Nonlinear Spin-Up of a Rotating Stratified Fluid: Theory

1998 ◽  
Vol 120 (4) ◽  
pp. 662-666 ◽  
Author(s):  
Richard E. Hewitt ◽  
Peter W. Duck ◽  
Michael R. Foster ◽  
Peter A. Davies
Keyword(s):  
Boundary Layer ◽  
Finite Time ◽  
Normal Component ◽  
Stratified Fluid ◽  
Considerable Effect ◽  
Asymptotic Results ◽  
Initial State ◽  
Similarity Type ◽  
Boundary Layer Equations ◽  
Schmidt Numbers

We consider the boundary layer that forms on the wall of a rotating container of stratified fluid when altered from an initial state of rigid body rotation. The container is taken to have a simple axisymmetric form with sloping walls. The introduction of a non-normal component of buoyancy into the velocity boundary-layer is shown to have a considerable effect for certain geometries. We introduce a similarity-type solution and solve the resulting unsteady boundary-layer equations numerically for three distinct classes of container geometry. Computational and asymptotic results are presented for a number of parameter values. By mapping the parameter space we show that the system may evolve to either a steady state, a double-structured growing boundary-layer, or a finite-time breakdown depending on the container type, rotation change and stratification. In addition to extending the results of Duck et al. (1997) to a more general container shape, we present evidence of a new finite-time breakdown associated with higher Schmidt numbers.

scholarly journals On the boundary layer arising in the spin-up of a stratified fluid in a container with sloping walls

1997 ◽  
Vol 335 ◽  
pp. 233-259 ◽  
Author(s):  
P. W. DUCK ◽  
M. R. FOSTER ◽  
R. E. HEWITT
Keyword(s):  
Boundary Layer ◽  
Layer Structure ◽  
Outer Region ◽  
Stratified Fluid ◽  
Main Body ◽  
Similarity Type ◽  
Cone Apex ◽  
Uniform Solution ◽  
Steady Solutions ◽  
Finite Time Singularity

In this paper we consider the boundary layer that forms on the sloping walls of a rotating container (notably a conical container), filled with a stratified fluid, when flow conditions are changed abruptly from some initial (uniform) state. The structure of the solution valid away from the cone apex is derived, and it is shown that a similarity-type solution is appropriate. This system, which is inherently nonlinear in nature, is solved numerically for several flow regimes, and the results reveal a number of interesting and diverse features.In one case, a steady state is attained at large times inside the boundary layer. In a second case, a finite-time singularity occurs, which is fully analysed. A third scenario involves a double boundary-layer structure developing at large times, most significantly including an outer region that grows in thickness as the square-root of time.We also consider directly the nonlinear fully steady solutions to the problem, and map out in parameter space the likely ultimate flow behaviour. Intriguingly, we find cases where, when the rotation rate of the container is equal to that of the main body of the fluid, an alternative nonlinear state is preferred, rather than the trivial (uniform) solution.Finally, utilizing Laplace transforms, we re-investigate the linear initial-value problem for small differential spin-up studied by MacCready & Rhines (1991), recovering the growing-layer solution they found. However, in contrast to earlier work, we find a critical value of the buoyancy parameter beyond which the solution grows exponentially in time, consistent with our nonlinear results.

Nonlinear Spin-Up of a Rotating Stratified Fluid: Experimental Method and Preliminary Results

1998 ◽  
Vol 120 (4) ◽  
pp. 667-671 ◽  
Author(s):  
Richard E. Hewitt ◽  
Peter A. Davies ◽  
Peter W. Duck ◽  
Michael R. Foster ◽  
Fraser W. Smith
Keyword(s):  
Boundary Layer ◽  
Stratified Fluid ◽  
Boundary Layer Problem ◽  
Similarity Type ◽  
Analysis Of Similarity ◽  
Finite Time Singularity ◽  
Rotating Stratified Fluid ◽  
General Method ◽  
Good Agreement ◽  
Final Rotation

We consider the nonlinear spin-up of a rotating stratified fluid in a conical container. An analysis of similarity-type solutions to the relevant boundary-layer problem (Duck et al, 1997) has revealed three types of behavior for this geometry. In general, the boundary-layer evolves to either a steady state, a growing boundary-layer, or a finite-time singularity depending on the initial to final rotation rate ratio, and a “modified Burger number.” We emphasize the experimental aspects of our continuing spin-up investigations and make some preliminary comparisons with the boundary-layer theory, showing good agreement. The experimental data presented is obtained through particle tracking velocimetry. We briefly discuss the qualitative features of the spin-down experiments which, in general, are dominated by nonaxisymmetric effects. The experiments are performed using a conical container filled with a linearly stratified fluid, the generation of which is nontrivial. We present a general method for creating a linear density profile in containers with sloping boundaries.

On laminar boundary-layer blow-off. Part 2

1971 ◽  
Vol 48 (2) ◽  
pp. 209-228 ◽  
Author(s):  
D. R. Kassoy
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Stokes Equations ◽  
Analytical Description ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Navier Stokes Equations ◽  
Similarity Type ◽  
Boundary Layer Equations ◽  
Wall Shear

Several examples of incipient blow-off phenomena described by the compressible similar laminar boundary-layer equations are considered. An asymptotic technique based on the limit of small wall shear, and the use of a novel form of Prandtl's transposition theorem, leads to a complete analytical description of the blow-off behaviour. Of particular interest are the results for overall boundarylayer thickness, which imply that, for a given large Reynolds number, classical theory fails for a sufficiently small wall shear. A derivation of a new distinguished limit of the Navier–Stokes equations, the use of which will lead to uniformly valid solutions to blow-off type problems for Re → ∞, is included. A solution for uniform flow past a flat plate with classical similarity type injection, based on the new limit, is presented. It is shown that interaction of the injectant layers and the external flow results in a favourable pressure gradient, which precludes the classical blow-off catastrophy.

scholarly journals Spin-up of stratified rotating flows at large Schmidt number: experiment and theory

1999 ◽  
Vol 389 ◽  
pp. 169-207 ◽  
Author(s):  
R. E. HEWITT ◽  
P. A. DAVIES ◽  
P. W. DUCK ◽  
M. R. FOSTER
Keyword(s):  
Boundary Layer ◽  
Finite Time ◽  
Schmidt Number ◽  
Quantitative Agreement ◽  
Time Singularity ◽  
Boundary Layer Problem ◽  
Similarity Type ◽  
Rotation Rates ◽  
Finite Time Singularity ◽  
Layer Problem

We consider the nonlinear spin-up/down of a rotating stratified fluid in a conical container. An analysis of axisymmetric similarity-type solutions to the relevant boundary-layer problem, Duck, Foster & Hewitt (1997), has revealed three types of behaviour for this geometry. In general, the boundary layer evolves to either a steady state, or a gradually thickening boundary layer, or a finite-time singularity depending on the Schmidt number, the ratio of initial to final rotation rates, and the relative importance of rotation and stratification.In this paper we emphasize the experimental aspects of an investigation into the initial readjustment process. We make comparisons with the previously presented boundary-layer theory, showing good quantitative agreement for positive changes in the rotation rate of the container (relative to the initial rotation sense). The boundary-layer analysis is shown to be less successful in predicting the flow evolution for nonlinear decelerations of the container. We discuss the qualitative features of the spin-down experiments, which, in general, are dominated by non-axisymmetric effects. The experiments are conducted using salt-stratified solutions, which have a Schmidt number of approximately 700.The latter sections of the paper present some stability results for the steady boundary-layer states. A high degree of non-uniqueness is possible for the system of steady governing equations; however the experimental results are repeatable and stability calculations suggest that ‘higher branch’ solutions are, in general, unstable. The eigenvalue spectrum arising from the linear stability analysis is shown to have both continuous and discrete components. Some analytical results concerning the continuous spectrum are presented in an appendix.A brief appendix completes the previous analysis of Duck, Foster & Hewitt (1997), presenting numerical evidence of a different form of finite-time singularity available for a more general boundary-layer problem.

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