Oscillations, Waves, and Interactions - GWDG

88 D. Ronneberger et al.
e. g. temporal, is taken of the equation, as was applied in the decomposition of the
field quantities. As a result the mean transport of momentum which is described by
the tensor ρuu t is composed of two parts: ρuu t = ρ u ′ u ′ t + ρ u u t wherein the
fluctuation of the density is disregarded. So the fluctuation of the velocity contributes
to the mean transport of momentum by the covariance of the velocity components.
The term −ρ u ′ u ′ t is also known as Reynolds stress tensor because it contributes to
the balance of forces between the volume elements in the same way as the pressure
and the viscous friction. Particularly the Reynolds shear stress τ ′′ := −ρu ′ v ′ which
exceeds the viscous shear stress τ µ := µ ∂ u/∂y by orders of magnitude in turbulent
flow, plays the dominant role in the development of the mean flow profile.
On the other hand, the mean velocity profile is crucial to the development of the
Reynolds tensor. To understand this we consider the hypothetical case that the
fluctuating part of the flow consists of a single mode of a small-amplitude oscillation
so that the Navier Stokes equation can be linearized. Additionally we confine our
consideration to the simple incompressible flow u = [U(y), 0, 0] t in a 2D channel
where we can use the wave ansatz
{u ′ , v ′ , p ′ }(x, y, t) = {û, ˆv, ˆp}(y) · exp[i(αx − ωt)] (3)
(with angular frequency ω, wavenumber α, axial and wall-normal space coordinates
x and y). Then the Orr-Sommerfeld equation is obtained for the amplitude ˆv(y) of
the wall-normal component of the flow velocity. This is a fourth-order differential
equation which reduces to the second order Rayleigh equation if the effects of viscosity
are disregarded:
d2ˆv =
dy2 �
α 2 + d2 U/dy 2
U − ω/α
�
·ˆv ; û = i
dˆv
α dy
�
i
; ˆp = ρ
α
dU
�
ˆv − i(ω − αU)ρû
dy
and for the sake of completeness the Rayleigh equation has been supplemented by
relations for û and ˆp. Together with the boundary conditions at the walls an eigenvalue
problem is established which generally has a number of solutions ω = Ω n(α).
Each of these solutions is the dispersion relation of a mode which may contribute to
the fluctuating part of the flow. It is obvious from the Rayleigh equation (4) that
the dispersion relations and consequently (û, ˆv)n as well as the Reynolds shear stress
τ ′′ n = −ρ 1
2 ℜ{û · ˆv∗ }n depend on the mean flow profile, in particular on d 2 U/dy 2 .
Furthermore it should be noted that τ ′′ n crucially depends on the phase difference
between û and ˆv and hence can appear also as a negative-friction force.
3.1.2 Exchange of energy between mean flow and flow oscillation
The Reynolds shear stress is essential for the exchange of energy between the mean
flow and the flow oscillation. The rate at which energy per volume is transferred
from the mean to the oscillating part of the flow is given by the product of shear
stress and shear rate τ ′′ · dU/dy (angular momentum × angular velocity). From the
incompressible Navier Stokes equations, one obtains
τ ′′ · dU
dy = −ρu′ v ′ · dU
dy
(4)
d ρ
= U
dx 2 u′2 + v ′2 + d
dy p′ v ′ + ɛµ . (5)