Chapter 4 Heat Transfer in Turbulent Flow

1. Turbulence

♦ Nature of turbulence

o Irregularity

-statistical method instead of deterministic approach

o Diffusivity

- rapid mixing and increased rate of transfer

o Large Re

- interaction of viscous term and nonlinear inertia term, and instability

o 3-dimensional vorticity fluctuations

- rotation and 3-dimensional

o Dissipation

- viscous loss and continuous energy supply

o Continuum

- smallest scales in turbulent flow >> molecular length scale

o Turbulent flow are flows.

The magnitude of instantaneous quantities ;

[eqn: v ~~=~ bar { v} ~~+~v' # T~=~ bar{T} ~~+~T' # p ~~=~ bar{p} ~~+~p']                                     (1)

Governing equations for turbulent flow

[eqn: {Partial v_x } over {partial t} ~+ ~{Partial v_x^2 } over {partial x}+ {Partial v_x v_y } over {partial y} ~~=~ { 1} over {rho } { partial} over {partial x }(mu { partial v_x} over { partial x} )+ { 1} over {rho }{ partial} over {partial y }(mu { partial v_x} over { partial y}) - { 1} over {rho }{ partial p} over {partial x }]

[eqn: {Partial T } over {partial t} ~+ ~{Partial v_x T } over {partial x}+ {Partial v_y T} over {partial y} ~~=~ { 1} over {rho c_p } LEFT [ { partial} over {partial y }(k { partial T} over { partial y} )+ { partial} over {partial x }(k { partial T} over { partial x}) RIGHT ]]                 (2)

Time-averaged mean values

① solve instantaneous differential equations directly and integrate

its solutions for proper time interval.

② time-average the differential equations themselves and solve.

Consider the heat transfer in the fully developed turbulent flow     between two parallel plates.  Eq.(2) can be expressed with Eq. (1) as,

[eqn: {Partial} over {partial x} [( bar { v_x} +v'_x )^2 ] +{Partial} over {partial y} [( bar { v_x}+v'_x )( bar { v_y} +v'_y ) ] ~~=~ { 1} over {rho } { partial} over {partial x }[mu { partial } over { partial x}( bar { v_x} +v'_x ) ] # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+ { 1} over {rho } { partial} over {partial y }[mu { partial } over { partial y}( bar { v_x} +v'_x ) ] - { 1} over {rho }{ partial } over {partial x } ({ bar p + p')](3)

[eqn: {Partial } over {partial x}[(bar{v_x}+v'_x )(bar{T} +T')]+ {Partial } over {partial y}[(bar{v_y}+v'_y )(bar{T} +T')] #~~~~~~~~~~~~~~~~~ ~~=~ { 1} over {rho c_p } LEFT [ { partial} over {partial y }[k { partial } over { partial y}(bar{T} +T') ]+ { partial} over {partial x }[k { partial } over { partial x} (bar{T} +T') ] RIGHT ]]         (4)

Note that

☞ [eqn: bar v ,~bar T ,~bar p] (x,y, t) ,  but  v', T', p'(x,y,z,t)

☞ [eqn: bar { bar { v} v' } ~~=~ bar v bar v' ~~=~0]   and  [eqn: bar { bar { v} T' } ~~=~ bar v bar T' ~~=~0]

☞ But [eqn: bar { v'_x v'_y } ~~ !=~0]

☞ same sign of [eqn: v'_x v'_y]

Instantaneous and time-averaged continuity equations

[eqn: {Partial} over {partial x} ( bar { v_x} +v'_x ) +{Partial} over {partial y} ( bar { v_y} +v'_y ) ~~=~0]

[eqn: { partial { bar {v_x } } } over {partial x} + { partial { bar {v_y } } } over {partial y}~~=~0]                                             (5)

