Investigation of secondary ow in low aspect ratio turbines using CFD. Henrik Orsan Royal Institute of Technology, , Stockholm

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1 Investigation of secondary ow in low aspect ratio turbines using CFD Henrik Orsan Royal Institute of Technology, , Stockholm March 23, 2014

2 ii Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Abstract In this thesis, secondary ow in a two stage, low aspect ratio turbine is investigated using CFD. A parameter study is carried out to investigate how the turbine performance is aected by the choice of aspect ratio. This is done in two steps, rst by changing the blade height and then the blade size. The study shows that increasing the aspect ratio will lead to a signicant increase of eciency, but the eect diminishes for large aspect ratios, at which the eciency moves towards an asymptotic value. Furthermore it is shown that increasing the aspect ratio to a certain value by changing the blade height results in a higher eciency compared to changing the blade size, which is due to the dierence in hub-to-tip ratio. An attempt to quantify the secondary losses is also made by looking at the radial kinetic energy at the outlet of a blade row. It turns out though, that the radial kinetic energy does not follow the same trend as the total pressure loss coecient, which implies that it can not be used to quantify the secondary losses. Lastly, an eort to improve the method used for generating blade proles is made, and the updated method is used to redesign rotor 2 to reduce losses.

3 iii Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Preface Ever since CFD became mature enough to be used in the industry, its applications have increased steadily. The benets of using CFD are many, and today it is used in many dierent industries, from life science to aerospace. A major benet is that it is possible to perform studies of many dierent geometries much faster and cheaper compared to doing an experiment. Despite the fact that turbines have been around for a long time, the ow in turbines with low aspect ratios is still not fully understood. So in an eort to get a better understanding of these types of turbines, CFD was used to investigate secondary ow in low aspect ratio turbines. This thesis was performed at the aerothermodynamics department at Volvo Aero Corporation in Trollhättan during the winter of 2011 and spring of NOTE: This version of the thesis has been censored for sensitive information.

4 iv Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Contents 1 Introduction 1 2 Objectives 1 3 Method of attack 2 4 Theory Fluid mechanics Governing equations Boundary layer Turbulence Computational uid dynamics (CFD) Introduction The numerical approach Turbulence modeling Wall functions Performance of turbines Turbine eciency Stage design parameters Bladerow performance What is secondary ow? Why aspect ratio? Current loss models 16 6 Setting up the simulations The turbine Meshing Boundary conditions and gas properties

5 v Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 7 Results from the simulations Single stage turbine Two stage turbine Inuence of aspect ratio Introduction Changing the blade height Changing the blade size Putting it all together Eect of reducing the mean radius Blade design Introduction Improving the prole design Redesigning rotor Some thoughts on the sources of error Conclusions Future work References 46

6 vi Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 List of Figures 1 Typical velocity prole for a boundary layer Illustration of the law of the wall. [CFD Online, 2013] Example of the ow eld in a turbine. [Denton, 1994] Formation of a horseshoe vortex in a turbine. [Sieverding, 1984] Traces in the snow caused by a horseshoe vortex. [Vogt, 2009] Passage vortex in a turbine. [Sieverding, 1984] Separation of a laminar boundary layer. [Lakshminarayana, 1996] Tip leakage in a turbine. [Denton, 1993] Tip clearance eect over a shrouded rotor. [Denton, 1994] Geometry of the turbine used Complete mesh of the turbine Radial velocity component at the rotor trailing edge Streamlines in the rotor Radial velocity component at rotor 1 TE for single stage (left) and two stages (right) Streamlines in the two stage turbine Mach number (left) and total pressure (right) for the two stage turbine Total temperature (left) and entropy (right) for the two stage turbine Turbine eciency as function of blade height Flow coecient (left) and loading coecient (right) for stage 1 and Eciency of stage 1 and Total pressure loss coecient for all blade rows Relative friction losses for all blade rows Relative radial kinetic energy at the outlet of all blade rows Entropy in the reference turbine (left) and a blade height of 860 % of the reference height (right) Radial velocity at the trailing edge of rotor 2 for the reference turbine (left) and a blade height of 860 % of the reference height (right) Entropy at the trailing edge of rotor 2 for the reference turbine (left) and a blade height of 860 % of the reference height (right)

7 vii Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 27 Span wise total pressure at the trailing edge of rotor 2 for the reference turbine (left) and a blade height of 860 % of the reference height (right) Turbine eciency as function of blade size Inuence of blade size on the ow coecient (left) and loading coecient (right) Eciency of the two stages as function of blade size Total pressure loss coecient for all blade rows Relative friction losses (left) and relative radial kinetic energy at the trailing edge (right) for all blade rows Entropy in the reference turbine (left) and for a blade size of 200 % (right) Turbine eciency as function of aspect ratio (based on turbine length) Interface of the MATLAB program used to create blade proles Pressure distribution for the original rotor 2 (left) and redesigned rotor 2 (right) at 50 % span

8 viii Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 List of Tables 2 Data for the turbine Cell data for the turbine Operating conditions for the o-design points Performance of the rst stage (normalized with data for the ADP) Performance of rotor (normalized with data for the ADP) Radial kinetic energy at the stator trailing edge Radial kinetic energy at the rotor trailing edge Performance of the two stage turbine (normalized with data for the single stage turbine) Performance of stages (normalized with data for the single stage turbine) Performance of blade rows Performance data for the modied turbine (normalized with data for the reference turbine) Performance data for the modied turbine at all operating points (normalized with data for the ADP) Performance of stages for the modied turbine at the ADP (normalized with data for the reference turbine) Total pressure loss coecient for the dierent blade rows at the ADP (normalized with data for the reference turbine) Total pressure loss coecient for a turbine with redesigned rotor 2 (normalized with data for the reference turbine) Performance of a turbine with redesigned rotor 2 (normalized with data for the reference turbine)

9 ix Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Nomenclature Symbols X = (x, y, z) Position vector [m] ρ Density [kg/m 3 ] t Time/Blade thickness [s]/[m] V = (u, v, w) Velocity vector [m/s] V = (u, v, w) Mean velocity [m/s] v = (u, v, w) Instantaneous velocity [m/s] p Pressure [P a] µ Dynamic viscosity [P a s] c p Specic heat [J/kg K] T Temperature [K] k Thermal conductivity/turbulent kinetic energy [W/m K]/[m 2 /s 2 ] R Specic gas constant/degree of reaction [J/kg K]/[-] η Kolmogorov length scale/eciency [m]/[-] ν Kinematic viscosity [m 2 /s] ɛ Turbulent dissipation rate [J/kg s] y + Dimensionless wall distance [-] u τ Friction velocity [m/s] u + Dimensionless velocity [-] δ ij Kronecker delta [-] C µ Constant in the k ɛ model [-] d DES limiter [m] d Distance from the wall to the grid cell (DES) [m] C DES Constant used in DES [-] = max( x, y, z) Largest dimension of the grid cell [m] κ Von Karman constant (law of the wall) [-] B Constant used in the law of the wall [-] W Work [J] h Enthalpy [J] Φ Flow coecient [-] c Velocity in turbine [m/s] U Blade speed [m/s] Ψ Loading coecient [-] Y T otal Total pressure loss coecient [-] Z Zweifel coecient [-] ṁ Mass ow [kg/s] b Axial chord [m] H Blade height [m] AR Aspect ratio [-]

