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On the growth of Taylor-Görtler vortices along highly concave walls
Abstract
The primary objective of this study has been to prepare a chart for computing the growth of Taylor-Görtler vortices in laminar flow along walls of both high and low concave curvature. Taylor-Görtler vortices are streamwise vortices having alternate right- and left-hand rotation that may develop in the laminar boundary layer along a concave surface.
... Les tourbillons longitudinaux sont organisés par paires et la distance entre deux paires de tourbillons correspond à la longueur d'onde λ de l'instabilité (Görtler, 1940, Smith, 1955. Les tourbillons d'une même paire sont contrarotatifs induisant ainsi des mouvements dans la direction normale à la paroi à leur frontière. ...
... Les premières stries sont en effet visibles et proviennent de la présence de mouvements transverses et normaux de faible intensité. Cette région de l'instabilité correspond à sa croissance linéaire, région de l'écoulement que l'on peut décrire théoriquement par analyse de stabilité linéaire (Görtler, 1940, Smith, 1955, Bippes et Görtler, 1972. Les prémices des tourbillons longitudinaux sont alors développés. ...
... Suivant les travaux de Görtler (1940), Smith (1955) et Floryan et Saric (1982), on peut exprimer les équations de Navier-Stokes linéarisées pour les perturbations u * = (u * , v * , w * ) et p * . On obtient alors l'équation de conservation de la masse (3.6) et les équations de conservation de quantité de mouvement suivant ξ * (3.7), suivant ψ * (3.8) et suivant z * (3.9) ...
L'enjeu de ce projet vise à mieux comprendre et modéliser la dynamique des courants gravitaires sur pente, qui contrôlent la circulation de l'océan, de l'atmosphère et des zones côtières. Le rôle du mélange turbulent induit par les instabilités hydrodynamiques qui s'y développent est primordial. En présence de pentes courbes, il existe une compétition entre l'instabilité de cisaillement (Kelvin-Helmholtz) et l'instabilité centrifuge (Görtler).Les conditions d'émergence de ces instabilités, leur interaction et leur effet sur la dynamique globale de l'écoulement sont encore mal compris à ce jour. Dans ce contexte, une étude numérique et théorique de la dynamique d'écoulements de couche limite sur paroi courbe est effectuée.Dans un premier temps, on caractérise l'instabilité hydrodynamique centrifuge, appelée instabilité de Görtler, par une approche théorique qui consiste à résoudre les équations du mouvement linéarisées (analyse de stabilité linéaire). On montre ainsi l'influence du nombre d'onde ainsi que du nombre de Görtler G sur la dynamique de l'écoulement.Dans un second temps, une série de simulations numériques instationnaires est réalisée pour des nombres de Reynolds modérés (faiblement turbulent) en utilisant les techniques de simulation des grandes échelles (SGE). Ces simulations permettent d'aller plus loin dans l'analyse de l'instabilité de Görtler car elle permettent de décrire la région non-linéaire de l'écoulement, ainsi que la transition à la turbulence. On étudie ainsi l'influence du forçage de la longueur d'onde de l'instabilité, ainsi que le comportement de l'écoulement quand l'instabilité se déclenche naturellement. Une comparaison avec les résultats de l'analyse de stabilité linéaire est présentée.Dans un troisième temps, une simulation numérique d'une couche limite turbulente sur paroi courbe est effectuée. On montre l'influence de la courbure sur la topologie de l'écoulement ainsi que sur la turbulence qui s'y développe. On montre également que des hétérogénéités spatiales persistent dans la région turbulente ce qui constitue un résultat majeur de la thèse.Enfin, on présente une série de simulations numériques visant à décrire une physique semblable à celles des écoulements géophysiques auxquels on s'interesse. Dans cette configuration, les forces de flottabilité impactent la dynamique de l'écoulement. Les impacts sur le développement de l'instabilité de Görtler, sur la topologie de l'écoulement ainsi que sur les propriétés turbulentes de l'écoulement sont étudiés. Une analogie entre les effets de courbure et les effets de flottabilité est préésentée.
... It has been shown, through stability diagrams, that a maximum growth of the Görtler vortices occurs when Λ is in the range of 200-270 [7,25,26]. Based on those results, a nondimensional wavelength of Λ = 250 has been preset in the experimental studies of Mitsudharmadi et al. [13] and Tandiono et al. [15,16] as well as in the numerical receptivity studies of Schrader et al. [17]. ...
... As shown by linear stability analysis, the selected dimensionless parameter produced the most amplified Görtler vortices [7,25,26]. Moreover, the utilized excitation method allowed to preset the wavelength of the Görtler vortices that emerged as primary instabilities in the laminar-to-turbulent transition process. ...
... The obtained results are plotted in It is observed that transition starts earlier for the cases within the range from Λ = 162 to 250. These results are in accordance with the ones obtained from linear stability theory where it was found that a maximum growth of the Görtler vortices occurs when Λ is within the range of 200-270[7,25,26].Another point to highlight is that the spanwise wavelength of the Görtler vortices has an impact on the type of secondary instability that develops. The baseline case with Λ = 250 only allowed to excite the varicose mode that is linked to the appearance of horseshoe vortices (see Section 4.2). ...
The laminar-to-turbulent transition process over a concave surface is studied using direct numerical simulations (DNS). It is found that the flow passing over such surface is highly susceptible to develop centrifugal instabilities in the form of Görtler vortices. Transition is triggered by means of wall-roughness elements which also serve to preset the wavelength of the Görtler vortices that remains constant during their streamwise development. This allows to obtain a clear spanwise characterization of the Görtler boundary layer, and its breakdown into turbulence. The different regions encountered in the transition process, i.e., linear, nonlinear, transition and fully turbulent, are identified and characterized. Parametric studies are presented showing that the transition starting point is delayed when the radius of curvature is increased, however, it occurs at the same critical Görtler number. Additionally, an increase of the height of the wall-roughness elements advances the transition onset upstream. Moreover, a linear relation between the critical Görtler number and the Reynolds number based on the wall-roughness streamwise location has been found. Furthermore, it is observed that, compared to a wall-roughness bump element, a bump-dimple geometry is more efficient for exciting the transition process.
https://authors.elsevier.com/a/1XHgfAQO4bjDd
... It is observed that transition starts earlier for the cases within the range from Λ = 162 to 250. These results are in accordance with the ones obtained from linear stability theory where it was found that a maximum growth of the Görtler vortices occurs when Λ is within the range of 200-270 (Floryan 1991, Smith 1955, Meksyn 1950. ...
... Their wavelength, Λ = 250, was chosen such as it matches previous numerical and experimental studies (Tandiono et al. 2008, Schrader et al. 2011, Tandiono et al. 2013. As shown by linear stability analysis, the selected dimensionless parameter produced the most amplified Görtler vortices (Floryan 1991, Smith 1955, Meksyn 1950. Moreover, the utilized excitation method allowed to preset the wavelength of the Görtler vortices that emerged as primary instabilities in the laminar-to-turbulent transition process. ...
Direct numerical simulations (DNS) are computed in order to study the complete laminar-to-turbulent transition process of a boundary layer developing over a concave surface. It is found that the flow passing through such geometry is prone to develop centrifugal instabilities in the form of Görtler vortices. Transition is triggered by means of wall-roughness elements that are also utilized to preset the spanwise wavelength of the Görtler vortices. The different regions encountered in the transition process, i.e. linear, nonlinear, transition, and fully turbulent, are identified and characterized. Primary and secondary (varicose and sinuous) instabilities are identified and analyzed as well. Parametric studies showing the effect of several physical parameters (radius of curvature, the vortices wave-length, the perturbation amplitude and streamwise location, and the wall-roughness perturbation geometry) on the transition starting point are presented. Furthermore, thermal analyses are conducted in order to study the modification of the thermal boundary layer due to the Görtler vortices swirl motion. The streamwise evolution of the surface heat transfer is investigated finding that it is considerably enhanced in the non-linear region surpassing the turbulence-region values. It is also found that the Reynolds analogy between streamwise-momentum and heat transfer is followed throughout the whole transition process.
... which uses an eddy viscosity ν T = 0.0234U ∞ θ (Tani 1962). The definition of the momentum thickness, θ, for curved wall flow and associated discussion are provided in Appendix A. Near the onset of curvature, the Görtler number is G t 4, which is large enough to promote instability (Smith 1955;Tani 1962), and it is larger in the FRC case. Based on this metric, free-stream turbulence enhances the curvature effects. ...
... In our configuration, at ξ = 50, the local Görtler number G t = 5.4 (c.f. figure 5b) and the most energetic spanwise wavenumber is k z θ = 0.43 which is also the most unstable mode according to linear theory (Smith 1955;Tani 1962). An important observation is the extent to which that wavenumber persists, or remains energetic, deep into the boundary layer, which is suggestive of a stronger modulation of the near-wall region than in flat plates, perhaps even a direct influence. ...
Direct numerical simulations are performed to contrast turbulent boundary layers over a concave wall without and with free-stream turbulence. Adverse pressure gradient near the onset of curvature leads to sharp decrease in skin friction and intermittent separation. The presence of free-stream turbulence reduces the probability of reverse flow, accelerates the recovery of the boundary layer in the downstream zero-pressure gradient region, and leads to a sustained and appreciable increase in the skin friction. The forcing also promotes the amplification of coherent G\"ortler structures in the logarithmic layer of the curved-wall boundary layer. Statistically, the spanwise and wall-normal Reynolds stresses intensify and the radial distance between their peaks increases downstream as the G\"ortler structures expand. The Reynolds shear stress coefficient also increases in the logarithmic layer in contrast to a decrease when a flat-plate boundary layer is exposed to free-stream turbulence. In addition, the more coherent and energetic roll motions in the forced flow promote mixing of free-stream and boundary-layer fluids, where the former is seen more often deep within the buffer layer.
... where U e , θ and R are the free-stream velocity at the edge of the boundary layer, the local momentum thickness and the radius of curvature of the mean flow, respectively, exceeds 0.3 (Görtler 1954;Smith 1955). The most amplified wavelength in the spanwise direction is found to be λ T , corresponding to U e λ T /ν √ λ T /R ≈ 220-270 (Smith 1955;Floryan & Saric 1982;Luchini & Bottaro 1998) or λ T ≈ δ − 2δ (Smits & Dussauge 2006). ...
... where U e , θ and R are the free-stream velocity at the edge of the boundary layer, the local momentum thickness and the radius of curvature of the mean flow, respectively, exceeds 0.3 (Görtler 1954;Smith 1955). The most amplified wavelength in the spanwise direction is found to be λ T , corresponding to U e λ T /ν √ λ T /R ≈ 220-270 (Smith 1955;Floryan & Saric 1982;Luchini & Bottaro 1998) or λ T ≈ δ − 2δ (Smits & Dussauge 2006). However, the above scaling laws are for laminar flows and their applicability to turbulent flows is unclear. ...
The spatio-temporal dynamics of separation bubbles induced to form in a fully developed turbulent boundary layer (with Reynolds number based on momentum thickness of the boundary layer of 490) over a flat plate is studied via direct numerical simulations. Two different separation bubbles are examined: one induced by a suction–blowing velocity profile on the top boundary and the other by a suction-only velocity profile. The latter condition allows reattachment to occur without an externally imposed favourable pressure gradient and leads to a separation bubble more representative of those occurring over airfoils and in diffusers. The suction-only separation bubble exhibits a range of clearly distinguishable modes, including a high-frequency mode and a low-frequency ‘breathing’ mode that has been observed in some previous experiments. The high-frequency mode is well characterized by classical frequency scalings for a plane mixing layer and is associated with the formation and shedding of spanwise-oriented vortex rollers. The topology associated with the low-frequency motion is revealed by applying dynamic mode decomposition to the data from the simulations and is shown to be dominated by highly elongated structures in the streamwise direction. The possibility of Görtler instability induced by the streamwise curvature on the upstream end of the separation bubble as the underlying mechanism for these structures and the associated low frequency is explored.
... Regardless of the mechanism that leads to instability, transition occurs when the disturbance waves approach a range of critical amplitudes [40]. Smith [41] demonstrated that, in the case of Görtler vortices, the critical amplification factor is of the order of 10. Following Schrijer [21], the critical amplification factor of can be defined as:Ñ ln ΔSt ...
... In the present case, λ ∼ 6.8 mm and the boundary-layer height δ R 2.4 mm is calculated at the reattachment point. For λ∕δ ∼ 2.8, N is close to 6 according to [40] and 10 according to [41]. Figure 25 shows the extent of the transition region for the present experiment. ...
This work presents an experimental and numerical study of hypersonic transitional shock-wave–boundary-layer interaction, wherein transition occurs between separation and reattachment in the detached shear layer. Experiments were conducted in a free-piston compression-heated Ludwieg tube that provided a Mach 5.8 flow at a freestream Reynolds number of 7×106 m−1. A shock generator deflected the flow by 10°, resulting in an oblique shock impinging on a flat plate. The shock triggered transition in the boundary layer and the formation of Görtler-like vortices downstream of reattachment. Heat flux and pressure distributions on the plate were measured globally using infrared thermography and pressure-sensitive paint. Oil film visualization was employed to evaluate the boundary-layer reattachment. Numerical results consist of Reynolds-averaged Navier–Stokes and fully laminar steady-state three-dimensional simulations. Shock-induced transition is considered to be the cause of the overshoot in peak pressure and peak heating of approximately 15%, in agreement with previous studies. Görtler instability, triggered by the concave nature of the bubble at separation, is identified as the main mechanism leading to boundary-layer transition, resulting in heat-flux variations of less than 30%. By comparing numerical results against thermographic values it is possible to delineate the extent of transition. Within this region, the disturbance amplification factor was estimated to be approximately between 6 and 10, in reasonable agreement with other relevant numerical and experimental data.
... where U e , θ and R are the free-stream velocity at the edge of the boundary layer, the local momentum thickness and the radius of curvature of the mean flow, respectively, exceeds 0.3 (Görtler 1954;Smith 1955). The most amplified wavelength in the spanwise direction is found to be λ T , corresponding to U e λ T /ν √ λ T /R ≈ 220-270 (Smith 1955;Floryan & Saric 1982;Luchini & Bottaro 1998) or λ T ≈ δ − 2δ (Smits & Dussauge 2006). ...
