Multiscaled inviscid Taylor-Green vortex flow for examining energy conservation error in incompressible flows

This study investigates kinetic energy conservation errors in incompressible flows using multiplexed inviscid Taylor-Green vortices. Fourth-order precision Runge-Kutta methods, specifically five- and six-step methods, were employed and an inviscid two-dimensional periodic flow field was used as a test case. This method allows the energy conservation error to be suppressed to a very low level. Recent previous studies have dealt with turbulence generation using a multi-scale turbulence grid. In this study, the Taylor-Green vortex of the analytical solution is multiplexed to generate a flow field with a number of wavenumbers. Visualisation results of the initial field and the energy distribution over time were obtained. It was observed that the errors in the time evolution of the fluid energy were extremely small, and that the errors actually increased with time. However, the influence of the time step range was limited and the analytical and numerical solutions were in general agreement. As a result, it was confirmed that the inviscid Taylor-Green vortex is effective in examining the energy conservation error for incompressible flows.


Introduction
In the field of fluid mechanics, the advantages of numerical analysis are its availability as an alternative method when accurate theoretical resolution is not available, its lower cost compared to experimental methods, and the fact that its accuracy can be validated [1].Accurate validation of the kinetic energy of a fluid is a very important research topic.In particular, the study of its energy conservation error affects the accuracy of fluid analysis.A Taylor-Green vortex [2,3] is used in numerical analyses.In this study, the use of such a canonical flow is considered necessary to validate the fluid analysis.
When analysing incompressible flows, the energy conservation properties should be well held [4][5][6][7].Previous studies have used an inviscid periodic flow to investigate the energy conservation properties of a flow field (e.g., [7]).When analysing these flow fields, it is necessary to set an initial field for the flow.In the previous study, a random field is often set as the initial field.There are two types of random numbers: uniform and Gaussian.The Gaussian random number can be calculated from the uniform random numbers.A characteristic feature of random fields is that they contain waves of all wavelengths homogeneously.The power spectrum of the random field is therefore considered to be constant with respect to the wavenumber.An analytical solution field is also often used [8].The analytical solution is based on trigonometric functions.In contrast to the random field, the analytical solution field contains a single wavenumber.Note that these initial fields satisfy the continuity equation.Thus, within the two governing equations, the initial fields satisfy the law of mass conservation.The analytical solution field can satisfy the Navier-Stokes equation in addition to the continuity equation, however contains only a single wavenumber.The random fields, on the other hand, contain waves over many wavenumbers that do not satisfy the Navier-Stokes equations.This research attempts to solve a challenge faced by both.In recent years, turbulence generation using a multi-scale turbulence grid [9][10][11][12][13][14][15][16][17], which has a view of non-equilibrium turbulence [18,19], has been used in previous studies.This turbulence grid consists of multiplexing a single turbulence grid.Using this multiplexing technique, this study attempts to multiplex the initial field of the inviscid field.Specifically, by focusing on the Taylor-Green vortex, which is given as an analytical solution, and multiplexing this single flow, a flow field containing many wavenumbers satisfying the governing equations can be obtained.This multiplexed initial flow field is then used to investigate the energy conservation error.
The aim of this study is to investigate the kinetic energy conservation error of incompressible flows using multiplexed inviscid Taylor-Green vortex.Fourth-order Runge-Kutta methods [20] are used as the time integration method.The inviscid two-dimensional periodic flow field is used to investigate this point.This flow field allows the analytically expected conditions of the energy conservation law in the inviscid state to be used and its conservation properties to be rigorously investigated.Two fourth-order Runge-Kutta schemes are used in this study.Specifically, the five-step and six-step Runge-Kutta methods are used.By using these two methods, the kinetic energy conservation error can be reduced to a level sufficiently small to be negligible.Using these computational methods, the energy conservation properties of the method can be investigated by studying the time evolution of the multiplexed Taylor-Green vortex.

