12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

IMPACT OF A DEAN VORTEX-CHARACTERIZED INFLOW STREAM ON
THERMAL MIXING IN A T-JUNCTION: A SENSITIZED-RANS MODELINGSTUDY

Ivan Joksimović1 , Tarik Čorbo2 , Suad Jakirlić1
1 Institute of Fluid Mechanics and Aerodynamics, Technical University Darmstadt

Allarich-Weiss Straße 10, 64287 Darmstadt, Germany
2 Faculty of Mechanical Engineering, University of Sarajevo,

Vilsonovo setaliste 9, 71000 Sarajevo, Bosnia and Herzegovina
joksimovic@sla.tu-darmstadt.de, jakirlic@sla.tu-darmstadt.de

ABSTRACT

numbers. To overcome the above-mentioned problems, a recently proposed ’scale-adaptive’ turbulence modeling strategy,
Menter & Egorov (2010), has been applied on an industrially relevant moderately high Reynolds number case which
involves mixing of two fluid streams with different temperatures. The dynamics of relevant sub-scales is modeled byan adequately sensitized anisotropy-resolving Reynolds-stressmodel. The model applied is capable of reproducing lowrank turbulent structures and their temporal dynamics at lower
cost, comparing to the LES frameworks. The database originating from the experimental investigation by Hosseini et al.
(2008) and detailed LES study by Tunstall et al. (2016) of
the thermal mixing in a T-Junction configuration is adopted
presently as a reference for computation and results comparison. Schematic of the computational domain with indicatedboundaries is shown in Fig. 2. Configuration dimensions andoperating parameters are listed in Table 1.

A computational study focusing on the thermal mixing of
two fluid streams in a T-junction configuration investigated by
Tunstall et al. (2016) has been presently performed by applying a RANS-based eddy-resolving model of turbulence. Main,
low-temperature stream enters the T-junction at a Reynolds
number corresponding to Re = 107893 after passing througha 90◦ bend. The secondary, high-temperature stream is injected perpendicularly into main branch at a Reynolds number of 5422. Substantial difference between flow rates of two
streams causes formation of a characteristic mixing process resembling a re-attached jet configuration. Of particular interest
is its interference with the secondary vortices - the so-called
Dean vortices - that are convected from the curved main inlet,
producing a low-rank mixing structures in its wake. The computational framework relying on the scale-adaptive turbulencemodeling approach, introduced by Menter & Egorov (2010),
has been used to simulate dynamics of large-scale turbulent
structures with moderate spatial and temporal resolution requirements. The motion of unresolved sub-scale structures aredescribed by an appropriately sensitized near-wall differential
Reynolds-stress model according to Jakirlic & Maduta (2015).

INTRODUCTION
The study of momentum and heat exchange in the turbulent mixing processes involving two or more fluid streams
of different temperatures within the pipelines of different geometric complexity is one of the core areas of fluid mechanics
and thermodynamics. The performance capabilities and efficiency of relevant thermal-hydraulic equipment is based to
the greatest extent on the understanding of the flow and thermodynamic processes and their interactions operating therein.
Mixing processes in industrial pipeworks involve often highReynolds number flows characterized by complex associatedphenomena such as spatially and temporally evolving structural flow dynamics, secondary currents and high level of turbulence anisotropy. This makes them extremely demanding
for industry-scale simulations. Computationally more suitable
RANS (Reynolds-Averaged Navier-Stokes) models are often
incapable to properly capture flow anisotropy and highly unsteady flows dynamics, whereas the current LES (Large-Eddy
Simulation) models require, despite their advantageous accuracy in resolving the major portion of underlying turbulence
spectrum, enhanced computational resources at high Reynolds

Table 1.

