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Petr Knobloch Editor
Boundary and Interior Layers, Computational and Asymptotic Methods – BAIL 2014
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Editor Petr Knobloch Faculty of Mathematics and Physics Charles University in Prague Praha, Czech Republic
ISSN 1439-7358 ISSN 2197-7100 (electronic) Lecture Notes in Computational Science and Engineering ISBN 978-3-319-25725-9 ISBN 978-3-319-25727-3 (eBook) DOI 10.1007/978-3-319-25727-3 Library of Congress Control Number: 2015957960 Mathematics Subject Classification (2010): 34-XX, 35-XX, 65Lxx, 65Mxx, 65Nxx, 76Dxx. Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)
Preface
These Proceedings contain contributions reflecting a selection of the lectures presented at the conference BAIL 2014: Boundary and Interior Layers – Computational and Asymptotic Methods, which was held from 15 to 19 September 2014 at Charles University in Prague, Czech Republic. Apart from two hiatuses, the BAIL conferences have been held every 2 years and BAIL 2014 was already the 14th conference in this series. The first three BAIL conferences were organized by Professor John Miller in Dublin (1980, 1982, 1984). The next ten were held in Novosibirsk (1986), Shanghai (1988), Colorado (1992), Beijing (1994), Perth (2002), Toulouse (2004), Göttingen (2006), Limerick (2008), Zaragoza (2010), and Pohang (2012). BAIL 2016 is planned for Beijing. The conference BAIL 2014 attracted 54 participants from 15 countries in three continents. They presented 50 lectures, five of which were invited. Almost half of the speakers present the contents of their lectures in these Proceedings. The conference included two mini-symposia devoted to stabilized methods in aerodynamics modeling and tailored finite point method for problems with nonsmooth solution. As usual, both mathematicians (pure and applied) and engineers participated in this BAIL conference, which led to a very fruitful exchange of ideas. A positive aspect was also the participation of several PhD students, showing that the topics of the conference remain attractive also for young scientists. The lectures at the conference comprised both rigorous mathematical results on problems with layers and their numerical solution and complicated computational techniques for advanced applications. The contributions in these Proceedings are devoted to theoretical and/or numerical analysis of problems involving boundary and interior layers and of methods for solving these problems numerically. Almost half of the contributions study fluid flow problems, whereas the remaining contributions contain results on convection– diffusion, convection–diffusion–reaction, and reaction–diffusion equations. Both steady and transient problems are considered. Most of the numerical schemes presented in these Proceedings are based on the finite element method, closely followed by the finite difference method. Nevertheless, in three cases, also solvers based on the finite volume method are applied. The topics covered by the contributions v
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include theoretical analyses of various (stabilized) numerical schemes, typically represented by stability results and error estimates; analyses for various adapted meshes, in particular, uniform error estimates on Shishkin meshes; a posteriori error estimation; various approaches to turbulence modelling; studies of the influence of finite precision arithmetic; computational techniques for advanced applications; asymptotic theory of boundary layers; and investigations of layer phenomena in convection–diffusion problems with fractional derivatives. Almost all contributions contain numerical results illustrating the respective theoretical considerations. In view of the wide variety of topics treated in the contributions, the Proceedings provide a very good overview of current research into the theory and numerical solution of problems involving boundary and interior layers. All contributions in the Proceedings were subject to a standard refereeing process. I would like to thank the authors of the contributions for their cooperation and the unnamed referees for their valuable suggestions that contributed to the high quality of this volume. I also wish to thank the organizers of the mini-symposia at BAIL 2014, all the attendees for their active participation in the conference, and my colleagues Hana Orosová, Jiˇrí Felcman, Jan Lamaˇc, and Petr Lukáš for their help with the organization of the conference. Praha, Czech Republic August 2015
Petr Knobloch
Contents
A Note on the Stabilised Q1 P0 Method on Quadrilaterals with High Aspect Ratios .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Gabriel R. Barrenechea and Andreas Wachtel
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A Posteriori Error Estimation of a Stabilized Mixed Finite Element Method for Darcy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Tomás Patricio Barrios, José Manuel Cascón, and María González
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A Local Projection Stabilized Lagrange-Galerkin Method for Convection-Diffusion Equations.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Rodolfo Bermejo, Rafael Cantón, and Laura Saavedra
