2) For a two-dimensional steady turbulent flow in Cartesian coordinates (x,y), with mean velocity components (U,V), write

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1 058:68 Turbulent Flows 004 G. Constantnescu HOMEWORKS: Assgnment I - 01/6/04, Due 0/04/04 1) A cubcal box of volume L 3 s flled wth flud n turbulent moton. No source of energy s present, so that the turbulence decays. Because the turbulence s confned to the box, ts length scale may be assumed to be equal to L at all tmes. Derve an expresson for the decay of the netc energy 3u / as a functon of tme. As the turbulence decays, ts Reynolds number decreases. If Re=uL/ν becomes smaller than 10 3 say, the nvscd estmate ε = u / L should be replaced by an estmate of the type ε = c νu / L, because the wea eddes remanng at low Reynolds number lose ther energy drectly to vscous dsspaton. Compute the coeffcent c requred for the dsspaton rate to be contnuous at Re=10. Derve an expresson for the decay of the netc energy when Re<10 (ths s called the fnal perod of decay). If L=1m, ν=15*10-6 m/s and u=1m/s at tme t=0, how long does t tae before the turbulence enters the fnal perod of decay? Assume that the effects of the walls of the box on the decay of the turbulence may be gnored. ) For a two-dmensonal steady turbulent flow n Cartesan coordnates (x,y), wth mean velocty components (U,V), wrte (a) the contnuty and momentum equatons; (b) the mean nternal energy equaton; (c) the mean vortcty equaton; (d) the turbulent netc energy equaton; Smplfy these for fully-developed flow between two parallel plates, a dstance h apart, under the nfluence of a pezometrc pressure gradent dp/dx. Consder the boundary condtons that apply n ths case. 3) TASK 1

2 Assgnment II - 0/04/04, Due 0/16/04 1) Derve the vortcty equaton.11 from the Naver Stoes equatons. Hence derve the mean vortcty equaton (.8). Wth ths expressed n Cartesan coordnates, dscuss the connecton between the Reynolds stresses and secondary flow n steady-fully developed flow n a straght duct. Reference: Su and Fredrch (1994), ASME. Fluds Engrg., 116, pp ) Derve the Posson equaton for the Reynolds averaged pressure feld 1 U U uu Show that - P = where U = U u, etc. ρ x x x x 3) Denote the average of a quantty by <>. Let φ( x, t) be a conserved scalar quantty that satsfes the conservaton equaton: t r ( Uφ) = Γ φ φ It s very easy to show that the Reynolds averaged equaton for ths scalar s gven by < φ > r < U t > < φ >= Γ r < φ > ( < uφ' > ) r r r where φ =< φ > φ', U =< U > u a) Show that the transport equaton for the scalar varance s < φ > r < U t ' > < φ' >= Γ < φ' r > ( < uφ' r > ) < uφ' > < φ > Γ < φ' φ' > b) Identfy the producton term, the dsspaton term, the turbulent scalar flux term and the vscous transport term r c) Introduce the gradent dffuson model < u φ' >= Γt < φ > nto the scalar varance equaton d) Assumng there s a balance between producton and dsspaton n the scalar varance equaton, and the eddy dffusvty s nown, provde a model for the scalar dsspaton rate ε φ (whch s one of the terms n the equaton for the scalar varance) e) In the -ε model, transport equatons are solved for and ε, so that these are nown quanttes. Assumng the tme scale of the turbulence decay /ε s equal to the tme scale of the scalar varance decay defned analogously, and usng the model for ε φ, provde a model for the scalar varance.

