ANALYSIS OF TURBULENT FLOW IN CLOSED AND OPEN CHANNELS WITH APPLICATION IN ELECTROMAGNETIC VELOCIMETRY C. Stelian 1,2

MAGNETOHYDRODYNAMICS Vol. 48 (2012), No. 4, pp. 637 649 ANALYSIS OF TURBULENT FLOW IN CLOSED AND OPEN CHANNELS WITH APPLICATION IN ELECTROMAGNETIC VELOCIMETRY C. Stelian 1,2 1 Institute of Thermodynamics and Fluid Mechanics, Ilmenau University of Technology, PO Box 100565, 98684 Ilmenau, Germany 2 Department of Physics, West University of Timisoara, Bd. V. Parvan, No. 4, 300223 Timisoara, Romania Electromagnetic velocimetry based on Lorentz force measurements is a new technique developed for liquid metal flow measurements. Numerical calibration of Lorentz force flowmeters requires accurate computations of the velocity distribution for high Reynolds number turbulent flows in ducts and open channels. In this paper, numerical modelling is used to investigate velocity profiles in internal and external turbulent flows at high Reynolds numbers. The numerical results obtained by using the k-ɛ turbulence model are compared to theoretical and experimental data from the literature. It is found that the logarithmic wall law can approximate the velocity profile in closed channel flows. The friction velocities in the are estimated by solving numerically the friction equation. In the case of open channel flows, the dip phenomenon is investigated at various aspect ratios of the channels. After the numerical model validation, flow computations are included in the numerical calibration procedure of the Lorentz force flowmeters. Numerical results show small variations of Lorentz forces due to the nonuniformity of velocity profiles in the case of high Reynolds number turbulent flows. 1. Introduction. The Lorentz force flowmeter (LFF) is a device used for contactless measurements of the flow rate in electrically conducting fluids at high temperatures [1 4]. The sketch of a LFF with two permanent magnets is presented in Fig. 1. A liquid metal of electrical conductivity σ flows at a mean velocity u 0 in a magnetic field of induction B produced by two magnet poles. Eddy currents of electrical density j induced in the fluid interact with the applied magnetic field and produce a Lorentz force F, which acts in the opposite to the flow direction. The volume density of this force is roughly given by f σu 0 B 2. (1) Because both the magnetic field and the velocity are non-uniform across the conducting fluid, the Lorentz force is computed by integrating the volume density f over the fluid volume: F = (j B)dV. (2) The secondary magnetic field created by eddy currents interacts with the applied magnetic field giving rise to an opposite electromagnetic force F, which acts on the magnet system. By measuring this force, the mean flow velocity u 0 can be estimated through expression (2), which can be simplified as F = σcu 0. (3) 637

C. Stelian Yoke Liquid metal Magnet Fig. 1. Principle of the Lorentz force flowmeter. Colour map represents the velocity distribution in a closed channel. The LFF calibration coefficient C characterizes the non-uniformity of the magnetic field and flow velocities in the channel and can be estimated experimentally or by numerical modelling. Previous numerical simulations performed with COMSOL Multiphysics [5] have shown that the LFF numerical calibration requires very accurate computations of the magnetic field and resulting Lorentz forces [6]. All those numerical computations have been performed by considering a constant flow velocity u 0. In this paper, the non-uniformity of the velocity profiles for internal and external turbulent flows at high Reynolds numbers is accounted for in the numerical computations of Lorentz forces. The application of the k ɛ model implemented in COMSOL Multiphysics is tested for modelling turbulent flows in closed and open channels. The numerical results are compared to theoretical predictions and experimental data from the literature. 2. Closed channel turbulent flow. In pipe steady turbulent flows, three regions are identified in the velocity profiles near the bounding walls. Within the viscous layer adjacent to the wall, the mean streamwise velocity u varies linearly with the distance y from walls [7]: u + = y +, (4) where u + = u/u and y + = yu /ν. The friction velocity is defined as u τw = ρ, where τ w is the wall shear stress, and ρ is the fluid density. The linear viscous relation describes the velocity profile from the wall to about y + =5. In the outer region y + > 25, the velocity distribution is described by the [7]: u + = 1 κ ln y+ + C, (5) where κ is the von Kármán constant (κ 0.41), and C is an integration constant (C 5). 638

