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

NUMERICAL SIMULATIONS OF NON-NEWTONIAN JETS

Giovanni Soligo
Complex Fluids and Flows Unit
Okinawa Institute of Science and Technology Graduate University
1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, 904-0495, Japansoligo.giovanni@oist.jp

Marco Edoardo Rosti
Complex Fluids and Flows Unit
Okinawa Institute of Science and Technology Graduate University
1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, 904-0495, Japan
marco.rosti@oist.jp

ABSTRACT

we will focus on a relatively unexplored region of the parameter space: the effect of increasing Weissenberg numbers at
low Reynolds number will be investigated. We consider threedifferent fluid models: Carreau (shear-thinning), Giesekus(shear-thinning and viscoelastic) and Oldroyd-B (viscoelastic). A reference simulation of a Newtonian fluid completes
the database.
We compute the classical jet statistics, namely the centerline velocity and jet thickness; previous works with a FENE-P
model (shear-thinning and viscoelastic fluid) showed that thesebulk quantities recover the power-law scalings obtained for a
Newtonian fluid in laminar (Parvar et al., 2020) and turbulent(Guimarães et al., 2020) planar jets. The universal validityof these scaling laws can be traced back to their derivation: no
hypothesis on the fluid model is taken, hence it is expected thatthey still hold true for non-Newtonian fluids and apply as wellto different non-Newtonian fluid models. We then proceed toquantitatively characterize the different flowing patterns observed at the highest value of the Weissenberg number. It isclear that when the fluid is characterized by flow elasticity, adifferent power-law scaling appears in the energy spectrum of
the turbulent kinetic energy.
In the following, the numerical method we use for our
simulation and the simulation setup are reported in the firstsection. We then present the main results from our numerical simulations, starting from a qualitative visualization of thestream-wise velocity field at steady-state, moving then to thebulk statistics and lastly to the turbulent kinetic energy spectrum. In the last section, the main findings are summarized.

Jets are a type of flow representative of more complex
flow configurations commonly found in nature and in industrial applications. Their importance led to a significative
progress in the understanding of the dynamics of jet flows
via analytic solution of simplified equations, numerical simulations and experiments. Most of these studies however focused on Newtonian fluids, while very few works involved
non-Newtonian fluids. These latter fluids are characterizedby complex fluid properties, as for instance fluid elasticity or
shear-dependent viscosity, which lead to the onset of a richer
flowing behavior. Our aim is to advance the current understanding of jets of non-Newtonian fluids through direct numerical simulations of different non-Newtonian fluids at low
Reynolds numbers. We observe an earlier transition to a turbulent regime, here intended in a more general meaning as disordered fluid motions. This regime is qualitatively different
from Newtonian turbulence; turbulent kinetic energy spectra
allow us to provide as well a quantitative characterization ofthis flowing regime.

INTRODUCTION
The flow of non-Newtonian fluids exhibits very different
flowing patterns depending on the flowing conditions; recent
experiments (Yamani et al., 2021) showed the onset of substantially different flowing pattern when the Reynolds (ratio ofinertial over viscous contributions) and Weissenberg (ratio of
the polymer relaxation time over a flow time scale) numbers
were changed. The observed flow is the result of the competition of inertial, viscous and elastic contributions, ranging
from laminar flow, Newtonian turbulence, elasto-inertial filaments and elasto-inertial turbulence. The onset of disordered
fluid motions in non-Newtonian fluids is also observed at a
Reynolds number much lower for what observed for Newtonian fluids in the very same flowing configuration. This finding hints at the existence of a different pathway to turbulence;
it has been proven that fluid elasticity is indeed a key elementin triggering flow instabilities (Joo & Shaqfeh, 1994; Kumar
et al., 2021; Walkama et al., 2020; De et al., 2017; Grilli et al.,2013; Joo & Shaqfeh, 1992; Khalid et al., 2021). In this work

COMPUTATIONAL SETUP
To simulate the dynamics of the planar jet we use direct numerical simulations of the Navier-Stokes and continuity equations, respectively equations 1 and 2, coupled with atransport equation for the non-Newtonian extra-stress tensor,
equation 3. We use u as the fluid velocity, t is time, ρ is thefluid density (homogeneous in space and constant in time), ηsand η p respectively are the solvent and polymeric dynamic viscosity, p is the pressure, τ is the non-Newtonian extra-stresstensor, λ is the polymer relaxation time scale and α is the mo-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Giesekus
Carreau
Oldroyd-B(a)Giesekus
Carreau
Oldroyd-B

