scholarly journals Unsteady turbulent buoyant plumes

2016 ◽  
Vol 794 ◽  
pp. 595-638 ◽  
Author(s):  
M. J. Woodhouse ◽  
J. C. Phillips ◽  
A. J. Hogg
Keyword(s):  
Similarity Solution ◽  
Axial Velocity ◽  
Shape Factor ◽  
Numerical Solutions ◽  
Integral Model ◽  
Governing Equations ◽  
Shape Factors ◽  
Buoyant Plumes ◽  
Radial Profiles ◽  
Steady Solutions

We model the unsteady evolution of turbulent buoyant plumes following temporal changes to the source conditions. The integral model is derived from radial integration of the governing equations expressing the evolution of mass, axial momentum and buoyancy in the plume. The non-uniform radial profiles of the axial velocity and density deficit in the plume are explicitly captured by shape factors in the integral equations; the commonly assumed top-hat profiles lead to shape factors equal to unity. The resultant model for unsteady plumes is hyperbolic when the momentum shape factor, determined from the radial profile of the mean axial velocity in the plume, differs from unity. The solutions of the model when source conditions are maintained at constant values are shown to retain the form of the well-established steady plume solutions. We demonstrate through a linear stability analysis of these steady solutions that the inclusion of a momentum shape factor in the governing equations that differs from unity leads to a well-posed integral model. Therefore, our model does not exhibit the mathematical pathologies that appear in previously proposed unsteady integral models of turbulent plumes. A stability threshold for the value of the shape factor is also identified, resulting in a range of its values where the amplitudes of small perturbations to the steady solutions decay with distance from the source. The hyperbolic character of the system of equations allows the formation of discontinuities in the fields describing the plume properties during the unsteady evolution, and we compute numerical solutions to illustrate the transient development of a plume following an abrupt change in the source conditions. The adjustment of the plume to the new source conditions occurs through the propagation of a pulse of fluid through the plume. The dynamics of this pulse is described by a similarity solution and, through the construction of this new similarity solution, we identify three regimes in which the evolution of the transient pulse following adjustment of the source qualitatively differs.

Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem of Cess and Koh): Steady and Unsteady Solutions of the Full 2D Problem

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
S. Kulkarni ◽  
A. Narain ◽  
S. Mitra
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Threshold Value ◽  
Steady Solution ◽  
Forced Flow ◽  
Film Condensation ◽  
Horizontal Plate ◽  
Governing Equations ◽  
Long Time ◽  
Steady Solutions

Accurate steady and unsteady numerical solutions of the full 2D governing equations—which model the forced film condensation flow of saturated vapor over a semi-infinite horizontal plate (the problem of Cess and Koh)—are obtained over a range of flow parameters. The results presented here are used to better understand the limitations of the well-known similarity solutions given by Koh. It is found that steady/quasisteady filmwise solution exists only if the inlet speed is above a certain threshold value. Above this threshold speed, steady/quasisteady film condensation solutions exist and their film thickness variations are approximately the same as the similarity solution given by Koh. However, these steady solutions differ from the Koh solution regarding pressure variations and associated effects in the leading part of the plate. Besides results based on the solutions of the full steady governing equations, this paper also presents unsteady solutions that characterize the steady solutions’ attainability, stability (response to initial disturbances), and their response to ever-present minuscule noise on the condensing-surface. For this shear-driven flow, the paper finds that if the uniform vapor speed is above a threshold value, an unsteady solution that begins with any reasonable initial-guess is attracted in time to a steady solution. This long time limiting solution is the same—within computational errors—as the solution of the steady problem. The reported unsteady solutions that yield the steady solution in the long time limit also yield “attraction rates” for nonlinear stability analysis of the steady solutions. The attraction rates are found to diminish gradually with increasing distance from the leading edge and with decreasing inlet vapor speed. These steady solutions are generally found to be stable to initial disturbances on the interface as well as in any flow variable in the interior of the flow domain. The results for low vapor speeds below the threshold value indicate that the unsteady solutions exhibit nonexistence of any steady limit of filmwise flow in the aft portion of the solution. Even when a steady solution exists, the flow attainability is also shown to be difficult (because of waviness and other sensitivities) at large downstream distances.

