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Turbulence, Heat and Mass Transfer 8
c
2015 Begell House, Inc.
DNS of a Jet in Cross Flow with Passive Scalar Mixing
Z. Wu1, D. Laurence1and I. Afgan1
1Modelling and Simulation Centre, School of Mechanical, Aerospace and Civil Engineering,
The University of Manchester
Manchester M13 9PL, zhao.wu@manchester.ac.uk
Abstract — Direct numerical simulation (DNS) with high-order compact scheme (Incompact3d) is used for
simulating a jet in cross flow (JICF) with passive scalar mixing and wall heat transfer. The low momentum
laminar circular jet is orthogonal to the walls of a turbulent channel flow generated using recycling technology.
Instantaneous and time-averaged profiles from DNS confirm important dynamic flow structures of JICF, including
the shear layer vortices, horseshoe vortex formation, the 3D near-wall back-flow after the jet, and the formation
of counter-rotating vortex pair. Four different Reynolds Averaged Navier Stokes (RANS) models are also tested
for the JICF in comparison to the DNS data. The velocity and temperature predictions from all RANS models
are found to produce only qualitative agreement with the DNS, however the predictions of kinetic energy and wall
heat transfer produced by all four RANS models are unsatisfactory. Wall temperature fluctuation variance from
the DNS is also presented.
1. Introduction
The heat transfer below a JICF is crucial for a combustor or a turbine blades wall cooling, de-
icing of the wings or for the nuclear power-plant vessel emergency cooling. JICF generates
several coherent structures: the shear layer vortice above the jet, horseshoe vortex formation
in front, 3D near-wall back-flow after the jet, and the formation of counter-rotating vortex pair
on the sides. JICF has been widely studied in the past based on computational fluid dynamics
methods: Reynolds Averaged Navier Stokes (RANS) turbulence models (Roth et al. (1992),
Acharya et al. (2001), Albugues (2005)); Large Eddy Simulations (LES) (Yuan et al. (1999),
Frohlich et al. (2004), Ziefle and Kleiser (2009)); Direct Numerical Simulations (DNS) (Mup-
pidi (2006), Hattori et al. (2014)). Most of these studies looked at the mean heat transfer profiles
which were found to be in good agreement with the experimental data. However, little or no
focus was given to the temperature fluctuations especially at the surface of the wall and inside it,
which can lead to severe thermal stress fatigue particularly in the context of cooling injections in
power plant safety procedures. In relation with the latter application, the current study focuses
on a low momentum injection such that the jet is almost attached to the wall which would have
a complex thermal effect on the pipe or vessel wall and for which there is virtually no detailed
experimental data. DNS of JICF with passive scalar mixing is carried out to observe the mixing
phenomenon in this fully-turbulent flow. To investigate the effect of wall thermal condition two
different kinds of wall thermal boundary conditions are tested; fixed wall temperature (isoT) and
adiabatic wall (isoQ). The results include instantaneous and time-averaged profiles which are
invaluable for the development and validation of engineering turbulence models. Four different
RANS models are also tested and the results are compared to the DNS data.
2. Physical and numerical parameters
2.1. Flow configuration
Fig. 1 shows the schematic view of the flow configuration of JICF and the coordinate system.
Hereafter, the subscript notation “c” and “j” represent the cross flow and the jet, respectively,
and h·i denotes the time-averaging operator. For all the simulations the channel width is kept at
2Turbulence, Heat and Mass Transfer 8
Recycling in DNS
10.59H6.75H
24H
2H
x
y
Oz
U∞
Uj
2H
Figure 1: Schematic view of the flow configuration of jet in channel flow in DNS. The centre of
jet-exit is the origin point. A recycling method is used before the jet to generate a fully-turbulent
inflow
2H, where His the radius of the jet. The dimensions of the complete computational domain
are 24H,2Hand 6Hin the x,y, and zdirections respectively. The origin point Ois located
at the centre of jet exit at the bottom wall. The Reynolds number based on the cross flow bulk
velocity ucand half channel width His set at 3,333 and is defined as
Re = ucH
ν= 3333,(1)
where νis the kinematic viscosity. The corresponding friction Reynolds number Reτis 211.