Then, time-averaged turbulent equations are,

[eqn: bar{v _{x}} {Partial bar{v _{x}}} over {Partial x} + bar{v _{y}} {Partial bar{v _{y}}} over {Partial y} ~~=~ {1} over {rho } LEFT [ {Partial } over {Partial x} ( mu {Partial bar{v _{x}}} over {Partial x} - bar{rho v prime _{x} ^{2}} )+ {Partial } over {Partial y} ( mu {Partial bar{v _{x}}} over {Partial y} - bar{rho v prime _{x} v prime _{y}} ) RIGHT ] - {1} over {rho } {Partial bar{p}} over {Partial x}]    (6)

[eqn: bar{v _{x}} {Partial bar{T} } over {Partial x} + bar{v _{y}} {Partial bar{T} } over {Partial y} ~~=~ {1} over {rho c _{p}} LEFT [ {Partial } over {Partial y} (k {Partial bar T} over {Partial y} - rho c_p bar {v'_y T' } ) + {Partial } over {Partial x} [k {Partial bar T } over {Partial x} - rho c_p bar {v'_x T' }) RIGHT ]]       (7)

=> laminar flow eqs. + additional terms (for turbulent motion)

Increased total transfer flux

[eqn: (tau_yx )_t ~~=~ - LEFT [ mu {Partial bar{v _{x}}} over {Partial x} - bar{rho v prime _{x} v prime _{y}} RIGHT ]]

[eqn: LEFT ( { q} over {A } RIGHT )_t ~~ =~ - LEFT [ k {Partial bar T} over {Partial y} - rho c_p bar {v'_y T' } RIGHT ]]                           (8)

=> Rate equations for turbulent flow system

(1) Eddy diffusivities of momentum and heat

[eqn: bar{ v prime _{x} v prime _{y}} ~~ ==~ - epsilon_m { partial bar v_x} over {partial y}]                                    (9)

Then,

[eqn: (tau_yx )_t ~~=~ tau_mol + tau_eddy ~~=~ - ( mu + rho epsilon_m ) {Partial bar{v _{x}}} over {Partial y}]

Or,      [eqn: {(tau_yx )_t} over rho ~~=~ ( nu + epsilon_m ) {Partial bar{v _{x}}} over {Partial y}]                                (10)

Similarly,

[eqn: LEFT ( { q} over {A } RIGHT )_t ~~ =~LEFT ( { q} over {A } RIGHT )_mole + LEFT ( { q} over {A } RIGHT )_eddy ~~=~ - (k + rho c_p epsilon_h ) {Partial bar T} over {Partial y}]

Or,

[eqn: {( q/A } RIGHT )_t} over {rho c_p} ~~ =~ - (alpha + epsilon_h ) {Partial bar T} over {Partial y}]                               (11)

(2) Distribution of [eqn: tau_y ~~and ~~ (q/A)_t]

For steady-state and fully-developed flow between parallel plates,

[eqn: bar v_y ~~=~0~, ~~~ {partial bar v_x} over { partial x}~~=~0]

Also,    [eqn: bar { {partial (rho v'_x^2 ) } over {partial x } } ~~=~0]

And      [eqn: { partial bar p} over { partial x } ~~=~ { d bar p} over{ d x }]

Then, from Eqs. (6) and (8),

[eqn: { dp} over {dx } ~~=~ { d tau_t} over {dy}]

If [eqn: tau_t ~~=~tau_o] at y=0

[eqn: { tau_t} over { tau_o} ~~=~ (1- { y} over {y_o } )]                          (12)

Also for the energy equation,

[eqn: rho c_p bar v_x { partial bar T} over {partial x } ~~=~ - { partial (q/A)_t} over { partial y}]

By integrating,

[eqn: { (q/A)_t} over {(q/A)_o } ~~=~ M(1- { y} over {y_o } )]                      (13)

,where

[eqn: M~~=~ {{1} over {y _{o} -y} int _{y} ^{y _{o}} {bar{v _{x}} {Partial bar{T}} over {Partial `x}}dy} over {{1} over {y _{o}} int _{0} ^{y _{o}} {bar{v _{x}} {Partial bar{T}} over {Partial `x}}dy}]

For circular tube,

[eqn: M~~=~ {{1} over {pi `r ^{2}} int _{r} ^{r _{o}} {bar{v _{x}} {Partial bar{T}} over {Partial `x}} 2 pi rdr} over {{1} over {r _{o}} int _{0} ^{r _{o}} {bar{v _{x}} {Partial bar{T}} over {Partial `x}} 2 pi rdr}]

For uniform wall heat flux,  [eqn: partial T/ partial x ~~=~ partial T_m /partial x] is independent of r.

