Gortler Vortices Research Articles

Görtler Vortices in the Shock Wave/Boundary-Layer Interaction Induced by Curved Swept Compression Ramp

This study builds on previous research into the basic flow structure of a separated curved swept compression ramp shock wave/turbulence boundary layer interaction (CSCR-SWBLI) at the leading edge of an inward-turning inlet. We employ the ice-cluster-based planar laser scattering (IC-PLS) technique, which integrates multiple observation directions and positions, to experimentally investigate a physical model with typical parameter states at a freestream Mach number of 2.85. This study captures the fine structure of some sections of the flow field and identifies the presence of Görtler vortices (GVs) in the CSCR-SWBLI. It is observed that due to the characteristics of variable sweep angle, variable intensity interaction, and centrifugal force, GVs exhibit strong three-dimensional characteristics in the curved section. Additionally, their position is not fixed in the spanwise direction, demonstrating strong intermittence. As the vortices develop downstream, their size gradually increases while the number decreases, always corresponding to the local boundary layer thickness. When considering the effects of coupling of bilateral walls, it is noted that the main difference between double-sided coupling and single-sided uncoupling conditions is the presence of a large-scale vortex in the central plane and an odd number of GVs in the double-sided model. Finally, the existence of GVs in CSCR-SWBLI is verified through the classical determine criteria Görtler number (GT) and Floryan number (F) decision basis.

Görtler vortices behavior and prediction in dual-incident shock-wave/turbulent-boundary-layer interactions

Görtler vortices (GVs) in dual-incident shock-wave/turbulent-boundary-layer interactions (dual-ISWTBLIs) are experimentally investigated in a Mach 2.48 flow. A double-wedge shock generator with two deflection angles of 8° and 5° is used to produce two incident shock waves (ISWs). Flow structures of the experiments with three different shock-wave distances were visualized by the ice-cluster-based planar laser scattering technique at two orthogonal planes (x–y and x–z planes). The images in the x–y plane present three types of flow patterns of dual-ISWTBLIs corresponding to the first type with a triangle-like separation, the second type with a quadrilateral-like separation, and the third type with two isolated interactions induced by the two ISWs. The images in the x–z plane indicate that the GVs exist in the first type of dual-ISWTBLI originating in the vicinity of the apex of the separation region and cover nearly the whole spanwise range of the reattachment region. By comparison, the GVs intermittently occur in the limited spanwise range of the reattachment region in the second type of dual-ISWTBLI. No GVs are observed in the third type of dual-ISWTBLI because no visible separation is induced under the experimental conditions considered in this situation. In addition, based on the wall-pressure distribution in the former two types of dual-ISWTBLIs, this paper proposes a method to estimate the mean-flow streamline curvature in the reattachment region, thereby obtaining the criteria for the existence of GVs, according to which reasonable explanations for the different distributions of GVs in the two types of dual-ISWTBLIs are provided.

Long wavelength streamwise vortices caused by wall curvature or roughness

Long wavelength instabilities of boundary layers caused by centrifugal effects or wall roughness are investigated. The wall roughness is modelled by small amplitude surface waviness. The instability is described in the nonparallel regime where it develops on the same length scale as the unperturbed flow. It is shown that instabilities initiated by disturbances close to the leading edge initially deform rapidly into algebraically growing eigensolutions but then deform into exponentially growing disturbances. The disturbances ultimately develop in a quasi-parallel manner and then pass successively through the high Gortler number, or equivalent large roughness parameter, regimes first described by Denier et al. (Nasa Contractor ICASE Report 90-31, 1990; Philos Trans R Soc A 334:51–85, 1991). It is shown that the mode which develops downstream is the most rapidly growing one available and not the second most unstable mode as claimed in a recent paper.

