Chapter 2 Modelling of Fluid Flow and Heat Transfer in Rotating-Disk Systems

Chapte 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems 2.1 Diffeential and Integal Equations 2.1.1 Diffeential Navie Stokes and Enegy Equations We will conside hee stationay axisymmetic fluid flow ove disks otating with a sufficiently high angula velocity so that effects of gavitational foces on momentum tansfe ae athe low. In a stationay cylinical coodinate system aanged in such a way that a disk o a system of disks otate aound its axis of symmety coinciding with the axis, while the point = is located on a suface of the disk Fig. 2.1), lamina fluid flow and heat tansfe ae descibed by Eqs. 1.31), 1.32), 1.33), 1.34) and 1.35) simplified accounting fo the conditions F = Fϕ = F = [41, 138, 139]: + v v2 ϕ = 1 p 2 ρ + ν 2 + 1 v ) 2 + 2 2, 2.1) v ϕ + v v ϕ + v ϕ = ν 2 v ϕ 2 + 1 v ϕ v ) ϕ 2 + 2 v ϕ 2, 2.2) v + v v = 1 p 2 ρ + ν v 2 + 1 ) v + 2 v 2, 2.3) + + v =, 2.4) T t + v T + v T = a1 T ) + a 2 T 2. 2.5) Fo tubulent flow with account fo the conditions F = Fϕ = F =, Eqs. 1.36), 1.37), 1.38) and 1.39) take the following fom [41, 138, 139]: I.V. Shevchuk, Convective Heat and Mass Tansfe in Rotating Disk Systems, Lectue Notes in Applied and Computational Mechanics 45, DOI 1.17/978-3-642-718-7_2, C Spinge-Velag Belin Heidelbeg 29 11

12 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems Fig. 2.1 Geometical aangement and main paametes of the poblem of fluid flow and heat tansfe ove a otating disk in still ai. + v v2 ϕ = 1 p ρ + ν + 1 v ϕ + v v ϕ + v ϕ v + v v = 1 ρ T t + v T + v T = 1 p + ν [ 2 v ) 2 ) v 2 + = ν 2 v ϕ v ϕ 2.6) ) v v 1 ) v 2 ϕ, ) 2 + ) v v ϕ + ) v ϕv + 2 ) 2.7) v v ϕ, ) 2 v + 1 ) v v + ) v 2, 2.8) a T v T )] + a T v T ). 2.9) In a otating coodinate system connected with a disk, Eqs. 2.1), 2.2) and 2.3) fo lamina flow can be tansfomed as follows [138, 139]: + v v2 ϕ 2ωv ϕ ω 2 = 1 p 2 ρ + ν 2 + 1 v ) 2 + 2 2, 2.1) v ϕ + v v ϕ + v v ϕ 2 v ϕ + 2ω = ν 2 v + v v = 1 p ρ + ν 2 v 2 + 1 v ϕ v ϕ 2 + 2 v ϕ 2 + 1 v + 2 v 2 ), 2.11) ). 2.12) The tems 2ωv ϕ and 2ω ae the pojections of the Coiolis foces onto the axes and ϕ, espectively, while the tem ω 2 is the pojection of centifugal foce onto the axis all divided by ρ). Fo tubulent flow, Eqs. 2.6), 2.7) and 2.8) can be ewitten in the same way [138, 139].

2.1 Diffeential and Integal Equations 13 2.1.2 Diffeential Bounday Laye Equations In the bounday laye appoximation, it is assumed that [41, 138, 139] a) the velocity component v is by ode of magnitude lowe than the components and v ϕ ; b) the ate of vaiation of the velocity, pessue and tempeatue in the diection of the axis is much lage than the ate of thei vaiation in the diection of the axis ; c) static pessue is constant in the -diection. The continuity equation 2.4) emains invaiable, while the othe equations of the system 2.6), 2.7), 2.8) and 2.9) take the following fom [41, 138, 139]: + v v2 ϕ = 1 ρ v ϕ + v v ϕ + v ϕ 1 p ρ T t + T + v p + 1 ρ = 1 ρ τ, 2.13) τ ϕ, 2.14) =, 2.15) T = 1 q ρc p, 2.16) τ = μ ρv v ϕ, 2.17) τ ϕ = μ v ϕ ρv ϕv, 2.18) q = λ T ρc pt v ). 2.19) In Eqs. 2.17), 2.18) and 2.19), only those components of the tubulent shea stesses and heat fluxes ae taken into account, whose input into momentum and heat tansfe is the most impotant. It is natual that the pessue in the bounday laye is assumed to be equal to its value outside of the bounday laye in the egion of potential flow p=p. The equation of the stationay themal bounday laye looks as follows: T + v T = 1 q ρc p. 2.2) The system of Eqs. 2.13), 2.14), 2.15), 2.16), 2.17), 2.18) and 2.19) is completed by an equation in the egion of potential flow outside of the bounday laye, whee velocity components and pessue p ae consideed invaiable in the -diection:

14 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems 1 dv 2, v2 ϕ, = 1 dp 2 ρ. 2.21) 2.1.3 Integal Bounday Laye Equations These equations which ae in fact integal diffeential equations) fo stationay fluid flow and heat tansfe can be deived fom Eqs. 2.13), 2.14), 2.15), 2.17), 2.18), 2.19) and 2.2) with allowance fo Eqs. 2.4) and 2.21) in the following fom [41, 138, 139]: o d δ d 2 ) v, d + d, δ ) vϕ v ϕ, d + d 2 δ δ ṁd 2πρ v ϕ d d + v ϕ, δ v, ) d ) v 2 ϕ, v2 ϕ d = τ w /ρ, 2.22) d ) vϕ, = 2 τ wϕ /ρ, 2.23) ṁd ) = 2 τ wϕ /ρ, 2.24) 2πρ d δ T T T ) d + dt ṁd,t 2πρ = q w/ρc p ). 2.25) Equations 2.22), 2.23) and 2.25) can be ewitten as follows [41, 138, 139, 18]: d v 2, δδ ) d [δ2 ω ) 2 δ ϕ ] + +, δ d, δ v2 ϕ, δδ ϕ = τ w/ρ, 2.26) ṁd 2πρ d v ϕ, ) = 2 τ wϕ /ρ, 2.27) d [ω 2 δδ T T w T )] + dt ṁd,t 2πρ = q w/ρc p ), 2.28)

