Ching Jen Chen Lecture in Turbulent Flows and Convective Heat Transfer

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C J Chen Lecture in Turbulent Flows

C J Chen Lecture

CJ Chen Lecture in Convective Heat Transfer

Viewing with Supplement Notes Historical Development

Fundamental Equations

#Turbulent Phenomenon Turbulence Modeling

Order of Magnitude Analysis

Hydrodynamic Instability

Statistical Analysis of Turbulence

Fractals, Chaos & Strange Attractors

Ching Jen Chen, Dean and Professor Emeritus

College of Engineering

Florida A&M University-Florida State University

cturbu!~nt Phenomenon It is impoi'Wit w recognize t~at tb~ iYrbuli!m.~ How motion is always three-dimensionali unsteady, rotational, and, mo$~ irnpeirctant. i~a~o Th~ irregularity of turbulent motion 8~ thie to the inherent nonlinear nature of the Navier-S~o;xes equations when the Reynolds nurmber is beyond the critical value. Thus, contrary to laminar ftow, which is regular and, deterministic, turbulent flow is stochastic and chaotic. In order to predict ~he gross or aYerage behavior of turbulent fiOW, a mathematical model must be established. ',

o "'"o"' a_f T ,,_.,, i..-~ ~~4--.:- ~ '\S 3 -t P~roJ.. :r-t~t' .. t-~e"' ( t>.4+--~-ui.)

c ... t-y~l...-~.:o .. $"""' ~~r'"-l~ ""'""""t-... .u .. A

3 - 2. . Lo..,...i.. t H~+..:o"" . A- . ~ ........... +-.-.... ..,._,l.:1-~ ...t.:.. ..

Du... il . JJ~.t + pq r D ~- ~ - ll'/C.,,.; + ~ b~i6~,

N~ .. u,,._. .. .,..,,.~. ~f"'" ..:t..'. () .. , c 1:1,;..-k.... &.~4.{ J ~........ co ..... rt....-< +ei ~ 'lw.&~ov.

... V~c.~u~ ~~ ~ j--....b.:-..... s 1, s~~~J-.~ I 2..Z...~~. c

No...".:.i...c - s.+-ok.ti l ,;6 ...t.=..,. .. i: _ ........... ~

Turbulent Wake Far Flow Behind a Projectile

151. Turbulent wake far behind a projectile. A bullet has been shot through the atmosphere at supersonic speed, and is now several hundred wake diameters to the left. This short-duration shadowgraph shows the remark-able sharpness of the irrL-gular boundary between the

88

highly turbulent wake produced by the bullet and ti almost quiescent air in irrotational motion outside;.Phoc graph made at Ballistic Research Laboratories, Aberdeen Pro ing Ground, in Carrsin & Kistler 1954

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..

Ill Cll c ::; Cll E j:: r:: .

.. r, i:" . ~.:-:-. . ;. .. ~ ,

... :, . ,l. : .: 1 . ;

' ..

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..: .. ,

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' t..~.I _;:. \..J o I 1 I ~

- - --. ........ - .. z-=- _, :::- ~ -i

~z $fK ~ ~ G ~ ,,~J

I!~ . _, ~ (>:::., . :s: .:...4 .~ ~ :"'

r.. f I.

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Boundary Layer Flow

-~~~}:.~~:.;~~:.,}~:>. .- .. .. : . .- .:

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109. Emmoas turbu11m opot. On a fiat plate, transidon from a lamin;ir tO a cutbul random appeu an.:.e or spo

Wake Flow Behind the Cylinder (Karman Vortex)

47. Circular cylinder at R=2000. Ac chis Reynolds number one may properly speak of a boundary layer. It is laminar over the from, separares, and breaks up inro a tur bulcnc wake. The sepnr:icion points, moving for\\'ard a>

48. Circular cylinder at R=l0,000. Ar five times the fK'

Boundary Layer Flow on Curved Surface

... ; . is6: Comparison oC lam.inar and turbulent boundary

l~ycrs. The lamin~r boundary layer in the upper photo-g~aj)h separn~cs from the crest of a convex surface (cf. figure 38); w~er~as the turbulent layer in the second

photograph remains attached; similar behavior is shown below for a sharp comer. (Cf. figures 55-58 for a sphere.) Titanium tetrachloride is painted on the forepart of che model in a wind tunnel. Head 1982

