Pressure power spectrum in high-Reynolds number wall-bounded flows

Highlights

• Pressure fluctuations are small-scale quantities compared to velocity fluctuations in a wall-bounded flow (Tsuji, Marusic, Johansson, Int. J. Heat Fluid Flow, vol. 61, 2016, pp. 2–11). The differing scales of pressure and velocity pose a challenge to modeling, and the scaling of the pressure spectrum in wall-bounded flows remains an unsolved issue from both a theoretical and measurement standpoint. To address this unresolved issue, we incorporate Kolmogorov’s theory within the framework of Townsend’s attached eddy hypothesis to account for the small scale nature of pressure fluctuations, leading to the first derivation that is consistent with both theories.

Abstract

We study the behaviors of pressure fluctuations in high Reynolds number wall-bounded flows. Pressure fluctuations are small-scale quantities compared to velocity fluctuations in a wall-bounded flow (Tsuji, Marusic, & Johansson, Int. J. Heat Fluid Flow, vol. 61, 2016, pp. 2–11.): at a given wall-normal distance y, the premultiplied velocity spectrum peaks at a streamwise wavelength on the order of the boundary layer thickness (${\lambda }_x=O\left(\delta \right)$), whereas the premultiplied pressure spectrum peaks at λ_x < O(y). The differing scales of pressure and velocity pose a challenge to modeling, and the scaling of the pressure spectrum in wall-bounded flows remains an unsolved issue from both a theoretical and measurement standpoint. To address this unresolved issue, we incorporate Kolmogorov’s theory (K41) within the framework of Townsend’s attached eddy hypothesis to account for the small scale nature of pressure fluctuations, leading to the first derivation that is consistent with both theories. Our main result is that at a wall-normal distance in the logarithmic layer the premultiplied pressure power spectrum scales as $\left[k_x{E}_{pp}\right]\sim {\lambda }_x^{n-1}{y}^{-(3+n)/4},$ for λ_x < y/tan (θ), and as $\left[k_x{E}_{pp}\right]\sim {\lambda }_x^{(3n-7)/4},$ for λ_x > y/tan (θ). Here, θ is the attached-eddy inclination angle, k_x is the streamwise wavenumber, the velocity spectrum follows a $k^{-1}$ scaling for 1/k_x > y/tan(θ) and a $k^{-5/3}$ scaling for 1/k_x < y/tan (θ), and n is a Reynolds-number-dependent constant. This result conforms to Kolmogorov’s theory of small scale turbulence, i.e., it yields a −7/3 scaling for the small scales at high Reynolds numbers, and also yields the anticipated −1 scaling for the logarithmic layer scales. Detailed analysis shows that pressure and spanwise velocity have differently statistical properties: while an outer peak emerges in the premultiplied spanwise velocity spectrum at high Reynolds numbers, no outer peak is expected in the premultiplied pressure spectrum. The derived scalings are confirmed using data from a direct numerical simulation of a channel flow at friction Reynolds number $R{e}_{\tau }=5200$.

Introduction

Pressure is an important flow quantity for many engineering and environmental flow applications, including vibration and fatigue. According to Kolmogorov’s theory of small-scale turbulence (Kolmogorov, 1941, Obukhov, 1949, Corrsin, 1951), the pressure power spectrum scales as $E_{pp}\sim k^{-7/3}$ in the inertial range. Evidence of this $-7/3$ scaling was later found in, e.g., Gotoh and Fukayama (2001) and Tsuji and Ishihara (2003), in the context of isotropic turbulence at Taylor microscale Reynolds number Re_λ > 600. Predicting pressure statistics in the context of wall-bounded flows is more difficult, and the scaling of the pressure spectrum is an unresolved issue from a theoretical standpoint.

Directly measuring pressure fluctuations in a laboratory experiment is challenging (Lauchle, Daniels, 1987, Tsuji, Imayama, Schlatter, Alfredsson, Johansson, Marusic, Hutchins, Monty, 2012), as pressure signals in wind tunnels are contaminated by noise from the tunnel fan and flow in the return circuit. Tsuji et al. (2012) compared laboratory measurements and direct numerical simulation (DNS) data and concluded that background noise in wind tunnels affect the pressure in the boundary layer, which in turn result in pressure profiles that are different from DNS. Having this caveat in mind, laboratory measurements of Elliott (1972) and Albertson et al. (1998) suggest a $-1.7$ scaling and a $-1.5$ scaling, respectively, instead of the expected $-7/3$ scaling. Because the small scales are approximately isotropic at high Reynolds numbers, in general, we expect Kolmogorov’s theory to be valid at small scales even for flows with mean shear (Pope, 2001).

