NUMERICAL MODELING OF TURBULENT WAKES
G.G. Chernykh*, A.G. Demenkov**, B.B. Ilyushin**, V.A. Kostomakha***,
N.P. Moshkin*, O.F. Voropayeva*
* Institute of Computational Technologies SB RAS, 630090, Novosibirsk, Russia
** Kutateladze Institute of Thermophysics SB RAS, 630090, Novosibirsk, Russia
*** Lavrentiev Institute of Hydrodynamics SB RAS, 630090, Novosibirsk, Russia
Аннотация Выполнено численное моделирование динамики плоских и осесимметричных турбулентных следов с варьируемыми значениями суммарного избыточного импульса в однородной жидкости. Построена математическая модель динамики закрученных турбулентных следов с варьируемыми значениями суммарного избыточного импульса и момента количества движения. Результаты расчетов хорошо согласуются с известными экспериментальными данными. Выполнен численный анализ автомодельности течения в дальнем закрученном следе за самодвижущимся телом.
Проведен численный анализ эволюции безымпульсных турбулентных следов и генерируемых ими внутренних волн за телами вращения в устойчиво стратифицированной среде, основанный на иерархии современных полуэмпирических моделей турбулентности. Результаты расчетов достаточно хорошо согласуются с известными экспериментальными данными. Разработаны численные модели анизотропного вырождения турбулентности в дальнем следе в линейно стратифицированной среде.
Построены численные модели динамики дальнего турбулентного следа и генерируемых им внутренних волн за буксируемым телом в устойчиво стратифицированной среде. Показано, что турбулентный след за буксируемым телом генерирует внутренние волны существенно большей амплитуды, чем безымпульсный турбулентный след.
The numerical analysis of dynamics of swirling turbulent wakes with varied values of total excess momentum and angular momentum was carried out [1-5]. To describe the flow the following system of averaged equations for the motion and continuity in the thin shear layer approach is used:
The closed system of equations is written for two different formulations of the closure relations. Mathematical Model 1 includes the following equations for determination of Reynolds stresses in addition to Eqs. (1)-(3):
To determine the values of the rate of dissipation we make use of the relevant differential equation
The value P is the energy production caused by averaged motion
The turbulent shear stress is derived from the nonequilibrium algebraic approximations introduced by Rodi :
In Model 2 we determine the Reynolds shear stresses by the Rodi’s nonequilibrium relations:
The normal Reynolds stresses are obtained from the transport differential equations (4)-(6) and from relationship (10). The quantities are well known empirical constants. Their values are taken to be equal 0.22, 0.17, 0.93, 0.6, 2.2, 1.45, 1.92.
The results of the numerical experiments made on the basis of Models 1,2 are given below. The experimental data were obtained in . The experiments were carried out at the Reynolds number where is the coefficient of kinematic viscosity.
Fig. 1a. Experimental and calculated distributions of the
defect of the longitudinal
Fig. 1b. Experimental and calculated distributions of the tangential velocity component .
In Fig. 1a the calculated distributions of the defect of the longitudinal velocity component are compared with the experimental data. Both models show a rather good fit of the calculation results to the experimental data in the near-axis zone of the wake. Agreement between the calculation results and the experimental data in the peripheral region of the wake is somewhat worse. This is because of lack of second-order models which do not allow for the intermittency of the flow in the external regions of the wake. Noteworthy is a substantial difference in the behavior of the calculated profilesfor small values of . This is because equation (1) in Model 2 is of diffusion type.
In Fig. 1b the calculated distributions of the tangential velocity component are compared with the experimental data. Both models describe satisfactorily the experimental data. As in the case of comparison between the calculated and experimentally measured distributions , the best agreement is obtained when Model 2 is used. We can also see that Model 1 yields larger (in magnitude) values of W than those obtained experimentally. Model 2 yields smaller (in magnitude) values of the peripheral velocity, which is also due to the diffusion character of equation (2) in this case.
The transverse distributions of the turbulent fluctuation intensities of the velocity components , , are shown in Fig. 2. The calculation results are shown by lines, the experimental data by points.