10 x Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Re Reynolds number [-] χ T e Trailing edge coecient (Ainley & Mathieson) [-] χ i Incidence coecient (Ainley & Mathieson) [-] β Relative ow angle [º] C L Lift coecient [-] C Constant used in Ainley & Mathieson [-] τ Radial tip clearance [m] ah Annulus height [m] χ Re Reynolds correction factor (Kacker & Okapuu) [-] N Number of revolutions [rpm] P R Pressure ratio [-] M Molar mass/mach number [kg/kmol]/[-] C f Skin friction coecient [-] A wet Wet area [m 2 ] A annulus Annulus area [m 2 ] q Dynamic pressure [P a] r m Mean radius [m] C p Pressure coecient [-] Subscripts and superscripts ' Blade 0 Total/Stagnation 1 Blade row inlet 2 Blade row outlet ijk Index (tensor notations) is Isentropic tt Total-to-total m Meridional θ Circumferential y y-direction P Prole S Secondary Tl Tip leakage Te Trailing edge ts Total-to-static abs Absolute rad Radial rel Relative

11 xi Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Abbreviations ADP CAD CAM CAE CFD DES DNS KE LE LES ODE OP PDE RANS SGS TE VAC Aerodynamic design point Computer aided design Computer aided manufacturing Computer aided engineering Computational uid dynamics Detached eddy simulation Direct numerical simulation Kinetic energy Leading edge Large eddy simulation Ordinary dierential equation Operating point Partial dierential equation Reynolds average Navier-Stokes Subgrid-scale Trailing edge Volvo Aero Corporation

12 1 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 1 Introduction Turbines are an important part of our modern society. They are used in jet- and rocket engines as well as to produce electricity, to name a few examples. One of the many challenges engineers face during the development of new turbines is the fact that the eciency is not known beforehand, and because of this, the design has to be iterative to achieve the desired performance. Some important tools used during the design process are various models used for estimating the losses. Part of the loss in a turbine is due to a phenomenon called secondary ow, which is dened as any part of the ow that is not in the direction of the core ow, e.g. vortices. The interaction of the secondary ow with the core ow will result in mixing and dissipation (generation of entropy), which account for the losses. Secondary ow exists in all turbines, but if the aspect ratio of the blades is large, it will only have a limited impact on the ow eld. If this is the case, the current loss models used during the design of a turbine (see for example [Ainley & Mathieson, 1951]) work well and provide good estimates of the losses. Unfortunately, the same can not be said about turbines with low aspect ratios. Here, the secondary ow will have a large impact on the ow eld, and the current models do a poor job of predicting the losses. Volvo Aero Corporation (VAC) has an interest in the design of low aspect ratio turbines. In order to improve the design process and make it more ecient, VAC strives to develop new mathematical models for the secondary ow and the associated losses, which can be used in the early stages of development to get a good estimate of the performance of the nished turbine. A rst step in the process to develop these new models - and the objective of this thesis - is to get an understanding of secondary ow, how it is aected by the choices made during the design process and operating the turbine at o-design conditions. 2 Objectives The main objective of this Master Thesis is to get an understanding of how changes in dierent design parameters aect the secondary ow in turbines, and thus also the performance. Specically, the inuence of aspect ratio and the eects of operating the turbine at dierent o-design conditions will be investigated. The study is divided into several steps, ˆ performance of a single stage turbine will be evaluated at dierent operating points, and the secondary ow will be visualized ˆ a two stage turbine will be evaluated at its aerodynamic design point (ADP), and the results will be compared to the results for the single stage turbine ˆ the inuence of aspect ratio will be investigated by comparing two stage turbines with dierent blade heights and sizes ˆ the eects of reducing the mean radius will be investigated.

13 2 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 At the end of this thesis, the knowledge gathered in the earlier steps will be used to try to design blades with good performance and reduced secondary losses. Changes to the current work ow when designing blades will also be proposed, in an eort to make the design process more ecient. This is a numerical study and the results from the simulations are assumed to be exact. Obviously this is not completely true, there will always be some dierences between numerical simulations and experimental data. In a rst study of secondary ow like this though, the results from the simulations are suciently accurate to draw preliminary conclusions and recommend future research topics. It should also be noted that it is a generic study. It is based on typical turbine data, not specic for an actual machine. 3 Method of attack Doing accurate simulations of the ow in a turbine is not an easy task due to the complexity of the ow eld and the fact that the ow is viscous, three-dimensional, compressible and unsteady. Or in other words, the complete Navier-Stokes equations have to be solved, and for this purpose a high-end CFD solver is needed. In this study, the software that will be used is CFX. It is a general purpose uid dynamics solver developed by Ansys, Inc. 1 capable of solving a wide range of dierent uid ow problems. The turbulence model that will be used is the classical k ε model, a two equation model that adds two transport equations for the turbulent kinetic energy k and the turbulent dissipation ɛ. When designing blades, an Excel-sheet provided by VAC will be used to generate the basic prole of the blade. The prole will then be imported to NX, an advanced CAD/CAM/CAE software package developed by Siemens PLM Software 2 where a script will be used to generate the complete geometry. Meshing the geometry will be done in Icem CFD, an advanced meshing software developed by Ansys, Inc. A script will be used to generate a hexa mesh which is used in the CFD simulations. To save time, changes that involve scaling the geometry in one or more directions without modifying the blade prole, such as changing the height of the blade, will be made in MATLAB 3. The mesh le generated in ICEM CFD is an ASCII le, and the coordinates of the nodes can easily be imported to MATLAB and modied

14 3 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 4 Theory 4.1 Fluid mechanics Governing equations All ows are governed by the Navier-Stokes equations (named after C. L. M. H. Navier and Sir George G. Stokes), which are derived by applying Newton's second law to a uid particle. The compressible version of these equations for a Newtonian uid (constant viscosity) are often written together with the continuity equation, the energy equation, and the ideal gas law (body forces and bulk viscosity are neglected) [Alfredsson & Burden, 1998], ρ + (ρv) = 0 (1) t ρu t +u ρu ρu ρu +v +w x y z = p x + ( 2µ u x x 2 ) 3 µ V + ( ( u µ y y + v )) + ( ( u µ x z z + w )) x (2) ρv t +u ρv ρv ρv +v +w x y z = p y + ( ( v µ x x + u )) + ( 2µ v y y y 2 ) 3 µ V + ( ( v µ z z + w )) y (3) ρw t +u ρw x ρw ρw +v +w y z = p z + ( ( w µ x x + u )) + ( ( w µ z y y + v )) + ( 2µ w z z z 2 ) 3 µ V (4) DT ρc p Dt = Dp ( k T ) + Φ (5) Dt p = ρrt, (6) where Φ is a collection of terms that describes the viscous dissipation of kinetic or mechanical energy to inner energy. The six unknowns are u, v, w, p, ρ and T, which fully dene the state of the uid at any point in the domain. Unfortunately, these equations form a system of nonlinear partial dierential equations (PDEs), which lacks an analytical solution except for a few special cases. To solve the complete Navier-Stokes equations for arbitrary ows, numerical methods have to be employed.

