Transport Equation for Dissipation Rate

Transport equation for dissipation rate

Exact transport equations for dissipation rates of turbulent kinetic energy, temperature variance and turbulent heat ﬂux are derived using Navier-Stokes and energy equations below. The individual terms in these equations are obtained and Reynolds and Prandtl number dependencies are examined using high spatial resolution DNS (Kozuka et al., 2008).

Nomenclature . k turbulent kinetic energy, u0 u0 2 ±i i 0 0 kθ temperature variance, θ θ 2 p pressure t time ui, u, v, w velocity component x1, x streamwise direction x2, y wall-normal direction x3, z spanwise direction Greek 0 2 ε dissipation rate of turbulent kinetic energy, ν (∂ui/∂xj) 0 2 εθ dissipation rate of temperature variance, κ (∂θ /∂xj) 0 0 εiθ dissipation rate of turbulent heat flux, (ν + κ) · (∂ui/∂xj) · (∂θ /∂xj) κ thermal diffusivity θ temperature ν kinematic viscosity ρ density Superscripts ()0 fluctuation component ( ) statistically averaged component

1, Transport equation for ε 0 Navier-Stokes equation on velocity ﬂuctuation ui can be given as follows: 0 0 ³ ´ 0 2 0 ∂ui 0 ∂ui ∂ui ∂ 0 0 0 0 1 ∂p ∂ ui + uk + uk + uiuk − uiuk = − + ν 2 . (1) ∂t ∂xk ∂xk ∂xk ρ ∂xi ∂xk

By diﬀerentiating Eq. (1) with xm, following equation can be obtained:

µ 0 ¶ 0 2 0 2 0 ∂ ∂ui ∂uk ∂ui 0 ∂ ui ∂uk ∂ui ∂ ui + + uk + + uk ∂xm ∂t ∂xm ∂xk ∂xm∂xk ∂xm ∂xk ∂xm∂xk

2 ³ ´ 2 0 3 0 ∂ 0 0 0 0 1 ∂ p ∂ ui + uiuk − uiuk = − + ν 2 . (2) ∂xm∂xk ρ ∂xm∂xi ∂xm∂xk 0 By multiplying both sides in Eq. (2) by 2ν (∂ui/∂xm), ensemble-averaged terms in Eq. (2) can be expressed as below: µ ¶ µ ¶ Ã ! ∂u0 ∂ ∂u0 ∂u0 ∂ ∂u0 ∂ ∂u0 ∂u0 ∂ε ・ 2ν i i = 2ν i i = ν i i = ∂xm ∂xm ∂t ∂xm ∂t ∂xm ∂t ∂xm ∂xm ∂t 2

0 0 0 0 µ ¶ 0 0 µ ¶ ∂u ∂u ∂ui ∂u ∂u ∂ui ∂ui ∂u ∂u ∂ui ∂uk ・ 2ν i k = ν i k + = ν i k + ∂xm ∂xm ∂xk ∂xm ∂xm ∂xk ∂xk ∂xm ∂xm ∂xk ∂xi 0 2 0 2 ∂ui 0 ∂ ui 0 ∂ui ∂ ui ・ 2ν uk = νuk ∂xm ∂xm∂xk ∂xm ∂xm∂xk 0 0 0 0 µ ¶ 0 0 µ ¶ ∂u ∂uk ∂u ∂u ∂u ∂uk ∂uk ∂u ∂u ∂uk ∂um ・ 2ν i i = ν i i + = ν i i + ∂xm ∂xm ∂xk ∂xm ∂xk ∂xm ∂xm ∂xm ∂xk ∂xm ∂xk Ã ! 0 2 0 0 0 ∂ui ∂ ui ∂ ∂ui ∂ui ∂ε ・ 2ν uk = uk ν = uk ∂xm ∂xm∂xk ∂xk ∂xm ∂xm ∂xk ( ) 0 2 ³ ´ 0 2 ¡ ¢ 2 0 0 ∂ui ∂ 0 0 0 0 ∂ui ∂ 0 0 ∂ uiuk ・ 2ν uiuk − uiuk = 2ν uiuk − ∂xm ∂xm∂xk ∂xm ∂xm∂xk ∂xm∂xk ( ) 0 µ 0 0 ¶ 2 0 0 ∂ui ∂ 0 ∂ui 0 ∂uk ∂ uiuk = 2ν uk + ui − ∂xm ∂xm ∂xk ∂xk ∂xm∂xk Ã ! 0 0 0 0 0 ∂ui ∂uk ∂ui ∂ 0 ∂ui ∂ui = 2ν + νuk ∂xm ∂xm ∂xk ∂xk ∂xm ∂xm Ã ! ∂u0 1 ∂2p0 2 ∂ ∂u0 ∂p0 ・ − 2ν i = − k ∂xm ρ ∂xm∂xi ρ ∂xk ∂xm ∂xm