2.2 Diffeential Methods of Solution 15 whee 1 δ = 1 ṽ ) dξ, δ = 1 ṽ 1 ṽ ) dξ, 1 δ ϕ = ) 1 v2 ϕ v 2 dξ, ϕ, 2.29) 1 δ ϕ = v ϕ v ϕ, ) ω ) 2 dξ, ṽ = /,. 2.3) 2.2 Diffeential Methods of Solution 2.2.1 Self-Simila Solution Fo lamina flows ove a single otating disk, exact solutions of the Navie Stokes and enegy equations wee obtained in woks [33, 41, 55, 58, 8, 16, 138, 139, 158, 199] using the following self-simila vaiables: = a + ω)fζ ), v = a + ω)νhζ ), v ϕ = a + ω)gζ ), p = ρνωpζ ), θ = T T )/T w T ), ζ = a + ω)/ν, 2.31) unde the bounday conditions ζ :, = a, v, = 2a, v ϕ, =, β = /ω = const, θ =, 2.32) ζ = : F = H =, G = 1, θ = 1, 2.33) ζ = : T w = T ef + c w n, T = T ef + c n o T = T ef + βc w n, 2.34) whee c, c w, c and n ae constants. Bounday conditions 2.34) can be tansfomed as follows: T = T w T = c n fo c = c w c ), 2.35)

16 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems o T = c w 1 β) n. 2.36) Taking into account Eq. 2.21) fo the adial pessue gadient in the egion of potential flow, Eqs. 2.1), 2.2), 2.3) and 2.4) and 2.2) take the following selfsimila fom: F 2 G 2 + F H = N2 β 2 1 + N) 2 + F, 2.37) 2FG + G H = G, 2.38) HH = P + H, 2.39) 2F + H =, 2.4) θ P n Fθ + Hθ ) =, 2.41) whee N=a/ω=const and, natually, tubulence components wee neglected). It is impossible to assign simultaneously non-eo values of β and N in Eq. 2.37), because in this case the deivative of the component Fζ) does not tend to eo on the oute bounday of the bounday laye. Howeve, Eq. 2.37) still holds eithe fo N = and β= ofoβ = and N=. In the past, solutions of the Eqs. 2.37), 2.38), 2.39), 2.4) and 2.41) have been obtained with the help of individually developed compute codes based on expansions in powe o exponential seies [8], use of the shooting method [58, 16, 138, 199], etc. Cuently, standad compute mathematics softwae like MathCAD, etc. allows pogamming solutions of the systems of equations like Eqs. 2.37), 2.38), 2.39), 2.4) and 2.41) with the help of the use inteface options. As shown in woks [41, 138, 139], a self-simila fom of the enegy equation with account fo dissipation effects imposes estiction onto the bounday conditions 2.34), 2.35) and 2.36): in this case, one can use only the value of the exponent n =2. Since effects of adial heat conduction and enegy dissipation in ai cooling systems at sub-sonic speeds ae negligible, the advantage to use abitay n values in the themal bounday laye equation 2.41) by fa compensates vey mino losses involved because of neglecting the afoementioned tems in the enegy equation. Exact solutions of Eqs. 2.37), 2.38), 2.39), 2.4) and 2.41) povide a eliable database useful, among othe applications, in validations of CFD codes and models aimed at solving much moe complicated physical poblems. Use of the self-simila solutions also enables obtaining appoximate analytical solutions fo poblems with bounday conditions diffeent fom Eqs. 2.32), 2.33), 2.34), 2.35) and 2.36).

2.2 Diffeential Methods of Solution 17 2.2.2 Appoximate Analytical Methods fo Lamina Flow Based on Appoximations of Velocity Pofiles The method of Slekin-Tag was used in the wok [41] to model lamina fluid flow ove a fee otating disk at β= and N=, as well as fo the case of N=const and β=. Velocity pofiles deived fo the case of β= and N= wee descibed by the sixth-ode powe polynomials, which in view of the necessity to futhe develop the method would inevitably esult in obtaining inconvenient and cumbesome elations fo the Nusselt numbe. Deceasing the ode of the appoximating polynomial to the thid ode esulted, in the case of N=const and β=, in noticeable eos in calculations of suface fiction, which wee equal to appoximately 25% at N=5 and inceased shaply with the futhe inceasing values of N. The autho of the wok [4] obtained an appoximate solution fo the velocity components in lamina fluid flow ove a fee otating disk in a fom of a athe complex combination of exponential and logaithmic functions. The method was not extended to include the heat tansfe poblem as well as to take into consideation the bounday condition 2.32). It should be expected that the development of the method [4] in this diection would esult in obtaining even moe inconvenient and cumbesome elations, in paticula, fo the Nusselt numbe, than those esulting fom the appoach of Slekin-Tag [41]. The appoximate solution [83] fo poous injection though a otating disk has a fom of a combined expansion in powe and exponential seies. The authos did not genealie thei method fo moe complex cases; howeve, it is obvious that thei appoach has the same deficiencies as the methods of [4, 41]. It can be thus concluded that velocity, pessue and tempeatue pofiles in lamina bounday layes ove a otating disk ae so much complicated fom the mathematical point of view that a seach of thei athe accuate analytical appoximation is inexpedient. As shown below, a combination of an integal method with the data of self-simila solutions can esult in obtaining athe simple and accuate appoximate analytical solutions fo suface fiction coefficients and Nusselt numbes. 2.2.3 Numeical Methods Authos of [136] solved bounday laye equations 2.13), 2.14), 2.15), 2.16), 2.17), 2.18), 2.19) and 2.2) with the help of a finite-diffeence method employing a modified algebaic model of tubulent viscosity by Cebeci-Smith [22]. Fo the case of lamina steady-state heat tansfe with tangential non-unifomity of heating of the disk suface, Eq. 1.3) was educed to a two-dimensional equation in woks [25, 26] using modified vaiables 2.31). A steady-state axisymmetic poblem with a localied heat souce was modelled in the wok [137] with the help of Eq. 2.9). In both afoementioned cases, a finite-diffeence technique was used. Equations 2.6), 2.7), 2.8) and 2.9) wee used to model both lamina and tubulent flow and heat tansfe in cavities between paallel otating disks [78, 79, 145, 148,

