Dipartimento di Ingegneria Civile e Industriale
Sezione Ingegneria Aerospaziale
Tesi di Laurea Magistrale
A PASSIVE METHOD FOR ENHANCING THE
PERFORMANCE OF A SYMMETRIC DIFFUSER IN
TURBULENT FLOW REGIME
Relatori Allievo
Prof. Ing. Maria Vittoria Salvetti Guglielmo
Prof. Ing. Guido Buresti Giambartolomei
Ing. Alessandro Mariotti
The object of the present work is testing the capabilities of a passive method for improving the efficiency of a plane symmetric diffuser. The passive control method consists in introducing small localized flow recirculations. These localized flow separations are obtained by introducing couples of appropriately contoured cavities in the diffuser diverging walls. The turbulent flow regime is considered. The Reynolds number, based on the diffuser width at the inlet section and on the inlet velocity on the axis at a given section is Re = 20000. The diffuser area ratio in 4.7 and the divergence angle is 5◦. The inlet flow is fully-developed turbulent. A preliminary mesh sensitivity analysis for two turbulence models (SSTk-ω and RSM) is performed for the flow inside the reference diffuser configuration (without cavities). Based on this analysis the SST k-ω model is chosen for an optimization procedure in order to identify the cavity geometries that maximize the pressure recovery in the diffuser and minimize the main boundary separation extent. Three different config-urations have been considered characterized by one, two and three couples of subsequent couples of general contoured cavities. In the optimized diffuser configuration with one couple of contoured cavities there are not small localized flow recirculations, while in the other two optimized diffuser configurations with two and three couples of contoured cavities small localized flow recirculations are present. The passive control increases the performance of the diffuser in all the considered con-figurations (the maximum pressure recovery is reached by the optimized diffuser configuration with three couples of cavities), leading to an increase of the mean pressure recovery coefficient by reduc-ing the extent of the main flow separation region. In particular, the success of the passive control is due both to a virtual geometry modification of the flow inside the diffuser and to a favourable effect of the cavities in reducing the momentum losses near the wall. A classical shape optimization with Bézier curve is also carried out with the same goal of maximizing the pressure recovery. The resulting optimized diffuser shape with Bézier curves does not show small local flow recirculations but produces a diverging wall shape similar to the one optimized with contoured cavities. Since the optimized diffuser with Bézier curve provides a pressure recovery very similar but not equal to the one of the optimized diffuser with three couples of cavities it seems that the "optimum" configuration can be given by the union of a high degrees of freedom Bézier curve and the presence of small localized flow recirculation placed at suitable locations.
1 Introduction 1
2 Flow analysis for the reference diffuser configuration 7
2.1 Geometry definition . . . 7 2.2 Simulation set-up and numerical methodology . . . 8 2.3 Flow features and validation . . . 12
3 Diffuser with optimized general-contour couples of cavities 18
3.1 Description of the optimization procedure . . . 18 3.2 Optimization of single and multiple couples of general-contour cavities . . . 22 3.2.1 Geometry definition and results for a single couple of general-contour cavities 22 3.2.2 Geometry definition and results for two couples of general-contour cavities . . 27 3.2.3 Geometry definition and results for three couples of general-contour cavities . 31
4 Diffuser optimized with Bézier curves 37
4.1 Notes on the Bézier curves . . . 37 4.2 Shape optimization of the divergent walls and results . . . 40
5 Conclusions 46
2.1 Diffuser geometry and reference frame . . . 7
2.2 Computational grid in the reference diffuser . . . 11
2.3 Particular of the calculation grid in the reference diffuser . . . 11
2.4 Pressure and x-velocity steadiness within the reference diffuser . . . 12
2.5 Cpx comparison for the SST k-ω model . . . 14
2.6 Cpx comparison for the RSM model . . . 15
2.7 Streamlines inside the diffuser (SST k-ω turbulence model) . . . 15
2.8 Longitudinal velocity field inside the diffuser (SST k-ω turbulence model) . . . 16
2.9 Mean pressure coefficient and separated region extent for the reference diffuser . . . . 16
3.1 Scheme of the automatic optimization procedure . . . 18
3.2 Example of the grid topology in the diffuser with couple of cavities . . . 19
3.3 Geometry of the general contour cavity . . . 22
3.4 Streamlines in the diffuser with one couple of optimized cavities . . . 24
3.5 Mean pressure coefficient, wall geometry and separated region extent: comparison between the diffuser without cavities and the diffuser with one couple of optimized cavities . . . 25
3.6 Streamlines in the diffuser with one couple of non optimum cavities . . . 26
3.7 Geometry of two subsequent general-contour cavities . . . 27
3.8 Streamlines in the diffuser with two subsequent couples of optimized cavities . . . 29
3.9 Longitudinal velocity field and streamlines inside the diffuser with two couple of opti-mized cavities . . . 30
3.10 Mean pressure coefficient, wall geometry and separated region extent: comparison between the diffuser without cavities and the the diffuser with two couple of optimized cavities . . . 31
3.11 Geometry of three subsequent general-contour cavities . . . 32
3.12 Streamlines in the diffuser with three subsequent couples of optimized cavities . . . . 34
3.13 Longitudinal velocity field inside the diffuser with three couples of optimized cavities . 34 3.14 Mean pressure coefficient, wall geometry and separated region extent: comparison between the diffuser without cavities and the the diffuser with three couple of optimized cavities . . . 36
4.1 8 Bézier points area of movement for the upper side . . . 41 4.2 Diffuser with optimized Bézier curves . . . 43 4.3 Mean pressure coefficient, wall geometry and separated region extent: comparison
between the reference diffuser and the diffuser with optimized Bézier curves . . . 43 4.4 Longitudinal velocity field inside the optimized diffuser with Bézier curves . . . 44 4.5 Mean pressure coefficient, wall geometry and separated region extent: comparison
between the diffuser without cavities and the diffuser with three couple of optimized cavities . . . 45
2.1 Summary of the simulation settings . . . 10
2.2 Cp for the different turbulence models and the different computational grids . . . 13
2.3 Computational costs and iterations for the SST k-ω model . . . 14
2.4 Computational costs and iterations for the RSM model . . . 14
2.5 Location of the flow separation in the reference diffuser configuration . . . 17
3.1 Optimization parameters ranges for a single couple of cavities . . . 23
3.2 Parameters range of optimum for a single couple of cavities . . . 24
3.3 Optimization parameters ranges for two couples of cavities . . . 28
3.4 Parameters range of optimum for two couple of cavities . . . 28
3.5 Optimization parameters ranges for three couples of cavities . . . 33
3.6 Parameters range of optimum for three couples of cavities . . . 33
Introduction
In most technological applications of fluid dynamics a crucial issue is the development of method-ologies for the control of external and internal flows, in order to achieve desired design objectives (see e.g. [5, 17, 18, 25, 47]). In particular boundary layers control is almost always of paramount importance. The present work focuses on the control of internal flows. In internal flow systems it is often necessary to decelerate the fluid, that is, to pass the flow through a diffuser. The primary purpose of a diffuser is to convert as large a fraction as possible of the dynamic pressure into static pressure. In internal subsonic flows this result can be easily obtained by means of channel opening (increasing of the section area). As a matter of fact, in subsonic flows, to an increasing of the section area corresponds to a deceleration of the flow (think of the mass balance). Furthermore in many cases it is at least equally important to have steady and symmetric flows.
