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Parallel implicit DNS of temporally-evolving turbulent shear layer instability

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Parallel Implicit DNS of Temporally-Evolving Turbulent

Shear Layer Instability

Ilyas Yilmaza,1,∗, Firat Oguz Edisb, Hasan Sayginc, Lars Davidsond

aEnergy Institute, Istanbul Technical University, Maslak, 34469, Istanbul, TurkeybDepartment of Astronautical Engineering, Faculty of Aeronautics and Astronautics,

Istanbul Technical University, Maslak, 34469, Istanbul, TurkeycDepartment of Mechanical Engineering, Faculty of Engineering and Architecture,

Istanbul Aydin University, Florya, 34295, Istanbul, TurkeydDivision of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of

Technology, Goteborg, SE-412 96, Sweden

Abstract

In this study, temporally-evolving incompressible and compressible Turbu-lent Shear Layer (TSL) instability problem is solved using an all-speed (all-Mach), implicit, non-dissipative and kinetic energy conserving algorithm. Anin-house, fully parallel, finite-volume Direct Numerical Simulation (DNS)solver was developed using PETSc. Convergence characteristics at low-Machnumbers were also improved using a relaxation procedure. We aim here to as-sess the performance and behavior of the present algorithm for complex flowswhich contain multi- scale physics and gradually evolve into turbulence. Theresults show that the algorithm is able to produce correct physical mecha-nisms and capture the evolution of the turbulent fluctuations for both incom-pressible and compressible cases. It is observed that the non-dissipative andkinetic energy conserving properties make the algorithm powerful and appli-cable to challenging problems. For higher Mach numbers, a shock-capturingor a dissipative mechanism is required for robustness.

Keywords: DNS, TSL, parallel, implicit, all-speed, compressibility

∗Corresponding author. Tel: +90 212 285 31 50Email addresses: yilmazily@itu.edu.tr, ilyasyilmaz@aydin.edu.tr (Ilyas

Yilmaz), edis@itu.edu.tr (Firat Oguz Edis), hasansaygin@aydin.edu.tr (HasanSaygin), lada@chalmers.se (Lars Davidson)

1Present address: Department of Mechanical Engineering, Faculty of Engineering andArchitecture, Istanbul Aydin University, Florya, 34295, Istanbul, Turkey

Preprint submitted to Journal of Computational and Applied MathematicsDecember 18, 2013

1. Introduction

We present a three-dimensional Direct Numerical Simulation (DNS) studyof mixing in temporally-evolving Turbulent Shear Layer (TSL) instability.For this purpose, an in-house, fully parallel, finite-volume DNS solver, iDNS,was developed based on a fully implicit, non-dissipative, and discrete kineticenergy conserving algorithm recently proposed by Hou & Mahesh [1]. ThePETSc library was utilized for an efficient parallelism [2].

Our aim here is two-fold: first, to demonstrate the algorithm’s abilityto evolve perturbations into turbulent mixing for the base flow with theconvective Mach number, Mc = 0.3, and, second, to analyze the effectsof compressibility on the growth rate of the instability by increasing theconvective Mach number.

The following sections include a brief review of the algorithm with afurther improvement in convergence, the solver details, the definition andthe numerical setup of the problem. The results obtained are analyzed andcompared to those from previous experimental and numerical studies.

2. Numerical Method

The algorithm solves the set of time-dependent, three-dimensional com-pressible Navier-Stokes equations. These are non-dimensionalized by us-ing the reference values. Unlike in many other methods, pressure is non-

dimensionalized, as p =pd−prefρrefu

2

ref

,, which is known as incompressible scaling

or low-Mach number scaling [3, 4]. The resulting non-dimensional formsof the equations satisfy the incompressibility condition, and allow to studylow-Mach number flows with the same algorithm without encountering theacoustic stiffness problem, when the Mach number goes to zero. The algo-rithm is second-order in both space and time on orthogonal grids. It is stableand robust at high-Reynolds numbers [1]. It is a pressure-correction type ofalgorithm which uses an iterative predictor-corrector approach to update theflow variables. The variables are stored at the cell centers. A fully implicit,finite-volume discretization is used. The discretization is centered in bothspace and time to ensure the non-dissipativeness. Thermodynamic quanti-ties such as density, pressure, and temperature are also staggered in time byhalf a time step relative to the velocities [5]. The values of the variables at

