipes and gas turbines. Investigations on the JIC, focusing in mixing of chimney plumes, started in the 1930s (Ooms, 1967; List, 1982). A more systematic analysis of the JIC started in the 1970s, culminating in the detailed measurements of Andreopoulos and Rodi (1984) using a three sensor hot-wire probe. Design correlations known for their practical relevance have been developed by Holdeman et al. (1997). Alvarez et al. (1993) has shown that steady-state simulations predict the ⇑ Corresponding author. Tel.: +49 721 608 42808 (Germany); fax: +49 721 608 43805. E-mail address: [email protected] (F.C.C. Galeazzo).
turbulent mixing of the JIC in poor agreement with experimental data. Margason (1993) reviews numerous investigations of the JIC configuration, much of them focusing on the complex system of vortices and their contribution to the stability of the flow field. Laser diagnostic tools have made tremendous progress in recent years, also benefiting studies about the JIC. Some contemporary references employing laser diagnostics on the JIC have been performed under weakly turbulent conditions, with low Reynolds numbers that do not represent the majority of the applications of the JIC (Su and Mungal, 2004; Cárdenas et al., 2007, 2010). The newly developed high-resolution laser diagnostic system using simultaneously Particle Image Velocimetry and Laser Induced Fluorescence has been realized at the Engler-Bunte-Institute and its application to the jet in crossflow configuration shows that it is able to deliver high quality correlated velocity-scalar data for this system under highly turbulent conditions (Galeazzo et al., 2011). The high-resolution measurements enabled the quantification of the known weaknesses of steady-state simulations in cases where large-scale structures dominate the turbulent mixing. 2. Experimental setup Some of the experimental results presented in this work were already published in Galeazzo et al. (2011). In contrast to other literature data which employs mostly laminar or weakly turbulent flows to study the JIC, these experiments were conducted under highly turbulent conditions. Table 1 shows the flow boundary conditions. The velocity ratio R is defined as the square root of the
0142-727X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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was used for the PIV evaluation. The statistics were evaluated from a total of 6200 samples per measuring area.
Table 1 Boundary conditions. Crossflow
Jet
Inlet Bulk velocity Turbulence intensity Re
9.08 m/s 1.5% 6.24 104
Inlet Bulk velocity Turbulence intensity Re
37.72 m/s 7% 1.92 104
momentum ratio R ¼ ðqjet U 2jet =qcross U 2cross Þ1=2 , and takes the value 4.15 for the current flow configuration. The experimental facility consists of a channel with square cross section (108 108 mm) in which a round jet (inner diameter D = 8 mm) is mounted flush to the wall. The pipe used to feed the jet is long enough to ensure a fully developed velocity profile. The center of the jet is placed 328 mm downstream of the beginning of the channel, where a plug flow velocity profile is generated by a specially built contraction nozzle. Optical access to the channel is given by four fused quartz windows placed at each side of the test section. The measurement technique is illustrated in Fig. 1. It consisted of a combination of two laser diagnostic methods: Particle Image Velocimetry (PIV) and Laser Induced Fluorescence (LIF). A frequency doubled Nd: YAG laser with an excited wavelength of 532 nm is used as light source. A Galilean telescope fans out the laser light beam, which is then guided through the test section. This light sheet provides two different signals in the mixing region in the JIC arrangement. The scattered light (PIV) comes from aerosol particles added to both flows at a wavelength of 532 nm, while the fluorescence signal (LIF) is generated by excited NO2 molecules added only to the jet flow at a range of wavelengths between 550 nm and 690 nm (Gulati and Warren, 1994). These two signals are spectrally separated and acquired by two different cameras. A CCD-camera (DantecÒ 80C60 HiSense PIV-Camera, 1280 1024 pixels) is used for the scattered light while an intensified CCD-camera (Roper ScientificÒ 512 512 pixels) is used for the fluorescence light. A commercial program (Dantec FlowManagerÒ Version 1.10) was used to post-process the PIV signals. The LIF signals were postprocessed using a program developed in-house. Both signals were acquired simultaneously, allowing spatial measurements of the instantaneous velocity and concentration fields and their correlation. An imaging area of 27.3 27.3 mm was used to completely resolve the flow phenomena including the high velocity gradients. An interrogation area of 32 32 pixels with an overlap of 50%
