CFD models of transitional flows

Transcription

1 Department of Engineering CFD models of transitional flows By Davide Di Pasquale Department of Engineering, University of Leicester, Leicester, LE1 7RH, UK. Thesis submitted to the University of Leicester in accordance with the requirements for the degree of Master of Philosophy Supervisor: Dr. Aldo Rona Co-supervisor: Dr. Shian Gao June

2 Abstract CFD models for transitional flows Davide Di Pasquale Department of Engineering University of Leicester In favour of the more widely investigated laminar and turbulent regimes, the study of the transitional flow regime has received a lower research effort. The important effect of laminar-turbulent transition is not included in the majority of today s engineering CFD simulations. This thesis deals with the problem of modelling a low Reynolds number zero pressure gradient boundary layers in an incompressible flow without any heattransfer effects. The zero-pressure gradient transitional boundary layer is one of the canonical shear flows important in many applications and of large theoretical interest. An overview of the more widely used approaches to model transition in Computational Fluid Dynamics (CFD), has shown the challenge of modelling transition by CFD. The approaches are compared to one another, highlighting their respective advantages and drawbacks. This work then progress to document some of the precautions that are required to interpret and use the European Research Community on Flow, Turbulence and Combustion (ERCOFTAC) dataset to calibrate CFD codes. Finally, this work implements and tests the transition model of Suzen and Huang and the laminar kinetic energy transition method by Walter and Leylek in the in-house computational fluid dynamics scheme Cosmic. Cosmic is a finite-volume in-house CFD code written in FORTRAN90. It is a multi-block Navier-Stokes solver, which uses MPI(Message Passing Interface) and is capable of handling complex geometries. The test cases are simple flat plate experiments. The test cases are used to test the predicting capabilities of two different transitions model, under various stream turbulence intensity, Reynolds number variations. After having investigated both models it is possible to state that the laminar kinetic energy method is more reliable with respect to the intermittency transport method when the flow-field is subjected to a lower free-stream turbulence intensity case, but both models have shown similar behaviour in case of higher free-stream turbulence intensity. 2

3 Acknowledgements I would like to express my profound gratitude to my supervisor Dr. A. Rona for his support and suggestions over the period of my studies. I would also like to especially thank Prof. Paul Gostelow for his invaluable help and for many interesting discussion throughout the years. I cannot forget my colleagues and friends at the University of Leicester, above all Marco Grottadaurea, Mohammed Fekry Farah El-Dosoky, Ivan Spisso, Manuele Monti, and Pietro Ghillani. Useful discussions took place in front of a lovely Italian coffee or just over my desk. I wish to thank Gian Franco Marras, Cristiano Calonacci and Nicola Varini, Cineca, Casalecchio di Reno, Italy, who provided me with useful suggestions on how porting the code in the Cineca new machine that is based on CPUs having a completely different architecture (IBM Power6) with respect to the clusters previously used (based on standard x86-64 processors) and with useful advices that made possible to improve the parallelization algorithm. This work was supported by the HPC-Europa2 Transnational Access programme, financed by the European Community - Research Infrastructure Action Structuring the European Research Area of the Seventh Framework Programme. This gave me access to the SP6 High Performance Computing facilities at Cineca. Finally, I would like to thank my family for the never ending support they rendered to me throughout my life. They gave me the opportunity to travel as well as study a subject that I was passionate about, without regrets and without questioning my choice. I would also like to thank all my friends for their cooperation during my stay here at University of Leicester. I also wish to thanks my girlfriend Daniela for her support and patient. This research project has been supported by a Marie Curie Early Stage Research Training Fellowship of the European Community s Sixth Framework Programme under contract number MEST CT The grant offered me the opportunity to network at international conferences and find new friends and colleagues among the AeroTraNet Marie Curie Early Stage Training fellows. 3

4 Contents 1. Introduction Motivation of research Background Transition Overview Different mechanisms of transition Natural transition Bypass transition Separated flow transition Wake-induced transition Reverse transition Parameters affecting transition Free stream turbulence Effect of pressure gradient Effect of surface roughness Effect of compressibility Effect of curvature Effect of heat transfer Summary remarks on transition flow physics Review of transition models for CFD Theoretical framework Transitional methods for CFD Stability theory approach Low Reynolds number turbulent closure approach The intermittency transport method with empirical correlations The laminar fluctuation energy method DNS for transition LES for transition The model The intermittency and vorticity Reynolds number approach Summary remarks on transition modelling approaches Review of experimental transition data for benchmarking CFD schemes Introduction Database integrity Summary remarks on the benchmark data for transition methods

5 4. Numerical scheme Introduction Governing equations Short-time Reynolds averaged Navier-Stokes equations Space discretization Calculation of inviscid fluxes MUSCL data reconstruction TVD scheme Entropy correction for the Roe scheme Calculation of viscous fluxes Time integration Runge-Kutta scheme Boundary Conditions Inviscid wall Non-slip wall Far-field boundary condition Symmetry plane Subsonic outflow Subsonic inflow Implementing and testing two transition models Turbulence closure without any transition model ܓ turbulence model Source term The Menter Shear Stress Transport (SST) model Turbulence closure with transition model The classical correlation method of Suzen and Huang Model implementation in the 2D version of the in-house CFD code Cosmic Test cases Zero pressure gradient transitional boundary layer results The Laminar Kinetic Energy Approach Transport equation for the turbulent kinetic energy Transport equation for the laminar kinetic energy Transport equation for the specific turbulent dissipation rate ω Model implementation in the 3D version of the in-house CFD code Cosmic Zero pressure gradient transitional boundary layer results Comparison between the two models

6 6. Code parallelization using MPI Single domain decomposition Recursive domain decomposition Input/output General strategy Input of parameters from external text file Use of external cgns file Parallelization performance Conclusion and Recommendations for future Work References

7 List of Figures 1.1 Simplified turbulent spot structure Schematic of transition process Schematic of disturbance level affecting transition, adapted from Morkovin et al. (1994) Illustration of wall-limiting concept leading to splat mechanism" for production of kl Cross-stream velocity measurements for T3A (a) and T3AM (b) Pressure vs. distance in x 1 for T3A case Pressure vs. distance in x 1 for T3AM case Free-stream velocity along the flat plate for the T3A test case Free-stream velocity along the flat plate for the T3AM test case Streamwise pressure gradient evaluated by the Thwaites method for the T3A test cases in the laminar part of the boundary layer Streamwise pressure gradient evaluated by the Thwaites method for the T3AM test cases in the laminar part of the boundary layer The interface between two adjacent cells The four-cell stencil used to build the MUSCL scheme The TVD second order scheme region The constructed control volume for the diffusive fluxes calculation The ghost cells (dashed line) around the computational domain (the solid thick line) Flow chart of the in-house code Cosmic without intermittency transport model Transition model low chart Computational mesh: For clarity, one point every 12 in both the x and y directions is shown, and the x-axis to y-axis ratio is

8 5.4 Inflow velocity profile for the T3A test case Inflow velocity profile for the T3AM test case Free stream turbulence intensity decay for test case T3A Comparison of the experimental shape factor coefficient against computational results for the T3A test case Comparison of the experimental shape factor coefficient against computational results for the T3AM test case Comparison of the experimental and skin friction coefficient against computational results for the T3AM test case Computational mesh. The total length of the test section is 1850mm; 150mm before the leading edge Comparison of the experimental skin friction coefficient against computational results for the T3A test case Comparison of the experimental skin friction coefficient against computational results for the T3AM test case Comparison of the predicted skin friction coefficient for the T3A case Comparison of the predicted skin friction coefficient for the T3AM case Variation of Reynolds number based on momentum thickness along the flat plate for T3A case Variation of Reynolds number based on momentum thickness along the flat plate for T3AM case Example of the decomposition of one block model via SDD MPI Cartesian communicators Exchanged cells between processor 0 and processor 1 in green

9 6.3 Example of the decomposition of a two blocks model via MPI Cartesian communicators. 114 List of Tables 3.1 Fluctuating velocity and their ratio at two different boundary layer thicknesses for the two test cases The values of MUSCL parameters Table of stretching factor RRD code variables SSD performance on the CINECA cluster. 121 List of abbreviations CFD Computational Fluid Dynamics CGNS CFD General Notation System DES Detached Eddy Simulations DNS Direct Numerical Simulation ERCOFTAC European Research Community on Flow, Turbulence and Combustion FST Free-Stream Turbulence HPC High Performance Computing LES Large Eddy Simulation LPT Low Pressure Turbine MPI Message Passing Interface OpenMP Open Multi-Processing PDE Partial Differential Equations RANS Reynolds-Averaged Navier-Stokes RDD Recursive domain decomposition SDD Single domain decomposition SIG Special Interest Group SMC Second-Moment-Closure T-S Tollmien-Schlichting 9

10 1. Introduction 1.1. Motivation of research Understanding, predicting and controlling laminar-to-turbulent flow transition is of great interest to engineers because of the wide range of practical applications in which transition is significant. Despite the enormous amount of research effort devoted to it, the current understanding of the problem is still far from complete. Most of the difficulty in the understanding lies in the large number of inter-linked factors that affect transition. In order to better understand this area, research in this topic is essential. While the importance of transition phenomena for aerodynamic and heat transfer simulations is widely accepted, it is difficult to include all of these effects in a single model. If the whole process can be understood much better, namely if it is possible to evaluate the influence of each parameter on the transition, it may be possible in the future to design specific system that control transition. A turbulent boundary layer flow has typically one order of magnitude higher skin friction and heat transfer than an equivalent laminar boundary layer due to increased mixing between the boundary layer and free-stream flow. The increased wall shear stresses due to turbulence result in significantly larger viscous drag on aircraft wings. It is estimated that fuel savings of up to 25% would be possible for a large commercial aircraft if laminar flow could be maintained over the wings (Thomas 1985, Saric 1994). On the other hand, separation and stall of low Reynolds number aerofoils and turbine blades can be significantly improved if the boundary layer is turbulent. Turbomachinery designers have often tended to induce transition to a turbulent boundary layer at a fixed location in order to avoid working with intermittent flow. An accurate prediction of flow transition is imperative in designing more efficient turbomachines. Losses have been observed on turbine blades working in a low static pressure environment, particularly affecting aircraft at cruise altitude or the later stages of steam turbines. Some of these reductions in performance are attributed to separation of laminar boundary layers (Suzen and Huang, 2004). The better mixing properties of a turbulent flow are an asset for designing efficient combustion systems and the increased heat transfer coefficient of a turbulent flow is desirable in some heat exchanger designs. In gas turbine engines, an accurate estimation of the thermal loads on turbine blades strongly depends on the prediction of the onset of turbulence. Wind turbines for renewable energy generation stand to benefit from this 10

11 research, as it is may be possible to improve the relative fraction of time in which a wind turbine blades work in the laminar flow regime. Specifically, off-shore aerogenerators can have a blade span of 70m. The large tangential velocity gradient along the span enables the blade to work over a large range of chord based 'local' Reynolds numbers. This implies that different transitional flows may co-exist along the blade span, with competing transitional modes changing the flow during one revolution. The opportunity for increasing the performance of these aerogenerators, in terms of lift to drag ratio, is therefore attractive, given the relative maturity of conventionally designed blades by the established RANS CFD methods. The heat-shielding requirement of hypersonic re-entry vehicles and the performance and detection of submarines and torpedoes are other applications for which predicting and controlling transition is essential. Therefore, a better understanding of transition will lead to better prediction of a wide range of wall-bounded flows with an overall improvement of the design Background Transition Overview Whenever a fluid flows over a solid body, such as the hull of a ship or an aircraft, frictional forces retard the motion of the fluid in a thin layer close to the solid body. A boundary layer is formed by the retardation of particles near to a surface, generating a cross-stream velocity gradient. The development of this layer is a major contributor to flow resistance and is of great importance in many engineering problems. The concept of a boundary layer is due to Prandtl (1904) who showed that the effects of friction within the fluid (viscosity) are significant only in a very thin layer close to the surface. Fluid flows are generally referred to as being either laminar or turbulent. Transition is a complex phenomenon, defined as the whole process of change from laminar to turbulent flow. Laminar flow is ordered and layered while turbulent flow is chaotic and not predictable. Almost all laminar flows eventually develop into turbulent flows. If the flow velocity is high enough, the flow in this layer will eventually become unordered, swirling and chaotic or simply described as being turbulent. One simple example is water streaming out from the water tap; at low velocities, the flow is laminar but at higher velocities, the flow becomes turbulent. The transition from laminar to turbulent flow state was first investigated by Reynolds (1883) who made experiments on the flow of water in glass tubes, visualizing the flow 11

12 state using ink as a passive marker. He found that the flow state was determined solely by a non-dimensional parameter that is since then called the Reynolds number, where ܮ ߩ = ߤ V is the mean velocity of the object relatively to the fluid (m/s) L is a characteristic linear dimension (m) μ is the dynamic viscosity of the fluid (kg/ms) ρ is the density of the fluid (kg/m 3 ) (1.1) The Reynolds number is a measure of the ratio between the inertial and the viscous forces in the flow, so that a high Reynolds number flow is dominated by inertial forces. Inertial forces associated with fluid mass try to amplify disturbances in the flow. Viscous forces try to damp disturbances. Depending on the flow characteristics and the fluid properties, like density, viscosity, velocity gradient, proximity to the boundaries, etc., disturbances can be damped or amplified. If these disturbances are damped the result is a laminar flow, otherwise the flow is turbulent. The origin of turbulence and the accompanying transition from a laminar to a turbulent regime is of fundamental importance for the whole science of fluid mechanics. Flow transition is an important phenomenon in many aspects of fluid flow, compared to the more widely modelled laminar and turbulent regimes; the study of this topic received comparatively less attention. Transition may be directly relevant to those physical situations where a flow is observed to change from laminar to turbulent, as it often happens, for example, on aircraft wings or past turbine blades. Transition affects strongly the evolution of losses and other factors of practical significance, such as the distributions of wall shear stress and surface heat transfer (Gostelow, J. P et al., 1989). It is accompanied by many changes in flow characteristics. Specifically, an abrupt change in the law of resistance occurs with transition and both the skin friction and heat transfer increase considerably in the turbulent flow. The most important feature of the phenomenon of transition is the increased diffusivity in the flow. Transition is a complex phenomenon. The main reason of its complexity is that, besides the simultaneous presence of turbulent and laminar flow, there is also the interaction between the two phases. To put it simply, during transition, the two phases of a laminar and turbulent flow exist together and alternate as a function of time. Furthermore, transition in boundary layers is affected by several parameters, such as the stream-wise 12

13 pressure gradient, noise, the Reynolds number, the surface roughness, the wake unsteadiness and turbulence intensity, making its prediction very difficult. Moreover, transition occurs through different mechanisms in different applications Different mechanisms of transition There are a number of different transition mechanisms documented in the literature. These depend on the turbulence level of the external flow, the pressure gradient along the laminar boundary layer, the geometrical details, and the surface roughness. Transition to turbulence in boundary layer flows may follow different routes depending on the flow situation. In all cases, where disturbances that enter the boundary layer grow in amplitude, transition may occur. The different modes of transition are natural transition, bypass transition, separatedflow transition, periodic-unsteady transition, and reverse transition. Bypass transition and periodic-unsteady transition are caused by the disturbances in the external flow, such as free stream turbulence and pressure gradient, and they are observed in majority of the turbomachinery applications. Separated-flow transition occurs in the free shear layer and may or may not involve Tollmien-Schlichting (T-S) waves. This mode of transition is observed in compressors. Reverse transition is observed in nozzles with highly accelerating flows. Natural transition The early research on transition was based on inviscid stability theory, which suggested that all boundary layer flows are only unstable if there is an inflexion point in the velocity profile. It was later predicted physically by Prandtl and then proven mathematically by Tollmien that a laminar boundary layer can be destabilized by the presence of viscous instability waves, often referred to as Tollmien-Schlichting (T-S) waves. In aerodynamic flows, transition is typically the result of flow instabilities, such as T-S waves or cross-flow instability. This type of transition begins with a weak instability in the laminar boundary layer at a critical value of the momentum thickness Reynolds number. These instabilities are one-dimensional. These weak instabilities proceed through various stages of amplifications and amplify into two-dimensional instabilities and finally into three-dimensional instabilities. These instabilities further grow and form loop vortices and then, during the growth of the waves, spanwise distortions and three- 13

14 dimensional non-linear interactions become relevant. Finally, areas of turbulence, denoted as turbulent spots, start to develop in the streamwise direction. Turbulent spots, discovered by Emmons in 1951, initiate with an irregular shape and grow in the streamwise direction during which the initial shape is preserved. The turbulent spot are the the key point of transition. The amount and size of the spots can be affected by changing the velocity of the flow or by placing disturbances in the flow. These spots grow in the streamwise direction, while the leading edge velocity of the spot is larger than the trailing edge velocity. The spots also grow in the lateral direction as they travel downstream. Due to their growth, the spots start to overlap each other and thus coalesce until a complete turbulent boundary layer is obtained and thus transition is completed. This instability becomes three-dimensional (3D) and non-linear by the formation of a vortex loop and eventually results in a non-linear breakdown to turbulence. Therefore, spot appears in a streamwise very narrow region in the boundary layer (BL). For simplification, as illustrated in figure 1.1, the turbulent spot is supposed to have a triangular shape. Fig. 1.1: Simplified turbulent spot structure. This process is often referred to as natural transition. It occurs at low free-streamturbulence (FST) level of less than 0.5%. This process of transition gradually appears after a critical value of Re is exceeded. The instability that initiates natural transition is via a subtle mechanism, whereby viscosity destabilizes the waves and they begin to grow very slowly. In a lowdisturbance environment, such as the one found in free flight and in some marine applications, quasi-two-dimensional T-S waves precede transition in a flat-plate boundary layer and similar flow fields. Physically, T-S waves are streamwise-travelling structures of spanwise-oriented vorticity. Various receptivity processes are responsible for generating these instabilities, which are probably initially three-dimensional and randomly distributed. However, quasi-two-dimensional T-S waves are preferentially 14

15 amplified in the boundary layer. Further downstream, the amplitude of the most amplified waves becomes sufficiently large for non-linear effects to become important. This process is depicted in figure 1.2 Fig 1.2: Schematic of transition process. Bypass transition Although the T-S transition scenario is now well understood, transition to turbulence does not follow this path when the initial disturbances are large. Morkovin (1969) first coined the term bypass transition for cases in which known instability mechanisms (at the time, T-S waves only) are bypassed. Transition triggered by large-amplitude surface roughness and free-stream turbulence is the prototypical bypass scenario. In turbomachinery applications, the main transition mechanism is bypass transition imposed on the boundary layer by high levels of turbulence in the free-stream (due to the combustion process). In this case, low-frequency oscillations in the streamwise velocity appear in the boundary layer. These oscillations are due to a streamwise streak of alternating high and low velocity and flow visualization studies have shown that the streaks meander slowly sideways and thereby give rise to the observed low-frequency variations. If the energy of the streamwise velocity fluctuations is measured in the boundary layer, it is found to have its maximum in the centre of the boundary layer and to exhibit an initial amplification that is linear with downstream distance, in contrast to amplified T-S waves that grow exponentially. The idea behind bypass transition is that the disturbances in the flow cause laminar fluctuations in the boundary layer that initiate spots, or that disturbances are strong enough to enter the boundary layer and initiate turbulent spots immediately. For transition at high free-stream turbulence levels of above 1%, the first and possibly the second and third stages of the natural transition 15

16 process are bypassed, such that turbulent spots are directly produced within the boundary layer by the influence of the free-stream disturbances. As the spot is born, it grows in the streamwise direction, merging with neighbouring spots, until a complete turbulent boundary layer is obtained. The occurrence of Tollmien-Schlichting waves, spanwise vorticity, and three-dimensional breakdowns is bypassed, which explains the name bypass transition. Linear stability theory is irrelevant, as the turbulent spots are generated more towards the leading edge of a plate (or turbine blade) compared to natural transition. As a means of categorizing transition scenarios, Morkovin (1969) introduced a transition roadmap. This roadmap was updated by Morkovin et al. (1994) and this updated roadmap is shown in figure 1.3. The transition process can be categorized into three main stages: receptivity, disturbance growth and breakdown. Receptivity is the means by which disturbances enter the boundary layer and provide the initial conditions for disturbance growth. Disturbances like freestream turbulence, surface roughness and sound enter the boundary layer as steady and/or unsteady perturbations. This stage is not yet well understood in all situations but is critical because it provides the initial amplitudes, phases and frequencies of the disturbances. Once disturbances enter the boundary layer, they proceed along one of the paths A-E on figure 1.3 depending not only on their initial amplitudes (as first put by Morkovin) but also, as recent thinking suggests, on other characteristics such as spatial wave numbers and temporal frequencies. They grow or decay depending on the stability characteristics of the flow and multiple types of disturbances may coexist and possibly interact once they reach large amplitudes. The final stage is breakdown to turbulence. In the case of TS waves (primary modes, path A) once the disturbances reach an amplitude near 1% of the freestream velocity, laminar flow can no longer be sustained and a more complex flow arises. This instability of the large-amplitude TS waves is referred to as a secondary instability because the primary disturbance is so large that it supports a second instability growth at frequencies that are not unstable in the original, unperturbed basic state. The secondary instability rapidly breaks down, turbulent bursts appear and these coalesce into fully developed turbulent flow. In the case of bypass transition, the disturbances break down without going through secondary instability and form turbulent bursts directly. 16

17 Fig 1.3: Schematic of disturbance level affecting transition, adapted from Morkovin et al. (1994). Separated flow transition Another important transition mechanism is separation-induced transition (Mayle, 1996), by which a laminar boundary layer separates under the influence of a pressure gradient and transition develops within the separated shear layer (which may or may not reattach) as a result of an inviscid instability mechanism. Where the flow does reattach, this reattachment forms a laminar-separation/turbulent-reattachment bubble on the surface. The bubble length depends on the transition process within the shear layer and may involve all of the stages listed in Figure 1.2 for natural transition. Because of this, it is generally accepted that the free-stream turbulence level plays a large role in determining the length of the separation bubble. Traditionally, separation bubbles have been classified as long or short, based on their effect on the pressure distribution around an aerofoil (Mayle, 1991). Short bubbles reattach shortly after separation and only have a local effect on the pressure distribution. Long bubbles can completely alter the pressure distribution around an aerofoil. Since long bubbles produce large losses and large deviations in exit flow angles, they should be avoided (Mayle, 1991). Short bubbles, on the other hand, can be used to trip the boundary layer and thus allow larger adverse pressure gradients downstream of the reattachment point. One of the major challenges lies in determining whether or not a separation bubble will be long or short. This is aggravated by the fact 17

