Reynolds averaged Navier-Stokes equation

Keywords Computational fluid dynamics (CFD) Reynolds averaging Closure of Reynolds-averaged Navier-Stokes equations Two-equation model Reynolds stress model 

Turbulent kinetic energy, m s Time-averaged pressure, kg m s  

Instantaneous velocity, ms Fluctuating velocity, ms Frictional velocity, ms  

Yu and X. Yuan, Introduction to Computational Mass Tranter, Heat and Mass Transfer, DOI 10.1007/978-3-642-5391 Springer-Verlag Berlin Heidelberg 2014 

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Some early spray models were based on the combination of a discrete droplet model with a multidimensional gas flow model for the prediction of turbulent combustion of liquid fuels in steady flow combustors and in direct injection engines. In an improved spray model,[438] the full Reynolds-averaged Navier-Stokes equations were... 

Consider a single-phase homogeneous stirred-tank reactor with a time-invariant velocity field U(x, y, z ) a single reaction of the form /) —> B. (This approach can be extended to the case of time-dependent velocity fields. If the flow in the tank is turbulent, then the velocity field is the solution of the Reynolds averaged Navier-Stokes equations). The tank is divided into a three-dimensional network of n spatially fixed volumetric elements, or n-interacting... 

Reynolds-averaged Navier-Stokes Equations (RANS)... 

RANS Reynolds-averaged Navier-Stokes equations... 

RANS, under which the Reynolds-averaged Navier Stokes equations are solved using some type of closure assumption to account for the Reynolds stress terms. RANS provides the values of the mean wind velocity and estimates of the turbulence statistics within the model domain. 

When dealing with turbulent flows all the relevant dimensionless numbers are evaluated with the available quantities. For example, in DNS, the fluid and particle instantaneous velocities will be employed, whereas in large-eddy simulation (LES) or in Reynolds-average Navier-Stokes-equations (RANS) simulations the filtered or Reynolds-average values will be used. 

RANS Reynolds-average Navier-Stokes equation... 

The Reynolds-averaged Navier-stokes equations (RANS) are obtained by averaging the instantaneous... 

Reynolds-averaged Navier-Stokes equations with K-e closure make it possible to estimate the characteristic mixing times of reactants of various scales. 

These equations have to be solved simultaneously with the Reynolds averaged Navier-Stokes equations and the Ergun-equation given in Section 11.7.5 to account for the radial void profile and generate both axial and radial flow velocity components, Ur and Uz. In these equations it is assumed that the heat transfer through the fluid and solid phase occur in parallel [de Wasch and Froment, 1971 Dixon and Cresswell, 1979 Dixon, 1985]. and AV can be calculated from the correlations of de Wasch and Froment [1972] and Zehner and Schliinder [1972], The internal diffusion limitations appear in those equations by means of the effectiveness factor r , which is obtained by numerical integration of the diffusion and reaction equations inside the particles, as discussed in Chapter 3. 

Use the Reynolds-averaged Navier-Stokes equations as described in Section 12.5. Account for the effects of turbulence via the turbulent viscosity and conductivity, as described in Sections 12.5.1 and 12.5.2, and using the standard k-s model and wall functions. The set of coupled partial differential equations can be solved with a CFD code. Verify the grid independency of the results. 

For the turbulent flow, according to the basic concept of Reynolds-Averaged Navier-Stokes Equation (RANS), any instantaneous quantity can be resolved into two parts the time-averaged quantity and the fluctuating quantity, the latter is oscillating positively and negatively around the former. Thus, m,- and p can be expressed as follows ... 

An advanced turbulence modeling based on hybridization of large eddy simulation (LES) and Reynolds-averaged Navier-Stokes equations (RANS) allowing one to use the best of both worlds ... 

Reynolds-averaged Navier-Stokes equations (RANS) could not predict with a sufficient (for the engineering purposes) accuracy such growth characteristics as the melt/crystal phase boundary shape or the oxygen concentration in the sihcon crystal that are critical for the crystal quahty. 

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