1 Introduction

Due to the ubiquity and importance in turbomachinery, the wall-bounded turbulent flow in a rotating reference frame has received a great deal of attention. Ever since the experimental investigations of a spanwise rotating plane Poiseuille flow (RPPF) conducted by Johnston et al.^[1], it has been well accepted that the existence of the Coriolis force due to the system rotation will alter both the mean flow and the turbulent fluctuations^[2]. In order to unravel the physics of turbulence in rotating wall-bounded flows, many studies were conducted on some typical flow categories with simple flow geometries, among which RPPFs and spanwise rotating plane Couette flows (RPCFs) are two canonical prototypes^[3]. In an RPPF, since the mean vorticity changes its sign across the channel, the Coriolis force destabilizes the flow in part of the channel while stabilizes it in the other part. In contrast, the Coriolis force may either stabilize or destabilize the entire flow in the RPCF, since the mean velocity increases monotonically from one wall to the other in most situations.

In the past decades, many experimental and numerical studies have been carried out to investigate the effects of spanwise rotations on RPPFs and RPCFs. The experimental work for RPPFs^{[1, 4-5]} and RPCFs^[6-9] and the direct numerical simulation (DNS) for RPPFs^[10-18] and RPCFs^[19-24] are examples. Many flow characteristics, e.g., mean flow profiles, turbulence intensities, Reynolds shear stress, correlation coefficients, power spectra, and flow structures, were reported, which made them popular test cases for both the Reynolds-averaged Navier-Stokes (RANS) communities and the large-eddy simulation (LES) communities.

The mean energy dissipation rate per unit volume ϕ for incompressible flows has been widely studied. It is defined as follows^[25]:

(1)

where the standard Cartesian tensor notation and summation on repeated indices are used. ⟨·⟩ denotes the quantities averaged in the wall parallel directions as well as in the time. μ is the dynamic viscosity. S_ij, D_ij, and d_ij are the total, mean, and fluctuating parts of the velocity deformation tensor, respectively. The dissipation function Φ, which is defined as the integral of ϕ over the channel cross section, also has a mean part Φ_M (the contribution from the mean shear) and a fluctuating part Φ_T (the contribution from the turbulent fluctuations). It has been shown by Laadhari^[26] and Abe and Antonia^[27] that, in the turbulent plane Poiseuille flow (PPF) at a sufficiently high Reynolds number, Φ_M⁺=Φ_M/(ρu_τ³), where is a wall friction velocity, τ_w is the mean wall-shear stress, and ρ is the fluid density, reaches a constant value while Φ_T⁺=Φ_T/(ρu_τ³) follows a logarithmic law on the friction Reynolds number Re_τ=u_τh/ν, where h is the channel half-width, and ν is the kinematic viscosity, directly resulting in a logarithmic skin friction law without any assumption on the mean velocity distribution. Abe and Antonia^[27] argued that the approach based on Φ provided an interesting insight into the logarithmic skin friction law.

The aim of the present paper is to investigate the effects of the spanwise system rotation on the energy dissipation function in both RPPFs and RPCFs with available DNS databases. The remainder of the paper is organized as follows. The relationships for the dissipation function Φ in a RPPF, a turbulent plane Couette flow (PCF), and a RPCF are derived in Section 2. The results of Φ in a RPPF, a PCF, and a RPCF are presented and discussed in Section 3. A summary is given in Section 4.

2 Relationships for the dissipation function Φ in RPPF, PCF, and RPCF

The turbulent Poiseuille flow, RPPF, PCF, and RPCF are unidirectional, where the mean flow can be denoted as U_i(y)=(U(y), 0, 0). In the present work, x, y, and z or x₁, x₂, and x₃ denote the streamwise, wall-normal, and spanwise coordinate directions, respectively, and u₁, u₂, and u₃ are the velocity fluctuations in the corresponding directions; u, v, and w are used interchangeably with u₁, u₂, and u₃. Following the derivation by Laadhari^[26], we have

(2)

Here, the first and second terms on the rightmost part of Eq. (2) are the mean part Φ_M and the fluctuating part Φ_T of the dissipation function Φ. According to the balance equation for the total turbulent kinetic energy across the channel, it is easy to obtain that

(3)

In order to distinguish these two evaluations of Φ_T, we denote the original definition Φ_{T, 1} and the calculation based on the turbulent kinetic energy production Φ_{T, 2}. Then, we have

(4)

In a turbulent PPF without system rotations, the total shear has the following form:

where τ_w is the mean wall shear stress. Thus,

Applying the integration by parts and the no-slip boundary conditions at the walls, we have