Operating conditions

Parameters

Main line

Branch line

Diameter [mm]

108

21

Bulk velocity [m/s]

0.89

0.23

Reynolds number

107,8935422

Temperature (dimensionless)

1

0

COMPUTATIONAL METHODOLOGY
Following the LES-reference, the incompressible flow
conditions are assumed. The equation system comprising
the continuity-, momentum- and temperature-evolution equations, as well as the equations governing the turbulent quantities, Reynolds stress components ui u j and the specific dissipation rate ω h , is implemented into the open-source C++ Library OpenFOAM, with which all simulations have been performed. Simplified-gradient-hypothesis (SGH) has been applied for calculation of sub-grid heat fluxes since the dominant portion of fluctuating energy would be attained at re-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
mainly activated only near the walls, thus returning the RANS
solution in their vicinity. Calibration of all constants is performed in Maduta (2013).

solved scales. Formally, second-order accurate schemes areapplied for discretization of both temporal and spatial derivatives. Adaptive time stepping with the criterion CFL < 0.7 isused.
As previously stated, choice of adequate turbulencemodel, capable of accounting for the complex flow phenomena, represents a nontrivial task. Since the most energetic
coherent structures in the flow are characterized by lowfrequency dynamics, only a top-level portion of turbulent spectrum needs to be resolved, and the remaining part may bemodeled. This relativizes the requirements for fine grid resolution. For that purpose, the ’eddy-resolving’ version of differential Reynolds Stress Model (RSM) by Jakirlic & Hanjalic(2002), denoted as Instability-Sensitive RSM - IS-RSM, hasbeen used. Detailed derivation and scrutiny of the model canbe found in Jakirlic & Maduta (2015). Modeled turbulence isaccounted for by the evolution equation for Reynolds StressTensor ui u j , including the effects of production Pi j , dissipation εi j , redistribution Φi j and diffusion Di j , with the latter
consisting of molecular and turbulent diffusion.
Dui u j= Pi j − εi j + Φi j + Dti j + Dνi j
Dt

T2 = 3kmax

i
1
2
∇ω
ω2

(4)

RESULTS
Geometry of the computational domain with appliedboundary conditions are adopted in accordance with the LES
study by Tunstall et al. (2016), with exception of handlingthe inflow turbulence. Namely, turbulence at the main inlet
is mapped from the parallel-running precursor simulation in astraight pipe. Precursor simulation also serves as a good indication of required mesh resolution, which is optimized insense of correct prediction of wall shear stress (see e.g. ∆Uτ ).
Applied metrics is given in Table 2.

(1)
Table 2.

Meshing metrics

∆r+

∆Θ+

∆L+

∆Uτ [%]

0.5 − 70

70

210

2.51

The results of the first- and second order statistics are presented in Fig. 1, and compared with DNS data for circularpipes by Pirozzoli et al. (2021). Concerning the velocity profile, its predicted near-wall shape follows closely the reference,
which is naturally expected since the value of wall-shear stress
was predicted accurately. Past the distance of y+ ≈ 100, slightdeviations are notable although the general agreement with thedata is good. This deviation can be explained by analysing the
profiles of second-moments, where again, in the vicinity of thewall, both the anisotropy of turbulence, as well as the overalllevel of turbulent kinetic energy are predicted correctly. However, in the zone further away from the wall, a certain deviationfrom the DNS data is pronounced, being at strongest with respect to the stream-wise Reynolds stress component. According to the original work of Maduta (2013), the model tends todissipate residual turbulence almost completely far away fromthe wall, hence the general level of turbulence is lower than expected. In spite of their somewhat smaller magnitude, second
moments do exhibit a positive influence on the resolved scales
since the general trend is correctly predicted and the simulation conditions don’t correspond to a ’coarse DNS’. Mesh resolution with similar characteristics as in precursor case is afterwards used in the main simulation. Total number of cells
is approx. 7.8 millions, which represents a substantial saving
compared to approx. 80 million cells used in the reference
LES case.
After eliminating the initial transient, the simulation has
been run up to the convergence of second order statistics. Contours of mean velocity field are given in Fig. 3. Since thebranch flow possesses a significantly lower momentum, it isforced to separate at the leading edge of the junction, creating
a ’reattached-jet’ configuration. We recall the back-bendingof the mean dividing streamline at the point of reattachment
being the characteristic feature of the redistribution term in(2)

By doing so, the correspondingly enhanced dissipation ofmodeled turbulent quantities suppresses their intensity towards
the sub-scale level (not sub-grid scale level; the present model
formulation is grid-spacing free). The turbulence production
is shifted to the resolved mean flow, forcing the emergence ofresolved flow structures in the regions where PSAS is active.
Since the sub-scale turbulence is adequately captured by using an advanced Reynolds-Stress Model, the spectral cut-off
can be positioned more towards the low-frequency part of thespectrum, thus enabling coarser mesh resolutions. Exact mathematical formulation of PSAS is non-unique and depends on theauthor’s interpretation of Menter and Egorov’s work. In thiscase, a simple relation promoting a stable solution process isbeing adopted in the form depending dominantly on the second
derivative of the underlying velocity field ∇2U:√
PSAS = C1 max( k∇2U −C2 T2 , 0))

k

∇k2 ,
2

.