15 4 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Boundary layer When the Navier-Stokes equations were rst derived, they were considered to be too complex, and could not really be used to analyze arbitrary ows. But this changed in 1904 when Ludwig Prandtl published a paper where he argued that for uids with small viscosity (e.g. water and air), it is possible to split the ow into a nearly inviscid outer region and a thin viscous layer close to solid surfaces (walls), a so called boundary layer [White, 2008]. The boundary layer is characterized by a velocity deciency compared to the free stream, and its thickness is dened as the normal distance from the wall to the point where the local velocity is 99 % of the free stream. A typical velocity prole for a boundary layer is shown in Figure 1. Figure 1: Typical velocity prole for a boundary layer. Turbulence If the Reynolds number for a laminar ow is increased, the ow will eventually transition from laminar to turbulent. This is an instability phenomenon, caused by small disturbances in the ow eld which grow by extracting energy from the laminar ow instead of being damped out by the ow. Turbulent ows are characterized by an unsteady and three-dimensional ow eld with seemingly random motion. Due to the chaotic nature of turbulent ows, they are very good at mixing momentum, thus leading to increased losses compared to laminar ows. Turbulent ows also exhibit a large separation of scales, both in terms of time and length, and as the ow evolves, turbulence is constantly produced and destroyed in parallel processes. The turbulent structures - so called eddies - are unstable, and break down into smaller and smaller sizes, until they are so small that they dissipate into heat, causing losses. The smallest scales present in turbulent ows are called Kolmogorov scales, and the length scale is dened as [Pope, 2000] ( ) 1 ν 3 4 η =. (7) ɛ

16 5 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Although the Navier-Stokes equations fully describe the motion of turbulent ows, it is convenient to simplify the equations and look at the mean motion of the ow due to the random nature of turbulence. This is done by assuming that the instantaneous velocity can be expressed as uctuations around a mean velocity, known as Reynolds decomposition, V = V + v. (8) It is then possible to derive the Reynolds-averaged Navier-Stokes (RANS) equations by inserting the above relation into the Navier-Stokes equations and taking the mean. equations are written as (using Einstein notations), V i t + V j V i x j The incompressible RANS = 1 p + ν 2 V i v iv j. (9) ρ x i x j As seen above, the RANS equations look the same as the Navier-Stokes equations, except the term v i v j, which are called Reynolds stresses. The problem with this approach to turbulent ows is that even though the equations have been simplied by averaging the dierent ow properties, the appearance of the Reynolds stresses introduces six more unknowns to the system, but no extra equations. This is known as the closure problem. The RANS equations are unclosed, and can not be solved unless the Reynolds stresses are determined by providing additional information. It should also be noted that even though the Reynolds stresses aect the ow as if they were true stresses, they are not (much like the dynamic pressure, which is not really a pressure, but rather kinetic energy per unit volume). The physics behind the Reynolds stresses is not the same as behind other (true) stresses. Viscous stress can be related to other ow properties through equations which depend only on uid properties (viscosity in this case), while the Reynold stresses arise from the ow itself, due to the uctuating velocity eld. When talking about turbulent boundary layers, a dimensionless length scale y +, is dened, y + = u τ y ν. (10) The reason for this is that it turns out that the relative importance of the viscous- and Reynolds stresses show a strong correlation to the value of y + for a wide range of dierent ows. Close to the wall, in the so called viscous sublayer (y + < 5), the dominating contribution to the total stress is from the viscous stress. On the other hand, in the log-law region (y + > 30), it is the Reynolds stresses that are the dominating contribution. The region in between is a transition region, called the buer layer. Alongside the dimensionless length scale, a dimensionless (average) velocity, u +, is dened, u + = u u τ. (11)

17 6 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/ Computational uid dynamics (CFD) Introduction As mentioned earlier, the Navier-Stokes equations form a system of nonlinear PDEs which lack an analytical solution. One way of approaching this problem is to try to nd ways to simplify the equations to make them easier to deal with. The risk of doing this however is that too simplied equations may not be able to account for important physical properties of the ow. If the ow is assumed to be inviscid, steady and irrotational (potential ow), the Navier-Stokes equations are reduced to the Laplace equation, and Bernoulli equation, which both can be solved analytically. Unfortunately, these equations have limited use in most engineering applications since they are too simple to account for the presence of vortices and viscous eects of the ow, such as boundary layers. A much more common simplication used in the industry today, which still holds if vortices are present, is to only assume inviscid ow. This reduces the Navier-Stokes equations to the so called Euler equations. Although simpler, these equations are still quite formidable since the highly nonlinear convective term is still present. What this shows is that in order to get accurate results, the complete Navier-Stokes equations (sometimes Euler) have to be solved, and the only way of doing that with the knowledge we have today is to employ numerical methods, which is the goal of CFD. The numerical approach Most CFD codes use a nite volume approach to solve the governing equations [Rizzi, 2011], which transforms the PDEs into algebraic equations for steady problems. The domain is discretized into a nite number of nodes, which form a mesh, and the small volume surrounding each node is referred to as a nite volume (or cell). In each cell, the integral form of the governing equations are evaluated, and the values of all ow properties are stored in the center node. In order to make sure that the balance of mass, momentum and energy is maintained in an integral sense, any divergence terms in the equations are converted to surface integrals via the divergence theorem, and evaluated as uxes at the cell faces. The values of the dierent uxes are obtained by interpolating or extrapolating values at the nodes. In today's modern CFD codes, many dierent schemes are used depending on the type of ow problem and geometry. Careful consideration of stability must also be made when choosing a scheme. For unsteady problems, the nite volume method transforms the PDEs into a system of ordinary dierential equations (ODEs). In this case, further discretization (in time) is needed to transform the ODEs into a system of algebraic equations. Two dierent types of schemes are available for this purpose, explicit- or implicit schemes. When using an explicit scheme (such as Euler forward), the values at the dierent nodes for the next time step can be calculated for one node at a time, independent of the other nodes. But when an implicit scheme is used (such as Euler backward), the values at the dierent nodes for the next time step have to be solved simultaneously for all nodes, resulting in a system of equations (increased computational cost). Just like the case with spatial discretization, many dierent schemes are used depending on the type of ow problem.

18 7 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Turbulence modeling In most problems of interest today, the ow eld is predominantly turbulent. It is possible to solve these types of problems exact through the Navier-Stokes equations, disregarding any numerical errors caused by the discretization. This is known as Direct Numerical Simulation (DNS), and the advantage is that the solution contains full information about the ow eld since no approximations have been made. Unfortunately, because of the large separation of scales in turbulent ows, a very ne mesh and a high temporal resolution are needed to resolve all eddies in the ow. As an example, using DNS for a full scale airplane requires roughly nodes [Wallin, 2011]. It is apparent that using DNS is extremely costly, and it will take many years before computers are powerful enough to allow practical use of the method for engineering applications. For now, turbulence models have to be used when solving the Navier-Stokes equations to reduce the computational cost. Two main classes of models are used today, Large-Eddy Simulation (LES) and RANS models. But hybrid LES-RANS models are also used, with the most common method being Detached-Eddy Simulation (DES). For turbulent ows, it is the large eddies, which are dependent on the geometry, that contain the most energy and inuence the ow eld the most. The smaller eddies are more universal and are not aected much by dierent geometries. This is the basis for all LES models, for which the large eddies are resolved directly (as in DNS), but the smaller eddies are modeled by using a so called subgrid-scale (SGS) model. In practice, this is accomplished by ltering the Navier-Stokes equations to remove the smaller scales from the solution. Instead of using a low-pass lter, most commercial CFD codes use the mesh to lter the equations [Wallin, 2011]. Only eddies that are larger than the cells in the mesh are resolved, the rest are modeled. LES may not be as expensive as DNS, but the spatial- and time resolutions needed to accurately resolve boundary layers at high Reynolds number are still very high, and it will take some time before practical use of LES for these types of problems is possible [Wagner et al. 2007]. As mentioned earlier, the RANS equations are unclosed due to the appearance of the Reynolds stresses, and the goal of a RANS model is to determine these stresses. The most well-known and validated RANS model - and the model used in this thesis - is the k ɛ model. It is a two equation model that adds two transport equations, one for the turbulent kinetic energy, and one for the dissipation. It is based on the (rather crude) Boussinesq eddy viscosity assumption, which states that the Reynolds stresses can be modeled in the same way as the viscous stresses by the introduction of a so called eddy viscosity (µ t ), ( vi ρ v i v j = µ t + v j 2 x j x i 3 v k δ ij x k In the standard k ɛ model, the eddy viscosity is modeled as ) 2 3 ρkδ ij. (12) µ t = ρc µ k 2 ɛ, (13) where C µ is set to RANS models are much cheaper compared to LES/DNS since only the time-