The fundamental parameter, as in many other fluid dynamics problems, is the Reynolds number (Re = l · u/ν where l is a reference length of the problem, u is a reference velocity and ν is the kinematic viscosity). If the flow inside a generic diffuser is characterized by a sufficient low Reynolds number the flow can be laminar, otherwise it is turbulent. Different geometries of the divergent part can be used to realize a diffuser, the most common are rectangular, annular and conic geometries. Due to their simpler geometry, straight walls are commonly used in the diffuser divergent part and many studies have been carried out to highlight the parameters that affect the performance of this type of diffuser, viz. the length, the area ratio (the ratio between the outlet area and the inlet area) and the divergence angle (see e. g. [21, 60, 73, 79, 82, 83, 86] for the laminar flow regime and [24, 41, 68, 85] for the turbulent regime).
Very different flow topologies can be found in a diffuser. When the geometric expansion is gradual, with low divergence angle and low area-ratio, the flow is attached to the side walls and is symmetric to the centreline. When the divergence angle or the area ratio increases, an asymmetric boundary layer detachment and a large flow recirculation can be observed along the diffuser, with separation occurring only on one side wall, while the flow on the other wall remains attached. By further increasing the divergence angle or the area ratio, first a longer flow recirculation occurs and then a smaller secondary recirculation zone can be found on the opposite wall to that of the larger one, until at very large divergence angles and area ratios, two massive recirculation regions occur in the diffuser: the flow separates at the beginning of the diffuser diverging walls on both sides, the
sizes of the two recirculation regions are nearly identical and the flow remains symmetric. The most extreme situation is represented by the sudden expansion where the diverging walls are replaced simply by a vertical step (see e.g. [2, 15, 21]). When the adverse pressure gradient is too intense (large divergence angles or high area ratio) unsteady phenomena may occur like vortex shedding (see e.g. [79]) and the flow can be no longer stable or steady. The Reynolds number is also crucial to determine which type of flow is present inside a given diffuser configuration and the transition from one configuration to another (see e.g. [21, 60, 75]). Both the performance level and the exit flow conditions are intimately related to the presence or absence of flow separation (or stall) in the diffuser. Regions of stalled flow in a diffuser block the flow, cause low pressure recovery, and usually result in severe flow asymmetry, severe unsteadiness, or both. Therefore zones of separation of the fluid are deleterious for both external and internal flows (even though the latter are often underestimated).
Diffusers are used in many engineering applications, as for example:
• Jet engines: the flow in front of the engine must be decelerated in order to have an effective combustion. This occurs thanks to a diffuser shape air intake and a compressor. Also in radial compressors there are diffuser vanes.
• Wind tunnels: after the test chamber there is always a diffuser in order to decrease the velocity of the flow, since the power used to maintain the tunnel operative is directly proportional to the cube of the velocity (see e.g. [57]).
• Turbomachines (see e.g. [28, 40]). As in jet engines the flow must necessarily be decelerated to have efficient combustion.
• Automotive: in piston engines before the carburetor there is a diffuser in order to homogenize the mixture air-fuel (see e.g. [62, 64]). Also in the exhaust system the catalyst needs a diffuser for similar reasons (see e.g. [7, 43]).
• Hydropower applications: there are diffusers for the same reasons of wind tunnels (see e.g [14, 23]).
• Microfluidic (see e.g. the review papers [29, 71, 73]); examples are the design of micropumps (see e.g. [74] [86], electronic cooling systems and other heat-exchanging devices (see e.g. [50, 81]).
• Biomedical problems (flow configurations similar to the previous point) (see e.g. [36]). The first four points of the previous list typically concern flows at high Reynolds number and therefore turbulent flow regimes, while the last two concern flows at low Reynolds number and therefore laminar flow regimes.
Many engineering devices have been studied to increase the performance of diffusers (in laminar and in turbulent conditions) and these methods of control can be divided into two groups: active and passive methods of control. The active methods often require external power supply and almost
always they do increase the weight and complexity of the diffuser on which they are mounted. On the other hand the passive methods of control do not require outside intervention and thus they do not increase much the complexity of the design. For these reasons, at equal beneficial effects, passive methods of control are often preferred. The methodologies for improving the performance of internal flows are similar to those involving external flows and typically concern, as said earlier, boundary layer controls. The following is a list of attempts to improve the performance of diffusers: • Studies on possible modifications of diffusers wall shape have been carried out both in laminar and in turbulent regime. The effect of contouring the wall shape of a two-dimensional diffuser was analysed in [13] by comparing the performance of trumpet-shaped and bell-shaped dif-fusers to the one of a straight-wall diffuser. More recently, systematic shape design methods using mathematical theories have been developed, based on the gradient algorithms (see e.g. [89]), or the response surface techniques (see e.g. [51]). The optimum conditions obtained by [46, 78, 89] in turbulent regime corresponded to a skin friction equal to zero along the diffuser walls. This coincides with the physical intuition that the design of the diffuser walls maximizing the pressure recovery in the divergent part is the one corresponding to zero skin friction along the diverging walls (see [76, 77]). Indeed, the optimum diffuser designs proposed in [76, 77] are characterized by continuous incipient detachment of the boundary layers along their walls.
• Blowing flows are often used in order to improve diffuser performance energizing the fluid so that it can better resist to the adverse pressure gradients. Blowing flows can be continuous or intermittent and they can be active (see e.g. [39, 42, 49]) or passive (see e.g. [67]). In active blowinga pump (and thus energy) is necessary while passive methods are more simple. In passive blowing it is sufficient to connect diffuser areas of different pressure.
• Vortex generation is another method to improve diffuser performance through the same phys-ical mechanisms of the previous point: to energize the flow and in particular the boundary layers. Also in this case the generation of vortex can be active (see e.g. [26, 30]) or passive (by means of micro-vanes or micro-ramps).
• Directed synthetic jets (DSJ) can be used to energize the boundary layers and are described in [56]. This method uses a device that produces acoustic pulses (pressure waves) in order to energize the boundary layers.
• The effect of movable surfaces are studied in [72, 80]. In these studies rotating cylinders are used in order to transfer momentum to the fluid.
• In recent years a new method has been developed to improve the performance of plane diffusers both in laminar (see e.g. [27, 48, 52, 54, 55]) that in turbulent regime (see e.g.[6, 52, 53]). The more commonly accepted idea in the optimization of diffusers performance, based on the works [46, 76, 77, 78, 89], is that the optimum diffuser shape implies an attached boundary layer at the edge of separation along the diverging walls. This new method goes against this idea
as tried to obtain an increase of the diffusers efficiency by means of local flow recirculations (then this is a passive method).
As this work will focus on this last method, it is necessary to deepen the last point of the previous list. The considered strategy is based on the use of single or multiple local flow recirculations along the diffuser diverging walls. This approach was originally inspired by the combination of the ideas of trapped vortices (see e.g. [16, 20, 35, 44, 63, 69]) and multistep afterbodies (see e.g. [12, 38, 84]). An example of these two ideas is shown in the next figures.