Numerical Methods
This study focuses on Taylor's analytical solution.In Taylor's analytical solution, the velocity and pressure fields, u, v, and p, are given as follows: u(x,y,t) = -2 cos(x) sin(y) exp(-2t/Re), v(x,y,t) = 2 sin(x) cos(y) exp(-2t/Re), and p(x,y,t) = (1/2) (cos(2x) + cos(2y) ) exp(-4t/Re). (1) Here, x, y, t are nondimensional coordinates and time, respectively, and Re in the equation is the Reynolds number.As shown in the above equation, the velocity and pressure fields can be expressed as both trigonometric and exponential functions.Here the exponential function in the above equation can be seen as determining the amplitude of the trigonometric function.This amplitude is constant in the inviscid state where the Re number goes to infinity.This Taylor analytic solution is often used in previous studies investigating numerical analysis.For example, previous studies have observed the property that in this Taylor analytic solution, the inertial term is independently balanced by the pressure term and the viscous term by the unsteady term.This property has been used to investigate the effect of numerical analysis errors on the flow field.
Previous studies [11,14,16] have used a multi-scale turbulence grid to generate decaying turbulence in wind tunnel experiments.This turbulence grid consists of a combination of single-scale turbulence grids.The present study applies this idea of multiplexing to generate the initial field.In the present study, the analytical Taylor solution presented in the paragraph above [8] is multiplexed.Specifically, the velocity field obtained by multiplexing this single-scale analytical solution, where the amplitudes of each scale are multiplexed with the same amplitude weights.The setting of these weights is identical to that of the multiplexed turbulence grids of previous studies.As seen in previous studies, the configuration of the wavenumber used for multiplexing is varied.This may be reflected in the multiplexed flow field of the present study.Specifically, the range of wavenumbers used for multiplexing can be varied using a parameter included in the above equation.In the present study, the value of this parameter is set to the maximum value that can be set on the computational grid.
In this study, a multi-scale inviscid Taylor-Green vortex was used to detect the energy conservation error in incompressible fluids.The governing equations for this study were the continuity equation and the Navier-Stokes equation of incompressible fluids, which were subjected to non-dimensionalisation. Prior to the analytical calculations, we set up the initial field and visualised the initial field energy distribution of the two-dimensional flow.We then varied the computational domain to observe its effect on the results.Then, we performed analytical calculations to follow the time evolution of the continuity equation error and visualised the energy distribution of the two-dimensional flow within the specified time step.We conducted analytical calculations to study the variation of the flow energy over time, focusing on the changes in energy when the computational domain is changed.Finally, based on the results obtained, we discussed the dependence of the fluid energy error on Δt.
In order to minimise discretization errors, this study adopted a high-precision discretization method in the time domain.Specifically, for the control equations (continuity equation and Navier-Stokes equation), we used two fourth-order accurate Runge-Kutta methods with 5 and 6 stages, respectively [20].These methods effectively suppressed energy conservation errors.Regarding the computational conditions of this study, we visualised the fluid energy K in the range of 0 to 5, with the computational domain set to 2π and 4π, and the grid resolution Nx = Ny = 32.The time points considered were t = 0, 10 and 30, with a time interval Δt = 0.03.For the non-viscous analysis we used time intervals of Δt = 0.03, 0.01, 0.003 and 0.001, with a grid resolution of Nx = Ny = 16 and computational domains of 2π and 4π.We performed a time evolution in the time range from t = 0 to 30.