2

cell-faces are obtained by symmetric interpolation, which is the simple aver-aging of two adjacent cell-center values. This also ensures the conservation ofkinetic energy discretely at the low-Mach number limit [6]. The face-normal

velocity, Vn, is treated as a separate flow variable like the velocity and isupdated in time [7, 8]. Vn is stored at the center of the cell-faces directly tomimic the staggered grids to prevent the odd-even decoupling in pressure inspace. In order to give a fully implicit treatment of pressure, the algorithmworks with a weighted sum of previous, current, and predicted values of thepressure, instead of the pressure itself directly [5]. An additional correction topressure is also adopted in the algorithm to enhance the convergence proper-ties by preventing possible successive oscillating pressure corrections in time[9]. This procedure provides a time- and space-dependent relaxation factorfor the pressure at the end of each time-step for low-Mach number flows.

3. Solver Details

In order to perform this study, an in-house, fully parallel DNS solver,iDNS, was developed based on the algorithm mentioned above and thePETSc parallel library. iDNS is a single-block, structured, fully implicit,finite-volume solver working on uniform Cartesian grids. Fig.1 shows theflowchart of iDNS and the iterative solution procedure as well. It consists ofthree parts: pre-processing, core and post-processing. In pre-processing, thePETSc framework is initialized. Grid related quantities and initial conditionsfor the simulations are generated. Core performs all steps for the solutionof the discretized equations. Post-processing calculates necessary quantitiesfor analyzing and comparison, and also stores them into files. iDNS waswritten in a modular fashion using Fortran syntax. The linear systems aris-ing from the discretization are first pre-conditioned by incomplete-LU withzero-filling (ILU(0)), and are then solved using GMRES. During the parallelperformance tests, very good speed-up and efficiency results were obtained.Tests also showed that the solver is scalable, since it maintains the efficiencywhen increasing the problem size and the number of cores.

4. Problem Definition and Setup

Turbulent shear layers are of interest for many physical and engineeringflows such as jets, wakes and mixing in combustion. They are induced by

3

Figure 1: Flowchart of iDNS.

4

strong gradients in the shear stress. Although they have a relatively sim-ple configuration, complex physical phenomena are included in these flows.They are mostly dominated by large-scale, quasi-2D, organized structures.However, they can rapidly undergo transition to turbulence in 3D. The firsttheoretical analyses were presented in the 50’s and 60’s [10]. Experimentalinvestigations began in 70’s [11, 12], and this was followed by early DNS sim-ulations in the 80’s and 90’s [13, 14] with the help of the advances in computertechnology. Recently, well-resolved numerical simulations were performed[15, 16]. There has remained an ongoing interest in turbulent shear layersand need for better understanding of their physical mechanisms, due to therequirements of flight (such as in wake control) and combustion technolo-gies (such as in supersonic mixing) for developing more efficient systems [17].They can be studied both spatially and temporally. In spatially-evolvingmixing layers, mixing layer thickness of two emerging streams of fluids de-velop in the streamwise direction. Such development requires a large extentof the domain. This is computationally much more expensive to track thanthe temporally-evolving mixing layers where the thickness of the layer in-creases as a function of time rather than as a function of the streamwisecoordinate.