18 that small changes in either the Reynolds number or the angle of attack of an aerofoil can cause a bubble to change from short to long (Mayle and Schulz, 1997). The sudden change in bubble length is often referred to as bursting. It can result in a significant loss of lift and could even cause the aerofoil to stall if the bubble fails to reattach. Separation induced transition can also occur around the leading edge of an aerofoil if the leading edge radius is small enough. The size of the leading edge separation bubble is a strong function of the free-stream turbulence intensity, the leading edge geometry, the angle of attack, and, to a lesser extent, of the Reynolds number (Walraevens and Cumpsty, 1995). Tain and Cumpsty (2000) found that the size of the leading edge bubble has a profound effect on the downstream boundary layer. They concluded that the larger the leading edge separation, the thicker the downstream boundary layer and thus the more likely it is to separate under an adverse pressure gradient. Consequently, compressor blade losses at off-design conditions are thought to be strongly influenced by separation-induced transition near the leading edge. Wake-induced transition Another type of transition process is called wake-induced transition or periodic unsteady transition (Kyriakides et al., 1999). This process is caused by the periodic passing of wakes from upstream aerofoils or obstructions. Transition induced by wakes or shocks, compared to the stages of transition listed in figure 1.2, appear to bypass the first stage of natural transition. The turbulent spots are formed and convect downstream. Turbulent spots immediately coalesce after their formation and immediately grow and propagate downstream to form a fully turbulent boundary layer. The wake influences the boundary layer and thus the transition process. This flow phenomenon is unsteady as, during a certain part of the blade-passing period, the flow along turbomachinery blades is the flow with the background turbulence, and during the remaining part of the period, the flow is a flow with the wakes. Therefore, transition of this type is denoted as unsteady transition. For bypass transition, the high background turbulence level drives the transition process, while for unsteady transition the turbulence in the wake has even higher intensity levels compared to the background turbulence. In turbomachines, viscous wakes from the proceeding stator or rotor blade row pass through the next rotor or stator blade row, generating unsteady pressure, surface heat transfer, and affecting the boundary layers. This is called unsteady wake/blade interaction (Fan and Lakshminarayana, 1996). 18

19 Reverse transition Transition from turbulent to laminar flow also exists and is called reverse transition or re-laminarisation. Reverse transition happens in cases of very strong acceleration as occurring near the leading edge or along the rear part of the pressure side of a Low Pressure Turbine (LPT) blade. It can also happen in flows through nozzles with strong acceleration (Mayle, 1991). This is because the acceleration on the pressure side of most aerofoils near the trailing edge, in the exit ducts of combustors, and on the suction side of turbine aerofoils near the leading edge is generally higher than that for which reverse transition occurs. Few details are known about the reverse transition process. It is assumed that streamwise vortex lines associated with the turbulence in the boundary layer become highly stretched as a result of the great acceleration and that vorticity dissipates through viscous effects. Re-laminarisation involves a balance between convection, production, and dissipation of the turbulent kinetic energy within the boundary layer Parameters affecting transition Transition is influenced by various factors, such as the free stream turbulence level, the pressure gradient, surface roughness, curvature, compressibility, heat transfer, film cooling, and acoustic disturbances. Although transition is affected by various other secondary factors, the above-mentioned factors often determine the transition process. The effect of above mentioned parameters on transition is detailed in the following section. Free stream turbulence In a low pressure gas turbine, the key parameter affecting the onset of transition and the spot production rate is the free-stream turbulence. The Reynolds number does not affect the transition process itself and has no effect on the spot production rate. It corresponds rather to a sensitization of the boundary layer to perturbations, such that the onset location can alter. Free stream turbulence increases the heat transfer rate in the turbulent boundary layer and the rate of heat transfer seems to be more sensitive to free-stream turbulence. Increasing the free-stream turbulence reduces the Reynolds number at which transition onset occurs. By increasing the free-stream turbulence, the production of turbulent spots increases and thus the transition length decreases. At higher turbulence levels, transition 19

20 occurs in a bypass mode and is completely independent of the Tollmien-Schlichting instability. Effect of pressure gradient The pressure gradient has a large effect on both the transition onset and the transition length. The general trend is that a favourable pressure gradient has a stabilizing effect on the flow, and therefore the transition onset is more downstream. The trailing edge velocity of the spots is found to be larger compared to the velocity in a zero pressure gradient flow. In combination with the observation that the leading edge velocity remains nearly unchanged, the result is that the transition length becomes larger. An adverse pressure gradient has a destabilizing effect on the flow. Therefore, the transition start is situated more upstream while the transition length decreases. Very strong adverse pressure gradients may result in boundary layer separation. The acceleration parameter is an equivalent measurement of the pressure gradient for favourable pressure gradient flows. With an increase in the acceleration parameter, the transition Reynolds number increases, indicating a delay in transition. At low freestream turbulence levels, the effect of acceleration is significant, while for flows in gas turbines where the turbulence level is high, the effect of acceleration is negligible as the onset of transition is controlled by the free-stream turbulence level. For adverse pressure gradient cases, an increase in negative acceleration decreases the transition Reynolds number, bringing an early transition onset. The effect of the turbulence level on transition is less significant for adverse pressure gradient flows than for favourable pressure gradient flows (Mayle, 1991). It is found in literature (e.g. Abu-Ghannam and Shaw [1980]) that at high free-stream turbulence levels the effect of the pressure gradient on boundary layer transition diminishes. This also supports the flat plate approach as proposed in the abstract of this thesis. The pressure gradient can be characterised in terms of the pressure gradient parameter at the onset of transition, which is defined as: ௧ ଶ ߠ = ఏ௧ ߣ ߥ ݔ (1.2) where θ t is the boundary layer momentum thickness, ν is the flow kinematic viscosity, and U is the free-stream velocity that changes with the tangential distance x along the solid boundary. 20

21 Effect of surface roughness In general, surface roughness eases the transition from laminar to turbulent flow. Surface roughness plays a dominant role on the transition process over an aerofoil. Increasing the surface roughness decreases the transition Reynolds number and thus the transition occurs more upstream on a rough surface when compared to a smooth surface aerofoil. At a high free stream turbulence level, a very rough aerofoil surface decreases the transition length by 60% when compared to a hydraulically smooth aerofoil surface. For a smaller roughness height, a smaller effect can be observed. The change in the transition Reynolds number affects the heat transfer rate at the surface. Guo et al. (1998) calculated that, for a roughness height of 25 μm, the heat transfer rate increases by up to 30%. For roughness heights from 0.8 μm to 2.3 μm, which are representative values for turbine blades, Abuaf et al. (1998) showed that surface polishing improves the aerodynamic performance and reduces the heat transfer load. However, they concluded that, on the basis of the heat load alone, the addition of the polishing steps that represent additional time and cost provide only a limited benefit. Surface roughness promotes transition by producing additional large-amplitude disturbances that require less amplification to break down into a turbulent spot. However, if the roughness elements are very small, the resulting perturbations are below the characteristic level of those generated by free-stream turbulence and the roughness will have no great impact on transition. From the works of Feindt (1972) and Mick (1987), Mayle (1991) concluded that transition usually takes place due to external disturbances and so the effect of surface roughness can be neglected for most turbomachine engine sizes. Nevertheless, the work of Bons (2002) suggests that, when turbulence and roughness are both present, synergies are generated that create larger effects on the skin friction factor, C f, and on the Stanton number, St, than by simply adding their individual effects. Effect of compressibility The majority of the flows in gas turbines are compressible. Measurements that study the effect of compressibility on bypass transition are still scarce. From the experiments available in the literature, it follows that the bypass transition mechanism remains unchanged compared to low Mach numbers. Compressibility has only a very slight influence on stability and transition at subsonic speeds. The Mach number has an effect on the onset, the production rate, and spreading angle of the turbulent spot. Aside from where transition forced by a shock wave interacting with a boundary layer, an increase 21

22 of the Mach number delays the onset of transition and decreases the spot production rate. Two models can be found in literature to account for the compressibility effects on the spot production rate: Chen & Tyson (1971) proposed a spot production rate evolution proportional to ( +. ۻૠૡ. ) (1.3) while Narasimha (1999) gave the spot production rate as proportional to ( +. ۻૡ. ). (1.4) Clark et al. (1993, 1994, 1996) showed from heat transfer measurement by means of thin film that, although the characteristics spot boundary velocities were unaffected by the Mach number, the spreading angle was varying over the Mach number range 0.24 to They noticed an important reduction of the spreading angle, by up to 30%, when the Mach number increases. However, at higher supersonic speeds, the compressibility has a more complex effect on transition. Thus, the effect of Mach number on the transition onset and on the spot production rate has to be taken into account along with the effect of shock waves. As the Mach number increases, the onset of transition is delayed and the spot production rate is decreased, thus increasing the transition length roughly by between 8% and 30%. A passing shock wave from an upstream aerofoil induces a small concentrated vortex on the pressure side of the aerofoil near the leading edge and this can cause transition as the shock moves along the surface (Mayle, 1991). Effect of curvature It is generally accepted that even a relatively small longitudinal surface or streamline curvature has a significant effect on the transport of momentum and heat in turbulent boundary layers. By a linear perturbation analysis of the conservation of momentum equation in which the viscous terms are neglected, it is possible to explain the tendency for a boundary layer to be stabilized when it develops along a convex wall and the opposite effect along concave surface (Gortler 1940). The effects of curvature on the onset of transition under low disturbances conditions are clear; a concave curvature leads to an early and more rapid transition and the opposite is true for a convex curvature. This was previously known but little documentation of the transport process in the flow was available. The effects of curvature surface on transition were studied by Görtler (1940) and Liepmann (1943). The effect of surface curvature on the stability of 22

23 a laminar boundary layer against two-dimensional disturbances is very small for the range of surface curvature that is likely to occur in practice. The Tollmien-Schlichting waves are thus expected to behave almost the same on a concave or convex surface as on a flat plate. However, the flow on a concave surface exhibits a different instability due to centrifugal pressure gradient, producing a three-dimensional system of alternating vortices with axes in the streamwise direction within the layer, as studied theoretically by Görtler. Liepman showed that transition might happen substantially earlier on a concave surface while it is only slightly delayed on a convex surface. Effect of heat transfer Heating or cooling the flow affects the transition at low levels of free stream turbulence intensity. The wall heat transfer influences the stability and transition because viscosity depends on temperature. A heated or cooled wall also heats/cools the fluid in its vicinity and thus changes the fluid viscosity. The stabilizing or destabilizing effect of the wall heat transfer is essentially due to the dependency of the dynamic viscosity μ on temperature. As shown by Schlichting (1979) in a gas flow, the heat transfer from the boundary layer to the wall stabilizes the boundary layer and leads to an increase in the transition Reynolds number. The reduced near-wall viscosity stabilizes the flow owing to the increased velocity gradient and decreased shape factor that affect the flow transition. Taking the temperature dependence of the viscosity into account, the contribution of the viscosity gradient to the curvature of the velocity at the wall of a flat plate is: ቆ ଶ ଶቇ ௪ ݕ ߤ൬ 1 = ൰ ൬ ݕ ௪ ߤ ൰ ௪ ௪ ݕ (1.5) If the wall is warmer than the gas outside the boundary layer, T w > T, the temperature gradient at the wall is negative, ( (ݕ ௪ < 0, and as the viscosity grows with increasing temperature for gases, then also ݕ ߤ ) ) ௪ < 0. Since the velocity gradient at the wall is positive, it therefore follows that ݕଶ ଶ ) ) ௪ > 0. Consequently, there is a point of inflection of the velocity profiles where the curvature vanishes, leading to an unstable boundary layer according to the point of inflection criterion. At high free stream turbulence levels, it was observed that heat transfer has a negligible effect on spot production rate and thus on the transition length (Mayle, 1991). 23

24 1.3. Summary remarks on transition flow physics Despite the amount of research effort devoted to it, the current understanding of the transition flow physics is still far from complete. Most of the difficulty in the understanding lies in the large number of inter-linked factors that affect transition. In order to better understand this area, further research in this topic is essential. While the importance of transition phenomena for aerodynamic and heat transfer simulations is widely accepted, it is difficult to include all of these effects in a single model. If the whole process can be understood much better, namely if it is possible to evaluate the influence of each parameter on the transition process, it may be possible in the future to design specific systems that control transition. Delaying transition, hence maintaining as much as possible laminar flow, it would lead to a significant reduction of drag and of other losses and an improvement in engineering component performance. On the other hand, inducing an early transition can suppress flow separation and reduce shape drag in selected engineering applications. 24

25 2. Review of transition models for CFD 2.1. Theoretical framework The theoretical framework for understanding transition is based on the stability of small perturbations in a base flow governed by the Navier-Stokes equations. Rayleigh (1880) derived simplified governing equations for the evolution of a small disturbance in a parallel flow neglecting viscous and non-linear terms. Rayleigh assumed a wave-like solution for the perturbation in both space and time of the form (2.1) (௧ (ఈ௫ ఉ௭ ఠ (ݕ) ݑ = (ݐ,ݖ,ݕ,ݔ) ᇱ ݑ By using a Fourier transform, Rayleigh reduced the simplified governing equations to an eigenvalue problem for exponentially growing or decaying disturbances. He showed that a necessary but not sufficient condition for inviscid instability is that the basic velocity profile has an inflection point. However, flows without inflection points, like boundary layers with a favourable pressure gradient, are observed to be unstable at finite Reynolds numbers. Following a procedure similar to Rayleigh s (1880), Orr (1907) and Sommerfeld (1908) included the viscous terms that resulted in the Orr- Sommerfeld (OS) equation that describes the evolution of two-dimensional velocity disturbances. Squire (1933) derived a similar equation for the wall-normal disturbance vorticity to describe three-dimensional disturbances and proved that two-dimensional (2D) disturbances grow faster than and become unstable upstream of three-dimensional (3D) disturbances. This result is known as Squire s Theorem. This result led researchers to focus on 2D disturbances for detecting the onset of transition until recently. However, 3D disturbances are also found to cause transition through the transient growth mechanism that will be explained in detail later in this chapter. The first solutions of the OS equation for a Blasius boundary layer were obtained by Tollmien (1929) and these were later refined by Schlichting (1933). Tollmien and Schlichting analytically predicted the existence of two-dimensional unstable waves that are the eigen-mode solutions that grow or decay exponentially. For a zero-pressure-gradient flat plate boundary layer, these waves are referred to as Tollmien-Schlichting (T-S) waves. The existence of T-S waves was experimentally verified by Schubauer and Skramstad (1947) once wind tunnels with a sufficiently low free-stream turbulence level were developed that allowed the detection of these waves. Although the T-S transition scenario is now well understood, transition to turbulence does not follow this path when the initial disturbances are large. 25

26 2.2. Transitional methods for CFD The modelling efforts by different research groups have resulted in a range of turbulence models that can be used in different applications, while balancing the accuracy requirements and the computational resources available to the CFD user. However, the important effect of laminar-turbulent transition is not included in the majority of today s engineering CFD simulations. The reason for this is that transition modelling does not offer the same wide spectrum of CFD-compatible model formulations that is currently available for turbulent flows, even though a large body of publications is available on the subject. There are several reasons for this unsatisfactory situation. Firstly, the transition process involves a wide range of scales, with energy and momentum transfer predominately affected by non-linear (inertial) processes between eddies of different scales and it is very sensitive to physical flow features such as pressure gradients and the free-stream turbulence level. Secondly, transition occurs through different mechanisms in different applications, such as natural transition, bypass transition, and separation-induced transition. The third complication arises from the fact that conventional Reynolds Averaged Navier-Stokes (RANS) procedures do not lend themselves easily to the description of transitional flows, where both linear and non-linear effects are relevant. RANS averaging eliminates the effects of linear disturbance growth and it is therefore difficult to apply to the transition process Stability theory approach An established transition modelling approach is based on stability theory that avoids the aforementioned RANS limitation. Stability theory is based on the study of the behaviour of small flow disturbances to see whether they grow or not. An input disturbance is assumed of the form (2.1) ( (ఈ௫ ఉ௧ (ݕ) = (ݐ,ݕ,ݔ) and any arbitrary two-dimensional disturbance is assumed to be representable in the form of equation (2.1) by expanding it in a Fourier series. If the disturbance amplitude grows, the flow is unstable and transition to turbulent flow is expected. The advantage of this approach is that the equations can be linearised, which makes this problem amenable to an analytical approach. Making use of the continuity and momentum equations for two-dimensional, incompressible, unsteady flow and neglecting quadratic terms in the disturbance velocity components results in the Orr-Sommerfeld equation 26

27 ( )( ᇱᇱ ߙ ଶ ) ᇱᇱ = (2.2) ) ସ ߙ + ᇱᇱ ଶ ߙ 2 ( ᇱᇱᇱᇱ ߙ where the primes denote differentiation with respect to the dimensionless coordinate y/δ, α is the wavelength of the disturbance, c is the wave speed, Ф is the eigenvector and R is the Reynolds number of the mean flow. The term on the left hand side are inertia terms (2 nd order), those on the right hand side are viscous terms (4 th order). The problem of stability thus reduces to an eigenvalue problem: for a given Reynolds number and wavenumber pair, equation (2.2) has eigenvalues ω k with corresponding eigenfunctions e k. The stability of each eigenmode is given by the imaginary part of ω k. If this is positive, the amplitude of the corresponding disturbance grows exponentially and if it is negative, the amplitude decays. It could be argued that it is sufficient to study the stability of each individual mode in order to determine whether the flow stays laminar or not. This is however not the case and the reason is that the eigenfunctions are non-orthogonal. Due to this fact, a sum of eigenmodes might show an initial growth even if linear stability predicts that all eigenmodes decay exponentially. This is known as the transient growth mechanism. Based on recent advances in hydrodynamic instability theory, it has been recognised that the dynamics of many wall-bounded shear flows is better described by a superimposition of normal modes rather than by a single (the least stable) mode. Even though the Laplace transform resides entirely in the stable half-plane, transient effects can cause energy amplification that may subsequently trigger non-linear saturation followed by secondary instabilities. The same viewpoint should hold for global modes, that is, the superimposition of mutually non-orthogonal global modes may result in a substantially different perturbation dynamics that the one that is predicted by the global spectrum. In addition, by means of the stability equation, a theoretical critical Reynolds number is obtained that indicates the point on the wall at which amplification of some individual disturbances begins and proceeds downstream of it. The transformation of such amplified disturbances into turbulence takes place over some finite streamwise distance. It must therefore be expected that the observed position of the point of transition will be downstream of the calculated one, in other words, the experimental critical Reynolds number exceeds its theoretical value. Because the growth is so slow, transition to turbulence might not be complete until a streamwise distance that can be as large as 20 times farther downstream from the leading edge than the initial starting position of linear instability (Durbin and Jacobs, 2002)Moreover, since the transition model is based on the linear stability theory, it cannot predict the 27

28 transition due to non-linear effects, such as high free-stream turbulence or surface roughness. Where the exponential growth of two-dimensional waves results in a finiteamplitude wave, the linear theory ceases to be valid. In fact, during the growth of the waves, spanwise distortion and three-dimensional non-linear interactions become relevant. The more widely used method based on the stability theory is the so-called e n method of Smith & Gamberoni (1956) and Van Ingen (1956). They proposed to correlate the onset of transition with the amplification rate of the most unstable wave at each position to determine the amplitude disturbance ratio ௫ ܣ (2.4) [ ]ݔ = ݔ ߙ නݔ = ܣ ௫ బ where A 0 is the initial disturbance amplitude at the first neutral stability point. e n methods are not compatible with general-purpose CFD methods as typically applied to complex geometries. The reason is that these methods require a priori knowledge of the geometry and of the grid topology. In addition, they involve numerous non-local operations, such as tracking the disturbance growth along each streamline, that are difficult to implement into today s CFD methods (Stock and Haase, 2000) and the typical industrial Navier Stokes solutions are not accurate enough to evaluate the stability equation. However, even the e n method is not free from empiricism. This is because the transition n-factor is not universal and depends on the wind tunnel and on the free-stream environment. This means that it works well for flows that are not too different from the ones used for its calibration. The main obstacle to the use of the e n model is that the required infrastructure needed to apply the model is complex. The stability analysis is typically based on velocity profiles obtained from highly resolved boundary layer codes that must be coupled to the pressure distribution of a RANS CFD code. The output of the boundary layer method is then transferred to a stability method, which then provides information back to the turbulence model in the RANS solver. The complexity of this set-up is mainly justified for special applications where the flow is designed to remain close to the stability limit for drag reduction, such as in laminar wing design. The problem is considerably more complex in 3D flows, in which streamwise and cross-flow disturbances can coexist. Moreover, when bypass transition occurs, this method does not work at all. 28

29 Low Reynolds number turbulent closure approach The second way to predict transition is by using low Reynolds number turbulence models. However, the ability of these turbulent models to predict transition is questionable. They typically suffer from a close interaction between the transition capability and the viscous sub-layer modelling and this can prevent an independent calibration of both phenomena (Savill, 1993). At best, low Reynolds number models can only be expected to simulate bypass transition, which is dominated by diffusion effects from the free-stream. This is because standard low Reynolds number models rely exclusively on the ability of the wall damping terms to capture the effects of transition. Realistically, it would be very surprising if these models that were calibrated for viscous sub-layer damping could faithfully reproduce the physics of transitional flows. It should be noted that there are several low Reynolds number models where transition prediction was considered specifically during the model calibration (Wilcox 1994, Langtry & Sjolander 2002, Walters & Leylek 2004). However, these model formulations still exhibit a close connection between the sub-layer behaviour and the transition calibration. The re-calibration of one functionality also changes the performance of the other. It is therefore not possible to introduce additional experimental information without a substantial re-formulation of the entire model. Models like the Launder- Sharma model (1978), where the near-wall behaviour is described by the turbulence Reynolds number (Re t = k 2 /νε), perform better than those that use the local wall distance. However, no model gives a reliable result for any arbitrary combination of Reynolds number, Free-Stream Turbulence (FST) level, and pressure gradient. Moreover, the model predictions are sensitive to the initial conditions, boundary conditions, and to numerical aspects such as the grid resolution and the computational domain extent. Westin and Hankes (1997) and Craft et al. (1997) improved the low Reynolds number closure approach for bypass and separation-induced transition by means of a non-linear eddy viscosity model that in general produces better results than a linear eddy viscosity model, but the non-linear eddy viscosity model is still sensitive to boundary conditions and numerical aspects. Some improvements have been obtained with the low Reynolds number Second-Moment-Closure (SMC) as used by Hanjalic et al. (1998). The benefits are located in the provision to account for the anisotropy of the free-stream and of the near-wall Reynolds stress field, particularly in the ability to reproduce the normal-to-the-wall velocity fluctuations. Another merit of this model is its exact treatment of the turbulent production and of the effects of streamline curvature. 29

30 These characteristics help also in handling other forms of non-equilibrium phenomena, such as separation and re-attachment that are frequently encountered with different forms of transition. With this model, Hanjalic (1998) was able to predict the onset of transition for several known test cases without having to use any empirical triggering, as recommended by the Special Interest Group on transition (SIG10) of ERCOFTAC. This technique has been successfully applied at high levels of FST intensity but not to flows with a FST intensity level lower than 3%. In addition, it is more complex to implement and also more computationally expensive than more empirical models (Wilcox 1994, Langtry & Sjolander 2002, Walters & Leylek 2004) The intermittency transport method with empirical correlations The third approach to predict transition, which is favoured by the turbomachinery community, is to use the concept of intermittency, as introduced by Dhawan and Narasimha (1959). This approach consists in blending together laminar and turbulent flow regimes as done by Abu-Ghannam and Shaw (1980), Mayle (1991), and Suzen & Huang (2000), based on empirical values of the critical Reynolds number. Around this critical value of Reynolds number, the flow becomes intermittent, which means that it alternates in time between being laminar and turbulent. The physical nature of this flow can be properly described with the aid of the intermittency factor γ, which is defined as the fraction of time during which the flow at a given position remains turbulent, or in other words, it is the fraction of time that the flow is turbulent during the transition phase. By letting the intermittency grow from zero to unity, the start and the evolution of transition can be imposed. This is commonly done by multiplying the eddy viscosity in a two-equation turbulence model by the intermittency factor. In other words, once γ is determined, it is multiplied by the eddy viscosity in the mean-flow equations. In the pre-transitional regime, γ is set to zero and γ assumes a positive value only where the model is required to initiate transition. Suzen & Huang (2000) developed an intermittency transport model that can produce both the experimentally observed streamwise variation of intermittency and a realistic profile in the cross-stream direction. The model combines the transport equation models of Steelant & Dick (1996) and Cho & Chung (1992). Specifically, the transport of intermittency, γ, is given by 30

31 ߛߩ (ݏ)ߚ ݑ ݑඥߩ ܥ(ܨ (1 (ߛ (1 = ߛഥߩ + ݐ ଷ ଶ ߛଵ ܥ ቌ ܨ + ഥ ݑ ଶ ߩߛ ଶ ܥ ߝ ଷ ܥ + ቍߛ ഥ ) ଵ ݑ ݑ) ߛ ߛ ߝ ଶ + ൬ቀ(1 ߪߛ(ߛ ఊ௧ ߪߛ(ߛ (1 + ߤ ఊ௧ ߤ ௧ ቁ ߛ൰ (2.3) where the modelling constant are σ γt = 1.0, C0 = 1.0; C1 = 1.6; C2 = 0.16; and C3 = This approach neglects the interaction between the turbulent and non-turbulent parts of the flow during transition. In order to capture this interaction, a conditional averaging technique leading to a set of turbulent and a set of non-turbulent equations for mass, momentum, and energy is necessary, as used by Steelant & Dick (1996). The conditional averaging is usually seen as too computationally expensive for engineering applications, as the number of equations doubles. Therefore, the intermittency concept is typically used in combination with globally averaged Navier-Stokes equations and the loss of some physical information is accepted. Despite its inability to capture the essence of the actual transition mechanism, single-point RANS turbulence closures offer more flexibility and better prospects for predicting a real flow with transition than the classical linear stability theory. Although much more limited in capturing the real physics than DNS or LES, statistical modelling is still the only viable method to compute complex flows with transition phenomena. It is worth noting that natural transition is much rarer in industrial flows than bypass and separation induced transition. The RANS intermittency statistical models typically correlate the transition momentum thickness Reynolds number to local free-stream conditions, such as the turbulence intensity and the pressure gradient. These models are relatively easy to calibrate and are often sufficiently accurate to capture the major effects of transition. In addition, correlations can be developed for the different transition mechanisms, ranging from bypass to natural transition as well as cross-flow instability or surface roughness. The main shortcoming of these models lies in their inherently non-local formulation. They typically require information on the integral thickness of the boundary layer and the state of the flow outside the boundary layer. While these models have been used successfully in special-purpose turbomachinery codes, the non-local operations involved with evaluating the boundary layer momentum thickness and with determining the free-stream conditions have precluded their implementation into general-purpose 31

32 CFD codes. Still, statistical RANS models can adequately capture the effects of transition in situations where most of the natural transition development stages are bypassed by some strong external disturbance The laminar fluctuation energy method A new and interesting class of transition models is based on the description of the laminar fluctuation energy in the pre-transitional region of a boundary layer. The pretransitional region of boundary layers subject to free-stream turbulence resembles a laminar boundary layer in terms of the mean velocity profile. As the FST level is increased, the profile becomes noticeably distorted from the typical Blasius profile, with an increase in momentum in the inner region and a decrease in the outer region, even for a FST level as low as about 1%. This shift in mean velocity profile is accompanied by the development of relatively high-amplitude streamwise fluctuations, which can reach intensities several times that of the free-stream turbulence. This process results in an increase in skin friction and heat transfer in the pre-transitional region and eventually leads to bypass transition through the breakdown of the streamwise fluctuations (Jacobs and Durbin, 2001). It is important to note that these streamwise fluctuations are not turbulence in the usual sense of that word. This distinction was made for modelling purposes by Mayle and Schulz (1997), who proposed a laminar kinetic energy k l equation to describe the development of such fluctuations upstream of transition. Structurally, these fluctuations are very different from turbulent fluctuations, since the energy is almost entirely contained in the streamwise component of the fluctuating velocity. Their dynamics is also considerably different. The familiar cascade of energy from larger to smaller scales is not present. Instead, fluctuations are amplified at certain scales determined by the boundary layer itself and remain at a relatively low wavenumber. Dissipation is therefore also expected to be relatively low, except very near the wall due to the no-slip condition. All of these considerations have led to adopt a second kinetic energy equation by Mayle and Schultz (1997) to describe these fluctuations. The growth of k l has been shown experimentally (Volino and Simon, 1997) and analytically (Leib et al., 1999) to correlate with low-frequency normal velocity fluctuations (v ) in the free-stream. The scale selectivity of the boundary layer was clearly demonstrated by Johnson and Ercan (1999), who plotted the amplification of six frequency bands in a pre-transitional boundary layer. The reasons for this selectivity and amplification are not yet completely understood. Volino (1998) considered the 32

33 possibility that the growth of k l is due to a splat mechanism, similar to the one discussed by Bradshaw (1994). It is thought that the wall redirects the normal velocity fluctuation into a streamwise component, at the same time as creating local pressure gradients in the boundary layer, leading to disturbance amplification. This mechanism is decidedly different from typical turbulence production and is adopted herein as a reasonable explanation of the development and amplification of k l. Splats are likely to occur only for eddies with a large length-scale relative to the wall distance. Therefore, the turbulent energy spectrum can be divided into wall-limited (large scales) and nonwall-limited (small scales) sections in the near-wall region (see Fig. 2.1), where the cutoff eddy size is designated by λ eff, which is the effective turbulent length scale threshold for the small scale turbulence. Fig. 2.1: Illustration of wall-limiting concept leading to splat mechanism" for production of k l. Scales smaller than λ eff interact with the mean flow as typical turbulence does and larger scales contribute to the production mechanism for k l. The laminar kinetic energy represents the magnitude of the non-turbulent streamwise fluctuations in the pretransition boundary layer. Another region of interest is the transition zone itself. Jacobs and Durbin (2001) showed that bypass transition is initiated by an instability of the upstream fluctuations, which leads to turbulent spot development and progression to full turbulence. It is not clear what initiates the instability. In the k l model, a local transition parameter is implemented that depends on the specific turbulent kinetic energy, the effective length scale, and on the fluid viscosity, based in part on measurements by Andersson et al. (1999). Once this parameter reaches a certain value, transition is assumed to begin, which results in a 33

34 transfer of energy from the streamwise fluctuations k l to the turbulent fluctuations k t. This is accompanied by a change in the length scale of the turbulence, as it would occur in an actual spot breakdown process. Downstream of transition, the model predicts a fully turbulent boundary layer. Almost all of the fluctuation energy is turbulent but a small amount of k l is still present within the viscous sub-layer. This agrees qualitatively with experimental observations indicating the presence of streamwise-oriented streaky structures in the viscous sub-layer and in the buffer region that bear a resemblance to those in the pre-transitional region. Although bypass transition is recognised to occur for a FST level greater than 1%, the downstream location of transition is in fact shortened by turbulence intensities greater than about 0.1% (Schlichting, 1979).This suggests that there is a mixed transition regime involving elements of both natural and bypass transition. In order to include natural transition and mixed mode transition into the k l model, modifications must be made to both the k l production terms and the transition production term that governs the transfer of energy between the streamwise fluctuations and the kinetic energy of turbulence. These modifications do not depend directly on turbulence quantities but depend instead on the local mean flow and the laminar kinetic energy k l. Recent examples of such models were formulated by Walters & Leylek (2004) and Laurdeau et al. (2004). A one-equation system is used to describe the non-turbulent fluctuations prior to transition, ݑߩ൫ ത ߩ൫ +൯ ݐ ഥ൯= ߩ ߩ ௧ ߩ ܦߩ + ߤ) ത ) (2.4) Where is the production of laminar kinetic energy by large-scale turbulent fluctuations, ܦ is near-wall dissipation that arises from the no-slip condition on laminar fluctuations, R represents the averaged effect of the breakdown of streamwise fluctuations into turbulence during bypass transition. The breakdown to turbulence due to instabilities is included as a separate natural transition production term ௧. This equation lacks the usual shear-stress/strain related generation term, but it contains a source term that is argued to arise from the pressure-diffusion correlation. Thus, equation (2.4) returns, on calibration, the requisite rise in the fluctuation energy level in the laminar regime, despite the absence of a shear stress k l production term, which is presumed to be zero. Information from this system is used to start and let grow the turbulent kinetic energy in a conventional two-equation k-ω RANS model. The model is based on an eddy viscosity coefficient, determined by using three transport equations for the turbulent kinetic energy k, the laminar kinetic energy k l, and the specific 34

35 turbulent kinetic energy dissipation rate ω. The model automatically predicts the onset of transition without any intervention from the user and is based strictly on local variables; therefore it does not require the evaluation of any integral parameter. The principle is physically sound, but the technique is still too new to allow a judgement on its quality. It has not been extensively validated except for a few flat plate test cases and a turbine blade case (Walters & Leylek 2004, 2005, Walters & Cokljat 2008). However, the initial results from this model are promising and indicate that the model appears to have the correct sensitivity to the free-stream turbulence level. It remains to be seen how accurately this model can predict the effects of pressure gradient and separation on transition, particularly at a low FST level, below < 1%. A recent Large Eddy Simulation (LES) performed by Lardeau and Leschziner (2007) shed some light on the validity of the assumption underlying the RANS closure for the kinetic energy fluctuation level observed upstream of the transition onset. This simulation has shown that, from a statistical point of view, shear-stress/strain-induced production is mainly responsible for the elevation of the pre-transitional laminar fluctuation energy, a process that is akin to that observed in the turbulent state, although here it is mainly confined to the upper part of the boundary layer. Indeed the ratio <uv>/k is quite high over a significant portion of the pre-transitional boundary layer, that contradicts the base assumption of the model, that is the shear stress term in the production of k l is zero. This indicates that further studies should be done on this model to improve its representation of pre-transitional flow DNS for transition Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) are suitable tools to predict transition (e.g. Durbin and Jacobs, 2002), although the proper specification of the external disturbance level and structure at the computational domain boundaries poses substantial challenges. In principle, laminar flow breakdown, the development of turbulent spots, and transition to fully turbulent flow can be simulated very accurately using DNS. A DNS computation is performed by solving the full timedependent Navier-Stokes equations. Since there is no Reynolds averaging, then there is no requirement for turbulence closure by a turbulence model. In order to capture the small scales of turbulence, a DNS computation requires a very fine computational mesh. Unfortunately, these methods are far too costly for typical engineering applications. For instance, a DNS simulation by Zheng et al. (1998) of a flat plate transitional boundary 35

36 layer used approximately 50 million grid points and was performed in about four weeks on a parallel computer with 64 processors. Due to its large computational requirements, DNS is clearly not yet at the stage where it can serve as a practical tool for engineering design. This will be the case for a long time, simulations of industrial flows usually involve larger and more complex geometries that require even higher computing resources. DNS simulations are currently used mainly as research tools and as a substitute for controlled experiments LES for transition Because of the significant computational costs associated with DNS, a number of researchers have applied the concept of Large Eddy Simulation to transitional flow. In LES computations, only the large scale eddies are resolved, while the small scale eddies are modelled using an eddy viscosity approach such as that proposed by Smagorinsky (1963). One of the main problems with LES is that the predicted transition location is very sensitive to the choice of the Smagorinsky constant that is used to calibrate the sub-grid eddy viscosity (Germano et al., 1991). Germano et al. (1991) have since proposed the dynamic sub-grid-scale model to compute the Smagorinsky constant locally. The dynamic model has the advantage that, in laminar boundary layers, the subgrid eddy viscosity is automatically reduced to zero. Consequently, it is believed that this model should be more appropriate for predicting transitional flows. Nevertheless, the dynamic LES model is not a complete solution to the issues associated with applying LES to predict transitional flows. LES computations performed by Michelassi et al. (2003) on a low-pressure turbine blade with periodically impinging wakes have indicated that, while the dynamic LES model was in good qualitative agreement with DNS results, noticeable differences were observed in the quantitative comparison of results from the two simulations The തതത model A Large Eddy Simulation (LES) for bypass transitional flow (Yang et al., 1994) suggested that v, the turbulence fluctuation in the wall-normal direction, plays an important role in the transition process. A wall-normal velocity disturbance slowly increases from close to the wall to regions of higher velocity. This motivated the use of the ߥ തതത ଶ model (Durbin, 1995), without the inclusion of γ, for transitional flows. The ݒ തതത ଶ model consists of three transport equations for the turbulent kinetic energy k, the 36

37 specific turbulent kinetic energy dissipation rate ω, and the flow-normal component of the kinetic energy ߥ ଶ ത along the streamlines. In addition, the model includes a Helmholtz type equation for a quantity f which models the pressure-strain term. The turbulent velocity and time scale are calculated from the standard k-ε equations. The ߥ ଶ ത transport equation is where തതത ݐ ߥ +ߥ൬ + 2 ߝതതത ݒ = 2 തതത ݒ + 2 ݒ ݐ ݒ ൰ തതത 2 ൨ (2.7) ߪ = ଶଶ ߝ ଶଶ + ଶݒ തതത (2.8) ߝ represents the redistribution of turbulent kinetic energy from the streamwise component of the velocity fluctuations. The asymptotic behaviour of ଶଶ and ߝ ଶଶ near a wall are: ଶଶ = 2 ଶݒ തതത ଶݒ തതത = 4 ଶଶ ߝ ;ߝ The Boussinesq approximation is used for the stress-strain relation. The ߥ തതത ଶ is solved separately from the mean flow and the k and ε transport equations. Results have shown that, with this model, the onset of transition at a low FST level is predicted early and that the peak of the streamwise fluctuation u is under-predicted. This could be due to the use of the Boussinesq stress-strain relation, in which the Reynolds stresses are assumed of the same shape as the viscous stresses. The results suggest that the dissipation rate transport equation for ε and, possibly, the elliptic relaxation equation for f, which is responsible for the energy distribution among the Reynolds stress components, require further calibration, particularly in the transition region. The introduction of an intermittency factor γ into the model is likely to increase the sensitivity of the model to a number of flow features, such as pressure gradients and the FST level, depending on how γ is coupled with the transport equations for k, ε, ߥ- and ଶ ത The intermittency and vorticity Reynolds number approach Based on the success of the intermittency concept in predicting transitional flows, a novel approach to avoid the need for non-local information in correlation-based models was introduced by Menter et al. (2002). In this formulation, only local information is used to activate the production term in the intermittency transport equation. The link between the correlation and the intermittency transport equation is achieved by using the vorticity Reynolds number Re ν (Van-Driest and Blumer, 1963). Since the vorticity Reynolds number depends only on the local density, viscosity, wall distance, and 37

38 vorticity of the flow, it is a local property and can be easily computed at each grid point. The model is based on two transport equations. One is the intermittency transport equation for γ. The intermittency function γ is coupled with the SST (Shear Stress Transport) k-ω turbulence model (Menter, 1992), in which γ is used to turn on the production term of the turbulent kinetic energy downstream of the transition point in the boundary layer. The second one is a transport equation formulated to avoid additional non-local operations introduced by the quantities used in experimental correlations. These correlations are typically based on free-stream values, like the turbulence intensity or the pressure gradient outside the boundary layer. The additional transport equation is formulated in terms of the transition onset Reynolds number Re θt, which is function of the boundary layer momentum thickness θ. Outside the boundary layer, the transported variable is forced to follow the value of Re θt provided by an experimental correlation. This information is then diffused into the boundary layer by a standard diffusion term. By this mechanism, the strong variations of the turbulence intensity and the pressure gradient in the free-stream that are typically observed in industrial flows can be taken into account. This transport equation essentially takes a non-local empirical correlation and transforms it into a local quantity, which can then be compared to the local vorticity Reynolds number in order to determine where in the flow the transition criteria are satisfied. At every location in the flow where the vorticity Reynolds number exceeds the local transition Reynolds number, a source term in the intermittency equation is activated and turbulent kinetic energy is produced. This is the central mechanism by which the transition model operates. To predict transition in separated flows, a separated flow modification is applied to the model (Menter et al., 2006) Summary remarks on transition modelling approaches There are a number of transition models that are either available or are under development. Testing and validating transition models is as important as developing the models themselves. Transition models can be calibrated for a particular test case. By doing so, the model gives good results for that particular case and may not perform well for other cases. What it is really required is a transition model that gives good results for a wide range of flows and that can be used for the design of efficient turbines for gas turbine engines. To this end, it would be useful to test the existing transition models 38

39 over the same adequately wide range of flows, relevant to the turbomachinery community. Closer inspection shows that many of the current transition models are not fully CFD compatible. Specifically, most formulations suffer from non-local operations that cannot be carried out (with reasonable effort) in general-purpose CFD codes. This is because modern CFD codes use mixed elements and this does not provide the platform for computing integral boundary layer parameters or allowing the integration of quantities along the direction of external streamlines. Even where structured boundary layer grids are used, like hexahedra, the code often retains the data structure of an unstructured mesh, as for instance, the Rolls Royce CFD code Hydra (Lapworth, L., 2009). The information in the body-normal grid direction is therefore not easily available. In addition, most industrial CFD simulations are carried out on parallel computers using a domain decomposition approach. This means, in the most general case, that boundary layers can be split and distributed across different processors, prohibiting any search or integration algorithm. Consequently, the main requirements for a fully CFD-compatible transition model are: 1. Allow the calibrated prediction of the onset and the length of transition; 2. Allow the inclusion of different transition mechanisms; 3. Be formulated locally (no search or line-integration operations); 4. Avoid multiple solutions (same solution for initially laminar or turbulent boundary layer); 5. Not affect the underlying turbulence model in the fully turbulent regime; 6. Allow a robust integration down to the wall with similar convergence as the underlying turbulence model; 7. Be formulated independent from the coordinate system; 8. Be applicable to three-dimensional boundary layers. In order for a transition model to be useful for industrial predictions of transitional flows, it must be accurate and robust and it must be based on a local formulation that is applicable to unstructured and massively parallelized CFD codes. The transition model of Menter satisfies most of the requirements for a fully CFD compatible transition model, but it is not coordinate-system independent. This is a consequence of the fact that transition correlations are based on non-galilean invariant parameters, such as the turbulence intensity, which is defined based on the local freestream velocity. This is because the empirical correlations require the free-stream 39

40 velocity in order to determine the turbulence intensity level. For stationary or rotating reference frames, this is not a problem as long as the relative velocity is used to compute the free-stream turbulence intensity. However, when moving walls are present, like sliding walls, deforming walls, or walls rotating at a different speed compared to the rotating reference frame, as in the casing a turbine rotor half stage, then the freestream velocity relative to the wall will be in error. Galilean invariance is an important criterion for general turbulence models and future work should therefore focus on improving this aspect of the model. The main limitation of the Menter transition model right now is thought to be the accuracy of the empirical correlations, in which the physics of transition is entirely contained. As more experimental data become available, which can be used to calibrate the empirical correlations, the model accuracy should also improve. With this in mind, DNS results are becoming more and more important as numerical test cases for transition because they eliminate a lot of the uncertainty that is present in transition experimental data. The proposed transport equations do not attempt to model the physics of the transition process unlike turbulence models, but form a framework for the implementation of correlation-based models into general-purpose CFD methods. The other promising development route in modelling transition is the laminar kinetic energy approach proposed by Mayle & Schultz (1997). This correlationfree technique predicts the onset of the transition process and is based on local variables, but so far it has not been extensively validated. However, the preliminary results are good and indicate that the model appears to have the correct sensitivity to the free-stream turbulence level. Still, it is an open question as to whether this model can be extended to predict additional effects such as surface roughness or free-stream largeamplitude flow disturbances. Still, it is expected that the concept of laminar-kineticenergy will be a very active area of transition research in the near future. To date, none of the transition models have been shown to satisfy all the requirements stated previously and there is clearly a need in industry for an accurate and robust transition model, based on local state variables. Despite its complexity, transition should not be viewed as outside the range of RANS methods. In many applications, transition is constrained to a narrow area of the flow due to geometric features, pressure gradients, and/or flow separation. Even relatively simple models can capture these effects with sufficient engineering accuracy. The challenge to a proper engineering transition model is therefore mainly in the formulation of a model that can be implemented into a general RANS environment. 40

41 3. Review of experimental transition data for benchmarking CFD schemes 3.1. Introduction Turbulence model developers rely on established databases for benchmarking advances in turbulence closure methods for Computational Fluid Dynamics (CFD). The European Research Community on Flow, Turbulence and Combustion (ERCOFTAC) provides databases for benchmarking different flow regimes. In particular, sub-group 5 of the Special Interest group of Ercoftac (SIG10) on Transition Modelling focuses on experimental real-flow data to provide a sufficiently robust experimental database to validate the predictive capabilities of transition codes. With any experimental data, there is a legitimate expectation for the database user to apply some judgment to overcome acceptable accuracy and consistency issues in the dataset. This chapter aims to document some of the precautions that are required to interpret and use the ERCOFTAC dataset to calibrate CFD codes. This chapter aims to demonstrate the use of the ERCOFTAC database to validate a two-dimensional CFD transitional solver, giving more details of the approach adopted. It aims to demonstrate how to use the ERCOFTAC experimental data to generate the inflow condition and, in doing so, to inform the community on the consistency of the information in this dataset by testing it for the known trends in the mean velocity components and their statistical fluctuations. The objectives are to expose some limitations of the velocity measurement data from two boundary layer test cases under a zero streamwise pressure gradient (Roach, P.E. & Brierley, 1990) Database integrity The ERCOFTAC database is a valuable tool for the transitional flow community. Still, in the T3A and T3AM datasets, some discrepancies were found from what is stated in the description of the experiment and in the database information, specifically with respect to the normal velocity component, the turbulence isotropy, and the pressure gradient. The test cases chosen to provide initial profiles and to validate the results of the transitional flow solver of chapter 5 are the ERCOFTAC Transitional Flow Benchmark cases T3AM and T3A of Roach & Brierley (1990), featuring hot-wire traverses of a 1.7m long flat plate transitional two-dimensional isothermal boundary layer with zero pressure gradient. These experiments were especially designed to test the ability of turbulence models to predict the effect of freestream turbulence on the 41