19 8 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 averaged inuence of turbulence is considered. The drawback of using a RANS model is that it does not perform well in ows with massive separation, e.g. ow around blunt bodies [Wallin, 2011]. In an eort to nd solutions to the problems associated with LES and RANS models, Spalart et al. developed the DES methodology in 1997 [Wallin, 2011]. The idea is to use a RANS model in the boundary layer close to the wall, and LES far from the wall. A so called DES limiter is used to determine when RANS or LES should be used, d = min(d, C DES ), (14) where C DES is set to A major advantage of using this approach is that DES works well in ows with both thin boundary layers (high Reynolds number) and massive separation. DES was rst developed for the Spalart-Allmaras model, but it is possible to implement other RANS models. Wall functions Although the use of a turbulence model reduces the computational cost, it is usually not enough for ows with high Reynolds number. Due to the large accelerations in the normal direction close to the wall, a very ne mesh is needed to accurately resolve the boundary layer. One way of avoiding this problem is to use wall functions, which approximate the velocity prole in the boundary layer between the wall and the rst node. Most wall functions are based on the law of the wall, in which the viscous sublayer is estimated using a linear function, and the log layer is estimated using a logarithmic function (see Figure 2). Figure 2: Illustration of the law of the wall. [CFD Online, 2013]

20 9 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 When using wall functions, it is normally required that the rst node away from the wall is located in the log-law region, 20 < y+ < 200 [CFD Online, 2013]. Unfortunately, this may prove dicult to achieve for all nodes close to the walls in a domain, depending on ow properties and geometry. For this purpose, scalable wall functions have been developed, which remove the lower limit regarding the location of the rst node. As always, it is important to know the strengths and weaknesses of the models being used. By employing wall functions, the computational cost may be reduced signicantly for ows with high Reynolds number. But on the other hand, the approximations may be poor for certain types of ows, e.g. boundary layer with an adverse pressure gradient and separation. 4.3 Performance of turbines Turbine eciency Turbines are necessarily not designed with maximum possible eciency in mind, but rather to provide enough torque/power for a given pressure dierence. However, this is not to say that eciency is not considered during the design process, as it has an impact on the output torque. In this thesis though, eciency will be one of the main parameters used when evaluating the performance of dierent turbines due to the fact that the losses associated with the secondary ow will have a direct impact on it. The general denition of isentropic eciency for a turbine is, η is = output work ideal work output specific work = ideal specific work. (15) This is a rather simple expression, but how it should be used in practice is not as straightforward as it rst may seem. If the change in potential energy is neglected and the turbine is assumed to be adiabatic (which is typically the case), the output specic work can be expressed as W = h 0,turbine. (16) How to express the ideal specic work requires some thinking though, and several dierent eciencies are dened based on how it is expressed. The main issue here is whether the outlet kinetic energy should be treated as a loss or not. Since this thesis deals with secondary ow, which leads to internal losses (generation of entropy), the kinetic energy at the outlet is of no interest and it will not be treated as a loss. The ideal specic work is now expressed as W is = h 0is,turbine. (17) The corresponding eciency is called the total-to-total eciency, dened as η tt = h 0,turbine h 0is,turbine. (18)

21 10 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 A more detailed description of the dierent eciencies can be found in any textbook on turbines, see for example [Dixon & Hall, 2010]. Stage design parameters When designing the dierent stages in a turbine it is convenient to work with dimensionless parameters. Three of those are considered especially important, namely the ow coecient, the loading coecient and the degree of reaction. The ow coecient is dened as φ = c m U. (19) It is directly related to the relative ow angles, and a high value implies that the ow is more axial. The second parameter, the loading coecient, expresses how highly loaded a stage is. It is dened as ψ = h 0 U 2. (20) The change in total enthalpy in an adiabatic axial turbine with constant mean radius can be written h 0 = U c θ by using the Euler turbine equation. This means that a highly loaded stage will have a large ow turning. In words, the degree of reaction expresses how much the ow is expanded in the rotor relative to the total expansion over the stage. It is normally dened using the static enthalpy, however in this thesis it will be dened using the static pressure, R = p rotor p stage. (21) Bladerow performance The overall performance of a turbine is a result of the performance of the individual bladerows, which in turn is directly related to the aerodynamics. One way of quantifying the aerodynamic losses in a turbine is to dene a total pressure loss coecient, Y total = p 01 p 02 p 02 p 2. (22) It is important to realize that this loss coecient is a measure of the overall losses, caused by dierent sources, including friction, shock waves, and secondary ow, which is the main issue in this thesis. In general, when making eorts to reduce losses, there may be situations where one type of loss is actually increased, but that is not a problem as long as the overall losses are reduced. An example of this are the dimples on the surface of a golf ball, which will trigger turbulence and increase the friction losses. But at the same time, separation is delayed, leading to a reduction of pressure drag and overall losses.

22 11 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 One of the most fundamental ways of reducing losses in a bladerow is to choose a good space-chord ratio. If the spacing of the blades is small, the friction losses will be very large, but on the other hand, if the spacing is too large, there will be large losses due to separation instead. In 1945, Zweifel stated that the ratio of the actual to an ideal tangential blade loading has an approximately constant value for minimum losses [Dixon & Hall, 2010]. In the case of compressible ow, this ratio can be written as, Z = ṁ(c y1 + c y2 ) (p 01 p 2 )bh. (23) For low Mach numbers, the Zweifel coecient should be approximately 0.8 to minimize losses, but as the Mach number is increased, this value is decreased. But it also turns out that this criterion is only valid for outlet angles of 60 70, and many modern turbines have values that are larger than What is secondary ow? Secondary ow was dened in the introduction as any part of the ow that is not in the direction of the core ow. And the interaction of the secondary ow with the core ow will result in mixing and dissipation (generation of entropy), which account for the losses. Despite the somewhat ambiguous denition, it is well known what cause secondary ow, namely the inlet boundary layer, end-walls, separation and tip leakage. An example of what the ow eld in a turbine looks like can be seen in Figure 3. Figure 3: Example of the ow eld in a turbine. [Denton, 1994]