The flow is initialized with a hyperbolic tangent function for the mean

streamwise velocity, u = ∆u2tanh

(

− y2δθ(0)

)

, where δθ(0) is the initial mo-

mentum thickness. This defines two parallel streams moving in oppositedirections with the same velocity, ∆u

2, where ∆u is the velocity difference

between the lower and the upper streams. It is also taken as the referencevelocity. The normalwise (y) and the spanwise (z) mean velocities are set tozero. The initial densities are equal for the two streams and are set to unity.The initial pressure is uniform as well. It is set to give the speed of sound asunity and is chosen as the reference pressure. The Reynolds number basedon the initial momentum thickness, Reθ, the velocity difference, and the av-erage dynamic viscosity is 160 and the Reynolds number based on the initialvorticity thickness (δω(0)), Reω, is 680. The ratio of the specific heats, γ, is1.4 and the Prandtl number, Pr, is 0.72. The density ratio between streamsis equal to 1. Different convective Mach numbers are studied. The idea ofthe convective Mach number was first introduced in order to identify theeffects of compressibility on the shear-layer growth rate. For equal densitiesand specific heats, it can be written as Mc =

∆uc1+c2

, where c1 and c2 denotethe speeds of sound for each of the streams.

5

The initial momentum thickness is taken as the reference length scale.The reference time scale is, τ = δω(0)

∆u.

The boundary conditions are periodic in the streamwise (x) and spanwise(z) directions. Inviscid slip-wall boundary conditions are used at bottom andtop boundaries in the normalwise (y) direction. The face-normal velocitiesrequire special treatment and must be set to zero on inviscid slip-walls.

The non-dimensional time-steps are set to 6× 10−3 for the quasi-incom-pressible case where Mc is 0.3, 1 × 10−2 for the mildly-compressible caseswhere Mc = 0.5, 0.7, and 5 × 10−3 for the highly compressible case whereMc = 0.9. The flow was followed up to the very late non-linear stage, andthen the simulation was stopped.

The same problem domain as was considered by Pantano & Sarkar [16]is used for our computations. The domain is very large, especially in thestreamwise direction, in order to allow the flow to evolve into a self-similarstate. This resolution is also sufficient for representing the large scale struc-tures in the flow, as shown by the same authors using a calculation of inte-gral length scales (which are sufficiently small compared to the domain sizein homogeneous directions) and two-point correlations (which are decorre-lated over half the domain size in homogeneous directions) in the self-similarstate. The domain size is given in terms of the initial momentum thicknessas Lx × Ly × Lz = 345δθ(0)× 172δθ(0)× 86δθ(0) with corresponding resolu-tion Nx × Ny × Nz = 512 × 256 × 128 for the cases considered. The initialmomentum thickness is set to 0.093. A uniform orthogonal grid is used.

In addition to these mean values, turbulent three-dimensional velocityfluctuations are superimposed on the initial mean velocity components. Thefluctuations are generated on the basis of a technique described by Davidson[18] using the isotropic turbulence energy spectrum, E(κ) = u′2

rms(κκ0

)4 exp(−2( κκ0

)4),where κ0 is the peak wave number. κ0 is adjusted so as to have 48 peak wave-lengths in the streamwise direction for the simulations. u′

rms denotes the rms

of the velocity fluctuations and is given as u′rms = (ti ·∆u) where ti is the

turbulent intensity which is set to 0.1. The initial turbulent velocity fluctu-ating field obtained is limited to the shear layer by multiplying with a shapefunction in the form exp(−( y

2δθ(0))2). The runs were performed on distributed

memory architectures with Intel Xeon dual-core and quad-core processors.256 cores with 2 GB of RAM per core were used for the simulations. Thetotal wall-clock time was 350 − 500 hours, depending on Mc. Since the im-plicit algorithm allows larger time steps as given above, this computationaltime is quite acceptable and possibly much shorter than an explicit method.

6

5. Results and Discussions

5.1. The quasi-incompressible case

The quasi-incompressible case (Mc = 0.3) is chosen as the base case. Thiscase is also used for the code verification, and the comparison as well. Inorder to analyze the results, the momentum thickness, δθ, is first introducedas

δθ =1

ρ0∆u2

∫ +∞

−∞ρ(1

2∆u− u1)(

1

2∆u+ u1)dy (1)

where u1 denotes the Favre averaged streamwise velocity component and ρis the Reynolds averaged density.

