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.

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.

The Classical Nusselt Problem of Film Condensation: Direct Steady and Unsteady Computational Simulations That Yield Results on Stability and Wave Effects

2005 ◽  
Author(s):  
L. Phan ◽  
S. L. Post ◽  
A. Narain
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Vertical Wall ◽  
Flow Instabilities ◽  
Film Condensation ◽  
Linear Wave ◽  
Governing Equations ◽  
Transfer Rates ◽  
Transverse Vibrations ◽  
Wave Effects

Accurate steady and unsteady numerical solutions of the full 2D governing equations for the Nusselt problem (film condensation of quiescent saturated vapor on a vertical wall) are presented and related to known results. The problem, solved accurately up to film Reynolds number of 60 (Reδ ≤ 60), establishes various features of the well known steady solution and reveals the interesting phenomena of stability, instability and non-linear wave effects. The wave effects are shown to arise from the intrinsic flow instabilities as well as sensitivity to ever present minuscule transverse vibrations of the condensing surface. The results also suggest ways to enhance wave fluctuations and heat transfer rates.

Nonlinear Stability of the Classical Nusselt Problem of Film Condensation and Wave Effects

2006 ◽  
Vol 74 (2) ◽  
pp. 279-290 ◽  
Author(s):  
L. Phan ◽  
A. Narain
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Vertical Wall ◽  
Film Condensation ◽  
Governing Equations ◽  
Transfer Rates ◽  
Laminar To Turbulent Transition ◽  
Turbulent Transition ◽  
Transverse Vibrations ◽  
Wave Effects

Accurate steady and unsteady numerical solutions of the full two-dimensional (2D) governing equations for the Nusselt problem (film condensation of quiescent saturated vapor on a vertical wall) are presented and related to known results. The problem, solved accurately up to film Reynolds number of 60 (Reδ⩽60), establishes various features of the well-known steady solution and reveals the interesting phenomena of stability, instability, and nonlinear wave effects. It is shown that intrinsic flow instabilities cause the wave effects to grow over the well-known experiments-based range of Reδ⩾30. The wave effects due to film flow’s sensitivity to ever-present minuscule transverse vibrations of the condensing surface are also described. The results suggest some ways of choosing wall noise—through suitable actuators—that can enhance or dampen wave fluctuations and thus increase or decrease heat transfer rates over the laminar-to-turbulent transition zone.

Film Condensation Over a Horizontal Cylinder for Combined Gravity and Forced Flow

1985 ◽  
Vol 107 (3) ◽  
pp. 687-695 ◽  
Author(s):  
M. di Marzo ◽  
M. J. Casarella
Keyword(s):  
Froude Number ◽  
Numerical Solutions ◽  
Driving Forces ◽  
Horizontal Cylinder ◽  
Forced Flow ◽  
Film Condensation ◽  
Vapor Flow ◽  
Governing Equations ◽  
Laminar Film Condensation ◽  
Wide Range

The problem of laminar film condensation of a saturated vapor flowing over a cold horizontal cylinder is investigated. A rigorous formulation of the governing equations for the vapor boundary layer and the condensed liquid film, including both the gravity-driven body forces and the imposed pressure gradient caused by the vapor flow, is presented. A generalized transformation of the governing equations allows a wide range of Froude numbers to be investigated. A unique value of the Froude number is defined which allows a distinction between the gravity-dominated flow (Fr→0) and the forced flow (Fr→∞) and basically defines the overlap region for the two solution domains. Numerical solutions are obtained in the merging flow regions controlled by both driving forces. The effects of density/viscosity ratio at the liquid-vapor interface, Prandtl number, Jakob number, and Froude number on the heat transfer characteristics are presented.

Similar Solutions for Laminar Film Condensation with Adverse Pressure Gradients

1973 ◽  
Vol 95 (2) ◽  
pp. 268-270 ◽  
Author(s):  
P. M. Beckett
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Film Condensation ◽  
Pressure Gradients ◽  
Two Dimensional ◽  
Laminar Film ◽  
Approximate Models ◽  
Laminar Film Condensation ◽  
Similar Solutions

Steady two-dimensional laminar film condensation is investigated when the saturated vapor has the Falkner–Skan mainstream. Numerical solutions and approximate models are discussed with reference to other published work.