(c)

ηe

10−1

Giesekus
Carreau
Oldroyd-B

(b)

10−2

10−3
10−2

10−1

100

101

102 10−2

10−1

100

γ̇

101

γ̇

102 10−2

10−1

100

101

102

γ̇

Figure 1. Rheology of the different fluid models for increasing Weissenberg numbers; from left to right panel (a) Wi = 1, panel (b)Wi = 10 and panel (c) Wi = 100.

artificial diffusivity can be added to the equation to damp the
disturbances (Alves et al., 2000; Mompean & Deville, 1997).
This however reduces the accuracy of the calculations (Dubief et al., 2005). We opt instead for the matrix-logarithm
formulation of the transport equation of the non-Newtonian
extra-stress (Fattal & Kupferman, 2004; Hulsen et al., 2005);
this methodology, while being computationally expensive, as
it requires solving an eigenvalues/eigenvectors problem on theconformation tensor, is however exact.
The equations are discretized on a staggered, uniform,
Cartesian grid; velocity data is stored at the cell faces, while
all other variables at the cell center. A second-order finite
difference scheme is used to approximate spatial derivative;
for the advection term of the extra-stress transport equationa fifth-order WENO scheme is instead adopted (Shu, 2009;
Sugiyama et al., 2011). The equations are discretized in timewith a second-order, explicit Adams-Bashforth scheme. Timeadvancement is performed via a fractional-step method (Kim
& Moin, 1985), coupled with a fast pressure solver (Dodd &Ferrante, 2014) for the Poisson equation for the pressure.
All simulations have been performed using the in-housecode Fujin, which has been extensively used and validated
in the past (Brizzolara et al., 2021; Mazzino & Rosti, 2021;
Olivieri et al., 2020; Rosti et al., 2021a,b; Rosti & Brandt,
2020; Rosti et al., 2019, 2020). Additional validation tests areavailable at https://groups.oist.jp/cffu/code.
To investigate non-Newtonian effects on the dynamic ofa planar jet we perform numerical simulations of three different non-Newtonian fluid models, namely Carreau, Giesekus
and Oldroyd-B. This way the effects of shear-thinning, fluid
elasticity and of their combined effect can be sorted out. The
rheology of the different fluid models is reported in figure 1
for all the simulated Weissenberg numbers. The Reynolds
number, computed using the inlet velocity vin , the half-heightof the slit through which the fluid is issued h and the zeroshear viscosity, is Re = 20. An additional, reference simulation of a Newtonian fluid is also performed; at this value of
the Reynolds number we observe a laminar flowing regime forthis latter case. We perform two-dimensional simulation at low
Weissenberg number in a domain of size 160h × 240h (streamwise × jet-normal directions) discretized with 1536 × 2304
grid points. The simulations at the highest Weissenberg number are performed on a three-dimensional domain of size
160h × 240h × 13.3h (stream-wise × jet-normal × span-wise
directions), discretized using 1536 × 2304 × 128 grid points.
In the following the x coordinate denotes the span-wise direction, y the stream-wise direction and z the jet-normal direction.
The jet is issued from a slit of height 2h on the left sideof the domain into a pool of the same fluid at rest. No-slip andno-flux boundary conditions are applied at the left boundary,
except at the slit through which the fluid is issued. Free-slip

bility parameter of the Giesekus fluid (equal to zero for the
Oldroyd-B fluid model).

ρ


h 
i
∂u
+ u · ∇u = −∇p + ∇ ηs ∇u + ∇uT + ∇ · τ (1)∂t∇·u = 0∇τ +λτ +αλ
τ · τ = η p ∇u + ∇uT
ηp

(2)

(3)

∇

The upper-convected derivative, τ, is defined as follows.
∇

τ=



∂τ
+ u · ∇τ − ∇uT · τ + τ · ∇u∂t

(4)