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Outflow Conditions for the Navier-Stokes Equations with Skew-Symmetric Formulation of the Convective Term . . . . . . . . . . . . . . . . Malte Braack
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Finite Element Approximation of an Unsteady Projection-Based VMS Turbulence Model with Wall Laws . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Tomás Chacón Rebollo, Macarena Gómez Mármol, and Samuele Rubino
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Spatial Semidiscretizations and Time Integration of 2D Parabolic Singularly Perturbed Problems . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Carmelo Clavero and Juan Carlos Jorge
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Boundary Layers in a Riemann-Liouville Fractional Derivative Two-Point Boundary Value Problem .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . José Luis Gracia and Martin Stynes
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On the Application of Algebraic Flux Correction Schemes to Problems with Non-vanishing Right-Hand Side . . . . . . .. . . . . . . . . . . . . . . . . . . . Petr Knobloch
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Investigation of Numerical Wall Functions Based on the 1D Boundary-Layer Equations for Flows with Significant Pressure Gradient .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 111 Tobias Knopp, Fabian Spallek, Octavian Frederich, and Gerd Rapin Modified SUPG Method on Oriented Meshes . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 121 Jan Lamaˇc On Numerical Simulation of Transition to Turbulence in Turbine Cascade.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135 Petr Louda, Karel Kozel, and Jaromír Pˇríhoda Understanding the Limits of Inf-Sup Stable Galerkin-FEM for Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147 Gert Lube, Daniel Arndt, and Helene Dallmann A Posteriori Optimization of Parameters in the SUPG Method for Higher Degree FE Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171 Petr Lukáš A Parameter-Uniform First Order Convergent Numerical Method for a System of Singularly Perturbed Second Order Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 183 Manikandan Mariappan, John J.H. Miller, and Valarmathi Sigamani Numerical Simulation of Air Jet Attachment to Convex Walls and Application to UAV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 197 Nikola Mirkov and Boško Rašuo Cholesky Factorisation of Linear Systems Coming from Finite Difference Approximations of Singularly Perturbed Problems .. . . . . . . . . . . . 209 Thái Anh Nhan and Niall Madden Numerical Experiments with a Linear Convection–Diffusion Problem Containing a Time-Varying Interior Layer . . . .. . . . . . . . . . . . . . . . . . . . 221 Eugene O’Riordan and Jason Quinn Second Order Uniformly Convergent Numerical Method for a Coupled System of Singularly Perturbed Reaction-Diffusion Problems with Discontinuous Source Term . . . . . . . . . . . . 233 S. Chandra Sekhara Rao and Sheetal Chawla A Multiscale Sparse Grid Technique for a Two-Dimensional Convection-Diffusion Problem with Exponential Layers .. . . . . . . . . . . . . . . . . . . 245 Stephen Russell and Niall Madden On the Delay and Inviscid Nature of Turbulent Break-Away Separation in the High-Re Limit . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 257 Bernhard Scheichl
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Use of Standard Difference Scheme on Uniform Grids for Solving Singularly Perturbed Problems Under Computer Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 269 Grigorii Shishkin Difference Schemes of High Accuracy Order on Uniform Grids for a Singularly Perturbed Parabolic Reaction-Diffusion Equation . . . . . . . 281 Lidia Shishkina Blow-Up of Solutions and Interior Layers in a Caputo Two-Point Boundary Value Problem .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293 Martin Stynes and José Luis Gracia On Finite Element Approximation of Fluid-Structure Interactions with Consideration of Transition Model . . . .. . . . . . . . . . . . . . . . . . . . 303 Petr Sváˇcek
Numerical Simulation of Air Jet Attachment to Convex Walls and Application to UAV Nikola Mirkov and Boško Rašuo
Abstract In this paper, we present a numerical study of the wall jet flow over a convex surface, viz. the Coanda wall jet, and its application to a conceptual Unmanned Aerial Vehicle (UAV) design which uses the Coanda effect as a basis of lift production. This configuration is important in a way that it considers the Coanda wall jet over a smooth convex wall with non-constant curvature, in contrast to most of the previous situations where only constant curvature walls were considered e.g. the Coanda wall jet over circular cylinder. To enable the mathematical representation of this complex geometrical configuration, we propose a form of a parametric representation of the conceptual geometry, based on Bernstein polynomials, which is universal in character and spans a complete design space. It is shown how dynamically changing the flow picture enables smooth change of net forces on the body. Capability to control the direction of the net force is shown to be useful for maneuvering the UAV. All simulations are done using an open-source finite-volume computational fluid dynamics code based on Reynolds-averaged Navier-Stokes equations. Turbulence is accounted for using the k! Shear Stress Transport model.