... where U e , θ and R are the free-stream velocity at the edge of the boundary layer, the local momentum thickness and the radius of curvature of the mean flow, respectively, exceeds 0.3 (Görtler 1954;Smith 1955). The most amplified wavelength in the spanwise direction is found to be λ T , corresponding to U e λ T /ν √ λ T /R ≈ 220-270 (Smith 1955;Floryan & Saric 1982;Luchini & Bottaro 1998) or λ T ≈ δ − 2δ (Smits & Dussauge 2006). However, the above scaling laws are for laminar flows and their applicability to turbulent flows is unclear. ...
The spatio-temporal dynamics of separation bubbles induced to form in a fully-developed turbulent boundary layer (with Reynolds number based on momentum thickness of the boundary layer of 490) over a flat plate are studied via direct numerical simulations. Two different separation bubbles are examined: one induced by a suction-blowing velocity profile on the top boundary and the other, by a suction-only velocity profile. The latter condition allows reattachment to occur without an externally imposed favourable pressure gradient and leads to a separation bubble more representative of those occurring over airfoils and in diffusers. The suction-only separation bubble exhibits a range of clearly distinguishable modes including a high-frequency mode and a low-frequency "breathing" mode that has been observed in some previous experiments. The high-frequency mode is well characterized by classical frequency scalings for a plane mixing layer and is associated with the formation and shedding of spanwise oriented vortex rollers. The topology associated with the low-frequency motion is revealed by applying dynamic mode decomposition to the data from the simulations and is shown to be dominated by highly elongated structures in the streamwise direction. The possibility of G\"{o}rtler instability induced by the streamwise curvature on the upstream end of the separation bubble as the underlying mechanism for these structures and the associated low frequency is explored.
... In opposite local non-parallel computations provide monotonically decreasing Görtler numbers when β decreases. One should also note a perfect matching between the present local non-parallel curve at N = 0 and the one from [4] based on Smith's model [12] for all the values taken by β . At identical intensity of rotation (abs(N) = 0.5), the increase in Go n for a negative rotation number is bigger than the decrease in Go n for a positive rotation number. ...
... (a) Geometry and (b) neutral curves in the plane (β , Go) for four values of the rotation number (N = −0.5, 0, 0.5 and 2). Present results based on (-) local parallel theory, (--) local non-parallel theory, (•) local non-parallel results from[15], (+) local non-parallel result from[4] based on Smith's model[12] and (×) based on their own model and ( ) one of the non-local non-parallel curves from[5]. ...
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In this paper, spatial linear stability analyses are performed on three flow configurations where rotational effect is present. The first two configurations are the two-dimensional Blasius boundary layer and the flow along a curved plate, both with a rotation vector along the spanwise direction. The third configuration is the three-dimensional flow due to a rotating disk with and without axial inflow. These flows are used as a verification of the extension of the stability analysis code NOLOT to rotating frames. For all these three flows, a perfect matching is observed in comparison with results from the literature for the neutral curves and the eigenfunctions. The present results show that rotation has a considerable effect on the stability of boundary layers. Depending on the intensity and orientation of the rotation vector, stabilization or destabilization of the boundary layer is observed.
... The decrease of the rolls wavelength for increasing Rayleigh number is supported by the proposed analogy with G€ ortler vortices since an increase in Ra F would lead to a higher G€ ortler number (see equation (25)), thus to a greater wave number as shown in Fig. 10. This figure, in fact, presents the wave number of the longitudinal rolls normalized with respect to d T .The dashed line is the neutral stability curve defined by Smith [29], while the solid line represents the neutral stability curve defined by G€ ortler [30]; data for these curves are taken from Ref. [19]. The increase of the normalized wave number with G is not linear but shows that kd T increases with increasing Ra F , thus the wavelength of the rolls decreases faster than the boundary layer thickness. ...
... (22)) versus normalized wave number for: Â Ra F0 ¼ 1:6x10 11 , +Ra F0 ¼ 6:0x10 11 , > Ra F0 ¼ 1:3x10 12 . The dashed line corresponds to the neutral stability curve defined by Smith [29] and the solid line is the neutral stability curve defined by G€ ortler [30]. ...
The convective flow driven by a differential heating along a horizontal boundary, commonly referred to as horizontal convection, is investigated. In particular, the inception of the convection cell following impulsive application of heating, i.e. the onset of horizontal convection, is studied experimentally in a rectangular box filled with water. Piecewise boundary conditions of uniform/constant heat flux and temperature are imposed along the box length at its bottomwall. The convective heat transfer coefficient on the heated half of the base is evaluated using the heated thin foil sensor modified to account for the unsteady term of the energy balance. The experiments are carried out over a range of Rayleigh numbers (based on heat flux input, box length and initial fluid properties) from 1.6 � 1011 to 1.3 � 1012 and for a Prandtl number (based on the initial water temperature) of approximately 6. Flow visualizations with the shadowgraph technique are also performed to complement heat transfer measurements. Thermocouple data are acquired inside the domain for validation purposes. A scaling for the characteristic time of the transient is proposed and verified. In the range of the investigated Rayleigh and Prandtl numbers, three subsequent phases in the onset process are identified: pure heat conduction through the fluid layer, Rayleigh-B�enard convection with transition of the boundary layer; onset and time evolution of longitudinal rolls. The presence of longitudinal rolls is justified via an analogy with G€ortler vortices theory and results show a Nusselt number enhancement on the heated side of the order of 200% with respect to that on the cold one.
... Second adding various suction/injection parameters W Ã with a fixed axial flow that Hussain obtained at T s ¼ 0:5. By applying a triple-deck structure flow, originally established by Blasius, over a flat plate, 38 we scaled the lower branch disturbance. Moreover, regarding the timeindependent stationary modes investigation condition, the effective wall shear (aU 0 r þ bV 0 ) at the leading order is considered to be zero. ...
In this paper, we investigated the effectiveness of the strength of axial flow and suction/injection for the viscous mode (type II) instability within the boundary layer of a rotating disk. To investigate the lower branch, we scaled the viscous mode by using a familiar triple-deck structure analogous to that which was found for Blasius flow over a flat plate. We also analyzed the linear stability behavior of high-Reynolds laminar-turbulent transition. To describe the stationary type II wavenumber and waveangle, we conducted an asymptotic analysis followed by a comparison with the type I mode. We found that a positive axial flow had a stabilizing effect and vice versa a negative axial flow exhibited a destabilizing effect. The results were consistent with previous studies in the literature for positive axial flow, as disturbances were advected downstream in the radial direction. Regarding the suction/injection study, we found suction to be stabilizing, which aligns with previous results in the literature although, conversely, the injection was found to be destabilizing. For the numerical analysis, we found that parameters which result in an increase or decrease of the critical Reynolds number led to a stabilization or destabilization of the flow, respectively. Finally, we compared the asymptotic and numerical stability results for both types I and II followed by the critical Reynolds numbers comparisons, which were found to be consistent in general with results in the literature.
... Curvature is computed as δ 0 /R, where δ 0 is the boundary layer thickness at the trailing edge of the body (x = 0) and R is the local radius of curvature. The Görtler number in the original studies of the instabilities over the concave walls (Görtler 1954;Smith 1955) is defined as ...