18 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems 151]. Authos of [148] applied a finite-diffeence method fo a two-dimensional lamina steady-state poblem. In woks [25, 78, 79, 9, 145, 225], an in-house compute code based on the finite-volume method was used to solve Eqs. 2.6), 2.7), 2.8) and 2.9) closed with low Reynolds numbe k ε models of tubulence [11, 128, 129]. Lage eddy simulation LES) appoach was used in the wok [224] to model a thee-dimensional stationay tubulent flow ove a otating disk, and tubulent flow and heat tansfe ove a single disk in ai flow paallel to the disk suface non-symmetical flow) in woks [22, 221]. Autho of [38] used a finite-diffeence method to solve the Navie Stokes equations closed with a k ε tubulence model by [1] as applied to thee-dimensional flow of a caying phase ai) in a otating-disk ginde of solid paticles. Authos of [151, 152] used a commecial CFD code Fluent to solve Eqs. 2.6), 2.7), 2.8) and 2.9) closed with an RNG k ε tubulence model. Commecial CFD code Phoenics was used by the authos of [188] to simulate stationay tubulent one-phase flow in a otating-disk ai cleane using a standad k ε tubulence model. In the wok [61], modelling of lamina conjugate tansient heat tansfe of a fee otating disk was pefomed using a commecial CFD code CFX. Numeical methods ae the most univesal tool of mathematical modelling inevitably used in simulations of poblems with complex geomety, abitay bounday conditions, etc. Relative complications in use of such methods ae significant time consumption fo one un often tens of hous), geneation of a computational mesh often weeks), sometimes lack of convegence of the numeical solution, etc. Theefoe, use of such methods is not justified fo elatively simple poblems that can be solved by means of simple appoaches. A geneal disadvantage of the diffeential methods which one can nevetheless comply with) is obtaining solution in a numeical fom that is usually a cetain inconvenience in compaison with analytical solutions. 2.3 Integal Methods of Solution 2.3.1 Momentum Bounday Laye The essence of integal methods consists in solving Eqs. 2.22), 2.23), 2.24), 2.25), 2.26), 2.27) and 2.28) closed with models fo the velocity pofiles and shea stess components on the wall fo the momentum bounday laye, as well as fo the tempeatue pofiles o enthalpy thickness) and wall heat flux fo the themal bounday laye. The most pefect integal method among those known in the liteatue is the method of [138, 139], which developed ideas of the authos of [41, 8]. A key point of the method [138, 139] consists in use of a genealied fom of the models that takes into account existence of lamina o tubulent flow via assigning paticula numeical values to cetain paametes of the model. This is a confimation of the idea expessed by Loytsyanskiy still in the yea of 1945 [113], who said that thee

2.3 Integal Methods of Solution 19 exists an analogy between basic chaacteistics of lamina and tubulent bounday layes. In the bounday laye, the adial and tangential v ϕ velocity components coelate with each othe. This coelation is descibed by the fomula [163] = v ϕ tanϕ. 2.42) Fo flows, whee adial velocity of the flow outside of the bounday laye can be neglected, i.e. fo, =, authos of [138, 139] specified the velocity pofiles as v ϕ = 1 gξ), = α f ξ), 2.43) whee functions gξ) and fξ) expessed in tems of the independent vaiable ξ=/δ wee assumed to be self-simila independent of the coodinate ). Fo a lamina bounday laye, functions gξ) and fξ) can be expessed as gξ) = G ξ), f ξ) = F ξ)/α, 2.44) whee functions G ξ) and F ξ) ae tabulated [41, 138, 139] being a solution of the poblem of fluid flow ove a fee otating disk in the self-simila fom of Eqs. 2.37), 2.38), 2.39) and 2.4) at N= and β=. Fo a tubulent bounday laye, the following powe-law appoximations wee used in the woks [41, 138, 139]: gξ) = 1 ξ n, 2.45) f ξ) = ξ n 1 ξ), tanϕ = α1 ξ), 2.46) whee n=1/5 1/1. The models 2.45) and 2.46) wee fomulated by von Kaman [8] in 1921 fo the fist time by analogy with the model fo tubulent flow in a ound pipe and ove a flat plate [158]. The exponent n is assumed to be known and selected depending on the chaacteistic Reynolds numbe Figs. 2.2, 2.3 and 2.4). Model 2.45) fo the tubulent bounday laye was also used by the authos of [7, 122, 13]. A moe accuate appoximation of the function fξ) in tubulent flow is the expession used by the authos of [32, 7, 118, 196]: f ξ) = ξ n 1 ξ) 2, tanϕ = α1 ξ) 2, 2.47) which, howeve, has not been used so widely because of somewhat inceased complexity in integating the tems in the left-hand side of Eqs. 2.22), 2.23) and 2.24). FoflowsatN=const, the following elation was used in [212]: Authos of [41, 69] used the following expessions: tanϕ = α + N α) ξ. 2.48)

2 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems Fig. 2.2 Pofiles of the non-dimensional tangential velocity in the tubulent bounday laye ove a fee otating disk. Calculation by Eq. 2.45) [163], 1 n=1/7, 2 1/8,3 1/9. Expeiments: 4 Re ω =.4 1.6) 1 6 [111], 5.6 1.) 1 6 [63], 6.65 1.) 1 6 [46]. Hee δϕ = b v ϕ v ω 1 ϕ ) ω d definition of [26, 46, 47, 63, 111]). 14 12 1 8 /δ ϕ 6 4-1 - 2-3 - 4-5 - 6 2.4.6.8 1. v ϕ /ω Fig. 2.3 Pofiles of the non-dimensional adial velocity in the tubulent bounday laye ove a fee otating disk. Calculation by Eq. 2.47) o Eq. 2.63)) [163], 1 n=1/7, 2 1/8,3 1/9, 4 1/7,vonKaman s Eq. 2.46) [8], 5 1/7,Eq. 2.5) fo b=.7, c=1.2, α=.23. Expeiments: 6 Re ω =.4 1 6, 7.65 1 6, 8.94 1 6, 9 1.6 1 6 [111], 1.6 1 6, 11 1. 1 6 [63] /δ ϕ 1 8 6-6 - 7-8 - 9-1 - 11 4-1 - 2 2-3 - 4-5,,4,8,12 /ω f ξ) = ξ n 1 ξ n/m ), tanϕ = α1 ξ n/m ), 2.49) whee constants n and m could be vaied independently. Howeve, model 2.49) has not been futhe developed because of its excessive complexity. Authos of [1, 2] used a tigonometic function to appoximate the tangent of the flow swil angle

2.3 Integal Methods of Solution 21 /ω,1-1 - 2-3 - 4-5 - 6,5-7 - 8-9 - 1-11 - 12-13 - 14,,,2,4,6,8 1, ω v ϕ )/ω Fig. 2.4 Coelation between the adial and tangential components in the bounday laye. Calculation by Eq. 2.64) at L=2 cuves 1 4) ol=1 cuve5) [163]: 1 n=1/7, 2 1/8,3 1/9, 4 1/1, 5 1/7, von Kaman s method [8], 6 1/7, Eq. 2.5) fo b=.7, c=1.2, α=.23. Expeiments: 7 Re ω =.4 1 6, 8.65 1 6, 9.94 1 6, 1 1.6 1 6 [111], 11.4 1 6, 12.6 1 6, 13 1. 1 6 [63], 14 2. 1 6 [21]. tanϕ = α[1 sin b cξ)], 2.5) whee constants b and c take the values b=.7, c=.12 at n=1/7, and b=.697, c=.117 at n=1/8. Authos [1, 2] asseted that the model 2.5) allows to attain bette ageement with expeiments fo a fee otating disk than with von Kaman s appoach 2.45), 2.46). Analysis shows, howeve, that the values of the constants b and c given in [1] ae eoneous. Fo example, keeping the value b=.7 at n=1/7 invaiable, it is necessay to use the value of c=1.2 Figs. 2.3 and 2.4). Howeve, the model [1, 2] has not been futhe developed to include heat tansfe, appaently, because esulting expessions fo the Nusselt numbe obtained on the base of Eq. 2.5) would have been too cumbesome. In woks [73 75], it was assumed fo N=const that tanϕ = α 1 ξ) + κ, 2.51) whee κ = ṁ/ [2πρs1 β)ω]. This appoximation is wose than Eq. 2.48), because it does not agee with the appaent condition tanϕ w = α and complicates the solution of Eqs. 2.22) and 2.23). The autho of [72] used a elation tan ϕ = cn, whee N = ṁ/2πρs), with the constant c vaying fom the value 1. at the inlet to a cavity between paallel co-otating disks to 1.22 in the egion of stabilied flow. This elation was used fo lage values of, and v ϕ,. The disadvantages of this appoach ae lack of ageement with the bounday condition tanϕ w = α and involvement of a new empiical constant c. Thus, models 2.49), 2.5) and 2.51) have shown wose pefomance than Eqs. 2.42), 2.43), 2.44), 2.45), 2.46), 2.47) and 2.48).

22 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems Afte integation of the tems in the left-hand side of Eqs. 2.22) and 2.23) with account fo Eqs. 2.42), 2.43), 2.44), 2.45), 2.46), 2.47) and 2.48), the unknowns to be found ae paametes α) and δ) fo a pescibed distibution of β) o α) and β) fo a pescibed δ). Unde conditions N=const o β=const, the value of α is also constant, while the value of δ is eithe also constant in lamina flow o has a fom of a powe-law function of in tubulent flow [41, 138, 139]. Apat fom the woks [138, 139], no othe autho has developed an integal method fo lamina flow fo an abitay distibution of β). In fames of the appoach with powe-law appoximation of the velocity pofiles, shea stesses τ w and τ wϕ in the ight-hand side of Eqs. 2.22), 2.23) and 2.24) ae detemined by the following equations [41, 138, 139]: τ w = ατ wϕ, τ wϕ = sgn1 β)τ w 1 + α 2 ) 1/2, 2.52) c f = C 2/n+1) n Re 2n/n+1) V, 2.53) C n = 2.28 +.924/n. 2.54) The constant C n takes values 8.74, 9.71, 1.6 and 11.5 fo n=1/7, 1/8, 1/9 and 1/1, espectively [41, 8, 13, 138, 139]. Appoximation 2.54) was obtained in the wok [69]. In woks [73 75], a modified velocity V = ω β 1 [1 + α + κ) 2 ] 1/2 was used that, with no eal justification of such a choice, only complicated all the mathematical deivations. One should also mention the model [52], which employed logaithmic appoximations fo the velocity pofiles. In fames of this appoach, one has nea the wall = αω + 2.5αV τ 1 + α 2 ) 1/2 ln ξ), v ϕ = 2.5V τ 1 + α 2 ln ξ). 2.55) ) 1/2 Appoach 2.55) was validated only fo the fee disk case; the heat tansfe poblem has not been solved. An algebaic equation fo the moment coefficient C M obtained in the wok [52] see Sect. 3.3) is tanscendental. Namely because of its excessive complexity and inconvenience, the logaithmic appoach [52] has not been widely used futhe. 2.3.2 Themal Bounday Laye In the majoity of the known woks [41, 68, 72 77, 8, 114, 13, 133, 135, 138, 139, 196], heat tansfe modelling in fames of the integal method has been pefomed with the help of a theoy of local modelling. Fo the fist time, this theoy was applied to otating-disk systems by Dofman [41], who used fo this pupose the method of Loytsyanskiy [113]. Accoding to the theoy of local modelling, a heat