Ã ! Ã !2 0 3 0 0 2 0 0 2 0 0 2 0 ∂ui ∂ ui 2 ∂ui ∂ ui ∂ui ∂ ∂ui ∂ui ∂ ui ・ 2ν ν 2 = 2ν 2 = ν 2 ν − 2 ν ∂xm ∂xm∂xk ∂xm ∂xk ∂xm ∂xk ∂xm ∂xm ∂xm∂xk Ã !2 2 2 0 ∂ ∂ ui = ν 2 (ε) − 2 ν ∂xk ∂xm∂xk Therefore，exact transport equation for dissipation rate ε of turbulent energy can be given as follows: Dε = P (1) + P (2) + P (3) + P (4) + T + V + π − γ , (3) Dt ε ε ε ε ε ε ε ε Mixed Production:

0 0 µ ¶ (1) ∂ui ∂uk ∂ui ∂uk Pε = ν + (4) ∂xm ∂xm ∂xk ∂xi Production by mean velocity gradient:

0 0 µ ¶ (2) ∂ui ∂ui ∂uk ∂um Pε = ν + (5) ∂xm ∂xk ∂xm ∂xk Gradient production:

0 2 (3) 0 ∂ui ∂ ui Pε = νuk (6) ∂xm ∂xm∂xk Turbulent production:

0 0 0 (4) ∂ui ∂uk ∂ui Pε = 2ν (7) ∂xm ∂xm ∂xk 3

Turbulent diﬀusion: Ã ! 0 0 ∂ 0 ∂ui ∂ui Tε = νuk (8) ∂xk ∂xm ∂xm

Viscous diﬀusion: Ã ! 2 0 0 ∂ ∂ui ∂ui Vε = ν 2 ν (9) ∂xk ∂xm ∂xm

Pressure diﬀusion: Ã ! 0 0 2 ∂ ∂uk ∂p πε = − (10) ρ ∂xk ∂xm ∂xm

Dissipation:

Ã !2 2 0 ∂ ui γε = 2 ν (11) ∂xm∂xk

2, Transport equation for εθ 0 By diﬀerentiating energy equation on temperature ﬂuctuation θ expressed by Eq. (12) with xm and 0 multiplying by 2κ (∂θ /∂xm), ensemble-averaged terms in Eq. (12) can be given as below.

0 0 0 0 0 2 0 ∂θ 0 ∂θ ∂θ 0 ∂θ ∂ukθ ∂ θ + uk + uk + uk − = κ 2 (12) ∂t ∂xk ∂xk ∂xk ∂xk ∂xk µ ¶ ( µ ¶ ) 0 0 0 2 ∂θ ∂ ∂θ ∂ ∂θ ∂εθ ・ 2κ = κ = ∂xm ∂xm ∂t ∂t ∂xm ∂t µ ¶ µ ¶ 0 0 0 2 ∂θ ∂ 0 ∂θ ∂θ ∂uk ∂θ 0 ∂ θ ・ 2κ uk = 2κ + uk ∂xm ∂xm ∂xk ∂xm ∂xm ∂xk ∂xm∂xk

