Unsteady Aerodynamic Characteristics Simulations Of Rotor Airfoil Under .
applied sciences Article Unsteady Aerodynamic Characteristics Simulations of Rotor Airfoil under Oscillating Freestream Velocity Qing Wang 1, * 1 2 * and Qijun Zhao 2 College of Energy and Power Engineering, Lanzhou University of Technology, Lanzhou 730050, China National key Laboratory of Science and Technology on Rotorcraft Aeromechanics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; zhaoqijun@nuaa.edu.cn Correspondence: wangqing lut@foxmail.com Received: 19 January 2020; Accepted: 3 March 2020; Published: 6 March 2020 Abstract: The dynamic stall characteristics of rotor airfoil are researched by employing unsteady Reynolds-Averaged Navier-Stokes (RANS) method under oscillating freestream velocity conditions. In order to simulate the oscillating freestream velocity of airfoil under dynamic stall conditions, the moving-embedded grid method is employed to simulate the oscillating velocity. By comparing the simulated dynamic stall characteristics of two-dimensional airfoil and three-dimensional rotor, it is indicated that the dynamic stall characteristics of airfoil under oscillating freestream velocity reflect the actual dynamic stall characteristics of rotor airfoil in forward flight more accurately. By comparing the simulated results of OA209 airfoil under coupled freestream velocity/pitching oscillation conditions, it is indicated that the dynamic stall characteristics of airfoil associate with the critical value of Cp peaks (i.e., the dynamic stall characteristics of OA209 airfoil would be enhanced when the maximum negative pressure is larger than 1.08, and suppressed when this value is smaller than 1.08). By comparing the characteristics of vortices under different oscillating velocities, it indicates that the dissipation rate of leading edge vortex presents as exponent characteristics, and it is not sensitive to different oscillating velocities. Keywords: rotor; dynamic stall; oscillating freestream velocity; airfoil; RANS equations 1. Introduction In forward flight conditions, the freestream velocity of a helicopter rotor blade in the advancing side is larger than that of the retreating side due to the effect of forward flight. As a result, the inflow velocity of rotor airfoil varies with the rotor azimuth. Therefore, investigations on unsteady aerodynamic characteristics of rotor airfoil under oscillating freestream velocity are important, especially for the dynamic stall phenomenon [1,2]. To discover the physical essence of a rotor airfoil dynamic stall, a lot of research under steady freestream velocity (SFV) conditions has been accomplished by using the experimental method [3–6], theoretical method [7–12], and numerical method [13–15]. Hence, the dynamic stall of airfoil under SFV conditions is well-studied at present. However, the dynamic stall of rotor airfoil under oscillating freestream velocity (OFV) conditions was seldom taken into account in early researches. In order to discover the physical phenomenon of dynamic stall under OFV condition, Favier [16] preliminarily researched dynamic stall characteristics of the NACA0012 under the condition of oscillating velocity by using the experimental method. After that, the aerodynamic loads of the NACA0012 airfoil were measured by Gursul [17,18] at a fixed angle of attack (AoA) under unsteady freestream conditions in a vertical unsteady water tunnel with a cross-sectional area of 45.7 45.7 cm. Based on this experiment, it was found that the lift coefficient (Cl ) can be one order of magnitude higher under OFV conditions than that under SFV conditions. Gharali [19] investigated the dynamic Appl. Sci. 2020, 10, 1822; doi:10.3390/app10051822 www.mdpi.com/journal/applsci
