Abstract
We present numerical simulations of the three-dimensional Galerkin truncated incompressible Euler equations that we integrate in time while regularizing the solution by applying a wavelet-based denoising. For this, at each time step, the vorticity field is decomposed into wavelet coefficients, which are split into strong and weak coefficients, before reconstructing them in physical space to obtain the corresponding coherent and incoherent vorticities. Both components are multiscale and orthogonal to each other. Then, by using the Biot-Savart kernel, one obtains the coherent and incoherent velocities. Advancing the coherent flow in time, while filtering out the noiselike incoherent flow, models turbulent dissipation and corresponds to an adaptive regularization. To track the flow evolution in both space and scale, a safety zone is added in wavelet coefficient space to the coherent wavelet coefficients. It is shown that the coherent flow indeed exhibits an intermittent nonlinear dynamics and a k^{-5/3} energy spectrum, where k is the wave number, characteristic of three-dimensional homogeneous isotropic turbulence. Finally, we compare the dynamical and statistical properties of Euler flows subjected to four kinds of regularizations: dissipative (Navier-Stokes), hyperdissipative (iterated Laplacian), dispersive (Euler-Voigt), and wavelet-based regularizations.
Highlights
A major challenge in computational fluid dynamics is the numerical simulation of high Reynolds number turbulence and in particular the numerical solution of the threedimensional (3D) incompressible Euler equations
The values of the corresponding Taylor-microscale λ are listed in Table II and we find very similar values for Coherent Vorticity Simulation (CVS), NS and hyperviscous regularization (HV)
Of the CVS, HV and Euler–Voigt regularization (EV) flows agree well with those observed for NS, which is not the case for what we find for EE
Summary
A major challenge in computational fluid dynamics is the numerical simulation of high Reynolds number turbulence and in particular the numerical solution of the threedimensional (3D) incompressible Euler equations. Since numerical schemes are limited to a finite number of modes, or grid points, the numerical integration of Euler equations requires to apply some kind of regularization to obtain a physically relevant solution for a given resolution. Such techniques should preserve the flow’s nonlinear dynamics and the solution’s properties. To obtain a physically relevant solution, typically hyperdissipative ( known as hyperviscous) regularizations are applied, which correspond to a Laplace operator which is iterated a certain number of times, as introduced in [3, 4] and applied in, e.g., [5, 6]. Setting the viscosity equal to zero yields the Euler–Voigt equations which formally correspond to adding the term α2∂t∆u to the momentum equation, where α > 0 is a length scale that represents the width of the spatial filter, see, e.g., the discussion in [12]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
