Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue5.2000:

Estimation of di€usion and 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 606-8502, Japan

b

Institute of Irrigation and Drainage Engineering, Mehran University of Technology and Engineering, Jamshoro, Sindh, Pakistan

c

Division of Biological Production System, Faculty of Agriculture, Gifu University, 1-1 Yanagido, Gifu 501-1112, Japan Received 23 March 1999; accepted 16 October 1999

Abstract

Di€usion and 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 and 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 and 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 and 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 hy-draulic jump is usually accompanied by an aerated re-circulating roller on the top, where intensive turbulence results in energy dissipation and the mixing of air also takes place. The development mechanism of such a roller has been investigated using high speed video cameras [7], and it is revealed that the roller is made up of several vortices which are generated at the toe of the hydraulic jump and 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, and temporally averaged quantities are examined. The steady state ap-proach is rather signi®cant for evaluating re-oxygena-tion e€ect in polluted rivers, energy dissipare-oxygena-tion e€ect in stilling basins of spillways, and 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

dif-fusivity and 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 grav-ity, 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 prob-lem is posed to estimate its di€usion and convection coecients from observed speci®c gravity data. For the sake of simplicity, considerations are restricted to ver-tically 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 re-quires particular attention to the convection term. Since the approaches to inverse coecient problems for con-vection 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.

The minimum principle includes the adjoint equation of the convection di€usion equation, and 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 and the convection coecients. To con®rm the validity of the methodology a sample computation is demonstrated by using ob-served 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 symbolX

is used for a domain which is a bounded open set in 2-D Euclidean space. The inner product between two 2-D vectorsuandvis denoted byuv. The absolute value of a scalaruis represented byjuj, and the Euclidean norm of a 2-D vectoruis also denoted byjuj. The class of all measurable scalar valued functionsu, de®ned onX, for whichR

Xjuj 2

dX<1 is denoted by L2

1. Similarly, L22 is

de®ned for 2-D vector valued functions u for which

R The norms de®ned in L2₁ and L2₂ from their respective inner products are commonly denoted by k k. For a scalar functionu, another normkuk₁is de®ned as

kuk1ˆ

all twice continuously di€erentiable functions u such thatkuk1<1. The completion ofCB2with respect to the

norm Eq. (1) is the Sobolev spaceH1_{, equipped with the}

norm Eq. (1). Let C₀1 be the class of all in®nite times continuously di€erentiable functions which have com-pact support in the domainX. The closure ofC₀1in the Sobolev space H1 is another Sobolev space H₀1. The Sobolev spaceH₀1 is equipped with a norm de®ned by

kukV ˆ kruk …2†

Considering mass conservation of air and 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† ˆdef ÿ r arc‡ r cbˆ0 …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 and air bubble rise velocity vector; andc

is the speci®c gravity in the domain Xbounded by its boundaryC. Thex- andy-coordinates are taken as the horizontal and vertical ones, respectively.

A boundary value problem of Dirichlet type is con-sidered prescribing a boundary condition

cˆ^c …5†

wherec^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† ˆdefh…aruÿub†;rui ˆ ÿl…a;b; ^ce;u† …6†

where u is the state variable, l the bilinear form of the convection di€usion equation, and^c_e is a particular ar-bitraryH1function that accords with^con the boundary

C, for all test functionsu2H₀1. Thus, the state variable

u is determined as cÿ^c_e, the residual of the speci®c gravity after subtracting ^c_e.

Generally speaking, if the coecients a2L2₁ and

b2L2₂ are predetermined, then a forward problem is solved to ®nduwhich satis®es Eq. (6) inH1

0. However, if

the coecients aandbare not given but observed data corresponding to the solution are available, then an in-verse 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 vectorvsatis®es

r cvˆ0 …7†

which is regarded as the governing continuity equation of air-mixed water, then both of l…a;b;u;u† ˆ ÿl…a;b; ^c_e;u†and l…a;b‡v;u;u† ˆ ÿl…a;b‡v; ^c_e;u†

will hold. Another example is multiplication of a con-stant k because l…ka;kb;u;u† ˆ ÿl…ka;kb; ^ce;u† is

al-ways satis®ed ifu 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 andb, 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 func-tionalJ…a;b†is de®ned by

where N is the number of observation points in the domainX,uithe observed value ofuati-th observation

Consid-ering physical property of di€usion, the di€usion coef-®cientais 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cientsa

and b is implicitly determined so that the problem is meaningful.