[eqn: THEREFORE ~~~M~~=~ {bar{v _x}}_avg(0-r) over {bar{v _x}}_avg(0-r_0)]

(3) Analysis of heat transfer in turbulent flow using the momentum-heat

flux analogy

- heat and momentum transfer analogy

; friction factor data  =>  heat transfer coefficient

From Eqs. (10), (11), (12) and (13),

[eqn: {tau_o} over rho (1- { y} over {y_o } )~~=~ - ( nu + epsilon_m ) {Partial bar{v _{x}}} over {Partial y}]                     (14)

[eqn: {LEFT. (q/A RIGHT ) _{o}} over {rho c _{p}} (1- {y} over {r _{o}} )~~=~-( alpha + epsilon _{h} ) {Partial bar{T}} over {Partial y}]                  (15)

① Reynold Analogy

Renolds assumed  [eqn: { TRIANGLE M } over {M} ~~=~ { TRIANGLE q } over {q_max }].

For single phase fluid,      [eqn: TRIANGLE M~~ -> `force( tau dA)`,~~M~~ -> ~~wV# # TRIANGLE q~~ -> `h( TRIANGLE T)dA`,~~q _{max} ~~ -> ~~wc _{p} TRIANGLE T]

[eqn: THEREFORE ~~ { taudA} over { wV} ~~=~ { h( TRIANGLE T)dA } over {wc_p TRIANGLE T }]

By dividing both sides with [eqn: rhoV] ,

[eqn: {h} over {c _p ``rho`V}} ~~=~ { tau} over {rho V } ~~=~ { f} over {2 }]                             (16)

⇨ Renold Analogy (relate h and f)

Assumption ;

- highly turbulence ( [eqn: nu ~<< ~epsilon_m ~~~and ~~ alpha ~<<~ epsilon_h] )

o momentum-heat analogy ([eqn: epsilon_m = epsilon_h])

[eqn: { dV} over {dT }~~=~ { tau_o c_p} over {(q/A)_o }]                                   (17)

o Laminar flow  ([eqn: epsilon_m = epsilon_h]=0)

[eqn: { dV} over {dT }~~=~Pr { tau_o c_p} over {(q/A)_o }]                                 (18)

② Prantdl-Taylor analogy

Integrate Eq. (17) from y= δ to y (v=V & T=Tm) and Eq.(18) from y=0 to δ to obtain

[eqn: { tau_o c_p} over {(q/A)_o }~~={V-v_ delta } over {T_ delta -T_m }]

[eqn: {tau _{o} c _{p}} over {(q/A) _{o}} ~~= {v _{delta }} over {T _o -T _{delta}]

By eliminating Tδ ,

[eqn: { h} over {c_p G } ~~=~ { f/2} over { 1+ (v_delta /V)(Pr -1)}]

By Hoffman, this agrees well with data for Pr >0.7, if

[eqn: {v _{delta }} over {V} ~~=~7.54Pr ^{-1/6} sqrt {f/2}]

③ Von-Karman analogy

- consider the resistance to heat transfer separately in the laminar

sublayer, the buffer zone, and the turbulent core.

Assume [eqn: E~~== ~ epsilon_h /epsilon_m] = constant,

[eqn: dT~~=~ {q/A} over {rho c _{p} E sqrt {tau _{o} / rho }} {(1-y/y _{o} `)`dy ^{+}} over {LEFT [ {1} over {E`Pr} + {1-y/y _{0}} over {dv ^{+} /dy ^{+}} -1 RIGHT ]} `# # # ~,where` ~~~`v ^{+} ~~=~v/ sqrt {tau _{o} / rho } `````~~~and` ~~y^+ = (y/nu) sqrt { tau_o /rho }]

Using universal velocity distributiobn, [eqn: v^+ = f(y^+ )],

[eqn: v^+ ~~=~ y^+ ~~~~~~~~~~~~~~~~~~~~at~0<y^+ <5 ## v^+ ~~=~5.0 ~ln y^+ -3.05 ~~~~~~~~~~at~5<y^+ <30 ## v^+ ~~=~2.5 ~ln y^+ +5.5 ~~~~~~~~`~~at ~~y^+ >30]

we can obtain the temperature distribution.

Finally,

[eqn: St~~=~ {NU } over {Re`Pr} ~~=~ {E sqrt {{f} over {2}} LEFT ( {T _{0} -T _{c}} over {T _{o} -T _{m}} RIGHT )} over {5[E~Pr`+`ln(1+5E~Pr)`+` {F} over {2} ln {Re} over {60} sqrt {{f} over {2}} ]}]

,where    [eqn: {T _{o} -T _{m}} over {T _{o} -T _{c}} ~~ == ` {1} over {pi r _{o} ^{2}} int _{0} ^{r _{o}} {{v} over {V}({T _{o} -T _{m}} over {T _{o} -T _{c}})2pi rdr }]

F = f(Re, Pr)

With E=1 ( [eqn: epsilon_m = epsilon_h] )

[eqn: rm N_u ~~=~ 0.023 Re^0.8 ~+~Pr^0.4 ~~~~~~~~~~~~~for ~Pr>0.6 ## rm N_u ~~=~7+ 0.25 Re^0.8 ~+~Pr^0.8 ~~~~~~~~~~~for ~0.005<Pr<0.05]

★ Some factors affecting the turbulent motion

① viscosity (e.g. viscous sublayer)

② high Mach # (>5)

③ combustion

④ high swirl rate

⑤ external force field

(4) Scales in Turbulent Flow

o Smallest scale and largest scale

- macroscales

. transport of momentum and contaminants

. energy production

. as big as the width of the flow

- Kolmogorov microscales

. viscous dissipation

. kinematic viscosity, [eqn: nu] , and dissipation rate per unit mass, [eqn: epsilon]

o Spectral analysis

- problems of multiple scales

- spatial and spectral analysis

. Fourier transform and frequency domain

- the energy cascade

o Kolmogorov's universal equilibrium theory

- small scale motions are statistically independent of the large scale

turbulence.

- production rate in large scale  ~ dissipation rate in small scale

[eqn: u ^{2} `` {u} over {l} ``` SIM `` epsilon]

o Quantities

￭ Turbulent velocity characteristic scale:

￭ Dissipation rate:

￭ Length scale:

￭ Time scale:

￭ Energy cascade:

￭ Turbulent Reynolds number:

￭ Dissipating (Kolmogorov) scale by kinematic viscosity() and

dissipation rate()

- Length scale :

- Time scale:

- Velocity scale:

2.  Turbulence models

[eqn: mu_T ~~=~rho v_T l]                                       (2-1)

How to determine  [eqn: v_T ~``, ```l]

Turbulent models

① Category 1

- Turbulent viscosity models with

the Bossinesq approximation

- Zero-Equation (Algebraic), Two-

Equation models

② Category 2

- Reynolds stress or stress-equation models without the Bossinesq

approximation

③ Category 3

- not based entirely on the Reynolds equations

- Large-eddy simulations or DNS

(1) Zero equation model

By Prandtl,

[eqn: mu_T ~~=~ rho~ l^2 LEFT | { partial u} over { partial y} RIGHT |]

where l ; a transverse distance over which the particle maintain their

original momentum

[eqn: l~ LEFT | { partial u} over { partial y} RIGHT |] ; characteristic velocity of turbulence, [eqn: v_T]

For 3D thin shear layers,

[eqn: mu _{T} ~~=~ rho `l ^{2} ` LEFT [ LEFT ( {Partial u} over {Partial y} RIGHT ) ^{2} `+ LEFT ( {Partial w} over {Partial y} RIGHT ) ^{2} `` RIGHT ] ^{1/2}]

Van Driest damping factor ;  [eqn: l _{m`} ```=` kappa `y`(1-exp(- {y ^{+}} over {A} ))]

(2) One-equation model

★ weakness of algebraic viscosity model

[eqn: mu _{T`````} =````k _{T`} ````=``0```````````````````````````` rm `when````````` Partial u`/` Partial `y`=`0]

Prandtl and Kolmogorov suggested

[eqn: mu _{T} `````=``C _{k} `` rho ``l``( bar{k} ) ^{1/2}]

where   the kinematic energy of turbulence ;  [eqn: bar{k`} ``` == `` {1} over {2} bar{u' _{i}u'_j}]

How to predict [eqn: bar k] ?

Transport PDE from NS equation for incompressible 2D thin shear layer  flow

- Note that a length parameter, [eqn: l], needs to specified algebraically.

- Prk  ; a Prandtl number for turbulence KE (~1.0)

- Inner BC for [eqn: bar k]

o the use of wall function (Launder and Spalding)

o neglect convection and diffusion terms

(3) Two-equation model

o In one eq. model, [eqn: l] has still algebraic expression and depends on

only local flow condition

o two-equation model  -  transport PDE for variation of [eqn: l]

o Two equation models are stable and numerically robust and is consi-

dered the industry standard model.

o While standard two equation models, such as the k-ε model, provided

good prediction for many flows of engineering interest, these are

applications for which these models may not be suitable. Among these

are:

. Flows with boundary layer separation.

. Flows with sudden changes in mean strain rate

. Flows in rotating fluids

. Flows over curved surfaces

o Shortcoming

- For the case when the Boussinesq assumption is not proper, the

two-eq. model is also not applicable.

- need to make assumptions in evaluating the various terms

①	k-ε Model

- k is turbulence kinetic energy [L2T-2] and is defined the variance

of the fluctuation in velocity.

- ε is the turbulence eddy dissipation[L2T-3]  [eqn: == ``C```( bar{k} ) ^{2/3} /l _{epsilon }]

o Standard k-ε Model

Kinematic Eddy Viscosity

Turbulent Kinetic Energy

Dissipation Rate

Closure Coefficients and Auxiliary Relations

o RNG k-ε Model

Use a modified coefficient, Cε2, defined by:

,where the mean strain rate tensor is

The closure coefficients for the RNG k-ε Model:

o The k-ω Model

- One of the main problems in turbulence modelling is the accurate prediction of flow separation  from a smooth surface. In general, turbulence models based on ε-equation predict the onset of sepa- ration too late and under-predict the amount of flow separation.

- Wilcox k-ω Model

; solves two transport equations, one for the turbulent kinetic

energy, k, and one for the turbulent frequency, ω.

. Kinematic Eddy Viscosity

. Turbulence Kinetic Energy (k-equation)

. Specific Dissipation Rate (ω-equation)

. Closure Coefficients and Auxiliary Relations

☞ The main problem is its strong sensitivity to free-stream

condition. Depending on the value specified for ω at the

inlet, a significant variation in the results of the model can

be obtained. This is undesirable and in order to solve the

problem, a blending between the k-ω model near the surface

and the k-ε model in the outer region was developed.

- Baseline (BSL) k-ω Model

;  combines the advantages of the Wilcox model and the k-ε

model, but still fails to properly predict the onset and amount

of flow separation from the smooth surfaces.

- Shear Stress Transport (SST) k-ω based Model

; k-ε model, Wilcox model and BSL model fail to properly predict

the onset and amount of flow separation. The main reason is

that all of that models do not account for the transport of the

turbulent shear stress. This results in an over-prediction of the

eddy-viscosity.

The proper transport behavior can be obtained by limiter to the

formulation of the eddy-viscosity:

(4) Reynold stress equation model

- Reynold stress(Eq. 2-1) is obtained directly from the transport

equation, not from assuming the eddy-viscosity.

- apply to anisotropic turbulent flow, swirling flow, strongly curved

flow, rotational flow, and the buoyant flow

- high degree of complexity

The equation for the Renolds stress tensor

(5) Direct Numerical Simulation (DNS)

- directly solve the Navier-Stokes

- resolve the eddy of all scale in the turbulent flow

. microgrid for Kolmogorov scale, [eqn: eta]

. sufficient size for largest scale, [eqn: l]

- Since [eqn: l/ eta `` SIM `Re^3/4] ,  number of necessary grid for 3D calculation

[eqn: rm N`` APPROX `Re ^{9/3} ``````````````````````````````````````````````(`10 ^{8} ````for``Re``=`5,000`)`]

Re         104      105       106        108       1010

N         109      1011      1013       1018       1022

- If the size of mesh isn't sufficiently small, it is difficult to precisely predict the dissipation in small scale.

- Thus, it may mislead the flow due to the failure of proper simulation for the energy cascade process between the production and dissipation in large scale.

- nearly impossible to analyze the high Re flow

- numerical method

￭ flow in simple geometry: Spectral method

￭ flow in complex geometry ; FDM, FVM

-  Application of DNS용

￭ Precisely provide velocity and pressure within an error range of numerical method in the function of 3D space and time

￭ contribute to finding the structure of turbulence and the

mechanism of turbulent flow

￭ evaluation of new turbulent control techniques which is extremely difficult to be observed in experiments

￭ provide detailed data of pressure-strain rate correlation and dissipation rate

(6) Large Eddy Simulation

- compromised method between DNS(accuracy) and RANS(efficiency)

￭ obtain the large eddy, which has the dynamic characteristics of turbulent flow, by DNS and model the small eddy where the isotropy can be assumed

￭ While widely-used turbulent models model all small and large eddy except the average flow, which results in inevitable error, LES models only small eddy and thus more precisely predict the turbulence than RANS.

- 4-Steps in LES

1) define Filtering operation

; separate  with  which is the sum of filtered (or        resolved) part and  of residual(or subgrid-scale, SGS)       part

- eliminate the residual part

2) establish filtered Navier-Stokes equations

- include residual-stress(SGS stress) tensor in momentum

equation

3) make a closure problem with modeling the residual-stress tensor

4) numerically solve  from filtered equations

① Filtering

: process to eliminate the fluctuation part of smaller scale than a

given characteristic length

- Definition of filtering for an arbitrary function f

where  : integration interval where the filter is applicable,

: filter function in i direction

※ Δ in Gaussian filter denote the

with width of a filter in the r-

direction, which gives a measure

concerning the minimum scale of

fluctuation in the simulation.

Then, the quantity f is expressed as,

[eqn: f``=` bar{f} ``+``f`']

,where [eqn: bar f ```, ```f`'] are called resolvable grid

-scale and subgrid-scale(SGS)

component, repectively.

Note that

[eqn: bar{{Partial `f} over {Partial x _{j}}} ``=`` {Partial bar{f}} over {Partial x _{j}} ``,```````````````````` bar{bar{f}} `` != ` bar{f`} ```,```````and`````` bar{bar{f} f prime } `` != `0]

compared to  <<f>> =  <f>   and  <<f>f'> = 0

Filtered the Navier-Stokes continuity and momentum equations is

given by

,where the subgrid-scale stress is

grid-subgrid scale interaction term

② LES models

- how to model Leonard, cross-term and SGS Renolds stresses

-  : strongly depend on filter의 type and width

※ Filtered velocity fields depends on the type of filter and filter width , yet these quantities do not appear directly in the equations – they appear only indirectly through the model for .