42 development and subsequent transition of laminar boundary layer under zero pressure gradient conditions. The wind tunnel geometry was identical to the T3A, and T3AM cases. The case T3AM has a free-stream velocity of 19.8 m/s and a Free Stream Turbulence (FST) level of 0.9%. The test case T3A has a free-stream velocity of 5.1 m/s and a higher FST level of 3.3%. The different freestream turbulence levels were imposed by inserting different turbulence grids into the wind tunnel test section upstream of the flat plate. T3AM and T3A are classified in the database as the low and moderate FST intensity cases respectively. In the ERCOFTAC data of cases T3A and T3AM, the sign of the cross-stream velocity u 2 is negative, as shown in figures 3.1(a) and 3.1(b) respectively. However, u 2 should be positive definite, as the displacement thickness grows monotonically along the flat plate, giving a negative u 1 / x 1 and a positive u 2 / x 2 so that the incompressible continuity equation (3.1) is satisfied. ݑ ݔ = 0, = 1, 2 (3.1) a) u 2 [m/s] 14 x 2 [mm] b) x 2 [mm] u 2 [m/s] Fig. 3.1: Cross-stream velocity measurements for T3A (a) and T3AM (b). 2 42

43 Moreover, the first data points closest to the wall in figures 3.1(a) and 3.1(b) are suspect; the velocity magnitude above these points decreases monotonically, whereas at these points the trend reverses. It is only possible to assume that the odd behaviour of u 2 near the wall is associated with measurement problems of the hot-wire anemometer when this was very close to the flat plate surface. Therefore, they should not be inserted in the dataset. The wall-normal velocity data was re-interpreted invoking the principle of conservation of mass and using the streamwise velocity profile. It was concluded that, by reversing the sign of the wall-normal velocity and removing suspect measurements close to the wall, this field can be made to be physically consistent with the measured streamwise velocity profile. As far as the free-stream turbulence is concerned, it is possible to verify that the turbulence is not isotropic, contrary to what is stated in the companion documentation of the database. The companion documentation reports that the generated turbulence is extremely homogeneous and isotropic, with a streamwise-to-normal fluctuating velocity ratio of about obtained by means of turbulence generating grids. But looking at table 3.1 that reports the data taken for the database away from the wall, it is noticeable that the ratio of streamwise-to-normal fluctuating velocity is quite far from ratio of 1.005, based on the Cross-Wire (CW) anemometry data. X. [mm] δ[mm] Y [mm] u [m/s] v [m/s] w [m/s] u /v [m/s] u /w [m/s] T3A T3AM Table 3.1: Fluctuating velocity and their ratio at two different boundary layer thicknesses for the two test cases. Table 3.1 shows that the inflow Reynolds stresses have an important statistical anisotropy that may lead to poor agreement with predictions from turbulence models that use the Boussinesq approximation and lead to an improved agreement with numerical results from using an explicit Reynolds stress model. The next item that was examined is the pressure gradient, which should be dp/dx 1 = 0 for the two test cases. In the wind tunnel where the measurements were taken, the bottom wall can be inclined to produce a zero pressure gradient on the test surface, 43

44 which is hung from the ceiling of the working section. Figure 3.2 and 3.3 show the streamwise pressure distribution along the flat plate, for the test case T3A and T3AM respectively, using the experimental values of density and temperature and applying simply the equation of state for the ideal gas p = ρrt. Pressure [Pa] x 1 [mm] Fig. 3.2: Pressure vs. distance in x 1 for T3A case Pressure [Pa] x 1 [mm] Fig. 3.3: Pressure vs. distance in x 1 for T3AM case. Figures 3.2 and 3.3 show that there is a noticeable streamwise variation in static pressure in both test cases. By application of the principle of linear momentum conservation at the top of a steady boundary layer, the free-stream velocity and pressure distributions are related by ஶ ݑ ஶ ߩ ஶ ݑ ଵ ݔ = ଵ ݔ (3.2) 44

45 therefore a non-zero pressure gradient boundary layer is expected to exhibit a nonconstant streamwise velocity profile. In fact, the experimental data show a non-constant value of the free-stream velocity u 0 along the flat plate for both test cases, as shown in figures 3.4 and 3.5. This supports the thesis of a non-zero streamwise pressure gradient being present during the tests u E E E E E E E+05 Re x Fig. 3.4: Free-stream velocity along the flat plate for the T3A test case u E E E E E E+06 Re x Fig. 3.5: Free-stream velocity along the flat plate for the T3AM test case. The streamwise pressure gradient was also examined over the predominantly laminar portion of the boundary layer using the integral method of Thwaites. This is an 45

46 empirical relationship, based on the observation that a laminar boundary layer obeys the following relationship: hence ଶ ߠ ݑ ݑ ଶߠ = (3.3) ଵ ݔ ߥ ଵ ݔ ߥ ߥ 0.45 = ଶ ߠ ݑ ௫ න ݑ ହ ݔ) ଵ ݔ ( ଵ (3.4) where u e is the tangential velocity above the boundary layer. The above equation may be analytically integrated to find the momentum thickness θ. After θ is found, the following relations are used to compute the shape factor H = δ * /θ (3.5) 0.1 ߣ 0 ଶ for ߣ ߣ = ܪ = ܪ ߣ (3.6) 0 ߣ 0.1 for where ߠ) = ߣ ଶ ݑ )(ߥ ݔ ଵ ) is the dimensionless pressure gradient and δ * is the boundary layer displacement thickness dp/dx Fig. 3.6: Streamwise pressure gradient evaluated by the Thwaites method for the T3A test cases in the x 1 laminar part of the boundary layer. 46

47 dp/dx x 1 Fig. 3.7: Streamwise pressure gradient evaluated by the Thwaites method for the T3AM test cases in the laminar part of the boundary layer. This regression displayed a departure from the design test conditions of a zero streamwise pressure gradient. This departure may be due to the flat plate leading edge effect propagating some distance downstream Summary remarks on the benchmark data for transition methods The ERCOFTAC database remains an established and valuable tool for advancing transition model research. The data from the two test cases T3A and T3AM have to be used with care, so the practitioners should be aware of the issues encountered. Perhaps an extension of the explanatory notes accompanying the database should be done. This investigation helps turbulence model developers in their validation activities by showing how the data can be re-interpreted in a way that is consistent with the boundary layer governing equations and the known physics of these near-wall flows. 47

48 4. Numerical scheme 4.1. Introduction Air flows are governed by second-order partial differential equations (Navier-Stokes) that represent the conservation laws for the mass, momentum (Newton s 2 nd law), and energy (first law of thermodynamics). The Navier-Stokes equations are a system of nonlinear Partial Differential Equations (PDE), hence no general closed form solution exits to date for complex flows. Computational Fluid Dynamics (CFD) is the method of replacing such PDE systems by a set of algebraic equations that are solved using digital computers. CFD is a widely used tool for predicting internal and external flows. CFD enables the user to model flows with complex physics and complex geometries and gives an insight into flow patterns that are difficult, expensive or impossible to study using traditional experimental techniques. CFD is a powerful technique to predict how a flow develops with time. To date, CFD is unable to replace experimentation completely, but the amount of experimentation and the overall cost can be significantly reduced by the use of CFD Governing equations The governing equations of a continuous Newtonian fluid flow are the Navier-Stokes equations. These partial differential equations represent the conservation laws of physics and can be used to describe the state of a system. The equation generated by applying the conservation of mass to a system is called the continuity equation. The second equation derived from applying Newton s second law is called the conservation of momentum. The last one is the conservation of energy, which represents the application of the first law of thermodynamics. The unsteady compressible threedimensional Navier-Stokes equations can be written in Cartesian tensor form as follows: ρ t + (4.1) 0 = (ρ ) ݐ (4.2) 0 = ( + ߩ) + (ߩ) ( + ) = hߩ + ߩ ݐ (4.3) The variables u,,, h, and are the velocity vector, viscous stress tensor, the specific total energy, the specific stagnation enthalpy, and the conduction heat flux 48

49 vector. These can be expressed in terms of the magnitude and gradient of the velocity vector and of temperature as follows: (4.4) ൰ 2 + ൬ ߤ = 3 where ߤ is the molecular viscosity estimated from Sutherlands law for air and ଷ ଶ = ߤ ( ) + = + ௩ = 2 2 (4.5) (4.6) ߩ + h = (4.7) ߤ = (4.8) ݎ where, ௩, and Pr are the gas constant pressure specific heat, the constant volume specific heat, and the Prandtl number respectively. To complete a closed set of equations, the static pressure is estimated, assuming a calorically perfect gas, as: where is the specific ideal gas constant (4.9) ߩ = = ௩ (4.10) From equations (4.6) and (4.9), the pressure can be related to the specific total energy and the velocity vector as:. = ߛ where ቂ ߩ( 1 ߛ) = 2 ቃ (4.11) The Navier-Stokes equations can be written in compact conservative form as follows: = 0 [() ௩ + () ] + ݐ (4.12) Equation 4.12 contains the conservative variables vector U, the inviscid flux vector F c and viscous flux vector F v, which are defined as: 49

50 ߩ = ߩ = ߩ + ߩ = ௩ 0 (4.13) + (ߩ + )ߩ ߩ 4.3. Short-time Reynolds averaged Navier-Stokes equations In case of laminar flow, the governing equations are closed by specifying the algebraic relations for the viscosity and the thermal conductivity coefficient as functions of pressure and temperature. In case of turbulent flow, most CFD codes do not solve the instantaneous governing equations directly due to limitations in RAM capacity and processor time. Instead, the flow variables, varying with time, are divided into their short-time mean and fluctuating components. The principle of this technique is called short-time Reynolds-averaging and applies to a given vector variable as follows: The mean component ഥ is defined as න ݐ = 1 ഥ (4.14) ᇱ + ഥ = ௧ ( ଵ) ௧ ݐ (ݐ,ݖ,ݕ,ݔ) (4.15) where and areݐ the time level and the time step in the CFD computation and the time interval ݐ chosen to be long enough with respect to the fluctuations of the turbulent flow and short compared to the time of variations not related to turbulence. By applying this technique to the Navier-Stokes equations, the short-time Reynolds averaged Navier-Stokes equations can be written as follows: ߩ (4.16) 0 = ( ഥ ߩ) + ݐ = 0 തതതതതതതത) ᇱ ᇱ ߩ + ത + ഥ ഥ ߩ) + ( ഥ ߩ) ݐ ഥ൫hത ߩ + ത൯ ത൯+ ൫ ߩ + ݐ = ൫ ഥ ߩ തതതതതത ᇱ h ᇱ + ത ഥ ߩ തതതതതതതത ᇱ ᇱ ഥ൯ (4.17) (4.18) The short-time averaged turbulent kinetic energy,, is defined as: 50

51 (4.19) ᇱ ᇱ = 1 2 The form of the continuity equation has not changed after averaging, but the momentum equation has an additional term, തതതതതതതതߩ ᇱ ᇱ which is the Reynolds stress tensor ഥ. This term represents the influence of turbulence on the momentum equations and depends on unknown fluctuating velocity components. It is far from being a constant fluid property, such the molecular viscosity, and its magnitude and shape depend on the flow pattern. Therefore, the short-time Reynolds averaged Navier-Stokes equations with the equation of state become an open set of equations and it is necessary to develop a turbulence closure model. The purpose of this model is to replace the Reynolds stress ത with an equation related to the mean flow variables. Using the Boussinesq relationship, the Reynolds stress ത can be written by analogy with the viscous stress as: (4.20) ߩ 2 3 ഥ൰ 2 3 ഥ + ൬ ഥ ௧ ߤ ᇱ = തതതതതതതതߩ ᇱ = ത where μ ୲ is the eddy viscosity estimated from a turbulence model, as defined in chapter 5. Becauseത has the same form as the viscous stress tensor, this allows to write an effective viscosity as: (4.21) ௧ ߤ + ߤ = ߤ Short-time Reynolds averaging introduces a new term in energy equation due to the influence of turbulence, which is the turbulent heat flux vector ߩ തതതതതത ᇱ h ᇱ. The turbulent enthalpy transport by turbulent motion ߩ തതതതതതത h is modelled as being proportional to the short-time averaged temperature gradient, following Wilcox (1993). In the short-time Reynolds averaged Navier-Stokes equations, this gives the turbulent heat flux vector, ௧ which is modelled by: ௧ ߤ = ௧ = h തതതതതതത ߩ ത (4.22) ௧ ݎ where ݎ ௧ is the turbulent Prandtl number ݎ ௧ = ఓ conductivity. and k t is the fluid thermal 51

52 4.4. Space discretization The physical domain is discretized on a structured non-orthogonal multi-block bodyfitted mesh. There are several techniques available to discretize the governing equations, such as Finite-Difference Methods (FDM), Finite-Element Methods (FEM), and Finite-Volume Methods (FVM). The FDM are certainly the oldest methods employed for the numerical solutions of partial differential equations. They are essentially based upon the properties of the Taylor series expansion, and consist of the direct applications of numerical derivative to the differential operator. The FDM methods need structured grids; the physical domain has to be regularized via a mapping procedure into a series of rectangular regions. Finite difference formulas of any order of accuracy can be built providing that a sufficiently large support exists. However, the bandwidth of the algebraic system of equations, which ultimately has to be inverted to obtain the solution, is related to the number of points involved in the support. Hence, increasing the accuracy of the formula will unavoidably lead to larger difficulties in the solution procedure. The accuracy of FDM is strongly dependent to the grid qualities; abrupt changes in the mesh size or excessing skewing imply loss of accuracy. (Hoffman, 1982) Historically, the finite element method was developed for applications in structural and solid mechanics. The FEM use a physical domain subdivided in cells (the elements), which constitute the grid. The elements can have any shape and the grid does not need to be structured. Furthermore, the solution of the discrete problem has an a priori determined character. The order of the function in the nodal points is predetermined in the sense that it can be linear, quadratic or of higher orders. The representation of the solution therefore depends upon the type of element as well as upon the geometrical representation of the domain (Zienkiewicz, 1977). The fundamental approach of the FEM is to develop local element equations on each element, based on an optimization technique to minimize the error of the solution, and to subsequently patch all of the element equations together into a global system of equations that results in a system of linear algebraic equations. In recent years the FEM have been put into a stronger and rigorous formal framework, which has led to precise definitions of accuracy and existence criteria. This is the case for most linear operator in structural mechanics, which can be always replaced by an equivalent variational formulation. A variational principle consists in expressing the problem under consideration as the minimum of a functional. In fluid mechanics, a variational formulation is in general not possible, and 52

53 this explains, to some extent, the delay in assessment of the FEM with respect to aerodynamics applications. The FVM method is used in this work for its ability to handle near discontinuous flow features. This enables to extend the transitional flow model implemented in this work to compressible flows, in future work. The integration of the governing equations (4.12) over any arbitrary control volume gives: න U ݐ + න [F c + F v ] = 0 (4.23) By using the Gauss divergence theorem, making the assumption that the control volume V is time-invariant to bring the time derivative outside the first term, equation (4.23) can be rewritten as: න U ݐ (4.24) 0 = v + ර F c + ර F where is the inward normal unit vector to the closed surface bounding the control volume V. The circular integral in equation (4.24) is taken anticlockwise positive. Let = 1 න U (4.25) where V i is a topologically rectangular control volume and the subscript i indicates the i th control volume in the non-uniform mesh. Then ೞ ර F c = ۴,,, ଵ (4.26) ೞ ර F v = ۴ ௩,,, ଵ (4.27) where N faces is the number of faces in the control volume V i, S i,k is the k th face of V i and n i, k is its inwards normal. Equation (4.24) can be written in a compact form as: (4.28) 0 = + R ݐ 53

54 where U i is the mean value of the conservative variables vector averaged over the cell volume V i and R i is the residual generated from the discretized terms and it is given by the sum of the two terms: ೞ R = ۴,,, + ଵ ೞ ۴ ௩,,, ଵ (4.29) By combining eq. (4.28) and eq. (4.29), the finite-volume approximation to the flow governing equations is (4.30) 0 = + (F c k ) + (F v k ) ݐ ଵ ଵ The interpretation of equation (4.30) states that the rate of change of volume-averaged conservative variables vector in the control volume is equal to the summation of the area-averaged convective and diffusive fluxes through the k discrete boundary faces. Equation (4.30) shows also that the spatial discretization and time integration are independent and this represents one of the advantages of the finite-volume method. To solve the finite-volume approximation of the Navier-Stokes equations, the first term in the residual operator R i in equation (4.29) needs to be linearized. Specifically, the flux vector F c has to be linearized with respect to U i. The Godunov method, or Flux Difference Splitting, is used. Interface fluxes normal to the finite-volume unit cell boundaries are estimated by a method based on the approximate Riemann solver of Roe (1981). The Roe (1981) approximate Riemann solver is first-order accurate in space, since the solution is projected on each cell as a piecewise constant state (Hirsch, 1990). To reduce the excessive artificial dissipation of this first-order method, Van Leer et al. (1987) replaced the piecewise constant state assumption with a quadratic reconstruction, leading to a higher order spatial reconstruction, the Monotone Upwind Scheme for Conservation Laws (MUSCL) interpolation. Following Manna (1992), the coefficients in the reconstruction are chosen to give a third-order accurate reconstruction of the spatial gradients in regions of smooth flow. This reconstruction uses four contiguous cells in the direction of the reconstruction, thus, to connect two computational blocks, a minimum of two layers of ghost cells are required to make the flow solver block indipendent. Sweby (1984) proved that, to achieve a numerically stable scheme, this needs to be Total Variation Diminishing (TVD). As a monotonic scheme is TVD, then 54

55 the MinMod limiter is introduced to achieve a monotonic behaviour in regions of model flow discontinuities. At the computational domain boundaries, a frame of one ghost cell deep is used to preserve the second-order accurate reconstruction in the domain interior. To discretize the viscous fluxes, an estimate of the velocity vector gradients is required. To compute this, a staggered grid is built across the cell interfaces where these gradients are estimated. The flow state at the surface boundary of the new control volume and its normals are obtained from the mesh geometry and then the velocity vector gradient is estimated using the Gauss divergence theorem Calculation of inviscid fluxes The Flux Difference Splitting (FDS) method calculates the fluxes at each cell interface by determining an approximate solution to a Riemann problem in the direction of each independent computational coordinate (i, j, k). The scheme is built based on Roe s approximate Riemann solver, which is a popular approximate Riemann solver among the CFD community. Roe s scheme provides a method of calculating the convective fluxes across a face of a control volume using the eigensystem of a Jacobian matrix, A. Since the Reynolds averaged Navier-Stokes equations are non-linear, and Roe s scheme is based on a linear one-dimensional formulation, the equations are linearized through the Jacobian matrix A. For a multi-dimensional problem, the convective fluxes are calculated in each independent spatial direction using the 1-D method, solving the Riemann problem across each cell interface along the interface normal. direction By reference to equation (4.12), the Jacobian matrix A can be defined as follows: () A = F c(u) (4.31). U Let U L and U R be the conservative variables vectors (i.e. the flow states) to the left and to the right of any cell interface,. Roe s approximate Riemann solver replaces the Jacobian matrix A(U) by a constant square matrix A(U L, U R ) that satisfies the following properties: The matrix A(U L, U R ) maintains the hyperbolic nature of the system, and has real eigenvalues and a complete set of eigenvectors. The matrix A(U L, U R ) is consistent with the exact Jacobian, A (i.e. as U L U R U, then, A(U L, U R ) A). 55

56 For any U L and U R, F(U L ) F(U R ) = A(U L U R ) (i.e. the conservation of the state properties is maintained). Let the eigenvectors of A(U L, U R ) be Ƹ and the corresponding eigenvalues መߣ be. By projecting the difference in the flow states onto the eigenvectors, the following equations can be written. U R U L = ߙ (4.32) ଵ F R F L = ߙ መߣ ଵ (4.33) where ߙ is the wave strength of the ௧ wave of wave shape and characteristic speed መߣ and m is the number of the eigenvalues of A which equals the rank of this square matrix. By considering the interface between two cells at + ଵ ଶ as shown in fig. 4-1, the fluxes at the interface are computed by the summation over the negative and positive wave speeds, starting from either the right or the left flow state: F + ଵ ଶ F + ଵ ଶ ఒ ஹ = F R ߙ መߣ ଵ ఒ ஸ = F L + ߙ መߣ ଵ (4.34) (4.35) By taking the arithmetic mean of equations (4.34) and (4.35), a first-order estimate of the interface fluxes is obtained as F + ଵ ଶ = 1 2 (F L + F R ) ߙ หߣመ ห ଵ (4.36) The Riemann problem at every cell interface, is solved in the interface-normal direction, as defined in fig The unit vectors are denoted by,, and which represent the unit normal to the interface surface, the first tangential unit vector, and the second tangential unit vector to the interface surface respectively. The ortho-normal unit vectors must satisfy the following cross=product, Manna (1992): (4.37) = and =, = 56

57 The velocity components in the ortho-normal directions are: = ݑ = ௧భ ݑ = ௧మ ݑ (4.38) R [i+1,j,k] - ሬ x 2 x 1 L [i,j,k] x 3 Fig. 4.1: The interface between two adjacent cells. In the present study, Roe s approximate Riemann solver is used to calculate the convective fluxes for the three-dimensional short-time Reynolds averaged Navier- Stokes equations. The eigenvalues of the approximate Jacobian matrix are obtained by solving the linear system of equations ห መߣ ۷ห= 0 where ۷is the identity matrix. These are given in El-Dosoky (2009) MUSCL data reconstruction In the finite-volume approximation, the continuous change in flow state in the physical domain is replaced by an assembly of control volumes with a constant flow state volume average in each volume and a discontinuous step change in flow state across the volume average boundaries,. A zeroth-order approach in interpreting the flow state inside each control volume is to take the flow state as uniform and equal to the finitevolume average. The evolution in time of the flow is obtained by solving a Riemann problem across all k boundaries, of. The Roe approximate Riemann solver used to estimate the convective fluxes at the cell interface is first-order space accurate because the solution is projected on each cell as a piecewise constant state, Hirsch (1990). Using 57

58 of this low-order spatial discretization leads to excessive diffusion. Following Van Leer (1979), second-order spatial accuracy or higher can be achieved in regions of smooth flow by using more upwind points and replacing the piecewise constant by a linear or quadratic reconstruction of the conservative variables distribution in each cell. This method is known as the Monotone Upwind Scheme for Conservation Laws (MUSCL) approach. In the present study, up to a third-order spatial accuracy is achieved in the estimation of the flow variables at cell interfaces by using a four-cell stencil, as shown in fig భ మ ۺ భ మ భ మ ۺ భ మ i i-1 i i+2 i+2 i-1,j,k i,j,k i+1,j,k i+2,j,k ଵ Fig. 4.2: The four-cell stencil used to build the MUSCL scheme. Based on a simple Taylor expansion, any scalar variable in can be interpolated at the cell interface + ଵ with an accuracy up to third-order on a uniform mesh. The cell ଶ interface interpolated variables at left and right of the location + ଵ are defined as: ଶ ۺ ଵ ଶ ଵ ଶ (ߟ (1 ߝ + 4 = ଵ ଶ (ߟ (1 ߝ 4 ଵ = ଷ ଶ ଵ (ߟ + (1 + ൨ (4.39) ଶ ଵ (ߟ + (1 + ൨ (4.40) ଶ 58

