Virtual Force Measurement of POD Modes for A Flat Plate in Low Reynolds Number Flows

Transcription

1 Vrtual Force Measurement of POD Modes for A Flat Plate n Low Reynolds Number Flows Zongxan Lang and Habo Dong Department of Mechancal & Aerospace Engneerng, Unversty of Vrgna, Charlottesvlle, VA 904 A POD Mode Force Survey Method (POD-FSM) s presented to measure vrtual forces of POD modes actng on a flat plate n low Reynolds number flows. In partcular, a ptchng-plungng plate s used to examne major aspects of the POD- FSM. These nclude the effect of the number of POD modes and dependence of computatonal domans. The results have shown that the superposton of vrtual forces of full POD modes can accurately predct the aerodynamc forces actng onto the plate, whereas the vrtual forces of sx hgh-energy contaned POD modes can also measure the thrust wth only 5% error. It s also found that the vrtual forces of POD modes are senstve to the sze of the computatonal doman. F = aerodynamc force actng on bodes F = force term of the th POD mode j Nomenclature F = force term caused by the nteracton of the th and j th POD mode F B, = force term caused by the nteracton of the th POD mode and bodes r = poston vector U = velocty snapshot and exact velocty feld U = mean (tme averaged) velocty over snapshots u = fluctuatng velocty feld = temporal coeffcent of the th POD mode = POD base element of velocty = vortcty of POD modes, curl of I. Introducton T s attractve to researchers and engneers to effectvely and effcently conduct physcs-based analyss I n a varety of flud dynamcs studes, e.g. flappng wng aerodynamcs. Among all methods, Drect Numercal Smulaton (DNS) has been valdated [-6] and demonstrated as an effectve and accurate approach. However, the tme cost of DNS s expensve for any computer-ntensve applcatons, especally for three-dmensonal problems. Thus, to accurately predct the dynamcs of flow problems such as flow control at a low computatonal cost, Reduced-Order Models (ROMs) [7], specfcally Proper Orthogonal Decomposton (POD) based ROMs [8], have been wdely used. In prevous studes [9, 0], a pressure corrected ROM method has been developed and proved vald for flappng flght. However, the aerodynamc performance of the ROMs was hardly dscussed. Ths s manly because there s a lack of tools relatng the aerodynamc force of flappng plates wth the POD Ph.D. Student, AIAA student member, zl3ra@vrgna.edu Assocate Professor, AIAA Assocate Fellow, habo.dong@vrgna.edu Amercan Insttute of Aeronautcs and Astronautcs Paper

2 modes of correspondng unsteady flows. In general, POD modes are consdered as statstcally representatve of the average knetc energy of flows []. Hence, tradtonal analyses of POD modes are based on the sense of energy capture []. As requred by a large number of ROM applcatons whch focus on optmzng aerodynamc force of flows [3], a method descrbng the drect connecton between the aerodynamc forces and POD modes of unsteady flows s necessary. To ths end, we present a POD Mode Force Survey Method (POD-FSM) that can be used to measure vrtual forces of POD modes of low Reynolds number flows obtaned by DNS. The results have shown the superposton of these vrtual forces of full POD modes can accurately reconstruct the aerodynamc forces of a ptchng-plungng plate n Reynolds number 00 flows. An outlne of the chapter s gven below. In Secton II, a bref ntroducton to the methodology of DNS and POD s gven frst, followed by the theorem of the mpulse equaton [4] and the POD-FSM. In Secton III, we frst present a flow past a statonary plate to valdate the mpulse equaton. Next, the POD-FSM s appled to flows past a plate undergong ptchng-plungng motons. Effect of the number of POD modes to the POD-FSM and computatonal doman dependence of the POD-FSM are studed. Vrtual forces of each ndvdual POD modes are also examned. II. Methodology A. Drect Numercal Smulaton The non-dmensonal ncompressble Naver-Stokes equatons, as wrtten n Eq. (), u 0 x u uu j p u t x x Re x x j j j are dscretzed on a Cartesan mesh usng second-order central dfference scheme n space. A secondorder accurate fractonal-step method for tme advancement s employed. Its key feature s that smulatons wth complex boundares can be carred out on statonary non-body conformal Cartesan grds, elmnatng the need for complcated re-meshng algorthms that are usually employed wth conventonal Lagrangan body-conformal methods. The boundary condtons on the mmersed body are mposed through a ghost-cell procedure [5]. The pressure Posson equaton s solved usng the geometrc multgrd method ntegratng wth the mmersed-boundary methodology. B. Proper Orthogonal Decomposton (POD) Proper Orthogonal Decomposton s a method used to represent large data felds wth a relatvely small number of elements. POD creates a set of bass functon whch spans the orgnal data set by capturng the characterstc components. The system can be represented by the frst few domnant modes. Let U x : N, x be a set of N snapshots of a doman Ω. The mean (tme averaged) N velocty of the snapshots s gven by U U x. Thus the fluctuatng velocty s N () u U U,,, N. To form a set of the modes n the context of proper orthogonal decomposton, t s requred to maxmze the quantty u, subjected to a constrant for each mode n the set. Here, () f, g s an nner product defned as Amercan Insttute of Aeronautcs and Astronautcs Paper

