Orr-Sommerfeld equation ( $[(- iω + i αU) (D^{2} - κ^{2}) - i α U^{''} - R e^{- 1} (D^{2} - κ^{2})^{2}] v = 0$ )

The Orr-Sommerfeld equation governs the wall-normal velocity of normal mode disturbances to an incompressible viscous parallel shear flow. Here

D = d / d y

κ^{2} = α^{2} + β^{2}

, and

R e

is the Reynolds number. It follows by eliminating pressure from the linearized Navier-Stokes equations. At a rigid no-slip wall,

v = D v = 0

. In the inviscid limit it reduces to the Rayleigh equation for inviscid shear flow.

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Squire's theorem

 1  0

For temporal normal-mode stability of a viscous parallel shear flow, any unstable three-dimensional mode maps by the Squire transformation to an unstable two-dimensional mode at a no larger Reynolds number. Hence the first temporal modal instability occurs among two-dimensional disturbances. The theorem does not exclude three-dimensional transient growth or forced responses.

Squire transformation ( $k^{'} = k^{2} + l^{2}, R^{'} = k R / k^{'}, c^{'} = c$ )

 0  0

For positive streamwise wavenumber

k

, replace a three-dimensional Orr-Sommerfeld equation by a two-dimensional one with the displayed parameters. Both the transverse operator and the coefficient

1/ (ik R)

remain unchanged, preserving the eigenfunction and phase-speed eigenvalue. This proves the modal comparison in Squire's theorem.

Airy reduction of the Orr-Sommerfeld equation in constant shear ( $Z = e^{iπ /6} (y - c - i δ^{3}) / δ, δ = R e^{- 1/3}$ )

 0  0

For nondimensional constant shear

U = y

and streamwise wavenumber one, put

W = (D^{2} - 1) v

. The Orr-Sommerfeld equation gives

W^{''} = [1 + i R e (y - c)] W

. The displayed substitution transforms it into the Airy ordinary differential equation, because

(e^{iπ /6})^{3} = i

. The scale

R e^{- 1/3}

balances shear advection against viscous diffusion.

Velocity reconstruction from the Orr-Sommerfeld vorticity ( $v = C_{+} e^{y} + C_{-} e^{- y} + \int_{y_{0}}^{y} sinh (y - s) W (s) d s$ )

 0  0

The equation

(D^{2} - 1) v = W

is inverted by the displayed formula: differentiating twice produces the endpoint contribution

W (y)

, since

sinh 0 = 0

and

(sinh)^{'} (0) = 1

. If

W = a_{1} Ai (Z) + a_{2} Bi (Z)

, there are four independent constants

C_{+}, C_{-}, a_{1}, a_{2}

. This reconstructs the velocity from the Airy reduction of the Orr-Sommerfeld equation in constant shear; four no-slip and no-flux conditions determine the boundary eigenproblem.

Squire equation ( $[(- iω + i αU) - R e^{- 1} (D^{2} - κ^{2})] η = - i β U^{'} v$ )

 0  0

The Squire equation governs wall-normal vorticity

η = \partial_{z} u - \partial_{x} w

in the same normal mode convention as the Orr-Sommerfeld equation. It follows by taking

\partial_{z}

of the streamwise momentum equation minus

\partial_{x}

of the spanwise momentum equation. At a rigid no-slip wall,

η = 0

Squire mode

 0  0

A Squire mode has zero wall-normal velocity and nonzero wall-normal vorticity, satisfying the homogeneous Squire equation. For real base velocity, positive finite Reynolds number, and zero boundary terms,

Im ω = - \frac{\int ( ∣ η ^{'} ∣ ^{2} + κ ^{2} ∣ η ∣ ^{2} ) d y}{R e \int ∣ η ∣ ^{2} d y} < 0

(1)

for a nontrivial mode in a finite channel with no-slip walls. This follows by multiplying the homogeneous equation by

\overline{η}

, integrating by parts and taking the real part. Such modes are damped even when the coupled velocity-vorticity system can exhibit transient growth from non-normal modes.

Dissipation identity for a homogeneous Squire mode ( $Im ω = - \frac{∥ η ^{'} ∥ _{2}^{2} + ( k ^{2} + l ^{2} ) ∥ η ∥ _{2}^{2}}{R ∥ η ∥ _{2}^{2}}$ )

 0  0

For real base-flow velocity and positive Reynolds number, multiply the homogeneous Squire equation by the conjugate mode and integrate with zero endpoint values. Taking real parts gives the displayed negative temporal growth rate. This proves decay of homogeneous Squire modes; the coupled shear-flow system can still be unstable or show transient growth because wall-normal velocity can force the Squire variable.

Orr-Sommerfeld mode

 0  0

An Orr-Sommerfeld mode has nonzero wall-normal velocity satisfying the Orr-Sommerfeld equation. Its wall-normal vorticity satisfies the accompanying forced Squire equation. The triangular coupling separates the velocity eigenproblem from the vorticity forcing; the vorticity need not vanish in three dimensions.

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The Orr–Sommerfeld equation is a fundamental equation in fluid dynamics that describes the stability of an incompressible flow, particularly in the context of boundary layer theory. It is named after William Richard Orr and Arnold Sommerfeld, who contributed to its development. The equation arises when analyzing small disturbances or perturbations in a basic flow profile. It is particularly important in studying the stability of laminar flows and understanding transition to turbulence.

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