Submitted to the Journal of Hydraulic Engineering, ASCE, January, 2006 NOTE ON THE ANALYSIS OF PLUNGING OF DENSITY FLOWS

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1 Submitte to the Journal of Hyraulic Engineering, ASCE, January, 006 NOTE ON THE ANALYSIS OF PLUNGING OF DENSITY FLOWS Gary Parker 1, Member, ASCE an Horacio Toniolo ABSTRACT This note is evote to the correction of an error in a calculation containe in a classical ublishe aer on the lunging of river flows as they enter lakes or reservoirs. The correction of the error rovies a felicitous result, accoring to which not only the ratio of unerflow thickness just after lunging to eth just before lunging, but also the ensimetric Froue numbers both just before an just after lunging are all secifie as a function of a single arameter. This single arameter characterizes the tenency of lunging to entrain ambient water into the ensity unerflow. INTRODUCTION When relatively enser river water meets relatively lighter ambient flui as it flows into a lake or reservoir the ense river flow often lunges to form a bottom unerflow. When the ensity ifference is meiate by the resence of susene mu in the water column of the river, the resulting unerflow is terme a turbiity current. The henomenon of lunging has been stuie by a number of authors, incluing Singh an Shah (1971), Savage an Brimberg (1975), Jain (1981), Farrell an Stefan (1986) an Bournet et al. (1999). Here attention is focuse on the classical aer ue to Akiyama an Stefan (1984), abbreviate to A&S in the text below. A&S offer a 1D formulation of lunging base on consierations of conservation of volume, mass an momentum. Momentum balance is imose on two control volumes, referre to as Control Volume I an Control Volume II below. The imosition of the conservation of volume, mass an momentum within Control Volume I results in a relation for the ratio H /H of the layer thickness of the unerflow just after lunging H to the eth just before lunging H as a function of a) the ensimetric Froue number just before lunging Fr an b) a imensionless arameter γ that escribes the egree to which lunging entrains ambient water into the unerflow. The further imosition of momentum balance on Control Volume II as a secon relation, so that both Fr an H /H, as well as the ensimetric Froue number Fr just ownstream of lunging are all secifie as functions of γ alone. A&S, however, misse this conclusion ue to an error in their calculation. The error consists of the assumtion of a critical value of Fr (i.e. a value of unity when a relevant shae factor is also set equal to unity) in the case of lunging on 1 1 Prof., Det. of Civil & Environmental Engineering an Det. of Geology, University of Illinois, Ven Te Chow Hyrosystems Laboratory, Urbana IL Asst. Prof., Det. of Civil Engineering, University of Alaska, Fairbanks, AK 75551

2 a stee sloe, an a normal value of Fr in the case of lunging on a mil sloe. These assumtions are not only erroneous but also unnecessary, for the elegant formulation of A&S alreay secifies Fr ineenently. In the resent note the error is correcte, an a new set of relations for lunging is resente. The new relations are lace in both imlicit analytical an exlicit grahical form to allow for easy alication. INCOMPRESSIBILITY AND MASS BALANCE The steay 1D lunging flow from a river into a boy of water escribe in Figure 1 is consiere. Dense river water flows from left to right into the ambient water of a lake, e.g. a reservoir. At oint l in the figure the river water lunges to form a ense unerflow. The ensity of the river water, ρ r is slightly greater than that of the ambient water in the lake, ρ a. The eth an eth-average flow velocity in the river at section just ustream of the lunge oint are H an U, resectively. Just ownstream of the lunge oint at section the layer thickness an layer-average velocity of the ense unerflow are H an U, resectively. Plunging raws ambient water towar the lunge oint at layeraverage velocity U a. The ensity of the unerflow at section is ρ, a value that is greater than ρ a but less than ρ r ue to mixing of the ambient water an river water in the rocess of lunging. Points an are sufficiently close to each other to aroximate the thickness of the ambient water at section as equal to H H. Here lunging is treate as a shock in the same way as a hyraulic jum. The conitions of incomressibility (volume conservation) an mass conservation require the following resective conitions with resect to Control Volume I; U H + Ua(H UH (1) ρ r UH + ρaua(h ρuh () The ensities ρ r an ρ are written as ρ r = ρa( 1+ εr ), ρ = ρa(1+ ε) (a,b) where ε r << 1, ε << 1 an ε < ε r. Defining a coefficient of mixing γ of ambient water into the unerflow as Ua(H γ = UH (4) it follows that Eq. (1) can be rewritten as U H (1+ γ) = UH (5) Eq. () reuces with Eqs. (), (4) an (5) to the result εr ε = 1+ γ (6) The above formulation an results are ientical to those aearing in A&S.

3 MOMENTUM BALANCE: CONTROL VOLUME I Momentum balance over Control Volume I is now ursue. For simlicity the vertical rofiles of ensity an velocity are assume to be uniform in the river flow at section just ustream of the lunge oint, the ense unerflow at section just ownstream of the lunge oint, an the flow of ambient water at section just ownstream of the lunge oint. These assumtions correson to setting the shae factors S 1 an S of A&S equal to unity, a conition that is easily relaxe if esire. In oint of fact, Parker et al. (1987) foun exerimental values for S 1 an S of 1.00 an 0.99, resectively, for the case of turbi unerflows. The values, however, o not ertain to unerflows immeiately after lunging. Momentum balance over Control Volume I requires that ρ a ( 1+ εr )UH ρa(1+ ε)uh ρaua(h H ) + F F (7) where F an F enote the ressure forces on the left- an right-han sies of the control volume, resectively. These ressure forces are assume to be hyrostatic, an are given by the relations H H F = y F, = 0 y (8a,b) 0 where y enotes an uwar vertical coorinate from the be an enotes ressure. The relations for ressure at sections an are, resectively = ρ g(1 + ε )(H y) a r ρag(h y), H < y < H (9a,b) = ρag(h H ) + ρa(1+ ε)(h y), 0 < y < H where g enotes the acceleration of gravity. Evaluating Eqs. (8a,b) with Eqs. 9(a,b), it is foun that 1 F = ρag(1 + εr )H (10a,b) 1 1 F = ρag( H H ) + ρag(1 + ε)h Substituting Eqs. (10a,b) into Eq. (7), an alying the stanar Boussinesq aroximation following from the assumtions ε r << 1 an ε << 1, it is foun that 1 1 U H UH Ua(H + εrgh εgh (11a) Reucing Eq. (11a) further with Eqs. (5) an (6), momentum balance in Control Volume I takes the form UH(1 + γ) UHγ 1 1 εr U H + εrgh gh (11b) H H H 1+ γ

4 The ensimetric Froue number Fr just ustream of the lunge oint, ensimetric Froue number Fr just ownstream of the lunge oint an the ratio ϕ are efine as follows; U U H Fr =, Fr =, ϕ = (1a,b,c) εrgh εgh H Eq. (11) can be rewritten with Eqs. (1a) an (1c) in the form ( 1+ γ) γ 1 1 ϕ Fr Fr Fr + (1) ϕ (1 ϕ) (1+ γ) Between Eqs. (1a,b,c), (5) an (6) it is foun that (1 + γ) Fr = Fr (14) ϕ Eq. (1) allows for the comutation of the ratio ϕ as a function of the ensimetric Froue number Fr just ustream of lunging an the mixing ratio γ. That is, if ε r, U, H an γ are secifie, the values ε, U an H just ownstream of lunging can be comute from Eq. (1) an Eqs. (5), (6), (1a) an (1c). Eq. (1) secifies a fourth-orer olynomial for ϕ. In the limiting case γ 0 it reuces to a thir-orer olynomial in ϕ, which is easily factore an solve to yiel the result Fr ϕ = (15) i.e. a relation of recisely the same form as the one for conjugate eths of a hyraulic jum. MOMENTUM BALANCE: CONTROL VOLUME II As note above, Eq. (1) allows the comutation of the flow just ownstream of the lunge oint if Fr an γ are secifie. A&S go one ste farther, however, to aly momentum balance to Control Volume II of Figure 1. Note that the left-han sie of Control Volume II is locate at the oint l, i.e. lunge oint, rather than at just ustream of it in Figure 1. As a result the flow velocity across the left-han sie of Control Volume II can be set equal to zero. In aition, A&S assume a hyrostatic ressure istribution on both sies of Control Volume II. This assumtion is not likely to be comletely accurate on the left-han sie, because the ownwelling there creates a non-hyrostatic contribution to ressure. Assuming, however, that this non-hyrostatic comonent can be neglecte, the following form for momentum balance is obtaine; 1 1 ρa Ua(H + ρa(1+ εr )g(h ρag(h (16a) or uon reuction 1 Ua (H + εrg(h (16b) 4