2.4 Integal Method fo Modelling Fluid Flow and Heat Tansfe in Rotating-Disk Systems 23 tansfe law detemining vaiation of the Stanton numbe has the following fom [41, 82]: St = M s Re T σ P n s, 2.56) whee M s, σ and n s ae univesal constants independent of the Pandtl numbe and tempeatue distibution T w on the disk suface. Fo tubulent flow, the constants in Eq. 2.56) have the values σ=.25, n s =.5 and M s =7.246 1 3 ; fo lamina flow, these constants ae equal to σ=1., n s =1. and M s =.733 [41, 89, 138, 139]. Relation 2.56) allows closing the themal bounday laye equation 2.28), whee the value δt becomes the only unknown. Instead of the enthalpy thickness δt, the authos of the wok [138, 139] employed the Reynolds analogy paamete χ defined as q w = χ c pt T w ). 2.57) τ wϕ ω1 β) Using model 2.56) and definition 2.57), the authos of [138, 139] solved Eq. 2.28) to find the unknown value of χ. Fo a long time, the wok [122] had been the only one that involved a powe-law appoximation of the tempeatue pofile in tubulent flow = T T w T T w = ξ n T T, θ = T T T w T = 1 = 1 ξ n T T 2.58) at n T =1/5. Authos of [122] used an unjustified additional coelation = δ T /δ = 6atT w =const and did not offe a model to deive dependence of the paamete on the factos affecting heat tansfe, which could become an altenative to the appoach based on Eq. 2.56). 2.4 Integal Method fo Modelling Fluid Flow and Heat Tansfe in Rotating-Disk Systems 2.4.1 Stuctue of the Method A seies of oiginal esults of investigations into fluid flow and heat tansfe in otating-disk systems pesented in this monogaph wee obtained with the help of an impoved integal method developed by the autho of the monogaph [163 181, 184, 189, 19]. This method will be efeed to as the pesent integal method thoughout the monogaph. The pesent integal method is based on using the following: a) integal equations 2.22), 2.23), 2.24), 2.25), 2.26), 2.27) and 2.28); b) impoved models of the tubulent velocity and tempeatue pofiles; c) a novel model fo the enthalpy thickness in lamina flow;

24 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems d) closing elations fo the shea stesses and heat fluxes on the disk suface; e) bounday conditions fo the velocity and tempeatue outside of the bounday laye, as well as fo the disk suface tempeatue. The key point of the pesent integal method consists in using the same appoach to model both lamina and tubulent flow egimes. The only diffeence consists in the paticula values of the cetain numeical coefficients involved in the model. Such an appoach is in fact a development of the idea of Loytsyanskiy [113], who noticed an analogy between the main paametes of the lamina and tubulent bounday layes subjected to the same bounday conditions. This idea has aleady poved its fuitfulness in the poblems of convective heat tansfe in otating-disk systems. It has aleady been used by the authos of the woks [41, 8, 138, 139]; howeve, the appoach employed in these woks in view of its intinsic impefectness lead to significant inaccuacy in modelling of some heat tansfe egimes see Sect. 3.2). An impotant featue of the pesent integal method is undestanding the fact that any powe-law appoximation of the velocity and tempeatue pofiles in lamina flow equies using polynomials of not less than the seventh ode. In its tun, this leads to deiving cumbesome and inconvenient elations fo the Nusselt numbe and the fiction coefficient. On the othe hand, simple powe-law appoximations of the velocity and tempeatue pofiles fo tubulent flow esult in quite simple and lucid elations fo the est of the bounday laye chaacteistics including the Nusselt numbe and the enthalpy thickness. Having obtained, on this basis, the mathematical fom of the necessay paametes fo tubulent flow, it is athe easy to extend these fomulas onto lamina flow unde an assumption that cetain coefficients ae deemed to be unknowns to be found empiically fom compaisons with the self-simila exact solution. As shown below, this method poved to be the most accuate among all the known integal methods fo the otating-disk systems. Theefoe, the logical sequence of developing the integal method below is as follows. Fist, the integal method fo tubulent flow will be developed and thooughly validated. Second, the integal method will be extended and validated fo lamina flow conditions. 2.4.2 Tubulent Flow: Impoved Appoximations of the Velocity and Tempeatue Pofiles Fo modelling of the velocity pofiles, we will use powe-law appoximations, namely, Eq. 2.42) fo, Eqs. 2.43) the fist one) and 2.45) fo v ϕ. The function tanϕ is specified in the fom of a quaatic paabola, whose coefficients a, b and c can be found using the bounday conditions on a disk and on the oute bounday of the bounday laye: tanϕ = a + bξ + cξ 2, 2.59) ξ =, tanϕ = tanϕ w = α, 2.6)