(a) Trapped vortex (b) Multi-step afterbody
The trapped-vortex idea is based on the observation that the vortices which are formed inside the cavities increase the amount of momentum present in the downstream boundary layer in order to delay his separation. In this control strategy the cavities are typically much larger than the boundary layer thickness. The trapped-vortex concept was first introduced by Ringleb [69] who proposed it for boundary layer control in both external and internal flows. Since then several applications of the trapped-vortex concept have been considered, particularly as regards flow control around wings. The trapped-vortex concept has been analyzed trough theoretical and numerical studies [16, 20, 35, 63] and experimental investigations [44, 63]. The trapped-vortex control applied to airfoils usually requires a steady large structure with almost constant vorticity to form in the cavity and to create a flow recirculation region separated from the outer flow by a thin and stronger shear layer [20]. The main difficulty of this approach is to obtain a strong and stable vortex inside the large cavity. In fact, it has been observed experimentally [44, 63] and shown by stability analyses [35] that stable vortices form only rarely inside the cavity and an active control is therefore almost always necessary. For this reason some control strategies have been proposed and tested. For instance the possibility of controlling the flow over an airfoil trough a trapped vortex combined with flow blowing and suction for its stabilization was studied experimentally in [44]. As regards the trapped-vortex control applied to diffusers, a vortex-controlled configuration that relies on the aerodynamic design of large cusps on the diffuser walls in which to place the vortices is described in [69]. Even though auxiliary devices, such as lips, vanes, secondary walls, are also used to aid the flow reattachment and the vortex formation, only limited success has been obtained, due to the difficulty in having a stable vortex system. Indeed it is demonstrated experimentally in [1] that the pressure recovery in a short diffuser can be enhanced only by an actively-stabilized trapped vortex (Cranfield diffuser). On the other hand the multi-step afterbody idea (see e.g. [84]) allowed a reduction of the base drag of a bluff body to be obtained. By definition, a bluff body is a body in which the boundary layer sooner or later separates and the base drag is largely predominant. It consists in a number of
backward facing step placed upstream of the base; at each step a small toroidal flow recirculation region derives from the separation and reattachment of the flow. These flow recirculations determine a configuration with a gradual reduction in base area and in base suctions. This physical mechanism combines the concept of the boat-tail afterbody to that of energizing the boundary layer. To achieve the flow separation and reattachment to each afterbody step, a small step height must be chosen, of the same order of the boundary layer thickness. Therefore these small vorticity regions are smaller and weaker compared to the trapped vortices. The use of steps along with curved contours to control separation and improve the performance of boat-tailed shapes was proposed in [12]. In spite of the preliminary nature of the numerical analysis, it was shown that efficient multi-step afterbodies may be obtained, provided the geometry of the steps is suitably chosen.
The use of step or cavities with contoured shapes in internal flows has been first proposed by the fluid dynamics group of the DICI (Dipartimento di ingegneri civile e industriale) department of the university of Pisa. As previously mentioned in the last point of the previous list, contoured cavities have been added to diffusers. Modified symmetric diffusers in laminar flow regime (see e.g. [27, 48, 52, 54, 55]) and an asymmetric one in turbulent flow regime (see e.g.[6, 52, 53]) with various types of contoured cavities in order to improve their performance have been studied. In particular in these works a strategy based on the use of single or multiple local flow recirculations along the diffuser diverging walls was considered. The chosen method to create these flow recirculations was the introduction of suitable optimized-contour cavities in the diffuser solid walls.
Since the case of symmetric diffusers only in laminar regime were studied and in the turbulent regime only an asymmetric diffuser, for which detailed experimental data were available, was studied, the natural consequence is the study of the performance improvement of a symmetric diffuser in turbulent regime. A symmetric diffuser produces, in fact, a larger pressure recovery compared to the asymmetric one for the same area ratio and length of the diverging walls. This is because the symmetric diffuser has an angle of divergence of the walls which is half of that the asymmetric one. This leads to a lesser adverse pressure gradient, a smaller separation zone and a higher efficiency.
This work, hence, focuses on the flow control in a symmetric diffuser configuration at relatively high Reynolds number. Indeed turbulent flows through geometric expansions are of interest for numerous engineering applications as previously remarked.
In the works relating to the laminar case a symmetric diffuser with an area ratio of 2was studied with different divergence angles and there was no previous test-cases. In the works relating to the turbulent case instead, the choice of the configuration was made on the basis of some articles avail-able in literature. In particular the test-case used in that study was an asymmetric planar diffuser with a diverging angle of the inclined wall equal to 10 degrees and an area ratio equal to 4.7. This diffuser configuration was investigated experimentally in [8, 9, 61]. In those experiment the Reynolds number based on the width of the diffuser and on the centreline velocity at the reference section upstream of the divergent part of the diffuser, at which the flow was fully-developed turbulent, was 20000. Several numerical studies of the same flow configuration or of similar ones are also docu-mented in literature, e.g. [4, 22, 31, 32, 34, 37, 88]. In particular large-eddy simulations (LES) were carried out in [37, 88], while the RANS approach was used in [4, 22, 31, 32, 34], together with a number of different turbulence models. Therefore, the well consolidated experimental and numerical
database available for that test-case permitted to carry out a detailed validation and calibration of numerical simulations in which a RANS solver was used. Particularly in [6, 52, 53] a preliminary study was carried out aimed at identifing a RANS turbulence model providing sufficiently accurate results with costs that could be acceptable for the subsequent numerical optimization of the diffuser with cavities.
Since in literature there is not a suitable test-case to validate the numerical simulations, in the present work it has been chosen to use a similar geometry to the previous turbulent case but symmetric, the same numerical models and general set-up validated and calibrated in the asymmetric case. The main results of the previous studies in turbulent regime [6, 52, 53] show that the use of optimized-shape cavities leads to an increase of the pressure recovery of 6.9 % with one cavity and 9.6 % with two cavities, compared to the original diffuser. Moreover the cavities were small compared to the boundary layer thickness. This suggested that this passive control device is more closely related to the multi-step afterbody concept than the use of large trapped vortices. The gain in pressure recovery is obtained by two effects: a virtual geometry modifications and a reduction of the loss of momentum in the near-wall region. In detail, the improvement of the diffuser performance seemed to be due both to the modification of the streamlines of the flow outside the recirculation zones and to the relaxation of the no-slip boundary condition along the contour of the recirculation region, which leads to lower momentum losses. It is worth pointing out that, in the asymmetric turbulent case, the first effect (virtual geometry modifications) is more predominant.
The present work continues the investigation started in [6, 52, 53] (turbulent flow condition). Also in this case, the main goal is the assessment of the possible improvement of the symmetric diffuser efficiency that can be obtained through the control of flow separation by means of one, two or three small local flow recirculations, obtained by introducing appropriately-shaped cavities in the diffuser divergent walls. In particular, an optimization procedure is carried out to find the optimal cavity shape and location, i.e. the ones producing the largest reduction in flow separation and, hence, the largest increase in diffuser efficiency. A complete optimization is carried out considering different parameters characterizing the location and the geometry for each couple of cavities. One of the main objectives of this activity is to compare the results obtained in the present application with the outcomes of the previous investigations, in the laminar flow regime case ([27, 48, 52, 54, 55]) and (maybe more similar) in the turbulent flow regime case ([6, 52, 53]). In particular, the comparison concerns the shape, the locations, and the dimensions of the optimal cavities, as well as the obtainable level at efficiency improvement. Moreover, a question at issue is whether the physical mechanism leading to the successfully control strategy identified in the asymmetric geometry are still effective for the present diffuser symmetric geometry.
The present manuscript is organized as follows. In the second Chapter the geometries of the reference diffuser are defined, together with the description of the computational grid and the validation of the numerical methodology used in the simulations. In the third Chapter the procedure used for the cavity shape optimization is described together with the results for one, two and three subsequent couples of contoured cavities. In the fourth Chapter the results of a classical shape optimization of the diffuser, obtained by using Bézier curve are described. Finally, concluding remarks are drawn in the last Chapter.
Flow analysis for the reference diffuser
configuration
2.1
Geometry definition
The considered diffuser geometry is rather similar to the one used in [6, 52, 53]. That one was asymmetric, while the present one is a symmetric configuration with the same area ratio and only the diverging part different. The present diffuser is divided into three sections: an inflow channel having constant width, a symmetric linearly diverging channel and an outflow channel having again constant width (see Fig. 2.1). The inlet widthh is used here as the reference length, while the outlet width is k = 4.7h. The diffuser area ratio, i.e. the ratio between the outlet and inlet cross-areas is hence AR = 4.7. The divergence angle α (the angle between the axis of the diffuser and the divergent wall) is 5◦. The three diffuser parts have lengths equal tol1= 65h, l2= 21h and l3 = 60h
respectively and they are connected through sharp edges. The upstream channel (l1) is sufficiently
long to obtain a fully-developed turbulent flow at the inlet of the diffuser diverging section as in [22]]. The downstream constant width channel (l3) is sufficiently long to let the flow reattach and
allows a regular and stable flow at the outlet. The second portion of the diffuser (l2) is the one in
which the cavities will be applied.
