Multiphase Science and Technology, Vol. 19, No. 1, pp. 1 48, 2007 A MATHEMATICAL MODEL FOR TWO-PHASE STRATIFIED TURBULENT DUCT FLOW

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1 Multiphase Science and Technology, Vol. 9, No., pp. 48, 27 A MATHEMATICAL MODEL FOR TWO-PHASE STRATIFIED TURBULENT DUCT FLOW Dag Biberg Scandpower Petroleum Technology, P.O.Box 3, Gåsevikvn. 2-4 N-227 Kjeller, Norway Abstract. An algebraic-logarithmic model for two-phase stratified turbulent channel flow is presented. The various aspects of the model compare favorably with a wide range of data from the literature. The model is characterized by its ability to reproduce the effect of interfacial waves and momentum transfer, associated with large pressure drop increases in stratified flows. Its analytic nature allows for the derivation of relatively simple preintegrated integrated analytically prior to run time) expressions for the mean wall and interfacial friction. The corresponding pipe flow expressions, ideally suited for -D computer models, are obtained by hydraulic similarity. The pipe flow friction formulae open up for the simulation of large pipeline systems with the consistency and accuracy of a cross sectional description, while maintaining the evaluation speed of a -D model. The work described in this article was motivated by the multiphase flow challenges met in the development of gas-condensate fields like Ormen Lange and Snøhvit on the Norwegian continental shelf.. INTRODUCTION Stratified flow is a frequently encountered flow regime in multiphase pipeline transport of oil and gas. Estimating pressure drop and liquid inventory is important for optimal Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim

2 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 2 BIBERG design and operation. Pipelines are normally simulated using -D models to keep reasonable simulation times. Wall and interfacial friction constitute the basic closure issues, mostly represented by simple nonrelated heuristic models or correlations. These phenomena are, however, in reality interrelated via the velocity distribution. Consistent modeling, yielding proper extrapolation properties, does thus require a cross-sectional flow description as a minimum. Engineering models, however, must be efficient. Look-ahead predictions for online applications are, e.g., required to run several times faster than real time to be meaningful. Design studies for large pipeline systems also require a certain evaluation speed to be practical. The simulation times associated with conventional multi-dimensinal CFD- models does thus preclude their use for all but the simplest cases at least for the time being. The modeling dilemma of speed versus consistency may to some degree be resolved by use of an analytical description for a cross section of the flow that may be integrated prior to run time preintegrated) without loss of generality, yielding a consistent set of interrelated near-algebraic friction models perfectly suited for -D multiphase pipe flow simulators. This approach opens up the possibility for simulating large pipeline systems with the consistency and accuracy of a cross-sectional model, while maintaining the computer speed of a traditional -D model. The flow of either fluid in a nondispersed stratified flow is essentially a single-phase wall-bounded flow subject to certain boundary conditions at the interface, representing the presence of the other fluid. Fairly simple models do thus sufficiently accurately describe the flow within either fluid for given interfacial conditions. The essential closure issue is related to the description of the complex interaction of the fluids at the interface, yielding the correct boundary values for the single-phase) flow within either fluid. No sufficiently general solution to this problem seems to be available at present. Contemporary 3-D CFD models will thus not in general without tuning) yield correct flow predictions. Obtaining more general solutions for the interfacial closure issue is the most pressing obstacle toward obtaining more reliable models. The transparent nature of the analytic model presented here may be helpful for focusing on this problem. A flat and smooth interface may be represented as a solid wall moving with interfacial velocity, yielding a particularly simple solution to the interfacial closure issue in this case. Wavy interfaces associated with large increases in frictional pressure drop will, however, always be present beyond a certain flow rate combination. This phenomenon must thus be properly accounted for to obtain correct predictions in the general case. Any model, however sophisticated, that does not properly account for this effect will be in error at the outset. A correct description of the interfacial wave phenomenon may thus be regarded as the key to a successful model. The main objective of the present work has been to obtain a consistent set of friction formulae, suitable for use in a -D pipe flow simulator, from a crosssectional flow description. Accounting for the effect of interfacial waves and momentum transfer, associated with large pressure drop increases, has been regarded as particularly important. Experimental studies of interfacial waves in stratified channel and pipe flows have been conducted by, e.g., Hanratty and Engen 957), Ellis and Gay 959), Akai et al. 98, 98), Strand 993), and Espedal 998). Wu 968, 98) studied

3 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 3 the closely related phenomenon of ocean waves. An interesting observation regarding wind stress on a water surface is given by Charnock 955). One-dimensional closure relations for wall and interfacial friction, some of which account for the effect of interfacial waves, have been considered by a large number of workers; see, e.g., Espedal 998) for a comprehensive review. Cross-sectional models considering the effect of interfacial waves have been developed by, e.g., Ellis and Gay 959), Akai et al. 98), Shoham and Taitel 984), Issa 988), Benkirane et al. 99), and Meknassi et al. 2). The roll wave phenomenon occurring in the transition to slug flow, as considered by, e.g., Watson 989), is not covered by the present analysis. This is also the case for the effect of secondary flows, as studied by, e.g., Suzanne 985). The present model is based on an algebraic expression for the eddy viscosity obtained from classic mixing length theory, as developed by Prandtl 925) and von Kármán 93). The more recent nonconstant stress layer effect considered by Glowacki and Chi 972), Reeves 974), and Galbraith et al. 977) is also accounted for. Channel flow geometry is utilized for simplicity. Viscous effects are neglected. The corresponding logarithmic velocity distribution yields wall friction laws that may be written in the form of the classic Prandtl/Kármán pipe flow formulae based on an effective flow diameter. The no-slip condition at the interface yields the corresponding interfacial shear stress. Pipe flow expressions are obtained by use of hydraulic similarity. The following sections contain the development of the channel flow model, from which the pipe flow friction formulae are obtained in Section 9. Holdup and pressure drop computations are discussed in Section 9.3. The corresponding computational algorithm is given in Appendix D. Data comparisons and closure relations are considered in the discussion in Section. 2. CHANNEL FLOW A definition sketch for the two-phase stratified channel flow to be considered here is given in Fig.. A less dense fluid f = g is flowing on top of a more dense fluid f = l. The flow rates, fluid depths, densities, and viscosities are Q f, h f, ρ fr, and µ fr respectively. The channel is inclined at an angle θ to the horizontal. The flow is referred to a Cartesian coordinate system x, y), in which the x-axis is parallel to the channel walls at y = ±h f, and y = defines the mean position of the fluid-fluid interface i. Corresponding equations for the fluids f = g or l will be written as a single expression for fluid f. Opposite signs in corresponding terms will be accounted for by use of the ± or operators, in which the top and bottom signs are to be associated with the fluids g and lr respectively reflecting their vertical position in the pipe). The fluids are considered to be in fully developed turbulent steady state motion. The time-averaged velocities in the x- and y-directions are u f = u f y) and v f, respectively. Variables u f and v f denote the corresponding turbulent fluctuations. The streamwise x-component) of the Reynolds equation is thus given by dτ f,xy dy = dp dx + ρ f g sin θ )