3. Numerical setup The simulations were performed using the open-source CFD toolbox OpenFOAM version 2.0 (OpenCFD, 2011). The computational domain was chosen in a way to save computational time, while capturing all important phenomena of the JIC. The domain extended 100 mm in the upstream crossflow direction and 200 mm in the downstream direction, as depicted in Fig. 2. The flow was modeled as incompressible. Three simulation approaches for turbulence have been studied: large-eddy simulation (LES), unsteady Reynolds-averaged Navier Stokes (URANS) and steady-state RANS. For the LES, a dynamic version (Lilly, 1992) of the Smagorinsky sub-grid scale turbulence model (Smagorinsky, 1963) was employed, while the SST turbulence model (Menter, 1994) was used for the RANS simulations. For the RANS simulations, the turbulent Schmidt number rt is constant and equal to 0.9, while the LES employed rt equal to 1.0, when not otherwise noted. The Reynolds stress tensor ðu0i u0i Þ of the RANS simulations has been calculated using the Boussinesq approximation (Wilcox 1998), with the Reynolds flux vector ðu0i c0 Þ being calculated in the same way, using a turbulent diffusivity Dt and the mean scalar gradient. As the SST turbulence model does not predict the root mean square value of the passive scalar crms directly, the following transport equation for the variance of the passive scalar cvar = (crms)2 has been used
@ðqcv ar Þ ~ i cv ar Þ rðqDt rcv ar Þ þ rðqu @t
e
¼ C 1 qDt rcv ar rcv ar C 2 q cv ar ; k
ð1Þ
with C1 = 2.8 and C2 = 2. Tilde denotes a Favre-averaged quantity. For the URANS simulation the contributions of the turbulence modeling have been summed up with the contribution of the velocity and passive scalar fluctuations calculated directly during the simulation. In contrast to RANS and URANS, the LES resolves the largest part of the turbulent fluctuations directly. Nevertheless, the small fraction of the turbulence being modeled by the sub-grid scale model has been added to all appropriate results. The convection terms in the RANS simulations have been discretized using the Gamma scheme developed by Jasak (1996). Being based on the central differencing scheme, it preserves overall second-order accuracy. In the LES, the filtered linear scheme
Fig. 1. Overview of the 2D-PIV/LIF measurement technique.
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 2. Overview of the computational domain with dimensions and the coordinate system employed, top. The fine grid at the symmetry plane near the jet inlet, bottom.
(OpenCFD, 2011) has been used for the momentum equation, which is a low dissipation, second order scheme based on central differencing. The Gamma scheme was employed in the transport equation of the passive scalar in LES context, with the blending factor kept to the minimum value of 0.1 as proposed by Jasak (1996), limiting the adverse effects of the added numerical diffusion. The pressure correction was performed using the PISO algorithm proposed by Issa (1986). For the unsteady simulations the time was discretized using a second-order accurate backward method. The averaging time was 0.15 s, which translates into 170 characteristic time units D/Ucross. Two grids have been developed for the simulations, employing hexahedral-shaped elements: a coarse grid composed of 1.5 million elements, with the diameter of the jet inlet having a resolution of 27 elements, and a fine grid composed of 7.5 million elements, with an increased jet inlet resolution of 39 elements (see Fig. 2). A no-slip boundary condition was applied at the walls, with typical y+ values at the crossflow walls about 15 for the coarse and 12 for the fine grid, and at the pipe walls about 12 for the two grids. The grid dependence of the results was explored. Fig. 3 shows line plots of the mean velocity component U, mean passive scalar C, specific Reynolds stress component u0 u0 and root mean square of the passive scalar crms at the symmetry plane (y/D = 0) and axial position x/D = 1.0 for RANS simulations using the coarse and fine grids. Profiles of U and u0 u0 from Galeazzo et al. (2011) for the same conditions are also included, which employed a very fine tetrahedral grid with more than 10 million elements and element sizes in the near-jet region equivalent to 1/40D. The velocity profiles are in good agreement with each other, with a small velocity deficit in the simulation using the coarse grid. The same trend could be observed for the Reynolds stress, with good agreement between the fine grid and the results from Galeazzo et al. (2011), while the coarse grid influenced negatively the results. Analyzing the profiles of mean passive scalar and passive scalar rms showed that the coarse grid is also impacting the results of these variables. The
Fig. 3. Line plots of mean velocity component U/Ucross, mean passive scalar C, specific Reynolds stress component u0 u0 =U 2cross and passive scalar rms crms at the symmetry plane (y/D = 0) and x/D = 1.0 for RANS simulations using different grids and results from Galeazzo et al. (2011).