The wake flow past an axisymmetric body of revolution at a diameter-based Reynolds number $Re=u_{\infty }D/\nu =5000$ is investigated via a direct numerical simulation. The study is focused on identification of coherent vortical motions and the dominant frequencies in this flow. Three dominant coherent motions are identified in the wake: the vortex shedding motion with the frequency of $St=fD/u_{\infty }=0.27$ , the bubble pumping motion with $St=0.02$ , and the very-low-frequency (VLF) motion originated in the very near wake of the body with the frequency $St=0.002$ – $0.005$ . The vortex shedding pattern is demonstrated to follow a reflectional symmetry breaking mode, whereas the vortex loops are shed alternatingly from each side of the vortex shedding plane, but are subsequently twisted and tangled, giving the resulting wake structure a helical spiraling pattern. The bubble pumping motion is confined to the recirculation region and is a result of a Görtler instability. The VLF motion is related to a stochastic destabilisation of a steady symmetric mode in the near wake and manifests itself as a slow, precessional motion of the wake barycentre. The VLF mode with $St=0.005$ is also detectable in the intermediate wake and may be associated with a low-frequency radial flapping of the shear layer.
... 8 When the viscous boundary layer flows past along a sufficiently concave surface, the combined effect of the viscosity, centrifugal forces, and adverse pressure gradient (APG) destabilizes the boundary layer, thereby inducing the GVs. [9][10][11][12] Although the GVs were first observed in the incompressible laminar flow, the same boundary-layer instability mechanism is also applicable for compressible and turbulent flows. [13][14][15][16][17] For hypersonic flow, GVs often occur in the laminar-turbulent transition of the concave-wall boundary layer. ...
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.
... Taylor-Görtler vortices appear as streamwise counter-rotating vortices. The formation of these vortices occurs in flow over concave curvature and also when the dividing streamline of the recirculation region separates at an angle to the main flow direction (Smith 1955;Drazin & Reid 2004). Unlike the flow topology in Ghia et al. (1989), in the present set-up a secondary recirculation region on the top wall is absent in the steady base flow where instability occurs. ...
Motivated by an interest in inciting instabilities and mixing for heat transfer enhancement in ducts, flow in a channel with repeated wedge-shaped protrusions is considered for various blockage ratios (wedge height to duct height), pitch (distance between wedges) and wedge angles. The stability of the two-dimensional base flow and its dependence on the geometric parameters is elucidated through a global linear stability analysis. A linearly unstable two-dimensional mode was found, contrasting similar confined flow set-ups. However, the primary instability is a three-dimensional mode manifesting as counter-rotating streamwise vortices over the wedge tip. Analysis of the kinetic energy budget indicates a lift-up mechanism leading to instability, with the dominant energy gain of the global three-dimensional mode due to shear in the base flow. Structural sensitivity and receptivity of the instability to momentum forcing identifies the core of the instability and locations important for flow control. An endogeneity approach is used to show that the local perturbation pressure gradient component dominates the distribution of the local contribution to the growth rate of the linear global eigenmode in most cases considered, despite its net contribution being identically zero. Weakly nonlinear Stuart–Landau analysis reveals that the primary bifurcation is supercritical across all tested geometric parameter combinations. This is consistent with the finding of low linear transient growth amplifications at subcritical Reynolds numbers, being orders of magnitude lower than in similar channel flow set-ups.
... It is suggested that if the Görtler number is larger than a critical value, the flow becomes unstable, and hence the occurrence of Görtler instability [19]. In general, it is widely accepted that the critical Görtler number G T is about 0.3 [20]. Görtler vortices may be expected to exist in SBLI when the curvature of the mean streamlines is concave. ...
The frequency properties and the typical flow features within the interaction zone in conical shock-wave/turbulent boundary-layer interactions (CSBLIs) are analyzed by performing proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) on a time-resolved direct numerical simulation database. The streamwise velocity and pressure fluctuations are processed according to their energy and frequency, respectively. As is the case of planar shock-wave/boundary-layer interactions, fluctuations in some bands of frequencies are more energetic downstream of the flow separation zone. The energetic low-frequency modes of the streamwise velocity fluctuations are reminiscent of Görtler-like vortices, whereas those of pressure fluctuations are connected with the “breathing” motion of the separation bubble. Higher-frequency modes, on the other hand, are related to wrinkling of the impinging conical shock. Although they bear resemblance with those in planar shock/turbulent boundary-layer interactions, it was found that the flow structures herein are less coherent owing to spanwise inhomogeneity of the impinging shock strength. The POD and DMD results of CSBLI support the notion that Görtler-like vortices might be an intrinsic mechanism of the interaction zone, which is probably responsible for the low-frequency unsteadiness of CSBLI. The present results may serve as a basic guidance for the design of fluidic control strategies of shock-induced flow separation.
... Most early studies were conducted within the framework of incompressible flow. The analysis of the G€ ortler instability was extended by Smith 5 and Floryan and Saric, 6 who sought to include part of the nonparallel-flow effects but continued to treat the instability as an eigenvalue problem. Linear and nonlinear theoretical studies of the G€ ortler instability were conducted by Hall. ...
The hypersonic boundary layer transition over a concave wall is investigated in a Mach 6.5 quiet wind tunnel using temperature sensitive paint (TSP), CO2-enhanced filtered Rayleigh scattering flow visualization, PCB fast-response pressure sensors, and a high-frequency schlieren technique. The TSP shows that low- and high-temperature streaks are distributed in the spanwise direction. The wavelengths of naturally developing Görtler streaks are randomly distributed, with an average of approximately 7 mm, and change little as the unit Reynolds number increases. More importantly, three-dimensional waves are clearly visualized and quantitatively measured inside the Görtler streaks. This is the first time that the entire evolution of the Görtler instability has been visualized using the Rayleigh-scattering flow visualization in hypersonic flow. The results demonstrate that three-dimensional waves are amplified as a result of the Görtler instability, resulting in a localized high-shear layer around the interface of the three-dimensional waves, which contributes to the formation of hairpin vortices and mushroom-like structures. The three-dimensional waves grow and play an important role in Görtler instability-induced boundary layer transitions.
... Concerning λ 10 mm, the amplification factor is evaluated at the centerline and it peaks at N 11 in the reattachment region, and is thus above the critical N cr 10 for which boundary-layer transition generally occurs [36]. Regarding λ 5 mm, N is evaluated along the plate at z 5 mm and it plateaus just below N cr at approximately 50 mm downstream of the reattachment line. ...
This work discusses the design, measurement, and simulation of an oscillating shock-wave/boundary-layer interaction on a flat plate at Mach 5.8 and Re∞ = 7 × 10^6 m−1. The shock generator is free to pitch and oscillates with a frequency of 42 Hz, resulting in a shock that varies in intensity and impingement point, with a maximum flow-deflection angle of approximately 10 deg. Transition appears to take place downstream of the separated region for both static (with a fixed flow-deflection angle) and dynamic experiments; however, heat-flux values are typically between laminar and turbulent solutions, thus suggesting that a complete transition to a fully turbulent boundary layer is delayed because of the favorable pressure gradient induced by the impinging expansion wave originating from trailing edge of the shock generator. Peak pressure is typically overpredicted by laminar simulations for large deflection angles. Starting from the reattachment point, heat-flux measurements show that the boundary layer gradually deviates from the laminar solution towards a fully turbulent boundary layer. Vortices are observed in the reattachment region, and their distribution is solely a function of the boundary-layer properties at the separation point. Transient effects induced by the shock motion result in a maximum bubble length variation of 30%. For the static cases, the separated region amplified disturbances with a frequency of approximately 200 Hz. In the dynamic experiment, harmonics induced by the pseudo sinusoidal motion of the shock generator were measured everywhere on the plate.