Analysis of turbulent flow in closed and open channels with application... The inner wall law (4) and the (5) overlap in an intermediate region 5 <y + < 25. The viscous and the intermediate layers have small dimensions and can be neglected at high Reynolds numbers. It is generally assumed that the, instead to be applicable only in the wall region, actually describes the entire velocity profile over the pipe cross-section [7]. Several methods are available to estimate the friction velocity in Eq. (5). Accurate estimations of u are difficult and require experimental measurements of velocity profiles or Reynolds stresses [8]. Usually, the friction velocity is estimated by fitting the measured velocity profiles to the. In the following, the friction velocities in turbulent duct flows are estimated by solving numerically the friction equation. 2.1. Comparison of theoretical and experimental velocity profiles. For turbulent flows in smooth-walled circular pipes of radius R, the is given in the cylindrical coordinates (r, z) by u u = 1 (R r)u ln + C. (6) κ ν The mean velocity of the flow u 0 can be obtained by integrating Eq. (6) over the cross-sectional area R [ ] 1 2π u (R r)u ln + C r dr = πr 2 u 0. (7) 0 κ ν After integrating Eq. (7), we obtain the following equation for u : u 0 u = 1 Ru ln + C 3 κ ν 2κ. (8) This equation can be solved numerically for a given mean velocity u 0 = νre/d, where Re is the Reynolds number and d is the pipe diameter. Eq. (8) can be also expressed in terms of the Darcy s friction factor f: f = 8τ ( ) w u 2 ρu 2 =8, (9) 0 u 0 giving a Prandtl s friction equation 1 f =2log(Re f 1/2 ) 0.8, (10) 1/2 where the coefficients 2 and 0.8 were slightly adjusted in order to fit better the experimental data. Fig. 2 and Fig. 3 compare the with the measured velocity profiles in circular pipes [9] and, respectively, in rectangular ducts [10, 11]. The friction velocity was estimated by solving numerically Eq. (8). The agreement is seen to be very good, so it is concluded that the can predict velocity profiles in internal turbulent flows at high Reynolds numbers (Re = 10 4 10 6 ). The procedure used to estimate u does not require previous experimental measurements of the velocity profiles. 2.2. Comparison of theoretical and numerical velocity profiles. The k-ɛ model implemented in the finite element software COMSOL Multiphysics is used for modelling turbulent closed and open channel flows. The Reynolds averaged Navier Stokes equations written for incompressible Newtonian fluids are given by ρ u t + ρ(u )u = [ pi +(η + η t ) ( u +( u) T )] + F, (11) 639

C. Stelian Laufer 1954 Laufer 1954 u/u0 Re = 4.0 10 4 Re = 4.28 10 5 r/r Fig. 2. Comparison of the measured dimensionless velocity profiles in circular pipes [9] with the : Re=4.02 10 4, ν =15 10 6 m 2 /s, R =0.1235 m, u 0 =2.44 m/s, u =0.133 m/s; Re=4.28 10 5, ν =5 10 6 m 2 /s, R =0.1235 m, u 0 =25.51 m/s, u =1.087 m/s. r/r Niederschulte 1989 Laufer 1951 u/u0 Re = 3.67 10 4 Re = 1.13 10 5 x/a Fig. 3. Comparison of the measured dimensionless velocity profiles in rectangular ducts [10, 11] with the : Re=3.67 10 4, ν =10 6 m 2 /s, a =0.0244 m, u 0 =0.663 m/s, u =0.037 m/s; Re=1.13 10 5, ν =15 10 6 m 2 /s, a =0.0635 m, u 0 =13.35 m/s, u =0.646 m/s. x/a u =0, (12) where u is the average part of the flow velocity, p is the pressure, I is the identity matrix, η is the dynamic viscosity, η t is the turbulent viscosity, and F is the volumetric density of the body force applied to the fluid. The k-ɛ model introduces two additional transport equations for the turbulent kinetic energy k and dissipation rate of turbulence energy ɛ: ρ k [( t η + η ) ] t k + ρu k = 1 σ k 2 η ( t u +( u) T ) 2 ρɛ, (13) ρ ɛ [( t η + η ) ] t ɛ + ρu ɛ = 1 σ ɛ 2 C ɛ ɛ1 k η ( t u +( u) T ) 2 ɛ 2 ρcɛ2 k. (14) The turbulent viscosity is given by 640 η t = ρc μ k 2 ɛ. (15)