As an integral quantity, the momentum thickness is less sensitive to sta-tistical noise than the vorticity thickness, δω = ∆u

( ∂u∂y

)max, and evolves smoothly

in time[14]. The growth rate can be found as the slope of a linear curve fit,δθ =

1∆u

dδθdt.

The time evolution of the momentum thickness is given in Fig. 2. Afteran initial settling period, which mainly depends on the initial fluctuations,an approximate linear growth is observed, in consistency with the previousresults. The growth rate is 0.0182 which is calculated by taking the slopeof the curve in the region where the flow is self-similar. Pantano & Sarkar[16] obtained 0.0184 for the quasi-incompressible case from a DNS database.They also obtained 0.016 in their DNS study at Mc = 0.3. Our value is ingood agreement with Pantano & Sarkar [16].

The Reynolds stress transport equation is introduced as in [16] to calcu-late the Reynolds stress tensor and the turbulent kinetic energy budget

∂(ρRij)

∂t+

∂(ρukRij)

∂xk

= ρ(Pij − ǫij)−∂Tijk

∂xk

+Πij (2)

where Rij , Pij , ǫij , Tijk ,Πij are turbulent stress, production, dissipation,transport and pressure-strain terms, respectively. Explicit forms and calcu-lations of these terms can be found in [16].

Fig. 3 shows the time evolution of the rms of the velocity fluctuations inthe streamwise direction. In consistency with the previous studies, it startsfrom an initial value, reaches a peak and then decreases to its specific value.The comparisons of the Reynolds stresses are given in Fig. 4–compared withthose form experiments and other DNS studies. A procedure of averagingover the time period between 200 and 600 was applied. As can be seen, the

7

t/τ

δ θ/δθ,

0

0 200 400 600 800 10000

5

10

15

Figure 2: Time evolution of momentum thickness for Mc = 0.3.

y/δω

√R11

/∆u

-1 -0.5 0 0.5 10

0.05

0.1

0.15

0.2

t//τ=0t//τ=50t//τ=100t//τ=150t//τ=200t//τ=300t//τ=600

Figure 3: Time evolution of√R11.

8

Table 1: Comparison of the peaks of the Reynolds stresses for Mc = 0.3.

√R11/∆u

√R22/∆u

√R33/∆u

√R12/∆u

Bell and Mehta [19] (EXP, Mc = 0) 0.180 0.140 0.146 0.100Pantano and Sarkar [16] (DNS) 0.155 0.134 0.143 0.103Hadjadj et. al. [20] (LES) 0.170 0.134 0.143 0.106Foysi and Sarkar [21] (LES) 0.174 0.129 0.143 0.106Present DNS 0.176 0.133 0.141 0.102

agreement is good. The streamwise component of the Reynolds stress is largerthan the other components. Table 1 compares the peak turbulent intensitiesof the Reynolds stresses to those from some prior numerical studies. Our DNSgives similar values to the others. Turbulent production and dissipation arepresented in Fig.’s. 5a and 5b. Our values are in good agreement with thosefrom the other DNS studies [16, 14]

Figure 5c shows the mean velocity in the streamwise direction and it iscompared to those from previous experimental and numerical studies. umean

is again obtained via an averaging between t/τ = 200 and t/τ = 600. Theagreement with other data is good.

The anisotropy of the Reynolds stresses shows the character and the de-pendence of the velocity fluctuations on the direction in turbulent flows. OnceReynolds stress terms are calculated, it can be obtained via bij =

Rij

2K− 1

3δij,

where K is the turbulent kinetic energy and δij is the Kronecker delta. It iscalculated by integrating over the shear layer [16], which is approximated bysummation.

Time evolutions of the anisotropy of the Reynolds stress terms are alsogiven in Fig. 5d. Table 2 compares their peak values to those from the Large-Eddy Simulation (LES) of Foysi & Sarkar [21] and DNS of Pantano & Sarkar[16]. In the self-similar region (i.e., after a sufficiently long time), the valuesof the diagonal components of bij reach a constant value. The present peakvalues are very close to those of Pantano & Sarkar [16]. The anisotropy inthe streamwise is stronger than in the other directions.

