TRANSIENT FLOW AROUND A VORTEX RING BY A VORTEX METHOD

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1 Proceedings of The Second International Conference on Vortex Methods, September 26-28, 21, Istanbul, Turkey TRANSIENT FLOW AROUND A VORTEX RING BY A VORTEX METHOD Teruhiko Kida* Department of Energy Systems Engineering, Osaka Prefecture University, Sakai, Osaka , Japan / kida@energy.osakafu-u.ac.jp Tekanori Take Department of Mechanical Systems Engineering, Shiga Prefecture University, Hikone, Siga , Japan Abstract The transient flow around a ring is simulated by a vortex blob method, which is derived from the integral equation with respect to vorticity. In this paper lengths and velocities are normalized with a typical length of the ring vortex and a typical circulation of the ring, respectively. From the normalized Navier- Stokes equations the vorticity equation is derived and the governing integral equation is constructed from it. As the discrete form of the vorticity field a vortex blob method is proposed and numerical calculations are carried out for the flow around an elliptic ring vortex. This method has an advantage that we can hold the distance between the nascent vortices within a given allowance. INTRODUCTION The present paper treats a transient flow around a vortex ring placed in unbounded fluid flow, in order to know the validity of the vortex blob method derived in this paper. In 3-D Euler flows, the high stretching of the vortex stick induces the divergence of the velocity field (see Maffott (2)), thus, a tool such as reconnection of the vortex tube is used to continue the numerical calculation of the vortex method in this case. The breakdown of smooth solutions of the 3-D Euler equations signifies the onset of the turbulent behavior and the formulation of the singularity is related to the concentration of vorticity on successively smaller sets. Beale et al. (1984) and Ponce (1985) proved that the maximum vorticity necessarily grows without bound as the critical time approaches and it is not possible for other kinds of singularities to form before the vorticity becomes unbounded. Constantin & Fefferman (1993) shows that large curvatures in vortex lines are necessary if a singularity is to form. Gibbon et al. (2) shows by using Kerr s picture (1993) that the structure of vortex stretching and compression around the singular point is very complicated and there is a strong switch from stretching to compression like a vortex jump. Pelz (1997) studies the finite-time singularity of a high-symmetric flow with dodecapole by using a vortex filament method and shows the existence of this singularity in 3-D Euler flows. In this calculation, a cubic Gaussian core function for the vorticity distribution and a second-order Runge-Kutta scheme for time marching are used. Further remeshing algorithm and cubic splines are used and he shows the finite time singularity. However, for the viscous flow Boratav & Pelz (1994) use a Fourier pseudospectral method and show: The six vortex dipoles deform one another, their cores become more and more flattened, and the distances shrink exponentially. There is a short local equilibrium interval when the viscous force and vortex stretching forces balance each other. And viscous dissipation forces will play a role against the singularity formation as rapidly as the convective forces play toward a singularity. The vortex method is one of powerful numerical method for knowing the mechanism of the onset of turbulent as shown by Pelz (1997). The present paper aims to construct an alternative vortex blob method based on the Navier-Stokes equations. First the Navier-Stokes equations are depicted by normalizing with a typical global length and a typical circulation, using the idea of Pelz (1997). Second the integral expression is derived by considering the expression derived by Nakajima & Kida (1997) and a vortex blob method is introduced as the discrete form of this 1

2 expression. Finally, numerical results are shown for flows around an elliptic ring with constant circulation initially. 1 Governing equations We here consider 3-D incompressible fluid flow. The governing equations are given as Navier-Stokes equations: u t ( u ) u = P ν 2 u, (1) u =, (2) where P is the pressure divided by ρ. Lengths and velocities are normalized with a typical global length of the vortex tubes l(t) and a typical circulation around the tubes. Normalized position vector and velocity vector are respectively denoted as X and v: X = S(t) x, (3) v = S(t) u, (4) where S(t) =1/l(t), and x and u are the position vector and the velocity vector in the physical state, respectively. Then, Eqs.(1) and (2) become as v ( v ) v SṠ t 3 i=1 X i v = Ṡ S2 v S Γ 2 P S2 ν 2 v, (5) o v =, (6) where Ṡ = ds dt. We here consider the relative coordinate systems given by ˆX = X 1 2 S2 ( e 1 e 2 e 3 ), (7) where e i is the unit vector of the i direction. Furthermore, we denote ˆX as x again. Then, the Navier- Stokes equations, Eqs.(5) and (6), are given as v Ṡ S2 ( v ) v = v t S Γ 2 P S2 ν 2 v, o (8) v =. (9) Here, we introduce the vorticity as Then, we have from Eq.(8), ω t ω = rot v. (1) Ṡ ( v ) ω = ω ( ω ) v S S2 ν 2 ω. (11) Furthermore, we again define the vorticity ˆω = ω/s(t) and we put ˆω as ω again. Then, we have ω t ( v ) ω =( ω ) v S2 ν 2 ω. (12) We note that the relation between the original physical field and the normalized field is given by Ω = S ω, (13) x = 1 ( X 1 ) S 2 S2 ( e 1 e 2 e 3 ). (14) where Ω is the vorticity in the physical state. 1.1 Integral Expression The construction of an integral equation with respect to vorticity from the Navier-Stokes was carried out by Nakajima and Kida (1997). Their expression, however, is not proved rigorously. From the approach by Nakajima and Kida (1997), the vorticity field is supposed to be expressed as the following integral form: ω i ( X,t)= ω oj ( a) g X X(),β(t) a i j( a, t, )da t ds g X(s) X,β(t s) a i j ( a, t, t s)f j( X,a,s)dadX, (15) where ω o is an initial vorticity field, X(s) = X a t s v a( a, s)ds, v a is a velocity vector, and g, β are defined as β(t) = 4S 2 νt, (16) g( X,σ)= ( σ ) 3/2 ( exp σ X π 2). (17) In this expression Eq.(15), f is unknown vector function. Substituting Eq.(15) into Eq.(12), we have the following relation: f j ( X, a, t)a i j( a, t, t)da [ = ω oj ( a) ( v( X,t) v a ( a, t)) g X(),β(t) a i j ( a, t, ) t ai j( a, t, ) ε ki ( X,t)a k j ( a, t, ) g( X(),β(t)) ] da 2

3 t ds f j ( [ X, a, s) ( v( X,t) v a ( a, t)) g X(s) X,β(t s) a i j ( a, t, s) t ai j ( a, t, s) ɛ ki( X,t)a k j ( a, t, s) g( X(s) X ],β(t s)) dx da (18) ( where ε ij = 1 vi 2 X j vj X i ). Therefore, we have f j ( X, a, [ t)a i j ( a, t, t) =ω oj( a) ( v( X,t) v a ( a, t)) g X(),β(t) a i j( a, t, ) t ai j ( a, t, ) ɛ ki( X,t)a k j ( a, t, ) g( X(),β(t)) ] t ds f j ( [ X, a, s) ( v( X,t) v a ( a, t)) g X(s) X,β(t s) a i j ( a, t, s) t ai j( a, t, s) ɛ ki ( X,t)a k j ( a, t, s) g( X(s) X ],β(t s)) dx. (19) Here, we suppose that a i j satisfies the following relation: t ai j ( a, t, s) =ε ik( a Φ( a, t),s)a k j ( a, t, s), (2) where Φ( a, t) = t v a ( a, s)ds. From Eq.