2.4 Integal Method fo Modelling Fluid Flow and Heat Tansfe in Rotating-Disk Systems 25 ξ = 1, tanϕ = tanϕ =, /ω v ϕ, ) = N/1 β) = κ, 2.61) Based on this, one can obtain that ξ = 1, dtanϕ)/dξ =. 2.62) a = α, b = 2α κ), c = α κ, tanϕ κ)/α κ) = 1 ξ) 2. 2.63) Pofiles of the velocity components and v ϕ fo a fee otating disk κ=) computed by Eqs. 2.42) and 2.63) ae depicted in Figs. 2.2 and 2.3. Wall values of the tangent of the flow swil angle α depending on the exponent n wee detemined with the help of the pesent integal method and documented below in Table 3.4 devoted to a fee otating disk. Also pesented in Table 3.4 ae numeical data fo α obtained using von Kaman s method [8]. Advantages of the powe-law model fo the and v ϕ pofiles jointly with the quaatic appoximation of tanϕ in the fom of Eq. 2.63) ae obvious: it povides computational esults that agee well with the expeimental data in the oute egion of the bounday laye, with the pofiles at n=1/9 being in best ageement with the expeimental data [63, 111] Figs. 2.2 and 2.3). The same conclusion follows fom Fig. 2.4, whee the adial and tangential velocities ae inteelated using an equation obtained fom Eqs. 2.42), 2.43), 2.45) and 2.63) [163] = α v ϕ 1 v 1 /n ϕ )L, 2.64) whee L=2 fo the pesent integal method and L=1 fo the von Kaman s method [8] although, in the nea-wall egion, the closest ageement between the pofiles is obseved fo n=1/7 1/8). Fom Eq. 2.64), a maximum in the dependence of on v ϕ can also be obtained in the following fom [163]: v ϕ, max = ξ n max, ξ max = n/n + L). 2.65) Equation 2.63) is a genealiation of the quaatic elation 2.47) poposed fo the case κ=. Somewhat wose ageement of Eq. 2.47) with expeimental data mentioned in the woks [118, 196] may have been caused by a less accuate choice of the constants n and α. Tempeatue pofiles in the bounday laye have been appoximated with the powe law, Eq. 2.58), which agees well with known expeiments see Fig. 2.5). 2.4.3 Models of Suface Fiction and Heat Tansfe Relations fo the shea stesses τ wϕ, τ w and wall heat flux q w will be found with the help of a two-laye model of the velocity and tempeatue pofiles in the wall

26 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems Fig. 2.5 Pofiles of the non-dimensional tempeatue θ in the tubulent bounday laye ove a fee otating disk. Expeiments [46], q w =const, Re ω =1. 1 6 : 1 inne heate on, 2 inne heate off. Calculations by Eq. 2.58): 3 n T =1/5, 4 1/4. θ 1..8.6.4.2-1 - 2-3 - 4...2.4.6.8 1. /δ T coodinates. It is obvious that in the egion whee the powe-law pofiles 2.45) and 2.58) ae in foce, one can ewite these pofiles as follows: V + = ξ n / c f /2, T + = ξ n T T cf /2/St. 2.66) Since these powe-law equations do not hold in the viscous sub-laye, they ae eplaced hee with the following equations: V + = +, T + = P +. 2.67) Splicing elations 2.66) and 2.67) at the bounday of the viscous sub-laye at the coodinate + 1 ) and heat conduction sub-laye at the coodinate + 1T ) and tansfoming, one can finally obtain fomulas fo the suface fiction coefficient and the Stanton numbe c f /2 = + 1 )2n 1)/n+1) Re 2n/n+1) V, 2.68) St = + 1 )n T 1 Re n T V c / f 2) 1 n T )/2 n T + / 1T + 1 )n T 1 P n T. 2.69) The value of + 1 most often used in a modified fom of the coefficient C n = + 1 )1 n is a constant depending only on the exponent n see Eq. 2.54)) [41, 13, 138, 139]. The inteelation among shea stesses τ w, τ wϕ and τ w is defined by Eqs. 2.52). Taking this into account τ w /ρ = Cn 2/n+1) sgn1 β)ν/δ) 2n/n 1) ω 1 β ) 2/n 1) α1 + α 2 ).51 n) /1+n), τ wϕ /ρ = Cn 2/n+1) sgn1 β)ν/δ) 2n/n 1) ω 1 β ) 2/n 1) 1 + α 2 ).51 n) /1+n). 2.7) The atio + / 1T + 1 ) is a function of the Pandtl numbe only; the quantity the unknown to be found) depends on the bounday condition fo T w ) and the Pandtl numbe P. Let us denote + / 1T + 1 )n T 1 P n T = P n p, whee the constant n p emains unknown as yet.

2.4 Integal Method fo Modelling Fluid Flow and Heat Tansfe in Rotating-Disk Systems 27 In the majoity of the solutions below, an assumption p T =n will be used. In this case, fomulas fo the Stanton and the Nusselt numbe take the following fom: / St = c f 2) n P n p, 2.71) Nu = St V ν P = StRe ωp β 1 1 + α 2 ) 1/2. 2.72) 2.4.4 Integal Equations with Account fo the Models fo the Velocity and Tempeatue Pofiles Integation of Eqs. 2.22) and 2.23) with espect to the coodinate with allowance fo elations 2.42), 2.43), 2.44) and 2.45) and 2.63) yields the following esult [163]: d { [ δω) 2 1 β) 2 κ A 1 α + A 2 κ) B 1 α 2 + B 2 ακ + B 3 κ 2)]} + δω 2 1 β) d Nω) [κ A 1 α + A 2 κ)] + ρδ ω) 2 C 1 + C 2 β + C 3 β 2) = τ w /ρ, 2.73) d { δω 2 4 1 β) [αd 1 + βd 2 ) + κ D 3 + βd 4 )] } ω) 2 β d [ δω 2 1 β)a 1 α + A 2 κ) ] = 2 τ wϕ /ρ, 2.74) whee A 1 = 1/n + 1) A 2 ; A 2 = 2/n + 2) 1/n + 3); B 1 = 1/2n + 1) 2/n + 1) + 6/2n + 3) 2/n + 2) + 1 / 2n + 5); B 2 = 2/n + 1) 1 / 2n + 3) + 4/n + 2) 2/2n + 5); D1 = A1 D2 ; C 1 = 2/n+1) C 3, C 2 = 2 1/2n + 1) 1/n + 1)); C 3 = 1+1 / 2n + 1); D1 = A1 D2 ; D2 = 1/2n+1) D4 ; D3 = A2 D4 ; D4 = 1/n+1) 1/2n+3). The themal bounday laye equation 2.25), being integated with espect to the coodinate with allowance fo Eqs. 2.42), 2.43), 2.45), 2.58) and 2.63), can be educed to the following fom [168]: d [ δω 2 1 β) F 1 T T w ) ] + dt δω 2 1 β) F 2 = = StV n T P n, 2.75) P T T w ) whee F 1 = E 1, F 2 = E 2 at 1; F 1 = E 3, F 2 = E 4 at 1; E 1 = n+1 aa T + bb T + cc T 2 ), a T = 1 / 1 + n + n T ) 1 / 1 + n), b T = 1 / 2 + n + n T ) 1 / 2 + n), c T = 1 / 3 + n + n T ) 1 / 3 + n), E 2 = n+1 [a / n + 1) + b / n + 2) + c 2/ n + 3)], E 3 = E 5 + κe 6,