Fig. 2. The transverse distributions of the turbulent fluctuation intensities of the velocity components.
Agreement between the calculation results obtained by Models 1, 2 and the experimental data can be considered to be satisfactory. The calculation results obtained by both models essentially coincide. The axial values of the intensities , , are sufficiently close. However, the distributions of these values in a vicinity of the wake axis differ rather essentially, which is due to a difference in the source terms in equations (4)-(6).
In Fig. 3 the calculated tangential stress are compared with the experimental data. Agreement between the calculation results and the experimental data is sufficiently good.
Fig. 3. Experimental and calculated distributions of the tangential stress .
Fig. Fig. 4. The behavior of the characteristic scales of turbulence depending on the distance from the body.
Figure 4 demonstrates a change in the characteristic scales of turbulence depending on the distance from a body. Here is the axial value of the defect of the longitudinal velocity component, is the maximum value of the tangential velocity component at the given cross-section of the wake, is the value of turbulent energy along the wake axis, and is the characteristic wake width specified by the relation . At large distances from the body, the dependence of all the scale functions on is a power dependence (solid thin straight lines in Fig. 4); within the framework of the mathematical model used, this is one necessary indication that the self-similarity of the turbulent motion in the wake is reached.
Fig. 5. The self-similar profiles of the defect of the longitudinal velocity component, the tangential velocity component and the turbulence energy.
The other feature of self-similarity is the affine similarity of the transverse distributions of various turbulence characteristics in the wake, which are normalized to the corresponding scales. Figure 5 presents self-similar profiles of the defect of the longitudinal velocity component, the tangential velocity component, and turbulent energy, which are examples of the realization of this flow regime in the wake. Figure 5a corresponds to Model 1, Figure 5b to Model 2. As is seen, just as the asymptotic decay is attained (see Fig. 4), the similarity of the transverse distributions is attained in the wake for x/D > 1000. The flow at these distances can be considered to be self-similar (in the framework of the adopted mathematical models).
Fig. 6. The behavior of the ratio of the turbulent energy production to the rate of dissipation.
Fig. 7. The decay of axial value of the turbulent Reynolds number.
In the course of calculations, the ratio of the turbulence-energy generation owing to the averaged-motion gradients to the dissipation rate versus was analyzed. We obtained < 0.3 for (see Fig. 6). It is known that the classical -model of turbulence is suitable only for flows characterized by; therefore, here we use a more complicated mathematical model of turbulence.
Despite the closeness of the indicated degeneration laws at all the distances studied, the flow in the wake was a developed turbulent flow. This is supported by the value of the turbulent Reynolds number calculated according to the Taylor microscale . It follows from the calculations that the axial values of in the range of ; here we have for large x/D (See Fig. 7) . Simplified models of far turbulent wake have been constructed .
The numerical simulation of plane and axisymmetric wakes with varied values of total excess momentum has been performed. The mathematical model of the flow is based on the unclosed system of the motion and continuity equations written in terms of the thin shear layer approach. The closure relation for this system is formulated with the help of nonequilibrium algebraic Reynolds stress model. The numerical solution of the problem is performed with the use of the finite difference algorithm realized on moving grids. The results of numerical experiments are in a good agreement with known experimental data .
^ 2. TURBULENT WAKES IN STRATIFIED MEDIA
A numerical analysis of the evolution of momentumless turbulent wakes in continuously stratified fluid has been carried out by using a hierarchy of semi-empirical turbulence models. In order to describe the flow in a far turbulent wake of a body of revolution in a stratified medium the three-dimensional parabolized system of the averaged equations for the motion, continuity and incompressibility in the Oberbeck-Boussinesq approach was used. This system of equations is nonclosed. The most complicated model implies the use of differential equation to determine the triple correlation for vertical velocity fluctuations and modified algebraic relations for other triple velocity correlations . In the present work we consider the hierarchy of five closed mathematical flow models. In the Model 1 the quantities of the components of the Reynolds stresses tensor are approximated by the "isotropic" relationships :
Model 2 is based on locally equilibrium approximations for the determination of . In Model 3 the quantities ( ) are computed by solving the differential equations :
We used the simplified relations
The turbulent fluxes and the dispersion of the density fluctuations in Models 1-3 are replaced by locally equilibrium approximations:
The difference between Model 4 and Model 3 consists in using the nonequilibrium algebraic relationships  for the determination of the quantities .