1 Introduction Turbulent air jets have a natural tendency to attach to the walls when blown close to them. After the air jet is issued into the quiescent surrounding, it entrains the air between the jet and the wall. Low pressure region is formed causing the jet to deflect towards the wall. This phenomenon known as the Coanda-effect has found diverse applications ever since it was discovered. Most applications came out of an observation that Coanda wall jet is an efficient way of generating aerodynamic
N. Mirkov Institute of Nuclear Sciences – Vinca, University of Belgrade, Mike Alasa 12–14, Belgrade, Serbia e-mail:
[email protected] B. Rašuo () Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade, Serbia e-mail:
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force. Underneath the attached wall jet static wall pressure falls below the ambient pressure causing the aerodynamic force on the body. Conceptual designs such as no tail rotor (NOTAR) helicopter, and aerodynamic devices like circulation control airfoils on vertical and short take-off and landing aircaft (V/STOL) are examples of successful application of Coanda wall jets. The use of Coanda wall jets as a primary mechanism for producing the lift force of the aircraft mostly remained in the blind spot of the aerodynamic community at least until the wider popularity of Unmanned Aerial Vehicles (UAV). Owing to the UAV, interest in the aircraft that produce lift with the aid of the Coanda-effect has been renewed in recent times. The need to address the aerodynamics of these aircraft in a systematic way has become present. The power available for the hover and maneuvering should be used as economically as possible. A need for thorough experimental and numerical tests is obvious. Wygnanski and co-workers did an experimental study of the curved wall jets [1– 5] observing that curved wall jet had greater spreading rate than the plane wall jet. They were observing large stream-wise vortices that increased turbulent momentum exchange which lead to faster spreading rates. In [6] more experimental evidence was given to support the view of wall curvature being responsible for maintenance the stream-wise coherent structures. Numerical studies of the Coanda wall jets using direct numerical simulation (DNS), as well as large eddy simulation (LES) are scarce, mostly due to a large range of scales present in these flows. In [7, 8] DNS is used to simulate configuration from the experiments of Wygnanski and coworkers confirming their observations related to the appearance and role of stream-wise vortices. Numerical studies relying on Reynolds Averaged Navier-Stokes approach with linear eddy-viscosity models (LEVM) were employed in several cases, such as [9, 10]. Most of the difficulties faced when using LEVM resulted from neglecting the influence of curvature in derivation of these models. LEVM models led to under-prediction of wall jet spreading rate and over-prediction of skin friction coefficient. All simulations in [9] showed great dependency on jet inflow conditions and transition process. Results from [9] indicated that Menter’s k ! SST model gave relatively good results when the wall pressure coefficient was predicted, which is significant for the present study. Because of the high computational cost of LES or DNS simulations for this type of flows, and because of our interest primarily in pressure distribution responsible for lift generation, we decide to use the SST model for our study interpreting the results cautiously, having in mind previous general criticism of LEVM models. Among other things, this paper is focused on conceptualizing the geometric shape of the aircraft which uses Coanda wall jet for lift production. Consequently, the Coanda wall jet studied in this paper differs from the previous ones since the wall jet is attached to a wall with no constant curvature, as postulated in other works [7–10] where wall jet attaches to the surface of the circular cylinder.