3 f, g f x g xdx, f f, f, and f s a tme average operaton on the data ensemble. The POD method n ths scenaro produces the POD modes that maxmze the turbulent knetc energy. The optmzaton problem can be transformed nto an egenvalue problem by usng the snapshot method [6], wrtten as: AV V (3) where the entry of matrx A s aj u, u j. After solvng the egenvalue problem, the egenvalue and correspondng egenvector V are usually reorganzed n descendng order. Each POD bass element element of the th egenvector V. Based on s gven by N au j j j 3 Amercan Insttute of Aeronautcs and Astronautcs Paper where a j s the j th, any snapshot can then be reconstructed usng a lnear n n n combnaton of the POD bass elements U U where un. N C. The Impulse Equaton and the POD-FSM Noca et al. [4] presented an mpulse equaton to measure nstantaneous forces on mmersed bodes wth only nformaton of velocty and vortcty, d F r dv n V ( t) S ( t) mpds N dt (4) d r n u ds n u u Sb( t) Sb( t) S uds N dt mp u I uu u us r r u N N r T I r T T N where n s the normal vector of surfaces, I s the unt tensor, u S s velocty of the control surface and T s T the vscous stress tensor T u u. The control surface ntegral term mp n (4) ncludes the nformaton of external boundares. The frst two terms n mp represent the Kutta-Zhukovsky or vortex force. If we consder POD modes as a specal type of vortex structure n terms of knetc energy and assume that each POD mode s assocated wth a vrtual force that can be measured by the mpulse equaton, we can have a force measurement for POD modes. Ths method, termed POD Mode Force Survey Method (POD-FSM), can be derved by substtutng the POD expressons of velocty and vortcty nto the mpulse equaton (4). Because the body surface s mpermeable, the surface ntegral term n u us uds can be omtted. As a result, one obtans the force expresson n terms of Sb () t Nm Nm Nm B, j (5) F F F F 0 0 j0 where F are force of the th POD mode, F j are force caused by the nteracton between the th and j th POD mode and F B, are the force caused by the nteracton between the th POD mode and the body. F s a non-nteracton term consstng of one volume ntegral term related to the frst moment of POD vortcty and three surface ntegral terms related to vscous stress tensor T, wrtten as

4 d F r dv n V S () t ds N dt where s the curl of POD modes r r, (6), r T I r T T N (7) T T r r Note that the mean velocty s denoted as mode zero (=0), whose temporal coeffcent 0 s equal to one. The nteracton terms F j nclude products of dfferent POD modes representng the orgnal terms n the mpulse equaton. Thus, F j are the force caused by the nteracton between the th and j th POD mode at the external surface of control volume. Fj j n S() t jds (8) where j j I j us r j (9) N r j N Immersed bodes may move n fluds. Ths movement nteracts wth each POD mode and generates addtonal forces, wrtten as d F, r nt ds B N dt Sb () t (0) If the mmersed body s a zero-thckness plate, F B, s equal to zero because the flow velocty s the ui, uu, u us r and r u same at the upper and lower surfaces as the movng velocty of the plate. The normal vectors of the upper and lower surfaces are opposte. As a result, the surface ntegral around the plate s zero. III. Results A. Valdaton of the Impulse Equaton The mpulse equaton [4] s valdated by a flow past a two-dmensonal statonary membrane plate. A two-dmensonal plate (zero thckness) s placed n flow felds wth angle of attack equal to 30. Incomng free stream of velocty U =. The chord length of the statonary membrane plate s one. The Reynolds number of the flow s 00. The sze of flow doman s 8 (X) 5 (Y) wth the grd number equal to The velocty boundares are of Drchlet boundary condton except at the rght-hand boundary where t s outflow. Homogeneous Neumann boundary condton s specfed on all the boundares for pressure. The hstory of drag and lft coeffcents, whch begns from an mpulse start and lasts for 40 unts of non-dmensonal tme, s shown n Fg.. Table lsts tme-average and root mean square (RMS) of the drag and lft coeffcents. The DNS result s computed by surface ntegral of the pressure and vscous terms around the plate, whle the computaton of the mpulse equaton uses a doman sze equal to the smulaton doman. Smaller computatonal domans of the mpulse equaton may change the force magntude up to 0%, whch s n a range smlar to the results gven n [7]. In general, despte that the mpulse method slghtly underestmates the lft and overestmates the drag, t shows a good agreement wth the DNS result n both lft and drag curves. 4 Amercan Insttute of Aeronautcs and Astronautcs Paper

