ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence November 9, 2016, Axel Brandenburg

Handout 16: The bottleneck in turbulence

At large wavenumbers k, the energy spectrum

E(k) = ϵ^{2/3} k^{5/3} f(k)  (1)

is expected to have a viscous cutoff that is described by the function f(k). However, f(k) decreases not necessarily monotonically with k.

1 Navier–Stokes equation in Fourier space

For an incompressible fluid, the Fourier-transformed Navier–Stokes equation for û_{k}(t) can be written in the form

\frac{d}{dt}û_{k} = −ikP̂_{k} − ∑_{k=p+q}(û_{p} · iq) û_{q} − νk^{2}û_{k}.  (2)

The pressure satisfies a Poisson-type equation, so

k^{2}P_{k} = ik ∑_{k=p+q}(û_{p} · iq) û_{q}  (3)

and therefore

(ikP_{k})_{i} = −\frac{k_{i}k_{j}}{k^{2}} ∑_{k=p+q}(û_{p} · iq) û_{q}  (4)

or

\frac{d}{dt} ˆû_{k} = −P(k) ∑_{k=p+q}(û_{p} · iq) û_{q} − νk^{2}û_{k},  (5)

where P_{ij} = δ_{ij} − k_{i}k_{j}/k^{2} is the projection operator, which projects out the non-solenoidal (irrotational) components.

An important point here is the fact that all nonlinear inactions proceed via triads in k space where k = p+q. It turns out that viscosity suppresses those triads that reach deep into the dissipative subrange. This makes nonlinear energy transfer less efficient and can lead to a pileup of energy in the inertial range shortly before the dissipative subrange (Falkovich, 1994). This is referred to as the bottleneck effect. To understand why it has not been a prominent effect in wind tunnel and atmospheric turbulence, we have to realize that most observed spectra have been obtained using hot-wire velocimetry and the Taylor hypothesis. We thus have to understand the relation between 1-D and 3-D energy spectra.

2 Relation between 1-D and 3-D energy spectra

To derive the relation between the three-dimensional spectrum E(k) and the total one-dimensional spectrum E_{1D}(k) ≡ E_{L}(k) + 2E_{T}(k), we consider a periodic box of volume V = L_{x}L_{y}L_{z} with a turbulent velocity field u(x), which has the Fourier transform

û(k) = \frac{1}{\sqrt{(2π)^{3}V}} ∫_{V} e^{ik·x}u(x) dx^{3},  (6)

with the inversion

u(x) = \sqrt{\frac{V}{(2π)^{3}}} ∫ e^{−ik·x}û(k) dk^{3}.  (7)

The one-dimensional kinetic energy spectrum is

E_{1D}(k_{z}) = 2 ∫∫ \frac{⟨|û(k)|^{2}⟩}{2} dk_{x} dk_{y},  (k_{z} ≥ 0),  (8)

where ⟨·⟩ denotes an ensemble average, and k = (k_{x}, k_{y}, k_{z}). The factor 2 in Eq. (8) accounts for the fact that E_{1D} does not distinguish between positive and negative k_{z}. Normalization of E_{1D}(k_{z}) is such that

∫_{0}^{∞} E_{1D}(k_{z}) dk_{z} = \frac{u_{rms}^{2}}{2} ≡ \frac{1}{V} ∫_{V} \frac{⟨|u(x)|^{2}⟩}{2} dx^{3}.  (9)

Equation (8) can also be written as the xy-average

E_{1D}(k_{z}) = \frac{1}{L_{x}L_{y}} ∫ ⟨|ũ(x, y, k_{z})|^{2}⟩ dx dy  (10)

and is for homogeneous turbulence equal to ⟨|ũ(x, y, k_{z})|^{2}⟩ at any point (x, y).

The three-dimensional velocity energy spectrum is given by

E(k) ≡ ∫_{4π} \frac{⟨|û(k)|^{2}⟩}{2}k^{2} dΩ_{k},  (11)

where dΩ_{k} denotes the solid angle element in k-space. E(k) satisfies the relation

∫_{0}^{∞} E(k) dk = \frac{u_{rms}^{2}}{2} .  (12)

If u is statistically isotropic in the sense that the ensemble average of the spectral energy of the velocity ⟨|u(k)|^{2}⟩ is only a function of k = |k|, then E(k) becomes

E(k) = 4πk^{2} \frac{⟨|û(k)|^{2}⟩}{2}.  (13)

To evaluate E_{1D} in this case, we introduce cylindrical coordinates (k_{r}, ϕ, k_{z}) in k-space and write the double integral (8) in the form

E_{1D}(k_{z}) = 2 ∫_{0}^{∞} \frac{⟨|û(k)|^{2}⟩}{2} 2πk_{r} dk_{r}
= 4π ∫_{k_{z}}^{∞} \frac{⟨|û(k)|^{2}⟩}{2} k dk,  (14)

since k^{2} = k_{r}^{2} + k_{z}^{2}, and therefore k_{r}^{2} = k^{2} − k_{z}^{2}. Comparing with Eq. (13), we see that

E_{1D}(k_{z}) = ∫_{k_{z}}^{∞} \frac{E(k)}{k} dk,  (15)

the inversion of which gives

E(k) = −k \frac{dE_{1D}(k)}{dk} = −E_{1D} \frac{d ln E_{1D}(k)}{d ln k} .  (16)

References

Dobler, W., Haugen, N. E. L., Yousef, T. A., & Brandenburg, A., “Bottleneck effect in three-dimensional turbulence simulations,” Phys. Rev. E 68, 026304 (2003).

Falkovich, G., “Bottleneck phenomenon in developed turbulence,” Phys. Fluids 6, 1411-1414 (1994).

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