Analysis of turbulent flow in closed and open channels with application... Fig. 4. Geometry of the axisymmetric two-dimensional simulation domain. The velocity field is given by arrows and colour map. The constants in Eqs. (13) (15) are determined from the experimental measurements: C μ =0.09, C ɛ1 =1.44, C ɛ2 =1.92, σ k =1,andσ ɛ =1.3. The boundary conditions are specified as follows: At the inlet boundary (see Fig. 4), the normal inflow velocity is set to u 0 (the flow mean velocity ). In addition, two turbulent quantities must be specified: the turbulence intensity I t (I t 0.05 in our simulations) and the turbulent length scale L t. This length is a measure of the distance, where eddies are not solved, and is given by L t =0.07 d (d is the pipe diameter or the channel width). At the opposite face of the pipe, an outflow boundary condition is imposed by setting the pressure to zero (p =0). At the solid walls of the pipe, the logarithmic wall functions are applied to specify the flow velocities parallel to the wall. This condition replaces the velocity boundary layer adjacent to the wall, which is very thin at high Reynolds numbers. It is assumed that the computational domain begins at a distance δ W from the real wall, where the velocity is given by the : ( 1 u w = u κ ln δ wu ) + C. (16) ν The distance δ w has the order of magnitude of 25 30 units y + and must be specified in the simulation model. In our computations, it is estimated as δ w =25 ν/u, where the friction velocity was found from Eq. (8). Numerical computations of the velocity profiles are performed at different Reynolds numbers: 10 4,10 5 and 10 6. The axisymmetric two-dimensional simulation domain is shown in Fig. 4. Since in our applications the turbulent flow is fully developed, the domain length must be larger than the entrance length L>L e.in smooth walls turbulent flows, the entrance length is given by L e 4.4d Re 1/6, (17) where d is the pipe diameter. Fig. 4 shows the development of the viscous boundary layer in a pipe flow at Re = 10 4 (diameter d =0.02 m). The flow parameters and the pipe dimensions used in the simulations are summarized in Table 1. The kinematic viscosity is ν =3 10 7 m 2 /s, κ =0.41 and C =5. Table 1. Flow parameters and pipe dimensions used in simulations. Re R [m) L [m] u 0 [m/s] u [m/s] δ w [m] 10 4 0.01 0.4 0.15 0.00975 0.0008 10 5 0.05 3 0.3 0.01475 0.0005 10 6 0.05 4.4 3 0.1178 0.00006 641

C. Stelian numeric u/u0 Re = 10 4 r/r numeric numeric u/u0 Re = 10 5 Re = 10 6 (c) r/r r/r Fig. 5. Comparison of the numerical dimensionless velocity profiles to the logarithmic law for turbulent flows in circular pipes: Re=10 4 ; Re=10 5 ;(c) Re=10 5. In Fig. 5, the numerical results are compared to the theoretical velocity profiles given by the. The agreement between the numerical results and the theoretical predictions is very good, which means that the COMSOL k-ɛ turbulence model can be successfully used for modelling turbulent flows in circular pipes at high Reynolds numbers (10 4 10 6 ). The numerical simulations performed for rectangular ducts have shown the same good agreement with the. 3. Open channel turbulent flow. Many experimental, theoretical and numerical studies have been dedicated to the open channel turbulent flow. Some empirical formulas were proposed to predict vertical velocity profiles in rectangular channels [12 14]. The formula proposed in [14] can predict the dip phenomenon, which appears in narrow open channels of aspect ratios A s = l/h 5(l is the channel width and h the liquid height): u = u κ [ ln y y 0 +2Π sin 2 ( πy 2h ) ( + α ln 1 y ) h y αππ y 0 y h y ] sin(πy/h) dy. (18) h In this formula, the first term is the classical (y 0 is the distance from the boundary, at which the velocity vanishes) and the second term is the Coles wake function, which contains a parameter Π. The last two terms were introduced in order to predict the velocity dip-phenomenon. The parameter α is 642