(a) SGS(sub-grid) model

￭ Most SGS models are based on the eddy viscosity model which assume the proportional relation between the anisotropic part  of  SGS stress tensor and the deformation rate  of large scale.

￭ From dimensional analysis, the eddy viscosity  is given as multiplication of the velocity scale and length scale.

- Smogorinsky's model

: adopt eddy viscosity model to SGS stress part in LES

Joseph Smagorinsky, Ph.D.Former DirectorGeophysical Fluid Dynamics Lab. Nat’l Oceanic and AtmosphericAdministration (NOAA)

￭ Definition

where ,

- Schuman model

. Idea ; The main component is greatly influenced by external

boundary conditions like walls, compared with the

fluctuating component.

. Instead of SGS eddy viscosity, he modeled [eqn: tau_ij^r] with dividing the

resolvable velocity strain into the mean part and the fluctuating

component.

- Moin and Kim's model

o Since Smagorinsky model() can be applied in high-Re

turbulent flow in the inertial subrange, but is not proper in the

viscous wall region, they use the van Driest damping function near

the wall region to obtain

o The cross term was model.

[eqn: tau _{ij} ^{r} ```=` nu _{SGS} ``( bar{e _{alpha beta }} `-< bar{e _{alpha beta }} `>)``+` nu_SGS^* < bar{e _{alpha beta }} `>]

,where [eqn: nu _{SGS} ^{*} `````=`[`C _{M} ^{*} TRIANGLE _{3} f(x _{2} )] ^{2} (< bar{e _{alpha beta }} `> ^{2} /2] ^{1/2}]

Here  f(x2) = a wall-damping function to impose the no-slip

boundary condition

(b) Dynamic SGS model

￭ not model the eddy-viscosity, but determine as spacial function

during calculation

￭ with two kind of filters, determine the model constant by using the similarity between two subgrid-scale

- Grid filter:  (width=)

- Test filter:  (width=  )

￭ Navier-Stokes equations grid filtered and test filtered

where

￭ apply Smagorinsky model to   and

- 상수 C를 구하기 위해 최소자승오차법을 사용 (Lilly, 1992)

→ minimize the difference between  (resolved stress

part) and (real stress part)

￭ determination of constant, C

- calculate  from  which is obtained from filtered N-S equation

- obtain C after calculating  and  from applying those values to the filter

③ Application of LES

￭ More economic than DNS, and capable to treat the flow of much higher Re

￭ provide much accurate solution than RANS

- flow having the strong streamline curvature effect which varies the separation point by time

￭ the only analysis tool for high Re turbulent flow in a complex geometry

■ Summary

￭ RANS

- determine only the time-averaged velocity by modeling the Renolds stress in the inertia term

- Most popular turbulent analysis method

- so powerful since it can qualitatively and quantitatively simulate very complicated flow characteristics only with PC or workstation

- The turbulent flow has the various size of eddy from a large scale corresponding to whole flow field to a small scale of Kolmogorov scale. But RANS methods(mixing length, one-equation, two-equation 모델) are based on the model constant for a specific flow type and thus are limited to predict all turbulent flow precisely.

- Flow types which is difficult for RANS to be applied

‧ flow wher the separation point varies with time

‧ flow having the strong curvature effect

‧ transient boundary flow, etc

- Velocity, pressure and Renolds stress obtained by RANS can't provide sufficient data to estimate the time- and space-varying flow structure and phenomena.

￭ DNS

- directly solve the Navier-Stokes without using the turbulent model

- Exact method to simulate even most small scale of turbulent flow

- need the number of grid corresponding to

→ large computer and much computation time

- limit to apply DNS in practical engineering problems

￭ LES

- use DNS to calculate the large eddy which is affected by the boundary condition and flow, but only model the small eddy which not affect on the dynamic behaviour of flow

- model the eddy of less size of grid

- Smagorinsky model is a typical SGS model but has a limitation to apply the general flow field since the model constant varies with flow field.

- Recently, the dynamic model for Smagonisky constant is developing by calculating as spacial function at every time step by the change of flow field.

