Sugerencias para futuros estudios 112 

The presence of two instability mechanisms raises the question of when each of them will dominate the instability process (Floryan, 2015). To approch this problem, we have defined the global critical Reynolds number (𝑅𝑒_{𝑔,𝑐𝑟}) as the minimum critical Reynolds number for a fixed S over all 𝛼’s. Figure 3-28 illustrates variation of 𝑅𝑒_{𝑔,𝑐𝑟} as a function of S for travelling wave and vortex instabilities. The curves cross each other at (𝑆, 𝑅𝑒_{𝑔,𝑐𝑟})=(0.0045,14740) for 𝑅₁ = 1 and at (0.006,10600) for 𝑅₁ =10.

The critical Reynolds number in the case of flows in smooth channels is 𝑅𝑒𝑐 =11544 (S. A. Orszag 1971). Comparing Figure 3-28A and 3-28B, it can be seen that 𝑅𝑒_𝑐 for the TS waves in case of small rib amplitude (𝑆 → 0 ) approaches the smooth channel value for 𝑅1 → ∞ and this serves as one of validations of the method used in the stability analysis.

(A) (B)

Figure 3-28: Variations of the global critical Reynolds number 𝑅𝑒𝑐,𝑔𝑙𝑜𝑏𝑎𝑙 describing the traveling-wave instability and the vortex instability as a function of the rib amplitude 𝑆. Figure 3-28A-B show results for 𝑅1 = 1 and 𝑅1 = 10, respectively.

Figure 3-29 and Figure 3-30 clearly illustrate the competition between travelling wave and vortex instabilities. One may note that although the critical Reynolds number for both modes decreases with an increase of S, this increase is more rapid in case of vortex mode than travelling waves. The changes in the critical Reynolds number are such that 𝑅𝑒𝑐 for both modes become equal at certain points. The position of these points is presented by a thick curve in the figures which separates the zones where vortex and TS wave instabilities are dominant. It can be seen that for geometries with high rib amplitudes which are located in the upper zone, the critical Reynolds number for vortices is significantly lower than TS waves. This would therefore suggest a rib geometry which leads to formation of vortices, yet, is not affected by travelling wave instabilities. In other words, accidental occurrence of TS waves can be avoided due to large difference between 𝑅𝑒_𝑐 of the two modes. Comparison between Figure 3-29 and Figure 3-30 also indicates

that increase of 𝑅1 leads to significant decrease of 𝑅𝑒𝑐 for TS waves while 𝑅𝑒𝑐 of vortex mode is not considerably affected by change in 𝑅₁.

(A) (B)

Figure 3-29: Variations of 𝑅𝑒𝑐 as a function of 𝛼 and 𝑆 for the onset of the travelling wave (solid lines) and the vortex instabilities (dashed lines) for ribs described by Eq. (2-49) with 𝑅₁ = 1. Figure 3-29A,B display results for 𝜑 = 0, π/2, respectively. The thick line separates zones of dominance of the travelling wave and the vortex instabilities.

(A) (B)

Figure 3-30:The same as in Figure 3-29, but for 𝑅₁ = 10. Figure 3-29A,B display results for 𝜑 = 0, π/2, respectively. The thick line separates zones of dominance of the travelling wave and the vortex instabilities.

Variations of 𝑅𝑒_𝑐 as a function of 𝑅₁ and 𝛼 presented in Figure 3-31 also confirms previous results. Unlike vortex mode, TS waves are sensitive to change of annuli radii, showing a significant decrease in 𝑅𝑒𝑐 with an increase in 𝑅1. Moreover, increase of 𝛼 or 𝜑 destabilizes the flow by reducing 𝑅𝑒_𝑐.

(A) (B)

Figure 3-31: Variations of 𝑅𝑒_𝑐 as a function of 𝑅₁ and 𝛼 for the onset of the travelling wave (solid lines) and the vortex instabilities (dashed lines) for ribs described by Eq. (2-49) with 𝑆 = 0.015. Figure 3-31A,B display results for 𝜑 = 0, π/2, respectively.