Appl. Sci. 2020, 10, 1822 2 of 18 stall of NACA0012 airfoil under variational velocity by employing the ANSYS Fluent software, and the results were indicated that the leading edge vortex (LEV) formation could be obviously affected by the phase angle between the oscillating pitch and the oscillating freestream velocity. However, the maximum AoA and maximum velocity of airfoil were superimposed in this research, which is much different from the actual working environment of a helicopter rotor in forward flight. As a result, the conclusions of the dynamic stall characteristics of two-dimensional (2D) airfoil may not coincide with that of a helicopter rotor airfoil. At the same time, a new experiment [20,21] was performed by the researchers of Ohio State University (OSU) to test the dynamic stall characteristics of SSC-A09 airfoil under oscillating velocity conditions. By comparing the measured Cl and Cm (pitching moment coefficient) under OFV conditions and SFV conditions, it was found that the lift curve slope and stall angle increased under the OFV conditions. However, due to the restriction of the experimental equipment, the maximum oscillating velocity (0.08 Ma) was much smaller than the normal forward flight velocity of the helicopter (Mach number of 0.1–0.3). Moreover, the aspect ratio of the blade model used in these experiments was only 1.0, which would generate obvious three-dimensional (3D) effects, and the characteristics of dynamic stall vortices would be influenced seriously. What is more, the added-mass effect due to the velocity variation would also influence the measured results in this research as mentioned in another study [22]. At the same time, Jones et al. [23–25] researched the dynamic stall characteristics of an airfoil under OFV conditions in horizontal free-surface water tunnel of the U.S. Air Force Research Laboratory. However, these studies were focused on aerodynamic characteristics of airfoil at fixed AoAs, which was much different from the working environment of a helicopter rotor. Meanwhile, the freestream velocities were also smaller than the rotational velocity of the helicopter rotor due to the restriction of water tunnel. Therefore, the dynamic stall characteristics of the rotor airfoil under the real environment of helicopter are still not included in these studies, and this issue is valuable to research more deeply. As mentioned above, the dynamic stall characteristics of rotor airfoil under an actual working environment of a helicopter are not well studied. Previous research seldom focused on the characteristics of a dynamic stall vortex under coupled freestream velocity/pitching oscillation conditions directly. Therefore, the purpose of this study is to explore the effects of OFV on dynamic stall characteristics of rotor airfoil. In this research, the moving-embedded grid method [26,27] is adopted to simulate the oscillating velocity and pitching of an airfoil. The unsteady RANS equations coupling with third-order Roe-MUSCL spatial discretization scheme are chosen as the governing equations to predict the unsteady flowfield of an airfoil, and the highly-efficient implicit scheme of lower-upper symmetric Gauss–Seidel (LU-SGS) is adopted for temporal discretization. To capture the separated airflow and vortex more accurately, the SST k ω turbulence model is employed in the Computational Fluid Dynamics (CFD) code. Based on these methods, the dynamic stall characteristics of airfoil are researched under coupled conditions. It is indicated from the simulated results that the dynamic stall characteristics of airfoil associate with a critical value of Cp peak. By studying the vortices characteristics, it is illustrated that the dissipation rate of the LEV presents as exponent characteristics. However, this dissipation rate is not sensitive to oscillating velocity. 2. Aerodynamic Environment of Helicopter Rotor The Figure 1a illustrates the velocity distribution of a helicopter rotor with blade tip, Mach number of 0.6, and advance ratio of 0.3. It is indicated that the rotor blade velocity varies with the rotor azimuth in forward flight. As a result, the relative freestream velocity of airfoil at different spanwise sections of rotor blade could be expressed as Ma MB MF sin(ωt) (1) where, MB represents the rotational velocity of the blade, MF denotes the forward flight velocity of the helicopter, ω and t represent the angular velocity and time, respectively.
Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 18 Appl. Sci. 2020, 10, 1822 3 of 18 where, M B represents the rotational velocity of the blade, M F denotes the forward flight velocity of the helicopter, ω and t represent the angular velocity and time, respectively. Aimingatatillustrating illustratingthe therelationship relationshipbetween betweenthe thebasic basicfreestream freestreamvelocity velocityand andthe theoscillating oscillating Aiming velocity, a normalized oscillating velocity λ is defined as velocity, a normalized oscillating velocity λ is defined as MM FF MM BB λλ (2)(2) The TheAoA AoAvariation variationinduced inducedby bycyclic cyclicpitching pitchingofofthe therotor rotorblade bladecan canbe bewritten writtenas as α α m Δα sin (ω t π ) (3) α αm α sin(ωt π) (3) where, α m and Δα represent the mean AoA and amplitude of pitch oscillation, respectively. In this o o pitch oscillation, respectively. In this where, αm and representofthe mean amplitude of 8.0and environment, the α variations AoA ( α mAoA and Δα 6.0 ) and freestream Mach number (MB environment, the variations of AoA (αm 8.0 and α 6.0 ) and freestream Mach number (MB 0.4 0.4 and MF 0.18) are shown in Figure 1b. and MF 0.18) are shown in Figure 1b. 18 0.9 α (αm 8.0Ο) 14 AoA (deg) 0.8 Mach number (MF 0.18) 0.7 12 0.6 10 0.5 8 0.4 6 0.3 4 0.2 2 0.1 0 0 60 120 180 240 300 Mach Number 16 0.0 360 Azimuth (deg) (a) (b) Figure Figure1.1.Mach Machnumber numberdistribution distributionofofrotor rotordisk diskand andangle angleofofattack attack(AoA) (AoA)variation variationwith withrotor rotor azimuth. azimuth.(a) (a)Mach Machnumber numberdistribution. distribution.(b) (b)AoA AoAand andfreestream freestreamvelocity velocityvariations. variations. NumericalSimulation SimulationMethod Method 3.3.Numerical orderto to generate generate the airfoil, thethe Poisson equations [28] InInorder the computational computationalgrids gridsaround aroundthe theOA209 OA209 airfoil, Poisson equations are chosen as the governing equations under 2D conditions. By solving these equations, the C-topology [28] are chosen as the governing equations under 2D conditions. By solving these equations, the Cgrids with 459with 80459 points generated. To satisfy requirement of OFV of simulation, the far-field topology grids 80are points are generated. Tothe satisfy the requirement OFV simulation, the boundary of airfoil fixedisatfixed 25 times airfoil chord,chord, and the of ythe nearnear the wall ofgrids far-field boundary of grids airfoilisgrids at 25 the times the airfoil andy the the grids the surface is smaller than 1.2. In this research, the background grid is composed of 300 300 points with wall surface is smaller than 1.2. In this research, the background grid is composed of 300 300 points a far-field boundary of 50ofc 50 (airfoil chord), and and the inverse mapmap method [29] is employed in this code with a far-field boundary c (airfoil chord), the inverse method [29] is employed in this to search donor element. As shown in Figure 2, the oscillating velocity is simulated by by moving the code to search donor element. As shown in Figure 2, the oscillating velocity is simulated moving airfoil back and forth periodically [30,31]. the airfoil back and forth periodically [30,31].
Appl. Sci. 2020, 10, 1822 4 of 18 Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 18 Figure 2. 2. Schematic of moving-embedded moving-embedded grids. grids. Figure In In order order to to simulate simulate the the unsteady unsteady compressible compressible flowfield flowfield around around an an airfoil, airfoil, the the integral integral form form of of the Navier–Stokes equations is employed in this work, i.e., the Navier–Stokes equations is employed in this work, i.e., x y Wd Ω S 0 0 WdΩ ((FFcc FFv v))ddS t Ω Ω t Ω (4) (4) Ω where W represents the conserved variables, Fc and Fv denote convective fluxes and viscous where represents i.e., the conserved variables, Fc and Fv denote convective fluxes and viscous fluxes, fluxes,W respectively, respectively, i.e., ρ ρVr 0 0 ρ ρV r n τ n τ ρu ρuV n p r n xp x xx y xy n y τxy ρu ρuV n τ r x x xx W , , F F (5) (5) W τ n τ , F c ρvV n p , Fv v n ρv ρcv ρvV x yx y yy n p n τ n τ r y x yx y yy r y Θ nyxΘ y n y Θ y ρE ρE ρHV V p ρ HrV x x nx Θn r Vtt where V VV , V is absolute velocity, V is contravariant velocity. τij denotes the viscous Vrr V τ ij denotes whereV the stresses, viscous t ,t V is absolute velocity,t Vt is contravariant velocity. and Θ is the term describing the heat condition in the fluid. i stresses, and Θi is the term describing the heat condition in the fluid. Meanwhile, the