Turbulent Flows between Concentric Cylinders

129. Taylor vortices between spheres. With a radius ratio of 0.95, the outer sphere fixed, and the inner one rotating at R=7600, alu-minum particles in silicone oil show laminar vortices near the equator. Sawatzki & Zierep 1970

130. Spiral turbulence between counter-rotating cylinders. This "barber-pole" pattern of alternating laminar and turbulent spirals, a phenomenon discovered by Coles, was formed by first rotating the outer cylinder from rest to R=l0,000 (based on outer radius) and then ac-celerating the inner one slowly to R= 4200 (based on inner rad.ius). Photograph by M. Gomuzn and H. L. Swinney

131. Axisymmetric turbulent Taylor vortices. The conditions are as in the pair of photographs on the opposite page, but at 1625 times the critical speed. A sudden start produces chaotic motion at first, but this reglar permanent turbulent pattern emerges within a minute. Koschmieder 1979

77

]

= Cj

Gonera1ion of turbulence by a grid. Smoke wires JJ 2 uniform lami nar str,am passing through a 1/ 16lnch : ":th v,.inch square perforations. The Reynolds num

omogeneous turbulence behind a grid. &hind '-:-.d cha~ above, the merging unsc;;ible w:akcs k:rn a hvm~ncous field . As it dt."Ca ys down-

ber is 1500 based on 1he I-inch mesh size. Instability of th shear layers leads to turbulent flow downmeam. Photo graph by Thomas Corke arvl Hassan Nagib

stream, it provides :a useful approximouion co the idealita tion of 1s0tropic turbulence. Ph~ograph bJ Thomas C.Orkc and Hanan Nat1b

154. Growth of moitcrial Jines in isotropic turbulence. A fine platinum wire at the left is Stretched across a water tunnel 18 mesh lengths behind ~ turbutcncegener:ning ~riJ, The ltcyoolds l\Uinbcr 1s 1J60 h:1~cJ on ~riJ rc.x.I c.Jiam-

etcr. Periodic electrical pulses gcnc:r:ttc: OOublc hydrogen bubbles that are sh by ~I. ). K11n1..: 11 , ~I. S. i Johru Hopkins Uniu. , 1%$

;

76. Largtscalc struc ture in a turbulent mixing layer. -.J itrotcn ~bovc llowing ;1t 1000 cm/s mixes \Yi th :'I helium 1rgc:"1 mixnirt below ar the same density newing :I{ 380 mh under a pressure o( 4 oumosphcrcs. Spark shadow ~hc~ograph\ shows simultaneous edge 01nd pl;m view~. !croonstrat ing the spanwi.sc organiiation of the large

77. Coherent structure ;;r; C higher Reynolds number. "h i~ ilow is as above but at twice the p ressure. Doublin~ 1..: f\..:ynolJ) number hos pruJu1.:cJ more sm&1ll~c.:ale suuc.:-

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eddies. The strcamwisc urea ks in the plan view (of w' half the sp::m is shown) corrt:sponJ to~ system of :lt...'Cllj ary vortex pairs oriented in the scrcamwisc directi1 Their spacing at the downstream ~idc of the layer is la than nc:u 1hc hct:inning. Phnrngmf>h hy ). H. Knnrml, Pi

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4. 1.2 NAVIER-STOKES (N-S) EQUATIONS: VALIDl1Y FOR 'IURBULENCE

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The foundation of turbulence modeling is the Navier-Stokes (N-S) equations. Therefore, it is necessary to understand them and to validate that they are capable of describing turbulent fluid motion. 1.2.1 Stokes's Postulations for a Viscous Fluid Model Consider the postulations used in deriving the N-S equations:

1. The fluid motion can be considered to be a continuous medium (continuum). 2. The viscous diffusion of U1 (or pU1) is proportional to the gradient of U; (rate of

strain). 3. The fluid is isotropic. 4. The fluid is homogeneous; that is, r11 = :F(U,, P, p, T). 5. When the fluid is at rest, the stress is hydrostatic. 6. When the ~ow is a pure dilatation, the average stress is equal to the pressure (Stokes's

hypothesis). 7. Viscous fluid model constants (or coefficients) require experimental determination,

namely, p, ., .2 (second viscosity).

There is nothing said in these postulations about excluding turbulent flows. As long as the flow does not violate the postulations, the N-S equations should be valid for turbulent flows.