5 Fgure. Force coeffcents hstory from t/t=0 to 40. The legend Impulse means the mpulse equaton. Table : Comparson of drag and lft coeffcents between DNS and the mpulse equaton DNS Impulse Average RMS Average RMS C D C L B. Flow past a membrane plate undergong a ptchng-plungng moton Drect numercal smulatons were conducted for flows past a two-dmensonal, membrane plate undergong the ptchng and plungng moton. The knematcs of the plate can be prescrbed by y( t) H cos( f t) () ( t) Asn( f t) () where H s the heave ampltude, A s the ampltude of the snusodal ptch angle varaton and f s frequency. The ptch angle ampltude A s equal to 30º. Strouhal number and Reynolds number are defned, respectvely, as St=Hf/U and Re=U c/ν, where U s the unform flow velocty, c s the chord length of the plate and ν s the knematcs vscosty of fluds. In the current study, the heave ampltude H s equal to 0.5, the unform flow velocty U s one, and the frequency s n a range of 0.4 f 0.6. Thus St ranges from 0.4 to 0.6. Wthout loss of generalty, we specfed a representatve low Reynolds number equal to 00 for the study. The confguraton of the flow doman s the same as the statonary case. The plate s placed at the mddle of the flow feld, as shown n Fg.. The number of POD modes and the sze of computatonal doman are two factors that can affect the aerodynamc force calculated by the Fgure. Computatonal domans for the mpulse equaton and the POD-FSM. The plate s at the lowest poston and s movng upward. POD-FSM. Ther effect on the accuracy of the POD-FSM are nvestgated n the current study. The flow reached a perodc sheddng state n approxmately -3 perods after startup. We took the 6 th perod as POD data ensemble to extract 96 snapshots every 0 frames (960 frames per perod) and generate, 6,, 4, 48 and 96 POD modes. Two regons were used n the POD analyss and the POD-FSM. Regon R s 5 Amercan Insttute of Aeronautcs and Astronautcs Paper R R

6 a rectangular doman wth the lower left corner beng (-3, -4) and the upper rght corner beng (5, 4) (see Fg. for reference). Regon R s defned by movng the upper rght corner to (3, 4). In addton, to examne the usefulness of the POD-FSM to the ptchng-plungng flows at a range of Strouhal numbers, we chose three representatve frequency f=0.4, 0.5 and 0.6. Table lsts a number of selected cases wth dfferent combnatons of the number of POD modes, the regons and St. Fg. shows a comparson of vortcty contour of the flow at St=0.6 between the DNS result and the reconstructon usng 96 POD modes. It can be seen that the major features of the reconstructed flow are smlar to the DNS result, except that near the tralng edge of the flappng plate the shape of the vortex on the upper surface s slghtly dstorted. Table : Summary of cases and parameters St # of POD Modes Regon RMS of Thrust Error of RMS % % % R: [-3,5] [-4,4] % % % % % % R: [-3,3] [-4,4] % %.65-7.% R: [-3,5] [-4,4].4-5.8% R: [-3,5] [-4,4] % a) DNS b) Reconstructon wth 96 POD modes Fgure. Vortcty contours at t/t= for the flow at St=0.6. Contrbuton of ndvdual egenvalue to the total turbulent knetc energy can be expressed as normalzed egenvalues n a form of represented by N m k. Captured knetc energy by the frst modes can be k N m k k The egenvalue spectrum of the flows at St=0.4, 0.5 and 0.6 s shown n Fg. k k. 3. The decay of egenvalues of three cases follows the same trend. Frst two modes contan nearly 88% of 6 Amercan Insttute of Aeronautcs and Astronautcs Paper

