The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged
equations of motion for
fluid flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
. The idea behind the equations is
Reynolds decomposition
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity u the decomposition would be
u(x,y,z ...
, whereby an instantaneous quantity is decomposed into its time-averaged and fluctuating quantities, an idea first proposed by
Osborne Reynolds. The RANS equations are primarily used to describe
turbulent flow
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between t ...
s. These equations can be used with approximations based on knowledge of the properties of flow
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
to give approximate time-averaged solutions to the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
.
For a
stationary flow of an incompressible
Newtonian fluid, these equations can be written in
Einstein notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
as:
The left hand side of this equation represents the change in mean momentum of a 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 In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules. In fluid dynamics Reynolds operators are often encountere ...
.
Derivation of RANS equations
The basic tool required for the derivation of the RANS equations from the instantaneous
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
is the
Reynolds decomposition
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity u the decomposition would be
u(x,y,z ...
. 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 In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules. In fluid dynamics Reynolds operators are often encountere ...
, it has a set of properties. One of these properties is that the mean of the fluctuating quantity is equal to zero
. Thus,
where
is the position vector. Some authors prefer using
instead of
for the mean term (since an overbar is sometimes used to represent a vector). In this case, the fluctuating term
is represented instead by
. This is possible because the two terms do not appear simultaneously in the same equation. To avoid confusion, the notation
,
, and
will be used to represent the instantaneous, mean, and fluctuating terms, respectively.
The properties of
Reynolds operator In fluid dynamics and invariant theory, a Reynolds operator is a mathematical operator given by averaging something over a group action, satisfying a set of properties called Reynolds rules. In fluid dynamics Reynolds operators are often encountere ...
s 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):
where
is a vector representing external forces.
Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged,
to yield:
The momentum equation can also be written as,
On further manipulations this yields,
where,
is the mean rate of strain tensor.
Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving:
Equations of Reynolds stress
The time evolution equation of
Reynolds stress is given by:
This equation is very complicated. If
is traced,
turbulence kinetic energy is obtained.
The last term
is turbulent dissipation rate. All RANS models are based on the above equation.
Applications (RANS modelling)
A model for testing performance was determined that, when combined with the vortex lattice (VLM) or boundary element method (BEM), RANS was found useful for modelling the flow of water between two contrary rotation propellers, where VLM or BEM are applied to the propellers and RANS is used for the dynamically fluxing inter-propeller state.
Notes
See also
*
Favre averaging
References
{{DEFAULTSORT:Reynolds-averaged Navier-Stokes equations
Fluid dynamics
Turbulence
Turbulence models
Computational fluid dynamics