scholarly journals On the consistency of two-phase local/nonlocal piezoelectric integral model

2021 ◽  
Vol 42 (11) ◽  
pp. 1581-1598
Author(s):  
Yanming Ren ◽  
Hai Qing
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Integral Model ◽  
Governing Equations ◽  
Two Phase ◽  
Differential Model ◽  
Piezoelectric Beam ◽  
General Strain

AbstractIn this paper, we propose general strain- and stress-driven two-phase local/nonlocal piezoelectric integral models, which can distinguish the difference of nonlocal effects on the elastic and piezoelectric behaviors of nanostructures. The nonlocal piezoelectric model is transformed from integral to an equivalent differential form with four constitutive boundary conditions due to the difficulty in solving intergro-differential equations directly. The nonlocal piezoelectric integral models are used to model the static bending of the Euler-Bernoulli piezoelectric beam on the assumption that the nonlocal elastic and piezoelectric parameters are coincident with each other. The governing differential equations as well as constitutive and standard boundary conditions are deduced. It is found that purely strain- and stress-driven nonlocal piezoelectric integral models are ill-posed, because the total number of differential orders for governing equations is less than that of boundary conditions. Meanwhile, the traditional nonlocal piezoelectric differential model would lead to inconsistent bending response for Euler-Bernoulli piezoelectric beam under different boundary and loading conditions. Several nominal variables are introduced to normalize the governing equations and boundary conditions, and the general differential quadrature method (GDQM) is used to obtain the numerical solutions. The results from current models are validated against results in the literature. It is clearly established that a consistent softening and toughening effects can be obtained for static bending of the Euler-Bernoulli beam based on the general strain- and stress-driven local/nonlocal piezoelectric integral models, respectively.

Steady “Attractors” for Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem in Cess and Koh) and Their Unsteady Responses to Initial Disturbances and Noise

2007 ◽  
Author(s):  
S. Kulkarni ◽  
A. Narain ◽  
S. Mitra
Keyword(s):  
Similarity Solution ◽  
Flow Problem ◽  
Leading Edge ◽  
Forced Flow ◽  
Film Condensation ◽  
Finite Domain ◽  
Horizontal Plate ◽  
Governing Equations ◽  
Exit Pressure ◽  
Steady Solutions

Accurate steady and unsteady numerical solutions of the full 2-D governing equations – that model the film condensation of saturated vapor flowing over a horizontal plate (the problem of Cess [1] and Koh [2]) – are obtained and new results on the solutions’ unsteady response to disturbances are presented. The computations reveal important features of this classical condensing flow problem. The results highlight the scope and limitations of the well-known similarity solution given by Koh [2]. For the steady problem formulation, the paper discusses the similarities and differences between the solution obtained by solving the full 2-D governing equations and the one obtained semi-analytically by the similarity solution approach of Koh [2]. It is shown that the pressure variations in the vapor domain near the leading edge, though small, are important in deciding condensation dynamics (steady and unsteady) and cannot, in general, be neglected, as is the case with the similarity solution. For this shear driven flow, by considering the unsteady solutions, the paper finds that any initial guess leads to an unsteady solution which is attracted to a long-term steady solution (which is same as the solution as the steady problem). However, the attraction rates gradually diminish with increasing distance from the leading edge and decreasing inlet speed. The steady solutions for this external flow problem are generally found to be stable to initial disturbances at the interface or in the interior of the flow domain. However, since these flows can only be physically realized on suitable finite length portions of the plate, the issue of their stability and sensitivity to exit pressure disturbances and ever-present noise (through exit pressure or bottom plate) is also considered. For example, for the finite domain realization of this problem, it is found that the flows are stable to small initial disturbances to the nearly uniform value of exit pressure. These finite domain realizations of the flow are unique in the sense that they allow different non-uniform steady pressure prescriptions leading to different steady solutions – particularly near the exit zone. As a result, near the exit of a long plate, large unsteadiness is expected due to sensitivity to small exit pressure noise/fluctuations. The exit pressure noise for finite domain realization of these flows is expected because of practical difficulties in maintaining constant uniform pressures at downstream locations of the top and exit boundaries. It is shown that the transverse component of gravity does not affect the solution or its dynamic response except for the expected changes in the nature of hydrostatic pressure variations.

