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\begin{center}{\large CHAPTER 5. RUDIMENTS OF HYDRODYNAMIC INSTABILITY} \end{center}
{\bf References}: \\
Drazin:  {\bf Introduction to Hydrodynamic Stability}\\
Chandrasekar:  {\bf Hydrodynamic and Hydromagnatic Instability}\\
Stuart: {\bf Hydrodynamic Stability},  in Rosenhead (ed) : {\bf Laminar boundary layers}\\
Lin : {\bf Theory of Hydrodynamic Stability}\\
Drazin and Reid : {\bf Hydrodynamic Stability}\\
Schlichting: {\bf Boundary Layer Theory}\\
C.S. Yih, 1965, {\bf  Dynamics of Inhomogeneous Fluids}, MacMillan.\\
Kundu : {\bf Fluiid Mechanics}. \\
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Various factors can be crucial to hydrodynamic instability.: shear, gravity, surface tension, heat, centrifugal force, etc.   Some instabilities lead to a different flow; some to turbulence.

We only discuss the linearized analysis of  instability of a few sample problems. More examples will be discussed in later chapters.\\
 
Let us take the parallel flow as the  example and outline the standard procedure of linearized analysis   as follows: 
\begin{enumerate}
\item Assume the  flow domain to be of infinite  extent in the  $x$ direction. 
  Let the   basic flow have velocity $U(z)$ in the $x$ direction.
\item Derive the linearized equation for infinitestimal perturbations and homogeneous boundary conditions. 
\item Consider sinusoidal wave- like disturbances so that all unknowns  are of the form 
 \[ f_i=\bar f_i(y,z)\, e^{i(kx-\omega t)}\]
\item Deduce the partial differential equation(s) in the transverse plane $(y,z)$ and  obtain an eignevalue problem.
\item Solve for  the eigenvalue $\omega $ for   given  real $k$. 
\item If  the eigenvalue is complex, $\omega=\omega_r+i\omega_i$, find   the condition  under which $\omega_i>0$.  Since 
\[ e^{-i\omega t}=e^{-i(\omega_r+i\omega_i ) t}=e^{-i\omega_r t} e^{\omega_i  t},\]
 the flow is   unstable $\omega_i>0$ as time grows, stable if $\omega_i<0$, and neutrally stable if $\omega_i=0$.  
\end{enumerate}

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\section{Kelvin-Helmholtz instability of flow with discontinuous shear and stratification.}
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\subsection{Nonlinear formulation}
Consider two immiscible fluids. The upper $(z>0)$ and lower $(z<0)$ fluids have different densities  $\rho_1$ and  $\rho_2 $. Before any perturbations appear the lower fluid is stationary, while the upper fluid moves at the steady and uniform velocity $\bar q_o=U\vec i $. With gravity, this is called the Kelvin-Helmholtz instability  problem. For the time being, let us assume that there is no gravity.

 
Now let there be a  disturbance on the interface whose vertical displacement  is 
$z=\zeta(x,t)$. The corresponding disturbances of velocity and pressure are $\vec q_1$ and $ p_1$ in the upper fluid and $\vec q_2$ and $ p_2$ in the lower fluid.

Let us first formulate the nonlinear problem.  In the upper fluid, we have 
\be \vec q_1=\nabla \Phi_1, ~~~\mbox{and}~~~\nabla^2 
\Phi_1=0, ~~~z>0\label{eq5-1-1}\ee
and the Bernoulli equation,
\be  \f{\p \Phi_1}{\p t}+\f{(\nabla^2 \Phi_1)^2}{2}=-\f{p_1}{\rho_1}+C_1\label{eq5-1-2}\ee  
In the lower fluid, we have :
\be \vec q_2=\nabla \Phi_2, ~~~\mbox{and}~~~\nabla^2\Phi_2=0~~~z<0\label{eq5-1-3}\ee
with the  Bernoulli equation, 
\be  \f{\p \Phi_2}{\p t}+\f{(\nabla^2 \Phi_2)^2}{2}=-\f{p_2}{\rho_2}+C_2\label{eq5-1-4}\ee 


