Direct Numerical Simulation of Turbulent Planar Jets with

Polymer Additives

Nuno Filipe Cabral Pimentelnuno.pimentel@tecnico.ulisboa.pt

Instituto Superior Tecnico, Universidade de Lisboa, Portugal

November 2018

Abstract

In this study it was performed Direct Numerical Simulations (DNS) of a spatial turbulent planarjet with polymer additives to further understand the mechanism of polymer interaction on jet flows.These simulations were achieved with the implementation of the visco-elastic fluid numerical model,represented by the rheological constitutive FENE-P model. The numerical model for the conformationtensor transport equation was adapted for the non-periodic boundary conditions of the flow, for whichno references are present in the literature. The development of the numerical algorithm took intoconsideration the computational performance of the simulations, in which it was implement a ghost cellmechanism to decrease the computation time. The developed algorithm has been extensively verified toensure the correct resolution of the governing equations for turbulent jet flow with polymer additives.The numerical model was tested in direct numerical simulation for different polymer molecules physicalcharacteristics, namely the polymeric concentration and the relaxation time. The observed resultsverified a decrease in the jet width on the presence of a visco-elastic flow, together with a decreaseof the viscous energy dissipation rate. It should be noted that the implemented numerical modelrepresents a major progress on the numerical simulations of turbulent spatial jet flow with polymeradditives, providing the first numerical results for this kind of study.Keywords: Visco-elastic turbulence, DNS, FENE-P, Spatial turbulent planar jet

1. Introduction

Since Toms (1948) [1] reported turbulent drag re-duction by the addition of long-chain polymers byup to 80 percent, this effect has been studied forboth practical and theoretical purposes. Currently,industrial applications regarding polymer drag re-duction are mostly related to long-distance liquidtransportation pipeline systems.

This work will focus on the study of this phe-nomenon on turbulent jet flows, in order to fur-ther understand the mechanism of polymer inter-action. The framework of this study has potentialapplications on the aerospace industry, namely onheat transfer reduction by injection of micro-jetsinto turbine blades [2]. There is a potential ap-plication on combining viscoelastic fluids with themicro-jet structures, as the heat transfer behavior ofviscoelastic fluids has been studied recently, namelyfocused on cooling applications for turbine disks [3].

Moreover, it is possible to make a formal anal-ogy between the Finite Extensible Nonlinear Elas-tic - Peterlin (FENE-P) set of equations for vis-coelastic flows and magneto-hydrodynamics (MHD)equations, as mentioned in [4]. Thus the numeri-

cal model considered here to simulate polymer solu-tions (FENE-P) will have a strong resemblance andapplication to the methodology used in numericalsimulations of MHD flows. As for aerospace indus-try applications, in recent years plasma flows haverapidly become an important set of new technolo-gies that find application to hypersonic propulsion,being evident a growing interest in plasma-basedflow manipulation through MHD forces [5].

2. Background

Turbulence is a characteristic of the flow, beinggenerally associated with flows that are highly un-steady with chaotic variations of velocity and pres-sure in space and time.

The concept of energy transfer through scales inturbulence was translated into a quantitative theoryby Kormogorov for a Newtonian fluid. For the caseof a visco-elastic fluid, the polymer interaction withthe flow leads to a steepening of the kinetic energyspectrum beyond a wavenumber lp, which is theLumley length-scale. In [6], it is presented a modelspectrum for the visco-elastic fluid (supported byexperimental results) consisting on 3 major regions:(I) Inertial cascade, (E) Elastic subrange and (V)

1

Viscous dissipation. This model spectrum is illus-trated on Figure 1.

Figure 1: Model spectrum for turbulence in poly-mer solutions indicating three regions. From [6].

The (I) Inertial cascade on Figure 1 consists ona Newtonian inertial cascade at low wavenumbers.

According to [6], the (E) Elastic subrange is aspectral region separated from the inertial cascadeby the Lumley scale lp, which is determined by theelastic properties of the fluid and the turbulencedissipation rate. In this region, fractions of theturbulence kinetic energy arriving from the iner-tial subrange are converted into elastic energy bypolymer stretching. Subsequent relaxation of thestretched polymers dissipates a part of this elas-tic energy due to the viscous drag of the polymermolecules in the solvent and interactions betweenmonomers of a single polymer. The remaining elas-tic energy is transformed back to turbulence kineticenergy. Consequently, the energy flux on this re-gion is continuously reduced from higher to lowerwavenumbers.

Experimental data [6] suggested that this re-gion follows a power-law with a slope of −3, wherethe polymer stresses overcome the viscous stresses.The energy spectrum in a turbulent structure ofwavenumber k decreases according to:

E(k) = Ckε2/30 l5/3p (lpκ)

−3(1)

where ε0 is equal to the energy flux from the New-tonian inertial cascade.