The principal ’eddy-resolving’ strategy behind the IS-RSM
aiming at appropriately sensitizing the model is to introduce anadditional production term PSAS into the equation of the scalesupplying variable: ω h = ε h /k, in line with the Scale-Adaptive
Simulation (SAS) proposal by Menter & Egorov (2010).

Dω h  Dω h =+ PSAS
Dt
Dt RANSh1(3)Here, the value of the second derivative leads to the very intuitive behaviour: in case the flow exhibits a strong streamline curvature, which can either be inherently present (as e.g.
in separation and re-circulation zones as well as in secondary
flow events) or externally inflicted (by e.g. super-imposing theartificial perturbation) magnitude of the second derivative of
the velocity field will naturally increase. If the mesh is appropriate, second-derivative can be accurately reconstructed andthe eddy-resolving capability of the model will
√ be amplified
by increasing the PSAS . On the other hand, k serves as asafeguard, prohibiting excessive destruction of modeled turbulence. T2 stems from the transformation of equation and is2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
since the curvature-based phenomena are highly pressure- andless model dependent. Temporal dynamics of the secondary
flows can be investigated by using the snapshot POD (Proper
Orthogonal Decomposition) analysis, as described in Weiss(2019). With changes in the representation basis, generic flow
variable Ψ, representing either velocity or temperature, is decomposed into the sorted sum of spatially coherent structuresΦi (⃗x) with the corresponding time dynamics ai (t):Eq. (1) in conventional RANS closure models (see e.g., Hanjalic & Jakirlic (1998)), indicating that the near-wall region isstrongly affected by the modeled turbulence. The overall sizeof this ’primary’ separation bubble corresponds approximately
to 2Db , which is in accordance with the reference data. Additionally, due to the presence of an adverse pressure gradient
within the elbow, main stream exhibits the ’secondary’ separation at the location corresponds to 55o behind the elbow entrance. Interestingly enough, separation wake reattaches andseparates once again directly downstream of the elbow, creating a third separation zone which will be of decisive interestfor this case. It is still unclear whether this deviation from theexperiment is caused by the boundary conditions, or by themodel itself. A more detailed discussion of this flow feature
will follow in the following.

+∞Ψ(⃗x,t) = Φ0 (⃗x) + ∑ Φi (⃗x)ai (t)

(5)

