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Visualization of an annular viscoplastic jet

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Abstract

The current study investigates the global flow field characteristics of a submerged annular viscoplastic jet. The mass and momentum conservation equations, governing the steady laminar flow field, along with the Bingham rheological model, are numerically solved using a finite-difference scheme. Central and outer recirculation regions typically characterize the flow of a Newtonian annular jet. However, the current visualizations demonstrate the existence of new and unique-to-viscoplastic-fluids flow features. When the yield numbers are small, central and outer recirculation regions exist. However, the extent of the outer region and recirculation intensity of the outer and central regions are found to substantially diminish with the yield number. At intermediate yield numbers, a stagnant, attached-to-the-wall region replaces the outer recirculation while a central, yet weaker one, exists. At high yield numbers, stagnant regions replace the recirculating ones, i.e., flow recirculation throughout the whole flow field is eliminated once a critical yield number is exceeded.

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Abbreviations

d :

Inner nozzle diameter

D :

Outer nozzle diameter

D o :

Outer confinement diameter, D o = 50D

D h :

Hydraulic diameter of annular nozzle, D h = D – d

L r :

Radial extent of the computational domain, L r = D o/2

L x :

Axial extent of the computational domain

P :

Non-dimensional pressure, \( {{P^{*} } \mathord{\left/ {\vphantom {{P^{*} } \rho }} \right. \kern-0pt} \rho }u_{\text{b}}^{2} \)

r :

Non-dimensional radial distance, r*/R

R :

Outer jet radius, D/2

Re :

Reynolds number, \( {{\rho D_{\text{h}} u_{\text{b}} } \mathord{\left/ {\vphantom {{\rho D_{\text{h}} u_{\text{b}} } {\eta_{\text{p}} }}} \right. \kern-0pt} {\eta_{\text{p}} }} \)

u :

Non-dimensional streamwise velocity, u*/u b

u b :

Initial jet streamwise bulk velocity, \( 8\int_{d/2}^{D/2} {{{u^{*} r^{*} {\text{d}}r^{*} } \mathord{\left/ {\vphantom {{u^{*} r^{*} {\text{d}}r^{*} } {\left( {D^{2} - d^{2} } \right)}}} \right. \kern-0pt} {\left( {D^{2} - d^{2} } \right)}}} \)

x :

Non-dimensional streamwise distance, x*/R

Y :

Yield number, \( {{\tau_{\text{y}} D_{\text{h}} } \mathord{\left/ {\vphantom {{\tau_{\text{y}} D_{\text{h}} } {\eta_{\text{p}} u_{\text{b}} }}} \right. \kern-0pt} {\eta_{\text{p}} u_{\text{b}} }} \)

\( \dot{\gamma }_{\text{ij}} \) :

Rate of deformation tensor, \( {{\partial u_{\text{i}} } \mathord{\left/ {\vphantom {{\partial u_{\text{i}} } {\partial x_{\text{j}} }}} \right. \kern-0pt} {\partial x_{\text{j}} }} + {{\partial u_{\text{j}} } \mathord{\left/ {\vphantom {{\partial u_{\text{j}} } {\partial x_{\text{i}} }}} \right. \kern-0pt} {\partial x_{\text{i}} }} \)

\( \dot{\gamma }_{\text{II}} \) :

Second invariant of rate of deformation tensor, \( \dot{\gamma }_{\text{ij}} \) \( \dot{\gamma }_{\text{ij}} \)

η p :

Plastic viscosity

μ eff :

Non-dimensional effective viscosity, \( {{\mu_{\text{eff}}^{ * } } \mathord{\left/ {\vphantom {{\mu_{\text{eff}}^{ * } } {\eta_{\text{p}} }}} \right. \kern-0pt} {\eta_{\text{p}} }} \)

ρ :

Density

τ ij :

Stress tensor element

τ y :

Yield stress

*:

Dimensional quantities

b :

Bulk properties of initial jet

c :

Centerline properties

i :

Initial jet properties, i.e., properties at x = 0

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Acknowledgments

The author acknowledges the support received through a 2012–2013 CSU-AAUP Research Grant.

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  1. Department of Engineering, Central Connecticut State University, New Britain, CT, 06050, USA

    Khaled J. Hammad

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Correspondence to Khaled J. Hammad.

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Hammad, K.J. Visualization of an annular viscoplastic jet. J Vis 17, 5–16 (2014). https://doi.org/10.1007/s12650-013-0190-1

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