Turbulence closure problem

In numerical fluid mechanics , one speaks of the closure problem of turbulence when the general Navier-Stokes equations are to be solved following time averaging. It turns out that in this case an additional unknown term occurs, which means that the system can no longer be solved without additional information.

root cause

This problem can be mathematically justified by the non-linearity of the Navier-Stokes equations.

Additional term

The momentum for a continuous Newtonian fluid takes the following form in the x-direction:

${\displaystyle \rho {\frac {Du}{Dt}}=\rho g_{x}-{\frac {\partial p}{\partial x}}+\eta \left[{\frac {{\partial }^{2}u}{\partial x^{2}}}+{\frac {{\partial }^{2}u}{\partial y^{2}}}+{\frac {{\partial }^{2}u}{\partial z^{2}}}\right]}$.

In general, no general solutions for the Navier-Stokes equations are known so far according to the Millennium problem . Often only the time averages of the flow variables are of interest. It is therefore a common attempt to average unknown quantities using Reynolds averaging . This results in the following form of the momentum equation:

${\displaystyle \rho {\frac {D{\bar {u}}}{Dt}}=\rho g_{x}-{\frac {\partial {\bar {p}}}{\partial x}}+\eta \left[{\frac {{\partial }^{2}{\bar {u}}}{\partial x^{2}}}+{\frac {{\partial }^{2}{\bar {u}}}{\partial y^{2}}}+{\frac {{\partial }^{2}{\bar {u}}}{\partial z^{2}}}\right]-\rho \left[{\frac {\partial {\bar {u'}}^{2}}{\partial x}}+{\frac {\partial {\bar {u'v'}}}{\partial y}}+{\frac {\partial {\bar {u'w'}}}{\partial z}}\right]}$.

The new term added at the end is also known as the Reynolds stress tensor and now contains new unknowns, which can be interpreted as the averaging over time of the instantaneous values.

See also

• Turbulence model

Individual evidence

1. ↑ Heinz Herwig: Fluid Mechanics. ISBN 978-3-540-32441-6 .
2. ↑ Bruno Kistner: Modeling and numerical simulation of the wake structure of turbo machines using the example of an axial turbine stage. dissertation