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Timestamp: 2019-04-25 04:21:34+00:00

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Master Thesis: In the present thesis flow around a low-pressure turbine outlet guide vane (LPT-OGV) is studied using Large Eddy Simulations (LES). Effort is made to capture the boundary layer transition over the guide vane, which has a crucial impact on the heat transfer characteristics at the vane walls. The results are compared with experimental studies performed on the OGV at the Fluid Mechanics Department in Chalmers University of Technology. Two different subgrid scale (SGS) models are used for performing comparative studies – the Smagorinsky-Lilly model and the Germano-Lilly dynamic model.
Previous simulations based on this experimental study was performed via RANS (Reynolds averaged Navier Stokes) modelling approach. guide vane. FST. Keywords: CFD. but either they were not big enough to trigger transiiton or there was no continuous forcing provided by the free-stream turbulence to trigger transition. bypass. Mesh independence study showed that the spanwise resolution and domain width of the mesh hardly played any role in transition. but it doesn’t get distributed into the other components. The pressure distribution corroborates with the experimental data. is used for performing simulations. Effort is made to capture the boundary layer transition over the guide vane. The solver of CALC-LES uses a geometric multigrid algorithm to solve the pressure (poisson) equation. multigrid method. LES. Better result from a mesh with finer streamwise resolution in the transition zone indicated that probably a finer streamwise mesh throughout the surface is required for transition prediction. The results are compared with experimental studies performed on the OGV at the department. Further inspection of the turbulent kinetic energy peak showed that probably the streamwise streaks grow. heat transfer. Two different subgrid scale (SGS) models are used for performing comparative studies – the Smagorinsky-Lilly model and the Germano-Lilly dynamic model.Abstract Engineering flows in the practical world are chaotic and comprise of many different scales of motion due to the influence of ambient noise or disturbances. Such disturbances have a significant impact on the nature of flow close to the boundaries/walls. To simulate such flows near the boundaries correctly constitutes an important aspect of designing and product development processes. In the present thesis flow around a low-pressure turbine outlet guide vane (LPT-OGV) is studied using Large Eddy Simulations (LES). but the same cannot be said for the heat transfer at the walls. boundary layer transition. called G3DMESH. Studying the resolved Reynolds stress components in the boundary layer revealed that the streamwise stress component increased in magnitude significantly. this implementation is studied thoroughly and modifications are made to render it operable for boundary conditions similar to that of the present problem. called CALC-LES. It is the first time that LES is being used. The computational grid is created using another inhouse meshing utility. As a result transition is not observed. iii . In the present thesis. which has a crucial impact on the heat transfer characteristics at the vane walls. The present simulations fail to capture the transition in the boundary layer over the OGV. A finite volume method based in-house solver in Fortran.
Throughout the thesis.Acknowledgements To start with. for all the discussions we had on various subjects. I would also like to thank Asst. he has been extremely patient with me. Prof. I would like to acknowledge my colleagues. 2014/12/15 Ansuman Pradhan v . It was always enjoyable and insightful. which has allowed me to learn things at my own pace. I am also thankful to Himanshu for providing his valuable critique on my thesis report. G¨oteborg. Niklas Andersson. Without their financial support over the two years of master studies. I also wish to thank Svenska Institutet. Manan and Sankar. first I would like to express my deepest gratitude towards my supervisor. Finally. Lars Davidson. for sharing some of his profound knowledge in turbulence modelling with me. be it fluid dynamics or mathematics or other abstract topics. this thesis wouldn’t have been possible. Prof. I would like to express my gratitude towards my family back home. for helping me in understanding the script for mesh-generation. whose constant support and love helped me in succeeding.
Nomenclature Abbreviations DNS Direct Numerical Solution FAS Full Approximation Storage FMG Full Multi-grid FST Free-stream Turbulence GMG Geometric Multi-grid GS Gauss-Seidel HPT High Pressure Turbine LES Large Eddy Simulations LKE Laminar Kinetic Energy LPT Low Pressure Turbine MG Multi-grid NS Navier-Stokes OGV Outlet Guide-vane PPE Pressure Poisson Equation RANS Reynolds-averaged Navier-Stokes equations rms Root mean square SGS Sub-grid scale TDMA Tridiagonal matrix algorithm vii .
[ν/P r] αr Residual/SGS thermal eddy-diffusivity ¯ ∆ Grid filter-width δij Kronecker-delta κ von K´arm´an constant ν Kinematic viscosity νr Residual/SGS eddy-viscosity ρ Density τθj Residual/SGS heat flux τijr Residual/SGS stress tensor θ Temperature θin Inlet Temperature e ¯ ∆ Test filter-width Cs Smagorinsky coefficient lsmg Smagorinsky lengthscale Cp Coefficient of pressure cp Heat capacity at constant pressure cx Axial chord I Turbulent Intensity lp Pitch of the cascade ls Turbulent length scale p Pressure Pr Molecular Prandtl Number P rt Turbulent Prandtl number Roman Symbols .TKE Turbulent Kinetic Energy Greek Symbols α Thermal diffusivity.
U0 Inlet Velocity v Velocity V Volume h  1 2 Subscripts (.) Spatial Grid-filtering f (.)i Einstein-notation Superscripts (.) Spatial Test-filtering ∂vi ∂xj + ∂vj ∂xi i .q Heat flux rate Sij Strain-rate tensor.
2 Solving for the temperature field . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . 1 2 3 3 2 Boundary Layer Transition Theory 2. . . . . . . . . .3 Aim and scope . . . . . . . . . . . . . . . . . . . . . . . . . . 2. . . . . . 3. . . . . . . . . . . .2 Bypass-transition mechanisms . . .1 Full Approximation Storage: FAS . . . . . . . . . . . . . . . . .1 Mechanisms based on transient growth (paths C and 2. . . . . . . . . . . . . . . . . . .4. . . 3. . . . . . . . . . . . . . . . . . . 3. . . . . . . . . .2. 4 Geometry and Mesh Generation 21 4. . . . . .2 Flow conditions and boundary specific restrictions . . . . . . . . . . . . 1. . . .1 Large Eddy Simulations . . . . . . . . .4 Geometric Multigrid Method . . . . . . . . .1 Geometry and computational domain . . . . . . . . . . . . . . . 21 4. 1.2 Dynamic model . .3 Previous studies using LES and mesh resolution issues . . . . . . .2. . . . . . . 14 15 15 16 18 18 19 19 . . . . . . . . . . . . . . . . . .3 Mesh generation . . . . . . . . . . . . . . . . .2 Mechanism ‘bypassing’ transient growth (path E) . .1 Smagorinsky model . . . . . . . .1.3 Solving for the pressure field: PPE . . . . . . . D) . . . . . . 3. . . . .1 Paths of transition to turbulence . . . . . . . . . 23 x . . .1. . . . . . . . . . . . . . . . . . . . . . . . . .2 Why LES? . . . 22 4. . . . . . . . . . . . 3.1 Previous work . . . . . . . . . . . . . . . . . . . . . 3. . . . . .Contents Abstract iii Acknowledgements v Nomenclature ix 1 Introduction 1. . . . . . . . . . . . . . . . . . . . . . . 5 6 7 8 12 13 3 Governing Equations and Solution Methods 3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 7. . . 26 5. . . . . . . . . . . . . . . . . . . . . . . .2 Heat transfer distribution .2. . . . . . . .1 Pressure distribution . . . . . . .2 Comparison with the experiments . . .CONTENTS 5 Case Set-up and Implementation 26 5. . . . . . . . . .2 Scope for future work . . . . . . . . . .1 Resolved stresses . . . .2. . . . . . . . .2. . . . . . . . .1 Post processing of results . . . . . . . . .1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 6. 54 Appendix A Script for mesh-generation 56 Appendix B Mesh resolution in wall-units 62 Bibliography 67 xi . . 6. . . . . . .4. . . . . . . .4. . . . . . . . . . . . . . . . . .2 Multigrid algorithm implementation . . . . . . . . 6. fields . . . . . . . . . . . . . . .2 Momentum and thermal fluxes . . 28 5. . . . . . .1 Structure of the multigrid code . . . . . . 6. 6. . 6. . . .1 Boundary conditions . . . . . . . . . . . . . . .4 Boundary layer study . . . 6. .3 Turbulent kinetic energy and rms . 33 33 33 34 36 36 38 39 45 49 7 Conclusions and scope for Future Work 53 7. . .3 Mesh independence study . . . . . . . . . . . . .4. . . . . . . . . . . . . . . . . . . . . 29 6 Results and Discussion 6. 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
if the blade metal temperature prediction is off by just 10◦ C (50◦ F ) . The blade life may be reduced to half. Immediately downstream of the LPT. The LPT comes at the rear end of the engine case next to the high pressure turbine (HPT) cascade(e. Heat transfer study of the vanes thus becomes the core of the design process. Numerous studies on this mechanism have been carried out till date. the main engine carcass and the aircraft attachment point. the swirl angle could vary significantly leading to off-design conditions. It has a significant effect on the heat transfer at the vane walls. flow separation and reattachment etc. analysing the flow during on-design conditions requires capturing the boundary layer transition mechanism accurately. Moreover.g. it is very difficult to capture the transition at high freestream turbulence (FST) levels. Computationally. The efficiency of gas turbine engines and the performance characteristics is said to increase with an increase in the core allowable temperature . the OGVs are positioned. While flow separation and reattachment are phenomena prominent in flow during off-design conditions. The motivation to push this temperature up requires to constantly look for better materials and cooling technologies for the rotor/guide vanes. computational and numerical study of the flow around the guide vanes and heat transfer prediction is of great importance. Off-design conditions lead to a different thermal load and heat transfer prediction could be completely different. Thus. Factors affecting heat transfer at the guide vane walls include boundary layer transition behaviour.1 Introduction The subject of the present study is the flow around a low pressure turbine (LPT) outlet guide vane (OGV). governed by non-linear interactions between boundary 1 . In aerodynamic terms. from the design point of view. This is the structural function of the OGVs. depending upon the operation point of the engine.. their function is to de-swirl the swirling hot exhaust gas coming from the turbine rotors into an axial flow. both experimental and numerical. which provide a structural connection between the aft bearing support. in case of a general two-spool engine).
1. due to the myriad of factors that come into play (discussed later). at the department . researchers have been attracted towards performing LES over complex geometries. though only the on-design condition is studied. ). In this thesis. since. OGVs in actual engines are arranged in a circular cascade and exhibit rotational periodicity. it is the first time a computational study based on LES is carried out on these guide vanes. the experiments done so far. The linear stability theory only concerns exponentially growing instabilities in the boundary layer and does not consider the non-linear interaction due to FST. 1. . are on linear cascades and they provide a fairly good approximation to the circular one. Historically. Through the present thesis. since the aim had been to understand the physics/mechanism behind the transition. Recently.1 Previous work The LPT OGV in the present study is designed by GKN Aerospace (previously Volvo Aero). This is the reason why. and the basis of comparison of the current work. with 25◦ and −25◦ incidence angle of the guide vanes respectively. Few of these studies will be discussed in Chapter 2. and it is the interaction of free-stream turbulence with the other instability inducing factors in the boundary layer that contributes to the growth of the non-linear modes predominantly.1. The experimental studies by Hj¨ arne  on such linear cascades at the department have so far served as the basis of subsequent experimental and computational studies. Many of these aforementioned efforts were based on either full/direct numerical simulations (DNS) or Large eddy simulations (LES). in the presence of such high levels of free-stream turbulence. non-modal linear stability analysis. as well (. It was observed that such a transition takes place when two or more of these instability inducing factors interact amongst themselves. In the present study a similar effort has been made to simulate the transition in boundary layer over the turbine OGV using LES. linear and non-linear optimization techniques for largest energy growth of instability modes etc. however. PREVIOUS WORK layer and FST disturbances. like turbine blades. 2 . The current study is based on the experimental data obtained by Chenglong et al. Corresponding to this experiment. simulations based on RANS modelling approach was performed at GKN Aerospace . they have been performed only on simple geometries. One of the reasons being that the mechanisms governing the transition are not fully understood yet. efforts were made to isolate these governing factors and understand their individual effects. A universal mathematical model describing the (non-linear) growth mechanisms leading to transition is hence difficult to contrive. transition was reported to take place for flows with Reynolds number (Re) even lower than the critical Re computed using the classical linear stability theory. performed experiments at two operational settings – on-design and off-design conditions. a linear cascade of OGVs is studied. To get around this problem. termed as ‘bypass transition’. Chenglong et al. with the advancement in computational capability.
an intermittency based model. In the present studies. No model gives a reliable result for various combinations of FST intensity. RANS based modelling approaches have been employed to simulate transitional flows. it is better than the RANS approach on the grounds of predicting transition more accurately. WHY LES? 1. The main 3 . Broadly speaking. All spectral effects are lost in the time averaging process involved in RANS based approaches.5% in the experiments and the turbulent length scale was increased 20 times to 0. There are however some models that work well for specific problems. but not to the extent as mentioned above.1. is solved. None of the models consider all the individual mechanisms and stages of transition. LES is used for simulations in a hope to achieve a more realistic view of transition. based on the above mentioned approaches. Reynolds number and pressure gradients.2. In this respect. In the second one. will be provided in Chapter 2. on the other hand. The onset of transition location did not match exactly and was off by a margin of 20%. models only the smallest scales of motion. the LKE (Laminar Kinetic Energy) model and the second. and are found to be very sensitive to initial conditions. preserving the spectral nature of the larger energy containing eddies. However. LKE represents the ‘non-turbulent’ stream-wise fluctuations in the pre-transitional boundary layer. once turbulence intensity at the inlet was scaled up to 8% from 3. The first one is based on a simple ‘point transition’ approach whereby a switch is made (forced) between laminar and turbulent computations at a point determined either empirically or by an established correlation of relevant experimental or DNS data. The intermittency factor represents the fraction of time the flow is turbulent. boundary conditions etc. the bigger issue at hand is that the exact mechanism of transition is very debatable. γ. 1. A good overview of the early developments on these methods is given by Savill . Hence it seems very unlikely that RANS based approaches could cope with transitional problems accurately. Recently. for most of the engineering applications.2 Why LES? Traditionally. an extra transport equation for what is called an ‘intermittency factor’. a third approach based on solving a separate transport equation for what is called as the ‘Laminar Kinetic Energy’ (LKE) has also been reported . LES. based on previous research. A good insight on the mechanism. consider only a few mechanisms or stages of transition. due to their computational cost effectiveness. while it is necessary to understand the physical mechanism of transition for better modelling of simulations. The LKE model showed better agreement with the experiments.3 Aim and scope Simulations at GKN Aerospace are done using two different models – one. ‘bypass transition’ is supposed to be very sensitive with respect to the spectral nature of the imposed FST.024m. they can be divided into two categories. The turbulent characteristics at the inlet are similarly scaled up. γ ranges from 0 to 1 while going from fully laminar to fully turbulent regime. However. These models.
transition onset location is not of immediate concern in the present study. the SGS models to be used. based on the current boundary conditions. discussion and conclusion. and the pressure poisson equation. • Mesh generation. The main motivation comes from the LES study of boundary layer transition over a flat plate by Voke and Yang . responsible for solving the pressure equation. – Chapter 3. – Chapter 2. who captured transition using a surprisingly coarse mesh. • Results. A major part of the thesis includes modifications made to the multigrid section of the solver. case setup. – Chapter 4 and 5.1. and implementation of multigrid method in the code. – Chapter 6 and 7. AIM AND SCOPE aim is to capture transition based on the scaled up turbulence characteristics.3. The report can be divided broadly into four parts: • Elucidating the various mechanisms and stages involved in ‘bypass transition’. 4 . • Elaborating on LES.