The (V) Viscous subrange consists on a regiondominated by viscous stresses which transform tur-bulence kinetic energy into heat, similarly to thedissipation range in the Newtonian spectrum.

2.1. Turbulent Planar JetsThe turbulent plane jet flow is a type of flow be-longing to the free shear flow type, which are char-acterised by developing far away from the interac-tion with boundaries, and that advances along apreferred direction.

Figure 2: Turbulent planar jet regions. From [7].

This type of flow is characterised by being statis-tically two-dimensional, with a dominant directionof mean flow across the streamwise x direction anda mean velocity in the spanwise z direction of zero.The plane jet can be split across different regions:the potential core region near the jet inlet, origi-nated by the high velocity gradient between the flowincoming from the jet inlet and that of the remain-ing environment; the transition region which com-prises the area in which the flow experiences transi-tion into a fully developed turbulent state; and theself-similarity region, in which the flow characteris-tics can be collapsed seamlessly after being properlyscaled following the Kolmogorov’s hypothesis.

2.2. Turbulence in dilute polymer solutions: numer-ical simulations

To model the polymer in numerical computations,the FENE-P model is the most widely used. Thepolymer orientation is represented as a continuoussecond-order tensor field, the so-called conforma-tion tensor. This is defined as the normalised sec-ond moment of the end-to-end vector between thetwo beads [8], described as:

Cij =〈rirj〉

13 〈r2〉eq

(2)

where ri is the instantaneous orientation of a poly-mer dumbbell, r2

eq is the square of the equilibriumseparation distance, and the angle brackets implyan ensemble average over the configuration space ofthe dumbbell.

The conformation tensor Cij is a symmetric andpositive definite (SPD) matrix. While mathemat-ically this constraint is satisfied by the governingequations, these properties can be lost due to cu-mulative numerical errors. Several attempts at nu-merical simulation of visco-elastic turbulence wereplagued by Hadamard instabilities, loss of conserva-tion or over dissipation [9]. A recent approach hasovercome these shortcomings [8], consisting on analgorithm based on the method of Kurganov andTadmor (KT). This second-order scheme guaran-

2

tees that a positive scalar will remain so at all pointsand it was generalized to guarantee that a SPDtensor also remains SPD. Furthermore, the methoddissipates less elastic energy than methods based onartificial diffusion, resulting in stronger polymerflowinteractions.

3. Governing equations and NumericalMethods

It is important to note that the starting point ofthe numeric work here presented is a DNS code forturbulent jet described in [7, 10]. Both thesis wereconcerned about DNS of Newtonian fluids, present-ing extensive validation of the DNS of turbulentjets. The main added features were the introduc-tion of the Conformation tensor transport equation,the coupling of the Navier-Stokes momentum equa-tion with the Polymer Stress Tensor and the use ofghost cells data on streamwise slab configuration.

3.1. Velocity FieldOn the velocity field governing equations, it is as-sumed that the fluid is incompressible and that itsatisfies the continuity and momentum equationswith an additional term related with divergence ofpolymer stress.

∂ui∂xi

= 0 (3)

∂ui∂t

+ uk∂ui∂xk

= −1

ρ

∂p

∂xi+

1

ρ

∂Tij∂xj

(4)

where ui is the i component of the velocity vector, ρis the constant fluid density, p is the local pressureand Tij is the combination of the viscous and poly-mer stress tensors. The latter can be expressed asa linear sum of contributions from the Newtonianstress (T

[s]ij ) and the polymer stress (T

[p]ij ):

Tij = T[s]ij + T

[p]ij (5)

where the Newtonian stress T[s]ij is given by:

T[s]ij = 2ν[s] ∂Sij

∂xj(6)

being ν[s] the kinematic viscosity of the fluid. TheSij is the Strain Rate Tensor defined as:

Sij =1

2

(∂ui∂xj

+∂uj∂xi

)(7)

3.2. The FENE-P Constitutive ModelFENE-P model constitutive equation of the Poly-mer Stress Tensor yields the following relationship:

T[p]ij =

ρν[p]

τp[f(Ckk)Cij − δij ] (8)

where Cij is the conformation tensor, ν[p] the zeroshear-rate polymeric viscosity, δij is the Kronecker

delta, L is the maximum possible extension of poly-mers and τp is the Zimm relaxation time of the poly-mer. The polymer viscosity ν[p] is included in themodel by a non-dimensional parameter β, whichrepresents the ratio between the solvent and thezero-shear-rate viscosity:

β =ν[s]

ν[p] + ν[s](9)