4 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 4 BIBERG y wg S wg y A g g i x S i l wl S wl A l g Figure left) Stratified channel or pipe flow; right) pipe flow front view. in which τ f,xy = µ f du f dy ρ f u f v f 2) is the apparent shear stress in the x-direction for a given y. The boundary conditions are no slip at the channel walls u f y = ±h f ) = 3) no slip at the interface and continuous shear stress across the interface u g y = ) = u l y = ) 4) τ g,xy y = ) = τ l,xy y = ) τ i 5) in which τ i denotes the interfacial shear stress. Integrating Eq. ) yields the shear stress distribution ) dp τ f,xy = dx + ρ f g sin θ y + τ i 6) after application of Eq. 5). Evaluating Eq. 6) at the channel walls at y = ±h f yields the fluid momentum balances given by ) dp h f dx + ρ f g sin θ ± τ i + τ fw = 7) in which τ fw = τ f,xy y = ±h f ) denotes the wall shear stress. Combining Eqs. 6) and 7) yields the shear stress distribution in terms of wall and interfacial shear stress τ f,xy = τ i τ fw ± τ i ) y h f 8)

5 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 5 The two-phase stratified flow is essentially a coupling of two single-phase wallbounded flows through the interfacial boundary conditions Eqs. 4) and 5). The symmetric nature of the problem allows us to represent both flows using a single generic model. This is clear by the fact that the equations for fluid l become identical to the equations for fluid g by reversing the positive direction of the vertical coordinate y and interfacial shear stress τ i, i.e., by letting which also implies that y y τ i τ i 9) τ l,xy τ l,xy v l v l ) since these variables are defined to be positive in the positive y-direction. We solve the equations for fluid g, which we chose to represent the generic flow model. The corresponding solution for fluid l is then obtained by switching the fluid indices g l and introducing the transformation Eqs. 9) and ). The fluid indices f = g or l are required to distinguish between the fluids in the two-phase flow. They are, however, not needed when considering the generic single-phase) flow model in the following sections and will thus be suppressed in this case. 3. EDD VISCOSIT AND MIXING LENGTH In this section, we consider the basic eddy viscosity and mixing length relations applied in the development of the generic flow model, representing the flow in either fluid. The eddy viscosity µ t is defined by the Boussinesq assumption, given by ρu v µ t du dy We neglect the viscous term in Eq. 2). The model will thus not be valid in the thin viscous regions occurring at the walls and possibly at the interface. Combining Eqs. 2) and ) now yields τ xy = µ t du 2) dy suppress fluid index f). Prandtl s mixing length hypothesis states that ) µ t = ρlv t 3) in which l is the mixing length and v t a characteristic turbulent velocity scale. Prandtl assumed that v t = l du dy 4) This model does not apply for du/dy =, for which it yields µ t = in combination with Eq. 3). Prandtl accounted for this fact by introducing a more complicated expression based on the statistical mean of du/dy in the vicinity of such points see,

6 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 6 BIBERG e.g., Schlichting 955). Disregarding situations in which du/dy = and combining Eqs. 2), 3), and 4), however, yields τ xy = l du ρ dy 5) v t τ xy = 6) ρ The logarithmic velocity distribution for wall bounded flows constitutes a cornerstone in turbulence theory. We have u = sgn τ w ) u κ ln y ) + C 7) h near the wall at y = h. The sgnτ w ) function accounts for the possibility of a flow reversal near the wall; u = τ w /ρ is the wall friction velocity, and C = sgn τ w ) u κ ) ln u h ν + κb ) h ln k s + κa 8) for hydraulic smooth and rough flows, respectively. Variable k s is the equivalent sand roughness. The constants κ, B 5.5, and A 8.5 were originally determined by Nikuradse s 932) pipe flow experiments. Prandtl derived the logarithmic velocity distribution Eq. 7) from the mixing length hypothesis using Eq. 5), assuming the mixing length to be proportional to the distance from the wall and the shear stress to be constant and equal to the wall shear stress, i.e., l = κh y) and τ xy τ w. It is, however, an experimental fact that the logarithmic velocity distribution may prevail in nonconstant stress layers at wall distances well beyond the point where these assumptions are valid. The logarithmic velocity distribution was later established without reference to mixing length by Millikan 939) and on an even more general basis by Rotta 962). It may therefore be argued that the logarithmic velocity distribution is of a more universal nature than the mixing length hypothesis. It does thus make more sense to determine the mixing length from the logarithmic velocity distribution, than to determine the logarithmic velocity distribution from the mixing length. Combining Eqs. 7) and 5) yields l = κh y h ) τ xy τ w 9) which reduces to l = κh y) for τ xy τ w see Glowacki and Chi 972, Reeves 974, Galbraith et al. 977, White 99). Equation 9) may be perceived as the mixing length representation of the logarithmic velocity distribution Eq. 7).