28 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems E 4 = αa 1 + κ 1) + κa 2, E 5 = α A 1 + n T D 2T ), D 2T = 1 / 1 + n + n T ) D 4T, E 6 = n T ) / n T + 1) + 1 A 2 + n T D 4T, D 4T = 2 / 2 + n + n T ) 1 / 3 + n + n T ). Mass flowate in the bounday laye can be calculated fom the following elation: ṁ d /ρω 3 ) = 2π1 β)a 1 α + A 2 κ)δ/. 2.76) As applied to the poblems unde consideation, Eqs. 2.73), 2.74) and 2.75) contain thee unknowns: a) in the entaining bounday layes: α, δ and fo given β and T ; b) in the Ekman-type bounday layes: α, β fo a given distibution of δ i.e. fo a constant mass flowate ṁ d ) and T fo a given constant value. In the fome case fo paticula bounday conditions 2.32), 2.33), 2.34), 2.35) and 2.36) and N=const), Eqs. 2.73), 2.74) and 2.75) can be solved analytically at constant values α and and powe law of adial vaiation of the bounday laye thickness δ. Fo abitay bounday conditions, Eqs. 2.73), 2.74) and 2.75) ae solved numeically; fo the sake of this, they ae educed to a fom convenient fo integation using the Runge-Kutte method [164, 169]: { α = 1 4 + 2 ) / 1 1 3 ), δ = 2 3 + 4 ) / 1 1 3 ), 2.77) = S 1 S 2 S 3 ) / S 4. 2.78) Hee 2 ={[sgn1 β) / / cf 2 3 Re 2 V / δ 2 Z 1 δ G 1 δ G 2 ]/ δ ) Q 2 2 } Q 1 ; 4 ={ sgn1 β) / cf ϕ 2 2 Re 2 V / δ 2 δ[αq 3 + Q 4 + βre ω ) αq 5 + Q 6 )]/Q 7 }; 1 = Z 1 / δq 1 ); 3 = δq 3 /Q 7 ; G 1 = Re 2 ω C 1 + C 2 β + C 3 β 2 ); Z 1 = Re 2 ω 1 β)2 [ B 1 α 2 + ακa 1 B 2 ) + κ 2 A 2 B 3 )]; G 2 = Re 2 ω 1 β) δ[ A 1 α + κ1 A 2 )]v, ; Q 1 = Re 2 ω 1 β)2 [ 2αB 1 + κa 1 B 2 )]; Q 2 = Re 2 ωi { α2 B 1 [ 2 1 β) 2 ] + αa 1 B 2 )[ 1 β), ] + A 2 B 3 )v 2, ) }; Q 3 = Re 2 ω 1 β)2 D 1 ; Q 4 = Re 2 ω 1 β), D 3 / ; Q 5 = Re ω 1 β)a 1 ; Q 6 = Re ωi, A 2 ; / Q 7 = αq 3 + Q 4 ;, =, ωa); Reωi = ωi 2 / / ν; cf 2 / / = / c f 2)α / 1 / + α 2 ) 1 /2 ; cf ϕ 2 = cf 2) 1 + α 2 ) 1 /2 δ = δ/ i ; = / i.

2.5 Geneal Solution fo the Cases of Disk Rotation in a Fluid Rotating 29 In Eq. 2.78), one has at 1: S 1 = Re ω 1 β 1 + α 2 ) 1 /2 St T T w ); S 2 = T δre ω n+1 [α1 β)/n + 1) 2 α1 β) N)/n + 2) + 2 α1 β) N)/n + 3)]; S 3 = n+1 L 1 + n+2 L 2 + n+3 L 3; S 4 = L 1 n + 1) n + L 2 n + 2) n+1 + L 3 n + 3) n+2 ; L 1 = L a T α1 β); L 2 = L b T 2)[α1 β) N]; L 3 = L c T [α1 β) N]; L = δre ω T T w ). The value of S 1 is the same fo both 1 and 1. At 1: S 2 = T δre ω [α 1 β) A 1 + NA 2 + N 1)]; S 3 = L 1 C6T + L 2 C7T ; S 4 = n T n T 1 D 2T L 1 + [1 + n T n T 1 )/n T + 1) D 4T n T n T 1 1]L 2 ; L 1 = L 1 β) α; L 2 = L N; C6T = A 1 + n T D 2T ; C7T = n T )/n T + 1) + 1 A 2 + D 4T n T ; T = T / T ef. The pimes denote hee deivatives with espect to the adial coodinate d / d ; i is a chaacteistic adius vey often, the inlet adius). In the Ekman-type layes α = c f α 2 αβ 1)Re ω1 + α 2 ) 1 /2 4πA 1 i { β = c f 2 1 β)2 Re ω 1 + α 2 ) 1 /2 4πA 1 i D 1 C w b 2 d T d = [ St V ν 2π i.5c w b [ β β 1)αB 1 ) ]}/ ᾱ, 1 A 1 D 1 1 B 1 C w b + dβ d β 1 C 3[β+n/n+1)] ] 1 ) d T w T T w + K H d 1 A 1 D 1 ), 2.79) K H K H 1. 2.8) Accoding to the ecommendations of [138, 139], paamete K H is consideed to be constant in the Ekman-type layes [164, 167, 17] K H = 1 D 2T / A1 ) n T = const o = const. 2.81) 2.5 Geneal Solution fo the Cases of Disk Rotation in a Fluid Rotating as a Solid Body and Simultaneous Acceleating Imposed Radial Flow Let us conside the case whee β=const, N=const and κ>. The condition β=const means solid-body otation, which takes place between otos and statos; the condition N=const means acceleating adial flow that exists in the stagnation egion

