Non-linear response of kinetic energy in an inviscid Taylor-Green flow obtained by OpenFOAM-LES with respect to time increments

This study investigates the conservation error of kinetic energy with respect to time increment using OpenFOAM LES. The energy conservation error is verified using an inviscid Taylor-Green vortex and the effect on energy conservation is checked by varying the time increment by a factor of 30. Differences in the response of turbulent energy conservation between large and small time increments are also verified. Higher order accuracy for time increments is expected when the kinetic energy conservation accuracy is high, however the actual behaviour in OpenFOAM needs to be investigated: as the implicit time integration method is used in OpenFOAM, a non-linear response is expected. As a result, it is evident that kinetic energy is not fully conserved in the inviscid flow of OpenFOAM. The kinetic energy distribution is similar for different time increments, with a different relationship between the conservation error observed in the range of large and small time increments.


Introduction
Direct Numerical Simulation (DNS), which directly calculates the governing equations, and Large-eddy Simulation (LES), which models small-scale eddies, are used for the numerical analysis of turbulent flows.LES is widely used because such a setup is rarely possible.
Canonical flows have often been used to validate flow fields obtained by numerical analysis.Channel flow turbulence and circular pipe turbulence have often been used to validate analyses of turbulent flow fields [2].The Taylor-Green vortex is often used as a benchmark for decaying turbulence [3].Taylor's analytical solution is one of the analytical solutions for 2D flow fields and is widely used to validate fluid analysis (e.g., [4]).Two-dimensional flow fields given by random numbers are also widely used to validate fluid analysis.The energy of this flow is analytically obtained to be constant under inviscid conditions.A spatial discretisation method with good conservation properties has been validated in previous studies using a two-dimensional inviscid field [4].
Fluid flow analysis can be set to sufficiently small time step widths.In particular, it is usually set to satisfy the CFL condition.On the other hand, the kinetic energy conservation error also varies with this time increment.When the kinetic energy conservation accuracy is high, explicit time integration methods are used and are also expected to be high-order accurate with respect to the time increments.Whether this property is established in generic fluid analysis, such as OpenFOAM [6], needs to be thoroughly investigated.A property expected for high kinetic energy conservation accuracy, as OpenFOAM uses an implicit time integral method and this study sees the possibility that the response of the conservation error to time increments is not linear.The study finds that the response of the conservation error to time increments may not be linear.
In the inviscid flow field, the energy of the velocity fluctuation field is conserved analytically constant.This allows the energy conservation properties of this OpenFOAM-LES to be strictly investigated.In order to clarify the influence of time increments on the energy conservation, the time increment width is varied by a factor of 30.This study will also show whether there is a difference in the response of the turbulence energy conservation property to time increments between the cases with large and small time increments.

Numerical Methods
In this study, LES is also used to analyse the vortex motion: large scale vortex motion that can be captured by the mesh is calculated directly as in DNS, while small scale vortex motion that cannot becaptured by the mesh is modelled by SGS stresses, which generally act as turbulent viscous stresses.The analysis in this study is of transient, incompressible flows.In this analysis, the Crank-Nicolson method is used as the time integration method and the second order central difference method is used as the spatial difference method.The Crank-Nicolson and central difference methods are appropriate in this study.A smagorinsky model constant value is varied to approach the inviscid conditions.In this study, the magnitude of the eddy viscosity was set to Cs = 0.01  for Ck = 10 -4 and C = 10 -4 , Cs = 0.001 for Ck = 10 -5 and C = 10 -3 , and Cs = 0.0001 for Ck = 10 -6 and C = 10 -2 .The kinematic viscosity coefficient was set to 0 to produce a inviscid field.In order to investigate the dependence of the time step Δt in the inviscid analysis, five conditions were set with Δt = 0.015, 0.01, 0.005, 0.005, 0.0015 and 0.0005 and the time evolved up to t = 20.

