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Notes: Summary  The nonlinear dynamics of axisymmetric, inviscid, incompressible, thin, annular liquid jets subjected to fluctuating body forces is studied numerically by means of an adaptive finite difference method which maps the time-dependent, curvilinear geometry of the jet into a unit square. The fluctuating body forces may arise from fluctuations in the gravitational acceleration in inertial frames or from the acceleration of a non-inertial frame of reference which translates parallelly to an inertial one. It is shown that both the pressure coefficient and the axial location at which the annular jet becomes a solid one are periodic functions of time with a period equal to that of the imposed body force fluctuations, and that their magnitude increases as the amplitude of the body force fluctuations is increased. It has also been shown that, for both intermittent, sinusoidal or rectangular excitations, increases in the frequency of the excitation result in the creation of superharmonics, broad, albeit peaked, spectra, and closed phase planes with many loops.

Notes: Abstract A domain-adaptive technique which maps a time-dependent, curvilinear geometry into a unit square is used to determine the steady state mass absorption rate and the collapse of annular liquid jets. A method of lines is used to solve the one-dimensional fluid dynamics equations written in weak conservation-law form, and upwind differences are employed to evaluate the axial convective fluxes. The unknown, time-dependent, axial location of the downstream boundary is determined from the solution of an ordinary differential equation which is nonlinearly coupled to the fluid dynamics and gas concentration equations. The equation for the gas concentration in the annular liquid jet is written in strong conservation-law form and solved by means of a method of lines at high Peclet numbers and a line Gauss-Seidel method at low Peclet numbers. The effects of the number of grid points along and across the annular jet, time step, and discretization of the radial convective fluxes on both the steady state mass absorption rate and the jet's collapse rate have been analyzed on staggered and non-staggered grids. The steady state mass absorption rate and the collapse of annular liquid jets are determined as a function of the Froude, Peclet and Weber numbers, annular jet's thickness-to-radius ratio at the nozzle exit, initial pressure difference across the annular jet, nozzle exit angle, temperature of the gas enclosed by the annular jet, pressure of the gas surrounding the jet, solubilities at the inner and outer interfaces of the annular jet, and gas concentration at the nozzle exit. It is shown that the steady state mass absorption rate is proportional to the inverse square root of the Peclet number except for low values of this parameter, and that the possible mathematical incompatibilities in the concentration field at the nozzle exit exert a great influence on the steady state mass absorption rate and on the jet collapse. It is also shown that the steady state mass absorption rate increases as the Weber number, nozzle exit angle, gas concentration at the nozzle exit, and temperature of the gases enclosed by the annular liquid jet are increased, but it decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-to-radius ratio at the nozzle exit are increased. It is also shown that the annular liquid jet's collapse rate increases as the Weber number, nozzle exit angle, temperature of the gases enclosed by the annular liquid jet, and pressure of the gases which surround the jet are increased, but decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-toradius ratio at the nozzle exit are increased. It is also shown that both the ratio of the initial pressure of the gas enclosed by the jet to the pressure of the gas surrounding the jet and the ratio of solubilities at the annular liquid jet's inner and outer interfaces play an important role on both the steady state mass absorption rate and the jet collapse. If the product of these ratios is greater or less than one, both the pressure and the mass of the gas enclosed by the annular liquid jet decrease or increase, respectively, with time. It is also shown that the numerical results obtained with the conservative, domain-adaptive method of lines technique presented in this paper are in excellent agreement with those of a domain-adaptive, iterative, non-conservative, block-bidiagonal, finite difference method which uncouples the solution of the fluid dynamics equations from that of the convergence length.

Notes: A finite-difference study of a steady, incompressible, viscous, magnetohydrodynamic (MHD) channel flow which has direct application to dc electromagnetic pumps is presented. The study involves the numerical solution of the coupled Navier–Stokes and Maxwell equations at low magnetic Reynolds numbers. It is shown that the axial velocity profiles have a characteristic M shape as the fluid approaches and passes the electrode. The electric potential varies almost linearly from the channel centerline to the channel wall. The current shows a steep gradient near the electrodes. Comparison between the finite-difference solution and a quasi-one-dimensional approach are presented. The two-dimensional numerical calculations predict a larger pressure rise, a smaller net current, and a smaller pump efficiency than the quasi-one-dimensional model.

Notes: Abstract An adaptive, block-bidiagonal finite difference method is used to study the response of annular liquid jets to the injection of mass into the volume enclosed by the annular jet. It is shown that the annular jet's response is characterized by damped oscillations in both the convergence length and the pressure of the gases enclosed by the jet, and that these oscillations are similar to those of an underdamped mass-spring-dashpot system. It is also shown that the maximum amplitude and the number of these oscillations increase as the nozzle exit angle, the initial pressure ratio across the annular jet, the pressure of the gases surrounding the jet, the Weber number, and the rate and duration of the mass injection are increased, are nearly independent of the annular jet's thickness-to-radius ratio at the nozzle exit, and decrease as the Froude number is increased. The numerical calculations presented in this paper indicate that the pressure of the gases enclosed by the annular liquid jet responds instantaneously to the mass injection, whereas there is a lag in the response of the convergence length. This lag is due to the inertia of the jet and assumption that the gases enclosed by the jet are isothermal, and decreases as the injection duration is increased. The calculations also indicate that the critical pressure coefficient of unity determined from the solution of the steady state governing equations can be exceeded without affecting the stability of the annular jet.

