Hydrodynamic Turbulence: Sweeping Effect and Taylor’s Hypothesis via Correlation Function

Abstract

We demonstrate the sweeping effect in turbulence using numerical simulations of hydrodynamic turbulence without a mean velocity. The velocity correlation function, $C(\mathbf{k},\tau )$, decays with time due to the eddy viscosity. In addition, $C(\mathbf{k},\tau )$ shows oscillations due to the sweeping effect by “random mean velocity field” ${ \tilde{\mathbf{U}}}_0$. We also perform numerical simulation with mean velocity $\mathbf{U}_0= 10\hat{z}$ (10 times the rms speed) for which $C(\mathbf{k},\tau )$ exhibits damped oscillations with the frequency of $|\mathbf{U}_0| k$ and decay time scale corresponding to the $\mathbf{U}_0=0$ case. For $\mathbf{U}_0=10\hat{z}$, the phase of $C(\mathbf{k},\tau )$ shows the sweeping effect, but it is overshadowed by oscillations caused by $\mathbf{U}_0$. We also demonstrate that for $\mathbf{U}_0=0$ and $10\hat{z}$, the frequency spectra of the velocity fields measured by real-space probes are respectively $f^{-2}$ and $f^{-5/3}$; these spectra are related to the Lagrangian and Eulerian space-time correlations respectively.

Appendices

Appendix 1: Sweeping Effect and Renormalization in Eulerian Framework

In this section we extend iterative renormalisation group (i-RG) of McComb (1990) and Zhou (2010) to include the effects of the mean velocity field $\mathbf{U}_0$. We show that the renormalised viscosity is independent of $\mathbf{U}_0$. However, this scheme fails to capture the sweeping effect. This issue was first raised by Kraichnan (1964) in direct interaction approximation (DIA) framework. Note that the above computations are based on Eulerian framework. Since the above RG scheme is covered in detail in many references, such as McComb (1990), Zhou (2010) and Verma (2001, (2004), here we highlight the changes induced by $\mathbf{U}_0$.

In Fourier space, the Navier–Stokes equations in the presence of $\mathbf{U}_0$ are (McComb 1990)

$$\begin{aligned} (-i\omega + i {\mathbf {U}_0 \cdot \mathbf {k}} + \nu k^{2} )u_{i}(\hat{{k}})= & {} -\frac{i}{2}P_{ijm}(\mathbf {k})\int _{\hat{{p}}+\hat{{q}}=\hat{{k}}}d\hat{{p}}\left[ u_{j}(\hat{{p}})u_{m}(\hat{{q}})\right] + f_i(\hat{k}) , \end{aligned}$$

(38)

$$\begin{aligned} k_i u_i(\mathbf{k})= & {} 0, \end{aligned}$$

(39)

where

$$\begin{aligned} P_{ijm}(\mathbf {k})= & {} k_{j}P_{im}(\mathbf {k})+k_{m}P_{ij}(\mathbf {k}),\nonumber \\ \hat{k}= & {} (\omega , \mathbf {k}), \hat{p} = (\omega ', \mathbf {p}), \mathrm {and} \, \, \hat{q} = (\omega '', \mathbf {q});~~ \hat{k} = \hat{p} + \hat{q} . \end{aligned}$$

(40)

We compute the renormalized viscosity in the presence of a mean velocity $\mathbf {U}_0$. In the renormalization process, the wavenumber range $(k_{N},k_{0})$ is divided logarithmically into N shells. The nth shell is $(k_{n},k_{n-1})$ where $k_{n}=h^{n}k_{0}\,\,(h<1)$ and $k_N = h^N k_0$. In the first step, the spectral space is divided in two parts: the shell $(k_{1},k_{0})=k^{>}$, which is to be eliminated, and $(k_{N},k_{1})=k^{<}$, set of modes to be retained. The velocity modes in the $k^{>}$ regime are averaged. The averaging procedure enhances the viscosity, and the new viscosity is called “renormalized viscosity”. The process is continued for other shells that leads to larger and larger viscosity.