unsteady condition simulation is accomplished by employing the dual Meanwhile, the unsteady condition simulation is accomplished by employing the dual timetime-stepping approach. Then Equation (4) would be changed as stepping approach. Then Equation (4) would be changed as n 1 * Ωn 1 W dd((Ω W )) R ( W * ) 0 R (W ) 0 dτ dτ (6) (6) where W * is the approximation for Wn n 11 , τ represents a pseudo time variable, and the unsteady where Wis is the approximation for W , τ represents a pseudo time variable, and the unsteady residual defined as residual is defined as n 1 3 W * 4 Wn n Wn 1 * * ) 3W 4W W RR((W (7) R R(W( W ) W )) (7) 2Δt 2 t Toimprove improvethe the computational computational efficiency, efficiency,the theimplicit implicitLU-SGS LU-SGS[32,33] [32,33]scheme schemeis is employed employed in in this this To work. The two equation SST turbulencemodel modelisisemployed employedto tosimulate simulatethe theviscous viscous stresses stresses of of work. SST kk ω turbulence the flowfield. the flowfield. In order order to to verify verify the the accuracy accuracy of of the the present present grid grid system system and and flowfield flowfield solver, solver, aa case case of of the the In NACA4412 airfoil airfoil is is presented presented in in this this section. section. In In this this case, case, the the pressure pressure coefficient coefficient (Cp) (Cp) simulated simulated by by NACA4412 the present CFD method compared with the test data [34] is shown in Figure 3a. In this case, three the present CFD method compared with the test data [34] is shown in Figure 3a. In this case, three different grids grids (699 (699 100, 100, 459 459 80, 80, 319 319 60) are used used to to analyze analyze the the grid grid sensitivity. sensitivity. By By comparing comparing different 60) are with the test data, it illustrated that the Cp simulated by the CFD method correlates well test with the test data, it illustrated that the Cp simulated by the CFD method correlates well with with test data data for a different grid. However, the convergence of a larger grid is lower than coarse grid, but it for a different grid. However, the convergence of a larger grid is lower than coarse grid, but it still still converges at least five orders of magnitude. It is indicated that theused gridin used this research converges at least five orders of magnitude. It is indicated that the grid thisin research could could satisfy the calculation requirement. Meanwhile, it is illustrated in Figure 3b that the velocity satisfy the calculation requirement. Meanwhile, it is illustrated in Figure 3b that the velocity profiles profiles are also close the test As ita is result, it is indicated thatused the grid used in thisisresearch are also close to the testtodata. As data. a result, indicated that the grid in this research suitable.is suitable.
Appl. Sci. FOR PEER REVIEW Appl. Sci. 2018, 2020, 8, 10,x1822 Appl. Sci. 2018, 8, x FOR PEER REVIEW 55 of 18 of 18 5 of 18 0.0 -4 -2.0 -3.0 -2.5 -3.5 Cp 0.025 0.020 6 αRe 1.52 13.82 10 -4.0-5.0 0 o -2 Re 1.52 10 -2 2000 4000 6000 8000 1000012000 -4.5 6 -5.0 0 X/c 0.62 X/c 0.62 0.030 0.025 -3.0 -4.0 -3.5 -4.5 α 13.82 Ma 0.08 -4 Cp -1.0 -2.0 -1.5 -2.5 o X/c 0.455 X/c 0.372 X/c 0.455 0.040 0.035 Test data Numerical results 0.035 0.030 Test data Numerical results y/c -6 X/c 0.372 0.040 -0.5 0.0 -1.0 -0.5 -1.5 y/c -6 Test data Grid with 699 100 Test Griddata with 459 80 Grid Gridwith with699 100 319 60 Grid with 459 80 Grid withMa 0.08 319 60 2000 4000 6000 8000 1000012000 0.020 0.015 0.015 0.010 0.010 0.005 0 0.005 0.000 0 0.0 0.0 0.2 0.2 0.4 0.4 0.6 x/c 0.6 0.8 0.000 -0.005 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 -0.005 V/V V/V V/V 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 1.0 0.8 1.0 x/c (b)V/V (b) V/V (a) (a) V/V Figure 3. Comparisons of pressure coefficient (Cp) and velocity profiles of NACA4412 airfoil with Figure 3.3.Comparisons of coefficient (Cp)and andvelocity velocityprofiles profiles NACA4412 airfoil with test data. (a) Cp distribution. (b) Velocity profiles. Figure Comparisons of pressure pressure coefficient (Cp) of of NACA4412 airfoil with test test data. distribution. Velocity profiles. data. (a) (a) CpCp distribution. (b)(b) Velocity profiles. In order to verify the accuracy of the present CFD method used in the dynamic stall characteristic In the ofofthe CFD used ininthe stall Inorder orderto verifyairfoil, theaccuracy accuracy thepresent present CFDmethod method used thedynamic dynamicairfoil stallcharacteristic characteristic simulations oftoverify rotor a typical case of dynamic stall about