7 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 7 In some cases, the interface may be represented as a solid wall moving with interfacial velocity. In this case, we also have a logarithmic velocity distribution near the interface at y =, given by u u i = sgn τ i ) u i y ) κ ln + C i 2) h in which u i is the interfacial velocity; u i = τ i /ρ is the corresponding friction velocity, and ) C i = sgn τ i ) u i ln u i h ν + κb ) 2) κ h ln k i + κa for hydraulic smooth and rough flows, respectively. Variable k i is the equivalent sand roughness of the interface. Modeling k i, or more precisely, a length scale proportional to k i, has been considered by, e.g., Charnock 955) and Wu 968, 98) in the study of the wind stress on an ocean surface. Modeling interfacial waves using an equivalent sand roughness k i will not be considered in the present approach. In a situation in which the velocity distribution at the interface follows Eq. 2), we also have τ xy l = κy τ i 22) by the same argument leading to Eq. 9). Finally, we note that Reichardt 959) found that the eddy viscosity in the Couette flow case, in which τ xy = constant = τ w = τ i, may be represented by µ t ρu h = κ y y ) 23) h h The corresponding mixing length is thus given by l = κy y ) 24) h since v t = τ w /ρ = u in this case see Eqs. 3) and 6)). 4. AN ALGEBRAIC EDD VISCOSIT MODEL We apply the dimensionless variables = y h R = τ i τ w 25) The flow is now confined in and subject to an interfacial drag represented by R. Note that R =,, and in Poiseuille flow, free surface flow, and Couette flow, respectively. We apply Prandtl s mixing length hypothesis Eq. 3) to model the eddy viscosity. Considering Eqs. 9) and 22) leads us to propose a mixing length model of the form

8 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 8 BIBERG l = LF 26) in which F represents the effect of the nonconstant shear stress distribution, i.e., τ xy F ) = τ xy τ w F ) = τ i 27) near the wall and interface, respectively. We thus also have F = at the wall and interface. L should agree with Eq. 24) in the Couette flow case, where τ xy = constant = τ w = τ i, yielding F and L l. This leads us to consider the parabolic representation L = a + a + a ) for the general case. The wall condition Eq. 9) implies that L ) = κh ) yielding L = and dl/d = κh at the wall =. At the interface, we simply require the mixing length model Eq. 26) to yield l = ) = l i 29) in which l i is the value of the mixing length at the interface, representing the corresponding turbulence level. We thus also have L = l i for =, for which F =. Determining the constants a,..., a 2 in Eq. 28) by the conditions considered above yields L = κh ) + K )) 3) in which the dimensionless interfacial turbulence parameter K = l i κh represents l i. The mixing length model Eq. 26) as given by the combination of Eqs. 27) and 3) now agrees with the Couette flow model Eq. 24) in the Couette flow case for which K = and F. We also have l ) = κh ) τ xy τ w + O ) 2) near the wall at = in agreement with the wall condition Eq. 9) corresponding to the logarithmic velocity distribution Eq. 7). Zero interfacial turbulence, i.e., K =, yields l ) = κh τ xy τ i 3) + O 2) 32) near the interface at =, in agreement with the interface condition Eq. 22) corresponding to the logarithmic velocity distribution Eq. 2). The interface is thus modeled as if it were a solid wall moving with interfacial velocity for zero interfacial turbulence. Nonzero interfacial turbulence K > may be associated with agitated wavy interfaces. The mixing length model Eq. 26) simply yields l = l i at the interface = in this case. The interfacial mixing length l i is the central closure parameter in the

9 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 9 present approach. Closure relations are considered in Section. Combining Eqs. 3) and 26) now yields in which µ t = ρlv t 33) V t Fv t 34) Introducing Prandtl s model Eq. 6) for v t and F from Eqs. 27) yields V t ) = τ xy u ρ Vt ) = τ xy u i ρ 35) near the wall and interface, respectively, in which τ xy is given by Eqs. 8), yielding in terms of u and R. We apply the interpolation τ xy = sgn τ w ) ρu 2 R ) ) 36) V t = b + b 2 + b b b 5 + b ) to obtain a complete representation for V t and determine the constants b,..., b 6 by the value of V t and its first and second derivatives at the wall and interface, given by V t = = u V t = = u i V ț V ț = = u + R) V ț = = = = u i + /R) Vț = = by application of Eq. 35) and u i = R u. Solving the equations for the constants yields u 3 + R 5/2 ) 3) V t = R 2 ) 2 38) + R ) + 2 Figure 2 shows the interpolation Eq. 38) for two typical R-values. It is seen to follow closely the continuation of the conditions Eqs. 35) at the wall and interface, as represented by the dashed lines. The eddy viscosity is now given by the combination of Eqs. 3), 33), and 38), yielding µ t κ ) + K )) ρhu = 3 + R 5/2 ) 3) R 2 ) 2 + R ) )

10 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim BIBERG V t u V t u R R Figure 2 Interpolation V t Eq. 38) for R = 2 and R = -.5; dashed lines are continuation of conditions Eq. 35). Note that Eq. 38) reduces to V tp u = 3 )) )) 4) V tf u = 4) V tc u = 42) in the Poiseuille, free surface, and Couette flow cases, respectively. The corresponding eddy viscosity distributions are obtained by combination with Eqs. 3) and 33), noting that K = in the Poiseuille and Couette flow cases, which both have a logarithmic velocity distribution at the interface, while K = K F > in the free surface flow case, which does not. The eddy viscosity Eq. 39) is thus given by

