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Research Papers: Fundamental Issues and Canonical Flows

# Eddy Heat Transfer by Secondary Görtler Instability

[+] Author and Article Information
L. Momayez

Thermofluids Complex Flows and Energy Research Group-LTN-CNRS-UMR 6607, Ecole Polytechnique, Université de Nantes, BP 50609, 44306 Nantes, France; LRPMN, IUT d’Alençon, Université de Caen, 61250 Damigny, France

G. Delacourt

Thermofluids Complex Flows and Energy Research Group-LTN-CNRS-UMR 6607, Ecole Polytechnique, Université de Nantes, BP 50609, 44306 Nantes, France

P. Dupont

LGCGM, EA3913, INSA de Rennes, Campus Beaulieu, 35043 Rennes, France

H. Peerhossaini1

Thermofluids Complex Flows and Energy Research Group-LTN-CNRS-UMR 6607, Ecole Polytechnique, Université de Nantes, BP 50609, 44306 Nantes, Francehassan.peerhossaini@univ-nantes.fr

1

Corresponding author.

J. Fluids Eng 132(4), 041201 (Apr 15, 2010) (10 pages) doi:10.1115/1.4001307 History: Received October 14, 2008; Revised February 19, 2010; Published April 15, 2010; Online April 15, 2010

## Abstract

Experimental measurements of flow and heat transfer in a concave surface boundary layer in the presence of streamwise counter-rotating Görtler vortices show conclusively that local surface heat-transfer rates can exceed that of the turbulent flat-plate boundary layer even in the absence of turbulence. We have observed unexpected heat-transfer behavior in a laminar boundary layer on a concave wall even at low nominal velocity, a configuration not studied in the literature: The heat-transfer enhancement is extremely high, well above that corresponding to a turbulent boundary layer on a flat plate. To quantify the effect of freestream velocity on heat-transfer intensification, two criteria are defined for the growth of the Görtler instability: $Pz$ for primary instability and $Prms$ for the secondary instability. The evolution of these criteria along the concave surface boundary layer clearly shows that the secondary instability grows faster than the primary instability. Measurements show that beyond a certain distance the heat-transfer enhancement is basically correlated with $Prms$, so that the high heat-transfer intensification at low freestream velocities is due to the high growth rate of the secondary instability. The relative heat-transfer enhancement seems to be independent of the nominal velocity (global Reynolds number) and allows predicting the influence of the Görtler instabilities in a large variety of situations.

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## Figures

Figure 1

Evolution of Stanton number as a function of Görtler number: (a) numerical results of Liu (13) without secondary instability (●) and with secondary instability (○); (b) present experimental results (▲) for U0=3 m/s and R=0.65 m. The curves correspond to the equations of the flat-plate laminar and turbulent boundary layers.

Figure 2

Schematic diagram of the concave-convex model (values in mm)

Figure 3

Thermal instrmentation of the wall

Figure 8

Hot wire signals on both sides of up-wash zone at position x=32 cm, Y/δ=0.5 for a nominal velocity U0=3 m/s

Figure 9

Axial evolution of the energy signal of the most amplified mode of secondary instability

Figure 10

Axial evolution of parameter Pz (a) representing the strength of primary Görtler instability and Prms (b), representing the strength of secondary Görtler instability

Figure 4

Evolution of Upw as a function of x in the region of up-wash flow (▲) and down-wash flow (x) for U0=3 m/s

Figure 5

Isocontour lines of U/Up: (a) at x=15 cm, (b) x=29 cm, and (c) x=49 cm; plotted as a function of spanwise coordinate z; U0=3 m s−1 in the absence of vortex-triggering grid

Figure 6

Isocontour lines of urms/U0(%): (a) at x=15 cm, (b) x=29 cm, and (c) x=49 cm; plotted as a function of spanwise coordinate z; U0=3 m s−1, in the absence of vortex-triggering grid

Figure 7

The photographs of flow visualizations of horseshoe vortices due to varicose mode of secondary instability in (x-y) plane for nominal velocity U0=3 m/s(19)

Figure 11

Comparison of the relative increases of heat transfer (St(x)−Stfp(x)/(St−Stfp)max), primary instability (Pz/Pz max) growth and secondary instability (Prms/Prms max) growth along the concave wall for U0=3 m/s

Figure 12

Influence of U0 on (Gθ/Gθ max), relative primary instability (Pz/Pz max) growth, relative secondary instability (Prms/Prms max) growth and relative increases of heat-transfer enhancement (St−Stflatplate/(St−Stflatplate)max), at x/R=0.45 (x=29 cm and R=65 cm)

Figure 13

Relative heat-transfer transfer enhancement (St/St0) as a function of Görtler number for different nominal velocities (St0=critical Stanton number)

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