0 0 0 2 ∂θ ∂uk ∂θ 0 ∂θ ∂ θ = 2κ + 2κuk ∂xm ∂xm ∂xk ∂xm ∂xm∂xk µ ¶ µ ¶ 0 0 0 0 2 0 ∂θ ∂ ∂θ ∂θ ∂uk ∂θ ∂ θ ・ 2κ uk = 2κ + uk ∂xm ∂xm ∂xk ∂xm ∂xm ∂xk ∂xm∂xk µ ¶ 0 0 0 0 0 0 ∂θ ∂θ ∂uk ∂ ∂θ ∂θ ∂θ ∂θ ∂uk ∂εθ = 2κ + uk κ = 2κ + uk ∂xm ∂xk ∂xm ∂xk ∂xk ∂xk ∂xm ∂xk ∂xm ∂xk µ ¶ µ ¶ 0 0 0 0 0 0 0 ∂θ ∂ 0 ∂θ ∂θ ∂uk ∂θ ∂ 0 ∂θ ∂θ ・ 2κ uk = 2κ + κ uk ∂xm ∂xm ∂xk ∂xm ∂xm ∂xk ∂xk ∂xm ∂xm 0 ³ ´ ∂θ ∂ ∂ 0 0 ・ 2κ ukθ = 0 ∂xm ∂xm ∂xm µ ¶ ( µ ¶ ) Ã !2 ∂θ0 ∂ ∂2θ0 ∂ ∂ ∂θ0 2 ∂2θ0 ・ 2κ κ = κ2 − 2κ2 ∂xm ∂xm ∂xk∂xk ∂xk ∂xk ∂xm ∂xk∂xk 4

Thus, transport equation for dissipation rate εθ of temperature variance can be derived as follows:

Dεθ (1) (2) (3) (4) = P + P + P + P + Tε + Vε − γε (13) Dt εθ εθ εθ εθ θ θ θ Production by mean temperature gradient:

∂θ0 ∂u0 ∂θ P (1) = 2κ k (14) εθ ∂xm ∂xm ∂xk Production by mean velocity gradient:

∂θ0 ∂θ0 ∂u P (2) = 2κ k (15) εθ ∂xm ∂xk ∂xm Gradient production:

∂θ0 ∂2θ P (3) = 2κu0 (16) εθ k ∂xm ∂xm∂xk Turbulent production:

∂θ0 ∂u0 ∂θ0 P (4) = 2κ k (17) εθ ∂xm ∂xm ∂xk Turbulent diﬀusion: µ ¶ 0 0 ∂ 0 ∂θ ∂θ Tεθ = κ uk (18) ∂xk ∂xm ∂xm Viscous diﬀusion: ( µ ¶ ) 0 2 ∂ ∂ 2 ∂θ Vεθ = κ (19) ∂xk ∂xk ∂xm

Dissipation:

Ã !2 2 0 2 ∂ θ γεθ = 2κ (20) ∂xk∂xj

3, Transport equation for εiθ 0 By diﬀerentiating Eq. (1) with xj and multiplying by (∂θ /∂xj), following equation can be obtained.

0 µ 0 ¶ µ 0 ¶ 0 µ 0 ¶ 2 ∂ ∂ui ∂θ ∂θ ∂uk ∂ui ∂θ 0 ∂ ui + + uk ∂t ∂xj ∂xj ∂xj ∂xj ∂xk ∂xj ∂xj∂xk

µ 0 ¶ 0 µ 0 ¶ 2 0 µ 0 ¶ 2 ³ ´ ∂θ ∂uk ∂ui ∂θ ∂ ui ∂θ ∂ 0 0 0 0 + + uk + uiuk − uiuk ∂xj ∂xj ∂xk ∂xj ∂xj∂xk ∂xj ∂xj∂xk µ ¶ µ ¶ µ ¶ 1 ∂θ0 ∂2p0 ∂θ0 ∂ ∂2u0 = − + ν i (21) ρ ∂xj ∂xj∂xi ∂xj ∂xj ∂xk∂xk 0 By diﬀerentiating Eq. (12) with xj and multiplying by (∂ui/∂xj), following equation can be obtained. µ 0 ¶ 0 µ 0 ¶ 0 µ 0 ¶ 2 ∂ ∂ui ∂θ ∂ui ∂uk ∂θ ∂ui 0 ∂ θ + + uk ∂t ∂xj ∂xj ∂xj ∂xj ∂xk ∂xj ∂xj∂xk 5

