ME-662 CONVECTIVE HEAT AND MASS TRANSFER

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A. W. Date

Mechanical Engineering Department Indian Institute of Technology, Bombay

Mumbai - 400076 India

LECTURE-27 TURBULENCE MODELS-2

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LECTURE-27 TURBULENCE MODELS-2

1 Low Ret Two-Eqn model

2 High Ret Stress-Eqn model

3 Low Ret Stress-Eqn model

4 Algebraic Stress-Eqn model

5 Scalar Transport model

1 Eddy Diffusivity model

2 Turbulent Flux Model

6 Modeling Combustion and Turbulence Interaction

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Low Re

_t

e- model L27(

₁₉¹

)

For low Ret =νt/ν ρ De

∂t = ∂

∂xi

(µ+ µt

σ_e) ∂e

∂xi

+µ_t

∂ui

∂xj

+ ∂uj

∂xi

∂ui

∂xj

−ρ ^∗ ρ D^∗

∂t = ∂

∂xi

(µ+µ_t σ)∂^∗

∂xi

− ^∗ e

C1µt

∂ui

∂xj

+∂uj

∂xi

∂ui

∂xj

−C2ρ ^∗

+ 2ν µ_t ( ∂²ui

∂xk ∂xl

)² µ_t = C_D^∗ (ρe²

^∗ ), C_D^∗ =CD exp

−3.4 (1+Ret/50)²

C₁^∗ = C1, C₂^∗ =C2

1−0.3 exp

−Re²_t ^∗ = −2ν(∂e^0.5

∂xi

)²

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Comments - L27(

₁₉²

)

1 Model constants¹ are sensitised to low Ret region near the wall. They tend to high Ret values beyond sub-layers

2 The correction to CD is chosen to give values ofν_t in agreement with the Van-Driest mixing length formula

3 The correction to C2 is selected from exptl. data on the decay of isotropic turbulence at low Ret ( at large times, e∝t⁻ⁿ where n'2.5 to 2.8).

4 The correction tois introduced to account for the non-isotropic contribution to the dissipation.

5 Wall-functions are no longer necessaryand e andEqns can be solved with ewall =^∗_wall =0. However, to capture the low Ret effects, very fine mesh (>60 grid nodes ) become necessary in the y⁺<100 region.

1Jones W P and Launder B L The Prediction of Laminarisation with a Two-Equation Model of Turbulence, Int. Jnl. of Heat and Mass Transfer, vol.

15, p 301, 1972

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Stress Eqn Model- L27(

₁₉³

)

Six transport equations for the one-point correlation u_i⁰ u⁰_j are derived from equation for Bij by setting separationξ_k =0 ( lecture 23 )

D u_i⁰u_j⁰

Dt = −

u_j⁰u_k⁰ ∂ui

∂xk

+u⁰_i u_k⁰ ∂uj

∂xk

{Pij}

− ∂

∂xk

"

u⁰_i u_j⁰ u_k⁰ +p⁰ ρ

n

u_i⁰ δ_jk +u⁰_j δ_iko

#

{D_ij} + p⁰

ρ

(∂u_i⁰

∂xj

+ ∂u_j⁰

∂xi

)

−2ν ∂u_i⁰

∂xk

∂u⁰_j

∂xk

{PS_ij} {_ij}

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Modeling u

_i⁰

u

_j⁰

Eqn - L27(

₁₉⁴

)

1 Invoking the idea of local isotropy at high Ret the destruction rate is equally distributed among all its components. Hence_ij = (2/3) δ_ij whereis obtained from its eqn.

2 Pressure-Strain Correlation PSij acts in two ways: Firstly, it sustains the division of TKE ( e ) into its three components u⁰²_i and secondly, it destructs the absolute magnitude of the shear stresses. Hence, without further elaboration

−PSij = Cp1

e (u_i⁰ v_j⁰ −2

3 eδ_ij) +Cp2(Pij − Pii

3 ) + Cp3(P_ij⁰ − 2

3P δij) +Cp4e(∂ui

∂xj

+∂uj

∂xi

) +PSw

PSw = e^3/2 LB

C_p1⁰

e (u_i⁰ v_j⁰ −2

3 eδij) +C_p2⁰ (Pij −P_ij⁰)