In order to determine the rate of dissipation the differential transport equation was used
In Models 1 – 4 we determined the diffusion terms in (13) by relationships:
And finally Model 5 differs from the Model 3 by using the simplified variant of algebraic model of triple velocity field correlations  instead of simplest Daly and Harlow approximations . This model takes into account the anisotropic damping effect by a stable stratification on the third-order moment values:
In addition, Model 5 includes modified equation of the rate of turbulent kinetic energy dissipation , which based on relationships:
We take . All these values are empirical constants.
A comparison with the experimental data of Lin and Pao  has been carried out. It has been shown  that the Model 3 is the best of Models 1-4. Models 3, 4 produce close results. But Model 3 is characterized by a significantly large deviations in description of the evolution of wake height. In the present paper the dynamics of a momentumless turbulent wake in a linearly stratified medium are illustrated by data that presented in Figs. 8-13; (- the unperturbed fluid velocity, T – the Vaisala-Brunt period, D- the body diameter). Figure 8 illustrates the behavior of the horizontal wake size In Fig. 9 we compare the vertical wake size computed by using Models 3, 5 with the results of the laboratory measurements of Lin and Pao . The quantities and have been determine from the relationships
Model 5 describes well the behavior of the wake vertical extent. There are considerable deviations while using Model 3. One succeeds in obtaining a satisfactory description of the anisotropic decay of wake characteristics only by using the mathematical Model 5 (see Figs. 8, 9). In Fig. 10 the axial values of the intensities of turbulent fluctuations of the longitudinal velocity component are compared with experimental data of Lin and Pao. The computed temporal variation of the intensity of turbulent fluctuations (Fig. 10) weakly depends on the applied mathematical model. Figure 11 shows the decay of the axial values of the intencities of turbulent fluctuations of the vertical velocity component. Comparisons with measurements demonstrate satisfactory agreement.
Simplified models of a far turbulent wake and internal waves generated by its collapse are constructed [15, 16].
A numerical model of a far turbulent wake behind the towed body has been presented. The modified model of turbulence is used the "isotropic" relationships for the Reynolds stresses . The turbulent fluxes and the dispersion of the density fluctuations are replaced by locally equilibrium approximations (see Rodi ).
To determine the values of the turbulence energy , the rate of dissipation and the shear Reynolds stress we make use of the differential equations . We determine the coefficients of turbulent viscosity by simplified relation (12).
The choice of this model of turbulence is due to the following reasons: it is close to the standard model of turbulence and we can take into account the anisotropy of the turbulence characteristics in the wakes in stratified fluid. A weak dependence of the internal waves characteristics on the applied mathematical model has been shown as well (see, for example  ).
Fig.12. Density profiles for the time ; The solid and dashed lines correspond to the momentumless wake and a drag wake, respectively (linearly stratified fluid)
Fig. 13. Time dependence of the (1-4) total turbulence energy and (5-8) total internal-wave energy. Curves 1, 3, 5 and 7 correspond to linear stratification and curves 2, 4, 6 and 8 correspond to pycnocline. The solid and dashed lines are for the self-propelled body and towed body, respectively.
The pattern of the internal waves generated by turbulent wakes in a linearly stratified fluid depicted in Figure 12 where the dynamics of curves is given for the moments of time
The calculation results show that the turbulent wake behind a towed body generates the waves of essentially greater amplitude then behind the self-propelled body. This phenomenon is illustrated in the Figure 13 where
are dimensionless values of the total turbulence energy and of the internal waves energy in cross section of wake, respectively. Some physical explanation of this effect has been made .
The work was supported by the Russian Foundation for the Basic Research (01-01-00783, 04-01-00209).
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