5 Further reucing Eq. (16b) with Eq. (4), UH γ 1 = εrg(h (H H ) (17) Between Eqs. (1a, 1c) an (17) the ensimetric Froue number just ustream of the lunge oint is evaluate as 1 Fr = (1 ϕ) (18) γ Substituting Eq. (18) into Eq. (1), the following relation is foun for the ratio ϕ; 1 1 (1 ϕ) ϕ (1 ϕ) (1+ γ) (1 ϕ) + 1 (19) γ γ ϕ (1+ γ) The aition of the momentum balance of Control Volume II is seen to change the character of the roblem so as to rovie extra information. In the case of Eq. (1), it was necessary to secify both Fr an γ (or equivalently Fr an γ) in orer to comute ϕ. It is seen from Eqs. (18) an (19), however, that both Fr an ϕ can be comute once γ is secifie. That is, the aition of Eq. (17) allows for irect comutation of the ensimetric Froue number just ustream of lunging. Rewriting Eq. (19) with the efinition χ = 1 ϕ (0) it is foun that χ γ χ (1+ γ) χ γ + γ (1 χ) 1 χ 1+ γ (1) again yieling a fourth-orer olynomial for χ. In the limit as γ 1 the only solution to Eq. (1) is χ () or thus ϕ = 1, H = H (a,b) i.e. no lunging. That is, Eq. (1) imlies that lunging is imossible in the absence of mixing. Substituting this result irectly into Eq. (18) yiels an ineterminate result for the ensimetric Froue number Fr just ustream of lunging. The ineterminancy can be resolve by seeking an asymtotic exansion in small γ of the solution to Eq. (1) of the form χ = aγ m +... (4) Substituting Eq. () into Eq. (1), the aroriate form is foun to be / χ = 1/ γ +... (5) which when substitute into Eq. (18) yiels the result l imfr = 1 (6) γ 0 5

6 RELATIONS FOR PLUNGING The authors were not able to factor, an thus solve Eq. (1) analytically for the case of finite γ. The equation is, however, reaily solve numerically. Figure shows the results of this solution, augmente with Eqs. (14) an (18), to yiel reictions of ϕ, Fr an Fr as functions of the mixing coefficient γ. The results aear eminently reasonable. For the case γ the execte result is obtaine: Fr an Fr are both equal to unity. Fr then ecreases monotonically below unity as γ increases. Over the range 0 < γ < 0.14 it is seen that Fr is moestly less than unity. At a value of γ near 0.14 it again rises to unity. Over the range γ > 0.14 Fr increases monotonically with increasing γ above unity. The analysis inicates that lunging to suercritical flow is not ossible unless the mixing coefficient γ is sufficiently large. DISCUSSION It is of value to recount in recise etail how the above results iffer from A&S. The resent aer is comletely equivalent to A&S in regar to volume conservation [resent Eq. (1) A&S Eq. (4)], mass conservation [resent Eq. () A&S Eq. (5)], momentum conservation on Control Volume I [resent Eq. (11a) A&S Eq. (16)] an momentum conservation on Control Volume II (resent Eq. (16a) A&S Eq. (17)]. In the analysis of A&S, the resent Eq. (16b) [A&S Eq. (17)] is use to eliminate only the term containing the ambient flow velocity U a in Eq. (11a) [A&S Eq. (16)] to obtain the result 1 1 U H UH + εrghh gεr 1+ H (7) 1+ γ [A&S Eq. (18)]. A&S then solve for U in terms of U using Eq. (5), an reuce using the efinition of Eq. (1b) for the ensimetric Froue number just ownstream of lunging an Eq. (6) to obtain the olynomial 1 + γ 1 1 Fr + Fr + (8) ϕ 1+ γ ϕ 1+ γ [A&S Eq. (0), in which they enote 1/ϕ as the arameter K]. The above equation is comletely consistent with the resent analysis. A&S then make one of two assumtions in Eq. (8): that the ensimetric Froue number just ownstream of lunging Fr is equal to the critical value on stee sloes an the value Fr n associate with normal flow on mil sloes, i.e. the choices Fr = 1 (9a) or Fr = Fr n (9b) 6