Analysis of turbulent flow in closed and open channels with application... given by α(z) =1.3exp( z/h), (19) where z is the distance from the channel sidewall. The parameters Π and α are dependent on the Reynolds number and channel roughness, and no general rule to estimate them has been found yet [14]. Good agreements between empirical formulas and experimental data have been obtained only after previous estimation of the parameters Π and α by fitting the measured velocity profiles. Therefore, the numerical modelling can be an useful tool to predict the velocity distribution in open channel flows. In this work, a two phase k ɛ turbulence model is used to simulate free surface turbulent flows in rectangular open channels. Eqs. (11) (14) are solved for a domain containing the fluid and an adjacent gas layer above it. At the free surface, the normal stress is set to zero. At the solid walls, the logarithmic wall functions given by Eq. (16) are used as boundary conditions. Numerical computations are performed for a rectangular channel of width l = 0.1 m at different liquid heights. The physical properties of aluminum are used for these computations: ρ = 2370 kg/m 3 and ν = 4.2 10 7 m 2 /s. The velocity profiles on the horizontal x-axis and the vertical direction for two aspect ratios of the channel A s = 1 and, respectively, A s = 5 are plotted in Fig. 6. A pronounced dip phenomenon at the small aspect ratio can be observed (Fig. 6a). In the case of wide channel geometry, the maximum velocity is reached close to the free surface (Fig. 6b). 4. Velocity effects on the Lorentz force computations. The influence of the velocity distribution on the Lorentz forces computations is analyzed for liquid metal flows in closed and open channels. The Lorentz forces are computed by using the COMSOL AC/DC electromagnetic module. The computations are performed velocity, [m/s] x, [m] h, [m] Fig. 6. Velocity profiles on the x-axis and vertical direction for two aspect ratios of the channel: A s =1; A s =5. 643

C. Stelian Lorentz force (solid) Lorentz force (flow) F/Fs Fig. 7. Dimensionless Lorentz forces F/F s versus the Re number for circular pipes (round symbols), rectangular ducts (square symbols) and for open rectangular channels (half-square symbols). Re for the magnetic system shown in Fig. 1, which was used in industrial tests of the Lorentz force flowmeter [15]. The Lorentz forces computed at different Reynolds numbers are compared to the computations performed for a solid conductor (F s ) moving at the mean flow velocity (u 0 ). The solid body computations are preferred for the LFF numerical calibration because are fast and simple. The dimensionless Lorentz forces F/F s are plotted versus the Re number in Fig. 7 for circular pipes (round symbols), rectangular ducts (square symbols) and for open rectangular channels (half-square symbols). The solid body computations corresponding to the open channel flow are performed by considering a mean velocity u m found by integrating the numerical computed velocities over the liquid metal volume exposed to the magnetic field. The relative error of the Lorentz forces solid body computations δf =(F F s )/F s varies between δf 5% at small Reynolds numbers (Re = 500) and δf 1% at high Reynolds numbers (Re = 10 5 10 6 ). In the case of laminar flows, this error is dependent on the channel-magnet geometry and can increase up to δf = 50%. In high Reynolds number flows typical for LFF industrial applications, the velocity effects on the Lorentz force computations are very small and can be neglected. Therefore, in this case, the complex turbulent flow simulations can be replaced by performing simple solid body computations. In order to explain the velocity effects on the measurable Lorentz forces, laminar and, respectively, turbulent flow computations are performed for the same simplified configuration shown in Fig. 8. The rectangular channel has a square cross-section of dimensions l = h =0.04m and length L = 1 m. Two magnets of dimensions 0.02 m 0.04 m 0.04 m are positioned at a distance of 0.08 m from each other and have the magnetization M x = 10 6 A/m. In order to have a full developed flow in the region exposed to the magnetic field, the center of the magnetic system (origin of the coordinate system) is positioned at the 0.8 m distance from the inflow surface. The magnetic induction at this point B 0 =0.08 T is considered as the reference value. The Lorentz forces are computed in the case of laminar flow at Re = 500 and, respectively, turbulent duct flow at Re = 3 10 4. By introducing in computations the Lorentz forces acting on the flowing liquid, a significant effect on the velocity profiles is found, in particular, in the case of laminar flow. This effect depends on 644