2 Boundary Layer Transition Theory Whether it is the smoke coming out of a chimney or the flow in a river. the closest one coming to rest instantaneously (no-slip condition). consider the flow over a semi-infinite flat 2-D plate. an initially laminar flow has the tendency to change into a more random or chaotic type of flow. This distance keeps increasing in the stream wise direction. boundary conditions and material (surface) properties. is referred to as laminar-turbulent transition. this nature – laminar or turbulent – can be very sensitive to perturbations in initial conditions. Flow in 5 . especially the triggering mechanisms leading to turbulence has not been achieved. In most of the applications involving fluids. open). Depending upon the inertial and viscous properties of the wall-bounded flow. This kind of evolution of flow. It is referred to as the Boundary Layer. additional layers parallel to the surface start getting affected due to viscosity. Such a flow comes under the category of bounded shear flows in open systems. even after more than a century of research. from laminar to fully turbulent. At a certain distance from the wall in normal direction the velocity reaches the free-stream value. as more and more layers of fluid start getting affected by the wall. To understand the behaviour of such boundary layers (bounded. the ability to control the transition would greatly increase engineering efficiency and performance. At the leading edge. As the fluid moves further downstream. most of the flows found in nature or in different engineering applications are chaotic or turbulent in nature. This results in the sudden increase in static pressure at the edge. The present study is focussed around the investigation of laminar-turbulent transition in the boundary layer of the flow in a turbine guide vane cascade. which is characteristically different from free shear flows and closed systems. In the presence of such perturbations. the fluid particles close to the wall are slowed down due to viscosity. However. a complete understanding of this evolution process.
marking the onset of transition.  based on some earlier studies. More importance is given to the mechanism that the author thinks is crucial for the present study. resulting in creating instabilities. Saric et al.1. Up until now. After a certain distance downstream however.2. no mathematical model has been able to predict the transition Reynolds number for the flat plate accurately. perturbations in initial conditions. The development of such instabilities in a laminar flow is the first step towards the transition of turbulence.1: The paths from receptivity to transition  interaction between external disturbances and the boundary layer instabilities is referred to as receptivity. This is characterized by an increase in wall shear stress and heat transfer coefficient. the flow gets completely turbulent. PATHS OF TRANSITION TO TURBULENCE the boundary layer near the leading edge is essentially laminar. In the following sections. The reason as stated earlier is the poor understanding of the triggering mechanisms.1 Paths of transition to turbulence As mentioned in the beginning of this chapter. boundary conditions or the inhomogeneities in surface (roughness) may enter the boundary layer and get amplified. based upon previous studies. 2. different mechanisms and paths leading to this transition are reviewed briefly. Further downstream. gave a simplified scenario 6 . the flow in the boundary layer starts getting chaotic. The Forcing environmental disturbances amplitude Receptivity Transient Growth A B C D E Bypass Primary Modes Secondary Mechanisms Breakdown Turbulence Figure 2.
Most of the mechanisms of bypass transition proposed so far follow the path D. surface roughness or initial conditions and one of the five different paths shown is followed.2. and are three-dimensional in nature. These interactions are then explained using the secondary instability theory. 2. Kachanov  has reviewed the theoretical and experimental work based on this type of transition. transition will go into the non-linear interaction stage directly.2 Bypass-transition mechanisms Occurrence of streamwise elongated structures with alternating positive and negative streamwise disturbance velocities in laminar boundary layers subjected to FST was first observed by Klebanoff . This is followed by the appearance of small-scale motions and the final stages of transition. After the first step. contrary to the modal analysis performed for the evolution of TS waves. Consider a scenario with very low free stream turbulence (FST) level at the inlet. Studies on transition via transient growth mechanisms are based on non-modal analysis of the linearized NS equations. Figure 2. In such situations.generally equal to or below 1% of the mean flow. wall curvature. A second scenario arises when the free stream turbulence (FST) level is high (generally higher than 1%). path A is followed.1 depicts the same. i. the interactions take place on convective time scales (much smaller than viscous time scales). which can be predicted by the modal/eigenvalue analysis of the linearized Navier-Stokes(NS) equations. This will be discussed more extensively in the next section. C and D are based on the evolution of the initial boundary layer instability via what is called as the transient growth mechanisms. These are exponentially growing instabilities evolving on large viscous time scales. The first stage in such type of transition is the development of two-dimensional Tollmien-Schlichting (TS) waves. According to his observations. An extensive review on the nonmodal stability analysis can be found in an article by Schmidt . Such a kind of transition is also called natural transition. receptivity. It is characterized by the appearance of streamwise elongated streaky structures of alternating high and low streamwise velocity in the laminar boundary layer. Such a transition is called bypass transition (path D and E).2. As the diagram in Figure 2. BYPASS-TRANSITION MECHANISMS for turbulence transition in external flows.1 depicts.e. Paths B. the initial external disturbance amplitude increases from left to right schematically. amplitude of the peak 7 . and the linear instability stage marked by the formation and growth of TS waves is “bypassed”. These streamwise structures (commonly referred to as Klebanoff modes) are low frequency oscillations and are very different from the exponentially growing perturbations (TS waves). which grow in magnitude as they move downstream and finally lead to complete breakdown resulting in turbulence . a number of different instabilities can occur independently or together depending on the Reynolds number. As amplitude grows. In such a case.
Such growth is larger for disturbances mainly exhibiting spanwise periodicity.  on bypass transition in a zero pressure gradient boundary layer. A detailed explanation can be found in the colloquium by Grossmann on shear flow turbulence .1 Mechanisms based on transient growth (paths C and D) Receptivity is the interaction of free-stream disturbances with a laminar boundary layer and is the first step towards transition to turbulence.2. A good description can be found in the introduction of the article by Butler and Farrell . the final effect can be seen as streaks with high and low stramwise velocity fluctuations. They are not modes in strict mathematical sense as they do not represent solution to an eigenvalue problem. Mathematically. called nonmodal growth as well. The initial disturbance able to induce maximum transient growth at a given time is referred to as optimal. which is used extensively by other researchers for comparative studies. Butler and Farrell  using variational methods. They found that perturbations in the form of streamwise/longitudinal vortices (correspond to zero-streamwise wavenum8 . A thorough experimental study was performed by Roach et al.2. hence. It is caused by the superposition of several modes. It is believed to be caused by the non-normal nature of the linear operator pertaining to the linearized disturbance equation. Later. the streamwise velocity can grow linearly in time in the presence of a distubance with no streamwise variation. Since the disturbance is elongated in the streamwise direction due to the mean shear. The transient growth mechanism is linear in nature and has been studied extensively using optimization techniques. This phenomenon is termed as ‘transient growth’. such inviscid amplifications eventually decay after a short time or short streamwise distance. found the optimal perturbation responsible for the maximum transient growth in a boundary layer for Couette and Poiseuille flows. Klebanoff modes. BYPASS-TRANSITION MECHANISMS response of such structures increased in proportion to the FST amplitude and boundary layer thickness. that is having low frequency or streamwise wavenumbers. this transient energy amplification is not due to the behaviour of a single eigenmode of the linearized disturbance equation as found in exponentially growing TS waves. He termed these structures.2. to trigger non-linear interactions and cause breakdown to turbulence. Kendall too observed large spanwise variations in streamwise velocity in the pre-transitional boundary layer subjected to FST in his experimental study . before decay sets in. They are usually depicted in terms of rms profiles of streamwise velocity in the boundary layer. The normal vorticity increases in time due to the tilting of cross-stream (spanwise) vorticity by the perturbation strain rate in spanwise direction (this disturbance can be viewed as a streamwise vortex). For inviscid flows. it is associated with the appearance of streamwise streaks in the boundary layer. It is then possible for a sufficiently amplified disturbance. This is ascribed to the lift-up and vortex-stretching mechanism or more accurately to the vortex tilting mechanism. 2. In case of ‘bypass’ transition. It is believed that in the presence of viscosity though.
2. multiplied by wall-normal Orr-Sommerfeld modes. actual frequency selection takes place inside the boundary layer and is not sensitive to the details of inlet disturbance spectrum. They had observed that disturbances convected with free-stream velocity did not couple to the slower fluid in the shear layer near the wall. In the DNS study . Andersson et al. the spanwise scale of the streaks inside the shear layer was found to be highly insensitive towards the details of inlet turbulence (FST). they also found that the optimal perturbation consisted of a pair of counter rotating streamwise vortices outside the boundary layer. linear mechanism is most rel- 9 . Based on an earlier proposition . they conclude that streaks are an implicit property of the boundary layer.  performed experiments on flat plate boundary layer. Jacobs and Durbin had explained the shear sheltering mechanism. i.2. Their experiments corroborated well with the linear non-modal growth mechanism of the boundary layer streaks. Though the amplitude of the FST is a crucial input for transition.  did an extensive review of experimental studies on disturbance growth inside the boundary layer.  and Luchini  in separate studies formulated a spatial instability problem unlike the temporal instability problem by Butler and Farrell. Several experimental and numerical studies have observed transition due to the presence of such streaks.e. the breakdown phase. In an earlier study . So. Only the continuous spectrum of the latter was considered due to their inherent property of being sinusoidal in the free-stream and approaching to zero near the wall. Brandt et al. that can be related to the Klebanoff modes found in the pre-transitional boundary layer in the ‘bypass’ transition case. Luchini also found that the shape of the streaks in the pre-transitional boundary layer tends to be attracted towards the shape of the optimal perturbation (Klebanoff modes). Penetration depth of such disturbances was found to be inversely proportional to their frequency and the Reynolds number based on the distance from the leading edge. BYPASS-TRANSITION MECHANISMS ber) are responsible for inducing the greatest energy growth in the laminar boundary layer in the form of powerful streamwise streaks. even for non-optimal initial perturbations.  in their DNS study of a flat plate boundary layer. they constructed the turbulence inflow by expanding the FST as a sum of spanwise and temporal fourier modes. like in the Jacobs and Durbin study . also used the continuous spectrum of the Orr-Sommerfeld equation to generate inflow boundary condition similar to Jacobs and Durbin. Jacobs and Durbin  performed Direct Numerical Simulation (DNS) of flow over a flat plate. and also concluded that the onset of transition moved upstream with increasing FST length scale. Brandt et al. The process is non-linear since the dominant frequencies found inside the boundary layer were much smaller than the ones prescribed at the inlet. However. Matsubara et al. The shape was found to be insensitive to a wide range of wavenumbers. frequencies and shape of the initial perturbation.  also conclude that for receptivity. Jon´ aˇs et al. probably the scales at the inlet spectrum are important in the next phase of transition.. They found that the transition onset moved upstream with increasing the FST length scale. but also included the Squire modes for the wall-normal vorticity.