fine grid, producing results comparable to the grid from Galeazzo et al. (2011) with the advantage of being composed of hexahedral elements, was used in the remaining work for all simulations. The jet trajectory in the JIC is influenced by the velocity profiles of the jet and the crossflow (Muppidi and Mahesh, 2005). In this work, the JIC fills a large part of the channel cross section, which increases the importance of the description of the crossflow boundary layer, as the velocity in the middle of the channel increases as the flow develops. Unfortunately, the crossflow boundary layer thickness could not be resolved using laser diagnostics because of the interference caused by the laser light reflected by the walls. Galeazzo et al. (2011) employed an alternative method to evaluate the development of the boundary layer, using the fact that the developing velocity profile of the crossflow depends on
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Fig. 4. Profiles of mean velocity components U/Ucross and W/Ucross and specific Reynolds stress components u0 u0 =U 2cross and w0 w0 =U 2cross , 1D above the jet exit, z/D = 1.0, y/D = 0.
the growth of the boundary layer. RANS simulations over the full length of the channel were compared to measurements using Laser Doppler Velocimetry (LDV) of the crossflow velocity at the center of the channel along the x direction. The profiles are in good agreement, indicating that the growth of the boundary layer was simulated correctly. To include the correct boundary layer thickness in the current simulations using the computational domain depicted in Fig. 2, velocity profiles of RANS simulations over the full length of the channel have been applied at the inflow boundaries, naturally including the predicted boundary layer thickness. The jet velocity and turbulence profiles are depicted in Fig. 4, which shows comparisons between the measurements and simulations 1D above the jet inlet (z/D = 1) and varying ±1D along the xdirection. The measurements are consistent, showing peaks of specific Reynolds stress components in the same regions where large mean velocity gradients occur. The velocity profiles of the unsteady simulations (LES and URANS) show a good agreement with the measurements, while the RANS simulation predicted the jet to be narrower than in the measurements. The turbulence is highly anisotropic in this region, with the peak values of the w0 w0 Reynolds stress component being almost three times higher than the u0 u0 component. Anisotropic turbulence is naturally simulated using the LES formulation, which is reflected in the good agreement of the predicted Reynolds stress levels. On the other hand, the RANS and URANS simulations had difficulties predicting the w0 w0 level correctly. The description of turbulence at the inlet boundary conditions for LES is a known challenge. For that purpose a boundary condition based on the work of Klein et al. (2003) has been used. Turbulent fluctuations have been applied to the mean velocity profiles at the inflow boundaries of the LES. The spurious pressure fluctuations generated at the inlet are one drawback of this approach. The solution is to apply a correction to the velocity field, to ensure that the mass flow remains constant. In every time step, after the turbulence fluctuations are superimposed on the mean velocity profiles, the mass flow is evaluated and compared to the reference mass flow. Then the velocity field is multiplied by a factor, typically smaller than 0.5%, correcting the value of the mass flow. With this correction, the pressure fluctuations were drastically reduced, followed by a significant improvement in the stability of the simulation. The boundary conditions of the mixing were defined with the pure jet flow having a passive scalar concentration C value of unity, while the pure crossflow has a value of zero. 4. Intermittency One of the characteristics of the jet in crossflow is that it has intermittent regions, meaning that in these regions fluid from the jet and from the crossflow can be found in an alternating matter.
Intermittency is present in all turbulent free shear flows (Libby 1996). The quantitative description of the intermittency is possible using the intermittency function I(xi, t), which assumes value unity in the jet flow, and zero in the crossflow. The first applications of the intermittency function were to distinguish between non-turbulent and turbulent flows, as in a free jet that flows into quiescent surroundings (Libby 1996). In this case, the intermittency function could be determined from a careful interpretation of velocity signals, as velocity fluctuations could be related to the jet flow. In contrast, the jet in crossflow is composed of two turbulent flows, making the discrimination strategy based on velocity fluctuations more subject to ambiguity. An alternative definition of the intermittency function is possible, as the jet flow can be associated with fluid having finite values of tracer concentration. Defining also a small threshold value for the concentration, an intermittency function I(xi, t) has been evaluated without ambiguity. The function I(xi, t) assumes value unity when the tracer value is higher than the threshold, and zero otherwise. As observed by Anderson et al. (1979), the value of the intermittency function is found to be relatively insensitive to the precise value of the threshold value within a reasonable range. They have studied a turbulent helium jet in an air coflow, with threshold values varying between C = 0.0075 and 0.026, and found a very limited effect on the evaluation of the intermittency. For simplicity, a fixed threshold value of C = 0.01 has been used in this work. The intermittency factor c(xi) represents the fraction of time a point is in the jet fluid, and is defined as the time average of I(xi, t)
cðxi Þ ¼ Iðxi ; tÞ:
ð2Þ
5. Results The ability of the different simulation strategies in predicting the formation and evolution of coherent structures is depicted in Fig. 5, showing the isosurface of C = 0.01 in a single time step of the LES and of the URANS and for the steady-state result of the RANS. The isosurface of the LES is very contorted, indicating a high turbulence level. However, it is difficult to discern any specific coherent structure in this picture. Large-scale structures have been also predicted by the URANS, resulting in the sinuous isosurface observed in Fig. 5. It is evident, however, that the fluctuation level is much lower in comparison to the LES. The steady-state RANS simulation does not predict large-scale structures at all, which results in a smooth passive scalar distribution. The two dimensional plots in Fig. 6 are a good starting point for the discussion of the results. In this figure, PIV measurements and the results of the simulations for the mean velocity component U and the specific Reynolds stress component u0 u0 at the symmetry plane (y/D = 0) are shown. The jet is mounted in the z-direction, so the jet fluid that enters in the channel has a very high velocity component (W) in this direction and a nearly zero velocity component in crossflow direction (U). As the jet flows into the crossflow, it is bent in the crossflow direction and U increases, creating the region of high U situated at x/D = 1 and z/D = 4. With the development of the jet, crossflow fluid is entrained, and U decreases continuously. On the lee side of the jet, a recirculation zone is formed. This region has negative U values, which is represented by the blue color in the contours. It is interesting to note that the recirculation region and the region with maximum U are very close to each other. The velocity gradient in this region is therefore very high, and very high production of turbulence is expected. The contour of u0 u0 confirms this expectation, with the locus of maximum u0 u0 situated in the region of high velocity gradient. Comparing the results of the simulations with the measurements in Fig. 6, it is clear that the LES has the better agreement. Both velocity and turbulence fields are very well reproduced,
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 5. Isosurface of passive scalar, C = 0.01. From top to bottom: LES, URANS and RANS simulations.