23 12 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 As the ow approaches the leading edge of a blade, the (inlet) boundary layer generated upstream in the turbine separates due to the adverse pressure gradient close to the leading edge. The separated boundary layer rolls up and wraps itself around the blade, creating a horseshoe vortex. Due to the pressure distribution of the blade, the pressure side leg of the horseshoe vortex is pushed towards the suction side of the adjacent blade, where it starts to interact with the suction side leg of the horseshoe vortex belonging to that blade. This is shown in Figure 4. Figure 4: Formation of a horseshoe vortex in a turbine. [Sieverding, 1984] This phenomenon is not unique for turbomachines, it can be observed in other situations as well. During winter, it is possible to see traces in the snow of the horseshoe vortex around a tree, which is created as the boundary layer along the ground is rolled up (see Figure 5). Figure 5: Traces in the snow caused by a horseshoe vortex. [Vogt, 2009] Because of the ow turning in a blade passage, there exists a pressure gradient normal to the stream lines. This pressure gradient has to be balanced by the centripetal acceleration of the ow at any given point in the passage. The boundary layer approximation of the Navier-Stokes equations states that

24 13 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 the pressure gradient normal to a wall is zero, which means that the pressure in the boundary layer is the same as the pressure in the free stream. Because of this, the pressure gradient normal to a stream line located in the boundary layer is equal to the pressure gradient normal to a stream line located outside the boundary layer. And since the boundary layer at the end-walls in a blade passage has a lower velocity/kinetic energy compared to the rest of the ow, this can only be true if the turning radius of the boundary layer is smaller than the turning radius of the free stream. The consequence of this is that as the boundary layer approaches the suction side of a blade, it rolls up and forms a so called passage vortex, this can be seen in Figure 6. Figure 6: Passage vortex in a turbine. [Sieverding, 1984] There is always a risk that the ow around a blade will separate due to the adverse pressure gradient on the aft part of the suction side. If the boundary layer is laminar when it separates, it may reattach as a turbulent boundary layer, leading to the presence of a separation bubble. This process is shown in Figure 7. In addition to increased losses due to mixing between the recirculating ow in the separation bubble and the core ow, the pressure drag of the blade will also increase due to the low pressure in the separated region.

25 14 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Figure 7: Separation of a laminar boundary layer. [Lakshminarayana, 1996] In a turbine, there is a small gap between the tip of the rotor and the casing, where part of the ow will pass through. This is called tip leakage, and results in the formation of a vortex (see Figure 8), which will lead to increased losses. The magnitude of the leakage ow is strongly aected by the tip clearance and the pressure ratio over the blade row, which of course is directly related to the blade loading. Another aspect of this to keep in mind is that the leakage ow does not undergo turning in the rotor, and thus does not produce any work. Figure 8: Tip leakage in a turbine. [Denton, 1993]

26 15 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 The choice of using a shrouded, or an unshrouded rotor is also an important factor. Traditionally, it has been assumed that shrouded turbines always are more ecient compared to unshrouded, since it is possible to use multiple seals. However, it has recently been shown that for a suciently small tip clearance, the two types of turbines have the same eciency, and if the clearance is reduced further, the unshrouded turbine is actually more ecient, [Yoon et al. 2010]. Unshrouded turbines also have the advantage of being lighter, thus reducing stress in the blades, allowing the rotor to run faster (increased power output). An example of tip clearance eect over a shrouded rotor with a tip seal is shown in Figure 9. Figure 9: Tip clearance eect over a shrouded rotor. [Denton, 1994] This part of the report is based on explanations given in [Wei, 2000] and [Saha, 2011]. 4.5 Why aspect ratio? The performance of a turbine depends on many dierent parameters, and it is up to the designer to make the right choices to achieve a turbine with the desired performance. In practice, some of the parameters will be locked due to geometric- or other constraints, but the number of choices that have to be made may still be quite large. The ideal would be to make a study of every single parameter and investigate how isolated changes aect the secondary ow and performance of the turbine. But because of the sheer number of parameters it is not feasible in a study like this, there is simply not enough time. A much better approach in this case would be to try to identify which parameter has the most impact on the secondary ow and work with that. In this thesis, the main parameter that was chosen was the aspect ratio. For a turbine blade, the aspect ratio (AR) is dened as, AR = H b. (24) Of course, the choice of aspect ratio is not just a wild guess, but rather the obvious choice when thinking about the nature of secondary ow. Secondary ow is characterized by three-dimensional

27 16 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 ow elds, a turbine with much secondary ow will have a strong tree-dimensional ow eld. One way to make the ow eld more two-dimensional, and thus reduce the secondary ow is to increase the aspect ratio of the blades in the turbine. The idea that aspect ratio has a major inuence on the ow is not unique for the turbomachine industry, the same thinking exists in every industry involved in uid dynamics in one way or another. A classical example is the induced drag on a wing, which is modeled as inversely proportional to the aspect ratio. 5 Current loss models Typically, loss models are used in an early stage of the development of a turbine to estimate the losses in the dierent blade rows. There are many dierent models to choose from, all with varying accuracy and complexity. Two of the most widely used models are the correlations of Ainley & Mathieson and Kacker & Okapuu, which will be described briey here. Ainley & Mathieson presented their method for predicting losses in 1951, and it has been shown since that the method provides a good estimate of the losses for axial turbines with conventional blades over a large part of its operating range. The performance data of turbines used to develop the method was obtained for Re = (based on mean chord and exit ow conditions), and compressibility eects are neglected, which limits the use of the method somewhat. It is assumed that the total loss can be expressed as the sum of the prole loss, secondary loss, and tip leakage loss, times a coecient, Y total = (Y P + Y S + Y T l )χ te. (25) The trailing edge coecient χ T e can be obtained from a gure given by Ainley & Mathieson. The prole loss coecient can be calculated from Y P = χ i Y P (i=0), (26) where the incidence coecient χ i can also be obtained from a gure. For conventional blades, the zero incidence prole loss coecient Y p(i=0) can be calculated from ( Y P (i=0) = Y P (β 1 =0) + β 1 β 2 ) 2 [ Y P (β ] 1 =β2) Y P (β 1 =0) ( tmax/b 0.2 ) β 1 /β 2. (27) The other values of Y p needed in the above equation are obtained from correlations between the prole loss coecient and pitch-to-chord ratio provided by Ainley & Mathieson. To nd an equation for the secondary loss coecient, Ainley & Mathieson did performance measurements on conventional blades. They based it on the blade loading, which is a main function of the blade turning, Y S = λ ( ) 2 CL cos 2 β 2 t/b cos 3. (28) β m

28 17 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 λ is a function of the ow acceleration in the blade row and can be found in a gure. An equation for the tip leakage loss coecient was obtained in a similar manner as for the secondary loss coecient, Y T l = C τ ah 4 (tan β 1 tan β 2 ) 2 ( cos 2 β 2 cos β m ). (29) It is now possible to calculate the total pressure loss coecient simply by using eq. 25. A more thorough description of this method can be found in [Ainley & Mathieson, 1951]. This method was later updated by Dunham & Came to better predict losses in turbines with lower aspect ratios. More recent turbine performance data was used, and some of the loss correlations were changed. See [Dunham & Came, 1970] for more information. In 1982, Kacker & Okapuu presented a new loss model, which is based on Ainley & Mathieson and Dunham & Came. In this model, the loss system is restructured and compressibility and shock losses are considered when calculating the prole and secondary loss coecients. It has been shown that this method is able to predict the eciencies of a wide range of axial turbines with an error margin as low as ±1.5 %. The equation used to calculate the total pressure loss coecient is Y total = χ Re Y P + Y S + Y T l + Y T e. (30) It is assumed that the Reynolds number only aects the prole loss, and the trailing edge loss has been separated from the other losses. The Reynolds number is based on the true chord and exit ow conditions, and the Reynolds correction factor χ Re can be calculated as, ( ) 0.4 Re 2 10 Re χ Re = Re (31) ( Re ) Re Providing a detailed explanation of how the dierent loss components are calculated is a quite daunting task and will not be done here. In principle, the equations used are based on those found in Ainley & Mathieson and Dunham & Came, but with some extra terms added to account for the compressibility eects of the ow. A thorough review can be found in [Kacker & Okapuu, 1982].