The same can be said for the numerical studies discussed so far. Reynolds number. Not all penetrating modes generated streaks. both mechanisms produce streaks of similar strength. They conclude that the coupling leading to boundary layer streaks and transition is local. Goldstein and Wundrow  in their analysis consider the leading edge and show that FST containing 10 . it is only the appearance of streaks. At moderate disturbance levels. It represented the penetration ability of the modes in the boundary layer and also its ability to generate streaks. that could be linear or non-linear. since the flow around a leading edge is non-parallel.  analyzed the Orr-Sommerfeld/Squire eigenvalue problem and showed that the penetration depth of a free-stream disturbance inside a boundary layer shear depends on four parameters. Earlier Berlin and Henningson  had proposed a non-linear receptivity mechanism based on their spectral DNS study of Blasius flow with both temporal and spatial formulations. Leading edge effects were not considered in the studies discussed so far. non-linear mechanism takes over if the FST contains high-frequency disturbances. the ones with high decay rate failed to do so. and occurs downstream of the leading edge. Most of the studies on transient growth assumed parallel flow. which had the inflow plane for simulations situated downstream of the leading edge. whereas non-linear mechanism plays important role at higher FST levels. As a result they could not capture the effect of leading edge. According to them. Disturbances with low frequencies and small wall-normal wavenumbers had higher coupling coefficients. A coupling coefficient was defined that depicted the local interaction of continuous Orr-Sommerfeld modes and the boundary layer. but argue that at the leading edge. It should be noted that the growth of streaks is still supposed to be based on linear transient mechanism. They also conclude that for receptivity. BYPASS-TRANSITION MECHANISMS evant if the FST contains low-frequency disturbances. and the local mean shear at the wall. disturbances with high wall-normal wavenumber. but since Rex → 0 (Re based on distance from leading edge). But they also pointed out that by increasing the wall-normal wavenumber. they also decay rapidly. The non-linear method discussed by Berlin and Henningson  forced streaks in the boundary layer locally even downstream of the leading edge. at low FST intensity levels linear mechanism dominate.g. Zaki et al. which on interacting with the boundary layer shear produce the streamwise streaks.2. Zaki and Durbin  consider a leading edge. as boundary layer thickness δ → 0. the decay rate also increases and the disturbance does not persist far downstream of the leading edge. The importance of wall-normal velocity in the free-stream for inducing transition was also emphasized by Voke and Yang  in their LES study of flow over a flat plate. It increases with increasing wall-normal wavenumber. But owing to the non-linear mechanism it is possible for the free-stream to continuously force streaks inside the boundary layer even downstream of the leading edge. non-linear mechanism is responsible for generating wall-normal perturbations associated with streamwise vortices inside the boundary layer. all modes act as low frequencies and become penetrating. e. whereas.2. The penetration is inversely proportional te frequency of disturbance.
 in their DNS study used a pair of inflow modes. which are then acted upon by fluctuations due to the latter. Brandt et al. But instead of transition being delayed or averted. was observed more frequently than the varicose mode of instability.  in their numerical simulations observed that adverse pressure gradient (APG) results in a higher shear close to the boundary as compared to zero or favourable pressure gradient cases (ZPG and FPG respectively). As a result. The reason is that APG induces stronger streaks in the boundary layer than in ZPG or FPG flows. are responsible for the formation of turbulent spots. Streak break-down and secondary instabilities: The streamwise elongated unsteady streaks in the boundary layer are the main driving source of bypass transition. the (high-frequency) weakly coupled mode. which is driven by the wall-normal shear. Mechanisms driving the conversion of streaks into turbulent spots and subsequent break-down have been an important aspect of bypass transition studies. and the other a weakly coupled mode (high frequency) and observe that it was sufficient to trigger transition. The sinuous mode. They are present predominantly at the upper boundary of the shear layer. The Pressure gradient has also a significant effect on the receptivity process. thus making it more succeptible to the FST forcing. a blunter leading edge results in a completely different mechanism to transition and will be discussed later. exhibits earlier onset and completion of transition. it is earlier and faster than in the latter two cases. The former generates streaks inside the boundary layer.2. These vortices then penetrate in the boundary layer to produce streaks. predominant close to the surface of the plate. But according to them. the mechanism to transition still remains the same. They observe that a blunter leading edge. The 11 . corresponding to wall-normal vortical structures. Jacobs and Durbin  observe that streaks with negative streamwise fluctuation (u′ < 0). BYPASS-TRANSITION MECHANISMS wake like disturbances. implying that the coupling of these jets to the freestream eddies is essential. one with high coupling coefficient (presumably low frequency). Zaki et al. the Orr-Sommerfeld modes constituting the inflow FST are expected to penetrate less in the APG case than in the other cases. Only some of these streaks develop to spots.  observe that streak breakdown and turbulent spot formation is caused by either of two types of instability modes of low-speed streaks (negative jets). They also find the same mechanism in play around the leading edge as observed by Goldstein and Wundrow– stretching and tilting of vortices around the leading edge. at given free stream conditions. The sinuous mode instability is similar to the backward jet mechanism of Jacobs and Durbin . termed as negative jets. owing to shear sheltering . characterized by streak oscillations in the spanwise direction.2. do not undergo instability. Positive jets. Zaki et al. Leading edge just enhances the growth of Klebanoff modes inside the boundary layer. Nagarajan et al.  carried out simulations on a flat plate boundary layer with a blunt elliptical leading edge. and are continuously in contact with the high frequency free-stream disturbances outside the boundary layer. get stretched and tilted around the leading edge and transform into low frequency modes corresponding to streamwise aligned vortices. The low-speed streaks (negative jets) lift towards the edge of the boundary layer.
boundary layer streaks appeared to be merely a kinematic feature. Spot precursors responsible for transition were identified to be wavepacket like disturbances originating at the leading edge (blunt). This is also similar to the mechanism observed by Jacobs and Durbin. corresponding to the low-speed streaks. There are many more studies based on streak growth which are not discussed here.2 Mechanism ‘bypassing’ transient growth (path E) Nagarajan et al. The possible explanation for such difference was attributed to the higher Re (based on FST integral length scale) in the study of Ovchinnikov et al. the vortices subsequently reorient themselves partially in streamwise direction. The non-linear development of obliquely oriented Λ-vortices into hairpin packets was observed to be responsible for the breakdown into transition. Recently. appearing as a result of lift up mechanism. However. 2. The result is either a horseshoe/hairpin/Λ vortex (two legs) or quasi-streamwise vortex (one-leg). They are associated with spanwise vortical structures in boundary layer in contrast to streamwise ones in the simulations of Nagarajan et al. They observe that wavepacket like disturbances in streamwise direction act as spot precursors that lead to transition. Also. For FST length scales comparable to the boundary layer thickness at the onset of transition. and that they are not confined to the lower part of the boundary layer close to the wall. BYPASS-TRANSITION MECHANISMS latter is unable to penetrate the shear layer and is present in the free-stream. which grow as they convect downstream. They also find that wavepackets appear in wallnormal velocity component as compared to the spanwise component in the simulations of Nagarajan et al. Wu and Moin  performed DNS for a spatially evolving turbulent zero pressure gradient boundary layer over a smooth flat plate. which were found in the upper part. Initially in spanwise direction. the transition followed a different path. The point the author wants to make is that linear growth or transient growth of streaks is not alone sufficient for triggering transition. Ovchinnikov et al. as was observed by the former. δ99 . in contrast to the sinuous streak instabilities discussed earlier. They inhabited the lower part of the boundary layer. Nonlinear forcing provided by the FST and its interaction with the streaks is an important condition to induce transition.2.  proposed 12 . and also the higher FST intensity in their study. Some of these jets get intensified and burst into turbulent spots. . though they seem to be necessary. According to their flow visualizations.2. Cherubini et al.  observe in their DNS study that the same inflow turbulence field leads to different paths to transition depending upon the leading edge geometry. the transition followed the Klebanoff mode mechanism discussed earlier. comparable to almost 7δ99 . receptivity of such disturbances inside the boundary layer was not discussed. Vortices aligned normal to the wall get stretched around the leading edge resulting in localized regions of streamwise vorticity inside the boundary layer. there is a change in the location of transition onset consequently.  studied the effect of FST length scales and leading edge on bypass transition. But for higher FST length scales. They were not found to be responsible for transition. These structures finally develop into spots and subsequently break down to turbulence.2.
the transition was said to follow the traditional streak mechanism (transient growth and secondary instability leading to break-down). lower intensities and length scales of the FST. justifying the preponderance of hairpin structures observed by Wu and Moin  in their transitional boundary layer. Their mesh resolution for structured LES was also fairly coarse in the streamwise direction. responsible for inducing turbulence in the boundary layer. The wall units were computed based on friction velocity just after transition. between 130 and 13 near leading and trailing edge respectively.  conclude that transition in a flat plate boundary layer in the presence of FST. when turbulent intensity and turbulent length scale of the FST are very high (for I > 4. They observed transition on a very coarse mesh with resolution based on wall units as. follows a purely non-linear route characterized by the formation of hairpin/Λ vortices. ∆x+ ranging from.  did heat transfer studies on high pressure turbine blades in the presence of high free stream turbulence levels using LES and ovserved transition in boundary layer. and will be discussed in Chapter 4. For. The optimal mesh resolution required for studying transitional flows is therefore debatable. Medic et al. Using non-linear optimization of the energy growth at short time.5% and l > 20). ∆z + ∼ 25 and y + ∼ 1 at the wall. called the minimal seed perturbation. 13 . Cherubini et al. They had streamwise resolution. 2. In the present simulations the resolution is similar to that of Collado Morata et al. (∆s+ < 50. approximately.2. also performed a similar heat transfer study on the turbine cascade using LES. They had ∆x+ ∼ 150. ∆z + = 14 and ∆y + = 1 at the wall to 80 beyond the boundary layer. Kaltenbach and Choi  performed LES of flow around an airfoil. spanwise resolution).3. ∆x+ = 80. 400 at the leading edge to 100 near the trailing edge and spanwise resolution. PREVIOUS STUDIES USING LES AND MESH RESOLUTION ISSUES a purely non-linear scenario of (bypass) transition in boundary layer. streamwise resolution and ∆z + < 10. In a more recent article.3 Previous studies using LES and mesh resolution issues One of the earliest works on boundary layer transition using LES includes the simulations on flat plate boundary layer by Voke and Yang . ∆z + . though. they identified the smallest flow structure. did not study transition. More recently Collado Morata et al. Their streamwise and spanwise resolution was fairly lower than that of Collado Morata et al.
One way is to solve the time averaged equations. leads to the Navier-Stokes (NS) equations. The NS equations in conservative form in the absence of body forces for an incompressible and isotropic Newtonian fluid can be written as: ∂vi =0 ∂xi (3. the extra terms appearing out of averaging are modelled: the RANS method. These two ways are the two extremes of the spectrum based on computational cost. The exact analytic solution of these equations is extremely difficult to achieve. momentum conservation and energy conservation equations (θ: temperature) respectively.1) ∂vi ∂vi vj 1 ∂p ∂ 2 vi + =− +ν ∂t ∂xj ρ ∂xi ∂xj ∂xj (3. wherein. along with energy and mass conservation on a portion of fluid volume.3) The three equations above represent the continuity. called control volume.2) The energy equation (heat transfer equation) in the absence of heat sources for such a fluid. without the need for modelling: called the DNS. This requires a very fine grid and for larger Reynolds number the computational cost goes up so high that it is practically not possible for a lot of problems. The possible way to get around the solution is to solve it numerically using iterations performed on the discretized equations. neglecting the viscous heating effect (dissipation function) can be expressed as: ∂2θ ∂θ ∂vj θ + =α ∂t ∂xj ∂xj ∂xj (3. Another way is to resolve the whole range of temporal and spatial scales.3 Governing Equations and Solution Methods Application of conservation of momentum. There are many 14 .
it assumes the Boussinesq hypothesis and the deviatoric part of the residual-stress tensor is proportional to the filtered strain rate tensor (S ij ).1.4) ∂τijr vi v¯j 1 ∂ p¯ ∂ 2 v¯i ∂¯ vi ∂¯ + =− +ν − ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj (3. after the filtering operation give rise to a set of equations with certain additional terms arising due to filtering. a ¯ is simply taken as simple volume-average box filter is used where the filter-width (∆) 1/3 ¯ the local grid size [∆ = (∆VIJK ) ]. The first step in LES is filtering in which the flow variable (say φ) is decomposed into ¯ and the residual/SGS(sub-grid scale) comthe sum of a filtered/resolved component (φ) ′ ponent (φ ). in LES. In this study two such models are used for performing LES – the Smagorinsky model and the Dynamic model. 3. 1 r δij = −2νr S ij τijr − τkk 3 (3. By analogy to the mixing-length hypothesis. The next step is to obtain closure by modelling the residual flux terms. Like all other eddy viscosity models used for RANS. consider the residual flux term in the momentum equation (eq.1)–(3.1 Smagorinsky model This model forms the basis of all other eddy-viscosity models.3). It is modelled based on certain assumptions. LARGE EDDY SIMULATIONS middle ways. ∂¯ vi =0 ∂xi (3.7) The constant of proportionality νr is the residual viscosity. Details about filtering operations can be found in .1. In this study. One of them is LES. First.6) where.(3. wherein the larger energy containing motions are resolved and the effect of smaller ones are modelled. .5) ¯vj ∂τθj ∂ θ¯ ∂ θ¯ ∂ 2 θ¯ + =α − ∂t ∂xj ∂xj ∂xj ∂xj (3. The majority of them are based on the eddy-viscosity modelling approach – Boussinesq assumption in case of RANS modelling. 3. τijr represents the residual-stress tensor. also called the SGS viscosity and needs be modelled.5)).1 Large Eddy Simulations As said earlier. defined as τijr ≡ vi vj − v¯i v¯j and τθj represents the residual heat flux and defined analogous to τijr . Equations (3.3. the eddy-viscosity is modelled as. the larger scales are resolved and smaller ones are modelled.
νr = lsmg q 2 = lsmg 2S ij S ij .8) 15 . (3.
LARGE EDDY SIMULATIONS where. Cs is the Smagorinsky coefficient.1. where. lsmg represents the Smagorinsky lengthscale and taken as proportional to the ¯ such that filter-width ∆.3. This model . ¯ lsmg ≡ Cs ∆.
gives a high residual viscosity at the wall. owing to high velocity gradients (hence. high .