Notes: Abstract A simplified analysis of the laminar boundary layer along an isothermal liquid curtain falling under gravity is presented. The analysis uses a von Kármán-Pohlhausen integral method and includes the effects of gravity, pressure differences, surface tension and nozzle exit geometry on the convergence length of liquid curtains which have applications as chemical reactors and as protection systems in laser fusion reactors. It is shown that the effects of the surrounding gases on the curtain shape and convergence length are small, and that good approximations to the liquid curtain shape can be obtained by using inviscid flow analyses.

Notes: Summary A study of isothermal gas absorption by underpressurized, axisymmetric, thin, inviscid, incompressible, annular liquid jets which form enclosed volumes, where hazardous wastes may be burned, is presented. The study considers the nonlinear dynamical coupling between the fluid dynamics of, and the gases enclosed by, the annular liquid jet. It assumes equilibrium conditions at the interfaces, and employs Sievert's solubility law to determine the gas concentration at the gas-liquid interfaces. Both steady-state and transient conditions are considered. Under steady-state conditions, the fluid dynamics and mass transfer phenomena are uncoupled, and the rate of generation of combustion gases is equal to the mass absorption rate by the liquid. The transient behaviour of the annular jet is determined from initial conditions corresponding to steady-state operation, once there is no gas generation by the combustion of hazardous wastes. It is shown that, for most of the conditions considered in this paper, there is no leakage of gaseous combustion products through the jet's outer interface, and that the amount of gases dissolved in the liquid at the nozzle exit and the solubility ratio play a paramount role in determining the mass fluxes of hazardous combustion products at the annular jet's interfaces.

Notes: Summary The response of annular liquid membranes to sinusoidal mass flow rate fluctuations at the nozzle exit is analyzed as a function of the amplitude and frequency of the axial velocity fluctuations at the nozzle exit and thermodynamic compression of the gas enclosed by the membrane. It is shown that both the pressure of the gases enclosed by the annular membrane and the axial distance at which the annular membrane merges on the symmetry axis are periodic functions of time which have the same period as that of the mass flow rate fluctuations at the nozzle exit. They are also nearly sinusoidal functions of time for small amplitudes of the mass flow rate fluctuations at the nozzle exit, and exhibit delay and lag times with respect to the sinusoidal axial velocity fluctuations at the nozzle exit. Both the delay and the lag times are functions of the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit and the polytropic exponent. The amplitudes of both the pressure of the gases enclosed by the annular liquid membrane and the convergence length increase and decrease, resp., as the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit, resp., are increased. They also increase as the polytropic exponent is increased.

Notes: Summary The nonlinear dynamics of axisymmetric, inviscid, incompressible, thin, annular liquid jets subjected to fluctuating body forces is studied numerically by means of an adaptive finite difference method which maps the time-dependent, curvilinear geometry of the jet into a unit square. The fluctuating body forces may arise from fluctuations in the gravitational acceleration in inertial frames or from the acceleration of a non-inertial frame of reference which translates parallelly to an inertial one. It is shown that both the pressure coefficient and the axial location at which the annular jet becomes a solid one are periodic functions of time with a period equal to that of the imposed body force fluctuations, and that their magnitude increases as the amplitude of the body force fluctuations is increased. It has also been shown that, for both intermittent, sinusoidal or rectangular excitations, increases in the frequency of the excitation result in the creation of superharmonics, broad, albeit peaked, spectra, and closed phase planes with many loops.

Notes: Summary  A study of isothermal gas absorption by underpressurized, axisymmetric, thin, inviscid, incompressible, annular liquid jets which form enclosed volumes, where hazardous wastes may be burned, is presented. The study considers the nonlinear dynamical coupling between the fluid dynamics of, and the gases enclosed by, the annular liquid jet. It assumes equilibrium conditions at the interfaces, and employs Sievert’ solubility law to determine the gas concentration at the gas-liquid interfaces. Both steady-state and transient conditions are considered. Under steady-state conditions, the fluid dynamics and mass transfer phenomena are uncoupled, and the rate of generation of combustion gases is equal to the mass absorption rate by the liquid. The transient behaviour of the annular jet is determined from initial conditions corresponding to steady-state operation, once there is no gas generation by the combustion of hazardous wastes. It is shown that, for most of the conditions considered in this paper, there is no leakage of gaseous combustion products through the jet’ outer interface, and that the amount of gases dissolved in the liquid at the nozzle exit and the solubility ratio play a paramount role in determining the mass fluxes of hazardous combustion products at the annular jet’ interfaces.

Notes: Three adaptive finite element methods based on equidistribution, elliptic grid generation and hybrid techniques are used to study a system of reaction-diffusion equations. It is shown that these techniques must employ sub-equidistributing meshes in order to avoid ill-conditioned matrices and ensure the convergence of the Newton method. It is also shown that elliptic grid generation methods require much longer computer times than hybrid and static rezoning procedures. The paper also includes characteristic, Petrov-Galerkin and flux-corrected transport algorithms which are used to study a linear convection-reaction-diffusion equation that has an analytical solution. The flux-corrected transport technique yields monotonic solutions in good agreement with the analytical solution, whereas the Petrov-Galerkin method with quadratic upstream-weighted functions results in very diffused temperature profiles. The characteristic finite element method which uses a Lagrangian-Eulerian formulation overpredicts the flame front location and exhibits overshoots and undershoots near the temperature discontinuity. These overshoots and undershoots are due to the interpolation of the results of the Lagrangian operator onto the fixed Eulerian grid used to solve the reaction-diffusion operator, and indicate that characteristic finite element methods are not able to eliminate numerical diffusion entirely.