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--- lecture7:reynoldsdecomp [2014/07/01 09:33] – created - external edit 127.0.0.1
+++ lecture7:reynoldsdecomp [2026/03/25 06:00] (current) – matt
@@ Line 14: / Line 14: @@
 **Reynolds decomposition** is a useful mathematical representation elaborated by Osborne Reynolds (1842–1912) //[[http://en.wikipedia.org/wiki/Osborne_Reynolds|Wikipedia>Reynolds]]// for analyzing the dynamics of a turbulent fluid through the conceptual and analitical separation of its **average regular motion** and of the **chaotic turbulent flutuations**. \\
 \\
-The basic concept consists in the decomposition of all the dynamic and thermodynamic instantaneous variables (eg, the velocity component $\textcolor{black}{u}$) that characterize the state of the the fluid in two components: \\
+The basic concept consists in the decomposition of all the dynamic and thermodynamic instantaneous variables (eg, the velocity component ${u}$) that characterize the state of the the fluid in two components: \\
-the //average// value $\textcolor{black}{\bar{u}}$ and the //fluctuations//, or deviations, $\textcolor{black}{u'}$ around the average.\\
+the //average// value ${\bar{u}}$ and the //fluctuations//, or deviations, ${u'}$ around the average.\\
 This decomposition allows to interpret the complex turbulent dynamics as the superposition of a so called "steady" average component that represent the laminar motion and that can be easily studied in a quantitative fashion, and the contribution of the turbulent eddies and fluctuations which are characterized by a high degree of randomness. The latter can hence be measured as deviations from the mean flow, and studied through their statistical properties such as the moments that characterize their probability distributions.\\
-In absence of turbulence (low Reynolds numbers) we expect then $\textcolor{black}{\vec{u}=\bar{\vec{u}}}$.\\
+In absence of turbulence (low Reynolds numbers) we expect then ${\vec{u}=\bar{\vec{u}}}$.\\
 \\
 --------
@@ Line 27: / Line 27: @@
 - average //in time// (so called "mean"):\\
 \begin{equation*}
-\textcolor{black}{ ^t\bar{u}=\frac{1}{\Delta t} \int_{0}^{\Delta t} u(t,\vec{x}) dt }
+{ ^t\bar{u}=\frac{1}{\Delta t} \int_{0}^{\Delta t} u(t,\vec{x}) dt }
 \end{equation*}
 - average //in space//:\\
 \begin{equation*}
-\textcolor{black}{ ^s\bar{u} =\frac{1}{ V} \int \int \int u(t,\vec{x}) d^3x}
+{ ^s\bar{u} =\frac{1}{ V} \int \int \int u(t,\vec{x}) d^3x}
 \end{equation*}
 - //ensemble// average:\\
 \begin{equation*}
-\textcolor{black}{ ^a\bar{u} = \frac{1}{N} \sum_{i=1}^{N} u_i}
+{ ^a\bar{u} = \frac{1}{N} \sum_{i=1}^{N} u_i}
 \end{equation*}
-where an //ensemble// is defined as a set of $\textcolor{black}{N}$ realizations ($\textcolor{black}{N \rightarrow \infty}$) of the event corresponding to the same macroscopic thermodynamic state, in other words a set of repetitions of the same experiment or simulation.\\
+where an //ensemble// is defined as a set of ${N}$ realizations (${N \rightarrow \infty}$) of the event corresponding to the same macroscopic thermodynamic state, in other words a set of repetitions of the same experiment or simulation.\\
-When working with real observations, it's often straightforward to define the average as the mean, where the mean is the average in time of the measured variable (eg, the velocity of the water flow) over a single measurement of length $\textcolor{black}{\Delta t}$.\\
+When working with real observations, it's often straightforward to define the average as the mean, where the mean is the average in time of the measured variable (eg, the velocity of the water flow) over a single measurement of length ${\Delta t}$.\\
 \\
 For an //ergodic// //[[http://en.wikipedia.org/wiki/Ergodicity|Wikipedia>Ergodicity]]// process (a process that is assumed to explore the whole phase space in a finite amount of time) the three definitions correspond to each other and we can hence use any of the three to define the average values of the quantities of interest. In the case of turbulent flow, it is reasonable and common to assume the process to be ergodic.