... Boundary layer flows are ubiquitous in both nature and engineering applications and are subject to instabilities, which generally lead to the transition to turbulence. When the flow develops over a curved boundary, this transition may occur through the development of the so-called Görtler instability, [1][2][3][4][5] which results from a local unbalance between the centrifugal force and the normal pressure gradient. The boundary layer then exhibits unstable flow motion in the form of pairs of counter-rotating streamwise vortices that give rise to longitudinal streaks, leading to strongly distorted velocity profiles and normal-to-wall flow motion correlated with local low and high momentum fluxes. ...
We present results from a highly resolved large-eddy simulation of a freely developing Blasius profile over a concave boundary in a large spanwise domain. Due to the large initial Reynolds and Görtler numbers (Reθ,0 = 1175, Gθ,0 = 75), we observe the onset of two dominant wavelengths: the first dominating in the linear/transition region, λ1, and the second dominating in the turbulent region, λ2. Extending previous linear stability analysis (LSA) to higher Görtler numbers and non-dimensional wavenumbers, both dominant wavelengths of the Görtler instability correspond to predictions of LSA, the latter comparable to laminar theory by replacing the kinematic viscosity with the turbulent viscosity in the definition of the Görtler number. The predicted spatial modes compare well with the computed profiles for both λ1 and λ2. The skin friction coefficient Cf is found heterogeneous in the spanwise direction according to the emerging wavelengths λ1 and λ2 of the Görtler instability. We report a smooth increase in Cf from the theoretical predictions of a laminar boundary layer to those for a turbulent boundary layer over a flat plate. The values only slightly overshoot these predictions in the domain of existence of the second dominant wavelength λ2, very different from that reported at lower Reynolds numbers.
... The instability causes the formation of spanwise-periodic and streamwise-elongated vortices. The analysis of Görtler was extended by Smith (1955), Ragab & Nayfeh (1980) and Floryan & Saric (1982), who sought to include part of the non-parallel-flow effects, but continued to treat the instability as an eigenvalue problem. These studies showed that Görtler vortices with a fixed wavelength decay near the leading edge, but start to amplify from some distance downstream. ...
Görtler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability - Volume 900 - Dongdong Xu, Jianxin Liu, Xuesong Wu
... The presence of pressure gradient in the radial direction of the concavity leads to the initiation of flow instabilities in the fluid flow. Under the action of the centrifugal forces, these disturbances manifest into counter rotating vortices which also grow with the streamwise distance [139,140]. The stability characteristics and laminar to turbulent boundary layer transition or the critical Reynolds number are a function of the Göertler number and the curvature of the concavity [141,142]. ...
This doctoral thesis focuses on mechanisms of heat transfer enhancement in plate and fin heat sink geometries. First part of the thesis is dedicated to study an academic configuration using numerical simulations to achieve an improvement in conjugate heat transfer by modifying only the geometrical shape (through punching) of the conductive plane fins. An in-depth local analysis of the flow and thermal fields was carried out with the local synergy principle, velocity and thermal gradients, to understand the effect of geometric modifications. This thesis also presents the development of heat sinks with increased thermo-hydraulic performance for on-board electronic box cooling applications. The intensification of the heat transfer is obtained by the generation of secondary flows which cause an intensive mixing of fluid and reduces the thermal resistance to the wall by disrupting the development of the thermal boundary layer. Different heat sink geometries with two types of secondary flow generators : delta winglet pair and protrusions were numerically studied using RANS approach. The thermo-hydraulic performances of the geometries equipped with vortex generators were compared with that of a smooth reference heat sink. The prototypes were also manufactured and tested on an experimental bench specifically designed to perform global performance measurements in terms of thermal power and pressure drops. Experimental and numerical results were compared to qualify the simulations performed. Subsequently, an optimization study using Taguchi factorial analysis was used to optimize the geometrical parameters of the chosen dissipaters. Two objective functions were considered : maximization of either iso-pumping power performance criteria (PEC) or average wall temperature of the dissipaters compared to the reference case. The global thermo-hydraulic performance analysis of the studied geometries was completed by a qualitative analysis of local flow and thermal fields, in particular with the local field synergy principle.
... In view of this shortcoming, Hämmerlin [39] reconsidered the Görtler problem and reintroduced some of the terms associated with effects of curvature (that had previously been neglected in favour of simplification). The analysis of Smith [79], additionally included some of the terms associated with the boundary layer growth. In both of these studies it was discovered that the wavenumber associated with the least stable disturbance is nonzero. ...
In this thesis the instability of two viscous incompressible flows is discussed by using numerical and analytical methods. The first problem concerns the steady streaming flow, that is contained within a hollow stationary cylinder and induced by the transverse oscillation of a solid inner cylinder. The small gap limit is taken so that a series solution in odd powers of the angular variable is possible. From the studies by Hall & Papageorgiou [37] and Watson et al. [97], it is known that the leading order equation has solutions that are steady, quasi-periodic and chaotic (period doubling). Since all the higher order equations are driven by the solution at leading order; the series solution for the steady streaming flow is investigated with an interest to determine any chaotic structures. The second problem concerns the flow in a horizontal circular pipe, that is subject to torsional oscillations about a vertical axis that passes symmetrically through the pipe. The onset of a new axisymmetric roll-type instability, as observed experimentally by Bolton & Maurer [10] for the corresponding rectangular tank problem (of small width), is sought in the high-frequency (Phi >> 1) and small-amplitude limit (alpha <<1). A perturbation of the WKBJ type is imposed upon the basic state, so that the slow angular variation of the disturbance is accounted for in the linear stability equations. Accordingly, a dispersion relation for the dimensionless frequency parameter Phi is derived. In order to identify the most dangerous disturbance, it is necessary to minimise the eigenvalue B = alpha/Phi^(1/4). The theory of Soward & Jones [82] is used to show that an acceptable solution of the governing eigenvalue problem, cannot be obtained for real values of the latitudinal variable theta; instead, the correct minimum is found in the complex theta-plane.
... High Dean numbers in the current study (Table 1) suggest some turbulence generation may be present in the 135° curve upstream of the hot-surface igniter. Görtler [53] and Smith [54] have shown boundary layer disturbances are damped for Görtler number less than 10. All conditions examined in the current study are at Görtler numbers less than 1. ...
... According to Malik [113], the transition occurs when the disturbance waves approach a range of critical amplitudes regardless of the mechanism that leads to instability. Smith [114] demonstrated that, in the case of Görtler vortices, the critical amplification factor is on the order of ݁ ே with ܰ ~ 10. Schrijer [105] investigated the Görtler vortices on a double compression ramp at ܯ = 7.5. ...
This work investigates fundamental fluid-structure interaction (FSI) experiments
performed in a short duration hypersonic wind tunnel at Mach 6. The thesis aims to
discuss and quantify the relationship between structural deformations and viscous
aspects such as transition, separated flow and shock wave – boundary layer
interactions (SWBLI). Part of the findings will be used to describe the impact of
deployment and deformation on the performance of control surfaces.
The main experiment involves a shock impinging on cantilevered elastic plate. The
pressure increase, determined by the shock reflection on the plate, causes the
cantilevered plate to oscillate. The plate motion affects the salient feature of the SWBLI
in terms of length of the separated region, transition, peak heating and peak pressure.
The problem is broken down into its main features and driving phenomena. In fact,
preliminary experiments involve a cantilevered plate without impinging shock and a
shock impinging on a rigid plate, in order to separate the phenomena purely due to FSI
or SWBLI.
Measurements consist of time-resolved sparse pressure and temperature
measurements as well as surface measurements. Pressure-sensitive paint (PSP) and
IR thermography are used to investigate and quantify the impact of three-dimensional
effects on pressure and thermal distributions. In this work, three-dimensional effects
are the result of limited plate width and/or Görtler boundary layer instability. The
experimental data is compared against transient fully laminar and fully turbulent
numerical solutions.
Among the major findings, the thesis demonstrates that the boundary layer
displacement thickness cannot always be considered a point function of the local plate
inclination and speed. The numerical solutions significantly underestimate peak
heating and peak pressure when boundary layer transition takes place within the separated region. Gortler boundary layer instability triggers the transition resulting in
peak hating fluctuations close to 10%. Concerning control surfaces, transition in the
separated region can lead to levels of heating 100% higher than the laminar values.
Finally, a 1 % control surface deformation at the trailing edge due to fluid-structure
interaction results in a 2 - 3% loss in efficiency.
... However, the definition ofx 0 still remains arbitrary and one has to carefully verify the conditions associated with the calculation of the N-factor when using it for transition prediction. The rigorous mathematical formulation of the Görtler flow based on the receptivity due to free-stream disturbances does not help improve the reliability of the N-factor approach for the transition prediction, in line with previous works revealing problems with this methodology (Smith 1955;Malik et al. 1999;Boiko et al. 2017). Our work on the neutral curve can be further generalized by taking into account the influence of convex curvature and by introducing the nonlinear effects studied by Xu et al. (2017), therefore relaxing the assumption of small-amplitude perturbations (Ustinov 2013). ...
The neutral curves of the boundary layer Görtler-vortex flow generated by free-stream disturbances, i.e., curves that distinguish the perturbation flow conditions of growth and decay, are computed through a receptivity study for different Görtler numbers, wavelengths, and low frequencies of the free-stream disturbance. The perturbations are defined as Klebanoff modes or strong and weak Görtler vortices, depending on their growth rate. The critical Görtler number below which the inviscid instability due to the curvature never occurs is obtained and the conditions for which only Klebanoff modes exist are thus revealed. A streamwise-dependent receptivity coefficient is defined and we discuss the impact of the receptivity on the $N$ -factor approach for transition prediction.
... Flow in a channel over wavy walls can become turbulent due to instability of the shear layer caused by the main flow interacting with a Kelvin-Helmhotz type vortex or a centrifugal instability over concave surface (H. Görtler, 1954;Smith, 1955;Gschwind et al., 1995) which takes the form of counter-rotating streamwise vortices called Görtler vortices with a form of a symmetrical mushroom-like shape over its cross section (Bakchinov et al., 1995;Mitsudharmadi et al., 2004Mitsudharmadi et al., , 2006Tandiono et al., 2009;Winoto et al., 2011;Budiman et al., 2015). There are two regions in the flowfield of these vortices: the so-called upwash region where low velocity fluid is lifted from the surface and the so-called downwash region where higher velocity fluid is pushed back towards the surface resulting in thinner boundary layer and higher shear stress (Winoto et al., 2011). ...
Particle Image Velocimetry (PIV) has been used to characterize the evolution of counter-rotating streamwise vortices in a rectangular channel with one sided wavy surface. The vortices were created by a uniform set of saw-tooth carved over the leading edge of a flat plate at the entrance of a flat rectangular channel with one-sided wavy wall. PIV measurements were taken over the spanwise and streamwise planes at different locations and at Reynolds number of 2500. Two other Reynolds numbers of 2885 and 3333 have also been considered for quantification purpose. Pairs of counter-rotating streamwise vortices have been shown experimentally to be centered along the spanwise direction at the saw-tooth valley where the vorticity . It has also been found that the vorticity of the pairs of counter-rotating vortices decreases along the streamwise direction, and increases with the Reynolds number. Moreover, different quantifications of such counter-rotating vortices have been discussed such as their size, boundary layer, velocity profile and vorticity. The current study shows that the mixing due to the wall shear stress of counter-rotating streamwise vortices as well as their averaged viscous dissipation rate of kinetic energy decrease over flat and adverse pressure gradient surfaces while increasing over favourable pressure gradient surfaces. Finally, it was also demonstrated that the main direction of stretching is orientated at around 45° with the main flow direction.
... For instance, in [24,32], the Görtler number of neutral point increases as the span-wise wave-number decreases below 0.1 (when it is normalized by the boundary-layer displacement thickness δ * ) and approaches 10 at wave-number about 0.01. Meanwhile, in calculations of [31,55], the same Görtler number remains small and nearly constant (about 0.7-0.8), while in [20] it continues to decrease at least down to 0.1. ...
A combined theoretical and numerical analysis of an experiment devoted to the excitation of Görtler 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 Görtler vortices depends substantially on the span-wise scale.
Mechanisms causing instability and change are commonplace in nature. Similar phenomena exacerbate the tasks of devising, building and maintaining systems for efficiently manufacturing paper to a tight specification and high standard of uniformity. In the case of a machine calender stack, CD control can impart stability to an otherwise unstable system. Instability of the headbox or approach flow system can however seriously affect the MD control of basis weight. Some types of instability lie outside the reach of control, and must if possible be eliminated through improved equipment design. For this reason a good understanding of instability is necessary. This paper considers some specific examples of unstable behaviour, including eddy and vortex formation, waves and other amplification mechanisms on Fourdrinier wires, corrugation growth, uneven wear, self-excited vibration, and thermal deformation affecting calender stacks.
Modal instabilities in a compressible flow through a channel at high Reynolds numbers are studied for three-dimensional (3D) perturbations. In addition to the Tollmien–Schlichting (TS) mode, there exist compressible modes in a channel flow that do not have a counterpart in the incompressible limit. The stability characteristics of these compressible modes, obtained through numerical calculations, are compared with boundary layer and Couette flows that have been previously studied. The dominant compressible instabilities in a channel flow are shown to be viscous in nature, in contrast to compressible boundary layer modes. For general compressible bounded-domain flows, a necessary condition for the existence of neutral modes in the inviscid limit is obtained, and this criterion is used to determine critical Mach numbers below which the compressible modes remain stable. This criterion also delineates a range of wave-angles which could go unstable at a specified Mach number. Asymptotic analysis is carried out for the lower and upper branches of the stability curve in the limit of high Reynolds number for both the T-S and the compressible modes. A common set of relations are identified for the scaling exponents, and the leading order eigenvalues for the unstable modes are obtained through an adjoint-based procedure. The asymptotic analysis shows that the stability boundaries for 3D perturbations at high Reynolds numbers can be calculated from the strain rate and the temperature of the base flow at the wall.