59 where, the volume averaged flow variable difference, is defined as: ଵ (4.41) ଵ = ଶ ଵ (4.42) ଵ = ଶ ଷ ଶ (4.43) ଵ ଶ = The parameter ߝ is the switch between the first-order and the higher spatial discretization accuracy, if = ߝ 0, the piecewise constant (first-order) interpolation is recovered, if = ߝ 1, the higher spatial discretization order is obtained. The other parameter ߟ specifies the order of the MUSCL scheme. Table 4.1 details the scheme type that is obtained using different values of,ߟ Lee (2006). For the present study, the values of ε and ߟ are 1 and 1/3 respectively. The scheme ߟ ߝ 0 N/A First order piece-wise constant scheme -1 Second order fully upwind biased scheme 1 0 Second order upwind biased scheme 1/2 QUICK method of Leonard 1/3 3 rd order asymmetric biased scheme 1 Three point central difference scheme Table 4.1 The values of MUSCL parameters TVD scheme Although the upwind schemes appear to appropriately account for the flow physics more than central difference schemes, the numerical results show that the higher order spatial discretization schemes exhibit instabilities in regions of rapidly changing flow, such as across shock waves. These instabilities derive from a lack of monotonicity preservation in the scheme. To preserve the numerical stability, a total variation concept is applied. The total variation of the conservative variables vector is given by: = න ( ) ஶ ஶ ฬ ݔฬ ݔ (4.44) where the integration extends over the full physical domain. The total variation for the discrete case is 59

60 ) ) = ଵ (4.45) where n is the time level and i denotes the i th finite-volume cell in the computational domain. The numerical scheme is called to be total variation diminishing (TVD) if ). ) ) ଵ ) Harten (1983) proved that: 1. All monotone schemes are TVD. 2. All TVD schemes are monotonicity preserving. To construct a high resolution upwind TVD scheme, a limiter is applied to the dependent variables or fluxes. By applying a non-linear limiter on the MUSCL approach, the fluxes at cell interface become: ۺ ଵ ଶ ଵ (ߟ (1 ߝ + 4 = ଶ ଵ ଶ ۺ ଵ (ߟ + (1 + ଶ ଵ ଶ ൨ (4.46) ଵ ଶ ۺ ଷ (ߟ (1 ߝ 4 ଵ = ଶ ଷ ଶ ଵ (ߟ + (1 + ଶ ଵ ଶ ൨ (4.47) where is the limiting function, which is dependent on the gradient slope asݎ follows: ଵ = ൬ ଵ ݎ) ൰= ) ۺ ଶ ଵ ۺ ଵ = ൬ ଵ ൰= ൬ ൰ ۺ ݎ ଵ ଶ ۺ ଷ = ൬ ଵ ݎ) ൰= ) ଶ ଶ ଵ (4.48) ଵ = ൬ ଶ ଵ ൰= ൬ ൰ ݎ ଵ ଶ The limiting functions (4.48) should be selected to satisfy the following TVD condition, Sweby (1984): = 0 (ݎ) [ݎ 1,2 ] (ݎ) ݎ [ݎ [2, (ݎ) 1 0 ݎ 1 ݎ < 0 (4.49) > 1 ݎ 60

61 that represents the confined area (the area between the solid and the dashed lines) in fig There are several limiter functions used by the CFD community. The most common limiter functions applied to the Roe method are the MinMod limiter, and the SuperBee limiter (4.50) [(ݎ max[0,min(1, = (ݎ minmod(1, = (ݎ) Φ (4.51) [(ݎ, min(2,(ݎmax[0,min(1,2 = (ݎ) Φ In the present study, the minmod limiter function, which represents the lower bound (the dashed line) of the TVD region shown in fig. 4-3, is used to calculate the cell interface fluxes. The formulations for the cell interface fluxes, after applying the minmod limiter function, defined in Manna (1992), are: ധധധധ ଵ (ߟ (1 ߝ + = ଵ ۺ ଶ 4 ଶ (4.52) ଵ൨ ധധധധ (ߟ + (1 + ଶ where ധധധധ ଷ (ߟ (1 ߝ ଵ = ଵ ଶ 4 ଶ (4.53) ଵ൨ തതതത (ߟ + (1 + ଶ ധധധധ ଵ ଶ = ଵ ଵ ଶ ଶ = minmod ൬ ଵ ଶ ଵ൰, ଶ ധധധധ ଵ ଶ ധധധധ ଷ ଶ = ۺ ଵ ଵ ଶ ଶ = ۺ ଷ ଷ ଶ ଶ = minmod ൬ ଵ ଶ = minmod ൬ ଷ ଶ ଵ൰, ଶ ଵ൰, ଶ (4.54) തതതത ଵ ଶ = ଵ ଵ ଶ ଶ = minmod ൬ ଵ ଶ ଷ൰, (4.55) [(.() ݏ, ) min max[0,.() ݏ = (,) minmod ଶ f (r ) r Fig. 4.3: The TVD second order scheme region. 61

62 4.8. Entropy correction for the Roe scheme The Roe scheme has a very low amount of diffusion on oblique grids and it is considered a non-diffusive scheme for grid aligned flows, Kermani and Plett [2001]. So, the scheme can exhibit non-physical solutions such as expansion shocks. To avoid these, a positive entropy gradient condition must be satisfied. A lot of effort has been devoted toward the solution of the entropy violation inherent in the Roe scheme. A numerical solution to this problem is given by Harten and Hyman [1983]. They noted that the entropy violation is due to the vanishing numerical viscosity in the Roe scheme. Therefore, they replaced the small values of numerical viscosity with larger values through the following formulation: መଶߣ + ଶ ߝ መ ߣ ߝ < ߣ ߝ 2 መ൯൧ߣ R ߣ൫ L ൯, ߣ ൫ߣመ = max 0, ߝ (4.56) Kermani and Plett [2001] modified the above equation by enlarging the applicable band over which the entropy fix formulation is applied. The modified formulation is: መଶߣ + ଶ ߝ መߣ ߝ < ߣ ߝ 2 (4.57) )] L ߣ R ߣ) max[0, = 2.0 ߝ This modification was made to prevent the occurrence of expansion shocks in the vicinity of a sonic expansion and is the entropy fix used in the Roe scheme. 62

63 4.9. Calculation of viscous fluxes j E F i k D C i+1,j+1,k i+2,j+1,k i-1,j+1,k i,j+1,k i+1,j,k i+2,j,k i-1,j,k i,j,k B i+1,j-1,k i+2,j-1,k i-1,j-1,k A i,j-1,k Fig. 4.4: The constructed control volume for the diffusive fluxes calculation. The calculation of the convective fluxes in the Reynolds averaged Navier Stokes equations is done on a ( ǡ ǡ ) structured topologically orthogonal mesh that discretises the computational domain. The conservative flow variables are defined as the finite- 63

64 volume averages in each volume V ୧ at the cell centre. The convective fluxes calculation is carried out over the six faces S, of each control volume bounded by solid lines in figure 4.4. In the case of the diffusive fluxes calculation, a staggered control volume is generated across cell interfaces, where the viscous fluxes need to be estimated. The new control volume is shown as shaded and bounded by dash lines in figure By considering for example the interface + ଵ between the cells i,j,k and i+1,j,k, the new ଶ control volume ABCDEFGH has eight vertices located at the mid positions on the four faces of the two cells containing the interface + ଵ. The coordinates of theses vertices ଶ are calculated from the coordinates of the original cell vertices. After calculating the coordinates of the vertices of the new control volume, the volume, the surface areas, and the normal to its faces are calculated. Next, the primitive flow variables along each face are estimated by ଵ,, ଵ ൯ ݑ +,, ଵ ݑ + ଵ,, ݑ +,, ݑ൫ = 1 4 ଵ,, ݑ = ଵ,, ଵ ൯ ݑ +,, ଵ ݑ + ଵ,, ݑ +,, ݑ൫ = 1 4 = u ୧,୨,୩ = 1 4 ൫u ୧,୨,୩ + u ୧ ଵ,୨,୩ + u ୧,୨ ଵ,୩ + u ୧ ଵ,୨ ଵ,୩ ൯ (4.58) = 1 4 ൫u ୧,୨,୩ + u ୧ ଵ,୨,୩ + u ୧,୨ ଵ,୩ + u ୧ ଵ,୨ ଵ,୩ ൯ The other variables are evaluated similarly. The gradients of the flow variables are calculated by applying the Gauss divergence theorem on the generated control volume as follows: (4.59) = 1 ර The diffusive fluxes, which are represented by the third term in equation (4.30), can be estimated by substituting these gradients into the viscous flux vector equation, which is defined in (4.13). 64

65 4.10. Time integration The majority of the numerical schemes for the approximate solution of the Euler equations and of the Navier-Stokes equations apply the method of lines, i.e., a separate discretization in space and in time. This approach offers the greatest flexibility, since different levels of approximation can be easily selected for the convective and the diffusive fluxes, as well as for the time integration, as required by the problem being solved. The explicit numerical scheme used in the present study first estimates the finitevolume fluxes of either the Euler or the short-time Reynolds averaged Navier-Stokes equation and then advances the flow solution in time. The governing equations (4.30) can be written in the following compact form by replacing the discretized spatial differential terms by R. t + R = 0 (4.60) where R denotes the residual vector generated from the summation of the discretized spatial differential term Runge-Kutta scheme The numerical method represented by equation (4.60) is a set of first order ordinary differential equations that can be advanced in time using an explicit time integration scheme. An explicit multi-stage time stepping scheme is used because it is computationally cheap and requires a small amount of computer memory. The explicit multi-stage Runge-Kutta time stepping scheme integrates equation (4.60) in time and preserves the properties of the TVD scheme. This scheme is: = 1, = ݐ = ߙ ଵ R (4.61) ଵ = where and denote the number of stages of Runge-Kutta scheme and the time level respectively. The weighing coefficients ߙ ଵ to ߙ are defined according to: 65

66 = ଵ ߙ 1 + 1, = 1, 2,, (4.62) The scheme does not use any under-relaxation factors to enhance the computational stability, but the stability of the scheme in the computation is achieved by the use of the two-steps in the Runge-Kutta integration. (Cockburn B. and Shu C.W., 1998). The results are obtained using an, m=2 two-step Runge-Kutta scheme in ߙwhich ଵ = 0.5 ߙ and = 1, which gives a second-order accurate time integration, as reported by Bennett (2005). The stability of the explicit scheme is restricted by the time step. To reduce the computational cost, the local time stepping technique to accelerate the solution of the governing equations was implemented. The explicit Runge-Kutta scheme remains stable only up to a certain value of the time step Δt. To be stable, a time-stepping scheme has to fulfil the Courant-Friedrichs-Lewy (CFL) condition, which states that the domain of dependence of the numerical method has to include the domain of dependence of the partial differential equation. The CFL condition for the Runge-Kutta explicit scheme dictates that the time step should be equal to or smaller than the time required to transport information across the stencil of the spatial discretization scheme. Hence, in a one-dimensional problem, the condition for the time step would read for the linear advection equation ݔ ߪ =ݐ Λ (4.63) where ݔΔ Λ represents the time necessary to propagate information over the cell size Δx with the velocity Λ. In a one-dimensional problem governed by Euler equations, the velocity Λ corresponds to the maximum eigenvalue of the convective flux Jacobian. The positive coefficient σ denotes the CFL number. The magnitude of the CFL number depends on the type and the parameters of the time-stepping scheme, as well as on the form of the spatial discretization scheme. Local time stepping accelerates the convergence by calculating Δݐat each cell i based on the local numerical stability limit. The expression for the local time step is, Blazek (2001): ܮܨܥ ݐ ൫l ௫ + l ௬ + l ௭ ൯+ ൫l ௫ఔ + l ௬ఔ + l ௭ఔ ൯ (4.64) 66

67 V is the volume of cell i, and l i ௫, l ௬, and l ௭ are the spectral radii of the convective flux Jacobian matrix. The spectral radii for three dimensional structured grids are calculated as follow: l ௫ = ݑ ) + ) ௫ (4.65) ௬ l ௬ = ݒ ) + ) (4.66) l ௭ = ݓ ) + ) ௭ (4.67) where is the local speed of sound, and ௫, ௬, and ௭ are the projected areas of the cell in,ݔ the,ݕ and, directionsݖ is a constant that has been set to 4.0, and l ௫ఔ, l ௬ఔ, and l ௭ఔ are the spectral radii of the viscous flux Jacobians which calculated from: l ௫ఔ = ൬ݔ 4 ߤ ൰൬ ߛߩ, ߩ 3 ௧ ߤ + ൰ ( ௫ ) ଶ ௧ ݎ ݎ (4.68) and similarly for l ௬ఔ and l ௭ఔ. This method allows the solution to advance at each cell using its maximum ݐ, instead of using whichݐ is equal to the minimum ݐ calculated all over the control volumes. However, if a time-accurate solution is required, mustݐ be fixed and taken equal to the minimum ݐ over all the control volumes to maintain numerical accuracy Boundary Conditions Boundary conditions have a crucial role in the numerical accuracy of any CFD scheme. Their role differenciates one case study from another, in flows that are governed by the same Navier-Stokes equations. Therefore, the ill-treatment of any boundary condition has a negative effect on the scheme stability and convergence speed and leads to an inaccurate solution. The boundary conditions are imposed using an exterior frame one cell deep all around the physical domain, as shown in figure 4.5. The boundary condition values are function of the second interior cell variables and of the imposed flow state on the exterior ghost cell. This function is specified according to the physical boundary condition defined at the interface between the first interior cell and the ghost cell. In this study, many types of boundary condition are used in the numerical solution of both Euler and RANS equations, which are inflow, outflow, solid wall, far-field, symmetry, periodic boundary, and inter-block boundary. 67

68 Fig. 4.5: The ghost cells (dashed line) around the computational domain (the solid thick line) Inviscid wall The inviscid wall boundary condition models a flow slipping over a surface. This is also known as the non-penetration boundary condition. Due to this condition, the near- wall velocity vector must be tangent to the surface. In other words, the wall-normal velocity component must vanish. This corresponds to u b n = 0 at the boundary S between the first interior cell and the ghost cell, where n is the inward normal vector to S. This is numerically achieved by imposing: (4.69) ௧ ( ) = ݑ (4.70) ௧ ( ݑ) 2 = where the subscript I denotes the flow state at the first interior cell and b the flow state at the ghost cell. The imposed pressure gradient is equal to zero by setting డ = 0. Accordingly, the pressure at the boundary cell is put equal to the pressure at the first interior cell. Also, the temperature is set equal to the first interior cell temperature by assuming an adiabatic wall from which డ = 0. Since the pressure and temperature of the boundary డ cell are equal to the respective first interior cell values, the density is taken equal to the. ߩ = state first interior cell value, by the equation of Non-slip wall For viscous flows, the fluid at the solid surface has a zero relative velocity. The non-slip condition corresponds to u = 0 at the boundary S i,k between the first interior cell and its ghost cell. This is numerically achieved by imposing u b డ = u int to impose a zero velocity vector at the solid surface. By imposing a zero pressure gradient normal to the wall and no heat flux through the wall (adiabatic solid wall), the pressure and 68

69 temperature can be set equal to the first interior cell values, so that = int and = int Far-field boundary condition The simulation of the flow around bodies such as wings, aerofoils, and vehicles requires a boundary far enough from these configurations so that free-stream conditions can be assumed at this boundary. The far-field boundary condition is applied in such cases. The far-field boundary condition should be far enough from the body to be able to assume that an unperturbed field is reached. The far-field boundary should prevent outgoing waves from reflecting back in the interior domain. The numerical treatment of the farfield boundary is based upon the locally fixed and extrapolated Riemann invariants. The Riemann invariants are defined normal to the boundary and they are evaluated as follows: The incoming Riemann invariant is calculated from the free stream conditions as:. = 2 1) ൨ ஶ ߛ) (4.71) The outgoing Riemann invariant is calculated from the interior conditions as:. = 2 1) ൨ ௧ ߛ) (4.72) The local normal velocity and speed of sound at the boundary are given by: + = ݑ 2 1 ߛ =, 4 ( ) (4.73) The boundary is classified as inflow or outflow according to the sign ݑ of. Since the unit sign of the Rienman invariant is defined as positive for a wave directed to the exterior of the physical domain, the positive sign of ݑ denotes outflow and the negative sign denotes inflow. At an outflow boundary, the set of boundary condition variables are: (4.74) ݑ + ௧ ((. ) ) = (4.75) ௧ ݏ = ݏ 69

70 ఊ ଵ ߩ = ቆ ߩ ఊ ଵ ௧ ଶ ቇ ௧ ߛ ଶ ߩ = ߛ At an inflow boundary, the set of boundary condition variables are: (4.76) (4.77) (4.78) ݑ + ஶ ((. ) ) = (4.79) ஶ ݏ = ݏ ఊ ஶ ଶ ߩ = ቆ ߩ ஶ ߛ ቇ ଵ ఊ ଵ (4.80) ଶ ߩ = ߛ (4.81) Symmetry plane A symmetry plane can be created if the model exhibits no flux across that plane. Using a symmetry plane reduces the size of the computational domain and consequently the computational cost and time. The symmetry plane has the following conditions: =. zero, The gradients of the pressure and density normal to the plane are 0, ρ. = 0, therefore. (4.82) ௧ = (4.83) ௧ ߩ = ߩ The gradient of the tangential velocity normal to the plane is zero. (4.84) 0 = /( ) ௧௧ ݑ = ௧ ݑ (4.85) The normal velocity should vanish at the symmetry plane Subsonic outflow (4.86) 0 = In subsonic outflow, only one boundary condition should be imposed because there is only one characteristic wave directed toward the interior of the computational domain. The static pressure at exit is defined and the remaining primitive variables are extrapolated from interior domain. The back pressure equation, as a function of the boundary normal Mach number component, is obtained from Manna (1992): ܯଵ ଷ where the coefficients ଵ, ଶ, ଷ, and ସ are: + ܯଶ ଶ + ܯଷ + ସ = 0 (4.87) 70

71 1 ߛ = ଵ ߛ 2 = ଶ + 3 ߛ = ଷ (4.88) 1 ߛ ) ଶ ൬ ܯ + 1 ) ߩቈቆ ସ = 2 ߛ + ߛ 2 ൰ቇ / ௧ The normal component of the boundary condition Mach number ܯ can be obtained by solving the first equation using a Newton-Raphson method. By substituting the value of ܯ into the following equation [ ఘ ସ (1 + ܯ ଶ )] ௧ = [ ఘ ସ (1 + ܯ ଶ )] ௨ ௬, the speed of sound at the boundary can be obtained: = ߛ] ܯ) + 1) ଶ ] + 1) ଶ ] ௧ ܯ) ߩ] (4.89) Then, the boundary condition primitive variables are determined as follows: Subsonic inflow ߛ = ߩ ଶ (4.90) (4.91) ܯ = ݑ (4.92) ݑ + ௧ ((. ) ) = In case of three dimensional inviscid flows, since there are four characteristic waves (λ 2 to λ 5 ) moving towards the domain interior, the number of variables to be defined is four. The selected variables imposed at the boundary interface depend on the available experimental data. Commonly, the imposed variables are the stagnation temperature T 0, stagnation pressure 0, and two inlet flow angles,ߙ and β. The imposed conditions are used in addition to the interior flow variables to calculate the intermediate boundary states following these steps: The negative Riemann invariant is defined as = 2 1) ൨ ௧ ߛ) where,ܖ are the normal unit vector to the boundary cell face and the speed of sound based on the interior conditions. The interior tangential velocity component is (4.93) (4.94) ௧ = ௧ ݑ The entropy value at the boundary and the total enthalpy using the imposed conditions are 71

72 = The positive Riemann invariant is defined as ߛ = ) ఊ ; h ߩ) 1 ߛ (4.95) = 1 + 4ඨh ௧ 3) ߛ) 1 ߛ ଶ 1 ߛ 2 2 ( ) ଶ (4.96) Now, the normal velocity and the speed of sound at the boundary can be calculated using the Riemann invariants + = ݑ The boundary velocity is ݑ = ݑඥ ଶ 2 ௧ ݑ + ଶ 1 ߛ = ; 4 ( ) The velocity components at the inlet are obtained by decomposing the boundary velocity u t according to the two prescribed flow angles ߙ cos ݑ = ݑ ߙ cos ߙ sin ݑ = ݒ ߚ sin ߙ sin ݑ = ݓ The density, static pressure, and temperature at the boundary are evaluated as follows: (4.97) (4.98) = ቆ ଶ ߩ ቇ ݏߛ ଶ ߩ = ߛ ଵ ఊ ଵ (4.99) = ൬ ൰ 72

73 5. Implementing and testing two transition models 5.1. Turbulence closure without any transition model turbulence model ܓ Subsection 4.3 showed that the short-reynolds averaged Navier-stokes equations contain unknown variables such as ߩ തതതതതതത h as a consequence of averaging. Therefore, additional mathematical relations are needed to close the system of mean flow equations (4.17 to 4.19) with the equation of state. These mathematical relations can be algebraic, ߝ the such as the Baldwin and Lomax model, or differential, such as the or models. In this flow solver, the model of Wilcox (2002) is used as the main turbulence closure model. A brief description of this turbulence model is given in this section. The Wilcox turbulence model was originally formulated by Wilcox (1993) for the Reynolds averaged Navier-Stokes equations, in which the flow averaged variables ഥ of Equation (4.15) are constant in time and ݐ in Equation (4.16). This model has since been applied by Rona and co-workers (1997) to the short-time averaged Navier- Stokes equations to model time- dependent flows characterized by large-scale coherent instabilities embedded in a background flow of small-scale random turbulence. These applications show that it is possible to apply the Wilcox (1993) turbulence closure to the short-time Reynolds averaged Navier-Stokes equations, provided the time scale of the resolved motion is much greater than that associated with the fluctuations in the small-scale turbulence. This is the driving principle for selecting inݐ Equation (4.16). The derivation of the turbulent kinetic energy transport equation starts by evaluating the scalar product of the Navier-Stokes momentum equations with the fluctuating component of the velocity vector. By Reynolds averaging the result and rearranging the notation, the turbulent kinetic energy transport equation is obtained: + ᇱ ഥ തതതതതതതതതത+ ᇱ ᇱ തതതതതതതതതത+ ᇱ ᇱ ഥ : ഥ = ഥ ൯ ߩ൫ + ൯ ߩ൫ ݐ (5.1) തതതതതതതതതതതത തതതതതതതതቆ ᇱ ᇱ തതതതതത ᇱ ᇱ ߩ ᇱ ᇱ. ᇱ ቇ 2 The turbulent kinetic energy transport equation consists of the following terms from left to right: 73