@@ Line 47: / Line 47: @@
 The next rules are true for all the previous definitions of average, given their //linear// //[[http://en.wikipedia.org/wiki/Linear_transformation|Wikipedia>LinearMap]]// character:
 \begin{equation*}
-\textcolor{black}{ \overline{\lambda a} = \lambda \overline{a}}\qquad \qquad \textcolor{black}{ \overline{a+b} = \overline{a} + \overline{b}}\qquad \qquad \textcolor{black}{ \overline{\overline a} = \overline a } \qquad \qquad \textcolor{black}{\overline{a \cdot b}=\overline{a}\cdot\overline{b}+\overline{a'\cdot b'}} \qquad \qquad \textcolor{black}{\overline{\frac{\partial a}{\partial t}}=\frac{\partial \overline a}{\partial t}}
+{ \overline{\lambda a} = \lambda \overline{a}}\qquad \qquad { \overline{a+b} = \overline{a} + \overline{b}}\qquad \qquad { \overline{\overline a} = \overline a } \qquad \qquad {\overline{a \cdot b}=\overline{a}\cdot\overline{b}+\overline{a'\cdot b'}} \qquad \qquad {\overline{\frac{\partial a}{\partial t}}=\frac{\partial \overline a}{\partial t}}
 \end{equation*}
-for any scalar $\textcolor{black}{\lambda}$ and any variables $\textcolor{black}{a=\bar a + a'}$ and $\textcolor{black}{b=\bar b + b'}$.
+for any scalar ${\lambda}$ and any variables ${a=\bar a + a'}$ and ${b=\bar b + b'}$.
 \\
 \\
@@ Line 55: / Line 55: @@
 \\
 From the previous equations it is possible to derive some simple properties:\\
-- the //mean// of the fluctuations is zero: $\textcolor{black}{u = \bar u + u' \rightarrow \overline{u} = \overline{\bar u + u'} \rightarrow \bar u = \bar u + \bar u' \rightarrow \bar u'= 0}$\\
+- the //mean// of the fluctuations is zero: ${u = \bar u + u' \rightarrow \overline{u} = \overline{\bar u + u'} \rightarrow \bar u = \bar u + \bar u' \rightarrow \bar u'= 0}$\\
-- both the average and fluctuations vector fields have zero divergence: $\textcolor{black}{\vec \nabla \vec u = \vec \nabla (\bar{\vec u} + \vec u') = 0 \quad \forall (\vec x, t) \rightarrow \vec \nabla \bar{\vec u} = \vec \nabla \vec u' = 0}$\\
+- both the average and fluctuations vector fields have zero divergence: ${\vec \nabla \vec u = \vec \nabla (\bar{\vec u} + \vec u') = 0 \quad \forall (\vec x, t) \rightarrow \vec \nabla \bar{\vec u} = \vec \nabla \vec u' = 0}$\\
 The fluctuations are hence //stochastic variables// with mean equal to zero.
 \\
@@ Line 67: / Line 67: @@
 \begin{equation*}
 \boxed{
-\textcolor{black}{
+{
 \begin{cases}
 \dfrac{\partial \rho}{\partial t}=\vec \nabla (\rho \vec u)} \qquad \qquad &\text{(continuity eqn: mass conservation)}\\
@@ Line 75: / Line 75: @@
 }
 \end{equation*}
-where $\textcolor{black}{p}$ is pressure, $\textcolor{black}{\rho}$ represent density, the advective and temporal evolution terms are included in the operator $\textcolor{black}{\frac{D}{Dt} = \frac{\partial}{\partial t} + \vec \nabla \cdot \vec u}$ with $\textcolor{black}{\vec \nabla = (\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})}$ and $\textcolor{black}{\vec F_g}$ is the external body force per unit mass, in our case gravity: $\textcolor{black}{\vec F_g = - \vec g}$. \\
+where ${p}$ is pressure, ${\rho}$ represent density, the advective and temporal evolution terms are included in the operator ${\frac{D}{Dt} = \frac{\partial}{\partial t} + \vec \nabla \cdot \vec u}$ with ${\vec \nabla = (\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z})}$ and ${\vec F_g}$ is the external body force per unit mass, in our case gravity: ${\vec F_g = - \vec g}$. \\
-The term $\textcolor{black}{\vec \nabla T}$ represent the //shear stress tensor// and depends on the properties of the fluid.
+The term ${\vec \nabla T}$ represent the //shear stress tensor// and depends on the properties of the fluid.