We solve the extra-stress transport equation only for the
Giesekus and Oldroyd-B fluid models. The Carreau fluid, on
the other hand, does not include any polymer-induced extrastresses; the solvent viscosity ηs however depends on the local
shear rate, γ̇, as reported in equation 5.
h
i n−1
2
ηs = η∞ + (η0 − η∞ ) 1 + (λ γ̇)2(5)Here the viscosities at zero shear rate and at infinite shear rate
are respectively η0 and η∞ . For the Carreau fluid the parameter
λ is the fluid consistency index and carries no information on
flow elasticity. However, in the sake of simplicity, we will
still refer to the ratio of the fluid consistency index over a flow
time-scale as the Weissenberg number. Lastly, the power-law
index, n, defines whether the fluid is shear-thickening (n > 1)or shear-thinning (n < 1). In this work we consider a shearthinning fluid, with power-law index n = 0.2.
A common issue that arises when solving the transport
equation for the extra-stress tensor, equation 3, is the highWeissenberg number problem (Keunings, 1986), which leads
to the unphysical loss of positive-definiteness of the conformation tensor. This issue occurs at large Weissenberg numbers
and manifests as disturbances amplifying over time in the conformation tensor (Dupret & Marchal, 1986; Min et al., 2001;
Sureshkumar & Beris, 1995). Different solutions have beensuggested to address the high-Weissenberg number problem:2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
120

(a)
80
40
z

0

−40
−80
−120

0

40

−0.5

0

80
y

v

120

0.5

20
10
z 0
−10
−20
20
10
z 0
−10
−20
20
10
160
z 0
−10
1 −20

0

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

(j)

20

40

y

60

80

1000

20

40

y

60

80

1000

20

40

y

60

80

100

Figure 2. Stream-wise fluid velocity v at steady-state conditions. The Newtonian reference case is reported in panel (a) on the fulldomain; the red box highlights the region of the domain that is shown in the panels (b)-(j). Each column refers to a different fluid
model, Carreau in panels (b), (e), (h), Giesekus in panels (c), (f), (i) and Oldroyd-B in panels (d), (g), (j). Each row correspond to adifferent Weissenberg number: from top to bottom Wi = 1 in panels (b)-(d), Wi = 10 in panels (e)-(g) and Wi = 100 in panels (h)-(j).

boundary conditions are imposed at the top and bottom boundaries. A non-reflective boundary condition (Orlanski, 1976) is
enforced at the outlet section, right boundary.

ological curve to the Carreau fluid, however fluid elasticity
plays an important role as well. The lower value of the local
viscosity leads to an increase of the local Reynolds number,thus justifying the transition to a turbulent-like regime. On thecontrary, the viscosity of the Oldroyd-B fluid is constant, thus
the disordered fluid motions we observe can be traced backto the fluid elasticity alone. Despite the very different flowing regime and rheological curves of the various fluid models,
it is interesting to note that the bulk statistics still obey thepower-law scalings computed for planar jets of a Newtonian
fluid in both laminar and turbulent regime. This result may be
unexpected at first, however no assumption on the actual fluid
model is made in the derivation of these scaling laws (Parvar
et al., 2020; Guimarães et al., 2020). Figures 3 and 4 report respectively the decay of the centerline velocity and the increasein the jet thickness along the stream-wise direction. The jet
thickness, δ0.5 , is here defined as the distance from the jet centerline at which the time- and space-averaged velocity equals
half the centerline velocity at the same stream-wise location.
At the lowest Weissenberg number, Wi = 1, non-Newtonian
effects are weak and viscous contributions dominate the flow.
The qualitative visualization shows no difference between the
Newtonian reference case and the three non-Newtonian fluid
models. This finding is confirmed also by the bulk statistics:
the centerline velocity, figure 3(a), and the jet thickness, figure 4, fall on top of each other for all cases, following thelaminar power-law scalings, vc ∝ y−1/3 and δ0.5 ∝ y2/3 .

RESULTS
The stream-wise velocity for all simulated cases is reported in figure 2. It can be immediately noticed the effectof the Weissenberg number, increasing from the top to thebottom row, and of the fluid model. Only a portion of thedomain is reported in panels (b)-(j), bounded by the red box
in panel (a). At low Weissenberg number, top row, the nonNewtonian contribution is weak and all cases exhibit a laminarflowing regime. As will be quantitatively shown in the following, no appreciable difference is seen among the various cases.
As the Weissenberg number is increased to Wi = 10, middle
row, the onset of an instability can be seen for the Oldroyd-B
model; at this intermediate Weissenberg number fluid elasticity starts playing a dominant role over viscous contributions.
For the Carreau fluid model, the effect of shear-thinning is
still too weak to trigger any flow instability; similarly for the
Giesekus fluid model. The laminar flowing regime displayedby the Giesekus fluid at this value of the Weissenberg number hints that while viscoelasticity promotes the transition to a
turbulent-like regime, shear-thinning, competing against fluid
elasticity, hinders or at least delays such transition. As wemove to the highest Weissenberg number, Wi = 100, we canimmediately notice a striking difference among the three nonNewtonian fluid models: while all cases are characterized bydisordered, turbulent-like fluid motions, the flowing pattern isvery different. Indeed, the mechanisms generating these disordered fluid motions are different: for the Carreau fluid (shearthinning) it is the competition between inertial and viscous
contributions, for the Giesekus fluid (shear-thinning and viscoelastic) by the competing effect of inertial, elastic and viscous contributions, while for the Oldroyd-B (viscoelastic) by
the competition of fluid elasticity and viscous terms.
The flow visualization, figure 2, clearly shows importantdifferences among the various fluid models, in particular at
higher Weissenberg numbers, where the contribution from thenon-Newtonian component is stronger. The Carreau fluid, being a shear-thinning fluid, exhibits turbulence motions moresimilar to classical, Newtonian turbulence as the local viscosity is decreased. The Giesekus fluid has a very similar rhe-

