RECENT ADVANCES in APPLIED MATHEMATICS
Spectral Differentiation Operators And Hydrodynamic Models For Stability Of Swirling Fluid Systems
DIANA ALINA BISTRIAN Department of Electrical Engineering and Industrial Informatics Engineering Faculty of Hunedoara,“Politehnica” University of Timisoara Str. Revolutiei Nr.5, Hunedoara, 331128 ROMANIA [email protected]
FLORICA IOANA DRAGOMIRESCU Department of Mathematics “Politehnica” University of Timisoara Victoriei Square Nr.2, Timisoara, 300006 ROMANIA [email protected]
GEORGE SAVII Department of Mechatronics Mechanical Engineering Faculty, “Politehnica” University of Timisoara Mihai Viteazu Nr.1, Timisoara, 300222 ROMANIA [email protected]
Abstract: In this paper we develop hydrodynamic models using spectral differential operators to investigate the spatial stability of swirling fluid systems. Including viscosity as a valid parameter of the fluid, the hydrodynamic model is derived using a nodal Lagrangean basis and the polynomial eigenvalue problem describing the viscous spatial stability is reduced to a generalized eigenvalue problem using the companion vector method. For inviscid study the hydrodynamic model is obtained by means of a class of shifted orthogonal expansion functions and the spectral differentiation matrix is derived to approximate the discrete derivatives. The models were applied to a Q-vortex structure, both schemes providing good results.
Key-Words: hydrodynamic stability, swirling flow, differentiation operators, spectral collocation.
1 Introduction substantiated mathematically by E. Hopf [2] for The role of the hydrodynamic stability theory in systems of nonlinear equations close to Navier- fluid mechanics reach a special attention, especially Stokes equations. C.C. Lin, a famous specialist in when reaserchers deal with problem of minimum hydrodynamic stability theory, published his first consumption of energy. This theory deserves special paper on stability of fluid systems in which the mention in many engineering fields, such as the mathematical formulation of the problems was aerodynamics of profiles in supersonic regime, the essentially diferent from the conservative treatment construction of automation elements by fluid jets [3]. The Nobel laureate Chandrasekhar [4] presents and the technique of emulsions. in his study considerations of typical problems in The main interest in recent decades is to use the hydrodynamic and hydromagnetic stability as a theory of hydrodynamic stability in predicting branch of experimental physics. Among the subjects transitions between laminar and turbulent treated are thermal instability of a layer of fluid configurations for a given flow field. R.E. Langer heated from below, the Benard problem, stability of [1] proposed a theoretical model for transition based Couette flow, and the Kelvin-Helmholtz instability. on supercritical branching of the solutions of the Many publications in the field of hydrodynamics Navier-Stokes equations. This model was are focused on vortex motion as one of the basic
ISSN: 1790-2769 328 ISBN: 978-960-474-138-0 RECENT ADVANCES in APPLIED MATHEMATICS
states of a flowing continuum and effects that vortex cross-stream coordinate and also zero mean radial can produce. Mayer [5] and Korrami [6] have velocity. The linearized equations are obtained after mapped out the stability of Q-vortices, identifying substituting the expressions for the components of both inviscid and viscous modes of instability. The the velocity and pressure field into the Navier mathematical description of the dynamics of Stokes equations and only considering contributions swirling flows is hindered by the requirement to of first order in delta. For