Estimation of di usion and convection coe cients in an aerated hydraulic jump

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1 Advances in Water Resources 23 (2000) 475±481 Estimation of di usion convection coe cients in an aerated hydraulic jump K. Unami a, T. Kawachi a, M. Munir Babar b, H. Itagaki c a Graduate School of Agricultural Science, Kyoto University, Kitashirakawa-oiwake-cho, Sakyo-ku, Kyoto , Japan b Institute of Irrigation Drainage Engineering, Mehran University of Technology Engineering, Jamshoro, Sindh, Pakistan c Division of Biological Production System, Faculty of Agriculture, Gifu University, 1-1 Yanagido, Gifu , Japan Received 23 March 1999; accepted 16 October 1999 Abstract Di usion convection coe cients of the aerated ow in a hydraulic jump are estimated in the framework of an optimal control problem. The speci c gravity of air±water mixture is governed by a steady state convection di usion partial di erential equation. The governing equation includes the di usion coe cient the convection coe cient, which are taken as the control variables to minimize a domain observation type functional which evaluates the error between observed speci c gravity the solution to the governing equation. The minimum principle is deduced to characterize the optimal control variables using the adjoint partial di erential equation. Solvability of the partial di erential equations is discussed in terms of variational problems. An iterative numerical procedure including a nite element scheme to solve the partial di erential equations is developed demonstrated by using the observed data. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Hydraulic jump; Aerated ow; Di usion coe cient; Convection coe cient; Estimation; Optimal control problem 1. Introduction In a wide horizontal rectangular channel of constant width, a hydraulic jump is formed when the state of ow changes from supercritical to subcritical one. The hydraulic jump is usually accompanied by an aerated recirculating roller on the top, where intensive turbulence results in energy dissipation the mixing of air also takes place. The development mechanism of such a roller has been investigated using high speed video cameras [7], it is revealed that the roller is made up of several vortices which are generated at the toe of the hydraulic jump subsequently travel downstream growing by pairing. Thus, unsteady behaviors in the ow structure of the hydraulic jump are evident. Despite such microscopically dynamic nature, the hydraulic jump is treated as a macroscopically steady phenomenon for engineering purposes, temporally averaged quantities are examined. The steady state approach is rather signi cant for evaluating re-oxygenation e ect in polluted rivers, energy dissipation e ect in stilling basins of spillways, so on. In this context, governing laws of the air-mixed water are di cult to be identi ed because averaged ow properties such as diffusivity residual stress cannot be obtained from the theoretical inductions. Thus, investigations based on observations are inevitable. Especially, when measuring the concentration of contained air is relatively easy, it is interesting to clarify what is possible to know from observed air concentrations [2]. In the present work, assuming that the speci c gravity, which is de ned by the ratio of the density of air± water mixture to that of pure water, is governed by a steady convection di usion equation, an inverse problem is posed to estimate its di usion convection coe cients from observed speci c gravity data. For the sake of simplicity, considerations are restricted to vertically 2-D ow structure. Being underdetermined, the inverse problem is formulated as an optimal control problem. This type of problem, referred to as an inverse coe cient problem [4], is often studied in the area of groundwater [8,9], but the problem considered here requires particular attention to the convection term. Since the approaches to inverse coe cient problems for convection dominant equations are not very well established [5], therefore in the present work the minimum principle which characterizes the coe cients such as to give a speci c gravity eld closest to speci ed data is deduced /00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S (99)