 \\
 ** Newtonian fluids, incompressibility and Boussinesq approximation**\\
 \\
-Water in a //Newtonian fluid//: this not only means that it is assumed to be //isotropic// (which means that it has the same mechanical properties in every direction), but it also implies that the //viscous stresses// generated by its flow are proportional to the local strain, in other words to the local change of the fluid's velocity vector. The proportionality constant is called //viscosity// $\textcolor{black}{\mu}$ and for newtonian fluids this quantity does not depend on the velocity of the flow. \\
+Water in a //Newtonian fluid//: this not only means that it is assumed to be //isotropic// (which means that it has the same mechanical properties in every direction), but it also implies that the //viscous stresses// generated by its flow are proportional to the local strain, in other words to the local change of the fluid's velocity vector. The proportionality constant is called //viscosity// ${\mu}$ and for newtonian fluids this quantity does not depend on the velocity of the flow. \\
 //Non-newtonian fluids// such as ketchup //[[https://www.youtube.com/watch?v=KB43fM_ozKQ|YT_video>Ketchup]]// or Oobleck //[[http://fuckyeahfluiddynamics.tumblr.com/tagged/oobleck|FYFD>Oobleck]]// behave sometimes in a pretty counter-intuitive fashion, showing phenomena such as //shear thickening or thinning// due to their non constant viscosity //[[http://en.wikipedia.org/wiki/Non-Newtonian_fluid|Wikipedia>NonNewtonian]]//.\\
 \\
-Water is also assumed to be //incompressible// (a pretty realistic assumption for a fluid) so that $\textcolor{black}{\rho = \rho_0}$ constant in time and space, which implies for the continuity equation:
+Water is also assumed to be //incompressible// (a pretty realistic assumption for a fluid) so that ${\rho = \rho_0}$ constant in time and space, which implies for the continuity equation:
 \begin{equation*}
-\textcolor{black}{\frac{\partial \rho}{\partial t} = 0, \quad \textcolor{black}{\vec \nabla \rho = 0 \quad \Rightarrow \quad \boxed{\vec \nabla \cdot \vec u = 0}}
+{\frac{\partial \rho}{\partial t} = 0, \quad {\vec \nabla \rho = 0 \quad \Rightarrow \quad \boxed{\vec \nabla \cdot \vec u = 0}}
 \end{equation*}
 that means that there are no sources or wells in the velocity field, or in other words no convergence or divergence of the flow. This approximation is equivalent to the so called //Boussinesq approximation// //[[http://en.wikipedia.org/wiki/Boussinesq_approximation_%28buoyancy%29|Wikipedia>BoussinesqApprox.]]// for gases, and denies the existence of sound waves that propagate through the medium via compression.\\
 \\
-For an incompressible newtonian fluid, the shear stress tensor $\textcolor{black}{\vec \nabla T}$ can then be written component by component as:
+For an incompressible newtonian fluid, the shear stress tensor ${\vec \nabla T}$ can then be written component by component as:
 \begin{equation*}
-\textcolor{black}{T_{i,j}=\mu \big( \frac{\partial u_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \big)}
+{T_{i,j}=\mu \big( \frac{\partial u_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \big)}
 \end{equation*}
 so that the last term of the momentum conservation equation can be expressed in the form:
 \begin{equation*}
-\textcolor{black}{\frac{1}{\rho} \vec \nabla T = \frac{\mu}{\rho} \nabla^2 \vec u}
+{\frac{1}{\rho} \vec \nabla T = \frac{\mu}{\rho} \nabla^2 \vec u}
 \end{equation*}
@@ Line 105: / Line 105: @@
 The 3D **Instantaneous Navier-Stokes Equation** for the motion of a newtonian incompressible turbulent fluid is:
 \begin{equation*}
-\textcolor{black}{
+{
 \boxed{
 \frac{\partial \vec u}{\partial t} + \bigg( u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z} \bigg) \vec u = - \frac{\vec \nabla p}{\rho_0} + \vec g + \nu \nabla^2 \vec u
@@ Line 111: / Line 111: @@
 }
 \end{equation*}
-where $\textcolor{black}{\nu=\frac{\mu}{\rho_0}}$ is called //kinematic viscosity// and $\textcolor{black}{\vec g = (0,0,-g)}$.\\
+where ${\nu=\frac{\mu}{\rho_0}}$ is called //kinematic viscosity// and ${\vec g = (0,0,-g)}$.\\
 \\
-Let's consider the 1D equation, for example for the **x-component**, and apply Reynolds decomposition to the variables $\textcolor{black}{u = \bar{\vec u} + \vec u'}$ and $\textcolor{black}{p = \bar p + p'}$. \\
+Let's consider the 1D equation, for example for the **x-component**, and apply Reynolds decomposition to the variables ${u = \bar{\vec u} + \vec u'}$ and ${p = \bar p + p'}$. \\
 The whole equation will then be averaged according to the rules:
 \begin{equation*}
-\textcolor{black}{
+{
 \overline{ \frac{\partial}{\partial t}(\bar u + u') + \bigg( (\bar u + u') \frac{\partial}{\partial x} + (\bar v + v') \frac{\partial}{\partial y} + (\bar w + w') \frac{\partial}{\partial z}\bigg)(\bar u + u') } = \overline{ - \frac{1}{\rho_0} \frac{\partial (\bar p + p')}{\partial x} + \nu \nabla^2 (\bar u + u') }
 }
 \end{equation*}
-where $\textcolor{black}{\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}}$. \\
+where ${\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}}$. \\
 Given the properties of the two vector fields we have:
 \begin{equation*}
-\textcolor{black}{
+{
 \overline{ \frac{\partial}{\partial t}(\bar u + u')} = \frac{\partial \bar u}{\partial t}  \qquad \qquad \overline{\frac{1}{\rho_0} \frac{\partial (\bar p + p')}{\partial x} } = \frac{1}{\rho_0}\frac{\partial \bar p}{\partial x}    \qquad \qquad   \overline{ \nu \nabla^2 (\bar u + u') } = \nu \nabla^2 \bar u
 }
@@ Line 129: / Line 129: @@
 while only the non-linear advective term produces some cross-terms which depend on the turbulent fluctuations
 \begin{equation*}
-\textcolor{black}{
+{
 \overline{ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} } & = \bigg( \bar u \frac{\partial \bar u}{\partial x} + \bar v \frac{\partial \bar u}{\partial y} + \bar w \frac{\partial \bar u}{\partial z} \bigg) + \bigg( \overline{u' \frac{\partial u'}{\partial x}} + \overline{v' \frac{\partial u'}{\partial y}} + \overline{w' \frac{\partial u'}{\partial z}}  \bigg)}
 }
 \end{equation*}
 \begin{equation*}
-\textcolor{black}{
+{
 = \bigg( \bar u \frac{\partial \bar u}{\partial x} + \bar v \frac{\partial \bar u}{\partial y} + \bar w \frac{\partial \bar u}{\partial z} \bigg) + \Bigg[ \bigg( \overline{\frac{\partial u'^2}{\partial x}} + \overline{\frac{\partial (u'v')}{\partial y}} + \overline{\frac{\partial (u'w')}{\partial z}} \bigg) - \bigg( \overline{u' \frac{\partial u'}{\partial x}} + \overline{u' \frac{\partial v'}{\partial y}} + \overline{u' \frac{\partial w'}{\partial z}} \bigg) \Bigg]
 }
 \end{equation*}
-where the last term is zero, being  $\textcolor{black}{\vec \nabla \vec u' = 0}$. \\
+where the last term is zero, being  ${\vec \nabla \vec u' = 0}$. \\
 \\
 For the **x-component of the averaged Navier-Stokes equation ** we obtain:
 \begin{equation*}
-\textcolor{black}{
+{
 \boxed{
 \frac{D \bar u}{D t} = \frac{\partial \bar u}{\partial t} + \bigg( \bar u \frac{\partial \bar u}{\partial x} + \bar v \frac{\partial \bar u}{\partial y} + \bar w \frac{\partial \bar u}{\partial z} \bigg) = - \frac{1}{\rho_0} \frac{\partial \bar p}{\partial x}+ \nu \nabla^2 \bar u - \bigg( \frac{\partial \overline{u'^2}}{\partial x} + \frac{\partial \overline{u'v'}}{\partial y} + \frac{\partial \overline{u'w'}}{\partial z} \bigg)
@@ Line 152: / Line 152: @@
 The whole **//Reynolds Averaged Navier-Stokes Equation//** (RANS) can be expressed as follow:
 \begin{equation*}
-\textcolor{black}{
+{
 \boxed{
 \frac{D \bar{\vec u}}{D t} = - \frac{\vec \nabla \bar p}{\rho_0} + \nu \nabla^2 \bar u - \frac{1}{\rho_0} \vec \nabla \hat \tau
@@ Line 158: / Line 158: @@
 }
 \end{equation*}
-where $\textcolor{black}{\hat \tau}$ is the Reynolds Stress Tensor, that represents the influence of the turbulent fluctuations on the mean flow. \\
+where ${\hat \tau}$ is the Reynolds Stress Tensor, that represents the influence of the turbulent fluctuations on the mean flow. \\
 \\
 **Reynolds stress tensor**\\
 \\
-The Reynolds stress tensor $\textcolor{black}{\hat \tau}$ is defined as follows:
+The Reynolds stress tensor ${\hat \tau}$ is defined as follows:
 \begin{equation*}
-\textcolor{black}{
+{
 \hat \tau = \rho_0 \cdot
 \begin{pmatrix}
@@ Line 171: / Line 171: @@
 }
 \end{equation*}
-where the terms $\textcolor{black}{\overline{u_i' u_j'}}$ can be interpreted as the correlation between the two velocity components $\textcolor{black}{\overline{u_i'}}$ and $\textcolor{black}{\overline{u_j'}}$.
