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.

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.

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.

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.

Laminar Free Convection on a Curved Wall

1971 ◽  
Vol 38 (4) ◽  
pp. 1081-1083
Author(s):  
K. W. McAlister
Keyword(s):  
Heat Transfer ◽  
Free Convection ◽  
Temperature Variation ◽  
Newtonian Fluid ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Governing Equations ◽  
Transfer Rates ◽  
Arbitrary Temperature ◽  
Free Parameters

Laminar free convection of a Newtonian fluid passing over a curved wall having arbitrary temperature variation is considered. The governing equations are presented and the method of free parameters is used to investigate the existence of similarity solutions. It is found that similarity solutions do exist when the wall inclination and temperature are required to be certain functions of the coordinate parallel to the wall. Numerical solutions to several example cases are presented which indicate that higher heat-transfer rates are possible on a wall which is concave with respect to the fluid.

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.

Prediction of f and Nu During Laminar-to-Turbulent Transition: Case of Flow in a Tube Containing a Twisted Tape

2013 ◽  
Author(s):  
A. W. Date ◽  
Pankaj I. Jagad
Keyword(s):  
Reynolds Numbers ◽  
Twisted Tape ◽  
Duct Flow ◽  
Governing Equations ◽  
Straight Pipe ◽  
Transition Range ◽  
Laminar To Turbulent Transition ◽  
Turbulent Transition ◽  
Characteristic Behaviour ◽  
Laminar And Turbulent Regimes

In a straight pipe (or, duct) flow, the f and Nu vs Re characteristics over transition range of Reynolds numbers (typically, 2300 ≤ Re ≤ 4000) demonstrate abrupt and sharp departures from tendencies under fully laminar and fully turbulent regimes. This characteristic behaviour is quite competently captured by the low Turbulence Reynolds number (Ret) two-equation turbulence model(s) over all flow regimes without requiring any regime-dependent modifications to the governing equations. In contrast, the experimentally determined f and Nu characteristics of flow in a tube containing twisted tape do not demonstrate sharp departures from tendencies in laminar and turbulent regimes. As a result, the so called transition regime of Reynolds numbers cannot be easily identified. The purpose of the present paper is to explore if these very special characteristics can be predicted by low Ret, k-ε model of turbulence by computing over a large range 500 ≤ Re ≤ 30000.

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.

Heat Transfer Features of a Horizontal Slot Jet Impinging in Mixed Convection

2006 ◽  
Author(s):  
Massimo Corcione ◽  
Emanuele Habib ◽  
Gianpaolo Ruocco
Keyword(s):  
Heat Transfer ◽  
Vertical Wall ◽  
Target Surface ◽  
Computer Code ◽  
Governing Equations ◽  
Transfer Rates ◽  
Average Heat ◽  
Target Wall ◽  
Jet Nozzle ◽  
Target Spacing

The mutual interactions occurring between a laminar confined slot jet impinging on a isothermal vertical wall and the natural upward draft promoted by the same target wall, as well as their effects on the local and average heat transfer rates at the target surface, are studied numerically. A specifically developed computer code based on the SIMPLE-C algorithm is used for the solution of the governing equations. Results are presented for different values of the Reynolds number based on the nozzle width in the range between 0 and 400, the Grash of number based on the target wall length in the range between 0 and 108, and a complete range of geometry parameters (jet nozzle width, length of the target wall, and nozzle-to-target spacing).

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.

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