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.