2 476 K. Unami et al. / Advances in Water Resources 23 (2000) 475±481 The minimum principle includes the adjoint equation of the convection di usion equation, the solvability of these two equations are discussed in terms of variational problems. A numerical method using a nite element scheme for the resolution of the equations is proposed to iteratively estimate the di usion the convection coe cients. To con rm the validity of the methodology a sample computation is demonstrated by using observed data. 2. Mathematical de nitions In order to follow up the subsequent sections smoothly a brief introduction of the mathematical concepts used in these sections is delineated here. For details to these de nitions, reference is made to the book of Sobolev spaces by Adams [1]. Note that the following notations are pertaining to real functions. The symbol X is used for a domain which is a bounded open set in 2-D Euclidean space. The inner product between two 2-D vectors u v is denoted by u v. The absolute value of a scalar u is represented by juj, the Euclidean norm of a 2-D vector u is also denoted by juj. The class of all measurable scalar valued functions u, de ned on X, for which R X juj2 dx < 1 is denoted by L 2 1. Similarly, L2 2 is de ned R for 2-D vector valued functions u for which X juj2 dx < 1. Inner products are de ned as hu; vi ˆ RX uv dx for u; v 2 L2 1 as hu; vi ˆ u v dx RX for u; v 2 L 2 2, respectively, using the same notation h ; i. The norms de ned in L 2 1 L2 2 from their respective inner products are commonly denoted by kk. For a scalar function u, another norm kuk 1 is de ned as q kuk 1 ˆ kuk 2 kruk 2 1 where r is the 2-D del operator. Let CB 2 be the class of all twice continuously di erentiable functions u such that kuk 1 < 1. The completion of CB 2 with respect to the norm Eq. (1) is the Sobolev space H 1, equipped with the norm Eq. (1). Let C0 1 be the class of all in nite times continuously di erentiable functions which have compact support in the domain X. The closure of C0 1 in the Sobolev space H 1 is another Sobolev space H0 1. The Sobolev space H0 1 is equipped with a norm de ned by ˆkruk 2 for u 2 H0 1 an inner product hu; vi V ˆ hru; rvi 3 for u; v 2 H Problem formulation Considering mass conservation of air neglecting compressibility e ect, the governing convection di usion equation of the transport phenomena in steady aerated ows is written as L a; b; c ḓef rarc rcb ˆ 0 4 where L is the di erential operator of the convection di usion equation, a the di usion coe cient, b the convection coe cient, which is the vector sum of water velocity vector air bubble rise velocity vector; c is the speci c gravity in the domain X bounded by its boundary C. The x- y-coordinates are taken as the horizontal vertical ones, respectively. A boundary value problem of Dirichlet type is considered prescribing a boundary condition c ˆ ^c 5 where ^c is the speci ed boundary value on the boundary C. The partial di erential equation system Eq. (4) with (5) is rearranged in its equivalent variational form l a; b; u; u ḓef h aru ub ; rui ˆ l a; b; ^c e ; u 6 where u is the state variable, l the bilinear form of the convection di usion equation, ^c e is a particular arbitrary H 1 function that accords with ^c on the boundary C, for all test functions u 2 H0 1. Thus, the state variable u is determined as c ^c e, the residual of the speci c gravity after subtracting ^c e. Generally speaking, if the coe cients a 2 L 2 1 b 2 L 2 2 are predetermined, then a forward problem is solved to nd u which satis es Eq. (6) in H0 1. However, if the coe cients a b are not given but observed data corresponding to the solution are available, then an inverse problem to reconstruct the coe cients from knowledge of the data arises. This inverse problem is underdetermined because its solution is not unique. For example, when a water velocity vector v satis es rcv ˆ 0 7 which is regarded as the governing continuity equation of air-mixed water, then both of l a; b; u; u ˆ l a; b; ^c e ; u l a; b v; u; u ˆ l a; b v; ^c e ; u will hold. Another example is multiplication of a constant k because l ka; kb; u; u ˆ l ka; kb; ^c e ; u is always satis ed if u is a solution to Eq. (6). Therefore, the inverse problem is approached in the framework of an optimal control problem to minimize a prescribed functional of the coe cients a b, which are considered as the control variables constrained in a certain admissible set where only convexity is assumed. In the present case, a domain observation type functional J a; b is de ned by X J a; b ˆ1 N 2 u a; b u i 8 2 where N is the number of observation points in the domain X, u i the observed value of u at i-th observation point, u a; b is the solution u of Eq. (6). Consid-