having the field maxima at the same position and having almost the same intensity as acquired in the measurements. Although the position of the jet is also reproduced well by the other type of simulations, the maximum values of U are slightly underpredicted. The predictions of u0 u0 , on the other hand, differ substantially between the different simulations. While the measurements show a maximum value of u0 u0 =U 2cross at the symmetry plane of 1.89, the prediction of the LES is 1.60, of the URANS is 1.03 and of the RANS simulation is 1.02. This fact can be explained by the presence of large-scale coherent structures in the jet in crossflow. The measurements were evaluated using simple time averages, so that the Reynolds stress tensor represents the whole velocity fluctuation about the time averaged mean velocity field. This implies that these fluctuations contain the contribution of two different phenomena: the turbulent fluctuations and the unsteadiness of the mean flow about the time-average created by the coherent structures, if such structures exist. The very good agreement of the fluctuation level of the LES with the measurements confirms the ability of the LES of solving the major coherent structures of the jet in crossflow. It is argued that the reason for the low u0 u0 levels predicted by the URANS simulation is the insufficient description of the coherent structures. This simulation could resolve a fraction of the unsteady character of the flow, however a significant fraction remained unresolved. It is the contribution of these unresolved coherent structures that makes the predicted fluctuation level to be lower than the measurements. As the coherent struc-
5
tures are not resolved at all by the steady-state RANS simulation, its fluctuation level contains only the contribution of the turbulence and is consequently lower than both unsteady simulations and measurements. Line plots allow the quantitative comparison between measurements and simulations. Fig. 7 shows line plots of the mean velocity component U/Ucross and the specific Reynolds stress component u0 u0 =U 2cross at the symmetry plane, y/D = 0. One of the most important parameters of the jet in crossflow for engineering applications is the jet penetration, which is defined in this work as the locus of maximum mean velocity component U. The jet penetration is well represented by all simulations. Comparing the profiles for U in more detail, the LES shows very good agreement in all positions, even near the bottom wall, a region known to be difficult to simulate accurately due to the recirculating flow present there. The two RANS simulations show similar results, with velocity magnitudes that are slightly lower than the measured ones in the region where the jet is located. The specific Reynolds stress component u0 u0 , on the other hand, shows a different picture. The agreement of the LES results with the measurements is very good in both magnitude and location, with only a slightly overpredicted u0 u0 level at x/D = 0. The RANS and URANS simulations show values that are consistently lower than both measurements and LES. Fig. 8 shows two dimensional plots of the passive scalar C, the intermittency factor c and the root mean square value of the passive scalar crms for the measurements and simulations at a horizontal plane, z/D = 1.5. Unfortunately, experimental data about the intermittency factor c is not available. The agreement of C predicted by the LES is very good, both in shape and level. The inner side of the kidney shape fits very well to the experimental data. Considering the URANS simulation, the gradients of the scalar are steeper than in the LES, which makes the agreement with the measurements slightly worse. The kidney shape of the jet agrees well to the experimental data. One part of the differences in the prediction of C can be explained by the contours of c. The c function represents the fraction of time in which each point remains in the jet fluid, and its distribution is closely related to the amount of largescale coherent structures which have been resolved by the simulation (Libby, 1996; Pope, 2000). Steady-state RANS simulations do not resolve coherent structures at all, and the c profiles assume the shape of a step function, because in this case the intermittency factor c is equal to the intermittency function I, assuming the value zero when C is smaller than 0.01 and the value unity otherwise. The LES is able to resolve a larger fraction of the large-scale coherent structures than the URANS, which results in c contours that are distributed over a larger area than the contours of the URANS simulation; in other words, the LES predicts less steep gradients of c than the URANS, which can also be seen in the profiles in Fig. 9. The agreement between the predictions of the LES and the URANS is worse exactly in the regions with poor agreement of c. The results of the steady-state RANS simulation are close to the results of the URANS, with the agreement of the scalar field C being not as favorable as was shown for the velocity fields. The core flow is reasonably well predicted, however with a more pronounced inner side of the kidney shape. This difficulty is expected, as the steadystate RANS simulation does not resolve coherent structures and the influence of the intermittency is more pronounced in this region. The crms is predicted by the LES in very good agreement with the measurements in both level and shape. The agreement of the URANS and the RANS results is satisfactory, however with very low levels of crms in the jet core, which does not correspond to the measurements. The peak levels predicted by the URANS are slightly higher than the RANS, which can also be seen in Fig. 9 at z/D = 1.5. The turbulent mixing can be analyzed in more detail using the line plots at the symmetry plane (y/D = 0) of the mean passive
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 6. Two dimensional maps of mean velocity component U/Ucross and specific Reynolds stress component u0 u0 =U 2cross at the symmetry plane, y/D = 0. From top to bottom: PIV measurements, LES, URANS and steady-state RANS.