∆ (3. e ¯ = r∆. The filtered NS equation at the second level (testfiltered) is written as. It can be different for different flow regimes . 3. To achieve ¯ this. The This filter is called the test filter . (3.1.7)) can be used for parametrizing both Tij and τij . the last term in both the equations are different. the τijr in (3. ¯ where r > 1. For simplicity. n o ¯ = min (∆VIJK )1/3 . except that the variables here are test filtered.13) (3.5) will be referred to as τij from now on. 3 2 δij e e¯ S e¯ .10) This is similar to (3. The residual stress tensors from both equations are. In order to dampen the eddy-viscosity. n is the distance to the nearest wall. it is difficult to represent different flow regimes. This is not accurate since the SGS turbulent fluctuations go to zero near a wall. let Mij and mij represent the deviatoric part of the stresses Tij and τij respectively. ¯ |S| Tkk = Mij = −2C ∆ ij 3 16 (3.14) . κn . ∆ optimal value is found to be 2 .11) v¯ie v¯j Tij = vg i vj − e (3.9) where. The Dynamic Model provides a way to determine the coefficient locally and dynamically (variable of time and space) as opposed to the constant value of Smagorinsky. the RANS length scale is used as an upper limit in the present solver. a second filtering operator is introduced with a larger width than the grid filter (∆).5).) in this region. The test filter-width. τij = vi vj − v¯i v¯j . Thus.12) and Assuming that the same functional form (as in Smagorinsky model (3. Thus.2 Dynamic model With a single universal value for the Smagorinsky constant Cs . Also. τij − and Tij − δij 2 τkk = mij = −2C∆ |S|S ij . v¯ie v¯j ∂Tij ∂e v¯i ∂ e p¯ 1 ∂e ∂ 2e v¯i + =− +ν − ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj (3.
it was necessary to consider the backscatter effects (the inverse transfer of energy from small to large scales). The solver in the present simulations use the formulation provided by Lilly. 3 (3.13) and (3. wherein. Germano proposed a method.18) To calculate Mij . might lead to numerical instabilities. iv (3. In order to achieve this. Lij − δij Lkk = 2CMij . which is locally and dynamically optimal. In order to avoid this. As mentioned earlier.16) where.16) and assuming negligible variation of C in the test-filter volume. Lij = Tij − τeij = v¯g ¯j − e v¯ie v¯j . however. the Dynamic SGS model treats the backscatter effect well. and since. Lilly suggested the least-squares approach.3.1. This in turn. So. Lij represents the resolved components of turbulent stress associated with scales of motion between the test scale and the grid scale and it can be computed explicitly. Piomelli et al. the whole model boils down to finding that C. The consistency between (3. 17 . which yields a computationally more stable value of C was developed shortly by Lilly . and C is evaluated as C= 1 Lij Mij 2 Mij Mij (3.17) where. C is averaged along the homogeneous directions of the flow.15) It was recognized by Germano that Tij and τij are related by a relation (Germano identity).it can be seen that.14) in (3. it was decided to use the model for the present study as well. C is equivalent to Cs2 in the Smagorinsky model. since it is an instantaneous and local quantity.13) and (3. Substituting equations (3. to model transitional channel flow appropriately. According to them.14) are substituted in (3. A better method.14). 2 2 ¯ ¯ e e¯ S e¯ ¯ |S| S|Sij − ∆ Mij = ∆ |^ ij (3. depends upon the proper choice of C. q e¯ S e¯ 2S ij ij (3.16) and then is contracted wtih S ij . LARGE EDDY SIMULATIONS where e¯ = 1 S ij 2  ∂e v¯j ∂e v¯i + ∂xj ∂xi  .13) and (3. e¯ = |S| Here. (3.  studied the budget terms of the resolved energy equation that included the SGS dissipation term from DNS of transitional and turbulent channel flow.19) It was realized that the computed value of C may vary wildly in time and space. the present work aims to study the boundary layer transition over a turbine guidevane.
a Geometric Multigrid (GM) algorithm is employed. 3. Drawing analogy from the eddy-viscosity model in equation 3. both the momentum and thermal equations are solved.3 Solving for the pressure field: PPE In case of an incompressible flow. The effectiveness of the afore mentioned fractional step projection method mentioned is based on the efficiency of the PPE matrix solver. The consistent PPE is derived by taking divergence over the momentum equation (3. the residual flux term in the temperature equation can be modelled as. in SIMPLE algorithm a pressure correction equation derived from the continuity equation is solved and used to update the velocity field.2) and in general. an intermediary velocity field from the NS equations devoid of the implicit Pressure gradient term is first solved. it is solved indirectly by making use of the continuity equation (3. a turbulent Prandtl number P rt is introduced as the ratio between the two residual diffusivities.7 to 0. There are ways of modelling it by writing a separate transport equation involving the term as in . τθj = −αr ∂ θ¯ . Pressure is involved in the momentum equations (3. there is no explicit equation for solving pressure.20) Similar to the definition of molecular Prandtl number P r.21) By applying proper boundary conditions. SOLVING FOR THE TEMPERATURE FIELD 3.9 in general. In the present case. by modelling only the residual/SGS viscosity. The present work doesn’t delve into this. assuming that the thermal boundary layer will develop similar to the velocity boundary layer.2): ∂ 2 τijr ∂ 2 v¯i v¯j ∂ 2 v¯i ∂ 3 v¯i 1 ∂ 2 p¯ =− − +ν − ρ ∂xi ∂xi ∂xi ∂t ∂xi ∂xj ∂xi ∂xj ∂xj ∂xi ∂xj (3. pressure is computed directly by solving a Pressure Poisson equation (PPE). A similar PPE as above is formulated by imposing the requirement of continuity on the correct velocity field.6. It is taken to be around 0. P rt = νr αr Thus.2. More about this formulation and the whole projection method to solve the system of equations is explained in . 18 .3. the residual flux term is modelled based on the residual/SGS viscosity.7 and by introducing a residual thermal eddy-diffusivity αr .2 Solving for the temperature field Now. consider the residual-heat flux term τθj in equation 3. which is explained in brief in the next section. In the present numerical method. and imposing the divergence of velocity field to go to zero – the continuity eqation – the above equation can be solved. For example. ∂xj (3. Instead. νr . To solve the PPE.1).
3. The subsequent subsections explain the theory in short.4. Φ is the true solution of the system and ψ. GEOMETRIC MULTIGRID METHOD 3. Of theses modes. rendering the process computationally expensive. In the present case/problem. This technique is very efficiently implemented in the CALC-LES code. especially non-linear. the solution consists of two kinds of errors. 3. The first kind of error is reduced either by using a higher order discretization scheme or by refining the mesh. the equation becomes A(e) = r. a geometric multigrid technique is used to solve the Pressure Poisson equation (PPE). The second kind of error reduces with the number of iterative sweeps. A(Φ) − A(ψ) = r (3. which is the difference between the exact solution and the iterative solution. while the longer wavelength components require a lot of iterations. The other one is called algebraic error.24) It can be easily seen that for a linear A. This step removes the high frequency error components. the modes with shorter wavelength smooth out within a few iterations. arising due to discretization scheme used or the quality of mesh used (coarse or fine). The algorithm to solve the linear problem can be found in the book by Versteeg and Malalasekera . The fine grid residual rh is computed. But in this case (non-linear). error can be said to be consisting of different components/modes of different wavelengths. This step is also called Pre-smoothing. (3.4. 19 . Upon Fourier decomposition. The smoothing effects on long and short wavelength error modes are neatly explained in . The algebraic error e is then defined as e = Φ − ψ.1 Full Approximation Storage: FAS The Full Approximation Storage (FAS) scheme is a multigrid algorithm formulated for a non-linear problem. when solved using iterative methods. A two-grid version of the FAS scheme can be explained in the following steps: • Fine grid iterations: In general it starts with a few iterations at the finest grid with mesh spacing h. but it can be generalized to include a linear problem.23) and the residual equation can be written as. One is truncation error. A(Φ) = f .22) where.4 Geometric Multigrid Method To any system of equations. consider a non-linear system of equations. Multigrid techniques are precisely used for the latter kind of errors – to accelerate the rate at which the error can be reduced. To start with FAS scheme. this doesn’t hold true. an intermediate solution to it after a couple of iterations/sweeps. (3.
Hence. e2h is computed.25) And similarly. the prolongated error vector e eh is used to correct the intermediate fine grid solution ψ and a few more iterations are performed to smoothen out the error that might have occurred due to restriction and prolongation. e eh = Ih2h e2h (3.29) The difference between e eh and eh is the lower frequency components.24) gives. e2h is stored). • Correction and final iterations: Finally. Then using the error equation (3.3. a lot of recursive schedules/schemes have been developed.30) This is a typical two-grid multigrid method. • Prolongation: The error e2h found in this way at the coarse mesh is transferred back (called Prolongation) to the fine mesh using an interpolation operator Ih2h . here the system is not solved for e2h directly.23) for the coarse grid.4. h ψ 2h = I2h h ψ (3.27) is solved for Φ2h . unlike in the linear case (where only the correction. GEOMETRIC MULTIGRID METHOD • Restriction: Both the fine grid residual and the intermediate solution vector are restricted to the coarse grid mesh using the restriction operator I2h h . the details of which are not discussed here. W-cycles.27) h e2h = Φ2h − I2h h ψ (3. Since a full approximation (of Φ2h ) is stored at each level. this method is named as full approximation storage.26) • Relaxation at the coarser level: Substituting the above two terms in the coarse grid formulation of equation (3. The implementation of the FAS scheme is explained in Chapter 5. F-cycles scheme or the FMG scheme. Based upon this method. h 2h h h h A2h (Φ2h ) = A2h (I2h h ψ ) + Ih (f − A (ψ )) =e f 2h (3. like V-cycles. they are absent in the former. 20 .28) Equation (3. h r2h = I2h h r h h h = I2h h (f − A (ψ )) (3. it is called Post-smoothing. Unlike in the linear case. ψh ← ψh + e eh (3.
The profile of cross-section of the guide-vane is shown in Figure 4. The 0.2.22 0.2 Figure 4. which shows the geometry magnified near the trailing edge. It is not supposed to affect the transition as it doesn’t provide a substantial change in the pressure gradient.08 0. The length of the extrapolated part is around 10% of the chord length.15 0. The extrapolated part can be seen in red in Figure 4.1: guide-vane 0.25 0.05 0.06 0.02 −0.27 x x Geometry profile of the Figure 4.21 0.1 0.06 −0. which is made sharp by extrapolating the suction side and pressure side curves further downstream till they coincide.04 −0.24 0. which 21 .01 0 y y 0.26 0.02 0 −0.2: Trailing edge extrapolation.2 0.08 0 0.23 0.01 −0.02 −0.4 Geometry and Mesh Generation 4.04 0. extruded in the third direction. given geometry . This operation is driven by the need of having a well-defined edge (or point in 2-D) to define the periodic boundaries of the cascade domain.1.03 0. extrapolated part trailing edge of this profile is circular.02 0.1 Geometry and computational domain The given geometry is a typical aerofoil profile in two-dimensions.
The angle is defined with respect to the longitudinal direction of the geometryaxes. 4. Other properties at the inlet were assumed to be ambient.e.2 elaborates the boundaries specific to each label in Figure 4.2. The lines joining the leading edges to the inlet section are based on the incident flow angle (mentioned later). The pitch lp of the linear cascade.5% Turbulent length scale ls 0. i. Table 4. Inlet flow conditions used in the experiments are depicted in Table 4.2 Flow conditions and boundary specific restrictions As per the design operation conditions. Reynolds number is calculated based on the axial-chord.1. There is a 22 . the ones joining the trailing edge to the outlet section are roughly based on slope of the camber line at the trailing edge. the choice is arbitrary. The axialchord cx . the projection of the actual chord in the longitudinal direction (analogous to axial direction in case of a circular cascade) is 243mm.2: Boundary specifications Figure 4. the clearance between the blades/vanes in transverse direction is 240mm.3 shows the flow domain with all the boundaries labelled as B1 to B8. Similarly. The span of the vane is 240mm.4. i. FLOW CONDITIONS AND BOUNDARY SPECIFIC RESTRICTIONS might have altered the transition location.1: Inlet flow conditions B1 Inlet B2 Outlet B3 and B4 Periodic B5 and B6 Walls B7 amd B8 Periodic Table 4.e. the angle of incidence of the flow is 25 degrees. The control volume for the simulations is chosen as the volume enclosed by the pressure side of one blade and the suction side of the adjacent blade.3. Inlet velocity U0 20m/s Reynolds number Re 3 × 105 Turbulent Intensity I 3.0012 Table 4. The inlet and outlet sections are situated at distances greater than the chord length on either side of the leading and trailing edges.
as explained earlier. MESH GENERATION 0. These sections have different sets of boundary conditions in the j-direction. where n is the desired number of levels. an in-house code is used called G3DMESH.2 B7 B3 0.1 0. 4.2 −0. separated by lines marked by the leading and trailing edge of the walls in the mesh i-direction (see Figure 4. These sections are blocked separately and then combined to a single block.1 y I 0 ilast II ifirst III B6 B1 −0. obviously for ease in meshing. whereas section II has wall boundaries in j.3). The mesh under consideration.4 0.1 B2 B8 B4 −0.3. The (i.3 B5 0. ‘ifirst’ and ‘ilast’ represent the start and end of the walls respectively.3 Mesh generation To generate the mesh.II and III certain restriction on the number of cells the mesh can have in each direction.3 0. Section I and III are cyclic in j-direction. where n is the number of levels in the desired multigrid. This is done for two reasons: one.6 x Figure 4. as explained in Chapter 3. 23 .3: Control Volume containing the three sub-blocks I. consists of three sections. The generated mesh is two-dimensional with quad cells and later while running simulations. The flow domain (in 2-D).2 −0. This is imposed by the Multigrid method used to solve the pressure poisson equation (PPE). and second. While meshing.1 0 0. has another restriction though.2 0. more importantly. to control the number of cells in each section in the i-direction. emphasis should be laid on the fact that in each section separately.4. these 2-D layers are stacked in the third direction (spanwise) with suitable distance between them. The script can be found in Appendix A.3 −0.5 0.j)-plane of the grid can be thought of as comprising of three sections/sub-blocks.4 0. Since the course mesh has a grid spacing which is twice as that of the fine one. number of cells in the i-direction should be a multiple of 2n . the mesh should have cells equal to a multiple of 2n in each direction.