11 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim µ tp ρhu = TWO-PHASE DUCT FLOW κ ) 3 )) )) 43) µ tf ρhu = κ ) + K F ) ) 44) µ tc ρhu = κ ) 45) in the Poiseuille, free surface and Couette flow cases, respectively. We obtain a reasonable estimate for K F by considering free surface flow to be analogous to the flow between one wall and the symmetry plane of a Poiseuille flow. The mixing length l should in this case be zero at the opposite wall positioned at =, yielding K F =.5 by application of Eq. 3). Further discussions and data comparisons are given in Section. 5. VELOCIT DISTRIBUTION The velocity distribution u is determined by Eq. 2), i.e., u = τxy Introducing Eqs. 36) and 39) and integrating yields dy 46) µ t u = sgn τ w ) u κ + C 47) in which K 3 + R 3) ln + K )) = ln ) + 48) R 5/2 K 3 R + ) 3 R R ln + R 5/6 ) ) + 3 K R 5/6) R + ) K R ) + 2 R 5/6 3 R ln 2 ) ) R 5/6) R 5/6) 6 K 2 + R 5/6 K + R 5/3) K R + ) 3 R R 2 ) R 5/3 + 2 ) R 5/6 ) K2 + R 5/6 K + R 5/3) tan 3 R 5/6 see Appendix A for details). The velocity distribution Eq. 47) is not valid at the wall =. The corresponding no-slip condition Eq. 3) does thus not apply. The integration constant C may, however, be determined by matching with the logarithmic velocity distribution near the wall, Eq. 7), i.e., by the condition that u ) = sgn τ w ) u κ ln ) + C 49)

12 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 2 BIBERG yielding C = C + sgn τ w ) u κ Ψ 5) where C is given by Eq. 8) and Ψ is given by K R + ) ) 3 R R + 2 Ψ = 3 K2 + R 5/6 K + R 5/3) tan R 5/6 5) 3 It may be demonstrated that ΨR ) = and ΨK ) = 52) Taking the limit of Eq. 47) for K thus yields u A = sgn τ w ) u κ ln ) + sgn R) R ln 53) 3 + sgn R) R ) ln where C = C, as given by Eq. 8). We also find that 3 + ) 3 R 5/2)) + C u P = sgn τ w ) u κ u F = sgn τ w ) u κ ln ) + ln 2 3 ln 3 + ) 3)) + C 54) ln ) ln + K F ) )) + C 55) u C = sgn τ w ) u κ ln ) ln ) + C 56) for Poiseuille, free surface, and Couette flow, respectively. Data comparisons and discussions are given in Section. The mean velocity U is given by 6. MEAN VELOCIT U = h h udy 57) Introducing the velocity distribution u from Eq. 47) and integrating yields in which U = sgn τ w ) u κ Λ + C 58)

13 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 3 Λ = 5 R + R R ) 5/2 + K 2 + K ) R 2 ln R ) 6 K 3 R 5/2) R 5/2 ) K K 3 + R 3) lnk) K ) K 3 R 5/2) R + ) ) 3 R R R 5/3 + R 5/6 + ) K 2 + R 5/6 K + R 5/3) K R 5/3) 2 tan R 5/6 3 K + K + ) R 5/6) 2 R R 5/6 tan 5/6 ) ) + 59) 3 see Appendix A for details). It may be demonstrated that Λ, as given by Eq. 59), reduces to Λ A = R + ) R R /3 2 3π R 5/6 ) 3 R 5/3 ln R 5/2)) 9 R 5/2 ) 6) for K =. We also have Λ P = π + 9) 6) Λ F = KF ln K F K F 62) Λ C = 63) for Poiseuille, free surface, and Couette flow, respectively. The corresponding mean velocities are thus given by 2 ) π + 9) + C 64) U P = sgn τ w ) u κ U F = sgn τ w ) u κ K F ln K F K F ) + C 65) U C = C 66) since C = C, as given by Eq. 8), for R = and K = see Eqs. 5) and 52)). 7. WALL FRICTION The expression for the mean velocity Eq. 58) relates the mean velocity U to the wall friction τ w through u ) for given R, K, h, ν, and k s. This expression may thus be perceived as a wall friction law. A convenient nondimensional form is obtained by introducing Eqs. 8) and 5) and multiplying with h/ν, yielding

14 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 4 BIBERG ln Re h ) + κb Re h = Re h ) h + Ψ + Λ 67) κ ln + κa k s for hydraulic smooth and rough flows respectively. Note that Ψ and Λ, as given by Eqs. 5) and 59), depend on R and K only. This expression relates the Reynolds number based on the fluid depth Re h = Uh/ν to the corresponding wall friction Reynolds number Re h = sgnτ w )u h/ν. The Reynolds numbers as defined here) may be negative or positive depending on sgnu) and sgnτ w ), respectively. We do, however, have sgnτ w ) = sgnu) in the parameter range to be associated with turbulent flow, see Appendix B). We are interested in relating the wall friction to the classic Prandtl/Kármán friction laws for hydraulic smooth and rough pipes, given by ) = 2 log Re D λ.8 68) λ ) D = 2 log ) λ 2k s respectively, in which D is the pipe inner diameter and Re D = U D/ν. The friction factor λ is related to the wall shear stress τ w by τ w = λ 4 ρ U U 2 The Colebrook-White interpolation formula, given by 2.5 = 2 log + k ) s λ Re D λ 3.7D reduces to Eqs. 68) and 69) for smooth and rough pipes, respectively, and agrees with commercial pipe friction data in the smooth to rough transition. Note that 2 log 2.5).8 and 2 log /2 3.7)).74.) An accurate explicit approximation to Eq. 7) is given in Appendix E. We have u = λ U /2 2. Combining Eqs. 58) and 7) and introducing the effective flow diameter D e, defined by thus yields = λ 2 2κ ln ln 7) 7) D e C De h 72) U D e λ ν De k s C De 2 2C De ) + κa ) + κb + Λ + Ψ since U/ U = sgnu) = sgnτ w ). Introducing base- logarithms and ln ) / 2 2κ ) , as in the original development of Eqs. 68) and 69) see, e.g., Schlichting 955), now yields 73)

