Gortler Vortices Research Articles

Effects of curvature variations on the nonlinear evolution of Goertler vortices

The nonlinear development of Goertler vortices over walls of variable curvature is studied. The parabolized disturbance equations governing the problem for small curvature, high Reynolds number, and order unity Goertler number are integrated numerically. Cases with concave, convex, and zero curvatures are analyzed. The ‘‘mushroom-shaped’’ distributions of low-momentum fluid riding above high-momentum fluid that are subject to secondary instability are predicted. The results show significant stabilization of disturbances introduced from a concave into a convex region, where new sets of vortices are successively created with opposite rotation to the preceding set. The convex curvature tends to eliminate the inflection points from the spanwise and normal profiles of the streamwise velocity and, hence, suppresses the oscillatory secondary instability that leads to turbulence. Decay of Goertler vortices in a flat region is found to be less rapid than in a convex region.

ON SPATIAL EVOLUTION OF LONG-WAVELENGTH GÖRTLER VORTICES GOVERNED BY A VISCOUS-INVISCID INTERACTION PART 1: THE LINEAR CASE

Abstract : The generation of long-wavelength, viscous-inviscid interactive Gortler vortices is studied in the linear regime by numerically solving the time-dependent governing equations. It is found that time dependent surface deformations, which assume a fixed nonzero shape at large times, generate steady Gortler vortices that amplify in the downstream direction. Thus, the Gortler instability in this regime is shown to be convective in nature, contrary to the earlier findings of Ruban and Savenkov. The disturbance pattern created by steady and streamwise-elongated surface obstacles on a concave surface is examined in detail, and also contrasted with the flow pattern due to roughness elements with aspect ratio of order unity on flat surfaces. Finally, tile applicability of the Briggs-Bers criterion to unstable physical systems of this type is questioned by providing a counterexample in the form of the inviscid limit of interactive Gortler vortices.

Nonlinear evolution of Görtler vortices in the near-wall part of a boundary layer

The evolution of Gortler vortices with wavelength smaller than the thickness of the boundary layer on a concave surface is modelled asymptotically at high Reynolds and Gortler numbers. It is known that in the initial linear stage of their evolution such vortices have the largest increment of amplitude growth. Numerical results demonstrate that taking the nonlinear interaction of the flow parameters into account considerably reduces the growth rate and leads to the forming of a perturbed vortex region core; profiles of the flow characteristics in the different stages of vortex evolution are presented.

Concave Wall Heat Transfer Characteristics with Longitudinal Pressure Gradients and Discrete Wall Jets.

The effects of free stream velocities, longitudinal pressure gradients and discrete wall jets on concave wall heat transfer characteristics have been examined in wind tunnel and water channel flows at momentum thickness Gortler numbers from 5 to 30 at the initial station. It was shown that longitudinal vortices became supressed with increasing free stream velocities. The pressure gradient parameters of up to 0.75 × 10-6 resulted in a regular vortex structure and caused mean heat transfer coefficient to increase. The pressure gradient parameter of 1.8 × 10-6 was sufficent to suppress the vortex development so that there was no heat transfer augmentation above that of flat plate values. Measurements with blowing parameters of up to unity showed that the streamwise vortices and the wall temperature patterns were controlled by the jets so that maximum Stanton numbers exceeded minimum values by a factor of around three. With blowing parameters larger than unity, the interaction between the stream-wise vortices and the wall jets caused irregularities in wall temperature patterns so that the streamwise variation of Stanton number remained nearly unchanged.

Goertler vortices with system rotation: Linear theory

The equations which describe a boundary layer on a curved wall in a rotating system are derived and a linear stability analysis of a basic Blasius velocity profile is performed. Rotation can be either stabilizing or destabilizing corresponding to whether a rotation number is negative or positive, respectively. The stability boundaries at different rotation numbers and curves of constant positive growth rates are presented. It is shown that the flow is completely stabilized when the rotation number is ≤−1 in agreement with an inviscid Rayleigh-type analysis. For the cases examined growth rates increase linearly, in the initial phase, with streamwise distance.

Unusual hydrodynamic structures in a rotating channel

Visualizing the flow in the meridional plane of a rotating circular pipe has revealed two interesting three-dimensional hydrodynamic effects. The first is the unsteady process that follows the sudden stopping of the rotation, including the appearance on the inner surface of periodic localized inhomogeneities. These grow differently in the axial and radial directions with the eventual formation of mushroom-like structures; this process can be qualitatively explained in terms of the theory of unsteady Gortler vortices [1]. The second effect is the establishment of an unusual extended stable periodic structure as the rotating channel, initially open at one end, is closed off by a fixed plate, leaving a small gap between the plate and the end of the pipe for air to flow into the pipe from the outside. This periodic structure consists of volumes with a closed circulating flow alternating with “open” volumes hydrodynamically communicating with each other and the end of the pipe through wall flow zones. Simple relations are proposed for some of the characteristics of the periodic structure detected.