Analysis of turbulent flow in closed and open channels with application... Fig. 8. Configuration with a rectangular channel of dimensions 0.04 m 0.04 m 1m and two magnets of dimensions 0.02 m 0.04 m 1m. the magnetic interaction parameter N, which represents the ratio of the Lorentz forces and the inertial forces acting on the fluid [1]: N = σdb2 0, (20) ρu 0 where d is the characteristic length of the flow. At low intensities of the magnetic field, which are specific for the Lorentz force velocimetry (B 0 < 0.1 T), this parameter is small for turbulent flows, but can increase significantly in the case of laminar flows due to small mean velocities u 0. In the following, the Lorentz forces are computed for laminar and turbulent flows characterized by the parameter N having the same order of magnitude. For this reason, the electrical conductivity of the liquid was significantly reduced in the laminar flow case (σ =2 10 5 Ω 1 m 1 ) if compared to the turbulent flow case (σ =2 10 6 Ω 1 m 1 ). The values obtained for N are relatively small: N lam =4.32 for the laminar flow, and N turb =1.08 in the case of turbulent flow. In Figs. 9,10 and 11, the velocity profiles, eddy currents and, respectively, the Lorentz forces are plotted on the x-, y- andz-axis. The numerical profiles are computed for solid conductor, laminar flow and turbulent duct flow (c). The effect of the Lorentz forces on the velocity distribution is illustrated in Fig. 9. The magnetic effects are more obvious in the case of laminar flow (Fig. 9b) if compared to the turbulent flow case (Fig. 9c). In the laminar flow case, the velocity profiles are distorted in the x-andz-directions. In the flow direction (the z-axis), the velocity increases over the entrance length, then decreases by 25% in the region with the magnetic field, due to the Lorentz forces damping effect. In the case of duct turbulent flow, the magnetic effects are significantly reduced (only 1% decrease of the maximum velocity) and a small oscillation is observed on the z-axis (Fig. 9c). In this case, the velocity profiles in the x-andy-directions are slightly affected by the magnetic field. The influence of the velocity distribution on the measurable Lorentz forces is explained by different patterns of the induced currents in the case of laminar and, respectively, turbulent flows (Fig. 10). The measurable force in the Lorentz force velocimetry corresponds to the damping Lorentz force, which has the negative z- direction and is produced by the interaction between the eddy currents of negative 645

C. Stelian velocity, [m/s] velocity, [m/s] velocity, [m/s] (c) x, [m] y, [m] z, [m] Fig. 9. Velocity profiles on the x-, y-andz-axis for: ( a) solid conductor (v 0 =0.3 m/s); laminar flow (v 0 =0.005 m/s); (c) turbulent duct flow (v 0 =0.3 m/s). jy, [A/m 2 ] jy, [A/m 2 ] jy, [A/m 2 ] (c) x, [m] y, [m] z, [m] Fig. 10. Current density j y on the x-, y- andz-axis for: solid conductor (σ = 2 10 6 Ω 1 m 1 ); laminar flow (σ =2 10 5 Ω 1 m 1 ); (c) turbulent duct flow (σ = 2 10 6 Ω 1 m 1 ). 646