Results and Discussion
Figure 1 (a)-(d) shows the visualisation results of the initial field with a computational domain of Lx = Ly = 2π.Under the initial conditions, a time interval of Δt = 0.03 was used and two different time integration methods were applied: The 4th order accurate 5-step Runge-Kutta method and the 6-step Runge-Kutta method.The observation range for the fluid energy K was set from 0 to 5. By observing the results in the figure, we can see that the energy distribution of the 2D fluid exhibits a symmetric state, and as the grid resolution increases, the fluid energy at symmetric positions also gradually increases.The upper part shows the results obtained using the 4th order accurate 5-step Runge-Kutta method, while the lower part shows the results obtained using the 4th order accurate 6-step Runge-Kutta method.As clearly shown in the figure, the change in time integration method did not affect the distribution of fluid energy in the initial field.
Figure 1 (e)-(h) shows the visualisation results of the initial field obtained by changing the computational domain from 2π to 4π.The initial conditions are almost identical to the previous 2π case, with a time interval of Δt = 0.03 and grid resolutions of Nx = Ny = 16, 32, 64 and 128.Similarly, we used the 5-stage and 6-stage Runge-Kutta methods with fourth-order accuracy, and set the observation range of the fluid energy K from 0 to 5. From the results shown in the figure, we observed that the energy distribution of the initial fluid field remains in a symmetric state, and the energy of the symmetric part also gradually increases as the grid resolution increases.In addition, we also observed that changing the time integration method did not change the fluid energy distribution in the initial field.Figure 3 shows the visualisation results of the two-dimensional distribution of fluid energy at different time intervals (t = 0, 10 and 30) during the analytical time evolution.The upper part of the figure uses the 4th order accurate 5 stage Runge-Kutta method, while the lower part uses the 4th order accurate 6 stage Runge-Kutta method.In terms of initial conditions, the computational domain is set to Lx = Ly = 2π, with a time interval of Δt = 0.03 and a grid resolution of Nx = Ny = 32.The range of the fluid energy K is set from 0 to 5. Observing the results in the figure, the two-dimensional distribution of the fluid energy can be seen to change with time.In particular, the visualisation results obtained from the two different time integration methods show significant similarity, indicating minimal differences in the energy distribution at the same positions.The upper part of Figure 4, (a) and (b) shows the evolution of the fluid energy with time in the computational domain of Lx = Ly = 2π.In (a), we used the 4th order accurate 5-stage Runge-Kutta method, while in (b) we used the 4th order accurate 6-stage Runge-Kutta method.Both figures have a grid resolution of Nx = Ny = 16.The red line represents the case with a time interval of Δt = 0.03, the orange line corresponds to Δt = 0.01, the green line represents Δt = 0.003 and the blue line represents Δt = 0.001.As the analytical calculation results show that K values tend to approach 1, the horizontal axis of the figure is set to |K − 1| to better observe the variations.From the figure, it can be seen that as the time interval decreases, the magnitude of the change in fluid energy also decreases.However, even in the case of the red line with the largest variation, the change in energy is in the range of 10 -8 , indicating that the differences in fluid energy over time are very small and can be neglected.
The lower part of Figure 4 shows the evolution of fluid energy with time in the computational domain of Lx = Ly = 4π.In (c), we used the 4th-order accurate 5-stage Runge-Kutta method, while in (d), we used the 4th-order 6-stage Runge-Kutta method.Both figures have a grid resolution of Nx = Ny = 16.The red line represents the case with a time interval of Δt = 0.03, the orange line corresponds to Δt = 0.01, the green line represents Δt = 0.003, and the blue line represents Δt = 0.001.Similarly, we set the horizontal axis to |K − 1| to better observe the variations.From the figure, it can be observed that the change in computational domain leads to periodic variations of fluid energy over time, and as the time interval increases, the magnitude of the variations also increases.However, even in the case of the largest variation, the change in fluid energy remains within the range of 10 -9 , indicating that the differences in fluid energy with a computational domain of 4π are still very small and can be neglected.
Figure 5 shows the investigation of the fluid energy error dependence on Δt.In (a) and (b), the computational domain is Lx = Ly = 2π, while in (c) and (d), Lx = Ly = 4π.These four plots have been generated from the data obtained by the least squares method based on Figure 4. From the figure, it can be seen that in the two plots with a computational domain of Lx = Ly = 2π the analytical and numerical solutions are in excellent agreement.In the case of the computational domain Lx = Ly = 4π and using the 4th order accurate 5-stage Runge-Kutta method, the analytical and numerical solutions also show good agreement.For the plots using the 6-stage Runge-Kutta method, although there is a relatively larger error in the numerical solution compared to the analytical solution at Δt = 0.001, the error is still in the range of 10 -15 to 10 -16 , indicating a reasonably good agreement.In summary, based on the results in Figure 5, it can be concluded that the dependence of the fluid energy on time t is generally in line with expectations.