The radius of the jet exit rj=Hequals to half of the channel width. The jet bulk velocity uj
is set to be uc/6, which leads to the Reynolds number based on the jet bulk velocity and jet-exit
diameter as 1,111. Thus, the jet is considered as laminar, and a parabolic profile can safely be
imposed. The jet velocity profile is defined as
uj(r) = 2uj(1 −r2
r2
j
), r = [0, H],(2)
where ris the distance to the jet exit centre.
The Prandtl number Pr = ν/α is 0.7 in this study, where αis the fluid thermal diffusivity.
All temperatures are normalised by the temperature differences between the jet and the cross
flow (θ= (T−Tc)/(Tj−Tc)).This leads to the dimensionless temperature to become zero
in the cross flow and unity for the jet. Two types of wall thermal boundary condition were
investigated: adiabatic (isoQ) and fixed temperature (isoT). For both the DNS and RANS, cold
flow simulations were first run. Once the flow was fully developed, thermal boundary conditions
were then imposed. The initial temperature was set to zero for the whole domain.
In DNS, the computational domain is discrete by 513 ×193 ×256 grid points in the x,y,
and zdirections, respectively. The grid points are uniformly distributed in xand zdirections,
but stretched in ydirection such that the near wall region is well refined. The grid spacing in
wall unit is ∆x+= 10,∆y+= 0.7∼7, and ∆z+= 5. The time step is 10−4H/uc. The
time-averaged DNS data was accumulated in time over 400H/ucFor the RANS simulations, a
block structured mesh is generated with O-mesh pattern near the jet leading to a total cell count
of 2.58 million. For this RANS mesh the height of boundary layer cells is 6.6×10−3at the
wall, with a growth rate of 1.5 until reaching the maximum cell size 0.06H. Some important
flow and mesh parameters are presented in Tab. 1.
Z. Wu et al. 3
Table 1: Main parameter of JICF simulations
Calculations DNS RAN
Inlet condition Recycling tech. Precursor RANS simulations
Outlet condition Convective b.c. Pressure outlet
Thermal b.c. Adiabatic or T= 0 Adiabatic or T= 0
Grid Spacing in wall unit [∆x+,∆y+,∆z+] = [10,0.7∼7,5] 1.39 at wall; 12.66 max
2.2. Numerical treatment
In the present study, the fluid properties are constant so that the fluid is incompressible and
the temperature is regarded as a passive scalars. The governing equations are the continuity
equation, the Navier-Stokes (NS) equation and the energy equation:
∇ · u= 0,(3)
∂u
∂t +1
2[∇(u×u)+(u· ∇)u] = −∇p+1
Re∇2u,(4)
∂θ
∂t +u· ∇θ=1
RePr∇2θ, (5)
where uis the dimensionless velocity vector, tis the time, pis the dimensionless pressure, and
θis the dimensionless temperature. All variables are normalized by cross flow bulk channel
velocity ucand channel half width H.
In this study the DNS simulations are performed with Incompact3d; an open-source code de-
veloped by Laizet and Lamballais (2009) and Laizet and Li (2011) from Universit´
e de Poitiers
and Imperial College London. All governing equations are solved on a collocated velocity grid
via the six-order central compact finite different scheme (Lele 1992), while the pressure is on a
staggered grid. A second-order Adams-Bashforth (AB) scheme is used for time-advancement
and continuity is verified at the end of each sub-time step by solving a pressure Poisson equa-
tion. This Poisson equation is solved through a spectral solver to avoid the expensive cost of
applying high-order scheme combined with iterative techniques. The solver is highly accurate
and has been benchmarked in the past; Laizet and Lamballais (2009), Flageul (2014) .
RANS calculations are carried out with the commercial CFD software, STAR-CCM+ v10.02.
Four kinds of RANS models are used: they are realizable k−(Shih 1994), SST k−ω(Menter
1994), EB-RSM (Lardeau and Manceau 2014), and EB-EVM k−(Billard and Laurence 2012,
Billard 2012). For the turbulent heat flux modelling, it is simply < uiθ >=−αt∂θ/∂xi, where
the turbulent heat diffusivity αtis modelled as αt=νt/Prtwith νtbeing the turbulent viscosity
and Prtthe turbulent Prandtl number. As seen later, the velocity and turbulence kinetic energy
predictions by RANS are so far from DNS, there is no point in trying more elaborate turbulent
heat flux models. Their predictions of the mean profiles will be compared with DNS data.