The function f(Ckk) is the Peterlin function givenby:

f(Ckk) ≡ L2 − 3

L2 − Ckk(10)

where Ckk = Cxx + Cyy + Czz is the trace of theconformation tensor, which represents the extensionlength. This function ensures finite extensibility, asit gives rise to a non-linear spring force that divergesas√Ckk → L, preventing the spring from extending

beyond L [8].To complement the continuity equation and the

conservation of momentum equation, a transportequation for the conformation tensor Cij is re-quired. The conformation tensor evolution equationis given by:

∂Cij∂t

+uk∂Cij∂xk

=∂ui∂xk

Cjk+∂uj∂xk

Cik−1

ρ

T[p]ij

ν[p](11)

The former equation is solved simultaneouslywith the velocity field flow equations, ensuringpolymer-flow interaction as the polymer moleculesare deformed by the velocity field and, in turn,the resulting conformation tensor introduces a poly-meric stress on the flow structure.

3.3. Computational domainThe computational domain for the turbulent jetspatial simulation consists of a box with uniformgrid and with Lx,Ly and Lz dimensions in each ofthe x (streamwise), y (normal) and z (spanwise) di-rections (see Figure 3). The y and z directions haveperiodic boundary conditions, whilst the stream-wise x is a non-periodic direction with inflow andoutflow boundaries.

Figure 3: View of the computational domain withreference frame and notation. From [7].

3

3.4. Numerical schemeThe code employs the following structure: Momen-tum equation terms highly accurate pseudo-spectralschemes to solve derivative on the normal and span-wise directions, and uses a compact scheme on thestreamwise direction. As for the Conformation Ten-sor Transport equation, it is employed the 2nd or-der scheme Kurganov-Tadmor and finite differencemethod.

The spatial discretization for the momentumequation is presented with detail on [10].

The Conformation Tensor Transport equationconvection term is discretized with a second-orderKT scheme, described in [8]. Spatial derivativesare second-order accurate everywhere, except forthe grid points losing SPD property. Where it oc-curs the scheme automatically reverts to first-orderaccurate for that grid point to maintain the SPDproperty is maintained. This scheme is given by:

u · ∇C =Hxi+1/2,j,k −Hx

i−1/2,j,k

∆x+

Hyi,j+1/2,k −Hy

i,j−1/2,k

∆y+

Hzi,j,k+1/2 −Hz

i,j,k−1/2

∆z(12)

where the convective flux tensor H in each directionis given by:

Hxi+1/2,j,k =

1

2ui+1/2,j,k(C+

i+1/2,j,k + C−i+1/2,j,k)−

1

2|ui+1/2,j,k|(C+

i+1/2,j,k −C−i+1/2,j,k)

(13)

Hyi,j+1/2,k =

1

2vi,j+1/2,k(C+

i,j+1/2,k + C−i,j+1/2,k)−

1

2|vi,j+1/2,k|(C+

i,j+1/2,k −C−i,j+1/2,k)

(14)

Hzi,j,k+1/2 =

1

2wi,j,k+1/2(C+

i,j,k+1/2 + C−i,j,k+1/2)−

1

2|wi,j,k+1/2|(C+

i,j,k+1/2 −C−i,j,k+1/2)

(15)

The conformation tensor C at the interface isconstructed from the following second-order linearapproximations:

C±i+1/2,j,k =Ci+1/2±1/2,j,k

∓(

∆x

2

)(∂C

∂x

)i+1/2±1/2,j,k

(16)

C±i,j+1/2,k =Ci,j+1/2±1/2,k

∓(

∆y

2

)(∂C

∂y

)i,j+1/2±1/2,k

(17)

C±i,j,k+1/2 =Ci,j,k+1/2±1/2

∓(

∆z

2

)(∂C

∂z

)i,j,k+1/2±1/2

(18)

The spatial derivatives of the conformation tensorare:

(∂C

∂x

)i,j,k

=

Ci+1,j,k−Ci,j,k

∆xCi,j,k−Ci−1,j,k

∆xCi+1,j,k−Ci−1,j,k

2∆x

(19)

It is selected the derivative approximation thatcan yield SPD results for C+

i−1/2 and C−i+1/2. When

two or more candidates satisfy the criterion, it is se-lected the one which maximizes the minimum eigen-value for these two tensors. When none of themmeet this criterion, the derivative is set to zero, re-ducing to first-order accurate. This slope limitingprocedure will ensure that all C± are SPD.