+where the terms ${\overline{u_i' u_j'}}$ can be interpreted as the correlation between the two velocity components ${\overline{u_i'}}$ and ${\overline{u_j'}}$.
 \\
@@ Line 180: / Line 180: @@
 To predict the average quantities associated to the turbulent flow the main problem we have to face is the fact that //the number of equations that describe the problem is lower then the number of unknown variables//. \\
 \\
-Reynolds stresses $\textcolor{black}{\overline{u_i' u_j'}}$ are in fact supplementary unknown quantities, not easy to measure and determine.\\
+Reynolds stresses ${\overline{u_i' u_j'}}$ are in fact supplementary unknown quantities, not easy to measure and determine.\\
-Even if we try to resolve the problem multiplying the RANS equations by the fluctuations $\textcolor{black}{u_i'}$ and mediating again, what we obtain is nothing more than a new equation in the third moments $\textcolor{black}{\overline{u_i' u_j' u_k'}}$ of the distribution of the fluctuations: trying to eliminate the second moments creates again a system with more unknowns than equations.\\
+Even if we try to resolve the problem multiplying the RANS equations by the fluctuations ${u_i'}$ and mediating again, what we obtain is nothing more than a new equation in the third moments ${\overline{u_i' u_j' u_k'}}$ of the distribution of the fluctuations: trying to eliminate the second moments creates again a system with more unknowns than equations.\\
 For this reason the system for the coupled evolution of the statistical moments of the fluctuations is //non-closed//, and to solve it it's necessary to make some hypothesis that link the unknown variables to measurable quantities. This hypothesis can be based for example on empirical observations.\\
 \\
 The simplest way to close the system of equations is linking the second moments to the mean values, the so-called **K-Theory** model.\\
-The assumption is based on the concept of //eddy viscosity// introduced by Boussinesq in 1887 //[[http://en.wikipedia.org/wiki/Turbulence_modeling#Eddy_viscosity|Wikipedia>EddyViscosity]]//. A new quantity $\textcolor{black}{k}$ called //turbulent viscosity// is defined in analogy to the molecular viscosity and the turbulent stresses are supposed to be proportional to the gradient of the average velocity, as it happens in the laminar flow. The turbulent viscosity is strongly dependent on the local degree of turbulence of the flow.\\
+The assumption is based on the concept of //eddy viscosity// introduced by Boussinesq in 1887 //[[http://en.wikipedia.org/wiki/Turbulence_modeling#Eddy_viscosity|Wikipedia>EddyViscosity]]//. A new quantity ${k}$ called //turbulent viscosity// is defined in analogy to the molecular viscosity and the turbulent stresses are supposed to be proportional to the gradient of the average velocity, as it happens in the laminar flow. The turbulent viscosity is strongly dependent on the local degree of turbulence of the flow.\\
 The Reynolds stress tensor takes this form
 \begin{equation*}
-\textcolor{black}{
+{
 \overline{u_i' u_j'} = k \bigg(\frac{\partial \bar u_i}{ \partial x_j} + \frac{\partial u_j}{ \partial x_i} \bigg) \Rightarrow \frac{1}{\rho_0} \vec \nabla \hat \tau = \frac{k}{\rho_0} \nabla^2 \bar{\vec u}
 }

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