P_ij⁰ = −u⁰_i u_k⁰ ∂Uk

∂xj

−u⁰_j u_k⁰ ∂Uk

∂xi

(see next slide)

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Contd . . . - L27(

₁₉⁵

)

This algebraic expression for PSij is derived from its exact Eqn². The term containing Cp1 is calledreturn-to-isotropy. The PSw

term is called thewall-reflection termwhich accounts for the effects of pressure reflections from the wall. The recommended constants are: Cp1 =1.5, C_p1⁰ =0.12, Cp2 =0.764, C_p2⁰ = 0.01, Cp3 =0.109, Cp4=0.182, LB =wall distance. Finally the Triple Velocity correlation u_i⁰ u_j⁰ u⁰_k in theDiffusion term Dij is modeled from its exact Eqn and(p⁰/ρ) n

∂u_i⁰/∂xj+∂u_j⁰/∂xi

o'0.

−u_i⁰u_j⁰ u_k⁰ =Cs

e

(

u_i⁰u_l⁰ ∂u_j⁰u_k⁰

∂xl

+u⁰_j u_l⁰ ∂u_k⁰ u⁰_i

∂xl

+u⁰_ku_l⁰ ∂u_i⁰ u⁰_j

∂xl

)

where Cs '0.08 to o.11 ( from num expts )

2Hanjalic K. and Launder B. E. A Reynolds Stress Model of Turbulence and its Application to Thin Shear Flows, JFM.,52(4), p 609-638, 1972

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Algebraic Models ( ASMs ) - L27(

₁₉⁶

)

1 Implementation of Stress-Eqn model requires solution of 6 differential eqns for u_i⁰ u⁰_j, 2 Eqns for e andcoupled with the 3 RANS Eqns. This is a formidable problem.

2 The modeled forms presented above show that spatial gradients of u_i⁰ u_j⁰ occur only in thediffusion and convection - these terms make the Eqns differential ones.

3 Alg. Stress Models are developedusing the idea that u_i⁰ u_j⁰

e '

D u_i⁰u_j⁰

Dt −Diff(u_i⁰u_j⁰)

D e

D t −Diff(e) = −(2/3) (1−Cp1)δ_ij + (P/ )F (P/ )−1+Cp1

F = (1−Cp2) Pij

P −Cp3

P_ij⁰

P −Cp4

e P (∂ui

∂xj

+∂uj

∂xi

) + 2

3 (Cp2+Cp3)δij ( computational expense reduced)

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Low Re

_t

ASM - L27(

₁₉⁷

)

u⁰_i u_j⁰ = −(2/3)eδij +e×F F = ν_t

e Sij +C1

ν_t

e (SikSjk − 1

3Skl Sklδ_ij) + C2

νt

e (Ωik Sjk+ ΩjkSjk) +C3

νt

e (ΩikΩjk− 1

3 ΩklΩklδij) + C4

ν_t e

(^∗)² (SklΩ_lj +SkjΩ_li)Skl

+ C5

νt e

(^∗)² (Ωil ΩlmSmj+Sil ΩlmΩmj− 2

3 SlmΩmnΩnl δij) + ν_t e

(^∗)² (C6Sij Skl Skl +C7Sij Ω_klΩ_kl) ( see next slide )

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Low Re

_t

ASM Contd - L27(

₁₉⁸

)

Ω_ij = (∂ui/∂xj−∂uj/∂xi) Sij = (∂ui/∂xj+∂uj/∂xi) µ_t = fµC_D^∗ e²/^∗ → fµ=1−exp

−(Ret

90 )^0.5−(Ret

90 )²

C_D^∗ = 0.3×(1+0.35 n

max(S,Ω) o1.5

)⁻¹

×

1−exp

− 0.36

exp(−0.75 max(S,Ω)

S = (e/^∗) q

0.5 Sij Sij Ω = (e/^∗)p

0.5Ω_ij Ω_ij

Constants are: C1 = -0.1, C2= 0.1, C3= 0.26, C4 = - 10 (C_D^∗)², C5= 0, C6= - 5 (C_D^∗)² and C7 = 5 (C_D^∗)². The model is tested for very complex strain fields - swirling flows, curved channels and jet-impingement on a wall ( Craft T. J., Launder B. L. and Suga K, IJHFF, 17(12), p 108, 1996 )