Figure 2.1: Diffuser geometry and reference frame
The adopted frame of reference is shown in Fig. 2.1. The abscissa axis start at the inlet of the diffuser diverging section, it is placed at the symmetry axis of the diffuser and it is pointed in the direction of the flow. The ordinate axis start at the symmetry axis of the diffuser and it
is orthogonal to the abscissa. Note that in all the figures dimensionless coordinates are used, i.e X = x/h and Y = y/h (capital letters are used for dimensionless parameters and lowercase letters for dimensional quantities). Note that in the following graphics of figures the ratio between the abscissa axis and the ordinate axis is often less than one.
2.2
Simulation set-up and numerical methodology
The simulation are carried out at Re = h · u/ν = 20000, where u is the x-velocity on the centreline at X = −6.5, which is chosen as the diffuser reference cross-section because therein the flow has already become fully-developed turbulent. This is witnessed by a velocity profile characterized by a ratio between the centreline velocity and the bulk velocityucl/ub = 1.14 (see [8, 19]). A Re = 20000
with the choice of the inlet section as the reference length, corresponds to a Re of about 2.5 · 106
if the reference length had been the total length of the diffuser (l1+l2 +l3). The flow regime is
then surely turbulent. This definition of the Reynolds number assured, for the asymmetric diffuser work, a matching with experimental conditions in [8, 9, 61].
Since the flow atX = 0 is fully-developed turbulent and, hence, the boundary layers thicknesses areh/2 high it reasonable to think that the influence of the inlet roundings is negligible.
The numerical methodologies used in this work are briefly described below. Two-dimensional nu-merical simulation of the flow inside the diffuser were carried out by solving the Reynolds-Averaged Navier-Stokes (RANS) equations. These equations are based on the Reynolds decomposition: each velocity vector is decomposed in his time-average value and in his fluctuating value. If this de-composition is inserted into the equations of motion and the average operator is applied, RANS equations are obtained. In these equations, the mean values of the velocity components and the pressure appear, together with the so-called Reynolds stress tensor, which described the effect of the fluctuations on the mean flow. Consequently, in order to predict the mean field in a turbulent flow using the RANS equations, the Reynolds stresses must be derived from an appropriate turbulence model. This is the so-called closure problem and several turbulence models of increasing complex-ity have been developed for the analysis of different types of flow by means of numerical codes. This approach allows to solve the Navier-Stokes equations, even if averaged, with the addition of a turbulence model in order to define the Reynolds stress tensor. In the present work two different turbulence models are used and compared:
• the Shear-Stress Transport k-ω (SST k-ω) (low Reynolds version); • the Reynolds Stress Model (RSM) (low Reynolds version).
The SSTk-ω turbulence model derives from the standard k-ω model which is one of the most common turbulence (probably after the k- model, where turbulence dissipation rate, that, how-ever, is not appropriate for flow characterized by massive boundary layers separations). The SST k-ω turbulence model is an eddy viscosity closure model. The eddy viscosity closure models hy-pothesizes the Reynolds stress tensor proportional to the tensor average strain rate by means of an eddy-viscosity coefficient µt. This is the Boussinesq hypothesis (see e.g [33]). A closure problem
with this approach is, from the point of view of computational costs, less onerous because it is needed to add only a small number (in this case two) of partial differential equation (additional transport equations) to the Navier-Stokes averaged equations. The standardk-ω model is based on the Wilcox model (see. [87]) which incorporates modifications for low-Reynolds numbers effects and compressibility. The turbulent eddy viscosity is modeled by means of the two functions k and ω i.e. the turbulent kinetic energy and the specific dissipation rate. The disadvantage of the Boussinesq hypothesis as presented, is that it assumes µt is an isotropic scalar quantity, which is not strictly
true. The SST k-ω turbulence model was developed by Menter (see [58]) and it is also based on the Boussinesq hypothesis. It has the robust formulation of thek-ω model in near wall regions with the free-stream independence of the k- model in the far field. The SST k-ω model incorporates a damped cross-diffusion derivative term in the ω equation and the definition of the turbulent eddy viscosity is modified to account for the transport of the turbulent shear stress. Moreover the mod-eling constants are different. These features make the SST k-ω model more accurate and reliable for a wider class of flows (e.g., adverse pressure gradient flows, airfoils, transonic shock waves) than the standardk-ω model. Other modifications include the addition of a cross-diffusion term in the ω equation and a blending function to ensure that the model equations behave appropriately in both the near-wall and far-field zones. For these reasons in the present work the SST k-ω turbulence model is used.
The alternative approach, embodied in the RSM, is to solve transport equations for each of the terms in the Reynolds stress tensor, abandoning the Boussinesq hypothesis. An additional scale-determining equation (normally for ) is also required. This means that five additional transport equations are required in 2D flows and seven additional transport equations must be solved in 3D. In many cases, models based on the Boussinesq hypothesis perform very well, and the additional computational expense of the Reynolds stress model is not justified. However, the RSM is clearly superior for situations in which the anisotropy of turbulence has a dominant effect on the mean flow. Such cases include highly swirling flows and stress-driven secondary flows. The low Reynolds stress model used in this work is described in [45].
The numerical simulations of the flow inside the diffusers are carried out using the commercial computational code FLUENT R (release 6.3.26) (see e.g [3]) that is based on the finite-volume
discretization method. This choice has been made against the results of the preliminary simulations in [52, 54, 55] which have shown that the use of this commercial closed source code implies the lowest computational costs and it is simpler to configure with respect to the open source codes AERO and OpenFoam.
The Finite Volume Method (FVM) is one of the most versatile discretization techniques used in CFD. Based on the control volume formulation of analytical fluid dynamics, the first step in the FVM is to divide the domain into a number of control volumes (aka cells, elements) where the variable of interest is located at the centroid of the control volume. The next step is to integrate the differential form of the governing equations (very similar to the control volume approach) over each control volume. Interpolation profiles are then assumed in order to describe the variation of the concerned variable between cell centroids. The resulting equation is called the discretized or discretization equation. In this manner, the discretization equation expresses the conservation principle for the
variable inside the control volume. The most compelling feature of the FVM is that the resulting solution satisfies the conservation of quantities such as mass, momentum, energy, and species. This is exactly satisfied for any control volume as well as for the whole computational domain and for any number of control volumes. Even a coarse grid solution exhibits exact integral balances. FVM is the ideal method for computing discontinuous solutions arising in compressible flows. Any discontinuity must satisfy the Rankine-Hugoniot jump condition which is a consequence of conservation. Since finite volume methods are conservative they automatically satisfy the jump conditions and hence give physically correct weak solutions. FVM is also preferred while solving partial differential equations containing discontinuous coefficients. For these reasons the code FLUENT R is a very
suitable tool when many simulations have to be executed.
In agreement with the numerical ingredients chosen in the previous works (see [52, 54, 55]), in the present work the Navier-Stokes equations are discretized by using the p-v coupled algorithm together with the second order centred discretization scheme for pressure and the second order upwind discretization scheme for momentum equations. The main p-v coupling algorithms are the coupled p-v algorithm and the segregated p-v algorithms. The former solves a coupled system of equations comprising the momentum equations and the continuity equation; in the latter the governing equations are solved sequentially and all of these de-coupled schemes are based on the predictor-corrector approach (there are several de-coupled algorithms). The full-implicit coupling is achieved through an implicit discretization of the pressure gradient terms in the momentum equations and an implicit discretization of the face mass flux, including the Rhie-Chow pressure dissipation terms (see e.g. [3]). Obviously the simulations with these settings can be steady or unsteady.