the NACA0012 is presented simulations ofofrotor airfoil, typical case of stall about the airfoil isispresented simulations rotor airfoil, typical caseThe ofdynamic dynamic stall about theNACA0012 NACA0012 airfoil presented under the SFV condition inaathis section. freestream Mach number is 0.283, the variation of AoA o under SFV The freestream Mach number 0.283, thevariation variation under the SFV condition in Mach 0.283, the ofofAoA 14.91 condition 9.88o sin(ωin t ) ,this is α the andsection. the reduced frequency (k) number is 0.151.isisAs shown in Figure 4,AoA it is o o 14.91 9.88 , and the reduced frequency (k) is 0.151. As shown in Figure 4, it is indicated sin ( ωt ) α 14.91 9.88 sin( ω t isα , and the reduced frequency (k) is 0.151. As shown in Figure 4, it indicated that the simulated Cl by the present CFD method is correlated well with the test data isof that the simulated Cl by method is correlated well withwell the test data of results National indicated that the simulated Cpresent l byAdministration theCFD present CFD method is correlated the test data ofof National Aeronautics andthe Space (NASA) [35]. Meanwhile, thewith calculated Aeronautics and Space Administration (NASA) [35]. Meanwhile, the calculated results of the present National Aeronautics and Space Administration [35].by Meanwhile, the calculated results the present CFD method are better than that(NASA) calculated Fluent software with SST k of ω CFD method are better than that calculated by Fluent software with SST k ω turbulence model. the present model. CFD method are better than that calculated by Fluent software with SST k ω turbulence Since we turbulence Since model. we lack lack appropriate appropriate test test data data of of rotor rotor airfoil airfoil under under dynamic dynamic stall stall conditions conditions coupled coupled with with variational velocity, the theory of Isaacs [36] is employed to verify the accuracy of the present CFD Since wevelocity, lack appropriate data of[36] rotor under conditions variational the theorytest of Isaacs is airfoil employed to dynamic verify thestall accuracy of thecoupled presentwith CFD method OFV in Figure 5. It is5.illustrated that thethat numerical results correlate variational velocity, the theoryas ofshown Isaacs [36] employed the accuracy of numerical the present CFD method under under OFVcondition, condition, as shown inis Figure Ittoisverify illustrated the results well with the theoretical data at λ 0.4, and the numerical results with λ 0.8 are basically close method as shown in λFigure It isthe illustrated that the numerical results correlateunder well OFV with condition, the theoretical data at numerical results with λ 0.8 are 0.4 , 5.and to the theoretical value. The deviation of the C /C at λ 0.8 could be attributed to the fact that the l0 , and correlate theoretical dataThe at λdeviation theCnumerical with l 0.4 λ 0.8 are basically well closewith to thethe theoretical value. of the l/Cl0 at λ results be attributed to 0.8 could airflow compressibility and viscidity are neglected in the Isaacs theory, where C represents the lift l0 basically close to the theoretical value. The deviation of the C l /C l0 at could be attributed to λ 0.8 the fact that the airflow compressibility and viscidity are neglected in the Isaacs theory, where Cl0 force at that steady it is indicated that theinpresent CFDtheory, method is suitable the fact thefreestream airflow and viscidity are neglected Isaacs where Cl0 represents the lift force compressibility atvelocity. steady Consequently, freestream velocity. Consequently, itthe is indicated that the present for simulating unsteady characteristics of a rotor airfoil under dynamic stall conditions coupled with represents the is liftsuitable force atfor steady freestream velocity. Consequently, it is indicated thatdynamic the present CFD method simulating unsteady characteristics of a rotor airfoil under stall oscillating freestream velocity. CFD method is suitable for simulating unsteadyvelocity. characteristics of a rotor airfoil under dynamic stall conditions coupled with oscillating freestream conditions coupled with oscillating freestream velocity. 3.0 Test data Numerical Test data data Numericaldata by Fluent Numerical 3.0 2.5 2.5 2.0 Numerical Fluent α 14.91 9.88by sin( ωt) o 2.0 1.5 Cl Cl o o o αMa 0.283 14.91 9.88 sin(ωt) k 0.151 Ma 0.283 1.5 1.0 k 0.151 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 0 0 5 5 10 10 15 20 AoA(deg) 15 20 AoA(deg) 25 25 30 30 Figure 4. 4. Comparison of simulated Cll with test data. Figure Figure 4. Comparison of simulated Cl with test data.
