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Reynolds transport theorem can be expressed as follows: [1] [2] [3] = + () in which n(x,t) is the outward-pointing unit normal vector, x is a point in the region and is the variable of integration, dV and dA are volume and surface elements at x, and v b (x,t) is the velocity of the area element (not the flow velocity).
Reynolds Stress equation models rely on the Reynolds Stress Transport equation. The equation for the transport of kinematic Reynolds stress = ′ ′ = / is [3] = + + + Rate of change of + Transport of by convection = Transport of by diffusion + Rate of production of + Transport of due to turbulent pressure-strain interactions + Transport of due to rotation + Rate of dissipation of .
One class of models, closely related to the concept of turbulent viscosity, are the k-epsilon turbulence models, based upon coupled transport equations for the turbulent energy density (similar to the turbulent pressure, i.e. the trace of the Reynolds stress) and the turbulent dissipation rate .
The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged [a] equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition , whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by Osborne Reynolds . [ 1 ]
The projection method was introduced to enable an approximate solution of the algebraic transport equation of the Reynolds-stresses. In contrast to the approach of the tensor basis is not inserted in the algebraic equation, instead the algebraic equation is projected. Therefore, the chosen basis tensors does not need to form a complete ...
As an example for pipe flows, with the Reynolds number based on the pipe diameter: =. Here l is the turbulence or eddy length scale, given below, and c μ is a k – ε model parameter whose value is typically given as 0.09;
The transport theorem (or transport equation, rate of change transport theorem or basic kinematic equation or Bour's formula, named after: Edmond Bour) is a vector equation that relates the time derivative of a Euclidean vector as evaluated in a non-rotating coordinate system to its time derivative in a rotating reference frame.
This equation is called the mass continuity equation, or simply the continuity equation. This equation generally accompanies the Navier–Stokes equation. In the case of an incompressible fluid, Dρ / Dt = 0 (the density following the path of a fluid element is constant) and the equation reduces to: