DYNAMICS OF CYLINDRICAL TURBULENT SPOT IN A LONGITUDINAL SHEAR FLOWOF A PASSIVE STRATIFIED FLUID
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Abstract (English):
A numerical model is constructed and dynamics of the cylindrical localized area of turbulent disturbances (turbulent spot) in a longitudinal horizontally homogeneous shear flow of a passive stratified fluid is studied. The results of calculations show a significant turbulent energy generation by shear flo . The problem of flow similarity with respect to the shear Froude number for sufficiently large values of this parameter is considered.

Keywords:
mathematical model of a turbulent spot in a shear flow, e⁓ε turbulent model, numerical simulation
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INTRODUCTION Evolution of localized regions of turbulent fluid (spots of turbulent) has a decisive influence on formation of a fine microstructure of hydrophysical fields in the ocean [1]. A fairly detailed analysis of studies in the field of the turbulent spot dynamics can be found in [2-4]. The problem of evolution of the turbulent mixing zone in momentumless turbulent wake in a transverse shear flow of a homogeneous fluid is considered in [4]. It is shown that a shear flow may lead to significant deformation of a turbulent area and generate a substantial turbulent energy prolonging the wake lifetime. Flow similarity with respect to the shear Froude number, equal to the ratio of the product of characteristic velocity of turbulent perturbations in the initial time and the characteristic time, caused toby shear flo , to the initial size of the turbulent region is shown D. Dynamics of turbulent spot in a transverse linear shear flow of linearly stratified fluid is studied in [5]. It was found that a shear flow in comparison with the case of a homogeneous fluid causes a further significant distortion of the pattern of internal waves generated by the turbulent spot. This paper describes a plane non-stationary problem of dynamics of turbulent disturbances in the longitudinal shear flow of passively stratified fluid. At the initial moment, a turbulent area represents an infinitely long cylinder directed along the axis x. Plane section of a cylinder (y,z) is shown in Fig.1; the figure also shows a linear shear flow directed along the axis of the cylinder. A numerical flow model based on the two-parameter semi-empirical turbulent model is built. The results of calculations illustrate the dynamics of turbulent fluid area, accompanied by a significant turbulent energy caused by the effect of a shear flo . The problem of a flow similarity with respect to the shear Froude number is discussed. Fig. 1. Flow diagram at the initial moment. Please cite this article in press as: Chernykh G.G., Fomina A.V. Dynamics of cylindrical turbulent spot in a longitudinal shear flow of a passive stratified fluid. Science Evolution,, 2016, vol. 1, no. 2, pp. 102-107. doi:10.21603/2500-1418-2016-1-2-102-107. Copyright © 2016, KemSU. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix, transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license. This article is published with open access at http://science-evolution.ru/ PROBLEM STATEMENT To describe the flo , a mathematical model based on the eε⁓model of turbulent is used: In the calculations the boundary conditions corresponding to r   , were shifted to borders of a sufficiently large rectangle whose dimensions were chosen based on the results of numerical experiments. U   K U   K U , (1) For reasons of symmetry the solution is sought in the first t y u y z u z quadrant of the plane ( y , z ). The boundary conditions on e   K e   K e  P  , (2) the coordinate axes were taken as follows: t y e y z e z U   e   2  1     0, y  0, z  0;           2 y y y y y t y K y z K z c1 e P c 2 e , (3) e   2         d K U  1  z  z  z  0, z  0, y  0. 1  K 1  K 1  s  , (4) t y  2  y  z  2  z dz z   2 Variables of the problem can be nondimensionalized with the use of characteristic scales of length D, t  y K1 y  z K1 z  velocity U 0  e0 and density aD0 . With this (5) nondimensionalization, a characteristic flow time is    2  2K      2       cT 2  . a quantity Tc  D e0. In such a case, the form of  y   z   e equations (1)-(5) does not change; in the future, if necessary, dimensionless quantities will be designated s s s s 0 0 In equations (1)-(5), the value U  U (t, y, z) as*. The problem has another characteristic time is a longitudinal horizontal velocity component;  dU dz 1   e D1 . By analogy with the 1 is an averaged density defect: 1   s ,    1 az  , a  const  0 ; 2 is a dispersion of well-known density Froude number; let us introduce a shear Froude number F  e T D  1 (see also s s s s s 0 s 0 s the density fluctuations; e is an turbulent energy,  is a dissipation rate; P is a turbulent energy generation by gradients of averaged motion; is a sign of averaging; terms containing factors in the form of molecular [4]). In this case, U *  F . is valid. In equations (2), (3), as mentioned above, the quantity P - is an energy, generated by turbulent gradients of the averaged flow viscosity and diffusion coefficie ts were rejected on the assumption of their smallness. P   u ' v ' U  y u ' w ' U  z The turbulent viscosity and diffusion coefficients are defined as follows e2 K e2 e2  U 2      Ku   2 2  y  U 2  z z               Ku  Ke  c , K  e , K  c , K1  c1  .   K  Ud   Ud  dU s  2   , U  U U . Values c  0.136,  1.3, c  0.208, c  0.087, u  u   z dz  d s   1      cT  1.25 , c1  1.44 , c2  1.92 are known empirical * * constants [6, 7]. On the assumption of the concept of dimensionless For t  t0 , the following initial conditions were quantities Us  U s e0 , z  z*D, we obtain specified dU  e0 dU *  e0 1  dU  1 * * e(t , y, z)   (r) , (t , y, z)   (r) , s s s . 0 1 0 2 dz D dz* D F dz* F U (t , y, z)  z e D  U (z) ,   2 s s  0 , 0 0 s 1 Thus, dimensionless equations (2), (3) (as well as   y   ,   z   , t  t0 . in (1)) have a quantity 1 Fs , characterizing presence Here,  const, D - is a diameter of turbulent of a background linear shear longitudinal flow in the problem. mixing zone in the initial moment e0  e(t0 , 0,0) . 2 2 2 2 2 2 Functions 1(r) , 2 (r) , r  y  z are finite bell- shaped functions, consistent with the experimental data of Lin and Pao (Lin, Pao) on decay of momentumless turbulent wake in a homogeneous fluid [8]. For r 2  y 2  z 2   the following conditions FINITE-DIFFERENCE SOLUTION ALGORITHM Finite-difference solution algorithm is based on the consistent time integration of the differential equation system (1) - (5) in each layer. Let us give an example of 1 1 e     2 established.  0 , U  Us (z) , t  t0 were the finite-di ference analogue of equation (2) based on an implicit splitting scheme [7, 9] en1/ 2  en  K n  en1/ 2  en1/ 2    K n  en1/ 2  en1/ 2  i, j i, j  e i1/ 2, j i1, j i, j e i1/ 2, j i, j i1, j  P   , y y t h2 i, j i, j en1  en1/ 2  K n  en1  en1    K n  en1  en1  i, j i, j  e i, j 1/ 2 i, j 1 i, j e i, j 1/ 2 i, j i, j 1 , z z t h2 e e i1/ 2, j i1/ 2, j  K n   0.5 K n   K n  ,  K n   0.5 K n    K n  , e e i1, j i1, j e e i, j i, j e e i, j1/ 2 i, j1/ 2 e e i, j1 i, j1 e e i, j i, j i, j i, j 0 0 en1  et  n 1t,i  h , j  h , i  0,..., N ; j  0,..., N ; n  0,..., N . y z y z t t, hy , hz  time and space variable pitches of a uniform difference grid. Finite difference equations are solved by the sweep method. The numerical model was tested by comparing the results of calculations on a sequence of grids with By analogy with [4] in the case of a linear shear flow  U  z , t ≫ 1, F ≫1 solution should be sought s 0 D s in the form t Tc  t Ts Ts Tc  t Ts  Fs  e  t y z  numerical calculations of the problem of a wake behind F 1.54  G  s e s e U 2 , , T DF 0.23 DF 0.23  , self-propelled and towed bodies in a homogeneous liquid 0  s s s  in a one-dimensional formulation and by comparing the calculation results in detail with the experimental data D F 2.54  G  t 3 s   3 s   U T y z  , , , , , , DF 0.23 DF 0.23  [7, 10]. Calculations of this work were carried out on a 0  s s s  uniform difference grid with the following parameters. 1 F 0.23  G  t , y , z  , t*  0.1, h*  h*  0.02. Grid area dimensions aD s 1  T DF 0.23 DF 0.23  0  y* Y *, y z 0  z* Z *, Y * Z * 8. 4-fold increase of the 0 '2  s s s   t y z  (8) 2 2 time pitch while halving hy , hz led to deviations in values F 0.46  G  , ,  , of grid solutions, not exceeding 1% of the uniform norm. aD 2 s  T DF 0.23 DF 0.23  0 s s s RESULTS OF CALCULATION em F 1.54    t  , OF THE TURBULENT SPOT DYNAMICS U 2 s e  T  IN THE LONGITUDINAL LINEAR SHEAR FLOW 0  s  L  t  L  t  Before presenting the results of calculations, y F 0.23     , z F 0.23    . D s 1L T D s 2 L T following [4, 11], let us give the considerations concerning similarity with respect to the shear Froude  s   s  Flow generated at the evolution of a localized U T 1  dU 1 turbulent area in a longitudinal shear flow is illustrated number Fs  0 s  , Ts   s  . It is well known by change in the characteristic dimensions L , L , D  [2, 3, 11] in  dz  shear Us (z)  0 at determined from ratios y z e(t, L , 0)  e(t,0 that the absence of