scholarly journals Unsteady turbulent line plumes

2018 ◽  
Vol 856 ◽  
pp. 103-134
Author(s):  
Andrew J. Hogg ◽  
Edward J. Goldsmith ◽  
Mark J. Woodhouse
Keyword(s):  
Numerical Solutions ◽  
Similarity Solutions ◽  
Stable Model ◽  
Governing Equations ◽  
Shape Factors ◽  
Piecewise Constant ◽  
Horizontal Variation ◽  
Model Account ◽  
Ill Posed ◽  
Well Posed

The unsteady ascent of a buoyant, turbulent line plume through a quiescent, uniform environment is modelled in terms of the width-averaged vertical velocity and density deficit. It is demonstrated that for a well-posed, linearly stable model, account must be made for the horizontal variation of the velocity and the density deficit; in particular the variance of the velocity field and the covariance of the density deficit and velocity fields, represented through shape factors, must exceed threshold values, and that models based upon ‘top-hat’ distributions in which the dependent fields are piecewise constant are ill-posed. Numerical solutions of the nonlinear governing equations are computed to reveal that the transient response of the system to an instantaneous change in buoyancy flux at the source may be captured through new similarity solutions, the form of which depend upon both the ratio of the old to new buoyancy fluxes and the shape factors.

Thermal Post-Buckling Analysis of Functionally Graded Composite Plates with Nonlinear Aerodynamic Forces

2010 ◽  
Vol 123-125 ◽  
pp. 280-283
Author(s):  
Chang Yull Lee ◽  
Ji Hwan Kim
Keyword(s):  
Material Properties ◽  
Numerical Solutions ◽  
Virtual Work ◽  
Shear Deformation Theory ◽  
Composite Plates ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Governing Equations ◽  
Functionally Graded Composite ◽  
Post Buckling

The post-buckling of the functionally graded composite plate under thermal environment with aerodynamic loading is studied. The structural model has three layers with ceramic, FGM and metal, respectively. The outer layers of the sandwich plate are different homogeneous and isotropic material properties for ceramic and metal. Whereas the core is FGM layer, material properties vary continuously from one interface to the other in the thickness direction according to a simple power law distribution in terms of the volume fractions. Governing equations are derived by using the principle of virtual work and numerical solutions are solved through a finite element method. The first-order shear deformation theory and von-Karman strain-displacement relations are based to derive governing equations of the plate. Aerodynamic effects are dealt by adopting nonlinear third-order piston theory for structural and aerodynamic nonlinearity. The Newton-Raphson iterative method applied for solving the nonlinear equations of the thermal post-buckling analysis

scholarly journals Consistent Weighted Average Flux of Well-Balanced TVD-RK Discontinuous Galerkin Method for Shallow Water Flows

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Thida Pongsanguansin ◽  
Montri Maleewong ◽  
Khamron Mekchay
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Order Approximation ◽  
Weighted Average ◽  
Flat Bottom ◽  
Flux Function ◽  
Dg Method ◽  
Average Flux ◽  
Steady Solutions

A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified well-balanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.