5.2. The compressible cases

In the previous section it was demonstrated for the base quasi-incompressiblecase that the algorithm and the solver produce the correct physical mecha-nisms behind the TSL instability. In the following cases, the behavior of thealgorithm in the presence of compressibility effects is investigated. For this

9

y/δω

√R11

/∆u

-1 -0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Spencer an Jones (EXP)Bell and Mehta (EXP)Rogers and Moser (DNS)Pantano and Sarkar (DNS)Present DNS

(a)

y/δω

√R12

/∆u

-1 -0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

Spencer an Jones (EXP)Bell and Mehta (EXP)Rogers and Moser (DNS)Pantano and Sarkar (DNS)Present DNS

(b)

Figure 4: Comparison of the Reynolds stresses for Mc = 0.3 (cont.)

10

y/δω

√R22

/∆u

-1 -0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Spencer and Jones (EXP)Bell and Mehta (EXP)Rogers and Moser (DNS)Pantano and Sarkar (DNS)Present DNS

(c)

y/δω

√R33

/∆u

-1 -0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Spencer and Jones (EXP)Bell and Mehta (EXP)Rogers and Moser (DNS)Pantano and Sarkar (DNS)Present DNS

(d)

Figure 4: Comparison of the Reynolds stresses for Mc = 0.3.

11

y/δθ

Pro

du

ctio

n

-4 -2 0 2 4

0

0.0005

0.001

0.0015

0.002

Pantano and Sarkar (DNS)Rogers and Moser (DNS)Present DNS

(a)

y/δθ

ε

-4 -2 0 2 4-0.0014

-0.0012

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

Rogers and Moser (DNS)Pantano and Sarkar (DNS)Present DNS

(b)

Figure 5: Comparison of the production (a), dissipation (b), mean streamwisevelocity (c), and time evolution of anisotropy terms (d) for Mc = 0.3 (cont.)

12

y/δω

um

ean

-1 -0.5 0 0.5 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5 Bell and Mehta (EXP)Spencer and Jones (EXP)Pantano and Sarkar (DNS)Present DNS

(c)

t/τ

bij

0 200 400 600-0.2

-0.1

0

0.1

0.2

0.3b11

b12

b22

(d)

Figure 5: Comparison of the production (a), dissipation (b), mean streamwisevelocity (c), and time evolution of anisotropy terms (d) for Mc = 0.3.

13

Table 2: Comparison of the peaks of the Reynolds stress anisotropies forMc = 0.3.

b11 b22 b12Foysi & Sarkar[21] (LES) 0.35 −0.20 0.21Pantano & Sarkar[16] (DNS) 0.26 −0.16 0.19Present DNS 0.25 −0.16 0.17

purpose, variations of Mc between 0.3 to 0.9 are studied. The results arepresented and are compared with the previous experimental and numericalstudies. Compressibility effects are also analyzed.

Fig. 6a compares the time evolutions of the momentum thicknesses fordifferent convective Mach numbers. A linear fit for each curve is plotted inthe self-similar region for calculation and comparison of the growth rates. Itis observed that the time required to obtain to a self-similar state increaseswith increasing Mc. The growth rate values are normalized by the baseincompressible growth rate value and are plotted against Mc in Fig. 6b.The filled circles represent our results. As is clearly seen, the compressibilityreduces the growth rate of the instabilities, agreeing well with many previousresults and the widely accepted Langley experimental curve represented bysolid line [16, 22, 23, 24, 25, 26].

Figure 7 shows that the compressibility effects reduce the Reynolds stressterms, production and dissipation, which is consistent with the previous stud-ies [16, 14, 21, 27]. The reduction in production is the source of the decreasein the growth rate, as confirmed by the other DNS studies [16, 14, 21]. Theturbulent dissipation is less affected by the compressibility than the otherterms, as previously noted in [16, 21]. A slight increase is observed in thepeak values of the Reynolds stress anisotropies at early times. The diagonalcomponents are more affected than the off-diagonal ones. The overall effectof increasing Mc on the values of bij’s is not large, which is consistent withprevious numerical studies [16, 21].