3: Mesh specifications – ni.4: Complete mesh. I.j)-plane. nk denote the number of cells in each direction. MESH GENERATION This gives rise to a three-dimensional single-block mesh ((i. the domain in k-direction (zmax ) is chosen to be less than 20% of the chord-length (cx ).4 depicts the 2-D mesh corresponding the (i.4. the spanwise direction). .3.3. made four times coarser for clarity Figure 4. II and III denote the 3 sub-blocks. MESH ni(I) ni(II) ni(III) nj nk zmax MESH-1 40 128 40 112 32 0.2cx ).05144cx MESH-2 40 128 40 112 64 0.j)-plane denoting the 2-D cross-sectional plane and k. Their mesh for structured 24 . The resolution at the walls is similar to that used in the study by Collado Morata et al.05144cx MESH-3 40 128 40 112 128 0. In the present study. Figure 4. last column denotes the domain size in k-direction. the mesh is made four times coarser in both directions.10288cx Table 4. For the sake of clarity.05144cx (< 0. The three types of meshes to be used in the study are specified in Table 4. for achieving a good resolution for LES. nj. Most of the study is done on a mesh with zmax equal to 0.
∆z + < 15 and y + < 1. MESH GENERATION LES had resolution (in wall-plus units) at walls as: ∆x+ ∼ 150.4. ∆z + ∼ 25 and y + ∼ 1. 25 . ∆x+ is close to 150 mostly and goes up to 250 in the initial laminar region on suction side.3. In the present simulations based on the local wall friction velocity. The streamwise variation of these values can be found in the appendix B. ∆z + and y + also go a little higher than mentioned in the initial laminar region.
5. the latter are then non-dimentionalized.1 illustrates the fluid properties and inlet 26 .5 Case Set-up and Implementation The simulations are carried out using CALC-LES.1 Boundary conditions fluid properties Density ρ Prandtl no. whereas van-Leer scheme is used for the temperature equation. is based on an implicit two-step time advancement technique with a multigrid pressure Poisson solver and a non-staggered grid system. To compare the results with the experimental ones. The numerical method explained by Davidson and Peng . Crank-Nicolson method is used for temporal dicretization of all the equations.7 Inlet velocity Uin 1 Reynolds no.1: Flow conditions The variables used in the simulation are all non-dimentional quantities. a cell-center based finite-volume in house code at the department. 1 Pr 0. central-differencing scheme is used for all the velocity equations. generally using inlet velocity and axial chord cx . Re 3 × 105 Inlet Temperature θin 0 inlet conditions Table 5. Table 5. For space discretization.
1. so that the intensity at the leading edge is close to that mentioned in the experimental studies. The foremost aim is however not to mimic the transition location from experiments exactly. These data are not provided in the experimental studies. BOUNDARY CONDITIONS flow conditions used for the simulation. Even in actual flows. . Rather.2 illustrates the various boundary conditions used in the simulation. Dirichlet(Uin .2: Boundary conditions.The fact that larger length scales decay at a slower rate  provided motivaiton to use an integral scale larger than that reported in the experiments. Inflow Boundary: To define the inflow boundary. The intensity at the inlet thus. for this reason. which is crucial in triggering transition. it is to observe transition first and then make changes in the inflow turbulence to match with the exact location 27 . They are generated using the method specified by Davidson in . al. v. synthetic isotropic fluctuations are used. It was found that the turbulence intensity specified at the inlet dropped considerably within a very short distance downstream. and w denote x. θ No-slip(u=v=w=0) Constant heat flux (qw = 10) Table 5. Inlet Fluctuations: For prescribing the inlet turbulence conditions at the inflow boundary. This will characterize the different scales of motion in the FST. at least the integral length scale and the intensity. v.1) and a fluctuation field with zero mean is superimposed on it. Table 5. The types of boundaries for the 2-D flow domain are explained in Chapter 4. had to be raised to a suitable value. which is not specified in the experiments corresponding the present studies.5. a decay in turbulence is observed and turbulence evolution depends upon the origin of tubulence . to match with the exact experimental data. Velocity fluctuation field is based on the turbulence intensity and turbulent length-scales mentioned at the inlet. As stated by Ovchinnikov et. before adopting a more physical decay rate. at the leading edge. it is important to know the turbulence characteristics. Use of zero Neumann boundary condition for pressure is customary to the solving of the PPE – both at inlet and outlet. u. z components of velocity respectively. fixed values of velocity and temperature are used (specified in Table 5. w Neumann for p. y. θin ) Inflow ∂p = 0) Neumann( ∂n Forced inlet fluctuations Outflow Walls Convective BC for u.
zero Neumann condition is prescribed at the walls (as at inlet and outlet). In such cases. when a nonreflective boundary condition is much demanded. 28 .3 depicts the turbulence specifications of both experiments and simulations. It is defined as: ∂φ ¯n ∂φ = 0.2 Multigrid algorithm implementation The current multigrid-code based on the Geometric Multigrid (GMG) method for Poisson equations was first implemented by Emvin for LES in a ventilated enclosure . If the outflow contains unsteadiness or vortical structures. convective boundary condition serves the purpose. MULTIGRID ALGORITHM IMPLEMENTATION of transition.1) ∂t ∂xn where xn corresponds the coordinate in the mean flow direction and Un corresponds the mean flow velocity. experiments Turbulent Intensity I 3. +U (5. qw specified in Table 4.5. Moreover.8 × 10−3 m Integral time scale τ 2. DNS or LES predictions are inherently unsteady and might contain dynamic vortical structures at the outflow. For pressure.2 × 10−3 m Turbulent Intensity I 10% Integral length scale ls 4.2. in the present study the extent of the wake (which is dominated by such vortices) is unknown. Hence the presence of such unsteadiness is quite possible at the boundary. 5. Un is taken as the bulk velocity based on the global mass influx. Convective boundary condition is used for describing velocity.3: Inlet turbulence: experiments and simulations Walls: No slip boundary and constant heat flux is specified at the walls.2 for the walls is the ratio of heat flux (q) to density (ρ) times isobaric heat capacity (cp ).5% Turbulent length scale ls 1. In the present case.1 × 10−3 s synthetic fluctuations Table 5. qw = ρcqp . it is important for the boundary condition to be specified in such a way that the vortices approach and pass the outflow boundary without much disturbance or reflection back into the domain. Outflow Boundary: At the outflow boundary. Table 5.
5. It is the most efficient method for meshes with high aspect ratio in a particular direction. the multigrid code is used for solving the pressure poisson equation (PPE).2)-plane is first solved exactly.1)-plane. MULTIGRID ALGORITHM IMPLEMENTATION The FAS scheme discussed in the Chapter 4 along with the FMG scheme is employed in the code. For smoothing. The third is an ‘alternating plane relaxation GS smoother’.3)-plane and each (3. The other in the subroutine ‘calcpe’ where the PPE is solved. which is best suited for an isotropic mesh.1: Flowchart of the multi-grid algorithm in CALC-BFC Each of these subroutines are described below: • MAIN: As mentioned above. One is the normal ‘symmetric point Gasuss-Siedel (GS) smoother’. For both restriction and prolongation. As mentioned earlier.1-2-3 directions are decided based on the mesh density in the different directions of the mesh. More details on it can be found in . wherein each (1. the MG code is initialized in this subroutine by 29 . The coarse grid assumes a grid spacing that is twice the fine grid in each direction. Here. one of the several methods included in the code can be used. followed by each (2. 5.1. A coarse grid cell is obtained by merging eight fine grid cells into one.1 Structure of the multigrid code Flow-chart depicting the flow of information amongst the concerned subroutines in the CALC-LES code is shown in Figure 5. MAIN CALCPE PETER INIT PETER MULTI KEY PETER RELAX PETER CYCLIC MG 2D KEY2 PETER 2D RELAX PETER 2D CYCLIC Figure 5.2.2. linear projection is used. One in the subroutine ‘main’ where it is initialized. The other one is using an ‘alternating line GS smoother’ (2D smoother). which has good smoothing properties for a stretched 2D grid. It is invoked at two regions in the solver.
The arguments passed include information about the number of cells in each direction. this subroutine helps in sorting the elements in a 3-D matrix based storage lexically into a pointer based array/list. the logical constants for cyclic conditions in each direction. that the PPE is solved. • KEY: As explained above. the pressure field. their x. the j-direction boundary condition for the current problem (part cyclic and part wall BC) is modified based on the location of the ifirst and ilast nodes.2. Subroutine ‘peter cyclic’ is called to ensure cyclic conditions on the boundaries before and after each sweep. It is here. boundary conditions at all the levels are specified. Subroutine ‘key’ is called which helps storing the variables lexically into an array/list. Following this. It should be noted that 30 .3).5. total number of sweeps desired and the logical constants for cyclic conditions in each direction. y and z coordinates.1)-cycle multigrid is performed without postsmoothing at the finest level. a ‘symmetric-point smoothing GS sweep’ is performed starting at the finest level – first a forward smooth and then a backward one. where the MG code was initialized. the first being ‘main’. Then at the same MG level. For the coarser levels. The ones in the coarser level are initialized to zero. number of nodes in the three directions. the source term at the next coarser level is computed. consummating the second step of the FAS scheme – restriction. MULTIGRID ALGORITHM IMPLEMENTATION calling ‘peter init’. this corresponds to the third step of the FAS scheme – relaxation. Cells defining the extent of the wall in i-direction (ifirst and ilast) are identified. Then the cell aspect ratios are computed in all the directions. In the end. For that. • CALCPE: This is the second instance in the CALC-LES code where the MG function is evoked. First. with the help of the subroutine ‘key’. then subroutine ‘peter multi’ is called. the source terms and the pressure field values are stored in a pointer based array at each MG level. This is done for all the MG levels. which decides the choice of the type of smoother. The input arguments include the MG grid-level and the number of nodes in each direction for that level. and the first and last cell number marking the extent of the walls in i-direction of the mesh-grid (region II in Figure 4. Next. Next. which differs at different MG grid-levels. As has been the case throughout the code. the grid is divided into required MG levels. the source term is computed for the given MG level and it consists of the residual correction from the earlier relaxation step and the multigrid source (that is nothing but the restricted source term from the previous finer level). If the variable multig is set to true. This corresponds to the first step of the FAS scheme – fine grid iterations – discussed earlier (Chapter 3). ‘peter relax’ is called to solve the PPE. to be used. explained in the beginning of this section. • PETER MULTI: The input/return arguments include the initial source term in the PPE (comprising of the velocity correction term). • PETER INIT: Based upon the restrictions described in Chapter 4. a V(1. The coefficients for the PPE at all MG levels are computed.
and the logical variable discussed earlier. Once the coarsest 3D MG level is reached.k)directions of the grid are relaxed one by one.k)-plane of the grid. It is explained in more detail in . Based upon the aspect ratios computed in ‘peter init’. First leg of the V(1. Subroutine ‘peter 2d cyclic’ is called to ensure correct boundary conditions. the logical variables related to cyclic boundaries. For example. This is followed by postsmoothing at each level leaving out the finest level. • MG 2D: This subroutine performs a 2D multigrid operation on the given plane passed as input argument and is similar to the 3D multigrid operation performed by ‘peter multi’.5. the pressure field is interpolated to the next finer levels – prolongation.1) cycle ends here. It has three subtypes. More information regarding the prolongation method can be found in . After one of these smoothing operations is performed. The third is an ‘alternating-plane GS smoother’. with smoothing performed along grid lines corresponding to that line GS. ‘X-plane’.j)-planes have cyclic boundary condition in the j-direction and some have wall or no-slip boundary condition. this subroutine performs a 2D relaxation on the planes pertaining to that MG level and returns the pressure field. while performing X-plane GS smoothing. variables denoting the overall 3D MG level and the choice of plane-smoother (X. The second type is an ‘alternating-line GS smoother’.Y or Z). But instead. This is performed by calling the subroutine ‘mg 2d’. The input/return arguments to this subroutine include the coefficients and the source terms of the equation pertaining to the concerned plane. it is initialized to zero at the coarser levels. the prefix indicating the normal direction of the chosen plane. For all the other intermediate levels. cell number defining the extent of wall in that direction for that MG level (related to ifirst and ilast). the type of solver is decided. MULTIGRID ALGORITHM IMPLEMENTATION pressure is not exactly restricted similarly as explained earlier in the steps of FAS scheme. The above procedure is followed regressively.1)-cycle 2D multigrid. It 31 . an ‘X-line GS’. ‘wallj’.2. A logical variable ‘wallj’ is defined to maintain discrepancy in passing arguments to ‘mg 2d’ regarding the kind of boundaries possessed by that particular (j. Starting at this level. In the present case. a ‘Y-line GS’ and a ‘Z-line GS’ smoother. X-plane GS smoother means that all the planes corresponding the (j. It also has three subtypes depending upon the chosen plane of smoothing. • PETER RELAX: This subroutine is invoked in ‘peter multi’ and it serves the purpose of the smoother or solver for the multigrid method. some (i. comprising of forward and backward sweeps. ‘peter cyclic’ is called to ensure cyclic boundary conditions when appropriate. it performs a V(1. until the coarsest level is reached. Restriction is not performed for the coarsest level. It uses another subroutine ‘key2’ to store the initial source term in a lexically converted pointer based array. ‘Y-plane’ and ‘Z-plane’ GS smoother. The first one is a ‘symmetric point GS smoother’.