(5)

where is the bulk velocity. Equation (5) is exactly the energy conservation relation, i.e., the total energy dissipated in a statistically stationary channel equals the total energy input via the mean pressure gradient^[26-27]. When normalized with the friction velocity u_τ and density ρ,

(6)

where Re_b=U_b2h/ν is the bulk Reynolds number. Laadhari^[26] founded that Φ_M⁺=18.27 while Φ_T⁺=5.2 ln Re_τ-14.17 when Re_τ > 500. Therefore, U_b⁺ follows a logarithmic low when U_b⁺=A ln Re_τ+B in which A=2.6 and B=2.05, and a relationship between Re_b and Re_τ can be obtained as follows: Re_b=2Re_τ(A ln Re_τ+B). Abe and Antonia^[27] found a similar logarithmic law for Φ_T⁺ for Re_τ≥300. However, the slope (5.08) and the intercept (-13.44) were different, and so were the constants A and B.

In the RPPF, the total shear has a different form as follows^{[10, 14, 16]}:

(7)

where the mean pressure gradient

is constant at a fixed rotation number. Equations (4) and (7) lead to

(8)

which is the same as that in the PPF. In the RPPF, -dP/dx is directly related to the global wall friction velocity, i.e.,

Thus, Eq. (6) can be achieved too if the global wall friction velocity is used.

In the PCF and RPCF, the total shear is constant across the channel^{[19, 21]}, i.e.,

(9)

Thus,

(10)

where U_w is half of the wall velocity difference, which is approximately equal to the bulk velocity U_b in the PCF and RPCF. Again, Eq. (10) has the same expression as Eq. (5), which is a direct relationship coming from the energy conservation in a statistically stationary wall-bounded unidirectional turbulent flow. The nondimensionalized form is

(11)

where Re_w=2hU_w/ν.

3 Results and discussion 3.1 Dissipation function Φ in PCF

In this subsection, we would like to present the dissipation function Φ in the PCF. The DNS data from Avsarkisov et al.^[28] are used for the analysis. Avsarkisov et al.^[28] presented a new set of DNS results in a large computational box at four Reynolds numbers, i.e., Re_τ = 125, 180, 250, and 550. Figure 1(a) shows the mean shear Φ_M⁺ and turbulent Φ_T⁺ contributions to the nondimensionalized (using the friction velocity u_τ) dissipation function Φ⁺=Φ/u_τ³ of a PCF at five different Re_τ in which the data are from Refs. [23] and [28]. It can be seen from the DNS data that Φ_M⁺=18.0 is also a constant for all the five Reynolds numbers, while Φ_T⁺, estimated from the integrations of both the turbulent energy production and the turbulent energy dissipation rate, follows a logarithmic law

(12)

The slope 4.67 is smaller than those for the PPFs estimated by Laadhari^[26] and Abe and Antonia^[27]. Furthermore, the values of Φ_T⁺ from the PCF are larger than that from the PPF at the same Re_τ if Re_τ≤ 9.7×10⁵. We attribute these deviations on the slope and values to the flow differences between the PPF and PCF. Avsarkisov et al.^[28] showed the root mean square (RMS) velocity fluctuation profiles from the PPF and PCF at Re_τ=550, and turned out that the off-wall intensities in all the three directions from the PCF are larger than those from the PPF for y⁺≥ 15. Higher turbulent intensities correspond to higher Φ_T⁺.

It is interesting to notice that the constant behavior of Φ_M⁺ exists at a much lower Reynolds number range in the PCF. This is not surprising, since the mean velocity follows a log law in which κ=0.41 and B=5.1, and no dependence of B on Re_τ in the range from 125 to 550 has been found^[28]. In contrast, due to the low-Reynolds number effect, the apparent log law at Re_τ=180 in the PPF has a larger intercept (B=5.5) than those in higher Reynolds number flows^[29-32].

Figure 1(b) shows the distributions of (dU⁺/dy⁺)² and -⟨uv⟩⁺ dU⁺/dy⁺ from the PCF at five different Re_τ in wall units. It is clearly seen that, when y⁺≤30, the profiles of (dU⁺/dy⁺)² and exhibit universal behaviors. Above this location, (dU⁺/dy⁺)² decreases faster than (y⁺)^-1, which provides a negligible contribution to Φ_M⁺, whereas the decay of -⟨uv⟩⁺ dU⁺/dy⁺ is close to (y⁺)^-1 in a region whose extent increases with Re_τ, which leads to the logarithmic dependence of Φ_T⁺ on Re_τ. The behaviors of (dU⁺/dy⁺)² and -⟨uv⟩⁺ dU⁺/dy⁺ are similar to those of the PPF^[26].