i=1

where Φ0 (⃗x) represents the mean.
Visualisation of two most energetic modes Φ1 (⃗x) andΦ2 (⃗x) with their temporal dynamics at location 2Dm upstreamof T-Junction is presented in Fig. 7. It can be seen that theswitching frequency of the dominant vortex pair corresponds
roughly with St ≈ 0.21 which is in accordance with reference
data. However, a second frequency peak, non-existent in thereference, with switching frequency of St ≈ 0.5 appears in thefrequency signature as well. There are many possible reasonscausing this behaviour, from uncertainties in boundary conditions (e.g., the pressure field is directly mapped from the precursor simulation) to the intrinsic behaviour of the model. We
speculate that this additional vortex pair stems primarily fromthe secondary flow appearance formed in additional (third)
separation zone at the elbow exit. Since the separation zone
is confined by the presence of the Dean vortex pair, it is expected that the switching frequency will be increased. However, since the energy is primarily attained at St ≈ 0.21, weconclude that the behaviour of the flow is less dependent of
the pressure boundary conditions and is more a model-related
problem.
Further downstream, at the position 1Db behind the junction, this additional separation wake evolves in a more complex structure, with several higher harmonics in the range between St ≈ 0.1 and St ≈ 1.0. Spatially, three distinct zones ofhorizontally oriented vortex switching directions are identified
in Fig. 8, stemming from the elbow-related Dean vortex pair,junction-related separation bubble vortices and third separation wake from the elbow exit. Despite correctly reproducing
the natural frequency response of the vortex switching, which
is dictated by the pressure-related phenomena, it seems that the
Dean vortex pair together with the elbow wake exhibits multiple contractions and expansions along the downstream coordinate, which would explain higher harmonics in the data. Dueto the incapability of the POD method to discriminate between
separate frequencies for a given mode, it was not clear what
physical phenomenon is correlated with the frequency peaks.
Therefore, we additionally analyse the temperature field to correlate the temperature switching frequency with the underlying
flow structure.
The POD analysis of the temperature field 1Db behindthe junction is given in Fig. 9. As for the most energetic POD
mode, the principal directions of temperature pulsations areoriented, as expected, perpendicularly to both the direction of
the branch flow, as well as the curvature radius. Although thefrequency signature of St ≈ 0.21, originating from the Dean
vortex pair, is present in the signal, it doesn’t decisively influence the temperature pulsations. Dominant peak of St ≈ 0.5
shows that thermal mixing in the junction is dictated mostly
by the dynamics of the additional separation wake, formed at
the elbow exit. Although the natural switching frequency ofthe separation bubble is expected to lie around St ≈ 0.1, it isProfiles of the mean streamwise velocity are plottedagainst the LES reference in Fig. 4. Again, it can be seenthat the overall agreement is very good, with small deviationsat position 7Db upstream, stemming from the additional, third
separation that has been described previously. Although themean flow is pushed away from the wall immediately past the
elbow exit, it quickly recovers and almost complete agreementwith the data is reached 2Db upstream of the junction. At all
locations, the results related to the near wall zone shows a very
high level of agreement with the LES data. This is especiallypronounced immediately downstream of the junction where
the complex flow conditions involving the separation bubble
are met.
Due to the lower flow mass, mixing of cold and hot
streams doesn’t influence the temperature distribution on theoutlet dominantly. Therefore, of primary interest for this workis the distribution of temperature fluctuations on the surrounding wall surface, since it will dictate the propagation of thermally induced fatigue in the material. Projection of RMS(’Root Mean Square’) values of the temperature field on to theupper wall of the junction is presented in Fig. 5. Regions ofhigh amplitudes are established immediately behind the junction and are correlated with the boundary of the mixing zone,
due to the presence of high amplitude, low rank structures inthe separation wake. As will be demonstrated later on, energetic flow structures which are advected from the elbow exhibit a strong oscillating behaviour, which is oriented perpendicularly towards the branch flow, and the intensity of the RMS
temperature is strongly correlated with the correct capturingof mixing dynamics in the wake. Next to the two symmetric belt-shaped contours downstream the branch pipe, zones of
the highest variance are found at the leading edge of the junction. Based on the authors’ experience with similar cases, thisflow feature can be attributed to to the branch stream. Consequently, the temperature field undergoes the high-amplitude,
low frequency pulsations that is dictated by the incoming flow,
hence its high variance. the poor mixing process between thetwo fluid streams, which promotes a suppression of the fluctuations perpendicularly
As shown in Fig. 6, due to the longitudinal streamline
curvature, region of an outwardly directed pressure gradientforms in the elbow, forcing sunsequently the fluid in near-wallzone to flow towards the centre of the curvature. Secondary
structures typical of those resembling the Dean vortices areformed and advected downstream. Due to the continuity condition, stagnation point forms on the inner wall and promotes
the decceleration of the flow, which separates at around 55o behind the elbow entrance. Mean position of the Dean vortices
is slightly more spread in relation to the LES-reference andcan be attributed to the uncertainties in boundary conditions

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
ory and model description. Flow, Turbulence and Combustion 85, 113–138.
Pirozzoli, S., Romero, J., Fatica, M., Verzicco, R. & Orlandi,
P. 2021 One-point statistics for turbulent pipe flow up toReτ ≈ 6000. Journal of Fluid Mechanics 926, 355–377.
Tunstall, R., Laurence, D., Prosser, R. & Skillen, A. 2016
Large eddy simulation of a T-junction with upstream elbow:
The role of Dean vortices in thermal fatigue. Applied Thermal Engineering 107, 672–680.
Weiss, Julien 2019 A tutorial on the Proper Orthogonal Decomposition. AIAA Paper 2019-3333 .

somehow obscured by the dynamics of the more energetic inflow flow structures, and can not be distinguished in the signal. The second POD mode seems to be non-symmetrical inregard to the vertical coordinate, and stems from the separationof the branch flow at the trailing edge of the junction, which
produces pulsations in the temperature field at the edge of themixing wake.

