The objective of this paper is to study the effect of an unsteady moving heat source on the aerodynamic performance of an NACA 0012 airfoil section, with particular focus on the lift and drag coefficients. The compressible Navier–Stokes equations are solved using a finite volume method as well as Spalart-Allmaras Model for turbulence simulation. The heat source periodically moves over the lower surface of the airfoil in the downstream direction. The numerical results show how the drag and lift coefficient strongly depend upon the velocity of the source. For a constant source power, a progressive improvement in the mean values of lift and drag coefficients is observed as velocity increases

Méndez Soto, M.A.

Porfiriev, D.P.

Samara University

NUMERICAL STUDY OF THE AERODYNAMIC PERFORMANCE OF NACA 0012 IN THE PRESENCE OF AN UNSTEADY HEAT SOURCE. M.A. Méndez Soto, D.P. Porfiriev Samara State Aerospace University, Samara, Russia The objective of this paper is to study the effect of an unsteady moving heat source on the aerodynamic perfor-mance of an NACA 0012 airfoil section, with particular focus on the lift and drag coefficients. The compressi-ble Navier–Stokes equations are solved using a finite volume method as well as Spalart-Allmaras Model for turbulence simulation. The heat source periodically moves over the lower surface of the airfoil in the down-stream direction. The numerical results show how the drag and lift coefficient strongly depend upon the velocity of the source. For a constant source power, a progressive improvement in the mean values of lift and drag coef-ficients is observed as velocity increases. Keywords: aerodynamic performance, heat source, lift coefficient, drag coefficient, computational fluid dy-namics. 1. Introduction Optimizing aerodynamic performance and increasing the reliability of flying machines has led scientists to find new methods that simultaneously provide increase in the lift and reduce drag. One of those approaches has been using procedures based on heat transfer effects. Over the last decades, several numerical and analytical methods to study heat effects have been developed. Generally, the effect is studied by imposing steady temperature differences between the airfoil surface and the freestream. For example, Norton et al. (1973) [1] studied the case of NACA 0012 heated at different ratios and considering both laminar and turbulent flows over the surface. Their results showed a destabilization of the boundary layer, earlier transition and separation for temperature ratio bigger than unity. The paper showed a reduction in the value of Clmax and an increase in drag as the airfoil was heated. Similar findings were found by Blohm and Marchman (1974) [2], who studied the heat transfer effects in subsonic flow around a delta wing. In this case, considering the effect on leading edge vortices. Fleming and Taylor (1992) [3] implemented a computed model of incompressible turbulent flow and heat transfer over rough surfaces introducing it into an existing Navier-Stokes algo-rithm. Similarly, Allen and Look [4] created a method to calculate heat transfer rate for wings and bodies of revolution over the surface as a chord wise distribution.  The effects of heat transfer on boundary layer stability and laminar-turbulent transition were pointed out by investigations such as the one carried out by Landrum and Macha (1987) [5], which experimentally presents the effect of heating the nose of a NACA 0012 on turbulence disturbances. Transition location was practically unaffected, but transition zone resulted larger and the amplitude of disturbances increased with heating. A later analytical investigation con-ducted by Kazakov et al. (1995) [6] showed the possibility of delaying transition in airfoils by heating a tiny area near the leading edge and an improvement in lift generation was achieved by varying the pressure distribution. More effects on shear layer development for attached and separated layers are discussed by Mabey (1990) [7], who also includes analysis of velocity pro-Информационные технологии и нанотехнологии-2016 805 files, thickness and transition of the boundary layer. The author points out that heat transfer has powerful effects on aerodynamic measurements that should not be neglected.  Motivated by the appearance of unmanned and micro aerial vehicles, numerous researchers have investigated heat transfer effects on flow with very low Reynolds and Mach numbers. By cooling the extrados and heating the intrados, Kim et al. (2003) [8] achieved to enhance lift and reduce drag, especially significant for small-scale airfoils. The same results were obtained by Bekka et al. (2009) [9] using a numerical computation of the flow around microscale wing for MAV with and without thermal effect. A more specific study of influence of heat transfer on the aerodynamic performance of a plunging and pitching NACA0012 airfoil at low Reynolds numbers can be found in Hinz et al. (2013) [10].  