Orr–Sommerfeld equation

The Orr-Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. A parallel, laminar flow that solves the Navier-Stokes equations can become unstable if certain conditions on the flow are satisfied, and the Orr-Sommerfeld equation determines precisely what these conditions are.

Formulation

A schematic diagram of the base state of the system. The flow under investigation represents a small perturbation away from this state. While the base state is parallel, the perturbation velocity has components in both directions.

The equation is derived by solving a linearized version of the Navier-Stokes equation for the perturbation velocity field

$\mathbf {u} =\left(U(z)+u'(x,z,t),0,w'(x,z,t)\right)$,

where $(U(z),0,0)$ is the unperturbed or basic flow. The perturbation velocity has the wave-like solution $\mathbf {u} '\propto \exp(i\alpha (x-ct))$ (real part understood). Using this knowledge, and the streamfunction representation for the flow, the following dimensional form of the Orr-Sommerfeld equation is obtained:

${\frac {\mu }{i\alpha \rho }}\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)^{2}\varphi =(U-c)\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)\varphi -U''\varphi$,

where $\mu$ is the dynamic viscosity of the fluid, $\rho$ is its density, and $\varphi$ is the streamfunction. The equation can be written in non-dimensional form by measuring velocities according to a scale set by some characteristic velocity $U_{0}$, and by measuring lengths according to channel depth $h$. Then the equation takes the form

${1 \over i\alpha \,Re}\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)^{2}\varphi =(U-c)\left({d^{2} \over dz^{2}}-\alpha ^{2}\right)\varphi -U''\varphi$,

where

$Re={\frac {\rho U_{0}h}{\mu }}$

is the Reynolds number of the basic flow. The relevant boundary conditions are the no-slip boundary conditions at the channel top and bottom $z=z_{1}$ and $z=z_{2}$,

$\alpha \varphi ={d\varphi \over dz}=0$ at $z=z_{1}$ and $z=z_{2}$,

The eigenvalue parameter of the problem is $c$ and the eigenvector is $\varphi$. If the imaginary part of the wave speed $c$ is positive, then the base flow is unstable, and the small perturbation introduced to the system is amplified in time.

Solutions

For all but the simplest of velocity profiles $U$, numerical or asymptotic methods are required to calculate solutions. Some typical flow profiles are discussed below.

The spectrum of the Orr--Sommerfeld for Poiseuille flow at criticality.

Dispersion curves of the Poiseuille flow for verious Reynolds numbers.

For Poiseuille flow, it has been shown that the flow is unstable (i.e. one or more eigenvalues $c$ has a positive imaginary part) for some $\alpha$ when $Re>Re_{c}=5772.22$ and the neutrally stable mode at $Re=Re_{c}$ having $\alpha _{c}=1.02056$, $c_{r}=0.264002$. To see the stability properties of the system, it is customary to plot a dispersion curve, that is, a plot of the growth rate ${\text{Im}}(\alpha {c})$ as a function of the wavenumber $\alpha$.

The first figure shows the spectrum of the Orr--Sommerfeld equation at the critical values listed above. This is a plot of the eigenvalues (in the form $\lambda =-i\alpha {c}$) in the complex plane. The rightmost eigenvalue is the most unstable one. At the critical values of Reynolds number and wavenumber, the rightmost eigenvalue is exactly zero. For higher (lower) values of Reynolds number, the rightmost eigenvalue shifts into the positive (negative) half of the complex plane. Then, a fuller picture of the stability properties is given by a plot exhibiting the functional dependence of this eigenvalue; this is shown in the second figure.

On the other hand, the spectrum of eigenvalues for Couette flow indicates stability, at all Reynolds numbers . However, in experiments, Couette flow is found to be unstable to small perturbations. This is explained through the non-normality of the eigenvalue problem associated with Couette (and indeed, Poiseuille) flow . That is, the eigenfunctions of the operator are complete but non-orthogonal. Thus, if a number of modes are excited, it is possible for the system energy to be transferred to the most dangerous of these modes. This cascade of energy destabilizes the system. Nevertheless, in this case, the Orr--Sommerfeld analysis still demonstrates that uni-model disturbances are stable.