As the Weissenberg number is increased to Wi = 10(intermediate Weissenberg number), effects from the nonNewtonian component, namely shear-thinning, fluid elasticity or both combined, become stronger. For the Carreau fluid
the reduction in the local viscosity is not sufficient to destabilize the flow and transition to a turbulent-like regime: further away from the inlet the laminar scalings are recovered forboth the centerline velocity and the jet thickness. It is interesting to note that memory of the inlet conditions is carried overa longer stream-wise distance: while for the Newtonian casethe laminar solution is recovered from about y = 5, for the
non-Newtonian cases transition towards the power-law scalings occurs further away once the non-Newtonian contribution
becomes dominant. Similarly, the Giesekus fluid follows thelaminar power-law scalings for both the centerline velocity andthe jet thickness at this intermediate Weissenberg number. On
the contrary, the Oldroyd-B fluid shows an initial turbulent-3

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

(b)

(c)

vc

1

0.5
Newtonian
Giesekus
Carreau
Oldroyd-B
y −1/3100

y −1/3y −1/2
101

102

100

y −1/2
101

y

102

100

101

y

102

y

Figure 3. Centerline velocity for different Weissenberg numbers, panel (a) Wi = 1, panel (b) Wi = 10 and panel (c) Wi = 100.
The analytic power-law scaling are reported with a dashed line (laminar scaling, vc ∝ y−1/3 ) and dash-dotted line(turbulent scaling,vc ∝ y−1/2 ).
y 2/3yNewtonian
Giesekus
Carreau
Oldroyd-B
y 2/3

δ0.5

101

y 2/3
y

100

(a)

100

(b)
101

102 100

y

(c)
101

102 100

y

101

102

y

Figure 4. Jet thickness for different Weissenberg numbers, panel (a) Wi = 1, panel (b) Wi = 10 and panel (c) Wi = 100. The analytic
power-law scaling are reported with a dashed line (laminar scaling, δ0.5 ∝ y2/3 ) and dash-dotted line(turbulent scaling, δ0.5 ∝ y).
like flowing regime closer to the inlet, at 10 < y < 40, followed by a transition towards a laminar scaling beyond y = 40.
We computed the Deborah number (ratio of the polymer timescale over a local flow time scale, not shown here) using local
quantities: the relaxation time of the polymer, λ , and the local
flow time scale δ0.5 /vc . The value of the Deborah number islarger than one up to y = 40 and then reduces below one beyond this stream-wise location. This indicates that fluid elasticity is predominant up to roughly y = 40 (Deborah number
larger than one), then viscous contributions become dominant(Deborah number lower than one) and the flow transitions to alaminar-like regime. These different flowing regimes can also
be seen in figure 2(g) where the initial disordered fluid motions are dissipated roughly beyond y = 40. This result hints
at a competition between shear-thinning and fluid elasticity:
at the intermediate Weissenberg number, the Oldroyd-B fluid
shows an initial transition towards a turbulent regime, while
the Giesekus fluid model does not exhibit any flow instability,
suggesting that shear-thinning hinders, or at least delays, thetransition to a turbulent-like regime.