2.3. Outlet conditions
For the DNS a simple 1-D convective boundary condition is imposed at the outlet for both the
velocity and the temperature:
∂u
∂t +uo(y)∂u
∂x = 0,(6)
∂θ
∂t +uo(y)∂θ
∂x = 0.(7)
4Turbulence, Heat and Mass Transfer 8
As suggested by Ol’shanskii and Staroverov (2000), the choice of a laminar Poiseuille flow
profile for uo(y)has little effect and can be numerically quite robust, despite the differences
between a laminar Poiseuille profile and the mean fully-turbulent profile, which is the case for
the current study. Thus uo(y)is defined as:
uo(y) = 3
2(1 −y2
H2)uc.(8)
2.4. Inlet conditions
In present simulation, the Reynolds number of cross flow (based on the cross flow bulk velocity
ucand half channel width H) is set as 3,333, which is larger than the critical value for turbulent
flow. In industrial applications, the cross flow Reynolds number could be even larger. Thus, a
fully-developed turbulent flow is expected before the jet exit. Several methods can be applied
to achieve this, but most of them have significant drawbacks. Once could use a long upstream
channel before jet exit to make the flow transition from laminar to turbulent. However, this is
computationally very expensive as natural transition is a slow process. Another option is to
run a pre-cursor or parallel DNS simulation of a periodic channel flow to obtain the turbulent
flow profiles, which is again computationally very expensive and requires a lot of data transfer
at each time step. For the current simulations the most viable option seemed to be a internal
recycling approach as proposed by Lamballais (2014). As illustrated in Fig. 1, the velocity at a
cross section xris copied to the inlet plane to be used as the Dirichlet inlet boundary condition.
This process can be mathematically expressed as:
un+1(0) = un(xr),(9)
where the superscript notation denotes the time steps. Thus, un+1 (0) is the velocity at location
x= 0 at time step n+ 1, whereas un(xr)is the velocity at location x=xrat time step n; This
technique quite simple and easily achievable. To numerically achieve this only a short upstream
length is required to ensure mimicking of a periodic channel flow. In present study, the stream-
wise length of the recycling region is set to 10.59H= 3.37πH, which was found sufficient for
performing DNS of a periodic channel flow. One may notice the strong interactions of cross
flow and jet at the windward side of jet exit might “contaminate” the recycling region, if the
jet exit is too close to the end section of recycling region. Preliminary tests have shown that
this upstream interaction only exists within 2D= 4Hupstream. So the jet exit is safely set at
6.75Hdownstream of end section of recycling. Further details and the validation of this internal
recycling technology can be found in Lamballais (2014).
For the RANS simulations, the inlet condition is straightforward. Precursor periodic channel
flow simulations using four different RANS models were performed. After desired convergence
was achieved, these four precursor mean profiles were then applied to the corresponding final
JICF-RANS simulations. For all RANS precursor simulations and DNS recycling domain the
flow is driven by imposing a constant mass flow rate.
3. Dynamic structures of JICF from DNS
Owing to the interaction of jet and cross flow, there are several dynamic structures in JICF,
including the horseshoe vortex, shear layer vortices, a counter-rotating vortex pair (CVP), and
the recirculation behind the jet. To understand the mixing phenomena of fluid flow and tem-
perature in JICF, these structures are studied and presented in Figs. 2 and 3. In Fig. 2a, six
vortices are identified at the upper jet and cross flow interface. They are termed as shear layer
Z. Wu et al. 5
(a)
(b)
Figure 2: Instantaneous profiles: (a) temperature contour on symmetry plane, (b) iso-surface
of Q-criterion Q= 25 coloured by temperature. Both plots are at the same time step (t=
16.8H/uc).
vortices, and labelled as (a)-(f). These vortices are also pointed out in Fig. 2b, which shows the
instantaneous iso-surface of Q-criterion at the same time step as that of Fig. 2a. The Q-criterion
iso-surfaces are coloured by temperature. In the near field, the temperature of the shear layer
vortices is changing quickly: inside a particular vortex, sharp change of the temperature from
low (dark blue) to high (red) are seen, then further downstream mid-range value (green) colours
the vortices; this indicates a strong mixing there.