3 K. Unami et al. / Advances in Water Resources 23 (2000) 475± ering physical property of di usion, the di usion coef- cient a is explicitly constrained under the condition of a > 0. However, since the solvability of variational problems is guaranteed only under certain conditions as discussed later, the admissible set of both coe cients a b is implicitly determined so that the problem is meaningful. 4. Minimum principle The optimal a b that minimize J a; b are characterized by the minimum principle. Henceforth, J a; b is abbreviated as J. Now, suppose that a b are optimal ones u ˆ u a; b. Variations in a b are denoted by da db, respectively, it is assumed that a e ˆ a eda b e ˆ b edb are admissible for any positive e smaller than 1. The G^ateau di erential du of u is de ned by u a e ; b e u du ˆ lim 9 e! 0 e then the G^ateau di erential dj of J is deduced as J a e ; b e J dj ˆ lim ˆ XN u u i du 10 e! 0 e from Eq. (8). Since J a e ; b e is not less than the minimized functional J, the minimum principle is stated as the inequality dj P 0 but reduced to a convenient form using the adjoint variable p 2 H0 1 which is the solution of the adjoint equation l a; b; p; u ḓef harp; rui hrp; ubi ˆ XN u u i u 11 where l is the adjoint bilinear form of l, for all test functions u 2 H0 1. The bilinear forms l l satisfy l a; b; u; p ˆl a; b; p; u 12 for any pair of p 2 H 1 0 u 2 H 1 0. Since du is in H 1 0 can be substituted into u in Eq. (11), the right h side of Eq. (10) is rewritten transformed as dj ˆ l a; b; p; du ˆl a; b; du; p ˆ l da; db; u; p ˆ h rp ru; dai hurp; dbi 13 because l a e ; b e ; u a e ; b e ; p l a; b; u; p lim e!0 e ˆ l a; b; du; p l da; db; u; p ˆ0 14 holds for any p. Thus, the minimum principle is obtained as dj ˆ hs a ; dai hs b ; dbip 0 15 where S a S b are the sensitivities de ned by rp ru urp, respectively. This minimum principle also suggests that optimal control variables may be searched in the negative directions of the sensitivities. 5. Solvability of variational problems The variational problems Eqs. (6) (11) have to be solved in order to approach optimal control variables a b. Due to the constraint a > 0 imposed on the diffusion coe cient a, the equivalent normalized forms of these equations are written as ˆ a ; ^c e; u 16 1; b a ; p; u ˆ XN l 1 a u u i u 17 for any u 2 H0 1. However, it is not self-evident that solutions to Eqs. (16) (17) uniquely exist. A su cient condition to guarantee the solvability of Eqs. (16) (17) is given by the Lax-Milgram's theorem [6], in which the bilinear form is assumed to be bounded coercive. In the present case, the bilinear forms l l are said to be bounded if there exists positive M such that 6 M 18 l 1; b 6 M 19 for any pair of u 2 H0 1 u 2 H 0 1, whereas they are said to be coercive if there exists positive l such that P lkuk 2 V 20 l 1; b P lkuk 2 V 21 for any u 2 H0 1. The conditions of Eqs. (18) (19) are always satis ed when M is taken as M ˆ 1 b k 1 a 22 where k 1 is a positive nite number which is equal to inf u2h 1 0 ;u6ˆ0 =kuk according to the Poincare's inequality, because

4 478 K. Unami et al. / Advances in Water Resources 23 (2000) 475±481 ru u b ; ru a 6 jhru; ruij u b a ; ru 6 kru ruk b a kuruk 6 b a kuk 6 1 b k 1 a l ˆ M 1; b ˆ 6 M ˆ M hold. A su cient condition to satisfy Eqs. (20) (21) is given as follows. 1. There exists a decomposition b=a ˆr 0 r 1 such that hr 0 ; rui ˆ 0 for all u 2 H The decomposition b=a ˆr 0 r 1 is supposed to be done so as to minimize kr 1 k, l de ned by l ˆ 1 r 1 k 1 25 is positive. Indeed, the left h sides of Eqs. (20) (21) are reduced to ˆ l 1; b ˆkuk 2 V u b a ; ru ˆkuk 2 V Z ˆkuk 2 V r 0; r u2 2 X r 1 uru dx P kuk 2 V kr 1kkuruk P kuk 2 V r 1 kuk2 V ˆ lkuk 2 V k 1 hur 1 ; rui 26 under this condition, which is satis ed if b=a is nearly divergence free, so that r 1 is small enough. However, the positiveness of l in Eq. (25), thus the solvability of the variational problems, are not guaranteed in general. 6. Numerical method 6.1. Finite element solver for the partial di erential equations The nite element method is used for numerically solving the variational problems Eqs. (6) (11). The domain X is discretized into a mesh comprising of nite number of triangular elements their nodes, which are chosen in such a way that there exists a one-to-one correspondence between the set of all the observation points including ones on the boundary C the set of the nodal points in the meshing. The speci ed boundary value ^c is piecewise linearly interpolated from the observed data on the boundary C. The unknowns u p, whose nodal values are numerically resolved, are interpolated by elementwise linear