Chapter 4

4

Conclusions

An analysis of flows in ribbed annuli has been carried out to determine conditions which can lead to generation of vortices. The annuli have been modified by axisymmetric ribs of arbitrary shapes placed at the cylinders.

Investigation of the stationary state suggests that addition of ribs results in an increase of the pressure losses for all rib wave numbers. Reduction of the rib amplitude or the rib wave number results in the reduction of these losses. Depending on which cylinders the ribs are placed at, change of the annulus’ radius 𝑅₁ can affect the losses differently. Losses increase as 𝑅1 increases if the ribs are placed at the inner cylinder; they decrease, however, if the ribs are placed at the outer cylinder. Increase of phase shift 𝜑 was shown to increase pressure losses with small rib wave numbers being more effective.

To determine circumstances under which vortices are created, vortex instability analysis has been carried out. Since we would like to avoid transition to turbulence, travelling wave instabilities have been also studied. It has been demonstrated that the addition of ribs always results in flow destabilization and an increase of the rib amplitude S reduces the critical Reynolds number. Moreover, increasing phase shift between ribs of the inner and the outer cylinders contributes to instability of the flow in both types of instabilities considered. To assess validation of method used for stability analysis, results for the limiting cases of 𝑅₁ → ∞ and 𝑆 → 0 were shown to match with results in channel flow and smooth annulus, respectively.

In case of vortex instability, centrifugal effect due to flow modulation results in formation of axial vortices. Use of the long wavelength ribs reduces the sreamwise flow modulation which, in turn, weakens the vortex instability. Use of high values of 𝛼 also stabilizes the flow by lifting up the stream above the rib’s peak and reducing flow modulation. It was shown that increase of 𝑅1 may have slight stabilizing or destabilizing effect regarding the value of 𝛼. On the other hand, in case of travelling wave instabilities

which can occur in the form of axisymmetric or oblique waves, it was found that critical Reynolds number decreases significantly with an increase of 𝑅1.

Investigation and comparison of the critical Reynolds numbers in travelling waves and vortex instabilities provided useful information about conditions where each of these instabilities is dominant. Generally, use of ribs with higher amplitudes will contribute to formation of vortices without interference with travelling waves. The minimum rib amplitude required for formation of vortices decreases with an increase in rib wave number. Although better mixing is expected under such conditions, one should notice that increase of 𝑆 and 𝛼 also results in higher pressure losses. So, proper specifications should be selected depending on the design requirements.

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Appendix

Stability of flow in a smooth annulus

The linear disturbance equations (3-16) – (3-17) reduce in the limit of 𝑆_𝑖𝑛 → 0 to the form of

ℒ_𝑂𝑆(𝑛)𝐺_𝑣(𝑛)+ 𝒞_𝑂𝑆(𝑛)Ω(𝑛) = 0, ℒ_𝑆𝑄(𝑛)Ω(𝑛)+ 𝒞_𝑆𝑄(𝑛)𝐺_𝑣(𝑛)= 0 . (A-1a, b) Each pair of the above equations describes disturbances of periodicity 2𝜋 (𝛿 + 𝑛𝛼)⁄ in the axial direction and 2𝜋 𝑀⁄ in the circumferential direction. In the case of vortices (𝛿 = 0, 𝑛 = 0) the above operators reduce to