29 18 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 6 Setting up the simulations 6.1 The turbine In this thesis, the turbine that will be used is a two stage low aspect ratio turbine. The mean radius is constant throughout the turbine, and all blades are prismatic. Normally, turbine blades are twisted and/or use dierent airfoils at dierent spans to account for the radial velocity changes. This is not done here however, since the blade height is very small compared to mean radius, making the radial velocity changes negligible. Data for the turbine can be seen in Table 2, and the geometry used in the simulations can be seen in Figure 10. Table 2: Data for the turbine. Blade height/mean Radius [-] Stator Rotor Stator Rotor Figure 10: Geometry of the turbine used. 6.2 Meshing The mesh is generated in ICEM through blocking (hexa mesh). Due to the periodic symmetry, it is only necessary to mesh a single blade in each row, which will greatly reduce the computational cost.

30 19 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Each blade is meshed separately and then joined together in CFX through a mixing plane. Usually when two meshes are joined together in this way, the interfaces are not conformal, thus leading to interpolation errors. However, this is fortunately not a big problem in this case since the domain is open, and most of the error will be convected out of the system as the solution progresses. The mesh used here does not have any tip clearance, and the advantage of this is that no small cavities have to be meshed, reducing the mesh size. Of course, the complete physics of the ow will not be accounted for, but the tip clearance eects will be present in all turbines regardless of aspect ratio, and this thesis is mainly concerned with how the aspect ratio aects the ow. It should also be noted that normally a mesh study is carried out to nd the optimal mesh settings before the actual simulations are performed, but this is very time consuming and it will not be performed here. Instead, the mesh is generated using the same settings as VAC have used previously. Information about the number of cells and how they are distributed can be seen in Table 3. The average values for y + are also shown. The complete mesh is shown in Figure 11. Table 3: Cell data for the turbine. No. of cells Average y + Stator Rotor Stator Rotor Total Figure 11: Complete mesh of the turbine.

31 20 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/ Boundary conditions and gas properties During the development of a turbine, it is common to dene an operating point at which the turbine is designed to operate. This is known as the aerodynamic design point (ADP), and the ow conditions at that point are chosen to try to maximize the turbine eciency when operating under those conditions. It is of course not possible to operate a turbine at its ADP all the time, and care must be taken to make sure that the performance does not diminish drastically when operating the turbine at o-design conditions. To evaluate the overall performance of the turbine, four o-design operating points (OP) have been dened, in addition to the ADP. Operating conditions for these four points are shown in Table 4. Table 4: Operating conditions for the o-design points. Operating point N/N ADP [-] P R ts /P R ts,adp [-] When setting up the simulations in CFX, pressure boundary conditions will be used. Previous research at VAC have shown that for this type of simulation, the results will be more accurate if a non zero reference pressure is used and the pressures at the inlet and outlet are dened relative to this value. Thus, throughout this thesis a reference pressure of kpa will be used. Only steady state simulations will be performed, which of course means that no transient eects are accounted for in the solution. But since the start up and stopping phase of the turbine are of no interest, the only transient eect present is the inuence of the bladerow wakes on the down stream ow, which is ok to neglect in a rst study like this. Every time a property is averaged (total pressure, velocity e.g.), a mass ow average is used unless stated otherwise. The material properties of the gas used to drive the turbine are assumed to be constant, with c p /c v = 1.4 and dynamic viscosity µ = kg/m s. 7 Results from the simulations 7.1 Single stage turbine Initially, only the rst stage of the turbine is simulated. The purpose of this is to get a feel for the performance of the turbine, and how to visualize the secondary ow in a good way. But it is also interesting to see how the dierent operating points aect the performance. For this single stage turbine, a total-to-static pressure ratio of 1.20 is used for the boundary conditions. The overall performance of the rst stage (and complete turbine in this case) is presented in Table 5.

32 21 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Table 5: Performance of the rst stage (normalized with data for the ADP). ADP OP1 OP2 OP3 OP4 N [-] P R ts [-] P R tt [-] Ψ[-] Φ[-] η is [-] ṁ in [-] Torque [-] Power [-] One interesting thing to note is that it is actually not the ADP that has the highest eciency, but OP2. The trade-o is that the mass ow is lower, thus leading to a lower torque/power output. This implies that the ADP has been chosen so that the turbine provides enough torque for what it has been designed to do, rather than maximum eciency, which was discussed in an earlier section. It can also be seen that the turbine is almost as ecient at OP3 compared to OP2, and the torque is only slightly less than at the ADP, whilst the power output is actually higher. The downside is that the rotor spins at a higher speed, which may cause problems in the blade roots due to the higher loads. Performance of the rotor is shown in Table 6 (the stator is almost not aected at all). Table 6: Performance of rotor (normalized with data for the ADP). Rotor ADP OP1 OP2 OP3 OP4 Z [-] Y total [-] β 1 [-] β 2 [-] β 1,abs [-] β 2,abs [-] M 1,abs [-] M 2,abs [-] V max U out [-] The rst thing to note is that the total pressure loss is not aected much by the change of operating point, typically less than 10 % for the rotor, and even less for the stator. Interestingly enough, OP2 has the highest pressure loss for the stator, even though the turbine eciency is the highest. On the other hand, the pressure loss for the rotor is the lowest. And since the pressure losses for the rotor is roughly a factor 3 larger compared to the stator, it explains the high eciency for OP2. It is somewhat surprising that despite the fact that the rotor has a value of the Zweifel coecient that is closer to the correct value, the total pressure loss is much higher. This implies that the friction losses is only responsible for a small part of the total pressure loss. In this case it is not a very farfetched guess that the secondary ow is responsible for a very large part of the total pressure loss. The question then arises how the secondary ow should be quantied. As mentioned earlier, secondary ow is

33 22 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 characterized by a 3-dimensional ow eld, and the velocity component that is normally not present (or at least very small) and accounts for the 3D-eects is the radial one. Thus, by looking at the radial kinetic energy (KE) and comparing it to the total kinetic energy it should be possible to estimate the relative importance of the secondary ow. This is shown at the stator and rotor trailing edges (TE) in Tables 7 and 8 for the single stage turbine. Table 7: Radial kinetic energy at the stator trailing edge. Stator TE ADP OP1 OP2 OP3 OP4 Vrad 2 [J/Kg] 114,5 136,8 91,7 111,9 118,0 V 2 rad V [-] 2 0, , , , , Table 8: Radial kinetic energy at the rotor trailing edge. Rotor TE ADP OP1 OP2 OP3 OP4 V 2 [J/kg] , rad V 2 rad Vrel 2 [-] 0, , , , , Just as expected, the radial kinetic energy at the rotor trailing edge is much higher compared to the stator trailing edge, almost by a factor 10. It can also be seen that the relative contribution is higher, meaning that the secondary ow is more dominant further downstream in a turbine. This is not very surprising though, as the vorticity generated upstream in the stator is convected downstream into the rotor. A plot of the radial velocity component at rotor trailing edge (ADP) is shown in Figure 12. Figure 12: Radial velocity component at the rotor trailing edge. The pattern of the radial velocity component indicates that there are vortices present, which is con-