74 1) The unsteady term 2) The convection term 3) Production 4) Dissipation 5) Pressure dilatation 6) Pressure work 7) Molecular diffusion 8) Pressure diffusion 9) Turbulent transport The first two terms i.e., the unsteady and the convective terms are the Reynolds transport derivatives of the turbulent kinetic energy in the Eulerian frame of reference. Terms 1 and 2 can be rearranged by the use of the continuity equation into the material transport of turbulent kinetic energy withݐܦ/ ҧ ൯ߩ൫ܦ ഥ being the transport velocity in the material operator ܦ. Some terms represent additional unknowns and must be modelled. Following the analysis of Wilcox (2002), the dissipation term തതതതതതതതത, which represents the viscous dissipation of turbulent shear stresses, can be written in terms of the short-time averaged dissipation per unit,ߝ, mass as തതതതതതതതത =.ߝҧߩ The specific dissipation of turbulent kinetic energy is modelled as: ) ᇱ ) : ᇱ തതതതതതതതതതതതതതത ߥ =ߝ (5.2) Wilcox (1994) related the specific dissipation of the turbulent kinetic energy to the specific dissipation rate of turbulent kinetic energy,, by the following form: (5.3) ߚ ߩ =ߝ ߩ ᇱ ᇱ = തതതതതതതതതത Equation (5.3) models term 4 in Equation (5.1). Unfortunately, there is no straightforward analogy for the pressure diffusion term, term 8 in Equation (5.1). So, the pressure diffusion and turbulent transport terms, terms 8 and 9 in Equation (5.1), are grouped together and assumed to behave as a gradient-transport process, from Wilcox (2002). A recent Direct Numerical simulation shows that this term is quite small, as stated by Wilcox (2002). Thus, the pressure diffusion and turbulent transport are modelled as: (5.4) ത ௧ ߤ ߪ = /. തതതതതതതതതതതതതതത ߩ തതതതത where ߪ ߤ and ௧ are the model closure coefficient and the eddy viscosity, respectively. The molecular diffusion term represents the mixing and transport of energy by natural fluid molecular motion and is related to the spatial gradient of the turbulent kinetic energy as: ߤ ᇱ ᇱ = തതതതതതതത (5.5) 74

75 The pressure work, term 6 in equation (5.1), is taken as zero because of the definition of = ഥ 0. The pressure dilatation term, term 5 in equation (5.1), is dropped because of the lack of a widely accepted model for this term, Wilcox (2002). Also, the Mach number in this study is below the hypersonic range which gives the possibility to neglect the pressure dilatation, Wilcox (2002). The dissipation term in the turbulent kinetic energy equation is modelled as a function of the specific dissipation rate of turbulent kinetic energy,, which is an additional unknown in the short-term Reynolds averaged Navier-Stokes equations. Therefore, another partial differential equation similar to the turbulent kinetic energy equation must be derived to estimate. The derivation of the transport equation for the specific dissipation rate of turbulent kinetic energy is similar to the derivation of the turbulent kinetic energy transport equation. It starts with taking the double scalar product of the spatial gradient details are available in McKeel (1996). of the Navier-Stokes momentum equations with ߥ 2. Further The complete set of partial differential equations, which consists of Reynolds averaged Navier-Stokes equations and the turbulence model, can be written as: ߩ (5.6) 0 = ( ഥ ߩ) + ݐ ( ത + ത) = ( + ഥ ഥ ߩ) + ( ഥ ߩ) ݐ (5.7) ௧ ) ൧ ߤ ߪ + ߤ) + ഥ ) ത + ത) + ) ഥ + ഥ) ത൯= ഥ൫hത ߩ + ത൯+ ൫ ߩ + ݐ (5.8) ௧ ) ൧ ߤ ߪ + ߤ) + ߩ ߚ ഥ ഥ = ഥ ൯ ߩ൫ + ൯ ߩ൫ ݐ (5.9) ) ] ௧ ߤߪ + ߤ)] + ଶ ߩߚ ഥ ഥ ( ഥ ߩ) = + ( ߩ) ݐ (5.10) The closure coefficients, cross-diffusion modification, compressibility correction and auxiliary relations of Wilcox k-ω model, Wilcox (2000), are defined as follows: = 13 = 1 2 ߪ, 2 = 1 ߪ, 25 (5.11) ߩ = ௧ ߤ (5.12) 75

76 (5.13) )] ௧ ܯ)ܨ ߦ + ఉ [1 ߚ = ߚ ) ௧ ܯ)ܨ ߦ ఉ ߚ ߚ = ߚ = 9 ߚ = 9 ߚ, 100 = 3 2 ߦ, 125 (5.14) (5.15) 1 0 ఉ = ( ଶ )/( ଶ ) > 0 (5.16) = 1 ଷ (5.17) (5.18) ൯ ௧ ܯ ௧ ܯ൫ܪ൧ ௧ ଶ ܯ ଶ ௧ ܯ = ) ௧ ܯ)ܨ = 1 4 ௧ ܯ (5.19) = 2 ௧ ܯ ௧ଶ (5.20) where ܯ൫ܪ ௧ ܯ ௧ ൯is the Heaviside step function defined as zero for ܯ ௧ < ܯ ௧ and 1 for ܯ ௧ ܯ ௧. ௧ = + ҧߩҧߛ൫ 2 3 ഥ൯.ହ is the speed of sound. The system of partial differential equations for the conservation of mass, momentum, and energy, along with the partial differential equations for the transport of turbulent kinetic energy and specific dissipation rate, can be written in the following compact form: where U is the vector of conservative variables (5.21) S = [( ) + F t ( ) [۴ + ݐ 76

77 is the inviscid flux vector ۴ ߩ ഥ ߩ ൯ ൫ ߩ + = ߩ ߩ ഥ ߩ + ഥ ഥ ߩ ൯ ഥ൫hߩ + = ۴ ഥ ߩ ഥ ߩ (5.22) (5.23) F t is the viscous flux vector 0 ) ത + ത) F t = + ௧ ) ௧ ߤ ߪ + ߤ) ഥ ) ത + ത) ߪ + ߤ) ) ௧ ߤ ) ௧ ߤߪ + ߤ) (5.24) and S is the source term ഥ S= 0 = ߩ ߚ ഥ ݏ ఠ ݏ ഥ ഥ ߩߚ ଶ (5.25) For sake of simplicity, the overline to indicate short-time averaged values will be omitted in the next sections; also will indicate the total stress tensor, which is the sum of the viscous stress tensor and the Reynolds stress tensor( ҧ+ ҧ ). The integration of the governing equations (5.21) over an arbitrary control volume gives: (5.26) = න S t ර F + c න U + ර F ݐ Let the physical space be dived into an assembly of control volumes and let be the ௧ control volume in this assembly. The convective and diffusive flux integrations in equation (5.26) can be considered as the summation of the contributions over all 77

78 discrete faces,, bounding the computational control volume. The discretized shorttime Reynolds averaged Navier-Stokes equations are obtained as: + (F c k ) + (F t k ) = ݐ (5.27) ଵ ଵ The vector of conservative variables orthonormal to S k is ߩ ݑ ߩ ݑ ߩ ௧ଵ ݑߩ = ௧ଶ ) + ( ߩ ߩ ߩ (5.28) ݑ where, ݑ ௧ଵ, ݑ ௧ଶ are the normal, tangential, and binormal velocity components on the closed surface S k of the control volume V i given by equation (4.41). The convective and diffusive fluxes through S k are: = c = F ) ) F c ߩ + ߩ ) + (h ߩ ߩ ߩ (5.29) 0 = t = F ) ) F t ) ௧ ߤ ߪ + ߤ) ( ) ߪ + ߤ) ) ௧ ߤ ) ௧ ߤߪ + ߤ) (5.30) where n, as defined in chapter 4, is the inward normal unit vector to the closed surface S k bounding the control volume V i. The interpretation of equation (5.27) states that the rate of change of the volume averaged conservative variables in the ௧ control volume is equal to the summation of the area-averaged convective and diffusive fluxes through the discrete boundary faces, S k, plus the source term. Applying the Roe approximate Riemann solver to evaluate the convective fluxes through S k, the resulting eigenvalues of the approximate Jacobian matrix = డ۴ ) ) డ are: 78

79 ݑ ௧ = ଵ መߣ ݑ = ଶ መߣ ݑ = ଷ መߣ ݑ = ସ መߣ (5.31) ݑ + ௧ = ହ መߣ ݑ = መߣ ݑ = መߣ By substituting the eigenvalue መߣ into Ƹ = Ƹ መߣ and solving for the eigenvector Ƹ, the corresponding eigenvectors are obtained: ଵ = 1 ݑ ௧ ݑ ௧భ ݑ ௧మ h ݑ ௧ ൨ Ƹ ଶ = ቈ1 ݑ ݑ ௧భ ݑ ௧మ h ௧ ଶ Ƹ ଷ = [ ݑ ௧భ 0 0] Ƹ ସ = [ ݑ ௧మ 0 0] (1 ߛ) Ƹ ହ = 1 ݑ + ௧ ݑ ௧భ ݑ ௧మ h ݑ ௧ ൨ (5.32) Ƹ = ߛ 3 (1 ߛ) 3 = [ ] 1 0൨ The symbols with hat indicate that their values are computed using the Roe averaging method by Roe (1981): R ݑ R ߩඥ L + ݑL ߩඥ ݑ = ߩඥ L + ߩඥ R ௧భ R ݑ R ߩඥ ௧భ L + ݑL ߩඥ ݑ ௧భ = ߩඥ L + ߩඥ R ௧మ R ݑ R ߩඥ ௧మ L + ݑL ߩඥ ݑ ௧మ = ߩඥ L + ߩඥ R (5.33) h = ߩඥ Lh L + ߩඥ R h R ߩඥ L + ߩඥ R 79

80 = ߩඥ L L + ߩඥ R R ߩඥ L + ߩඥ R = ߩඥ L L + ߩඥ R R ߩඥ L + ߩඥ R R R ߩඥ L + L ߩඥ = Ƹ ߩඥ L + ߩඥ R R ߩ L ߩඥ = ߩ ௧ = ඨ൬ Ƹߛ ߩ ൰ Where subscripts L and R refer to the flow states to the left and to the right of S k respectively. In order to calculate the Roe numerical flux F భ + using equation (4.36) the మ wave strengths ߙ must be evaluated. The relations to calculate the wave strengths can be derived by projecting the conservative variables difference U R U L onto the eigenvectors as follow: ߩ ݑߩ ௧భ ݑߩ ௧మ ݑߩ = ߙ Ƹ ߩ ଵ ߩ ߩ (5.34) where represents the difference across the cell interface S k, for example ߩ = ߩ R ߙ, the L. By solving the above linear system of simultaneous equations for ߩ characteristic wave strengths are estimated as: 80

81 = 1 ଵ ߙ 2 ௧ ଶ ൫ߩ + ߩ൯ ߩ ௧ ݑ ൨ 1 ߩ = ଶ ߙ ଶ + 2 ൫ߩ + ߩ൯൨ ௧ 3 ௧భ ݑ ߩ = ଷ ߙ ௧మ ݑ ߩ = ସ ߙ (5.35) = 1 ହ ߙ 2 ௧ ଶ ൫ߩ + ߩ൯+ ߩ ௧ ݑ ൨ ߩ = ߙ ߩ = ߙ 5.2. Source term The source term vector defined in equation (5.25) includes the production and destruction terms of the equations. The production term is evaluated using equation (4.20) to calculate the Reynolds stress tensor r and by applying the Gauss divergence theorem to calculate the velocity gradients. The coefficients ߚ and ߚ of the turbulence model including the compressibility correction are defined by equations (5.13) and (5.14). Evaluating the source term completes the estimation of the linear terms in equation (5.27) and the final step to evaluate the solution of the volumeaveraged conservative variables vector is by the time integration of the differential This is performed by the Runge-Kutta method with local time-stepping detailed.ݐ / in section The Menter Shear Stress Transport (SST) model The shear stress transport (SST) model, developed by Menter (1992), combines the best qualities of the k-ω and the k-ε models. Specifically, the k-ε model is not able to capture the proper behaviour of turbulent boundary layers up to separation. The k-ω model is reported by Wilcox (2002) as being more accurate than k-ε model in boundary layers under favourable, zero and adverse moderate pressure gradients in the near wall layers and has therefore been successfully applied to flows with moderate adverse pressure gradients. In the modelling of shear flows, the ω-equation shows a strong sensitivity to the values of ω in the free-stream outside a boundary layer. The free-stream sensitivity 81

82 has largely prevented the ω-equation from replacing the ε-equation as the standard scale-equation in turbulence modelling, despite its superior performance in the nearwall region. This was one of the main motivations for the development of the SST model. The SST model zonal formulation is based on blending functions, which ensure a proper selection of the k-ω and k-ε zones without user interaction. Menter (1992) showed that the SST model exhibits an improved agreement with experiments compared to other two-equation RANS turbulence models for a variety of test cases. The SST model gives more accurate predictions in regions of separation in complex flow with a strong adverse pressure gradient, Menter (1992). The transport equations for k and ω are: ത, ൯= ഥ ത ߩ൫ ത ߩ൫ ൯+ ݐ ௧, ൯ ത ൧ ߤ ߪ + ߤ൫ ത ߩ ഥ + ߚ ഥ = ( ഥ ഥ ߩ) + ( ഥ ߩ) ௧ ߩߛ ௧, ߤ ݐ ௧, ൯ ωഥ൧ ߤ ఠ ߪ + ߤ൫ ҧ ഥߩ ଶ + ߚ ഥܝ ఠ ଶ ߪ[ ଵ ܨ 1 ] ߩ ഥ ത ഥ (5.36) (5.37) where, is the turbulent shear stress and is modelled by: (5.38) ۷ ത ߩ 2 3 ഥ൨ ۷ 2 3 ( ഥ) ഥ + ௧ ߤ =, The blending function ܨ ଵ is defined by: (5.39) ) ଵସ ݎ ) anhݐ = ଵ ܨ where, with ଵ ݎ = min max ൭ ඥ ത ߤ 500 ; ݕ 0.09 ഥ ఠ ଶ ത ߪߩ 4 ; ଶ൱ ݕഥ ߩ (5.40) ൩ ଶ ݕ ఠ ܦܥ ఠ ଶ ߪߩ2 ఠ = max ܦܥ 1 ഥ ൫ ത ഥ൯; 10 ଶ ൨ (5.41) where ݕ is the distance to the nearest wall and ܦܥ ఠ represents the positive part of the cross-diffusion term in equation (5.37). The constants of the model are obtained by 82

83 blending the constants in the k-ω and k-ε models using the coupling function F 1 as follows: ଵ ߪ ߪ ଶ ߪ ఠ ଵ ߪ ఠ ߪ ఠ ଶ ߪ ൦ ߚ ൪= ܨ ଵ ൦ ߚ ൪+ (1 ܨ ଵ ) ൦ ଵ ߚ ൪ (5.42) ଶ ଶ ߛ ଵ ߛ ߛ F 1 is equal to zero away from the surface (k-ε model) and switches over to one inside the boundary layer (k-ω model). The constants are ߪ ଵ = 0.85, ߪ ఠ ଵ = 0.5, ߚ ଵ = 0.075, ߛ, 0.09 = ߚ ଵ = ఉ భ ఉ ఙ ഘ భ మ ඥఉ, and = ߢ 0.41 while the ߝ constants are ߪ ଶ = 1.0, ߪ ఠ ଶ = 0.856, ߚ ଶ = ଶ = ఉ మ ߛ 0.09, = ߚ , ఙ ഘ మ మ ఉ ඥఉ = ߢ, and By enforcing the Bradshaw s assumption that the turbulent shear stress in the boundary layer is equal to ρα 1 k RANS, Menter s SST turbulent eddy viscosity can be obtained from = ௧, ߤ ఘഥఈ భത ಲ ௫[ఈ భ ఠഥ; మ ] ߙ where ଵ = 0.31, S is the strain rate tensor: and its magnitude (5.43) ( ( ഥ ) + ഥ ) = 1 2 (5.44) : 2 = and F 2 is a second blending function defined by: ଶ =tanh max ൭2 ඥ ത ܨ ߤ 400 ; ݕ 0.09 ഥ ଶ ଶ൱൩ ݕഥ ߩ (5.45) A production limiter is used in the SST model, as suggested by Menter (1992) to prevent the build-up of unrealistic turbulence in stagnation regions: = min ത ܣ,ݎ ߚ 20 ; ഥ ഥ ത ߩ ൧ (5.46) The limiter bounds the production term up to 20 times the destruction term and replaces equation. ഥ in the transport, ത 83

84 5.4. Turbulence closure with transition model The classical correlation method of Suzen and Huang This approach to predict transition, which is favoured by the turbomachinery community, uses the concept of intermittency, as introduced Dhawan and Narasimha, to blend together laminar and turbulent flow regimes, based on empirical correlations. Many detailed investigations of the process of transition reveal that, over a certain range of Reynolds numbers around the critical value, the flow becomes intermittent, which means that it alternates in time between being laminar and turbulent. The physical nature of this flow can be properly described with the aid of the intermittency factor γ, which is defined as the fraction of time during which the flow at a given position is turbulent. This approach to predict transition uses one additional transport equation for the intermittency factor γ. It replaces the empirical correlation of Dhawan and Narasimha (1958) of Eq for the intermittency distribution along the flow direction, to blend together laminar and turbulent flow regimes, (5.47) ] ଶ ௫ )ߪ ]ݔ ௫௧ ) 1 = ߛ where x is the curvilinear coordinate along the wall, ௫ = ߥ ݔݑ ஶ, = ߥ ଶ ஶ ଷ ஶ is the ݑ non-dimensional production rate parameter of turbulent spots, ν is the kinematic viscosity, and σ is the Emmos parameter which depends on the shape and velocity of the turbulent spots. In Eq. (5.47), the spot production rate parameter ߪ is computed as: = ߪ ݑଵଵ ସ (5.48) where Tu is the free-stream turbulence intensity. The Dhawan and Narasimha (1958) correlation provides the distribution of the intermittency factor along the normal direction to the wall. Suzen & Huang developed an intermittency transport model that can produce both the experimentally observed streamwise variation of intermittency and a realistic γ profile in the cross-stream direction. To this end, it combines the model of Steelant and Dick (1996) with the model of Cho and Chung (1992) employing a blending function. The transport equation for the intermittency factor,,ߛ is: (ߛ ߩ) ఊ + ఊ ܦ = ߛഥ ߩ + ݐ (5.49) where ܦ ఊ and ఊ are the diffusion and source term respectively: 84

85 (5.50) {ߛ ൧ ௧ ߤ ఊ௧ ߪ(ߛ (1 + ߤ ఊ ߪ(ߛ (1 ఊ = ܦ ఊ = (1 1 )](ߛ (ܨ + ଵ )ܨ ଶ)] + ଷ (5.51) and F is a blending function. The first term, T 0, derives from the model of Steelant and Dick (1996), (5.52) (ݔ)ߚഥ ഥ ߩ ܥ = where β(x) = 2f(x)f (x) and the function f(x) is the following polynomial interpolation function for ߪ( ௫ ௫௧ ) ଶ around the point x t of transition onset: with x = x- x t, and + ݔ + ଶ ݔ + ଷ ݔ + ସ ݔ = (ݔ) h ଵ ݔ + h ଶ (5.53) ߪ 50ට =.ହ ߪቀ = 0.204, =, ቁ, = 0, =.ସ ଽ ቀఙ ቁ ଵ.ହ, h ଵ = 50, h ଶ = 10 where U is the boundary layer stream-wise velocity. The T1, T2 and T3 terms are derived from the model of Cho and Chung (1992) and are given respectively as: ߛଵ ܥ = ଵ (5.54) ഥ : ଶ = ܥ ଶ ଵ ߩߛ ଶ ഥ (5.55) ߛ ഥ ഥ ഥ ߚ The shear stresses are defined as: (5.56) ߛ ߛ ߩ ଷ ܥ = ଷ ߚ (5.57) ߩ 2 ൨ ௧ ߤ = 3 85

86 The blending function, F, provides a smooth passage between the two models in the transition region. It is evaluated as a function of the ratio,(ߥ )/ with from equation 5.44 (ߥ ) = tanh ସ ቈ ܨ.ଵ ).ଷ ߛ 200(1 (5.58) The above equation is based on the correlation due to Klebanoff for the distribution of γ in the normal direction to the wall. It switches from the model of Steelant and Dick (1996) close to the wall to the model of Cho and Chung (1992) in the outer region. The values of the closure coefficients employed for the present model are: = 1 ఊ௧ ߪ 1, = ఊ ߪ = 0.15 ଷ ܥ 0.16, = ଶ ܥ 1.6, = ଵ ܥ 1, = ܥ (5.59) By letting the intermittency grow from zero to unity, the start and the evolution of transition can be imposed. This is done by multiplying the eddy viscosity in a twoequation turbulence model by the intermittency factor γ. In other words, once γ is determined, it is multiplied by the eddy viscosity in the short-time Reynolds averaged Navier-Stokes equations. In the pre-transitional regime, γ is set to zero and γ assumes a positive value only where the model is required to initiate transition. This approach neglects the interaction between the turbulent and non-turbulent parts of the flow during transition. In order to capture this interaction, a conditional averaging technique leading to a set of turbulent and a set of non-turbulent equation for mass, momentum and energy is necessary, as used by Steelant & Dick (1996). The conditional averaging is usually seen as too computationally expensive for engineering applications, as the number of equations doubles. Therefore, the intermittency concept is typically used in combination with globally averaged Navier-Stokes equations and the loss of some physical information is accepted. Whereas the intermittency transport equation defines the intermittency distribution for transitional flows in the simulation, the onset of transition is defined by correlations. The onset of attached flow transition is determined as a function of the turbulence intensity, Tu. The onset of attached flow transition is determined by the Abu-Ghannam & Shaw (1980) correlation (5.60) (ݑ exp(6.91 ఏ =

87 this relation implies the transition onset always occurs at a momentum thickness Reynolds number which is 163 or larger. This lower limit stems from the Tollmien- Schlichting stability limit. The length of the transition region is obtained from the Suzen et al. (2000) correlation as ߪ = ቆ ݒଶ ଷ 1.8 = ߪቇ ݑ 10 ଵଵ ସ (5.61) where U is the boundary layer stream-wise velocity. The Suzen correlation is based on extensive experimental analysis covering a good range of key turbulent parameters. Both these relations are suitable for a zero pressure gradient boundary layer model Model implementation in the 2D version of the in-house CFD code Cosmic The Suzen and Huang intermittency transport model has been implemented in the finitevolume scheme Cosmic detailed in section 4. The intermittency factor was accommodated into Cosmic through expanding the flow properties array by one further element to 7, so that in 2D there are seven conservative variables (ρ, ρu, ρv, ρe 0, ρk, ρω, ργ). The new transport equation, balancing its production, destruction, convection and diffusion, has been implemented explicitly in the scheme. Through factoring γ into the eddy viscosity, the new solver alters the balance of flow field kinetic energies, thus locally turning a laminar Reynolds Averaged Navier Stokes (RANS) method into a k-ω turbulence model at runtime. Specifically, the upwind Riemann solver of Roe of section 4.5 convects γ through the flow field. The γ-diffusion term has been implemented using Gauss' divergence theorem around cell interfaces, detailed in section 4.9. All derivatives in the source terms are discretized by second-order central differencing. Using the numerical method described in this section, the convective and viscous fluxes of the short-time averaged Navier- Stokes equations, including the k-ω model transport equation and the γ transport equation, are computed. In each computational cell, the resultant fluxes are combined with a volume integral of the turbulent source terms. The resulting solution in each cell is then integrated in time using the Runge-Kutta method of section The intermittent behaviour of the transitional flows is modelled by modifying the eddy viscosity, μ t by the intermittency factor, γ. The transport equations for the 2D RANS with Suzen and Huang intermittency model are: 87

