Reynolds stresses
Encyclopedia
In fluid dynamics
, the Reynolds stress (or, the Reynolds stress tensor
) is the stress tensor in a fluid
obtained from the averaging operation over the Navier-Stokes equations to account for turbulent
fluctuations in fluid momentum
.
, the flow velocities
are split into a mean part and a fluctuating part using Reynolds decomposition
:
with
being the flow velocity vector having components 
in the 
coordinate direction (with 
denoting the components of the coordinate vector 
). The mean velocities 
are determined by either time averaging
, spatial averaging or ensemble averaging
, depending on the flow under study. Further
denotes the fluctuating (turbulence) part of the velocity.
The components τij of the Reynolds stress tensor are defined as:
with ρ the fluid density
, taken to be non-fluctuating for this homogeneous fluid.
Another – often used – definition, for constant density, of the Reynolds stress components is:
which has the dimensions of velocity squared, instead of stress.
:
Given the fluid velocity
as a function of position and time, write the average fluid velocity as 
, and the velocity fluctuation is 
. Then 
.
The conventional ensemble
rules of averaging are that
One splits the Euler equations
or the Navier-Stokes equations
into an average and a fluctuating part. One finds that upon averaging the fluid equations, a stress on the right hand side appears of the form
. This is the Reynolds stress, conventionally written 
:
The divergence
of this stress is the force density on the fluid due to the turbulent fluctuations.
, Newtonian fluid
, the continuity and momentum
equations—the incompressible Navier–Stokes equations—can be written as
and
where
is the Lagrangian derivative or the substantial derivative,
Defining the flow variables above with a time-averaged component and a fluctuating component, the continuity and momentum equations become
and
Examining one of the terms on the left hand side of the momentum equation, it is seen that
where the last term on the right hand side vanishes as a result of the continuity equation. Accordingly, the momentum equation becomes
Now the continuity and momentum equations will be averaged. The ensemble rules of averaging need to be employed, keeping in mind that the average of products of fluctuating quantities will not in general vanish. After averaging, the continuity and momentum equations become
and
Using the chain rule on one of the terms of the left hand side, it is revealed that
where the last term on the right hand side vanishes as a result of the averaged continuity equation. The averaged momentum equation now becomes, after a rearrangement:
where the Reynolds stresses,
, are collected with the viscous normal and shear stress terms, 
.
. A transport equation for the Reynolds stress may be found by taking the outer product
of the fluid equations for the fluctuating velocity, with itself.
One finds that the transport equation for the Reynolds stress includes terms with higher-order correlations (specifically, the triple correlation
) as well as correlations with pressure fluctuations (i.e. momentum carried by sound waves). A common solution is to model these terms by simple ad-hoc prescriptions.
It should also be noted that the theory of the Reynolds stress is quite analogous to the kinetic theory of gases, and indeed the stress tensor in a fluid at a point may be seen to be the ensemble average of the stress due to the thermal velocities of molecules at a given point in a fluid. Thus, by analogy, the Reynolds stress is sometimes thought of as consisting of an isotropic pressure part, termed the turbulent pressure, and an off-diagonal part which may be thought of as an effective turbulent viscosity.
In fact, while much effort has been expended in developing good models for the Reynolds stress in a fluid, as a practical matter, when solving the fluid equations using computational fluid dynamics, often the simplest turbulence models prove the most effective. One class of models, closely related to the concept of turbulent viscosity, is the so-called
model(s), 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 
.
Typically, the average is formally defined as an ensemble average as in statistical ensemble theory. However, as a practical matter, the average may also be thought of as a spatial average over some lengthscale, or a temporal average. Note that, while formally the connection between such averages is justified in equilibrium statistical mechanics by the ergodic theorem, the statistical mechanics of hydrodynamic turbulence is currently far from understood. In fact, the Reynolds stress at any given point in a turbulent fluid is somewhat subject to interpretation, depending upon how one defines the average.
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, the Reynolds stress (or, the Reynolds stress tensor
Stress tensor
Stress tensor may refer to:* Stress , in classical physics* Stress-energy tensor, in relativistic theories* Maxwell stress tensor, in electromagnetism...
) is the stress tensor in a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
obtained from the averaging operation over the Navier-Stokes equations to account for turbulent
Turbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
fluctuations in fluid momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
.
Definition
For a homogeneous fluid and an incompressible flowIncompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid...
, the flow velocities
Flow velocity
In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...
are split into a mean part and a fluctuating part using Reynolds decomposition
Reynolds decomposition
In fluid dynamics and the theory of turbulence, Reynolds decomposition is a mathematicaltechnique to separate the average and fluctuating parts of a quantity.For example, for a quantity \scriptstyle u the decomposition would be...