The numerical options used in the present work are summarized in Tab. 2.1.
Code P-V scheme Pressure Momentum
FLUENT R Coupled 2nd order 2nd order upwind
Table 2.1: Summary of the simulation settings
For all the considered turbulence models the low Reynolds number option is used and therefore no wall functions are employed. For this reason a suitable grid refinement is adopted in order to have y+ ≤ 1 at the walls. The fluid considered is air (ν = 1.46 · 10−5 m2/s) and the boundary
conditions in the diffuser scheme are:
• Inlet (X = −65): velocity inlet with uniform profile (about 0.25 m/s) and turbulence intensity of 1 % and a turbulent length scale of 0.001h;
• Walls: no-splip conditions;
• Outlet (X = 81): outlet pressure: gauge pressure; backflow turbulent intensity 1 % and backflow turbulent length scale of 0.001h.
These setting of the turbulence are however pleonastic because the inlet channel is long enough to ensure a fully-developed turbulent flow at X = −6.5. Even a laminar flow at the inlet would
surely become turbulent at X = −6.5.
The computational grid is made with the software GAMBIT (release 2.3.16) and it is a structured grid made of quadrilateral elements (see Fig. 2.2) of uniform length in the streamwise direction. The boundary layer quadrilateral elements are also of uniform height in the streamwise direction.
Figure 2.2: Computational grid in the reference diffuser
The most critical regions are the near-wall zones because in the considered turbulence models no wall functions are employed and the boundary layer near the wall must be well characterized. In order to have y+ ≤ 1 at the walls the quadrilateral elements of the grid have a height smaller
and smaller as they get closer to the wall (see Fig. 2.3). To ensure this condition the first row of the boundary layer discretization is 1.375 · 10−3h high. Then a suitable growth rate is imposed on the grid elements in the lateral direction.
Figure 2.3: Particular of the calculation grid in the reference diffuser
First of all, a preliminary unsteady simulation with an appropriate grid was carried out to check that a steady solution was obtained, as found in the asymmetric turbulent case ([4, 6, 22, 31, 32, 34, 37, 52, 53, 88]). In this preliminary simulation, unsteady time advancing was chosen together with the same simulation settings of Tab. 2.1. The adopted dimensionless time step was ∆T = ∆t/(l/u) = 8.78·10−3and represents the ratio between the dimensional simulation advancing
total lengthl = l1+l2+l3. The time step then chosen to 5 seconds. The Courant-Friedrichs-Lewy
(CFL) number was set to 40 and the number of iterations per time step 20.
Since after a numerical transient the flow inside the diffuser was found to become steady after 10000 time steps as it is shown in the following Fig. 2.4 where at each color corresponds sections each distant at 10h of the diffuser, the section at X = −6.5 and the inlet and outlet sections, in the following of the work only steady-state simulations were carried out. The above-described pressure-based coupled algorithm and second-order upwind scheme are used to solve the steady-version of the Navier-Stokes equations.
0 2000 4000 6000 8000 10000 −1 0 1 2 3 4 5 6
Convergence history of static pressure
Number of time steps
Area−weighted average static pressure
0 2000 4000 6000 8000 10000 0.05 0.1 0.15 0.2 0.25 0.3
Convergence history of x−velocity
Number of time steps
x−velocity
Figure 2.4: Pressure and x-velocity steadiness within the reference diffuser
Grid sensitivity analyses for the two considered turbulence models have been carried out. For each turbulence model four calculation grids have been tested having 292 x 65 (coarse grid), 584 x 100 (medium grid), 1168 x 200 (fine grid) and 2920 x 400 (extra-fine grid) nodes respectively (the first number is the number of nodes in the streamwise direction, the second in the lateral direction). The more refined grids are generated approximately halving the dimension of the cells in the streamwise and lateral directions and consequently decreasing their growth rate in the lateral dimension (1.2, 1.1, 1.05, 1.025 of the coarse, medium, fine and extra-fine grid respectively). The same lateral dimension of the first cell adjacent to the wall is used for all the grids to have the same value of the wall y+.
2.3
Flow features and validation
The diffuser performance is evaluated through the mean pressure recovery coefficient Cp, which is
defined as: Cp = pout− pin 1 2ρuin2 , (2.1)
where pin and uin are the area-weighted average pressure and x-velocity (equal to the bulk
pressure at the diffuser outlet (X = 81).
The performance of the diffuser is also evaluated through the efficiencyη, defined as:
η = Cp
Cpideal
, (2.2)
where the ideal pressure recovery coefficientCpideal is calculated as:
Cpideal = 1 − uout uin 2 = 1 − Ain Aout 2 = 1 − 1 AR 2 = 1 − h k 2 , (2.3)
where in the second passage the balance of mass in steady and incompressible condition has been used:
uinAin=uoutAout. (2.4)
Thus, for this diffuser (with area ratioAR = Aout/Ain = 4.7) Cpideal' 0.95.
The simulations are also compared by evaluating the mean pressure coefficient Cpx at different
sections along the length of the diffuser, which is defined as:
Cpx =
px− pin 1 2ρuin2
, (2.5)
wherepx is the area-weighted averaged pressure at the considered X section of the diffuser.
Grid independence is checked, on the most important parameter: the mean pressure recovery coefficientCp. It was assumed that grid independence was reached when two different refined grids
showed a difference of less than 2% in the mean pressure recovery coefficient. The main results of the grid sensitivity analyses are summarized in Tab. 2.2.
Grid SSTk-ω RSM
295 x 65 (coarse) 0.768 0.694
584 x 100 (medium) 0.769 0.739
1168 x 200 (fine) 0.769 0.784
2920 x 400 (extra-fine) 0.769 0.799
Table 2.2: Cp for the different turbulence models and the different computational grids
The SST k-ω model shows grid independence already for the coarse mesh, i.e. the 295 x 65 nodes grid. Conversely the RSM model reach grid independence only for the fine grid 1168 x 200 with a consequent increase of the required computational time for the simulations. Obviously the differences between two successive grids will be less marked when they will be more refined. Note that, at grid convergence, the predictions of Cp given by the two turbulence models differ by less
than 4%. Moreover with the SST k-ω model, the difference between the medium mesh and the coarse mesh in terms of Cp is less of 0.15%.
It is noteworthy that the performance of the symmetric diffuser respect to the asymmetric one (with the same AR, divergence angle and length) is higher: Cp= 0.768 (symmetric) and Cp= 0.716
(asymmetric).
In the following are shown the trends of theCpx inside the reference diffuser (−6.5 ≤ X ≤ 81) and
the computational costs (approximate times) for the two considered turbulence models to achieve the convergence of residuals (a simulation can be considered to convergence when the curves of the residuals are flat). The computer used is a 4 processor with 4 cores each; it has been used 8 processes. Calculation times are however indicative because they depend on the use of computer.
0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 X
C
px Coarse mesh Medium mesh Fine mesh Extra−fine meshFigure 2.5: Cpx comparison for the SST k-ω model
Grid Cp Computational cost Number of iterations
295 x 65 (coarse) 0.768 <1 minute 500
584 x 100 (medium) 0.769 1 minute 750
1168 x 200 (fine) 0.769 1 hour 3500
2920 x 400 (extra-fine) 0.769 7 hours 14000
Table 2.3: Computational costs and iterations for the SST k-ω model
Regarding the RSM turbulence model simulations it is necessary to specify that the four simula-tions (one for each grid) have been initialized from the SST k-ω in order to obtain the convergence first and to have robust simulations.