15 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 5 = λ in which Re De = U D e /ν and ) 2 log Re De λ.8 ) log De 2k s + ε 74) where K = ε = 2 2κ K + Λ + Ψ ln C De ) 75) { κb ln 2 2 ) κ ) κa + ln 2) κ ) } { } 2.9 for smooth and rough flows. Comparing Eq. 74) with Eqs. 68) and 69), we find that pipe flow similarity is obtained by equating ε Eq. 75)) to zero, yielding C De = expk + Λ + Ψ) 76) We may replace Eq. 74) by the Colebrook Eq. 7). The wall shear stress τ w for a given mean velocity U, shear ratio R, and interfacial turbulence level K is now thus given by application of Eq. 7), in which the friction factor λ is computed from Colebrook s Eq. 7) after replacing the pipe diameter D with the effective flow diameter D e, as given by Eqs. 72) and 76). We have Ψ = for Poiseuille, free surface, and Couette flow. The corresponding Λ- values are given by Eqs. 6) 63). The C De values for these flows are thus particularly easy to compute. Applying the estimated K F =.5 for free surface flow yields the values in Table for smooth and rough flows. The corresponding effective diameters as defined by Eq. 72) are thus given by D P e 2h D F e 4h D C e 8h 77) The effective diameters for Poiseuille and free surface flow D P e and D F e agree with the corresponding hydraulic diameters, defined by D h = 4 Flow area Wetted perimeter The hydraulic diameter concept does not, however, apply for Couette flow and may thus not be used to check D C e. 78) 8. INTERFACIAL FRICTION We now consider the interaction of the fluids at the interface and thus reintroduce the fluid indices f = g and l. The interfacial shear stress is determined by the noslip condition u g = u l ) at the interface Eq. 4). The velocity distribution in either fluid u f as represented by Eq. 47) is, however, singular at the interface f =

16 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 6 BIBERG Table Effective flow diameters scaled by fluid depth. C De Poiseuille Free surface Couette Smooth Rough for an hydraulic smooth interface, i.e., for zero interfacial turbulence K f =. This problem may be solved by evaluating the interfacial velocity in either fluid at the onset f = f s = ν f exp κb)/u if h f ) of the velocity distribution in the immediate vicinity of the interface, where u f u i = in Eq. 2). We solve this problem, however, by introducing the small nonzero K f = Kf min, given by K min f = ν f exp κb) u if h f i.e., Kf min = f s. Applying K f = Kf min moves the onset of the velocity distribution to the interface at f = for an hydraulic smooth interface see Appendix C). The difference in the interfacial velocity obtained at f = with K f = Kf min to the value obtained at f = f s with K f = is negligible. We associate Kf min with a smooth interface and apply K f Kf min as a lower limit for general interfaces to avoid singularity in the velocity distribution. This allows us to evaluate the no-slip condition Eq. 4) at the interface f =. Replacing all occurrences of the wall shear stress τ wf in the resulting expression using the momentum balances Eq. 7) on the form ) dp τ wf = h f dx + ρ f g sin θ τ i 8) yields, formally, F 79) τ i ; dp ) dx, ɛ l, K g, K l = 8) Note in particular that R f = ±τ i /τ wf, in which τ wf should be represented by Eq. 8). The no-slip condition Eq. 4) formulated as Eq. 8) does thus implicitly determine the interfacial shear stress τ i for a given holdup ɛ l and pressure drop dp/dx, in a given channel containing a given pair of fluids, assuming appropriate values have been assigned to the interfacial turbulence parameters K f. The analogous expression for laminar flow is explicit in terms of holdup and pressure drop see Biberg and Halvorsen 2). There do not, however, seem to be any explicit solutions for Eq. 8), which thus must be solved numerically. We obtain an alternative formulation of the no-slip condition Eq. 4) by eliminating the integration constant Eq. 5) representing the wall effect in the velocity distribution Eq. 47) using the expression for the mean velocity of Eq. 58), yielding U g U l = sgn U g ) u g κ Λ g ig ) sgn U l ) u l κ Λ l il ) 82)

17 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 7 in which if represents the value of f at the interface = and sgnτ fw ) has been replaced by sgnu f ). We use Eq. 82) to represent the no-slip condition in computations, see Appendix D). Some insight into the nature of the interfacial shear stress as given by the present model may be obtained by introducing u f = τ fw /ρ f = τ i / ρ f R f in Eq. 82) and solving for τ i, yielding κ 2 ρ g ρ l R g R l U g U l ) 2 τ i = sgn U g ) Λ g ig ) ρ l R l sgn U h ) Λ l il ) ) 2 83) ρ g R g This expression displays the perfect symmetry in structure of the interfacial shear stress with respect to fluids g and l. The actual magnitude of the contribution to the shear stress from either fluid does of course depend on the flow parameters. The interfacial shear stress is, e.g., mainly determined by the flow in the less dense fluid if the density difference is large, i.e., if ρ g /ρ l. The laminar channel and pipe flow expressions corresponding to Eq. 83) may be found in Biberg 24). 9. PIPE FLOW The channel flow friction formula, developed in the preceding sections, may be utilized to obtain the corresponding pipe flow expressions. Figure contains a definition sketch for stratified pipe flow. The pipe flow momentum balances are given by A f dp dx + ρ f g sin θ ) + τ wf S wf ± τ i S i = 84) We denote the pipe inner diameter and area by D and A, respectively. A given wetted angle, e.g., δ l, allows for an explicit computation of the central parameters defining the flow geometry. We have δ g = π δ l. The wall wetted and interfacial perimeters S wf and S i are thus given by S wf = Dδ f S i = D sin δ f 85) The phase fractions ɛ f are given by ɛ f = π δf 2 sin 2δ f ) 86) and interrelated by ɛ g + ɛ l =. The corresponding flow areas are given by A f = ɛ f A 87) We apply hydraulic diameters as defined by Eq. 78) as effective pipe flow diameters for the closed duct flow case R = Poiseuille flow) and the free surface flow case R =, i.e., Def P 4A f Def F 4A f 88) S wf + S i S wf

