scholarly journals Compressible integral representation of rotational and axisymmetric rocket flow

2016 ◽  
Vol 809 ◽  
pp. 213-239 ◽  
Author(s):  
M. Akiki ◽  
J. Majdalani
Keyword(s):  
Numerical Simulations ◽  
Closed Form ◽  
Mass Flux ◽  
Mathematical Formulation ◽  
Circular Cross Section ◽  
Ideal Gas ◽  
Injection Rate ◽  
Dimensional Solution ◽  
Abel Transformation ◽  
Injection Speed

This work focuses on the development of a semi-analytical model that is appropriate for the rotational, steady, inviscid, and compressible motion of an ideal gas, which is accelerated uniformly along the length of a right-cylindrical rocket chamber. By overcoming some of the difficulties encountered in previous work on the subject, the present analysis leads to an improved mathematical formulation, which enables us to retrieve an exact solution for the pressure field. Considering a slender porous chamber of circular cross-section, the method that we follow reduces the problem’s mass, momentum, energy, ideal gas, and isentropic relations to a single integral equation that is amenable to a direct numerical evaluation. Then, using an Abel transformation, exact closed-form representations of the pressure distribution are obtained for particular values of the specific heat ratio. Throughout this effort, Saint-Robert’s power law is used to link the pressure to the mass injection rate at the wall. This allows us to compare the results associated with the axisymmetric chamber configuration to two closed-form analytical solutions developed under either one- or two-dimensional, isentropic flow conditions. The comparison is carried out assuming, first, a uniformly distributed mass flux and, second, a constant radial injection speed along the simulated propellant grain. Our amended formulation is consequently shown to agree with a one-dimensional solution obtained for the case of uniform wall mass flux, as well as numerical simulations and asymptotic approximations for a constant wall injection speed. The numerical simulations include three particular models: a strictly inviscid solver, which closely agrees with the present formulation, and both $k$–$\unicode[STIX]{x1D714}$ and Spalart–Allmaras computations.

scholarly journals Analogue tuning of particle focusing in elasto-inertial flow

Meccanica ◽  
2021 ◽  
Author(s):  
I. Banerjee ◽  
M. E. Rosti ◽  
T. Kumar ◽  
L. Brandt ◽  
A. Russom
Keyword(s):  
Numerical Simulations ◽  
Cross Section ◽  
Immersed Boundary Method ◽  
Circular Cross Section ◽  
Finite Size ◽  
Reynolds Numbers ◽  
Immersed Boundary ◽  
Particle Focusing ◽  
Inertial Flow ◽  
Particle Behaviour

AbstractWe report a unique tuneable analogue trend in particle focusing in the laminar and weak viscoelastic regime of elasto-inertial flows. We observe experimentally that particles in circular cross-section microchannels can be tuned to any focusing bandwidths that lie between the “Segre-Silberberg annulus” and the centre of a circular microcapillary. We use direct numerical simulations to investigate this phenomenon and to understand how minute amounts of elasticity affect the focussing of particles at increasing flow rates. An Immersed Boundary Method is used to account for the presence of the particles and a FENE-P model is used to simulate the presence of polymers in a Non-Newtonian fluid. The numerical simulations study the dynamics and stability of finite size particles and are further used to analyse the particle behaviour at Reynolds numbers higher than what is allowed by the experimental setup. In particular, we are able to report the entire migration trajectories of the particles as they reach their final focussing positions and extend our predictions to other geometries such as the square cross section. We believe complex effects originate due to a combination of inertia and elasticity in the weakly viscoelastic regime, where neither inertia nor elasticity are able to mask each other’s effect completely, leading to a number of intermediate focusing positions. The present study provides a fundamental new understanding of particle focusing in weakly elastic and strongly inertial flows, whose findings can be exploited for potentially multiple microfluidics-based biological sorting applications.

scholarly journals A Theoretically Derived Probability Distribution of Scour

2018 ◽  
Author(s):  
Salvatore Manfreda ◽  
Oscar Link ◽  
Alonso Pizarro
Keyword(s):  
Probability Distribution ◽  
Closed Form ◽  
State Of The Art ◽  
Mathematical Formulation ◽  
Gumbel Distribution ◽  
The State ◽  
Bridge Scour ◽  
Flood Peak ◽  
Hydraulic Conditions ◽  
Model Extension

Based on recent contributions regarding the treatment of unsteady hydraulic conditions into the state-of-the-art of scour literature, the theoretically derived probability distribution of bridge scour is introduced. The model is derived assuming a rectangular hydrograph shape with a given duration, and random flood peak following a Gumbel distribution. A model extension for a more complex flood event is also presented, assuming a synthetic exponential hydrograph shape. The mathematical formulation can be extended to any flood-peak probability distribution. The aim of the manuscript is to move forward the current approaches adopted for the bridge design coupling hydrological, hydraulic, and erosional models in a mathematical closed form.

