Reynolds equations

The Reynolds equations or Reynolds-averaged Navier-Stokes equations (after Osborne Reynolds ) are a simplification of the Navier-Stokes equations that are used in numerical fluid mechanics to approximate turbulent flows . Because of the English term Reynolds-Averaged Navier-Stokes equation s, they are also referred to as RANS equations .

Basic principle

Since the Navier-Stokes equations cannot be solved numerically with reasonable effort for turbulent flows with technically relevant Reynolds numbers (see direct numerical simulation ), the variables are divided into a mean value and a fluctuation value. Here the mean value is chosen so that the fluctuation has the mean value zero.

One possibility is Reynolds averaging , which averages over a short period of time, or ensemble averaging for unsteady flows. As a result, additional terms appear in the equations that have to be described using a turbulence model .

The RANS equations contain partial time derivatives, but since only the mean values ​​of the quantities (pressure, speed) appear in the equations, the solution should be stationary. After a sufficient length, the simulation should converge towards the steady state. Only the steady state at the end of a sufficiently long simulation (averaging period) is meaningful. Of course, the turbulence itself is not stationary. Solving the RANS equations only provides information about the mean values ​​of the quantities. Alternatively, you can also solve the stationary equations directly using iterative methods.

There are also unsteady RANS models. Because of the English term unsteady , these are often referred to as URANS. The basic assumption in these models is that the time scale on which turbulence occurs is significantly smaller than the time scale with which the mean values ​​of the variables change. Only if this assumption is correct will URANS models provide meaningful solutions. With URANS, the averaging period must therefore be greater than the time scale of the turbulence, but less than the time scale of the mean values.

Incompressible problems

In the incompressible Navier-Stokes equations

• the instantaneous values ​​of the speed components and the pressure are replaced by the respective sum of the mean value and statistical fluctuation:

${\displaystyle u_{i}={\bar {u_{i}}}+u_{i}'}$

${\displaystyle p={\bar {p}}+p'}$

• Density and viscosity fluctuations neglected:

${\displaystyle {\frac {\partial \rho }{\partial t}}=0\Leftrightarrow \rho =konst.}$

${\displaystyle {\frac {\partial \eta }{\partial t}}=0\Leftrightarrow \eta =konst.}$

Then the momentum equation gives the Navier-Stokes equations

${\displaystyle \rho {\frac {\partial u_{i}}{\partial t}}+\rho \left({\frac {\partial u_{i}u_{j}}{\partial x_{j}}}\right)=f_{i}-{\frac {\partial p}{\partial x_{i}}}+\eta {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)}$

noted here in Einstein's sum convention , the incompressible Reynolds-averaged momentum equation:

${\displaystyle \rho {\frac {\partial {\bar {u}}_{i}}{\partial t}}+\rho \left({\frac {\partial {\bar {u}}_{i}{\bar {u}}_{j}}{\partial x_{j}}}\right)={\bar {f}}_{i}-{\frac {\partial {\bar {p}}}{\partial x_{i}}}+{\frac {\partial }{\partial x_{j}}}\left(\eta \left({\frac {\partial {\bar {u}}_{i}}{\partial x_{j}}}+{\frac {\partial {\bar {u}}_{j}}{\partial x_{i}}}\right)-\rho \underbrace {\overline {u_{i}'u_{j}'}} _{RST}\right)}$

The resulting from the averaging new term consequence of the non-negligible speed correlation: . This tensor is called the Reynolds Stress Tensor (RST) . ${\displaystyle {\overline {u_{i}'u_{j}'}}}$

Compressible problems

In the compressible Navier-Stokes equations, the so-called Favre averaging is also used in order to avoid products of mean values. In addition to the Reynolds stress tensor, the turbulent kinetic energy is another unknown term.

literature

• Joel H. Ferziger, Milovan Perić: Numerical Fluid Mechanics . 1st edition. Springer, Berlin 2008, ISBN 978-3-540-67586-0 .

• Hermann Schlichting , Klaus Gersten: boundary layer theory . 10th edition. Springer, Berlin 2006, ISBN 978-3-540-23004-5 .