The linear stability of a compressible flow in a pipe is examined using a modal analysis. A steady fully developed flow of a calorifically perfect gas, driven by a constant body acceleration, in a pipe of circular cross section is perturbed by small-amplitude normal modes and the temporal stability of the system is studied. In contrast to the incompressible pipe flow that is linearly stable for all modal perturbations, the compressible flow is unstable at finite Mach numbers due to modes that do not have a counterpart in the incompressible limit. We obtain these higher modes for a pipe flow through numerical solution of the stability equations. The higher modes are distinguished into an “odd” and an “even” family based on the variation of their wave-speeds with wave-number. The classical theorems of stability are extended to cylindrical coordinates and are used to obtain the critical Mach numbers below which the higher modes are always stable. The critical Reynolds number is calculated as a function of Mach number for the even family of modes, which are the least stable at finite Mach numbers. The numerical solution of the stability equations in the high Reynolds number limit demonstrates that viscosity is essential for destabilizing the even family of modes. An asymptotic analysis is carried out at high Reynolds numbers to obtain the scalings, and solutions for the eigenvalues in the high Reynolds number limit for the lower and upper branches of the stability curve.
Graphical abstract
p>The objectives of this thesis are to (i) investigate and further validate the Viscous Cell Boundary Element Method for more complex fluid mechanics problems and moving bodies and (ii) to use fundamental ideas of image processing and spectral methods to investigate the ability to inverse model the physical phenomena of data generated by the fluid code or any experimental data and to reduce successfully the system to a much smaller substitute that encapsulates the main physics of the original system. Lastly we aim to design a controller to use in open-loop flow control coupled with the fluid solver. The controller is designed by reduction of the equations describing the system.</p
View Video Presentation: https://doi.org/10.2514/6.2021-2588.vid Slotted, natural-laminar-flow airfoils can produce high lift coefficients with low sectional drag. This is done by using favorable pressure gradients to maintain laminar flow on the fore element and using the aft element for the pressure recovery. However, the configuration creates a concave region at the rear of the fore element that is susceptible to Goertler instabilities. This work seeks to investigate and characterize the growth of Goertler instabilities on slotted, natural-laminar-flow airfoils. Three airfoils are considered: the S414.LSLTT, S103, and S207. Transition is predicted to be unlikely on the S414.LSLTT, but likely on the S103 and S207. Reynolds number is also seen to have a significant influence on estimates of the S103 amplification ratios. The effects of Goertler transition on the S207 are also briefly explored and discussed.
Cambridge Core - Thermal-Fluids Engineering - Theory and Computation in Hydrodynamic Stability - by W. O. Criminale
Measurements are presented of the streamwise velocity variation within a laminar boundary layer on a concave surface of 4 m radius of curvature for which the free-stream velocity gradient factor (ν/U0 2 )dU0 /dx was approximately 1 × 10−6 . The stream velocity variation was consistent with the presence of counterrotating vortices resulting from the Görtler instability. The vortices exhibited exponential growth over the streamwise extent of the measurements to a disturbance amplitude of approximately 13 percent of the local free-stream velocity. The vortex growth rates were found to be less than those for a zero velocity gradient factor, indicating that a favorable pressure gradient stabilizes the flow with respect to the Görtler instability. Boundary layer profiles at local upwash and downwash positions are compared with the linear theory for which the mean flow was modeled using the Pohlhausen approximation to the solution of the boundary layer equations. The agreement between the measured and predicted profiles indicates that the linear stability theory can provide a fair approximation to the small amplitude growth of the Görtler instability.
The approximate theory presented, including the effect of the boundary layer, permits a much more accurate determination of pressure and drag of wavy surfaces in subsonic and supersonic flow (for M ± 2.0) than is possible with inviscid flow theory. The B.L. (boundary layer) theory shows that the effective wave height and associated pressure decrease and eventually approach zero as the ratio of boundary-layer thickness to wave length increases. In supersonic flow, the decrease in effective wave height due to the boundary layer is accompanied by an aft phase shift in the B.L. wave, further decreasing the pressure drag due to the surface waves. The analysis correlates the theory with the results of surface waviness experiments conducted at the Langley Research Center and includes an estimate of drag contribution due to Gortler vortices generated by multiple waves. Application of the theory to a subsonic wing shows that surface waviness of appreciably smaller magnitude than that commonly found on military aircraft causes a significant detrimental effect on the drag-rise characteristics. Additional experimental research is needed to better evaluate the phase shift in supersonic flow, and the effect of Gortler vortices in both subsonic and supersonic flow. © 1974 American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
We study the instability of a concave boundary layer (Görtler instability) on a concave-convex model, in a water channel. Two different geometries were used either with (no streamwise pressure gradient) or without a counter-profile (favorable streamwise pressure gradient). In the first case Görtler vortices were present and randomly spaced. In the second case no vortex was detectable. The two flows exhibite different responses to localized forcings. We argue that these results may be interpreted by the non-linear convective instability concept, with a subcritical bifurcation when in presence of a favorable pressure gradient (accelerating flow).
Die Stabilitätstheorie laminarer Grenzschichten, deren Entwicklung mit den Namen Tollmien, Schlichtung, Lin u. a. verknüpft ist, untersucht das Verhalten einer instabilen laminaren Grenzschicht gegenüber kleinen und zweidimensionalen Störungswellen.
Im folgenden soll über Stabilitätsuntersuchungen bei inkompressiblen Flüssigkeiten berichtet werden, denen die Navier-Stokesschen Gleichungen zugrunde liegen. Es handelt sich um verschiedene Arbeiten aus dem Institut für Angewandte Mathematik und Mechanik von Herrn Prof. Görtler in Freiburg. Es sei zunächst die Problemstellung der einzelnen Beiträge kurz angegeben:
1.
Beitrag zur nichtlinearen Stabilitätstheorie endlicher Störungen im Anschluß an ein allgemeines Stabilitätskriterium von Serrin.
2.
Numerische Berechnung dreidimensionaler Störungen der Blasiusschen Plattengrenzschicht.
3.
Stabilitätsrechnung zur Entstehung von Längswirbeln in der voll turbulenten Grenzschichtströmung an gekrümmter Wand.
4.
Theoretische und numerische Behandlung eines Stabilitätsproblems unter Berücksichtigung temperaturabhängiger Stoffbeiwerte, insbesondere temperaturabhängiger Oberflächenspannung.
First-order nonlinear interactions of streamwise vortices with different amplification rates, spanwise wave numbers, and initial amplitudes in boundary layer flows on concave walls are studied by using the method of multiple scales. Numerical results show that the effect of interaction between two vortices in harmonic resonance is strong when the amplitude of the secondary wave (higher harmonic) has reached certain % of the mean flow. In combination resonance between three waves, it is found that the two primary waves with moderate initial amplitudes have a strong influence on their difference harmonic wave. The resonant interaction between waves of higher wave numbers causes strong amplification at earlier stages (at lower values of G) than the resonant interaction between waves of lower wave numbers.
By towing a concave wall of 500mm radius of curvature at a constant speed in a tank filled with an aqueous solution of polythylene oxide of about 4×106 molecular weight, the neutral stability of Görtler vortices was determined for the 2 and 5wppm solutions with the hydrogen bubble technique and the streamwise velocity of basic flow in boundary layer along the wall was also measured.
In this chapter, we consider the linear stability of some flows over a curvilinear wall, which have close relations to the plane shear layers considered in previous chapters: the Couette flow between concentric rotating cylinders, the flow in a curved channel, and the self-similar boundary layer over a curved wall.