In i-RG scheme, after $(n+1)$st step, the renormalized equation appears as

$$\begin{aligned}&\bigl [ -i\omega + i {\mathbf {U}_0 \cdot \mathbf {k}} + ( \nu _{(n)}(k) +\delta \nu _{(n)}(k)) k^2 \bigr ] u_{i}^{<}(\hat{k}) = \nonumber \\&\qquad -\frac{i}{2}P_{ijm}(\mathbf{k }) \int _{\hat{{p}}+\hat{{q}} = \hat{k}} \frac{d \mathbf {p} d\omega '}{(2\pi )^{d+1}} [u_{j}^{<}(\hat{p})u_{m}^{<}(\hat{k}-\hat{p})] + f^<_i(\hat{k} ) \end{aligned}$$

(41)

with

$$\begin{aligned} \delta \nu _{(n)}(\hat{k}) k^2 = \frac{1}{d-1}\int _{\hat{p}+\hat{q}=\hat{k}}^{\Delta } \frac{d\mathbf {p} d\omega '}{(2\pi )^{d+1}} [B(k,p,q)G(\hat{q})C(\hat{p})]. \end{aligned}$$

(42)

In the above expression,

$$\begin{aligned} B(k,p,q)=k p [(d-3)z+2z^{3}+(d-1)x y], \end{aligned}$$

(43)

where d is the space dimensionality, x, y, z are the direction cosines of ${{\mathbf {k}}, {\mathbf {p}}, {\mathbf {q}}}$, and $G(\hat{q}), C(\hat{p})$ are respectively Green’s and correlation functions that are defined as (McComb 1990; Zhou 2010; Verma 2004)

$$\begin{aligned} G(\hat{q})= & {} \frac{1}{-i \omega '' + i {\mathbf {U}_0 \cdot \mathbf {q}} + \nu _{(n)}(q) q^2}, \end{aligned}$$

(44)

$$\begin{aligned} C(\hat{p})= & {} \frac{C(\mathbf{p})}{-i \omega ' + i {\mathbf {U}_0 \cdot \mathbf {p}} + \nu _{(n)}(p) p^2}. \end{aligned}$$

(45)

Using $\omega = \omega ' + \omega ''$, we obtain

$$\begin{aligned} \delta \nu _{(n)}(\omega , k) k^2= & {} \frac{1}{d-1} \int _{\hat{p}+\hat{q}=\hat{k}} \frac{d\mathbf {p} d\omega '}{(2\pi )^{d+1}} B(k,p,q) C(\mathbf {p}) \nonumber \\&\times \frac{1}{\bigl [ -i\omega + i\omega ' + i {\mathbf {U}_0 \cdot \mathbf {q}} + \nu _{(n)}(q) q^2 \bigr ] \bigl [-i\omega ' + i {\mathbf {U}_0 \cdot \mathbf {p}} + \nu _{(n)}(p)p^2 \bigr ]} \nonumber \\= & {} \frac{1}{d-1} \int _\mathbf{{p + q = k}}^{\Delta } \frac{d\mathbf {p}}{(2\pi )^d} \frac{B(k,p,q) C(\mathbf {p})}{ \bigr [ -i(\omega - {\mathbf {U}}_0 \cdot {\mathbf {k}}) + \nu _{(n)}(p) p^2 + \nu _{(n)}(q) q^2 \bigr ]} \nonumber \\= & {} \frac{1}{d-1} \int _\mathbf{{p + q = k}}^{\Delta } \frac{d\mathbf {p}}{(2\pi )^d} \frac{B(k,p,q) C(\mathbf {p})}{ \nu _{(n)}(p) p^2 + \nu _{(n)}(q) q^2 }. \end{aligned}$$

(46)

Note that $\omega - {\mathbf {U}}_0 \cdot {\mathbf {k}}= \omega _D$ is the Doppler-shifted frequency in the moving frame, where the frequency of the signal is reduced. It is analogous to the reduction of frequency of the sound wave in a moving train when the train moves away from the source. For $\mathbf{U}_0=0$, it is customary to assume that $\omega \rightarrow 0$ since we focus on dynamics at large time scales (McComb 1990; Zhou 2010; McComb 2014). The corresponding assumption for $\mathbf{U}_0\ne 0$ is to set $\omega _D \rightarrow 0$ because $\omega _D$ is the effective frequency of the large scale modes in the moving frame. The approximation $\omega \rightarrow \omega _D$ essentially takes away the effect of Galilean transformation and provides inherent turbulence properties. Note that in Taylor’s frozen-in turbulence hypothesis, $\omega = \mathbf{U}_0 \cdot \mathbf{k}$ that yields $\omega _D = 0$ (Tennekes and Lumley 1972).