18 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 8 BIBERG The hydraulic diameters Eq. 88) are assumed to be very good approximations to the actual unknown) effective flow diameters. Smooth interface calculations using Colebrook s single-phase) friction formula Eq. 7) based on Def P and DF ef are very accurate see, e.g., Biberg 999c). Also, the difference between Def F and the corresponding effective diameter in laminar pipe flow is only a few percent, see Biberg 999a. This may also be demonstrated to be the case for Def P. The effective Couette flow diameter corresponding to the hydraulic diameters Eq. 88), obtained by the procedure described in the following section, is given by Def C 4A f + S ) i 89) S wf S wf The effective/hydraulic diameters Eqs. 88) and 89) reduce to the corresponding channel flow expressions as given by Eq. 77), in which S wf = S i = b and A f = bh f, where b denotes the infinite) channel width. 9. Wall friction in a pipe The channel flow wall friction model derived in Section 7 is given in terms of an effective flow diameter Eqs. 72) and 76)), i.e., D e = h expk + Λ + Ψ) 9) to be applied in the classic pipe flow friction laws Eqs. 68) and 69), as represented by the Colebrook formula Eq. 7). A reasonable generalization of these expressions to account for the pipe flow geometry should thus be obtained by determining the effective pipe flow diameter corresponding to Eq. 9). We start by dividing Eq. 9) by its free surface flow value, yielding D e De F = expλ + Ψ Λ F ) 9) since Ψ = and Λ = Λ F ; as given by Eq. 62) for free surface flow. We are now considering the generic flow model and thus suppress the fluid indices f, see Section 3.) We also have Ψ = and Λ = Λ P, as given by Eq. 6) for Poiseuille flow. The ratio of the Poiseuille and free surface flow values of the effective flow diameter Eq. 9) thus yields ) D P ln e De F = Λ P Λ F 92) Combining Eqs. 9) and 92) yields in which D e = D F e D P e D F e ) F R,K) 93) F R, K) Λ + Ψ ΛF Λ P Λ F 94)

19 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 9 We now let the effective flow diameter for stratified pipe flow be given by Eq. 93), after replacing the effective hydraulic) channel flow diameters D P e and D F e with the corresponding pipe flow expressions Eqs. 88), yielding finally D e = 4A f S w Sw S w + S i ) F R, K) 95) The fluid index f is retained in the flow area A f to avoid confusion with the pipe area A.) The corresponding wall friction is now given by introducing Eq. 95) into Colebrook s formula Eq. 7), i.e., by setting D = D e. Note that Eq. 95) yields D e = D P e and D e = D F e for Poiseuille and free surface flow, for which F R =, K = ) = and F R =, K = K F ) =, respectively. In Couette flow, we have F R =, K = ) = ΛF Λ P 96) ΛF for K F =.56 see Eqs. 52), 6), and 62)). The estimated K F =.5 yields F R =, K = ) =.889. The effective Couette flow diameter is thus given by Eq. 89). A physical interpretation of the wall friction law as given by Eqs. 7) and 95) may be obtained from the alternative formulation λ = Single-phase free surface flow Two-phase coupling { }} ){{ log + k }}{ s Re D F e λ 3.7De F 2 log + S i S w ) F R, K) 97) in which the first term equals Colebrook s formula Eq. 7) based on the hydraulic diameter for free surface flow De F Eq. 88)) and the second term represents the effect of the interfacial drag and turbulence, as represented by R and K. Note that /De P /De F ) = + S i /S w. The wall friction is thus given by a nonlinear combination of a single-phase free surface flow term and a two-phase coupling term, accounting for the presence of the other fluid. The 2 log + S i /S w ) factor in the two-phase coupling term accounts for the holdup dependency in the shape of the stratified pipe flow geometry. We have S i /S w. This term thus acts to reduce the impact of a given interfacial drag and turbulence on the wall friction, depending on the holdup. This effect is not present in channel flow in which S i /S w and thus 2 log + S i /S w ) = 2 log 2) = constant. The influence of the conditions at the interface on the wall friction is thus smaller in the pipe flow case. An analogous expression to Eq. 97) may be found in laminar flow see Biberg 24).

20 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 2 BIBERG 9.2 Interfacial friction in a pipe We obtain the pipe flow version of the interfacial friction law Eq. 8) by introducing the wall friction from the pipe flow momentum balances Eq. 84), i.e., τ wf = A ) f dp S wf dx + ρ S i f g sin θ τ i 98) S wf We also replace the channel flow fluid depths h f by the corresponding average value in a pipe cross section h f, given by h f = h f s) ds = A f 99) S i S i S i in which h f s) denotes the local depth at a given interfacial position s. The pipe flow interfacial turbulence parameters K f Eq. 3)) are thus given by K f = l if κh f ) An outline of an algorithm applying the alternative no-slip condition Eq. 82) in pipe flow holdup and pressure drop computations is given in Appendix D. 9.3 Holdup and pressure drop in a pipe We now consider computing the holdup and pressure drop in a given pipe for a given flow rate and fluid combination. The flow rates Q f, fluid properties ρ f and µ f, and pipe-specific parameters D, k s, and θ are thus known input parameters. The equation determining the holdup is given by eliminating the pressure gradient between the momentum balances Eq. 84), yielding ɛ l τ g S g ɛ g τ l S l + τ i S i ɛ g ɛ l ρ l ρ g ) ga sin θ = ) A symmetric expression for the corresponding pressure drop may be obtained by eliminating the interfacial shear stress between the momentum balances Eq. 84), yielding dp dx = τ gs g A + τ ls l A + ɛ gρ g + ɛ l ρ l ) g sin θ 2) The mean velocities are given by U f = Q f A f 3) A given wetted angle δ l allows for an explicit computation of the parameters describing the flow geometry see Section 9. Simple friction models are also often explicit for a given δ l and the known input parameters specified above. The holdup Eq. ) may thus in this case be considered to be a function of the form F δ l ) = 4)

