The mathematical analysis of the coupling of a turbulent kinetic energy equation to the navier-stokes equation with an eddy viscosity

1997 ◽  
Vol 28 (2) ◽  
pp. 393-417 ◽  
Author(s):  
Roger Lewandowski
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Stokes Equation ◽  
Mathematical Analysis ◽  
Eddy Viscosity ◽  
Energy Equation ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Turbulent Kinetic Energy Equation

Scaling the Turbulent Kinetic Energy Equation

2015 ◽  
pp. 239-242
Author(s):  
Jordi Vila-Guerau de Arellano ◽  
Chiel C. van Heerwaarden ◽  
Bart J. H. van Stratum ◽  
Kees van den Dries
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Energy Equation ◽  
Turbulent Kinetic Energy Equation

scholarly journals Scaling and modelling of turbulence in variable property channel flows

2017 ◽  
Vol 823 ◽  
Author(s):  
Rene Pecnik ◽  
Ashish Patel
Keyword(s):  
Mach Number ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Energy Equation ◽  
Stokes Equations ◽  
Channel Flows ◽  
Alternative Formulation ◽  
Variable Property ◽  
Budget Analysis ◽  
Turbulent Kinetic Energy Equation

We derive an alternative formulation of the turbulent kinetic energy equation for flows with strong near-wall density and viscosity gradients. The derivation is based on a scaling transformation of the Navier–Stokes equations using semi-local quantities. A budget analysis of the semi-locally scaled turbulent kinetic energy equation shows that, for several variable property low-Mach-number channel flows, the ‘leading-order effect’ of variable density and viscosity on turbulence in wall bounded flows can effectively be characterized by the semi-local Reynolds number. Moreover, if a turbulence model is solved in its semi-locally scaled form, we show that an excellent agreement with direct numerical simulations is obtained for both low- and high-Mach-number flows, where conventional modelling approaches fail.

scholarly journals On the over-production of turbulence beneath surface waves in Reynolds-averaged Navier–Stokes models

2018 ◽  
Vol 853 ◽  
pp. 419-460 ◽  
Author(s):  
Bjarke Eltard Larsen ◽  
David R. Fuhrman
Keyword(s):  
Kinetic Energy ◽  
Surface Waves ◽  
Turbulent Kinetic Energy ◽  
Exponential Growth ◽  
Potential Flow ◽  
Eddy Viscosity ◽  
Surf Zone ◽  
Breaking Waves ◽  
Navier Stokes ◽  
Closure Models

In previous computational fluid dynamics studies of breaking waves, there has been a marked tendency to severely over-estimate turbulence levels, both pre- and post-breaking. This problem is most likely related to the previously described (though not sufficiently well recognized) conditional instability of widely used turbulence models when used to close Reynolds-averaged Navier–Stokes (RANS) equations in regions of nearly potential flow with finite strain, resulting in exponential growth of the turbulent kinetic energy and eddy viscosity. While this problem has been known for nearly 20 years, a suitable and fundamentally sound solution has yet to be developed. In this work it is demonstrated that virtually all commonly used two-equation turbulence closure models are unconditionally, rather than conditionally, unstable in such regions. A new formulation of the $k$–$\unicode[STIX]{x1D714}$ closure is developed which elegantly stabilizes the model in nearly potential flow regions, with modifications remaining passive in sheared flow regions, thus solving this long-standing problem. Computed results involving non-breaking waves demonstrate that the new stabilized closure enables nearly constant form wave propagation over long durations, avoiding the exponential growth of the eddy viscosity and inevitable wave decay exhibited by standard closures. Additional applications on breaking waves demonstrate that the new stabilized model avoids the unphysical generation of pre-breaking turbulence which widely plagues existing closures. The new model is demonstrated to be capable of predicting accurate pre- and post-breaking surface elevations, as well as turbulence and undertow velocity profiles, especially during transition from pre-breaking to the outer surf zone. Results in the inner surf zone are similar to standard closures. Similar methods for formally stabilizing other widely used closure models ($k$–$\unicode[STIX]{x1D714}$ and $k$–$\unicode[STIX]{x1D700}$ variants) are likewise developed, and it is recommended that these be utilized in future RANS simulations of surface waves. (In the above $k$ is the turbulent kinetic energy density, $\unicode[STIX]{x1D714}$ is the specific dissipation rate, and $\unicode[STIX]{x1D700}$ is the dissipation.)

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