On the interface $F=z-\zeta(x,t)=0$ the kinematic conditions are   \be 
\f{\p F }{\p t}+\nabla \Phi_1\cdot \nabla F=0\label{eq5-1-5}\ee
and 
\be\f{\p F }{\p t}+\nabla \Phi_2\cdot \nabla F=0\label{eq5-1-6}\ee
or, more explicitly,
\be  \f{\p \zeta }{\p t}+\f{\p \Phi_1}{\p x}\f{\p \zeta}{\p x}=\f{\p \Phi_1}{\p z}, ~~~z=\zeta,\label{eq5-1-7}\ee
\be  \f{\p \zeta }{\p t}+\f{\p \Phi_2}{\p x}\f{\p \zeta}{\p x}=\f{\p \Phi_2}{\p z}, ~~~z=\zeta,\label{eq5-1-8}\ee
The dynamic  condition is 
\be
 p_1= p_2, ~~~\mbox{or}~~z=\zeta.\label{eq5-1-9}\ee
\be  \rho_1\lp\f{\p \Phi_1}{\p t}+\f{(\nabla^2 \Phi_1)^2}{2}-C_1\rp=\rho_2\lp \f{\p \Phi_2}{\p t}+\f{(\nabla^2 \Phi_2)^2}{2}-C_2\rp, ~~~\mbox{or}~~z=\zeta.\label{eq5-1-10}\ee
Far away from the interface, 
\be \nabla \Phi_1 \to U_1\vec i, ~~z\to \infty,\label{eq5-1-11}
\ee
\be \nabla \Phi_2 \to U_2\vec i, ~~z\to -\infty.\label{eq5-1-12}\ee
Note that  in the absence of any transient disturbances, $\nabla\Phi_1=U_1\vec i, \nabla\Phi_2=U_2\vec i$, so that 
\be \rho_1\lp C_1-\f{U_1^2}{2}\rp = \rho_2\lp C_2-\f{U_2^2}{2}\rp\label{eq5-1-13}\ee 

\subsection{Linearization}
We now limit ourselves to infinitesimal disturbances. Specificlly,  the interface displacement is infinitesimal when compared to other length scales.  
\be \Phi_1=U_1 x +\phi_1, ~~and~~\Phi_2=U_2 x +\phi_2,\label{eq5-1-14} \ee
 with $\phi_n  \ll \Phi_n, n=1,2$ and linearize all the conditions. We then get
\be 
\nabla^2 
\Phi_1=0, ~~~z>0,\label{eq5-1-15} \ee
nad \be \nabla^2 
\Phi_2=0, ~~~z<0, \label{eq5-1-16}\ee
Note that 
\be f(x,\zeta,t) = f(x,0,t)+\zeta\f{\p f}{\p z}|_{z=0}+\cdots, \label{eq5-1-17}\ee
On the inteface we have ther kinematic conditions
\be  \f{\p \zeta }{\p t}+U_1\f{\p \zeta}{\p x}=\f{\p \phi_1}{\p z}, ~~~z=0,\label{eq5-1-18}\ee
\be  \f{\p \zeta }{\p t}+U_2\f{\p \zeta}{\p x}=\f{\p \phi_2}{\p z}, ~~~z=0,\label{eq5-1-19}\ee
and the dynamic condition
\be  \rho_1\lp\f{\p \phi_1}{\p t}+U_1\f{\p \phi_1}{\p x}\rp=\rho_2\lp \f{\p \phi_2}{\p t}+U_2\f{\p \phi_2}{\p x}\rp, ~~~z=0.\label{eq5-1-20}\ee
where (\ref{eq5-1-13}) has been used.

\subsection{Normal mode analysis}

 Assume a sinusoidal disturbance \be\zeta=Ae^{i(kx-\omega t)}\ee
We follow the usual convention that   the real part of the final solution is to be taken to represent  physical quaitities. 
To satisfy Laplace equation we take
\be \phi_1 = \bar \phi_1 \,e^{-kz}e^{i(kx-\omega t)}\label{eq5-1-21}\ee
\be  \phi_2= \bar \phi_2\, e^{kz}e^{i(kx-\omega t)}\label{eq5-1-22}\ee
Here $ A,  \bar \phi_1, \bar \phi_2$ are unknown constants.
 
Substituting into boundary conditions on the interface eqs (\ref{eq5-1-18}-\ref{eq5-1-20}) we get  three homogeneous  alegebric equations for  $A,  \bar \phi_1,   \bar \phi_2 $,
\be ( -i\omega  +ikU_1)A=-k\bar \phi_1 \label{eq:5.1.22}\ee
\be  (-i\omega  +ikU_2)A=k\bar \phi_2 \label{eq:5.1.23}\ee
\[ \rho_1(-i\omega +ikU_1)\phi_1=\rho_2(-i\omega +ikU_2)\phi_2\]
which can be written in matrix form
\[ \lb\begin{array}{ccc}
-i\omega +ikU_1 & k& 0\\
-i\omega +ikU_2 & 0&-k\\
0&\rho_1(-i\omega +ikU_1)& -\rho_2(-i\omega +ikU_2)
\end{array}\rb\lcb \begin{array}{c}
A\\ \bar \phi_1\\ \bar \phi_2\end{array}
\rcb=0\]
In order to get non trivial solutions the coefficient determinant of the algebraic equations must vanish,  
  leading to the eigenvalue condition 
\[ \rho_1(-\omega+kU_1)^2 + \rho_2(-\omega+kU_2)^2=0\] 
or
\[ (\rho_1+\rho_2)\omega^2-2k(\rho_1U_1+\rho_2U_2)\omega +k^2(\rho_1U_1^2+U\rho_2U_2^2)=0\]
The solution is 
\be \omega=k\f{\rho_1U_1+\rho_2U_2}{\rho_1+\rho_2}\pm ik\f{\sqrt{\rho_1\rho_2}|U_1-U_2|}{\rho_1+\rho_2}\label{eq:5.1.24}\ee
 The lower sign corresponds to instability. Thus a velocity discontinuity is always unstable. This  is the limiting case of the continuous velocity profile with a point of inflection.

 
\subsection{Physical interpretation} Let us use the mathematical results to see the physics (
Batchelor, p 516ff). For simlicity we choose $\rho_1=\rho_2=\rho$ and $U_1=-U_2=U>0$ so that (\ref{eq:5.1.24}) reduces to 
\be 
\omega = \pm ik  U \label{specialcase}
\ee
and is purely imaginary. The positive root (upper sign) signifies instability. 
 