• KEY2: This is similar to the subroutine ‘key’. It enforces periodic conditions at required boundaries after performing 2D-multigrid in ‘mg 2d’. and in addition to it. Next. 32 . Once the coarsest level in 2D multigrid is reached. the source terms are restricted to the next coarser level and the coefficients are computed at the same level. The input arguments include the variable indicating the MG level and the maximum number of nodes in each direction.2. Y or Z-plane GS) and the type of boundaries (periodic or not). and the pressure field over the 2D plane is returned to ‘peter relax’. • PETER 2D CYCLIC: It is similar in function to ‘peter cyclic’. Pressure at the coarser level is initialized to zero as in ‘peter multi’. followed by relaxation at that level by calling ‘peter 2d relax’. it performs a TDMA operation to solve the pressure field (cyclic TDMA in case of periodic boundaries). • PETER CYCLIC: This subroutine is called in ‘peter multi’ and ‘peter relax’.5. the pressure is prolongated to the next finer level. It creates pointer based storage of variables from a 2D matrix based storage. which is performed by calling the subroutine ‘peter 2d relax’ and the source term is computed at that level. MULTIGRID ALGORITHM IMPLEMENTATION starts with 2D-plane relaxation at the given level. Relaxation is not performed on the finest level though. It has the same input arguments as for ‘mg 2d’. to enforce periodic boundary conditions at the required boundaries. • PETER 2D RELAX: This subroutine performs relaxation on the 2D plane passed as input argument by ‘mg 2d’. Based upon the choice of smoother (X. the number of iteration sweeps to be performed as well.
p0 and U0 are the reference pressure and velocity. 6. ν(ρcp )/P r Nu = (6. cx is the axial chord of the guide vane. P r is the Prandtl number. As mentioned earlier. at the walls respectively. Pressure distribution over the guide vane and heat transfer at the guide vane walls are expressed by computing pressure coefficient.1 Post processing of results The simulations are run for an ample amount of time.2 Comparison with the experiments In this section. Cp is computed as: Cp = p − p0 .6 Results and Discussion 6. It is enough to ensure that a statistically steady state is achieved. 1 2 2 ρ(U0 ) (6. N u. and q is the heat flux at the walls. 7 to 10 through-flow times (cx /U0 ).1) where. before sampling is started. N u is computed as follows: h cx k q/(θw − θ∞ ) = cx . ν is the kinematic viscosity. respectively. and ρ is the density of the fluid. The averaging is then done for a time equivalent to 7 to 10 times of cx /U0 . Cp and Nusselt number. p denotes the pressure where Cp is computed. simulations are carried out using two differ33 . results from the experimental study () are compared with those from the simulations.3) where.2) (6. h and k are the convective heat transfer coefficient and thermal conductivity of the fluid respectively. at the inlet of the domain. θw and θ∞ are temperature at the wall and in the free-stream (same as at inlet) respectively.
2 x/cx Figure 6. which lies at the immediate vicinity of the leading edge on the pressure surface as reported in the experiments . The residual statistics can be found in the Appendix. Cp .  Smagorinsky. that can be attributed to the lack of pressure measurement stations in experiments as compared to simulations (grid points). The initial accelerating part on the pressure side (approximately.4 0. COMPARISON WITH THE EXPERIMENTS ent SGS modelling techniques.5 −1 0 0. It can be seen that both the models (Smagorinsky and dynamic-SGS) are in good agreement with the experimental data qualitatively. The experimental data were collected over the mid-span of the OGV. The upper curve represents the suction side.5 suction side 1 −Cp 0.5 0 pressure side −0.1) 34 . Though its exact location in simulations and experiment shows a very little discrepancy.1 depicts the coefficient of pressure. Dynamic-SGS. ‘smagorinsky’ and ‘dynamic SGS’. x/cx = 0 to 0. 6. and can be ignored.8 1 1.1: Coefficient of pressure at walls.2 0.2.6 0. Figure 6. distribution over the OGV. characterized by a low Cp and the lower one.1 Pressure distribution 1. Cp ≈ 1 represents the stagnation point. ◦ Experimental.6. characterized by a higher Cp . the pressure side.2. The x-coordinates are scaled with respect to the axial chord cx .
Both these observations can be attributed to the change in curvature and length of the trailing edge and hence.6. though the simulations show a slight higher acceleration. ◦ Experimental. that is absent in simulations. It is reported in the experimental study that this fact is corroborated by the presence of a transitional separation bubble at this point shown by the numerical analysis performed by the authors . COMPARISON WITH THE EXPERIMENTS is in very good agreement with the experiments. However. slight changes in the flow conditions downstream of it. the simulations show a higher deceleration as compared to the experiments. But then. from x/cx = 0.5 0 0. A distinct bump at x/cx = 0. Also.2: Nusselt number profile. The discrepancy at the trailing edge is again expected due to the modification of the trailing edge in the present 35 . there is deceleration in the local flow. there is a slight acceleration observed in the experiments after approximately. the local flow follows the trend shown by the experiments. approximately at x/cx = 0.5 1 x/cx Figure 6.4 on the suction surface is observed in the experiments.8.2.  Smagorinsky. no such bump or separation bubble is observed in the 1600 pressure side 1400 suction side Nu 1200 1000 800 600 400 −1 −0. Dynamic-SGS. showing a more smoother pressure profile. On the suction side.22. which marks the onset of the boundary layer transition to turbulence. x/cx = 0.1 onwards. accelerating till the throat of the cascade. Further downstream. present analysis. This can be attributed to the fact that in simulations the trailing edge is modified a little bit as discussed in Chapter 4.
As the flow accelerates on both the sides from the stagnation point.3 depicts the Cp and N u profile over the guidevane surface for these meshes. but it has more number of sample nodes than MESH-2 (twice). it can be observed that both Smagorinsky and Dynamic-SGS model predict a similar pressure distribution over the OGV. This rise is different from that observed in experiments.3). and MESH-3 has the same spanwise resolution as MESH-2. At the leading edge. the Nu number shows a gradual increase. further downstream of x/cx = 0. at around x/cx = 0. It can be seen that the pressure curve overlaps completely in all the three cases. as can be seen in figure 6.32 on the pressure side and at x/cx = 0. The x-coordinates on the pressure side are represented on the negative x-axis. Also. As in the experiments. The 36 . cannot be observed in the present simulations. Nusselt number (Nu) profile over the OGV surface is studied.36 (onset of transition predicted by experiments). Detailed analysis of the boundary layer will be done in the upcoming sections. both the magnitude and the rate of growth being too low or small as compared to the experiments. 6. Figure 6. indicating the onset of transition to turbulence. It can be seen from the figure that the profiles of Nusselt number (Nu) from simulations are not matching with the experimental data. The overall shape of the pressure coefficient profile in the simulations suggests that the local flow over the OGV is forced to remain laminar. further downstream of x/cx = −0. heat transfer analysis is performed. 6.3. the boundary layer thickness increases. since they are of very little importance to the present study. On the suction side though. starting at around x/cx = −0. Smagorinsky model constantly predicts a slightly higher value than dyanmic-SGS in this region.2 Heat transfer distribution To investigate the boundary layer transition further. The very high values of Nu are clipped in the plot. indicating that the boundary layer is still remaining laminar. further downstream. where the experiments showed a significant increase in Nu. the simulations on the other hand follow the prevailing decreasing trend in Nu and then get fairly constant towards the trailing edge. and the value of Nu drops as the boundary layer grows. This trend is not found in the simulations performed. MESH INDEPENDENCE STUDY simulations as discussed earlier. shown by a sudden increase first.32.3 Mesh independence study As mentioned in Chapter 4.45 in case of dynamic-SGS and x/cx = 0. the physical trend of transition.2. followed by a gradual decrease that testifies an expanding (turbulent) boundary layer.36 on the suction side. On the pressure side.2. Nu is very high owing to the smaller width of laminar boundary layer there. MESH2 has a better spanwise resolution than MESH-1. Thereafter.6. three types of meshes are studied (see Table 4. as reported in the experiments. Moreover.56 in Smagorinsky case. there is a sharp increase in Nu. evident of transition.
5 x/cx 0 0.025). since they have the same resolution.0125).5 1400 suction side 1 pressure side suction side 1200 Nu −Cp 0. as MESH-2.5 1 x/cx (a) Coefficient of Pressure.3.2 −1 −0. (nk = 32.6.0125). MESH-1  heat transfer profile for MESH-2 and MESH-3 are very close. Cp (b) Nusselt number.  MESH-2B (same A separate set of simulations were performed on a mesh with same number of cells as in MESH-2.4 0.5 1000 0 800 pressure side −0. It will be referred to as MESH-2B.4 0. z |max = 0. but with streamwise resolution refined in the transition zone.2 −1 −0.8 1 1.0125). But they have different spanwise domain size. which shows that the spanwise domain has no significant effect on the transition.5 1400 suction side 1 pressure side suction side 1200 Nu −Cp 0. 1600 1. z |max = 0. since MESH-1 and MESH-3 give very similar results.4 depicts the Cp and N u profile over the guidevane surface for these meshes. N u MESH-2 (nk = 64.5 1000 0 800 pressure side −0. Figure 6. Cp (b) Nusselt number.6 0. z |max = 0.5 600 −1 400 0 0. MESH INDEPENDENCE STUDY 1600 1.8 1 1. Figure 6. N u MESH-2 (nk = 64.3: ◦ Experimental.2 0. 37 .5 600 −1 400 0 0.5 1 x/cx (a) Coefficient of Pressure. It can be observed that in the simulations with MESH-2B. z |max = 0. △ MESH-3 (nk = 128.5 x/cx 0 0.4: ◦ Experimental. Figure 6. It can also be said that the spanwise resolution does not play an important role in the present simulations. refined in streamwise direction in the transition zone).2 0.6 0.
5 2 2. It shows an initial increase for a while.5 x/cx Figure 6.5 depicts the location of all the stations.5: stations for study of stresses and fluxes indicates that the non-linear interactions in the boundary layer have been triggered. In this section also. results from both Smagorinsky and dynamic-SGS model are presented. Resolved Reynolds stresses and heat fluxes are plotted at certain stations on the cascade surfaces and in the wake region downstream of trailing edge. once triggered.4 Boundary layer study It becomes imperative now to understand the flow in the boundary. A second axis is also shown most of the time depicting the y + values corresponding the normal distance. It could be possible that a very fine streamwise resolution is needed for the non-linear interactions to be sustained continuously.5 0 i1 i2 i5 −0.5 0 0. it should have continued to grow. but then drops down in the middle of the transition zone. All the variables plotted at these stations over the surface are transformed to the local coordinate system corresponding to the wall with one axis parallel to the wall in the streamwise direction (subscript ‘s’) and the other normal to the wall pointing into the domain (subscript ‘n’). 38 . But then.6. The stresses and fluxes will be plotted with respect to the normal distance from wall (∆n) scaled by the pitch of the cascade (lp ).5 1 1.4. 6.5 1 i1 i2 i3 i4 i5 i3 i4 i6 i7 i6 i7 y/lp 0. Figure 6. plotted are the maximum rms velocity and temperature profiles over the surface. so as to understand why the simulations do not comply with experiments. The momentary increase 1. Also.5 −1 −1 −0. BOUNDARY LAYER STUDY the N u profile follows the experimental curve very closely for a very short distance in the transition zone on both suction and pressure sides. All the results in this section pertain to the simulations carried out on MESH-2. The third axis (subscript ‘z’) is parallel to the spanwise direction in the global coordinate system.
5 4 4 3. the wall normal component hvn′ vn′ i of the stress is negligible very close to the wall.1 50 1.5 ∆n/lp 2 y+ ∆n/lp 0 1 2 3 80 3.5 120 5 0 0. Figure 6.6: Stresses at i1 .5 1 1 0.5 0 0.04 0 −3 0.002).5 40 40 3 80 3. the top row corresponding the pressure side and bottom.5 1 1.5 1 1.5 y+ 60 2. It can be seen in Figure 6. hvs′ vn′ i.5 1 1.02 120 5 0.5 4 4 100 4.04 0 0 Figure 6.5 20 1.6 illustrates the stress values at station i1 . negative values of νr can be seen near the walls (νtot /ν < 1). Legends: νtot /ν Resolved stresses Figure 6.5 20 1. hvn′ vn′ i.4.004).5 2 50 1.04 0 0. +wall shear stress. 39 . but contrarily the stresses seem to be more damped than in the Smagorinsky case.02 0.7: Total viscosity at i1 . hvz′ vz′ i.7 depicts the total viscosity (νtot = ν + νr ) scaled by kinematic viscosity (ν) at the points corresponding to those in the Figure 6.5 60 2. top: pressure side (x/cx = 0.004). indicating the presence of backscatter. BOUNDARY LAYER STUDY −3 smagorinsky dynamic SGS x 10 −3 0 0 0. normal stress on the pressure side in the spanwise direction.5 0. bottom: suction side (x/cx = −0. top: pressure side (x/cx = 0.4.5 1 2 100 3 2.002).5 0 0.6 that with both the models. the suction side. Legends: ◦ hvs′ vs′ i.5 0 Figure 6. This shows that this point is close to the stagnation point discussed in the previous section.5 3. On both sides.04 0 0.6. The local wall shear stress at that location is also depicted (‘+’ marker on the x-axis).5 −3 x 10 x 10 150 5 150 5 4.5 100 4.02 0.5 4. hvz′ vz′ i is largest.5 0 smagorinsky x 10 y+ 0 0 0.02 0.5 0 0.5 3 2.6. 6. bottom: suction side (x/cx = −0.5 ∆n/lp 100 y+ ∆n/lp dynamic SGS 0. In the dynamic-SGS simulation.5 1 1.