Elastic–plastic deformation and residual stresses in helical springs

2019 ◽  
Vol 16 (3) ◽  
pp. 448-475
Author(s):  
Vladimir Kobelev
Keyword(s):  
Residual Stresses ◽  
Closed Form ◽  
Cross Section ◽  
Circular Cross Section ◽  
Content Type ◽  
Closed Form Solutions ◽  
Helical Springs ◽  
First Case ◽  
Setting Process ◽  
The Wire

Purpose The purpose of this paper is to develop the method for the calculation of residual stress and enduring deformation of helical springs. Design/methodology/approach For helical compression or tension springs, a spring wire is twisted. In the first case, the torsion of the straight bar with the circular cross-section is investigated, and, for derivations, the StVenant’s hypothesis is presumed. Analogously, for the torsion helical springs, the wire is in the state of flexure. In the second case, the bending of the straight bar with the rectangular cross-section is studied and the method is based on Bernoulli’s hypothesis. Findings For both cases (compression/tension of torsion helical spring), the closed-form solutions are based on the hyperbolic and on the Ramberg–Osgood material laws. Research limitations/implications The method is based on the deformational formulation of plasticity theory and common kinematic hypotheses. Practical implications The advantage of the discovered closed-form solutions is their applicability for the calculation of spring length or spring twist angle loss and residual stresses on the wire after the pre-setting process without the necessity of complicated finite-element solutions. Social implications The formulas are intended for practical evaluation of necessary parameters for optimal pre-setting processes of compression and torsion helical springs. Originality/value Because of the discovery of closed-form solutions and analytical formulas for the pre-setting process, the numerical analysis is not necessary. The analytical solution facilitates the proper evaluation of the plastic flow in torsion, compression and bending springs and improves the manufacturing of industrial components.

Boundary-layer-induced potential flow on an elliptic cylinder

1974 ◽  
Vol 66 (1) ◽  
pp. 145-157 ◽  
Author(s):  
Stanley G. Rubin ◽  
Frank J. Mummolo
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Circular Cross Section ◽  
Elliptic Cylinder ◽  
Analytic Form ◽  
Slender Body ◽  
Dimensional Solution ◽  
Slender Body Theory ◽  
Simple Analytic Form ◽  
Body Theory

The application of slender-body theory to the evaluation of the three-dimensional surface velocities induced by a boundary layer on an elliptic cylinder is considered. The method is applicable when the Reynolds number is sufficiently large so that the thin-boundary-layer approximation is valid. The resulting potential problem is reduced to a two-dimensional consideration of the flow over an expanding cylinder with porous boundary conditions. The limiting solutions for a flat plate of finite span and a nearly circular cross-section are obtained in a simple analytic form. In the former case, within the limitations of slender-body theory, the results are in exact agreement with the complete three-dimensional solution for this geometry.

Motion of an array of drops through a cylindrical tube

1998 ◽  
Vol 358 ◽  
pp. 1-28 ◽  
Author(s):  
C. COULLIETTE ◽  
C. POZRIKIDIS
Keyword(s):  
Stokes Flow ◽  
Apparent Viscosity ◽  
Numerical Solutions ◽  
Initial Period ◽  
Mathematical Formulation ◽  
Axisymmetric Flow ◽  
Circular Cross Section ◽  
Cylindrical Tube ◽  
Boundary Integral ◽  
Tube Radius