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2 Definition of Geometry In this section, the geometry chosen for the current investigations and its parametric representation are described. The way geometry is represented strongly affects the optimization process. In [11–13], it is shown how different airfoils, axially symmetric bodies, and other more complex three dimensional shapes (wings, fuselages, etc.) can be represented by means of the class function, shape function representation, within the ‘CST’ approach. The class function is used to define a general class of geometries, while the shape function is used to define specific shapes within the geometry class. The main idea presented in these papers is to decompose the basic shape into scalable elements corresponding to discrete components by representing the shape function with a Bernstein polynomial. The Bernstein polynomial of order n is composed of the n C 1 terms of the form: Sr;n . / D Kr;n Kr;n
r
.1 /nr ;
! n nŠ ; rŠ.n r/Š r
(1) (2)
where factors Kr;n are binomial coefficients as shown above. Figure 1 shows an example of unit function decomposition using Bernstein polynomials of sixth degree. By scaling any component in the unit function representation we can make well localized, small variations of that function. The geometry of cross section of a Coanda-effect UAV is similar to elliptic-arc. Therefore, the class function will be the function defining an elliptic arc, . / D A
0:5
.1
/0:5 :
Fig. 1 The unit function decomposition using sixth degree Bernstein polynomial
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a)
c)
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← ζ (ψ) = 0.7ψ 0.5 (1 − ψ)0.5 , S(ψ) = 1, Cupper = ζ (ψ)S(ψ), Clower = 0.25Cupper .
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Fig. 2 Definition and universal parametric representation of the UAV shape: (a) Bernstein polynomial decomposition of an elliptic arc . /, (b) Cross-section defined by Cupper and Clower . (c) 6-DOF motion of the Coanda effect UAV
By adjusting the coefficient A in the class function representation, one can choose the aspect ratio of an elliptic arc. Taking A to be equal to two, gives a circle. By scaling the coefficients in the component representation of a shape function, we can derive appropriate variations of the basic shape. Figure 2a shows decomposition of an ellipse (A D 0:7) using Bernstein polynomials of sixth degree. We need to emphasize that the order of Bernstein polynomial, used for the function decomposition, is chosen freely. The higher order makes more localized variations of the basic shape possible. For our numerical studies, we used geometry defined by the following expressions. The upper contour is defined by Cupper . / D . /S. /;
(4)
where . / D 0:7
0:5
.1
/0:5 ;
(5)
and S. / D 1:
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The value of the parameter defined by
lies in the interval
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2 Œ0; 1. The lower contour is
Clower . / D 0:25Cupper . /:
(7)
The final cross-section shape is shown in Fig. 2b, and an illustration of complex rigid body motions the UAV with Coanda effect-based lift production can undergo is shown in Fig. 2c. Using the ‘CST’ method, we are able to define various contour shapes by scaling the components of Bernstein polynomial representation of the unit shape function. This technique has the following properties: • Present contour representation technique captures the entire design space of smooth geometries. • Every contour in the entire design space can be derived from the unit shape function contour. • Every contour in the design space is derivable from every other. With this technique it is easy to construct various shapes of the UAV based on Coanda-effect and compare their performances.
3 Numerical Method Simulations are performed using the authors’ own open-source code [14], implementing cell-centered, second-order accurate finite volume discretization of Reynolds-Averaged Navier-Stokes equations (RANS). The code is intended for block structured non-orthogonal geometries, with collocated variable arrangement. The pressure correction and momentum equations are solved iteratively using the SIMPLE algorithm [15]. The discretization schemes are second-order accurate. Turbulence is accounted using Menter’s k ! Shear-Stress Transport model. Automatic wall boundary condition approach [16] is used. It is particularly suited when, during geometric multigrid procedure, a wall-coarsening takes place, which requires a paradigm shift in treating the cells in the first layer next to the wall, because of the difference in yC values. O-type structured body-fitted meshes were generated with refined layers towards the wall. Aerodynamic forces acting on a body immersed in fluid are calculated in a usual way – pressure and viscous forces are integrated over the body surface to give net forces.