4. Minimum principle

The optimal a andb that minimize J…a;b†are char-acterized by the minimum principle. Henceforth,J…a;b†

is abbreviated asJ. Now, suppose thataand bare op-timal ones and uˆu…a;b†. Variations in a and b are denoted by da and db, respectively, and it is assumed that ae_{ˆa‡eda and b}e_ˆ

b‡edb are admissible for any positiveesmaller than 1. The G^ateau di€erentialdu

ofu is de®ned by

duˆ lim

e!‡0

u…ae;be† ÿu

e …9†

and then the G^ateau di€erentialdJ ofJ is deduced as

dJ ˆ lim

mized functionalJ, the minimum principle is stated as the inequalitydJP0 but reduced to a convenient form

using the adjoint variablep2H1

0which is the solution of

the adjoint equation

where l is the adjoint bilinear form of l, for all test functionsu2H1

0. The bilinear forms landl satisfy l…a;b;u;p† ˆl…a;b;p;u† …12†

for any pair ofp2H1

0 andu2H01. Sinceduis inH01and

can be substituted intouin Eq. (11), the right hand side of Eq. (10) is rewritten and transformed as

dJ ˆl…a;b;p;du† ˆl…a;b;du;p†

holds for any p. Thus, the minimum principle is ob-tained as

dJˆhSa;dai ‡hSb;dbiP0 …15†

whereSaandSbare the sensitivities de®ned byÿrp ru

and 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) and (11) have to be solved in order to approach optimal control variables a

and b. Due to the constrainta>0 imposed on the dif-fusion coecienta, the equivalent normalized forms of these equations are written as

l 1;b

0. However, it is not self-evident that

so-lutions to Eqs. (16) and (17) uniquely exist. A sucient condition to guarantee the solvability of Eqs. (16) and (17) is given by the Lax-Milgram's theorem [6], in which the bilinear form is assumed to beboundedandcoercive. In the present case, the bilinear formslandlare said to bebounded if there exists positiveMsuch that

l 1;b

to becoerciveif there exists positive lsuch that

l 1;b

The conditions of Eqs. (18) and (19) are always sat-is®ed whenMis taken as

M ˆ1‡ b

where k1 is a positive ®nite number which is equal to

inf_u2H1

0;u6ˆ0…kukV=kuk† according to the Poincar

ru

A sucient condition to satisfy Eqs. (20) and (21) is given as follows.

Indeed, the left hand sides of Eqs. (20) and (21) are reduced to

under this condition, which is satis®ed if b=a is nearly divergence free, so thatr1is small enough. However, the

positiveness oflin Eq. (25), and 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) and (11). The domainXis discretized into a mesh comprising of ®nite number of triangular elements and 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 boundaryCand the set of the nodal points in the meshing. The speci®ed boundary value ^c is piecewise linearly interpolated from the ob-served data on the boundaryC. The unknownsuandp, whose nodal values are numerically resolved, are inter-polated by elementwise linear functions, whereas the coecientsaandb, which are estimated in the iterative procedure, are taken as elementwise constants.

The standard Galerkin procedure is applied for dis-cretizing Eqs. (6) and (11). An elementwise linear func-tion whose nodal values are equal to^con the boundary

Cand vanish in the domainXis taken as^c_e. The Gauss± Jordan method is used for solving the resulting system of linear equations system because its matrix to be in-verted is not diagonal dominant. If the discretized ver-sion of one of the variational problems is not regular, then the method stops.