From the flow field a strong horseshoe vortex can also be seen. This comes from the reverse
flow of cross flow. As the jet exits from the hole, it behaves like a wall-mounted cylinder
obstacle. When the cross flow encounters this obstacle, an adverse pressure gradient builds
up there which forces the cross flow reversal very near the wall in the low inertia layer of the
channel flow and forms the horseshoe vortex. This mechanism of the formation of the horseshoe
vortex is also reported by Muppidi (2006). The horseshoe vortex is at low temperature proving
it originates from cross flow. Other large structures can also be seen in the snapshots. In the
downstream far field, the vortex temperature becomes mid-range leading to the conclusion that
the jet and the cross flow have already fully mixed. In present simulations, it was also noticed
that as the shear layer vortex rolls up, it has an entrainment effect on the horseshoe vortex.
That is to say, the roll-ups of shear layer vortex affects the unsteady characteristics of horseshoe
vortex, a feature previously reported by Perry et al. (1993), who noted that the unsteadiness of
the horseshoe vortices is also affected by the cycle of the jet shear layer.
Time-averaged profiles in Fig. 3 show two other main structures in JICF, the counter-rotating
vortex pair (CVP) and the recirculation behind the jet. In Fig. 3a, the vorticity in the stream-
wise X-direction hωxi=∂hvzi/∂y −∂hvyi/∂z is shown at several cross sections. It can be
clearly seen that there are two vortices with opposite signs of ωx. This opposite-sign vortex pair
is termed as the counter-rotating vortex pair (CVP). In the present simulations, though the jet
pipe is not computed, the CVP still appears. This supports Muppidi’s (2006) model claiming
that the jet pipe is not essential for the formation of a CVP. Since it is impossible to build a JICF
6Turbulence, Heat and Mass Transfer 8
(a) hωxicontours on several yz slices
(b) surface streamline on symmetry plane z= 0
(c) surface streamline at y= 0.2(d) 3D streamline with temperature contour
Figure 3: Time-averaged DNS data accumulated in time over 400H/uc
experiment without the jet pipe to confirm this model, DNS (numerical experiments as accurate
as experiments), may be the only tool available to examine this. To the best of the author’s
knowledge, this is probably the first DNS computational study confirm their model predictions.
In Fig. 3b, a small bubble is found behind the jet, then a long back-flow is seen to originate
from x= 3 all the way back to x= 1 which can also identified in Fig. 3c. The 3D streamlines
(Fig. 3d) would help explain that the back-flow behind the jet comes from the sides and thus
injects a thin layer of cold fluid which “protects” the wall. This recirculation is important in
temperature mixing of JICF which significantly affects the wall temperature variance which in
the longer run can lead to or prevent material thermal failure. Further flow physics is discussed
in the next section.
4. Comparison of RANS with DNS data
Another object of this study is to evaluate the performance of RANS model predictions in JICF.
Several quantities from RANS simulation are compared with DNS data. These comparisons are
presented through Figs. 4 to 7.
4.1. Mean profiles over whole domain
For velocity field, the jet trajectory, stream-wise velocity huiand turbulence kinetic energy hki
are indicated in Figs. 4, 5a and 5b, respectively.
Trajectory is the most basic characteristic of any jet. In this study, the jet trajectory is defined
as the streamline developing from the centre of jet exit i.e. the coordinate origin point O, as
Z. Wu et al. 7
Figure 4: Jet trajectory defined as streamline developing from centre of jet exit. Solid: DNS;
: EB-EVM; +: EB-RSM; ×: k-;: k-ω.
shown in 4. In DNS, the jet trajectory shows a local maximum at x= 2.5and local minimum at
x= 3.9, after which it shows an upwards trend. It is surprising to note that the best prediction
of the jet trajectory is given by k−, whose result fits well with the DNS data in quantitative.but
other aspects of Other this models predictions are only roughly agree with DNS.