µ 0 ¶ 0 µ 0 ¶ 2 0 µ 0 ¶ 0 0 µ 0 ¶ 2 0 ∂ui ∂uk ∂θ ∂ui ∂ θ ∂ui ∂uk ∂θ ∂ui 0 ∂ θ + + uk + + uk ∂xj ∂xj ∂xk ∂xj ∂xj∂xk ∂xj ∂xj ∂xk ∂xj ∂xj∂xk µ 0 ¶ 2 ³ ´ µ 0 ¶ µ 2 0 ¶ ∂ui ∂ 0 0 ∂ui ∂ ∂ θ + uiθ = κ (22) ∂xj ∂xj∂xk ∂xj ∂xj ∂xk∂xk By adding both sides in Eq. (21) to in Eq. (22) and multiplying by (ν + κ), ensemble-averaged terms are given as follows: ½ µ ¶ µ ¶ ¾ ( ) 0 0 0 0 0 0 ∂ ∂u ∂θ ∂ ∂u ∂θ ∂ ∂θ ∂u ∂εiθ ・ (ν + κ) i + i = (ν + κ) i = ∂t ∂xj ∂xj ∂t ∂xj ∂xj ∂t ∂xj ∂xj ∂t (µ ¶ µ ¶ ) ( ) 0 2 0 0 2 0 0 0 ∂θ ∂ ui ∂ui ∂ θ ∂ ∂θ ∂ui ・ (ν + κ) uk + uk = uk (ν + κ) ∂xj ∂xj∂xk ∂xj ∂xj∂xk ∂xk ∂xj ∂xj (µ ¶ µ ¶ µ ¶ µ ¶ ) 0 0 0 0 0 0 0 0 ∂θ ∂u ∂ui ∂θ ∂uk ∂u ∂u ∂uk ∂θ ∂u ∂u ∂θ ・ (ν + κ) k + i + i + i k ∂xj ∂xj ∂xk ∂xj ∂xj ∂xk ∂xj ∂xj ∂xk ∂xj ∂xj ∂xk ( ) ∂θ0 ∂u0 ∂u ∂θ0 ∂u0 ∂u0 ∂θ0 ∂u ∂u0 ∂u0 ∂θ = (ν + κ) k i + (ν + κ) i + i k + (ν + κ) i k ∂xj ∂xj ∂xk ∂xj ∂xk ∂xj ∂xk ∂xj ∂xj ∂xj ∂xk (µ ¶ µ ¶ ) 0 2 0 2 ∂θ 0 ∂ ui ∂ui 0 ∂ θ ・ (ν + κ) uk + uk ∂xj ∂xj∂xk ∂xj ∂xj∂xk ( ) 0 2 0 2 0 ∂θ ∂ ui 0 ∂ui ∂ θ = (ν + κ) uk + uk ∂xj ∂xj∂xk ∂xj ∂xj∂xk (µ ¶ µ ¶ ) 0 2 ¡ ¢ 0 2 ³ ´ ∂θ ∂ 0 0 ∂θ ∂ 0 0 ・ (ν + κ) uiuk − uiuk ∂xj ∂xj∂xk ∂xj ∂xj∂xk (µ ¶ µ ¶ µ ¶ ) 0 ∂u0 0 0 2 0 0 2 ³ ´ ∂ui k ∂θ ∂ui 0 ∂ θ ∂ui ∂ 0 0 + (ν + κ) + uk + uiθ ∂xj ∂xj ∂xk ∂xj ∂xj∂xk ∂xj ∂xj∂xk (µ ¶ µ ¶ ) µ ¶ 0 2 ¡ ¢ 0 0 0 0 2 0 ∂θ ∂ 0 0 ∂ui ∂uk ∂θ ∂ui 0 ∂ θ = (ν + κ) uiuk + + (ν + κ) uk ∂xj ∂xj∂xk ∂xj ∂xj ∂xk ∂xj ∂xj∂xk Ã ! Ã ! 0 0 0 0 0 0 0 ∂ 0 ∂ui ∂θ ∂uk ∂ui ∂θ ∂ui ∂θ = (ν + κ) uk + (ν + κ) + ∂xk ∂xj ∂xj ∂xj ∂xk ∂xj ∂xj ∂xk µ ¶ µ ¶ (ν + κ) ∂θ0 ∂2p0 (ν + κ) ∂ ∂p0 ∂θ0 (ν + κ) ∂2θ0 ∂p0 ・ − = − + ρ ∂xj ∂xj∂xi ρ ∂xi ∂xj ∂xj ρ ∂xj∂xi ∂xj ½ µ ¶ µ ¶ µ ¶ µ ¶¾ ∂θ0 ∂ ∂2u0 ∂u0 ∂ ∂2θ0 ・ (ν + κ) ν i + κ i ∂xj ∂xj ∂xk∂xk ∂xj ∂xj ∂xk∂xk ( µ ¶ Ã !) ∂ ∂θ0 ∂2u0 ∂ ∂u0 ∂2θ0 = (ν + κ) ν i + κ i ∂xk ∂xj ∂xj∂xk ∂xk ∂xj ∂xj∂xk Ã ! ∂2u0 ∂2θ0 ∂2u0 ∂2θ0 − (ν + κ) ν i + κ i ∂xj∂xk ∂xj∂xk ∂xj∂xk ∂xj∂xk Ã ! Ã ! ∂ ∂θ0 ∂2u0 ∂u0 ∂2θ0 ∂2u0 ∂2θ0 = (ν + κ) i + κ i − (ν + κ)2 i ∂xk ∂xj ∂xj∂xk ∂xj ∂xj∂xk ∂xj∂xk ∂xj∂xk 6