Appl. Sci. 2020, 10, 1822 6 of 18 Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 18 3.5 Numerical results (λ 0.4) Numerical results (λ 0.8) Isaacs theory (λ 0.4) Isaacs theory (λ 0.8) 3.0 Cl/Cl0 2.5 k 0.2 Ma 0.2(1 λ sin(ωt)) α 3.0 O 2.0 1.5 1.0 0.5 0 60 120 180 Azimuth (deg) 240 300 360 Figure 5. Comparison of simulated results Isaacs’s theory under the oscillating freestream Figure 5. Comparison of simulated results withwith Isaacs’s theory under the oscillating freestream velocity velocity (OFV) condition. (OFV) condition. 4. Analyses Analyses and and Discussions Discussions 4. 4.1. The Dynamic Stall Characteristics of Airfoil Under 3D Conditions 4.1. The Dynamic Stall Characteristics of Airfoil Under 3D Conditions Aiming at researching the dynamic stall characteristics of rotor airfoil with OFV under actual Aiming at researching the dynamic stall characteristics of rotor airfoil with OFV under actual rotor working conditions, a two-blade untwisted rotor model with a rectangular planform is designed rotor working conditions, a two-blade untwisted rotor model with a rectangular planform is based on the OA209 airfoil, and the aspect ratio of the rotor blade is 15.0. The blade grids with 356 designed based on the OA209 airfoil, and the aspect ratio of the rotor blade is 15.0. The blade grids 70 90 points are generated based on the 2D airfoil grids, and the background grids are 176 99 with 356 70 90 points are generated based on the 2D airfoil grids, and the background grids are 156. The numerical simulation is performed at an advance ratio of 0.3 with blade tip, Mach number 176 99 156. The numerical simulation is performed at an advance ratio of 0.3 with blade tip, Mach of 0.6, and the pitch angle is α 10.0 6.0 sino(ωt). In o this case, two different spanwise sections of number of 0.6, and the pitch angle is α 10.0 6.0 sin(ωt ) . In this case, two different spanwise blade, 0.6 R and 0.7 R (R is the radius of rotor), are selected to compare the numerical results under the sections of blade, 0.6 R and 0.7 R (R is the radius of rotor), are selected to compare the numerical 2D conditions, where the steady Mach numbers are 0.36 and 0.42, the unsteady Mach numbers are results under the 2D conditions, where the steady Mach numbers are 0.36 and 0.42, the unsteady verified based on the steady Mach number with λ of 0.5 and 0.43, respectively. The corresponding Mach numbers are verified based on the steady Mach number with λ of 0.5 and 0.43, respectively. airflow information of different spanwise sections is shown in Table 1. The corresponding airflow information of different spanwise sections is shown in Table 1. Blade Section 0.6 R 0.7 R Table 1. Airflow information at different blade sections. Table 1. Airflow information at different blade sections. MB λ Blade Section MB 0.7 R 0.36 0.42 0.36 0.6 R 0.42 λ 0.5 k 0.5 0.43 0.0556 0.43 0.0476 k 0.0556 0.0476 In order to verify the accuracy of the present CFD method used in the 3D rotor condition, a case of In order to verify the accuracy of the present CFD method used in the 3D rotor condition, a case SA349/2 rotor with an advance ratio of 0.26 is present in this section, just as shown in Figures 6 and 7. of SA349/2 rotor with an advance ratio of 0.26 is present in this section, just as shown in Figures 6 and The comparisons of normal force (Cn) and Cp distributions (0.75 r/R) are correlated well with the 7. The comparisons of normal force (Cn) and Cp distributions (0.75 r/R) are correlated well with the test data [37]. It is illustrated that the present CFD method can effectively simulate the unsteady test data [37]. It is illustrated that the present CFD method can effectively simulate the unsteady aerodynamic characteristics of a rotor. aerodynamic characteristics of a rotor.
