Transition of boundary layer flows in the presence of Goertler vortices

Further experimental results on transition of boundary layer flows in the presence of streamwise counter-rotating Goertler vortices were obtained on concave test surface of 3.0 m radius of curvature. The test surface was mounted in a perspex (plexiglass) curved rectangular section duct connected to a low speed, blow down type, wind tunnel for a free-stream velocity range of 2.0 to 13.1 m/s. Velocity measurements were made using a single sensor hot-wire anemometer and boundary layer flow transitions were detected by a frequency spectrum method. The experimental results show a simple linear relationship between transition start position and free-stream velocity, and lie well between the limit lines of Goertler number transition criteria. It is found that the onset of transition at the flow upwash region occurs when the Goertler number based on the boundary layer momentum thickness reaches a value of about 7.5.     

Numerical prediction of the formation of Goertler vortices on a concave surface with suction and blowing

This paper presents a numerical prediction of the formation of Goertler vortices on a concave surface with suction and blowing. Suction stabilizes the boundary layer flow on the surface, whereas blowing destabilizes the flow. The criterion on the position marking the onset of Goertler vortices is defined in the present paper. For facilitating the numerical study, the computation is carried out in the transformed x–η plane. The results show that the onset position characterized by the Goertler number depends on the local suction/blowing parameter, the Prandtl number and the wavenumber. The value of the critical Goertler number increases with the increase in suction, while the value of the Goertler number decreases with the increase in blowing. Both the experimental and the numerical data can be correlated by G_θ*=10.2(a′θ)*^3/2 without suction and blowing and by a simple relation G*_x=(G*_x)_γ=0 e^−γ with suction and blowing. The obtained critical Goertler number and wavenumber are in good agreement with the previous experimental data. Copyright © 1999 John Wiley & Sons, Ltd.     

Effects of free stream turbulence on the distribution of heat transfer around turbine blade sections

Local convective heat transfer coefficients to a number of modern gas turbine blade sections have been measured under a wide range of mainstream conditions, from notionally steady flows to highly perturbed turbulent flows. The paper discusses the results and, through a detailed analysis of the pertinent boundary layer flow parameters and their relation to the observed experimental results, tests criteria for the occurrence of transition from laminar to turbulent boundary layers, a factor which all the data from this work confirm as critical in predicting the quantitative effects of mainstream turbulence on heat transfer rates. Artificially induced mainstream turbulence, which is endemic in the flows in a real turbine, enhances significantly the heat transfer rates, especially to the leading edge regions and on the pressure surface, particularly when the acceleration is tending to suppress transition. The results presented here confirm existing criteria for laminarisation and the applicability of some of those available for predicting laminar-turbulent transition. The observations also demonstrate how surface geometry can influence the stability of the flows, and the uncertainties which remain in assessing the effect of Goertler vortices and their role in the convective heat transfer process.     

Effect of rotation on Goertler vortices in the boundary layer flow on a curved surface

This paper presents a numerical prediction of the formation of Goertler vortices on a curved surface with effect of rotation. The criterion of flow visualization marking the onset position of Goertler vortices is employed in the present paper. For facilitating the numerical study, the computation is carried out in the transformed x and ηplane. The results show that the onset position characterized by the Goertler number, depends on the rotation number Ro, the Prandtl number and the wave number. The value of critical Goertler number increases with the increase in negative rotation, while the value of Goertler number decreases with the increase in positive rotation on a concave surface. On the contrary, the value of critical Goertler number decreases with the increase in negative rotation on a convex surface. The obtained critical Goertler number and wave number are compared with the previous theoretical and experimental data. Copyright © 2002 John Wiley & Sons, Ltd.     

Görtler vortices in Falkner–Skan flows with suction and blowing

In this paper, we use nonlinear calculations to study curved boundary‐layer flows with pressure gradients and self‐similar suction or blowing. For an accelerated outer flow, stabilization occurs in the linear region while the saturation amplitude of vortices is larger than for flows with a decelerating outer flow. The combined effects of boundary‐layer suction and a favourable pressure gradient can give a significant stabilization of the flow. Streamwise vortices can be amplified on both concave and convex walls for decelerated Falkner–Skan flow with an overshoot in the velocity profile. The disturbance amplitude is generally lower far downstream compared with profiles without overshoot. Copyright © 2007 John Wiley & Sons, Ltd.     