functions, whereas the coe cients a b, which are estimated in the iterative procedure, are taken as elementwise constants. The stard Galerkin procedure is applied for discretizing Eqs. (6) (11). An elementwise linear function whose nodal values are equal to ^c on the boundary C vanish in the domain X is taken as ^c e. The Gauss± Jordan method is used for solving the resulting system of linear equations system because its matrix to be inverted is not diagonal dominant. If the discretized version of one of the variational problems is not regular, then the method stops Iterative estimation procedure The iterative procedure remedies the control variables a b to minimize the functional J. The estimates of the control variables at the k-th iteration are denoted by a k b k. The sensitivities S a ˆ rpru S b ˆ urp are evaluated from the numerical solutions for u p at each element, but the mean value is used for u in S b. Then, a k b k are calculated as a k ˆ a k 1 as a 27 b k ˆ b k 1 bs b 28 where a b are the relaxation parameters for a b, respectively. If a k is not greater than 0, then it is reset as a k 1 to ful ll the imposition of the constraint a > Initial estimates A constant value is used as an initial estimate for the di usion coe cient a in the whole domain, whereas the convection coe cient b is given as the vector sum of a predetermined water velocity vector v a constant air bubble rise velocity vector which has only positive vertical y-component. The eld of the water velocity vector v is supposed to comprise of continuous two parts of

5 K. Unami et al. / Advances in Water Resources 23 (2000) 475± mainstream roller. In the mainstream part of the eld uniform velocity distribution is assumed in the vertical y-direction, whereas in the roller part the velocity distribution is linearly taken in the same direction. The conservation of mass with respect to the discharge entering into the domain is taken into account. In order to nd such v numerically, a 1-D nite di erence method which assumes the hydrostatic pressure distribution is used rst for determining the interface between the mainstream the roller, then v in the mainstream is calculated. Next, a strip integral method is employed to obtain v in the roller ful lling the continuity requirement. The eld of the vector b=a constructed in this way is nearly divergence free. 7. Demonstrative example The domain X, which models a vertical plain of ow eld comprising of a hydraulic jump where the mainstream ows from the left to the right, is divided into 289 nite elements by 169 nodes as displayed in Fig. 1. The hydraulic jump has been experimentally studied by Chanson [3], the temporally averaged void fraction has been measured at every observation point, which corresponds to one of the 169 nodes, by means of a conductivity probe system. The measured void fraction is converted into the speci c gravity c, then it is used for the observed state variable u i. Fig. 2 shows vertical distributions of the speci c gravity at x ˆ 0.00, 0.10, 0.20, 0.30, 0.45, 0.65 m. From this plot of speci c gravity distribution it can be visualized that c-value decreases in the middle layer of vertical section, however this de ciency is overcome the distribution becomes almost uniform as the ow moves in the horizontal x- direction. The upstream ow depth before entering the hydraulic jump is m, the depth averaged velocity is 2.47 m/s, which works as another constraint imposed on the convection coe cient b. Initial estimates of the coe cients a b are portrayed in Fig. 3. The di usion coe cient a 0 the air bubble rise velocity are uniformly taken as 0.1 m 2 /s 1.0 m/s, respectively, the corresponding speci c gravity eld realized by the initial estimates is depicted in Fig. 4. It can be seen from this gure that there is no signi cant layer of diminished speci c gravity as is clearly found in the observed one. The iterative procedure is implemented setting the values of the relaxation parameters a b as , respectively. Larger values are also tested, but the iterative procedure ends in failure. First order convergence of the functional value is obtained as shown in Fig. 5. After iterations, the value of J is reduced to 5:8 10 4, the maximum di erence between the observed speci c gravity the calculated one is about , which is considered small enough for the practical purposes. Thus, the estimated coe cients a b at th iteration are regarded as an approximate solution to the inverse problem are plotted in Fig. 6. Fig. 1. Finite element mesh for domain X. Fig. 2. Observed speci c gravity eld.