The dissipation function Φ⁺ in the PCF also follows a logarithmic law

(13)

From Eq. (11), the dependence relationships of U_w⁺ and Re_w on Re_τ can be obtained, i.e.,

(14)

(15)

3.2 Dissipation functions Φ in the RPPF and RPCF

In this subsection, we focus on the effects of spanwise system rotations on the energy dissipation function Φ in the RPPF and RPCF. For the RPPF, the DNS data at Re_τ=180 based on the global friction velocity u_τ^[16] are used. The rotation number defined by the global friction velocity Ro_τ=2Ωh/u_τ, where Ω is the constant angular velocity, varies from 0 to 130. The corresponding rotation number defined by the bulk velocity, i.e., Ro=2Ωh/U_b, varies from 0.000 to 2.287. The flow is driven by an imposed constant streamwise pressure gradient. For the RPCF, the DNS data at Re_w=2 600^[23] are used. The rotation number Ro=2Ωh/U_w varies from 0.0 to 0.9. Different from the RPPF, no constant streamwise pressure gradient is imposed in the RPCF, and the flow is driven by the relative movements of the two walls. For more details about the DNS descriptions and their validations, please refer to Refs. [16] and [23].

Figure 2 shows the nondimensionalized dissipation function Φ⁺ and its two shares, i.e., the mean shear contribution Φ_M⁺ and the turbulent contribution Φ_T⁺, in the RPPF at different Ro_τ. The results of twice of the mean bulk velocity in wall unit, i.e., 2U_b⁺, are also shown for comparison. It is seen from the figure that the deviations between 2U_b⁺ and Φ⁺ are negligible, which verifies the derivation of Eq. (6). When the rotation number increases in the range Ro_τ≤ 2.5, Φ_M⁺ first increases, then decreases, reaches its minimum at about Ro_τ=10, and increases again. Φ_T⁺ shows an opposite tendency. It first decreases with the rotation number when Ro ≤ 2.5, and then increases with it, reaching its peak value at about Ro = 40. After that, Φ_T⁺ decreases with the rotation number.

Figure 3 shows Φ_T⁺, Φ_M⁺, Φ⁺, and U_w⁺ in the RPCF at different Ro. The negligible deviations between U_w⁺ and Φ⁺ verify the derivation of Eq. (11). Different from the results obtained from the RPPF shown in Fig. 2, Φ⁺ obtained from the RPCF first decreases with Ro when Ro ≤ 0.15, and then increases. This difference might come from the different driving methods used in both cases. In the RPPF, the flow is driven by a constant mean pressure gradient, making sure of a constant global friction velocity, whereas in the RPCF, the flow is driven by the moving walls, resulting in a constant mean bulk velocity. The two shares, Φ_M⁺ and Φ_T⁺, also show different behaviors as compared with those from the RPPF. When Ro increases in the range Ro≤ 0.02, Φ_M⁺ obtained from the RPCF first increases, then decreases rapidly until it reaches a plateau during 0.2≤ Ro≤0.6, and finally increases again. When Ro increases in the range Ro≤ 0.02, Φ_T⁺ obtained from the RPCF first decreases, then increases until Ro=0.8, and finally decreases.

4 Summary

The dissipation function has been studied in the turbulent PPFs without system rotations. It is believed that the approach based on dissipation can provide an interesting insight into the logarithmic skin friction law in PPFs. In this paper, we present the analysis on the dissipation function in the turbulent PPFs and PCFs subject to spanwise rotations.

Based on mathematical derivation, it is shown that the nondimensionalized forms of Φ are the same in the PPF and PCF with or without system rotations, although the total shear has different forms in the PPF and PCF. In all these problems, the dissipation function can be separated as two parts. One is the mean part Φ_M⁺, and the other is the turbulent fluctuating part Φ_T⁺. From this formula, we first study Φ in the PCF without system rotations. The results show that the mean part is constant while the fluctuation part follows a logarithmic law, resulting in a similar logarithmic skin friction law as the PPF. Furthermore, the log-law in the PCF is valid at about Re_τ=80, which is much lower than that in the PPF, where Re_τ is supposed to be larger than 300.

However, if the flow system rotates in the spanwise direction, no obvious dependence on the rotation number can be evaluated. In the RPPF, the dissipation function shows an increase in the rotation number, while in the RPCF, it first decreases and then increases with the rotation number. In the future, more data should be generated to study the dependence of the dissipation function on the rotation and Reynolds numbers.