Appl. Sci. 2018, 8, x FOR PEER REVIEW Appl. Sci. Sci.2018, 2020, 8, 10,x 1822 Appl. FOR PEER REVIEW 7 of 18 of 18 18 7 7of 1.4 0.8 Test data Numerical data Test data Numerical data 1.4 1.2 1.2 1.0 0.7 0.6 0.6 0.5 0.5 0.4 Cn Cn Cn Cn 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.3 0.3 0.2 0.4 0.2 0.2 0.0 Test data Numerical data Test data Numerical data 0.8 0.7 0.2 0.1 0 60 120 180 0 60 120 ψ180 () 0.0 240 300 360 240 300 360 0.1 0.0 o 0 60 120 180 0 60 120 ψ() 0.0 240 300 360 240 300 360 o 180 o o ψ() ψ() (a) (a) (b) (b) Figure 6. Comparison of normal force between the test data and numerical data. (a) r/R 0.75. (b) r/R of of normal force between the test and numerical data. (a) r/R (a) 0.75. r/R Figure 0.97. 6.6.Comparison Figure Comparison normal force between the data test data and numerical data. r/R(b) 0.75. (b) 0.97. r/R 0.97. -1.5 -1.5 -1.5 -1.0 -1.0 -1.0 -0.5 -0.5 Cp Cp Cp Cp Test data Numerical data Test data Numerical data -1.5 Test data Numerical data Test data Numerical data -1.0 -0.5 0.0 -0.5 0.0 0.0 0.0 0.5 0.5 0.5 0.5 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 0.2 0.4 0.2 0.4 x/c 0.6 0.8 1.0 0.6 0.8 1.0 0.2 0.4 0.2 0.4 x/c Test data Numerical data Test data Numerical data -1.5 -1.0 1.0 0.8 1.0 Test data Numerical data Test data Numerical data -1.5 -1.0 -1.0 -0.5 -1.0 -0.5 Cp Cp Cp Cp 0.8 0.6 (b) (b) -1.5 -2.0 -1.5 -0.5 0.0 -0.5 0.0 0.0 0.0 0.5 0.5 1.0 0.0 1.0 0.0 0.6 x/c (a) (a) -2.0 x/c 0.5 0.5 0.2 0.4 0.2 0.4 x/c 0.6 0.8 1.0 0.6 0.8 1.0 1.0 0.0 1.0 0.0 0.2 0.4 0.2 0.4 x/c (c)x/c (d)x/c (c) (d) 0.6 0.8 1.0 0.6 0.8 1.0 Figure CpCp between thethe testtest data andand numerical data.data. (a) ψ(a) ψ90 . (b) . ψ(b) ψ180 . (c) . Figure 7.7.Comparison Comparisonofof between data numerical 90 180 Figure 7. Comparison of Cp between the test data and numerical data. (a) 90 . (b) 180 . (c) ψ ψ 270 . (d) 360 . ψ ψ (c) ψ 270 . (d) ψ 360 . ψ 270 . (d) ψ 360 . The comparisons comparisons of of dynamic dynamic stall stall characteristics characteristics of of the the OA209 OA209 airfoil airfoil between between the the 2D 2D and and 3D 3D The conditions are shown shownin inFigure Figure8.8.It Itisisillustrated illustratedthat that simulated OFV condition l under The comparisons of dynamic stall characteristics ofthe the OA209 airfoil between the 2D and 3D conditions are the simulated Cl Cunder thethe OFV condition is is much closer to that simulated under the 3D rotor condition compared with the simulated results conditions in Figureunder 8. It isthe illustrated the simulated Cl under the simulated OFV condition is much closerare toshown that simulated 3D rotorthat condition compared with the results under closer SFV conditions conditions at sections sections of 0.6 0.6the and 0.7 R. R.condition Meanwhile, can be bewith noticed that the simulated simulated much to that simulated under 3D 0.7 rotor compared thethat simulated results under SFV at of RR and Meanwhile, itit can noticed the resultsSFV under the 2D 2D OFV OFV conditions areRsmaller smaller than that of of the the 3D 3D simulation between the azimuth under conditions at sections of 0.6 and 0.7than R. Meanwhile, it can be noticed that the simulated results under the conditions are that simulation between the azimuth results under the 2D OFV conditions are smaller than that of the 3D simulation between the azimuth