high Reynolds numbers a consider three-dimensional and nonlinear effects, restrictive hypothesis to neglect viscosity can be singularity and various instabilities as in [7, 8, 9]. imposed in some problems. The linearized equations The objective of this paper is to present new in operator form are instruments that can provide relevant conclusions on L⋅ S =0, S = ( v v v π )T (2) the stability of swirling flows, assessing both r θ z and the elements of matrix L being analytically methodology and numerical methods. The study involves new mathematical models and W 1 1 LU11 = ∂t + ∂θ + ∂z − − 2 simulation algorithms that translate equations into rRe r Re 2W 2 computer code instructions immediately following L = − + ∂ , L = 0 , L = ∂ , problem formulations. Classical vortex problems 12 r r 2 Re θ 13 14 r were chosen to validate the code with the existing W 2 LW=' + − ∂ , results in the literature. The paper is outlined as 21 r r 2 Re θ follows: Section 1 gives a brief motivation for the W 1 1 LU= ∂ + ∂ + ∂ − + , L = 0 , study of hydrodynamic stability using computer 22 t rθ z Re r 2 Re 23 aided techniques. The dispersion equation governing 1 the linear stability analysis for swirling flows against L = ∂ , LU= ' , 24 r θ 31 normal mode perturbations is derived in Section 2. W 1 In Section 3 a nodal collocation method is proposed L32 = 0 , LU33 = ∂t + ∂θ + ∂z − , for viscous stability investigations and in Section 4 a r Re 1 modal collocation method is developed, based on L = ∂ , L = ∂ + , shifted orthogonal expansions, assessing different 34 z 41 r r 1 boundary conditions. In Section 5 the hydrodynamic L = ∂ , L = ∂ , L = 0 , models are applied upon the velocity profile of a Q- 42 r θ 43 z 44 vortex and Section 6 concludes the paper. where ∂{}t,,, z r θ denote the partial derivative operators and primes denote derivative with respect to radial 2 Problem Formulation coordinate. In linear stability analysis the Hydrodynamic stability theory is concerned with the disturbance components of velocity are shaped into response of a laminar flow to a disturbance of small normal mode form, given here or moderate amplitude. If the flow returns to its {vvvz,,,,,,,, r θ π} = { FriGrHrPrEtz( ) ( ) ( ) ( )} ( θ ) (3) original laminar state one defines the flow as stable, i( kz+ mθ − ω t) whereas if the disturbance grows and causes the where E( t,, zθ ) ≡ e , FGHP,,, represent the laminar flow to change into a different state, one complex amplitudes of the perturbations, k is the defines the flow as unstable. Instabilities often result complex axial wave number, m is the tangential in turbulent fluid motion, but they may also take the integer wave number and ω represents the complex flow into a different laminar, usually more frequency. The hydrodynamic equation of complicated state. The equations governing the dispersion is obtained, where we have explicitly general evolution of fluid flow are known as the decomposed into operators that multiply ω and the Navier-Stokes equations [4]. They describe the different powers of k conservation of mass and momentum. The evolution 2 T ωM+ M + kM + k M2 ⋅( F G H P) = 0 .(4) equations for the disturbance can be derived by ( ω k k ) The non zero elements of matrices are given considering a basic state U= () u,,, u u p and a { z r θ } explicitly by M = 1, M= − i , M= − i , perturbed state {V= () vz,,, v r vθ π} , with the ω1,1 ω 2,2 ω3,3
M2 = i / Re , M2 = i / Re , M2 = i / Re , disturbance being of order 0≺δ ≺≺ 1 k 1,1 k 2,2 k 3,3
U= U( r),0, W( r) , P( r) + δ V (1) MUk1,1 = − , Mk 2,2 = Ui , Mk 3,3 = Ui ,
Consistent with the parallel mean flow Mk 3,4 = i , Mk 4,3 = i assumption is that the functional form for the mean and the elements of matrix M are part of the velocity components only involves the
