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# Article: A new criterion for the instability threshold of a square tube bundle subject to an air-water cross-flow

We investigate the fluid-elastic instability of a square tube bundle subject to two-phase cross-flow. A dimensional analysis is carried out, leading to a new criterion of instability. This criterion establishes a direct link with the instability thresholds in single-phase flows. In parallel to the dimensional analysis, experimental work is carried out to i) determine the instability thresholds in single-phase flows (new relation between the Scruton, Stokes and Reynolds number), ii) to test the validity of the two-phase flow instability criterion, derived from the dimensional analysis. The experiments are carried out on a square tube bundle (pitch ratio of 1.5) consisting on 5 rows of 3 tubes (plus two end-rows of half tubes). The central tube is mounted on two flexible blades allowing a vibration in the transverse direction only, whereas all the other tubes are rigid. The instability threshold in single-phase (water) flow is obtained from a method of direct measurement of the fluid-elastic forces, in which the motion of the central tube is imposed. The instability threshold in two-phase (air-water) flow is obtained from a method of indirect measurement, with an active system stability control, of the fluid-elastic forces, in which the central tube vibrates freely.

Three sets of blades with different stiffnesses are tested to investigate the stability of the central tube. The criterion of instability is in very good agreement with air-water experiments, predictive for all homogeneous void fractions, and so for all flow regimes (identified in our experiments with an optical probe). This new criterion is of theoretical interest for the understanding of complex two-phase flow excitations, as well as of practical significance for the predictive analysis of industrial components.

Full article: https://authors.elsevier.com/a/1hm3N3AMrTGrkA

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## Comments

## A very interesting work!

Dear Romain, I look through the paper and am impressed by the new way of obtaining fluidelastic instability thresholds via simple dimensional analysis. The single-phase equation recovers the Connors' equation widely used in engineering and academia. Just a quick question. How to determine the coefficients of the instability threshlods equation? Only via experiments?

Best,

Jinxiong

## Dear Jinxiong,

Dear Jinxiong,

Thanks very much for your interest in our work and your question.

From a first dimensional analysis we establish an expression of the fluidelastic instability threshold in two-phase cross-flows.

This threshold (see Eq. 9) is related to the single-phase instability thresholds, which therefore must be determined.

From a second dimensional analysis, we show that the single-phase instability threshold is a function of the Scruton number Sc, the Reynolds number Re and the Stokes number Sk. This function is expressed as a power law (see Eq. 16) of the form: Sc = a Re^b Sk^c (Connors' equation corresponding to the special case b = -c =2). The coefficients a, b and c are determined experimentally using the direct approach of Tanaka. In this approach, the tube is imposed a harmonic displacement. Measurements are performed for various flow rates, i.e. different Reynolds numbers, and different forcing frequencies, i.e. different Stokes numbers. From the measured fluid-forces, the critical Scruton number is determined. This method makes it possible to map the instability threshold by independently varying the Reynolds number and the Stokes number. Coefficients a, b and c are then extracted from experiments with the power law fit.

Best,

Romain

## Dear Romain, Thanks for your

Dear Romain, Thanks for your detailed explanation.