Appl. Sci. FOR PEER REVIEW Sci. 2018, 2020, 8, 10,x Appl. Sci. 2018, 8, x 1822 FOR PEER REVIEW 88of of 18 8 of 18 of because the of 240.0 240.0 to to 360.0 , 360.0 , the airflow airflow separation separation is is restricted restricted by by the the spanwise spanwise flow flow under under the the 3D 3D , because of 240.0 to 360.
simulated results of OA209 airfoil under coupled freestream velocity/pitching oscillation conditions, it is indicated that the dynamic stall characteristics of airfoil associate with the critical value of Cp peaks (i.e., the dynamic stall characteristics of OA209 airfoil would be enhanced when the maximum
Figure 1: Coupling of linear structural model and nonlinear unsteady aerodynamics within an aeroelastic CFD code such as FUN3D. A CFD-based aeroelastic system (such as the FUN3D code) consists of the coupling of a nonlinear unsteady aerodynamic system (ow solver)
on aerodynamic performance. The effect on the aerodynamic behaviour of the seam on the garments has not been well studied. The seam on sports garments plays a vital role in aerodynamic drag and lift. Therefore, a thorou gh study on seam should be undertaken in order to utilize its aerodynamic advantages and minimise its negative impact.
Buoyancy Pressure. SMIC, 2020 24 100 Simulations Velocity & Acceleration Friction. SMIC, 2020 25 100 Simulations Free falling / Projectile. SMIC, 2020 26 100 Simulations Circular motion Momentum and Energy. SMIC, 2020 27 100 Simulations Collision. SMIC, 2020 28 100 Simulations Harmonicmotion Thermodynamics. SMIC, 2020 29 100 Simulations
Keywords : Aeroelasticity, CFD, finite-volume discretization, unsteady aerodynamics, unstructured meshes Introduction Aeroelasticity can be defined as the science which studies the mutual interaction between aerodynamic and structural dyn
linearly time varying system. In order to extend this methodology to dynamic aeroelasticity, it is also necessary to model the unsteady aerodynamic loads over an airfoil. Accordingly, an unsteady aerodynamic panel method is developed using a distributed set of doublet p
widely used on the aerodynamic design; however, these jobs only focus on the static aerodynamic performance; the aero-dynamic optimization design combined with dynamic char-acteristics is rare. The dynamic aerodynamic characteristics of an aircraft originate from the motion of the aircraft affected by vari-ous factors.
An Analysis of Unsteady Flooding Processes: Varying . These limit solutions allow us to understand when rate-dependent . Simulations are run both on homogeneous models, on different layered models and on a more complex two-dimensional model. The rate-dependent simulations show smooth transitions between the low- and high-rate limits, and .
Std. XII : Commerce Adjustments for Reserve Fund, Partner’s Loan Account, Asset taken over by Partner and Contingent Liability *Q.5. A, B and C were partners sharing profits and losses in the ratio of 3 : 2 : 1. On 31st March, 2010, their Balance Sheet was as follows: Balance Sheet as on 31st March, 2010 Liabilities Amount Assets Amount Sundry Creditors 15400 Cash at Bank 3,500 Bills .