34 23 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 rmed by the streamlines in the rotor, shown in Figure 13. Figure 13: Streamlines in the rotor. 7.2 Two stage turbine The next step is to simulate the complete turbine and evaluate its performance. To save time, only the ADP is simulated, which is acceptable since the changes in total pressure loss are quite small. The overall performance of the two stage turbine is shown in Table 9. Table 9: Performance of the two stage turbine (normalized with data for the single stage turbine). ADP P R ts [-] N [-] ṁ in [-] Torque [-] Power [-] η is [-] Radial KE [-] Two stages It is obvious that the single stage turbine has a higher eciency, and also lower radial kinetic energy at the outlet, which is treated as a loss. Data for the stage performance is presented in Table 10. Table 10: Performance of stages (normalized with data for the single stage turbine). Two stages Stage 1 Stage 2 Ψ[-] Φ[-] η is [-] Stage 1 has signicantly higher eciency compared to stage 2, and also a higher work load. Stage 1 for the two stage turbine also has slightly lower eciency compared to stage 1 for the single stage

35 24 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 turbine. It can be interesting to look at the the radial velocity component at rotor 1 trailing edge for the two turbines side by side for comparison (see Figure 14). By doing that it can be seen that the magnitude of the radial velocity is larger for the two stage turbine. Figure 14: Radial velocity component at rotor 1 TE for single stage (left) and two stages (right). The performance of the dierent blade rows is shown in Table 11, and yet again, data for the single stage turbine is shown for comparison. Single stage Stator Rotor Y total [-] V 2 rad Vrel 2 [-] Table 11: Performance of blade rows. Two stages Stator 1 Rotor 1 Stator 2 Rotor 2 Y total [-] V 2 rad Vrel 2 [-] Rotor 2 has a very high total pressure loss, but it may not necessarily be due to a bad design, it can also be caused by the thick boundary layer at the inlet to the rotor. This large total pressure loss of course has a very negative impact on the eciency of the second stage (and turbine), which could be seen earlier. To visualize the ow in the complete turbine, streamlines have been plotted, which can be seen in Figure 15. The streamlines clearly show the presence of vortices at rotor 1 and 2 trailing edges.

36 25 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Figure 15: Streamlines in the two stage turbine. Other interesting properties to look at are the Mach number and total pressure, which are shown in Figure 16 (absolute values at 50 % span). Figure 16: Mach number (left) and total pressure (right) for the two stage turbine. Regions of higher Mach numbers are clearly visible at places with sharp curvature and contraction of the gas channel. Regions where one would expect to have losses (wakes, vortices e.g.) show a decrease

37 26 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 of total pressure, which is to be expected. Moving on, the total temperature and entropy can be seen in Figure 17 (absolute values at 50 % span). Figure 17: Total temperature (left) and entropy (right) for the two stage turbine. The stators do not produce any work, thus the total temperature is constant. But as energy is extracted from the ow in the rotors, the total temperature is decreased. The behavior of the entropy is quite interesting. As the ow moves from stator 1 to rotor 1, streaks with low entropy become visible. This is not a physically correct description of the ow, but rather a consequence of the method used by CFX to pass the information from one side to the other of the mixing plane. CFX calculates a circumferential average of the entropy upstream of the mixing plane, and in order to match this value downstream of the mixing plane, these streaks of low entropy have to be present. This means that the local values of the entropy just after the mixing plane may not be correct, but in an average sense it is still correct. Luckily, what is often interesting to look at is the change in entropy between the inlet and outlet of the turbine, and in that case these local entropy deciencies do not inuence the solution much. Since generation of entropy can be seen as losses, it is not surprising that the regions where the entropy increases coincide with the regions of total pressure loss.

38 27 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/ Inuence of aspect ratio Introduction As discussed earlier, it is believed that the aspect ratio has a signicant impact on the secondary ow in the turbine. And this part of the thesis will investigate how the performance of the turbine (focusing on total pressure losses and eciency) is aected by changes in the aspect ratio. The idea is to change the aspect ratio both by modifying the height of the blades, as well as the size. If the aspect ratio was the only parameter that aects the secondary ow, it would not matter if the height or the size of the blade is changed as long as the aspect ratio is the same. However, this is unlikely, and there are most certainly other parameters that also aect the secondary ow. And this is one of the reasons that making simple 1D loss models for low aspect ratio turbines is not a trivial task. The simulations in this part are all performed at the ADP. Changing the blade height The rst thing that will be investigated is how the turbine performance is aected by changes in the blade height. This is accomplished in practice by using MATLAB to scale the entire mesh in the radial direction (remember that the blade height is constant throughout the turbine) and then run a simulation with the new mesh in CFX. How the turbine eciency is aected by the blade height can be seen in Figure 28. Figure 18: Turbine eciency as function of blade height. For blade heights close to the reference value, there is almost a linear relationship between the eciency

39 28 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 and blade height, and the slope is quite large. This means that in this region, a small change of blade height will have a signicant impact on the eciency. But as the blade height is increased further, the relationship becomes non-linear at approximately double the reference height, and the turbine eciency moves towards an asymptotic value. So if a turbine designer is assigned the task of increasing the eciency of a turbine with a large aspect ratio, changing the blade height is not a very good way of doing it. The next thing to look at is of course the performance of the two stages. Figure 19 shows the ow coecient and loading coecient for stages 1 and 2. Figure 19: Flow coecient (left) and loading coecient (right) for stage 1 and 2. A similar trend as for the case with the turbine eciency can be observed here as well. For blade heights less than approximately double the reference height, both the ow coecient and loading coecient is aected signicantly by changes in the blade height. How the eciency of the stages is aected by the blade height can be seen in Figure 20. Figure 20: Eciency of stage 1 and 2.

40 29 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Just as before, the blade height has a quite large impact on the eciency at less than approximately double the reference height. It can also be seen that the second stage always has a lower eciency compared to the rst stage. This is probably due to the fact that the inlet boundary layer is thicker in stage 2, and that the vorticity generated upstream is convected downstream into stage 2. Another thing worth noting is that the reference turbine actually has the worst eciency out of all the cases for the second stage. Next it is time to look at the blade rows, and Figure 21 shows the total pressure loss at dierent blade heights. Figure 21: Total pressure loss coecient for all blade rows. The observations made earlier are conrmed here. As the blade height is increased, the total loss coecient for all blade rows moves towards an asymptotic value. This decrease of losses is of course to a large extent caused by the lower secondary losses as the blade height (and aspect ratio) is increased. Just as before, the losses are higher in the rotors compared to the stators. It can also be seen that the low eciency of stage 2 is a direct consequence of the high losses in rotor 2. For this specic turbine, much can be gained in terms of eciency by increasing (or even lowering!) the blade height in rotor 2. Instead of looking at the full losses, it is also interesting to look at the secondary and friction losses separately. The friction losses relative to the full losses for the dierent blade heights can be seen in Figure 22.