Due to the cross flow vein contracting effect of jet, higher velocities are observed in the
upper half channel (see Fig. 5a). In the upstream and the downstream near field (x < 2.5), all
RANS model velocity predictions are quite well; good quantitative agreements with DNS data.
However, further downstream the recovery after the recirculation bubble is much too slow with
RANS models. Only the k−and the EB-RSM show agreements, while some significant gaps
can be seen by other RANS models: for example, the EB-EVM and the k−ωshow slightly
negative velocities for a long range of 1.5<x<4.5while in DNS this is limited to x < 3(see
also Fig. 5a).
On the other hand Turbulence kinetic energy kis severely underestimated predicted by all
RANS models (see Fig. 5b). Again, the performance of the k−model seems to be not as bad
as that of the other models, especially in the downstream region where the mixing of the jet and
the cross flow is strong. The velocity is badly recovered after the recirculation as the a much
lower kinetic energy is predicted by all RANS models; these predictions of uand kfor RNAS
are typically observed for the transverse flow past a wall-mounted cylinder (Afgan et al. 2007).
The mean temperatures are compared in Figs. 5c and 5d. All RANS models are in slightly
better agreements with the DNS in both the isoT and isoQ cases. However, big differences can
still be observed at y= 0.6, x = 2.5, where hkiis also severely underestimated. Note that
in this passive scalar case the flow field is same in both cases so it is only the isoQ thermal
boundary condition that cools the fluid more efficiently than in the isoT case.
4.2. Wall statistics
For most of the industrial applications, wall friction and wall heat flux or temperature are often
the main objectives of the study and higher order statistics of wall quantities provide information
for e.g. evaluating potential material thermal failure risks. As the flow separation is an important
structure in the JICF flow, and makes a major impact to the wall temperature, the distributions
of friction coefficient hCfion the bottom wall is first checked. Skin friction is defined as
hCfi=hτwxi/(ρuc2/2), where τw x is the local stream-wise wall shear stress.
Fig. 6a shows the skin friction coefficient on the bottom wall at z= 0. The plot is piecewise
8Turbulence, Heat and Mass Transfer 8
(a) stream-wise velocity hui
(b) turbulence kinetic energy hki
(c) temperature hθiin isoT case
(d) temperature hθiin isoQ case
Figure 5: Time-averaged profiles on symmetry plane. Solid: DNS; : EB-EVM; +: EB-RSM;
×: k-;: k-ω.
Z. Wu et al. 9
(a) Skin friction coefficient hCfi
(b) Wall temperature hθwifor the isoQ case (c) Nu number for the isoT case
Figure 6: Time-averaged wall thermal statistics on bottom wall at z= 0. Solid: DNS; :
EB-EVM; +: EB-RSM; ×: k-;: k-ω.
continuous since the jet exit is located at x= [−1,1]. For the DNS, negative hCfivalues
indicate that the flow separates at three different locations: horseshoe vortex location (x=
[−1.36,−1]), a small recirculation bubble just behind the jet exit (x= [1,1.13]), and a dominant
backflow region at x= [1.64,3.66] due to 3D effects as discussed regarding Fig. 3b and 3d.
Except the EB-RSM that does not predict negative hCfibefore the jet, the three other RANS
models give good agreements in the horseshoe before the jet exit (x < 1), but all models are
relatively worse downstream of the jet. To be specific, k−has a good qualitative agreement
in the dominant backflow region, but under-predicts hCfiafter that. By contrast, EB-RSM
predicts significant smaller value of hCfiinside the recirculation region despite showing a good
recovery downstream. EB-EVM and k−ωresults are the worst, both of them showing a much
longer extent of backflow in opposition to the rapid recovery given with DNS, as commented
regarding the velocity profiles.
The time-averaged wall temperature hθwiand Nusselt number are shown in Figs. 6b and 6c
respectively. Here the reference temperature for Nu number is set as θref =θj= 1. One should
note that wall temperature is only available for the isoQ case whereas the Nu number is only
available for the isoT case.