7 corresoning to the assumtion of shae factors S 1 an S in A&S equal to unity [A&S Eq. (4) or (40), resectively with S 1 = 1 an S = 1]. In the case of Eq. (9a) this allows for the solution of Eq. (8) in close form, yieling γ + γ 4 = (0) ϕ (1 + γ) 1+ γ [A&S Eq. (46)]. The corresoning solution in the case of Eq. (9b) is A&S Eq. (45). The error in the analysis of A&S is the assumtion of Eqs. (9a,b), accoring to which the ensimetric Froue number Fr. just ownstream of lunging must be equal to either the critical or normal value eening on the sloe. In oint of fact the analysis of A&S itself oes not allow an ineenent secification of Fr. Rather, Fr must be relate to Fr accoring to Eq. (14), which must in turn be relate to ϕ an γ accoring to Eq. (18). Using Eqs. (14) an (18) in conjunction with Eq. (8) results in Eq. (19), which secifies ϕ as a function of γ alone. Once ϕ is comute from this relation, Fr is comute from Eq. (18) an Fr is comute from Eq. (14). It is seen from Figure that the only values of γ resulting in a value of Fr of unity are the two choices 0 an Correction of the error, however, allows a felicitous conclusion. The basic structure of the analysis of A&S (1984), in terms of the conservation of volume, mass an momentum in Control Volume I an the conservation of momentum in Control Volume II is in fact sufficient to secify not only the relation between unerflow thickness H just after lunging an the flow eth H just before lunging, but also the ensimetric Froue numbers Fr an Fr just before an just after lunging, all as functions of a single imensionless arameter γ characterizing mixing at the lunge oint. CONCLUSION The correction of an error in the comutations of Akiyama an Stefan (1984) has a most felicitous result. The basic structure of their analysis roves sufficient to secify all relevant arameters concerning 1D lunging, incluing: a) the ratio of unerflow thickness just ownstream to eth just ustream; an b) the ensimetric Froue numbers just ustream an ownstream, as functions of a single imensionless arameter γ. This arameter characterizes the entrainment of ambient flui into the unerflow at the oint of lunging. The research for this aer was fune by the National Center for Earthsurface Dynamics, which is in turn fune by the Science an Technology Centers (STC) rogram of the National Science Founation. Svetlana Kostic rovie several helful comments. APPENDIX I.--REFERENCES 7

8 Akiyama, J., an Stefan, H. (1984). Plunging flow into a reservoir: Theory. Journal of Hyraulic Engineering, ASCE, 110(4), Bournet, P., Dartus, D, Tassin, B., an Vincon-Leite B. (1999). Numerical Investigation of lunging ensity current. Journal of Hyraulic Engineering, ASCE, 15(6), Farrell, G. J., an Stefan, H. (1986). Mathematical moeling of lunging reservoir flows. Journal of Hyraulic Research, 6(5), Jain, S. C. (1981). Plunging hemonena in reservoirs. Proceeings, Symosium on Surface Water Imounments, ASCE, Minneaolis, MN. U.S.A., June 5, Parker, G., Garcia, M., Fukushima, Y. an Yu, W. (1987). Exeriments on turbiitry currents over an eroible be. Journal of Hyraulic Research, 5(1), Savage, S. B. an Brimberg, J. (1975). Analysis of lunging henomena in water reservoirs. Journal of Hyraulic Research, 1(), Singh, B., an Shah, C.R. (1971). Plunging henomenon of ensity currents in reservoirs. La Houille Blanche, 6(1), APPENDIX II.--NOTATION The following symbols are use in this aer: a = coefficient in Eq. 4; F = ressure force just ownstream of lunging, efine by Eq. 10b; F = ressure force just ustream of lunging, efine by Eq. 10a; Fr = ensimetric Froue number just ustream of lunging, efine by Eq. 1a; Fr = ensimetric Froue number just ownstream of lunging, efine by Eq. 1b; g = gravitational acceleration; H = thickness of ensity unerflow just ownstream of lunging; H = river eth just ustream of lunging; m = exonent in Eq. 4; = ressure; U = layer-average velocity of ensity unerflow just ownstream of lunging; U = layer-average velocity of ensity unerflow just ustream of lunging; y = coorinate efine uwar normal from the be; χ = 1 - ϕ = 1 (H /H ); ε = fractional ensity excess in the unerflow just ownstream of lunging; ε r = fractional ensity excess in the river water just ustream of lunging; γ = imensionless mixing arameter efine by Eq. 4; ϕ = the ratio H /H ; ρ a = ensity of the ambient water in the lake or reservoir; ρ ensity of the unerflow just ownstream of lunging; 8

9 ρ r = ensity of the river water just ustream of lunging. FIGURE CAPTIONS Figure 1. Definition iagram for lunging showing Control Volumes I an II. Figure. Preictions for ϕ, Fr an Fr, as functions of γ. 9

10 Figure 1 Figure 10