88 ۴ ௫ + ݐ ۴ ௬ + ݔ ௫ ߥ ۴ = ݕ ௫ ߥ ۴ + ݔ (5.62) + ݕ U - flow properties array in conservation form Fc - flux vectors in x and y Fv - viscous flux vectors in x and y S - turbulence and intermittency source terms ߩ ݑߩ ݒߩ ) + )ߩ = = ߩ ߩ ߛߩ ݑߩ ߩ + + ଶ ݑߩ ݒݑߩ ( ߩ + + ߩ)ݑ = ۴ ݑߩ ݑߩ ߛݑߩ ݒߩ ݒݑߩ ߩ + + ݒݑߩ ( ߩ + + ߩ)ݒ = ۴ ݒߩ ݒߩ ߛݒߩ 0 0 ௫௫ ௫௬ ௫௬ ௬௬ ۴ ௩ = ௫௫ ݑ + ௫௬ ݒ + ܥ ௫ + ܥ ଵ ௫ ۴ ௩ = ௫௬ ݑ + ௬௬ ݒ + ܥ ௬ + ܥ ଵ ௬ ܥ ௫ ௬ ܥ ܥ ଶ ௫ ܥ ଶ ௬ ௬ ߛ ଷ ܥ ௫ ߛ ଷ ܥ (ߛ (1 ܨ ߤ൬ ௧ ቀ ସ ݑቀ൫ ଷ ௫ ݒ ௬ ൯ ଶ + ݑ ௫ ݒ ௬ ቁ+ ݑ൫ ௬ + ݒ ௫ ൯ ଶ ቁ ଶ ݑ൫ ߩ ଷ ߩ ߚ ൯൰ ௬ ݒ + ௫ ቀ ఝఠ ߤቁ൬ ௧ቀ ସ ݑቀ൫ ଷ ௫ ݒ ௬ ൯ ଶ + ݑ ௫ ݒ ௬ ቁ+ ݑ൫ ௬ + ݒ ௫ ൯ ଶ ቁ ଶ ݑ൫ ߩ ଷ ௫ + ݒ ௬ ൯൰ ߩ ߚ ଶ (ݏ) ᇱ (ݏ) ଶ ݒ + ଶ ݑ ߩ ܪ 2 (ܨ (1 ଷ ݑቀ൫ ௧ ቀ ସ ߤ ௫ ݒ ௬ ൯ ଶ + ݑ ௫ ݒ ௬ ቁ+ ݑ൫ ௬ + ݒ ௫ ൯ ଶ ቁ ቀ భఊ ቁ ଶ ݑ൫ ߩ ଷ ௫ + ݒ ௬ ൯ ܥ ߩߛ ଶ బ.ఱ ߛ௫ ݑݑ൫ ఉ ఠ ඥ௨ మ ௩ మ ௫ + ݑݑ ௬ ߛ ௬ + ݒݒ ௫ ߛ ௫ + ݒݒ ௬ ߛ ௬ ൯ ߩ ଷ ܥ+ ߛ൫ ఉ ఠ ௫ ଶ + ߛ ଶ ௬ ൯ where C 0, C 1, and C 2 and C 3 are defined in equation The main computation steps for solving Eq are: a) Solve the mean flow equations: the momentum equations with a modified eddy viscosity, and the continuity equation. b) Solve the turbulence model: the k and ω-equations, and then compute the μ t. c) Compute the onset location of transition, by searching for the point where Re θ = Re θt. d) Solve the transition model: the γ-equation. 88

89 Figure 5.1 shows the program flow chart of the two-dimensional in-house finite-volume scheme Cosmic, which precedes the start of this study. Figure 5.2 shows the flowchart of the scheme modified to model transitional flow. main pre-processing Reads initial solution and test setup. Stores data to variables and crosschecks geometry files. scheme time-step Calculates time-step throughout the test domain based on the user-defined CFL. impose boundaries Apply pre-programmed boundary conditions to edge nodes, as specified in the initialisation file. calculate laminar & eddy viscosity Viscosities calculated for each computational cell based on the flow properties in each cell. convective fluxes LHS: upwind method computes fluxes from orthogonal cells, flux differences stored to residual. compute viscous fluxes RHS: central differencing on cell interface calculates viscous fluxes, flux difference added to residual. compute source terms RHS: central differencing Gauss divergence theorem employed in source terms. test for convergence Apply 2-step Runge- Kutta, test for convergence, continue for required number of iterations. pre-processing 6 flow parameters (conservative form) at each node written to file. Fig. 5.1: Flow chart of the in-house code Cosmic without intermittency transport model. 89

90 main pre-processing New conservative variable read in for each cell. scheme time-step impose boundaries Limit gamma & factory viscosity Conservative ϒ limited 0 < ϒ< ρ, primitive ϒ factored into eddy viscosity term. calculate laminar & eddy viscosity convective fluxes compute viscous fluxes LHS of transport model implemented in internal domain. ϒ diffusion term implemented by Gauss divergence theorem at each cell interfaces. compute source terms Implement gamma terms New source terms added with remaining ϒ terms, derivatives by central differencing. test for convergence pre-processing ϒ written to TecPlot file in primitive form and to solution file in conservative form. Legend: Amended source file New source file Fig. 5.2: Transition model low chart. 90

91 Test cases The test cases analysed are the ERCOFTAC flat plate test cases T3A and T3AM. They differ in the free-stream velocity and turbulence intensity as detailed in section 3.2. The T3 cases are particularly difficult because the freesteam turbulence is not constant and varies rapidly along the length of the flat plate. The boundary layers are modelled as two-dimensional flow by considering a rectangular control volume spanning from downstream of the flat plat leading edge to 1.7m downstream of this position in the streamwise direction, as shown in figure 5.3. The rectangular control volume extents from the flat plate surface to 0.15m above it. The control volume is discretized to an assembly of rectangular computational cells, with 1357 nodes points in the x-direction and 193 nodes in the y direction. For accuracy grid points are clustered in regions of large gradients, but spread out elsewhere for economy of computation. The stretching factor has been chosen in order to have a value of y+ < 0.2 for the first point away from the wall, and such that the viscous region of the boundary layer (y+ < 30) contains more than 40 cells. This is because wall-function grids cannot be used because they cannot properly resolve the laminar boundary layer. Moreover, a sufficient grid points in the streamwise direction are needed to resolve the transitional region. Fig. 5.3: Computational mesh: For clarity, one point every 12 in both the x and y directions is shown, and the x-axis to y-axis ratio is 0.4. In figure 5.3, the flow runs from left to right. The computational domain boundaries are subsonic inflow to the left, subsonic outflow to the right, non-slip along the wall to the bottom and far-field along the top. The subsonic inflow boundary is located 495mm downstream of the flat plate physical leading edge for the T3AM test case. For the T3A test case, the inflow boundary is located 95mm downstream of the physical leading edge. A laminar profile (γ=0) is imposed as the inflow of the computational domain. As far as the inflow boundaries in both the computations, these are derived from a [m] 91

92 numerical solution of the Blasius equation that is re-scaled to match the experimental laminar boundary layer integral parameters in the laminar part of the flow-field. experimental Blasius 1.0E E E E E-03 y [m] 5.0E E E E E E u [m/s] Fig. 5.4: Inflow velocity profile for the T3A test case. Experimental Blasius 1.4E E E-02 y[m] 8.0E E E E E u[m/s] Fig. 5.5: Inflow velocity profile for the T3AM test case. Transition is strongly affected by the decay of turbulence in the free-stream. Therefore, it is important in a transitional boundary layer simulation to assign the appropriate inlet boundary condition for k and ω. The inlet turbulent kinetic energy is fixed according to 92

93 the free-stream turbulence level and the specific dissipation rate for each case are adjusted according the downstream decay of the free-stream turbulence level to match the results reported in Blair (1983). See figure 5.6 for instance, keeping k the same and trying different values of epsilon. Fig. 5.6: Free stream turbulence intensity decay for test case T3A. Initially γ is set to zero throughout the flow field. On the solid wall boundary the value of the wall-normal γ gradient is kept as zero. At the freestream, a zero value of γ is assumed. At the outflow boundary, γ is extrapolated from inside the computational domain Zero pressure gradient transitional boundary layer results The ERCOFTAC T3A test case The in-house finite-volume scheme Cosmic described in section 4 was applied to model a zero pressure gradient boundary layer developing over a flat plate in air. The freestream turbulence intensity is 3.3% and the flat plate trailing edge Reynolds number is 6x10 ହ. This is the test case T3A of the Special Interest Group on transition of the European Research Community on Flow, Turbulence and Combustion (ERCOFTAC). Numerical predictions of this transitional flow using the Suzen and Huang (2000) transition model in the in-house code are shown in figure 5.7 and figure 5.8. The shape factor variation with Reynolds number Re x is compared with the 93

94 experimental data. Re x =xu /ν is the non-dimensional distance from the flat plate leading edge, as opposed to the distance from the computational domain inflow as defined in figure 5.3. The shape factor = ܪ ߜ ߠ indicates the region where the boundary layer tends to be turbulent. A reduction in H indicates transition is about to occurs. In a flow with zero pressure gradient, the shape factor before transition should be constant and equal to 2.61, which is the shape factor for the Blasius profile of a laminar boundary layer, as reported by Schliting (1979). The Blasius profile is a selfsimilar solution and is therefore not Reynolds number dependent. This results in a constant value for the shape factor with Reynolds number, as shown by the top horizontal dashed line in figure 5.7. The bottom horizontal dashed line in figure 5.7 shows the shape factor from a self-similar power law approximation of a fully turbulent boundary layer. It is Reynolds number independent and, just like the Blasius solution for a laminar flow; it gives a constant horizontal line of value of 1.4. The full square symbols in figure 5.7 show the shape factor as measured in experiment by Roach & Brierley (1990) along flat plate. The experimental values of the shape factor changes from about 2.61 at the first measurement location downstream of the leading edge to approximately 1.4, indicating an asymptotic approach to the analytic value from the power law fully turbulent boundary layer profile towards the flat plate trailing edge. The experimental data is essentially fully bounded between the shape factor for a fully turbulent boundary layer, which is the dataset lower bound, and the shape factor for a fully laminar profile, which is the upper bound. This confirms that the flow in the T3A experiment was transitional. The continuous line shows the numerical prediction from the in-house code with the Suzen and Hung (2000) transitional model. The experimental data show a decline of H from the laminar value right from the flat plate leading edge. From Re x =0, on the other hand, the computation shows a pure laminar flow before the transition onset over the range 0 < Re x < The very first measurements of the shape factor close the flat plate leading edge is marginally above the reference Blasius value of This indicates a possible leading edge effect in the experiment in form of a local non-zero pressure field, due to the leading edge stagnation point. This may have caused a non-zero streamwise pressure gradient in the upstream portion of the flat plate. The numerical results also show a small increase in shape factor from the computation domain inflow. This effect is localised and limited to the first flow computational cells inbounds from the computational inlet and is driven by local numerical adjustments in 94

95 the predicted flow state probably due to the Newton-Raphson algorithm of the inflow boundary condition detailed in sec Figure 5.8 shows the variation of the skin friction coefficients along the flat plate of the T3A experiment. The skin friction coefficient is defined as: = ܥ ௪ 1 2 ஶଶ ߩ (5.62) The streamwise coordinate is shown in the non-dimensional form of Re x, as in figure 5.7. Figure 5.8 uses the same notation as figure 5.7. Re x = 0 denotes the flat plate leading edge as opposed to the computational domain inflow, which is downstream of the leading edge by a stand-off distance of x=495mm, as stated in sec The friction coefficient at the flat plate leading edge, in laminar flows, is an integrable singularity and it is therefore not reported in fig 5.8. The dashed line towards the bottom of the graph is the Blasius results for a laminar boundary layer C f =0.664Re x -1/2. The top dashed line is the streamwise variation of the friction coefficients from a power law approximation for a fully turbulent boundary layer, C f =0.027Re 1/7, over the range 0 < Re x <10 5. The experimental data from the T3A test, as shown by the full square symbols, follow closely the laminar C f curve from Blasius upstream of Re x =10 5 whereupon C f is measured to increase towards the power law approximation. The measured values of skin friction are shown in figure 5.8 to overshoot the fully turbulent C f dashed line at approximately Re x =2.5x10 5, so that the experimental C f approaches the power law asymptotically from above the dashed line from the power law. The numerical prediction from the in-house finite-volume scheme with the Suzen and Huang (2000) transitional model gives a skin friction coefficient distribution that has an appreciable overlap with the experimental data. Over the range 10 4 < Re x < 10 5, the numerical prediction follows the laminar boundary layer C f from Blasius. Above Re x 10 5, the numerical prediction starts to depart from the laminar C f profile, indicating the onset of transition, in agreement with the measurements. Whereas the onset of transition is a statistical process that occurs over a finite length of the flat plate, it is common to indicate a transition onset point, or line, which is estimated as Re x = 1.35x10 5 in both experiment and computation. Thereafter, the numerical prediction follows the same trend as the measurements in that the C f overshoots the power law C f profile, asymptoting to it at high Re x. The model predicts a slight overshoot at the end of transition region versus the measured data, but relaxes toward the turbulent asymptote farther downstream. The reasons for this are not known, and Suzen and Huang (2000) 95

96 did not give any idea on the reason of this discrepancy. However, it might be related to the manner in which the model accounts for the transfer of the kinetic energy. Specifically, the distributed breakdown function in equation 5.53 was calibrated against the conditioned Navier-Stokes method. To allow a faster response to flow transition when coupled with the current approach may need to be modified. H T3A experiment Laminar Turbulent Computed E E E E E E E+05 Re x Fig 5.7: Comparison of the experimental shape factor coefficient against computational results for the T3A test case T3A experiment Computed Laminar Turbulent Cf E E E E E E E+05 Re x Fig 5.8: Comparison of the experimental skin friction coefficient against computational results for the T3A test case. 96

97 The ERCOFTAC T3AM test case The T3AM test is a transitional boundary layer developing over a flat plate under zero streamwise pressure gradient in air. The end plate Reynolds number Re L 2.5x10 6 and the free-stream turbulence intensity is 0.98%, which is lower than in the ERCOFTAC T3A test case of figure 5.7 and figure 5.8. Figure 5.9 shows the variation of the skin friction coefficient versus the normalised distance from the flat plate leading edge Re x. In figure 5.9 an extended abscissa is used compared to figures 5.7 and 5.8 to accommodate the delay in transition onset due to the lower free-stream turbulence level. The bottom dashed line is the C f =0.664Re -1/2 x result from the Blasius profile used in figure 5.7. The curve in figure 5.8 has been extended to Re x =2.5x10 6. The top dashed line is the C f =0.027Re 1/7 result from the self-similar 1/7 power law from a fully turbulent boundary layer. This is also the same curve as the top dashed lime in figure 5.8, extended to Re x =2.5x10 6. The full square symbols are measurements from Roach & Brierley (1990) archived in the ERCOFTAC experimental data bank. The experimental values for the T3AM test case follows the C f distribution for a laminar boundary layer over the range 0 < Re x < 1.8x10 6. The transition occurs over the range 1.8x10 6 < Re x < 2x10 6, which is significantly downstream than Re x = 1.35x10 5, which is the transition onset from the higher free-stream turbulence intensity test case T3A. The numerical prediction of the skin friction obtained by the in-house finite-volume scheme with the Suzen and Huang (2000) transition model are shown in figure 5.9 by the continuous line. The numerical scheme seems to respond correctly to the reduction in free-stream turbulence intensity from the T3A case to the T3AM case, predicting a delay in transition onset from Re x = 1.3x10 5 to approximately Re x = 1.5x10 6. There is however a noticeable difference between the transition length predicted by the numerical model and the one reported by the measurements. It is possible that the low free-stream turbulence intensity of the T3AM test case may have caused the flow to undergo natural transition and a fully developed turbulent boundary layer state is not reached before the end of flat plate in experiment. 97

98 Cf T3AM experimental Laminar Turbulent computed E E E E E E+06 Re x Fig. 5.9: Comparison of the experimental and skin friction coefficient against computational results for the T3AM test case The Laminar Kinetic Energy Approach The model is a three-equation eddy-viscosity type, including transport equations for specific turbulent kinetic energy ( ത), specific laminar kinetic energy ( ത l ), and specific turbulent kinetic energy dissipation rate ഥ. These are ( ത ߩ) ݐ ൰ ത൨ ߙߩ + ߤ൬ + ܦ ߩ ത ഥ ߩ ௧ ߩ + ߩ + ߩ = ( ഥ ത ߩ) + ߪ (5.63) ( ߩ) ത ߤ + ܦ ߩ ௧ ߩ ߩ ߩ = ( ഥ ߩ) + ݐ (5.64) ( ഥ ߩ) ݐ ഥ ഥ ߩ + ఠ ܥҧߩ + ҧ ఠߩ = ഥ ത ( + ௧) ܥҧߩ ఠ ଶ ഥ ଶ ߣ ቆ ߙ ఠ ଷ ఠ ܥ ߩ + ඥ ത ቇ P k, R, R Nat are defined in section ߣ ସ ଷ ݕ ଷ ߙ + ߤ൬ + ൰ ഥ ఠ ߪ Transport equation for the turbulent kinetic energy (5.65) The first term P k in equation (5.63) models the production of turbulent kinetic energy due to turbulent fluctuations (work done by the mean strain rate) and represents the rate 98

99 at which kinetic energy is transferred from the mean flow to the turbulence. This is modelled following the Boussinesq approximation as = ௦, ߥ ൬ ഥ + ( ഥ ) 2 3 : ഥ൰ ഥ 2 3,௦ ഥ (5.66) where ν T,s is the turbulent kinetimatic viscosity (5.67) ൰, 2.5 ߣ ఓ ඥ,௦ ܥ൬ ఓ ௦, ߥ = ଶ where S is the magnitude of the of mean strain rate tensor : ൫2, ൯ C μ the turbulent viscosity coefficient and λ eff is the effective turbulent length scale threshold for the small scale turbulence, as explained in sec The limit on the turbulent kinematic viscosity is imposed to limit turbulent kinetic energy production rate in the case of boundary layer separation and in highly strained free-stream regions. In equation (5.67) f μ and f INT are damping functions used to impose near-wall viscous effects and to prevent the over-prediction of momentum and scalar transport in the later stage of bypass transition (a weakness of the previous model Walters &Leylek 2004), respectively. ఓ = 1 ቆݔ ඥ,௦ ఓ ܣ ቇ (5.68) = ൬, 1൰ (5.69) ܥ,௦ =,௦ ଶ ߝߥ (5.70) Re T,s is the turbulent Reynolds number for the small, turbulence producing scales, A μ =6.75 and C INT =0.75. The turbulent kinetic energy k can be divided into small-scale energy, k T,s and largescale energy k T,l.,௦ = ቆ ߣ ቇ ߣ ଶ ଷ ;, = 1 ቆ ߣ ቇ ߣ ଶ ଷ (5.71) λ eff is the effective length scale. It is the minimum length scale of the eddies contributing to the production of the non-turbulent flow fluctuations. λ eff is estimated as (5.72) ( ߣ, ݕ ఒ ܥ) = ߣ where C λ =2.495, λ T =k 3/2 /ε is the turbulent length scale, ε=ωk is the turbulent dissipation rate and y n is the distance from the nearest wall. 99

100 One issue with predicting transition using a low Reynolds number turbulence model is that some model constants are calibrated on a fully developed turbulent flow. For this model C μ, is taken to be 0.09 in the fully developed turbulent boundary layer, otherwise the turbulent kinematic viscosity coefficient is calculated as with A 0 =4.04 and A S =2.12. = ఓ ܥ 1 ቀ ܣ + ܣ (5.73) ቁ ߝ The second term R in equation (5.63) represents the effect of the velocity streamwise fluctuation on the specific turbulent kinetic energy during bypass transition. It is specifically the rate of production of turbulent kinetic energy by laminar kinetic energy during the bypass transition process. It appears with opposite sign in equation (5.63) and equation (5.64), resulting in no net change of total fluctuation energy. Its presence in Equation (5.65) causes a decrease of turbulence length scale from the initial (approximately free-stream) value to a value dependent on the wall distance, since the length scale in the inertial range of a fully turbulent boundary layer scales linearly with the wall distance. Since the breakdown process is modelled as the production of turbulence by laminar fluctuations, the R term defined in equation (5.74) is assumed proportional to the laminar fluctuation energy k l and inverse to the effective turbulent time scale. It is intended to produce a reduction in turbulence length scale during the transition breakdown process ߣ ቆ ߚ ܥ = ߣ ቇ (5.74) The threshold function β BP controls the bypass transition process, defined as: where ൬ݔ 1 = ߚ ܣ ൰ (5.75) with the closing coefficients C R =0.05, A BP =3, and C BP,crit =12. = ቈቆݔ ݕ (5.76) 0,௧ቇ, ܥ It is assumed that the laminar fluctuations breakdown whenever the turbulent kinetic energy is greater than some threshold value, relative to the wall distance and fluid 100

101 kinematic viscosity. In other words, transition initiates when the laminar stream-wise fluctuations are transported at a certain distance from the wall, where that distance is determined by the energy content of the free stream, and by the kinematic viscosity. Additionally, the breakdown to turbulence due to instabilities is included as a separate natural transition production term R Nat in equation (5.63) R Nat has no dependence on the turbulent kinetic energy and is given by where: ௧ = ܥ, ௧ ߚ ௧ Ω (5.77) ൫ ௧ݔ ቈݔ 1 = ௧ ߚ.ହ.ଶହ ௫ ܥ ௧,௧, 0൯ (5.78) ௧ ܣ and ષ = ൫ඥ2ષ ષ ୧୨ ൯is the magnitude of mean rotation tensor. φ Nat is the natural transition parameter based on wall distance, strain rate, and kinematic viscosity with A Nat = 60. ߗ ଶ ݕ = ௧ ; ௫ = ඥ ݕ (5.79) The fourth term in equation (5.63) ω is the specific dissipation of turbulent kinetic energy. The fifth term D T in equation (5.63) is the turbulent near-wall dissipation term (5.80) ) )ߥ 2 = ܦ The last term ߤቂቀ + ఘఈ ఙ ቁ ቃmodels the diffusion of specific turbulent kinetic energy, where the turbulence scalar diffusivity ߙ is with C μ,std =0.09 and σ k = Transport equation for the laminar kinetic energy (5.81) ߣ ఓ,௦௧ ܥఓ = ߙ The first term P kl in equation (5.64) is the production of the specific laminar kinetic energy due to large-scale turbulent fluctuations: =, ߥ ഥߘ൬ + ( ഥߘ) 2 3 : ഥ൰ߘ ഥߘ 2 3 ߘ, ഥ (5.82) Where ν T,l is the large scale eddy kinematic viscosity, which is given by 101