For the isoQ case, wall temperatures hθwipredicted by all RANS models are acceptable for
10 Turbulence, Heat and Mass Transfer 8
(a) EB-EVM (b) EB-RSM (c) k−
(d) k−ω(e) DNS (f) colour map
Figure 7: Nu number in isoT case.
engineering applications. It is not surprising that as the k−model works well in solving wall
temperatures, as it gave better velocity predictions than the other models. However, regarding
to the k−’s bad recovery of hCfiin the far field, its temperature predictions are a bit off there.
The other three models predict the wall temperature fairly well across the symmetry plane.
For the isoT case, Nu number reaches a maximum of 10 at x= 3 in the DNS. All the RANS
models are significantly deviate; they under-predict Nu inside the backflow by factors of 2 to
3. Fig. 7 shows Nu over the whole wall. In fact Nu is higher (Nu = 20) in 2 steaks originating
from the sides of the jet. All RANS models under-predict somewhat the width of this high
heat transfer streak. The k−ωshows a high Nu value around the front half of the jet which is
not present in the DNS. The k−ωand especially the EB-EVM suggest some vortex shedding
although these are steady-state RANS. Unsteady RANS should be tested, but the economy
advantage of URANS over LES is reduced. The 3D streamline (Fig. 3d), shows that the the
colder cross flow wraps around the jet and injects a thin layer of cold fluid which “protects” the
wall on the symmetry axis.
The wall temperature variance (ht0t0i) (Fig. 8) is only available from the DNS results as
second moment closure for the thermal field was not attempted give the rather poor RANS
predictions on first moments. However this variable is very important for thermal stress fatigue
in the solid wall. Temperature fluctuations variance is similar to the Nu number map. Both are
quite symmetrical, indicating the DNS solutions are well converged. These footprint maps with
2 streaks starting from the sides of the jet can be associated with the horseshoe vortex visible on
hωxiin Fig. 3a. However the strength of the latter linked to the upstream boundary layer profile
inhomogeneity would be reduced at higher Re numbers.
5. Conclusions
To the best of the author’s knowledge, this is the first high-order DNS investigation involving
the JICF mixing phenomenon. One DNS and four RANS calculations of JICF have been carried
out. The RANS models results are compared with the DNS which is used as a reference.
Z. Wu et al. 11
Figure 8: Wall temperature variance ht0t0iin DNS
From an industrial perspective wall thermal stresses are a crucial issue in designing the T-
pipe junctions. For the present DNS case, a realistic turbulent inflow condition was obtained
using a simple recycling technology. The open source code “Incompact3d” allows the authors
to build up further data for the Jet in Cross Flow (JICF), which could be used as a benchmark
for developing and validating new RANS, LES or hybrid models for such flows.
The velocity fields obtained by all RANS models quantitatively agree with the DNS as re-
gards the Nu number map on the wall. Quantitatively however predictions of kinetic energy and
the temperature and particularly Nu profiles on the symmetry axis are found to be unsatisfactory.
The present transverse jet case is known to be highly challenging because it involves strong
mixing at the jet edges, which causes the flow near the jet-exit and in the downstream near-field
region to be far from equilibrium with a number of steady and unsteady large structures. Theses
features make the linear RANS models fail. The EB-RSM model is expected to predict the
flow better since anisotropy is taken into account in this model. However, the results show that
EB-RSM models predictions are hardly any better than other RANS models.
The lack of similarity between mean velocity and temperature (respectively parabolic and
top hat at the jet exit, Dirichlet and Neumann wall boundary conditions) probably makes it
harder than the wall mounded cylinder or wing body junction aerodynamic test cases. It can
safely be concluded from the results of this study that using steady state RANS models one can
only provide very qualitative estimations for such a complex flow, with errors on the wall heat
exchange coefficient exceeding 200% locally.
Acknowledgements
The authors acknowledge the use of the facilities of N8 HPC provided and funded by the N8
consortium and EPSRC (Grant No.EP/K000225/1), of ARCHER HPC computer supported by
UK EPSRC Turbulence Consortium. The authors are grateful to Cedric Flageul for help in
implementing heat transfer in the DNS code.
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