The update of the conformation tensor C requiresthe area-averaged velocity at the edge of the vol-ume surrounding each grid point. In [8], the veloci-ties u(x, y, z), v(x, y, z) and w(x, y, z) are obtainedfrom the inverse transform of the Fourier coeffi-cients u(kx, ky, kz), v(kx, ky, kz) and w(kx, ky, kz).However, since the streamwise direction x is notperiodic this method is not applicable. Thus, thearea-averaged velocities are computed consideringthe physical domain along the streamwise directionx and in the spectral domain along the normal y andspanwise z directions. The resulting expressions forthe cell velocities are:

ui+1/2,j,k =

FT−1

{u(x, ky, kz)

sin(ky∆y2 )

ky∆y2

sin(kz∆z2 )

kz∆z2

}(20)

vi,j+1/2,k =1

∆x

∫ xi+∆x2

xi−∆x2

FT−1

{v(x, ky, kz)e

iky∆y2

sin(kz∆z2 )

kz∆z2

}dx

(21)

wi,j,k+1/2 =1

∆x

∫ xi+∆x2

xi−∆x2

FT−1

{w(x, ky, kz)e

ikz∆z2

sin(ky∆y2 )

ky∆y2

}dx

(22)

where FT−1 is the inverse fast Fourier trans-form, u(x, ky, kz), v(x, ky, kz) and w(x, ky, kz) arethe Fourier coefficients at each grid point. Forthe streamwise direction it is required to calculatethe velocity and integrate along x in the physicalspace. Along the streamwise direction x, the ve-locity on the cell edge is interpolated with 5th or-der Lagrange Polynomials and the integration along

4

the streamwise direction is performed consideringGauss-Legendre 3 point rule. The area averagedvelocity for v component is then the following:

v(i, j + 1/2, k) =1

2

[AiF

(x+

∆x

2ξi

)](23)

F (x) = FT−1

{v(x, ky, kz)e

iky∆y2

sin(kz∆z2 )

kz∆z2

}(24)

where ξi and Ai are the Gauss nodes and weights,respectively. The area averaged spanwise velocityw is obtained analogously to v.

It should be noted that the formulation presentedis applicable to the velocity on positive edges i+1/2,j+1/2 and k+1/2. The area averaged velocities onnegative edges i− 1/2, j − 1/2 and k − 1/2 are ob-tained by shifting one position backwards the pos-itive edge velocity field in the respective velocitydirection. This shifting requires special treatmenton the boundaries. For the normal v and spanwisew velocities, the lower boundary for the negativeedge cell velocities is equal to the upper boundarypositive edge cell velocities, thus ensuring a periodbehaviour. Regarding the streamwise velocity u,the negative cell edge velocity at the inlet bound-ary is assumed to be the velocity u at the inlet.

The Conformation Tensor Transport equationstretching term is computed with a central differ-ence scheme second-order accurate everywhere inspace, except for the boundaries on the streamwisedirection. This scheme results in:

∂ui∂xk

Cjk +∂uj∂xk

Cik =(ui)m+1 − (ui)m−1

2∆xkCjk

+(uj)m+1 − (uj)m−1

2∆xkCik

(25)

where i, j and k are the computational indexesand m is an auxiliary variable that represents i, jand k for the ∂/∂x, ∂/∂y and ∂/∂z derivatives, re-spectively. On the inlet and outlet boundaries thecentral difference scheme is reverted to a forwardand backward finite difference first-order accuratescheme, respectively. The polymer stress couplingterm on the momentum equations also considers acentral difference scheme second-order accurate, re-verting to first-order accurate on the inlet and out-let boundaries.

The time advancement method of the momentumequation and for the conformation tensor trans-port equation is an explicit 3rd order Runge-Kuttascheme of numerical integration.

3.5. Boundary and Initial Conditions

The velocity profile imposed at the inlet at eachtimestep is given by:

u(x0, t) = Umed(x0) + Unoise (26)

where x0 is the inlet plane, Unoise is a superimposedrandom numerical noise and Umed is the mean inletprofile given by:

Umed(x0) =U2 + U1

2

+U2 − U1

2tanh

[h

4θ0

(1− 2|y|

h

)] (27)

In (27) θ0 is the momentum thickness of the initialshear layer, h is the jet inlet slot width and y is thedistance from the centre of the jet. U1 is the co-flowvelocity and U2 is the centreline velocity. The co-flow is required to allow the growth of the jet’s shearlayer, otherwise hampered by the lack of naturalentrainment from the periodic lateral boundaries.

The numerical noise, Unoise, is superposed to themean profile in (27), at each timestep t by imposinga three-component fluctuating velocity field (u, vand w), in order to exhibit the statistical charac-teristics of isotropic turbulence [11].