Compared to laboratory experiments (Tsuji, Fransson, Alfredsson, Johansson, 2005, Liu, Katz, 2006, Tsuji, 2007), getting pressure data from numerical simulations, e.g., DNS, is much more straightforward. For example, Kim (1989) and Abe et al. (2005) divided the pressure source term into a rapid part and a slow part and found that the slow pressure fluctuations are dominant in the channel except very near the wall. Patwardhan and Ramesh (2014) found in a DNS boundary layer flow that the pressure spectrum follows a −1 scaling near the wall (Bradshaw, 1967) and a −7/3 scaling in the outer region. In a recent work, Panton et al. (2017) found that

$$〈p^{\prime }{p}^{\prime }〉\sim \log (\delta /y)$$

in the logarithmic layer, where δ is the half-channel width, ′ indicates the fluctuation from the mean, and  <  ·  >  is the ensemble average.

A similar logarithmic scaling was previously found for the streamwise velocity variance (Smits, McKeon, Marusic, 2011, Marusic, Monty, Hultmark, Smits, 2013), i.e.,

$$〈u^{\prime }{u}^{\prime }〉\sim \log (\delta /y),$$

whose presence in high Reynolds number wall-bounded flows was predicted by Townsend as a result of his attached eddy hypothesis (AEH) (Townsend, 1976, Woodcock, Marusic, 2015, Yang, Abkar, 2018, Marusic, Monty, 2019). Despite their similar forms, the two scalings in Eqs. (1) and (2) are due to eddies at different scales, as pointed out by Tsuji et al. (2016). In their recent work, Tsuji et al. (2016) divided the instantaneous pressure into a large-scale component p_L and a small scale component p_S based on a cutoff length scale λ_c. They examined $〈p_L^2〉$ and $〈p_S^2〉$ and concluded that the logarithmic scaling in Eq. (1) is due to the smaller scales (λ_x ≪ cδ), in contrast to the logarithmic scaling in Eq. (2), which is due to the larger scales (λ_x > cδ). Here, λ_x is the streamwise wavelength, c is a constant (Christensen, Adrian, 2001, Marusic, Heuer, 2007).

Here, we examine the pressure and velocity spectra at a wall-normal distance where Eq. (1) is valid, i.e., in the logarithmic layer. This will give us a better idea of the scales that dominate velocity and pressure fluctuations. Fig. 1(a) shows the compensated pressure variance, i.e.,

$$\text{err}=〈p^{\prime }{p}^{\prime }〉-A_p\log (\delta /y)-B_p,$$

in a $R{e}_{\tau }=5200$ channel (Lee and Moser, 2015), where $A_p=2.6$ and $B_p=.24$. The DNS computational domain is 8πδ × 2δ × 3πδ. The grid resolution is $\Delta x^{+}\times \Delta y^{+}\times \Delta z^{+}=12.7\times 6.4\times 10.4$ at the centerline and is $\Delta y^{+}=0.071$ at the wall. Further details of the DNS can be found in Kim et al. (1987), Lee and Moser (2015), and Graham et al. (2016). According to Fig. 1(a), the variance of the pressure fluctuations follows a logarithmic scaling near $y/\delta =0.1$. Fig. 1(b) shows the premultiplied velocity and pressure spectra as a function of the streamwise wavenumber at the wall-normal height. It is clear that velocity fluctuations are most energetic at large scales whereas pressure fluctuations are most energetic at small scales. Specifically, as will become clear in the later sections, pressure fluctuations are most energetic at scales λ_x < y/tan (θ), and velocity fluctuations are most energetic at scales y/tan (θ) < λ_x (Hu et al., 2020). The differing scales of velocity and pressure fluctuations pose challenge to modeling. Specifically, AEH, which has been quite useful in providing scaling estimates of velocity statistics in high Reynolds number wall-bounded flows, is a model of the large-scale energetic motions not the small scales, and therefore, it does not directly provide scaling estimates for pressure power spectra.

In summary, the scaling of the pressure spectrum in wall-bounded flows is an unsolved problem from both a theoretical and measurement standpoint. The objective of this work is to address this unresolved issue. We do this by combining AEH and K41 to model pressure and velocity fluctuations in a unified framework. In addition to the pressure power spectrum, we will also explore the scaling of other pressure statistics. The rest of the paper is organized as follows. In Section 2, we review the basics of K41 and its estimates of pressure statistics for isotropic turbulence. In Section 3, we combine AEH and K41 and make estimates of pressure statistics in wall-bounded flows. The data are compared to our model in Section 4. We further extend the model in Sections 5 and 6. Concluding remarks are presented in Section 7.