7 total turbulent knetc energy. The accumulaton of frst sx modes ncreases up to 97%. Hgher order POD modes contan trval fluctuatng knetc energy that manly concentrates near the flappng plate. The thrust and lft coeffcents are defned as F C T T (3) U c C L F L Uc (4) where F T and F L are thrust and lft actng on the plate, respectvely. The force hstory of the flows at St=0.4, 0.5 and 0.6 s shown n Fg. 4. Comparng to the result n [] whch used an ellpsodal fol wth thckness rato equal to 0., the zero-thckness plate causes relatvely larger peak forces n both lft and drag. As St decreases, the peak force of thrust and lft decreases sgnfcantly. The tme averaged thrust coeffcent of the flows at St=0.4, 0.5 and 0.6 s 0.44, 0.9 and.56, respectvely; and ther RMS s 0.63,. and.99, respectvely. The mpulse equaton and the POD-FSM were appled to the flows and both methods show a good agreement wth the DNS result. When all 96 POD modes are used, Fgure 3. Egenvalue spectrum of the flows at St=0.4, 0.5 and 0.6. the forces calculated by the POD-FSM converge to the correspondng results of the mpulse equaton n all three cases. Comparng wth the DNS results, the RMS error of thrust s no greater than 6%, as ndcated n Table. a) St=0.4 b) St=0.5 c) St=0.6 Fgure 4. Hstory of thrust and lft coeffcents Fgs. 5a and 5b show, respectvely, the thrust and lft coeffcents of the DNS result and reconstructed forces wth dfferent number of POD modes n Regons R for the case at St=0.6. The thrust reconstructed wth or more modes agrees well wth the DNS result. The curve of 6 modes has a slghtly dfferent trend, comparng wth the DNS result. The curve of modes s trgonometrc-lke wave wth the ampltude and tmng of peaks and valleys greatly devatng from the DNS. In contrast, the reconstructed lft of 6 modes shows excellent agreement wth the DNS result and the -mode case stll follows the general trend. From the perspectve of statstcal measurement n one perod, the RMS thrust of 96 modes s.9, whch s 4% smaller than the DNS result. Decreasng the number of POD modes from 96 to slghtly reduces the error. The error of 6 modes ncrease to 5.0%, and the thrust of modes s 8% smaller than 7 Amercan Insttute of Aeronautcs and Astronautcs Paper

8 the DNS. It s notable that the dfference of the error of 96 modes and 6 modes s only %, despte that the force curves have vsble dfferences at peak locatons and ampltudes. a) Thrust coeffcents C T (R) b) Lft coeffcents C L (R) c) Thrust coeffcents C T (R) Fgure 5. Force coeffcents of St=0.6 wth dfferent number of POD modes n Regons R and R. Fg. 5c plots the thrust coeffcents of the DNS result and reconstructed forces wth dfferent numbers of modes n Regon R. RMS of thrust wth 96 modes s.04, whch s.5% larger than the DNS result. Decreasng the number of POD modes from 96 to does not change the RMS error. The error of 6 modes slghtly decreases to.0%, whch s even smaller than the error of 96 modes. The result of modes s -7.%, approxmately the same as the one wth Regon R. However, the tmng of peaks and valleys of modes of Regon R s dfferent to that of Regon R. Ths s because the POD modes and and ther temporal coeffcents are dfferent wth respect to dfferent szes of computatonal doman. In general, from these observatons, t concludes that a lmted number of POD modes can be suffcent to reconstruct the orgnal force and POD modes that assocate wth hgher energy are more mportant than those wth lower energy n force reconstructon. The reconstructed force approaches to the soluton of the mpulse equaton as the number of POD modes ncreases. However, results of the mpulse equaton may dffer from DNS results, as we have shown n the pror valdaton secton. The error of the POD-FSM caused by usng less amounts of POD modes may compensate the error of the mpulse equaton. Therefore, statstc force calculated by several modes can be more approxmate to DNS results. Eq. (5) contans non-nteracton and nteracton force terms. The non-nteracton term can be defntely accounted for a specfc POD mode. However, t s dffcult to dstrbute the nteracton terms, whch are caused by nteractons of two modes, nto one mode. Here, to make a comparson of the force of POD modes, we defne the force related to the th POD mode as N m NF F F (5) j j0 where NF means nteracton forces. The nteracton force hstory of the mean flow NF 0 and modes NF are shown n Fg. 6. For the force of the mean flow usng Regons R or R, the lft changes slghtly as the number of modes change. However, the thrust wth modes sgnfcantly devates from other curves. Ths mples that modes 3 and 4 play an mportant role n the thrust producng. From Fg. 6b, t can be seen that the lft curves wth dfferent mode number concde for mode. Ths means that the lft of mode s ndependent to the number of modes. It s notable that the trend of NF 0 and NF do not resemble the DNS result. Partcularly, the lft peak of NF at approxmately t=5.6 s much larger than the DNS result. Because NF 0 at t=5.6 s negatve, addng NF and NF 0 produces a value smaller than NF. Comparng Fg. 6a and 6c, t seems that there s a phase shftng between the curves of Regons R and R. Addtonally, the ampltude of peaks and valleys s dfferent. Because other flow parameters are the same, t can be concluded that the force calculated by the POD-FSM s not doman ndependent. Ths 8 Amercan Insttute of Aeronautcs and Astronautcs Paper