Heat Transfer in a Laminar Channel Flow Generated by Injection Through Porous Walls

2007 ◽  
Vol 129 (8) ◽  
pp. 1048-1057 ◽  
Author(s):  
Clarisse Fournier ◽  
Marc Michard ◽  
Françoise Bataille
Keyword(s):  
Channel Flow ◽  
Numerical Solutions ◽  
Scaling Law ◽  
Similarity Solutions ◽  
Temperature Profiles ◽  
Governing Equations ◽  
Temperature Boundary ◽  
Laminar Channel Flow ◽  
Porous Walls ◽  
Uniform Fluid

Steady state similarity solutions are computed to determine the temperature profiles in a laminar channel flow driven by uniform fluid injection at one or two porous walls. The temperature boundary conditions are non-symmetric. The numerical solution of the governing equations permit to analyze the influence of the governing parameters, the Reynolds and Péclet numbers. For both geometries, we deduce a scaling law for the boundary layer thickness as a function of the Péclet number. We also compare the numerical solutions with asymptotic expansions in the limit of large Péclet numbers. Finally, for non-symmetric injection, we derive from the computed temperature profile a relationship between the Nusselt and Péclet numbers.

Spray Cloud Formation Over Marine Vessels by a Water Breakup Model

2017 ◽  
Author(s):  
Saeed R. Dehghani ◽  
Greg F. Naterer ◽  
Yuri S. Muzychka
Keyword(s):  
Numerical Solutions ◽  
Reynolds Numbers ◽  
Cloud Formation ◽  
Sea Spray ◽  
Governing Equations ◽  
Fishing Vessel ◽  
Wave Impact ◽  
Droplet Sizes ◽  
Marine Vessels ◽  
Breakup Model

Water breakup affects the variety of droplet sizes and velocities in a cloud of spray resulting from a sea wave striking a vessel bow. The Weber and Reynolds numbers of droplets are the main parameters for water breakup phenomena. “Stripping breakup” is a faster phenomenon than “bag breakup” and occurs at higher velocities and with larger diameters of droplets. A water breakup model employs droplet trajectories to develop a predictive model for the extent of spray cloud. The governing equations of breakup and trajectories of droplets are solved numerically. Stripping breakup is found as the major phenomenon in the process of the formation of wave-impact sea spray. Bag breakup acts as a complementary phenomenon to the stripping breakup. The extent of the spray as well as wet heights, for a Mediumsized Fishing Vessel (MFV), are obtained by numerical solutions. The results show that bag breakup occurs at higher heights. In addition, there is no breakup when droplets move over the deck.

scholarly journals What can asymptotic expansions tell us about large-scale quasi-geostrophic anticyclonic vortices?

1995 ◽  
Vol 2 (3/4) ◽  
pp. 186-193 ◽  
Author(s):  
A. Stegner ◽  
V. Zeitlin
Keyword(s):  
Asymptotic Expansions ◽  
Large Scale ◽  
Characteristic Size ◽  
Governing Equations ◽  
Free Surface Elevation ◽  
Circular Profile ◽  
Frontal Dynamics ◽  
Rotating Shallow Water ◽  
Steady Solutions ◽  
Rotating Shallow Water Equations

Abstract. The problem of the large-scale quasi-geostrophic anticyclonic vortices is studied in the framework of the baratropic rotating shallow- water equations on the β-plane. A systematic approach based on the multiplescale asymptotic expansions is used leading to a hierarchy of governing equations for the large-scale vortices depending on their characteristic size, velocity and a free surface elevation. Among them are the Charney-Obukhov equation, the intermediate geostrophic model equation, the frontal dynamics equation and some new nonlinear quasi-geostrophic equation. We are looking for steady-drifting axisymmetric anticyclonic solutions and find them in a consistent way only in this last equation. These solutions are soliton-like in the sense that the effects of weak non-linearity and dispersion balance each other. The same regimes on the paraboloidal β-plane are studied, all giving a negative result in what concerns the axisymmetric steady solutions, except for a strong elevation case where any circular profile is found to be steadily propagating within the accuracy of the approximation.

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