Thus, practically no attention has been drawn to the effects that an unsteady heat source might cause on aerodynamic performance of airfoils. Similarly, most of the investigations focus on the influence produced specifically on the boundary layer. However, it is worth mentioning that there are experimental and theoretical studies that have shown an improvement in lift genera-tion by adding a moving unsteady heat source for the flow around a cylinder [11].  Given the lack of investigation in this area, the objective of the current study is to present the possible effect of an unsteady heat source in airfoil aerodynamics. As a framework for the nu-merical simulations the 2D NACA 0012 Validation Case of NASA Langley Research Centre is used here [12]. All the numerical calculations of this research were computed on the supercom-puter "Sergey Korolyov" at Samara State Aerospace University using the software ANSYS Fluent 15.0.  2. Theory and simulation parameters 2.1 Governing equations The governing equations are two-dimensional unsteady compressible Reynolds-averaged Na-vier-Stokes equations (RANS), which in an Einstein notation can be expressed as follows (e.g. Anderson et al (2010) [13] and Bekka (2009) [9]):  (1)  (2)  (3) where E is the total energy, is the stress tensor for molecular and Reynolds stresses, and  is the total heat flux.  The stress tensor is evaluated using Stokes hypothesis and Boussinesq assumption and can be expressed as: ( ) 0,iut   ˆ( )( ),i j ijij ju uu pt t x x          ( ) ( )( )ˆ ,j ji ij jj j jEu uEu qt x x x            ˆ ij jqИнформационные технологии и нанотехнологии-2016 806  (4)  (5) For simulations in this paper, the turbulent eddy viscosity is calculated using Spalart-Allmaras turbulence model. This model was designed and optimised for aerospace applications, especially for flows past wings and airfoil. Some of its advantages include: ease of implementa-tion for any type of grid (e.g. structured or unstructured, single-block or multi-block) and com-putational efficiency, since it only solves for only a single additional variable, which makes it quite stable and less memory-intensive than the other models [14]. This model solves a single transport equation, written as:  (6) where . A more detailed description of variables and constants included in this model can be found in Spalart and Allmaras (1994) [15]. Unsteady RANS are numerically solved in ANSYS Fluent by using the finite volume method, which rewrites a general scalar transport equation as an algebraic expression that can be calcu-lated for each control volume [16]. The base of the method is the integral form for the transport of a scalar quantity in an arbitrary volume , as follows:  (7) where – density, – velocity vector, – surface area vector, – diffusion coefficient for  and   source of  per unit volume. Applying discretization to (7), it yields:  (8) where  is the number of faces enclosing a cell and the values  of  and  are determined us-ing temporal and spatial discretization schemes, respectively.  In the spatial discretization, the Third-Order MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) scheme was used to evaluate the final solution for density, momentum, turbulent viscosity and energy. Similarly, Green-Gauss node-based and second-order upwind schemes are applied for gradient and pressure evaluation, respectively. In the temporal discreti-zation, a second-order implicit method is assigned.  ˆ 2 ,3kk ijij ij ijSS       2 2 .3 3kk ij ijij t ijS kS         t 22( )( )1,jvib vv j j jvuvGt xv vv C Y Sx x x                                1t vvf   V,V VdV v dA dA S dVt           v A  S 1 1,N Ni i i i i ii iV v A A S Vt            N t iИнформационные технологии и нанотехнологии-2016 807 Additional models, to solve the closure problem of unsteady RANS equations, include ideal-gas law, constant specific heat assumption and Sutherland’s model for dynamic viscosity.  