The inlet boundary for the polymer conformationtensor is imposed at each timestep and is based onthe assumption of steady flow, fully developed inthe streamwise direction x. Considering the inletvelocity profile without co-flow (see equation (27)),one obtains an analytical profile for the polymerconformation tensor, given by the real root of thefollowing cubic equation:

C223 +

L2

2τp2(∂U∂y

)2C22 −L2

2τp2(∂U∂y

)2 = 0 (28)

The lateral boundaries are periodic, which re-quires this periodicity to be stated for all three com-ponents of the velocity field, the pressure and thepolymer conformation tensor. For a given quantityφ(x, y, z, t) these are given by:

φ(x, y, z, t) = φ(x, y + Ly, z, t)φ(x, y, z, t) = φ(x, y, z + +Lz, t)

(29)

The outlet boundary is the most sensitive bound-ary condition of the planar jet. The fluid structuresthat leave the domain must exit unperturbed by theoutflow condition and ensuring that reflected wavesdo not go back into the domain. Therefore, the codeemploys a non-reflective boundary condition usedby [12], where all the terms of the Navier-Stokesequations are explicitly advanced.

5

Figure 4: View of the domain partitioning with ref-erence frame and notation. From [10].

3.6. Code architecture

The computational domain is implemented on amixed decomposition, with the co-existance ofstreamwise slabs and normal slices (see Figure 4).

The data field configuration is swapped accordingto the operation taking place. The direct and in-verse Fourier transforms are computed in indepen-dent yz planes, thus being intrinsically connectedto the normal slice configuration. The compactscheme for the streamwise derivatives is computedon the slab configuration, as it requires a completeline of data on the streamwise direction not avail-able on normal slice configuration. The streamwisederivatives for the conformation tensor transportequation do not require complete lines of data onthe streamwise x direction. Thus, it was opted toperform the calculation on the normal slice configu-ration with the aid of ghost cells, avoiding the needto perform intensive communications between pro-cessors to swap data configuration.

Ghost cells here consist in replicated data fromneighbouring partitions that is passed between pro-cessors. This process requires data exchange at eachRunge-Kutta sub-step. An illustrative example ofghost cell application is presented in Figure 5.

(a) Normal slice configuration withoutghost cell data on the xy plane.

(b) Normal slice configuration with ghostcell data on the xy plane.

Figure 5: Ghost cell configuration data exchange onthe streamwise direction for a domain partitioned in3 normal slices.

4. Code Verification

The Conformation Tensor Transport Equation wasverified with a frozen in time 2D Couette flow, thusallowing the comparison with a steady analytical so-lution. The velocity profile (see Figure 6) was keptconstant throughout the simulation by imposing itin each Runge-Kutta step.

Figure 6: u velocity profile in the y direction.

From Figure 6 it is noticeable that the veloc-ity profile is a mirrored Couette profile, due to thenumerical scheme and the need to ensure periodicconditions. Therefore, the discontinuity points ob-served on the profile (y/h = −5, 0, 5 for Figure 6)are not comparable to the analytical solution.

The conformation tensor Cij is a SPD matrix bydefinition. For this reason, Cij = Cji and, there-fore, the system of linear equations only encom-passes six independent variables.

As it is considered a steady and fully-developedflow in the streamwise x direction, the following as-sumptions are enforced:

∂

∂t= 0,

∂

∂x= 0 (30)

Given the two-dimensional flow (velocity profileu(y)) and the conservation of mass, it follows:

v = 0, w = 0,∂

∂z= 0 (31)

Considering the flow assumptions from Equations(30) and (31), the conformation tensor transportequation (11) can be rewritten in this case as:

uk∂Cij∂xk

=∂ui∂xk

Cjk +∂uj∂xk

Cik−1

τp[f(Ckk)Cij − δij ]

(32)

Having equation (32), each entry of the conforma-tion tensor can be organized as a function of C22.The solution for C22 can be obtained from the cu-bic polynomial described in equation (28), for which

6

the real root solution is:

C22 =

3

√√√√√ L2

4(∂u∂y

)2

τ2p

+

√√√√ L6

216(∂u∂y

)6

τ6p

+L4

16(∂u∂y

)4

τ4p

+

3

√√√√√ L2

4(∂u∂y

)2

τ2p

−√√√√ L6

216(∂u∂y

)6

τ6p

+L4

16(∂u∂y

)4

τ4p

(33)

The followings simulation parameters were con-sidered on the verification exercise:

Box height h 2πBox dimensions (terms of h) 0.9× 1× 0.118Mesh 128× 128× 128Max. velocity umax 1Zero shear-rate viscosity ν[p] 0.002Max. molecular extensibility L 10Relaxation time (s) τp 0.1Polymer concentration β 0.8

Table 1: Verification simulation parameters.

Having both analytical and numerical results, itis performed the comparison between both. Thisverification is presented in Figures 7 and 8 forthe diagonal and non-diagonal conformation tensorcomponents, respectively. In each figure the dashedlines correspond to the numerical solution, whilstthe solid lines refer to the analytical solution.

Figure 7: Numerical (n) and analytical (a) solu-tions for Cii considering a u velocity profile in they direction.

From Figures 7 and 8 it is possible to observe thatthe numerical and analytical solutions present thesame results, thus ensuring the verification of thetriggered equation components. For the remainingscenarios the same conclusion is obtained.

5. ResultsSeveral runs were performed in order to study theinfluence of polymer additives on the turbulent pla-

Figure 8: Numerical (n) and analytical (a) solutionsfor Cij , i 6= j considering a u velocity profile in they direction.

nar jet. It should be noted that the results wereproduced as part of an ongoing investigation withMateus C. Guimaraes [13], considering the numer-ical algorithm developed in this work. The com-putational and physical parameters of the runs arepresented in Tables 2 and 3.

nx ny nz Lx/h Ly/h Lz/h

512 512 128 18 18 4

Table 2: Computational domain parameters for thesimulations. Number of grid points (nx, ny, nz);non-dimensional grid size (Lx/h, Ly/h, Lz/h).

The Wi is given by the ratio between the polymerrelaxation time and Kolmogorov time scale:

Wi =τpτs

(34)

where τs represents the Kolmogorov time scale(τs = (ν[s]/ε[s])(1/2)). The Reynolds number basedon the Taylor microscale is defined as:

Reλ =u′λ

ν[s](35)

where u′ is the reference perturbation velocity andλ is the Taylor scale. The Taylor scale λ is given bythe viscous dissipation rate assuming local isotropyat the jet centreline:

λ =

√15ν[s]u′

2

ε[s](36)

In order to simulate the smallest scales that arepresent in the flow, the grid size has to be of theorder of the Kolmogorov scale η, defined in termsof the solvent turbulent viscous energy dissipationrate ε[s] and the solvent viscosity ν[s]:

η =

((ν[s])3

/ε[s])1/4

(37)

7

Case Wi Re Reλ τp(s) β ∆x/η14h

A 0 3500 100 0 1 4.0

B 0.33 3500 110 0.025 0.8 4.0

C 0.65 3500 110 0.05 0.8 4.0

D 1.19 3500 110 0.10 0.8 3.9

E 1.66 3500 140 0.20 0.8 3.4

F 0.20 3500 120 0.025 0.6 3.8

G 0.27 3500 160 0.20 0.6 3.3

H 1.35 3500 160 0.025 0.4 3.3

I 1.55 3500 180 0.20 0.4 3.1

J 2 3500 200 0.40 0.8 2.8

K 2.6 3500 220 0.60 0.8 2.5

L 3.3 3500 250 0.80 0.8 2.3

Table 3: Physical parameters for the simulations.Weissenberg number Wi; slot width, initial maxi-mum jets velocity, solvent viscosity based Reynoldsnumber Re; Averaged Taylor turbulence micro-scale centreline velocity solvent viscosity basedReynolds number on the x direction Reλ; polymerrelaxation time τp; polymer concentration β; meshresolution normalized by the Kolmogorov smallscale at x/h = 14. Polymer normalized maximumextensibility L of 100 for all cases.

According to [10], several authors state that theratio of the element size ∆x to the Kolmogorov scaleη should have a value of the order of 2.0 to 3.0. Itwas opted to select to analyse ∆x/η at x/h = 14,as this region refers to a fully developed turbulentregion of the jet. By having mesh resolution nor-malized by the Kolmogorov small scales within theliterature values, the mesh size was deemed ade-quate.

The simulations are started with variable timestep and constant Courant number (CFL = 1/6).Once self-similarity is attained, the time step is keptconstant at a lower level than the minimum reachedduring the first part of the simulation and statisticalquantities are computed by accumulating data.

The half-width or mean flow thickness δ is definedas follows according to [13]:

δ =

∫ ∞0

u− U2L

Uc − U2Ldy (38)

where Uc is the centreline velocity and U2L is thelocal co-flow velocity. The streamwise velocity isnon-dimensionalized according to the following ex-pression: (

U1 − U2

〈Uc〉 − U2

)2

(39)

where U1 is the inlet centreline velocity and U2 isthe inlet co-flow velocity.

All variables presented in this section are an av-erage on time and on the z spanwise direction.