For the p and . constants, the N-S equations and ~e energy equation are

and

where

au1 _ 0 ax, - DU; aP a"'-u,

P Dt = pG; - ax; + . ax1ax;

DT a2r pC,-D = k-;---a + .~. f CIXj X j

~ = (au1 + auj) au,. ax; ax, ax;

(l.6)

(1.7)

(1.8)

One may be concerned that the continuous cascade of turbulent eddies may violate the continuum assumption. Let us examine the limiting eddy and vortex stretching in turbulent ftows. For ex31llple, in turbulent .flows the pauems in Fig. 1.2 are often seen. The eddies tend to be smaller for larger Reynolds number (Re). The turbulent eddies arc always ill'egular, time dependent. and three-dimensional with the associated vortices always being stretched. One may aslc. will the cascade of an eddy evolve such that the eddies become indefinitely small?

1.2.2 Limits in Vortex Stretching Let us isolate a vortex and examine its stretching (see Fig. 1.3). Consider the curl of the N-S equations. Then we have the vorticity equation. where the vorticity (w) is the curl

--- ... ,.- .. --- .... ' ... ------. - -...... . --- ... _ .__ . - - .... ---- ... ..._ .. --- -

Tftec:h007 TF007--0I July 26. 1997 17:17 r--1 -,

INl'RODUCllON TO TURBULENCE 5

F1gurt 1.l Eddy patterns in jet flows.

of the velocity and is given as Dw aw - = - + (v v)w = (w v)v + vv2w, Dt ot (l.9) c.....v.

where

"'= V" x ...

Vectors an presented ID bold

( l.l 0) faced typuad not by arrows

, S~~ . We sec that when Dw/ Dt = 0, the voncx stretching will stop. From an order--of-'~ ~ magnitude analysis of Eq. 1.9, balancing the right-hand side will occur when cuu / l = \J~~ va>/l2 orul/v =I. - ~-t O(? Here. cu is the limiting vorticity, u is the eddy velocity, and I is the eddy size.

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We find that when ul/v (the eddy Re) is 0(1), D(J)Dt ~ 0, or, in other words, there will be no more vortex stretching. Thus, the smallest turbulent eddies should be at ul/v = 1.

Let us consider this from another point of view. The Kolmogorov scaling (see Hinze. 1915 (58))(11, E), which refers to the small eddies. gives u = (vE)l/4, / = (v3 /E)ll4, and

OID _ _:: -v,

F1pn 1.3 Bxamilling the siretcbinr of a voneit.

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8 FUNDAMENTALS OPTURBULSNCE MODEUNG

* u1 = total value u1 = mean value

ui = flactuating value

F1pre 1.6 Reynolds long-dme aveqgillg.

A similar proccdW'C may be applied lO the two- and three-dimensional cases. How-ever, the analysis is more complicated. Hence. although statistical analysis may reveal details of turbulent sttuctures. it is cumbersome to use for solving engin~ring problems.

Phenomenological analysis. In this approach, turbulence model postulations are made. The analysis loses some details of the turbulence physics, but it can provide solutions to engineering problems. The starting point of phenomenological analysis is the construc-tion of some process of averaging.

1.3.2 Reynolds Averaging Let uj = U; + u1, where u; is the total or instantaneous value, U1 is the mean value, and u, is the fluctuating value.

Long-timeaveraging(T - oo): - 1 fT d "1 = u, = r lo u, t.

In this case, U1 is not a function of time. Soc Fig. 1.6.

Shorttimeaveraging(ATsbort): a7 = U1(t) = - 1- r+"f ui(t + t')dt'. AT}_-

Here t' is the time variable in 6 T duration near t . In this case, U1 is a fuoction of time and t:.T. If tJ.T becomes large, the short-time average recovers the original Reynolds long-time average. A short-time average is somewhat similar to a signal that has been smoothed or has had a high-frequency component filtered out. It should also be noted that u, = 0 and u1u1 # 0 in general. See Fig. 1.7

1.3.3 Ensemble Averaging (Phase Averaging) In a short-time averaging process. the experiment needs to be performed only once. However, the averaging depends on the filter time interval. Ll. T , which is not known in

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IITTRODUCTION TO TURBULENCE 9

u1 = total value Ui mean v;ilue

ui flaccuacing value

u ( t)

FiggR 1.7 Reynolds shoiHimoaveraging.

advance. An alternative way of conslrUCting an average for a time-dependent flow is by obtaining an ensemble average, which requires one to perform the experiment repeatedly for N times, and then the averaging is taken over the entire range of experiments over N experiments, as indicated in Fig. 1.8.