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Scalar Transport - L27(

₁₉⁹

)

From Lecture 21,

ρ_mcpm

"

∂Tˆ

∂t +∂uˆjTˆ

∂xj

#

= − ∂qˆj

∂xj

+µΦˆ_v (Instantaneous) ρ_mcpm

∂T

∂t +∂ujT

∂xj

= − ∂

∂xj

(−km

∂T

∂xj

+ρ_mcpmu⁰_j T⁰) + µ_eff Φ_v +ρ_m (Time averaged) ρmcpmu_j⁰ T⁰ must be obtained from

1 Eddy Diffusivity model, or

2 Transport Eqn for u_j⁰ T⁰

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Eddy Diffusivity model - L27(

¹⁰₁₉

)

1 Analogous toµt, we defineTurbulent thermal conductivity kt

so that

−u_i⁰ T⁰ = ( kt

ρcp

)∂T

∂xi

=α_t ∂T

∂xi

= ν_t PrT

∂T

∂xi

where PrT = Turbulent Prandtl number simeq 0.9 when Ret

is high.

2 Hence, energy Eqn will read as D T

D t = ∂

∂xk

( ν

Pr + ν_t σt

) ∂T

∂xk

+ Qgen

ρcp

where Qgen =µeff Φv +ρm. Usually,ρm << µeff Φv.

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Comments on EDM - L27(

¹¹₁₉

)

1 The model is very convenient becauseν_t is obtained from mixing length, or one- or two-eqn models and PrT is an absolute constant

2 The disadvantage is thatαt =0 whereνt =0. In several flows, significant temperature gradients and hence heat transfer exist in regions whereν_t =0.

3 Likeν_t,α_t is also isotropic. But, measurement of decay of non-axi-symmetric temperature profiles in a fully developed turbulent flow in a pipe suggests that the ratio of tangential to radial diffusivities (α_t,θ/α_t,r )>>1 near the wall.

4 Therefore, in general, u_i⁰ T⁰ must be obtained directly from its differential transport equation.

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u

_i⁰

T

⁰

Eqn - L27(

¹²₁₉

)

Eqn for u_i⁰T⁰ is derived by multiplying Eqn forT by uˆ _i⁰ and Eqn foruˆi by T⁰ - addition and time-averaging gives.

∂u_i⁰ T⁰

∂t +uk

∂u_i⁰ T⁰

∂xk

= −

u_i⁰ u_k⁰ ∂T

∂xk

+u_k⁰ T⁰ ∂ui

∂xk

{PT}

− ∂

∂xk

"

u_i⁰ u⁰_kT⁰ +p⁰T⁰

ρ δ_ik −α ∂u_i⁰T⁰

∂xk

#

{D_T} + p⁰

ρ

∂T⁰

∂xi

− (ν+α) ∂u⁰_i

∂xk

∂T⁰

∂xk

{RDT} {DisT}

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Modeling u

_i⁰

T

⁰

Eqn - L27(

¹³₁₉

)

1 Like PSij,Redistribution term RDT is modeled as RDT = −C_{T 1}

e u_i⁰ T⁰ +CT 2u_k⁰ T⁰ ∂ui

∂xk

= −0.5

e u_n⁰ T⁰ e^3/2 LB

( for Pr >1)

= −

CT 1+0.5(Pr +1 Pr )

e u_i⁰ T⁰ ( for Pr << 1)

2 At high Ret or ( Peclet ),the task of Destruction is performed by RDT . Hence,DisT =0.

3 In thediffusion term, effect of p⁰ is either neglected or taken to be 0.2×u_i⁰ u_k⁰ T⁰ where

−u_i⁰ u_k⁰ T⁰ =CT

e

"

u_j⁰ u_k⁰ ∂u⁰_i T⁰

∂xj

+u_i⁰u_k⁰ ∂u_j⁰ T⁰

∂xj

#

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Solving u

_i⁰

T

⁰

Eqn - L27(

¹⁴₁₉

)