5 10 1 10 1 20 y+ 40 3 50 ∆n/lp 30 2 2.04 0.5 0 y+ 0 20 0. top: pressure side (x/cx = 0.5 times of its value near the wall at section i2 (Figure 6.5 4.5 100 4 3.06 80 1 −3 1.8: Stresses at i2 .5 1 1 20 0. Also. BOUNDARY LAYER STUDY −3 smagorinsky dynamic SGS x 10 −3 0 0.5 0 dynamic SGS 0 0.9: Total viscosity at i2 . It can be seen that hvs′ vs′ i is still very large as compared to the other components.5 2 0 Figure 6.06 0 0.5 3.04 70 5 80 0 60 4 0. This station lies in the locally accelerating part of the domain.5 ∆n/lp 80 3 y+ ∆n/lp 100 4 2 80 3 60 2.04 0.5 1. bottom: suction side (x/cx = 0.02 0.5 2 1 1.8).8 and Figure 6.126).5 2 40 40 1. It can be seen that the streamwise component of stress hvs′ vs′ i is the most dominating part and the other components are negligible.5 40 y+ 1. Legends: as in Figure 6. On the suction side.02 0. top: pressure side (x/cx = 0.02 0.6. This observation is more prominent in the dynamic-SGS simulation.9 represent the stresses and total viscosity at station i2 . bottom: suction side (x/cx = 0. Legends: as in Figure 6. In Figure 6.5 60 4 4.126).10 shows the stresses at station i3 .5 3.5 ∆n/lp smagorinsky x 10 3 50 3.5 20 1.5 0 0.9). This region of the boundary layer is probably laminar in nature.174).5 60 2.02 0. it has increased almost 1.6 1 1.174). it can be seen the boundary layer thickness has increased considerably.06 0 0.5 2 −3 x 10 x 10 120 120 5 5 4.5 70 5 0.4. as this streamwise component is much bigger than the corresponding value at section i2 even farther away 40 .5 4.5 2 1 1. probably due to the higher residual viscosity predicted by it (Figure 6.06 0 0 Figure 6.5 2 30 2.04 0.7 Figure 6.
6.5 ∆n/lp 2 y+ ∆n/lp 2 3 40 3. Figure 6.510).06 0 0.5 1. the boundary layer should have gone completely turbulent.04 0. the boundary layer is on the verge of attaining full turbulence.10: Stresses at i3 .06 0 0.02 0.2).04 0. It can be inferred that in the present simulations.5 50 4.11: Total viscosity at i3 . Now.5 50 4 4.482). This can be seen on both suction and pressure side walls.5 0 2 10 0.5 60 5 60 5 70 0 0. This will be discussed further in upcoming sections.02 0.5 30 2. the non-linear interactions responsible for distribution of stresses in different directions are absent.5 smagorinsky dynamic SGS x 10 0 0.5 ∆n/lp 3 −3 x 10 30 2.12 represents the stresses at station i4 .5 1 1 10 0.5 4 4 40 30 2 3 y+ y+ 3 2.5 50 4 4.5 0 0. the lower y + values show that the wall friction velocity has decreased from earlier. top: pressure side (x/cx = 0.7 from the wall.5 20 1. as per the experiments.06 1 −3 4 5 1 2 3 4 5 x 10 5 5 50 4. should have increased significantly. bottom: suction side (x/cx = 0.5 10 10 1 1 1. namely wall-normal (n direction) and spanwise (z direction). Legends: as in Figure 6. which is obvious as the local flow is decelerating. Legends: as in Figure 6. This implies that the other components. top: pressure side (x/cx = 0.04 0. This is supported further by the very low value of wall shear stress (a ‘+’ marker on the x-axis in the figure).02 0.02 0. Here.4.5 40 3.5 2 20 1. at this station.5 y+ 30 2. BOUNDARY LAYER STUDY −3 0 smagorinsky dynamic SGS x 10 −3 0 0 0.5 20 20 3 40 3. Moreover. That means an isotropic dis- 41 .482).5 ∆n/lp 3. bottom: suction side (x/cx = 0.6 1 2 3 4 5 1 2 3 4 5 0 Figure 6.04 70 0.06 0 0 Figure 6. according to the experiments (section 6.510).
025 0.025 2 250 ∆n/lp y+ 150 6 8 10 2 4 6 8 10 0.015 200 250 0.04 0 0. both compared to their corresponding values at station i3 (Figure 6. bottom: suction side (x/cx = 0.14.02 ∆n/lp 350 y+ 0 100 0. at station i5 .005 50 0 0.12: Stresses at i4 .01 y+ ∆n/lp 0.02 0. The pressure side too shows a similar increment. On the suction side though.2. BOUNDARY LAYER STUDY dynamic SGS smagorinsky 0 0 50 0. the boundary layer thickness has increased considerably both on suction and pressure sides.04 0 0.7 tribution of stresses in all directions except for the wall-normal component close to the wall.01 100 0.02 300 300 0.02 150 y+ 150 ∆n/lp 0. Legends: as in Figure 6.764).015 4 0.01 0. as mentioned earlier.4.764).005 0 350 300 0. Legends: as in Figure 6.02 0. Further downstream. This is not found in the present simulations. 42 .752).02 0. as emphasized by the Nusselt number plot in figure 6. The suction side clearly shows signs of growing turbulence.025 300 0.04 0 0 Figure 6. i4 . Here.02 200 0. This is almost the trailing edge of the OGV. bottom: suction side (x/cx = 0.04 0.02 0. the spanwise component hvz′ vz′ i shows a slight increase close to the wall and the wall-normal component hvn′ vn′ i little away from the wall. but very small in magnitude. hvn′ vn′ i and hvz′ vz′ i are significant. top: pressure side (x/cx = 0.015 150 0.10).6 50 2 4 6 8 10 2 4 6 8 10 0 Figure 6.752).015 200 250 0. the stresses are depicted in figure 6.13: Total viscosity at i4 . and here the geometry is different from the one in experiments.025 0.01 250 200 0. On the suction side as in the previous station.6. top: pressure side (x/cx = 0.005 dynamic SGS smagorinsky 0 0 50 0.005 100 100 0.
01 0. Legends: as in Figure 6.02 ∆n/lp 300 0. bottom: suction side (x/cx = 0. It can be seen that the disturbance is felt further away from the domain boundary at this station.14: Stresses at i5 . This can be expected since the mesh is quite coarser here.025 300 0.16 depicts stresses at i6 . ∆n on the y-axis corresponds to the distance from the corresponding domain boundaries.01 100 0.005 100 100 200 0. top: pressure side (x/cx = 0.02 0.01 200 0. the stress components are plotted in the global coordinate system as there is no wall. keeps on expanding in space.01 y+ ∆n/lp 0.02 0 0 Figure 6.025 0.6 50 2 4 6 2 4 6 0 Figure 6.18. Figure 6. it can also be seen that the SGS viscosity goes very high in this region as compared to the earlier stations. top: pressure side (x/cx = 0. The present study does not delve into the analysis of this wake region. Here.005 0.005 0 100 0.02 200 0.005 50 0 0.15: Total viscosity at i5 . 43 .976).01 0. resulting in the formation of large scale vortices.7 Stations i6 and i7 correspond to the wake region.973).015 4 0.015 150 y+ 150 ∆n/lp 0.015 250 250 0.01 0.02 0.02 300 350 0 0. Moreover.025 150 0. bottom: suction side (x/cx = 0.973).01 200 0. and only the very large scale eddies are being resolved.6. The two shear layers from the suction and pressure surface merge in this region. meaning. the wake region is expanding. BOUNDARY LAYER STUDY dynamic SGS smagorinsky 0 dynamic SGS smagorinsky 0 0 0 50 50 0.01 0. as can be seen in figure 6.4.976).015 y+ 0. which enhance the turbulence.02 0 0. Legends: as in Figure 6. This region.025 350 2 250 ∆n/lp y+ 150 6 2 4 6 300 250 0. called the wake.02 0 0.
′ ′ ′ vy vy .05 0. if they can act as sources of disturbance.272). BOUNDARY LAYER STUDY 0 0. since the SGS viscosity is very small (as predicted by both the models) in the region of interest (close to the wall).09 0 5 −5 0 5 −3 0. Summing up the analysis of Reynolds stresses in the boundary layer.02 0.05 0.09 0 5 −3 x 10 0 −5 dynamic SGS 0.04 5 10 x 10 0.08 0.08 0. whether they aid in transition from (orderly) laminar to turbulent is not certain.06 −5 ∆n/lp smagorinsky dynamic SGS ∆n/lp ∆n/lp smagorinsky 0 0 0 5 −3 x 10 Figure 6. bottom: suction side (x/cx = 1.04 0. the numerical oscillations created this way help in sustaining turbulence and not dampen out.04 0.08 0.02 0.272). Legends: as in Figure 6.01 0.06 ∆n/lp 0.272).01 −5 −3 x 10 15 20 5 10 15 20 5 10 15 20 5 10 15 20 0.01 0.07 0.08 0.03 0. vx vy . In other words. The growth in the streamwise component. predict similar behaviour. top: pressure side (x/cx = 1. since it is non-dissipative .4. side Legends: x = 1. top: pressure side (x/cx = 1. But.02 0. Both the models (Smagorinsky and dynamic-SGS). hvz vz i.07 0.7 Figure 6.07 0.01 0.03 0. whether to be attributed to the free-stream fluctuations or artificial (non-physical) fluctuations caused by the central differencing (CD) scheme has to be investigated. bottom: suction ′ ′ ◦ hvx vx i. while the same cannot be said for the wall-normal and spanwise components though. As a result.05 0.07 0.06 0. as provided 44 .05 0.17: Total viscosity at i6 . it can be said that there is a definite increase in the streamwise stress component as the flow moves downstream. The difference is not much.09 0.16: Stresses at i6 . CD scheme is generally used in LES for turbulent flows.272).02 0.6.
and the thermal flux in the same ′ ′ direction.03 0.4.05 0. P r ∂n . Most certainly. This can give an idea about the penetration of wall-normal disturbances into the boundary layer. bottom: suction side (x/cx = 1.05 0.18 0.7 by FST.14 0.19: Total viscosity at i7 .711).07 0.08 0.12 0.02 0.06 0.01 0 −2 0 2 4 −2 −3 x 10 0 2 0 0 4 −3 x 10 Figure 6.04 0.06 0.18: Stresses at i7 .07 0. they cannot. hv  n θ i. and invoke the non-linear modes is not known. Here subscripts ‘s’ and ‘n’ denote the wall-parallel 45 .06 0. bottom: suction side (x/cx = 1.01 0.08 0.09 −2 0 2 4 −2 0 2 −3 dynamic SGS 0 4 10 20 0 10 20 10 20 0 10 20 −3 x 10 x 10 0.08 0. hvn′ vs′ i. given by the Reynolds shear stress.04 0. the resolved part of turbulent momentum flux in wall normal direction. (θ  denotes the temperature) are compared along with the molecular ν ∂hθi heat flux.02 0.18 0. Legends: as in Figure 6.14 0. CD scheme is always preferable due to its non-dissipative nature.02 0.02 0.711).6.16 0.06 ∆n/lp ∆n/lp 0.08 0.16 0. BOUNDARY LAYER STUDY smagorinsky dynamic SGS 0 0.2 0.1 0.2 Momentum and thermal fluxes In this section.09 0.03 0.4. Nevertheless.16 Figure 6. Legends: as in Figure 6.1 0. top: pressure side (x/cx = 1. top: pressure side (x/cx = 1. transition is very selective to FST scales (as discussed in chapter 2) and that cannot be provided by artificial oscillations (through CD scheme or otherwise).04 ∆n/lp ∆n/lp smagorinsky 0 0.711).711). 6.12 0.04 0.
and the thermal fluxes are normalized by the local wall heat flux.5 80 −1 −3 −0.002). as a result of the opposite nature of velocity and temperature gradients at the wall.126).5 −0.5 −0.5 100 3.5 0 0.5 10 1 2 ∆n/lp 0 3.5 4.5 0 smagorinsky x 10 y+ 0 −0. bottom: suction side (x/cx = −0. A different discretization scheme for one of the variables (see beginning of Chapter 5) might also result in a difference.5 0 0.5 4 4 3. owing to numerical dissipation as a result of the particular discretization scheme.20 depicts the fluxes at the station i1 .5 0 −3 x 10 x 10 120 150 5 5 4.5 0.5 0 0 0.5 4. the turbulent flux terms mentioned will behave in the same way.5 1 1 0. They are of opposite signs if the plate is hotter than the fluid.20: Fluxes at i1 .5 y+ 80 ∆n/lp 60 3 20 1.004).5 1. On the suction side.5 ∆n/lp 2.5 70 120 5 5 −0. Figure 6.4. which in the present case is a fixed value qw throughout the surface.5 0 −1 −0. and normal directions respectively.5 2 80 3 60 2. ◦ h−vn θ i /qw .5 0.5 0 0.21: Fluxes at i2 . top: pressure side (x/cx = 0.174). ν ∂hθi ′ ′ /qw .5 100 3 y+ ∆n/lp dynamic SGS 0. bottom: suction side (x/cx = 0.6. Legends: hvs′ vn′ i /τw .5 2 40 50 1.5 0 −1 Figure 6.20. top: pressure side (x/cx = 0. the turbulent 46 . they are supposed to be equal in magnitude. BOUNDARY LAYER STUDY −3 smagorinsky dynamic SGS x 10 −3 0 0 20 1. in magnitude at least. On a flat plate with zero pressure gradient and unit turbulent Prandtl number (P rt ).5 40 2.5 40 y+ 1 3 50 3.5 0 0 Figure 6. P r ∂n 20 −0. Due to the similar nature of the non-dimensionalized momentum and thermal equations. Legends: same as in Figure 6. The momentum flux is normalized by the local wall shear stress τw .5 4 100 60 4 4.5 2 30 2.5 0 −1 −0.