Mathematical methods for free-surface flows

For Couette flow, it is possible to make mathematical progress in the solution of the Orr--Sommerfeld equation. In this section, a demonstration of this method is given for the case of free-surface flow, that is, when the upper lid of the channel is replaced by a free surface. Note first of all that it is necessary to modify upper boundary conditions to take account of the free surface. In non-dimensional form, these conditions now read

$\varphi ={d\varphi \over dz}=0,$ at $z=0$,

${\frac {d^{2}\varphi }{dz^{2}}}+\alpha ^{2}\varphi =0$, $\Omega \equiv {\frac {d^{3}\varphi }{dz^{3}}}+i\alpha Re\left[\left(c-U\left(z_{2}=1\right)\right){\frac {d\varphi }{dz}}+\varphi \right]-i\alpha Re\left(Fr+\alpha ^{2}S\right){\frac {\varphi }{c-U\left(z_{2}=1\right)}}=0,$ at $z=1$.

The first free-surface condition is the statement of continuity of tangential stress, while the second condition relates the normal stress to the surface tension. Here

$Fr={\frac {gh}{U_{0}^{2}}},\,\,\ S={\frac {\sigma }{\rho u_{0}^{2}h}}$

are the surface tension and gravity parameters respectively (or the Froude and inverse Weber numbers.)

For Couette flow $U\left(z\right)=z$, the four linearly independent solutions to the non-dimensional Orr--Sommerfeld equation are ,

$\chi _{1}\left(z\right)=\sinh \left(\alpha z\right),\qquad \chi _{2}\left(z\right)=\cosh \left(\alpha z\right)$,

$\chi _{3}\left(z\right)={\frac {1}{\alpha }}\int _{\infty }^{z}\sinh \left[\alpha \left(y-\xi \right)\right]Ai\left[e^{i\pi /6}\left(\alpha Re\right)^{1/3}\left(\xi -c-{\frac {i\alpha }{Re}}\right)\right]d\xi ,$

$\chi _{4}\left(z\right)={\frac {1}{\alpha }}\int _{\infty }^{z}\sinh \left[\alpha \left(y-\xi \right)\right]Ai\left[e^{5i\pi /6}\left(\alpha Re\right)^{1/3}\left(\xi -c-{\frac {i\alpha }{Re}}\right)\right]d\xi ,$

where $Ai\left(\cdot \right)$ is the Airy function of the first kind. Substitution of the superposition solution $\varphi =\sum _{i=1}^{4}c_{i}\chi _{i}\left(z\right)$ into the four boundary conditions gives four equations in the four unknown constants $c_{i}$. For the equations to have a non-trivial solution, the determinant condition

$\left|{\begin{array}{cccc}\chi _{1}\left(0\right)&\chi _{2}\left(0\right)&\chi _{3}\left(0\right)&\chi _{4}\left(1\right)\\\chi _{1}'\left(0\right)&\chi _{2}'\left(0\right)&\chi _{3}'\left(0\right)&\chi _{4}'\left(0\right)\\\Omega _{1}\left(1\right)&\Omega _{2}\left(1\right)&\Omega _{3}\left(1\right)&\Omega _{4}\left(1\right)\\\chi _{1}''\left(1\right)+\alpha ^{2}\chi _{1}\left(1\right)&\chi _{2}''\left(1\right)+\alpha ^{2}\chi _{2}\left(1\right)&\chi _{3}''\left(1\right)+\alpha ^{2}\chi _{3}\left(1\right)&\chi _{4}''\left(1\right)+\alpha ^{2}\chi _{4}\left(1\right)\end{array}}\right|=0$

must be satisfied. This is a single equation in the unknown c, which can be solved numerically or by asymptotic methods. It can be shwon that for a range of wavenumbers $\alpha$ and for sufficiently large Reynolds numbers, the growth rate $\alpha c_{\text{i}}$ is positive.