scalar C and the intermittency factor c, the root mean square value of the passive scalar crms and the u0 c0 component of the specific Reynolds flux vector depicted in Fig. 9 for three different z/D positions (1.5, 3 and 4.5). The C profiles show more pronounced differences than the mean velocity depicted in Fig. 7 for the same locations. The LES simulation shows good agreement with the measurements, with slightly underpredicted peak scalar values at z/D = 3 and 4.5. The C profiles of the RANS and URANS simulations agree well, although not as good as the profiles of the mean velocity in Fig. 7 do. The position of the jet shows good agreement, however the profiles of the simulations are slightly narrower than the one of the measurements. The LES shows less steep gradients of c than the URANS, which is particularly evident at z/D = 4.5. Comparing the c and C profiles, it can be seen that the broader C distribution of the LES correlates very well with the also broader c distribution. Comparing the RANS simulations, the C profile show good agreement with the measurements at z/D = 1.5, while at z/ D = 3 and 4.5 it had problems predicting the jet borders correctly. The LES predicts crms in good agreement especially at z/D = 1.5, with the two peaks at the jet boundaries being clearly discerned. However, the overall level is lower than in the measurements,
despite the fact that the Reynolds stresses and fluxes are well represented. The peak levels of crms predicted by the URANS and RANS simulations agree well with the measurements, while the agreement of the shape of the profiles is not as favorable. The LES slightly overpredicts u0 c0 at z/D = 1.5, while for the other two locations the agreement is good. For the other simulations the agreement of u0 c0 is good, however the location of the peaks at z/D = 3 and 4.5 are slightly moved in the downstream direction in the steady-state simulation. The comparison with the experimental data has shown that, while the mean velocity is well predicted by all simulations, the prediction of the mean passive scalar C depends heavily on the level of complexity of the simulation. The agreement of C is very good using the LES, and degrades significantly when moving to URANS and RANS simulations. This different behavior is attributed to the different way that the velocity and the passive scalar react to the presence of the jet. When portions of ambient fluid are entrained by the jet flow, they are accelerated before effectively meeting the jet flow, leading to velocity fluctuations. These fluctuations, that are not turbulent in nature, are induced by the pressure fluctuations associated with the turbulence (Libby, 1996), which
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Fig. 7. Line plots of mean velocity component U/Ucross and specific Reynolds stress component u0 u0 =U 2cross at the symmetry plane, y/D = 0.
explains their different behavior in flows dominated by large-scale structures. Fig. 10 shows a planar cut along the symmetry plane of a single time step of the LES. The jet flow region is marked with gray color, while the color of the ambient fluid is white. The boundary between the jet flow and the ambient fluid is marked by the isocontour of C = 0.01, the same definition used to calculate the intermittency function. The boundary is highly convoluted, which is a result of the various coherent structures of the flow. A small region is shown in more detail, with arrows representing the local velocity vector. In the top right hand corner of the detail, a region of ambient fluid (white) is being entrained by the jet flow (gray). The entrained region is characterized by the velocity vectors, which have a component in the vertical direction. In contrast, in other areas away from the entrained region, the velocity vectors of the ambient fluid have components mainly in the horizontal direction. Although the difference in the velocity magnitude is not very pronounced, the velocity vectors of the ambient fluid in the entrained region show that the fluid has been accelerated in the vertical direction. This finding corroborates with the results of Anderson et al. (1979) and Pope (2000), which show that the entrained fluid develops higher velocity than the surrounding fluid, even before being effectively entrained. It follows that the velocity gradient across the interface between the two flows is smaller than the scalar gradient, which explains their different behavior in flows dominated by large-scale structures. The difference between the gradients of the velocity and of the passive scalar can be quantified analyzing the diffusion across the superlayer, the instantaneous interface between the two flows, as the diffusion is directly related to the gradients. In order to calculate the diffusion from the LES, the superlayer was approximated by the surface determined by the isosurface of 1% passive scalar, which is the same used to define the intermittency function (see Fig. 5). To allow a direct comparison, the instantaneous velocity component w, non-dimensioned by the bulk jet velocity Ujet, is employed along the instantaneous passive scalar c, as both are transported into the crossflow by the jet and have values spanning between zero and one. The diffusion flux across the superlayer of velocity Jw and of the passive scalar Jc are defined by the dot product of the gradients and the normal vector ~ n