6 480 K. Unami et al. / Advances in Water Resources 23 (2000) 475±481 Fig. 3. Initial estimates of a b. Fig. 4. Speci c gravity eld for the initial estimates. in the result suddenly increases at the lower edge of the layer where the speci c gravity diminishes. 8. Conclusions Fig. 5. Convergence process. The speci c gravity eld for these estimated coe cients a b is shown in Fig. 7. Even though the estimated results are not unique one as mentioned in the formulation of the inverse problem, nonetheless they are considered to be close to the reality because the velocity at the upstream is constrained. The value of the di usion coe cient a ranges from 10 7 to It is large in the layer where the speci c gravity is relatively small. The convection coe cient b is not much di erent from its initial estimates approximately satis es the velocity constraint at the upstream, but its vertical y-component The inverse problem to estimate the coe cients in the convection di usion equation which is assumed to govern the air-mixed ow in a hydraulic jump is formulated as the optimal control problem to minimize the functional, as observed data of speci c gravity are available. Solvability of the convection di usion equation as well as its adjoint equation is guaranteed if the convection coe cient divided by the di usion coe cient is nearly divergence free. Using the iterative procedure which includes the nite element scheme for the resolution of the two equations, the estimation is numerically implemented. In the demonstrative example, the rst order convergence of the speci c gravity eld which the nite element scheme yields to the observed one is established without any failure in the resolution process, even though the inverse problem is underdetermined. The obtained coe cients are not unique but are considered close to the reality. It should be remarked that strong heterogeneity exists in the distribution of the di usion coe cient that the vertical component of the convection coe cient, which is considered as the air

7 K. Unami et al. / Advances in Water Resources 23 (2000) 475± Fig. 6. Estimates of a b at th iteration. Fig. 7. Speci c gravity eld for a b estimated at th iteration. bubble rise velocity, suddenly changes at the lower edge of the layer where the speci c gravity decreases. The knowledge obtained from estimated results shall be used in the development of mathematical numerical models of hydraulic jumps to analyze forward problems. For that purpose, it is still necessary to identify the absolute magnitude of the air bubble rise velocity, which depends on the bubble size the pressure distribution. If the coe cients are far from the divergence free condition, then the procedure presented here is not e ective. This may happen in other air mixed ows such as spillway chute ows, pump intakes, plunging jets, so on. Further investigations are required for equations which have such coe cients are not coercive. References [2] Babb AF, Aus HC. Measurement of air in owing water. Journal of the Hydraulics Division, ASCE 1981;107(HY12):1615±30. [3] H. Chanson, Air bubble entrainment in free-surface turbulent shear ows. San Diego: Academic Press, 1996, pp. 73±92. [4] Hasanov A. Inverse coe cient problems for monotonic potential operators. Inverse Problems 1997;13:1265±78. [5] Ito K, Kunisch K. Estimation of the convection coe cient in elliptic equations. Inverse Problems 1997;13:995±1013. [6] Lax PD, Milgram N. Parabolic equations. Contributions to the Theory of Partial Di erential Equations; Annals of Mathematical Studies 1954;33:167±90. [7] Long D, Ste er PM, Rajaratnam N, Smy PR. Structure of ow in hydraulic jumps. Journal of Hydraulic Research 1991;29(2):207± 18. [8] Sun NZ, Yeh WW-G. Coupled inverse problems in groundwater modeling 1. Sensitivity analysis parameter identi cation. Water Resources Research 1990;26(10):2507±25. [9] Sun NZ, Yeh WW-G. Coupled inverse problems in groundwater modeling 2. Identi ability experimental design. Water Resources Research 1990;26(10):2527±40. [1] R. Adams, Sobolev spaces. New York: Academic Press, 1975, pp. 1±64.