Flow Induced Vbraton Project Progress Report Date: 16 th November, 2005 Submtted by Subhrajt Bhattacharya Roll no.: 02ME101 Done under the gudance of Prof. Anrvan Dasgupta Department of Mechancal Engneerng, T Kharagpur Department of Mechancal Engneerng, Indan Insttute of Technology, Kharagpur 721302.
Certfcate Ths s to certfy that the report enttled Flow Induced Vbraton submtted by Subhrajt Bhattacharya to the Department of Mechancal Engneerng, T Kharagpur, s a bona fde record of work carred out under my supervson and gudance. Prof. Anrvan Dasgupta, Dept. of Mechancal Engneerng, T Kharagpur. Date:
1. Introducton: The problem of flud-structure nteracton s encountered n varous Engneerng applcatons. The present problem deals wth vbratons nduced nto structures due to flow takng place on ts surface, and hence analyzng the stablty of the flow. Though the present approach to the problem s grossly smplfed and no substantal smulaton or numercal soluton have yet been deduced, the works done tll now have great potental n applcatons lke flow over underwater vehcle, etc. 2. Problem defnton and am: Our present am s to analyze the stablty of flud flows over a flexble flat surface. By the term stablty we mean that we try to determne the crtcal Reynolds number for a gven flow profle over the structure. We wll consder flexblty of the plate and frame the correspondng equatons. However for the purpose of determnng the numercal soluton we have presently nvestgated only the case of flow over a rgd surface to check f we obtan the standard results for stablty of parallel flow over rgd surfaces avalable n standard text. 3. Orgn of Turbulence and ways to analyze stablty of a flow: Flow nduced vbraton s caused n structures by forcng due to tme varant pressure actng on the surface of the structure. If the flow s magned to be a lnear superposton of a steady state lamnar flow and a perturbaton flow feld, the lamnar flow won t cause the forcng on the structure as t acts as a tme nvarant pressure on the structure s surface. It s the perturbaton flow feld that vares wth tme and hence produces forcng on the structure s surface. Ths same perturbaton flow feld s the sole cause of Turbulence n the flow. The orgn of the perturbaton flow feld may be somethng lke a very small dsturbance caused n the lamnar flow feld. A flow s sad to be stable f for any small ntal dsturbance (.e. perturbaton) added to the lamnar flow feld, the perturbaton flow feld gradually des down wth tme. It wll be termed as a Turbulent flow f the perturbaton flow feld gets magnfed wth tme. Hence our prmary approach wll be to nvestgate the nature of varaton of a perturbaton flow feld wth tme.
. The Orr-Somerfeld equaton: The two-dmensonal ncompressble flow over the plate s governed by the Naver Stoke s equatons, 2 2 u u u 1 1 p u v F u u + + = X + ν + 2 2 t x y ρ ρ x x y (1a) 2 2 v v v 1 1 p v v + u + v = FY + ν + 2 2 t x y ρ ρ y x y (1b) and the contnuty equaton, u v + = 0 (2) x y Let us consder a steady-state lamnar flow over an nfntely long plate. Y fg - 1 X For ths flow, u = U(y) v = 0 (3) p = P(x) Now let the perturbaton feld be descrbed by the perturbaton components denoted by a prme upon the steady-state lamnar varables. Hence the fnal flow feld wll be descrbed by, u = U(y) + u (x, y, t) v = v (x, y, t) () p = P(x) + p (x, y, t) Now, both the steady-state lamnar feld (3) and the superposed feld () satsfes equaton (1) and (2). Hence by substtutng them n (1) and (2) and performng some smplfcaton, we obtan the equatons governng the perturbaton flow feld, u' u' U 1 p' 2 + U + v' + = ν u ' (5a) t x y ρ x v' v' 1 p' 2 + U + = ν v' (5b) t x ρ y u' v' + = 0 (5c) x y From these equatons, wth a known velocty profle U(y), we may obtan solutons for u', v', and p'. Our present am wll be to determne for a gven velocty profle U(y), the coeffcent of vscosty ν and the boundary & ntal condtons of the perturbaton felds, whether or not the perturbaton components de down wth tme.