The basic flow has a vortex sheet at $z=0$. Due to the disturbance,  the  vorticity distribution is changed. Across the  interface the total vorticity  is approximately 
\[ \int_{0_-}^{0_+}\f{\p u'}{\p z}dz =  u'_1(z=0_+)-u'_2(z=0_-)= 
\f{\p \phi_1}{\p x}|_0- \f{\p \phi_2}{\p x} |_0\]
Note that a positive vorticity vector points    into the paper (as does the $y$ axis); the circulation is clockwise  in the $x-z$ plane.  From the   solution (\ref{eq:5.1.22}) and (\ref{eq:5.1.23}) we get 
\[ \f{\p \phi_1}{\p x}|_0=ik\bar \phi_1e^{ikx-i\omega t}=-i(-i\omega +ikU_1)Ae^{ikx-i\omega t}=-i(-i\omega +ikU_1)\zeta\]
and 
\[ \f{\p \phi_2}{\p x}|_0=ik\bar \phi_2e^{ikx-i\omega t}= i(-i\omega +ikU_2)Ae^{ikx-i\omega t}= i(-i\omega +ikU_2)\zeta\]
Therefore the vorticity along the interface is 
\[ [-2\omega +k(U_1+U_2)]\zeta  =\mp 2 ikU\zeta
=2kU A\exp (ikx\mp i\pi/2)\pm kUt)\]
Consider only the upper signs for the unstable solution,
\[ -2ikU\zeta =2kUA\exp\lb ik\lp x-\f{\pi}{2k}\rp +kUt\rb=2kUA\exp\lb ik\lp x-\f{\lambda}{4}\rp +kUt\rb\]
since $k=2\pi/\lambda$ where $\lambda$ is th wave length. In real form the vorticity disturbance varies as 
\[  2kUA\cos \lb k\lp x-\f{\lambda}{4}\rp \rb e^{kUt}\]
whereas the interface varies as 
  \[ A\cos kx \, e^{kUt} \]
The vorticity disturbance varies along the interface and leads the interface displacement $\zeta$ in phase by  $\pi/2$ or one quarter of a wavelength. As sketched in the upper  part of Figure \ref{fig:Batchelor},   the   vorticity disturbance  is   at the positive   (clockwise) maximum at points like $A$ where $\zeta=0$ and $\zeta_x<0$ and at the negative (counterclockwise) maximum at points like $B$ where $\zeta=0$ and $\zeta_x>0$, being zero at the crests and troughs of the interface.  All these vortices     tend to lift the interface crests and supress the troughs. 
\begin{figure}[h] \begin{center}
\includegraphics[scale=0.5]{Batchelor.eps} \end{center}
\caption{Disturbance vorticity along a velocity dicontinuity. $U_1=U>0, U_2=-U<0$. }\label{fig:Batchelor}
\end{figure}


 Moreover  the disturbances are  convected by the  average  velocity  
\[  
\f{\p \phi_1}{\p x}|_0+ \f{\p \phi_2}{\p x} |_0=k(U_1-U_2)\zeta=2U\zeta\]
which  is  positive   near the crests and negative near the troughs. Therefore    clockwise vorticity disturbances   accumulate  around  $A$, while the counterclockwise disturbances are swept away from and thinned out around  $B$.  Because of this assymetry the wavy interface not only tend to amplify but to  curl up   in the forward direction  as sketched in the lower part of Figure \ref{fig:Batchelor}.
  
After some time the disturbance grows so large that nonlinearity must be accounted for. Numerical nonlinear analysis as well as  laboratory experiments show that the interface curls up, as can be seen in the famous photograph by Thorpe:  Figure\ref{fig:Thorpe}. 

\begin{figure}[h] \begin{center}
\includegraphics[scale=0.5]{Thorpe.eps} \end{center}
\caption{Kelvin-Helmholtz instability in the laboratory. }\label{fig:Thorpe}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf Homework} K-H instability:  Show that  when gravity and surface tension are both  present, the eigenvalue condition is 
\be \f{\omega}{k} = \f{\rho_1U_1+\rho_2U_2}{\rho_1+\rho_2}\pm
 \lb -\f{\rho_1\rho_2(U_1-U_2)^2}{(\rho_1+\rho_2)^2} + \f{g}{k}\f{\rho_2-\rho_1}{\rho_1+\rho_2}+\f{Tk}{\rho_1+\rho_2} \rb^{1/2} \ee
Discuss the condition for instability.  

{\bf Remark:} The special case of $U_1=U_2=0$ but $g\neq 0$, i.e.,  the instability of normally accelerated density discontinuity,   is called the Rayleigh-Taylor instability.  It is of interest when an interface is acelerated in the normal direction, and has a wide range of industrial and astrophysical applcations. 

{\bf Remark:} The K-H instabilty problem can be generalized in a number of ways, by including viscosity and  continuous density and/or velocity profiles,  see Chandrasekhar. 




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