N u; = U1(t) = lim ..!.. ~ u;(t, n),

N-ooN ~ iI

where N = number of experiments. Also, note that

I. Au[= Auj =AU; 2. Uf+ii1 = Vi + Ui 3. au; fax j = aut fax j 4. ujuj = U1U1 +u1u;.

(1.11)

Equation 1.11 includes the short-time Reynolds average and is potentially more general. However, it is more difficult to produce because it requires N experiments. N should be large enough so that the ensemble average is no longer dependent of N .

1.3.4 DemityWeighted Averaging (Compnsnble Fluids) The use of the Reynolds average for compressible ftuids is cumbersome.. For example,

puv ;.,. (p + p')(U + u)(V + v) = . is a messy operation. Alternatively, a 9ensity-weighted average (~r mass-weighted av-erage) ~ay be used.

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10 f'UHDAMENI'AU OFTIJRBULSNCE MODELING

+k Ui ~I t

+~ Oj ~!

* Ui

*

U11u

N: N

Lcttmgi be tho density-weighted averll8e for f, we define

Then. instead of

u; = U1 + u1 (Reynolds average), (\): ClG111illg qmitation or we have

doUle prime mark --------/"'\ uj = 01 + u;' (density-weighted average) . $Jlllbol'aneet p = p + p' (Reynolds average).

.. - - ---. - . .._.,....~, ---- ..... .. ,.... ___ ._

- . ----------------- . .... ___ ., .. ___ _ --------

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(1.12)

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INTRODOCl'ION TO TURBULENCE 11

From Eq. l .12. we have

But,

Thus,

pu~' = 0, or

(p + p' )u'f = 0. That is.

or

ii~' = -p'u/1 /p :F 0. In general,

u7 =F o. For example. the mass conservation equation (Eq. 1.6) under density-weighted averaging becomes

or

Hence.

a(~+ p') apll at + 0X1 1 =O.

ap a-po, _0 at+ ax, - '

which is simpler than the continuity equation with conventional Reynolds averaging: ap apO; au,p'

0 -+--+--= -a1 ax, ax,

1.3.5 Conditional Averaging (Sampling Average) In some problems such as intermittent phenomena or thermal spikes, a special averaging process may reveal mo~ physics of turbulence. Let l1 be the duration of the laminar flow and t; be the turbulent Bow duration. Then, we can define the inrermittency, y, as the ratio of the turbulent fiow duration to the total flow duration. that is,

E,11 r= . L:j 1j + E,1,

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------ -- r ' - --- --------- - - .... --- - -- ----- - . . ... - --------

' ' ',

' 1. nJ IVj/t\.D/'\ r~; t\DJVl'\.I M/~\...M r..>: t'U(MJJ\T V \J\...: T0e

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INTRODt.ICT!ON TO nJRSULENCE 13

Figure 1.11 lntennittait samplini-

Figure J.12 Thennat spikes.

(-, +) j 2

(-, -) j 3

(+, +) jl

(+, -) j - 4 Figure l.13 QiWmnt cooditional averaging.

Quadrant coadidonal averaging (sampling). This is used to monitor or investigate the details of the turbulent directional characlcristics (u, v ). Let j bo the j th quadrantat the time t (sec Fig. 1.13). Then

H1(t;) = 1 if (u, v) is in jtb quadrant at t = t; = 0 ifnoL

With u = U +u and v = V + v, we can define the jth-quadrant conditional average (see Fig. 1.14) as

...... - -~- -. . . ---- - ---- - -'

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14 FUNDAMENTALS OF 11JRBULENCE MOD!'LING

flow -

) ) 7 7 7 7 ) ? 7 7 7 7 /? 7

Data

Samplino

u

v

4 3 3 4 4 2

Figure 1.14 Data sampling in conditional averaging. 2 3 4 2

~ ~ ~ ""' 1.4 AVERAGED IN~SmLE 1URBULENCEEQUATIONS

1.4.1 Navfer-Stok.es and Energy Equations We start out by defining the turbulent quantity to be equal to the sum of the mean value of the quantity plus its fluctuating value. Thus, P* = P + p. T* = T +8, ui = U; +u;. and r:1j = r:11 + -rf1 Taking ensemble average of insWltaneous incompressible N-S and energy equations. Eqs. l.6-1.8, we have the following equations.