In order to satsfy eqn. (5c), we defne a stream functon ψ ( xyt,, ) such that ψ ψ u ' = and v ' = (6) y x It s now assumed that the dsturbances, and hence ψ s superposton of several perodc dsturbances (perodc n x) propagatng along the drecton of flow. Hence, we substtute, ( α x βt) ψ ( xyt,, ) = φ( ye ) (7) 2π ψ s sad to be perodc on x wth frequency α and wavelength λx =. α Though α can be assumed to be real, β beng the tme frequency should be assumed to be complex n order to keep the possblty of non-perodc magnfcaton or decay of ψ wth tme. The fnal ψ wll be superposton of all the solutons of ψ. We defne the complex velocty of propagaton of the dsturbance as β c= = cr + c (8) α [ ct ( x crt ψ φ( y) e α + )] = It may be noted that for c < 0, the soluton of ψ des down wth tme, and hence so does the perturbaton components. Thus the flow s stable for c < 0 and tends to become turbulent for c > 0. We put u' and v' n terms of φ, α and β n (5a) and (5b) and elmnate p' to obtan a sngle equaton. We non-dmensonalze the equatons by redefnng y as y δ, U as U U, c m c as U, where U m s the free-stream velocty of the flow and δ s the boundary layer m thckness. The non-dmensonalzed equaton hence obtaned s called the Orr- Sommerfeld equaton: where, R U δ ν 2 ( U c)( φ αφ) U φ= ( φ 2αφ 2 + αφ ) (9) α R m = s the Reynolds Number. A trval soluton to eqn.(9) s φ = 0. For non-trval soluton, for a gven α and R, we obtan an egenvalue of c and the correspondng egenfuncton φ. Hence our mmedate target s to fnd an egenvalue soluton of (9).
5. Boundary condtons for the Orr-Sommerfeld equaton: A. Boundary condtons for flow over a rgd, statc plate: For ths case, At y = 0, v' = 0 and u' = 0 φ (0) = 0 and φ (0) = 0 At y, v' = 0 and u' = 0 φ( ) = 0 and φ ( ) = 0 B. Boundary condtons for flow over a flexble plate modeled as a beam: p'(x,0,t) We model the plate as nfnte flexble beam. Hence, the dfferental equaton governng the moton of the beam s gven by, 2 fg - 2 λ w w = p ( x,0, t) EI (10) 2 t x Where, w denotes the dsplacement of the beam n Y drecton, λ s mass per unt length of the beam. The forcng on the beam s due to the tme-varant perturbaton pressure. It s to be noted n ths case that equatons (5), (9) and (10) gets coupled. One way of solvng the equaton wll be as follows. ( α x βt) As p' s of the form of e, we may wrte for eqn. (10), ( α x βt) wxt (, ) = ke (11) Substtutng ths w n (10) and hence substtutng the hence obtaned p'(x,0,t) nto (5a), and Substtutng v' n terms of φ n the same (5a) equaton, all at y = 0, we obtan, ρ αu φ α ν β φ νφ k + = α λβ Hence the boundary condtons become, At y = 0, 2 (0) (0) ( ) (0) (0) 2 ( EIα ) v' = w and u' = 0 β φ(0) = k and φ (0) = 0, where k s the non-dmensonalzed k. α At y, v' = 0 and u' = 0 φ( ) = 0 and φ ( ) = 0 Hence, as t can be seen, one of the boundary condtons s a bt more complex nvolvng φ, φ, φ and c. (12)
The numercal technque for solvng the Orr-Sommerfeld equaton for the egenvalues and egenfunctons n ether of the cases wll reman smlar, except that the boundary condtons are modfed. However tll date we have only nvestgated the case A,.e. the case of flow over a rgd, statc plate. 6. Numercal methods attempted for solvng the Orr-Sommerfeld equaton: A. Galerkn s Method: Ths method, though can handle the case of flow over a rgd, statc plate satsfactorly, ts applcaton n solvng the case of flexble plate s dffcult. The method for the case of flow over a rgd, statc plate s descrbed below n bref. We denote eqn.(9) by Γ ( φ) = 0, where Γ denotes the operator. We wrte φ as a lnear superposton of several functons φ that satsfy the boundary y condtons gven n 5.A. Such functons were chosen to be of the form, ye η φ = µ. Hence we wrte, We now defne an error, n φ( y) = ξφ( y) = 1 ey ( ) ( ) n =Γ ξφ y = 1 We need to choose the values of ξ n such a way that ths error s mnmzed. We do that by solvng the set of n equatons, e, φ = 0, = 1 to n (1) where, χ, χ denotes the nner product gven by, j e, φ e = φ dy The n equatons n (1) have ξ as the unknowns and can be represented n the matrx form as, ξ1 [ M ] = 0 (15) ξ n Where the matrx [M] contans the unknown c. For non-trval soluton of ths equaton, we must have, [ M ] = 0 (16) 0 (13)
In general we ll obtan n solutons for c. We should chose the one for whch the ξ s are n such that Γ ξφ ( y) s mnmum. = 1 Ths method was mplemented n Mathemata 5.1 and a rather unsatsfactory result was obtaned probably due to the followng reasons:. Due to lmtaton of computatonal power, n had to be lmted to n order to get a satsfactory and accurate ntegraton value of the nner product and soluton of c from (16).. The choce of η and µ n choosng the functonal forms were done arbtrarly. We determned the egenvalues c for dfferent α and R and plotted the contour n the α-r plane for whch c = 0. Ths contour wll mark the margn between the stable and unstable zones of α and R. However the contour c = 0 could not be found satsfactorly n the fst quadrant of α-r plane. But on plottng a contour plot of c, the followng was obtaned. The plot demonstrates that the basc shape of the standard results s beng approached by the soluton: 1 0.8 0.6 0. 0.2 0 0 200 00 600 800 1000 fg - 3 The horzontal axs denotes R and the vertcal axs denotes α.