Equation (46) indicates that the correction in viscosity, $\delta \nu _{(n)}$, is independent of $\mathbf{U}_0$. After this step, the derivation of renormalised viscosity with and without $\mathbf{U}_0$ are identical. Equation (46) however does not include any sweeping effect, which is a serious limitation of Eulerian field theory, as pointed out by Kraichnan (1964) in direct interaction approximation (DIA) framework. Kraichnan (1965) then formulated Lagrangian-history closure approximation for turbulence and showed consistency with Kolmogorov’s spectrum (also see Leslie 1973). Effectively, a consistent theory needs to include a term of the form $i\tilde{\mathbf{U}}_0 \cdot \mathbf{q}$ in the denominator of Eq. (44). A procedure adopted by Verma (1999) for “mean magnetic field” renormalisation in magnetohydrodynamic turbulence may come out to be handy for such computations, which may be attempted in future.

Appendix 2: Computation of Spatio-Temporal Correlations and Frequency Spectra of Turbulent Flow

Using the normalized correlation function of Eq. (22), we derive the following spatio-temporal correlation function:

$$\begin{aligned} C(\mathbf{r}, \tau )= & {} \int d\mathbf{k} C(\mathbf{k}) \exp (-\nu (k) k^2 \tau -i\mathbf{U_0 \cdot k} \tau ) \exp (-i \mathbf{k} \cdot \tilde{\mathbf{U}}_0(\mathbf{k}) \tau ) \exp (i \mathbf{k} \cdot \mathbf{r}). \end{aligned}$$

(47)

We time average $\tilde{U}_0$ over random ensemble (Kraichnan 1964; Wilczek and Narita 2012) that yields

$$\begin{aligned} C(\mathbf{r}, \tau )= & {} \int d\mathbf{k} C(\mathbf{k}) \exp (-\nu (k) k^2 \tau - i\mathbf{U_0 \cdot k} \tau ) \langle \exp (-i c k \tilde{U}_0( k)\tau ) \rangle \exp (i \mathbf{k} \cdot \mathbf{r}) \nonumber \\= & {} \int d\mathbf{k} C(\mathbf{k}) \exp (-\tau /\tau _\mathrm{{c}} - i\mathbf{U_0 \cdot k} \tau ) \exp (-k^2 [\tilde{U}_0( k)]^2 \tau ^2) \exp (i \mathbf{k} \cdot \mathbf{r}). \end{aligned}$$

(48)

In addition, we set $\mathbf{r}=0$ to compute the temporal correlation at a single point.

In the above integral, following Pope (2000), we replace the isotropic and homogeneous $C(\mathbf{k})$ with

$$\begin{aligned} C(\mathbf{k}) = \frac{E(k)}{4\pi k^2} = \frac{f_L(kL) f_\eta (k\eta ) K_\mathrm {Ko} \epsilon ^{2/3} k^{-5/3}}{4\pi k^2}, \end{aligned}$$

(49)

where $\epsilon$ is the energy dissipation rate, which is same as the energy flux, and

$$\begin{aligned} f_L(kL)= & {} \left( \frac{kL}{[(kL)^2 + c_L]^{1/2}} \right) ^{5/3+p_0}, \end{aligned}$$

(50)

$$\begin{aligned} f_\eta (k\eta )= & {} \exp \left[ -\beta \left\{ [ (k \eta )^4 + c_\eta ^4 ]^{1/4} - c_\eta \right\} \right] , \end{aligned}$$

