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TECHNICAL PAPERS

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

[+] Author and Article Information
L. Phan

Department of Mechanical Engineering—Engineering Mechanics, Michigan Technological University, Houghton, MI 49931

A. Narain

Department of Mechanical Engineering—Engineering Mechanics, Michigan Technological University, Houghton, MI 49931narain@mtu.edu

J. Appl. Mech 74(2), 279-290 (Jan 26, 2006) (12 pages) doi:10.1115/1.2198249 History: Received October 16, 2005; Revised January 26, 2006

## Abstract

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.

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## Figures

Figure 1

Cooled vertical plate in a quiescent (far field) vapor flow—geometry used for simulations

Figure 2

(a) For the case of R-113 (see Table 1) experiencing film condensation on a vertical plate, the figure above shows steady film thickness values under different approximations. Here, xc=Xc∕Yc=50 and c=Yc∕ΔN(Xc)=47. (b) For the case of R-113 (see Table 1) experiencing film condensation on a vertical plate, the figure above shows the streamline pattern and contour zones depicting a range of ∣uI∣ values. (c) For Curves 4 and 5 of the base flow in (a), the figure above shows the u(x*,y) versus y for x*=20. (d) For the case of R-113 (see Table 1) experiencing film condensation on a vertical plate, the figure above shows the contour zones for temperature and a representative plot of θI(x*,y) for x*=20. (e) For Curves 4 and 5 of the base flow in (a), the figure above shows the πI(x*,y) versus y for x*=20.

Figure 3

(a) For the base flow in Fig. 2, the figure above shows the stable response (Δt=7.5, t=150) of the film thickness δ(x,t) as a result of an initial disturbance δ(x,0)=δsteady(x)[1+εδ′(x,0)], where δ′(x,0)≡sin(2πx∕λo), ε=0.15 and λo=7. (b) For the base flow in Fig. 2, the figure above shows the unstable response (Δt=7.5, t=247.5) of the film thickness δ(x,t) as a result of an initial disturbance δ(x,0)=δsteady(x)[1+εδ′(x,0)], where δ′(x,0)≡sin(2πx∕λo), ε=0.15 and λo=15.

Figure 4

For the base flow in Fig. 2, the figure above shows the stable and unstable response (Δt=7.5, t=247.5) of the film thickness δ(x,t) as a result of an initial disturbance δ(x,0)=δsteady(x)[1+εδ′(x,0)], where δ′(x,0)≡sin(2πx∕λo), ε=0.15 and λo=5, 9, 15, and 23

Figure 5

For the base flow in Fig. 2, the figure above shows the effect of surface tension at the interface (Δt=7.5) on the flow exposed to initial disturbance δ(x,0)=δsteady(x)[1+εδ′(x,0)], where δ′(x,0)≡sin(2πx∕λo), ε=0.15 and λo=15

Figure 6

(a) For the base flow in Fig. 2 and initial disturbances defined in Fig. 3, the figure above shows the unstable response (Δt=7.5, t=247.5), for λo>λcr, of the film thickness δ(x,t) as a result of an initial disturbance δ(x,0)=δsteady(x)[1+εδ′(x,0)], where ε=0.15 and λo=15. (b) For the case shown in (a), the figure above shows the values of characteristic speed u¯ (steady value at t=0 and disturbed value at t=247.5) as function of x. (c) For the case shown in (a), the figure above shows the characteristic curves found by solving Eq. 21 through a fourth-order Runge-Kutta method. (d) For the cases shown in (a)–(c), the figure above shows values of a(t)∕a(0) for different values of initial disturbance wavelength λo for xc(t)=6.65.

Figure 7

Stability boundaries as obtained by Unsal and Thomas (14) and Spindler (15), and this work

Figure 8

Result showing waves while accounting for vapor motion and its fluctuations and, also, while neglecting vapor motion and its fluctuations (by not incorporating vapor domain calculations in the simulations). Here, Δt=7.5 and εb=0.15E-05.

Figure 9

(a) Nonresonance bottom wall noise that do not lead to growing waves. Here, Δt=7.5, εb=0.15E-05, and Tb=30. (b) Nonresonance bottom wall noise that lead to growing waves. Here, Δt=7.5, εb=0.5E-05 and Tb=250.

Figure 10

(a) Effects of resonant noise for λb=15>λcr and with Δt=7.5 and εb=0.35E-05. (b) Effects of resonant noise for λb=5<λcr and with Δt=7.5 and εb=0.5E-05.

Figure 11

(a) For the base flow in Fig. 2, the figure above shows the wall heat flux (W∕m2) for cases with and without initial disturbance δ(x,0)=δsteady(x)[1+εδ′(x,0)], where δ′(x,0)≡sin(2πx∕λo), ε=0.15, λo=15, and Δt=7.5. (b) For the base flow in Fig. 2, the figure above shows the wall shear stress for cases with and without initial disturbance δ(x,0)=δsteady(x)[1+εδ′(x,0)], where δ′(x,0)≡sin(2πx∕λo), ε=0.15, λo=15, and Δt=7.5.

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