Continuity equation:

(l.13)

Momentum equation:

(l.14)

wherein

( au, au1 ) "u = . ax, + ax, and

, ---r,, = - pu1u1.

Here, T:;j is modeled from the viscous fluid model, whereas r:[1 needs to be modeled from lhc turbulence model.

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-----,

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INTROOucrtON TO TI.llUIULENCE 15

Energy equation (C, = C,):

where

DT oU; aq, aqf ' pCp- = r,i- - - + - + 4> Dt ax1 ax; ax,

q1 = -KoT /0X1 qf = -Cppu,e ct>'= e. au,

1 ax1

, (au, au1 ) r:;; = , ax1 +ax, .

(1.15)

Here, q, is modeled from the Fourier postulations, whereas qf and ' need to be modeled from the turbulence model. Equations 1.7-1.9 give five equations for the five variables P. U, V, W, and T, whereas there are no equations for u1u;, u18 and '. Therefore, the set of equations is not closed. The problem of closing the turbulence mathematical model is called the turbulence closure problem.

Before the closure problem can be addressed. the ensemble-average quantities of u1u1, u16, and 1 must be derived. These additional unknowns, a total often (six from u1u1, thiee from u;IJ, and one from 4>') can be derived from the fluctuation equations for u, and 8. The fluctuation equations for"' and e are obtained from

[N-S] - [N-SJ-+ Equation for u1, or

(Oil/ 8u1 0U1 OU/) a} ( o1u; ) apu;U1

P at+ u, ax, +"'ax, +"'ax; = - ax, + , ax,ax, .!-ax;- (l.16) [Energy] - [Energy j -+ Equation for 8,

or

c ( ae u ae ar ae ) _ a20 a"Cpiiii;O _ , P , at+ 'ax,+"' ax1 +ui ax, -kax,ax, + ax, (1.17)

1.~.2 Turbulence 'lhlmport EquaUom In order to close the problem, we will derive u1u J> u16, and so forth from .Eqs. l.16 and 1.17. These are the s~nd-order. one-point correlation-based transport equations.

Reynokls-stnss transport equations. The equation for u, u 1 is obtained by the follow-ing method;

Multiply Eq. 1.16, for the ith variable, by u 1 Multiply F.q. 1.16, for the jth variable, by u1 Add the results of the above two operations. Take the average of the total addition.

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INTRODUCTION TO TURBULENCE 17

Rate of dWipation equation. The dissipation rate of turbulent kinetic energy, au, au,

=V--0X1 ax, directly affects the growth rate of k. It is thus one of the most important turbulent quantities. It appears naturally in the derivation of the k equation. The equation for the rate of dissipation of the turbulent kinetic energy, E, is obtained by the following approach:

Perform the operation a;ax, on Eq. 1.16 for the ith variable. Multiply theresulting equation by au;/8X1 Take rhe ensemble average of the product and multiply by two.

By going through these steps, the e equation is obtained as folJows:

DE a ( - 2v 8u1 ap oe ) Dt = ax1 -'ui - -;; axj axj + v ax,

-- 2 - 2vu, au; a u, - 2v aui . au, au, + au, au j

ax1 ax,ax1 ax1 ax; axi ax, ax, - 2v au, au, au J - 2(v a2u1 ) 2

axi ax, ax, ax,ax, (1.20) Here the fluctuation or instantaneous quantity of ds given by

I au1 au/ e = "ax, ax,

In order to understand the E equation, let us examine the various tenns on the right-hand side of the equation.

The first two tenns again represent the turbulent diffusion of. The third term represents the molecular diffusion of E. The fourth and fifth terms represent the production of E. The last two terms represent the destruction (source or sink) of the dissipation rate of

the turbulent kinetic energy.

Reynolds turbulent heatftux equation. If we define the fluctuating viscous dissipation term as

/ I au, -i. au,. 41 = -r,1 ax1 - '""ax,,.,

then the equation for the Reynolds turbulent heat fiux, u;IJ, can be obtained by using the following approach:

Multiply Eq. 1.16, for the ith variable, by 6. Multiply Eq. l.17 by u,.

.