6. Conclusions

An in-house, fully parallel DNS solver, iDNS, was developed based ona fully implicit, non-dissipative, discrete kinetic energy conserving, all-speedflow algorithm and was successfully applied to temporally-evolving TurbulentShear Layer (TSL) problem.

14

t/τ

δ θ/δθ,

0

0 500 1000 1500 2000 25000

5

10

15

20

25

30

Mc=0.3Mc=0.5Mc=0.7Mc=0.9

(a)

Mc

(dδ θ/d

t)/(

dδθ/d

t)in

c

0 0.2 0.4 0.6 0.8 1 1.2 1.40.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2 Clemens and Mungal (EXP)

Debisschop and Bonnet (EXP)

Hall et. al. (EXP)

Langley curve (EXP)

Pantano and Sarkar (DNS)

Papamoschou and Roshko (EXP)

Samimy and Elliot (EXP)

Present DNS

(b)

Figure 6: Comparison of the momentum thicknesses (a) and the growth rates(b) for different values of Mc. 15

Mc

√R11

/∆u

0 0.5 1 1.5

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Debisschop and Bonnet (EXP)Elliot and Samimy (EXP)Goebel and Dutton (EXP)Pantano and Sarkar (DNS)Wyganski and Fiedler (EXP)Present DNS

(a)

Mc

√R22

/∆u

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18Debisschop and Bonnet (EXP)Elliot and Samimy (EXP)Goebel and Dutton (EXP)Pantano and Sarkar (DNS)Wyganski and Fiedler (EXP)Present DNS

(b)

Figure 7: The effect of compressibility on the various terms of the Reynoldsstress transport equation (cont.) 16

y/δθ

Pro

du

ctio

n

-2 0 2 40

0.0005

0.001

0.0015

0.002

0.0025

Mc=0.3 Pantano and Sarkar (DNS)Mc=0.3 Present DNSMc=0.7 Pantano and Sarkar (DNS)Mc=0.7 Present DNSMc=0.9

(c)

y/δθ

ε

-4 -2 0 2 4-0.0014

-0.0012

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004Mc=0.3, Rogers and Moser (DNS)Mc=0.3, Pantano and Sarkar (DNS)Mc=0.3, Present DNSMc=0.7, Pantano and Sarkar (DNS)Mc=0.7, Present DNSMc=0.9, Present DNS

(d)

Figure 7: The effect of compressibility on the various terms of the Reynoldsstress transport equation. 17

TSL simulation results prove that the algorithm is capable of captur-ing the evolution of the perturbations into a non-linear hydrodynamic statewhich can be regarded as the onset of the turbulence. By doing this study, weperformed further assessment of the algorithm and examined its behavior andapplicability to these kinds of flows, including complex spatial and temporalmulti-scale physics. All the results compare well to the previous experimen-tal and numerical results. It seems that non-dissipativeness and discretekinetic energy conserving properties work well. The improved convergencecharacteristic obtained via a time and space-relaxed procedure for the pres-sure saves the computational resources and prevents the loss of accuracy atlow-Mach numbers. A dissipative mechanism is necessary for studying higherMach numbers. However, single, unified approaches are very attractive forsolving such complex problems.

7. Acknowledgements

Computing resources used in this work were provided by the NationalCenter for High Performance Computing of Turkey (UYBHM) under grantnumber 1001212011 and TUBITAK–ULAKBIM, High Performance and GridComputing Center (TRUBA Resources). Since an earlier part of the workwas carried out at Chalmers University of Technology during the first au-thor’s research visit, IY thanks Istanbul Technical University and The Sci-entific and Technological Research Council of Turkey (TUBITAK) for sup-porting his stay. The financial support of SNIC (Swedish National Infras-tructure for Computing) during this period for computer time at C3SE(Chalmers Center for Computational Science and Engineering) under theProject SNIC001-10-22 is also gratefully acknowledged.

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