We study the pressure-driven transient motion of a periodic file of deformable liquid drops through a cylindrical tube with circular cross-section, at vanishing Reynolds number. The investigations are based on numerical solutions of the equations of Stokes flow obtained by the boundary-integral method. It is assumed that the viscosity and density of the drops are equal to those of the suspending fluid, and the interfaces have constant tension. The mathematical formulation uses the periodic Green's function of the equations of Stokes flow in a domain that is bounded externally by a cylindrical tube, which is computed by tabulation and interpolation. The surface of each drop is discretized into quadratic triangular elements that form an unstructured interfacial grid, and the tangential velocity of the grid-points is adjusted so that the mesh remains regular for an extended but limited period of time. The results illustrate the nature of drop motion and deformation, and thereby extend previous studies for axisymmetric flow and small-drop small-deformation theories. It is found that when the capillary number is sufficiently small, the drops start deforming from a spherical shape, and then reach slowly evolving quasi-steady shapes. In all cases, the drops migrate radially toward the centreline after an initial period of rapid deformation. The apparent viscosity of the periodic suspension is expressed in terms of the effective pressure gradient necessary to drive the flow at constant flow rate. For a fixed period of separation, the apparent viscosity of a non-axisymmetric file is found to be higher than that of an axisymmetric file. In the case of non-axisymmetric motion, the apparent viscosity reaches a minimum at a certain ratio of the drop separation to tube radius. Drops with large effective radii to tube radius ratios develop slipper shapes, similar to those assumed by red blood cells in flow through capillaries, but only for capillary numbers in excess of a critical value.

scholarly journals Revisiting the Agung 1963 volcanic forcing – impact of one or two eruptions

2019 ◽  
Vol 19 (15) ◽  
pp. 10379-10390 ◽  
Author(s):  
Ulrike Niemeier ◽  
Claudia Timmreck ◽  
Kirstin Krüger
Keyword(s):  
Mass Flux ◽  
Radiative Forcing ◽  
Aerosol Particles ◽  
Injection Rate ◽  
Meridional Transport ◽  
The Third ◽  
Volcanic Clouds ◽  
Different Seasons ◽  
Future Emission ◽  
The Individual

Abstract. In 1963 a series of eruptions of Mt. Agung, Indonesia, resulted in the third largest eruption of the 20th century and claimed about 1900 lives. Two eruptions of this series injected SO2 into the stratosphere, which can create a long-lasting stratospheric sulfate layer. The estimated mass flux of the first eruption was about twice as large as the mass flux of the second eruption. We followed the estimated emission profiles and assumed for the first eruption on 17 March an injection rate of 4.7 Tg SO2 and 2.3 Tg SO2 for the second eruption on 16 May. The injected sulfur forms a sulfate layer in the stratosphere. The evolution of sulfur is nonlinear and depends on the injection rate and aerosol background conditions. We performed ensembles of two model experiments, one with a single eruption and a second one with two eruptions. The two smaller eruptions result in a lower sulfur burden, smaller aerosol particles, and 0.1 to 0.3 Wm−2 (10 %–20 %) lower radiative forcing in monthly mean global average compared to the individual eruption experiment. The differences are the consequence of slightly stronger meridional transport due to different seasons of the eruptions, lower injection height of the second eruption, and the resulting different aerosol evolution. Overall, the evolution of the volcanic clouds is different in case of two eruptions than with a single eruption only. The differences between the two experiments are significant. We conclude that there is no justification to use one eruption only and both climatic eruptions should be taken into account in future emission datasets.

Direct Numerical Simulations of Micro-Bubble Expansion in Gas Embolotherapy

2004 ◽  
Vol 126 (6) ◽  
pp. 745-759 ◽  
Author(s):  
Tao Ye ◽  
Joseph L. Bull
Keyword(s):  
Numerical Simulations ◽  
Bubble Dynamics ◽  
Moving Boundary ◽  
Vessel Diameter ◽  
Ideal Gas ◽  
Direct Numerical Simulations ◽  
Operating Conditions ◽  
Droplet Vaporization ◽  
Potential Development ◽  
Micro Bubble

We are currently developing a novel gas embolotherapy technique that involves the selective, acoustic vaporization of liquid perfluorocarbon droplets in or near a tumor as a possible treatment for cancer. The resulting bubbles can then stick within the tumor vasculature to occlude blood flow and “starve” the tumor. The potential development of high stresses during droplet vaporization is a major concern for safe implementation of this technique. No prior study, either experimentally or theoretically, addresses this important issue. In this work, the acoustic vaporization procedure of the therapy is investigated by direct numerical simulations. The nonlinear, multiphase, computational model is comprised of an ideal gas bubble surrounded by liquid inside a long tube. Convective and unsteady inertia, viscosity, and surface tension affect the bubble dynamics and are included in this model, which is solved by a novel fixed-grid, sharp-interface, moving boundary method. We assess the potential for flow-induced wall stresses to rupture the vessel or damage the endothelium during vaporization under a range of operating conditions by varying dimensionless parameters—Reynolds, Weber, and Strouhal numbers, inertial energy and initial droplet size. It is found that the wall pressure is typically highest at the start of the bubble expansion, but the maximum wall shear stress occurs at a later time. Smaller initial bubble diameters, relative to the vessel diameter, result in lower wall stresses.