Instability and transition in the boundary layer driven by a rotating slender cone

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Centrifugal/elliptic instabilities in slowly varying channel flows

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Investigation of the influence of a local change in surface temperature on the laminar boundary layer stability in a hypersonic nozzle

The influence of a local change in surface temperature of a contoured nozzle corresponding to the Mach number M = 6 on the boundary layer stability and laminar-turbulent transition is numerically studied. The profiles of the laminar boundary layer are obtained by solving the Navier-Stokes equations with the use of the Ansys Fluid software system. N-factors of the growth rates of the Goertler vortices and also disturbances of the first and second Mack modes are calculated in the approximation of the linear stability theory. It is demonstrated that local heating ensures lower growth rates of the amplitudes of the Goertler vortices and the first Mack mode as compared to the base case; the more intense the heating, the more expressed this effect. The growth rate of the amplitude of the second-mode disturbances decreases during local heating of the nozzle to a temperature close to the stagnation temperature and increases at higher temperatures of local heating. It is found that local cooling leads to an increase in the growth rates of the amplitudes of the Goertler vortices and second Mack mode. The amplitude of the first Mack mode in the cooling region is smaller than that in the base case; however, further downstream, it is much greater than that in the base case. It is found that the surface of contoured nozzles should be heated in the region of the maximum growth rates of the amplitudes of the Goertler vortices; the higher the temperature, the more pronounced the expected effect. However, the maximum possible temperature is determined by the growth of the second Mack mode. The optimal option is to use the temperature of local heating of the surface at which the growth rate of the amplitude of the second mode is smaller than that of the Goertler vortices.

Görtler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability

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Experimental study on shock and turbulent boundary layer interactions under different incident shock waves

In this paper, the interactions between shock wave and the turbulent boundary layer of an incoming wall were experimentally studied. The experiments were performed in a Mach 3.4 supersonic low-noise wind tunnel at the unit Reynolds number of 6.30×106 m−1. Under different incident shock wave conditions, fine flow field structure, wall pressure, and wall temperature distribution were determined. Relationship was also analyzed. Instantaneous fine structures of typical flow fields with shock wave and turbulent boundary layer interactions were investigated using the nano-tracer planar laser scattering technique, and the transient flow images were analyzed statistically. The oscillation position of the induced shock wave satisfies a normal distribution. Simultaneously, the wall pressure distribution under different incident shockwave conditions of the interaction region was determined using the flush air data sensing system, and the relationship between flow the structure of the separation region and the wall pressure distribution with different incident shock wave intensities is analyzed. Using a temperature sensitive paint technology, the wall temperature distribution under different incident shockwave conditions of the interaction region model was determined. Also, the relationship between the strip structure caused by Gortler vortices and the incident shock wave intensity in the shockwave/turbulent boundary layer interaction was analyzed.

Effects of streamwise wall curvature variation on the nonlinear evolution of different Görtler vortices wavelengths

In this paper, a high-order, pseudo-spectral numerical code (HOPe) is used to investigate the influence of curvature variations on the nonlinear evolution of different Gortler vortices wavelengths. The mechanism by which Gortler instability selects the wavelength of the vortex when considering boundary layer over variable curvature surface is not clearly understood. In order to investigate the influence of the wall curvature and the effects of the spanwise wavelength on the laminar–turbulent transition process, three spanwise wavelengths and four different streamwise wall curvature distributions were analyzed: constant, concave–convex, concave-flat, and periodic curvature wall. The results show concave–convex wall distribution is the most stable case. With the change from concave to convex curvature, the centrifugal effects were turned off and this tends to eliminate the inflection points from the spanwise and normal profiles of the streamwise velocity. Thus, there is evidence that the convex curvature can be useful for laminar flow control. Furthermore, the numerical results indicate that the spanwise wavelength of stationary disturbance has a significant effect on the instability phenomenon, where the largest spanwise wavelength is the most unstable case when the curvature distribution is changed.

Laminar boundary layer stability calculation for contoured Mach 6 nozzle

Stability of the laminar boundary layer on the surface of a hypersonic nozzle for the Mach number M = 6 of the Transit-M wind tunnel is calculated. The laminar boundary-layer profiles are obtained by solving the Navier-Stokes equations numerically within the framework of the Ansys Fluent software. N-factors of the Goertler vortices and of the first and second Mack modes are obtained in the approximation of the linear stability theory. It is demonstrated that the Goertler vortices are the most unstable disturbances for the nozzle under consideration. Empirical dependences of the local Reynolds number of the laminar-turbulent transition on the N-factor and unit Reynolds number are determined.

Wake/shear layer interaction for low-Reynolds-number flow over multi-element airfoil

Time-resolved particle image velocimetry (TR-PIV) and hydrogen bubble visualization are employed to study the effects of Reynolds number on the wake/shear layer interactions over multi-element airfoil (30P30N). The Reynolds number based on the stowed chord length (Rec) ranges from 9.3 × 103 to 3.05 × 104. According to the variation of dominated flow structures, a critical Rec interval from 1.27 × 104 to 1.38 × 104 is found, which is novel for the low-Reynolds-number flow over multi-element airfoil. The slat wakes can be divided into two types by this critical interval. When Rec is smaller than this critical interval, no roll-up occurs to the shear layer of slat cusp. Gortler vortices generated by a virtual curved wall dominate the slat wake. When Rec is larger than this critical interval, roll-ups occur to the shear layer of slat cusp, which is similar to the cases at high Reynolds number (Rec ~ 106). These roll-ups and their evolution result in the co-existence of spanwise vortices and streamwise vortices in the slat wake. Different kinds of slat wake result in different kinds of wake/shear layer interactions above the main element. The flow physics behind these complex interactions, especially the novel flow structures and their evolution, is analyzed in detail to contribute to the fundamental research of wake/shear layer interactions. When Gortler vortices dominate the slat wake, they could trigger streaky structures within the leading-edge separated shear layer of the main element. When spanwise vortices and streamwise vortices co-exist in the slat wake, novel spanwise “double secondary vortices” are triggered above the main element by the spanwise vortices of slat cusp shear layer.

Flow effects in the reattachment region of supersonic laminar separated flow

The structure of a laminar separation flow around a compression corner at a free-stream Mach number M∞ = 6 is considered. The existence of a new element in the reattachment area—a high total-pressure layer—is shown. This layer is located downstream from the reattachment line above the boundary layer, and its total pressure is higher than that in either the boundary layer beneath it or the supersonic flow above it. Its appearance is due to the difference in the total pressure losses of the streamlines passing through a compression-wave fan and a shock wave in the reattachment zone. The vortex structure in the area of attachment is considered and the formation of streamwise vortices is shown. The origin of these vortices is similar to that of Taylor–Goertler vortices in subsonic flow. The pressure-pulsation distribution in the reattachment zone is also analyzed. It is shown that the spectrum of wall-pressure pulsations in the separation zone has two local maxima, the first caused by longitudinal fluctuations in the separation region and the second by pulsations of the vortices.

Experimental investigation of Görtler vortices in hypersonic ramp flows

A sharp leading-edge ramp model with 15°, 20°, and 25° ramp angles is experimentally investigated at a freestream Mach number 7.7 and a unit Reynolds number 4.2 × 106 m−1 in the Aachen Shock Tunnel TH2. The objective of this paper is to analyze Gortler vortices in terms of shear layer length, flow curvature, and vortex related parameters. The spanwise heat flux variation caused by it and the streamwise heat flux enhancement during its tenure are also studied. Thermocouples, pressure sensors, schlieren imagery, and infrared imaging are used for the investigation. The boundary layer on the flat plate until separation is laminar. An important outcome is that the arc length at reattachment is constant irrespective of the ramp angle. Characteristic boundary layer thicknesses at different Mach number show that the momentum thickness is insensitive to compressibility and is used to determine the Gortler number. A model is presented to determine the Gortler number in terms of ramp angle and a constant based on separation angle, arc length, momentum thickness, and Reynolds number. The half of the vortex wavelength is equal to the boundary layer thickness just before reattachment. The length scale required for breakdown of Gortler vortices decreases with rising ramp angle and is analogous to peak heating length. The streamwise heat flux enhancement occurs during the tenure of Gortler vortices and the enhancement rises with ramp angle. Although the visibility of Gortler vortices through temperature variation is distinct, the spanwise heat flux variation is not too high. Moreover, the spanwise heat flux variation rises marginally with ramp angle.

Visualization of Görtler vortices in supersonic concave boundary layer

By employing a nanoparticle-based planar laser scattering technique, with inflow Mach number of 2.95, boundary layers formed over two concave walls with respective curvature radii of 113 and 908 mm are visualized. The formation and the breakup process of the Gortler vortices have been clearly captured, which has seldom been done before under the supersonic condition. Different from the incompressible case, the breakup of the Gortler vortices in the supersonic boundary layer is found to be mainly attributed to the varicose mode instability. The Gortler vortices are not stationary, instead they are moving in the spanwise direction.

On the odd and even secondary instabilities of Görtler vortices

Boundary layer flows over concave wall can be unstable to disturbances giving rise to streamwise counter-rotating vortices known as Gortler vortices. These vortices in its nonlinear form are responsible for a strong distortion of the streamwise velocity profiles in the wall-normal and spanwise directions. The resulting inflectional velocity profiles are unstable to unsteady disturbances. These disturbances are called secondary instabilities and can develop into horseshoe vortices or a sinuous motion of the Gortler vortices. These types of secondary instabilities are known as even (varicose) and odd (sinuous) modes, respectively. Although many studies focused this subject, it has not been stated which mode dominates the transition process. In the present study the secondary instability of Gortler flow is investigated using high-order spatial numerical simulation. Multi-frequency unsteady disturbances are introduced with the same spanwise wavelength as the Gortler vortices, but different spanwise phases. Three different spanwise phases are used and the effect on the secondary instability is analyzed. Both, even and odd secondary instabilities are observed, according to the relative spanwise position of the unsteady disturbances. The growth analysis for each secondary crossplane instability mode is made using a temporal Fourier analysis and the physics is explored with the aid of the flow structures visualization. The results introducing disturbances that give rise to odd and even modes simultaneously show that, for the spanwise wavelength analyzed, the odd modes grow first and dominate the transition process.

The minimal flow unit in complex turbulent flows

Direct numerical simulations of spanwise-rotating periodic turbulent channel flow were conducted for three rotation numbers: Rob = 0, 0.2, and 0.5 at a Reynolds number of 8000 based on laminar centerline mean velocity. An additional simulation was conducted at a Reynolds number of 27 000 and rotation number Rob = 0.2 to study higher-Reynolds number effects. In order to determine the proper computational box size for a minimal flow unit (MFU) at Rob = 0.5, spanwise arrays of Taylor–Gortler vortices in the highly turbulent pressure region were examined and complete realization of the vortices was demonstrated to be necessary for accurate MFU turbulence statistics requiring a minimum spanwise domain length Lz = π. An equivalent MFU model produced similar results for Rob = 0.2 but demonstrated decreased accuracy for the higher-Reynolds number simulation. Analysis of pressure statistics also showed that the MFU model could accurately approximate mean pressure but not fluctuating pressure distributions across t...

Excitation of unsteady Görtler vortices by localized surface nonuniformities

A combined theoretical and numerical analysis of an experiment devoted to the excitation of Gortler vortices by localized stationary or vibrating surface nonuniformities in a boundary layer over a concave surface is performed. A numerical model of generation of small-amplitude disturbances and their downstream propagation based on parabolic equations is developed. In the framework of this model, the optimal and the modal parts of excited disturbance are defined as solutions of initial-value problems with initial values being, respectively, the optimal disturbance and the leading local mode at the location of the source. It is shown that a representation of excited disturbance as a sum of the optimal part and a remainder makes it possible to describe its generation and downstream propagation, as well as to predict satisfactorily the corresponding receptivity coefficient. In contrast, the representation based on the modal part provides only coarse information about excitation and propagation of disturbance in the range of parameters under investigation. However, it is found that the receptivity coefficients estimated using the modal parts can be reinterpreted to preserve their practical significance. A corresponding procedure was developed. The theoretical and experimental receptivity coefficients are estimated and compared. It is found that the receptivity magnitudes grow significantly with the disturbance frequency. Variation of the span-wise scale of the nonuniformities affects weakly the receptivity characteristics at zero frequency. However, at high frequencies, the efficiency of excitation of Gortler vortices depends substantially on the span-wise scale.