2.2 Domain and boundary conditions An existing structured 1793 x 513 grid is recompiled from [1] and used as computational do-main. The grids have a farfield extent of about 500 chord lengths and no farfield point vortex boundary condition correction is being applied in this investigation. The flow domain is scaled twice in order to achieve chord Reynolds Number equal to 6 million. The same boundary conditions are imposed, as in the case of NASA validation case. These conditions include free stream Mach number of 0.15, chord Reynolds number equal to 6 Mil-lion and the angle of attack equal to 10 degrees. For Sutherland’s Model the reference viscosity and temperature are 1.865·10-5 kg/ms 300 K, respectively and at the far stream boundary the turbulent initial value is set using an initial turbulent viscosity ratio equal to 0.210438. The air-foil surface is modeled as a wall with no slip and adiabatic conditions. For the Spalart-Allmaras Model a Prandtl Number of 0.72 is used, as well as a value of 0.9 for Energy and Wall Prandtl Numbers, respectively.  A schematic of the modified flow domain and boundary conditions is presented in Fig. 1.  Fig. 1. Schematic of the modified computational domain and boundary conditions The unsteady heat source has a circular shape with a 0.005-meter radius. Time-dependent movement is set to be periodic and uniform along the lower surface in the downstream direc-tion. The power of source does not vary over time and its value is constant within the area of the source. 3. Results and discussion 3.1 Grid validation study In order to determine the results sensitivity to grid refinement, three different grids have been employed and tested for the case of no source present in the flow. CD and CL results are further compared with the solutions available in the open source for the Validation and Verification of Информационные технологии и нанотехнологии-2016 808 Turbulence Modelling of NASA Langley Center [1]. Their numerical study obtained values of 1.09094 and 0.01227275 for CL  and CD, respectively.  The details of the three grids are given in Table 1. Table 1. Description of grids over NACA0012 airfoil used for grid independence study Grid Total Number of Cells Points on the airfoil surface Grid 1 3729 65 Grid 2 57921 257 Grid 3 919809 1025 The values obtained for CD and CL for the three grids, as well as the corresponding errors are presented in Table 2. As the number of cells increase, the value for CD and CL decreases and asymptotically approaches the values numerically obtained and validated in [1]. For the purpose of numerical investigations and considering the available computational capacity, a grid of at least 919809 cells is selected, which provides errors of less than 0.45% for CD and 0.035% for CL. Thus, the results obtained using Grid 3 are further used as reference values for comparisons in the following sections of this paper.  Table 2. Comparison of CD and CL obtained using different grids Grid CD Value Error (%) CL Value Error (%) Grid 1 0.0157969 28.70 1.10224 1.04 Grid 2 0.0123895 0.95 1.09496 0.37 Grid 3 0.0123218 0.40 1.09064 0.03 The convergence of CD and CL as the number of cells increases is plotted in Fig. 2.  3.2 Thermal Effect Study To investigate the effect of an unsteady heat source around a NACA 0012, several possibilities for source movement were analyzed. Preliminarily, uniform clockwise movement around the entire surface and movement along only one of the surfaces were considered. The results showed a relative decrease in CL and increase in CD for different velocities. For instance, an application of a 15 kW/m3 source with a uniform velocity of 15 m/s caused a relative reduction of 11% in the lift-to-drag, which derived from a 9% increment in CD and a reduction of 4% in CL. In the case of 100 m/s as source velocity, the results did not change much with an 8% of CL decrease and a 2% CD increase.   Fig. 2. Convergence of CD and CL for different grid refinements Информационные технологии и нанотехнологии-2016 809  Fig. 3. Mean value of lift (a) and drag (b) coefficients for different source velocities A posterior analysis of the overall pattern of the data showed a slight improvement in the aero-dynamic characteristics as the source moved along the lower surface. Thus, a source movement only along the lower surface was chosen as focus of investigation. To see the effect of source velocity on the aerodynamic performance of NACA 0012, four sim-ulations are conducted for different velocities using a constant source power value of 30 kW/m3. Fig. 3a, b shows the change in CD and CL mean values in the presence of the unsteady heat source with different velocities. The data demonstrates a monotonic decrease of CD as velocity becomes higher, whereas CL monotonically increase for increasing source velocity. Besides, gradient of increment in CL seems to gradually diminish with increasing source velocity.  