Results
As shown in Figure 1, the turbulence energy decreases significantly in the initial time compared to the other model constant conditions.On the other hand, from about t = 10, the decrease of the turbulence energy becomes slower and the difference with the turbulence energy for the other two model constants becomes smaller.For the two conditions Cs = 0.001 and 0.0001 the turbulence energy values are identical at all times.The specific values have been varied to 0.0015 and 0.0005.This range is wide enough to examine the dependence on the time step width.This result shows that the turbulence energy can be successfully calculated in OpenFOAM even when the time step width is varied.In addition, the turbulence energy decreases with time for all model constant conditions, as was the case when Δt = 0.005 was set.Similarly, the decay is greater for Cs = 0.01 than for the other two conditions, and for Cs = 0.001 and 0.0001 the value of the turbulence energy is similar at all times.In general, the turbulence energy conservation error in the LES analysis tends to decrease with decreasing time step width.In contrast to the expected results, the present analysis shows little change in the distribution when the time step is changed.
Figure 3 shows the dependence of the turbulence energy on Δt in order to examine quantitatively the relationship between the turbulence energy and the time step width.The figure shows that the value of the turbulence energy is constant with respect to the change in time increments at t=10 and 20.As shown in Figure 3, there is a quantitative validation that there is little Δt dependence of the turbulence energy at any time point.These results show that the inviscid analysis in OpenFOAM does not save turbulence energy.As the time step width is reduced, the energy conservation error is expected to become very small, e.g. to the second or fourth power.However, in the OpenFOAM analysis, the error did not change proportionally to Δt even when the time step width was varied over a sufficient range.Therefore, the effect of the time step width on the energy conservation error is small.For all time intervals, the model approaches the inviscid state as the value of the model constant decreases.Comparing the results with respect to time interval, for Cs = 10 -2 the values on the vertical axis are quite close, but for smaller values of Cs = 10 -3 and 10 -4 the flow field approaches the inviscid state as the time increment becomes smaller.Figure 4 also shows the relationship between the model constant and the convergence to inviscidity for relatively small values of Δt = 0.005, 0.0015 and 0.0005.The green, blue and black lines show the results for Δt = 0.005, 0.0015 and 0.0005 respectively.

Discussion
An attempt is made to discuss the results of the present study in terms of time ticks.In the present study, the kinetic energy conservation error responds non-linearly to time increments.These previous studies have mainly used the partial step method.In contrast, this study uses the PISO method in OpenFOAM [10]; unlike the fractional step method, the PISO method is considered to implicitly determine the velocity field.There is a possibility that the implicit nature of the simultaneous equations approach may have caused this non-linearity with respect to time increments.
OpenFOAM has recently been widely used in both fundamental and applied research, often with OpenFOAM based LES studies.Similarly, LES is used in this study.The present study analyses unsteady decaying turbulence in a periodic computational domain without wall surfaces.This turbulence can be found in the outer region of the boundary layer in real flow fields.From this point of view, the universality of the results obtained in this study is expected to be sufficiently high.When setting up the computational conditions in a numerical analysis, the grid dependence of the results should be investigated.

Conclusion
This study investigates the nonlinear response of kinetic energy in inviscid flows to time increments in the context of OpenFOAM-LES.It is also realised that as the time step width is reduced, the energy conservation error decreases significantly, following a proportionality to the square, fourth power, etc.Based on these considerations, the study aims to investigate the correlation between kinetic energy and time step size in inviscid flows using OpenFOAM-LES.
The validation results confirm that the kinetic energy is not conserved in the inviscid flow in OpenFOAM at any time step width.When the kinetic energy distribution was derived by varying Δt from 0.005 to 0.0015 and 0.0005, the values decreased with time and were almost the same as before changing the time step width.The relationship between the model constant and the convergence to the inviscid state was then determined for large and small time intervals.In the large time interval range the model approaches the inviscid state as Δt is reduced, whereas in the small time interval range the relationship is not applicable.

Figure 1 .
Figure 1.Turbulent kinetic energy under the asymptote of model constant values to the inviscid condition.Here, the time increment is set to 0.005.

Figure 2 .
Figure 2. Effect of the value of the time increment on the time variation of the global turbulent kinetic energy.

Figure 3 .
Figure 3. Turbulent kinetic energy at two downstream time points compared with the value expected under inviscid condition.

Figure 4 .
Figure 4. Effect of the value of time increments on the time evolution of the global turbulent kinetic energy at the initial time.Here, the left and right figures show smaller and larger values of time increments, respectively.