ISSN: 1790-2769 329 ISBN: 978-960-474-138-0 RECENT ADVANCES in APPLIED MATHEMATICS
2 mW i i i( m +1) The eigenvalue problem describing the spatial M11 = − −drr − dr + , hydrodynamic stability for a viscous fluid system r Rer Re r 2 Re reads now 2W 2 im M = − + , M = 0 , M= d , MSS0 MMk 12 2 13 14 r 0 k2 (8) r r Re + k = 0 0 I −I 0 iW2 m Ψ Ψ M= iW ' + + , 21 r r 2 Re where the first row is the polynomial eigenvalue imW 1 1 m2 problem (6) and the second row enforces the M = −d − d + , M = 0 , 22 r Rerr r Re r r 2 Re 23 definition of Ψ . im The collocation method is associated with a grid M 24 = , M31 = iW ' , M 32 = 0 , of clustered nodes x and weights w j= 0,..., N . r j j ( ) imW 1 1 m2 The collocation nodes must cluster near the M = −d − d + , 33 r Rerr r Re r r 2 Re boundaries to diminish the negative effects of the Runge phenomenon [11]. Another aspect is that the i im M 34 = 0 , M41 = idr + , M 42 = , MM43= 44 = 0 , convergence of the interpolation function on the r r clustered grid towards unknown solution is where prime denotes differentiation with respect to extremely fast. We recall that the nodes x and x the radius and dr and drr mean the differentiation 0 N coincide with the endpoints of the interval [a, b] , operators of first and second order. and that the quadrature formula is exact for all polynomials of degree ≤2N − 1, i. e., 3 Nodal Collocation Approach For N b v x w= v x w x dx , (9) Spatial Stability Including Viscosity ∑ ()j j ∫ ()() When the complex frequency ω= ω +i ⋅ ω , j =0 a r i for all v from the space of test functions. ω= Re( ω ) , ω= Im( ω ) is determined as a function r i Let {Φ } a finite basis of polynomials of the real wave number k a temporal stability ℓ ℓ=0..N analysis is performed. Conversely, solving the relative to the given set of nodes, not necessary dispersion relation (4) for the complex wave number being orthogonal. If we choose a basis of non- orthogonal polynomials we refer to it as a nodal k= k + i ⋅ k , k = Re(k ) , k= Im( k ) , when ω is r i r i basis, Lagrange polynomials for example. In nodal given real leads to the spatial branches k(,)ω ϒ approach, each function of the nodal basis is where by ϒ we denoted the set of all other physical responsible for reproducing the value of the parameters involved. In both cases, the sign of the polynomial at one particular node in the interval. imaginary part indicates the decay or either the When doing simulations and solving PDEs, a major growth of the disturbance. The growth of the wave problem is one of representing a deriving functions solution in spatial case depends on the imaginary on a computer, which deals only with finite integers. part of the axial wavenumber, as described in the In order to compute the radial and pressure next formula derivatives that appear in our mathematical model, Fcos k z +Θ − Fsin k z +Θ + the derivatives are approximated by differentiating a −k z r( r ) i( r ) e i , Θ≡mθ − ω t. (5) global interpolative function built trough the i Frsin()() k r z+Θ + Ficos k r z +Θ collocation points. We choose {Φ } given by Here the flow is considered unstable when the i i=0.. N Lagrange’s formula disturbance grows, i.e. the imaginary part of k is ω (r) N negative. A given ω leads to a polynomial Φ()r = N , where ω =()r − r . i ′ N() r ∏ m eigenvalue problem of form ωN()()r i r− r i m=1 2 T We constructed the interpolative spectral (MkMkM0 +k + 2 )( FGHP) =0, M0 ≡ω MMω + .(6) k differentiation matrix , having the entries In general, the direct solution of polynomial ()()NN+1 × + 1 eigenvalue problems can be heavy. For this case, we 2N 2 + 1 2N 2 + 1 = , = − , can transform the polynomial eigenvalue problem 00 6 NN 6 into a generalized eigenvalue problem, using the −ξ = j , j=1,… , N − 1 , companion vector method, assessed also in [10]. We jj 2(1−ξ 2 ) augment the system with the variable j i+ j T ɵ T λ −1 Ψ ≡ kFkGkH, S≡ FGHP (7) i () … ( ) ( ) ij = , i≠ j , i, j= 1, , N − 1 , λj() ξ i− ξ j