A Theory of Rotating Condensation

1959 ◽  
Vol 81 (2) ◽  
pp. 113-119 ◽  
Author(s):  
E. M. Sparrow ◽  
J. L. Gregg
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Large Body ◽  
Rotating Disk ◽  
Disk Surface ◽  
Film Condensation ◽  
Navier Stokes ◽  
Torque Moment ◽  
Prandtl Numbers ◽  
Energy Equations

An analysis is made for film condensation on a rotating disk situated in a large body of pure saturated vapor. The centrifugal field associated with the rotation sweeps the condensate outward along the disk surface, and gravity forces need not be involved. The problem is formulated as an exact solution of the complete Navier-Stokes and energy equations. Numerical solutions are obtained for Prandtl numbers between 0.003 and 100 and for cpΔT/hfg in the range 0.0001 to 1.0. Results are given for the heat transfer, as well as for the condensate layer thickness, torque moment, and temperature and velocity profiles.

Direct Computational Simulations for Internal Condensing Flows and Results on Attainability/Stability of Steady Solutions, Their Intrinsic Waviness, and Their Noise Sensitivity

2004 ◽  
Vol 71 (1) ◽  
pp. 69-88 ◽  
Author(s):  
A. Narain ◽  
Q. Liang ◽  
G. Yu ◽  
X. Wang
Keyword(s):  
Numerical Solutions ◽  
Resonance Condition ◽  
Vertical Channel ◽  
High Growth ◽  
Film Condensation ◽  
Noise Sensitivity ◽  
Reported Study ◽  
Condensing Flows ◽  
Steady Solutions ◽  
Internal Condensing Flows

The paper presents a new two-dimensional computational approach and results for laminar/laminar internal condensing flows. Accurate numerical solutions of the full governing equations are presented for steady and unsteady film condensation flows on a sidewall inside a vertical channel. It is found that exit conditions and noise sensitivity are important. Even for stable steady solutions obtained for nearly incompressible vapor phase flows associated with unconstrained exit conditions, the noise sensitivity to the condensing surface’s minuscule transverse vibrations is high. The structure of waves, the underlying characteristics, and the “growth/damping rates” for the disturbances are discussed. A resonance condition for high “growth rates” is proposed and its efficacy in significantly enhancing wave motion and heat transfer rates is computationally demonstrated. For the unconstrained exit cases, the results make possible a separately reported study of the effects of shear, gravity, and surface tension on noise sensitive stable solutions.

Effects of Gravity and Surface Tension and Interfacial-Waves and Heat-Transfer Rates in Internal Condensing Flows

2003 ◽  
Author(s):  
Q. Liang ◽  
X. Wang ◽  
A. S. Barve ◽  
A. Narain
Keyword(s):  
Heat Transfer ◽  
Surface Tension ◽  
Numerical Solutions ◽  
Film Condensation ◽  
Vapor Flow ◽  
Transfer Rates ◽  
Condensing Flows ◽  
Steady Solutions ◽  
Internal Condensing Flows ◽  
Good Agreement

The paper presents accurate numerical solutions of the full 2D governing equations for steady and unsteady laminar/laminar internal condensing flows. The chosen geometry allows for film condensation on the bottom wall of a tilted (from vertical to horizontal) channel. It is found that it is important to know whether the exit conditions are constrained or unconstrained because incompressible vapor flows occur only for exit conditions that are unconstrained. For the incompressible vapor flow situations, a method for computationally obtaining the stable steady/quasi-steady solutions is given here and the resulting solutions are shown to be in good agreement with some relevant experimental data for horizontal channels. These solutions are shown to be sensitive to the frequency-content and strength of ever-present minuscule transverse vibrations of the condensing surface. The effects of noise-sensitivity, gravity (terrestrial to zero-gravity), and surface tension on the attainability of stable steady/quasi-steady solutions, structure of superposed waves, and heat-transfer rates are discussed. It is shown that significant enhancement in wave-energy and heat-transfer rates are possible by designing the condensing surface noise to be in resonance with the intrinsic waves.

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.

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