History and Significance of the Morton Number in Hydraulic Engineering

Forum History and Significance of the Morton Number in Hydraulic Engineering Michael Pfister Investigations on Bubble Rise Velocity Research and Teaching Associate, Laboratory of Hydraulic Constructions (LCH), Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 18, Dimensional Analysis CH-1015 Lausanne, Switzerland (corresponding author). E-mail: [email protected] The motion of gas bubbles in boiling water was of particular relevance in the early 19th century, as it affects the water circulation Willi H. Hager, F.ASCE in boilers of steamboats, steam engines, and pumps. Mechanical engineers focused on the latter to optimize their efficiency. Using Professor, Laboratory of Hydraulics, Hydrology and Glaciology (VAW), simplified approaches with stagnant water intended to allow for ETH Zurich, Wolfgang-Pauli-Str. 27, CH-8093 Zürich, Switzerland. analytical approaches, which were not successful according to E-mail: [email protected] Schmidt (1934a), however. He conducted a dimensional analysis of the bubble motion in stagnant fluids, finding that the related Forum papers are thought-provoking opinion pieces or essays processes may be described with the following dimensionless founded in fact, sometimes containing speculation, on a civil en- numbers gineering topic of general interest and relevance to the readership of the journal. The views expressed in this Forum article do not ¼ VbD ð1Þ necessarily reflect the views of ASCE or the Editorial Board of Rb ν the journal. V DOI: 10.1061/(ASCE)HY.1943-7900.0000870 F ¼ pffiffiffiffiffiffib ð2Þ b gD Introduction ρ 2 ¼ VbD ð3Þ Two-phase air-water flows occur frequently in hydraulic engineer- Wb σ ing. Their characteristics, particularly the depth-averaged air concentrations, are of relevance for design. These must be known, for Here, g = gravity acceleration, υ = kinematic viscosity, ρ = instance, to provide an adequate chute freeboard, or to avoid chock- density, σ = surface tension, V = velocity, D = bubble diameter, ing of steep tunnel flows. Currently, two main tools are available to and subscript b refers to a bubble. Schmidt mentioned that the bub- estimate the two-phase flow characteristics: physical modeling and ble (1) Reynolds number Rb describes the ratio of inertia to viscous numerical simulations. Both are, however, subjected to uncertain- forces, (2) Froude number Fb the ratio of inertia to gravity, and ties. Physical models are prone to scale effects, whereas numerical (3) Weber number Wb the ratio of inertia to surface forces. How- simulations are sensitive regarding the code, calibration, as well as ever, Eqs. (1)–(3) do not allow for explicitly deriving Vb. Schmidt the validation of the results. (1934a, b) proposed to rearrange the equations to provide only one A correct reproduction of the air-water flow properties is difficult, explicit term containing Vb, namely, Rb according to Eq. (1). The given that simultaneous modeling of the dominant forces is impos- other two terms are then sible if water is used both in the prototype and the model. For bubbles, these forces are expressed with the Weber W ,ReynoldsR , 2ρ b b Wb ¼ D g ð4Þ 2 and Froude Fb bubble numbers, as demonstrated by Schmidt σ Fb (1934a). His fundamental work and the validation by Haberman and Morton (1953) indicate that the fluid (described by the nondi- pffiffiffiffiffiffiffiffiffiffiffi mensional parameter M) affects the bubble features, at least as long as 3 F2R4 σ W and R are below a certain value. A similar observation was b b ¼ pffiffiffiffiffiffiffi ð5Þ b b W ρ 3 ν4 made for mixture air-water flows. Accordingly, scale effects related b g to air concentrations in modeled flows remain, but are small if these Downloaded from ascelibrary.org by Ecole Polytechnique Federale on 04/10/14. Copyright ASCE. For personal use only; all rights reserved. limits are respected. The parameter M, namely, the Morton number, Note that Eq. (4) includes D, g, and fluid constants, whereas is thus a key variable in modeling air-water two-phase flows. Eq. (5) consists exclusively of fluid properties and g. Consequently This forum paper identifies initial sources proposing the Morton Vb in a stagnant, infinite fluid body is explicitly expressed by non- number, focusing on the individual bubble behavior in a stagnant dimensional numbers and their combinations as (Schmidt 1934a, b) fluid. Related results are then transferred in the framework of an V D D2ρg σ analogy to the general modeling of high-speed two-phase air- b ¼ f ; pffiffiffiffiffiffiffi ð6Þ water flows. These approaches are detailed based on M, although ν σ ρ 3 gν4 this number was proposed for different conditions. Finally, short biographies of the three main persons involved in the development Schmidt (1934a) concluded his investigation with the remark of M are presented, namely, those of Ernst Schmidt, William L. that physical experiments are required to support the above