ISSN: 1790-2769 330 ISBN: 978-960-474-138-0 RECENT ADVANCES in APPLIED MATHEMATICS
2if i= 0, N rmax H( kUr max −ω) ± HWr max ± P =0 = 0, λi = . 1 otherwise rmax F( kUr max −ω) ± FWr max + kr max P = 0 , (20) derived in [12]. where U and W are the axial, respectively the We made use of the conformal transformation r max r max tangential velocity calculated at domain limit rmax . 1+b exp( − a) rmax 1−ξ r ()ξ = (10) A different approach is obtained by taking as 1−ξ 2 basis functions simple linear combinations of 1+b exp − a 2 orthogonal polynomials. These are called bases of that maps the standard interval ξ ∈[ −1,1] onto the modal type, i. e., such that each basis function provides one particular pattern of oscillation of physical range of our problem r∈ 0, r . Because [ max ] lower and higher frequency. We approximate the large matrices are involved, we numerically solved perturbation amplitudes as a truncated series of the eigenvalue problem using the Arnoldi type shifted Chebyshev polynomials algorithm [11], which provides entire eigenvalue N * and eigenvector spectrum. ()(F,,,,,, G H P= ∑ fk g k h k p k)⋅ T k , (21) k =1 where T * are shifted Chebyshev polynomials on the 4 Modal Collocation With Orthogonal k physical domain [0, r ] . The clustered Chebyshev Basis For Inviscid Stability Analysis max Gauss grid Ξ = ξ in −1,1 is defined by The collocation method that we present in this ( j )1≤j ≤ N [ ] section has the peculiar feature that can approximate relation the perturbation field for all types of boundary π(j+ N − 1) conditions, especially when the boundary limits are ξj+1 = cos , ξ j+1 ∈[] − 1,1 ,j = 0.. N − 1. (22) N −1 described by sophisticated expressions. We consider This formula has the advantage that in floating- the mathematical model of an inviscid columnar point arithmetic it yields nodes that are perfectly vortex derived in [13] whose velocity profile is symmetric about the origin, being clustered near the written as V( r) = U( r),0, W( r) . boundaries diminishing the negative effects of the G mH Runge phenomena [11, 12]. This collocation nodes G'+ + + kF = 0 , (11) r r are the roots of Chebyshev polynomials and mW 2WH distribute the error evenly and exhibit rapid ω − −kU G − + P ' = 0 , (12) convergence rates with increasing numbers of terms. r r In order to approximate the derivatives of the mW W mP −ω + +kU H + W' + G + = 0 ,(13) unknown functions, we express the derivative of the r r r * shifted Chebyshev polynomial Tn as a difference mW −ω + +kU F + U' G + kP = 0 . (14) between the previous and the following term r *′ rmax (n−1) * * We assume for for this model that the radial Tn () r= Tn−1 ()() r− Tn+1 r , n ≥ 2 . (23) 4 r r− r amplitude of the velocity perturbation at the wall is ()max Let us consider negligible, i.e. G( rmax )= 0 , for a truncated radius N distance r selected large enough such that the * * max F()()() r= f1 T 1 r + ∑ fk T k r . (24) numerical results do not depend on that truncation of k =2 By differentiating (24) results infinity. We have at r = 0 N |m |> 1 , F = G = H = P = 0 , (15) ′ *′ *′ ( ) F()()() r= f1 T 1 r + ∑ fk T k r . (25) k =2 (m=0) , G = H = 0, FPfinite , , (16) But T*′ () r = 0 and involving relation (23) results (m= ±1) , H ± G = 0, F = P = 0 . (17) 1 N r (k −1) and at r= r ′ max * * max F() r= ∑ fk T k−1 ()() r− T k +1 r . (26) k =2 4 r() r− r (|m |> 1) , F = G = H = P = 0 , (18) max The interpolative differentiation matrix D that 2WH ()m=0 , r max − P' = 0, G = 0, approximates the discrete derivatives has the rmax elements HkU−ω H =0, FkU −ω F + kP = 0 r max r max , (19) Dm, n= E n( r m ), m = 2.. N − 1, n = 2.. N − 1 , (27) 2WH ()m= ±1 , r max − P' = 0, G = 0, where for k=2.. N − 1 rmax