3 4. Derve the equaton for the fluctuatng velocty: u U t u u U ( u u u u ) = (1/ ρ ) p ν and then the Reynolds stress transport equaton: u u u t U u u = u u u ( 1/ ρ)( u p u p) ν u u ν u u u u U u u U startng from the momentum & contnuty equatons for the total velocty: t ( U u) ( U u) ( U u) = (1/ ρ ) ( P p) ν ( U u) ( U u) = 0, U = U, u = 0 Symbolcally the steps are [ NS( U u) NS( U u) ] u [ NS( U u) NS( U u ] RS = u ) (you are bascally taen the frst moment of the momentum equaton) How many ndependent equatons and how many unnowns are there? Why not also form the moment of the contnuty equaton? 5. Usng the N-S equatons: U t ( U ) U = (1/ ρ ) P ν U u u show that the rate at whch the mean energy (per unt mass) 1/UU s lost to turbulence s u u U = PROD vscous_contrbuton. Hnt: Terms that are conservatve (can be wrtten as dvergence of a functon) are ust redstrbutng energy, so they are not an energy loss. What ths exercse demonstrates s that the term producton s actually referrng to the transfer of energy from the mean flow to the turulence, and not to a net source of energy. 6. Ansotropy equaton: The Reynolds stress ansotropy tensor s defned as b = uu / /3δ. Usng the fact that the equaton for the te n homogeneous turbulence s t = P ε (P s the producton term) the equatons for the turbulent stresses t uu = φ P ε where φ s the pressure velocty correlaton (or redstrbuton) term, derve the followng equaton for b : φ ε tb = ( U U ) b U b U ( b /3δ )( P ε ) / 3 where P/= b U. For the other defntons please see pag 5-57 n Durbn s boo.

4 Assgnment III - 0/16/04, Due 0/3/04 1. Fnd the probablty dstrbuton functon, P(u), and the probablty densty functon, β(u), for the followng perodc functons wth perod T = : (a) The trangle wave: u(t) = t 0 < t < 1 u(t) = - t 1 < t < (b) The sne wave: u(t) = sn πt (wth T = ). Analyss of an ensemble of expermental data for a random varable u(t) reveals that a good approxmaton for ts probablty densty β(u) s { cos[ 0.68( 0) ]} β ( u) = u 15<u<5 (a) Plot ths functon (b) Calculate the mean and standard devaton for u (c) Determne the Gaussan probablty dstrbuton for a varable wth the same frst and second moments as n (b) above. Plot the Gaussan probablty densty on the same plot as (a), and comment on the comparson. 3. Determne the autocorrelaton functon ρ(τ) for the perodc sgnal u(t) = sn ωt usng the tme average. Repeat for u(t) = cos ωt, and comment on the results. 4. Show that, for a statonary random functon u(t), wth autocorrelaton coeffcent ρ(τ), (a) b b-a u(t) dt = u ρ ( τ ) b - a - τ a 0 dτ (b) du(t) dt = - u d ρ ( τ ) d τ τ = 0 5. What s the Fourer transform of the delta functon, δ(t)?

5 6. In a spray, the probablty densty functon of droplet dameter D s unform n the range 50<D<100 µm and zero otherwse. Determne the probablty densty functon β(d) and the Sauter mean dameter, defned as D 3 / D. 7. Consder a round ol et nected nto stll water. The two fuds are perfectly mmscble (.e., they do not mx at the molecular level but can be broen down nto small ol drops n water, and vce versa, much le salad ol). The ol has a densty of 600 g/m3 and the densty of water s 1000 g/m3. Let f represent the mass fracton of ol at any locaton. (a) Fnd an expresson for the probablty densty functon b(f) as a functon of f (hnt: snce the two fluds do not mx, b(f) comprses two delta functons). (b) Calculate the mean densty ρ as a functon of f. Recall that ρ = 1 0 ρ( f ) β ( f ) df 8. Most computer lbrares have a random number algorthm generator that generates values between 0 and 1 wth equal probablty.e. P( u ~ )=1 for 0<u ~ <1. Show that the mean u ~ =1/ and that u = 1/ 1, where u s the fluctuaton u=u ~ - u ~. Deduce that ξ = 1( u ~ 1/ ) has ξ = 0 and ξ = 1 (you have to proof that analytcally frst). A Gaussan random varable can be approxmated by summng N values ξ and normalzng by N. E.g., for N=16, ξ = / 4 ξ. Program ths and verfy by averagng a large number (10,000 say) of values that ξ = 0 and ξ 1. G G G =