:
with
being the flow velocity vector having components in the coordinate direction (with denoting the components of the coordinate vector ). The mean velocities are determined by either time averagingAverage
In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
, spatial averaging or ensemble averaging
Ensemble average
In statistical mechanics, the ensemble average is defined as the mean of a quantity that is a function of the micro-state of a system , according to the distribution of the system on its micro-states in this ensemble....
, depending on the flow under study. Further
denotes the fluctuating (turbulence) part of the velocity.The components τij of the Reynolds stress tensor are defined as:
with ρ the fluid density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
, taken to be non-fluctuating for this homogeneous fluid.
Another – often used – definition, for constant density, of the Reynolds stress components is:
which has the dimensions of velocity squared, instead of stress.
Averaging and the Reynolds stress
To illustrate, Cartesian vector index notation is used. For simplicity, consider an incompressible fluidIncompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid...
:
Given the fluid velocity
as a function of position and time, write the average fluid velocity as , and the velocity fluctuation is . Then .The conventional ensemble
Ensemble average
In statistical mechanics, the ensemble average is defined as the mean of a quantity that is a function of the micro-state of a system , according to the distribution of the system on its micro-states in this ensemble....
rules of averaging are that
One splits the Euler equations
Euler equations
In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass , momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically,...
or the Navier-Stokes equations
Navier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...
into an average and a fluctuating part. One finds that upon averaging the fluid equations, a stress on the right hand side appears of the form
. This is the Reynolds stress, conventionally written :The divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of this stress is the force density on the fluid due to the turbulent fluctuations.
Reynolds averaging of the Navier–Stokes equations
For instance, for an incompressible, viscousViscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
, Newtonian fluid
Newtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...
, the continuity and momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
equations—the incompressible Navier–Stokes equations—can be written as
and
where
is the Lagrangian derivative or the substantial derivative,Defining the flow variables above with a time-averaged component and a fluctuating component, the continuity and momentum equations become
and
Examining one of the terms on the left hand side of the momentum equation, it is seen that
where the last term on the right hand side vanishes as a result of the continuity equation. Accordingly, the momentum equation becomes
Now the continuity and momentum equations will be averaged. The ensemble rules of averaging need to be employed, keeping in mind that the average of products of fluctuating quantities will not in general vanish. After averaging, the continuity and momentum equations become
and
Using the chain rule on one of the terms of the left hand side, it is revealed that
where the last term on the right hand side vanishes as a result of the averaged continuity equation. The averaged momentum equation now becomes, after a rearrangement:
where the Reynolds stresses,
, are collected with the viscous normal and shear stress terms, .Discussion
The question then is, what is the value of the Reynolds stress? This has been the subject of intense modeling and interest, for roughly the past century. The problem is recognized as a closure problem, akin to the problem of closure in the BBGKY hierarchyBBGKY hierarchy
In statistical physics, the BBGKY hierarchy is a set of equations describing the dynamics of a system of a large number of interacting particles...
. A transport equation for the Reynolds stress may be found by taking the outer product
Outer product
In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix...
of the fluid equations for the fluctuating velocity, with itself.
One finds that the transport equation for the Reynolds stress includes terms with higher-order correlations (specifically, the triple correlation
) as well as correlations with pressure fluctuations (i.e. momentum carried by sound waves). A common solution is to model these terms by simple ad-hoc prescriptions.It should also be noted that the theory of the Reynolds stress is quite analogous to the kinetic theory of gases, and indeed the stress tensor in a fluid at a point may be seen to be the ensemble average of the stress due to the thermal velocities of molecules at a given point in a fluid. Thus, by analogy, the Reynolds stress is sometimes thought of as consisting of an isotropic pressure part, termed the turbulent pressure, and an off-diagonal part which may be thought of as an effective turbulent viscosity.
In fact, while much effort has been expended in developing good models for the Reynolds stress in a fluid, as a practical matter, when solving the fluid equations using computational fluid dynamics, often the simplest turbulence models prove the most effective. One class of models, closely related to the concept of turbulent viscosity, is the so-called
model(s), 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 .Typically, the average is formally defined as an ensemble average as in statistical ensemble theory. However, as a practical matter, the average may also be thought of as a spatial average over some lengthscale, or a temporal average. Note that, while formally the connection between such averages is justified in equilibrium statistical mechanics by the ergodic theorem, the statistical mechanics of hydrodynamic turbulence is currently far from understood. In fact, the Reynolds stress at any given point in a turbulent fluid is somewhat subject to interpretation, depending upon how one defines the average.