ISSN: 1790-2769 331 ISBN: 978-960-474-138-0 RECENT ADVANCES in APPLIED MATHEMATICS
(k −1) N NN * * kU r h+ ± W −ω r h ± p = 0 , (38) Ek () r = Tk −1 ()() r− Tk +1 r . (28) r max max ∑ k( r max max )∑k ∑ k r() rmax − r 1 1 1 The eigenvalue problem governing the inviscid N N N k Ur max r max ∑ fk + rmax ∑ pk +( ± W r max −ω r max )∑ fk = 0.(39) stability analysis appears now as a system of 4N 1 1 1 equations, when include the boundary conditions. A 1 special situation occur for the cases m = ±1, when Let us denote by [r] = diag() ri , = diag(1/ ri ) , r only 7 relations define the boundary conditions. To * th [η] = () ηij2≤ i ≤ N − 1, , ηij= T j() r i , [U] = diag( U ( ri )) regain the 8 equation we choose the third relation 1≤j ≤ N from the mathematical model and we compute it in [W] = diag( W ( ri )) , 2≤i ≤ N − 1. Written in matrix the extreme node r= r . max formulation, the hydrodynamic model reads We have chosen this relation for few reasons. kM+ω M + mM + M s = 0 , We observed that the equations that not contain the ( k ω m 0 ) T axial perturbation F are the second and the third. s= ( f1,..., fN , g1 ,..., gN , h1 ,..., hN , p1 ,..., pN ) , (40) The second equation contains the derivative of the where M , M , M and M are square matrices of pressure perturbation that cannot be computed in k ω m 0 dimension 4N and the elements being matrix blocks extreme nodes because the interpolative derivative M matrix may produce singularities since contains the M = k , k boundary conditions blocks expression Ek () r . The remain possibility is actually M the third equation symmetrized. M = ω , ω The hydrodynamic model reads, for j=2.. N − 1 boundary conditions blocks N N N 1 * m * * M m G'+ gTr+ hTrkfTr+ , (29) M m = , ∑k k() j ∑k k()() j ∑ k k j boundary conditions blocks rj k=1 rj k=1 k=1 M mW N 2W N M = 0 , ω − −kU g T* r− h T* r+ P' = 0, 0 ∑k k() j ∑ k k() j (30) boundary conditions blocks rj k=1 rj k=1 [r ] [η ] 0 0 0 N mW * 0 [][]U η 0 0 −ω + + kU h T r + M = ∑ k k() j k r k =1 0 0 [][]rU η 0 j U η 0 0 η N N [][][] W * m * +W' + ∑ gk T k() r j + ∑ pk T k() r j = 0, (31) 0 0 0 0 rj k =1 rj k =1 0 − []η 0 0 M = , mW N ω 0 0 − [][]r η 0 −ω + + kU f T* r + ∑ k k() j − η 0 0 0 rj k =1 [] N N 0 0 [η ] 0 * * +U'∑ gk T k()() r j + k ∑ pk T k r j = 0, (32) W 0 []η 0 0 k =1 k =1 r M = N N m 0 0 [][][]W η η kr U h+ mW − rω h + maxr max ∑ k ( r max max )∑ k W k =1 k =1 []η 0 0 0 r N N +(W + r W′ ) g+ m p = 0 , (33) 0 [η ] + [r] D 0 0 r max maxr max ∑k ∑ k k =1k = 1 W N N 0 0 2 η −D k +1 k +1 [] −1 g ± −1h = 0 , (34) M 0 = r ∑()()k ∑ k 1 1 0 [][][][]Wη+ rW ' η 0 0 N N k +1 k +1 0[][]U ' η 0 0 ∑()()−1 fk = ∑ −1pk = 0 , (35) 1 1 where D represents the interpolative 2W N 2 N 2(k − 1) 2 derivative matrix. r max ∑hk −p2 − ∑ pk ∑ 2 − rmax1r max 3rmax r= k − 1 k odd k even 5 Model Validation On a Q-Vortex N 2 2(k − 1) Profile −∑pk ∑ 2+ 1 = 0 , (36) 4rmax r= k − 1 Assuming the velocity profile of Q-Vortex, written k even k odd in form N g = 0 , (37) − r 2 q −r2 ∑ k U() r= a + e , W ()r=1 − e , (41) 1 r ()
ISSN: 1790-2769 332 ISBN: 978-960-474-138-0 RECENT ADVANCES in APPLIED MATHEMATICS