(51)

with $c_L, c_\eta , p_0, \beta$ as constants, and L as the large length scale of the system. We also substitute $\tau _\mathrm{{c}}(k) = 1/(\nu (k) k^2) = \epsilon ^{-1/3}k^{-2/3}$ and $\tilde{U}_0(k) = \epsilon ^{1/3}k^{-1/3}$ (from dimensional analysis). We ignore the coefficients in front of these quantities for brevity. After the above substitutions, we obtain

$$\begin{aligned} C(\tau )= & {} K_\mathrm {Ko} \epsilon ^{2/3} \int dk k^{-5/3} f_L(kL) f_\eta (k\eta ) \exp (-i\mathbf{U_0 \cdot k} \tau ) \times \nonumber \\&\exp (-\epsilon ^{1/3}k^{2/3}\tau ) \exp (-\epsilon ^{2/3}k^{4/3}\tau ^2). \end{aligned}$$

(52)

The above form of $C(\tau )$ is valid for any $\mathbf{U}_0$. The above integral is too complex, hence we perform asymptotic analysis in two limiting cases that are described below.

For $\mathbf {U}_0 \cdot \mathbf {k} \gg \nu (k) k^2$ and $\mathbf {U}_0 \cdot \mathbf {k} \gg k \tilde{U}_0(k)$

For this case $U_0$ dominates other velocity scales, hence we take $\tau \sim 1/(U_0 k)$ as the dominant time scale. For simplification, we make a change of variable, $\tilde{k} = U_0 k \tau$. In addition, we choose the z axis to be along the direction of $\mathbf{U}_0$. Under these simplifications, the integral becomes

$$\begin{aligned} C(\tau )\approx & {} K_\mathrm {Ko} (\epsilon U_0 \tau )^{2/3} \int d{\tilde{k}} \tilde{k}^{-5/3} f_L(\tilde{k} (L/U_0\tau )) f_\eta ( \tilde{k} (\eta / U_0 \tau ) \frac{\sin (U_0 k \tau )}{U_0 k \tau } \nonumber \\&\times \exp [- \tilde{k}^{2/3} (U/U_0)^{2/3} (\tau /T)^{1/3} - \tilde{k}^{4/3} (U/U_0)^{4/3} (\tau /T)^{2/3} ]. \end{aligned}$$

(53)

We focus on $\tau$ in the inertial range, hence $L/U_0\tau \gg 1$ and $\eta / U_0 \tau \ll 1$, consequently, $f_L(\tilde{k} (L/U_0\tau )) \approx 1$, and $f_\eta ( \tilde{k} (\eta / U_0 \tau ) \approx 1$. Therefore,

$$\begin{aligned} C(\tau )\approx & {} K_\mathrm {Ko} (\epsilon U_0 \tau )^{2/3} \int d{\tilde{k}} \tilde{k}^{-5/3} \frac{\sin \tilde{k}}{\tilde{k} } \exp [- \tilde{k}^{2/3} (U/U_0)^{2/3} (\tau /T)^{1/3} - \tilde{k}^{4/3} (U/U_0)^{4/3} (\tau /T)^{2/3} ] \nonumber \\\approx & {} B K_\mathrm {Ko}(\epsilon U_0 \tau )^{2/3}, \end{aligned}$$

(54)

where B is the value of the nondimensional integral. The Fourier transform of the above $C(\tau )$ yields the following frequency spectrum:

$$\begin{aligned} E(f)\approx & {} \int C(\tau ) \exp (i 2\pi f \tau ) d\tau = \int B K_\mathrm {Ko}(\epsilon U_0 \tau )^{2/3} \exp (i 2\pi f \tau ) d\tau \nonumber \\\sim & {} (\epsilon U_0)^{2/3} f^{-5/3}. \end{aligned}$$

(55)

The above frequency spectrum is the prediction of Taylor’s frozen-in turbulence hypothesis.

For $\mathbf {U}_0=0$

We set $\mathbf {U}_0 =0$ in Eq. (52). In the resulting equation, both the remaining exponential terms (the damping and sweeping effect terms) have the following time scale:

$$\begin{aligned} \tau (k) \sim 1/(k u_k) \sim \epsilon ^{-1/3} k^{-2/3}. \end{aligned}$$

(56)

Hence, for computing the integral $C(\tau )$, we make a change of variable:

$$\begin{aligned} k = \tilde{k} \epsilon ^{-1/2} \tau ^{-3/2} \end{aligned}$$

(57)

that transforms the integral to

$$\begin{aligned} C(\tau )\approx & {} K_\mathrm {Ko} \epsilon \tau \int d{\tilde{k}} \tilde{k}^{-5/3} f_L(\tilde{k} (L/U\tau )^{3/2}) f_\eta ( \tilde{k} (\tau _d/ \tau )^{3/2}) \exp (- \tilde{k}^{2/3} - \tilde{k}^{4/3} ), \end{aligned}$$

(58)

where U is the large-scale velocity, and $\tau _d$ is the dissipative time scale. We focus on $\tau$ in the inertial range, hence $L/U\tau \gg 1$ and $\tau _d/ \tau \ll 1$. Therefore, using Eqs. (50), (51), we deduce that $f_L(\tilde{k} (L/U\tau )^{3/2}) \approx 1$ and $f_\eta ( \tilde{k} (\tau _d/ \tau )^{3/2}) \approx 1$. Therefore,

$$\begin{aligned} C(\tau ) \approx K_\mathrm {Ko} \epsilon \tau \int d{\tilde{k}} \tilde{k}^{-5/3} \exp (- \tilde{k}^{2/3} - \tilde{k}^{4/3} ) \approx A K_\mathrm {Ko} \epsilon \tau , \end{aligned}$$

(59)

where A is the value of the integral of Eq. (59). The Fourier transform of $C(\tau )$ yields the following frequency spectrum:

$$\begin{aligned} C(f)= & {} \int C(\tau ) \exp (i 2\pi f \tau ) \mathrm{{d}}\tau = A K_\mathrm {Ko} \epsilon \int \tau \exp (i 2\pi f \tau ) \mathrm{{d}}\tau \sim \epsilon f^{-2}. \end{aligned}$$

(60)

Thus, the damping and sweeping terms yield frequency spectrum $E(f) \sim f^{-2}$.

We could also derive the above frequency spectra using scaling arguments Landau and Lifshitz (1987). From Eq. (23), we obtain the dominant frequency as

$$\begin{aligned} \omega = \mathbf {U}_0 \cdot \mathbf {k} +c k \tilde{U}_0(k) -i \nu (k) k^2. \end{aligned}$$

(61)

When $\mathbf {U}_0 \cdot \mathbf {k} \gg \nu (k) k^2$ and $\mathbf {U}_0 \cdot \mathbf {k} \gg k \tilde{U}_0(k)$, we obtain $\omega = U_0 k_z$. Therefore, using the formula for one-dimensional spectrum $E(k) = K_\mathrm {Ko} \epsilon ^{2/3} k^{-5/3}$, and $\omega = 2 \pi f$, we obtain

$$\begin{aligned} E(f)= & {} E(k) \frac{\mathrm{{d}}k}{\mathrm{{d}}f} \sim (\epsilon U_0)^{2/3} f^{-5/3}. \end{aligned}$$

(62)

On the contrary, when $\mathbf {U}_0 \cdot \mathbf {k} \ll \nu (k) k^2$ (for zero or small $U_0$), we obtain $\omega \approx \nu (k) k^2 = \nu _* \sqrt{K_\mathrm {Ko}} \epsilon ^{1/3} k^{2/3}$ and hence,

$$\begin{aligned} E(f)= & {} E(k) \frac{\mathrm{{d}}k}{\mathrm{{d}}f} \sim \epsilon f^{-2}, \end{aligned}$$

(63)

consistent with the formulas derived earlier.

The spectral exponent ($-2$) for Burgers equation matches the above exponent for the frequency spectrum (for the $U_0 = 0$ case). However, there are important differences between Burgers turbulence and hydrodynamic turbulence. Burgers turbulence exhibits $k^{-2}$ spectrum in wavenumber space (e.g. see  Verma 2000), but hydrodynamic turbulence for $U_0=0$ case shows $f^{-2}$ spectrum in frequency space. The $k^{-2}$ spectrum in the Burgers turbulence is related to shocks, but $f^{-2}$ spectrum for hydrodynamics has no connection to shocks.

