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

DIRECT NUMERICAL SIMULATIONS OF TURBULENT PLANAR JETS
AND WAKES WITH FENE-P FLUIDS

Mateus C. Guimarães
Departamento de Engenharia Mecânica
IDMEC/LAETA, Instituto Superior Técnico,
Universidade de Lisboa
Avenida Rovisco Pais,
1049-001 Lisbon, Portugal
mateus.carvalho@tecnico.ulisboa.pt

Fernando T. Pinho
Departamento de Engenharia Mecânica
CEFT, ALiCE, Faculdade de Engenharia,
Universidade do Porto
Rua Dr. Roberto Frias,
4200-465 Porto, Portugal
fpinho@fe.up.pt

Carlos B. da Silva
Departamento de Engenharia Mecânica
IDMEC/LAETA, Instituto Superior Técnico,
Universidade de Lisboa
Avenida Rovisco Pais,
1049-001 Lisbon, Portugal
carlos.silva@ist.utl.pt

ABSTRACTand the continuity equation isDirect numerical simulations of spatially evolving turbulent planar jets and wakes with viscoelastic fluids are performed in order to study the influence of viscoelasticity on
the flow dynamics and to develop a theory for free turbulent
shear flows with viscoelastic fluids. The DNS display excellent agreement with the new theoretical results.

∇ · u = 0.
The polymer stress tensor is given byσ [p] =INTRODUCTION
It is well-known that the addition of minute amounts ofpolymers into a Newtonian solvent can lead to drastic changes
in the turbulent flow dynamics resulting in e.g. massive dragreduction for pipe and channel flows, and the establishmentof an elastic range of the energy spectrum for isotropic turbulence. Different theories have been proposed to explain theobserved behaviour in wall turbulence or isotropic turbulence.
However, there is currently no theory for the description of freeturbulent shear flows such as jets and wakes with viscoelastic
fluids. The current work presents results from several direct
numerical simulations (DNS) of spatially evolving turbulent
planar jets and wakes with incompressible viscoelastic polymer solutions in a Newtonian solvent, in order to develop atheory for free turbulent shear flows with viscoelastic fluids.

ρν [p][ f (Ckk )C − I],τp(3)and the evolution equation for the conformation tensor C is1∂C+ u · ∇C = ∇uT ·C +C · ∇u − [ f (Ckk )C − I].
∂t
τp(4)

In the above, u is the velocity vector and p is the pressure,while polymer and solvent zero-shear kinematic viscosities areν [p] and ν [s] , respectively. τ p is the polymer relaxation timeand ρ is the density of the fluid. The Peterlin function usedin the present work is given by f (Ckk ) ≡ (L2 − 3)/(L2 −Ckk ),where L is the maximum length of the polymers normalized bytheir equilibrium radius. The ratio of zero shear rate viscosities
is β = ν [s] /(ν [s] + ν [p] ). The polymer concentration in thesolution is proportional to 1 − β .
The simulations were carried out using a highly accurate
code that employs a combination of pseudo-spectral and 6thorder ’Compact’ finite differences schemes, for the momentum
equations, and the Kurganov-Tadmor scheme for the equations
governing the conformation of the polymer chains (Kurganov& Tadmor, 2000; Vaithianathan et al., 2006). An explicit
3rd-order Runge-Kutta scheme is used for time-stepping andthe pressure-velocity coupling is treated with a fractional step
method. Inflow and outflow boundary conditions are used atDIRECT NUMERICAL SIMULATIONS
The rheology of the solutions is characterised by theFinitely Extensible Non-Linear Elastic constitutive model
closed with Peterlin’s approximation (FENE-P). The momentum equation is given by
1
1∂u+ u · ∇u = − ∇p + ν [s] ∇2 u + ∇ · σ [p] ,∂t
ρ
ρ

(2)

(1)

1

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

the boundaries facing the streamwise direction. More details
about the numerical methods and code validation can be foundin Guimarães et al. (2020, 2022) and references therein.

Several direct numerical simulations (DNS) of spatiallyevolving turbulent planar jets and wakes with viscoelastic
FENE-P fluid have been performed. Large viscoelastic effects on the flow dynamics and evolution have been observed
when the Weissenberg number is high. An analytical theory
for the far field fully turbulent region of the flow has been developed using similarity analysis and scaling arguments; excellent agreement is obtained between the results of the DNS
and the analytical predictions. We expect that the ideas usedin our theory of viscoelastic turbulent planar jets and wakes
will also be useful in the description of other spatially evolving canonical turbulent flows such as free shear layers in theaxisymmetric (round) configuration, isotropic turbulence decaying behind a grid of bars and the outer region of a flat plateboundary layer.