0 5 10 15 20

x/h

0.5

1

1.5

2

δ/h

(a)

δ = 0.110xδ = 0.083xτp = 0.0τp = 0.4τp = 0.6τp = 0.8

0 5 10 15 20

x/h

1

1.5

2

2.5

3

3.5

[(Uc−U2)/(U

1−U2)]−2

(b)

0.210(x/h − 1)0.158(x/h − 1)τp = 0.0τp = 0.4τp = 0.6τp = 0.8

Figure 9: Streamwise evolution of non-dimensional(a) jet’s half-width and (b) centreline velocity forcases A, J, K and L. From [13].

On Figure 9 it is possible to identify the self-similar region by fitting straight lines to the linearregion of each curve. A good linear fit was foundbetween x/h = 6 to x/h = 17 for the Newtoniancase, whilst for the case with τp = 0.8 a good linearfit was observed from x/h = 10 to x/h = 17.

From Figure 9 (a) it is noticeable a lower jetwidth on the visco-elastic cases in comparison tothe Newtonian case. Similarly, the centreline veloc-ity shows a larger value on the visco-elastic casesthan on the Newtonian case. Both these remarksare indicative of a lower energy dissipation rate forthe FENE-P solutions due to the interaction of thepolymer particles on the flow physics. This resultfollows the expected trend, as polymer particles areexpected to absorb elastic energy at a higher ratefrom the flow, reducing the overall energy dissipa-tion of the flow.

On Figure 10 it is shown statistical quantities fora fully turbulent streamwise cross-section (x/h =12). It is possible to observe that the streamwiseu velocity profiles collapse for different relaxationtime. Concerning the normal y velocity profile, it isnoted that a higher polymer relaxation time leadsto, typically, a lower average normal velocity mag-nitude. The spanwise w average velocity is not pre-sented, since it presents an average zero value for aplanar jet [10]. As for the velocity perturbation rootmean squared components, it is observed an over-all decrease of all components with increasing poly-mer relaxation time. Moreover, it is seen that allperturbation velocities converge to zero with higherdistances from the centreline, as the flow is movingtowards zones not perturbed by the jet. Concerningthe Reynolds shear stresses, the examination of Fig-ure 10 indicates that for higher polymer relaxationtime the shear stress reduces, namely for cases Kand L. This observation reinforces the previous re-mark, indicating a smaller energy dissipation ratefor the polymer solution case due to the energy-absorption effect of the polymers.

The conformation tensor Cij are plotted for the

8

0 0.5 1 1.5 2 2.50

0.5

1(u

−U2L)/(U

c−U2L)

(a)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2-0.06

-0.04

-0.02

0

0.02

v/(Uc−U2L) (b)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2 30

0.1

0.2

√

u′2/(Uc−U2L)

(c)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2 30

0.1

0.2

√

v′2/(Uc−U2L)

(d)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2 3

y/δ

0

0.1

0.2

√

w′2/(Uc−U2L)

(e)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

0 1 2 3

y/δ

0

0.01

0.02

0.03u′v′/(Uc−U2L)2 (f)

τp = 0.0τp = 0.4τp = 0.6τp = 0.8

x/h = 12

Figure 10: Normalized mean velocity: (a) stream-wise and (b) normal direction, (c-e) velocity rootmean squared components and (f) Reynolds shearstress profiles for cases A, J, K and L. From [13].

same streamwise cross-section (x/h = 12) in Fig-ure 11. The averaged conformation tensor com-ponents C13 and C23 are not presented in Figure11, as due to the nature of the turbulent flow isit possible to prove that the average value of thesecomponents is zero. This was observed on the sim-ulation results. Concerning the remaining confor-mation tensor components, it is seen that with in-creasing polymer relaxation time, the magnitude ofthe averaged Cij increases. Moreover, one noticesthat the average values converges to the polymerequilibrium state, that is value 1 for the diagonaltensor terms and 0 for the remaining tensor terms,on the non-perturbed region of the flow. As theelastic energy stored by a stretched polymer is pro-portional to the trace of the conformation tensor,the observed higher average values of the confor-mation tensor components with increasing polymerrelaxation time indicate that the polymer moleculeshave an increasing absorbed elastic energy.

On Figure 12 it is presented the Weissenbergnumber for the y = 0 cross-section. It is observedthat for higher polymer relaxation time with thesame β (especially clear on cases with β = 0.8), Wi

number increases. Simultaneously, the same caseswhen viewed from a turbulent kinetic energy dissi-pation perspective (see Figure 13) present a lowerviscous energy dissipation, which in turn would con-tribute to a lower Wi number. Thus, this indicatesthat the polymer elastic forces on the flow are farmore significant than the solvent viscous forces.

