Direct numerical simulation of turbulent heat transfer in annuli: Effect of heat flux ratio

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Direct numerical simulation of turbulent heat transfer in annuli: Effect of heat flux ratio

Meryem Ould-Rouiss, L. Redjem Saad, Guy Lauriat

To cite this version:

Meryem Ould-Rouiss, L. Redjem Saad, Guy Lauriat. Direct numerical simulation of turbulent heat transfer in annuli: Effect of heat flux ratio. International Journal of Heat and Fluid Flow, Elsevier, 2009, 30 (4), pp.579-589. �10.1016/j.ijheatfluidflow.2009.02.018�. �hal-00734060�

1 Shemati of the omputationaldomain. . . . . . . . . . . . . . . . . . . . . 18

2 Two-point orrelations intheaxial diretion.. . . . . . . . . . . . . . . . . . 19

3 Two-point orrelations intheazimuthal diretion. . . . . . . . . . . . . . . . 20

4 Meanveloity prole.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 RMSveloityutuations: (a) streamwise, (b)radial, ()azimuthal. . . . . 22

6 Positions of zerototal shear stressr0 ^{and maximum veloity}rmax^{. . . . . . 23}

7 Meantemperature proles. (a)innerylinber, (b)outerylinder . . . . . . 24

8 RMSof temperature utuations for q^∗= 1^{. . . . . . . . . . . . . . . . . . . 25}

9 RMS of temperature utuations for various heat ux ratios. (a) q^∗ ≤ 1^, (b)q^∗≥1 ^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26}

10 Turbulentheatuxesforq^∗ = 1^{. (a)streamwiseomponent,(b)}wall-normal omponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

11 Streamwise turbulent heatux for various heatux ratios. (a)q^∗ ≤1^,(b) q^∗≥1 ^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28}

12 Wall-normal turbulent heatuxfor variousheatuxratios. (a)q^∗≤1^,(b) q^∗≥1 ^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29}

13 Position ofzero wall-normal turbulent heatuxversus heatuxratio. . . . 30

R

L

R₁

2

Figure1: Shemati oftheomputational domain.

z/ δ R

(z )

0 2 4 6 8 10 12

-0.2 0 0.2 0.4 0.6 0.8 1

v_z Θ

inner (a)

z/ δ R

(z )

0 2 4 6 8 10 12

-0.2 0 0.2 0.4 0.6 0.8 1

v_z Θ

outer (b)

Figure2: Two-point orrelationsin theaxialdiretion.

r θ/δ R

( θ)

0 2 4 6 8

-0.2 0 0.2 0.4 0.6 0.8 1

v_z Θ

outer (b)

r θ/δ R

( θ)

0 0.02 0.04 0.06 0.08 0.1 0.12

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1

v_z Θ

inner (a)

Figure3: Two-point orrelationsintheazimuthal diretion.

y/δ v z/u b

0 0.5 1 1.5 2

0 0.5 1 1.5

v_z/u_b

Chung et al. (2002)

(a)

y⁺

v+ z

10⁰ 10¹ 10²

0 5 10 15 20 25

inner outer

v_z⁺=y⁺

v_z⁺=2.5 ln (y⁺)+5.5

(b)

Figure 4: Mean veloityprole.

y⁺ v’+ z(rms)

0 20 40 60 80 100

0 0.5 1 1.5 2 2.5 3

Interne Externe Chung et al (2002) (a)

y⁺ v’+ r(rms)

20 40 60 80 100

0 0.2 0.4 0.6 0.8 1

Interne Externe Chung et al (2002) (b)

y⁺ v’+ θ(rms)

0 20 40 60 80 100

0 0.4 0.8 1.2

Interne Externe Chung et al (2002) (c)

Figure 5: RMS veloityutuations: (a)streamwise, (b)radial,() azimuthal.

y/δ v’zv’r/(uτ)2 e

0.5 1 1.5

-1 -0.5 0 0.5 1 1.5

v’zv’r

v’_zv’_r+1/Re(dv_z/dy)

y/δ=0.61

y/δ vz/ub

0 0.5 1 1.5 2

0 0.5 1 1.5

y/δ=0.64 DNS

y/δ

vz/ub

0 0.5 1 1.5 2

0 0.5 1 1.5

y/δ=0.64 (a)

y/δ

v’zv’r/(uτ)2 e

0 0.5 1 1.5 2

-1 -0.5 0 0.5 1 1.5

v’_zv’_r

v’_zv’_r-1/Re(dv_z/dy)

y/δ=0.61

(b)

Figure6: Positions ofzero total shearstress r0 ^{and maximumveloity} rmax^.

y⁺

Θ+

10⁰ 10¹ 10²

0 5 10 15 20

q^*=1 q^*=0.01 q^*=100

Θ⁺=(1/0.362)ln(y⁺)+1.8

Θ⁺=Pr.y⁺

y⁺

Θ+

10⁰ 10¹ 10²

0 5 10 15 20

q^*=1 q^*=0.01 q^*=100

Θ⁺=(1/0.362)ln(y⁺)+1.8

Θ⁺=Pr.y⁺

y⁺

Θ’+ rms

0 20 40 60

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

inner outer inner outer

Redjem et al. (2007)

}

Figure 8: RMSoftemperature utuations for q^∗= 1^.

y/δ Θ’ rms

0 0.5 1 1.5 2

0 0.01 0.02 0.03

q^*=1 q^*=0.5 q^*=0.25 q^*=0.1 q^*=0.01

(a)

y/δ Θ’ rms

0.5 1 1.5 2

0 0.05 0.1 0.15

q^*=1 q^*=2 q^*=5 q^*=10 q^*=100

(b)

∗ ≤ 1

y⁺ v’ zΘ’+

0 20 40 60

0 2 4 6 8 10

inner outer inner outer

Redjem et al. (2007)

}

(a)

y⁺ v’ rΘ’+

0 20 40 60

0 0.5 1 1.5

inner outer inner outer

Redjem et al. (2007)

}

(b)

∗ = 1

y/δ v’ zΘ’

0 0.5 1 1.5 2

0 0.001 0.002 0.003

q^*=1 q^*=0.5 q^*=0.25 q^*=0.1 q^*=0.01

(a)

y/δ v’ zΘ’

0 0.5 1 1.5 2

0 0.006 0.012 0.018

q^*=1 q^*=2 q^*=5 q^*=10 q^*=100

(b)

∗ ≤ 1

y/δ v’ rΘ’

0 0.5 1 1.5 2

-0.0002 0 0.0002 0.0004

q^*=1 q^*=0.5 q^*=0.25 q^*=0.1 q^*=0.01

(a)

y/δ v’ rΘ’

0 0.5 1 1.5 2

-0.0015 -0.001 -0.0005 0 0.0005

q^*=1 q^*=2 q^*=5 q^*=10 q^*=100

(b)

Figure 12: Wall-normal turbulent heat ux for various heat ux ratios. (a) q^∗ ≤ 1^{, (b)} q^∗≥1

q^*

y/δ

10^-2 10^-1 10⁰ 10¹ 10²

10^-2 10^-1 10⁰ 10¹

y/δ

y/δ=0.46q^*0.53

Figure13: Positionof zerowall-normal turbulent heatux versusheat uxratio.