Therefore, exact transport equation for dissipation rate εiθ of turbulent heat ﬂux can be written as below.

Dεiθ (1) (2) (3) (4) = P + P + P + P + Tε + Vε + πε + φε − γε (23) Dt εiθ εiθ εiθ εiθ iθ iθ iθ iθ iθ Production by mean temperature gradient:

∂u0 ∂u0 ∂θ P (1) = (ν + κ) i k (24) εiθ ∂xj ∂xj ∂xk Production by mean velocity gradient: ( ) ∂θ0 ∂u0 ∂u ∂θ0 ∂u0 ∂u0 ∂θ0 ∂u P (2) = (ν + κ) k i + (ν + κ) i + i k (25) εiθ ∂xj ∂xj ∂xk ∂xj ∂xk ∂xj ∂xk ∂xj

Gradient production: ( ) ∂θ0 ∂2u ∂u0 ∂2θ P (3) = (ν + κ) u0 i + u0 i (26) εiθ k k ∂xj ∂xj∂xk ∂xj ∂xj∂xk

Turbulent production: Ã ! ∂u0 ∂u0 ∂θ0 ∂u0 ∂θ0 P (4) = (ν + κ) k i + i (27) εiθ ∂xj ∂xk ∂xj ∂xj ∂xk

Turbulent diﬀusion: Ã ! 0 0 ∂ 0 ∂ui ∂θ Tεiθ = (ν + κ) uk (28) ∂xk ∂xj ∂xj

Viscous diﬀusion: Ã ! 0 2 0 0 2 0 ∂ ∂θ ∂ ui ∂ui ∂ θ Vεiθ = (ν + κ) + κ (29) ∂xk ∂xj ∂xj∂xk ∂xj ∂xj∂xk

Pressure diﬀusion: µ ¶ (ν + κ) ∂ ∂p0 ∂θ0 πεiθ = − (30) ρ ∂xi ∂xj ∂xj Pressure gradient correlation:

(ν + κ) ∂2θ0 ∂p0 φεiθ = (31) ρ ∂xj∂xi ∂xj Dissipation: Ã ! 2 0 2 0 2 ∂ ui ∂ θ γεiθ = (ν + κ) (32) ∂xj∂xk ∂xj∂xk

Reference Kozuka, M., Seki, Y. and Kawamura, H., 2008, ”DNS of turbulent heat transfer in a channel ﬂow with a high spatial resolution”, Proc. of 7th Int. Symp. on Engineering Turbulence Modelling and Measurements(ETMM7), Vol. 1, pp. 163-168.