Analysis of turbulent flow in closed and open channels with application... j y density and the transverse magnetic field. The positive j y current density produces a positive (accelerating) F z force. In the case of a solid conductor moving at a constant velocity, the eddy currents are closed in the longitudinal plane (y, z), due to the magnetic field nonuniformity in the flow direction. There is no closure path of electrical currents in the transversal x-y section, where j y plots show only negative values (Fig. 10a). Because of the eddy currents, which turn in a region with a low magnetic field, the positive F z force is much smaller than the negative (measurable) force. In the case of laminar flow (Fig. 10b), the wall velocity boundary layers create an electric potential difference and the eddy currents can also be closed up in the transversal x-y section. The x plot in Fig. 10b shows a positive j y density at a wall boundary layer thickness of 2 mm. Therefore, in the case of laminar flow, the eddy currents can be closed up in both transversal and longitudinal planes. The induced currents, which turn in the transversal cross-section, create opposite Lorentz forces, so the resulting negative F z force is significantly reduced if compared to the solid conductor case. These results are depicted in Fig. 11b, which shows a positive Lorentz force density f z in the wall boundary layers (case of the x plot). In the case of turbulent duct flow, because of the thin wall velocity boundary layers, the currents close up in the transversal cross-section is negligible, with predominant loops in the longitudinal plane (Fig. 10c). Therefore, the induced currents pattern is similar in the turbulent flow and, respectively, in solid body computations, giving almost the same results for the computed Lorentz forces. The relative error for the integrated Lorentz force F z is δf =5%forthe laminar flow and δf = 2% for the turbulent flow computations. With the increased fz, [N/m 3 ] fz, [N/m 3 ] fz, [N/m 3 ] (c) x, [m] y, [m] z, [m] Fig. 11. Lorentz force density f z on the x-, y- andz-axis for: solid conductor; laminar flow; (c) turbulent duct flow. 647

C. Stelian Reynolds numbers Re = 10 5 10 6,thiserrorisverysmallδF < 1%. In the case of laminar flows, the interaction parameter N can increase significantly at low velocities or large dimensions of the channel (see Eq. 23). At high N, thevelocity profiles are very distorted with the appearance of M-shaped behaviour, and the Lorentz forces can decrease to δf = 50%. The eddy current pattern and the Lorentz forces are also significantly influenced by the geometry of the channelmagnet system. Conclusions. In this paper, the turbulence model implemented in COMSOL Multiphysics is validated by comparing numerical velocity profiles to theoretical predictions and experimental data from the literature. The numerical computations are performed for turbulent flows in circular pipes and rectangular closed and open channels. As the first step, the application of the to predict velocity profiles in circular pipes and rectangular ducts has been investigated. Theoretical predictions of the velocity distribution require accurate estimations of the friction velocities, which are usually determined from fitting the experimental velocity profiles. In this paper, a simple method to evaluate friction velocities from numerical solution of the friction equation was used. This method does not require previous experimental measurements of the flow velocities in the channel, giving the same results as from the fitting procedure. Finally, a good agreement between experimental data, and numerical velocity profiles has been found in the case of closed channel turbulent flows. In the case of open channel flow, the phenomena occurring at the free surface are simulated by using a two phase k-ɛ turbulence model. The computations performed at different aspect ratios of the channel show that the vertical velocity profiles and the dip phenomenon are strongly influenced when the aspect ratio is varied between A s =1andA s =5. After validation, flow computations are introduced into the numerical calibration procedure of the Lorentz force flowmeters. The Lorentz forces are computed for liquid metal laminar and turbulent flows at different Reynolds numbers. By comparing these computations to the simple case of a solid conductor moving at the mean flow velocity, significant differences are found only for the laminar flows. This is explained by a different pattern of the induced electrical currents in the case of the laminar and, respectively, turbulent flows. In the case of high Reynolds number turbulent flows, the velocity effects are very small, so the simple case of a moving solid conductor can be used in the LFF numerical calibration. Acknowledgements. The author is grateful to the German Research Foundation (Deutsche Forschungsgemeinschaft) for supporting the work in the frame of the Research Training Group (Graduiertenkolleg) for Lorentz force velocimetry and Lorentz force eddy current testing at Ilmenau University of Technology. The author also acknowledges helpful discussions with Prof. A. Thess at Ilmenau University of Technology and Prof. S. Molokov at Coventry University. REFERENCES [1] J. Shercliff. The Theory of Electromagnetic Flow Measurements (Cambridge University Press, Cambridge, UK, 1962). [2] A. Thess, E.V. Votyakov, and Y. Kolesnikov. Lorentz force velocimetry. Phys. Rev. Lett., vol. 96 (2006), pp. 164501. 648

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