@ðw=U jet Þ ~ n; @xi
ð3Þ
@c ~ n: @xi
ð4Þ
Jw ¼ m
Jc ¼ m
Integrating over the whole isosurface of a single time step of the LES resulted in a diffusion flux of velocity Jw of 2.64 106 m3/s and a diffusion flux of passive scalar Jc of 26.59 106 m3/s, a clear indicator that the passive scalar gradient is bigger than the velocity gradient across the superlayer. Another important parameter in the analysis of turbulent mixing is the turbulent Schmidt number rt. Within the RANS framework, there are two quantities that describe the turbulent diffusivity of momentum and passive scalar: the eddy viscosity mt and the turbulent diffusivity Dt, respectively. The eddy viscosity is the factor of proportionality between the specific Reynolds stress tensor u0i u0j and the mean strain-rate tensor Sij through the Boussinesq approximation (Wilcox, 1998)
2 u0i u0j ¼ 2mt Sij kdij ; 3
ð5Þ
where k is the turbulent kinetic energy and dij is the Kronecker delta. In the same way, the turbulent diffusivity Dt is the factor of proportionality between the specific Reynolds flux vector u0i c0 and the mean scalar gradient
u0i c0 ¼ Dt
@C ; @xi
ð6Þ
The ratio between the two diffusivities is the turbulent Schmidt number
rt ¼ mt =Dt :
ð7Þ
One approach often used to enhance the mixing in RANS simulations is to decrease the standard value of the turbulent Schmidt number from 0.9 and consequently scale the Reynolds flux vector with it. Fig. 11 shows the results of three simulations using the SST turbulence model, with rt of 0.3, 0.5 and 0.7. The passive scalar field of the simulation using rt = 0.3 does not have the steep
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 8. Two dimensional maps of mean passive scalar C and the intermittency factor c, z/D = 1.5. From top to bottom: LIF measurements, LES, URANS and RANS.
gradients of the standard simulation using rt = 0.9, which is consistent with the increased turbulent mixing. The above changes to rt affect the whole field simultaneously. The mixing in the jet core is well represented using the value of 0.9 while the borders appear to be better predicted using a lower value. This finding is consistent to the fact that the intermittency is higher at the jet borders; the deviations are expected to be higher in this region. It is tempting to apply the definitions of Eqs. (5) and (6) to the experimental data and to the LES simulation and calculate rt, as both evaluate the Reynolds stresses and fluxes along with the gradients of the mean quantities. However, the application of these definitions is not straightforward. Within the RANS framework the eddy viscosity is a scalar quantity, defined in Eq. (5). The turbulent diffusivity (Eq. (6)) and the turbulent Schmidt number (Eq. (7)) are also scalar quantities. On the other hand, when evaluating the results of the measurements, it becomes clear that the eddy viscosity assumes a different value for each component of the Reynolds stress tensor, being in fact a tensor quantity. The same applies to the turbulent diffusivity and to the turbulent Schmidt number, which are in fact vector quantities. The comparison of these tensor and vector quantities with the scalar results of RANS models demands great care.
Despite the shortcomings, much can be learned when analyzing the results of the measurements and the LES. The two-dimensional experimental data allows the calculation of one pair of eddy viscosity and turbulent diffusivity. For example, at the symmetry plane (y/D = 0), the Reynolds stress component u0 w0 and the Reynolds flux component u0 c0 , together with the gradients of the mean quantities, can be evaluated and the turbulent Schmidt number in the xdirection can be calculated as
rt;x
u0 w0 ¼ @U @W þ @x @z
,
u0 c0 @C @x
:
ð8Þ
Fig. 12 shows the component of rt,x obtained from the measurements and from the LES results at the symmetry plane together with the axial evolution at three vertical positions. To improve the readability of the picture, the plots were limited to regions with values of C between 0.05 and 0.95 and with absolute values of oC/ox above 0.04. The major difficulty in comparing the turbulent Schmidt numbers from the measurements or the LES with the definition from the RANS framework is actually the fact that the turbulence is not the only source of fluctuation in the jet in crossflow. Eqs. (5)
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Fig. 9. Line plots of passive scalar concentration C, intermittency factor c, passive scalar rms crms and specific Reynolds flux component u0 u0 =U 2cross at the symmetry plane, y/ D = 0.