41 30 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Figure 22: Relative friction losses for all blade rows. The theoretical values have been calculated by rst using CFX to nd a good value for the skin friction coecient C f, and then using the classical at plate theory to see if it provides a good estimate of the friction losses. The relative friction loss coecient is then calculated as Y wet = C f A wet q out q out A annulus cos(β out ). (32) The agreement is quite good except for rotor 1, where the maximum error is ~30 %. As the blade height is increased, the friction loss coecient is decreased. This is easy to understand by looking at the equation above, the annulus area increases faster than the wet area with increasing blade height. The radial kinetic energy relative to the total kinetic energy of the ow at the outlet of all blade rows is shown in Figure 23.

42 31 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Figure 23: Relative radial kinetic energy at the outlet of all blade rows. Interestingly enough, the radial kinetic energy does not follow the same trend as the total pressure loss coecient, which is somewhat surprising. This implies that the radial kinetic energy at the outlet is not a good way of quantifying the secondary losses. This is probably because the losses may have already been realized in the turbine, before the outlet. To look into it further, it is necessary to look at the entropy changes in the turbine. Figure 33 shows the entropy at 50 % span for the reference turbine and a blade height of 860 % of the reference height.

43 32 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Figure 24: Entropy in the reference turbine (left) and a blade height of 860 % of the reference height (right). Like before, the streaks of low entropy is clearly visible. In the reference turbine it is obvious that the entropy increases in the blade rows, and not only at the outlet, which supports the claim that the losses are realized earlier in the turbine. The increase of entropy in the reference turbine at 50 % span is caused by the vortices at the end walls, which occupies the entire gas channel since it is so small. On the other hand, in the case with a blade height of 860 % of the reference height, the gas channel is very large and the vortices are only located close to the end walls. This can be seen very clearly by looking at the radial velocity at the trailing edge of rotor 2, which is shown in Figure 25. Figure 25: Radial velocity at the trailing edge of rotor 2 for the reference turbine (left) and a blade height of 860 % of the reference height (right).

44 33 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 The absolute values are of no interest in this case since the radial velocity is only used to show the presence of vortices at the end walls. In the reference turbine there is a large vortex in the middle of the gas channel, whilst in the other case there are two smaller vortices at the end walls. How this aects the losses can be visualized by looking at the entropy instead, which is shown in Figure 26. Figure 26: Entropy at the trailing edge of rotor 2 for the reference turbine (left) and a blade height of 860 % of the reference height (right). It is obvious that when the two vortices coincide, they strengthen each other and generate more entropy compared to when they are separated. What is very interesting about all this is that the blade height required to fully separate the two end wall vortices is approximately double the reference height. Increasing the blade height more will just move the vortices further away from each other (the wake also becomes visible). So it is not surprising that the performance of the turbine (including stages/blade rows) is aected so much by changes in the blade height at less than approximately double the reference height, which could be seen very clearly earlier. Relating the entropy to losses can be a bit tricky for some, so in a last eort to make things even clearer, the span wise total pressure at the trailing edge of rotor 2 for the two cases presented above can be seen in Figure 27.

45 34 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Figure 27: Span wise total pressure at the trailing edge of rotor 2 for the reference turbine (left) and a blade height of 860 % of the reference height (right). This shows more or less the same thing as the entropy. In the reference turbine, the vortex in the middle of the gas channel generates losses, which is indicated by the lower values of the total pressure. In the other case, losses are generated at the end walls by the two separated vortices, which again is indicated by the lower values of the total pressure. Changing the blade size Evaluating the eects of changing the blade size is done in more or less the same way as for the blade height. But instead of scaling the mesh in the radial direction, it is scaled in the axial and tangential directions. This means that by increasing the size, the aspect ratio is reduced, and vice versa. Another important dierence is that when the blade size is changed, the number of blades have to be changed as well to keep the same solidity. The number of blades to be removed or added is always rounded to the nearest integer. How the turbine eciency is aected by the blade size is shown in Figure 28.

46 35 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Figure 28: Turbine eciency as function of blade size. Unlike previously when the blade height was changed, it is not possible to identify a linear region. The turbine eciency exhibits a non linear behavior for all blade sizes. Also, note that the eciency is decreased with increasing blade size, which corresponds to a reduction of the aspect ratio. In Figure 29, the inuence of the blade size on the ow coecient and loading coecient is shown. Figure 29: Inuence of blade size on the ow coecient (left) and loading coecient (right). The changes in these two parameters are much smoother compared to when the blade height was changed. There is no region with large changes like before. For blade sizes not too far from the

47 36 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 reference turbine, the loading coecient for stage 2 is almost constant, and for stage 1 it decreases with increasing blade size. This means that the work split is changing, the work in rotor 1 relative to the total work of the turbine is reduced. How the stage eciency varies with blade size can be seen in Figure 30. Figure 30: Eciency of the two stages as function of blade size. The eciency for the two stages follows the same trend as the loading coecient, which is not too surprising. What is somewhat surprising though, is that despite the fact that the turbine eciency shows no linear behavior, the eciency for stage 1 does (but not stage 2). The slope is also quite large, increasing the blade size to 200 % will result in a reduction of the eciency for stage 1 of about 6 percentage. Figure 31 shows the total pressure loss for all blade rows.

48 37 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 Figure 31: Total pressure loss coecient for all blade rows. Yet again, stage 1 shows a linear behavior. Other than that, the same observations made in the case with changing the blade height can also be made here. The rotors have higher losses compared to the stators, and much can be gained for rotor 2 by decreasing or even increasing (corresponds to decreasing the aspect ratio!) the blade size. Next comes the relative friction losses, and relative radial kinetic energy at the trailing edge, which can be seen in Figure 32. Figure 32: Relative friction losses (left) and relative radial kinetic energy at the trailing edge (right) for all blade rows. Here, the theory is not very good at predicting the values of the friction losses, none of the blade rows

49 38 Investigation of secondary ow in low aspect ratio turbines using CFD 23/03/14 show a good agreement. However, it does capture the trends quite well. Explaining the fact that the friction losses increase with increasing blade size is not very hard. As the blade size is increased, the wet area also increases, but the annulus area remains the same, thus leading to increased friction losses (see eq. 32). Regarding the relative radial kinetic energy at the trailing edge, it again seems to conrm that it is not a good way of quantifying the secondary losses as they have already been realized in the blade rows. Except for stator 1, the trends do not follow the total pressure loss coecient, and rotor 1 looks especially messy when it comes to the radial kinetic energy. Lastly, it is of interest to look at the entropy at 50 % span for the reference turbine and for a blade size of 200 %, which is shown in Figure 33. Figure 33: Entropy in the reference turbine (left) and for a blade size of 200 % (right). The streaks of low entropy occupies more of the gas channel when the blade size is 200 % compared to the reference turbine. Also, since the blade height is not changed as the blade size is increased, the end wall vortices always coincide to some extent, causing entropy to increase in the turbine at 50 % span. By looking at the entropy, it is again possible to see that the secondary losses have been realized (increase of entropy) in the turbine before the trailing edges of the blade rows. Putting it all together It is now time to put everything together and try to link the changes in blade height to the changes in blade size through the aspect ratio. This is done by plotting the turbine eciencies from the two cases in the same gure, with aspect ratio on the X-axis (see Figure 34). Since the axial length of the dierent blade rows varies, the length of the full turbine has been used to calculate the aspect ratio instead of the blade length.