6.2. Iterative estimation procedure

The iterative procedure remedies the control variables

a and b to minimize the functionalJ. The estimates of the control variables at thek-th iteration are denoted by

a…k† _{and b}…k†_{. The sensitivities S}

aˆ ÿrp ru and

Sbˆurpare evaluated from the numerical solutions for uandpat each element, but the mean value is used foru

in Sb. Then,a…k† andb…k†are calculated as

a…k†ˆa…kÿ1†ÿaSa …27†

and

b…k†ˆb…kÿ1†ÿbSb …28†

whereaandbare the relaxation parameters foraandb, respectively. Ifa…k†is not greater than 0, then it is reset as

a…kÿ1†to ful®ll the imposition of the constraint a>0.

6.3. Initial estimates

A constant value is used as an initial estimate for the di€usion coecientain the whole domain, whereas the convection coecient b is given as the vector sum of a predetermined water velocity vectorvand a constant air bubble rise velocity vector which has only positive ver-ticaly-component. The ®eld of the water velocity vector

mainstream and roller. In the mainstream part of the ®eld uniform velocity distribution is assumed in the vertical y-direction, whereas in the roller part the ve-locity 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 suchvnumerically, a 1-D ®nite di€erence method which assumes the hydrostatic pressure distribution is used ®rst for determining the interface between the mainstream and the roller, and thenvin the mainstream is calculated. Next, a strip integral method is employed to obtain v in the roller ful®lling the continuity re-quirement.

The ®eld of the vectorb=a constructed in this way is nearly divergence free.

7. Demonstrative example

The domainX, which models a vertical plain of ¯ow ®eld comprising of a hydraulic jump where the main-stream ¯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], and 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 gravityc, and then it is used for the observed state variableui. Fig. 2 shows vertical

distributions of the speci®c gravity at x ˆ 0.00, 0.10, 0.20, 0.30, 0.45, and 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 and the distribution becomes almost uniform as the ¯ow moves in the horizontal x -direction. The upstream ¯ow depth before entering the hydraulic jump is 0.017 m, and the depth averaged ve-locity is 2.47 m/s, which works as another constraint imposed on the convection coecientb.

Initial estimates of the coecientsa and b are por-trayed in Fig. 3. The di€usion coecienta…0†and the air bubble rise velocity are uniformly taken as 0.1 m2_{/s and}

1.0 m/s, respectively, and 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 parametersaandbas 10ÿ7_and

10ÿ6_{, respectively. Larger values are also tested, but the}

iterative procedure ends in failure. First order conver-gence of the functional value is obtained as shown in Fig. 5. After 5105_{iterations, the value ofJis reduced}

to 5:810ÿ4_{, and the maximum di€erence between the}

observed speci®c gravity and the calculated one is about 0.0104, which is considered small enough for the prac-tical purposes. Thus, the estimated coecientsaandbat 5105_{-th iteration are regarded as an approximate}

solution to the inverse problem and are plotted in Fig. 6.

Fig. 1. Finite element mesh for domainX.

The speci®c gravity ®eld for these estimated coecients

a andb is shown in Fig. 7. Even though the estimated results are not unique one as mentioned in the formu-lation 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cienta ranges from 10ÿ7 _{to 10}ÿ1_{. 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 and approximately satis®es the velocity constraint at the upstream, but its verticaly-component

in the result suddenly increases at the lower edge of the layer where the speci®c gravity diminishes.

8. Conclusions

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 for-mulated as the optimal control problem to minimize the functional, as observed data of speci®c gravity are available. Solvability of the convection di€usion equa-tion as well as its adjoint equaequa-tion 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 reso-lution of the two equations, the estimation is numeri-cally 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 con-sidered close to the reality. It should be remarked that strong heterogeneity exists in the distribution of the di€usion coecient and that the vertical component of the convection coecient, which is considered as the air

Fig. 3. Initial estimates ofaandb.

Fig. 4. Speci®c gravity ®eld for the initial estimates.

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 and 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 and the pressure distribution. If the coecients are far from the divergence free con-dition, 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, and so on. Further investigations are required for equations which have such coecients and are notcoercive.

References

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[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 and 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 and experimental design. Water Re-sources Research 1990;26(10):2527±40.

Fig. 6. Estimates ofaandbat 5105_{-th iteration.}