4 Simulation Setup and Results The case that we have studied has moderately small Reynolds number, Re D 63;775 based on mean velocity and inlet height. Velocity is uniform across the inlet (Ub D 20 m/s). Such velocity distribution at inlet is chosen in accordance to [9] where
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a top-hat velocity profile is used at inlet. We have not considered any particular method by which the inlet flow is generated. We have to remind, however, that in such case some additional information should be provided, such as swirl velocity component, if a fan is used for providing inflowing air. To generate lateral forces which would produce forward motion, we experimented with varying inlet velocity along the circumference of the inlet, resulting in an asymmetric Coanda effect. Single block O-type grid was used with roughly 50,000 cells. Three increasingly refined grids were used in geometric multigrid approach. Grid points are clustered toward to wall, and in the direction along the wall mesh is piecewise uniform, with increased grid density in the region of high wall curvature, i.e. where upper and lower aerodynamic surface meet, which is also, under such conditions, a region of wall jet separation. A detail of the mesh around the UAV body is shown in Fig. 3. In [17, 18] the symmetric configurations are studied to numerically estimate total lift force needed for hovering flight. In Figs. 4, 5 and 6 we show the development of velocity profile along the upper surface. The profiles are taken along surface normal, at positions defined by angle , which is the angle between the radius vector of the point at the body’s surface and the horizontal plane. Distribution of the pressure coefficient over the body’s surface is shown in Fig. 7. Next, we focus on changes in aerodynamic forces, needed for aircraft maneuvering by varying the inlet conditions. The nozzle air-speed ratio, e.g. 20/15, determines the nozzle air-speeds in Œm=s at opposite directions. In three-dimensional configurations nozzle air speed changes continuously along the circumference.
Fig. 3 Detail of the computational grid. Grid is successively refined toward the wall. Along the wall, mesh is piecewise uniform, with increased grid density in the region of high curvature. Grid density discontinuities are seen as ‘rays’ emanating from the body
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Fig. 4 Velocity magnitude profiles along surface normal at wall surface position defined by the point’s radius vector angle with horizontal plane, D 30ı
Fig. 5 Velocity magnitude profiles along surface normal at wall surface position defined by the point’s radius vector angle with horizontal plane, D 10ı
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Fig. 6 Velocity magnitude profiles along surface normal at wall surface position defined by the point’s radius vector angle with horizontal plane, D 0ı
Fig. 7 Pressure coefficient distribution for symmetric conditions needed for hovering
The streamlines and velocity contours are shown for the 20/15 case in Fig. 8. Figure 9 shows the pressure coefficient distribution change for asymmetric flow conditions as the nozzle air-speed ratio is varied. Similar to symmetric situation, in all cases the pressure coefficient has the lowest value at the upper surface edge, indicating maximum wall-jet velocity being reached there – a consequence of the Coanda effect. Finally, Fig. 10 shows the resulting pressure force components for asymmetric flow cases.
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Fig. 8 Velocity contours and streamlines for 20/15 nozzle air-velocity ratio -1
Unsymmetric case (20/19) Unsymmetric case (20/18) Unsymmetric case (20/17) Unsymmetric case (20/16) Unsymmetric case (20/15)
Cp
-0.5
0 -0.4
-0.2
0
0.2
0.4
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Fig. 9 Pressure coefficient distribution for various nozzle air-velocity ratios
Fig. 10 Total pressure force for various nozzle air-velocity ratios
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5 Discussion and Conclusion In this paper, we present the numerical study based on finite volume discretization algorithm of RANS equations closed by LEVM model of turbulent wall jet flow over convex surface with non constant curvature. The distinguished features of the flow are jet attachment to wall as a consequence of the Coanda effect, entrainment of surrounding air and jet spreading, massive separation in the region of high curvature, and formation of the large recirculation region. All these features combined make this type of flow very difficult to simulate accurately. The motivation to do so comes from an interesting and very useful application to UAV. Using numerical simulation we were able to tackle the fascinating and not fully exploited phenomena of the Coanda effect-based lift production. It is shown here that by gradually changing the flow picture around the UAV by varying jet inflow air-speed along the circumference of the revolving body, we change the resulting force in a smooth way. The UAV is tilted by asymmetric lift distribution under these conditions, the horizontal component of the resulting force appears, producing forward flight conditions. A small change in inlet velocity does not lead to great change in the resulting force, a favorable circumstance which indicates that maneuverability of such a configuration is possible using this mechanism. Future investigations will deal with non-stationary simulation and coupling of fluid and 6DOF solver to enable further insight into the characteristics of Coanda effect based UAV lift production, and investigate different mechanisms to achieve its efficient maneuverability. Acknowledgements The authors gratefully acknowledge the financial support of Serbian Ministry of Education, Science and Technological Development for the financial support of this research trough projects TR-33036 and TR-35006.