where q represents the swirl number and a the singularities and the spectral differentiation provides a measure of free-stream axial velocity, we matrix was derived to approximate the discrete perform a spatial stability analysis using the derivatives. The models were applied to a Q-vortex collocation method described above. The spectra of structure, the scheme based on shifted Chebyshev the eigenvalue problem governing the spatial polynomials providing good results. stability is depicted in Figure 1. It is noticeable that the eigenvalue with the References: largest imaginary part defines the most unstable [1]Langer R.E., On the stability of the laminar flow mode. In Table 1 we have compared the results of a viscous fluid, Bull. Amer. Math. Soc., 46, pp. obtained by this method with those of Olendraru et 257-263, 1944. al. [14], in the non axisymmetrical case m > 1. [2]Hopf E., On nonlinear partial differential equations, Lecture series of the Symposium on partial differential equations, University of California, pp.7-11, 1955. [3]Lin C.C., The theory of hydrodynamic stability, Cambridge University Press, Cambridge, 1955. [4]Chandrasekhar S., Hydrodynamic and hydromagnetic stability. Dover, NewYork, 1981. [5]Mayer E.W., On the Structure and Stability of Slender Viscous Vortices, PhD thesis, University of Michigan, Ann Arbor, MI, 1993. [6]Khorrami M. R., On the viscous modes of Fig.1 Spectra of the hydrodynamic eigenvalue instability of a trailing line vortex, Journal of problem computed at ω = 0.01 , m = −3 , a = 0 , Fluid Mechanics, 225, pp.197-212, 1991. q = 0.1, for N = 100 collocation nodes. [7]Leibovich S., Stewartson K, A sufficient condition for the instability of columnar vortices, Table 1. Comparative results of the most amplified Journal of Fluid Mechanics, 126, pp. 335-356, k-spatial wave at a = 0 , q = 0.1, ω = 0.01 for the 1983. [8]Khorrami M. R., Malik M.R., Ash R.L., case of the counter-rotating mode m = −3 : Application of spectral collocation techniques to k= k, k eigenvalue with largest imaginary part cr( r i ) the stability of swirling flows, J. Comput. Phys., and critical distance of the most amplified Vol. 81, pp. 206–229, 1989. perturbation rc . [9]Rotta J.C., Experimentalier Beitrag zur Shooting method [14] Entstehung turbulenter Strömung im Rohr, Ing. k =(0.506, − 0.139) r = 1.0005 Arch., 24, pp. 258-281, 1956. cr c [10]Parras L., Fernandez-Feria R., Spatial stability Collocation method and the onset of absolute instability of k =0.50819, − 0.14192 r = 0.971 cr ( ) c Batchelor’s vortex for high swirl numbers, J. Error 0.79% 2.94% Fluid Mech., Vol. 583, pp. 27– 43, 2007. [11]Trefethen L.N., Spectral methods in Matlab, 6 Conclusion SIAM, Philadelphia, 2000. [12]Canuto C. et al., Spectral methods - Evolution In this paper we developed hydrodynamic models to complex geometries and applications to fluid using spectral differential operators to investigate dynamics, Springer, New York, 2007. the spatial stability of swirling fluid systems, using [13]Bistrian D.A., Dragomirescu F. I., Muntean S., two different methods. Topor M., Numerical Methods for Convective When viscosity is considered as a valid Hydrodynamic Stability of Swirling Flows, parameter of the fluid, the hydrodynamic model is th Recent Advances in Systems, 13 WSEAS implemented using a nodal Lagrangean basis and the International Conference on Systems, pp. 283- eigenvalue problem describing the viscous spatial 288, Rodos, 2009. stability is solved using the companion vector [14]Olendraru C., Sellier A., Rossi M., Huerre P., method. The second model for inviscid study is Inviscid instability of the Batchelor vortex: assessed for the construction of a certain class of Absolute-convective transition and spatial shifted orthogonal expansion functions. The choice branches, Physics of Fluids, Vol. (11) 7, pp. of the grid and the choice of the trial basis eliminate 1805-1820, 1999.
ISSN: 1790-2769 333 ISBN: 978-960-474-138-0