21 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 2 General explicit solutions may not be obtained. Variable δ l must thus be determined by a numerical root search. The highly implicit formulae for the wall friction Eqs. 7), 7), and 95), depending on the shear ratios R f = ±τ i /τ wf, may, however, not be rendered explicit for a given δ l and known input parameters. We solve this problem by obtaining R g from the no-slip relation Eq. 4), modified for pipe flow conditions as described in Section 9, by application of a numerical root search. The corresponding R l = τ i /τ l is then given by the holdup Eq. ), i.e., R l = ɛ g R g S l ɛ l S g + R g S i ɛ g ɛ l G)) where An outline of the solution procedure is given in Appendix D. G = ρ l ρ g )Ag sin θ R g τ g 5). DISCUSSION The present model does not yield a detailed description of the agitated wavy turbulent) zone at the interface. Wavy interfaces are simply accounted for by introducing certain interfacial turbulence levels at the mean position of the interface, represented by the corresponding values of the interfacial mixing lengths l i in either fluid Eq. 29). The interfacial mixing lengths are considered to be the central closure parameters of the model. Closure relations for air-water flow are given in Section. The interfacial mixing lengths are represented by the corresponding dimensionless interfacial turbulence parameters K Eq. 3)). The symmetric nature of stratified flow allows us to represent the flow in either fluid using a generic single-phase model in which the action of the other fluid is represented by the dimensionless interfacial drag R and turbulence level K see Sections 2 and 4. The two-phase flow model is given by the coupling of two single-phase models through the interfacial boundary conditions, i.e., the no-slip Eq. 4) and continuous shear stress Eq. 5). It is reasonable to assume that there are conditions relating the interfacial mixing lengths also. This has, however, not been investigated here.) The generic flow model is based on the eddy viscosity concept, as defined by the Boussinesq assumption Eq. ). It has been a requirement that the corresponding wall friction should reduce in form to classic Prandtl/Kármán pipe flow friction laws for smooth and rough pipes Eqs. 68) and 69)) in the single-phase limits. The viscous contribution to the apparent shear stress in Eq. 2) has therefore been neglected as in the development of these expressions. The model is thus not valid in the thin viscous regions occurring at the walls and possibly at the interface. The algebraic eddy viscosity model Eq. 39) derived in Section 4 agrees with the wall condition Eq. 9), yielding a logarithmic velocity distribution Eq. 7) near the wall. The model also agrees with the corresponding condition at the interface Eq. 22) for zero interfacial turbulence, i.e., K =. It will thus in this case yield a logarithmic velocity distribution, as given by Eq. 2), near the interface as well. The interface is thereby modeled as if it were a solid wall moving with the interfacial velocity for zero interfacial turbulence K =. Nonzero interfacial turbulence K > is accounted for by simply assigning a certain value to the corresponding interfacial mixing length l i subject to model closure).

22 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 22 BIBERG The velocity distribution Eq. 47) is obtained from the algebraic eddy viscosity model Eq. 39) by use of the standard integral Eq. 46). The corresponding interfacial shear stress is then given by the no-slip condition at the interface Eq. 4), as in laminar flow. The velocity distribution Eq. 47) is, however, not valid at the interface below a certain threshold value of the interfacial turbulence parameter K = K min associated with an hydraulic smooth interface see Eq. 79)). We thus limit K K min. The mean velocity Eq. 58) is given by a second integration Eq. 57). The resulting expression relates the wall friction τ w represented by u = τ w /ρ to the flow rate Q represented by U = Q/h for a given shear ratio R and interfacial turbulence level K. The shear ratio R is determined by the total model. K is given by the closure relation for l i. The expression for the mean velocity Eq. 58) does thus determine the wall friction. It is recast, without loss of generality, into the form of the Colebrook-White pipe flow friction formula Eq. 7) based on the effective flow diameter D e, as given by Eqs. 72) and 76). The Colebrook-White formula accounts for the hydraulic smooth to rough transition. An accurate explicit computation for a given R and K is facilitated by use of Eq. 32) in Appendix E. Alternative wall friction laws suitable for analysis are given by Eqs. 67) and 26). The pipe flow friction formulae are obtained from the channel flow expressions by hydraulic similarity in Section 9. The corresponding holdup and pressure drop calculations are discussed in Section 9.3. A numerical algorithm is given in Appendix D.. General remarks The present model is in principle valid for a given Newtonian) fluid combination and may thus be applied for liquid-liquid as well as gas-liquid flows. The discussion will, however, to some degree focus on gas-liquid flows. Poiseuille- and Couette-type flows may occur in either phase. The velocity maximum will, however, typically be located in the gas, which then will be in a Poiseuille-type flow R >, pulling the liquid along in a Couette-type flow R <. We will thus occasionally plot the Couette-type flows upside down as a function of ), i.e., with the associated wall below the flow and the interface on top. The velocity defect given by Eqs. 47) and 58) may be written as u U u = F, R, K) 6) The actual shape of the velocity distribution is thus determined by the parameters R and K only. The velocity defect for K = only depends on R and is symmetrical with respect to interchanging the values of the wall and interfacial shear stress, i.e., τ w τ i. The symmetry is given by the corresponding symmetry in the eddy viscosity distribution Eq. 39) and boundary conditions Eqs. 9) and 22). The velocity defect flow rate is given by u U u d = 7) h

23 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 23 The area beneath the velocity defect distribution is thus preserved and equal to zero. Velocity defect plots are therefore well suited for examining the behavior of the velocity distribution for changes in the input parameters. The present model is based on Eq. 2). The positions of maximum velocity m and zero shear in a Poiseuille type flow R > do thus coincide. The corresponding R value may be obtained from Eq. 8), yielding R = m m 8) Shear ratios R < correspond to a situation in which the net driving force in the momentum balance Eq. 7) opposes the flow, i.e., dp/dx + ρ f g sin θ >. This situation may only occur in the more dense fluid in upwardly inclined flows or in the less dense fluid in downwardly inclined flows. The interfacial mixing lengths l i Eq. 29)) in either fluid) representing the corresponding interfacial turbulence levels are the central closure parameters of the present model see Section 4). Appropriate values for l i must be determined to obtain a predictive model. The dimensionless form given by interfacial turbulence parameter K = l i / κh) arising naturally in the equations) will be considered here. K = yields a logarithmic velocity distribution at the interface. Nonzero K > displaces the onset of the logarithmic velocity distribution toward the interface and eventually out of the physical domain beyond the interface. We have K = for Poiseuille and Couette flow, for which R = ± respectively, and K = K F > for free surface flow for which R =. Considering free surface flow to be analogous to half a Poiseuille flow yields the estimate K F =.5, by application of Eq. 3). We use single-phase data to determine K for flat, nonwavy interfaces and two-phase data to determine K for wavy interfaces. Non wavy interfaces and single phase data We start by examining the model behavior for the Poiseuille, free surface, and Couette flow cases. Figures 3 and 4 contain corresponding data comparisons for the eddy viscosity Eq. 39) and velocity distribution Eq. 47).. Poiseuille flow. The eddy viscosity Eq. 39) and velocity Eq. 47) reduce to Eqs. 43) and 54) in the Poiseuille flow case R = and K = ). The eddy viscosity is compared to the Hussain and Reynolds 975) eddy viscosity data Re h = 27, 6, 46, 4 and 64, 6) in Fig. 3. The model is seen to capture the characteristic double hump in the data. The velocity distribution is compared to the Nikuradse 932) pipe flow data Re D = ) in Fig. 4. The pipe and channel flow velocity profiles are equivalent in the Poiseuille flow case.).2 Free surface flow. The eddy viscosity Eq. 39) and velocity Eq. 47) reduce to Eqs. 44) and 55) in the free surface flow case R = and K = K F.5). The velocity distribution is compared to the Nezu and Rodi 986) free surface flow data Re h = ) in Fig. 4. Nezu and Rodi applied the Coles 956) log/wake law velocity distribution, given by