[ DNS vs. LES vs. RANS ]

3. Boundary Condition

(1) Law of the wall

- need fine mesh due to large

gradient of velocity profile

- viscous sublayer ; no turbulence

where  gradient Richardson # is

- logarithmic

; 난류에 의한 에너지의 생성과 소산이 균형.

y+ >  11.63

- wake region

(u+) = y+         ( y+ < 11.63 )

u+ =  ln (y+E)              ( y+  11.63 )

where   y+ =  ,    (friction velocity)

and  E = 9.7,   =0.42

o k,  in the logarithmic region

; 벽가까이에서는 난류에너지의 대류항과 확산항이 매우 작아 무시되므로

난류 생성항과 소산항이 대략적으로 평형을 이룬다.

And in this region,

From  ,

Or

Then the final form of wall shear stress is,

난류운동에너지 방정식의 생성항,  G

난류운동에너지 방정식의 소산항,  에 대한 적분

한편 난류소산율,  는 난류소산율 방정식에서 풀지 않고 식(  )로 부터,

From Eqs. (   ), (    ), and (   )

(2) Law of the wall in the energy equation

where

Or, by Jayatilleke,

where

4. Turbulence model in two-phase flows

- highly promoted by the existence of

bubbles

- roughly isotropic behavior for highly

turbulent bubbly flow

- increase of bubble-induced velocity

fluctuation ∝ [eqn: < alpha > ^{0.8}]

_ IAC is a function of turbulence

(Serizawa, Ishii 등)

(1) Liquid velocity profile and shear stress

distribution

- The two-phase liquid velocity profile becomes flattened by the

bubbles and is roughly approximated by a power law.

[eqn: -` rho _{l} `bar{uv}] become a maximum near the wall

The turbulent shear stress in pipe flows for given [eqn: < alpha >```and``````<j _{l``} >]

[eqn: tau / tau _{w} ```=```r ^{*} (1-B int _{0} ^{1} {alpha r ^{*} dr ^{*} ``)```+``(B/r ^{*} ) int _{0} ^{r ^{*}} {alpha r ^{*} dr ^{*}}}]

where   [eqn: r^* ```~=~ r/R````, ``````B```=```rho_l ``g R/tau_w]

By assuming that the momentum is transferred solely in the liquid phase,

[eqn: tau```=```rho_l```(1-alpha)(nu_l```+epsilon_M^'``+```epsilon_M^'' ```)d barU_l ``/dy]

where

[eqn: -rho_l```bar{u^' v^'} ````= `````rho_l``epsilon_M^' ```d barU_l ``/dy ## -rho_l```bar{u^'' v^''} ````= `````rho_l``epsilon_M^'' ```d barU_l ``/dy]

[eqn: epsilon _{M ^{}} ^{'} ,````epsilon _{M ^{}} ^{''} ,`]?

(2) Eddy diffusivity

[eqn: epsilon_M ```=```-rho_l```bar{u v} ````/ ```[``rho_l``(`d barU_l ``/dy)]]

Sato  ->  Table 1

[eqn: epsilon_M~~ =~~epsilon_M^'~~ + epsilon_M^''~~## epsilon_M^''~~ = K``alpha``L``U]

Two regions

y < dB   ;  wall region

R > y > dB  ;  core region  (more bubble interaction)

[eqn: l_m``(y) ```= ```l_o ``(y/R)^m ``,````````~~~`l_o ``,`m ~~ PROPTO ~~R ``, ~alpha]

By Lahey,

[eqn: u ~=~u^t ~+~u^b ~,~~~~v~=~v^t ~+~v^b]

[eqn: epsilon _{M} `=~k ^{b} ``l ^{2} `` LEFT | d bar{u} _{l} /dy RIGHT | ``+``k ^{b} ` alpha `l`u _{r}]

By Serizawa,

[eqn: epsilon _{HTP} / epsilon _{HSP} ````=``1+`462`X _{tt} ^{-1.27}]