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16 FUNDAMENTALS OFTIJRBUl..ENCE MODtilJNG

By following the above-mentioned procedure, the equations for u1u i obtained are

Du;uj = ~ (-u;UjU/ - !!..(Oj/Ui + Oi!Uj) + v ou;Uj) Dr ax, P ax1

(_auj _oU;) OU; ou; p (OU; au;) - u;ui ax, + u;ur ax, - Zv ax, ax, + p ax; + oX; (l.18) For understanding the physics of this equation, let us consider what the seven tenns on the right-hand side of this equation represent

The first two terms on the right represent the turbulent diffusion of the momentum. The third term represents the molccular diffusion of the momentum and is usually

negligible compami with the first two tenns. However, it is the term with the highest deriwtive. Hence, it must be retained.

Tiie founh and the fifth terms represent the production of stresses. u111. i by the inter-action of Reynolds stresses u;u 1 and the gradient of the mean values.

The si;t;th term represents the viscous dissipation of the Reynolds stresses. u1u I The seventh tenn represents the pressure-strain (PS) of the flow, which tends to restore

isotropicity of the flow.

'lbrbuleat ldnedc energy (k) eqaadon. From Eq. 1.18 for "'"i we can easily obtain the turbulent ldnetic energy equation by setting i = j. The mean turbulent kinetic energy is denoted by i:: = u1u; /2, whereas die ftuctUating turbulent kinetic et1ergy is denoted by k' = u.1u;/2. Then, the turbulent kinetic energy (k) equation can be obt.ained as

_,,,, _ -k'ur--+v- -u1u1--E'+O. Dk a ( - pu, ok) au, Dt ax, P ax, ax, (1.19)

Note that the PS term goes to zero because of the continuiry equation. The physics hidden in the equation is revealed when each of the tenns on the right-hand side is studied carefully.

The first two terms on the right represent the diffusion of the kinetic energy from the high intensity to the low intensity that is due to turbulent fluctuating motions.

The third tenn repn:sents the molecular diffusion of the turbulent kinetic energy. The fourth term reimsents the production of the turbulent kinetic energy that is due to

the in~on of turbulent stress and the gradient of the mean-fl.ow velocity. The fifth term. E, represents the rate of dissipation of the turbulent kinetic energy that

cc:nds to occur at the small-eddy scale. ' is always positive because

The PS term of the Reynolds stress ttansport equation in the last term vanishes. and, hence, its net contribution to the turbule_nt kinetic energy trans.fer rate is uro.

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18 FUNDAMENTALS OF TURBULENCE MODELING

Add the two products. Take the ensemble average of the resultant sum.

Thus, the u;O equation can be written as

Dlii8 a ( -- Pe -ae BUi) - = - -u1u18-011-+au1-+ve-

Dt ax, p ax1 ax1

(- ao --au;) au; ao pae -,-- u;ui ax,+ "18 ax, - (a+ v) ax, ax,+ p8X1 + u, . (l.21) Here a and v are the thermal diffUsivity and the kinematic viscosity, respectively.

J' l.S 'roRBULENCE CLOSURE PROBLEM The averaged N-S equations, as indicated by Eq. 1.14, is given as

8U; =O ax;

DU, a p O'Cjj chfj P Dt = pG, - ax, + ax + ax J 1

wherein

and tf1 = - pu;uj.

The above-mentioned four equations are for U1 and p. The problem of specifying un-knowns, u1u I is the turbulence closure problem. Historically the first attempt to model the Reynolds stress, -rfj, is to model it as a function of the mean-flow equations. Because the u1u 1 term is modeled directly without any additional differential equation, this model is also called the zero-equation model or the first-order closun model. For example,

1. -iii= v,au /aY. This is the Boussinesq eddy viscosity model. 2. - uv = l2 JdU /dYI au /8 Y. This is the Prandtl mixing length model. 3. -uv = KX8U /iJY. This is the Prandtl wake mixing length model. 4. -iiil = K21(dU/dY)/(d2U/dY2)12au1ar. This is the von Kinn8n mixing length

model.

In the early twentieth century, because of the lack of computing machines, the above-indicated turbulence models were very popular because they arc relatively simple and intuitive. However, models were very limited in the sense that the model could apply only to a particular problem or problems with similar geometry and were inappropriate for predicting other types of flows with different geometries. These simple models, in

general, have a low-prediction capability. Nevertheless. they possess the advantag~ of

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