Grid Cp Computational cost Number of iterations
295 x 65 (coarse) 0.694 5 minutes 2000 (+500)
584 x 100 (medium) 0.739 20 minutes 5000 (+750)
1168 x 200 (fine) 0.784 2 hours 10000 (+3500)
2920 x 400 (extra-fine) 0.799 40 hours 40000 (+14000)
0 10 20 30 40 50 60 70 80 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 X Cpx Coarse mesh Medium mesh Fine mesh Extra−fine mesh
Figure 2.6: Cpx comparison for the RSM model
It is clear that RSM model requires much higher computational costs respect to the SST k-ω model.
The visualization of the flow streamlines in Fig. 2.7 , obtained using the SST k-ω turbulence model (coarse grid), shows that the diffuser is characterized by a large asymmetric zone of separated flow which reattaches before the end of the diffuser. Moreover, as the geometric configuration is symmetrical, the flow configuration is bistable, i.e. the separation zone can develop on either side of the diffuser with the same probability.
Figure 2.7: Streamlines inside the diffuser (SST k-ω turbulence model)
The asymmetric flow inside the reference diffuser is also evidenced by the longitudinal velocity field in the reference diffuser shown in Fig. 2.8 obtained again using the SST k-ω turbulence model (coarse grid).
Obviously, also the other fields like lateral velocity field, pressure field, turbulent kinetic energy field, specific dissipation rate field, turbulent viscosity field and so on, are asymmetric.
The extent of the flow separated region predicted by the two considered turbulence models are compared in Fig 2.9.
Figure 2.8: Longitudinal velocity field inside the diffuser (SST k-ω turbulence model) −5 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 X
C
px SST κ−ω RSM(a) Mean pressure coefficient at different X sections
−5 0 5 10 15 20 25 30 35 40 45 50 −2 0 2 X Y SST κ−ω RSM
(b) Separated region extents
Figure 2.9: Mean pressure coefficient and separated region extent for the reference diffuser
The behaviour of Cpx along the diffuser axis for the two turbulence models is shown in Fig.
2.9(a). It can be seen that there is an initial expansion due to the growth of the boundary layer thickness ("virtual nozzle"). In the diverging part of the diffuser the pressure increase but the effect of separation is to slow down the rate of increase of Cpx (see Fig. 2.5, Fig. 2.6 and Fig. 2.9).
Moreover it can be seen from Fig. 2.9(b) that the RSM model predicts a smaller separation zone than the SST k-ω. For this reason the Cp is bigger with the RSM model. It must be emphasized
that the points of separation and reattachment of the separation zone predicted with the RSM model are quite atypical and perplexing.
The Tab. 2.5 summarize the prediction of the extent of the separation zone in the reference diffuser configuration with the two turbulence models.
Model X separation X reattachment
SSTk-ω 3.5 27.5
RSM 6.5 24.5
Table 2.5: Location of the flow separation in the reference diffuser configuration
Since the SSTk-ω turbulence model reaches grid convergence for a coarser grid resolution and, hence, it implies the lowest computational costs, the following simulations and the optimization of the configurations with cavities were carried out with the SST k-ω turbulence model (coarse grid). For a comparison between the used turbulence models and others with experimental data (asym-metric diffuser configuration) the reader can refer to [6, 52, 53].
To conclude this chapter may be relevant to recall that CDF simulations are useful to compare two ore more configurations made with the same calculation grid and with the same settings. This approach can reveal the variation in the considered objective function and this is the matter of interest despite the absence of a validation with experimental data. The main objective of most of the experiments is to verify if the change of a parameter leads to the improvement of the performance of the considered system. What really matters is the comparison, the variation, compared to the reference condition, not the exact values of the individual performances. For these reasons CFD methods are gaining more and more: it is not able to provide accurate results as the real testing but is capable of capturing the variation in performance between a reference configuration and the modified one. Moreover, obviously, "virtual" experimentations like CFD are much more cheaper than real investigation.
Diffuser with optimized general-contour
couples of cavities
3.1
Description of the optimization procedure
All the optimization described in this thesis were carried out through the automatic procedure shown in Fig. 3.1.
Figure 3.1: Scheme of the automatic optimization procedure
In each optimization loop, the diffuser geometry is defined, the computational grid is generated and the cost function is evaluated through the numerical simulation of the flow inside the diffuser. The CAD was constructed with the software CATIA R (version 5.19) with the addition of the
pa-rameters of the problems, the calculation grid generator used is GAMBIT (version 2.3.16), the CFD solver used is FLUENT R (version 6.3.26) and the optimization software used is modeFRONTIER R
(version 4.3.0 b20101110).
with an unstructured triangular mesh with the same cell growth rate of the reference structured mesh (see Fig. 3.2). Unstructured meshes are required within the cavities because the walls have a too small radius of curvature to have a structured meshes therein. The grid is, hence, structured-unstructured. The structured grid is connected with the unstructured one with proper streamwise growth rates. The size of the triangular cells within the cavities are refined enough to allow a good computational prediction of the flow also for the smaller ones. All the computational simulations of the optimization procedure will be performed with a sufficient number of iteration in order to ensure residuals convergence.
Figure 3.2: Example of the grid topology in the diffuser with couple of cavities
The numerical results are managed by the optimization algorithm, which determines the mod-ified configurations, finally leading to the optimized geometry. The optimization algorithm is the Multi-objective Genetic Algorithm MOGA-II [59, 65], used herein with a single-objective func-tion (see e.g. [66] for the use of MOGA-II in single-objective optimizafunc-tion). This algorithm has been chosen for its robustness, because the complexity of the optimization problem was not known beforehand. In all cases the parameter space is discretized in intervals of uniform size; an ini-tial population is generated through a pseudo-random Sobol sequence [59], by using a subset of the specified discrete parameters values. The population evolves through the following reproduction op-erators: directional crossover, mutation and selection [59]. The probability of directional crossover, of mutation and of selection are set to 0.5, 0.1 and 0.05 (default settings).
In simpler words, the Genetic Algorithms (GA) are based on the analogy with the natural evolution of species, i.e. natural selection and reproduction guide the evolution towards individuals that are better adapted to the environment, in order to lead to optimal choices.
It should be highlighted that the choice of the problem initial population and the choice of the amplitude of the space parameters have great importance to identify which input variables most affect the objective function and, so, to have a fast convergence of the optimization algorithm. If, for instance, the range of variation of a parameter is to wide, the optimization may not be able to
converge towards the absolute maximum or minimum of the objective functions.
It might be appropriate to spent a few words on the environment of the optimization software modeFRONTIER R. The most important elements are the "Application Nodes", each of which
represents a specific software. In the present work three "application nodes have been used: one for CATIA R, one for GAMBIT and one for FLUENT R. Each of these nodes is reached by "Input
Files" (the geometrical parameters of CATIA R designs) and by "Support Files" (the GAMBIT and
FLUENT R journals). The optimization cycle starts with the definition of the so called "Design
Of Experiments" (DOE) which defines the choice of the initial population. The DOE node is then connected with the "Logic node" "Scheduler" within which there is the optimization algorithm (the MOGA-II). The "Scheduler" is subsequently connected with a "Logic Node" "Logic If" whose function is to compare and eventually previously discard some configurations. The "Logic If" node reaches then the first CATIA R "Application node". From the first CATIA R "Application nodes"
leaves a "File Node" "Transfer Node" that is the files of the geometry (in this case more STEP files) that reaches the "Application node" GAMBIT that provides to build the calculation grid. When the calculation grid is ready, a "File Node" "Transfer Node" which contains the mesh file (.msh file) leaves the GAMBIT node and reaches the last "Application node", namely FLUENT R. When the
fluid dynamics calculations are ready, the "File node" "Output Files" with the quantities of interest leave the FLUENT R "Application node". These "Output Files" become "Variable Nodes" "Output
Variables" which are connected to the "Goal Node" "Design Objective" which determines the cost function (Cp) to maximize. It is also possible to create a "Networking Node" "SSH script" in order
to use an external more powerful computer. If this external computer has the ability to manage multiple processes, more than one configuration at a time can be managed by modeFRONTIER R.
