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Research Papers: Natural and Mixed Convection

# Numerical Investigation of the Linear Stability of a Free Convection Boundary Layer Flow Using a Thermal Disturbance With a Slowly Increasing Frequency

[+] Author and Article Information
Manosh C. Paul

Department of Mechanical Engineering, University of Glasgow, Glasgow G12 8QQ, UKm.paul@mech.gla.ac.uk

D. Andrew S. Rees

Department of Mechanical Engineering, University of Bath, Bath BA2 7AY, UK

J. Heat Transfer 130(12), 122501 (Sep 18, 2008) (8 pages) doi:10.1115/1.2976554 History: Received September 19, 2007; Revised May 22, 2008; Published September 18, 2008

## Abstract

Numerical simulations are performed to investigate the linear stability of a two-dimensional incompressible free convection flow induced by a vertical semi-infinite heated flat plate. A small-amplitude local temperature disturbance with a slowly increasing frequency is introduced on the surface near to the leading edge in order to generate disturbance waves within the boundary layer. The aim is to compare the response of the thermal boundary layer with that obtained by selecting discrete disturbance frequencies. In the present study, air is considered to be the working fluid for which the value of the Prandtl number is taken to be $Pr=0.7$. The computational results show that the disturbance decays initially until it reaches a critical distance, which depends on the current frequency of the disturbance. Thereafter the disturbance grows, but the growth rate also depends on the effective frequency of the disturbance. Comparisons with previous work using constant disturbance frequencies are given, and it is shown that the sine-sweep technique is an effective method for analyzing the instability of convectively unstable boundary layers.

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## Figures

Figure 1

Contours of θ̂ recorded at various values of the temporal frequency, λ, for c=5×10−5. Here ten contour levels are plotted based on the global extrema within each frame (see the text of the paper). The dashed contour lines represent negative values of θ̂ while the solid lines represent positive θ̂.

Figure 2

Time variation of the surface rate of heat transfer recorded at (a) x∗=16 and (b) x∗=207 where c=5×10−5

Figure 3

Envelope of the maximum heat transfer response, Q, against time for various locations along the heated surface where c=5×10−5

Figure 4

Envelope of the maximum heat transfer response, Q, against distance for various disturbance frequencies, where c=5×10−5

Figure 5

Comparison of Q for the three different values of c and the discrete frequency (5). The dashed lines are for c=10−4, the dashed-dotted lines are for c=5×10−5, the solid lines are for c=2.5×10−5, and the circles are for the discrete frequency results.

Figure 6

The envelope of the curves presented in Figs.  45. For legends, see Fig. 5. The solid line with square symbols denotes the results for c=2.5×10−5 obtained by the finer grid, 960×96.

Figure 7

(a) The values of λ, which correspond to Qmax(x∗) shown in Fig. 6. These curves show which frequency corresponds to the most amplified disturbance at each value of x∗. (b) Neutral curves. The solid line with square symbols denotes the results for c=2.5×10−5 obtained by the finer grid, 960×96.

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