1 The model constants are: CT 1= 3.6, CT 2= 0.266 and CT = 0.11.

2 Required correlations are taken as

−u_i⁰ T⁰ = ν_t PrT

∂T

∂xi

and − u⁰_i u_j⁰ =νt Sij

3 νt is determined from e andEqns

4 For complete range of Prandtl numbers, PrT is modeled as PrT =0.85+0.0309

Pr +1 Pr

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Algebraic Flux Model - L27(

¹⁵₁₉

)

1 Eqn forscalar fluctuationsis derived as D T⁰²/2

Dt = − ∂

∂xi

"

u_i⁰T⁰²

2 −α ∂

∂xi

(T⁰² 2

)#

− u_i⁰ T⁰ ∂T

∂xi

− α(∂T⁰

∂xi

)² whereα(^∂T_∂x⁰

i)² =T ∝ ^e T⁰²

2 The AFM is derived from D u_i⁰ T⁰

D t −Diff(u_i⁰ T⁰) =

"

(P−)e+ (P−)

T⁰²

2

# u_i⁰ T⁰ ep

T⁰² T⁰² = C_T⁰ e

u_i⁰ T⁰ ∂T

∂xk

prod = diss assumed where C_T⁰ '1.6 for Pr ≥1.

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Evidence from DNS - Pipe flow - L27(

¹⁶₁₉

)

1.5 3.0 4.5

10 Y⁺ 100

T 2

Pr = 2

Pr = 0.1 ( q w / ρCp Uτ)

Pr = 0.71

1 Mean T profiles for pipe flow agreed with DNS

2 Location of peak T⁰² shows that production shifts

towards larger y⁺ as Pr decreases.

0.2

−0.3

−0.2

− 0.1 0.1 0.3

LossGain

Production

Diffusion

Redistribution Dissipation

50 150

0.02 0.04

− 0.04

− 0.02

GainLoss

Redistribution

Diffusion

Production Dissipation

Budget v T Budget

50 150

( a ) u T ( b )

1 u⁰T⁰ budget is similar to e-budget

2 v⁰T⁰ budget resembles u⁰v⁰ budget justifying Eddy Diff model for this case.

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Combustion and Turbulence - 1 - L27(

¹⁷₁₉

)

1 InCombustionit is necessary to solve differential eqns for all participating species k.

∂(ρmωk)

∂t + ∂(ρmujωk)

∂xj

= ∂

∂xj

ρ_mDeff

∂ωk

∂xj

+Rk

whereRk = rate of species generation/consumption.

Deff =ν/Sc+ν_t/SCt and SCt '0.9.

2 The simplest postulate is called the Simple Chemical Reaction ( SCR ) that is written as

1 kg of Fuel+Rst kg of Oxidant= (1+Rst) kg of Product There are only three species Fuel, Oxidant air and Products and Rst =(A/F)stoich

3 Rox =Rst ×Rfu and Rpr =−(1+Rst)×Rfu. In laminar flow Rfu =−A exp(−E

RuT )ω^m_fuω_oxⁿ (A and E are fuel-specific)

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Combustion and Turbulence - 2 - L27(

¹⁸₁₉

)

1 Inturbulent combustion, however, it is observed thatouter edges of flames are very intermittent and jagged.

2 Experimentally it is observed thateven if time-averaged ω_fu andω_ox are high, Rfu rates are not as high as would be expected from the Arrhenius formula

3 This is because,the fuel and oxidant at a given point are present at different times. Clearly, therefore, time scales of chemical reaction and turbulence are important. These are characterised bySL/u_rms⁰ where SL is the laminar flame speed of the fuel.

4 These ideas are captured³in Rfu =−Cebuρ_m

q

(ω⁰_fu)²

e ' −Cebuρ_mω_fu e

3Spalding D. B. Development of Eddy-Breakup Model of Turbulent Combustion, 16th Symposium on Combustion, p 1657, 1976

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Combustion and Turbulence - 3 - L27(

¹⁹₁₉

)

1 In practical computing, the applicability of the EBU has been enhanced by the following variant

Rfu =−ρ_m e Min

Aω_fu, A Rst

ω_ox, A⁰ 1+Rst

ω_prod

where A = 4 and A⁰ '2.

2 In the next lecture, we shall discuss tow important aspects of turbulent flows: ( a ) Laminar-to-Turbulent Transition and ( b ) Effect of Wall Roughness

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