The Finite Dean Problem: Nonlinear Theory

The Dean problem models flows in curved pipes of square cross-sectionby considering pressure-driven axisymmetric flow between concentriccylinders. The inclusion of ends on these cylinders introducesa range of flows not seen in an infinite formulation. Weaklynonlinear analyses are made possible by considering end conditionssuch as those of Blennerhassett & Hall (1979). This paperextends to Dean flow this nonlinear work of Hall (1980). A newderivation of the amplitude equations will be given, which alsouses a perturbation from the intersection points of the oddand even curves of critical Taylor number against cylinder heightgiven by linear theory. Equilibrium solutions of the amplitudeequations are then found and their stability examined. Unlikethe Taylor-vortex results of Hall, the properties of these solutionsdepend on the particular intersection point under consideration.A comparison is given with previous work on the finite Deanproblem.     

Triple-deck Flows over Wavy Surfaces

Sinusoidally perturbed laminar flow over a flat plate is considered,in which the amplitude and wave number bear the triple-deckrelation to the Reynolds number R. To find the resulting flowfield and in particular the surface shear stress and pressureit is necessary to solve the non-linear lower-deck equation.Analytic solutions of the linearized equations are obtainedand the behaviour of the stresses near the leading edge of thewaviness is examined. As regions further downstream are consideredit is found that the linearized results predict the same surfacestress phase shifts as those derived by Benjamin (1959) forboundary-layer flows over wavy surfaces of amplitude and wavenumberindependent of R. For larger amplitudes, the full non-linearequations are solved numerically via a modification of the spectralmethod of Burggraf & Duck (1981). This modification is shownto significantly increase both the speed and accuracy of theoriginal method for lower-deck problems in general. Specialattention is paid to the problems associated with the semi-infiniteextent of the surface perturbation. The results of these non-linearcalculations of the stresses are examined to determine the effectsof increasing amplitude h and of increasing distance downstream.It is found that the surface stress extrema are phase-shifteddownstream with respect to the surface perturbation both asthe amplitude is increased and as regions further downstreamof the leading edge are considered. Moreover, the stress profilesbecome considerably distorted from sinusoidal as h increases.It is found that separation occurs at higher values of h thanpredicted by linear theory. Finally, differences between resultsfor positive and negative h are examined.