CONCLUSION
Thermal mixing of two fluid streams at a moderately
high Reynolds number has been simulated by using thescale-adaptive computational strategy.
Correspondingly
sensitized differential Reynolds-Stress Model is utilized todescribe the dynamics of the sub-scale turbulence. Substantial saving regarding the computational effort has beenachieved in comparison with the reference LES case. The
flow statistics has been predicted following closely the LES
reference, showing the capability of the model to reproduce
complex flow conditions at reasonable costs. Considering
the temporal dynamics of the mixing structures, the ProperOrthogonal-Decomposition analysis has been perfoemd toanalyze temporal response of the most energetic flow features.
Due to the strong influence of pressure-driven effects, thedynamics of an elbow-induced Dean vortex pair has beenadequately reconstructed by the model. However, at elbow
exit, another flow separation occurs, injecting additional
peaks into the eddy-switching spectrum. Source of thisadditional separation zone remains likely model-related.
This third separation wake is retained in the vicinity of theupper wall and has affecting decisively the dynamics of themixing structures, which is no longer dominantly influenced
by the Dean vortex pair at St ≈ 0.21, but by the dynamics ofthe vortex pair, present in the third separation zone, at St ≈ 0.5.
Acknowledgment Financial support from the German Academic Exchange Service (DAAD - Deutscher Akademischer
Austauschdienst) is gratefully acknowledged.

REFERENCES
Hanjalic, K & Jakirlic, S 1998 Contribution towards thesecond-moment closure modelling of separating turbulent
flows. Computers and Fluids 27 (2), 137–156.
Hosseini, S. M., Yuki, K. & Hashizume, H. 2008 Classificationof turbulent jets in a T-junction area with a 90-deg bend
upstream. Int. J. Heat and Mass Transfer 51, 2444–2454.
Jakirlic, S. & Hanjalic, K. 2002 A new approach to modellingnear-wall turbulence energy and stress dissipation. Journalof Fluid Mechanics 459, 139–166.
Jakirlic, S. & Maduta, R. 2015 Extending the bounds of‘steady’ RANS closures: Toward an instability-sensitive
Reynolds stress model. Int. J. Heat and Fluid Flow 51,
175–194.
Maduta, R. 2013 An eddy-resolving Reynolds stress model for
unsteady flow computations: development and application.
PhD thesis, Technische Universität Darmstadt, Darmstadt,
Germany.
Menter, F. & Egorov, Y. 2010 The scale-adaptive simulation
method for unsteady turbulent flow predictions. Part 1: the-Figure 1. Validation of the precursor pipe flow simulation :profiles of the mean-velocity (upper), normal Reynolds stresscomponents (middle) and turbulence kinetic energy (lower).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 2.

Figure 3.

Schematic of the solution domain with assigned boundary conditions.

Contours of the mean velocity field, colored by its magnitude. Note that the figure is not in scale with the schematic.

Figure 4.

Profiles of the streamwise velocity. Reference data from LES study are given in red dashed lines.

Figure 5.

Contours of the dimensionless RMS Temperature on the upper wall.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Figure 6. Streamlines of the secondary flow, respectively at locations 65o (left) and 90o (middle) behind the elbow entrance, as well
as 1Db (right) behind the junction. Backward coloring corresponds to the magnitude of the streamwise velocity component.

Figure 7. Streamlines of the first (left) and second (middle) most dominant POD mode of the velocity field at location 2Dm upstreamof the junction, with the FFT Analysis of their time coefficient (right). Backward coloring indicates the magnitude of the mode.

Figure 8. Streamlines of the first (left) and second (middle) most dominant POD mode of the velocity field at location 1Db downstreamof the junction, with the FFT Analysis of their time coefficient (right). Backward coloring indicates the magnitude of the mode.

Figure 9. Contours of the first (left) and second (middle) most dominant POD mode of the temperature field at location 1Db downstream of the junction, with the FFT Analysis of their time coefficient (right). Backward coloring indicates the magnitude of the mode.

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