In the present DNS the Reynolds numbers are Reh =UJ h/ν [s] = 3500 for jets and Red = U∞ d/ν [s] = 14000 for
wakes, where h is the inlet slot-width of the jet and d is thetransverse length scale of the wake generator object, UJ isthe inlet jet velocity and U∞ is the wake free-stream velocity. The Weissenberg number (Wi = τ p ∆U0 /d for wakes andWi = τ pUJ /h for jets) and polymer concentration parameter β
are varied in order to observe the influence of the fluid rheology in the dynamics of the resulting viscoelastic turbulent
flow. The number of grid points of the (uniform) computational mesh in each direction is given by nx , ny and nz , anddomain sizes are Lx , Ly and Lz . For the turbulent planar wakesimulations we have nx = 4032, ny = 1152 and nz = 288 with
Lx /d = 84, Ly /d = 24 and Lz /d = 6. For the turbulent planar jet DNS we have nx = 1152, ny = 1152 and nz = 144 with
Lx /d = 40, Ly /d = 40 and Lz /d = 5.

ACKNOWLEDGEMENTS
M.C.G acknowledges Fundação para a Ciência e Tecnologia (FCT) through contract UI/BD/151060/2021. C.B.S.
acknowledges Fundação para a Ciência e Tecnologia (FCT)
through IDMEC, under LAETA, project UIDB/50022/2020.
F.T.P. is thankful for funding by FCT and ALiCE throughproject LA/P/0045/2020 and FCT and CEFT through projects
UIDP/00532/2020.

The DNS show that the polymers in the solution lead todrastic changes in the evolution of several flow statistics suchas the spreading rates and the rates of decay of both the mean
velocity and passive scalar concentration, Reynolds stresses
and viscous dissipation rate (Guimarães et al., 2020, 2022).
Two-dimensional contours of the instantaneous vorticity modulus and the passive scalar concentration are shown in figures
1 and 2, respectively, to illustrate the simulations of jets andwakes. When the inlet Weissenberg number is large, the mean
flow structure at the fully turbulent far field can be split onto
two different regions; a first region where a large reductionof the spreading and decay rate and a suppression of the normalized turbulent velocity fluctuations is observed, and a subsequent region where the local Deborah number is low andviscoelastic effects are milder (Guimarães et al., 2022).

REFERENCES
Guimarães, M. C., Pimentel, N., Pinho, F. T. & da Silva, C. B.
2020 Direct numerical simulations of turbulent viscoelastic
jets. Journal of Fluid Mechanics 899.
Guimarães, M. C., Pinho, F. T. & da Silva, C. B. 2022 Turbulent planar wakes of viscoelastic fluids analysed by direct
numerical simulations. Journal of Fluid Mechanics, submitted .
Kurganov, A. & Tadmor, E. 2000 New high-resolution central
schemes for nonlinear conservation laws and convectiondiffusion equations. J. Comp. Phys. 160 (1), 241–282.
Pope, S. B. 2000 Turbulent Flows. Cambridge University
Press.
Tennekes, H. & Lumley, J. L. 1972 A first course in turbulence.
The MIT Press.
Vaithianathan, T., Robert, A., Brasseur, J. G. & Collins,
L. R. 2006 An improved algorithm for simulating threedimensional, viscoelastic turbulence. J. Non-Newt. Fluid
Mech. 140 (1), 3–22.

THEORYA new theory describing the far-field fully-developed region of turbulent viscoelastic jets and wakes is presented,
based on a scaling analysis of the equations governing themean flow and polymer conformation fields, and results inseveral scaling laws for the shear layer width, centreline
velocity/velocity-deficit, polymer stresses and polymer chain
conformation. Some results have already been published atGuimarães et al. (2020), where it is shown that Newtonian andviscoelastic shear layers follow the same scaling laws for thecentreline velocity decay and shear layer width, but with different values of the scaling law coefficients. The new theoryis also able to predict the scales that yield self-similarity of theprofiles of polymer stress and polymer chain elongation andthe condition for similarity of the profiles of Reynolds shear
stress. Additionally, the scaling laws for the streamwise evolution of the maximum polymer shear stress σc (x) at the initialhighly elastic region of the far field are derived. For jets, theresult is σc (x) ∼ x−5/2 while for wakes we have σc (x) ∼ x−2 .
Comparisons between the analytical predictions resulting fromthe new theory and the DNS results display excellent agreement (Guimarães et al., 2020, 2022). The new theory is thus anextension of the classical theory of free turbulent shear flowsof Newtonian fluids (Tennekes & Lumley, 1972; Pope, 2000)
to the more complex case involving non-Newtonian viscoelastic fluids.

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)
y/d

Newtonian(b)y/dViscoelastic

010

Figure 1.

2030

40x/d

60

50

70

84

Vorticity magnitude contours for (a) Newtonian wake and (b) viscoelastic wake with Wi = 4.2, L = 100 and β = 0.8.

(b)

(a)
Newtoniany/hViscoelastic

010

Figure 2.

20x/h

30

040

1020x/h30

Passive scalar contours for (a) Newtonian jet and (b) viscoelastic jet with Wi = 4, L = 100 and β = 0.8.

340