Transient wall-jet flowing over a circular cylinder

The transient flow of a two-dimensional wall-jet over a circular cylinder, following rapid initiation and termination, was investigated experimentally. Unsteady surface pressures and unsteady pressure-sensitive paint were used to gain a basic understanding of the flow physics. Jet initiation produced a starting vortex, upstream of which the Coandă flow developed, producing a large low-pressure peak. Immediately following jet termination, the pressure increased over the first quarter of the circumference, while the downstream separation region remained virtually unaffected. Simplifying analyses and dimensional arguments were used to show that the timescales characterizing the transient development of the integrated loads depend only on the square of the slot height and the kinematic viscosity and are thus independent of the jet velocity. Following jet initiation, the resulting loads varied according to a linear transient model, while small nonlinearities were observed following jet termination. Unsteady pressure-sensitive paint showed that the starting jet emerges from the slot in a two-dimensional manner and that streamwise streaks, identified as Gortler vortices, form well before the flow reaches steady state. During termination, the streamwise structures dissipate downstream initially, with the dissipation propagating upstream.

(Invited) Theoretical Analysis of Turbulent Mass Transfer with Rotating Cylinders

The data of Eisenberg1 and of Mohr2 provide an excellent basis for examining mass transfer in the system where the inner cylinder rotates within a larger stationary cylinder. The data are here correlated more carefully with respect to the dependence of the Stanton number on the Reynolds number, the Schmidt number, and the geometric ratio k of the inner cylinder radius to that of the outer cylinder. Three methods are deployed in this endeavor. Simple empiricism involves fitting the data according to the equation below where the parameters A, n, and p are functions of k. Enlightened empiricism expands on the traditional method of fitting the eddy viscosity as a function of the distance from the wall in the form of y + = (y/n)(t0/r)0.5, where t0is the shear stress at the solid wall. This method does not work well in a predictive manner for rotating systems or systems where the flow is in the direction of the curvature of the surface (as with Goertler vortices). Dissipation theory relates the local dissipation to its generation, transport, and decay, involving a parameter for the decay constant of the dissipation. The electrochemical data of Eisenberg (with 1-k values from 0.1723 to 0.871) seem to be in the regime of wide gaps, while Mohr focused on thin gaps (1-k ranging from 0.0172 to 0.0839) to explore the failure of Eisenberg’s data to show an expected behavior where thin gaps blend into turbulent Couette flow. The dissipation theorem should provide a new method of understanding turbulent shear flow and a way of predicting the effect of varying the radius ratio. *The author of this work is an Electrochemical Society Honorary Member

Theoretical Analysis of Turbulent Mass Transfer with Rotating Cylinders

Dissipation theory provides a new approach to understanding turbulent shear flow. The universal velocity profile and the law of the wall rely on the stress at a somewhat distant wall to provide a correlating parameter, but they are not adequately universal and general, as observed with rotating cylinder flow. Instead, the dissipation theorem expresses the governing physics by means of local laws describing interactions of statistical quantities of the flow. More work is needed to refine the method, but it can predict fluid flow and mass transfer with perhaps only one or two empirical parameters. The data of Eisenberg and of Mohr provide an excellent basis for examining mass transfer in the system where the inner cylinder rotates within a larger stationary cylinder. Neither author fully exploited the data, and this work correlates the data more carefully with respect to the dependence of the Stanton number on the Reynolds number, the Schmidt number, and the geometric ratio κ of the inner cylinder radius to that of the outer cylinder. Three methods are deployed in this endeavor. Simple empiricism involves fitting the data according to St = A(κReg)nScp, where the parameters A, n, and p are functions of κ. Enlightened empiricism expands on the traditional method of fitting the eddy viscosity as a function of the distance from the wall in the form of y+ = (y/ν)(τ0/ρ)0.5, where τ0 is the shear stress at the solid wall. This method does not work well in a predictive manner for rotating systems or systems where the flow is in the direction of the curvature of the surface (as with Goertler vortices). Dissipation theory relates the local dissipation to its generation, transport, and decay, involving a parameter for the decay constant of the dissipation. The electrochemical data of Eisenberg (with 1-κ values from 0.1723 to 0.871) seem to be in the regime of wide gaps, while Mohr focuses on thin gaps (1-κ ranging from 0.0172 to 0.0839) to explore the failure of Eisenberg's data to show an expected behavior where thin gaps blend into turbulent Couette flow. The dissipation theorem should provide a new method of understanding turbulent shear flow and a way of predicting the effect of varying the radius ratio.