6 Assgnment IV - 0/3/04, Due 03/03/04 1. Show that, n sotropc turbulence, (a) p ' ' = 0 v (b) u x = 1 v x, etc.. For homogeneous sotropc turbulence the two pont correlaton r r r r r R ( x, t) =< u ( x, t) u ( x, t) > can be wrtten as r r R = u' ( f g) gδ where <> s the average operator. r R r Show frst that = 0. Then usng ths show that g = f ' r f 3. Assumng that the turbulence s sotropc, estmate the value of the mcroscale λ correspondng to a decrease n rms turbulence ntensty n a body of water from 1 to 0.5 m/s n a tme of 0 seconds. 4. The rms dameter of a cloud of mared partcles n a feld of sotropc turbulence s found to grow ntally at a rate of 0.03 m/s. If the Lagrangan tme scale s 10 s, what wll be the rms dameter of the cloud at t = 60 s? 5. For homogeneous turbulence, show that the equaton for the turbulent netc energy reduces to < U > =< uu > ε t x Recall that n homogeneous turbulence fluctuatons are not a functons of space whle the mean velocty gradent n each drecton can assume a constant non-zero value. From the defnton of ε show exactly what terms are not equal to zero. For homogeneous sotropc turbulence, show that the equaton for the turbulence netc energy reduces to = ε t Recall that n sotropc homogeneous turbulence the mean flow feld s shear free.

7 Assgnment V - 03/03/04, Due 03/08/04 1. Consder the turbulent mxng layer between two streams of velocty U 1 and U. Show that the layer thcness grows lnearly wth dstance from the orgn. Descrbe how you would determne the velocty dstrbuton across the mxng layer.. Investgate a smlarty soluton for the axsymmetrc wae, n lamnar and turbulent flow, to determne the growth and decay laws, and the velocty profles. In the turbulent case, assume a constant eddy vscosty across the wae.

8 Assgnment VI - 03/03/04, Due 03/08/04 1) Consder vertcal axsymmetrc flow, e.g., a thermal plume from a pont heat source, or a boundary layer outsde a vertcal cylnder. Wth buoyancy present, the equatons of moton (under thn shear-layer approxmatons) n lamnar flow may be wrtten ( ru) ( rv) x r = 0 (1) ( u) ( u) u v x r ν u = β ( t t ) r r r r () ( t) ( t) u v x r α t = r r r r (3) where t s temperature, t s the ambent temperature, β s the coeffcent of thermal expanson, and α s the thermal dffusvty. () Investgate smlarty solutons for a lamnar axsymmetrc thermal plume to determne ts growth and decay laws. To do ths, you need to consder the ntegral constrants, ntroduce approprate velocty and length scales, and see condtons whch would yeld smlarty solutons. You wll fnd that t s useful to ntroduce a stream functon, and defne a dmensonless temperature dfference: q = (t -t )/(t o - t ), where t o s the temperature at the plume axs. Obtan the ordnary dfferental equatons that govern the velocty and temperature profles (do not solve unless you want to mae a proect out of t). () Next consder turbulent flow. A two-dmensonal turbulent plume s dscussed n Tennees and Lumley, sec. 4.6, pp Develop the equatons for the axsymmetrc turbulent plume. Then nvestgate smlarty solutons to determne the growth and decay laws. Agan, obtan the ordnary dfferental equatons that govern the velocty and temperature profles usng ether eddy-vscosty or mxng-length hypothess. ) The momentum equaton for the temporally evolvng turbulent mxng layer s gven by < U t > = < uv y > a) Introduce the gradent transport model, assumng constant eddy vscosty b) Show that by ntroducng the smlarty coordnate

9 η = y ν t t and the smlarty varable < U > f = U s where Us = Uh Ul and U l and U h are the velocty at y and y, the momentum equaton can be wrtten as df f η = dη dη c) Show that <U> s gven by 1 < U ( η ) >= U erf ( η) s U c

