On the existence of global-in-time weak solutions and scaling laws for Kolmogorov's two-equation model of turbulence

This paper is concerned with Kolmogorov's two-equation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators

On the existence of global-in-time weak solutions and scaling laws for Kolmogorovs two-equation model of turbulence

This paper is concerned with Kolmogorov's two-equation model for free turbulence in space dimension 3, involving the mean velocity u, the pressure p, an average frequency omega, and a mean turbulent kinetic energy k. We first discuss scaling laws for a slightly more general two-equation models to highlight the special role of the model devised by Kolmogorov in 1942. The main part of the paper consists in proving the existence of weak solutions of Kolmogorov's two-equation model under space-periodic boundary conditions in cubes with positive side length l. To this end, we provide new a priori estimates and invoke existence result for pseudo-monotone operators

Global-in-time existence of weak solutions to Kolmogorov's two-equation model of turbulence

We consider Kolmogorov's model for the turbulent motion of an incompressible fluid in ℝ3. This model consists in a Navier-Stokes type system for the mean flow u and two further partial differential equations: an equation for the frequency ω and for the kinetic energy k each. We investigate this system of partial differential equations in a cylinder Ω x ]0,T[ (Ω ⊂ ℝ3 cube, 0 < T < +∞) under spatial periodic boundary conditions on ∂Ω x ]0,T[ and initial conditions in Ω x {0}. We present an existence result for a weak solution {u, ω, k} to the problem under consideration, with ω, k obeying the inequalities and

On the existence of global‐in‐time weak solutions and scaling laws for Kolmogorov's two‐equation model for turbulence

This paper is concerned with Kolmogorov's two-equation model for turbulence in R 3 $\mathbb {R}^3$ involving the mean velocity u, the pressure p, an average frequency ω > 0 $\omega >0$ , and a mean turbulent kinetic energy k. We consider the system with space-periodic boundary conditions in a cube Ω = ( ] 0 , a [ ) 3 $\Omega =({]0,a[}){}^3$ , which is a good choice for studying the decay of free turbulent motion sufficiently far away from boundaries. In particular, this choice is compatible with the rich set of similarity transformations for turbulence. The main part of this work consists in proving existence of global weak solutions of this model. For this we approximate the system by adding a suitable regularizing r-Laplacian and invoke existence result for evolutionary equations with pseudo-monotone operators. An important point constitutes the derivation of pointwise a priori estimates for ω (upper and lower) and k (only lower) that are independent of the box size a, thus allow us to control the parabolicity of the diffusion operators.Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659Peer Reviewe

Detecting instruction effects. Deciding between covariance analytical and change-score approach

The article focuses on estimating effects in nonrandomized studies with two outcome measurement occasions and one predictor variable. Given such a design, the analysis approach can be to include the measurement at the previous time point as a predictor in the regression model (ANCOVA), or to predict the change-score of the outcome variable (CHANGE). Researchers demonstrated that both approaches can result in different conclusions regarding the reported effect. Current recommendations on when to apply which approach are, in part, contradictory. In addition, they lack direct reference to the educational and instructional research contexts, since they do not consider latent variable models in which variables are measured without measurement error. This contribution assists researchers in making decisions regarding their analysis model. Using an underlying hypothetical data-generating model, we identify for which kind of data-generating scenario (i.e., under which assumptions) the defined true effect equals the estimated regression coefficients of the ANCOVA and the CHANGE approach. We give empirical examples from instructional research and discuss which approach is more appropriate, respectively. (DIPF/Orig.