Factors influencing heat transfer to the pressure surfaces of gas turbine blades

It is suggested that heat transfer through the laminar boundary layer flowing over the concave pressure surface of a turbine blade is strongly influenced by the presence of Taylor-Goertler vortices, as well as by mainstream turbulence. Transition occurs when these factors in concert outweigh the tendency of the boundary layer to remain laminar in the favourable pressure gradients characteristic of flow over pressure surfaces.     

Effect of rotation on the formation of longitudinal vortices in mixed convection flow over a flat plate

This paper presents a numerical study of the effect of rotation on the formation of longitudinal vortices in mixed convection flow over a flat plate. The criterion on the position of marking the onset of longitudinal vortices is defined in this paper. The onset position characterized by the Goertler number G _δ depends on the Grashof number, the rotation number Ro, the Prandtl number Pr and the wave number. The results show that negative rotation stabilizes the boundary layer flow on the surface. On the contrary, positive rotation destabilizes the flow. The numerical data are compared with the experimental results.     

BY-PASS MECHANISM OF TRANSITION TO TURBULENCE

The by-pass mechanism of transition for a wall-bounded shear layer is explained for the case when an infinite row of convecting vortices migrate over a boundary layer at a specific speed range. Such a mechanism is important for noisy flows over bluff bodies, flows inside turbomachinery and flows over helicopter rotor blades. By solving the Navier–Stokes equation, it is shown that this by-pass transition is a consequence of vortices migrating at convection speeds that are significantly lower than the free-stream speed. This situation is commonly found in flows that are affected by the presence of periodic wakes. Whenever the speed of migrating vortices is in a certain critical range, there is a local instability of the underlying shear layer with a very high-growth rate as compared to the growth of pure Tollmien–Schlichting waves created by wall excitation. The above interpretation is supported by solving the linearized and full Navier–Stokes equation for disturbance quantities under the parallel flow approximation in two dimensions. Various ramifications of such a by-pass route of transition are discussed in this paper.     

Three-dimensional instability analysis of boundary layers perturbed by streamwise vortices

A parametric study is presented for the incompressible, zero-pressure-gradient flat-plate boundary layer perturbed by streamwise vortices. The vortices are placed near the leading edge and model the vortices induced by miniature vortex generators (MVGs), which consist in a spanwise-periodic array of small winglet pairs. The introduction of MVGs has been experimentally proved to be a successful passive flow control strategy for delaying laminar-turbulent transition caused by Tollmien–Schlichting (TS) waves. The counter-rotating vortex pairs induce non-modal, transient growth that leads to a streaky boundary layer flow. The initial intensity of the vortices and their wall-normal distances to the plate wall are varied with the aim of finding the most effective location for streak generation and the effect on the instability characteristics of the perturbed flow. The study includes the solution of the three-dimensional, stationary, streaky boundary layer flows by using the boundary region equations, and the three-dimensional instability analysis of the resulting basic flows by using the plane-marching parabolized stability equations. Depending on the initial circulation and positioning of the vortices, planar TS waves are stabilized by the presence of the streaks, resulting in a reduction in the region of instability and shrink of the neutral stability curve. For a fixed maximum streak amplitude below the threshold for secondary instability (SI), the most effective wall-normal distance for the formation of the streaks is found to also offer the most stabilization of TS waves. By setting a maximum streak amplitude above the threshold for SI, sinuous shear layer modes become unstable, as well as another instability mode that is amplified in a narrow region near the vortex inlet position.     

Direct Numerical Simulation of Hypersonic Boundary-Layer Flow on a Flared Cone

The forced transition of the boundary layer on an axisymmetric flared cone in Mach 6 flow is simulated by the method of spatial direct numerical simulation (DNS). The full effects of the flared afterbody are incorporated into the governing equations and boundary conditions; these effects include nonzero streamwise surface curvature, adverse streamwise pressure gradient, and decreasing boundary-layer edge Mach number. Transition is precipitated by periodic forcing at the computational inflow boundary with perturbations derived from parabolized stability equation (PSE) methodology and based, in part, on frequency spectra available from physical experiments. Significant qualitative differences are shown to exist between the present results and those obtained previously for a cone without afterbody flare. In both cases, the primary instability is of second-mode type; however, frequencies are much higher for the flared cone because of the decrease in boundary-layer thickness in the flared region. Moreover, Goertler modes, which are linearly stable for the straight cone, are unstable in regions of concave body flare. Reynolds stresses, which peak near the critical layer for the straight cone, exhibit peaks close to the wall for the flared cone. The cumulative effect appears to be that transition onset is shifted upstream for the flared cone. However, the length of the transition zone may possibly be greater because of the seemingly more gradual nature of the transition process on the flared cone. Received 20 March 1997 and accepted 21 May 1997     

Investigation of the spectrum of short-wave Görtler vortices in a gas

The linear stage of short-wave Görtler vortices in the boundary layer near a concave surface is studied for the regime of weak hypersonic viscid-inviscid interaction at high Reynolds and Görtler numbers. It is assumed that the gas is perfect and the viscosity is a linear function of the enthalpy. It is found that neutral vortices are located near the surface if it has zero temperature. When the surface is heated, the vortices move away from it, whereas all newly incipient vortices are located near the surface. It is shown that the growth rate of the vortices has a maximum and the heating of the surface has a stabilizing effect on the vortices.     

An experimental investigation of the combined effects of surface curvature and streamwise pressure gradients both in laminar and turbulent flows

Flow and heat transfer characteristics over flat, concave and convex surfaces have been investigated in a low speed wind tunnel in the presence of adverse and favourable pressure gradients (k), for a range of –3.6 × 10^–6 ≤ k ≤ +3.6 × 10^–6. The laminar near zero pressure gradient flow, with an initial momentum thickness Reynolds number of 200, showed that concave wall boundary layer was thinner and heat transfer coefficients were almost 2 fold of flat plate values. Whereas for the same flow condition, thicker boundary layer and 35% less heat transfer coefficients of the convex wall were recorded with an earlier transition. Accelerating laminar flows caused also thinner boundary layers and an augmentation in heat transfer values by 28%, 35% and 16% for the flat, concave and convex walls at k = 3.6 × 10^–6. On the other hand decelerating laminar flows increased the boundary layer thickness and reduced Stanton numbers by 31%, 26% and 22% on the flat surface, concave and convex walls respectively. Turbulent flow measurements at k = 0, with an initial momentum thickness Reynolds number of 1100, resulted in 30% higher and 25% lower Stanton numbers on concave and convex walls, comparing to flat plate values. Moreover the accelerating turbulent flow of k = 0.6 × 10^–6 brought about 29%, 30% and 24% higher Stanton numbers for the flat, concave and convex walls and the decelerating turbulent flow of k = –0.6 × 10^–6 caused St to decrease up to 27%, 25% and 29% for the same surfaces respectively comparing to zero pressure gradient values. An empirical equation was also developed and successfully applied, for the estimation of Stanton number under the influence of pressure gradients, with an accuracy of better than 4%.     

The effect of G?rtler instability on hypersonic boundary layer transition

The evolution of G?rtler vortices and its interaction with other instabilities are investigated in this paper.Both the Mack mode and the G?rtler mode exist in hypersonic boundary-layer flows over concave surfaces, and their interactions are crucially important in boundary layer transition. We carry out a direct numerical simulation to explore the interaction between the G?rtler and the oblique Mack mode.The results indicate that the interaction between the forced G?rtler mode and the oblique Mack mode promotes the onset of the transition. The forced oblique Mack mode is susceptible to nonlinear interaction.Because of the development of the G?rtler mode, the forced Mack mode and other harmonic modes are excited.     

Investigation of intermittency measurement methods for transitional boundary layer flows

Three turbulent intermittency methods, namely the , TERA (turbulent energy recognition algorithm), and M-TERA (modified turbulent energy recognition algorithm) methods, for identifying the intermittent flow characteristics associated with boundary layer transition from laminar to turbulent were considered and compared. The data used were obtained from hot-wire measurements in transitional boundary layer flows on a concave surface with a 2-m radius of curvature and on a flat plate. Comparisons show that the and TERA methods are more sensitive to the choice of threshold constants than the M-TERA method. In terms of the intermittency distribution across the boundary layer, the values obtained by the and TERA methods are unrealistically high in the near-wall region, while those obtained by the M-TERA method are more realistic. In the outer boundary layer region and outside the boundary layer, the and M-TERA methods give reasonable intermittency values, whereas the TERA method produces unrealistically high values in the region outside the boundary layer. In addition, the M-TERA method provides a sharper definition of theend of transition.     

Wake-boundary layer interaction subject to convex and concave curvatures and adverse pressure gradient

Measurements of mean velocity and turbulent quantities have been carried out when the wake of a symmetrical airfoil interacts with the boundary layer on the (i) walls of a straight duct/diffuser and (ii) convex and concave walls of a curved duct/diffuser. The effects of adverse pressure gradient and of curvatures on the interaction are studied separately and in combination. Six cases are considered, viz. with (i) neither pressure gradient nor curvature, (ii) adverse pressure gradient and no curvature, (iii) and (iv) convex curvature with zero and adverse pressure gradients, respectively, (v) and (vi) concave curvature with zero and adverse pressure gradients, respectively. For the flows with curvature, the curvature parameter δ/R is 0.023, and for the flows with adverse pressure gradient, the Clauser pressure gradient parameter β is 0.62. The individual influences of adverse pressure gradient and convex and concave curvatures on the boundary layer are similar to those observed by earlier investigations. It is further observed that the combined effect of concave/convex curvature and the adverse pressure gradient causes higher turbulence intensities than the sum of the individual effects. The effect of curvature is to make the wake asymmetric, and in combination with adverse pressure gradient the asymmetry increases. It is observed that the adverse pressure gradient causes faster wake–boundary-layer interaction. Comparing measurements in a straight duct, a curved duct, a curved diffuser and a straight diffuser, it is seen that the convex curvature reduces the boundary layer thickness. The asymmetry in wake development compensates for this effect and the wake–boundary-layer interaction on a convex surface is almost the same as that on a straight surface. In the case of interaction with the boundary layer on a concave surface, the curvature increases the boundary layer thickness and causes enhanced turbulence intensities. However, the asymmetry in wake is such that the extent of wake is lower towards the boundary layer side. As a result, the wake–boundary-layer interaction on concave surface is almost the same as on a straight surface. The interaction is faster in the presence of adverse pressure gradient. Received: 16 June 2000 / Accepted: 17 May 2001     

Effects of riblets upon flow stability

The onset of transition in the boundary layer over a grooved surface is studied by introducing the presence of grooves into the standarde ⁹ model with the aid of a previously obtained equivalent boundary condition. Under conditions of self-similarity and smallness of the grooves, Tollmien-Schlichting waves are found to be excited at a slightly smaller Reynolds number and Görtler vortices at a slightly larger one than on a smooth surface.     

Magnetic effect on the formation of longitudinal vortices in a rotating laminar boundary layer

This paper presents a numerical study of magnetic effect on the formation of longitudinal vortices in a rotating laminar boundary layer. The criterion for the position marking the onset of longitudinal vortices is defined in this paper. The onset position characterized by the rotational Goertler number G_δ,rot, depends on the local rotation number, Reynolds number, the magnetic field parameter, the Prandtl number and the wave number. The results show that positive rotation destabilizes the flow. The flow is found to become more unstable to the vortex mode of instability as the value of magnetic field parameter M increases. The numerical data shows good agreement with the experimental results. Copyright © 2005 John Wiley & Sons, Ltd.     

Effect of compressibility on the development of taylor-görtler vortices at high reynolds numbers

The development of disturbances in viscous compressible flows caused by centrifugal forces is investigated. On the basis of an asymptotic analysis of the Navier-Stokes equations at high Reynolds and Görtler numbers, mathematical models describing the development of three-dimensional unstable vortex structures are constructed. Various linear boundary-value problems are analytically solved. One type of boundary layer instability is that generated by a centrifugal force field. This kind of instability can manifest itself in the flow past concave surfaces or, in general, in flows with streamlines of positive curvature [1, 2]. Instability-driven Görtler vortices have been the subject of much research which was reviewed, for example, in [2–4].     

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

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

Interaction between longitudinal vortices and flat plate boundary layer

The interaction between longitudinal vortices and flat plate boundary layer has been studied numerically for both laminar and turbulent flow situations. The vortices are assumed to be placed in an otherwise two-dimensional boundary layer flow. The flow is assumed to be incompressible and steady. Considering the fact that the velocity, vorticity and temperature gradients in the transverse directions are much larger than the longitudinal (streamwise) gradients for these flows, the original Navier Stokes equations are parabolized in the streamwise direction. A simple model, based on Boussinesq hypothesis, is used for turbulent flow. The discretized equations are then solved step by step in the streamwise direction, using an iterative procedure at each station. Numerical solutions have been obtained for different parameters, such as the Reynolds number, the circulation and the initial position of the vortices. The computed flow patterns and the skin friction coefficient and Stanton number are found to be qualitatively consistent with available experimental results. It is shown that the interaction between the vortices and the boundary layer may severely disturb the boundary layer flow field and thus considerably increase the local skin friction and heat transfer rate on surface of an aircraft.