If one considers cases with an equal τp (e.g.τp =0.2s), it is noticeable that higher β leads to a higherWi number. Nonetheless, the variation on the Weis-senberg number is clearly smaller with the influence

0 1 2 3

0

20

40

Cij

(a)ij = 11

ij = 22

ij = 33

ij = 12

τp = 0.4

0 1 2 3

-50

0

50

100

150

Cij

(b)ij = 11

ij = 22

ij = 33

ij = 12

τp = 0.6

0 1 2 3

y/δ

0

100

200

Cij

(c)ij = 11

ij = 22

ij = 33

ij = 12

τp = 0.8

0 1 2 30

50

100

√

c′2 ij

(d)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.4

0 1 2 30

50

100

150

√

c′2 ij

(e)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.6

0 1 2 3

y/δ

0

100

200

√

c′2 ij

(f)ij = 11ij = 22ij = 33ij = 12ij = 13ij = 23

τp = 0.8

Figure 11: Mean (a-c) and root mean squared (d-f)components of the conformation tensor at x/h = 12for cases J, K and L. From [13].

of β than with the influence of τp.

0 5 10 15 20 25

x/h

0

1

2

3

Wi=

τ p/√

ν[s] /ε[s] τp = 0.025, β = 0.8

τp = 0.025, β = 0.6τp = 0.025, β = 0.4τp = 0.050, β = 0.8τp = 0.100, β = 0.8τp = 0.200, β = 0.8τp = 0.200, β = 0.6τp = 0.200, β = 0.4τp = 0.400, β = 0.8τp = 0.600, β = 0.8τp = 0.800, β = 0.8

Figure 12: Streamwise evolution of centreline Weis-senberg number for visco-elastic cases. From [13].

As for the viscous energy dissipation rate pre-sented in Figure 13, it is noted that the presenceof polymer additives on the flow leads to a lowerviscous energy dissipation rate. Moreover, for thesame τp a lower β leads to lower viscous energy dis-sipation. Similarly, for the same β a higher τp leadsto lower viscous energy dissipation.

0 5 10 15 20 25 30

x/h

0

2

4

6

8

10

ε[s]h/U

3 1

Newtonian fluidτp = 0.025, β = 0.8τp = 0.025, β = 0.6τp = 0.025, β = 0.4τp = 0.050, β = 0.8τp = 0.100, β = 0.8τp = 0.200, β = 0.8τp = 0.200, β = 0.6τp = 0.200, β = 0.4τp = 0.400, β = 0.8τp = 0.600, β = 0.8τp = 0.800, β = 0.8

Figure 13: Streamwise evolution of centreline tur-bulent kinetic energy dissipation of the solvent forall cases. From [13].

6. ConclusionsIn the present work it was implemented the FENE-P model within an existing DNS tool, resulting in

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a DNS tool for turbulent spatial jet with polymeradditives. It is important to stress that DNS orexperimental analyses of turbulent spatial jet havenot been found in the literature. As such, thiswork provides the first time a turbulent spatial jetis analysed with a FENE-P visco-elastic configura-tion, leading to a new ground breaking numericaltool with potential for new study areas.

During the development of the current work, ex-tensive care was taken concerning the numerical im-plementation in order to ensure impact minimiza-tion of the implementation of FENE-P on the over-all DNS code performance. Such was achieved withthe inclusion of the ghost cells mechanism and min-imization of the call of MPI commands, which al-lowed a relevant performance enhancement. Iter-ation times with FENE-P on the spatial jet coderesulted in an increase of 7-fold, whilst for com-parable homogeneous isotropic turbulent FENE-Psimulations a 10-fold increase was observed.

DNS analyses were performed, in which it wasanalysed the evolution of the jet’s characteristics,the conformation tensor components evolution andthe turbulent kinetic energy dissipation. Regardingthe jet’s characteristics, focus was given on the jet’swidth, velocity components and Reynolds shearstress profile. For all of the abovementioned, thedifference between the Newtonian fluid and FENE-P solution is clear. The jet width in the FENE-P solution is lower than the one observed for theNewtonian case, which indicates that the viscousenergy dissipation is smaller for the polymer solu-tion case. Similarly, the centreline velocity shows ahigher value for polymer solution than for the New-tonian case, confirming the previous remark. Asfor the remaining components, results show that ve-locity perturbations on turbulent regime are lowerin the visco-elastic solution and that the Reynoldsshear stress profiles decreases on the FENE-P so-lution. Both of the previous remarks confirm asmaller energy dissipation rate for the polymer solu-tion case due to the energy-absorption effect of thepolymers, which is indicative of the drag reductionphenomenon.

Acknowledgements

Firstly, I would like to start thanking Professor Car-los Silva for giving me the opportunity to work inthis area and for his advice during the thesis.

I would like to thank Mateus Guimaraes, for thedata post-processing and obtaining the first resultswith the the algorithm developed within this work.

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