(19), we can express f j as f j ( X, a, t) = F ( X, a, t; j) g X A( a, t) G( X, a, t; j)g X A( a, t), (21) where A( a, t) = a Φ( a, t). Let us assume that β(t) 1, that is, Then, we easily see ω i ( X,t) 1 4 t t ω oj ( a)g a i j( a, t, s)ds a i j ( a, t, s) β(t s) ds S 2 νt 1. ( X(),β(t) ) a i j( a, t, )da f j ( X(s), a, s)da 2 f j ( X(s), a, s)da. (22) We choose the approximate velocity field v a as v a ( a, t) = v( A( a, t),t). (23) Then, we easily see for X A 1: F ( X, a, t; j) ( X A( a, t)) F ( A, a, t; j) (24) G( X, a, t; j) ( X A( a, t)) G( A, a, t; j).(25) Therefore, we have ω i ( X,t) ω oj ( a)g X(),β(t) a i j( a, t, )da O(t/β). (26) Thus, for small time t we have the same expression as one given by Nakajima & Kida (1997): ω i ( X,t) ω oi ( a)g( X(),β(t))da t ω oj ( a)ε ij ( a)g( X(),β(t))da, (27) where X() = X a Φ( a, t) and Φ( a, t) v a ( a)t. 1.2 Vortex Blob Method Let us derive a vortex blob method from Eq.(27). We here assume that the vorticity field is approximately expressed as the following discrete form: ω i ( X,t)= Γ i k(t)g( X X k,σ). (28) Multiplying g( X X m,σ) to Eq.(28) and integrating with respect to X, we have from using Eq.(27): ω i ( X,t)g( X X m (t),σ)dx Γ i k() g( a a k,σ) g( X X m (t),σ)g( X A( a, t),β)dadx t Γ j k () g( a a k,σ) g( X X m (t),σ)g( X A( a, t),β) ε ij ( a)dadx. (29) Here, we use the following relation: g( a, σ)g( x a, β)da = g( x, σ β ), σ β = σβ σ β. (3) 3

4 Then, we have the relation with respect to Γ k : Γ i k(t)g( X X k (t),σ/2) = Γ i k() g( X X k (t),σ 2β ) t Γ j k ()E ij( X,k), (31) σβ where σ 2β = σ2β and E ij is defined as E ij ( x, k) = g( x A, σ β ) g( a a k,σ)ε ij ( a)da. (32) The velocity field is obtained by using the Biot- Savart formula. Substituting Eq.(28) into the Biot- Savart formula, we can obtain the velocity field as v l = 1 Γ i k Î o ( X 4π X X k,σ)ɛ ijl, (33) j where ɛ ijk is the Eddington tensor and Îo is defined as Î o ( x, σ) = 2π 1 σ 3/2 x Erf(σ1/2 x ), (34) Erf(x) = x exp( x 2 )dx. (35) Using Eq.(33), we can easily obtain the strain rate ε ij as ε ij = 1 [ Γ p 2 k Î o ( X 8π X j X X k,σ)ɛ pqi q 2 Î o ( X X i X ] X k,σ)ɛ pqj. (36) q We substitute Eq.(36) into Eq.(32), then we have E ij ( X,k)= 1 Γ p s 8π s=1 [ 2 Ĵ( X X sj X X k, X s X k )ɛ pqi sq 2 Ĵ( X X si X X k, X s ] X k )ɛ pqj, (37) sq where Ĵ is defined as Ĵ( x, y) = g( x a, σ β )g( a, σ) Îo( a y, σ)da. (38) In order to obtain Ĵ( x, y), we define the following function K( x, y, γ): K( x, y, γ) = g( a x, σ)g( a y, σ β ) Îo( a, γ 2 )da. (39) Here, we consider K γ : 2 K( x, y, γ) = g( a x, σ)g( a y, σ γ π 1/2 β ) exp( γ 2 a 2 )da. (4) Then, we have γ K = 2 ( ) 3/2 σσ β π 1/2 π(σ σ β γ 2 ) [ σ x σβ y 2 exp σ σ β γ 2 σ x 2 σ β y 2]. (41) σβ Since K( x, y, ) = and β 1, that is, σβ σ, we have Ĵ( x, y) 2 σ 1/2 ( σ 2 ) 3/2 π 1/2 π(2σ γ 2 ) [ σ x 2σ y 2 ] exp 2σ γ 2 dγ exp ( σ ( y 2 x y 2)). (42) Performing the above integration, we finally arrive at ( σ ) 2 1/2 2 1 Ĵ( x, y) π σ x 2 y ( exp σ ( (σ ) ) 1/2 2 x 2) Erf x 2 y.(43) 2 Thus, we can obtain E ij by substituting Eq.(43) into Eq.(37). Therefore, we can construct the linear equation with respect to Γ k (t). In the present vortex blob method, the global length of the vortex tubes is kept to be constant, so that we can choose the length between control points of Eq.(31) within a given allowance. 2 Numerical Results The present paper treats the transient flow around an elliptic vortex ring. The initial physical state is shown in Fig.1. In the present calculation, the vortex ring is constructed by M vortex tubes: x = r cos θ, y = br sin θ, z = r m sin δ k, (44) where r =1r m cos δ k ;(k =1, 2,.., M). The typical length and the typical circulation are taken as the initial total length of the vortex tubes constructing the ring and the initial circulation of the tubes, respectively. The total length of the vortex ring is given as l = l N o S = X i1 X i, (45) i=1 4

5 z r y x br Γ Figure 1: Physical model of an elliptic vortex ring 1 4 dl/dt b= t Figure 2: Global stretching. where l o is the initial total length of the vortex tubes and N is the total number of the vortex blobs, N = nm, where n is the number of vortex blobs of a vortex ring tube. The rate of the total length of the vortex ring is given by dl dt = l Ṡ o S 2 N ( X = i1 X i ) ( v i1 v i ) X i1 X. (46) i i=1 This rate, dl dt, is the global stretching rate of the vortex ring. Figure 2 shows the numerical result of the elliptic ring with ratio b =.1in the case where ν =1, n = 2, M =2,r m = ɛ/n, ɛ =1.5π/n, and δ = and δ 1 = π. Figure 3 shows the vortex blob distribution of the circular vortex ring. Within the present calculated time, the configuration of the ring is circular, which is moved downward. Figure 4(a) and (b) are the vortex blob distribution in the case of the elliptic ring with ratio b =.1. The vortex blobs near the largest curvature are moved rapidly at early stage of motion and the elliptic configuration is changed remarkably with lapse of time.

6 .3 Y.2 t=.2356 t=.2356 t= t= t= t= t= t= b= Z.3.2 b=.1 T Ω T Ω t X -1.5 X Figure 6: Total kinematic energy and enstrophy. Figure 4: Vortex blob distribution. ( 1 25 ) ω. e. ω 3 Conclusions 5-5 t= X,Y,Z X Y Z Figure 5: Global stretching. The present paper introduces an alternative vortex blob method derived from the Navier-Stokes equations normalized with the typical length of the vortex tubes and the circulation of the tubes. The numerical calculation is carried out for an elliptic vortex ring and the stretching and compression of vorticity field are simulated. This work was supported by Grant-in-Aid for Scientific Research (No ) from the Ministry of Education, Science and Culture of Japan. References [1] Moffatt, H.K. (2), The interaction of skewed vortex pairs; a model for blown-up of the Navier- Stokes equations, J. Fluid Mech., 49, p [2] Boratov, O.N. & Pelz, R.B. (1994), Direct numerical simulation of transition to turbulence from a high-symmetry initial condition, Phys. Fluids, 6, p [3] Gibbon, J.T., Galanti, B. & Kerr, R.M. (2), Stretching and compression of vorticity in the 3D Euler equations, Turbulence structure and vortex dynamics, ed. Hunt, J.C.R. & Vassilicos, J.C., Cambridge Univ. Press, p [4] Constantin, P. & Fefferman, C. (1993), Direction of vorticity and problem of global regularity for the Navier-Stokes equations, Indian Univ. Math. J., 42, p [5] Pelz, R.B. (1997), Locally self-similar, finite-time collapse in a high-symmetry vortex filament model, Physical Review E, 55, p [6] Kerr, R.M. (1997), Evidence for a singularity of the three-dimensional, incompressible Euler equations, Phys. Fluids A-5, p [7] Beale, J.T., Kato, T. & Majda, A. (1984), Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94, p [8] Ponce, G. (1985), Remaks on a paper by J.T. Beale, T. Kato, and A. Majda, Commun. Math. Phys., 98, p [9] Nakajima, T. & Kida, T. (1997), Threedimensional vortex methods derived from the Navier-Stokes equations, JSME International J., B, 4, p