9 dffers from the mpulse equaton, by whch the force calculated wth dfferent computatonal domans can be hghly smlar. a) NF 0 n Regon R b) NF n Regon R c) NF 0 n Regon R Fgure 6. Interacton force of the mean flow and POD modes of St=0.6 wth dfferent number of POD modes n Regons R and R. IV. Concluson A POD Mode Force Survey Method (POD-FSM) s presented to measure vrtual forces of POD modes of a ptchng-plungng plate. It shows that aerodynamc forces actng on the plate can be decomposed nto a lnear combnaton of the vrtual force of each ndvdual POD modes, the nteracton between the POD modes, and the nteracton between the POD modes and the plate. Several aspects of the POD-FSM, such as the effect of the number of POD modes and the senstvty of computatonal doman, are examned. It s found that the superposton of the vrtual forces of full POD modes can accurately predct the force actng on the plate wthn ±6% RMS errors for the flows n a range of 0.4 St 0.6. Sx hgh-energy contaned POD modes can obtan a good approxmaton of the thrust wthn ±5% RMS errors. Two energetc POD modes can approxmate the thrust wth ±8% RMS errors. Ths ndcates that POD modes wth hgher energy are more mportant than those wth lower energy from the perspectve of aerodynamc force. It s also found that the vrtual force of each ndvdual POD mode s doman dependent. Dfferent szes of computatonal domans produce dfferent vrtual force curves for POD modes. Acknowledgments Ths s work s supported under AFRL FA montored by Dr. Douglass Smth and NSF CBET-337. References Sun, M. and Lan, S.L., A computatonal study of the aerodynamc forces and power requrements of dragonfly (Aeschna juncea) hoverng. Journal of Expermental Bology.Vol. 07, No., 004, pp Dong, H., Mttal, R., Najjar, F. M., Wake topology and hydrodynamc performance of low aspect-rato flappng fols. Journal of Flud Mechancs.Vol. 556, No. 006, pp Narasmhan, M., et al., Optmal yaw regulaton and trajectory control of borobotc AUV usng mechancal fns based on CFD parametrzaton. Journal of Fluds Engneerng-Transactons of the Asme.Vol. 8, No. 4, 006, pp Lang, Z. and Dong, H. Computatonal Study of Wng-Wake Interactons between Ipslateral Wngs of Dragonfly n Flght. AIAA Paper Dong, H. and Lang, Z. Effects of Ipslateral Wng-Wng Interactons on Aerodynamc Performance of Flappng Wngs. AIAA Paper Dong, H., Lang, Z., and Harff, M., Optmal Settngs of Aerodynamc Performance Parameters n Hoverng Flght. Internatonal Journal of Mcro Ar Vehcle.Vol., No. 3, 009, pp We, M. and Yang, T. A global approach for reduced-order models of flappng flexble wngs. AIAA paper Amercan Insttute of Aeronautcs and Astronautcs Paper

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