ℒ_𝑂𝑆(0) = −𝑖𝜎 (𝐷2+𝐷 𝑟 − 𝑀2 𝑟2) − 1 𝑅𝑒[𝐷 4₊2 𝑟𝐷 3₋(2𝑀 2 _{+ 1)} 𝑟2 𝐷2+ (2𝑀2_{+ 1)} 𝑟3 𝐷 +𝑀 2_(𝑀2_{− 4)} 𝑟4 ] , 𝒞𝑂𝑆 (0) = 0, (A-2a) ℒ_𝑆𝑄(0) = −𝑖𝜎 − 1 𝑅𝑒(𝐷 2₊𝐷 𝑟 − 𝑀2 𝑟2), 𝒞𝑆𝑄 (0) = −𝑖𝐷𝑢0 𝑟𝑀 , (A-2b)

with 𝐺_𝑣(0) = −𝑖𝑟𝑔_𝑣(0), Ω(0)= − 𝑔_𝑢(0)⁄𝑀, and (A-1a) and (A-1b) assume the following forms {𝑅𝑒−1[𝐷4 +6𝐷 3 𝑟 + (5 − 2𝑀2_)𝐷2 𝑟2 − (2𝑀2_{+ 1)𝐷} 𝑟3 + ( 𝑀2− 1 𝑟2 ) 2 ] − 𝑖𝜎 (𝐷2+3𝐷 𝑟 − 𝑀2− 1 𝑟2 )} 𝑔𝑣(0) = 0, (A-3) (𝐷2+𝐷 𝑟 − 𝑀2 𝑟2 + 𝑖 𝑅𝑒 𝜎) 𝑔𝑢 (0) = 𝑅𝑒𝐷𝑢0 𝑔𝑣 (0) . (A-4)

𝐺_𝑣(0) = 𝐷𝐺_𝑣(0) = 0, 𝐺_𝑢(0) = 0 at 𝑟 = 𝑅₁ and 𝑟 = 1 + 𝑅₁. (A-5a, b) The original coupled problem (A-1) separates into two independent eigenvalue problems with (A-3)-(A-5a) describing the OS modes and the homogeneous part of (A-4) and (A- 5b) describing the Squire modes.

Solution for the Squire mode can be expressed in terms of Bessel functions in the form of

𝑔_𝑢(0) = 𝑐₁𝐽_𝑀(√−𝑅𝑒𝜎𝑖𝑟) + 𝑐2𝑌𝑀(√−𝑅𝑒𝜎𝑖𝑟) (A-6) where 𝐽𝑀 and 𝑌𝑀 denote the Bessel functions of the first and second kinds with index M, respectively, and 𝑐₁ and 𝑐₂ are arbitrary constants. The amplification rate 𝜎_𝑖 could be positive, zero, or negative. Positive 𝜎_𝑖 leads to the Bessel functions with imaginary arguments, these functions become complex and, thus, cannot represent real 𝑔_𝑢(0). It can be concluded that a positive 𝜎𝑖 is not physically relevant. Zero 𝜎𝑖 leads to infinite value of 𝑌_𝑀 and, thus, is not admissible. Negative 𝜎_𝑖 provides an ability to construct a real solution which can satisfy the boundary conditions. This demonstrates that only attenuated disturbances are admissible. Imposition of homogeneous boundary conditions on (A-6) leads to nontrivial solution only if

𝐽𝑀(√−𝑅𝑒𝜎𝑖𝑅1)𝑌𝑀[√−𝑅𝑒𝜎𝑖(1 + 𝑅1)] − 𝐽_𝑀[√−𝑅𝑒𝜎𝑖(1 + 𝑅1)]𝑌𝑀(√−𝑅𝑒𝜎𝑖𝑅1) = 0.

(A-7)

Numerical solution of (A-7) leads to the determination of the amplification rate for the specified 𝑅1, 𝑅𝑒 and M.

Curriculum Vitae

Post-secondary University of Tehran Education and Tehran, Iran

Degrees: 2009-2014 BSc.

The University of Western Ontario London, Ontario, Canada

2014-2015 MESc.

Related Work Teaching Assistant and Research Assistant Experience: The University of Western Ontario

2014-2015 Publications (Journals):

 Conceptual Design of a Super-Critical CO2 Brayton Cycle Based on The Stack Waste Heat Recovery for Shazand Steam Power plant in Iran

Journal of Energy Equipment and Systems. Volume 2, Issue 1, Winter and Spring 2014, Page 95-101