24 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 24 BIBERG R K R K K.5 K t hu a) R K t hu b) t hu c) Figure 3 Eddy viscosity model Eq. 39) vs. data: a) Hussein and Reynolds 975) Poiseuille flow data; b) Ueda et al. 977) free surface flow data; c) Reichardt 959) Couette flow data. u Coles = u κ π ) ) 2 ln ) + 2Π sin 2 ) + C 9) present notation) in the discussion of their data. Π is the Coles wake parameter. The constant C is given by Eq. 8). They observed Π to be a function of the Reynolds number and found that Π = for low Reynolds numbers, increasing to the constant value Π for Re h 5. Equating Eq. 9) and Eq. 55) at the free surface = yields K F = e 2Π. According to Nezu and Rodi 986), we thus have K F = for low Reynolds numbers, decreasing to K F.67 for Re h 5, i.e., a somewhat

25 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim TWO-PHASE DUCT FLOW 25 R K R K u - -5 max u u R K.657 R K u max u u u U wall Figure 4 Velocity distribution Eq. 47) vs. data: top left) Hanjalic and Launder 972) asymmetric one rough wall) flow data; top right) Nikuradse 932) Poiseuille pipe) flow data; bottom left) Nezu and Rodi 986) free surface flow data; bottom right) Reichardt 959) Couette flow data. Scaling as given by the authors; u max = maximum velocity, U wall = wall velocity. higher value than the estimated K F.5. We applied the reported Π = corresponding to K F =.657 in the data comparison. Note that both Eqs. 9) and 55) reduce to the pure log law for Π = corresponding to K F = ). The eddy viscosity Eqs. 39) or 44) is compared to the Ueda et al. 977) free surface flow data Re h = 344) in Fig. 3. K F =.8 corresponding to Π = ) was found to yield a reasonable fit. The predictions corresponding to K F =.5 and

26 Electronic Data Center, Downloaded from IP by Universitetsbibl. I Trondheim 26 BIBERG = dashed lines) are included for reference. The model is seen to capture the slightly asymmetric parabolic shape in the data, which have a maximum at.55. K F.3 Couette flow. The eddy viscosity Eq. 39) and velocity distribution Eq. 47) reduce to the symmetric parabola Eq. 45) and double logarithm Eq. 56) in the Couette flow case R = and K = ). These expressions are identical to the expressions considered by Reichardt 959) in his Couette flow study. The models are compared to the Reichardt data in Figs. 3 and 4. The reported Reynolds number Re Uw = 34,, based on the wall velocities U w, corresponds to Re h =.82 5 )..4 Asymmetric flow. The velocity distribution Eq. 47) reduces to Eq. 53) for asymmetric flows in which K =. Figure 4 contains a data comparison versus the Hanjalic and Launder 972) asymmetric one rough wall) flow data Re h = 2, ), using the shear ratio R corresponding to a velocity maximum at m =.78 and Eq. 8)..5 Mixed Poiseuille-Couette flows. The K values corresponding to more general R values may be determined from the El Telbany and Reynolds 98) data featuring velocity and shear stress measurements for 5 mixed Poiseuille-Couette flows covering the parameter range < R <. El Telbany and Reynolds found that the velocity at the low shear wall interface) remains logarithmic for R.. We have K = corresponding to a logarithmic velocity distribution. We thus assume that K = for R. The velocity distribution Eq. 47) is compared to the El Telbany and Reynolds 982) data in Fig. 5. The nonzero K values for R <. were obtained by the condition that the computed wall friction coefficients λ calc, as given by Eqs. 7), 72), and 76), should match the measured values λ meas. The resulting K = K meas are listed in Table 2. We applied the rounded values K A =.5,.8 and in the data comparison in Figure 5 and in the calculation of λ calc in Table. We note that the K values in the Couette-type flow cases 6, 7, and 8, which all have a shear ratio close to zero, agree with the estimated free surface flow value K F =.5. We used K =.8 for the lowest shear ratio R =.57) in case 9. The error in λ calc using K =.5 was only 4%. There was, however, some discrepancy in the predicted vs. measured velocity.) We used K = in the Poiseuille type flow cases and. There is some disagreement in the predicted versus measured velocity distribution in the Poiseuille-type flow cases 2, 3, and 4 in Fig. 5. These cases also have the largest error 3%) in λ calc in Table 2. This discrepancy is connected to the substantial separation between the measured) position of zero shear and maximum velocity m in these cases. The present model is based on the assumption m = and may thus not be expected to capture all aspects of these flows. Applying the R = R m values corresponding to the velocity maximum m obtained by data interpolation) and Eq. 8) yields the improved velocity predictions given by the dashed lines in Fig. 5. It also reduces the corresponding error in λ calc. The R m values are given in the plots.