Conclusion
The aim of this study is to analyse and validate the kinetic energy conservation error of an incompressible fluid using the multi-scale inviscid Taylor-Green vortex flow.To achieve this, this study first set up the initial field and visualise the two-dimensional fluid energy distribution at the initial stage and after time evolution to better observe its changes.For the time evolution of the fluid energy, the present study adopted high-order accurate time integration methods, specifically the 4th-order accurate 5-stage and 6-stage Runge-Kutta methods.By using these methods, we were able to effectively suppress errors in the kinetic energy conservation.In addition, we varied the computational domain and performed time evolution of both the continuity equation and the fluid energy over the time range from t = 0 to 30.We also investigated the dependence of the fluid energy error on the time interval Δt.Through the comprehensive analysis of the time evolution characteristics of the Taylor-Green vortex flow using the aforementioned research methods, we validate the effectiveness of the adopted approach in the examination for the conserving energy.
Through the above validation methods, the visualised results of the initial field and the timeevolved two-dimensional fluid kinetic energy distribution were successfully obtained.In the time evolution graph of the continuity equation error, we observed that different time intervals did not have much effect on the continuity equation error values, but the values gradually increased with time.As for the time evolution of the fluid kinetic energy, we found that the error values for different time intervals were extremely small, therefore they could be neglected.In addition, the error values showed an increasing trend with time.In the graph of the dependence of the fluid kinetic energy error on the time interval Δt, the analytical and numerical solutions were in good agreement, and these results were in line with our expectations.In summary, using the above validation methods, we have successfully verified the effectiveness of the inviscid Taylor-Green vortex flow in conserving kinetic energy for incompressible fluids.

Figure 1 .
Figure 1.Dependence of the multiplexed initial flow field on the computational domain and spatial resolution.

Figure 2 .
Figure 2. Time evolution of the error in the continuity equations shown for the two Runge-Kutta methods.
(a) N x = 16 for L x = 2p (b) N x = 32 for L x = 2p (h) N x = 128 for L x = 4p (g) N x = 64 for L x = 4p (f) N x = 32 for L x = 4p (e) N x = 16 for L x = 4p (d) N x = 128 for L x = 2p (c) N x = 64 for L x =

Figure 3 .
Figure 3.Time evolution of the flow field from the initial field presented for the two Runge-Kutta methods.

( a )Figure 2
Figure 2(a)  and (b) shows the evolution of the fluid continuity equation error with respect to time t.In this figure, the computational domain is set to Lx = Ly = 2π and the grid resolution is Nx = Ny = 16.We carried out the time evolution under four different time intervals: Δt = 0.03, 0.01, 0.003 and 0.001.The red line represents the case with a time interval of Δt = 0.03, the orange line corresponds to Δt = 0.01, the green line represents Δt = 0.003, and the blue line represents Δt = 0.001.In(a)  and (b), we have used 4th order accurate 5-stage and 6-stage Runge-Kutta methods, respectively, for time integration.From the figure, the continuity equation error under different time intervals gradually increases with time.In(a)  and (b), the continuity equation errors with time intervals of Δt = 0.03 and Δt = 0.01 increase more rapidly around t = 20.At the same time, the errors at different time intervals do not show significant differences overall and remain in a similar range.Figure3shows the visualisation results of the two-dimensional distribution of fluid energy at different time intervals (t = 0, 10 and 30) during the analytical time evolution.The upper part of the figure uses the 4th order accurate 5 stage Runge-Kutta method, while the lower part uses the 4th order accurate 6 stage Runge-Kutta method.In terms of initial conditions, the computational domain is set to Lx = Ly = 2π, with a time interval of Δt = 0.03 and a grid resolution of Nx = Ny = 32.The range of the fluid energy K is set from 0 to 5. Observing the results in the figure, the two-dimensional distribution of the fluid energy can be seen to change with time.In particular, the visualisation results obtained from the two different time integration methods show significant similarity, indicating minimal differences in the energy distribution at the same positions.

Figure 4 .
Figure 4. Absolute value of the deviation of the global kinetic energy from the initial value shown for the two Runge-Kutta methods.

Figure 5 .
Figure 5. Dependence of the absolute value of the deviation of the global kinetic energy on the rate of change of time with respect to time increments.