Simple Predictions for the Sonic Conditions in a Real Gas

1977 ◽  
Vol 99 (1) ◽  
pp. 217-225 ◽  
Author(s):  
P. A. Thompson ◽  
D. A. Sullivan
Keyword(s):  
Equation Of State ◽  
Closed Form ◽  
Thermodynamic Parameters ◽  
Ideal Gas ◽  
Accurate Prediction ◽  
Bernoulli Equation ◽  
Real Gas ◽  
Similarity Principle ◽  
Isentropic Flow ◽  
The Ideal

The steady isentropic flow of a fluid which satisfies an arbitrary equation of state is treated, with emphasis on the prediction of pressure, density, velocity, and massflow at the sonic state. The isentrope P(v) is described by a limited number of thermodynamic parameters, the most important ones being the soundspeed c and fundamental derivative Γ. Using this description, an application of the Bernoulli equation and appropriate thermodynamic relations yields simple closed-form predictions for the sonic state. These predictions are recognizable as generalizations of well-known ideal gas formulas, but are applicable to fluids very far removed from the ideal gas state, even including liquids. Comparisons in several cases for which precise independent solutions are available suggest that the methods found here are accurate. A derived similarity principle allows the accurate prediction of sonic properties from any single given sonic property.

Torsion of Composite Elastic Bars of Arbitrary Cross Section

1968 ◽  
Vol 90 (3) ◽  
pp. 435-440 ◽  
Author(s):  
E. M. Sparrow ◽  
H. S. Yu
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Elastic Properties ◽  
Cross Section ◽  
Excellent Agreement ◽  
Solution Method ◽  
Circular Cross Section ◽  
High Accuracy ◽  
Arbitrary Cross Section ◽  
Closed Form Solutions

A method of analysis is presented for determining closed-form solutions for torsion of inhomogeneous prismatic bars of arbitrary cross section, the inhomogeneity stemming from the layering of materials of different elastic properties. It is demonstrated that the solution method is very easy to apply and provides results of high accuracy. As an application, solutions are obtained for the torsion of a bar of circular cross section consisting of two materials separated by a plane interface. The results are compared with those of various limiting cases and excellent agreement is found to exist. Among the limiting cases, an exact solution was derived by Green’s functions for the problem in which the interface between the materials coincides with a diameter of the circular cross section.

Computation of Rotor/Stator Interaction With Hydrogen-Fuelled Combustion

2003 ◽  
Author(s):  
Masanori Sato ◽  
Takashi Nagumo ◽  
Kazuyuki Toda ◽  
Makoto Yamamoto
Keyword(s):  
Numerical Simulation ◽  
Numerical Simulations ◽  
Injection Rate ◽  
Hydrogen Combustion ◽  
Present System ◽  
Hydrogen Fuel ◽  
General Aspect ◽  
3 Dimensional ◽  
Low Emission ◽  
Rotor Stator Interaction

For the next-generation aircraft, a new propulsion system using hydrogen fuel has been proposed. In the present system, hydrogen fuel injected from a stator surface combusts in the turbine passages, accordingly, the conventional combustor can be cut out. The advantage of this system is that we can design a lighter and smaller engine with low emission. We have demonstrated the realizability of this system by using the cycle analysis and the numerical simulations. Through the previous studies, it was confirmed that the rotor/stator interaction has to be investigated, because the hydrogen combustion phenomena within the stator passage is so complex, and thus it would highly affect the rotor performance. In this paper, we focus on the rotor/stator interaction for the detailed investigation of realizability of this system. The 2- and 3-dimensional numerical simulations are performed for a single stage turbine with hydrogen-fuelled combustion. In the 2-dimensional study, the effects of the injection position and injection rate on the flow structure, the static temperature over the blades, and the blade performance are investigated. Furthermore, 3-dimensional numerical simulation is performed. The general aspect of 3-dimensional flow field is demonstrated, and the effect of hydrogen combustion on the components of turbine, for example hub, tip and blade, are investigated.

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