3 2 Modelling of Fluid Flow and Heat Tansfe in Rotating-Disk Systems of fluid flow impinging on an othogonal suface; the condition κ> means that eciculation flow does not emege on a otating disk [138, 139, 196]. Unde these conditions, the system of Eqs. 2.73) and 2.74) can be solved analytically in the most geneal fom [174, 18, 189]: δ = C δ m, C δ = γ ω/ν) 2n/3n+1), δ / = γ Re 2n/3n+1) ω, 2.82) α = const, m = 1 n)/3n + 1), 2.83) γ = γ 1 β 1 n)/3n+1), 2.84) C M = ε M Re 2n/3n+1) ϕ, 2.85) ṁ d /μ) = ε m Re n+1)/3n+1) ω, 2.86) c f /2 = A c Re 2n/3n+1) ω, 2.87) α = H 2 /2H 3 + [H 2 /2H 3 ) 2 H 1 /H 3 ] 1 /2, 2.88) γ = Cn 2/3n+1) 1 + α 2 ).51 n)/3n+1) H n+1)/3n+1) 9, 2.89) ε m = ε m 1 β 2n+1)/3n+1), ε m = 2πγA 1α + A 2 κ)sgn1 β), 2.9) 8π ε M = 5 4n/3n + 1) C n 2 n+1 n+1 γ 2n 1 β 2n 1) 3n+1 1 + α 2 ) 2n+1) 1 n sgn1 β), 2.91) A c = C 2/n+1) n γ 2n/n+1) 1 + α 2 ) n/n+1) β 1 2n/n+1), 2.92) whee H 1 = C 3 β C 5 ) + β 1)κ 2 H 4 ; H 2 = κβh 5 + H 6 ); H 3 = βh 7 + H 8 ; H 4 = 1+2+m)A 2 3+m)B 3 ; H 5 = A 1 2+m) B 2 3+m)+D 4 m+4) A 2 2+m); H 6 = A 1 2+m)+B 2 3+m)+D 3 4+m); / H 7 = 3+m)B 1 +4+m)D 2 2+m)A 1 ; H 8 = 3 + m)b 1 + 4 + m)d 1 ; C 5 = C 1 C3 ; H 9 = α[d 1 + βd 2 )4 + m) β2 + m)a 1 ] + κ[d 3 + βd 4 )4 + m) βa 2 2 + m)]. An analytical solution of Eq. 2.75) is possible only unde assumptions of =const, P=const, n=n T and bounday conditions 2.34), 2.35) and 2.36). It is evident that in this case D 2T =D 2 and D 4T =D 4. Substituting Eqs. 2.36), 2.72), 2.82) and 2.83) into Eq. 2.75) and solving it jointly with Eq. 2.74), one can obtain [168] that [ F 1 2 + m + n ) + βn ] β 1 F 2 n P n p = 4 + m)c 4 + 2β β 1 C 5. 2.93) The elations fo the functions F 1 and F 2 ae given in explanations to Eq. 2.75), while C 4 = αd 1 +κd 2 ), C 5 = 1/n+1)+1/n+2)+1/n+3). Equation 2.93) has diffeent solutions fo the cases 1 and 1 which is eflected by the diffeent foms of the functions F 1 and F 2 fo 1 and 1). Case 1 coesponds to

2.5 Geneal Solution fo the Cases of Disk Rotation in a Fluid Rotating 31 heat tansfe in gases at P 1. Case 1 coesponds, in geneal, to heat tansfe in liquids at P 1; moe details elevant to the case whee 1 can be found in Chap. 8. Solution 2.93) in its geneal fom fo simultaneously non-eo values of β and N is a tanscendental algebaic equation. A solution fo the paamete in an explicit fom can be obtained at N= and 1. The exponent n, which is consideed univesally the same fo all types of fluid flow, will be found below on analysis of the fee otating disk case. Expessions fo the Nusselt and Stanton numbes have the following fom: St = A c Re 2n/3n+1) ω n P n p, 2.94) Nu = A c 1 + α 2 ) 1/2 1 β Re n+1)/3n+1) ω n P 1 n p. 2.95) Validations of the pesent integal method ae pefomed in Chaps. 3, 5 and 6 fo ai flows. Extension of the pesent integal method onto the lamina flow case is pefomed in Chaps. 3 and 5. Chapte 3 epesents esults fo a fee otating disk β=, N=). Analysed in Chap. 5 ae the cases of a disk otating in a fluid that itself otates as a solid body β=const, N=) and a disk otating in a unifomly acceleating non-otating fluid β=, N=const). Chapte 6 epesents esults fo tubulent thoughflow between paallel co-otating disks. In Chap. 8, the method is validated fo cases of Pandtl o Schmidt numbes lage than unity.

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