The compressible Görtler problem in two-dimensional boundary layers

We investigate the growth rates of Gortler vortices in a compressible flow in the inviscid limit of large Gortler number. Numerical solutions are obtained for O(1) wavenumbers. The further limits of (i) large Mach number and (ii) large wavenumber with O(1) Mach number are considered. It is shown that two different types of disturbance mode can appear in this problem. The first is a wall layer mode, so named as it has its eigenfunctions trapped in a thin layer near the wall. The other mode investigated is confined to a thin layer away from the wall and termed a trapped-layer mode for large wavenumbers and an adjustment-layer mode for large Mach numbers, since then this mode has its eigenfunctions concentrated in the temperature adjustment layereigenfunctions concentrated in the temperature adjustment layer

Hydrodynamic and Thermal Measurements in a Turbulent Boundary Layer Recovering From Concave Curvature

The behavior of a boundary layer on a flat wall downstream of sustained concave curvature is documented. Experiments are conducted with negligible streamwise pressure gradient and a low free-stream turbulence intensity (0.6 percent). The turbulent boundary layer has a moderate strength of curvature (δ/R = 0.024) at the entry to the recovery section. Results show that the skin friction coefficient, which increases over the concave wall, decreases rapidly at first over the recovery wall, then slowly approaches flat-wall values. Stanton number values decrease rapidly, undershooting expected flat-wall values. A discussion of this behavior, supported by profile measurements, is given. Effects include destabilization in the concave-curved flow and rapid streamline readjustment (acceleration) at the end of the curved section. Goertler vortices established on the curved wall persist onto the recovery wall; however, their effects weaken.

On the development of G�rtler vortices in wall jet flow

The development of Gortler vortices in wall jet flow over curved surfaces is considered in both the linear and nonlinear growth regimes. It is shown, using asymptotic methods based on the largeness of the wavenumber of the vortices, that this hydrodynamic instability is prone to occur more readily on concave rather than convex surfaces. It is found that after passing the position of neutral stability, the flow develops a surprising structure quite unlike that produced in the Blasius boundary-layer. Once the flow is into the unstable regime, the effect of increasing the Gortler number is to move the vortices away from the wall.

The inviscid compressible G�rtler problem in three-dimensional boundary layers

In this paper we investigate numerical solutions for the growth rates of Gortler vortices in a compressible three-dimensional flow in the inviscid limit of a large Gortler number. We look at a range of Mach numbers and find that there are three different types of behaviour for the mode growth-rate, corresponding to whether the flow is incompressible, has a Mach number small enough so that temperature-adjustment-layer modes do not appear in the two-dimensional case, or has a Mach number large enough so that they do. We find that it takes a considerably greater crossflow to destroy the Gortler vortices for moderate Mach numbers than it did in the incompressible case looked at by Bassom and Hall (1991). From this we believe that Gortler vortices may well still be a cause of transition for many practical compressible inviscid three-dimensional flows.

On the growth of mushroomlike structures in nonlinear spatially developing Goertler vortex flow

Spatially developing longitudinal vortices, originating from upstream Goertler vortices, are studied by a finite-difference algorithm. Three-dimensional parabolized Navier–Stokes equations for a ‘‘slightly’’ concave wall and a thin boundary layer are used, in conjunction with the total (mean plus perturbation) flow quantities as dependent variables. The behavior of Goertler vortex development from the linear region to the fully nonlinear region was obtained just prior to the development of secondary instabilities and turbulence. Comparisons with the experiments are discussed, showing the development of mushroomlike features in the streamwise velocity component.

Controlling the spacing of streamwise vortices on concave walls

The wavelength of Gortler streamwise vortices changed by a factor of three by varying the amount of tunnel side wall boundary-layer removal just upstream of the concave wall test section's leading edge. Nondimensional wavelengths were achieved with the larger wavelengths resulting from increased suction. The minimum wavelength compares well with the maximum amplification curve based on past theoretical studies. Low-speed streaks appeared to merge with other low-speed streaks or were weakened to the point of disappearing as the suction level was increased. The downstream location of secondary instability onset changed with variation in suction level and/or wavelength. The Gortler number and amplification factor for secondary instability onset, however, was approximately independent of suction level and/or wavelength.

The effect of wall compliance on the Goertler vortex instability

The stability of the flow of a viscous incompressible fluid over a curved compliant wall to longitudinal Goertler vortices is investigated. The compliant wall is modeled by a particularly simple equation relating the induced wall displacement to the pressure in the overlying fluid. Attention is restricted to the large Goertler number regime; this regime being appropriate to the most unstable Goertler mode. The effect of wall compliance on this most unstable mode is investigated.

The secondary instability in Goertler flow

The ‘‘high-frequency’’ secondary instability of nonlinearity developed and strongly modified Goertler vortices is numerically studied by considering two kinds of secondary instability modes, i.e., sinuous modes and varicose modes. A spectral collocation method with Chebyshev polynomials is used for calculations. It is found that secondary instabilities are more correlated with the vertical vorticity and the sinuous type disturbance would prevail over the varicose one for the same frequency and streamwise wave number found in experiments.

On scalar transport in nonlinearly developing Görtler vortex flow

Abstract It is shown that for an incompressible, rotationless fluid, similarity exist between the total streamwise velocity in Gortler vortex flow and the transported scalars, heat and mass, for Prandtl and Schmidt numbers of unity. This analogy is developed for a “thin boundary layer” relative to the surface radius and the streamwise development length scale. Quantitative interpretation of mushroomlike structures from flow visualization and enhanced surface heat and mass transfer rates are discussed. Some results for Prandtl and Schmidt numbers other than unity, for a gas and a liquid system, are presented.

On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness

The receptivity problem for Görtler vortices induced by wall roughness is investigated. The roughness is modelled by small amplitude perturbations to the curved wall over which the flow takes place. The amplitude of these perturbations is taken to be sufficiently small for the induced Görtler vortices to be described by linear theory. The roughness is assumed to vary in the spanwise direction on the boundary-layer lengthscale, whilst in the flow direction the corresponding variation is on the lengthscale over which the wall curvature varies. In fact the latter condition can be relaxed to allow for a faster streamwise roughness variation so long as the variation does not become as fast as that in the spanwise direction. The function that describes the roughness is assumed to be such that its spanwise and streamwise dependences can be separated; this enables us to make progress by taking Fourier or Laplace transforms where appropriate. The cases of isolated and distributed roughness elements are investigated and the coupling coefficient which relates the amplitude of the forcing and the induced vortex amplitude is found asymptotically in the small wavelength limit. It is shown that this coefficient is exponentially small in the latter limit so that it is unlikely that this mode can be stimulated directly by wall roughness. The situation atO(1) wavelengths is quite different and this is investigated numerically for different forcing functions. It is found that an isolated roughness element induces a vortex field which grows within a wedge at a finite distance downstream of the element. However, immediately downstream of the obstacle the disturbed flow produced by the element decays in amplitude. The receptivity problem at larger Görtler numbers appropriate to relatively large wall curvature is discussed in detail. It is found that the fastest growing linear mode of the Görtler instability equations has wavenumber proportional to the one-fifth power of the Gortler number. The mode can be related to both inviscid disturbances and the disturbances appropriate to the right-hand branch of the neutral curve for Görtler vortices. The coupling coefficient between this, the fastest growing vortex, and the forcing function is found in closed form.

Transition of boundary layer flows in the presence of Goertler vortices

Transition of boundary layer flows in the presence of Goertler vortices

Concentration and heat transfer in nonlinear Gortler vortex flow and the analogy with longitudinal momentum transfer

The similarity between heat, mass and streamwise-momentum transfer for the nonlinear Gortler vortex problem is pointed out. Quantitative interpretation of enhanced surface heat and mass transfer rates for unity Prandtl and Schmidt numbers are given. The relation between the visualized mushroomlike concentration distribution and the similar streamwise velocity development is also quantitatively interpreted.

On the Instability of Görtler Vortices to Nonlinear Travelling Waves

Recent theoretical work by Hall & Seddougui (1989) has shown that strongly nonlinear high-wavenumber Gortler vortices developing within a boundary layer flow are susceptible to a secondary instability which takes the form of travelling waves confined to a thin region centred at the outer edge of the vortex. This work considered the case in which the secondary mode could be satisfactorily described by a linear stability theory, and in the current paper our objective is to extend this investigation of Hall & Seddougui (1989) into the nonlinear regime. We find that, at this stage, not only does the secondary mode become nonlinear, but it also interacts with itself so as to modify the governing equations for the primary Gortler vortex. In this case, then, the vortex and the travelling wave drive each other, and indeed the whole flow structure is described by an infinite set of coupled nonlinear differential equations. We undertake a Stuart-Watson type of weakly nonlinear analysis of these equations and conclude, in particular, that on this basis there exist stable flow configurations in which the travelling mode is of finite amplitude. Implications of our findings for practical situations are discussed, and it is shown that the theoretical conclusions drawn here are in good qualitative agreement with available experimental observations.

Görtler vortices in growing boundary layers: The leading edge receptivity problem, linear growth and the nonlinear breakdown stage

Görtler vortices are thought to be the cause of transition in many fluid flows of practical importance. In this paper a review of the different stages of vortex growth is given. In the linear regime nonparallel effects completely govern this growth and parallel flow theories do not capture the essential features of the development of the vortices. A detailed comparison between the parallel and nonparallel theories is given and it is shown that at small vortex wavelengths the parallel flow theories have some validity; otherwise nonparallel effects are dominant. New results for the receptivity problem for Gortler vortices are given; in particular vortices induced by free-stream perturbations impinging on the leading edge of the wall are considered. It is found that the most dangerous mode of this type can be isolated and its neutral curve is determined. This curve agrees very closely with the available experimental data. A discussion of the different regimes of growth of nonlinear vortices is also given. Again it is shown that, unless the vortex wavelength is small, nonparallel effects are dominant. Some new results for nonlinear vortices of O(l) wavelengths are given and compared with experimental observations. The agreement between theory and experiment is shown to be excellent up to the point where unsteady effects become important. For small wavelength vortices the nonlinear regime is of particular interest since a strongly nonlinear theory can be developed there. Here the vortices can be large enough to drive the mean state which then adjusts itself to make all modes neutral. The breakdown of this nonlinear state into a three-dimensional time dependent flow is also discussed.