In this latter case it is necessary to introduce "Logic Nodes" "Queue Start" before the CATIA R
"Application node" and a "Queue End" after the FLUENT R "Application node".
A critical point of this optimization cycle has been certainly the calculation grid construction. This because the mesh generator (GAMBIT) had to provide to create a new mesh for the new configuration coming out from the parametric CAD generator (CATIA R) with an unique and
general journal file. The problem is that when GAMBIT imports a geometry (for example a STEP or IGES file) it numbers the edges of the geometry with a proper orientation (south, east, west and north). If the geometry changes, the edges numbering may change too. Since the journal file is one with inside a precise association between the edges geometry numbers and the features associated with them it is clear that when the couples of cavities vary their position along the diffuser divergent walls, this leads to an error. In order to overcome this problem the CAD geometry has been split in several parts (7 and more STEP files) and then assembled one by one at the time of building the calculation grid. With this dodge the mesh generator GAMBIT numbers the edges of one part of the modified diffuser at the time and so the journal file can be adequate for all the possible geometries explored by the optimization algorithm.
Different indexes of the diffuser efficiency could be chosen as the objective function in the opti-mization procedure, namely the maxiopti-mization of the section-averaged pressure recovery coefficient Cp or the minimization of the total dissipation inside the diffuser. The parameter Cp, which is
evalua-tion has a smaller computaevalua-tional cost compared to that of the total dissipaevalua-tion and represents a good index of the diffuser performance. Nonetheless, after the optimization, the dissipation were also evaluated in order to have a further confirmation of the results. While Cp has been already
defined in equations 2.1, in incompressible regime, the dissipation function Φ is defined as follows:
Φ = τ :gradV = 2µ " ∂u ∂x 2 + ∂v ∂y 2 + ∂w ∂z 2# +µ " ∂u ∂y + ∂v ∂x 2 + ∂v ∂z + ∂w ∂y 2 + ∂w ∂x + ∂u ∂z 2# ≥ 0 (3.1)
where the constitutive equations for the components of the viscous stress tensor τ of a Newtonian fluid have already been introduced, together with the simplifications deriving from the incompress-ible flow regime (divV = 0). The dissipation function Φ represents the work per unit time on an elementary volume of fluid by the viscous stresses acting over its surface due to the non-isotropic part of the deformation. Furthermore, it corresponds to the irreversible (or dissipative) part of the work done in unit time by the surface stresses.
The role of the dissipation function becomes clear by referring to the following form of the integral equation of the kinetic energy balance:
Z V ∂ρV2 2 ∂t dV + Z S ρ V 2 2 V · ndS = − Z S pV · ndS + Z S τn· VdS − Z V ΦdV (3.2)
where the three integrals on the right-hand side represent, respectively, the work done on the fluid, in unit time, by the pressure and viscous forces acting on the volume boundary S and the total dissipation within the volumeV of fluid bounded by surface S.
Since the diffuser flow is steady (see Sec. 2.2) and the work done on the fluid by the viscous forces acting on the boundary is negligible compared to that of the pressure forces (V = 0 along the diffuser walls while the integral representing the total work of the viscous forces is very small at the inlet and outlet compared to the one connected with the pressure forces), the kinetic energy balance becomes: Z S ρ V 2 2 + p ρ V · ndS = − Z V ΦdV (3.3)
Equation (3.3) highlights that the dissipation is immediately connected with the balance of the kinetic and pressure energy in the flow, i.e. not all the kinetic energy variation is converted into pressure energy because of the viscous losses. Thus, the minimum of the total dissipation is expected to be found in the diffuser configuration with an optimized efficiency. For an incompressible flow, the total dissipation Φt in a volume V may be easily obtained as a function of the enstrophy ω2
Bobyleff-Forsyth formula (see e.g. [10, 11] and [70]): Φt= Z V ΦdV = µ Z V ω2dV + 2µ Z S a · ndS (3.4)
3.2
Optimization of single and multiple couples of general-contour
cavities
The possible improvements of the diffuser efficiency that can be obtained by introducing a single couple of contoured cavity in the diffuser divergent walls are first investigated. In particular, the optimized cavity shape that allows the diffuser efficiency to be maximized with one couple of cavities in the same operating conditions of Sec. 2.2 are described in Sec. 3.2.1. The same optimization procedure used for the single cavity is then applied to maximize the efficiency of a diffuser with two subsequent couples of cavities (Sec. 3.2.2) and with three subsequent couples of cavities (Sec. 3.2.3). In particular the differences between the asymmetric case and the present (symmetric) case will be emphasized.
3.2.1 Geometry definition and results for a single couple of general-contour cavities
One couple of general-contour cavities is positioned in the diverging walls of the diffusers in sym-metrical manner. The cavities start with a sharp edge, have an upstream part with a semi-elliptical shape and end with a spline up to a point that can be along the original diffuser diverging walls or inside the diffuser. The final points of the cavities are then connected to the end of the diffuser diverging parts through straight lines. The splines and the straight lines are tangent. Fig. 3.3 shows the upper part of the configuration of the diffuser with a single couple of general-contour cavities. The lower part is obviously symmetrical. Therefore, this cavity shape can end with a bulge inside the original diffuser profile, which is expected to promote the flow reattachment after the flow recirculation inside the cavity.
Figure 3.3: Geometry of the general contour cavity
From the CAD generator point of view, the cavities are constituted by a conic and by a spline. The portion of the conic more close to the axis of the diffuser starts with a condition of tangency with the horizontal; the portion of the conic more far to the axis of the diffuser starts with a condition of tangency with the diverging wall. The upstream portion of the spline has a condition
of tangency with the portion of the conic more far to the axis of the diffuser (they are connected); the downstream portion of the spline has a condition of tangency with the segment connecting it to the end of the diverging wall. The tensions spline parameter are kept fixed to the default values (1 and 1) not to increase too much the number of free parameters and to maintain the geometry consistent. The parameters that will be introduced in the following refer to an auxiliary frame of reference placed at (X, Y ) = (0, 0.5h) and with the abscissa axis coincident with the diffuser diverging wall.
Henceforth, for the sake of simplicity, we will refer to the characteristics of the cavity and to the diverging walls in the singular implying the presence of another symmetric cavity.
An optimization of the cavity shape is then carried out in order to maximizeCp and, hence, the
efficiencyη of the diffuser. The optimization parameter, shown in Fig. 3.3 are five:
• the distance from the beginning of the diffuser diverging part to the upstream edge of the cavity,s/h;
• the cavity total length, t/h;
• the ellipse axis parallel to the original diffuser diverging wall, a/h; • the ellipse axis normal to the original diffuser diverging wall, b/h;
• the normal distance from the end point of the cavity to the original diffuser diverging wall (i.e. the bulge magnitude), r/h.
The range of the optimization parameters has been chosen on the basis of the results of the previous works [6, 52, 53]. In particular it has been demonstrated that the values of b/h are small compared to the boundary layer thickness. The upstream edge of the cavity and its ending point are allowed to vary along the whole diffuser diverging wall, whereas the chosen range of variation of the cavity axes,a and b, are respectively chosen to be 0.02h−0.4h and 0.01h−0.2h. The normal distance from the end point of the cavity to the original diverging wall, r/h, is allowed to vary from 0 (i.e. with the spline tangent o the diffuser divergent wall) to 2h. The parameter space was discretized by using uniform intervals having a width of 0.02, 0.01, 0.2, 0.2 and 0.025 for a/h, b/h, s/h, t/h and r/h respectively. Since not all the geometric configurations created with the combinations of the previous parameter values are obtainable, further constrains are imposed, in order to reduce the number of unacceptable configurations: (s + t) 6 l2/(cos α) and r 6 0.75(s + t) tan α.