An essential limitation for the accurate prediction of heat transferring equipment is sparse information concerning the effect of the turbulence on convective heat, mass and momentum transfer. This is largely due to the fact that the local transport mechanisms are not yet fully understood, e.g. in laminar boundary layers along concave walls. It is well known that these boundary layers have a strong inviscid instability mechanism due to the presence of centrifugal forces. This centrifugal instability may result in a secondary flow in the form of counter-rotating vortices (Görtler vortices), whose axes are parallel to the main flow direction (Fig. 1).
There are nearly 1500 references in the following list. The size of this list might have two consequences. First, a student undertaking research in this subject could feel daunted by the task of making an original contribution. Second, a grant officer might conclude that all the important work has been done and further funding would have diminishing returns. A survey of the literature, and especially an attempt to classify it, shows that these consequences would be wrong. We have come to the laboratory and we have seen many enticing patterns of flow in this simple geometry, but we are far from conquering the subject. Much work remains before we gain a firm understanding of the connection between transitional flows and turbulence. Also, we have by no means exploited all the practical possibilities of what we have learned.
A good understanding of turbulent compressible flows is essential to the design and operation of high-speed vehicles. Such flows occur, for example, in the external flow over the surfaces of supersonic aircraft, and in the internal flow through the engines. Our ability to predict the aerodynamic lift, drag, propulsion and maneuverability of high-speed vehicles is crucially dependent on our knowledge of turbulent shear layers, and our understanding of their behavior in the presence of shock waves and regions of changing pressure. Turbulent Shear Layers in Supersonic Flow provides a comprehensive introduction to the field, and helps provide a basis for future work in this area. Wherever possible we use the available experimental work, and the results from numerical simulations to illustrate and develop a physical understanding of turbulent compressible flows. © 2006 Springer Science+Business Media, Inc. All rights reserved.
The paper is concerned with stability of the flow in a boundary layer with respect to three-dimensional longitudinal vortices (Goertler vortices) formed under the action of the centrifugal force. The vortices arc shown to be generated on a convex surface that moves along a curved trajectory in contrast to the case when a fixed curved surface is flowed around and the vortices are formed on the concave side. The linear stability of the flow is studied and the stability diagram is constructed. The critical Goertler number is found to far exceed the critical number of the flow over a fixed concave surface. The range of the wave numbers which characterize instability of the flow is narrower in comparison to the case of the flow over a surface. This suggests that the flow near a curved surface moving along a curved trajectory is more stable than the flow near a fixed curved surface. The systematic evaluation of the eigenfunctions is performed for various values of both the Goertler number Gr and the dimensionless wave number αθ, where d is a momentum thickness. These functions are plotted, the positions and the values of the functions' characteristic points (the maximum and minimum values, the points of intersection with the vertical axes) are found. The data presented in the paper allow all the components of both the disturbed velocity and the pressure to be determined approximately provided the information on only one component, e. g., the maximum value of the longitudinal component of the disturbed velocity, that can be measured experimentally, is known. The position of the regions of instability to three-dimensional longitudinal vortices over the body of a dolphin moving rectilinearly, whose stern moves up and down, is discussed.
A linear stability analysis has revealed the non-parallelism of flows as one of the factors causing the generation of longitudinal vortices in boundary layers. This has been experimentally verified through water tank tests for a concave plate and wind tunnel tests for a flat plate. Longitudinal vortices generated in the boundary layer on concave plates, i. e. , Goertler vortices, as they develop, generate horseshoe type vortices as a secondary instability. Nonlinear interaction between these vortices promotes collapse of the vortices or the transition to turbulence. Even after the transition, organized three-dimensional structures are recognized in the boundary layer.
It is a property of certain concentrations of glycerine-water solutions, when in a state of steady flow, that the planes of equal shear in the liquid become visible in ordinary light, if viewed along a path tangent to the shear plane. This phenomenon has been used successfully to study a fluid-flow problem at the University of Michigan. The nature and scope of application of this optical property is currently being studied further. This report gives details of what may prove to be a useful tool in certain phases of experimental fluid-flow problems.
The constant temperature hot-film technique has been used to investigate the transition region on a modified flat plate and a large two-dimensional wing. Time histories and r.m.s. values of the output signals together with the power spectral density (PSD) clearly indicate the build up of energy associated with the development of Tollmien-Schlichting waves prior to the onset of transition. A large increase in the low-frequency energy content of the spectrum occurs at the onset of transition. Measurements of skin friction through the transition region have been obtained based on the mean heat loss of the hot film.
The stability for three-dimensional disturbances of viscous flow over concave cylindrical surfaces is investigated by the method of asymptotic solutions. The results confirm Gortler's conclusions that these disturbances can only take place on concave surfaces, and that in the critical condition that the parameter Rdelta(delta /r)1/2 remains constant, where Rdelta is Reynolds number and delta /r is the ratio of the thickness of the boundary layer to the radius of curvature. Additionally, it is found that there are many steady states, and the values of Rdelta(delta /r)1/2 are evaluated for the first two states. For a fixed Rdelta(delta /r)1/2 there is a finite number of steady states.
Aus: Nachrichten aus d. Mathematik. Bd 2, Nr 1 = Nachrichten v. d. Ges. d. Wissenschaften zu Göttingen, Math.-phys. Kl. N. F., Fachgr. 1. Göttingen, Math.-Naturwiss. Diss., 1940 (Nicht f. d. Austausch.).
Tests of a Griffith Aerofoil in the 13 ft. X 9 ft. wind tunnel, British Reports and Memoranda No. 2148 A personal communication from the N.P.L. supplied much additional and more detailed data than available in the R & M. Furthermore, the factor 1
- E J Richards
- W S Walker
- J R Greening
E. J. Richards, W. S. Walker, J. R. Greening, Tests of a Griffith Aerofoil in the 13 ft. X 9 ft. wind
tunnel, British Reports and Memoranda No. 2148, March 1944
A personal communication from the N.P.L. supplied much additional and more detailed data than
available in the R & M. Furthermore, the factor 1.36, page 9 of R <fe M 2148 was changed to the
correct value 0.54
Improved solutions of the Falkner and Skan boundary layer equation, Sherman Fair-child Fund Paper No. FF-10
- A M O Smith
A. M. O. Smith, Improved solutions of the Falkner and Skan boundary layer equation, Sherman Fairchild Fund Paper No. FF-10, Inst. Aeronaut. Sci. (1954)
- H W Liepmann
H. W. Liepmann, Investigation of boundary layer transition on concave walls, NACA Wartime Report
ACR No. 4J28, Feb. 1945
Tests of a Griffith Aerofoil in the 13 ft. X 9 ft. wind tunnel
- E J Richards
- W S Walker
- J R Greening
E. J. Richards, W. S. Walker, J. R. Greening, Tests of a Griffith Aerofoil in the 13 ft. X 9 ft. wind
tunnel, British Reports and Memoranda No. 2148, March 1944
A personal communication from the N.P.L. supplied much additional and more detailed data than
available in the R & M. Furthermore, the factor 1.36, page 9 of R <fe M 2148 was changed to the
correct value 0.54
Tests of a Griffith Aerofoil in the 13𝑓𝑡.×9𝑓𝑡.. wind tunnel, British Reports and Memoranda No. 2148
- E J Richards
- W S Walker
- J R Greening