y , 0) 2 , a sufficiently long decay times, a flow generated at the evolution of a plane localized area of turbulent disturbances in homogeneous and passive stratified fluid becomes self-similar. In this regard, the following representations are valid: e  em f1 r L,  m f2 r L, e(t, 0, Lz )  e(t, 0, 2. . Fig. 2 (a), (b) show graphs of functions Ly (t), Lz (t) for various Us (z) . It can be seen that up to values t Tc  300 , dimensions are the same for all the above shears Us (z) and a shear-free flo . For large time values the flow is characterized by more intense expansion of a turbulent spot, caused by generation of the 1  1 m H1  y L , z L, (6) turbulent energy in the spot due to a shear flo . Given considerations of similarity with respect to the shear '2  '2 m H 2  y L , z L, Froude number Fs , Fig. 2 (c), (d) shows characteristic “universal” curves F 0.23L t T , F 0.23L t T  . A where r  y2  z 2, L - is still the characteristic size s y s z s of the turbulent spot. In accordance with the results of numerical experiments and analytical studies [7], the following representations are valid for values m m 1 m m m 1 m e ,  ,  , '2 m certain discrepancy between the results of calculations in the initial interval of dynamics of a turbulent spot is caused by a flow non-self-similarity at the initial time inter in shearless case occurs more later). Flow is also characterized by behavior of e  t  t 1.54  D  t  t 2.54 characteristic scale of a turbulent energy e(t, 0,0) m  B  0  , m  B  0  , depending on time and shear flow Us (z) (Fig. 3 (a)). U U U U   T T   2 2 2 1 3 0 c 0  Tc  It can be seen that up to the time t Tc  400 , values 1  t  t 0.23 e(t, 0,0) are practically the same (maximum deviation 1 m 1 m   B  0  , 3 3 corresponds to  0.004 and is approximately 7%); for aD0  Tc  (7) large time values the flow is characterized by generation 1  t  t 0.46 of the turbulent energy due to gradients of a longitudinal 0 0 '2 2 2  B4   , velocity component U (t, y, z) . The total turbulent m m aD0   Tc    L  t  t 0.23 energy Et (t)  e dy dz behaves similarly (Fig. 3 (b)).  B  0  . 0 0 c c D 5  T  The results of the corresponding simulation for the shear Science Evolution, 2016, vol. 1, no. 2 Froude number Fs are shown in Fig. 3 (c), (d). A certain discrepancy between the results of calculations in the initial stage of decay is also caused by a flow non-self- similarity in this time interval. (b) s y s s y s s z s s z s (c) (d) y y z z Fig. 2. Graphs of functions L (t) - (a); L (t) - (b); F 0.23L t T  - (c) ; F 0.23L t T  - (d) for various values Us (z) . (a) (b) (c) (d) Fig. 3. Graphs of functions e (t)  e(t, 0,0) - (a); E (t) - (b); F 1.54e t T  - (c); F 1.08E t T  - (d) for various values m of Us (z) . s m s s t s To illustrate similaritywithrespecttotheshear Froude number, Fig. 4 (a), (b), (c), (d), (e), (f) show also graphs of a turbulent energy, a defect density and a dispersion of density fluctuations along the vertical axis in the initial coordinates and flow area “similarity” coordinates for of a cylindrical localized area of turbulent disturbances in a longitudinal horizontally homogeneous shear flow of a passive stratified fluid is carried out. The results of numerical experiment show significant generation of turbulent energy caused by a shear flo . The latter t Ts  2 . It can be seen that with Fs  1000 graphs of may extend the turbulent spot life. The problem of flow functions do not differ much from each other, which also indicates the approximate similarity of flow at large Fs . Appropriate hydrodynamic fields are more fully illustrated by contour lines (Figure 5), corresponding to t Ts  2 . They were obtained for   0.001 . The main results are as follows. The mathematical model is built and numerical simulation of evolution similarity with respect to the shear Froude number for sufficiently la ge values of this parameter is considered. Numerical experiments confirm a flow similarity at large values of the Froude number. The study was supported by a grant of Leading Scientific Schools NSH - 7214.2016.9 and Russian Fundamental Research Fund (project 17-01-00332). (b) (c) (d) (е) (f) Fig. 4. Graphs of turbulent energy functions e , a density defect '2 and a dispersion of density fluctuations '2 along the vertical axis in the initial coordinates - (a), (c), (e) and flow area in “similarity” coordinates - (b), (d), (f) for t Ts2. Fig. 5. Universal contour lines of turbulent energy, density defect, dispersion of density fluctuations and dissipation rate, t Ts  2 .
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