destabilizes the flow and originates a turbulent-like flow. Fluid
elasticity is instead the only factor promoting flow destabilization in the Oldroyd-B fluid: the stretching and retracting of thepolymers induce disordered fluid motions; despite the pathway
to this turbulent-like regime being very different from that of
Newtonian turbulence, we observe that the centerline velocity
and jet thickness still obey the computed power-law scalings
for a Newtonian turbulent planar jet.
We then proceed to quantitatively characterize the different flowing patterns we observed at the highest Weissenberg
number. We compute the turbulent kinetic energy spectra forthe three fluid models at Wi = 100; we record velocity dataover time at a fixed stream-wise location, y = 40, at the jet centerline and compute the spectra in time. Several authors haveobserved scaling exponents within the inertial range different
from that characteristic of the Newtonian turbulent energy cascade, χ = −5/3 (Balkovsky et al., 2001; Fouxon & Lebedev,
2003; Perlekar et al., 2010; Vonlanthen & Monkewitz, 2013;
Zhang et al., 2021). While most of these studies agree on asteeper decay law, there is yet no general consensus on the actual value of the scaling exponent, with theoretical predictions
by Balkovsky et al. (2001) and Fouxon & Lebedev (2003)
finding a value for the exponent χ ≤ −3 and experimental
measurements by Vonlanthen & Monkewitz (2013) reportinga power-law exponent χ = −3. Figure 5 reports the computed
turbulent kinetic energy power spectra for all fluid models at
the highest Weissenberg number. The Carreau fluid follows
a clear χ = −5/3 power law exponent in the inertial range:
the turbulent fluid motion observed is indeed akin to classical Newtonian turbulence, being originated by the predominance of inertial over viscous contributions. The Giesekus andOldroyd-B fluids instead show a marked χ = −3 power law
exponent; both fluid models are characterized by fluid elasticity, and shear-thinning for the Giesekus fluid alone. The measured scaling exponent well agrees with the measurements by
Vonlanthen & Monkewitz (2013); it is also interesting to note

Lastly, at the highest Weissenberg number all fluid models
exhibit a turbulent-like flowing regime. Although the flowing
pattern is qualitatively different among the various fluid models, the same power-law scalings for Newtonian turbulent planar jets is recovered for all fluid models. For the Carreau fluid
the local viscosity is lower, as indicated in figure 1(c), thus reducing the contribution of the viscous terms and leading to ahigher local Reynolds number. Transition to turbulence is thenpromoted by the predominance of inertial over viscous terms.
Further away from the inlet the shear rate is lower, thus leading
to a higher value of the local viscosity; turbulent motions arethen dissipated. The Giesekus fluid, having a similar rheological curve to the Carreau fluid, experiences a similar reductionin the value of the local viscosity. In addition, the polymers
within the fluid are more stretched, as indicated by the conformation tensor. The combined effect of these two factors

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
like flowing regime is the Weissenberg number, rather thanthe Reynolds number, as it quantifies the strength of nonNewtonian effects. We note that viscoelasticity alone seems
to promote the transition to a turbulent-like regime, with the
Oldroyd-B fluid showing turbulent motion already at the intermediate Weissenberg number, while shear-thinning seems tohinder the action of fluid elasticity, as suggested by the laminarflowing regime of the Giesekus fluid at the same Weissenberg
number.

100

P

10−2
10−4
10−6
10−8
10−10
10−3

Giesekus
Carreau
Oldroyd-B
f −3
f −5/3
10−2

10−1

100

101

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f

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Figure 5. Turbulent kinetic energy spectra for the three nonNewtonian fluid models at the highest Weissenberg number,
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CONCLUSIONS
We performed numerical simulations of a planar jet at low
Reynolds number using different non-Newtonian fluid models
characterized by shear-thinning (Carreau), shear-thinning andviscoelasticity (Giesekus) and viscoelasticity alone (OldroydB). We changed the strength of the non-Newtonian contribution via a dimensionless parameter, the Weissenberg number, defined as the ratio of the polymer relaxation time scale(or the fluid consistency index for the shear-thinning Carreau
fluid) over the flow time scale. For increasing values of the
Weissenberg number we observe that the flow, initially laminar, transitions to a turbulent-like regime. Flow visualization at steady-state conditions shows that the flowing regimes
are substantially different among the various fluid models:
for the Carreau fluid turbulence is a result of the competition between inertial and viscous terms, for the Giesekus fluid
it is among inertial, elastic and viscous terms, while for the
Oldroyd-B fluid it between elastic and viscous terms. Thisdifference is also clear from the power spectra of the turbulent
kinetic energy: the Carreau fluid shows the same inertial-range
power law exponent χ = −5/3 as Newtonian turbulence, while
the Giesekus and Oldroyd-B fluid models follow a χ = −3
power law decay in agreement with Vonlanthen & Monkewitz(2013). It is interesting to note that despite the different flowing regimes and energy spectra observed, the bulk statistics,
namely the centerline velocity and jet thickness, still follow
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to turbulence as the Weissenberg number is increased; here
the key parameter determining the transition to a turbulent-512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
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