The ranges of the optimization parameters are summarized in Tab. 3.1.
a/h b/h s/h t/h r/h
Minimun Value 0.02 0.01 0.05 2 0
Maximum Value 0.4 0.2 18.05 20 2
Step 0.02 0.01 0.2 0.2 0.025
It is important to point out that some parameter values are dictated by the requirements of the calculation grid.
These discrete parameter values are explored by the MOGA-II algorithm. The initial population is composed of 50 individuals distributed in the discretized parameter space by means of the previ-ously cited Sobol sequence. Then, 15 additional generations (including unacceptable configurations) were created by the optimization algorithm, each one composed of 50 individuals. Starting from the fourteenth generation, most of the new individuals created by the algorithm are characterized by s/h = 0.025 − 1.025, a/h = 0.02 − 0.16, b/h = 0.01 − 0.02, t/h = 15.8 − 18 and r/h = 0.225 − 0.3 (8 over 50 individuals in the fourteenth generation). All these configuration give values of the objective functions which differ from the maximum one by less than 0.2%. To ensure the convergence of the residuals, the number of iterations for a single simulation was set to 2000.
The ranges of the optimization parameters that characterize the optimum configurations are summarized in Tab. 3.2.
s/h a/h b/h t/h r/h
Range of optimum 0.025-1.025 0.02-0.16 0.01-0.02 15.8-18 0.225-0.3
Table 3.2: Parameters range of optimum for a single couple of cavities One of the optimum configurations is shown in Fig. 3.4.
Figure 3.4: Streamlines in the diffuser with one couple of optimized cavities
It is clear that the optimum configurations are characterized by a very small semi-ellipse axis normal to the diffuser diverging wall b/h: the optimum values of b/h practically coincide with minimum value allowable (see Tab.3.1). The ellipse axis a/h have a negligible effect as in the previous works. Moreover the parameters t/h is very large. In this first case the only fundamental parameters are the sum (s/h + t/h) and r/h. The first one determine where the narrowing of the diffuser starts; the latter determines the magnitude of the bulge, i.e. how much the diffuser is locally narrowed. The main flow separation is however delayed and its extent reduced (compare with Fig. 2.7). The detailed comparison between the diffuser without cavities and the diffuser with one couple of optimized cavities in terms of mean pressure coefficient at different sections (Cpx, defined in Eq.
2.5, by using the area-weighted average pressure at the local section) and of separated region extent are shown in Fig. 3.5(a) and in Fig. 3.5(b) respectively. The starting point of the main separation zone moves from Xsep = 3.5 to Xsep = 16.5 and the reattachment point from Xreatt = 27.5 to
[13]) where the mild convex-inward curvature of the diverging walls causes a lower adverse pressure gradient respect to the original diffuser. On the other side the convex-inward curvature produces a narrowing of the divergent diffuser sections. This can be seen in Fig. 3.5(a): the Cpx curve of
the optimized diffuser in the first part of the divergent (from X = 2 to X = 15) lies beneath the Cpx curve of the reference diffuser. After X = 15 however, the Cpx curve of the optimized diffuser
surmounts the Cpx curve of the reference diffuser because the flow encounters a smaller zone of
separation and a bigger lateral section (even if the lateral walls are more close to the axis). The optimum values ofr/h are a compromise: too large values of r/h produce a too big local narrowing of the divergent walls which causes a local reduction of the pressure recovery. Too small values of r/h produce a too small local narrowing of the divergent walls which causes premature separation of the flow. −5 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 X
C
pxdiffuser without cavities
diffuser with optimized single couple cavities
(a) Mean pressure coefficient at different X sections
−5 0 5 10 15 20 25 30 35 40 45 50
−2 0 2
X
Y diffuser without cavities
diffuser with optimized single couple of cavities
(b) Wall geometry and separated region extent
Figure 3.5: Mean pressure coefficient, wall geometry and separated region extent: comparison be-tween the diffuser without cavities and the diffuser with one couple of optimized cavities
with a very long cavity (large values of t/h) in order to produce a "trumpet-shaped" diffuser configuration. Since the five optimization parameters are necessarily linked and the parametert/h is the most important, the real cavity which principal parameters areb/h and a/h is placed in a zone where the adversed pressure gradient is still too gentle (s/h small) and the main flow separation is still too far. In that place the main flow does not need a small flow recirculation (eventually established in the cavity) which would produce a relaxation of the no-slip boundary condition along the contour of the recirculation region. For this reason the parameters b/h and a/h are minimal (even if not all zero).
A pressure recovery increase of 5.9% is found compared to the sharp-edged reference diffuser (Cp increases from 0.768 to 0.813 in the configuration with optimum couple of cavities). The total
dissipation Φt inside the diffuser also confirms the improvement of the diffuser performance. A reduction of the dissipation of 1.4% is indeed found.
In the following Fig. 3.6(a) a non optimum configuration but with a non minimum values of the parametersb/h and a/h will be shown.
(a) Diffuser with non optimized couple of cavities
(b) Streamlines near to the optimized cavity
Figure 3.6: Streamlines in the diffuser with one couple of non optimum cavities
This non optimum configuration is characterized by the following values of the parameters: s/h = 1.85, a/h = 0.12, b/h = 0.04, t/h = 17.4 and r/h = 0.225. It can be seen that occurred a symmetrization of the flow: the main flow separation is split in two smaller separations. This alternative may be used in engineering applications which require a symmetric flow. In this configu-ration a small recirculation region forms in the cavity (see 3.6(b)). This non optimum configuconfigu-ration leads to a pressure recovery increase of 4.9% compared to the sharp-edged reference diffuser (Cp
increases from 0.768 to 0.806).
out to check that the flow is steady even after the introduction of the contoured couple of cavities in the diffuser diverging wall. A steady solution was reached, which coincides with the one of steady-state simulation.
3.2.2 Geometry definition and results for two couples of general-contour cavi-ties
In this section two subsequent couples of cavities are considered in order to investigate whether a second couple of cavity can further improve the diffuser efficiency and reduce the flow separation. The two couples of cavities are also introduced in the diverging walls of the diffuser. Both couple of cavities start with a sharp edge, have un upstream part with a semi-elliptical shape and end with a spline up to a point that can be along the original diverging wall of the diffuser inside it.
Figure 3.7: Geometry of two subsequent general-contour cavities
The starting point of the second cavity coincides with the ending point of the spline of the first cavity. The ending point of the second cavity is then connected to the end of the diffuser diverging part through a straight line which is tangent to the preceding spline (see Fig. 3.7). From the CAD generator point of view the situation is almost equal to the previous configuration with one couple of cavities. The only difference is that the downstream portion of the first spline has a condition of tangency with the reference diverging wall and not with the segment connecting it to the end of the diverging wall.
The same optimization carried out for the single couple of cavities (see Sec. 3.2.1) was used to identify the shape of the device that maximises the diffuser efficiency. The optimization parameters are seven:
• the distance from the beginning of the diffuser diverging part to the upstream edge of the first cavity,s/h;
• the first cavity total length, t1/h;
• the second cavity total length, t2/h;
• the ellipse axis normal to the diffuser diverging wall of the first cavity, b1/h;
• the ellipse axis normal to the diffuser diverging wall of the second cavity, b2/h;