5 1.5 y+ 30 2.5 20 1. Turbulent heat flux is lower at all stations inside the boundary layer as compared to the molecular/viscous one. Clearly. Legends: same as in Figure 6.482).21.510).20. top: pressure side (x/cx = 0.4) where subscripts ‘1’ and ‘2’ denote the streamwise and wall-normal directions respectively. top: pressure side (x/cx = 0. it can be seen that the wall-normal stress component decreases from station i1 to i2 (see 47 .5 30 2 20 1. the production term can be written as: P2θ = −v2′ v2′ ∂θ . it can be seen that the thermal boundary layer is getting thicker downstream.5 30 2.5 60 60 5 5 70 −3 −2 −1 0 −3 −2 −1 0 −3 −3 x 10 0 −3 −2 −1 0 5 50 4. ∂x2 (6.5 50 4 4.5 2 40 3 y+ y+ 3 ∆n/lp 3.4.5 0 −2 −3 10 0. bottom: suction side (x/cx = 0.5 ∆n/lp 2 ∆n/lp 10 1 3 40 3.23: Fluxes at i4 .20 and Figure 6. bottom: suction side (x/cx = 0.5 60 4.764).5 4 50 4 40 3.5 ∆n/lp −1 x 10 5 2.752). From the transport equation of the wall-normal heat flux ∂θ vector v2′ θ′ .5 smagorinsky dynamic SGS x 10 0 0. momentum and thermal fluxes are almost identical.5 −3 −2 −1 0 −3 −2 −1 0 0 0 Figure 6. assuming the streamwise temperature gradient ( ∂x ) to be negligible when 1 ∂θ compared to the wall-normal gradient ( ∂x2 ).22: Fluxes at i3 .5 50 4 4.20.5 20 20 2 y+ 3 40 3. This equation shows the importance of wall-normal fluctuations (vn′ vn′ ) in the generation of turbulent heat flux. from the stress plots in the previous section. BOUNDARY LAYER STUDY −3 0 smagorinsky dynamic SGS −3 x 10 0 0 0. Comparing Figure 6.5 10 1 1. Legends: same as in Figure 6. −3 −2 −1 0 −3 −2 −1 0 0 Figure 6.5 30 2.5 1 1 10 0.6.
side −vx′ vy′ /τw .01 0.25: Fluxes at i7 . The local flow is laminar here as there is no significant turbulent mixing in the boundary layer at these stations.05 0. This shows that the boundary layer is getting thicker 48 .973). At station i4 (Figure 6.01 0. Station i3 corresponds to the point where the boundary layer is about to get completely turbulent according to the experimental observations.2).06 40 3 y+ ∆n/lp 3.6 and 6.5 10 1 0. bottom: suction side (x/cx = 0.711). P r ∂y Figure 6. −3 0 smagorinsky smagorinsky dynamic SGS x 10 dynamic SGS 0 0 0.
in both dynamic-SGS and smagorinsky case. The aim is to study the evolution of the fluctuating fields over the walls. On the pressure side. It was discussed earlier that it is important for the free-stream turbulence to keep interacting with the boundary layer in order to force a ‘bypass’ transition.2). This was motivated by the fact that in ‘bypass transition’.32 and x/cx = 0. the fluctuations grow in amplitude inside the boundary layer.12) also show an increase in wall-normal and shear stress components from the earlier stations – again prominent on the suction side. It can also be seen that these peak fluctuations occur in the boundary layer. The stress plots at this station (see Figure 6. on the other hand. and that probably is due to the streamwise pressure gradient being low on this side. The flux components plotted here correspond to the global coordinates as was in case of plotting stresses in the previous section. but they cannot be sustained. the transition zone lies between x/cx = 0. Merging of the two shear layers from both pressure and suction side is evident from the two distinct peaks of different amplitudes in each of the flux terms (from pressure and suction side). where the peak value is attained. but then it decays slowly further downstream.4.56 on the pressure side and between x/cx = 0. BOUNDARY LAYER STUDY downstream. On the pressure side. the streaks are formed but unable to sustain due to the absence of forcing by FST. These can be associated to the appearance of streaks in the boundary layer. The local wall shear stress used for normalization is taken as the average of the corresponding values at the last cells of the trailing edge from both suction and pressure sides. The ∆n in y-axis is the distance from the corresponding domain boundaries. displays the maximum of TKE (kt ) over the cascade surfaces. 49 . It can be concluded that on the suction side. It might be that in the present case.26. 6.33 the peak in TKE starts to get bigger until around x/cx = 0. scaled by the pitch length (lp ). Figure 6. there is a slight increase in TKE just before the (experimentally) measured trigger point.53 on the suction side (see Figure 6. these fluctuations grow appreciably while going from laminar to turbulent phase. both the momentum and thermal fluxes are identical.24) corresponds to the point close to the trailing edge. As per the experiments.5. and the fluxes are similar to those at i4 . Plots from both smagorinsky and dynamic model are demonstrated. on the suction side. the normal velocity field (v) corresponding to the global coordinates and the temperature field (θ) are studied. Similarly. after which it drops steadily.3 Turbulent kinetic energy and rms fields In this section the maximum of turbulent kinetic energy (TKE) and rms fields over the surface is studied.25 corresponds to the second station in the wake (i7 ).36 and x/cx = 0.6. Station i5 (Figure 6. no such phenomenon is observed. Figure 6. On the pressure side. Along with the TKE (kt ).4. at about x/cx = 0. It also depicts the distance from the surface.
Figure 6.26: Peak TKE (kt ) over the surface; top: pressure side, bottom: suction side.
kt |max (right y axis); distance from wall (left y axis).
component in the global coordinates.
From the plots on the pressure side, it can be observed that the predicted peak fluctuations in the experimentally observed transition zone occur inside the boundary layer.
vrms |max (right y axis); distance from wall (left y axis).
there is no real transition in the boundary layer.
θrms |max (right y axis); distance from wall (left y axis).
remains almost constant, but it occurs further close to the boundary.
transition seem to never get triggered in the simulations.
force production in the other directions.
study the receptivity of FST generated by the synthetic isotropic turbulence generation method employed here. Two possibilities are concluded: one. The slight rise in the heat transfer showed by the Nu profile on the suction side can be explained by a similar rise in the thermal flux. But this growth is not sustained. showed a small spike locally. Then it fails to sustain the growth. It is therefore very important to study the evolution of turbulence from inflow boundary to the leading edge. then again it dropped to the initial levels and continued downstream. and second. It depends upon many factors like discretization schemes and SGS models. 7. 54 . The Nu profile followed the experimental data only about halfway through the transition zone. More about it can be understood by the nature of FST turbulence outside the boundary layer in this region. the concept of ‘shear sheltering’ is introduced. It shows probably the need of continuous forcing provided by the FST for these fluctuations to be maintained.2 Scope for future work In the literature. which proves the absence of non-linear interactions triggering transition. Studying the peak of TKE (kt ) showed that the fluctuations grow appreciably near the experimentally observed transition point. A mesh with a finer resolution just in the experimentally observed transition region showed signs of transition. The sheltering of high frequency disturbances from the free-stream by the wall shear layer is explained. it is merely the effect of numerical error arising out of the uneven mesh. But.2. there is a need for very fine streamwise resolution throughout the surface to make the continuous interaction of non-linear modes. It can also indicate a growth of streamwise streaks. at least on the suction side. First. and then move on to a more complex geometry like a guide-vane.7. A simulation with a mesh refined only in the transition region. was found to be inconclusive. and how the low frequency modes be allowed to enter the boundary layer. It will be interesting to first study a simple geometry like a flat plate. It would also be interesting to change the resolution in the streamwise direction. Another possibility is that the continuous forcing provided by the FST (high frequency components) is also absent in the present simulations. A mesh independence study showed that the spanwise resolution or domain size in the spanwise direction had no effect on the transition prediction. SCOPE FOR FUTURE WORK stantial rise in the boundary layer in the experimentally observed transition zone. The plot showing peak rms value of the normal component of velocity. One possibility in the present simulations is that the low frequency fluctuations are not present in the boundary layer. v. on the other hand. It would be interesting to see how the turbulence evolves from the inflow boundary till it reaches the leading edge.
A Script for mesh-generation !##################### !## NUMBER OF FACES ## !##################### 41 =i1 !face corresponding i-first 113 =j 1 =k 129 =i2 41 =i3 i1 i2 + 1 .=i12 i12 i3 + 1 .=i123 !face corresponding i -last !###################### !## NUMBER OF BLOCKS ## !###################### DEFMSH 3 i1 j k i2 j k i3 j k !################## !#### BLOCK 1 ##### !################## CURV2 56 .
0000000e+00 57 0.APPENDIX A.0036084e-03 0 CURV4 1 i1 1 1 3 j 2 2 305 90 0. 2 30 8. SCRIPT FOR MESH-GENERATION 1 1 1 1 1 i1 1 -2.7289594e-01 1 -1. 0.6697485e-02 0. 0.0036084e-03 .2 5e-6 .009 -2. 2e-05 5e-6 2.5330251e-01 0.7289594e-01 0 CURV2 1 1 j 1 4 j 2 5e-07 1 -2.1348296e-01 0.5330251e-01 0.3 1 -1.0036084e-03 4 CURV2 1 i1 j 1 2 i1 2 2e-05 1 -2.7289594e-01 2 -1.009 -1.6517036e-02 0.6517036e-02 0. 5e-07 90 !################## !#### BLOCK 2 ##### !################## COPYC 1 i1 1 1 3 j 2 1 1 1 3 j CURV1 2 1 1 1 1 i2 240 -1. -2.
6662408e-01 2. .0036084e-03 2. . (coordinates of the suction-side curve) . 2 358 90 CURV1 2 i2 j 1 2 i2 225 2.0000000e+00 1 225 5e-04 2e-04 0 !################## !#### BLOCK 3 ##### !################## COPYC 2 i2 1 1 3 j 3 1 1 1 3 j CURV2 58 0.6662408e-01 6. . (coordinates of the pressure-side curve) . . . . SCRIPT FOR MESH-GENERATION . -1.1348296e-01 0.4060003e-01 0.6662408e-01 4 5e-07 2.0002772e-04 0.3 1 2.0000000e+00 1 240 2e-04 5e-04 0 CURV4 2 i2 1 1 3 j 2 2 2 90 0. . . .0000000e+00 .3 5e-07 . . .4060003e-01 0.APPENDIX A. 2.
009 2.6495158e-01 0 CURV2 3 i3 1 1 3 j 2 5e-06 1 6.6662408e-01 0 -3.009 1 2.0405967e-01 0. SCRIPT FOR MESH-GENERATION 3 1 1 1 1 i3 2 2e-04 1 6. 0.APPENDIX A.4060003e-01 0. 5e-06 2. !######################## !## FILLING THE VOLUME ## !######################## FILLB 1 1 1 1 1 3 i1 j FILLB 2 1 1 1 1 3 i2 j FILLB 3 i3 1 1 2 3 i3 j !######################### !## SMOOTHING OPERATION ## !######################### GSMTHB 3 1 1 1 1 1 3 i1 j 2 1 1 1 1 3 i2 j 3 1 1 1 1 3 i3 j 59 2e-04 .6495158e-01 2 CURV2 3 i3 j 1 2 i3 2 0.5940326e-02 0.
bin 1 k 60 .bin 1 k 1 k !####################### !## MERGING OF BLOCKS ## !####################### i123 =nni j =nnj k =nnk DEFMSH 1 nni nnj nnk TDSR RETRIEVE 1 1 i1 1 j block1. SCRIPT FOR MESH-GENERATION 2 1 2 0 2 3 0 0 15 i1 1 1 1 1 1 3 1 3 1 j j i2 1 1 1 1 1 3 1 3 1 j j !################# !## SAVE BLOCKS ## !################# TDSR STORE 1 1 i1 1 j 1 k block1.bin TDSR STORE 2 1 i2 1 j block2.bin TDSR STORE 3 1 i3 1 j block3.APPENDIX A.
bin !############### !## SAVE MESH ## !############### SAVE volsol.APPENDIX A. SCRIPT FOR MESH-GENERATION TDSR RETRIEVE 1 i1 i12 1 j block2.bin RITFILE STOP 61 .bin TDSR RETRIEVE 1 i12 i123 block3.bin 1 k 1 j 1 k TDSR STORE 1 1 nni 1 nnj 1 nnk meshv5.
0.2 0 x/cx 0. 62 Smagorinsky.4 0.5 0.1: Wall y + values.5 2 2 y+ y+ Mesh resolution in wall-units 1.6 0.4 0.8 1 1.5 1.B 3 3 2.8 1 0 1.2 0.2 x/cx (a) pressure side Figure B.5 1 1 0.5 0 0 0. LEGENDS: are scaled by the axial-chord length.5 2.6 0.2 (b) suction side Dynamic-SGS. X-coordinates .
6 x/cx (a) pressure side (b) suction side Figure B. MESH RESOLUTION IN WALL-UNITS 250 200 200 150 150 x+ x+ 250 100 100 50 50 0 0 0. 63 .2 0.8 1 0 1.3: Wall z + values.APPENDIX B.4 x/cx 0.2: Wall x+ values.2 x/cx 0 0.2 x/cx (a) pressure side (b) suction side Figure B.6 0.6 0.2 0 0. LEGENDS: as in Figure B.4 0.4 0.2 0.2 0.8 1 1.4 0. 20 20 15 15 z+ 25 z+ 25 10 10 5 5 0 0 0.1.1.8 1 0 1.2 0.6 0. LEGENDS: as in Figure B.2 0.8 1 1.
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