Elliptic Approximation

In Section “Taylor’s Frozen-in hypothesis for $\mathbf{U}_0 \ne 0$, and frequency spectrum” we showed that the equal-time velocity correlation for $\mathbf{U}_0=0$ matches with unequal-space temporal correlation for nonzero $\mathbf{U}_0$ (see Eq. (33)). Elliptic approximation combines the Taylor’s frozen-in hypothesis with the sweeping effect. This task was performed by He et al. (2010) and He and Tong (2011). Here, we reproduce their arguments using Eq. (47).

We consider a fluid flow with a mean velocity of $\mathbf{U}_0$ along the z axis. We focus on the vertical velocities measured at two points z and $z+r$, but at times t and $t+\tau$ (see Fig. 8 for an illustration). For the same, the space-time correlation derived using Eq. (47) is

$$\begin{aligned} C(r, \tau )= & {} \int d\mathbf{k} C(\mathbf{k}) \exp (-\nu (k) k^2 \tau ) \exp \left\{ [r -i(U_0 +\tilde{U}_{0z}) \tau )] i k_z -i \tilde{\mathbf{U}}_{0\perp } \cdot \mathbf{k}_\perp \tau \right\} . \end{aligned}$$

(64)

Now suppose that

$$\begin{aligned} r \approx U_0 \tau \gg \nu (k) k^2 \tau , \end{aligned}$$

(65)

then

$$\begin{aligned} C(r,\tau ) = \int d\mathbf{k} C(\mathbf{k}) \exp \left\{ [r -(U_0 +\tilde{U}_{0z}) \tau )] i k_z -i \tilde{\mathbf{U}}_{0\perp } \cdot \mathbf{k}_\perp \tau \right\} . \end{aligned}$$

(66)

We can relate the above correlation function to an equal-time correlation function

$$\begin{aligned} C(\mathbf{r}_E, 0) = \exp [ i r_{Ez} k_z + i \mathbf{r}_{E\perp } \cdot \mathbf{k}_{\perp }] \end{aligned}$$

(67)

with

$$\begin{aligned} r_{Ez} = [r -(U_0 +\tilde{U}_{0z}) \tau )];~~ \mathbf{r}_{E\perp } = \tilde{\mathbf{U}}_{0\perp } \tau \end{aligned}$$

(68)

or

$$\begin{aligned} r_E^2 = r_{Ez}^2 + |\mathbf{r}_{E\perp }|^2 = (r-U\tau )^2 + (V\tau )^2, \end{aligned}$$

(69)

where

$$\begin{aligned} U= & {} U_0 +\tilde{U}_{0z} \end{aligned}$$

(70)

$$\begin{aligned} V= |\tilde{\mathbf{U}}_{0\perp }|. \end{aligned}$$

(71)

This is the statement of elliptic approximation (He et al. 2010; He and Tong 2011). Our derivation is slightly different from those of He et al. (2010) and He and Tong (2011).

Fig. 8

A and B represent respectively the velocity measurements at locations z and $z+r$ and at times t and $t+\tau$. The fluid element at B would be at $\mathrm{B}^\prime$ at time t, thus A and $\mathrm{B}^\prime$ would represent equal-time measurements. Note that $r_E = r-U_0 \tau$

Thus, the elliptic approximation includes both, the sweeping effect and Taylor’s frozen-in turbulence hypothesis. The velocities $U_0$ and $\tilde{U}_0$ yield the Eulerian and Lagrangian space-time correlations respectively, and they are related to the sweeping effect and Taylor’s hypothesis respectively. It is easy to see that the conventional Taylor’s hypothesis is applicable when $U_0 \gg \tilde{U}_0$ and it yields $f^{-5/3}$ spectrum, for which the physical interpretation is as follows. The velocity correlation for the velocity measurements at A and B of Fig. 8, $C(r, \tau ) = \langle \mathbf{u}(z,t) \mathbf{u}(z+r,t+\tau ) \rangle$, is same as those measured at A and $\mathrm{B}^\prime$ at the same time t, $C(r_E, 0) = \langle \mathbf{u}(z,t) \mathbf{u}(z+r- U_0 \tau ,t) \rangle$. This is because the fluid element at $\mathrm{B}^\prime$ at time t reaches B at time $t+\tau$.