On the non-parallel instability of the rotating-sphere boundary layer

Abstract

We present a new solution for the steady boundary-layer flow over the rotating sphere that also accounts for the eruption of the boundary layer at the equator and other higher-order viscous effects. Non-parallel corrections to the local Type I and Type II convective instability modes of this flow are also computed as a function of spin rate. Our instability results are associated with the previously observed spiral vortices and remarkable agreement between our predictions of the number of vortices and experimental observations is found. Vortices travelling at 70-80% of the local surface speed are found to be the most amplified for sufficient spin rates, also consistent with prior experimental observations.

Laminar boundary layer separation and reattachment on a rotating sphere

“…In contrast, Smith & Duck (1977), based on an analysis of a dual layer structure of overall size , conjectured a more extensive recirculation region of . This is opposed to the numerical study of Dennis, Ingham & Singh (1981) who found that this interaction zone is of ; however, no recirculation has been observed in any prior experiment or numerical simulation (Segalini & Garrett 2017), until a recent numerical study by Calabretto et al. (2019) observed a small area of reverse flow at significantly large Reynolds numbers, .…”

“…Further theoretical results for small were obtained by Lamb (1924), Bickley (1938), Collins (1955), Thomas & Walters (1964) and Takagi (1977); whilst for , Dennis, Singh & Ingham (1980) calculated series solutions of Gegenbauer functions, but these become more difficult to obtain as increases due to the nonlinearity of terms in the series. For large , boundary layer theory provides a suitable model of the flow near the surface where the sphere imparts the fluid with angular momentum (Segalini & Garrett 2017). The governing equations that describe the boundary layer flow were initially derived by Howarth (1951) who proposed two solution methods: a solution based on a Pohlhausen technique and a series solution based on the latitudinal angle where the latter, at leading order, recovered the von Kármán equations for the rotating disk flow, emphasising the strong connection between the flow at the poles with that of the rotating disk.…”

A consolidated method for predicting laminar boundary layers on rotating axi-symmetric bodies

“…Equations (20a,b) can be obtained from (8a,b) by using r = r 0 sin θ, s = rθ and adopting η as defined in equation 13d. Figure 6 presents a comparison of the velocity profiles computed using the present approach with those generated by Garrett (2002), Garrett and Peake (2002) and Segalini and Garrett (2017), who made use of the above formulation.…”

“…Comparison of the velocity profiles on a rotating sphere (in still air) at θ = 10 • → 80 • in 10 • increments (left to right), obtained using the present approach, with those reported byGarrett (2002),Garrett and Peake (2002) andSegalini and Garrett (2017) (•); η as defined in equation(13d).…”

Radially expanding/contracting and rotating sphere with suction

“…After the successful formulation and analysis of fluid flow phenomena due to a rotating sphere in early studies, such as Howarth (1951), Ovseenko and Ovseenko (1968), Banks (1976), Elshaarawi et al (1993) (see also the cousin problem of decelerating rotating disk, Watson and Wang, 1979), recent research has focused on further investigations of diverse physical mechanisms. These include entropy generation (Antar and ElShaarawi, 2009), hydrodynamic forces (Poon et al, 2013), Magnus effects (Kim et al, 2013), couple stress fluid around a rotating sphere (Ashmawy, 2016), blood viscosity estimations from a rotating sphere (Furukawa et al, 2016), instability (Garrett, 2002;Segalini and Garrett, 2017), a decelerating rotating sphere (Turkyilmazoglu, 2018) and a latitudinally stretching rotating sphere (Turkyilmazoglu, 2019).…”