10 Assgnment VII - Due 03//04 TASKS,3,4,5. Assgnment VIII - Due 04/05/04 TASKS 6,7.

11 Assgnment IX - 03/08/04, Due 04/1/04 1. The velocty dstrbuton n turbulent flow n a ppe of radus a may be approxmated by a power-law u y = U a 1 / m where y s measured from the wall. The power depends on the Reynolds number. The momentum equaton shows that the pressure vares lnearly along the ppe, and the total stress (molecular plus turbulent) vares lnearly across the ppe. Use ths nformaton to fnd an expresson for the dstrbuton of the mxng length across the ppe. Plot ths n an approprate non-dmensonal form for m = 7. δu. An experment s performed on fully turbulent channel flow at Re = = 10 ν 5 The 6 flud s water ( ν = 1.14*10 m / s ) and the channel half sze δ = cm. The sn frcton 3 coeffcent s found to be C = 4.4*10. Determne: U u / U,Re andδ δ where f, τ τ v / δv = ν / uτ s the vscous lengthscale. What are the thcnesses of the vscous wall regon (y <50) and of the vscous sublayer, both as fractons of δ and n mllmeters? 3. Carry out a seres expanson of the nstantaneous velocty components to determne the behavor of the Reynolds stress components, turbulent netc energy, and dsspaton rate near a free surface (zero stress boundary). Comment on the dfferences between ths and a sold wall. 4. Smplfy the -ε, -τ and -ω models for the logarthmc layer (neglect convecton, dffuson, and vscous terms). Determne the value of the Karman constant, κ, mpled by these models and compare wth the expermental value of Hnt: for all models, assume a soluton of the form: du dy Uτ Uτ =, = ' κy C µ ν = κu τ τ y 5. Begnnng wth the -ω model of Wlcox and σ* = σ, mae the formal change of varables ε = β ω and derve the mpled -ε model. Express your fnal results n

12 standard -ε model notaton and determne the mpled values of the closure coeffcents C µ, C ε1, C ε, σ and σ ε n terms of α, β, β, σ and σ. 6. Show that the expanson for the Reynolds shear stress at the wall can be wrtten as: τ < uv > / u = σy 3 where the nondmensonal coeffcent σ may be assumed to be ndependent of the Reynolds number. For the flow between two walls (fully developed channel flow) show usng the momentum equatons (remember that one can show that the total stress vares lnearly across the channel) that n wall coordnates the expanson for the streamwse velocty s: u y 1 = y σy Re 4 τ 4. Thus f Re τ >> 1 the expanson for the law of the wall s 1 u = y σ y Mxng length model wth Van Drest dampng: In a wall boundary layer, the log-law 99 apples n the regon 35<y <0. δ 99 ( δ s the thcness of the b.l. n wall unts). To extend the mxng length model all the way to y=0, Van Drest suggested that κy should be multpled by an exponental dampng functon. Thus the turbulence length scale y / A near the wall becomes l = κ y(1 e ), where the Von Karman constant s m Apply ths to the constant total stress layer, uv ν y U = u τ to fnd a formula for U. y Integrate ths numercally and mae a log-lnear plot of U vs. y. What s the addtve constant B that you obtan from the log law f A =6? What value of A gves B=5.5?

13 du Hnt: The formula you have to obtan s dy 1 4lm = lm 1 8. A shortcomng of the -ε model: Show that n ncompressble flow the eddy vscosty formula u u = ν S / 3δ gves the same rate of turbulent energy producton as t 1 P = ν tss rrespectve of the mean rate of rotaton tensor Ω = ( U U ). When turbulence s rotated (e.g., the case of the flow n turbomachnery) the centrfugal acceleraton can affect the turbulent energy. Dscuss whether the classcal -ε model can predct such effects. There s an analogy between rotaton and streamlne curvature, so your concluson should apply to effects of curvature on the turbulence, too.

14 Assgnment X - Due 04/8/04 TASKS 8,9,10,11 Assgnment XI - Due 05/05/04 TASKS 1,13