References 1. Neuendorf, R., Wygnanski, I.: On a turbulent wall jet flowing over a circular cylinder. J. Fluid Mech. 381, 1–25 (1999) 2. Neuendorf, R.: Turbulent wall jet along a convex curved surface. Ph.D. thesis, University of Berlin, Berlin (2000) 3. Cullen, L.M., Han, G., Zhou, M.D., Wygnanski, I.: On the role of longitudinal vortices in turbulent flow over convex surface. In: AIAA Paper, St. Louis, Missouri, 2002–2828 (2002) 4. Han, G., Zhou, M.D., Wygnanski, I.: On streamwise vortices in a turbulent wall jet flowing over a circular cylinder. In: AIAA Paper, Portland, 2004–2350 (2004) 5. Likhachev, O., Neuendorf, R., Wygnanski, I.: On streamwise vortices in a turbulent wall jet that flows over a convex surface. Phys. Fluid 13(6), 1822–1825 (2001) 6. Pajayakrit, P., Kind, R.J.: Streamwise vortices in the outer layer of wall jets with convex curvature. AIAA J. 37(2) 281–283 (1999) 7. Wernz, S.H., Valsecchi, P., Gross, A., Fasel, H.F.: Numerical investigation of transitional and turbulent wall jets over a convex surface. In: AIAA Paper, Orlando, 2003–3727 (2003)
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8. Wernz, S.H., Gross, A., Fasel, H.F.: Numerical investigation of coherent structures in plane and curved wall jets. In: AIAA Paper, Toronto, 2005–4911 (2005) 9. Gross, A., Fasel, H.F.: Coanda wall jet calculations using one- and two-equation turbulence models. AIAA J. 44(9) 2095–2107 (2006) 10. Swanson, R., Rumsey, C., Sanders, S.: Progress towards computational method for circulation control airfoils. In: AIAA Paper, Reno NV, 2005–2089 (2005) 11. Kulfan, B.M., Bussoletti J.E.: “Fundamental” parametric geometry representation for aircraft component shapes. In: AIAA, Portsmouth, Virginia, 2006–6948, (2006) 12. Kulfan, B.M.: Universal parametric geometry representation method. J. Aircr. 45(1), 142–158 (2008) 13. Kulfan, B.M.: Recent extensions and applications of the ‘CST’ universal parametric geometry representation method. RAeS J. 45(1), 157–176 (2010) 14. http://www.sourceforge.net/projects/caffasst 15. Peri´c, M., Ferziger, J.: Computational Methods for Fluid Dynamics. Springer, Berlin (2000) 16. Menter, F., et al.: The SST turbulence model with improved wall treatment for heat transfer predictions in gas turbines. In: Proceedings of the International Gas Turbine Congress, Tokyo, 2–7 Nov 2003 17. Mirkov, N., Rašuo, B. : Numerical simulation of air jet attachment to convex walls and applications. In: Proceedings of 27th International Congress of the Aeronautical Sciences, Nice, 2010 18. Mirkov, N., Rašuo, B. : Maneuverability of an UAV with coanda effect based lift production. In: Proceedings of 28th International Congress of the Aeronautical Sciences, Brisbane, 2012
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