Fig. 10. Symmetry plane (y/D = 0) of one realization of the LES. The solid line represents the boundary between jet flow and the ambient fluid. The detail shows also arrows representing the local velocity vector.
and (6) are derived assuming that turbulence is the only source of fluctuation in the flow, as all equations in RANS context. This is clearly not the case in the jet in crossflow, especially in regions where the coherent structures have already developed. The exper-
imental results support this argument. At the jet root, near the nozzle, the coherent structures have not evolved sufficiently to affect the flow. In this region rt,x assumes values between 0.3 and 1.3, which agrees well to the values recommended in the literature.
Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012
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Fig. 11. Two dimensional maps of mean passive scalar C, z/D = 1.5, for simulations using the SST turbulence model and turbulent Schmidt numbers rt of 0.3, 0.5 and 0.7.
Fig. 12. Two dimensional maps and line plots of turbulent Schmidt number component rt,x at the symmetry plane, y/D = 0, for the measurements (a) and the LES (b).
Further downstream the influence of the coherent structures is more pronounced, and the values of rt,x begin to fluctuate between very high and very low values, which is indicated by the sudden appearance of red and blue regions in the contour plots from z/D = 3 and downstream. This finding leads to the conclusion that, in the presence of coherent structures, Eqs. (5) and (6) do not adequately represent the phenomena present there.
6. Conclusions The results of different simulations, ranging from simple steadystate Reynolds-averaged Navier Stokes (RANS) to more sophisticated unsteady RANS and large-eddy simulation (LES), have been
compared to high quality experimental data acquired using Particle Image Velocimetry combined with Laser Induced Fluorescence. The quality of the agreement of the simulation results with the measurements is strongly coupled with the description of the coherent structures. The agreement of the LES results with the measurements is very good for both mean variables and fluctuation values. The unsteady RANS simulation using the SST turbulence model could resolve only a fraction of the coherent structures. The resulting fluctuation levels are systematically lower than the measurements, even though the agreement of the mean variables is good. The analysis of the results shows that the agreement of the LES results is better than the URANS results not only because the turbulence is better reproduced, but also because the LES simulated the coherent structures in much more detail. The steady-state RANS
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simulations do not resolve the coherent structures at all, neglecting this important source of unsteadiness. It follows that the fluctuation levels predicted by the steady-state simulations are significantly lower than the measurements. The agreement of the mean velocity is better than the mean passive scalar for the steady-state simulations. This fact could be explained by the intermittency, which has a more pronounced influence on the velocity than in the passive scalar. A detailed analysis has shown that in the flow regions dominated by coherent structures, the turbulent Schmidt number rt do not adequately represent the mixing phenomena present there. Different values of rt are needed to represent well the mixing in the jet core and along the jet borders, which is consistent to the fact that the intermittency is higher at the borders. The results have shown that steady-state RANS simulations provide good quantitative and qualitative agreement with experimental data when the flow is statistically stationary, i.e., when the influence of large-scale coherent structures is negligible. However, when the flow is not statistically stationary, with pronounced large-scale coherent structures, Reynolds averaged values do not converge to their time-averages. As the coherent structures are not turbulent in nature, their influence on the mean flow is not modeled by the turbulence models. Hence, to achieve high-fidelity results for all variables, time dependent simulations are mandatory, which increases the computational cost substantially. Fortunately, the increase in the computational cost of the simulations translates into higher accuracy, especially for the turbulent mixing. Acknowledgements This project is a research commissioned by the DLR for the state of Baden-Württemberg, Germany, which is funded by the Landesstiftung Baden-Württemberg gGmbH. One part of the computation time was kindly provided by the Steinbuch Centre for Computing (SCC) of the Karlsruhe Institute of Technology (KIT). References Alvarez, J., Jones, W.P., Seoud, R., 1993. Predictions of momentum and scalar fields in a jet in cross-flow using first and second order turbulence closures. In:
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Please cite this article in press as: Galeazzo, F.C.C., et al. Computational modeling of turbulent mixing in a jet in crossflow. Int. J. Heat Fluid Flow (2013), http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.012