“Reynolds-averaged Navier–Stokes equations”的意思、由来-开放百科全书

The Reynolds-averaged Navier–Stokes equations (or RANS equations) are time-averaged^[1]

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.^[2] The RANS equations are primarily used to describe turbulent flows. These equations can be used with approximations based on knowledge of the properties of flow turbulence to give approximate time-averaged solutions to the Navier–Stokes equations.

For a stationary, incompressible Newtonian fluid, these equations can be written in Einstein notation in Cartesian coordinates as:

The left hand side of this equation represents the change in mean momentum of fluid element owing to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent stress

owing to the fluctuating velocity field, generally referred to as the Reynolds stress. This nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving, and has led to the creation of many different turbulence models. The time-average operator

is a Reynolds operator.

Derivation of RANS equations

The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to separation of the flow variable (like velocity

) into the mean (time-averaged) component (

) and the fluctuating component (

). Because the mean operator is a Reynolds operator, it has a set of properties. One of these properties is that the mean of the fluctuating quantity is equal to zero

. Thus,

The properties of Reynolds operators are useful in the derivation of the RANS equations. Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid):

Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged, ^[4] to yield:

Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving:

Notes

1. ^The true time average (

) of a variable (

) is defined by:

For this to be a well-defined term, the limit (

) must be independent of the initial condition at

. In the case of a chaotic dynamical system, which the equations under turbulent conditions are thought to be, this means that the system can have only one strange attractor, a result that has yet to be proved for the Navier-Stokes equations. However, assuming the limit exists (which it does for any bounded system, which fluid velocities certainly are), there exists some

such that integration from

is arbitrarily close to the average. This means that given transient data over a sufficiently large time, the average can be numerically computed within some small error. However, there is no analytical way to obtain an upper bound on

.
2. ^Reynolds, Osborne, 1895: "On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion." Philosophical Transactions of the Royal Society of London. A, v. 186, pp. 123-164. Available online from [https://www.jstor.org/stable/90643 JSTOR].
3. ^{{cite book|last=Tennekes|first=H.|title=A first course in turbulence|year=1992|publisher=MIT Press|location=Cambridge, Mass. [u.a.]|isbn=978-0-262-20019-6|edition=14. print.|author2=Lumley, J. L. }}
4. ^Splitting each instantaneous quantity into its averaged and fluctuating components yields,:

Time-averaging these equations yields,:

Note that the nonlinear terms (like

) can be simplified to,

5. ^This follows from the mass conservation equation which gives,: