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.

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.

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.

scholarly journals Analytic Approximate Solution for Falkner-Skan Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Bogdan Marinca
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Boundary Layer Problem ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method ◽  
Good Agreement ◽  
Layer Problem

This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

Laminar Free Convection From a Downward-Projecting Fin

1968 ◽  
Vol 90 (1) ◽  
pp. 63-70 ◽  
Author(s):  
G. S. H. Lock ◽  
J. C. Gunn
Keyword(s):  
Boundary Layer ◽  
Free Convection ◽  
Prandtl Number ◽  
Experimental Program ◽  
Boundary Layer Problem ◽  
Theoretical Predictions ◽  
Theoretical Results ◽  
Good Agreement ◽  
Quiescent Fluid

A theoretical analysis of conduction through and free convection from a tapered, downward-projecting fin immersed in an isothermal quiescent fluid is presented. The problem is solved by assuming quasi-one-dimensional heat conduction in the fin and matching the solution to that of the convection system, which is treated as a boundary layer problem. For an infinite Prandtl number, solutions are derived which take the form of a power law temperature distribution along the fin. The effect of this power (n) on heat transfer, drag, and the corresponding boundary layer profiles is discussed. It is shown that n is independent of the fin profile and dependent on a single nondimensional group χ. The theoretical results for infinite Prandtl number are compared with corresponding results derived from previous work using a Prandtl number of unity. The effect of Prandtl number on the determination of n and consequently the fin effectiveness is found to be extremely small. The results of an experimental program are also presented. These consist of temperature profiles and the n — χ relation for different fin geometries and surrounding fluids. Comparison with the theoretical predictions reveals good agreement.

Quasi-steady flow of a rotating stratified fluid in a sphere

1976 ◽  
Vol 76 (2) ◽  
pp. 209-228 ◽  
Author(s):  
Susan Friedlander
Keyword(s):  
Boundary Layer ◽  
Simple Model ◽  
Surface Stress ◽  
Steady Motion ◽  
Stratified Fluid ◽  
Interior Solution ◽  
The Sun ◽  
Boussinesq Model ◽  
Parameter Ranges ◽  
Rotating Stratified Fluid

The steady and quasi-steady motion achieved in a rotating stratified sphere of fluid is studied in the context of a linearized Boussinesq model. In certain parameter ranges an explicit expression is obtained for the flow field as a functional of the surface stress. The non-singular interior solution is used to examine the behaviour of the boundary layer close to the equator. The results agree with previous conclusions about the behaviour of a rotating stratified fluid in simpler geometries. Viewing the problem as a simple model for the interior core of the sun, this work indicates a solar spin-down time that is within the lifetime of the sun.

Linear theory of rotating stratified fluid motions

1967 ◽  
Vol 29 (1) ◽  
pp. 1-16 ◽  
Author(s):  
V. Barcilon ◽  
J. Pedlosky
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Linear Theory ◽  
Stable Stratification ◽  
Ekman Layer ◽  
Stratified Fluid ◽  
Rotating Fluids ◽  
Ekman Layers ◽  
Steady Motions ◽  
Rotating Stratified Fluid

A linear theory for steady motions in a rotating stratified fluid is presented, valid under the assumption that ε < E, where ε and E are respectively the Rossby and Ekman numbers. The fact that the stable stratification inhibits vertical motions has important consequences and many features of the dynamics of homogeneous rotating fluids are no longer present. For instance, in addition to the absence of the Taylor-Proudman constraint, it is found that Ekman layer suction no longer controls the interior dynamics. In fact, the Ekman layers themselves are frequently absent. Furthermore, the vertical Stewartson boundary layers are replaced by a new kind of boundary layer whose structure is characteristic of rotating stratified fluids. The interior dynamics are found to be controlled by dissipative processes.

The boundary layer on a flat plate moving transversely in a rotating, stratified fluid

1971 ◽  
Vol 46 (4) ◽  
pp. 769-786 ◽  
Author(s):  
Larry G. Redekopp
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Entire Range ◽  
Stratified Fluid ◽  
Similarity Solutions ◽  
Horizontal Plate ◽  
Viscous Boundary Layer ◽  
Limited Region ◽  
Viscous Boundary ◽  
Rotating Stratified Fluid

The motion of a horizontal plate moving in its own plane in a rotating, stratified fluid is studied to establish the parameter conditions specifying the onset of boundary-layer blocking for the entire range of Rossby and Russell numbers. Régimes in Rossby–Russell number space defining the range of applicability of the inertia–viscous, buoyancy–viscous, and Coriolis–viscous boundary-layer balances are presented, and similarity solutions valid over a limited region of that space are derived. The plate drag and heat transfer are computed from the similarity solutions.

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 Boundary layer analysis for a 2-D Keller-Segel model

2020 ◽  
Vol 18 (1) ◽  
pp. 1895-1914
Author(s):  
Linlin Meng ◽  
Wen-Qing Xu ◽  
Shu Wang
Keyword(s):  
Boundary Layer ◽  
Diffusion Coefficient ◽  
Approximate Solution ◽  
Chemical Diffusion ◽  
Singular Perturbation Theory ◽  
Energy Estimates ◽  
Chemical Diffusion Coefficient ◽  
Boundary Layer Problem ◽  
Classical Energy ◽  
Layer Analysis

Abstract We study the boundary layer problem of a Keller-Segel model in a domain of two space dimensions with vanishing chemical diffusion coefficient. By using the method of matched asymptotic expansions of singular perturbation theory, we construct an accurate approximate solution which incorporates the effects of boundary layers and then use the classical energy estimates to prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.

Numerical Study of Boundary Layers With Reverse Wedge Flows Over a Semi-Infinite Flat Plate

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
R. Ahmad ◽  
K. Naeem ◽  
Waqar Ahmed Khan
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Numerical Study ◽  
Boundary Layer Equation ◽  
Wedge Angle ◽  
Adverse Pressure Gradient ◽  
Problem Solution ◽  
Weighted Residual Method ◽  
Boundary Layer Problem ◽  
Layer Problem

This paper presents the classical approximation scheme to investigate the velocity profile associated with the Falkner–Skan boundary-layer problem. Solution of the boundary-layer equation is obtained for a model problem in which the flow field contains a substantial region of strongly reversed flow. The problem investigates the flow of a viscous liquid past a semi-infinite flat plate against an adverse pressure gradient. Optimized results for the dimensionless velocity profiles of reverse wedge flow are presented graphically for different values of wedge angle parameter β taken from 0≤β≤2.5. Weighted residual method (WRM) is used for determining the solution of nonlinear boundary-layer problem. Finally, for β=0 the results of WRM are compared with the results of homotopy perturbation method.

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