102 Where is defined in equation (5.44) and, ߥ, ߥ, ߥ =, 0.5, ൨ (5.83) = ఛ, ܥ ଵ ቆ Ωλ ଶ (5.84) Ω ଶ ݕ ଶ ௧ ܥ ߚ + ߣቇඥ, ߥ The upper bound limit introduced in equation (5.83) ensures that realizability is not violated in a two-dimensional developing boundary layer. The left hand side of the min term in equation (5.83) contributes to production of fluctuations when only bypass transition is considered, while the right hand side contributes to the development of T-S waves. In equation (5.84) f τ,l is a time-scale based damping function ଶ ఛ, = 1 ܥ ቈݔ ఛ, ൬ ൰ (5.85). with τ m =1/Ω, τ T,l = ఒ ඥ,, C l,1 =3.4*10-6, and C τ,l =4360. The second term of the right hand side of equation 5.84 includes the followings:,,௧ ܥ 0൯ ଶ ൫ ௧ݔ ܯ ൭ݔ 1 = ߚ ൱ (5.86) ܣ The third term D L in equation (5.64) is the laminar near-wall dissipation term (5.87) ( ඥ ߘ ඥ ߘ)ߥ 2 = ܦ According to the definitions of D T and D L, the total specific dissipation of the fluctuation energy, ε TOT, is defined as the sum of the specific dissipation of turbulent kinetic energy ε and of the near-wall dissipation terms D T and D L Transport equation for the specific turbulent dissipation rate ω The first term P ω in equation (5.65) is the increase in inverse time scale of turbulence due to either turbulence production mechanisms or flow-field instabilities. It takes the form: ఠ = ഥ ത ܥ ఠ ଵ ఠ, ߥ ൬ ഥ + ( ഥ ) 2 3 : ഥ൰ ഥ 2 3,௦ + ഥ൨ ( ܥ. Ω) ഥ ത (5.88) The first term is due to the production of turbulent kinetic energy, where the effective turbulent kinetic viscosity (5.89) ߣ ఓ ඥ,௦ ܥఓ ఠ, ߥ = corresponds to the turbulent kinetic energy dissipation by the small-scales without any imposed limit. 102

103 The second term in equation (5.65) represents the increase in the inverse of the time scale of turbulence (ω) in unstable regions of the boundary layer to take into account the additional instabilities brought about due to adverse pressure gradients. This term is zero if the mean rotation Ω does not change with distance from the wall. The damping function f ΔP in equation (5.88) takes the form: = ఛ, Ω ݕ > 0 (5.90) with C ω1 =0.44 and C ΔP =0.15. = 0 Ω ݕ 0 (5.91) The coefficient C ωr in the second term of the equation (5.65) enforces a reduction of turbulent length scale during the transition breakdown and is given by ଶ ߣ ఠ = 1.5 ቆ ܥ ଷ ቇ 1 (5.92) ߣ The term C ω2 in the third term of equation (5.65) is assigned according the following functional form: ߣ ఠ ଶ = 0.92 ቆ ܥ ቇ ߣ ସ ଷ (5.93) This enforces a decrease in the turbulent length scale close to the wall. The use of ω as the scale-determining variable can lead to a reduced intermittency effect in the outer region of a turbulent boundary layer and, consequently, to the elimination of the wake region in the velocity profile. The third term on the right-hand side of equation (5.65) is included to rectify this. This third term includes a damping function that counter balance the dissipation term in equation (5.65). This dumping function is ఠ = ݔ ቆ ߣ ߣ ସ ቇ ൩ (5.94) and the closure contents in the third and last term in equation (5.65) are C ω3 =0.3 and σ ω =1.17. The influence of turbulent and laminar velocity fluctuations on the mean-flow momentum and energy equations is accounted for by defining a total eddy kinematic viscosity which is used to model the Reynolds- stress tensor per unit of mass (5.95) ഥ൰ ( ഥ ) + ൬ ഥ ߥ ఫ ݑ = തതതതതݑ ప 103

104 where ߥ = ௦, ߥ +, ߥ and = + =,௦ +, Model implementation in the 3D version of the in-house CFD code Cosmic The laminar kinetic energy approach has been implemented in the finite-volume scheme Cosmic detailed in section 4. The laminar kinetic energy model was accommodated into 3D parallel version of Cosmic through expanding the flow properties array by one further element to 8. The new equation has been implemented explicitly in the scheme. The upwind Riemann solver of Roe convects k L around the flow field. The new diffusion term has been implemented using Gauss divergence theorem around cell interfaces. All derivatives in the source terms are discretized by central differencing. At the start of the computation, k L is set to zero throughout the flow field. On solid wall boundaries the value of the gradient of k L is kept as zero, as well as the gradient of k and ω. In both the computations, the inlet condition is a constant velocity profile. The inlet turbulent kinetic energy is fixed according to the turbulence level and the inlet boundary conditions for k and ω are set as constant values, determined by matching the free stream turbulence (FST) intensity curve. The transport equations for the 3D RANS with the laminar kinetic energy approach are: ۴ ௫ + ݐ ۴ ௬ + ݔ ۴ ௭ + ݕ ௫ ߥ ۴ = ݖ ௫ ߥ ۴ + ݔ ௭ ߥ ۴ + ݕ (5.96) + ݖ U - flow properties array in conservation form Fc - flux vectors in x and y and z Fv - viscous flux vectors in x and y and z S - turbulence and intermittency source terms ݑߩ ௧௧ ߩ ଶ ݑߩ ݒݑߩ ݓݑߩ ݑ ߩߩ ݒߩ ଶ ݒߩ 3 ݓߩ ݓݒߩ = ܨ; = ߩ ߩ ܨ ; ߩ൬ݑ 3 = ߩ൬ݒ ௧௧൰ ߩ 3 ݒߩ ݑߩ ߩ ߩ ݑߩ ݒߩ ݑߩ ݒߩ ݒߩ ݒݑߩ 104

105 ݓߩ ݓݑߩ 0 ݓݒߩ ௫௫ ଶ + + ଶ ݓߩ ௫௬ ଷ ௧௧ ߩ ௫௭ = ܨ ௧௧ቁ ߩ + ଶ + ߩቀ ݓ = ఔ ܨ ; ଵ ௫ ܥ + ௧௧ ܥ + ௫௭ ݓ + ௫௬ ݒ + ௫௫ ݑ ଷ ݓߩ ଶ ௫ ܥ ݓߩ ߤ ݓߩ ଷ ௫ ܥ 0 0 ௫௬ ௫௭ ௬௬ ௬௭ ௬௭ ௭௭ ଵ ௬ ܥ + ௧௧ ܥ + ௬௭ ݓ + ௬௬ ݒ + ௫௬ ݑ = ఔ ܨ = ఔ ܨ ; ܥ + ௭௭ ݓ + ௬௭ ݑݒ + ௫௭ ݑ ௧௧ + ܥ ଵ ௭ ଶ ௬ ܥ ଶ ௭ ܥ ߤ ߤ ଷ ௭ ܥ ଷ ௬ ܥ ቃ ௫ ) ଶ ݓ + ௭ ݑ) + ଶ ௬ ൯ ݓ + ௭ ݒ൫ ௫ ൯ ଶ + ݒ + ௬ ݑቂ൫, ߥߩ ൯+ ௭ ݓ ௬ ݒ ௭ ݓ ௫ ݑ ௬ ݒ ௫ ݑ ௭ ଶ ݓ + ௬ ଶ ݒ + ௫ ଶ ݑ ൫, ߥߩ 3 ܦߩ ߩ ௧ ߩ + ߩ ൯+ ௭ ݓ + ௬ ݒ + ௫ ݑ൫ ௦, ߩ 6 = 4 ቃ ௫ ) ଶ ݓ + ௭ ݑ) + ଶ ௬ ൯ ݓ + ௭ ݒ൫ ௫ ൯ ଶ + ݒ + ௬ ݑቂ൫, ߥߩ ൯+ ௭ ݓ ௬ ݒ ௭ ݓ ௫ ݑ ௬ ݒ ௫ ݑ ௭ ଶ ݓ + ௬ ଶ ݒ + ௫ ଶ ݑ൫, ߥߩ 3 ܦߩ ௧ ߩ ߩ ൯ ௭ ݓ + ௬ ݒ + ௫ ݑ൫, ߩ 6 4 ߩ ఠ, ߥଵ ఠܥ ݑ൫ 3 ௫ ଶ + ݒ ଶ ௬ + ݓ ଶ ௭ ݑ ௫ ݒ ௬ ݑ ௫ ݓ ௭ ݒ ௬ ݓ ௭ ൯ ߩ+ ଶ ௬ ൯ ݓ + ௭ ݒ൫ ௫ ൯ ଶ + ݒ + ௬ ݑቂ൫ ఠ, ߥଵ ఠܥ ௫ ) ଶ ቃ ݓ + ௭ ݑ) + ݑ൫ ௦, ߩ 6 ௫ + ݒ ௬ + ݓ ௭ ܥ+ ൯ ఠ ( + ௧) ܥߩ ఠ ଶ ଶ + ܥߩ ఠ ଷ ఠ ߙ ൬ ߣ ସ ଷ ൰ ଷݕ Zero pressure gradient transitional boundary layer results The computational mesh is made of almost half million cells. For accuracy, grid points are clustered in regions of large gradients, close to the wall and at the leading edge of the flat plate, and are spread out elsewhere for computational economy. The stretching factor has been chosen in order to have a value of y+ < 0.2 for the first point away from the wall, and such that the viscous region of the boundary layer (y+<30) contains more than 40 cells, because the wall-function law cannot be used because using this, it is not possible properly resolve the laminar boundary layer. In figure 5.10, the flow runs from left to right. The computational domain streamwise boundaries are subsonic inflow to the left, subsonic outflow to the right. The subsonic inflow boundary is located 150mm upstream of the flat plate physical leading edge and 1700mm downstream of it for both test cases. ߣ 105

106 N. of cells Stretching ratio Stretching factor 0<x<l.e. 222 proportional l.e.<x< t.e. 888 proportional <z<top 143 proportional front<y<back 3 uniform 1.0 Table 5.1: Table of stretching factor. A slip wall before the physical leading edge and a non-slip boundary condition along the wall to the bottom and far-field along the top. At front and back, a symmetry boundary condition is applied. The inlet turbulent kinetic energy is fixed according to the free-stream turbulence level and the specific dissipation rate for each case are adjusted according the downstream decay of the free-stream turbulence level. The location of the upstream inflow boundary allowed to resolve the flat plate leading edge stagnation point. This allowed for a natural stagnation of the free-stream flow and boundary layer start. One advantage of this geometry model is the elimination of any need to prescribe an inflow boundary layer profile. A uniform inflow velocity is imposed across the upstream boundary and the boundary layer develops entirely without an imposed profile prescription. z y Fig 5.10: Computational mesh. The total length of the test section is 1850mm; 150mm before the leading edge. x The ERCOFTAC T3A test case The flat plate trailing edge Reynolds number is 6x10 ହ. Figure 5.11 shows the variation of the skin friction coefficients along the flat plate of the T3A experiment. The skin friction coefficient is defined as in equation (5.62). The streamwise coordinate is shown in the non-dimensional form of Re x. Figure 5.11 uses the same notation as figure 5.8. Re x = 0 denotes the flat plate leading edge as opposed to the computational domain inflow, which is upstream of the leading edge by a stand-off distance of x=150mm, as previously stated in this section. 106

107 As in figure 5.8, in figure 5.11 the green dashed line towards the bottom of the graph is the Blasius results for a laminar boundary layer C f =0.664Re -1/2 x. The top dashed line is the streamwise variation of the friction coefficients from a power law approximation for a fully turbulent boundary layer, reported C f =0.027Re 1/7. The experimental data from the T3A test, as shown by the blue full square symbols, follow closely the experimental C f curve from Blasius upstream of Re x =10 5 whereupon C f is measured to increase towards the power law approximation. The measured values of skin friction are shown in figure 5.11 to slightly overshoot the fully turbulent C f dashed line at approximately Re x =2.8x10 5, so that the experimental C f approaches the power law asymptotically from above the dashed line from the power law. The numerical prediction from the in-house finite-volume scheme with the Walter and Leylek (2004) transitional model gives a skin friction coefficient distribution that has an appreciable overlap with the experimental data. Over the range 10 4 < Re x < 10 5, the numerical prediction follows the laminar boundary layer C f from Blasius. Above Re x 10 5, the numerical prediction starts to depart from the laminar C f profile, indicating the onset of transition, in agreement with the measurements. Whereas the onset of transition is a statistical process that occurs over a finite length of the flat plate, it is common to indicate a transition onset point, or line, which is estimated as Re x 1.35x10 5 in both experiment and computation. Thereafter, the numerical prediction follows the same trend as the measurements Computed Experimental Turbulent Laminar Cf E E E E E E E+05 Re x Fig 5.11: Comparison of the experimental skin friction coefficient against computational results for the T3A test case. 107

108 The ERCOFTAC T3AM test case The end plate Reynolds number is Re L and the free-stream turbulence intensity is 0.98%. Figure 5.11 shows the variation of the skin friction coefficient versus the normalised distance from the flat plate leading edge Re x. In figure 5.12 it is also reported the streamwise variation of the friction coefficients from a fully turbulent simulation using the Menter SST turbulence closure. It is the continuous yellow line that it is in very good agreement with the turbulent analytical solution. The SST turbulence model of Menter (1992) always produces a fully turbulent feature and cannot detect any effect of transition. The experimental values for the T3AM test case follows the C f distribution for a laminar boundary layer over the range 0 < Re x < 1.5x10 6. The transition occurs over the range 1.5x10 6 < Re x < 2x10 6, which is significantly downstream than Re x = 1.35x10 5, which is the transition onset from the higher free-stream turbulence intensity test case T3A. The numerical prediction of the skin friction obtained by the in-house finite-volume scheme with the Walter and Leylek (2004) transition model are shown in figure 5.12 by the red continuous line. The numerical scheme respond correctly to the reduction in free-stream turbulence intensity from the T3A case to the T3AM case, predicting a delay in transition onset from Re x = 1.35x10 5 to approximately Re x = 1.5x10 6. Over this Re x value the calculated Cf is in good agreement with the experimental data SST Experimental Computed Laminar Turbulent Cf E E E E E E+06 Re x Fig 5.12: Comparison of the experimental skin friction coefficient against computational results for the T3AM test case. 108

109 5.6. Comparison between the two models After having investigated both models, it is possible to state that two transitional flow models have shown to have the capability of accurately predicting transitional flows under a moderate FST intensity, but the Walter and Leylek model provides the most accurate predictions for this T3A test case (see figure 5.13). Both transition model predicted the length of the transition quite well but the the skin friction plot shows that the intermittency model predicted slightly early transition to turbulent flow when compared to the experiment Experimental Turbulent Laminar S&H W&L Cf E E E E E E E+005 Fig 5.13: Comparison of the predicted skin friction coefficient for the T3A case. Moreover, the laminar kinetic energy method is more reliable with respect to the intermittency transport method when the flow-field is subjected to a lower free-stream turbulence intensity case. The laminar kinetic energy model predicts satisfactory both onset and length of transition compared to the experimental value. Using the intermittency transport method, the onset of transition from the computations is much earlier compared to the experimental value and the transition length is too short in this case as seen in figure Re x 109

110 experimental Laminar Turbulent S&H W&L Cf E E E E E E+06 Re x Fig 5.14: Comparison of the predicted skin friction coefficient for the T3AM case. Figure 5.15 and figure 5.16 show the predicted momentum thickness Reynolds number Re θ against the experimental value for the T3A and T3AM test cases respectively. This factor is directly related to the momentum thickness of the boundary layer which indicates the development of the boundary layer. In case of the boundary layer flow on a flat plate with zero pressure gradient, the boundary layer is laminar in the entrance region of the flat plate, and becomes transitional and then turbulent. The momentum thicknesses, from analysis, of laminar and turbulent boundary layers can be determined from ߠ =.ସ௫ ඥ and ߠ ௧௨ =.ଵହହହ௫ which are the upper and lower dash lines, భ ళ respectively, so that the values for the transitional boundary layer can be found in the region between both dash lines. In the profile of the momentum thickness Reynolds number along the flat plate, the transition starts at the point where the profile deviates from the laminar line, and ends at the point where the profile touches the turbulence line. The experimental data indicate that the transition starts at Reynolds numbers of 1.50x10 6, 1.35x10 5 and corresponding to the momentum thickness Reynolds numbers of 820 and 272 for T3AM and T3A cases, respectively. At the end of transition, the experimental data show that the transition ends at the Reynolds number of about 3 x10 5 while for the T3AM case the development of the boundary layer from laminar to turbulent flow is not yet complete within the length of the flat plate, 170 cm, and hence no ending point of transition appears. 110

111 Experimental Laminar Turbulent W&L S&H Re θ E E E E E E E+05 Re x Fig. 5.15: Variation of Reynolds number based on momentum thickness along the flat plate for T3A case. From figure 5.15 is clear again that both model give reasonable results compared with the experimental data for the moderate freestream turbulence intensity T3A test case Experimental Laminar Turbulent W&H S&H Re θ E E E E E E+06 Re x Fig. 5.16: Variation of Reynolds number based on momentum thickness along the flat plate for T3AM case. As can be seen also from figure (5.16), with the model of Suzen and Huang (2000), the deviation of the momentum thickness Reynolds number profile from the laminar line shows an early onset of transition in the case of low freestream turbulent intensity, 111

112 conversely to the laminar kinetic energy, which show a good agreement with the experimental value. The laminar kinetic energy approach has the advantage of an easier inflow condition set-up, in fact, an advantage of the model is the elimination of any need to prescribe initial boundary layer profiles, and indeed the boundary layer develops entirely from a free-stream prescription. 112

113 6. Code parallelization using MPI Multi-processors HPC clusters are nowadays extensively used for solving CFD problems. Specifically, medium ( 20 x 10 6 cells) and large (> 30 x 10 6 cells) CFD test cases cannot be run without modern HPC facilities. The two main-stream parallelization methods are Message Passing Interface (MPI) and Open Multi-Processing (OMP). MPI for code parallelisation is highly recommended with respect to OMP because it is more flexible and does not need a shared-memory cluster. The flow solver is parallelised using MPI and it was tested on two distributed-memory HPC clusters. MPI (Message- Passing Interface) is a message-passing library interface specification. MPI addresses primarily the message-passing parallel programming model, in which data is moved from the address space of one process to that of another process through cooperative operations on each process. (Extensions to the \classical" message-passing model are provided in collective operations, remote-memory access operations, dynamic process creation, and parallel I/O.) MPI is a specification, not an implementation; there are multiple implementations of MPI. This specification is for a library interface; MPI is not a language, and all MPI operations are expressed as functions, subroutines, or methods, according to the appropriate language bindings, which for C, C++, Fortran-77, and Fortran-95, are part of the MPI standard Single domain decomposition The computational domain is built as an assembly of individual three-dimensional, curvilinear and topologically orthogonal computational zones in (i, j, k) as shown in figure 6.1a. A two-cell deep layer of ghost cells surrounds each zone where outer zone connectivity information is updated at each Runge-Kutta time integration. Zones with updated ghost cells are independent and can be integrated in parallel. A first level of parallelization is relative to the communication of the processes within a single block called intra-communication (see figure 6.1b). The setup of these communications relies largely on functions, provided by MPI, that allow to automatically subdividing a structured mesh in smaller subsets of data assigned to different processors. While the domain is decomposed, a set of information is also provided to each process (essentially the position of the process itself within the structured block and its local rank) to allow the exchange of messages between neighbouring processes. 113

114 Consider any of 12 zones that make up the computational domain, shown in figure 6.1a. Given the number of processors available in the cluster, each zone is sliced into blocks along k, hence the name single domain decomposition (SDD). The k-direction is chosen as it coincides with the outermost variable pointer for a given zone in the FORTRAN programming language, such as for U (i, j, k). Variables belonging to a k-slice are contiguous in memory and are faster to be sent and received among other processors, as they do not need buffering. During the initialisation of a test-case, three MPI TYPE arrays are defined to reduce the overhead related to exchanging data across the distributed memory architecture, respectively for k, j and an i planes. SDD was implemented using MPI assuming the processor memory to be sufficient to include all the computational variables of all zones of the CFD problem during run time. SDD can also be implemented using OMP on a shared-memory cluster but also works for small CFD computations on the distributed memory platforms used in this work. a) 8 (0,2) 9 (1,2) 10 (2,2) 11 (3,2) 4 (0,1) 5 (1,1) 6 (2,1) 7 (3,1) b) 0 (0,0) 1 (1,0) 2 (2,0) 3 (3,0) Fig. 6.1: Example of the decomposition of one block model via SDD MPI Cartesian communicators. Figure 6.2 shows a simple 2D domain divided into three computational blocks. A blue dashed edge, a red continuous edge and a black dash-dot edge are used to identify these zones that are denoted as zone 1, 2 and 3. The k-direction coincides with the vertical direction in this example. The zones are evenly sized but have different aspect ratios, giving a different number of cells along the k-direction. In this example, only two 114

115 processors, 0 and 1, are used. A dotted line separates the computational blocks computed by processors 0 and 1 respectively. To compute the convective fluxes using the flow solver, as described in paragraph 4.5, four contiguous cells are needed in all directions. Green hashing is used in figure 6.2 to highlight the data that are transferred from processor 0 to processor 1 at each Runge-Kutta step (see section ). In this example, a symmetric situation characterises the data exchanged in processor 1. All the cells of zone 1, computed by processor 0, are transferred to processor 1 and all the cell of processor 1 of zone 2 are transferred to processor 0. This is a bottleneck situation for the SDD. SDD is generally representative of the best parallelisation strategy for a low number of zones that are characterised by a large number of cells. Fig. 6.2: Exchanged cells between processor 0 and processor 1 in green Recursive domain decomposition SDD requires a large memory allocation and its inter-block communication overhead limits it to a small CFD computation. Recent HPC clusters are assembled to minimise the communication time among processors but computer cores do not have more memory than an ordinary home desktop. Nowadays shared memory cluster are also being replaced by large distributed memory HPC clusters. To use these clusters to their full potential, a more complex parallelization strategy is implemented. The second level of parallelization aims to allow the communication between processes that are part of two separate abutting blocks. Due to the fact that the two connected zones can have different indexes this level of parallelization is much more problematic then the previous and its implementation can be performed in a number of ways following different approaches. 115