B. Automated search of egenvalues Integraton by Runge-Kutta: Ths method prmarly proposed by Betchov & Crmnale [2], deals the regons above and below the boundary layer separately. We frst nvestgate the Orr-Sommerfeld equaton (9) for y > 1. In ths regon, the non-dmensonalsed velocty s U = 1. Hence the equaton s modfed as, 2 2 (1 c)( φ αφ) = ( φ 2αφ + αφ) for y > 1 (17) α R Ths equaton beng th order lnear n φ wth constant coeffcent has smple analytcal soluton gven by, where, p j are the roots of, Hence, p1 j φ( y) = Aje j= 1 p y (18) 2 2 2 2 2 (1 c) ( p α ) = ( p α ) (19) α R = α, p1 R p3 = α 1 + (1 c) α 1 2, = α R p3 = α 1 + (1 c) α From boundary condton at, as y, φ = 0 and φ = 0. Hence we have, A 1 = A 3 = 0 for y > 1. p2y py φ( y) = A e + A e. 2 Now we argue that, snce for y > 1, the soluton s a lnear superposton of two modes p2 y p y e and e, the soluton for y < 1 wll also be lnear superposton of these two same modes. Hence now our task s to fnd the soluton of these two modes n the regon y < 1. We attan ths by performng two ntegraton passes from y = 1 to y = 0 ndependently. We used Runge-Kutta method for ths numercal ntegraton. Integraton Pass I: We start from y = 1 wth A 2 = 0 and A = 1(or some other value). p p 2 p Therefore we take the ntal values φ (1) = e, φ (1) = pe, φ (1) = pe & φ (1) = pe 3 p and move on ntegratng towards y = 0. Let the soluton obtaned n ths process be called φ ( y). Integraton Pass : Smlarly wth A 2 = 1(or some other value) and A = 0 we obtan the second pass ntegraton φ ( y). Hence the fnal soluton s of the form φ( y) = a φ ( y) + a φ ( y). I I I 1 2
From the boundary condtons at y = 0, φ(0) = aiφi (0) + aφ (0) = 0 and φ (0) = aiφ I (0) + aφ (0) = 0, for non-trval soluton of a I and a, we must have, φi(0) φ(0) = 0 φ (0) φ (0) I (20) It s to be noted that the only unknown n (20) for a gven α and R s c. Hence from 20 we obtan the egenvalue c. The correspondng egenvector gves a I and a, and hence the fnal soluton of φ ( y). Search for Egenvalue n c-plane: However t was not possble to solve c explctly from (20). Hence we assumed some c and determned the solutons from the two ntegraton passes, φ I ( y) and φ ( y). For those partcular solutons we defned, φi(0) φ(0) f() c = φ (0) φ (0) I As we know f(c) should converge to 0, we used an teraton scheme [2] as follows. We start wth an arbtrary value of c and some small value of δ c and go on updatng t usng the followng teraton, f( c+ δc) c= λ 1 δc f() c c c+ c δ c µ c where, λ = 1, 0.8 or 0.5 dependng on whether the convergence s fast, moderate or slow. 1 and, µ =. The teraton contnues tll c reaches a substantally small value. 1
Lke before, we once agan searched for the contour of c = 0 n the α-r plane. The results obtaned, though not satsfactory, s descrbed n the followng plot of 20 ponts: fg - The prmary reason for the devaton of ponts from a sngle smooth contour s the hgh oscllaton of the soluton φ ( y) as the second numercal ntegraton approaches y = 0. Ths fact s well demonstrated n the followng plot of the ampltude of φ ( y) vs. y for a partcular α and R: fg - 5
The small ampltude of φ ( y) s due to the ntal choce of a small A 2 for the second Integraton pass. C. Modfed second Integraton Pass usng Statons n between: Ths method, as explaned by Betchov & Crmnale [2], was mplemented n order to reduce the oscllaton of the second soluton. The man prncple of ths method s based on the choce of some statons n between y = 1 and y = 0. At these statons the second ntegraton s paused and t s updated by lnearly combnng wth the frst ntegraton φ I ( y) to make A = 0. We are presently workng on ths technque and hope to obtan some satsfactory result very soon. Once we obtan a soluton for the case of flow over a rgd, statc plate, we ll attempt the soluton for the case of flow over a flexble plate (modeled as an nfnte beam) on the smlar lne. References: [1] Dr. Hermann Schlchtng, Boundary Layer Theory, McGraw Hll, 1968. [2] Robert Betchov, Wllam O. Crmnale, Jr., Stablty of Parallel Flows, Academc Press, 1967.