men- Haberman, and of Rose K. Morton-Sayre. tioned hypotheses, in particular including different fluids. © ASCE 02514001-1 J. Hydraul. Eng. J. Hydraul. Eng. Experimental Validation In literature, the group is often simply referred to as the M-group or property group.” Rosenberg (1950) presented a literature survey on the motion of air bubbles in liquids, reporting a considerable scatter and uncertainty. He thus conducted model tests to determine the latter. Based on a preliminary dimensional analysis, he derived the relevant param- Relevance for Physical Modeling eters as the drag coefficient, the Reynolds number, and a third parameter involving only the fluid properties. By referring to Analogy to Physical Modeling Schmidt (1934a), Rosenberg denotes this parameter [derived from High-speed air-water two-phase flows are more complex than bub- Eq. (5)] with the symbol M as bles rising in stagnant water. Selected observations of the “simple” μ4 case are nevertheless interesting to describe the complex flow fea- ¼ g ð7Þ M ρσ3 tures. For instance, it is known from the literature that minimum Reynolds and Weber numbers (with the flow depths as reference) Here, μ = dynamic viscosity. The dependence of the bubble drag, limit scale effects related to air concentrations in two-phase flows. A similar observation was made by Haberman and Morton (1953) shape, and path on Rb was determined. Since the tests were con- fined to one single liquid only, the effect of M remained unascer- concerning the drag of a single bubble. Particular outcomes of “ ” tained. The results indicated that this parameter may have a the latter work are thus conceptually transferred from the simple — considerable effect on the motion and shape of a bubble. case to complex two-phase flows, knowing that these can strictly — Haberman and Morton (1953) continued the work of Rosenberg spoken not be applied one-to-one. by conducting experiments on the drag coefficient of rising air bubbles in various stagnant fluids and under different temperatures. Dynamic Similarity The following liquids were used: Water (6, 19, 21, and 49°C), glim solution, mineral oil, varsol, turpentine, methyl alcohol, olive oil, Physical models are scaled with the geometrical factor λ, resulting syrup, different corn syrup-water mixtures, glycerin-water mix- in the length ratio between a prototype (subscript P) and a model λ ¼ = tures, and an ethyl alcohol-water mixture. The explicit effect of (subscript M) reference length as lP lM. As to the hydraulic M on the bubble rise velocity was systematically studied parameters, these are “translated” from the model to the prototype for 0.3 × 10−11 ≤ M ≤ 0.2 × 10−2. with the dynamic similarity. The latter bases essentially on Isaac The rate of rise velocity was found to mainly depend on the Newton, according to whom the sum of all forces F acting on a liquid viscosity for spherical bubbles, and on surface tension for control volume is equal to its mass m times the related acceleration ellipsoidal bubbles. The drag coefficients of small spherical bub- a (i.e., inertia I ¼ ma). To achieve dynamic similarity between the bles overlapped with these of rigid spheres, whereas a general drag model and the prototype, the ratio of the dominant force F (relative I reduction occurs with increased bubble size and thus larger Rb to inertia ) should consequently be identical. [Fig. 1(a)]. This decrease was explained with the formation of an The dominant gravity, surface tension, and viscous forces of air circulation inside the bubble. For liquids of low M (<10−3), a two-phase air-water flow features are listed, e.g., by Schmidt F ¼ mg l = minimum in the drag curve is reached at Rb ≅ 250, i.e., near the (1934a). For gravity, , so that (with reference length, t = V = transition from spherical to ellipsoidal bubble shape. For Rb > reference time, and fluid velocity) 3 × 103 ≅ 2 6 results the constant drag coefficient . The transi- 3ρ 1 2 2 1 2 ≅ 20 I ¼ ma ¼ l l ¼ l l ¼ V ¼ 2 ð9Þ tion to spherical caps is completed at Wb [Fig. 1(b)]. The 3 2 2 3 F drag coefficient of spherical caps is independent of bubble size F mg l ρ t g t l g lg (i.e., of Rb and Wb) and has the constant value of 2.6 [as noted from Fig. 1(a)]. Hence, the fluid (in terms of M) has no more effect Gravity is the dominant force driving free surface flows, according to Froude similarity. The Froude number F shall thus be identical in on the drag if limit values of Rb and Wb are attained. These main results were published in the Proceedings and Transactions ASCE both the prototype and the model. F ¼ lσ (Haberman and Morton 1954, 1956). Considering surface forces ( ) as relevant, then Haberman (1956) submitted in the framework of his Ph.D. 3 2 I ma l ρ l l ρ 2 ρ thesis a mathematical analysis of the wall effect of spheres, based ¼ ¼ ¼ l ¼ V l ¼ W ð10Þ F lσ lσ t2 t2 σ σ on Stokes’ approximation for the hydrodynamic equations of slow flows.

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