Despite a relative increment of CD  mean values with respect to CD obtained with no source present, the overall aerodynamic performance tends to increase since CL also reaches higher values. Nevertheless, this overall improvement is only comparatively better for high velocities. These results suggests the possible existence of a minimum source velocity necessary to ob-serve an enhanced aerodynamic performance for a specific source power. The determination of this minimum velocity is beyond the scope of this article. Информационные технологии и нанотехнологии-2016 810 4. Conclusions The effect of an unsteady moving heat source on the aerodynamic performance of NACA 0012 was investigated using numerical simulation. Computations are performed varying velocity source and using a constant power. In all cases, the computed results are performed using sec-ond order implicit scheme for time integration and third order MUSCL and second-order up-wind schemes for spatial integration. Preliminarily, an investigation into the sensitivity of the results to grid refinement is conducted; furthermore, the results are verified based on solutions available in [1].  Simulation results reveal that velocity increment has positive effects on aerodynamic perfor-mance both increasing lift coefficient and decreasing drag coefficient.  In future papers, more attention has to be paid to the optimization of the source parameters in order to obtain a lift-maximized aerodynamic performance. A more detailed range of values for velocity and power could be considered as well as a further frequency study of the oscillations for CD and CL coefficients.   References 1. Norton, D.J., Macha, J.M., Young, J.C.. Surface temperature effect on subsonic stall. Journal of Spacecraft and Rockets 10,581–587; 1973. 2. Blohm,   R.W.,   Marchman, J.F.. Heat transfer effects on a delta wing in subsonic flow. In: Proceedings of the 1974 Heat Transfer and Fluid Mechanics Institute pp.220–235; 1974. 3. Fleming, K.R., and Taylor, R.P., Incompressible Navier-Stokes Algorithm for flow and Heat transfer over rough surfaces, AIAA 27th thermophysics conference, July-6-8, AIAA – 2925; 1992. 4. Allen J., Look. B.C., A method for calculating heat transfer in the Laminar flow region of bodies, NACA, Re-port no. 764, pp. 178-182. 5. Landrum, D.B., Macha, J.M.. Influence of a heated leading edge on boundary layer growth, stability and transi-tion. In: Proceedings of the 19th Fluid Dynamics, Plasma Dynamics and Laser Conference, Honolulu, Hawaii; 1987. 6. Kazakov, A.V., Kogan, M.N., Kuparev, V.A.. Optimization of laminar-turbulent transition delay by means of local heating of the surface. Fluid Dynamics 30,563–570; 1995. 7. Mabey, D.G.. Effects of heat transfer on aerodynamics and possible implications for wind tunnel tests. Progress in Aerospace Sciences27, 267–303; 1990. 8. Kim, J., Rusak, Z., Koratkar, N.. Small-scale airfoil aerodynamic efficiency improvement by surface tempera-ture and heat transfer. AIAA Journal 41, 2105–2113; 2003 9. Bekka, N.,Bessaih, R.,Sellam,M.. Numerical study of heat transfer around the small-scale airfoil using various turbulence models. Numerical Heat Transfer Part A: Applications 56,946–969; 2009. 10. Hinz, D.F., Alighanbari, H., Breitsamter, C.. Influence of heat transfer on the aerodynamic performance of a plunging and pitching NACA0012 airfoil at low Reynolds numbers. Journal of Fluids and Structures 37, 88–99; 2013. 11. Bitjurin B. A., Zavershinskij I. P., Klimov A.I., Molevich N. E., Moralev I.A., Munhoz D., Poljakov L.A., D.P. Porfiriev, Sugak S.S.. Obtekanie cilindra s jelektricheskoj dugoj, vrashhajushhejsja v magnitnom pole [Flow around a cylinder in the presence of an electric arc, rotating in a magnetic field]. Teplofizika Vysokikh Tem-peratur [High Temperature] 2016; 54(4). [In Russian] 12. 2D NACA 0012 Airfoil Validation Cases available at: http://turbmodels.larc.nasa.gov/naca0012numerics_val.html  13. Anderson, J.D., Gerard D., and Dick E. Computational Fluid Dynamics: An Introduction. Ed. John F. Wendt. Third ed. Berlin: Springer; 2010. 14. Argyropoulos C., Markatos N.. Recent advances on the numerical modelling of turbulent flows, Applied Mathematical Modelling. 39, 693–732; 2012. 15. Spalart, P. and Allmaras, S. A One-Equation Turbulence Model for Aerodynamic Flows // Recherche Aerospa-tiale. American Institute of Aeronautics and Astronautics, pp. 5-21; 1994. 16. ANSYS, Inc.. ANSYS Fluent Theory Guide. ANSYS, Inc., PA, USA;  2013.  

Numerical study of the aerodynamic performance of NACA 0012 in the presence of an unsteady heat source

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Numerical study of the aerodynamic performance of NACA 0012 in the presence of an unsteady heat source

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