A simple hydrodynamic model of a laminar free-surface jet in horizontal or vertical flight

Haustein, Herman D.; Harnik, Ron S.; Rohlfs, Wilko

doi:10.1063/1.4996771

Cited by 18 publications

(20 citation statements)

References 39 publications

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“…A subscript with respect to x or r denotes partial differentiation. We observe from equation (2.4) that the influence of gravity scales according to G, which is in agreement with previous studies (Duda and Vrentas, 1967;Massalha and Digilov, 2013;Haustein et al, 2017). For x > 0, the dynamic boundary condition at the free surface r = R(x) takes the form r•n = We À1 (r•n)n, where n is the normal vector with axial and radial components…”

Section: Governing Equations and Boundary Conditionssupporting

confidence: 91%

“…The Froude number turns out to be not very useful in the current formulation, as the influence of gravity scales the ratio of the hydrostatic pressure to the viscous shear stress. Duda and Vrentas (1967) and Haustein et al (2017) referred to this ratio, G, as the Stokes number. In addition, the Weber number, We, will also be used in this study.…”

Section: Problem Formulation and The Physical Domainmentioning

confidence: 99%

“…The centerline velocity and pressure are often measured or computed in the literature in an effort to capture the flow development and rate of relaxation upstream and downstream of the exit. Although the exponential type decay has been recognized by Bohr (1909) more than a century ago, the rate of decay remains not fully understood, as there has been no agreement on its value (see, for instance, Goren and Wronski, 1966;Duda and Vrentas, 1967;Rohlfs et al, 2014;Haustein et al, 2017). It is generally assumed that the decay can be represented by a one-constant rate as e -l x/Re (x > 0), which prompted Haustein et al (2017) to raise the question whether the flow decay can indeed be described by a constant, l .…”

Section: 1mentioning

confidence: 99%

“…Although the exponential type decay has been recognized by Bohr (1909) more than a century ago, the rate of decay remains not fully understood, as there has been no agreement on its value (see, for instance, Goren and Wronski, 1966;Duda and Vrentas, 1967;Rohlfs et al, 2014;Haustein et al, 2017). It is generally assumed that the decay can be represented by a one-constant rate as e -l x/Re (x > 0), which prompted Haustein et al (2017) to raise the question whether the flow decay can indeed be described by a constant, l . Our present formulation clearly suggests that this would be equivalent to keeping the dominant mode.…”

Section: 1mentioning

confidence: 99%

“…The model is based on a modified Bernoulli's equation and energy considerations, including the jet interfacial energy density and viscous losses. More recently, Haustein et al (2017) carried out a combined theoretical, numerical and experimental study for a wide range of parameters on jets emerging vertically and horizontally (negligible gravity) from a tube. Their theoretical model involves three empirical parameters and leads to a good agreement with experimental data and numerical simulation.…”

Section: Introductionmentioning

confidence: 99%

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An asymptotic-numerical comparative study for axisymmetric free surface liquid jet near and far from exit at high Reynolds number

Wang

Khayat

2020

HFF

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Purpose The purpose of this study is to examine theoretically the axisymmetric flow of a steady free-surface jet emerging from a tube for high inertia flow and moderate surface tension effect. Design/methodology/approach The method of matched asymptotic expansion is used to explore the rich dynamics near the exit where a stress singularity occurs. A boundary layer approach is also proposed to capture the flow further downstream where the free surface layer has grown significantly. Findings The jet is found to always contract near the tube exit. In contrast to existing numerical studies, the author explores the strength of upstream influence and the flow in the wall layer, resulting from jet contraction. This influence becomes particularly evident from the nonlinear pressure dependence on the upstream distance, as well as the pressure undershoot and overshoot at the exit for weak and strong gravity levels, respectively. The approach is validated against existing experimental and numerical data for the jet profile and centerline velocity where good agreement is obtained. Far from the exit, the author shows how the solution in the diffusive region can be matched to the inviscid far solution, providing the desired appropriate initial condition for the inviscid far flow solution. The location, at which the velocity becomes uniform across the jet, depends strongly on the gravity level and exhibits a non-monotonic behavior with respect to gravity and applied pressure gradient. The author finds that under weak gravity, surface tension has little influence on the final jet radius. The work is a crucial supplement to the existing numerical literature. Originality/value Given the presence of the stress singularity at the exit, the work constitutes a superior alternative to a computational approach where the singularity is typically and inaccurately smoothed over. In contrast, in the present study, the singularity is entirely circumvented. Moreover, the flow details are better elucidated, and the various scales involved in different regions are better identified.

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Section: Governing Equations and Boundary Conditionssupporting

confidence: 91%

Section: Problem Formulation and The Physical Domainmentioning

confidence: 99%