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Structure-function scaling of bounded two-dimensional turbulence

Profile image of G.H. KeetelsG.H. Keetels

2011, Physical Review E

https://doi.org/10.1103/PHYSREVE.84.026310
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10 pages

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Abstract

Statistical properties of forced two-dimensional turbulence generated in two different flow domains are investigated by numerical simulations. The considered geometries are the square domain and the periodic channel domain, both bounded by lateral no-slip sidewalls. The focus is on the direct enstrophy cascade range and how the statistical properties change in the presence of no-slip boundaries. The scaling exponents of the velocity and the vorticity structure functions are compared to the classical Kraichnan-Batchelor-Leith (KBL) theory, which assumes isotropy, homogeneity, and self-similarity for turbulence scales between the forcing and dissipation scale. Our investigation reveals that in the interior of the flow domain, turbulence can be considered statistically isotropic and locally homogeneous for the enstrophy cascade range, but it is weakly intermittent. However, the scaling of the vorticity structure function indicates a steeper slope for the energy spectrum than the KBL theory predicts. Near the walls the turbulence is strongly anisotropic at all flow scales.

Figures (13)
FIG. 1. (Color online) Total kinetic energy (top) and enstrophy (bottom) vs dimensionless time (where t = 1 corresponds to a typical eddy turnover time of the larger vortices) of a forced turbulence simulation in a square bounded domain. The integral-scale Reynolds number is approximately 25 000. In Fig. 1 the evolution of the total kinetic energy E = sto |u|? dx dy and enstrophy Z = to w’ dx dy are given for the simulation in the square domain. It can be deduced that the flow slowly converges toward a statistically steady state. The maximum value of the (time-dependent) Reynolds number Re(f) is approximately 25 000. A similar observation
FIG. 1. (Color online) Total kinetic energy (top) and enstrophy (bottom) vs dimensionless time (where t = 1 corresponds to a typical eddy turnover time of the larger vortices) of a forced turbulence simulation in a square bounded domain. The integral-scale Reynolds number is approximately 25 000. In Fig. 1 the evolution of the total kinetic energy E = sto |u|? dx dy and enstrophy Z = to w’ dx dy are given for the simulation in the square domain. It can be deduced that the flow slowly converges toward a statistically steady state. The maximum value of the (time-dependent) Reynolds number Re(f) is approximately 25 000. A similar observation
TABLE I. Numerical parameters for the simulations on a square domain and periodic channel domain with aspect ratio a = 4. Here, v is the kinematic viscosity, N and M are the number of grid points in the x and y (wall-normal for the periodic channel) direction, respectively, Ar the time step, A, the forcing amplitude, t. the forcing correlation time, ky the forcing wave number, and /; the typical forcing length scale. correlation time of t! he forcing. The viscosity and the forcing amplitude are chosen in such a way that the integral-scale Reynolds number o forcing is limited to t relates to a forcing f both simulations is about 20000. The he wave numbers 77 < |ky| < 97, which length scale of about one-eighth of the domain width. The forcing length scale must be large to obtain a long direct enstro phy cascade range, but has to be small enough to produce an isotropic forcing field. For the bounded square domain we ensure that Fourier modes with k, = 0 or ky = Oare not present in the forcing, such that the forcing does not exert a net torque on the fluid.
TABLE I. Numerical parameters for the simulations on a square domain and periodic channel domain with aspect ratio a = 4. Here, v is the kinematic viscosity, N and M are the number of grid points in the x and y (wall-normal for the periodic channel) direction, respectively, Ar the time step, A, the forcing amplitude, t. the forcing correlation time, ky the forcing wave number, and /; the typical forcing length scale. correlation time of t! he forcing. The viscosity and the forcing amplitude are chosen in such a way that the integral-scale Reynolds number o forcing is limited to t relates to a forcing f both simulations is about 20000. The he wave numbers 77 < |ky| < 97, which length scale of about one-eighth of the domain width. The forcing length scale must be large to obtain a long direct enstro phy cascade range, but has to be small enough to produce an isotropic forcing field. For the bounded square domain we ensure that Fourier modes with k, = 0 or ky = Oare not present in the forcing, such that the forcing does not exert a net torque on the fluid.
FIG. 3. Snapshots of the vorticity and the stream function in the periodic channel domain. The vorticity ranges from w = —120 to @ = +120 with 30 gray levels. The stream function is shown with an interval of 0.1. The black isolines correspond with positive values of w, and the gray isolines represent negative values of w. The spatial structure of the flow can be analyzed by considering the statistical properties of velocity differences du(x,r,t) = u(x +r,t) — u(x,t). For statistically stationary flows it is allowed to drop the time dependence; we are thus left with du(x,r,t) = du(x,r). The velocity difference can be decomposed in components parallel and perpendicular
FIG. 3. Snapshots of the vorticity and the stream function in the periodic channel domain. The vorticity ranges from w = —120 to @ = +120 with 30 gray levels. The stream function is shown with an interval of 0.1. The black isolines correspond with positive values of w, and the gray isolines represent negative values of w. The spatial structure of the flow can be analyzed by considering the statistical properties of velocity differences du(x,r,t) = u(x +r,t) — u(x,t). For statistically stationary flows it is allowed to drop the time dependence; we are thus left with du(x,r,t) = du(x,r). The velocity difference can be decomposed in components parallel and perpendicular
FIG. 2. Snapshots of the vorticity distribution and the stream function of forced 2D turbulence in the square domain taken at t = 90 (top row) andt = 110 (bottom row). The vorticity ranges from @ = —120 to w = +120 with 30 gray levels. The stream function is shown with an interval of 0.1. The solid isolines correspond with positive values of y, and the dashed isolines represent negative values of y.
FIG. 2. Snapshots of the vorticity distribution and the stream function of forced 2D turbulence in the square domain taken at t = 90 (top row) andt = 110 (bottom row). The vorticity ranges from @ = —120 to w = +120 with 30 gray levels. The stream function is shown with an interval of 0.1. The solid isolines correspond with positive values of y, and the dashed isolines represent negative values of y.
FIG. 4. (Color online) Transverse second-order structure function of the velocity, Sir) (solid line). The arrow denotes the length scale of the forcing. The dashed line corresponds with the isotropic estimate for the transverse second-order structure function based on the longitudinal structure function. The structure functions are computed in a square box with a width 1.0 in the center of the flow domain (the square computational domain has a width 2). Figure 4 shows the transverse second-order structure function of the velocity obtained from a forced 2D turbulence simulation in a square box (with width 2). The second-order structure function is either directly evaluated in a square box in the center of the domain or indirectly by applying Eq. (13). Various box sizes with a width ranging from 0.6 to 1.2 have
FIG. 4. (Color online) Transverse second-order structure function of the velocity, Sir) (solid line). The arrow denotes the length scale of the forcing. The dashed line corresponds with the isotropic estimate for the transverse second-order structure function based on the longitudinal structure function. The structure functions are computed in a square box with a width 1.0 in the center of the flow domain (the square computational domain has a width 2). Figure 4 shows the transverse second-order structure function of the velocity obtained from a forced 2D turbulence simulation in a square box (with width 2). The second-order structure function is either directly evaluated in a square box in the center of the domain or indirectly by applying Eq. (13). Various box sizes with a width ranging from 0.6 to 1.2 have
FIG. 6. Probability density function P(éq) of the vorticity incre- ments dw for separations r SJ, (r = 1/256, 1/128, 1/64, 1/32, 1/16, 1/8, and 1/4) in the interior of the square bounded domain. A Gaussian distribution is given in gray for comparison.
FIG. 6. Probability density function P(éq) of the vorticity incre- ments dw for separations r SJ, (r = 1/256, 1/128, 1/64, 1/32, 1/16, 1/8, and 1/4) in the interior of the square bounded domain. A Gaussian distribution is given in gray for comparison.
FIG. 9. ESS scaling exponent for y = 0 (black) and y = 0.99 (gray) for the structure functions of order p = 2, 4, 6, and 8. The dotted lines correspond with the p/3. All results are obtained from simulations of forced 2D turbulence in the periodic channel domain.
FIG. 9. ESS scaling exponent for y = 0 (black) and y = 0.99 (gray) for the structure functions of order p = 2, 4, 6, and 8. The dotted lines correspond with the p/3. All results are obtained from simulations of forced 2D turbulence in the periodic channel domain.
FIG. 10. (Color online) The second-order vorticity structure function S},, in the interior of the square domain. The second-order structure function of the vorticity, S,o(r) = ((6@)), is displayed in Fig. 10. This figure reveals that for the smallest scales ((Sw)*) « r!> and that it flattens for separations near the forcing scale /,. The structure function S2(r) = ((6@)*) provides more information about the scaling exponents of the energy spectrum, since the slope of the second-order structure function of the vorticity is well separated from the upper bound. From Fig. 10 we can deduce that the corresponding energy spectrum has a slope of k~? near the forcing scale and becomes steeper for higher wave numbers, viz., E(k) « k~*°, which is significantly steeper than the inertial range prediction in the enstrophy cascade range by
FIG. 10. (Color online) The second-order vorticity structure function S},, in the interior of the square domain. The second-order structure function of the vorticity, S,o(r) = ((6@)), is displayed in Fig. 10. This figure reveals that for the smallest scales ((Sw)*) « r!> and that it flattens for separations near the forcing scale /,. The structure function S2(r) = ((6@)*) provides more information about the scaling exponents of the energy spectrum, since the slope of the second-order structure function of the vorticity is well separated from the upper bound. From Fig. 10 we can deduce that the corresponding energy spectrum has a slope of k~? near the forcing scale and becomes steeper for higher wave numbers, viz., E(k) « k~*°, which is significantly steeper than the inertial range prediction in the enstrophy cascade range by
FIG. 11. (Color online) The mixed third-order structure function /§ separation r. with x denoting the ensemble-averaged enstrophy dissipation rate. It can be seen as an equivalent of the well-known 4/5 law of 3D turbulence. The mixed third-order structure function, given by (15), can be interpreted as a measure of the nonlinear transfer rate of enstrophy. The role of the longitudinal velocity increments can be related to the alignment between the rate of strain tensor and the vorticity gradient vector, which describes the nonlinear amplification of the vorticity gradients. In the
FIG. 11. (Color online) The mixed third-order structure function /§ separation r. with x denoting the ensemble-averaged enstrophy dissipation rate. It can be seen as an equivalent of the well-known 4/5 law of 3D turbulence. The mixed third-order structure function, given by (15), can be interpreted as a measure of the nonlinear transfer rate of enstrophy. The role of the longitudinal velocity increments can be related to the alignment between the rate of strain tensor and the vorticity gradient vector, which describes the nonlinear amplification of the vorticity gradients. In the
FIG. 12. (Color online) The mixed third-order structure function vs third-order structure function of the vorticity. Dashed line represents a slope of +1.
FIG. 12. (Color online) The mixed third-order structure function vs third-order structure function of the vorticity. Dashed line represents a slope of +1.

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